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ht_witness_witness_witness_right_right_left", "exact ht_witness_witness_witness_right_right_right" ], "script_sha256": "1cab201fb9e2c0b9bcaab84d60e737db555fc23e0641ed56da76005052f5fb5f", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_convolution_triangular_candidate.py", "sha256": "d53722e52ffb3f98d16d693c8cc28d605e62da8f36d5e6ecffe3df66179aa11f" }, "statement": "forall ab ac L AB AC K bb bc M N i j t. (exists pfc_gap_tri_term_old_length. pfc_gap_tri_term_old_length+(N)=(L)) -> (exists pfc_gap_tri_term_new_length. pfc_gap_tri_term_new_length+(N)=(K)) -> (forall mdr_i_pfp_tri_term_equal mdr_a_pfp_tri_term_equal. (exists mdr_gap_pfp_tri_term_equalb. mdr_gap_pfp_tri_term_equalb + S (mdr_i_pfp_tri_term_equal) = (N)) -> (((exists ff_h_mdr_pfp_tri_term_equalo. ff_h_mdr_pfp_tri_term_equalo + S (mdr_a_pfp_tri_term_equal) = S ((S (mdr_i_pfp_tri_term_equal)) * ac)) /\\ exists ff_q_mdr_pfp_tri_term_equalo. ab = ff_q_mdr_pfp_tri_term_equalo * S ((S (mdr_i_pfp_tri_term_equal)) * ac) + (mdr_a_pfp_tri_term_equal))) -> (((exists ff_h_mdr_pfp_tri_term_equaln. ff_h_mdr_pfp_tri_term_equaln + S (mdr_a_pfp_tri_term_equal) = S ((S (mdr_i_pfp_tri_term_equal)) * AC)) /\\ exists ff_q_mdr_pfp_tri_term_equaln. AB = ff_q_mdr_pfp_tri_term_equaln * S ((S (mdr_i_pfp_tri_term_equal)) * AC) + (mdr_a_pfp_tri_term_equal)))) -> (exists pfa_gap_tri_term_index. pfa_gap_tri_term_index + S (j) = (N)) -> (exists pfc_complement_tri_term_old pfc_left_tri_term_old pfc_right_tri_term_old. (((j)+pfc_complement_tri_term_old=(i)) /\\ ((((((exists pfa_gap_tri_term_oldleftinside. pfa_gap_tri_term_oldleftinside + S (j) = (L)) /\\ ((((exists ff_h_pfp_tri_term_oldleftentry. ff_h_pfp_tri_term_oldleftentry + S (pfc_left_tri_term_old) = S ((S (j)) * ac)) /\\ exists ff_q_pfp_tri_term_oldleftentry. ab = ff_q_pfp_tri_term_oldleftentry * S ((S (j)) * ac) + (pfc_left_tri_term_old)))))) \\/ (((exists pfc_gap_tri_term_oldleftoutside. pfc_gap_tri_term_oldleftoutside+(L)=(j)) /\\ (((pfc_left_tri_term_old)=0))))) /\\ ((((((exists pfa_gap_tri_term_oldrightinside. pfa_gap_tri_term_oldrightinside + S (pfc_complement_tri_term_old) = (M)) /\\ ((((exists ff_h_pfp_tri_term_oldrightentry. ff_h_pfp_tri_term_oldrightentry + S (pfc_right_tri_term_old) = S ((S (pfc_complement_tri_term_old)) * bc)) /\\ exists ff_q_pfp_tri_term_oldrightentry. bb = ff_q_pfp_tri_term_oldrightentry * S ((S (pfc_complement_tri_term_old)) * bc) + (pfc_right_tri_term_old)))))) \\/ (((exists pfc_gap_tri_term_oldrightoutside. pfc_gap_tri_term_oldrightoutside+(M)=(pfc_complement_tri_term_old)) /\\ (((pfc_right_tri_term_old)=0))))) /\\ (((t)=pfc_left_tri_term_old*pfc_right_tri_term_old)))))))) -> (exists pfc_complement_tri_term_new pfc_left_tri_term_new pfc_right_tri_term_new. (((j)+pfc_complement_tri_term_new=(i)) /\\ ((((((exists pfa_gap_tri_term_newleftinside. pfa_gap_tri_term_newleftinside + S (j) = (K)) /\\ ((((exists ff_h_pfp_tri_term_newleftentry. ff_h_pfp_tri_term_newleftentry + S (pfc_left_tri_term_new) = S ((S (j)) * AC)) /\\ exists ff_q_pfp_tri_term_newleftentry. AB = ff_q_pfp_tri_term_newleftentry * S ((S (j)) * AC) + (pfc_left_tri_term_new)))))) \\/ (((exists pfc_gap_tri_term_newleftoutside. pfc_gap_tri_term_newleftoutside+(K)=(j)) /\\ (((pfc_left_tri_term_new)=0))))) /\\ ((((((exists pfa_gap_tri_term_newrightinside. pfa_gap_tri_term_newrightinside + S (pfc_complement_tri_term_new) = (M)) /\\ ((((exists ff_h_pfp_tri_term_newrightentry. ff_h_pfp_tri_term_newrightentry + S (pfc_right_tri_term_new) = S ((S (pfc_complement_tri_term_new)) * bc)) /\\ exists ff_q_pfp_tri_term_newrightentry. bb = ff_q_pfp_tri_term_newrightentry * S ((S (pfc_complement_tri_term_new)) * bc) + (pfc_right_tri_term_new)))))) \\/ (((exists pfc_gap_tri_term_newrightoutside. pfc_gap_tri_term_newrightoutside+(M)=(pfc_complement_tri_term_new)) /\\ (((pfc_right_tri_term_new)=0))))) /\\ (((t)=pfc_left_tri_term_new*pfc_right_tri_term_new))))))))", "statement_sha256": "a82dffe6d2e770d80e223c1ce948eeb3b3230383db138389e6db2e1a459b9008", "summary": "A genuine antidiagonal term below a shared left prefix survives changing its code and its declared input length.", "summary_sha256": "1a48d7a93970f84a08c6e253232a49d8a569cbec9aa2311467b2c50ec91ce1c3" }, { "alpha_v33_frontier_enrollment": { "body_receipt_sha256": "53dfa7c21fca8a860b77784a8f6f0aff4d2561d6000ef5b55dc82241956a2b6d", "bundle_campaign": "polynomial-euclidean-division", "bundle_node_id": 256, "bundle_sha256": "6ae667d8518e4dbe722bb08ad1b08715a0d282c2893e533c8133d770fe861dcf", "campaign": "polynomial-euclidean-division", "first_enrolled_version": "v33", "parent_catalog_sha256": "41b9f387d88a5a4f0fe5ee2bd5578f37a27a4657b0a80f1a1a2cb5109f69a623", "rfc_sha256": "38d609ffe5dbf7c36cd0e5112dcfb219934bac3c8af09da5d96f30844ae44895", "source_sha256": "d53722e52ffb3f98d16d693c8cc28d605e62da8f36d5e6ecffe3df66179aa11f", "test_sha256": "e3d9c9f2439383a50dd1d9b5f9d1569c8e7e3bbef2a5a2ed64f3bb5f2384ef46" }, "body_checked": true, "body_receipt": { "command_count": 47, "dependency_count": 1, "dne_command_count": 0, "name": 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((((exists ff_h_pfp_tri_prefix_chosen_entry. ff_h_pfp_tri_prefix_chosen_entry + S (t) = S ((S (j)) * dc)) /\\ exists ff_q_pfp_tri_prefix_chosen_entry. db = ff_q_pfp_tri_prefix_chosen_entry * S ((S (j)) * dc) + (t))) /\\ ((exists pfc_complement_tri_prefix_chosen_term pfc_left_tri_prefix_chosen_term pfc_right_tri_prefix_chosen_term. (((j)+pfc_complement_tri_prefix_chosen_term=(i)) /\\ ((((((exists pfa_gap_tri_prefix_chosen_termleftinside. pfa_gap_tri_prefix_chosen_termleftinside + S (j) = (L)) /\\ ((((exists ff_h_pfp_tri_prefix_chosen_termleftentry. ff_h_pfp_tri_prefix_chosen_termleftentry + S (pfc_left_tri_prefix_chosen_term) = S ((S (j)) * ac)) /\\ exists ff_q_pfp_tri_prefix_chosen_termleftentry. ab = ff_q_pfp_tri_prefix_chosen_termleftentry * S ((S (j)) * ac) + (pfc_left_tri_prefix_chosen_term)))))) \\/ (((exists pfc_gap_tri_prefix_chosen_termleftoutside. pfc_gap_tri_prefix_chosen_termleftoutside+(L)=(j)) /\\ (((pfc_left_tri_prefix_chosen_term)=0))))) /\\ ((((((exists pfa_gap_tri_prefix_chosen_termrightinside. pfa_gap_tri_prefix_chosen_termrightinside + S (pfc_complement_tri_prefix_chosen_term) = (M)) /\\ ((((exists ff_h_pfp_tri_prefix_chosen_termrightentry. ff_h_pfp_tri_prefix_chosen_termrightentry + S (pfc_right_tri_prefix_chosen_term) = S ((S (pfc_complement_tri_prefix_chosen_term)) * bc)) /\\ exists ff_q_pfp_tri_prefix_chosen_termrightentry. bb = ff_q_pfp_tri_prefix_chosen_termrightentry * S ((S (pfc_complement_tri_prefix_chosen_term)) * bc) + (pfc_right_tri_prefix_chosen_term)))))) \\/ (((exists pfc_gap_tri_prefix_chosen_termrightoutside. pfc_gap_tri_prefix_chosen_termrightoutside+(M)=(pfc_complement_tri_prefix_chosen_term)) /\\ (((pfc_right_tri_prefix_chosen_term)=0))))) /\\ (((t)=pfc_left_tri_prefix_chosen_term*pfc_right_tri_prefix_chosen_term))))))))))", "specialize hd (j)", "apply hd", "exact hj", "cases ht", "cases ht_witness", "exists x", "split", "exact ht_witness_left", "specialize polynomial_diagonal_left_prefix_transport (ab)", "specialize polynomial_diagonal_left_prefix_transport (ac)", "specialize polynomial_diagonal_left_prefix_transport (L)", "specialize polynomial_diagonal_left_prefix_transport (AB)", "specialize polynomial_diagonal_left_prefix_transport (AC)", "specialize polynomial_diagonal_left_prefix_transport (K)", "specialize polynomial_diagonal_left_prefix_transport (bb)", "specialize polynomial_diagonal_left_prefix_transport (bc)", "specialize polynomial_diagonal_left_prefix_transport (M)", "specialize polynomial_diagonal_left_prefix_transport (N)", "specialize polynomial_diagonal_left_prefix_transport (i)", "specialize polynomial_diagonal_left_prefix_transport (j)", "specialize polynomial_diagonal_left_prefix_transport (x)", "apply polynomial_diagonal_left_prefix_transport", "exact hl", "exact hk", "exact he", "exact hj", "exact ht_witness_right" ], "script_sha256": "a8628f56031db7ae807b19d8a7530813e3ab5f622abd030fc15d0cb8345002c5", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_convolution_triangular_candidate.py", "sha256": "d53722e52ffb3f98d16d693c8cc28d605e62da8f36d5e6ecffe3df66179aa11f" }, "statement": "forall ab ac L AB AC K bb bc M N i db dc. (exists pfc_gap_tri_prefix_old_length. pfc_gap_tri_prefix_old_length+(N)=(L)) -> (exists pfc_gap_tri_prefix_new_length. pfc_gap_tri_prefix_new_length+(N)=(K)) -> (forall mdr_i_pfp_tri_prefix_equal mdr_a_pfp_tri_prefix_equal. (exists mdr_gap_pfp_tri_prefix_equalb. mdr_gap_pfp_tri_prefix_equalb + S (mdr_i_pfp_tri_prefix_equal) = (N)) -> (((exists ff_h_mdr_pfp_tri_prefix_equalo. ff_h_mdr_pfp_tri_prefix_equalo + S (mdr_a_pfp_tri_prefix_equal) = S ((S (mdr_i_pfp_tri_prefix_equal)) * ac)) /\\ exists ff_q_mdr_pfp_tri_prefix_equalo. ab = ff_q_mdr_pfp_tri_prefix_equalo * S ((S (mdr_i_pfp_tri_prefix_equal)) * ac) + (mdr_a_pfp_tri_prefix_equal))) -> (((exists ff_h_mdr_pfp_tri_prefix_equaln. ff_h_mdr_pfp_tri_prefix_equaln + S (mdr_a_pfp_tri_prefix_equal) = S ((S (mdr_i_pfp_tri_prefix_equal)) * AC)) /\\ exists ff_q_mdr_pfp_tri_prefix_equaln. AB = ff_q_mdr_pfp_tri_prefix_equaln * S ((S (mdr_i_pfp_tri_prefix_equal)) * AC) + (mdr_a_pfp_tri_prefix_equal)))) -> (forall pfc_index_tri_prefix_old. (exists pfa_gap_tri_prefix_oldbound. pfa_gap_tri_prefix_oldbound + S (pfc_index_tri_prefix_old) = (N)) -> exists pfc_value_tri_prefix_old. ((((exists ff_h_pfp_tri_prefix_oldentry. ff_h_pfp_tri_prefix_oldentry + S (pfc_value_tri_prefix_old) = S ((S (pfc_index_tri_prefix_old)) * dc)) /\\ exists ff_q_pfp_tri_prefix_oldentry. db = ff_q_pfp_tri_prefix_oldentry * S ((S (pfc_index_tri_prefix_old)) * dc) + (pfc_value_tri_prefix_old))) /\\ ((exists pfc_complement_tri_prefix_oldterm pfc_left_tri_prefix_oldterm pfc_right_tri_prefix_oldterm. (((pfc_index_tri_prefix_old)+pfc_complement_tri_prefix_oldterm=(i)) /\\ ((((((exists pfa_gap_tri_prefix_oldtermleftinside. pfa_gap_tri_prefix_oldtermleftinside + S (pfc_index_tri_prefix_old) = (L)) /\\ ((((exists ff_h_pfp_tri_prefix_oldtermleftentry. ff_h_pfp_tri_prefix_oldtermleftentry + S (pfc_left_tri_prefix_oldterm) = S ((S (pfc_index_tri_prefix_old)) * ac)) /\\ exists ff_q_pfp_tri_prefix_oldtermleftentry. ab = ff_q_pfp_tri_prefix_oldtermleftentry * S ((S (pfc_index_tri_prefix_old)) * ac) + (pfc_left_tri_prefix_oldterm)))))) \\/ (((exists pfc_gap_tri_prefix_oldtermleftoutside. pfc_gap_tri_prefix_oldtermleftoutside+(L)=(pfc_index_tri_prefix_old)) /\\ (((pfc_left_tri_prefix_oldterm)=0))))) /\\ ((((((exists pfa_gap_tri_prefix_oldtermrightinside. pfa_gap_tri_prefix_oldtermrightinside + S (pfc_complement_tri_prefix_oldterm) = (M)) /\\ ((((exists ff_h_pfp_tri_prefix_oldtermrightentry. ff_h_pfp_tri_prefix_oldtermrightentry + S (pfc_right_tri_prefix_oldterm) = S ((S (pfc_complement_tri_prefix_oldterm)) * bc)) /\\ exists ff_q_pfp_tri_prefix_oldtermrightentry. bb = ff_q_pfp_tri_prefix_oldtermrightentry * S ((S (pfc_complement_tri_prefix_oldterm)) * bc) + (pfc_right_tri_prefix_oldterm)))))) \\/ (((exists pfc_gap_tri_prefix_oldtermrightoutside. pfc_gap_tri_prefix_oldtermrightoutside+(M)=(pfc_complement_tri_prefix_oldterm)) /\\ (((pfc_right_tri_prefix_oldterm)=0))))) /\\ (((pfc_value_tri_prefix_old)=pfc_left_tri_prefix_oldterm*pfc_right_tri_prefix_oldterm))))))))))) -> (forall pfc_index_tri_prefix_new. (exists pfa_gap_tri_prefix_newbound. pfa_gap_tri_prefix_newbound + S (pfc_index_tri_prefix_new) = (N)) -> exists pfc_value_tri_prefix_new. ((((exists ff_h_pfp_tri_prefix_newentry. ff_h_pfp_tri_prefix_newentry + S (pfc_value_tri_prefix_new) = S ((S (pfc_index_tri_prefix_new)) * dc)) /\\ exists ff_q_pfp_tri_prefix_newentry. db = ff_q_pfp_tri_prefix_newentry * S ((S (pfc_index_tri_prefix_new)) * dc) + (pfc_value_tri_prefix_new))) /\\ ((exists pfc_complement_tri_prefix_newterm pfc_left_tri_prefix_newterm pfc_right_tri_prefix_newterm. (((pfc_index_tri_prefix_new)+pfc_complement_tri_prefix_newterm=(i)) /\\ ((((((exists pfa_gap_tri_prefix_newtermleftinside. pfa_gap_tri_prefix_newtermleftinside + S (pfc_index_tri_prefix_new) = (K)) /\\ ((((exists ff_h_pfp_tri_prefix_newtermleftentry. ff_h_pfp_tri_prefix_newtermleftentry + S (pfc_left_tri_prefix_newterm) = S ((S (pfc_index_tri_prefix_new)) * AC)) /\\ exists ff_q_pfp_tri_prefix_newtermleftentry. AB = ff_q_pfp_tri_prefix_newtermleftentry * S ((S (pfc_index_tri_prefix_new)) * AC) + (pfc_left_tri_prefix_newterm)))))) \\/ (((exists pfc_gap_tri_prefix_newtermleftoutside. pfc_gap_tri_prefix_newtermleftoutside+(K)=(pfc_index_tri_prefix_new)) /\\ (((pfc_left_tri_prefix_newterm)=0))))) /\\ ((((((exists pfa_gap_tri_prefix_newtermrightinside. pfa_gap_tri_prefix_newtermrightinside + S (pfc_complement_tri_prefix_newterm) = (M)) /\\ ((((exists ff_h_pfp_tri_prefix_newtermrightentry. ff_h_pfp_tri_prefix_newtermrightentry + S (pfc_right_tri_prefix_newterm) = S ((S (pfc_complement_tri_prefix_newterm)) * bc)) /\\ exists ff_q_pfp_tri_prefix_newtermrightentry. bb = ff_q_pfp_tri_prefix_newtermrightentry * S ((S (pfc_complement_tri_prefix_newterm)) * bc) + (pfc_right_tri_prefix_newterm)))))) \\/ (((exists pfc_gap_tri_prefix_newtermrightoutside. pfc_gap_tri_prefix_newtermrightoutside+(M)=(pfc_complement_tri_prefix_newterm)) /\\ (((pfc_right_tri_prefix_newterm)=0))))) /\\ 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((((exists ff_h_pfp_tri_coefficient_chosen_entry. ff_h_pfp_tri_coefficient_chosen_entry + S (t) = S ((S (j)) * x1)) /\\ exists ff_q_pfp_tri_coefficient_chosen_entry. x = ff_q_pfp_tri_coefficient_chosen_entry * S ((S (j)) * x1) + (t))) /\\ ((exists pfc_complement_tri_coefficient_chosen_term pfc_left_tri_coefficient_chosen_term pfc_right_tri_coefficient_chosen_term. (((j)+pfc_complement_tri_coefficient_chosen_term=(i)) /\\ ((((((exists pfa_gap_tri_coefficient_chosen_termleftinside. pfa_gap_tri_coefficient_chosen_termleftinside + S (j) = (L)) /\\ ((((exists ff_h_pfp_tri_coefficient_chosen_termleftentry. ff_h_pfp_tri_coefficient_chosen_termleftentry + S (pfc_left_tri_coefficient_chosen_term) = S ((S (j)) * ac)) /\\ exists ff_q_pfp_tri_coefficient_chosen_termleftentry. ab = ff_q_pfp_tri_coefficient_chosen_termleftentry * S ((S (j)) * ac) + (pfc_left_tri_coefficient_chosen_term)))))) \\/ (((exists pfc_gap_tri_coefficient_chosen_termleftoutside. pfc_gap_tri_coefficient_chosen_termleftoutside+(L)=(j)) /\\ (((pfc_left_tri_coefficient_chosen_term)=0))))) /\\ ((((((exists pfa_gap_tri_coefficient_chosen_termrightinside. pfa_gap_tri_coefficient_chosen_termrightinside + S (pfc_complement_tri_coefficient_chosen_term) = (M)) /\\ ((((exists ff_h_pfp_tri_coefficient_chosen_termrightentry. ff_h_pfp_tri_coefficient_chosen_termrightentry + S (pfc_right_tri_coefficient_chosen_term) = S ((S (pfc_complement_tri_coefficient_chosen_term)) * bc)) /\\ exists ff_q_pfp_tri_coefficient_chosen_termrightentry. bb = ff_q_pfp_tri_coefficient_chosen_termrightentry * S ((S (pfc_complement_tri_coefficient_chosen_term)) * bc) + (pfc_right_tri_coefficient_chosen_term)))))) \\/ (((exists pfc_gap_tri_coefficient_chosen_termrightoutside. pfc_gap_tri_coefficient_chosen_termrightoutside+(M)=(pfc_complement_tri_coefficient_chosen_term)) /\\ (((pfc_right_tri_coefficient_chosen_term)=0))))) /\\ (((t)=pfc_left_tri_coefficient_chosen_term*pfc_right_tri_coefficient_chosen_term))))))))))", "specialize hr_witness_witness_witness_left (j)", "apply hr_witness_witness_witness_left", "exact hj", "cases ht", "cases ht_witness", "exists x3", "split", "exact ht_witness_left", "specialize polynomial_diagonal_left_prefix_transport (ab)", "specialize polynomial_diagonal_left_prefix_transport (ac)", "specialize polynomial_diagonal_left_prefix_transport (L)", "specialize polynomial_diagonal_left_prefix_transport (AB)", "specialize polynomial_diagonal_left_prefix_transport (AC)", "specialize polynomial_diagonal_left_prefix_transport (K)", "specialize polynomial_diagonal_left_prefix_transport (bb)", "specialize polynomial_diagonal_left_prefix_transport (bc)", "specialize polynomial_diagonal_left_prefix_transport (M)", "specialize polynomial_diagonal_left_prefix_transport (N)", "specialize polynomial_diagonal_left_prefix_transport (i)", "specialize polynomial_diagonal_left_prefix_transport (j)", "specialize polynomial_diagonal_left_prefix_transport (x3)", "apply polynomial_diagonal_left_prefix_transport", "exact hl", "exact hk", "exact he", "specialize lt_of_lt_of_le (j)", "specialize lt_of_lt_of_le (S i)", "specialize lt_of_lt_of_le (N)", "apply lt_of_lt_of_le", "exact hj", "exact hi", "exact ht_witness_right", "split", "exact hr_witness_witness_witness_right_left", "exact hr_witness_witness_witness_right_right" ], "script_sha256": "b8a4a2c882d40bc6f8827583ed0b6443fffc9d334181fbc5024ffb2de574e076", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_convolution_triangular_candidate.py", "sha256": "d53722e52ffb3f98d16d693c8cc28d605e62da8f36d5e6ecffe3df66179aa11f" }, "statement": "forall p ab ac L AB AC K bb bc M N i r. (exists pfc_gap_tri_coefficient_old_length. pfc_gap_tri_coefficient_old_length+(N)=(L)) -> (exists pfc_gap_tri_coefficient_new_length. pfc_gap_tri_coefficient_new_length+(N)=(K)) -> (forall mdr_i_pfp_tri_coefficient_equal mdr_a_pfp_tri_coefficient_equal. (exists mdr_gap_pfp_tri_coefficient_equalb. mdr_gap_pfp_tri_coefficient_equalb + S (mdr_i_pfp_tri_coefficient_equal) = (N)) -> (((exists ff_h_mdr_pfp_tri_coefficient_equalo. ff_h_mdr_pfp_tri_coefficient_equalo + S (mdr_a_pfp_tri_coefficient_equal) = S ((S (mdr_i_pfp_tri_coefficient_equal)) * ac)) /\\ exists ff_q_mdr_pfp_tri_coefficient_equalo. ab = ff_q_mdr_pfp_tri_coefficient_equalo * S ((S (mdr_i_pfp_tri_coefficient_equal)) * ac) + (mdr_a_pfp_tri_coefficient_equal))) -> (((exists ff_h_mdr_pfp_tri_coefficient_equaln. ff_h_mdr_pfp_tri_coefficient_equaln + S (mdr_a_pfp_tri_coefficient_equal) = S ((S (mdr_i_pfp_tri_coefficient_equal)) * AC)) /\\ exists ff_q_mdr_pfp_tri_coefficient_equaln. AB = ff_q_mdr_pfp_tri_coefficient_equaln * S ((S (mdr_i_pfp_tri_coefficient_equal)) * AC) + (mdr_a_pfp_tri_coefficient_equal)))) -> (exists pfa_gap_tri_coefficient_index. pfa_gap_tri_coefficient_index + S (i) = (N)) -> (exists pfc_terms_code_tri_coefficient_old pfc_terms_scale_tri_coefficient_old pfc_natural_sum_tri_coefficient_old. ((forall pfc_index_tri_coefficient_olddiagonal. (exists pfa_gap_tri_coefficient_olddiagonalbound. pfa_gap_tri_coefficient_olddiagonalbound + S (pfc_index_tri_coefficient_olddiagonal) = (S (i))) -> exists pfc_value_tri_coefficient_olddiagonal. ((((exists ff_h_pfp_tri_coefficient_olddiagonalentry. ff_h_pfp_tri_coefficient_olddiagonalentry + S (pfc_value_tri_coefficient_olddiagonal) = S ((S (pfc_index_tri_coefficient_olddiagonal)) * pfc_terms_scale_tri_coefficient_old)) /\\ exists ff_q_pfp_tri_coefficient_olddiagonalentry. pfc_terms_code_tri_coefficient_old = ff_q_pfp_tri_coefficient_olddiagonalentry * S ((S (pfc_index_tri_coefficient_olddiagonal)) * pfc_terms_scale_tri_coefficient_old) + (pfc_value_tri_coefficient_olddiagonal))) /\\ ((exists pfc_complement_tri_coefficient_olddiagonalterm pfc_left_tri_coefficient_olddiagonalterm pfc_right_tri_coefficient_olddiagonalterm. (((pfc_index_tri_coefficient_olddiagonal)+pfc_complement_tri_coefficient_olddiagonalterm=(i)) /\\ ((((((exists pfa_gap_tri_coefficient_olddiagonaltermleftinside. pfa_gap_tri_coefficient_olddiagonaltermleftinside + S (pfc_index_tri_coefficient_olddiagonal) = (L)) /\\ ((((exists ff_h_pfp_tri_coefficient_olddiagonaltermleftentry. ff_h_pfp_tri_coefficient_olddiagonaltermleftentry + S (pfc_left_tri_coefficient_olddiagonalterm) = S ((S (pfc_index_tri_coefficient_olddiagonal)) * ac)) /\\ exists ff_q_pfp_tri_coefficient_olddiagonaltermleftentry. ab = ff_q_pfp_tri_coefficient_olddiagonaltermleftentry * S ((S (pfc_index_tri_coefficient_olddiagonal)) * ac) + (pfc_left_tri_coefficient_olddiagonalterm)))))) \\/ (((exists pfc_gap_tri_coefficient_olddiagonaltermleftoutside. pfc_gap_tri_coefficient_olddiagonaltermleftoutside+(L)=(pfc_index_tri_coefficient_olddiagonal)) /\\ (((pfc_left_tri_coefficient_olddiagonalterm)=0))))) /\\ ((((((exists pfa_gap_tri_coefficient_olddiagonaltermrightinside. pfa_gap_tri_coefficient_olddiagonaltermrightinside + S (pfc_complement_tri_coefficient_olddiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_tri_coefficient_olddiagonaltermrightentry. ff_h_pfp_tri_coefficient_olddiagonaltermrightentry + S (pfc_right_tri_coefficient_olddiagonalterm) = S ((S (pfc_complement_tri_coefficient_olddiagonalterm)) * bc)) /\\ exists ff_q_pfp_tri_coefficient_olddiagonaltermrightentry. bb = ff_q_pfp_tri_coefficient_olddiagonaltermrightentry * S ((S (pfc_complement_tri_coefficient_olddiagonalterm)) * bc) + (pfc_right_tri_coefficient_olddiagonalterm)))))) \\/ (((exists pfc_gap_tri_coefficient_olddiagonaltermrightoutside. pfc_gap_tri_coefficient_olddiagonaltermrightoutside+(M)=(pfc_complement_tri_coefficient_olddiagonalterm)) /\\ (((pfc_right_tri_coefficient_olddiagonalterm)=0))))) /\\ (((pfc_value_tri_coefficient_olddiagonal)=pfc_left_tri_coefficient_olddiagonalterm*pfc_right_tri_coefficient_olddiagonalterm))))))))))) /\\ (((exists fs_u_pfc_tri_coefficient_oldsum fs_v_pfc_tri_coefficient_oldsum. ((((exists fs_h_pfc_tri_coefficient_oldsum_body_start. fs_h_pfc_tri_coefficient_oldsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_tri_coefficient_oldsum)) /\\ exists fs_q_pfc_tri_coefficient_oldsum_body_start. fs_u_pfc_tri_coefficient_oldsum = fs_q_pfc_tri_coefficient_oldsum_body_start * S ((S (0)) * fs_v_pfc_tri_coefficient_oldsum) + (0))) /\\ ((((exists fs_h_pfc_tri_coefficient_oldsum_body_terminal. fs_h_pfc_tri_coefficient_oldsum_body_terminal + S (pfc_natural_sum_tri_coefficient_old) = S ((S (S (i))) * fs_v_pfc_tri_coefficient_oldsum)) /\\ exists fs_q_pfc_tri_coefficient_oldsum_body_terminal. fs_u_pfc_tri_coefficient_oldsum = fs_q_pfc_tri_coefficient_oldsum_body_terminal * S ((S (S (i))) * fs_v_pfc_tri_coefficient_oldsum) + (pfc_natural_sum_tri_coefficient_old))) /\\ forall fs_i_pfc_tri_coefficient_oldsum_body_steps. (exists fs_lt_pfc_tri_coefficient_oldsum_body_steps_bound. fs_lt_pfc_tri_coefficient_oldsum_body_steps_bound + S fs_i_pfc_tri_coefficient_oldsum_body_steps = S (i)) -> exists fs_a_pfc_tri_coefficient_oldsum_body_steps fs_r_pfc_tri_coefficient_oldsum_body_steps fs_s_pfc_tri_coefficient_oldsum_body_steps. ((((exists fs_h_pfc_tri_coefficient_oldsum_body_steps_summand. fs_h_pfc_tri_coefficient_oldsum_body_steps_summand + S (fs_a_pfc_tri_coefficient_oldsum_body_steps) = S ((S (fs_i_pfc_tri_coefficient_oldsum_body_steps)) * pfc_terms_scale_tri_coefficient_old)) /\\ exists fs_q_pfc_tri_coefficient_oldsum_body_steps_summand. pfc_terms_code_tri_coefficient_old = fs_q_pfc_tri_coefficient_oldsum_body_steps_summand * S ((S (fs_i_pfc_tri_coefficient_oldsum_body_steps)) * pfc_terms_scale_tri_coefficient_old) + (fs_a_pfc_tri_coefficient_oldsum_body_steps))) /\\ ((((exists fs_h_pfc_tri_coefficient_oldsum_body_steps_partial. fs_h_pfc_tri_coefficient_oldsum_body_steps_partial + S (fs_r_pfc_tri_coefficient_oldsum_body_steps) = S ((S (fs_i_pfc_tri_coefficient_oldsum_body_steps)) * fs_v_pfc_tri_coefficient_oldsum)) /\\ exists fs_q_pfc_tri_coefficient_oldsum_body_steps_partial. fs_u_pfc_tri_coefficient_oldsum = fs_q_pfc_tri_coefficient_oldsum_body_steps_partial * S ((S (fs_i_pfc_tri_coefficient_oldsum_body_steps)) * fs_v_pfc_tri_coefficient_oldsum) + (fs_r_pfc_tri_coefficient_oldsum_body_steps))) /\\ ((((exists fs_h_pfc_tri_coefficient_oldsum_body_steps_successor. fs_h_pfc_tri_coefficient_oldsum_body_steps_successor + S (fs_s_pfc_tri_coefficient_oldsum_body_steps) = S ((S (S fs_i_pfc_tri_coefficient_oldsum_body_steps)) * fs_v_pfc_tri_coefficient_oldsum)) /\\ exists fs_q_pfc_tri_coefficient_oldsum_body_steps_successor. fs_u_pfc_tri_coefficient_oldsum = fs_q_pfc_tri_coefficient_oldsum_body_steps_successor * S ((S (S fs_i_pfc_tri_coefficient_oldsum_body_steps)) * fs_v_pfc_tri_coefficient_oldsum) + (fs_s_pfc_tri_coefficient_oldsum_body_steps))) /\\ fs_s_pfc_tri_coefficient_oldsum_body_steps = fs_r_pfc_tri_coefficient_oldsum_body_steps + fs_a_pfc_tri_coefficient_oldsum_body_steps)))))) /\\ ((((exists pfa_gap_tri_coefficient_oldresiduebound. pfa_gap_tri_coefficient_oldresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_tri_coefficient_oldresiduecongruence pfa_offset_right_tri_coefficient_oldresiduecongruence. (pfc_natural_sum_tri_coefficient_old) + (p) * pfa_offset_left_tri_coefficient_oldresiduecongruence = (r) + (p) * pfa_offset_right_tri_coefficient_oldresiduecongruence))))))))) -> (exists pfc_terms_code_tri_coefficient_new pfc_terms_scale_tri_coefficient_new pfc_natural_sum_tri_coefficient_new. ((forall pfc_index_tri_coefficient_newdiagonal. (exists pfa_gap_tri_coefficient_newdiagonalbound. pfa_gap_tri_coefficient_newdiagonalbound + S (pfc_index_tri_coefficient_newdiagonal) = (S (i))) -> exists pfc_value_tri_coefficient_newdiagonal. ((((exists ff_h_pfp_tri_coefficient_newdiagonalentry. ff_h_pfp_tri_coefficient_newdiagonalentry + S (pfc_value_tri_coefficient_newdiagonal) = S ((S (pfc_index_tri_coefficient_newdiagonal)) * pfc_terms_scale_tri_coefficient_new)) /\\ exists ff_q_pfp_tri_coefficient_newdiagonalentry. pfc_terms_code_tri_coefficient_new = ff_q_pfp_tri_coefficient_newdiagonalentry * S ((S (pfc_index_tri_coefficient_newdiagonal)) * pfc_terms_scale_tri_coefficient_new) + (pfc_value_tri_coefficient_newdiagonal))) /\\ ((exists pfc_complement_tri_coefficient_newdiagonalterm pfc_left_tri_coefficient_newdiagonalterm pfc_right_tri_coefficient_newdiagonalterm. (((pfc_index_tri_coefficient_newdiagonal)+pfc_complement_tri_coefficient_newdiagonalterm=(i)) /\\ ((((((exists pfa_gap_tri_coefficient_newdiagonaltermleftinside. pfa_gap_tri_coefficient_newdiagonaltermleftinside + S (pfc_index_tri_coefficient_newdiagonal) = (K)) /\\ ((((exists ff_h_pfp_tri_coefficient_newdiagonaltermleftentry. ff_h_pfp_tri_coefficient_newdiagonaltermleftentry + S (pfc_left_tri_coefficient_newdiagonalterm) = S ((S (pfc_index_tri_coefficient_newdiagonal)) * AC)) /\\ exists ff_q_pfp_tri_coefficient_newdiagonaltermleftentry. AB = ff_q_pfp_tri_coefficient_newdiagonaltermleftentry * S ((S (pfc_index_tri_coefficient_newdiagonal)) * AC) + (pfc_left_tri_coefficient_newdiagonalterm)))))) \\/ (((exists pfc_gap_tri_coefficient_newdiagonaltermleftoutside. pfc_gap_tri_coefficient_newdiagonaltermleftoutside+(K)=(pfc_index_tri_coefficient_newdiagonal)) /\\ (((pfc_left_tri_coefficient_newdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_tri_coefficient_newdiagonaltermrightinside. pfa_gap_tri_coefficient_newdiagonaltermrightinside + S (pfc_complement_tri_coefficient_newdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_tri_coefficient_newdiagonaltermrightentry. ff_h_pfp_tri_coefficient_newdiagonaltermrightentry + S (pfc_right_tri_coefficient_newdiagonalterm) = S ((S (pfc_complement_tri_coefficient_newdiagonalterm)) * bc)) /\\ exists ff_q_pfp_tri_coefficient_newdiagonaltermrightentry. bb = ff_q_pfp_tri_coefficient_newdiagonaltermrightentry * S ((S (pfc_complement_tri_coefficient_newdiagonalterm)) * bc) + (pfc_right_tri_coefficient_newdiagonalterm)))))) \\/ (((exists pfc_gap_tri_coefficient_newdiagonaltermrightoutside. pfc_gap_tri_coefficient_newdiagonaltermrightoutside+(M)=(pfc_complement_tri_coefficient_newdiagonalterm)) /\\ (((pfc_right_tri_coefficient_newdiagonalterm)=0))))) /\\ (((pfc_value_tri_coefficient_newdiagonal)=pfc_left_tri_coefficient_newdiagonalterm*pfc_right_tri_coefficient_newdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_tri_coefficient_newsum fs_v_pfc_tri_coefficient_newsum. ((((exists fs_h_pfc_tri_coefficient_newsum_body_start. fs_h_pfc_tri_coefficient_newsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_tri_coefficient_newsum)) /\\ exists fs_q_pfc_tri_coefficient_newsum_body_start. fs_u_pfc_tri_coefficient_newsum = fs_q_pfc_tri_coefficient_newsum_body_start * S ((S (0)) * fs_v_pfc_tri_coefficient_newsum) + (0))) /\\ ((((exists fs_h_pfc_tri_coefficient_newsum_body_terminal. fs_h_pfc_tri_coefficient_newsum_body_terminal + S (pfc_natural_sum_tri_coefficient_new) = S ((S (S (i))) * fs_v_pfc_tri_coefficient_newsum)) /\\ exists fs_q_pfc_tri_coefficient_newsum_body_terminal. fs_u_pfc_tri_coefficient_newsum = fs_q_pfc_tri_coefficient_newsum_body_terminal * S ((S (S (i))) * fs_v_pfc_tri_coefficient_newsum) + (pfc_natural_sum_tri_coefficient_new))) /\\ forall fs_i_pfc_tri_coefficient_newsum_body_steps. (exists fs_lt_pfc_tri_coefficient_newsum_body_steps_bound. fs_lt_pfc_tri_coefficient_newsum_body_steps_bound + S fs_i_pfc_tri_coefficient_newsum_body_steps = S (i)) -> exists fs_a_pfc_tri_coefficient_newsum_body_steps fs_r_pfc_tri_coefficient_newsum_body_steps fs_s_pfc_tri_coefficient_newsum_body_steps. ((((exists fs_h_pfc_tri_coefficient_newsum_body_steps_summand. fs_h_pfc_tri_coefficient_newsum_body_steps_summand + S (fs_a_pfc_tri_coefficient_newsum_body_steps) = S ((S (fs_i_pfc_tri_coefficient_newsum_body_steps)) * pfc_terms_scale_tri_coefficient_new)) /\\ exists fs_q_pfc_tri_coefficient_newsum_body_steps_summand. pfc_terms_code_tri_coefficient_new = fs_q_pfc_tri_coefficient_newsum_body_steps_summand * S ((S (fs_i_pfc_tri_coefficient_newsum_body_steps)) * pfc_terms_scale_tri_coefficient_new) + (fs_a_pfc_tri_coefficient_newsum_body_steps))) /\\ ((((exists fs_h_pfc_tri_coefficient_newsum_body_steps_partial. fs_h_pfc_tri_coefficient_newsum_body_steps_partial + S (fs_r_pfc_tri_coefficient_newsum_body_steps) = S ((S (fs_i_pfc_tri_coefficient_newsum_body_steps)) * fs_v_pfc_tri_coefficient_newsum)) /\\ exists fs_q_pfc_tri_coefficient_newsum_body_steps_partial. fs_u_pfc_tri_coefficient_newsum = fs_q_pfc_tri_coefficient_newsum_body_steps_partial * S ((S (fs_i_pfc_tri_coefficient_newsum_body_steps)) * fs_v_pfc_tri_coefficient_newsum) + (fs_r_pfc_tri_coefficient_newsum_body_steps))) /\\ ((((exists fs_h_pfc_tri_coefficient_newsum_body_steps_successor. fs_h_pfc_tri_coefficient_newsum_body_steps_successor + S (fs_s_pfc_tri_coefficient_newsum_body_steps) = S ((S (S fs_i_pfc_tri_coefficient_newsum_body_steps)) * fs_v_pfc_tri_coefficient_newsum)) /\\ exists fs_q_pfc_tri_coefficient_newsum_body_steps_successor. fs_u_pfc_tri_coefficient_newsum = fs_q_pfc_tri_coefficient_newsum_body_steps_successor * S ((S (S fs_i_pfc_tri_coefficient_newsum_body_steps)) * fs_v_pfc_tri_coefficient_newsum) + (fs_s_pfc_tri_coefficient_newsum_body_steps))) /\\ fs_s_pfc_tri_coefficient_newsum_body_steps = fs_r_pfc_tri_coefficient_newsum_body_steps + fs_a_pfc_tri_coefficient_newsum_body_steps)))))) /\\ ((((exists pfa_gap_tri_coefficient_newresiduebound. pfa_gap_tri_coefficient_newresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_tri_coefficient_newresiduecongruence pfa_offset_right_tri_coefficient_newresiduecongruence. 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"intro N", "intro i", "intro r", "intro he", "intro hi", "intro hr", "specialize prime_field_convolution_coefficient_prefix_transport (p)", "specialize prime_field_convolution_coefficient_prefix_transport (ab)", "specialize prime_field_convolution_coefficient_prefix_transport (ac)", "specialize prime_field_convolution_coefficient_prefix_transport (N)", "specialize prime_field_convolution_coefficient_prefix_transport (AB)", "specialize prime_field_convolution_coefficient_prefix_transport (AC)", "specialize prime_field_convolution_coefficient_prefix_transport (S N)", "specialize prime_field_convolution_coefficient_prefix_transport (bb)", "specialize prime_field_convolution_coefficient_prefix_transport (bc)", "specialize prime_field_convolution_coefficient_prefix_transport (M)", "specialize prime_field_convolution_coefficient_prefix_transport (N)", "specialize prime_field_convolution_coefficient_prefix_transport (i)", "specialize prime_field_convolution_coefficient_prefix_transport (r)", 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(forall mdr_i_pfp_tri_append_equal mdr_a_pfp_tri_append_equal. (exists mdr_gap_pfp_tri_append_equalb. mdr_gap_pfp_tri_append_equalb + S (mdr_i_pfp_tri_append_equal) = (N)) -> (((exists ff_h_mdr_pfp_tri_append_equalo. ff_h_mdr_pfp_tri_append_equalo + S (mdr_a_pfp_tri_append_equal) = S ((S (mdr_i_pfp_tri_append_equal)) * ac)) /\\ exists ff_q_mdr_pfp_tri_append_equalo. ab = ff_q_mdr_pfp_tri_append_equalo * S ((S (mdr_i_pfp_tri_append_equal)) * ac) + (mdr_a_pfp_tri_append_equal))) -> (((exists ff_h_mdr_pfp_tri_append_equaln. ff_h_mdr_pfp_tri_append_equaln + S (mdr_a_pfp_tri_append_equal) = S ((S (mdr_i_pfp_tri_append_equal)) * AC)) /\\ exists ff_q_mdr_pfp_tri_append_equaln. AB = ff_q_mdr_pfp_tri_append_equaln * S ((S (mdr_i_pfp_tri_append_equal)) * AC) + (mdr_a_pfp_tri_append_equal)))) -> (exists pfa_gap_tri_append_earlier_index. pfa_gap_tri_append_earlier_index + S (i) = (N)) -> (exists pfc_terms_code_tri_append_earlier_old pfc_terms_scale_tri_append_earlier_old pfc_natural_sum_tri_append_earlier_old. ((forall pfc_index_tri_append_earlier_olddiagonal. (exists pfa_gap_tri_append_earlier_olddiagonalbound. pfa_gap_tri_append_earlier_olddiagonalbound + S (pfc_index_tri_append_earlier_olddiagonal) = (S (i))) -> exists pfc_value_tri_append_earlier_olddiagonal. ((((exists ff_h_pfp_tri_append_earlier_olddiagonalentry. ff_h_pfp_tri_append_earlier_olddiagonalentry + S (pfc_value_tri_append_earlier_olddiagonal) = S ((S (pfc_index_tri_append_earlier_olddiagonal)) * pfc_terms_scale_tri_append_earlier_old)) /\\ exists ff_q_pfp_tri_append_earlier_olddiagonalentry. pfc_terms_code_tri_append_earlier_old = ff_q_pfp_tri_append_earlier_olddiagonalentry * S ((S (pfc_index_tri_append_earlier_olddiagonal)) * pfc_terms_scale_tri_append_earlier_old) + (pfc_value_tri_append_earlier_olddiagonal))) /\\ ((exists pfc_complement_tri_append_earlier_olddiagonalterm pfc_left_tri_append_earlier_olddiagonalterm pfc_right_tri_append_earlier_olddiagonalterm. (((pfc_index_tri_append_earlier_olddiagonal)+pfc_complement_tri_append_earlier_olddiagonalterm=(i)) /\\ ((((((exists pfa_gap_tri_append_earlier_olddiagonaltermleftinside. pfa_gap_tri_append_earlier_olddiagonaltermleftinside + S (pfc_index_tri_append_earlier_olddiagonal) = (N)) /\\ ((((exists ff_h_pfp_tri_append_earlier_olddiagonaltermleftentry. ff_h_pfp_tri_append_earlier_olddiagonaltermleftentry + S (pfc_left_tri_append_earlier_olddiagonalterm) = S ((S (pfc_index_tri_append_earlier_olddiagonal)) * ac)) /\\ exists ff_q_pfp_tri_append_earlier_olddiagonaltermleftentry. ab = ff_q_pfp_tri_append_earlier_olddiagonaltermleftentry * S ((S (pfc_index_tri_append_earlier_olddiagonal)) * ac) + (pfc_left_tri_append_earlier_olddiagonalterm)))))) \\/ (((exists pfc_gap_tri_append_earlier_olddiagonaltermleftoutside. pfc_gap_tri_append_earlier_olddiagonaltermleftoutside+(N)=(pfc_index_tri_append_earlier_olddiagonal)) /\\ (((pfc_left_tri_append_earlier_olddiagonalterm)=0))))) /\\ ((((((exists pfa_gap_tri_append_earlier_olddiagonaltermrightinside. pfa_gap_tri_append_earlier_olddiagonaltermrightinside + S (pfc_complement_tri_append_earlier_olddiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_tri_append_earlier_olddiagonaltermrightentry. ff_h_pfp_tri_append_earlier_olddiagonaltermrightentry + S (pfc_right_tri_append_earlier_olddiagonalterm) = S ((S (pfc_complement_tri_append_earlier_olddiagonalterm)) * bc)) /\\ exists ff_q_pfp_tri_append_earlier_olddiagonaltermrightentry. bb = ff_q_pfp_tri_append_earlier_olddiagonaltermrightentry * S ((S (pfc_complement_tri_append_earlier_olddiagonalterm)) * bc) + (pfc_right_tri_append_earlier_olddiagonalterm)))))) \\/ (((exists pfc_gap_tri_append_earlier_olddiagonaltermrightoutside. pfc_gap_tri_append_earlier_olddiagonaltermrightoutside+(M)=(pfc_complement_tri_append_earlier_olddiagonalterm)) /\\ (((pfc_right_tri_append_earlier_olddiagonalterm)=0))))) /\\ (((pfc_value_tri_append_earlier_olddiagonal)=pfc_left_tri_append_earlier_olddiagonalterm*pfc_right_tri_append_earlier_olddiagonalterm))))))))))) /\\ (((exists fs_u_pfc_tri_append_earlier_oldsum fs_v_pfc_tri_append_earlier_oldsum. ((((exists fs_h_pfc_tri_append_earlier_oldsum_body_start. fs_h_pfc_tri_append_earlier_oldsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_tri_append_earlier_oldsum)) /\\ exists fs_q_pfc_tri_append_earlier_oldsum_body_start. fs_u_pfc_tri_append_earlier_oldsum = fs_q_pfc_tri_append_earlier_oldsum_body_start * S ((S (0)) * fs_v_pfc_tri_append_earlier_oldsum) + (0))) /\\ ((((exists fs_h_pfc_tri_append_earlier_oldsum_body_terminal. fs_h_pfc_tri_append_earlier_oldsum_body_terminal + S (pfc_natural_sum_tri_append_earlier_old) = S ((S (S (i))) * fs_v_pfc_tri_append_earlier_oldsum)) /\\ exists fs_q_pfc_tri_append_earlier_oldsum_body_terminal. fs_u_pfc_tri_append_earlier_oldsum = fs_q_pfc_tri_append_earlier_oldsum_body_terminal * S ((S (S (i))) * fs_v_pfc_tri_append_earlier_oldsum) + (pfc_natural_sum_tri_append_earlier_old))) /\\ forall fs_i_pfc_tri_append_earlier_oldsum_body_steps. (exists fs_lt_pfc_tri_append_earlier_oldsum_body_steps_bound. fs_lt_pfc_tri_append_earlier_oldsum_body_steps_bound + S fs_i_pfc_tri_append_earlier_oldsum_body_steps = S (i)) -> exists fs_a_pfc_tri_append_earlier_oldsum_body_steps fs_r_pfc_tri_append_earlier_oldsum_body_steps fs_s_pfc_tri_append_earlier_oldsum_body_steps. ((((exists fs_h_pfc_tri_append_earlier_oldsum_body_steps_summand. fs_h_pfc_tri_append_earlier_oldsum_body_steps_summand + S (fs_a_pfc_tri_append_earlier_oldsum_body_steps) = S ((S (fs_i_pfc_tri_append_earlier_oldsum_body_steps)) * pfc_terms_scale_tri_append_earlier_old)) /\\ exists fs_q_pfc_tri_append_earlier_oldsum_body_steps_summand. pfc_terms_code_tri_append_earlier_old = fs_q_pfc_tri_append_earlier_oldsum_body_steps_summand * S ((S (fs_i_pfc_tri_append_earlier_oldsum_body_steps)) * pfc_terms_scale_tri_append_earlier_old) + (fs_a_pfc_tri_append_earlier_oldsum_body_steps))) /\\ ((((exists fs_h_pfc_tri_append_earlier_oldsum_body_steps_partial. fs_h_pfc_tri_append_earlier_oldsum_body_steps_partial + S (fs_r_pfc_tri_append_earlier_oldsum_body_steps) = S ((S (fs_i_pfc_tri_append_earlier_oldsum_body_steps)) * fs_v_pfc_tri_append_earlier_oldsum)) /\\ exists fs_q_pfc_tri_append_earlier_oldsum_body_steps_partial. fs_u_pfc_tri_append_earlier_oldsum = fs_q_pfc_tri_append_earlier_oldsum_body_steps_partial * S ((S (fs_i_pfc_tri_append_earlier_oldsum_body_steps)) * fs_v_pfc_tri_append_earlier_oldsum) + (fs_r_pfc_tri_append_earlier_oldsum_body_steps))) /\\ ((((exists fs_h_pfc_tri_append_earlier_oldsum_body_steps_successor. fs_h_pfc_tri_append_earlier_oldsum_body_steps_successor + S (fs_s_pfc_tri_append_earlier_oldsum_body_steps) = S ((S (S fs_i_pfc_tri_append_earlier_oldsum_body_steps)) * fs_v_pfc_tri_append_earlier_oldsum)) /\\ exists fs_q_pfc_tri_append_earlier_oldsum_body_steps_successor. fs_u_pfc_tri_append_earlier_oldsum = fs_q_pfc_tri_append_earlier_oldsum_body_steps_successor * S ((S (S fs_i_pfc_tri_append_earlier_oldsum_body_steps)) * fs_v_pfc_tri_append_earlier_oldsum) + (fs_s_pfc_tri_append_earlier_oldsum_body_steps))) /\\ fs_s_pfc_tri_append_earlier_oldsum_body_steps = fs_r_pfc_tri_append_earlier_oldsum_body_steps + fs_a_pfc_tri_append_earlier_oldsum_body_steps)))))) /\\ ((((exists pfa_gap_tri_append_earlier_oldresiduebound. pfa_gap_tri_append_earlier_oldresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_tri_append_earlier_oldresiduecongruence pfa_offset_right_tri_append_earlier_oldresiduecongruence. (pfc_natural_sum_tri_append_earlier_old) + (p) * pfa_offset_left_tri_append_earlier_oldresiduecongruence = (r) + (p) * pfa_offset_right_tri_append_earlier_oldresiduecongruence))))))))) -> (exists pfc_terms_code_tri_append_earlier_new pfc_terms_scale_tri_append_earlier_new pfc_natural_sum_tri_append_earlier_new. ((forall pfc_index_tri_append_earlier_newdiagonal. (exists pfa_gap_tri_append_earlier_newdiagonalbound. pfa_gap_tri_append_earlier_newdiagonalbound + S (pfc_index_tri_append_earlier_newdiagonal) = (S (i))) -> exists pfc_value_tri_append_earlier_newdiagonal. ((((exists ff_h_pfp_tri_append_earlier_newdiagonalentry. ff_h_pfp_tri_append_earlier_newdiagonalentry + S (pfc_value_tri_append_earlier_newdiagonal) = S ((S (pfc_index_tri_append_earlier_newdiagonal)) * pfc_terms_scale_tri_append_earlier_new)) /\\ exists ff_q_pfp_tri_append_earlier_newdiagonalentry. pfc_terms_code_tri_append_earlier_new = ff_q_pfp_tri_append_earlier_newdiagonalentry * S ((S (pfc_index_tri_append_earlier_newdiagonal)) * pfc_terms_scale_tri_append_earlier_new) + (pfc_value_tri_append_earlier_newdiagonal))) /\\ ((exists pfc_complement_tri_append_earlier_newdiagonalterm pfc_left_tri_append_earlier_newdiagonalterm pfc_right_tri_append_earlier_newdiagonalterm. (((pfc_index_tri_append_earlier_newdiagonal)+pfc_complement_tri_append_earlier_newdiagonalterm=(i)) /\\ ((((((exists pfa_gap_tri_append_earlier_newdiagonaltermleftinside. pfa_gap_tri_append_earlier_newdiagonaltermleftinside + S (pfc_index_tri_append_earlier_newdiagonal) = (S N)) /\\ ((((exists ff_h_pfp_tri_append_earlier_newdiagonaltermleftentry. ff_h_pfp_tri_append_earlier_newdiagonaltermleftentry + S (pfc_left_tri_append_earlier_newdiagonalterm) = S ((S (pfc_index_tri_append_earlier_newdiagonal)) * AC)) /\\ exists ff_q_pfp_tri_append_earlier_newdiagonaltermleftentry. AB = ff_q_pfp_tri_append_earlier_newdiagonaltermleftentry * S ((S (pfc_index_tri_append_earlier_newdiagonal)) * AC) + (pfc_left_tri_append_earlier_newdiagonalterm)))))) \\/ (((exists pfc_gap_tri_append_earlier_newdiagonaltermleftoutside. pfc_gap_tri_append_earlier_newdiagonaltermleftoutside+(S N)=(pfc_index_tri_append_earlier_newdiagonal)) /\\ (((pfc_left_tri_append_earlier_newdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_tri_append_earlier_newdiagonaltermrightinside. pfa_gap_tri_append_earlier_newdiagonaltermrightinside + S (pfc_complement_tri_append_earlier_newdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_tri_append_earlier_newdiagonaltermrightentry. ff_h_pfp_tri_append_earlier_newdiagonaltermrightentry + S (pfc_right_tri_append_earlier_newdiagonalterm) = S ((S (pfc_complement_tri_append_earlier_newdiagonalterm)) * bc)) /\\ exists ff_q_pfp_tri_append_earlier_newdiagonaltermrightentry. bb = ff_q_pfp_tri_append_earlier_newdiagonaltermrightentry * S ((S (pfc_complement_tri_append_earlier_newdiagonalterm)) * bc) + (pfc_right_tri_append_earlier_newdiagonalterm)))))) \\/ (((exists pfc_gap_tri_append_earlier_newdiagonaltermrightoutside. pfc_gap_tri_append_earlier_newdiagonaltermrightoutside+(M)=(pfc_complement_tri_append_earlier_newdiagonalterm)) /\\ (((pfc_right_tri_append_earlier_newdiagonalterm)=0))))) /\\ (((pfc_value_tri_append_earlier_newdiagonal)=pfc_left_tri_append_earlier_newdiagonalterm*pfc_right_tri_append_earlier_newdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_tri_append_earlier_newsum fs_v_pfc_tri_append_earlier_newsum. ((((exists fs_h_pfc_tri_append_earlier_newsum_body_start. fs_h_pfc_tri_append_earlier_newsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_tri_append_earlier_newsum)) /\\ exists fs_q_pfc_tri_append_earlier_newsum_body_start. fs_u_pfc_tri_append_earlier_newsum = fs_q_pfc_tri_append_earlier_newsum_body_start * S ((S (0)) * fs_v_pfc_tri_append_earlier_newsum) + (0))) /\\ ((((exists fs_h_pfc_tri_append_earlier_newsum_body_terminal. fs_h_pfc_tri_append_earlier_newsum_body_terminal + S (pfc_natural_sum_tri_append_earlier_new) = S ((S (S (i))) * fs_v_pfc_tri_append_earlier_newsum)) /\\ exists fs_q_pfc_tri_append_earlier_newsum_body_terminal. fs_u_pfc_tri_append_earlier_newsum = fs_q_pfc_tri_append_earlier_newsum_body_terminal * S ((S (S (i))) * fs_v_pfc_tri_append_earlier_newsum) + (pfc_natural_sum_tri_append_earlier_new))) /\\ forall fs_i_pfc_tri_append_earlier_newsum_body_steps. (exists fs_lt_pfc_tri_append_earlier_newsum_body_steps_bound. fs_lt_pfc_tri_append_earlier_newsum_body_steps_bound + S fs_i_pfc_tri_append_earlier_newsum_body_steps = S (i)) -> exists fs_a_pfc_tri_append_earlier_newsum_body_steps fs_r_pfc_tri_append_earlier_newsum_body_steps fs_s_pfc_tri_append_earlier_newsum_body_steps. ((((exists fs_h_pfc_tri_append_earlier_newsum_body_steps_summand. fs_h_pfc_tri_append_earlier_newsum_body_steps_summand + S (fs_a_pfc_tri_append_earlier_newsum_body_steps) = S ((S (fs_i_pfc_tri_append_earlier_newsum_body_steps)) * pfc_terms_scale_tri_append_earlier_new)) /\\ exists fs_q_pfc_tri_append_earlier_newsum_body_steps_summand. pfc_terms_code_tri_append_earlier_new = fs_q_pfc_tri_append_earlier_newsum_body_steps_summand * S ((S (fs_i_pfc_tri_append_earlier_newsum_body_steps)) * pfc_terms_scale_tri_append_earlier_new) + (fs_a_pfc_tri_append_earlier_newsum_body_steps))) /\\ ((((exists fs_h_pfc_tri_append_earlier_newsum_body_steps_partial. fs_h_pfc_tri_append_earlier_newsum_body_steps_partial + S (fs_r_pfc_tri_append_earlier_newsum_body_steps) = S ((S (fs_i_pfc_tri_append_earlier_newsum_body_steps)) * fs_v_pfc_tri_append_earlier_newsum)) /\\ exists fs_q_pfc_tri_append_earlier_newsum_body_steps_partial. fs_u_pfc_tri_append_earlier_newsum = fs_q_pfc_tri_append_earlier_newsum_body_steps_partial * S ((S (fs_i_pfc_tri_append_earlier_newsum_body_steps)) * fs_v_pfc_tri_append_earlier_newsum) + (fs_r_pfc_tri_append_earlier_newsum_body_steps))) /\\ ((((exists fs_h_pfc_tri_append_earlier_newsum_body_steps_successor. fs_h_pfc_tri_append_earlier_newsum_body_steps_successor + S (fs_s_pfc_tri_append_earlier_newsum_body_steps) = S ((S (S fs_i_pfc_tri_append_earlier_newsum_body_steps)) * fs_v_pfc_tri_append_earlier_newsum)) /\\ exists fs_q_pfc_tri_append_earlier_newsum_body_steps_successor. fs_u_pfc_tri_append_earlier_newsum = fs_q_pfc_tri_append_earlier_newsum_body_steps_successor * S ((S (S fs_i_pfc_tri_append_earlier_newsum_body_steps)) * fs_v_pfc_tri_append_earlier_newsum) + (fs_s_pfc_tri_append_earlier_newsum_body_steps))) /\\ fs_s_pfc_tri_append_earlier_newsum_body_steps = fs_r_pfc_tri_append_earlier_newsum_body_steps + fs_a_pfc_tri_append_earlier_newsum_body_steps)))))) /\\ ((((exists pfa_gap_tri_append_earlier_newresiduebound. pfa_gap_tri_append_earlier_newresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_tri_append_earlier_newresiduecongruence pfa_offset_right_tri_append_earlier_newresiduecongruence. 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(exists pfc_complement_tri_last_old pfc_left_tri_last_old pfc_right_tri_last_old. (((N)+pfc_complement_tri_last_old=(N)) /\\ ((((((exists pfa_gap_tri_last_oldleftinside. pfa_gap_tri_last_oldleftinside + S (N) = (N)) /\\ ((((exists ff_h_pfp_tri_last_oldleftentry. ff_h_pfp_tri_last_oldleftentry + S (pfc_left_tri_last_old) = S ((S (N)) * ac)) /\\ exists ff_q_pfp_tri_last_oldleftentry. ab = ff_q_pfp_tri_last_oldleftentry * S ((S (N)) * ac) + (pfc_left_tri_last_old)))))) \\/ (((exists pfc_gap_tri_last_oldleftoutside. pfc_gap_tri_last_oldleftoutside+(N)=(N)) /\\ (((pfc_left_tri_last_old)=0))))) /\\ ((((((exists pfa_gap_tri_last_oldrightinside. pfa_gap_tri_last_oldrightinside + S (pfc_complement_tri_last_old) = (M)) /\\ ((((exists ff_h_pfp_tri_last_oldrightentry. ff_h_pfp_tri_last_oldrightentry + S (pfc_right_tri_last_old) = S ((S (pfc_complement_tri_last_old)) * bc)) /\\ exists ff_q_pfp_tri_last_oldrightentry. bb = ff_q_pfp_tri_last_oldrightentry * S ((S (pfc_complement_tri_last_old)) * bc) + (pfc_right_tri_last_old)))))) \\/ (((exists pfc_gap_tri_last_oldrightoutside. 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(((exists ff_h_pfp_tri_last_new_entry. ff_h_pfp_tri_last_new_entry + S (a) = S ((S (N)) * ac)) /\\ exists ff_q_pfp_tri_last_new_entry. ab = ff_q_pfp_tri_last_new_entry * S ((S (N)) * ac) + (a))) -> (((exists ff_h_pfp_tri_last_leading_entry. ff_h_pfp_tri_last_leading_entry + S (b) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_tri_last_leading_entry. bb = ff_q_pfp_tri_last_leading_entry * S ((S (0)) * bc) + (b))) -> (exists pfc_complement_tri_last_new pfc_left_tri_last_new pfc_right_tri_last_new. (((N)+pfc_complement_tri_last_new=(N)) /\\ ((((((exists pfa_gap_tri_last_newleftinside. pfa_gap_tri_last_newleftinside + S (N) = (S N)) /\\ ((((exists ff_h_pfp_tri_last_newleftentry. ff_h_pfp_tri_last_newleftentry + S (pfc_left_tri_last_new) = S ((S (N)) * ac)) /\\ exists ff_q_pfp_tri_last_newleftentry. ab = ff_q_pfp_tri_last_newleftentry * S ((S (N)) * ac) + (pfc_left_tri_last_new)))))) \\/ (((exists pfc_gap_tri_last_newleftoutside. pfc_gap_tri_last_newleftoutside+(S N)=(N)) /\\ (((pfc_left_tri_last_new)=0))))) /\\ ((((((exists pfa_gap_tri_last_newrightinside. pfa_gap_tri_last_newrightinside + S (pfc_complement_tri_last_new) = (S d)) /\\ ((((exists ff_h_pfp_tri_last_newrightentry. ff_h_pfp_tri_last_newrightentry + S (pfc_right_tri_last_new) = S ((S (pfc_complement_tri_last_new)) * bc)) /\\ exists ff_q_pfp_tri_last_newrightentry. bb = ff_q_pfp_tri_last_newrightentry * S ((S (pfc_complement_tri_last_new)) * bc) + (pfc_right_tri_last_new)))))) \\/ (((exists 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((((exists ff_h_pfp_tri_sum_old_last. ff_h_pfp_tri_sum_old_last + S (t) = S ((S (N)) * dc)) /\\ exists ff_q_pfp_tri_sum_old_last. db = ff_q_pfp_tri_sum_old_last * S ((S (N)) * dc) + (t))) /\\ (((exists fs_u_pfc_tri_sum_old_prefix fs_v_pfc_tri_sum_old_prefix. ((((exists fs_h_pfc_tri_sum_old_prefix_body_start. fs_h_pfc_tri_sum_old_prefix_body_start + S (0) = S ((S (0)) * fs_v_pfc_tri_sum_old_prefix)) /\\ exists fs_q_pfc_tri_sum_old_prefix_body_start. fs_u_pfc_tri_sum_old_prefix = fs_q_pfc_tri_sum_old_prefix_body_start * S ((S (0)) * fs_v_pfc_tri_sum_old_prefix) + (0))) /\\ ((((exists fs_h_pfc_tri_sum_old_prefix_body_terminal. fs_h_pfc_tri_sum_old_prefix_body_terminal + S (s) = S ((S (N)) * fs_v_pfc_tri_sum_old_prefix)) /\\ exists fs_q_pfc_tri_sum_old_prefix_body_terminal. fs_u_pfc_tri_sum_old_prefix = fs_q_pfc_tri_sum_old_prefix_body_terminal * S ((S (N)) * fs_v_pfc_tri_sum_old_prefix) + (s))) /\\ forall fs_i_pfc_tri_sum_old_prefix_body_steps. (exists fs_lt_pfc_tri_sum_old_prefix_body_steps_bound. fs_lt_pfc_tri_sum_old_prefix_body_steps_bound + S fs_i_pfc_tri_sum_old_prefix_body_steps = N) -> exists fs_a_pfc_tri_sum_old_prefix_body_steps fs_r_pfc_tri_sum_old_prefix_body_steps fs_s_pfc_tri_sum_old_prefix_body_steps. ((((exists fs_h_pfc_tri_sum_old_prefix_body_steps_summand. fs_h_pfc_tri_sum_old_prefix_body_steps_summand + S (fs_a_pfc_tri_sum_old_prefix_body_steps) = S ((S (fs_i_pfc_tri_sum_old_prefix_body_steps)) * dc)) /\\ exists fs_q_pfc_tri_sum_old_prefix_body_steps_summand. db = fs_q_pfc_tri_sum_old_prefix_body_steps_summand * S ((S (fs_i_pfc_tri_sum_old_prefix_body_steps)) * dc) + (fs_a_pfc_tri_sum_old_prefix_body_steps))) /\\ ((((exists fs_h_pfc_tri_sum_old_prefix_body_steps_partial. fs_h_pfc_tri_sum_old_prefix_body_steps_partial + S (fs_r_pfc_tri_sum_old_prefix_body_steps) = S ((S (fs_i_pfc_tri_sum_old_prefix_body_steps)) * fs_v_pfc_tri_sum_old_prefix)) /\\ exists fs_q_pfc_tri_sum_old_prefix_body_steps_partial. fs_u_pfc_tri_sum_old_prefix = fs_q_pfc_tri_sum_old_prefix_body_steps_partial * S ((S (fs_i_pfc_tri_sum_old_prefix_body_steps)) * fs_v_pfc_tri_sum_old_prefix) + (fs_r_pfc_tri_sum_old_prefix_body_steps))) /\\ ((((exists fs_h_pfc_tri_sum_old_prefix_body_steps_successor. fs_h_pfc_tri_sum_old_prefix_body_steps_successor + S (fs_s_pfc_tri_sum_old_prefix_body_steps) = S ((S (S fs_i_pfc_tri_sum_old_prefix_body_steps)) * fs_v_pfc_tri_sum_old_prefix)) /\\ exists fs_q_pfc_tri_sum_old_prefix_body_steps_successor. fs_u_pfc_tri_sum_old_prefix = fs_q_pfc_tri_sum_old_prefix_body_steps_successor * S ((S (S fs_i_pfc_tri_sum_old_prefix_body_steps)) * fs_v_pfc_tri_sum_old_prefix) + (fs_s_pfc_tri_sum_old_prefix_body_steps))) /\\ fs_s_pfc_tri_sum_old_prefix_body_steps = fs_r_pfc_tri_sum_old_prefix_body_steps + fs_a_pfc_tri_sum_old_prefix_body_steps)))))) /\\ ((u=s+t)))))", "specialize beta_sum_succ_decompose (db)", "specialize beta_sum_succ_decompose (dc)", "specialize beta_sum_succ_decompose (N)", "specialize beta_sum_succ_decompose (u)", "apply beta_sum_succ_decompose", "exact hu", "cases hold", "cases hold_witness", "cases hold_witness_witness", "cases hold_witness_witness_right", "have hnew : exists t s. ((((exists ff_h_pfp_tri_sum_new_last. ff_h_pfp_tri_sum_new_last + S (t) = S ((S (N)) * ec)) /\\ exists ff_q_pfp_tri_sum_new_last. eb = ff_q_pfp_tri_sum_new_last * S ((S (N)) * ec) + (t))) /\\ (((exists fs_u_pfc_tri_sum_new_prefix fs_v_pfc_tri_sum_new_prefix. ((((exists fs_h_pfc_tri_sum_new_prefix_body_start. fs_h_pfc_tri_sum_new_prefix_body_start + S (0) = S ((S (0)) * fs_v_pfc_tri_sum_new_prefix)) /\\ exists fs_q_pfc_tri_sum_new_prefix_body_start. fs_u_pfc_tri_sum_new_prefix = fs_q_pfc_tri_sum_new_prefix_body_start * S ((S (0)) * fs_v_pfc_tri_sum_new_prefix) + (0))) /\\ ((((exists fs_h_pfc_tri_sum_new_prefix_body_terminal. fs_h_pfc_tri_sum_new_prefix_body_terminal + S (s) = S ((S (N)) * fs_v_pfc_tri_sum_new_prefix)) /\\ exists fs_q_pfc_tri_sum_new_prefix_body_terminal. fs_u_pfc_tri_sum_new_prefix = fs_q_pfc_tri_sum_new_prefix_body_terminal * S ((S (N)) * fs_v_pfc_tri_sum_new_prefix) + (s))) /\\ forall fs_i_pfc_tri_sum_new_prefix_body_steps. (exists fs_lt_pfc_tri_sum_new_prefix_body_steps_bound. fs_lt_pfc_tri_sum_new_prefix_body_steps_bound + S fs_i_pfc_tri_sum_new_prefix_body_steps = N) -> exists fs_a_pfc_tri_sum_new_prefix_body_steps fs_r_pfc_tri_sum_new_prefix_body_steps fs_s_pfc_tri_sum_new_prefix_body_steps. ((((exists fs_h_pfc_tri_sum_new_prefix_body_steps_summand. fs_h_pfc_tri_sum_new_prefix_body_steps_summand + S (fs_a_pfc_tri_sum_new_prefix_body_steps) = S ((S (fs_i_pfc_tri_sum_new_prefix_body_steps)) * ec)) /\\ exists fs_q_pfc_tri_sum_new_prefix_body_steps_summand. eb = fs_q_pfc_tri_sum_new_prefix_body_steps_summand * S ((S (fs_i_pfc_tri_sum_new_prefix_body_steps)) * ec) + (fs_a_pfc_tri_sum_new_prefix_body_steps))) /\\ ((((exists fs_h_pfc_tri_sum_new_prefix_body_steps_partial. fs_h_pfc_tri_sum_new_prefix_body_steps_partial + S (fs_r_pfc_tri_sum_new_prefix_body_steps) = S ((S (fs_i_pfc_tri_sum_new_prefix_body_steps)) * fs_v_pfc_tri_sum_new_prefix)) /\\ exists fs_q_pfc_tri_sum_new_prefix_body_steps_partial. fs_u_pfc_tri_sum_new_prefix = fs_q_pfc_tri_sum_new_prefix_body_steps_partial * S ((S (fs_i_pfc_tri_sum_new_prefix_body_steps)) * fs_v_pfc_tri_sum_new_prefix) + (fs_r_pfc_tri_sum_new_prefix_body_steps))) /\\ ((((exists fs_h_pfc_tri_sum_new_prefix_body_steps_successor. fs_h_pfc_tri_sum_new_prefix_body_steps_successor + S (fs_s_pfc_tri_sum_new_prefix_body_steps) = S ((S (S fs_i_pfc_tri_sum_new_prefix_body_steps)) * fs_v_pfc_tri_sum_new_prefix)) /\\ exists fs_q_pfc_tri_sum_new_prefix_body_steps_successor. fs_u_pfc_tri_sum_new_prefix = fs_q_pfc_tri_sum_new_prefix_body_steps_successor * S ((S (S fs_i_pfc_tri_sum_new_prefix_body_steps)) * fs_v_pfc_tri_sum_new_prefix) + (fs_s_pfc_tri_sum_new_prefix_body_steps))) /\\ fs_s_pfc_tri_sum_new_prefix_body_steps = fs_r_pfc_tri_sum_new_prefix_body_steps + fs_a_pfc_tri_sum_new_prefix_body_steps)))))) /\\ ((v=s+t)))))", "specialize beta_sum_succ_decompose (eb)", "specialize beta_sum_succ_decompose (ec)", "specialize beta_sum_succ_decompose (N)", "specialize beta_sum_succ_decompose (v)", "apply beta_sum_succ_decompose", "exact hv", "cases hnew", "cases hnew_witness", "cases hnew_witness_witness", "cases hnew_witness_witness_right", "have hz : x=0", "specialize polynomial_diagonal_last_term_left_empty (ab)", "specialize polynomial_diagonal_last_term_left_empty (ac)", "specialize polynomial_diagonal_last_term_left_empty (bb)", "specialize polynomial_diagonal_last_term_left_empty (bc)", "specialize polynomial_diagonal_last_term_left_empty (S d)", "specialize polynomial_diagonal_last_term_left_empty (N)", "specialize polynomial_diagonal_last_term_left_empty (x)", "apply polynomial_diagonal_last_term_left_empty", "specialize polynomial_diagonal_prefix_entry (ab)", "specialize polynomial_diagonal_prefix_entry (ac)", "specialize polynomial_diagonal_prefix_entry (N)", "specialize polynomial_diagonal_prefix_entry (bb)", "specialize polynomial_diagonal_prefix_entry (bc)", "specialize polynomial_diagonal_prefix_entry (S d)", "specialize polynomial_diagonal_prefix_entry (N)", "specialize polynomial_diagonal_prefix_entry (db)", "specialize polynomial_diagonal_prefix_entry (dc)", "specialize polynomial_diagonal_prefix_entry (S N)", "specialize polynomial_diagonal_prefix_entry (N)", "specialize polynomial_diagonal_prefix_entry (x)", "apply polynomial_diagonal_prefix_entry", "exact hd", "specialize le_refl (S N)", "apply le_refl", "exact hold_witness_witness_left", "have ht : x2=a*b", "specialize polynomial_diagonal_last_term_left_append (AB)", "specialize polynomial_diagonal_last_term_left_append (AC)", "specialize polynomial_diagonal_last_term_left_append (bb)", "specialize polynomial_diagonal_last_term_left_append (bc)", "specialize polynomial_diagonal_last_term_left_append (d)", "specialize polynomial_diagonal_last_term_left_append (N)", "specialize polynomial_diagonal_last_term_left_append (a)", "specialize polynomial_diagonal_last_term_left_append (b)", "specialize polynomial_diagonal_last_term_left_append (x2)", "apply polynomial_diagonal_last_term_left_append", "exact ha", "exact hb", "specialize polynomial_diagonal_prefix_entry (AB)", "specialize polynomial_diagonal_prefix_entry (AC)", "specialize polynomial_diagonal_prefix_entry (S N)", "specialize polynomial_diagonal_prefix_entry (bb)", "specialize polynomial_diagonal_prefix_entry (bc)", "specialize polynomial_diagonal_prefix_entry (S d)", "specialize polynomial_diagonal_prefix_entry (N)", "specialize polynomial_diagonal_prefix_entry (eb)", "specialize polynomial_diagonal_prefix_entry (ec)", "specialize polynomial_diagonal_prefix_entry (S N)", "specialize polynomial_diagonal_prefix_entry (N)", "specialize polynomial_diagonal_prefix_entry (x2)", "apply polynomial_diagonal_prefix_entry", "exact hetable", "specialize le_refl (S N)", "apply le_refl", "exact hnew_witness_witness_left", "have hprefix : forall pfc_index_tri_sum_transported. (exists pfa_gap_tri_sum_transportedbound. pfa_gap_tri_sum_transportedbound + S (pfc_index_tri_sum_transported) = (N)) -> exists pfc_value_tri_sum_transported. ((((exists ff_h_pfp_tri_sum_transportedentry. ff_h_pfp_tri_sum_transportedentry + S (pfc_value_tri_sum_transported) = S ((S (pfc_index_tri_sum_transported)) * dc)) /\\ exists ff_q_pfp_tri_sum_transportedentry. db = ff_q_pfp_tri_sum_transportedentry * S ((S (pfc_index_tri_sum_transported)) * dc) + (pfc_value_tri_sum_transported))) /\\ ((exists pfc_complement_tri_sum_transportedterm pfc_left_tri_sum_transportedterm pfc_right_tri_sum_transportedterm. (((pfc_index_tri_sum_transported)+pfc_complement_tri_sum_transportedterm=(N)) /\\ ((((((exists pfa_gap_tri_sum_transportedtermleftinside. pfa_gap_tri_sum_transportedtermleftinside + S (pfc_index_tri_sum_transported) = (S N)) /\\ ((((exists ff_h_pfp_tri_sum_transportedtermleftentry. ff_h_pfp_tri_sum_transportedtermleftentry + S (pfc_left_tri_sum_transportedterm) = S ((S (pfc_index_tri_sum_transported)) * AC)) /\\ exists ff_q_pfp_tri_sum_transportedtermleftentry. AB = ff_q_pfp_tri_sum_transportedtermleftentry * S ((S (pfc_index_tri_sum_transported)) * AC) + (pfc_left_tri_sum_transportedterm)))))) \\/ (((exists pfc_gap_tri_sum_transportedtermleftoutside. pfc_gap_tri_sum_transportedtermleftoutside+(S N)=(pfc_index_tri_sum_transported)) /\\ (((pfc_left_tri_sum_transportedterm)=0))))) /\\ ((((((exists pfa_gap_tri_sum_transportedtermrightinside. pfa_gap_tri_sum_transportedtermrightinside + S (pfc_complement_tri_sum_transportedterm) = (S d)) /\\ ((((exists ff_h_pfp_tri_sum_transportedtermrightentry. ff_h_pfp_tri_sum_transportedtermrightentry + S (pfc_right_tri_sum_transportedterm) = S ((S (pfc_complement_tri_sum_transportedterm)) * bc)) /\\ exists ff_q_pfp_tri_sum_transportedtermrightentry. bb = ff_q_pfp_tri_sum_transportedtermrightentry * S ((S (pfc_complement_tri_sum_transportedterm)) * bc) + (pfc_right_tri_sum_transportedterm)))))) \\/ (((exists pfc_gap_tri_sum_transportedtermrightoutside. pfc_gap_tri_sum_transportedtermrightoutside+(S d)=(pfc_complement_tri_sum_transportedterm)) /\\ (((pfc_right_tri_sum_transportedterm)=0))))) /\\ (((pfc_value_tri_sum_transported)=pfc_left_tri_sum_transportedterm*pfc_right_tri_sum_transportedterm))))))))))", "specialize polynomial_diagonal_prefix_left_transport (ab)", "specialize polynomial_diagonal_prefix_left_transport (ac)", "specialize polynomial_diagonal_prefix_left_transport (N)", "specialize polynomial_diagonal_prefix_left_transport (AB)", "specialize polynomial_diagonal_prefix_left_transport (AC)", "specialize polynomial_diagonal_prefix_left_transport (S N)", "specialize polynomial_diagonal_prefix_left_transport (bb)", "specialize polynomial_diagonal_prefix_left_transport (bc)", "specialize polynomial_diagonal_prefix_left_transport (S d)", "specialize polynomial_diagonal_prefix_left_transport (N)", "specialize polynomial_diagonal_prefix_left_transport (N)", "specialize polynomial_diagonal_prefix_left_transport (db)", "specialize polynomial_diagonal_prefix_left_transport (dc)", "apply polynomial_diagonal_prefix_left_transport", "specialize le_refl (N)", "apply le_refl", "specialize le_succ (N)", "specialize le_succ (N)", "apply le_succ", "specialize le_refl (N)", "apply le_refl", "exact he", "intro j", "intro hj", "specialize hd (j)", "apply hd", "specialize le_succ (S j)", "specialize le_succ (N)", "apply le_succ", "exact hj", "have hs : exists fs_u_pfc_tri_sum_same_prefix fs_v_pfc_tri_sum_same_prefix. ((((exists fs_h_pfc_tri_sum_same_prefix_body_start. fs_h_pfc_tri_sum_same_prefix_body_start + S (0) = S ((S (0)) * fs_v_pfc_tri_sum_same_prefix)) /\\ exists fs_q_pfc_tri_sum_same_prefix_body_start. fs_u_pfc_tri_sum_same_prefix = fs_q_pfc_tri_sum_same_prefix_body_start * S ((S (0)) * fs_v_pfc_tri_sum_same_prefix) + (0))) /\\ ((((exists fs_h_pfc_tri_sum_same_prefix_body_terminal. fs_h_pfc_tri_sum_same_prefix_body_terminal + S (x1) = S ((S (N)) * fs_v_pfc_tri_sum_same_prefix)) /\\ exists fs_q_pfc_tri_sum_same_prefix_body_terminal. fs_u_pfc_tri_sum_same_prefix = fs_q_pfc_tri_sum_same_prefix_body_terminal * S ((S (N)) * fs_v_pfc_tri_sum_same_prefix) + (x1))) /\\ forall fs_i_pfc_tri_sum_same_prefix_body_steps. (exists fs_lt_pfc_tri_sum_same_prefix_body_steps_bound. fs_lt_pfc_tri_sum_same_prefix_body_steps_bound + S fs_i_pfc_tri_sum_same_prefix_body_steps = N) -> exists fs_a_pfc_tri_sum_same_prefix_body_steps fs_r_pfc_tri_sum_same_prefix_body_steps fs_s_pfc_tri_sum_same_prefix_body_steps. ((((exists fs_h_pfc_tri_sum_same_prefix_body_steps_summand. fs_h_pfc_tri_sum_same_prefix_body_steps_summand + S (fs_a_pfc_tri_sum_same_prefix_body_steps) = S ((S (fs_i_pfc_tri_sum_same_prefix_body_steps)) * ec)) /\\ exists fs_q_pfc_tri_sum_same_prefix_body_steps_summand. eb = fs_q_pfc_tri_sum_same_prefix_body_steps_summand * S ((S (fs_i_pfc_tri_sum_same_prefix_body_steps)) * ec) + (fs_a_pfc_tri_sum_same_prefix_body_steps))) /\\ ((((exists fs_h_pfc_tri_sum_same_prefix_body_steps_partial. fs_h_pfc_tri_sum_same_prefix_body_steps_partial + S (fs_r_pfc_tri_sum_same_prefix_body_steps) = S ((S (fs_i_pfc_tri_sum_same_prefix_body_steps)) * fs_v_pfc_tri_sum_same_prefix)) /\\ exists fs_q_pfc_tri_sum_same_prefix_body_steps_partial. fs_u_pfc_tri_sum_same_prefix = fs_q_pfc_tri_sum_same_prefix_body_steps_partial * S ((S (fs_i_pfc_tri_sum_same_prefix_body_steps)) * fs_v_pfc_tri_sum_same_prefix) + (fs_r_pfc_tri_sum_same_prefix_body_steps))) /\\ ((((exists fs_h_pfc_tri_sum_same_prefix_body_steps_successor. fs_h_pfc_tri_sum_same_prefix_body_steps_successor + S (fs_s_pfc_tri_sum_same_prefix_body_steps) = S ((S (S fs_i_pfc_tri_sum_same_prefix_body_steps)) * fs_v_pfc_tri_sum_same_prefix)) /\\ exists fs_q_pfc_tri_sum_same_prefix_body_steps_successor. fs_u_pfc_tri_sum_same_prefix = fs_q_pfc_tri_sum_same_prefix_body_steps_successor * S ((S (S fs_i_pfc_tri_sum_same_prefix_body_steps)) * fs_v_pfc_tri_sum_same_prefix) + (fs_s_pfc_tri_sum_same_prefix_body_steps))) /\\ fs_s_pfc_tri_sum_same_prefix_body_steps = fs_r_pfc_tri_sum_same_prefix_body_steps + fs_a_pfc_tri_sum_same_prefix_body_steps)))))", "specialize beta_sum_transport_prefix (db)", "specialize beta_sum_transport_prefix (dc)", "specialize beta_sum_transport_prefix (eb)", "specialize beta_sum_transport_prefix (ec)", "specialize beta_sum_transport_prefix (N)", "specialize beta_sum_transport_prefix (x1)", "apply beta_sum_transport_prefix", "exact hold_witness_witness_right_left", "specialize polynomial_diagonal_prefix_functional (AB)", "specialize polynomial_diagonal_prefix_functional (AC)", "specialize polynomial_diagonal_prefix_functional (S N)", "specialize polynomial_diagonal_prefix_functional (bb)", "specialize polynomial_diagonal_prefix_functional (bc)", "specialize polynomial_diagonal_prefix_functional (S d)", "specialize polynomial_diagonal_prefix_functional (N)", "specialize polynomial_diagonal_prefix_functional (db)", "specialize polynomial_diagonal_prefix_functional (dc)", "specialize polynomial_diagonal_prefix_functional (eb)", "specialize polynomial_diagonal_prefix_functional (ec)", "specialize polynomial_diagonal_prefix_functional (N)", "apply polynomial_diagonal_prefix_functional", "exact hprefix", "intro j", "intro hj", "specialize hetable (j)", "apply hetable", "specialize le_succ (S j)", "specialize le_succ (N)", "apply le_succ", "exact hj", "have hbase : x1=x3", "specialize beta_sum_functional (eb)", "specialize beta_sum_functional (ec)", "specialize beta_sum_functional (N)", "specialize beta_sum_functional (x1)", "specialize beta_sum_functional (x3)", "apply beta_sum_functional", "exact hs", "exact hnew_witness_witness_right_left", "have huvalue : u=x1", "trans x1+x", "exact hold_witness_witness_right_right", "rewrite hz", "simp", "trans x3+x2", "exact hnew_witness_witness_right_right", "congr", "trans x1", "symm", "exact hbase", "symm", "exact huvalue", "exact ht" ], "script_sha256": "d7e1a640b34310b6a4a92ca533187e18ef0beb30ae7e4b1471597fc65097ed55", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_convolution_triangular_candidate.py", "sha256": "d53722e52ffb3f98d16d693c8cc28d605e62da8f36d5e6ecffe3df66179aa11f" }, "statement": "forall ab ac AB AC bb bc d N a b db dc eb ec u v. (forall mdr_i_pfp_tri_sum_equal mdr_a_pfp_tri_sum_equal. (exists mdr_gap_pfp_tri_sum_equalb. mdr_gap_pfp_tri_sum_equalb + S (mdr_i_pfp_tri_sum_equal) = (N)) -> (((exists ff_h_mdr_pfp_tri_sum_equalo. ff_h_mdr_pfp_tri_sum_equalo + S (mdr_a_pfp_tri_sum_equal) = S ((S (mdr_i_pfp_tri_sum_equal)) * ac)) /\\ exists ff_q_mdr_pfp_tri_sum_equalo. ab = ff_q_mdr_pfp_tri_sum_equalo * S ((S (mdr_i_pfp_tri_sum_equal)) * ac) + (mdr_a_pfp_tri_sum_equal))) -> (((exists ff_h_mdr_pfp_tri_sum_equaln. ff_h_mdr_pfp_tri_sum_equaln + S (mdr_a_pfp_tri_sum_equal) = S ((S (mdr_i_pfp_tri_sum_equal)) * AC)) /\\ exists ff_q_mdr_pfp_tri_sum_equaln. AB = ff_q_mdr_pfp_tri_sum_equaln * S ((S (mdr_i_pfp_tri_sum_equal)) * AC) + (mdr_a_pfp_tri_sum_equal)))) -> (((exists ff_h_pfp_tri_sum_appended. ff_h_pfp_tri_sum_appended + S (a) = S ((S (N)) * AC)) /\\ exists ff_q_pfp_tri_sum_appended. AB = ff_q_pfp_tri_sum_appended * S ((S (N)) * AC) + (a))) -> (((exists ff_h_pfp_tri_sum_head. ff_h_pfp_tri_sum_head + S (b) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_tri_sum_head. bb = ff_q_pfp_tri_sum_head * S ((S (0)) * bc) + (b))) -> (forall pfc_index_tri_sum_old_table. (exists pfa_gap_tri_sum_old_tablebound. pfa_gap_tri_sum_old_tablebound + S (pfc_index_tri_sum_old_table) = (S N)) -> exists pfc_value_tri_sum_old_table. ((((exists ff_h_pfp_tri_sum_old_tableentry. ff_h_pfp_tri_sum_old_tableentry + S (pfc_value_tri_sum_old_table) = S ((S (pfc_index_tri_sum_old_table)) * dc)) /\\ exists ff_q_pfp_tri_sum_old_tableentry. db = ff_q_pfp_tri_sum_old_tableentry * S ((S (pfc_index_tri_sum_old_table)) * dc) + (pfc_value_tri_sum_old_table))) /\\ ((exists pfc_complement_tri_sum_old_tableterm pfc_left_tri_sum_old_tableterm pfc_right_tri_sum_old_tableterm. (((pfc_index_tri_sum_old_table)+pfc_complement_tri_sum_old_tableterm=(N)) /\\ ((((((exists pfa_gap_tri_sum_old_tabletermleftinside. pfa_gap_tri_sum_old_tabletermleftinside + S (pfc_index_tri_sum_old_table) = (N)) /\\ ((((exists ff_h_pfp_tri_sum_old_tabletermleftentry. ff_h_pfp_tri_sum_old_tabletermleftentry + S (pfc_left_tri_sum_old_tableterm) = S ((S (pfc_index_tri_sum_old_table)) * ac)) /\\ exists ff_q_pfp_tri_sum_old_tabletermleftentry. ab = ff_q_pfp_tri_sum_old_tabletermleftentry * S ((S (pfc_index_tri_sum_old_table)) * ac) + (pfc_left_tri_sum_old_tableterm)))))) \\/ (((exists pfc_gap_tri_sum_old_tabletermleftoutside. pfc_gap_tri_sum_old_tabletermleftoutside+(N)=(pfc_index_tri_sum_old_table)) /\\ (((pfc_left_tri_sum_old_tableterm)=0))))) /\\ ((((((exists pfa_gap_tri_sum_old_tabletermrightinside. pfa_gap_tri_sum_old_tabletermrightinside + S (pfc_complement_tri_sum_old_tableterm) = (S d)) /\\ ((((exists ff_h_pfp_tri_sum_old_tabletermrightentry. ff_h_pfp_tri_sum_old_tabletermrightentry + S (pfc_right_tri_sum_old_tableterm) = S ((S (pfc_complement_tri_sum_old_tableterm)) * bc)) /\\ exists ff_q_pfp_tri_sum_old_tabletermrightentry. bb = ff_q_pfp_tri_sum_old_tabletermrightentry * S ((S (pfc_complement_tri_sum_old_tableterm)) * bc) + (pfc_right_tri_sum_old_tableterm)))))) \\/ (((exists pfc_gap_tri_sum_old_tabletermrightoutside. pfc_gap_tri_sum_old_tabletermrightoutside+(S d)=(pfc_complement_tri_sum_old_tableterm)) /\\ (((pfc_right_tri_sum_old_tableterm)=0))))) /\\ (((pfc_value_tri_sum_old_table)=pfc_left_tri_sum_old_tableterm*pfc_right_tri_sum_old_tableterm))))))))))) -> (exists fs_u_pfc_tri_sum_old_actual fs_v_pfc_tri_sum_old_actual. ((((exists fs_h_pfc_tri_sum_old_actual_body_start. fs_h_pfc_tri_sum_old_actual_body_start + S (0) = S ((S (0)) * fs_v_pfc_tri_sum_old_actual)) /\\ exists fs_q_pfc_tri_sum_old_actual_body_start. fs_u_pfc_tri_sum_old_actual = fs_q_pfc_tri_sum_old_actual_body_start * S ((S (0)) * fs_v_pfc_tri_sum_old_actual) + (0))) /\\ ((((exists fs_h_pfc_tri_sum_old_actual_body_terminal. fs_h_pfc_tri_sum_old_actual_body_terminal + S (u) = S ((S (S N)) * fs_v_pfc_tri_sum_old_actual)) /\\ exists fs_q_pfc_tri_sum_old_actual_body_terminal. fs_u_pfc_tri_sum_old_actual = fs_q_pfc_tri_sum_old_actual_body_terminal * S ((S (S N)) * fs_v_pfc_tri_sum_old_actual) + (u))) /\\ forall fs_i_pfc_tri_sum_old_actual_body_steps. (exists fs_lt_pfc_tri_sum_old_actual_body_steps_bound. fs_lt_pfc_tri_sum_old_actual_body_steps_bound + S fs_i_pfc_tri_sum_old_actual_body_steps = S N) -> exists fs_a_pfc_tri_sum_old_actual_body_steps fs_r_pfc_tri_sum_old_actual_body_steps fs_s_pfc_tri_sum_old_actual_body_steps. ((((exists fs_h_pfc_tri_sum_old_actual_body_steps_summand. fs_h_pfc_tri_sum_old_actual_body_steps_summand + S (fs_a_pfc_tri_sum_old_actual_body_steps) = S ((S (fs_i_pfc_tri_sum_old_actual_body_steps)) * dc)) /\\ exists fs_q_pfc_tri_sum_old_actual_body_steps_summand. db = fs_q_pfc_tri_sum_old_actual_body_steps_summand * S ((S (fs_i_pfc_tri_sum_old_actual_body_steps)) * dc) + (fs_a_pfc_tri_sum_old_actual_body_steps))) /\\ ((((exists fs_h_pfc_tri_sum_old_actual_body_steps_partial. fs_h_pfc_tri_sum_old_actual_body_steps_partial + S (fs_r_pfc_tri_sum_old_actual_body_steps) = S ((S (fs_i_pfc_tri_sum_old_actual_body_steps)) * fs_v_pfc_tri_sum_old_actual)) /\\ exists fs_q_pfc_tri_sum_old_actual_body_steps_partial. fs_u_pfc_tri_sum_old_actual = fs_q_pfc_tri_sum_old_actual_body_steps_partial * S ((S (fs_i_pfc_tri_sum_old_actual_body_steps)) * fs_v_pfc_tri_sum_old_actual) + (fs_r_pfc_tri_sum_old_actual_body_steps))) /\\ ((((exists fs_h_pfc_tri_sum_old_actual_body_steps_successor. fs_h_pfc_tri_sum_old_actual_body_steps_successor + S (fs_s_pfc_tri_sum_old_actual_body_steps) = S ((S (S fs_i_pfc_tri_sum_old_actual_body_steps)) * fs_v_pfc_tri_sum_old_actual)) /\\ exists fs_q_pfc_tri_sum_old_actual_body_steps_successor. fs_u_pfc_tri_sum_old_actual = fs_q_pfc_tri_sum_old_actual_body_steps_successor * S ((S (S fs_i_pfc_tri_sum_old_actual_body_steps)) * fs_v_pfc_tri_sum_old_actual) + (fs_s_pfc_tri_sum_old_actual_body_steps))) /\\ fs_s_pfc_tri_sum_old_actual_body_steps = fs_r_pfc_tri_sum_old_actual_body_steps + fs_a_pfc_tri_sum_old_actual_body_steps)))))) -> (forall pfc_index_tri_sum_new_table. (exists pfa_gap_tri_sum_new_tablebound. pfa_gap_tri_sum_new_tablebound + S (pfc_index_tri_sum_new_table) = (S N)) -> exists pfc_value_tri_sum_new_table. ((((exists ff_h_pfp_tri_sum_new_tableentry. ff_h_pfp_tri_sum_new_tableentry + S (pfc_value_tri_sum_new_table) = S ((S (pfc_index_tri_sum_new_table)) * ec)) /\\ exists ff_q_pfp_tri_sum_new_tableentry. eb = ff_q_pfp_tri_sum_new_tableentry * S ((S (pfc_index_tri_sum_new_table)) * ec) + (pfc_value_tri_sum_new_table))) /\\ ((exists pfc_complement_tri_sum_new_tableterm pfc_left_tri_sum_new_tableterm pfc_right_tri_sum_new_tableterm. (((pfc_index_tri_sum_new_table)+pfc_complement_tri_sum_new_tableterm=(N)) /\\ ((((((exists pfa_gap_tri_sum_new_tabletermleftinside. pfa_gap_tri_sum_new_tabletermleftinside + S (pfc_index_tri_sum_new_table) = (S N)) /\\ ((((exists ff_h_pfp_tri_sum_new_tabletermleftentry. ff_h_pfp_tri_sum_new_tabletermleftentry + S (pfc_left_tri_sum_new_tableterm) = S ((S (pfc_index_tri_sum_new_table)) * AC)) /\\ exists ff_q_pfp_tri_sum_new_tabletermleftentry. AB = ff_q_pfp_tri_sum_new_tabletermleftentry * S ((S (pfc_index_tri_sum_new_table)) * AC) + (pfc_left_tri_sum_new_tableterm)))))) \\/ (((exists pfc_gap_tri_sum_new_tabletermleftoutside. pfc_gap_tri_sum_new_tabletermleftoutside+(S N)=(pfc_index_tri_sum_new_table)) /\\ (((pfc_left_tri_sum_new_tableterm)=0))))) /\\ ((((((exists pfa_gap_tri_sum_new_tabletermrightinside. pfa_gap_tri_sum_new_tabletermrightinside + S (pfc_complement_tri_sum_new_tableterm) = (S d)) /\\ ((((exists ff_h_pfp_tri_sum_new_tabletermrightentry. ff_h_pfp_tri_sum_new_tabletermrightentry + S (pfc_right_tri_sum_new_tableterm) = S ((S (pfc_complement_tri_sum_new_tableterm)) * bc)) /\\ exists ff_q_pfp_tri_sum_new_tabletermrightentry. bb = ff_q_pfp_tri_sum_new_tabletermrightentry * S ((S (pfc_complement_tri_sum_new_tableterm)) * bc) + (pfc_right_tri_sum_new_tableterm)))))) \\/ (((exists pfc_gap_tri_sum_new_tabletermrightoutside. pfc_gap_tri_sum_new_tabletermrightoutside+(S d)=(pfc_complement_tri_sum_new_tableterm)) /\\ (((pfc_right_tri_sum_new_tableterm)=0))))) /\\ (((pfc_value_tri_sum_new_table)=pfc_left_tri_sum_new_tableterm*pfc_right_tri_sum_new_tableterm))))))))))) -> (exists fs_u_pfc_tri_sum_new_actual fs_v_pfc_tri_sum_new_actual. ((((exists fs_h_pfc_tri_sum_new_actual_body_start. fs_h_pfc_tri_sum_new_actual_body_start + S (0) = S ((S (0)) * fs_v_pfc_tri_sum_new_actual)) /\\ exists fs_q_pfc_tri_sum_new_actual_body_start. fs_u_pfc_tri_sum_new_actual = fs_q_pfc_tri_sum_new_actual_body_start * S ((S (0)) * fs_v_pfc_tri_sum_new_actual) + (0))) /\\ ((((exists fs_h_pfc_tri_sum_new_actual_body_terminal. fs_h_pfc_tri_sum_new_actual_body_terminal + S (v) = S ((S (S N)) * fs_v_pfc_tri_sum_new_actual)) /\\ exists fs_q_pfc_tri_sum_new_actual_body_terminal. fs_u_pfc_tri_sum_new_actual = fs_q_pfc_tri_sum_new_actual_body_terminal * S ((S (S N)) * fs_v_pfc_tri_sum_new_actual) + (v))) /\\ forall fs_i_pfc_tri_sum_new_actual_body_steps. (exists fs_lt_pfc_tri_sum_new_actual_body_steps_bound. fs_lt_pfc_tri_sum_new_actual_body_steps_bound + S fs_i_pfc_tri_sum_new_actual_body_steps = S N) -> exists fs_a_pfc_tri_sum_new_actual_body_steps fs_r_pfc_tri_sum_new_actual_body_steps fs_s_pfc_tri_sum_new_actual_body_steps. ((((exists fs_h_pfc_tri_sum_new_actual_body_steps_summand. fs_h_pfc_tri_sum_new_actual_body_steps_summand + S (fs_a_pfc_tri_sum_new_actual_body_steps) = S ((S (fs_i_pfc_tri_sum_new_actual_body_steps)) * ec)) /\\ exists fs_q_pfc_tri_sum_new_actual_body_steps_summand. eb = fs_q_pfc_tri_sum_new_actual_body_steps_summand * S ((S (fs_i_pfc_tri_sum_new_actual_body_steps)) * ec) + (fs_a_pfc_tri_sum_new_actual_body_steps))) /\\ ((((exists fs_h_pfc_tri_sum_new_actual_body_steps_partial. fs_h_pfc_tri_sum_new_actual_body_steps_partial + S (fs_r_pfc_tri_sum_new_actual_body_steps) = S ((S (fs_i_pfc_tri_sum_new_actual_body_steps)) * fs_v_pfc_tri_sum_new_actual)) /\\ exists fs_q_pfc_tri_sum_new_actual_body_steps_partial. fs_u_pfc_tri_sum_new_actual = fs_q_pfc_tri_sum_new_actual_body_steps_partial * S ((S (fs_i_pfc_tri_sum_new_actual_body_steps)) * fs_v_pfc_tri_sum_new_actual) + (fs_r_pfc_tri_sum_new_actual_body_steps))) /\\ ((((exists fs_h_pfc_tri_sum_new_actual_body_steps_successor. 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(forall mdr_i_pfp_tri_append_prefix mdr_a_pfp_tri_append_prefix. (exists mdr_gap_pfp_tri_append_prefixb. mdr_gap_pfp_tri_append_prefixb + S (mdr_i_pfp_tri_append_prefix) = (N)) -> (((exists ff_h_mdr_pfp_tri_append_prefixo. ff_h_mdr_pfp_tri_append_prefixo + S (mdr_a_pfp_tri_append_prefix) = S ((S (mdr_i_pfp_tri_append_prefix)) * ac)) /\\ exists ff_q_mdr_pfp_tri_append_prefixo. ab = ff_q_mdr_pfp_tri_append_prefixo * S ((S (mdr_i_pfp_tri_append_prefix)) * ac) + (mdr_a_pfp_tri_append_prefix))) -> (((exists ff_h_mdr_pfp_tri_append_prefixn. ff_h_mdr_pfp_tri_append_prefixn + S (mdr_a_pfp_tri_append_prefix) = S ((S (mdr_i_pfp_tri_append_prefix)) * AC)) /\\ exists ff_q_mdr_pfp_tri_append_prefixn. AB = ff_q_mdr_pfp_tri_append_prefixn * S ((S (mdr_i_pfp_tri_append_prefix)) * AC) + (mdr_a_pfp_tri_append_prefix)))) -> (((exists ff_h_pfp_tri_append_actual_entry. ff_h_pfp_tri_append_actual_entry + S (a) = S ((S (N)) * AC)) /\\ exists ff_q_pfp_tri_append_actual_entry. AB = ff_q_pfp_tri_append_actual_entry * S ((S (N)) * AC) + (a))) -> (((exists ff_h_pfp_tri_append_actual_head. ff_h_pfp_tri_append_actual_head + S (b) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_tri_append_actual_head. bb = ff_q_pfp_tri_append_actual_head * S ((S (0)) * bc) + (b))) -> (exists pfc_terms_code_tri_append_previous_coefficient pfc_terms_scale_tri_append_previous_coefficient pfc_natural_sum_tri_append_previous_coefficient. ((forall pfc_index_tri_append_previous_coefficientdiagonal. (exists pfa_gap_tri_append_previous_coefficientdiagonalbound. pfa_gap_tri_append_previous_coefficientdiagonalbound + S (pfc_index_tri_append_previous_coefficientdiagonal) = (S (N))) -> exists pfc_value_tri_append_previous_coefficientdiagonal. ((((exists ff_h_pfp_tri_append_previous_coefficientdiagonalentry. ff_h_pfp_tri_append_previous_coefficientdiagonalentry + S (pfc_value_tri_append_previous_coefficientdiagonal) = S ((S (pfc_index_tri_append_previous_coefficientdiagonal)) * pfc_terms_scale_tri_append_previous_coefficient)) /\\ exists ff_q_pfp_tri_append_previous_coefficientdiagonalentry. pfc_terms_code_tri_append_previous_coefficient = ff_q_pfp_tri_append_previous_coefficientdiagonalentry * S ((S (pfc_index_tri_append_previous_coefficientdiagonal)) * pfc_terms_scale_tri_append_previous_coefficient) + (pfc_value_tri_append_previous_coefficientdiagonal))) /\\ ((exists pfc_complement_tri_append_previous_coefficientdiagonalterm pfc_left_tri_append_previous_coefficientdiagonalterm pfc_right_tri_append_previous_coefficientdiagonalterm. (((pfc_index_tri_append_previous_coefficientdiagonal)+pfc_complement_tri_append_previous_coefficientdiagonalterm=(N)) /\\ ((((((exists pfa_gap_tri_append_previous_coefficientdiagonaltermleftinside. pfa_gap_tri_append_previous_coefficientdiagonaltermleftinside + S (pfc_index_tri_append_previous_coefficientdiagonal) = (N)) /\\ ((((exists ff_h_pfp_tri_append_previous_coefficientdiagonaltermleftentry. ff_h_pfp_tri_append_previous_coefficientdiagonaltermleftentry + S (pfc_left_tri_append_previous_coefficientdiagonalterm) = S ((S (pfc_index_tri_append_previous_coefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_tri_append_previous_coefficientdiagonaltermleftentry. ab = ff_q_pfp_tri_append_previous_coefficientdiagonaltermleftentry * S ((S (pfc_index_tri_append_previous_coefficientdiagonal)) * ac) + (pfc_left_tri_append_previous_coefficientdiagonalterm)))))) \\/ (((exists pfc_gap_tri_append_previous_coefficientdiagonaltermleftoutside. pfc_gap_tri_append_previous_coefficientdiagonaltermleftoutside+(N)=(pfc_index_tri_append_previous_coefficientdiagonal)) /\\ (((pfc_left_tri_append_previous_coefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_tri_append_previous_coefficientdiagonaltermrightinside. pfa_gap_tri_append_previous_coefficientdiagonaltermrightinside + S (pfc_complement_tri_append_previous_coefficientdiagonalterm) = (S d)) /\\ ((((exists ff_h_pfp_tri_append_previous_coefficientdiagonaltermrightentry. ff_h_pfp_tri_append_previous_coefficientdiagonaltermrightentry + S (pfc_right_tri_append_previous_coefficientdiagonalterm) = S ((S (pfc_complement_tri_append_previous_coefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_tri_append_previous_coefficientdiagonaltermrightentry. bb = ff_q_pfp_tri_append_previous_coefficientdiagonaltermrightentry * S ((S (pfc_complement_tri_append_previous_coefficientdiagonalterm)) * bc) + (pfc_right_tri_append_previous_coefficientdiagonalterm)))))) \\/ (((exists pfc_gap_tri_append_previous_coefficientdiagonaltermrightoutside. pfc_gap_tri_append_previous_coefficientdiagonaltermrightoutside+(S d)=(pfc_complement_tri_append_previous_coefficientdiagonalterm)) /\\ (((pfc_right_tri_append_previous_coefficientdiagonalterm)=0))))) /\\ (((pfc_value_tri_append_previous_coefficientdiagonal)=pfc_left_tri_append_previous_coefficientdiagonalterm*pfc_right_tri_append_previous_coefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_tri_append_previous_coefficientsum fs_v_pfc_tri_append_previous_coefficientsum. ((((exists fs_h_pfc_tri_append_previous_coefficientsum_body_start. fs_h_pfc_tri_append_previous_coefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_tri_append_previous_coefficientsum)) /\\ exists fs_q_pfc_tri_append_previous_coefficientsum_body_start. fs_u_pfc_tri_append_previous_coefficientsum = fs_q_pfc_tri_append_previous_coefficientsum_body_start * S ((S (0)) * fs_v_pfc_tri_append_previous_coefficientsum) + (0))) /\\ ((((exists fs_h_pfc_tri_append_previous_coefficientsum_body_terminal. fs_h_pfc_tri_append_previous_coefficientsum_body_terminal + S (pfc_natural_sum_tri_append_previous_coefficient) = S ((S (S (N))) * fs_v_pfc_tri_append_previous_coefficientsum)) /\\ exists fs_q_pfc_tri_append_previous_coefficientsum_body_terminal. fs_u_pfc_tri_append_previous_coefficientsum = fs_q_pfc_tri_append_previous_coefficientsum_body_terminal * S ((S (S (N))) * fs_v_pfc_tri_append_previous_coefficientsum) + (pfc_natural_sum_tri_append_previous_coefficient))) /\\ forall fs_i_pfc_tri_append_previous_coefficientsum_body_steps. (exists fs_lt_pfc_tri_append_previous_coefficientsum_body_steps_bound. fs_lt_pfc_tri_append_previous_coefficientsum_body_steps_bound + S fs_i_pfc_tri_append_previous_coefficientsum_body_steps = S (N)) -> exists fs_a_pfc_tri_append_previous_coefficientsum_body_steps fs_r_pfc_tri_append_previous_coefficientsum_body_steps fs_s_pfc_tri_append_previous_coefficientsum_body_steps. ((((exists fs_h_pfc_tri_append_previous_coefficientsum_body_steps_summand. fs_h_pfc_tri_append_previous_coefficientsum_body_steps_summand + S (fs_a_pfc_tri_append_previous_coefficientsum_body_steps) = S ((S (fs_i_pfc_tri_append_previous_coefficientsum_body_steps)) * pfc_terms_scale_tri_append_previous_coefficient)) /\\ exists fs_q_pfc_tri_append_previous_coefficientsum_body_steps_summand. pfc_terms_code_tri_append_previous_coefficient = fs_q_pfc_tri_append_previous_coefficientsum_body_steps_summand * S ((S (fs_i_pfc_tri_append_previous_coefficientsum_body_steps)) * pfc_terms_scale_tri_append_previous_coefficient) + (fs_a_pfc_tri_append_previous_coefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_tri_append_previous_coefficientsum_body_steps_partial. fs_h_pfc_tri_append_previous_coefficientsum_body_steps_partial + S (fs_r_pfc_tri_append_previous_coefficientsum_body_steps) = S ((S (fs_i_pfc_tri_append_previous_coefficientsum_body_steps)) * fs_v_pfc_tri_append_previous_coefficientsum)) /\\ exists fs_q_pfc_tri_append_previous_coefficientsum_body_steps_partial. fs_u_pfc_tri_append_previous_coefficientsum = fs_q_pfc_tri_append_previous_coefficientsum_body_steps_partial * S ((S (fs_i_pfc_tri_append_previous_coefficientsum_body_steps)) * fs_v_pfc_tri_append_previous_coefficientsum) + (fs_r_pfc_tri_append_previous_coefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_tri_append_previous_coefficientsum_body_steps_successor. fs_h_pfc_tri_append_previous_coefficientsum_body_steps_successor + S (fs_s_pfc_tri_append_previous_coefficientsum_body_steps) = S ((S (S fs_i_pfc_tri_append_previous_coefficientsum_body_steps)) * fs_v_pfc_tri_append_previous_coefficientsum)) /\\ exists fs_q_pfc_tri_append_previous_coefficientsum_body_steps_successor. fs_u_pfc_tri_append_previous_coefficientsum = fs_q_pfc_tri_append_previous_coefficientsum_body_steps_successor * S ((S (S fs_i_pfc_tri_append_previous_coefficientsum_body_steps)) * fs_v_pfc_tri_append_previous_coefficientsum) + (fs_s_pfc_tri_append_previous_coefficientsum_body_steps))) /\\ fs_s_pfc_tri_append_previous_coefficientsum_body_steps = fs_r_pfc_tri_append_previous_coefficientsum_body_steps + fs_a_pfc_tri_append_previous_coefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_tri_append_previous_coefficientresiduebound. pfa_gap_tri_append_previous_coefficientresiduebound + S (c) = (p)) /\\ ((exists pfa_offset_left_tri_append_previous_coefficientresiduecongruence pfa_offset_right_tri_append_previous_coefficientresiduecongruence. (pfc_natural_sum_tri_append_previous_coefficient) + (p) * pfa_offset_left_tri_append_previous_coefficientresiduecongruence = (c) + (p) * pfa_offset_right_tri_append_previous_coefficientresiduecongruence))))))))) -> (exists pfc_terms_code_tri_append_actual_coefficient pfc_terms_scale_tri_append_actual_coefficient pfc_natural_sum_tri_append_actual_coefficient. ((forall pfc_index_tri_append_actual_coefficientdiagonal. (exists pfa_gap_tri_append_actual_coefficientdiagonalbound. pfa_gap_tri_append_actual_coefficientdiagonalbound + S (pfc_index_tri_append_actual_coefficientdiagonal) = (S (N))) -> exists pfc_value_tri_append_actual_coefficientdiagonal. ((((exists ff_h_pfp_tri_append_actual_coefficientdiagonalentry. ff_h_pfp_tri_append_actual_coefficientdiagonalentry + S (pfc_value_tri_append_actual_coefficientdiagonal) = S ((S (pfc_index_tri_append_actual_coefficientdiagonal)) * pfc_terms_scale_tri_append_actual_coefficient)) /\\ exists ff_q_pfp_tri_append_actual_coefficientdiagonalentry. pfc_terms_code_tri_append_actual_coefficient = ff_q_pfp_tri_append_actual_coefficientdiagonalentry * S ((S (pfc_index_tri_append_actual_coefficientdiagonal)) * pfc_terms_scale_tri_append_actual_coefficient) + (pfc_value_tri_append_actual_coefficientdiagonal))) /\\ ((exists pfc_complement_tri_append_actual_coefficientdiagonalterm pfc_left_tri_append_actual_coefficientdiagonalterm pfc_right_tri_append_actual_coefficientdiagonalterm. (((pfc_index_tri_append_actual_coefficientdiagonal)+pfc_complement_tri_append_actual_coefficientdiagonalterm=(N)) /\\ ((((((exists pfa_gap_tri_append_actual_coefficientdiagonaltermleftinside. pfa_gap_tri_append_actual_coefficientdiagonaltermleftinside + S (pfc_index_tri_append_actual_coefficientdiagonal) = (S N)) /\\ ((((exists ff_h_pfp_tri_append_actual_coefficientdiagonaltermleftentry. ff_h_pfp_tri_append_actual_coefficientdiagonaltermleftentry + S (pfc_left_tri_append_actual_coefficientdiagonalterm) = S ((S (pfc_index_tri_append_actual_coefficientdiagonal)) * AC)) /\\ exists ff_q_pfp_tri_append_actual_coefficientdiagonaltermleftentry. AB = ff_q_pfp_tri_append_actual_coefficientdiagonaltermleftentry * S ((S (pfc_index_tri_append_actual_coefficientdiagonal)) * AC) + (pfc_left_tri_append_actual_coefficientdiagonalterm)))))) \\/ (((exists pfc_gap_tri_append_actual_coefficientdiagonaltermleftoutside. pfc_gap_tri_append_actual_coefficientdiagonaltermleftoutside+(S N)=(pfc_index_tri_append_actual_coefficientdiagonal)) /\\ (((pfc_left_tri_append_actual_coefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_tri_append_actual_coefficientdiagonaltermrightinside. pfa_gap_tri_append_actual_coefficientdiagonaltermrightinside + S (pfc_complement_tri_append_actual_coefficientdiagonalterm) = (S d)) /\\ ((((exists ff_h_pfp_tri_append_actual_coefficientdiagonaltermrightentry. ff_h_pfp_tri_append_actual_coefficientdiagonaltermrightentry + S (pfc_right_tri_append_actual_coefficientdiagonalterm) = S ((S (pfc_complement_tri_append_actual_coefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_tri_append_actual_coefficientdiagonaltermrightentry. bb = ff_q_pfp_tri_append_actual_coefficientdiagonaltermrightentry * S ((S (pfc_complement_tri_append_actual_coefficientdiagonalterm)) * bc) + (pfc_right_tri_append_actual_coefficientdiagonalterm)))))) \\/ (((exists pfc_gap_tri_append_actual_coefficientdiagonaltermrightoutside. pfc_gap_tri_append_actual_coefficientdiagonaltermrightoutside+(S d)=(pfc_complement_tri_append_actual_coefficientdiagonalterm)) /\\ (((pfc_right_tri_append_actual_coefficientdiagonalterm)=0))))) /\\ (((pfc_value_tri_append_actual_coefficientdiagonal)=pfc_left_tri_append_actual_coefficientdiagonalterm*pfc_right_tri_append_actual_coefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_tri_append_actual_coefficientsum fs_v_pfc_tri_append_actual_coefficientsum. ((((exists fs_h_pfc_tri_append_actual_coefficientsum_body_start. fs_h_pfc_tri_append_actual_coefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_tri_append_actual_coefficientsum)) /\\ exists fs_q_pfc_tri_append_actual_coefficientsum_body_start. fs_u_pfc_tri_append_actual_coefficientsum = fs_q_pfc_tri_append_actual_coefficientsum_body_start * S ((S (0)) * fs_v_pfc_tri_append_actual_coefficientsum) + (0))) /\\ ((((exists fs_h_pfc_tri_append_actual_coefficientsum_body_terminal. fs_h_pfc_tri_append_actual_coefficientsum_body_terminal + S (pfc_natural_sum_tri_append_actual_coefficient) = S ((S (S (N))) * fs_v_pfc_tri_append_actual_coefficientsum)) /\\ exists fs_q_pfc_tri_append_actual_coefficientsum_body_terminal. fs_u_pfc_tri_append_actual_coefficientsum = fs_q_pfc_tri_append_actual_coefficientsum_body_terminal * S ((S (S (N))) * fs_v_pfc_tri_append_actual_coefficientsum) + (pfc_natural_sum_tri_append_actual_coefficient))) /\\ forall fs_i_pfc_tri_append_actual_coefficientsum_body_steps. (exists fs_lt_pfc_tri_append_actual_coefficientsum_body_steps_bound. fs_lt_pfc_tri_append_actual_coefficientsum_body_steps_bound + S fs_i_pfc_tri_append_actual_coefficientsum_body_steps = S (N)) -> exists fs_a_pfc_tri_append_actual_coefficientsum_body_steps fs_r_pfc_tri_append_actual_coefficientsum_body_steps fs_s_pfc_tri_append_actual_coefficientsum_body_steps. ((((exists fs_h_pfc_tri_append_actual_coefficientsum_body_steps_summand. fs_h_pfc_tri_append_actual_coefficientsum_body_steps_summand + S (fs_a_pfc_tri_append_actual_coefficientsum_body_steps) = S ((S (fs_i_pfc_tri_append_actual_coefficientsum_body_steps)) * pfc_terms_scale_tri_append_actual_coefficient)) /\\ exists fs_q_pfc_tri_append_actual_coefficientsum_body_steps_summand. pfc_terms_code_tri_append_actual_coefficient = fs_q_pfc_tri_append_actual_coefficientsum_body_steps_summand * S ((S (fs_i_pfc_tri_append_actual_coefficientsum_body_steps)) * pfc_terms_scale_tri_append_actual_coefficient) + (fs_a_pfc_tri_append_actual_coefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_tri_append_actual_coefficientsum_body_steps_partial. fs_h_pfc_tri_append_actual_coefficientsum_body_steps_partial + S (fs_r_pfc_tri_append_actual_coefficientsum_body_steps) = S ((S (fs_i_pfc_tri_append_actual_coefficientsum_body_steps)) * fs_v_pfc_tri_append_actual_coefficientsum)) /\\ exists fs_q_pfc_tri_append_actual_coefficientsum_body_steps_partial. fs_u_pfc_tri_append_actual_coefficientsum = fs_q_pfc_tri_append_actual_coefficientsum_body_steps_partial * S ((S (fs_i_pfc_tri_append_actual_coefficientsum_body_steps)) * fs_v_pfc_tri_append_actual_coefficientsum) + (fs_r_pfc_tri_append_actual_coefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_tri_append_actual_coefficientsum_body_steps_successor. fs_h_pfc_tri_append_actual_coefficientsum_body_steps_successor + S (fs_s_pfc_tri_append_actual_coefficientsum_body_steps) = S ((S (S fs_i_pfc_tri_append_actual_coefficientsum_body_steps)) * fs_v_pfc_tri_append_actual_coefficientsum)) /\\ exists fs_q_pfc_tri_append_actual_coefficientsum_body_steps_successor. fs_u_pfc_tri_append_actual_coefficientsum = fs_q_pfc_tri_append_actual_coefficientsum_body_steps_successor * S ((S (S fs_i_pfc_tri_append_actual_coefficientsum_body_steps)) * fs_v_pfc_tri_append_actual_coefficientsum) + (fs_s_pfc_tri_append_actual_coefficientsum_body_steps))) /\\ fs_s_pfc_tri_append_actual_coefficientsum_body_steps = fs_r_pfc_tri_append_actual_coefficientsum_body_steps + fs_a_pfc_tri_append_actual_coefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_tri_append_actual_coefficientresiduebound. pfa_gap_tri_append_actual_coefficientresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_tri_append_actual_coefficientresiduecongruence pfa_offset_right_tri_append_actual_coefficientresiduecongruence. (pfc_natural_sum_tri_append_actual_coefficient) + (p) * pfa_offset_left_tri_append_actual_coefficientresiduecongruence = (r) + (p) * pfa_offset_right_tri_append_actual_coefficientresiduecongruence))))))))) -> (((exists pfa_gap_tri_append_actual_productleft. pfa_gap_tri_append_actual_productleft + S (a) = (p)) /\\ (((exists pfa_gap_tri_append_actual_productright. pfa_gap_tri_append_actual_productright + S (b) = (p)) /\\ ((((exists pfa_gap_tri_append_actual_productresultbound. pfa_gap_tri_append_actual_productresultbound + S (t) = (p)) /\\ ((exists pfa_offset_left_tri_append_actual_productresultcongruence pfa_offset_right_tri_append_actual_productresultcongruence. 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(forall pfrep_power_equivalent_symmetric_old pfrep_left_equivalent_symmetric_old pfrep_right_equivalent_symmetric_old. ((exists pfrep_position_equivalent_symmetric_oldfirst. ((pfrep_position_equivalent_symmetric_oldfirst+S (pfrep_power_equivalent_symmetric_old)=(L)) /\\ ((((exists ff_h_pfp_equivalent_symmetric_oldfirstentry. ff_h_pfp_equivalent_symmetric_oldfirstentry + S (pfrep_left_equivalent_symmetric_old) = S ((S (pfrep_position_equivalent_symmetric_oldfirst)) * c)) /\\ exists ff_q_pfp_equivalent_symmetric_oldfirstentry. b = ff_q_pfp_equivalent_symmetric_oldfirstentry * S ((S (pfrep_position_equivalent_symmetric_oldfirst)) * c) + (pfrep_left_equivalent_symmetric_old)))))) \\/ (((exists pfrep_gap_equivalent_symmetric_oldfirstoutside. pfrep_gap_equivalent_symmetric_oldfirstoutside+(L)=(pfrep_power_equivalent_symmetric_old)) /\\ (((pfrep_left_equivalent_symmetric_old)=0))))) -> ((exists pfrep_position_equivalent_symmetric_oldsecond. ((pfrep_position_equivalent_symmetric_oldsecond+S (pfrep_power_equivalent_symmetric_old)=(M)) /\\ ((((exists ff_h_pfp_equivalent_symmetric_oldsecondentry. ff_h_pfp_equivalent_symmetric_oldsecondentry + S (pfrep_right_equivalent_symmetric_old) = S ((S (pfrep_position_equivalent_symmetric_oldsecond)) * e)) /\\ exists ff_q_pfp_equivalent_symmetric_oldsecondentry. d = ff_q_pfp_equivalent_symmetric_oldsecondentry * S ((S (pfrep_position_equivalent_symmetric_oldsecond)) * e) + (pfrep_right_equivalent_symmetric_old)))))) \\/ (((exists pfrep_gap_equivalent_symmetric_oldsecondoutside. pfrep_gap_equivalent_symmetric_oldsecondoutside+(M)=(pfrep_power_equivalent_symmetric_old)) /\\ (((pfrep_right_equivalent_symmetric_old)=0))))) -> pfrep_left_equivalent_symmetric_old=pfrep_right_equivalent_symmetric_old) -> (forall pfrep_power_equivalent_symmetric_new pfrep_left_equivalent_symmetric_new pfrep_right_equivalent_symmetric_new. ((exists pfrep_position_equivalent_symmetric_newfirst. ((pfrep_position_equivalent_symmetric_newfirst+S (pfrep_power_equivalent_symmetric_new)=(M)) /\\ ((((exists ff_h_pfp_equivalent_symmetric_newfirstentry. ff_h_pfp_equivalent_symmetric_newfirstentry + S (pfrep_left_equivalent_symmetric_new) = S ((S (pfrep_position_equivalent_symmetric_newfirst)) * e)) /\\ exists ff_q_pfp_equivalent_symmetric_newfirstentry. d = ff_q_pfp_equivalent_symmetric_newfirstentry * S ((S (pfrep_position_equivalent_symmetric_newfirst)) * e) + (pfrep_left_equivalent_symmetric_new)))))) \\/ (((exists pfrep_gap_equivalent_symmetric_newfirstoutside. pfrep_gap_equivalent_symmetric_newfirstoutside+(M)=(pfrep_power_equivalent_symmetric_new)) /\\ (((pfrep_left_equivalent_symmetric_new)=0))))) -> ((exists pfrep_position_equivalent_symmetric_newsecond. ((pfrep_position_equivalent_symmetric_newsecond+S (pfrep_power_equivalent_symmetric_new)=(L)) /\\ ((((exists ff_h_pfp_equivalent_symmetric_newsecondentry. ff_h_pfp_equivalent_symmetric_newsecondentry + S (pfrep_right_equivalent_symmetric_new) = S ((S (pfrep_position_equivalent_symmetric_newsecond)) * c)) /\\ exists ff_q_pfp_equivalent_symmetric_newsecondentry. b = ff_q_pfp_equivalent_symmetric_newsecondentry * S ((S (pfrep_position_equivalent_symmetric_newsecond)) * c) + (pfrep_right_equivalent_symmetric_new)))))) \\/ (((exists pfrep_gap_equivalent_symmetric_newsecondoutside. pfrep_gap_equivalent_symmetric_newsecondoutside+(L)=(pfrep_power_equivalent_symmetric_new)) /\\ (((pfrep_right_equivalent_symmetric_new)=0))))) -> pfrep_left_equivalent_symmetric_new=pfrep_right_equivalent_symmetric_new)", "statement_sha256": "3b88824d2c57b6363973fbc976c26ecc123535f061b929600351a9b8b926ef1a", "summary": "Formal coefficient equivalence is symmetric without choosing canonical raw beta codes.", 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((pfrep_position_equivalent_middle+S (k)=(M)) /\\ ((((exists ff_h_pfp_equivalent_middleentry. ff_h_pfp_equivalent_middleentry + S (z) = S ((S (pfrep_position_equivalent_middle)) * e)) /\\ exists ff_q_pfp_equivalent_middleentry. d = ff_q_pfp_equivalent_middleentry * S ((S (pfrep_position_equivalent_middle)) * e) + (z)))))) \\/ (((exists pfrep_gap_equivalent_middleoutside. pfrep_gap_equivalent_middleoutside+(M)=(k)) /\\ (((z)=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 (k)", "apply prime_field_polynomial_power_coefficient_exists", "cases hv", "trans x", "specialize he (k)", "specialize he (a)", "specialize he (x)", "apply he", "exact ha", "exact hv_witness", "specialize hf (k)", "specialize hf (x)", "specialize hf (r)", "apply hf", "exact hv_witness", "exact hr" ], "script_sha256": "466fb2ab3d319d3a6d9c82bec330075173d00334dbffef6bc2a8d5be28e3e702", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_representation_candidate.py", "sha256": "fc3b40a6ec88841b937251bfc2b4c2dcce55ddeec9932c2533e0f74e46fc5c6a" }, "statement": "forall b c L d e M f g N. (forall pfrep_power_equivalent_transitive_a pfrep_left_equivalent_transitive_a pfrep_right_equivalent_transitive_a. ((exists pfrep_position_equivalent_transitive_afirst. ((pfrep_position_equivalent_transitive_afirst+S (pfrep_power_equivalent_transitive_a)=(L)) /\\ ((((exists ff_h_pfp_equivalent_transitive_afirstentry. ff_h_pfp_equivalent_transitive_afirstentry + S (pfrep_left_equivalent_transitive_a) = S ((S (pfrep_position_equivalent_transitive_afirst)) * c)) /\\ exists ff_q_pfp_equivalent_transitive_afirstentry. b = ff_q_pfp_equivalent_transitive_afirstentry * S ((S (pfrep_position_equivalent_transitive_afirst)) * c) + (pfrep_left_equivalent_transitive_a)))))) \\/ (((exists pfrep_gap_equivalent_transitive_afirstoutside. pfrep_gap_equivalent_transitive_afirstoutside+(L)=(pfrep_power_equivalent_transitive_a)) /\\ (((pfrep_left_equivalent_transitive_a)=0))))) -> ((exists pfrep_position_equivalent_transitive_asecond. ((pfrep_position_equivalent_transitive_asecond+S (pfrep_power_equivalent_transitive_a)=(M)) /\\ ((((exists ff_h_pfp_equivalent_transitive_asecondentry. ff_h_pfp_equivalent_transitive_asecondentry + S (pfrep_right_equivalent_transitive_a) = S ((S (pfrep_position_equivalent_transitive_asecond)) * e)) /\\ exists ff_q_pfp_equivalent_transitive_asecondentry. d = ff_q_pfp_equivalent_transitive_asecondentry * S ((S (pfrep_position_equivalent_transitive_asecond)) * e) + (pfrep_right_equivalent_transitive_a)))))) \\/ (((exists pfrep_gap_equivalent_transitive_asecondoutside. pfrep_gap_equivalent_transitive_asecondoutside+(M)=(pfrep_power_equivalent_transitive_a)) /\\ (((pfrep_right_equivalent_transitive_a)=0))))) -> pfrep_left_equivalent_transitive_a=pfrep_right_equivalent_transitive_a) -> (forall pfrep_power_equivalent_transitive_b pfrep_left_equivalent_transitive_b pfrep_right_equivalent_transitive_b. ((exists pfrep_position_equivalent_transitive_bfirst. ((pfrep_position_equivalent_transitive_bfirst+S (pfrep_power_equivalent_transitive_b)=(M)) /\\ ((((exists ff_h_pfp_equivalent_transitive_bfirstentry. ff_h_pfp_equivalent_transitive_bfirstentry + S (pfrep_left_equivalent_transitive_b) = S ((S (pfrep_position_equivalent_transitive_bfirst)) * e)) /\\ exists ff_q_pfp_equivalent_transitive_bfirstentry. d = ff_q_pfp_equivalent_transitive_bfirstentry * S ((S (pfrep_position_equivalent_transitive_bfirst)) * e) + (pfrep_left_equivalent_transitive_b)))))) \\/ (((exists pfrep_gap_equivalent_transitive_bfirstoutside. pfrep_gap_equivalent_transitive_bfirstoutside+(M)=(pfrep_power_equivalent_transitive_b)) /\\ (((pfrep_left_equivalent_transitive_b)=0))))) -> ((exists pfrep_position_equivalent_transitive_bsecond. ((pfrep_position_equivalent_transitive_bsecond+S (pfrep_power_equivalent_transitive_b)=(N)) /\\ ((((exists ff_h_pfp_equivalent_transitive_bsecondentry. ff_h_pfp_equivalent_transitive_bsecondentry + S (pfrep_right_equivalent_transitive_b) = S ((S (pfrep_position_equivalent_transitive_bsecond)) * g)) /\\ exists ff_q_pfp_equivalent_transitive_bsecondentry. f = ff_q_pfp_equivalent_transitive_bsecondentry * S ((S (pfrep_position_equivalent_transitive_bsecond)) * g) + (pfrep_right_equivalent_transitive_b)))))) \\/ (((exists pfrep_gap_equivalent_transitive_bsecondoutside. pfrep_gap_equivalent_transitive_bsecondoutside+(N)=(pfrep_power_equivalent_transitive_b)) /\\ (((pfrep_right_equivalent_transitive_b)=0))))) -> pfrep_left_equivalent_transitive_b=pfrep_right_equivalent_transitive_b) -> (forall pfrep_power_equivalent_transitive_result pfrep_left_equivalent_transitive_result pfrep_right_equivalent_transitive_result. ((exists pfrep_position_equivalent_transitive_resultfirst. ((pfrep_position_equivalent_transitive_resultfirst+S (pfrep_power_equivalent_transitive_result)=(L)) /\\ ((((exists ff_h_pfp_equivalent_transitive_resultfirstentry. ff_h_pfp_equivalent_transitive_resultfirstentry + S (pfrep_left_equivalent_transitive_result) = S ((S (pfrep_position_equivalent_transitive_resultfirst)) * c)) /\\ exists ff_q_pfp_equivalent_transitive_resultfirstentry. b = ff_q_pfp_equivalent_transitive_resultfirstentry * S ((S (pfrep_position_equivalent_transitive_resultfirst)) * c) + (pfrep_left_equivalent_transitive_result)))))) \\/ (((exists pfrep_gap_equivalent_transitive_resultfirstoutside. pfrep_gap_equivalent_transitive_resultfirstoutside+(L)=(pfrep_power_equivalent_transitive_result)) /\\ (((pfrep_left_equivalent_transitive_result)=0))))) -> ((exists pfrep_position_equivalent_transitive_resultsecond. 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(forall mdr_i_pfp_equal_to_equivalent_input mdr_a_pfp_equal_to_equivalent_input. (exists mdr_gap_pfp_equal_to_equivalent_inputb. mdr_gap_pfp_equal_to_equivalent_inputb + S (mdr_i_pfp_equal_to_equivalent_input) = (L)) -> (((exists ff_h_mdr_pfp_equal_to_equivalent_inputo. ff_h_mdr_pfp_equal_to_equivalent_inputo + S (mdr_a_pfp_equal_to_equivalent_input) = S ((S (mdr_i_pfp_equal_to_equivalent_input)) * c)) /\\ exists ff_q_mdr_pfp_equal_to_equivalent_inputo. b = ff_q_mdr_pfp_equal_to_equivalent_inputo * S ((S (mdr_i_pfp_equal_to_equivalent_input)) * c) + (mdr_a_pfp_equal_to_equivalent_input))) -> (((exists ff_h_mdr_pfp_equal_to_equivalent_inputn. ff_h_mdr_pfp_equal_to_equivalent_inputn + S (mdr_a_pfp_equal_to_equivalent_input) = S ((S (mdr_i_pfp_equal_to_equivalent_input)) * e)) /\\ exists ff_q_mdr_pfp_equal_to_equivalent_inputn. d = ff_q_mdr_pfp_equal_to_equivalent_inputn * S ((S (mdr_i_pfp_equal_to_equivalent_input)) * e) + (mdr_a_pfp_equal_to_equivalent_input)))) -> (forall pfrep_power_equal_to_equivalent_result pfrep_left_equal_to_equivalent_result pfrep_right_equal_to_equivalent_result. ((exists pfrep_position_equal_to_equivalent_resultfirst. ((pfrep_position_equal_to_equivalent_resultfirst+S (pfrep_power_equal_to_equivalent_result)=(L)) /\\ ((((exists ff_h_pfp_equal_to_equivalent_resultfirstentry. ff_h_pfp_equal_to_equivalent_resultfirstentry + S (pfrep_left_equal_to_equivalent_result) = S ((S (pfrep_position_equal_to_equivalent_resultfirst)) * c)) /\\ exists ff_q_pfp_equal_to_equivalent_resultfirstentry. b = ff_q_pfp_equal_to_equivalent_resultfirstentry * S ((S (pfrep_position_equal_to_equivalent_resultfirst)) * c) + (pfrep_left_equal_to_equivalent_result)))))) \\/ (((exists pfrep_gap_equal_to_equivalent_resultfirstoutside. pfrep_gap_equal_to_equivalent_resultfirstoutside+(L)=(pfrep_power_equal_to_equivalent_result)) /\\ (((pfrep_left_equal_to_equivalent_result)=0))))) -> ((exists pfrep_position_equal_to_equivalent_resultsecond. ((pfrep_position_equal_to_equivalent_resultsecond+S (pfrep_power_equal_to_equivalent_result)=(L)) /\\ ((((exists ff_h_pfp_equal_to_equivalent_resultsecondentry. ff_h_pfp_equal_to_equivalent_resultsecondentry + S (pfrep_right_equal_to_equivalent_result) = S ((S (pfrep_position_equal_to_equivalent_resultsecond)) * e)) /\\ exists ff_q_pfp_equal_to_equivalent_resultsecondentry. d = ff_q_pfp_equal_to_equivalent_resultsecondentry * S ((S (pfrep_position_equal_to_equivalent_resultsecond)) * e) + (pfrep_right_equal_to_equivalent_result)))))) \\/ (((exists pfrep_gap_equal_to_equivalent_resultsecondoutside. pfrep_gap_equal_to_equivalent_resultsecondoutside+(L)=(pfrep_power_equal_to_equivalent_result)) /\\ (((pfrep_right_equal_to_equivalent_result)=0))))) -> pfrep_left_equal_to_equivalent_result=pfrep_right_equal_to_equivalent_result)", "statement_sha256": "9575ec49a2188d966e5c7d3e10a0cdcded5bd8ebbf549840f9cfb5a73817d3b5", "summary": "The inherited same-length decoded equality implies formal 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(((exists ff_h_pfp_equivalent_to_equal_lookup. ff_h_pfp_equivalent_to_equal_lookup + S (r) = S ((S (i)) * e)) /\\ exists ff_q_pfp_equivalent_to_equal_lookup. d = ff_q_pfp_equivalent_to_equal_lookup * S ((S (i)) * e) + (r)))", "specialize beta_at_exists (d)", "specialize beta_at_exists (e)", "specialize beta_at_exists (i)", "apply beta_at_exists", "cases hv", "have heq : a=x1", "specialize he (x)", "specialize he (a)", "specialize he (x1)", "apply he", "left", "exists i", "split", "exact hs", "exact ha", "left", "exists i", "split", "exact hs", "exact hv_witness", "rewrite heq", "rewrite heq", "exact hv_witness" ], "script_sha256": "d97152dd9301167fe02c8d5d322444e117f78335697ff4f1a01fcbf253dfd73a", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_representation_candidate.py", "sha256": "fc3b40a6ec88841b937251bfc2b4c2dcce55ddeec9932c2533e0f74e46fc5c6a" }, "statement": "forall b c d e L. (forall pfrep_power_equivalent_to_equal_input pfrep_left_equivalent_to_equal_input pfrep_right_equivalent_to_equal_input. ((exists pfrep_position_equivalent_to_equal_inputfirst. ((pfrep_position_equivalent_to_equal_inputfirst+S (pfrep_power_equivalent_to_equal_input)=(L)) /\\ ((((exists ff_h_pfp_equivalent_to_equal_inputfirstentry. ff_h_pfp_equivalent_to_equal_inputfirstentry + S (pfrep_left_equivalent_to_equal_input) = S ((S (pfrep_position_equivalent_to_equal_inputfirst)) * c)) /\\ exists ff_q_pfp_equivalent_to_equal_inputfirstentry. b = ff_q_pfp_equivalent_to_equal_inputfirstentry * S ((S (pfrep_position_equivalent_to_equal_inputfirst)) * c) + (pfrep_left_equivalent_to_equal_input)))))) \\/ (((exists pfrep_gap_equivalent_to_equal_inputfirstoutside. pfrep_gap_equivalent_to_equal_inputfirstoutside+(L)=(pfrep_power_equivalent_to_equal_input)) /\\ (((pfrep_left_equivalent_to_equal_input)=0))))) -> ((exists pfrep_position_equivalent_to_equal_inputsecond. ((pfrep_position_equivalent_to_equal_inputsecond+S (pfrep_power_equivalent_to_equal_input)=(L)) /\\ ((((exists ff_h_pfp_equivalent_to_equal_inputsecondentry. ff_h_pfp_equivalent_to_equal_inputsecondentry + S (pfrep_right_equivalent_to_equal_input) = S ((S (pfrep_position_equivalent_to_equal_inputsecond)) * e)) /\\ exists ff_q_pfp_equivalent_to_equal_inputsecondentry. d = ff_q_pfp_equivalent_to_equal_inputsecondentry * S ((S (pfrep_position_equivalent_to_equal_inputsecond)) * e) + (pfrep_right_equivalent_to_equal_input)))))) \\/ (((exists pfrep_gap_equivalent_to_equal_inputsecondoutside. pfrep_gap_equivalent_to_equal_inputsecondoutside+(L)=(pfrep_power_equivalent_to_equal_input)) /\\ (((pfrep_right_equivalent_to_equal_input)=0))))) -> pfrep_left_equivalent_to_equal_input=pfrep_right_equivalent_to_equal_input) -> (forall mdr_i_pfp_equivalent_to_equal_result mdr_a_pfp_equivalent_to_equal_result. 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(((forall pfp_repeat_index_left_pad_zerozeros. (exists pfa_gap_left_pad_zerozerosindex. pfa_gap_left_pad_zerozerosindex + S (pfp_repeat_index_left_pad_zerozeros) = (0)) -> (((exists ff_h_pfp_left_pad_zerozerosentry. ff_h_pfp_left_pad_zerozerosentry + S (0) = S ((S (pfp_repeat_index_left_pad_zerozeros)) * c)) /\\ exists ff_q_pfp_left_pad_zerozerosentry. b = ff_q_pfp_left_pad_zerozerosentry * S ((S (pfp_repeat_index_left_pad_zerozeros)) * c) + (0)))) /\\ ((forall pfrep_index_left_pad_zero pfrep_value_left_pad_zero. (exists pfa_gap_left_pad_zerobound. pfa_gap_left_pad_zerobound + S (pfrep_index_left_pad_zero) = (L)) -> (((exists ff_h_pfp_left_pad_zeroinput. ff_h_pfp_left_pad_zeroinput + S (pfrep_value_left_pad_zero) = S ((S (pfrep_index_left_pad_zero)) * c)) /\\ exists ff_q_pfp_left_pad_zeroinput. b = ff_q_pfp_left_pad_zeroinput * S ((S (pfrep_index_left_pad_zero)) * c) + (pfrep_value_left_pad_zero))) -> (((exists ff_h_pfp_left_pad_zerooutput. ff_h_pfp_left_pad_zerooutput + S (pfrep_value_left_pad_zero) = S ((S ((0)+pfrep_index_left_pad_zero)) * c)) /\\ exists ff_q_pfp_left_pad_zerooutput. b = ff_q_pfp_left_pad_zerooutput * S ((S ((0)+pfrep_index_left_pad_zero)) * c) + (pfrep_value_left_pad_zero)))))))", "statement_sha256": "e5aa36f642740d6d675c3941ba0c8ee630bfa98b482a06fe47e8f9a1e583361a", "summary": "Zero left padding uses the original code and changes no coefficient.", "summary_sha256": "0f8679d98b89d07c3619be5064a14ef8b7b96cd3624856c50b1ce822a8c612ce" }, { "alpha_v33_frontier_enrollment": { 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(exists pfa_gap_left_pad_previousbound. pfa_gap_left_pad_previousbound + S (pfrep_index_left_pad_previous) = (L)) -> (((exists ff_h_pfp_left_pad_previousinput. ff_h_pfp_left_pad_previousinput + S (pfrep_value_left_pad_previous) = S ((S (pfrep_index_left_pad_previous)) * c)) /\\ exists ff_q_pfp_left_pad_previousinput. b = ff_q_pfp_left_pad_previousinput * S ((S (pfrep_index_left_pad_previous)) * c) + (pfrep_value_left_pad_previous))) -> (((exists ff_h_pfp_left_pad_previousoutput. ff_h_pfp_left_pad_previousoutput + S (pfrep_value_left_pad_previous) = S ((S ((t)+pfrep_index_left_pad_previous)) * e)) /\\ exists ff_q_pfp_left_pad_previousoutput. d = ff_q_pfp_left_pad_previousoutput * S ((S ((t)+pfrep_index_left_pad_previous)) * e) + (pfrep_value_left_pad_previous)))))))", "exact IH", "cases hold", "cases hold_witness", "cases hold_witness_witness", "have ha : exists a. (((exists ff_h_pfp_left_pad_next_source. ff_h_pfp_left_pad_next_source + S (a) = S ((S (L)) * c)) /\\ exists ff_q_pfp_left_pad_next_source. b = ff_q_pfp_left_pad_next_source * S ((S (L)) * c) + (a)))", "specialize beta_at_exists (b)", "specialize beta_at_exists (c)", "specialize beta_at_exists (L)", "apply beta_at_exists", "cases ha", "have hn : exists d e. (((((exists ff_h_pfp_left_pad_next_entry. ff_h_pfp_left_pad_next_entry + S (x2) = S ((S (t+L)) * e)) /\\ exists ff_q_pfp_left_pad_next_entry. d = ff_q_pfp_left_pad_next_entry * S ((S (t+L)) * e) + (x2))) /\\ ((forall mdr_i_pfp_left_pad_preserved mdr_a_pfp_left_pad_preserved. 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(((forall pfp_repeat_index_left_pad_existszeros. (exists pfa_gap_left_pad_existszerosindex. pfa_gap_left_pad_existszerosindex + S (pfp_repeat_index_left_pad_existszeros) = (t)) -> (((exists ff_h_pfp_left_pad_existszerosentry. ff_h_pfp_left_pad_existszerosentry + S (0) = S ((S (pfp_repeat_index_left_pad_existszeros)) * e)) /\\ exists ff_q_pfp_left_pad_existszerosentry. d = ff_q_pfp_left_pad_existszerosentry * S ((S (pfp_repeat_index_left_pad_existszeros)) * e) + (0)))) /\\ ((forall pfrep_index_left_pad_exists pfrep_value_left_pad_exists. (exists pfa_gap_left_pad_existsbound. pfa_gap_left_pad_existsbound + S (pfrep_index_left_pad_exists) = (L)) -> (((exists ff_h_pfp_left_pad_existsinput. ff_h_pfp_left_pad_existsinput + S (pfrep_value_left_pad_exists) = S ((S (pfrep_index_left_pad_exists)) * c)) /\\ exists ff_q_pfp_left_pad_existsinput. b = ff_q_pfp_left_pad_existsinput * S ((S (pfrep_index_left_pad_exists)) * c) + (pfrep_value_left_pad_exists))) -> (((exists ff_h_pfp_left_pad_existsoutput. ff_h_pfp_left_pad_existsoutput + S (pfrep_value_left_pad_exists) = S ((S ((t)+pfrep_index_left_pad_exists)) * e)) /\\ exists ff_q_pfp_left_pad_existsoutput. d = ff_q_pfp_left_pad_existsoutput * S ((S ((t)+pfrep_index_left_pad_exists)) * e) + (pfrep_value_left_pad_exists)))))))", "statement_sha256": "8503ebeaced372c2e66a8aaa29b4464a24c4745fb7f661d12a119ada8b91f647", "summary": "Finite induction genuinely constructs the zero block and appends every actual input coefficient, including empty input and arbitrary encodings.", 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(exists pfa_gap_left_pad_entry_graphbound. pfa_gap_left_pad_entry_graphbound + S (pfrep_index_left_pad_entry_graph) = (L)) -> (((exists ff_h_pfp_left_pad_entry_graphinput. ff_h_pfp_left_pad_entry_graphinput + S (pfrep_value_left_pad_entry_graph) = S ((S (pfrep_index_left_pad_entry_graph)) * c)) /\\ exists ff_q_pfp_left_pad_entry_graphinput. b = ff_q_pfp_left_pad_entry_graphinput * S ((S (pfrep_index_left_pad_entry_graph)) * c) + (pfrep_value_left_pad_entry_graph))) -> (((exists ff_h_pfp_left_pad_entry_graphoutput. ff_h_pfp_left_pad_entry_graphoutput + S (pfrep_value_left_pad_entry_graph) = S ((S ((t)+pfrep_index_left_pad_entry_graph)) * e)) /\\ exists ff_q_pfp_left_pad_entry_graphoutput. d = ff_q_pfp_left_pad_entry_graphoutput * S ((S ((t)+pfrep_index_left_pad_entry_graph)) * e) + (pfrep_value_left_pad_entry_graph))))))) -> (exists pfa_gap_left_pad_entry_bound. pfa_gap_left_pad_entry_bound + S (i) = (L)) -> (((exists ff_h_pfp_left_pad_entry_source. ff_h_pfp_left_pad_entry_source + S 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(((((exists ff_h_pfp_left_pad_bound_entry. ff_h_pfp_left_pad_bound_entry + S (a) = S ((S (x)) * c)) /\\ exists ff_q_pfp_left_pad_bound_entry. b = ff_q_pfp_left_pad_bound_entry * S ((S (x)) * c) + (a))) /\\ ((exists pfa_gap_left_pad_bound_value. pfa_gap_left_pad_bound_value + S (a) = (p)))))", "specialize hc (x)", "apply hc", "exact ho_right_witness_left", "cases hv", "cases hv_witness", "exists x1", "split", "rewrite ho_right_witness_right", "rewrite ho_right_witness_right", "specialize h_right (x)", "specialize h_right (x1)", "apply h_right", "exact ho_right_witness_left", "exact hv_witness_left", "exact hv_witness_right" ], "script_sha256": "1e25bfd4b78be5519af15c0a37d6102b2c9afb4607f46858b391d40232130c8d", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_representation_candidate.py", "sha256": "fc3b40a6ec88841b937251bfc2b4c2dcce55ddeec9932c2533e0f74e46fc5c6a" }, "statement": "forall p b c L t d e. (~((p) = 1) /\\ forall pfa_factor_left_left_pad_bounded_prime pfa_factor_right_left_pad_bounded_prime. (p) = pfa_factor_left_left_pad_bounded_prime * pfa_factor_right_left_pad_bounded_prime -> pfa_factor_left_left_pad_bounded_prime = 1 \\/ pfa_factor_right_left_pad_bounded_prime = 1) -> (forall fom_index_pfp_left_pad_bounded_source. (exists fom_gap_pfp_left_pad_bounded_source_index_bound. fom_gap_pfp_left_pad_bounded_source_index_bound + S (fom_index_pfp_left_pad_bounded_source) = L) -> exists fom_value_pfp_left_pad_bounded_source. ((((exists fom_beta_height_pfp_left_pad_bounded_source_entry. fom_beta_height_pfp_left_pad_bounded_source_entry + S (fom_value_pfp_left_pad_bounded_source) = S ((S (fom_index_pfp_left_pad_bounded_source)) * c)) /\\ exists fom_beta_quotient_pfp_left_pad_bounded_source_entry. b = fom_beta_quotient_pfp_left_pad_bounded_source_entry * S ((S (fom_index_pfp_left_pad_bounded_source)) * c) + (fom_value_pfp_left_pad_bounded_source))) /\\ (exists fom_gap_pfp_left_pad_bounded_source_value_bound. fom_gap_pfp_left_pad_bounded_source_value_bound + S (fom_value_pfp_left_pad_bounded_source) = p))) -> (((forall pfp_repeat_index_left_pad_bounded_graphzeros. (exists pfa_gap_left_pad_bounded_graphzerosindex. pfa_gap_left_pad_bounded_graphzerosindex + S (pfp_repeat_index_left_pad_bounded_graphzeros) = (t)) -> (((exists ff_h_pfp_left_pad_bounded_graphzerosentry. ff_h_pfp_left_pad_bounded_graphzerosentry + S (0) = S ((S (pfp_repeat_index_left_pad_bounded_graphzeros)) * e)) /\\ exists ff_q_pfp_left_pad_bounded_graphzerosentry. d = ff_q_pfp_left_pad_bounded_graphzerosentry * S ((S (pfp_repeat_index_left_pad_bounded_graphzeros)) * e) + (0)))) /\\ ((forall pfrep_index_left_pad_bounded_graph pfrep_value_left_pad_bounded_graph. (exists pfa_gap_left_pad_bounded_graphbound. pfa_gap_left_pad_bounded_graphbound + S (pfrep_index_left_pad_bounded_graph) = (L)) -> (((exists ff_h_pfp_left_pad_bounded_graphinput. ff_h_pfp_left_pad_bounded_graphinput + S (pfrep_value_left_pad_bounded_graph) = S ((S (pfrep_index_left_pad_bounded_graph)) * c)) /\\ exists ff_q_pfp_left_pad_bounded_graphinput. b = ff_q_pfp_left_pad_bounded_graphinput * S ((S (pfrep_index_left_pad_bounded_graph)) * c) + (pfrep_value_left_pad_bounded_graph))) -> (((exists ff_h_pfp_left_pad_bounded_graphoutput. ff_h_pfp_left_pad_bounded_graphoutput + S (pfrep_value_left_pad_bounded_graph) = S ((S ((t)+pfrep_index_left_pad_bounded_graph)) * e)) /\\ exists ff_q_pfp_left_pad_bounded_graphoutput. d = ff_q_pfp_left_pad_bounded_graphoutput * S ((S ((t)+pfrep_index_left_pad_bounded_graph)) * e) + (pfrep_value_left_pad_bounded_graph))))))) -> (forall fom_index_pfp_left_pad_bounded_target. (exists fom_gap_pfp_left_pad_bounded_target_index_bound. fom_gap_pfp_left_pad_bounded_target_index_bound + S (fom_index_pfp_left_pad_bounded_target) = t+L) -> exists fom_value_pfp_left_pad_bounded_target. ((((exists fom_beta_height_pfp_left_pad_bounded_target_entry. fom_beta_height_pfp_left_pad_bounded_target_entry + S (fom_value_pfp_left_pad_bounded_target) = S ((S (fom_index_pfp_left_pad_bounded_target)) * e)) /\\ exists fom_beta_quotient_pfp_left_pad_bounded_target_entry. d = fom_beta_quotient_pfp_left_pad_bounded_target_entry * S ((S (fom_index_pfp_left_pad_bounded_target)) * e) + (fom_value_pfp_left_pad_bounded_target))) /\\ (exists fom_gap_pfp_left_pad_bounded_target_value_bound. fom_gap_pfp_left_pad_bounded_target_value_bound + S (fom_value_pfp_left_pad_bounded_target) = p)))", "statement_sha256": "9b06cee98d2a6366af11b50ad7a7f9dd9bd5042cf642af2c424d9abafe65eff7", "summary": "Actual leading-zero padding preserves canonical prime-field bounds at its exact enlarged length.", "summary_sha256": "64e390683c84440cc7a71fbcc2e57c43aca40d7ebf4ee3ab76f6e06e2315f7ba" }, { "alpha_v33_frontier_enrollment": { "body_receipt_sha256": 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(exists pfa_gap_left_pad_functional_firstbound. pfa_gap_left_pad_functional_firstbound + S (pfrep_index_left_pad_functional_first) = (L)) -> (((exists ff_h_pfp_left_pad_functional_firstinput. ff_h_pfp_left_pad_functional_firstinput + S (pfrep_value_left_pad_functional_first) = S ((S (pfrep_index_left_pad_functional_first)) * c)) /\\ exists ff_q_pfp_left_pad_functional_firstinput. b = ff_q_pfp_left_pad_functional_firstinput * S ((S (pfrep_index_left_pad_functional_first)) * c) + (pfrep_value_left_pad_functional_first))) -> (((exists ff_h_pfp_left_pad_functional_firstoutput. ff_h_pfp_left_pad_functional_firstoutput + S (pfrep_value_left_pad_functional_first) = S ((S ((t)+pfrep_index_left_pad_functional_first)) * e)) /\\ exists ff_q_pfp_left_pad_functional_firstoutput. d = ff_q_pfp_left_pad_functional_firstoutput * S ((S ((t)+pfrep_index_left_pad_functional_first)) * e) + (pfrep_value_left_pad_functional_first))))))) -> (((forall pfp_repeat_index_left_pad_functional_secondzeros. (exists pfa_gap_left_pad_functional_secondzerosindex. pfa_gap_left_pad_functional_secondzerosindex + S (pfp_repeat_index_left_pad_functional_secondzeros) = (t)) -> (((exists ff_h_pfp_left_pad_functional_secondzerosentry. ff_h_pfp_left_pad_functional_secondzerosentry + S (0) = S ((S (pfp_repeat_index_left_pad_functional_secondzeros)) * g)) /\\ exists ff_q_pfp_left_pad_functional_secondzerosentry. f = ff_q_pfp_left_pad_functional_secondzerosentry * S ((S (pfp_repeat_index_left_pad_functional_secondzeros)) * g) + (0)))) /\\ ((forall pfrep_index_left_pad_functional_second pfrep_value_left_pad_functional_second. (exists pfa_gap_left_pad_functional_secondbound. pfa_gap_left_pad_functional_secondbound + S (pfrep_index_left_pad_functional_second) = (L)) -> (((exists ff_h_pfp_left_pad_functional_secondinput. ff_h_pfp_left_pad_functional_secondinput + S (pfrep_value_left_pad_functional_second) = S ((S (pfrep_index_left_pad_functional_second)) * c)) /\\ exists ff_q_pfp_left_pad_functional_secondinput. b = ff_q_pfp_left_pad_functional_secondinput * S ((S (pfrep_index_left_pad_functional_second)) * c) + (pfrep_value_left_pad_functional_second))) -> (((exists ff_h_pfp_left_pad_functional_secondoutput. ff_h_pfp_left_pad_functional_secondoutput + S (pfrep_value_left_pad_functional_second) = S ((S ((t)+pfrep_index_left_pad_functional_second)) * g)) /\\ exists ff_q_pfp_left_pad_functional_secondoutput. f = ff_q_pfp_left_pad_functional_secondoutput * S ((S ((t)+pfrep_index_left_pad_functional_second)) * g) + (pfrep_value_left_pad_functional_second))))))) -> (forall mdr_i_pfp_left_pad_functional_result mdr_a_pfp_left_pad_functional_result. 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(exists pfa_gap_suffix_to_pad_suffixbound. pfa_gap_suffix_to_pad_suffixbound + S (pftrim_index_suffix_to_pad_suffix) = (L)) -> (((exists ff_h_pfp_suffix_to_pad_suffixsource. ff_h_pfp_suffix_to_pad_suffixsource + S (pftrim_value_suffix_to_pad_suffix) = S ((S ((t)+pftrim_index_suffix_to_pad_suffix)) * e)) /\\ exists ff_q_pfp_suffix_to_pad_suffixsource. d = ff_q_pfp_suffix_to_pad_suffixsource * S ((S ((t)+pftrim_index_suffix_to_pad_suffix)) * e) + (pftrim_value_suffix_to_pad_suffix))) -> (((exists ff_h_pfp_suffix_to_pad_suffixoutput. ff_h_pfp_suffix_to_pad_suffixoutput + S (pftrim_value_suffix_to_pad_suffix) = S ((S (pftrim_index_suffix_to_pad_suffix)) * c)) /\\ exists ff_q_pfp_suffix_to_pad_suffixoutput. b = ff_q_pfp_suffix_to_pad_suffixoutput * S ((S (pftrim_index_suffix_to_pad_suffix)) * c) + (pftrim_value_suffix_to_pad_suffix)))) -> (((forall pfp_repeat_index_suffix_to_pad_resultzeros. 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((((L)=(t)+(M)) /\\ (((forall fom_index_pfp_trim_to_pad_sourceinput. (exists fom_gap_pfp_trim_to_pad_sourceinput_index_bound. fom_gap_pfp_trim_to_pad_sourceinput_index_bound + S (fom_index_pfp_trim_to_pad_sourceinput) = L) -> exists fom_value_pfp_trim_to_pad_sourceinput. ((((exists fom_beta_height_pfp_trim_to_pad_sourceinput_entry. fom_beta_height_pfp_trim_to_pad_sourceinput_entry + S (fom_value_pfp_trim_to_pad_sourceinput) = S ((S (fom_index_pfp_trim_to_pad_sourceinput)) * c)) /\\ exists fom_beta_quotient_pfp_trim_to_pad_sourceinput_entry. b = fom_beta_quotient_pfp_trim_to_pad_sourceinput_entry * S ((S (fom_index_pfp_trim_to_pad_sourceinput)) * c) + (fom_value_pfp_trim_to_pad_sourceinput))) /\\ (exists fom_gap_pfp_trim_to_pad_sourceinput_value_bound. fom_gap_pfp_trim_to_pad_sourceinput_value_bound + S (fom_value_pfp_trim_to_pad_sourceinput) = p))) /\\ (((forall pfp_repeat_index_trim_to_pad_sourceremoved. (exists pfa_gap_trim_to_pad_sourceremovedindex. pfa_gap_trim_to_pad_sourceremovedindex + S (pfp_repeat_index_trim_to_pad_sourceremoved) = (t)) -> (((exists ff_h_pfp_trim_to_pad_sourceremovedentry. ff_h_pfp_trim_to_pad_sourceremovedentry + S (0) = S ((S (pfp_repeat_index_trim_to_pad_sourceremoved)) * c)) /\\ exists ff_q_pfp_trim_to_pad_sourceremovedentry. b = ff_q_pfp_trim_to_pad_sourceremovedentry * S ((S (pfp_repeat_index_trim_to_pad_sourceremoved)) * c) + (0)))) /\\ (((forall pftrim_index_trim_to_pad_sourcesuffix pftrim_value_trim_to_pad_sourcesuffix. (exists pfa_gap_trim_to_pad_sourcesuffixbound. pfa_gap_trim_to_pad_sourcesuffixbound + S (pftrim_index_trim_to_pad_sourcesuffix) = (M)) -> (((exists ff_h_pfp_trim_to_pad_sourcesuffixsource. ff_h_pfp_trim_to_pad_sourcesuffixsource + S (pftrim_value_trim_to_pad_sourcesuffix) = S ((S ((t)+pftrim_index_trim_to_pad_sourcesuffix)) * c)) /\\ exists ff_q_pfp_trim_to_pad_sourcesuffixsource. b = ff_q_pfp_trim_to_pad_sourcesuffixsource * S ((S ((t)+pftrim_index_trim_to_pad_sourcesuffix)) * c) + (pftrim_value_trim_to_pad_sourcesuffix))) -> (((exists ff_h_pfp_trim_to_pad_sourcesuffixoutput. ff_h_pfp_trim_to_pad_sourcesuffixoutput + S (pftrim_value_trim_to_pad_sourcesuffix) = S ((S (pftrim_index_trim_to_pad_sourcesuffix)) * e)) /\\ exists ff_q_pfp_trim_to_pad_sourcesuffixoutput. d = ff_q_pfp_trim_to_pad_sourcesuffixoutput * S ((S (pftrim_index_trim_to_pad_sourcesuffix)) * e) + (pftrim_value_trim_to_pad_sourcesuffix)))) /\\ (((M)=0 \\/ (exists pftrim_leading_trim_to_pad_sourcenormal. ((((exists ff_h_pfp_trim_to_pad_sourcenormalentry. ff_h_pfp_trim_to_pad_sourcenormalentry + S (pftrim_leading_trim_to_pad_sourcenormal) = S ((S (0)) * e)) /\\ exists ff_q_pfp_trim_to_pad_sourcenormalentry. d = ff_q_pfp_trim_to_pad_sourcenormalentry * S ((S (0)) * e) + (pftrim_leading_trim_to_pad_sourcenormal))) /\\ ((~(pftrim_leading_trim_to_pad_sourcenormal=0))))))))))))))) -> (((forall pfp_repeat_index_trim_to_pad_resultzeros. (exists pfa_gap_trim_to_pad_resultzerosindex. pfa_gap_trim_to_pad_resultzerosindex + S (pfp_repeat_index_trim_to_pad_resultzeros) = (t)) -> (((exists ff_h_pfp_trim_to_pad_resultzerosentry. ff_h_pfp_trim_to_pad_resultzerosentry + S (0) = S ((S (pfp_repeat_index_trim_to_pad_resultzeros)) * c)) /\\ exists ff_q_pfp_trim_to_pad_resultzerosentry. b = ff_q_pfp_trim_to_pad_resultzerosentry * S ((S (pfp_repeat_index_trim_to_pad_resultzeros)) * c) + (0)))) /\\ ((forall pfrep_index_trim_to_pad_result pfrep_value_trim_to_pad_result. (exists pfa_gap_trim_to_pad_resultbound. pfa_gap_trim_to_pad_resultbound + S (pfrep_index_trim_to_pad_result) = (M)) -> (((exists ff_h_pfp_trim_to_pad_resultinput. ff_h_pfp_trim_to_pad_resultinput + S (pfrep_value_trim_to_pad_result) = S ((S (pfrep_index_trim_to_pad_result)) * e)) /\\ exists ff_q_pfp_trim_to_pad_resultinput. d = ff_q_pfp_trim_to_pad_resultinput * S ((S (pfrep_index_trim_to_pad_result)) * e) + (pfrep_value_trim_to_pad_result))) -> (((exists ff_h_pfp_trim_to_pad_resultoutput. ff_h_pfp_trim_to_pad_resultoutput + S (pfrep_value_trim_to_pad_result) = S ((S ((t)+pfrep_index_trim_to_pad_result)) * c)) /\\ exists ff_q_pfp_trim_to_pad_resultoutput. b = ff_q_pfp_trim_to_pad_resultoutput * S ((S ((t)+pfrep_index_trim_to_pad_result)) * c) + (pfrep_value_trim_to_pad_result)))))))", "statement_sha256": "f3c6bead49de080d6c0606e475943610e8424961a60dc17e4b64c894a0c3eb58", "summary": "Actual trimming identifies its input as the retained coefficient prefix with exactly the removed 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(((forall pfp_repeat_index_pad_power_graphzeros. (exists pfa_gap_pad_power_graphzerosindex. pfa_gap_pad_power_graphzerosindex + S (pfp_repeat_index_pad_power_graphzeros) = (t)) -> (((exists ff_h_pfp_pad_power_graphzerosentry. ff_h_pfp_pad_power_graphzerosentry + S (0) = S ((S (pfp_repeat_index_pad_power_graphzeros)) * e)) /\\ exists ff_q_pfp_pad_power_graphzerosentry. d = ff_q_pfp_pad_power_graphzerosentry * S ((S (pfp_repeat_index_pad_power_graphzeros)) * e) + (0)))) /\\ ((forall pfrep_index_pad_power_graph pfrep_value_pad_power_graph. (exists pfa_gap_pad_power_graphbound. pfa_gap_pad_power_graphbound + S (pfrep_index_pad_power_graph) = (L)) -> (((exists ff_h_pfp_pad_power_graphinput. ff_h_pfp_pad_power_graphinput + S (pfrep_value_pad_power_graph) = S ((S (pfrep_index_pad_power_graph)) * c)) /\\ exists ff_q_pfp_pad_power_graphinput. b = ff_q_pfp_pad_power_graphinput * S ((S (pfrep_index_pad_power_graph)) * c) + (pfrep_value_pad_power_graph))) -> (((exists ff_h_pfp_pad_power_graphoutput. ff_h_pfp_pad_power_graphoutput + S (pfrep_value_pad_power_graph) = S ((S ((t)+pfrep_index_pad_power_graph)) * e)) /\\ exists ff_q_pfp_pad_power_graphoutput. d = ff_q_pfp_pad_power_graphoutput * S ((S ((t)+pfrep_index_pad_power_graph)) * e) + (pfrep_value_pad_power_graph))))))) -> ((exists pfrep_position_pad_power_source. ((pfrep_position_pad_power_source+S (k)=(L)) /\\ ((((exists ff_h_pfp_pad_power_sourceentry. ff_h_pfp_pad_power_sourceentry + S (a) = S ((S (pfrep_position_pad_power_source)) * c)) /\\ exists ff_q_pfp_pad_power_sourceentry. b = ff_q_pfp_pad_power_sourceentry * S ((S (pfrep_position_pad_power_source)) * c) + (a)))))) \\/ (((exists pfrep_gap_pad_power_sourceoutside. pfrep_gap_pad_power_sourceoutside+(L)=(k)) /\\ (((a)=0))))) -> ((exists pfrep_position_pad_power_result. 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((exists pfrep_position_pad_equivalent_resultfirst. ((pfrep_position_pad_equivalent_resultfirst+S (pfrep_power_pad_equivalent_result)=(L)) /\\ ((((exists ff_h_pfp_pad_equivalent_resultfirstentry. ff_h_pfp_pad_equivalent_resultfirstentry + S (pfrep_left_pad_equivalent_result) = S ((S (pfrep_position_pad_equivalent_resultfirst)) * c)) /\\ exists ff_q_pfp_pad_equivalent_resultfirstentry. b = ff_q_pfp_pad_equivalent_resultfirstentry * S ((S (pfrep_position_pad_equivalent_resultfirst)) * c) + (pfrep_left_pad_equivalent_result)))))) \\/ (((exists pfrep_gap_pad_equivalent_resultfirstoutside. pfrep_gap_pad_equivalent_resultfirstoutside+(L)=(pfrep_power_pad_equivalent_result)) /\\ (((pfrep_left_pad_equivalent_result)=0))))) -> ((exists pfrep_position_pad_equivalent_resultsecond. 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(exists pfa_gap_trim_equivalence_copyremovedindex. pfa_gap_trim_equivalence_copyremovedindex + S (pfp_repeat_index_trim_equivalence_copyremoved) = (t)) -> (((exists ff_h_pfp_trim_equivalence_copyremovedentry. ff_h_pfp_trim_equivalence_copyremovedentry + S (0) = S ((S (pfp_repeat_index_trim_equivalence_copyremoved)) * c)) /\\ exists ff_q_pfp_trim_equivalence_copyremovedentry. b = ff_q_pfp_trim_equivalence_copyremovedentry * S ((S (pfp_repeat_index_trim_equivalence_copyremoved)) * c) + (0)))) /\\ (((forall pftrim_index_trim_equivalence_copysuffix pftrim_value_trim_equivalence_copysuffix. (exists pfa_gap_trim_equivalence_copysuffixbound. pfa_gap_trim_equivalence_copysuffixbound + S (pftrim_index_trim_equivalence_copysuffix) = (M)) -> (((exists ff_h_pfp_trim_equivalence_copysuffixsource. ff_h_pfp_trim_equivalence_copysuffixsource + S (pftrim_value_trim_equivalence_copysuffix) = S ((S ((t)+pftrim_index_trim_equivalence_copysuffix)) * c)) /\\ exists ff_q_pfp_trim_equivalence_copysuffixsource. b = ff_q_pfp_trim_equivalence_copysuffixsource * S ((S ((t)+pftrim_index_trim_equivalence_copysuffix)) * c) + (pftrim_value_trim_equivalence_copysuffix))) -> (((exists ff_h_pfp_trim_equivalence_copysuffixoutput. ff_h_pfp_trim_equivalence_copysuffixoutput + S (pftrim_value_trim_equivalence_copysuffix) = S ((S (pftrim_index_trim_equivalence_copysuffix)) * e)) /\\ exists ff_q_pfp_trim_equivalence_copysuffixoutput. d = ff_q_pfp_trim_equivalence_copysuffixoutput * S ((S (pftrim_index_trim_equivalence_copysuffix)) * e) + (pftrim_value_trim_equivalence_copysuffix)))) /\\ (((M)=0 \\/ (exists pftrim_leading_trim_equivalence_copynormal. 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((((exists fom_beta_height_pfp_trim_equivalence_sourceinput_entry. fom_beta_height_pfp_trim_equivalence_sourceinput_entry + S (fom_value_pfp_trim_equivalence_sourceinput) = S ((S (fom_index_pfp_trim_equivalence_sourceinput)) * c)) /\\ exists fom_beta_quotient_pfp_trim_equivalence_sourceinput_entry. b = fom_beta_quotient_pfp_trim_equivalence_sourceinput_entry * S ((S (fom_index_pfp_trim_equivalence_sourceinput)) * c) + (fom_value_pfp_trim_equivalence_sourceinput))) /\\ (exists fom_gap_pfp_trim_equivalence_sourceinput_value_bound. fom_gap_pfp_trim_equivalence_sourceinput_value_bound + S (fom_value_pfp_trim_equivalence_sourceinput) = p))) /\\ (((forall pfp_repeat_index_trim_equivalence_sourceremoved. (exists pfa_gap_trim_equivalence_sourceremovedindex. pfa_gap_trim_equivalence_sourceremovedindex + S (pfp_repeat_index_trim_equivalence_sourceremoved) = (t)) -> (((exists ff_h_pfp_trim_equivalence_sourceremovedentry. ff_h_pfp_trim_equivalence_sourceremovedentry + S (0) = S ((S (pfp_repeat_index_trim_equivalence_sourceremoved)) * c)) /\\ exists ff_q_pfp_trim_equivalence_sourceremovedentry. b = ff_q_pfp_trim_equivalence_sourceremovedentry * S ((S (pfp_repeat_index_trim_equivalence_sourceremoved)) * c) + (0)))) /\\ (((forall pftrim_index_trim_equivalence_sourcesuffix pftrim_value_trim_equivalence_sourcesuffix. (exists pfa_gap_trim_equivalence_sourcesuffixbound. pfa_gap_trim_equivalence_sourcesuffixbound + S (pftrim_index_trim_equivalence_sourcesuffix) = (M)) -> (((exists ff_h_pfp_trim_equivalence_sourcesuffixsource. ff_h_pfp_trim_equivalence_sourcesuffixsource + S (pftrim_value_trim_equivalence_sourcesuffix) = S ((S ((t)+pftrim_index_trim_equivalence_sourcesuffix)) * c)) /\\ exists ff_q_pfp_trim_equivalence_sourcesuffixsource. b = ff_q_pfp_trim_equivalence_sourcesuffixsource * S ((S ((t)+pftrim_index_trim_equivalence_sourcesuffix)) * c) + (pftrim_value_trim_equivalence_sourcesuffix))) -> (((exists ff_h_pfp_trim_equivalence_sourcesuffixoutput. ff_h_pfp_trim_equivalence_sourcesuffixoutput + S (pftrim_value_trim_equivalence_sourcesuffix) = S ((S (pftrim_index_trim_equivalence_sourcesuffix)) * e)) /\\ exists ff_q_pfp_trim_equivalence_sourcesuffixoutput. d = ff_q_pfp_trim_equivalence_sourcesuffixoutput * S ((S (pftrim_index_trim_equivalence_sourcesuffix)) * e) + (pftrim_value_trim_equivalence_sourcesuffix)))) /\\ (((M)=0 \\/ (exists pftrim_leading_trim_equivalence_sourcenormal. ((((exists ff_h_pfp_trim_equivalence_sourcenormalentry. ff_h_pfp_trim_equivalence_sourcenormalentry + S (pftrim_leading_trim_equivalence_sourcenormal) = S ((S (0)) * e)) /\\ exists ff_q_pfp_trim_equivalence_sourcenormalentry. d = ff_q_pfp_trim_equivalence_sourcenormalentry * S ((S (0)) * e) + (pftrim_leading_trim_equivalence_sourcenormal))) /\\ ((~(pftrim_leading_trim_equivalence_sourcenormal=0))))))))))))))) -> (forall pfrep_power_trim_equivalence_result pfrep_left_trim_equivalence_result pfrep_right_trim_equivalence_result. ((exists pfrep_position_trim_equivalence_resultfirst. ((pfrep_position_trim_equivalence_resultfirst+S (pfrep_power_trim_equivalence_result)=(L)) /\\ ((((exists ff_h_pfp_trim_equivalence_resultfirstentry. ff_h_pfp_trim_equivalence_resultfirstentry + S (pfrep_left_trim_equivalence_result) = S ((S (pfrep_position_trim_equivalence_resultfirst)) * c)) /\\ exists ff_q_pfp_trim_equivalence_resultfirstentry. b = ff_q_pfp_trim_equivalence_resultfirstentry * S ((S (pfrep_position_trim_equivalence_resultfirst)) * c) + (pfrep_left_trim_equivalence_result)))))) \\/ (((exists pfrep_gap_trim_equivalence_resultfirstoutside. pfrep_gap_trim_equivalence_resultfirstoutside+(L)=(pfrep_power_trim_equivalence_result)) /\\ (((pfrep_left_trim_equivalence_result)=0))))) -> ((exists pfrep_position_trim_equivalence_resultsecond. 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(exists mdr_gap_pfp_pad_transport_outputb. mdr_gap_pfp_pad_transport_outputb + S (mdr_i_pfp_pad_transport_output) = (t+L)) -> (((exists ff_h_mdr_pfp_pad_transport_outputo. ff_h_mdr_pfp_pad_transport_outputo + S (mdr_a_pfp_pad_transport_output) = S ((S (mdr_i_pfp_pad_transport_output)) * e)) /\\ exists ff_q_mdr_pfp_pad_transport_outputo. d = ff_q_mdr_pfp_pad_transport_outputo * S ((S (mdr_i_pfp_pad_transport_output)) * e) + (mdr_a_pfp_pad_transport_output))) -> (((exists ff_h_mdr_pfp_pad_transport_outputn. ff_h_mdr_pfp_pad_transport_outputn + S (mdr_a_pfp_pad_transport_output) = S ((S (mdr_i_pfp_pad_transport_output)) * E)) /\\ exists ff_q_mdr_pfp_pad_transport_outputn. D = ff_q_mdr_pfp_pad_transport_outputn * S ((S (mdr_i_pfp_pad_transport_output)) * E) + (mdr_a_pfp_pad_transport_output)))) -> (((forall pfp_repeat_index_pad_transport_oldzeros. (exists pfa_gap_pad_transport_oldzerosindex. pfa_gap_pad_transport_oldzerosindex + S (pfp_repeat_index_pad_transport_oldzeros) = (t)) -> (((exists ff_h_pfp_pad_transport_oldzerosentry. ff_h_pfp_pad_transport_oldzerosentry + S (0) = S ((S (pfp_repeat_index_pad_transport_oldzeros)) * e)) /\\ exists ff_q_pfp_pad_transport_oldzerosentry. d = ff_q_pfp_pad_transport_oldzerosentry * S ((S (pfp_repeat_index_pad_transport_oldzeros)) * e) + (0)))) /\\ ((forall pfrep_index_pad_transport_old pfrep_value_pad_transport_old. (exists pfa_gap_pad_transport_oldbound. pfa_gap_pad_transport_oldbound + S (pfrep_index_pad_transport_old) = (L)) -> (((exists ff_h_pfp_pad_transport_oldinput. ff_h_pfp_pad_transport_oldinput + S (pfrep_value_pad_transport_old) = S ((S (pfrep_index_pad_transport_old)) * c)) /\\ exists ff_q_pfp_pad_transport_oldinput. b = ff_q_pfp_pad_transport_oldinput * S ((S (pfrep_index_pad_transport_old)) * c) + (pfrep_value_pad_transport_old))) -> (((exists ff_h_pfp_pad_transport_oldoutput. ff_h_pfp_pad_transport_oldoutput + S (pfrep_value_pad_transport_old) = S ((S ((t)+pfrep_index_pad_transport_old)) * e)) /\\ exists ff_q_pfp_pad_transport_oldoutput. d = ff_q_pfp_pad_transport_oldoutput * S ((S ((t)+pfrep_index_pad_transport_old)) * e) + (pfrep_value_pad_transport_old))))))) -> (((forall pfp_repeat_index_pad_transport_newzeros. (exists pfa_gap_pad_transport_newzerosindex. pfa_gap_pad_transport_newzerosindex + S (pfp_repeat_index_pad_transport_newzeros) = (t)) -> (((exists ff_h_pfp_pad_transport_newzerosentry. ff_h_pfp_pad_transport_newzerosentry + S (0) = S ((S (pfp_repeat_index_pad_transport_newzeros)) * E)) /\\ exists ff_q_pfp_pad_transport_newzerosentry. D = ff_q_pfp_pad_transport_newzerosentry * S ((S (pfp_repeat_index_pad_transport_newzeros)) * E) + (0)))) /\\ ((forall pfrep_index_pad_transport_new pfrep_value_pad_transport_new. (exists pfa_gap_pad_transport_newbound. pfa_gap_pad_transport_newbound + S (pfrep_index_pad_transport_new) = (L)) -> (((exists ff_h_pfp_pad_transport_newinput. ff_h_pfp_pad_transport_newinput + S (pfrep_value_pad_transport_new) = S ((S (pfrep_index_pad_transport_new)) * C)) /\\ exists ff_q_pfp_pad_transport_newinput. 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((((exists ff_h_pfp_pad_operation_oldleft. ff_h_pfp_pad_operation_oldleft + S (pfp_left_pad_operation_old) = S ((S (pfp_index_pad_operation_old)) * ac)) /\\ exists ff_q_pfp_pad_operation_oldleft. ab = ff_q_pfp_pad_operation_oldleft * S ((S (pfp_index_pad_operation_old)) * ac) + (pfp_left_pad_operation_old))) /\\ (((((exists ff_h_pfp_pad_operation_oldright. ff_h_pfp_pad_operation_oldright + S (pfp_right_pad_operation_old) = S ((S (pfp_index_pad_operation_old)) * bc)) /\\ exists ff_q_pfp_pad_operation_oldright. bb = ff_q_pfp_pad_operation_oldright * S ((S (pfp_index_pad_operation_old)) * bc) + (pfp_right_pad_operation_old))) /\\ (((((exists ff_h_pfp_pad_operation_oldtarget. ff_h_pfp_pad_operation_oldtarget + S (pfp_value_pad_operation_old) = S ((S (pfp_index_pad_operation_old)) * cc)) /\\ exists ff_q_pfp_pad_operation_oldtarget. cb = ff_q_pfp_pad_operation_oldtarget * S ((S (pfp_index_pad_operation_old)) * cc) + (pfp_value_pad_operation_old))) /\\ ((((exists pfa_gap_pad_operation_oldoperationleft. pfa_gap_pad_operation_oldoperationleft + S (pfp_left_pad_operation_old) = (p)) /\\ (((exists pfa_gap_pad_operation_oldoperationright. pfa_gap_pad_operation_oldoperationright + S (pfp_right_pad_operation_old) = (p)) /\\ ((((exists pfa_gap_pad_operation_oldoperationresultbound. pfa_gap_pad_operation_oldoperationresultbound + S (pfp_value_pad_operation_old) = (p)) /\\ ((exists pfa_offset_left_pad_operation_oldoperationresultcongruence pfa_offset_right_pad_operation_oldoperationresultcongruence. ((pfp_left_pad_operation_old) + (pfp_right_pad_operation_old)) + (p) * pfa_offset_left_pad_operation_oldoperationresultcongruence = (pfp_value_pad_operation_old) + (p) * pfa_offset_right_pad_operation_oldoperationresultcongruence)))))))))))))))) -> (((forall pfp_repeat_index_pad_operation_abzeros. (exists pfa_gap_pad_operation_abzerosindex. pfa_gap_pad_operation_abzerosindex + S (pfp_repeat_index_pad_operation_abzeros) = (t)) -> (((exists ff_h_pfp_pad_operation_abzerosentry. ff_h_pfp_pad_operation_abzerosentry + S (0) = S ((S (pfp_repeat_index_pad_operation_abzeros)) * AC)) /\\ exists ff_q_pfp_pad_operation_abzerosentry. AB = ff_q_pfp_pad_operation_abzerosentry * S ((S (pfp_repeat_index_pad_operation_abzeros)) * AC) + (0)))) /\\ ((forall pfrep_index_pad_operation_ab pfrep_value_pad_operation_ab. (exists pfa_gap_pad_operation_abbound. pfa_gap_pad_operation_abbound + S (pfrep_index_pad_operation_ab) = (L)) -> (((exists ff_h_pfp_pad_operation_abinput. ff_h_pfp_pad_operation_abinput + S (pfrep_value_pad_operation_ab) = S ((S (pfrep_index_pad_operation_ab)) * ac)) /\\ exists ff_q_pfp_pad_operation_abinput. ab = ff_q_pfp_pad_operation_abinput * S ((S (pfrep_index_pad_operation_ab)) * ac) + (pfrep_value_pad_operation_ab))) -> (((exists ff_h_pfp_pad_operation_aboutput. ff_h_pfp_pad_operation_aboutput + S (pfrep_value_pad_operation_ab) = S ((S ((t)+pfrep_index_pad_operation_ab)) * AC)) /\\ exists ff_q_pfp_pad_operation_aboutput. AB = ff_q_pfp_pad_operation_aboutput * S ((S ((t)+pfrep_index_pad_operation_ab)) * AC) + (pfrep_value_pad_operation_ab))))))) -> (((forall pfp_repeat_index_pad_operation_bbzeros. (exists pfa_gap_pad_operation_bbzerosindex. pfa_gap_pad_operation_bbzerosindex + S (pfp_repeat_index_pad_operation_bbzeros) = (t)) -> (((exists ff_h_pfp_pad_operation_bbzerosentry. ff_h_pfp_pad_operation_bbzerosentry + S (0) = S ((S (pfp_repeat_index_pad_operation_bbzeros)) * BC)) /\\ exists ff_q_pfp_pad_operation_bbzerosentry. BB = ff_q_pfp_pad_operation_bbzerosentry * S ((S (pfp_repeat_index_pad_operation_bbzeros)) * BC) + (0)))) /\\ ((forall pfrep_index_pad_operation_bb pfrep_value_pad_operation_bb. (exists pfa_gap_pad_operation_bbbound. pfa_gap_pad_operation_bbbound + S (pfrep_index_pad_operation_bb) = (L)) -> (((exists ff_h_pfp_pad_operation_bbinput. ff_h_pfp_pad_operation_bbinput + S (pfrep_value_pad_operation_bb) = S ((S (pfrep_index_pad_operation_bb)) * bc)) /\\ exists ff_q_pfp_pad_operation_bbinput. bb = ff_q_pfp_pad_operation_bbinput * S ((S (pfrep_index_pad_operation_bb)) * bc) + (pfrep_value_pad_operation_bb))) -> (((exists ff_h_pfp_pad_operation_bboutput. ff_h_pfp_pad_operation_bboutput + S (pfrep_value_pad_operation_bb) = S ((S ((t)+pfrep_index_pad_operation_bb)) * BC)) /\\ exists ff_q_pfp_pad_operation_bboutput. BB = ff_q_pfp_pad_operation_bboutput * S ((S ((t)+pfrep_index_pad_operation_bb)) * BC) + (pfrep_value_pad_operation_bb))))))) -> (((forall pfp_repeat_index_pad_operation_cbzeros. (exists pfa_gap_pad_operation_cbzerosindex. pfa_gap_pad_operation_cbzerosindex + S (pfp_repeat_index_pad_operation_cbzeros) = (t)) -> (((exists ff_h_pfp_pad_operation_cbzerosentry. ff_h_pfp_pad_operation_cbzerosentry + S (0) = S ((S (pfp_repeat_index_pad_operation_cbzeros)) * CC)) /\\ exists ff_q_pfp_pad_operation_cbzerosentry. CB = ff_q_pfp_pad_operation_cbzerosentry * S ((S (pfp_repeat_index_pad_operation_cbzeros)) * CC) + (0)))) /\\ ((forall pfrep_index_pad_operation_cb pfrep_value_pad_operation_cb. (exists pfa_gap_pad_operation_cbbound. pfa_gap_pad_operation_cbbound + S (pfrep_index_pad_operation_cb) = (L)) -> (((exists ff_h_pfp_pad_operation_cbinput. ff_h_pfp_pad_operation_cbinput + S (pfrep_value_pad_operation_cb) = S ((S (pfrep_index_pad_operation_cb)) * cc)) /\\ exists ff_q_pfp_pad_operation_cbinput. cb = ff_q_pfp_pad_operation_cbinput * S ((S (pfrep_index_pad_operation_cb)) * cc) + (pfrep_value_pad_operation_cb))) -> (((exists ff_h_pfp_pad_operation_cboutput. ff_h_pfp_pad_operation_cboutput + S (pfrep_value_pad_operation_cb) = S ((S ((t)+pfrep_index_pad_operation_cb)) * CC)) /\\ exists ff_q_pfp_pad_operation_cboutput. CB = ff_q_pfp_pad_operation_cboutput * S ((S ((t)+pfrep_index_pad_operation_cb)) * CC) + (pfrep_value_pad_operation_cb))))))) -> (forall pfp_index_pad_operation_new. (exists pfa_gap_pad_operation_newindex. pfa_gap_pad_operation_newindex + S (pfp_index_pad_operation_new) = (t+L)) -> exists pfp_left_pad_operation_new pfp_right_pad_operation_new pfp_value_pad_operation_new. ((((exists ff_h_pfp_pad_operation_newleft. ff_h_pfp_pad_operation_newleft + S (pfp_left_pad_operation_new) = S ((S (pfp_index_pad_operation_new)) * AC)) /\\ exists ff_q_pfp_pad_operation_newleft. AB = ff_q_pfp_pad_operation_newleft * S ((S (pfp_index_pad_operation_new)) * AC) + (pfp_left_pad_operation_new))) /\\ (((((exists ff_h_pfp_pad_operation_newright. ff_h_pfp_pad_operation_newright + S (pfp_right_pad_operation_new) = S ((S (pfp_index_pad_operation_new)) * BC)) /\\ exists ff_q_pfp_pad_operation_newright. 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(~((p) = 1) /\\ forall pfa_factor_left_pad_operation_prime pfa_factor_right_pad_operation_prime. (p) = pfa_factor_left_pad_operation_prime * pfa_factor_right_pad_operation_prime -> pfa_factor_left_pad_operation_prime = 1 \\/ pfa_factor_right_pad_operation_prime = 1) -> (forall pfs_index_pad_operation_old. (exists pfa_gap_pad_operation_oldindex. pfa_gap_pad_operation_oldindex + S (pfs_index_pad_operation_old) = (L)) -> exists pfs_left_pad_operation_old pfs_right_pad_operation_old pfs_result_pad_operation_old. ((((exists ff_h_pfp_pad_operation_oldleft. ff_h_pfp_pad_operation_oldleft + S (pfs_left_pad_operation_old) = S ((S (pfs_index_pad_operation_old)) * ac)) /\\ exists ff_q_pfp_pad_operation_oldleft. ab = ff_q_pfp_pad_operation_oldleft * S ((S (pfs_index_pad_operation_old)) * ac) + (pfs_left_pad_operation_old))) /\\ (((((exists ff_h_pfp_pad_operation_oldright. ff_h_pfp_pad_operation_oldright + S (pfs_right_pad_operation_old) = S ((S (pfs_index_pad_operation_old)) * bc)) /\\ exists ff_q_pfp_pad_operation_oldright. bb = ff_q_pfp_pad_operation_oldright * S ((S (pfs_index_pad_operation_old)) * bc) + (pfs_right_pad_operation_old))) /\\ (((((exists ff_h_pfp_pad_operation_oldresult. ff_h_pfp_pad_operation_oldresult + S (pfs_result_pad_operation_old) = S ((S (pfs_index_pad_operation_old)) * cc)) /\\ exists ff_q_pfp_pad_operation_oldresult. cb = ff_q_pfp_pad_operation_oldresult * S ((S (pfs_index_pad_operation_old)) * cc) + (pfs_result_pad_operation_old))) /\\ ((((exists pfa_gap_pad_operation_oldoperationleft. pfa_gap_pad_operation_oldoperationleft + S (pfs_right_pad_operation_old) = (p)) /\\ (((exists pfa_gap_pad_operation_oldoperationright. pfa_gap_pad_operation_oldoperationright + S (pfs_result_pad_operation_old) = (p)) /\\ ((((exists pfa_gap_pad_operation_oldoperationresultbound. pfa_gap_pad_operation_oldoperationresultbound + S (pfs_left_pad_operation_old) = (p)) /\\ ((exists pfa_offset_left_pad_operation_oldoperationresultcongruence pfa_offset_right_pad_operation_oldoperationresultcongruence. ((pfs_right_pad_operation_old) + (pfs_result_pad_operation_old)) + (p) * pfa_offset_left_pad_operation_oldoperationresultcongruence = (pfs_left_pad_operation_old) + (p) * pfa_offset_right_pad_operation_oldoperationresultcongruence)))))))))))))))) -> (((forall pfp_repeat_index_pad_operation_abzeros. (exists pfa_gap_pad_operation_abzerosindex. pfa_gap_pad_operation_abzerosindex + S (pfp_repeat_index_pad_operation_abzeros) = (t)) -> (((exists ff_h_pfp_pad_operation_abzerosentry. ff_h_pfp_pad_operation_abzerosentry + S (0) = S ((S (pfp_repeat_index_pad_operation_abzeros)) * AC)) /\\ exists ff_q_pfp_pad_operation_abzerosentry. AB = ff_q_pfp_pad_operation_abzerosentry * S ((S (pfp_repeat_index_pad_operation_abzeros)) * AC) + (0)))) /\\ ((forall pfrep_index_pad_operation_ab pfrep_value_pad_operation_ab. (exists pfa_gap_pad_operation_abbound. pfa_gap_pad_operation_abbound + S (pfrep_index_pad_operation_ab) = (L)) -> (((exists ff_h_pfp_pad_operation_abinput. ff_h_pfp_pad_operation_abinput + S (pfrep_value_pad_operation_ab) = S ((S (pfrep_index_pad_operation_ab)) * ac)) /\\ exists ff_q_pfp_pad_operation_abinput. ab = ff_q_pfp_pad_operation_abinput * S ((S (pfrep_index_pad_operation_ab)) * ac) + (pfrep_value_pad_operation_ab))) -> (((exists ff_h_pfp_pad_operation_aboutput. ff_h_pfp_pad_operation_aboutput + S (pfrep_value_pad_operation_ab) = S ((S ((t)+pfrep_index_pad_operation_ab)) * AC)) /\\ exists ff_q_pfp_pad_operation_aboutput. AB = ff_q_pfp_pad_operation_aboutput * S ((S ((t)+pfrep_index_pad_operation_ab)) * AC) + (pfrep_value_pad_operation_ab))))))) -> (((forall pfp_repeat_index_pad_operation_bbzeros. (exists pfa_gap_pad_operation_bbzerosindex. pfa_gap_pad_operation_bbzerosindex + S (pfp_repeat_index_pad_operation_bbzeros) = (t)) -> (((exists ff_h_pfp_pad_operation_bbzerosentry. ff_h_pfp_pad_operation_bbzerosentry + S (0) = S ((S (pfp_repeat_index_pad_operation_bbzeros)) * BC)) /\\ exists ff_q_pfp_pad_operation_bbzerosentry. BB = ff_q_pfp_pad_operation_bbzerosentry * S ((S (pfp_repeat_index_pad_operation_bbzeros)) * BC) + (0)))) /\\ ((forall pfrep_index_pad_operation_bb pfrep_value_pad_operation_bb. (exists pfa_gap_pad_operation_bbbound. pfa_gap_pad_operation_bbbound + S (pfrep_index_pad_operation_bb) = (L)) -> (((exists ff_h_pfp_pad_operation_bbinput. ff_h_pfp_pad_operation_bbinput + S (pfrep_value_pad_operation_bb) = S ((S (pfrep_index_pad_operation_bb)) * bc)) /\\ exists ff_q_pfp_pad_operation_bbinput. bb = ff_q_pfp_pad_operation_bbinput * S ((S (pfrep_index_pad_operation_bb)) * bc) + (pfrep_value_pad_operation_bb))) -> (((exists ff_h_pfp_pad_operation_bboutput. ff_h_pfp_pad_operation_bboutput + S (pfrep_value_pad_operation_bb) = S ((S ((t)+pfrep_index_pad_operation_bb)) * BC)) /\\ exists ff_q_pfp_pad_operation_bboutput. BB = ff_q_pfp_pad_operation_bboutput * S ((S ((t)+pfrep_index_pad_operation_bb)) * BC) + (pfrep_value_pad_operation_bb))))))) -> (((forall pfp_repeat_index_pad_operation_cbzeros. (exists pfa_gap_pad_operation_cbzerosindex. pfa_gap_pad_operation_cbzerosindex + S (pfp_repeat_index_pad_operation_cbzeros) = (t)) -> (((exists ff_h_pfp_pad_operation_cbzerosentry. ff_h_pfp_pad_operation_cbzerosentry + S (0) = S ((S (pfp_repeat_index_pad_operation_cbzeros)) * CC)) /\\ exists ff_q_pfp_pad_operation_cbzerosentry. CB = ff_q_pfp_pad_operation_cbzerosentry * S ((S (pfp_repeat_index_pad_operation_cbzeros)) * CC) + (0)))) /\\ ((forall pfrep_index_pad_operation_cb pfrep_value_pad_operation_cb. (exists pfa_gap_pad_operation_cbbound. pfa_gap_pad_operation_cbbound + S (pfrep_index_pad_operation_cb) = (L)) -> (((exists ff_h_pfp_pad_operation_cbinput. ff_h_pfp_pad_operation_cbinput + S (pfrep_value_pad_operation_cb) = S ((S (pfrep_index_pad_operation_cb)) * cc)) /\\ exists ff_q_pfp_pad_operation_cbinput. cb = ff_q_pfp_pad_operation_cbinput * S ((S (pfrep_index_pad_operation_cb)) * cc) + (pfrep_value_pad_operation_cb))) -> (((exists ff_h_pfp_pad_operation_cboutput. ff_h_pfp_pad_operation_cboutput + S (pfrep_value_pad_operation_cb) = S ((S ((t)+pfrep_index_pad_operation_cb)) * CC)) /\\ exists ff_q_pfp_pad_operation_cboutput. CB = ff_q_pfp_pad_operation_cboutput * S ((S ((t)+pfrep_index_pad_operation_cb)) * CC) + (pfrep_value_pad_operation_cb))))))) -> (forall pfs_index_pad_operation_new. (exists pfa_gap_pad_operation_newindex. pfa_gap_pad_operation_newindex + S (pfs_index_pad_operation_new) = (t+L)) -> exists pfs_left_pad_operation_new pfs_right_pad_operation_new pfs_result_pad_operation_new. ((((exists ff_h_pfp_pad_operation_newleft. ff_h_pfp_pad_operation_newleft + S (pfs_left_pad_operation_new) = S ((S (pfs_index_pad_operation_new)) * AC)) /\\ exists ff_q_pfp_pad_operation_newleft. AB = ff_q_pfp_pad_operation_newleft * S ((S (pfs_index_pad_operation_new)) * AC) + (pfs_left_pad_operation_new))) /\\ (((((exists ff_h_pfp_pad_operation_newright. ff_h_pfp_pad_operation_newright + S (pfs_right_pad_operation_new) = S ((S (pfs_index_pad_operation_new)) * BC)) /\\ exists ff_q_pfp_pad_operation_newright. BB = ff_q_pfp_pad_operation_newright * S ((S (pfs_index_pad_operation_new)) * BC) + (pfs_right_pad_operation_new))) /\\ (((((exists ff_h_pfp_pad_operation_newresult. ff_h_pfp_pad_operation_newresult + S (pfs_result_pad_operation_new) = S ((S (pfs_index_pad_operation_new)) * CC)) /\\ exists ff_q_pfp_pad_operation_newresult. CB = ff_q_pfp_pad_operation_newresult * S ((S (pfs_index_pad_operation_new)) * CC) + (pfs_result_pad_operation_new))) /\\ ((((exists pfa_gap_pad_operation_newoperationleft. pfa_gap_pad_operation_newoperationleft + S (pfs_right_pad_operation_new) = (p)) /\\ (((exists pfa_gap_pad_operation_newoperationright. pfa_gap_pad_operation_newoperationright + S (pfs_result_pad_operation_new) = (p)) /\\ ((((exists pfa_gap_pad_operation_newoperationresultbound. pfa_gap_pad_operation_newoperationresultbound + S (pfs_left_pad_operation_new) = (p)) /\\ ((exists pfa_offset_left_pad_operation_newoperationresultcongruence pfa_offset_right_pad_operation_newoperationresultcongruence. 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(((((exists ff_h_pfp_pad_operation_a. ff_h_pfp_pad_operation_a + S (a) = S ((S (x)) * ac)) /\\ exists ff_q_pfp_pad_operation_a. ab = ff_q_pfp_pad_operation_a * S ((S (x)) * ac) + (a))) /\\ (((((exists ff_h_pfp_pad_operation_r. ff_h_pfp_pad_operation_r + S (r) = S ((S (x)) * bc)) /\\ exists ff_q_pfp_pad_operation_r. bb = ff_q_pfp_pad_operation_r * S ((S (x)) * bc) + (r))) /\\ ((((exists pfa_gap_pad_operation_pointleft. pfa_gap_pad_operation_pointleft + S (k) = (p)) /\\ (((exists pfa_gap_pad_operation_pointright. pfa_gap_pad_operation_pointright + S (a) = (p)) /\\ ((((exists pfa_gap_pad_operation_pointresultbound. pfa_gap_pad_operation_pointresultbound + S (r) = (p)) /\\ ((exists pfa_offset_left_pad_operation_pointresultcongruence pfa_offset_right_pad_operation_pointresultcongruence. 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(~((p) = 1) /\\ forall pfa_factor_left_pad_operation_prime pfa_factor_right_pad_operation_prime. (p) = pfa_factor_left_pad_operation_prime * pfa_factor_right_pad_operation_prime -> pfa_factor_left_pad_operation_prime = 1 \\/ pfa_factor_right_pad_operation_prime = 1) -> (((exists pfa_gap_pad_operation_oldscalar. pfa_gap_pad_operation_oldscalar + S (k) = (p)) /\\ ((forall pfp_index_pad_operation_old. (exists pfa_gap_pad_operation_oldindex. pfa_gap_pad_operation_oldindex + S (pfp_index_pad_operation_old) = (L)) -> exists pfp_source_pad_operation_old pfp_value_pad_operation_old. ((((exists ff_h_pfp_pad_operation_oldsource. ff_h_pfp_pad_operation_oldsource + S (pfp_source_pad_operation_old) = S ((S (pfp_index_pad_operation_old)) * ac)) /\\ exists ff_q_pfp_pad_operation_oldsource. ab = ff_q_pfp_pad_operation_oldsource * S ((S (pfp_index_pad_operation_old)) * ac) + (pfp_source_pad_operation_old))) /\\ (((((exists ff_h_pfp_pad_operation_oldtarget. ff_h_pfp_pad_operation_oldtarget + S (pfp_value_pad_operation_old) = S ((S (pfp_index_pad_operation_old)) * bc)) /\\ exists ff_q_pfp_pad_operation_oldtarget. bb = ff_q_pfp_pad_operation_oldtarget * S ((S (pfp_index_pad_operation_old)) * bc) + (pfp_value_pad_operation_old))) /\\ ((((exists pfa_gap_pad_operation_oldoperationleft. pfa_gap_pad_operation_oldoperationleft + S (k) = (p)) /\\ (((exists pfa_gap_pad_operation_oldoperationright. pfa_gap_pad_operation_oldoperationright + S (pfp_source_pad_operation_old) = (p)) /\\ ((((exists pfa_gap_pad_operation_oldoperationresultbound. pfa_gap_pad_operation_oldoperationresultbound + S (pfp_value_pad_operation_old) = (p)) /\\ ((exists pfa_offset_left_pad_operation_oldoperationresultcongruence pfa_offset_right_pad_operation_oldoperationresultcongruence. ((k) * (pfp_source_pad_operation_old)) + (p) * pfa_offset_left_pad_operation_oldoperationresultcongruence = (pfp_value_pad_operation_old) + (p) * pfa_offset_right_pad_operation_oldoperationresultcongruence))))))))))))))))) -> (((forall pfp_repeat_index_pad_operation_abzeros. (exists pfa_gap_pad_operation_abzerosindex. pfa_gap_pad_operation_abzerosindex + S (pfp_repeat_index_pad_operation_abzeros) = (t)) -> (((exists ff_h_pfp_pad_operation_abzerosentry. ff_h_pfp_pad_operation_abzerosentry + S (0) = S ((S (pfp_repeat_index_pad_operation_abzeros)) * AC)) /\\ exists ff_q_pfp_pad_operation_abzerosentry. AB = ff_q_pfp_pad_operation_abzerosentry * S ((S (pfp_repeat_index_pad_operation_abzeros)) * AC) + (0)))) /\\ ((forall pfrep_index_pad_operation_ab pfrep_value_pad_operation_ab. (exists pfa_gap_pad_operation_abbound. pfa_gap_pad_operation_abbound + S (pfrep_index_pad_operation_ab) = (L)) -> (((exists ff_h_pfp_pad_operation_abinput. ff_h_pfp_pad_operation_abinput + S (pfrep_value_pad_operation_ab) = S ((S (pfrep_index_pad_operation_ab)) * ac)) /\\ exists ff_q_pfp_pad_operation_abinput. ab = ff_q_pfp_pad_operation_abinput * S ((S (pfrep_index_pad_operation_ab)) * ac) + (pfrep_value_pad_operation_ab))) -> (((exists ff_h_pfp_pad_operation_aboutput. ff_h_pfp_pad_operation_aboutput + S (pfrep_value_pad_operation_ab) = S ((S ((t)+pfrep_index_pad_operation_ab)) * AC)) /\\ exists ff_q_pfp_pad_operation_aboutput. AB = ff_q_pfp_pad_operation_aboutput * S ((S ((t)+pfrep_index_pad_operation_ab)) * AC) + (pfrep_value_pad_operation_ab))))))) -> (((forall pfp_repeat_index_pad_operation_bbzeros. (exists pfa_gap_pad_operation_bbzerosindex. pfa_gap_pad_operation_bbzerosindex + S (pfp_repeat_index_pad_operation_bbzeros) = (t)) -> (((exists ff_h_pfp_pad_operation_bbzerosentry. ff_h_pfp_pad_operation_bbzerosentry + S (0) = S ((S (pfp_repeat_index_pad_operation_bbzeros)) * BC)) /\\ exists ff_q_pfp_pad_operation_bbzerosentry. BB = ff_q_pfp_pad_operation_bbzerosentry * S ((S (pfp_repeat_index_pad_operation_bbzeros)) * BC) + (0)))) /\\ ((forall pfrep_index_pad_operation_bb pfrep_value_pad_operation_bb. (exists pfa_gap_pad_operation_bbbound. pfa_gap_pad_operation_bbbound + S (pfrep_index_pad_operation_bb) = (L)) -> (((exists ff_h_pfp_pad_operation_bbinput. ff_h_pfp_pad_operation_bbinput + S (pfrep_value_pad_operation_bb) = S ((S (pfrep_index_pad_operation_bb)) * bc)) /\\ exists ff_q_pfp_pad_operation_bbinput. bb = ff_q_pfp_pad_operation_bbinput * S ((S (pfrep_index_pad_operation_bb)) * bc) + (pfrep_value_pad_operation_bb))) -> (((exists ff_h_pfp_pad_operation_bboutput. ff_h_pfp_pad_operation_bboutput + S (pfrep_value_pad_operation_bb) = S ((S ((t)+pfrep_index_pad_operation_bb)) * BC)) /\\ exists ff_q_pfp_pad_operation_bboutput. BB = ff_q_pfp_pad_operation_bboutput * S ((S ((t)+pfrep_index_pad_operation_bb)) * BC) + (pfrep_value_pad_operation_bb))))))) -> (((exists pfa_gap_pad_operation_newscalar. pfa_gap_pad_operation_newscalar + S (k) = (p)) /\\ ((forall pfp_index_pad_operation_new. (exists pfa_gap_pad_operation_newindex. pfa_gap_pad_operation_newindex + S (pfp_index_pad_operation_new) = (t+L)) -> exists pfp_source_pad_operation_new pfp_value_pad_operation_new. ((((exists ff_h_pfp_pad_operation_newsource. ff_h_pfp_pad_operation_newsource + S (pfp_source_pad_operation_new) = S ((S (pfp_index_pad_operation_new)) * AC)) /\\ exists ff_q_pfp_pad_operation_newsource. AB = ff_q_pfp_pad_operation_newsource * S ((S (pfp_index_pad_operation_new)) * AC) + (pfp_source_pad_operation_new))) /\\ (((((exists ff_h_pfp_pad_operation_newtarget. ff_h_pfp_pad_operation_newtarget + S (pfp_value_pad_operation_new) = S ((S (pfp_index_pad_operation_new)) * BC)) /\\ exists ff_q_pfp_pad_operation_newtarget. 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"statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "38d609ffe5dbf7c36cd0e5112dcfb219934bac3c8af09da5d96f30844ae44895", "kind": "alpha_v33_frontier_campaign_rfc", "path": "research/arithmetic-library/prime-field-polynomial-euclidean-division-rfc-v1.md", "role": "reviewed_constructive_campaign_contract", "selector": "document" }, { "document_sha256": "6ae667d8518e4dbe722bb08ad1b08715a0d282c2893e533c8133d770fe861dcf", "kind": "alpha_v33_complete_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/prime-field-polynomial-euclidean-division-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=289]" }, { "document_sha256": "cea85e5c595a021061fb50997df7ad489c8905ab563695209dadc8235f6762cb", "kind": "alpha_v33_live_original_kernel_and_lean_receipt", "path": "research/arithmetic-library/artifacts/alpha-v33-research-receipt-v1.json", "role": "fresh_release_proof_verification", "selector": "families[slug=polynomial-euclidean-division]" }, { "document_sha256": "41b9f387d88a5a4f0fe5ee2bd5578f37a27a4657b0a80f1a1a2cb5109f69a623", "kind": "sealed_alpha_v32_parent", "path": "artifacts/peano-library/alpha/catalog-v32.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "polynomial-euclidean-division", "logical_spec_sha256": "bac6edc3eeba3c9a4591f79703a28b252981b220ed4208474fa470288dd12076", "membership": "alpha_only", "name": "prime_field_polynomial_constant_right_coefficient", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro ab", "intro ac", "intro L", "intro bb", "intro bc", "intro k", "intro i", "intro a", "intro r", "intro hp", "intro hA", "intro hB", "intro hk", "intro hi", "intro ha", "intro hr", "have hp0 : ~(p=0)", "intro hpzero", "specialize prime_nonzero (p)", "apply prime_nonzero", "exact hp", "exact hpzero", "have hc : exists c. (exists pfc_terms_code_constant_old_coefficient pfc_terms_scale_constant_old_coefficient pfc_natural_sum_constant_old_coefficient. ((forall pfc_index_constant_old_coefficientdiagonal. (exists pfa_gap_constant_old_coefficientdiagonalbound. pfa_gap_constant_old_coefficientdiagonalbound + S (pfc_index_constant_old_coefficientdiagonal) = (S (i))) -> exists pfc_value_constant_old_coefficientdiagonal. ((((exists ff_h_pfp_constant_old_coefficientdiagonalentry. ff_h_pfp_constant_old_coefficientdiagonalentry + S (pfc_value_constant_old_coefficientdiagonal) = S ((S (pfc_index_constant_old_coefficientdiagonal)) * pfc_terms_scale_constant_old_coefficient)) /\\ exists ff_q_pfp_constant_old_coefficientdiagonalentry. pfc_terms_code_constant_old_coefficient = ff_q_pfp_constant_old_coefficientdiagonalentry * S ((S (pfc_index_constant_old_coefficientdiagonal)) * pfc_terms_scale_constant_old_coefficient) + (pfc_value_constant_old_coefficientdiagonal))) /\\ ((exists pfc_complement_constant_old_coefficientdiagonalterm pfc_left_constant_old_coefficientdiagonalterm pfc_right_constant_old_coefficientdiagonalterm. (((pfc_index_constant_old_coefficientdiagonal)+pfc_complement_constant_old_coefficientdiagonalterm=(i)) /\\ ((((((exists pfa_gap_constant_old_coefficientdiagonaltermleftinside. pfa_gap_constant_old_coefficientdiagonaltermleftinside + S (pfc_index_constant_old_coefficientdiagonal) = (i)) /\\ ((((exists ff_h_pfp_constant_old_coefficientdiagonaltermleftentry. ff_h_pfp_constant_old_coefficientdiagonaltermleftentry + S (pfc_left_constant_old_coefficientdiagonalterm) = S ((S (pfc_index_constant_old_coefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_constant_old_coefficientdiagonaltermleftentry. ab = ff_q_pfp_constant_old_coefficientdiagonaltermleftentry * S ((S (pfc_index_constant_old_coefficientdiagonal)) * ac) + (pfc_left_constant_old_coefficientdiagonalterm)))))) \\/ (((exists pfc_gap_constant_old_coefficientdiagonaltermleftoutside. pfc_gap_constant_old_coefficientdiagonaltermleftoutside+(i)=(pfc_index_constant_old_coefficientdiagonal)) /\\ (((pfc_left_constant_old_coefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_constant_old_coefficientdiagonaltermrightinside. pfa_gap_constant_old_coefficientdiagonaltermrightinside + S (pfc_complement_constant_old_coefficientdiagonalterm) = (1)) /\\ ((((exists ff_h_pfp_constant_old_coefficientdiagonaltermrightentry. ff_h_pfp_constant_old_coefficientdiagonaltermrightentry + S (pfc_right_constant_old_coefficientdiagonalterm) = S ((S (pfc_complement_constant_old_coefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_constant_old_coefficientdiagonaltermrightentry. bb = ff_q_pfp_constant_old_coefficientdiagonaltermrightentry * S ((S (pfc_complement_constant_old_coefficientdiagonalterm)) * bc) + (pfc_right_constant_old_coefficientdiagonalterm)))))) \\/ (((exists pfc_gap_constant_old_coefficientdiagonaltermrightoutside. pfc_gap_constant_old_coefficientdiagonaltermrightoutside+(1)=(pfc_complement_constant_old_coefficientdiagonalterm)) /\\ (((pfc_right_constant_old_coefficientdiagonalterm)=0))))) /\\ (((pfc_value_constant_old_coefficientdiagonal)=pfc_left_constant_old_coefficientdiagonalterm*pfc_right_constant_old_coefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_constant_old_coefficientsum fs_v_pfc_constant_old_coefficientsum. ((((exists fs_h_pfc_constant_old_coefficientsum_body_start. fs_h_pfc_constant_old_coefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_constant_old_coefficientsum)) /\\ exists fs_q_pfc_constant_old_coefficientsum_body_start. fs_u_pfc_constant_old_coefficientsum = fs_q_pfc_constant_old_coefficientsum_body_start * S ((S (0)) * fs_v_pfc_constant_old_coefficientsum) + (0))) /\\ ((((exists fs_h_pfc_constant_old_coefficientsum_body_terminal. fs_h_pfc_constant_old_coefficientsum_body_terminal + S (pfc_natural_sum_constant_old_coefficient) = S ((S (S (i))) * fs_v_pfc_constant_old_coefficientsum)) /\\ exists fs_q_pfc_constant_old_coefficientsum_body_terminal. fs_u_pfc_constant_old_coefficientsum = fs_q_pfc_constant_old_coefficientsum_body_terminal * S ((S (S (i))) * fs_v_pfc_constant_old_coefficientsum) + (pfc_natural_sum_constant_old_coefficient))) /\\ forall fs_i_pfc_constant_old_coefficientsum_body_steps. (exists fs_lt_pfc_constant_old_coefficientsum_body_steps_bound. fs_lt_pfc_constant_old_coefficientsum_body_steps_bound + S fs_i_pfc_constant_old_coefficientsum_body_steps = S (i)) -> exists fs_a_pfc_constant_old_coefficientsum_body_steps fs_r_pfc_constant_old_coefficientsum_body_steps fs_s_pfc_constant_old_coefficientsum_body_steps. ((((exists fs_h_pfc_constant_old_coefficientsum_body_steps_summand. fs_h_pfc_constant_old_coefficientsum_body_steps_summand + S (fs_a_pfc_constant_old_coefficientsum_body_steps) = S ((S (fs_i_pfc_constant_old_coefficientsum_body_steps)) * pfc_terms_scale_constant_old_coefficient)) /\\ exists fs_q_pfc_constant_old_coefficientsum_body_steps_summand. pfc_terms_code_constant_old_coefficient = fs_q_pfc_constant_old_coefficientsum_body_steps_summand * S ((S (fs_i_pfc_constant_old_coefficientsum_body_steps)) * pfc_terms_scale_constant_old_coefficient) + (fs_a_pfc_constant_old_coefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_constant_old_coefficientsum_body_steps_partial. fs_h_pfc_constant_old_coefficientsum_body_steps_partial + S (fs_r_pfc_constant_old_coefficientsum_body_steps) = S ((S (fs_i_pfc_constant_old_coefficientsum_body_steps)) * fs_v_pfc_constant_old_coefficientsum)) /\\ exists fs_q_pfc_constant_old_coefficientsum_body_steps_partial. fs_u_pfc_constant_old_coefficientsum = fs_q_pfc_constant_old_coefficientsum_body_steps_partial * S ((S (fs_i_pfc_constant_old_coefficientsum_body_steps)) * fs_v_pfc_constant_old_coefficientsum) + (fs_r_pfc_constant_old_coefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_constant_old_coefficientsum_body_steps_successor. fs_h_pfc_constant_old_coefficientsum_body_steps_successor + S (fs_s_pfc_constant_old_coefficientsum_body_steps) = S ((S (S fs_i_pfc_constant_old_coefficientsum_body_steps)) * fs_v_pfc_constant_old_coefficientsum)) /\\ exists fs_q_pfc_constant_old_coefficientsum_body_steps_successor. fs_u_pfc_constant_old_coefficientsum = fs_q_pfc_constant_old_coefficientsum_body_steps_successor * S ((S (S fs_i_pfc_constant_old_coefficientsum_body_steps)) * fs_v_pfc_constant_old_coefficientsum) + (fs_s_pfc_constant_old_coefficientsum_body_steps))) /\\ fs_s_pfc_constant_old_coefficientsum_body_steps = fs_r_pfc_constant_old_coefficientsum_body_steps + fs_a_pfc_constant_old_coefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_constant_old_coefficientresiduebound. pfa_gap_constant_old_coefficientresiduebound + S (c) = (p)) /\\ ((exists pfa_offset_left_constant_old_coefficientresiduecongruence pfa_offset_right_constant_old_coefficientresiduecongruence. (pfc_natural_sum_constant_old_coefficient) + (p) * pfa_offset_left_constant_old_coefficientresiduecongruence = (c) + (p) * pfa_offset_right_constant_old_coefficientresiduecongruence)))))))))", "specialize prime_field_convolution_coefficient_exists (p)", "specialize prime_field_convolution_coefficient_exists (ab)", "specialize prime_field_convolution_coefficient_exists (ac)", "specialize prime_field_convolution_coefficient_exists (i)", "specialize prime_field_convolution_coefficient_exists (bb)", "specialize prime_field_convolution_coefficient_exists (bc)", "specialize prime_field_convolution_coefficient_exists (1)", "specialize prime_field_convolution_coefficient_exists (i)", "apply prime_field_convolution_coefficient_exists", "exact hp0", "cases hc", "have hc0 : x=0", "specialize prime_field_convolution_coefficient_zero_past_support (p)", "specialize prime_field_convolution_coefficient_zero_past_support (ab)", "specialize prime_field_convolution_coefficient_zero_past_support (ac)", "specialize prime_field_convolution_coefficient_zero_past_support (i)", "specialize prime_field_convolution_coefficient_zero_past_support (bb)", "specialize prime_field_convolution_coefficient_zero_past_support (bc)", "specialize prime_field_convolution_coefficient_zero_past_support (1)", "specialize prime_field_convolution_coefficient_zero_past_support (i)", "specialize prime_field_convolution_coefficient_zero_past_support (x)", "apply prime_field_convolution_coefficient_zero_past_support", "exact hp0", "exists 0", "simp [zero_add]", "exact hc_witness", "have hshort : exists pfc_terms_code_constant_short_coefficient pfc_terms_scale_constant_short_coefficient pfc_natural_sum_constant_short_coefficient. ((forall pfc_index_constant_short_coefficientdiagonal. (exists pfa_gap_constant_short_coefficientdiagonalbound. pfa_gap_constant_short_coefficientdiagonalbound + S (pfc_index_constant_short_coefficientdiagonal) = (S (i))) -> exists pfc_value_constant_short_coefficientdiagonal. ((((exists ff_h_pfp_constant_short_coefficientdiagonalentry. ff_h_pfp_constant_short_coefficientdiagonalentry + S (pfc_value_constant_short_coefficientdiagonal) = S ((S (pfc_index_constant_short_coefficientdiagonal)) * pfc_terms_scale_constant_short_coefficient)) /\\ exists ff_q_pfp_constant_short_coefficientdiagonalentry. pfc_terms_code_constant_short_coefficient = ff_q_pfp_constant_short_coefficientdiagonalentry * S ((S (pfc_index_constant_short_coefficientdiagonal)) * pfc_terms_scale_constant_short_coefficient) + (pfc_value_constant_short_coefficientdiagonal))) /\\ ((exists pfc_complement_constant_short_coefficientdiagonalterm pfc_left_constant_short_coefficientdiagonalterm pfc_right_constant_short_coefficientdiagonalterm. (((pfc_index_constant_short_coefficientdiagonal)+pfc_complement_constant_short_coefficientdiagonalterm=(i)) /\\ ((((((exists pfa_gap_constant_short_coefficientdiagonaltermleftinside. pfa_gap_constant_short_coefficientdiagonaltermleftinside + S (pfc_index_constant_short_coefficientdiagonal) = (S i)) /\\ ((((exists ff_h_pfp_constant_short_coefficientdiagonaltermleftentry. ff_h_pfp_constant_short_coefficientdiagonaltermleftentry + S (pfc_left_constant_short_coefficientdiagonalterm) = S ((S (pfc_index_constant_short_coefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_constant_short_coefficientdiagonaltermleftentry. ab = ff_q_pfp_constant_short_coefficientdiagonaltermleftentry * S ((S (pfc_index_constant_short_coefficientdiagonal)) * ac) + (pfc_left_constant_short_coefficientdiagonalterm)))))) \\/ (((exists pfc_gap_constant_short_coefficientdiagonaltermleftoutside. pfc_gap_constant_short_coefficientdiagonaltermleftoutside+(S i)=(pfc_index_constant_short_coefficientdiagonal)) /\\ (((pfc_left_constant_short_coefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_constant_short_coefficientdiagonaltermrightinside. pfa_gap_constant_short_coefficientdiagonaltermrightinside + S (pfc_complement_constant_short_coefficientdiagonalterm) = (1)) /\\ ((((exists ff_h_pfp_constant_short_coefficientdiagonaltermrightentry. ff_h_pfp_constant_short_coefficientdiagonaltermrightentry + S (pfc_right_constant_short_coefficientdiagonalterm) = S ((S (pfc_complement_constant_short_coefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_constant_short_coefficientdiagonaltermrightentry. bb = ff_q_pfp_constant_short_coefficientdiagonaltermrightentry * S ((S (pfc_complement_constant_short_coefficientdiagonalterm)) * bc) + (pfc_right_constant_short_coefficientdiagonalterm)))))) \\/ (((exists pfc_gap_constant_short_coefficientdiagonaltermrightoutside. pfc_gap_constant_short_coefficientdiagonaltermrightoutside+(1)=(pfc_complement_constant_short_coefficientdiagonalterm)) /\\ (((pfc_right_constant_short_coefficientdiagonalterm)=0))))) /\\ (((pfc_value_constant_short_coefficientdiagonal)=pfc_left_constant_short_coefficientdiagonalterm*pfc_right_constant_short_coefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_constant_short_coefficientsum fs_v_pfc_constant_short_coefficientsum. ((((exists fs_h_pfc_constant_short_coefficientsum_body_start. fs_h_pfc_constant_short_coefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_constant_short_coefficientsum)) /\\ exists fs_q_pfc_constant_short_coefficientsum_body_start. fs_u_pfc_constant_short_coefficientsum = fs_q_pfc_constant_short_coefficientsum_body_start * S ((S (0)) * fs_v_pfc_constant_short_coefficientsum) + (0))) /\\ ((((exists fs_h_pfc_constant_short_coefficientsum_body_terminal. fs_h_pfc_constant_short_coefficientsum_body_terminal + S (pfc_natural_sum_constant_short_coefficient) = S ((S (S (i))) * fs_v_pfc_constant_short_coefficientsum)) /\\ exists fs_q_pfc_constant_short_coefficientsum_body_terminal. fs_u_pfc_constant_short_coefficientsum = fs_q_pfc_constant_short_coefficientsum_body_terminal * S ((S (S (i))) * fs_v_pfc_constant_short_coefficientsum) + (pfc_natural_sum_constant_short_coefficient))) /\\ forall fs_i_pfc_constant_short_coefficientsum_body_steps. (exists fs_lt_pfc_constant_short_coefficientsum_body_steps_bound. fs_lt_pfc_constant_short_coefficientsum_body_steps_bound + S fs_i_pfc_constant_short_coefficientsum_body_steps = S (i)) -> exists fs_a_pfc_constant_short_coefficientsum_body_steps fs_r_pfc_constant_short_coefficientsum_body_steps fs_s_pfc_constant_short_coefficientsum_body_steps. ((((exists fs_h_pfc_constant_short_coefficientsum_body_steps_summand. fs_h_pfc_constant_short_coefficientsum_body_steps_summand + S (fs_a_pfc_constant_short_coefficientsum_body_steps) = S ((S (fs_i_pfc_constant_short_coefficientsum_body_steps)) * pfc_terms_scale_constant_short_coefficient)) /\\ exists fs_q_pfc_constant_short_coefficientsum_body_steps_summand. pfc_terms_code_constant_short_coefficient = fs_q_pfc_constant_short_coefficientsum_body_steps_summand * S ((S (fs_i_pfc_constant_short_coefficientsum_body_steps)) * pfc_terms_scale_constant_short_coefficient) + (fs_a_pfc_constant_short_coefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_constant_short_coefficientsum_body_steps_partial. fs_h_pfc_constant_short_coefficientsum_body_steps_partial + S (fs_r_pfc_constant_short_coefficientsum_body_steps) = S ((S (fs_i_pfc_constant_short_coefficientsum_body_steps)) * fs_v_pfc_constant_short_coefficientsum)) /\\ exists fs_q_pfc_constant_short_coefficientsum_body_steps_partial. fs_u_pfc_constant_short_coefficientsum = fs_q_pfc_constant_short_coefficientsum_body_steps_partial * S ((S (fs_i_pfc_constant_short_coefficientsum_body_steps)) * fs_v_pfc_constant_short_coefficientsum) + (fs_r_pfc_constant_short_coefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_constant_short_coefficientsum_body_steps_successor. fs_h_pfc_constant_short_coefficientsum_body_steps_successor + S (fs_s_pfc_constant_short_coefficientsum_body_steps) = S ((S (S fs_i_pfc_constant_short_coefficientsum_body_steps)) * fs_v_pfc_constant_short_coefficientsum)) /\\ exists fs_q_pfc_constant_short_coefficientsum_body_steps_successor. fs_u_pfc_constant_short_coefficientsum = fs_q_pfc_constant_short_coefficientsum_body_steps_successor * S ((S (S fs_i_pfc_constant_short_coefficientsum_body_steps)) * fs_v_pfc_constant_short_coefficientsum) + (fs_s_pfc_constant_short_coefficientsum_body_steps))) /\\ fs_s_pfc_constant_short_coefficientsum_body_steps = fs_r_pfc_constant_short_coefficientsum_body_steps + fs_a_pfc_constant_short_coefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_constant_short_coefficientresiduebound. pfa_gap_constant_short_coefficientresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_constant_short_coefficientresiduecongruence pfa_offset_right_constant_short_coefficientresiduecongruence. (pfc_natural_sum_constant_short_coefficient) + (p) * pfa_offset_left_constant_short_coefficientresiduecongruence = (r) + (p) * pfa_offset_right_constant_short_coefficientresiduecongruence))))))))", "specialize prime_field_convolution_coefficient_prefix_transport (p)", "specialize prime_field_convolution_coefficient_prefix_transport (ab)", "specialize prime_field_convolution_coefficient_prefix_transport (ac)", "specialize prime_field_convolution_coefficient_prefix_transport (L)", "specialize prime_field_convolution_coefficient_prefix_transport (ab)", "specialize prime_field_convolution_coefficient_prefix_transport (ac)", "specialize prime_field_convolution_coefficient_prefix_transport (S i)", "specialize prime_field_convolution_coefficient_prefix_transport (bb)", "specialize prime_field_convolution_coefficient_prefix_transport (bc)", "specialize prime_field_convolution_coefficient_prefix_transport (1)", "specialize prime_field_convolution_coefficient_prefix_transport (S i)", "specialize prime_field_convolution_coefficient_prefix_transport (i)", "specialize prime_field_convolution_coefficient_prefix_transport (r)", "apply prime_field_convolution_coefficient_prefix_transport", "exact hi", "specialize le_refl (S i)", "apply le_refl", "intro j", "intro z", "intro hj", "intro hz", "exact hz", "specialize le_refl (S i)", "apply le_refl", "exact hr", "have hm : exists z. (((exists pfa_gap_constant_chosen_productleft. pfa_gap_constant_chosen_productleft + S (a) = (p)) /\\ (((exists pfa_gap_constant_chosen_productright. pfa_gap_constant_chosen_productright + S (k) = (p)) /\\ ((((exists pfa_gap_constant_chosen_productresultbound. pfa_gap_constant_chosen_productresultbound + S (z) = (p)) /\\ ((exists pfa_offset_left_constant_chosen_productresultcongruence pfa_offset_right_constant_chosen_productresultcongruence. ((a) * (k)) + (p) * pfa_offset_left_constant_chosen_productresultcongruence = (z) + (p) * pfa_offset_right_constant_chosen_productresultcongruence)))))))))", "specialize prime_field_multiply_exists (p)", "specialize prime_field_multiply_exists (a)", "specialize prime_field_multiply_exists (k)", "apply prime_field_multiply_exists", "exact hp", "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 (a)", "apply matrix_rank_bounded_prefix_value", "exact hA", "exact hi", "exact ha", "specialize matrix_rank_bounded_prefix_value (bb)", "specialize matrix_rank_bounded_prefix_value (bc)", "specialize matrix_rank_bounded_prefix_value (1)", "specialize matrix_rank_bounded_prefix_value (p)", "specialize matrix_rank_bounded_prefix_value (0)", "specialize matrix_rank_bounded_prefix_value (k)", "apply matrix_rank_bounded_prefix_value", "exact hB", "exists 0", "simp [zero_add]", "exact hk", "cases hm", "have hs : ((exists pfa_gap_constant_actual_sumleft. pfa_gap_constant_actual_sumleft + S (x) = (p)) /\\ (((exists pfa_gap_constant_actual_sumright. pfa_gap_constant_actual_sumright + S (x1) = (p)) /\\ ((((exists pfa_gap_constant_actual_sumresultbound. pfa_gap_constant_actual_sumresultbound + S (r) = (p)) /\\ ((exists pfa_offset_left_constant_actual_sumresultcongruence pfa_offset_right_constant_actual_sumresultcongruence. ((x) + (x1)) + (p) * pfa_offset_left_constant_actual_sumresultcongruence = (r) + (p) * pfa_offset_right_constant_actual_sumresultcongruence))))))))", "specialize prime_field_convolution_coefficient_append (p)", "specialize prime_field_convolution_coefficient_append (ab)", "specialize prime_field_convolution_coefficient_append (ac)", "specialize prime_field_convolution_coefficient_append (ab)", "specialize prime_field_convolution_coefficient_append (ac)", "specialize prime_field_convolution_coefficient_append (bb)", "specialize prime_field_convolution_coefficient_append (bc)", "specialize prime_field_convolution_coefficient_append (0)", "specialize prime_field_convolution_coefficient_append (i)", "specialize prime_field_convolution_coefficient_append (a)", "specialize prime_field_convolution_coefficient_append (k)", "specialize prime_field_convolution_coefficient_append (x)", "specialize prime_field_convolution_coefficient_append (x1)", "specialize prime_field_convolution_coefficient_append (r)", "apply prime_field_convolution_coefficient_append", "intro j", "intro z", "intro hj", "intro hz", "exact hz", "exact ha", "exact hk", "exact hc_witness", "exact hshort", "exact hm_witness", "rewrite hc0 at hs", "rewrite hc0 at hs", "have heq : r=x1", "specialize prime_field_add_functional (p)", "specialize prime_field_add_functional (0)", "specialize prime_field_add_functional (x1)", "specialize prime_field_add_functional (r)", "specialize prime_field_add_functional (x1)", "apply prime_field_add_functional", "exact hs", "specialize prime_field_add_zero_left (p)", "specialize prime_field_add_zero_left (x1)", "apply prime_field_add_zero_left", "exact hp", "have hmc : ((exists pfa_gap_constant_product_copyleft. pfa_gap_constant_product_copyleft + S (a) = (p)) /\\ (((exists pfa_gap_constant_product_copyright. pfa_gap_constant_product_copyright + S (k) = (p)) /\\ ((((exists pfa_gap_constant_product_copyresultbound. pfa_gap_constant_product_copyresultbound + S (x1) = (p)) /\\ ((exists pfa_offset_left_constant_product_copyresultcongruence pfa_offset_right_constant_product_copyresultcongruence. ((a) * (k)) + (p) * pfa_offset_left_constant_product_copyresultcongruence = (x1) + (p) * pfa_offset_right_constant_product_copyresultcongruence))))))))", "exact hm_witness", "cases hmc", "cases hmc_right", "cases hmc_right_right", "exact hmc_right_right_left", "rewrite heq", "rewrite heq", "specialize prime_field_multiply_commutative (p)", "specialize prime_field_multiply_commutative (a)", "specialize prime_field_multiply_commutative (k)", "specialize prime_field_multiply_commutative (x1)", "apply prime_field_multiply_commutative", "exact hm_witness" ], "script_sha256": "a2797430f69c33d172619ab7ea77f3faa9521226f24db789490a8bb791a7b8fa", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_representation_candidate.py", "sha256": "fc3b40a6ec88841b937251bfc2b4c2dcce55ddeec9932c2533e0f74e46fc5c6a" }, "statement": "forall p ab ac L bb bc k i a r. (~((p) = 1) /\\ forall pfa_factor_left_constant_coefficient_prime pfa_factor_right_constant_coefficient_prime. (p) = pfa_factor_left_constant_coefficient_prime * pfa_factor_right_constant_coefficient_prime -> pfa_factor_left_constant_coefficient_prime = 1 \\/ pfa_factor_right_constant_coefficient_prime = 1) -> (forall fom_index_pfp_constant_coefficient_A. (exists fom_gap_pfp_constant_coefficient_A_index_bound. fom_gap_pfp_constant_coefficient_A_index_bound + S (fom_index_pfp_constant_coefficient_A) = L) -> exists fom_value_pfp_constant_coefficient_A. ((((exists fom_beta_height_pfp_constant_coefficient_A_entry. fom_beta_height_pfp_constant_coefficient_A_entry + S (fom_value_pfp_constant_coefficient_A) = S ((S (fom_index_pfp_constant_coefficient_A)) * ac)) /\\ exists fom_beta_quotient_pfp_constant_coefficient_A_entry. ab = fom_beta_quotient_pfp_constant_coefficient_A_entry * S ((S (fom_index_pfp_constant_coefficient_A)) * ac) + (fom_value_pfp_constant_coefficient_A))) /\\ (exists fom_gap_pfp_constant_coefficient_A_value_bound. fom_gap_pfp_constant_coefficient_A_value_bound + S (fom_value_pfp_constant_coefficient_A) = p))) -> (forall fom_index_pfp_constant_coefficient_B. (exists fom_gap_pfp_constant_coefficient_B_index_bound. fom_gap_pfp_constant_coefficient_B_index_bound + S (fom_index_pfp_constant_coefficient_B) = 1) -> exists fom_value_pfp_constant_coefficient_B. ((((exists fom_beta_height_pfp_constant_coefficient_B_entry. fom_beta_height_pfp_constant_coefficient_B_entry + S (fom_value_pfp_constant_coefficient_B) = S ((S (fom_index_pfp_constant_coefficient_B)) * bc)) /\\ exists fom_beta_quotient_pfp_constant_coefficient_B_entry. bb = fom_beta_quotient_pfp_constant_coefficient_B_entry * S ((S (fom_index_pfp_constant_coefficient_B)) * bc) + (fom_value_pfp_constant_coefficient_B))) /\\ (exists fom_gap_pfp_constant_coefficient_B_value_bound. fom_gap_pfp_constant_coefficient_B_value_bound + S (fom_value_pfp_constant_coefficient_B) = p))) -> (((exists ff_h_pfp_constant_coefficient_constant. ff_h_pfp_constant_coefficient_constant + S (k) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_constant_coefficient_constant. bb = ff_q_pfp_constant_coefficient_constant * S ((S (0)) * bc) + (k))) -> (exists pfa_gap_constant_coefficient_index. pfa_gap_constant_coefficient_index + S (i) = (L)) -> (((exists ff_h_pfp_constant_coefficient_entry. ff_h_pfp_constant_coefficient_entry + S (a) = S ((S (i)) * ac)) /\\ exists ff_q_pfp_constant_coefficient_entry. ab = ff_q_pfp_constant_coefficient_entry * S ((S (i)) * ac) + (a))) -> (exists pfc_terms_code_constant_coefficient_actual pfc_terms_scale_constant_coefficient_actual pfc_natural_sum_constant_coefficient_actual. ((forall pfc_index_constant_coefficient_actualdiagonal. (exists pfa_gap_constant_coefficient_actualdiagonalbound. pfa_gap_constant_coefficient_actualdiagonalbound + S (pfc_index_constant_coefficient_actualdiagonal) = (S (i))) -> exists pfc_value_constant_coefficient_actualdiagonal. ((((exists ff_h_pfp_constant_coefficient_actualdiagonalentry. ff_h_pfp_constant_coefficient_actualdiagonalentry + S (pfc_value_constant_coefficient_actualdiagonal) = S ((S (pfc_index_constant_coefficient_actualdiagonal)) * pfc_terms_scale_constant_coefficient_actual)) /\\ exists ff_q_pfp_constant_coefficient_actualdiagonalentry. pfc_terms_code_constant_coefficient_actual = ff_q_pfp_constant_coefficient_actualdiagonalentry * S ((S (pfc_index_constant_coefficient_actualdiagonal)) * pfc_terms_scale_constant_coefficient_actual) + (pfc_value_constant_coefficient_actualdiagonal))) /\\ ((exists pfc_complement_constant_coefficient_actualdiagonalterm pfc_left_constant_coefficient_actualdiagonalterm pfc_right_constant_coefficient_actualdiagonalterm. (((pfc_index_constant_coefficient_actualdiagonal)+pfc_complement_constant_coefficient_actualdiagonalterm=(i)) /\\ ((((((exists pfa_gap_constant_coefficient_actualdiagonaltermleftinside. pfa_gap_constant_coefficient_actualdiagonaltermleftinside + S (pfc_index_constant_coefficient_actualdiagonal) = (L)) /\\ ((((exists ff_h_pfp_constant_coefficient_actualdiagonaltermleftentry. ff_h_pfp_constant_coefficient_actualdiagonaltermleftentry + S (pfc_left_constant_coefficient_actualdiagonalterm) = S ((S (pfc_index_constant_coefficient_actualdiagonal)) * ac)) /\\ exists ff_q_pfp_constant_coefficient_actualdiagonaltermleftentry. ab = ff_q_pfp_constant_coefficient_actualdiagonaltermleftentry * S ((S (pfc_index_constant_coefficient_actualdiagonal)) * ac) + (pfc_left_constant_coefficient_actualdiagonalterm)))))) \\/ (((exists pfc_gap_constant_coefficient_actualdiagonaltermleftoutside. pfc_gap_constant_coefficient_actualdiagonaltermleftoutside+(L)=(pfc_index_constant_coefficient_actualdiagonal)) /\\ (((pfc_left_constant_coefficient_actualdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_constant_coefficient_actualdiagonaltermrightinside. pfa_gap_constant_coefficient_actualdiagonaltermrightinside + S (pfc_complement_constant_coefficient_actualdiagonalterm) = (1)) /\\ ((((exists ff_h_pfp_constant_coefficient_actualdiagonaltermrightentry. ff_h_pfp_constant_coefficient_actualdiagonaltermrightentry + S (pfc_right_constant_coefficient_actualdiagonalterm) = S ((S (pfc_complement_constant_coefficient_actualdiagonalterm)) * bc)) /\\ exists ff_q_pfp_constant_coefficient_actualdiagonaltermrightentry. bb = ff_q_pfp_constant_coefficient_actualdiagonaltermrightentry * S ((S (pfc_complement_constant_coefficient_actualdiagonalterm)) * bc) + (pfc_right_constant_coefficient_actualdiagonalterm)))))) \\/ (((exists pfc_gap_constant_coefficient_actualdiagonaltermrightoutside. pfc_gap_constant_coefficient_actualdiagonaltermrightoutside+(1)=(pfc_complement_constant_coefficient_actualdiagonalterm)) /\\ (((pfc_right_constant_coefficient_actualdiagonalterm)=0))))) /\\ (((pfc_value_constant_coefficient_actualdiagonal)=pfc_left_constant_coefficient_actualdiagonalterm*pfc_right_constant_coefficient_actualdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_constant_coefficient_actualsum fs_v_pfc_constant_coefficient_actualsum. ((((exists fs_h_pfc_constant_coefficient_actualsum_body_start. fs_h_pfc_constant_coefficient_actualsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_constant_coefficient_actualsum)) /\\ exists fs_q_pfc_constant_coefficient_actualsum_body_start. fs_u_pfc_constant_coefficient_actualsum = fs_q_pfc_constant_coefficient_actualsum_body_start * S ((S (0)) * fs_v_pfc_constant_coefficient_actualsum) + (0))) /\\ ((((exists fs_h_pfc_constant_coefficient_actualsum_body_terminal. fs_h_pfc_constant_coefficient_actualsum_body_terminal + S (pfc_natural_sum_constant_coefficient_actual) = S ((S (S (i))) * fs_v_pfc_constant_coefficient_actualsum)) /\\ exists fs_q_pfc_constant_coefficient_actualsum_body_terminal. fs_u_pfc_constant_coefficient_actualsum = fs_q_pfc_constant_coefficient_actualsum_body_terminal * S ((S (S (i))) * fs_v_pfc_constant_coefficient_actualsum) + (pfc_natural_sum_constant_coefficient_actual))) /\\ forall fs_i_pfc_constant_coefficient_actualsum_body_steps. (exists fs_lt_pfc_constant_coefficient_actualsum_body_steps_bound. fs_lt_pfc_constant_coefficient_actualsum_body_steps_bound + S fs_i_pfc_constant_coefficient_actualsum_body_steps = S (i)) -> exists fs_a_pfc_constant_coefficient_actualsum_body_steps fs_r_pfc_constant_coefficient_actualsum_body_steps fs_s_pfc_constant_coefficient_actualsum_body_steps. ((((exists fs_h_pfc_constant_coefficient_actualsum_body_steps_summand. fs_h_pfc_constant_coefficient_actualsum_body_steps_summand + S (fs_a_pfc_constant_coefficient_actualsum_body_steps) = S ((S (fs_i_pfc_constant_coefficient_actualsum_body_steps)) * pfc_terms_scale_constant_coefficient_actual)) /\\ exists fs_q_pfc_constant_coefficient_actualsum_body_steps_summand. pfc_terms_code_constant_coefficient_actual = fs_q_pfc_constant_coefficient_actualsum_body_steps_summand * S ((S (fs_i_pfc_constant_coefficient_actualsum_body_steps)) * pfc_terms_scale_constant_coefficient_actual) + (fs_a_pfc_constant_coefficient_actualsum_body_steps))) /\\ ((((exists fs_h_pfc_constant_coefficient_actualsum_body_steps_partial. fs_h_pfc_constant_coefficient_actualsum_body_steps_partial + S (fs_r_pfc_constant_coefficient_actualsum_body_steps) = S ((S (fs_i_pfc_constant_coefficient_actualsum_body_steps)) * fs_v_pfc_constant_coefficient_actualsum)) /\\ exists fs_q_pfc_constant_coefficient_actualsum_body_steps_partial. fs_u_pfc_constant_coefficient_actualsum = fs_q_pfc_constant_coefficient_actualsum_body_steps_partial * S ((S (fs_i_pfc_constant_coefficient_actualsum_body_steps)) * fs_v_pfc_constant_coefficient_actualsum) + (fs_r_pfc_constant_coefficient_actualsum_body_steps))) /\\ ((((exists fs_h_pfc_constant_coefficient_actualsum_body_steps_successor. fs_h_pfc_constant_coefficient_actualsum_body_steps_successor + S (fs_s_pfc_constant_coefficient_actualsum_body_steps) = S ((S (S fs_i_pfc_constant_coefficient_actualsum_body_steps)) * fs_v_pfc_constant_coefficient_actualsum)) /\\ exists fs_q_pfc_constant_coefficient_actualsum_body_steps_successor. fs_u_pfc_constant_coefficient_actualsum = fs_q_pfc_constant_coefficient_actualsum_body_steps_successor * S ((S (S fs_i_pfc_constant_coefficient_actualsum_body_steps)) * fs_v_pfc_constant_coefficient_actualsum) + (fs_s_pfc_constant_coefficient_actualsum_body_steps))) /\\ fs_s_pfc_constant_coefficient_actualsum_body_steps = fs_r_pfc_constant_coefficient_actualsum_body_steps + fs_a_pfc_constant_coefficient_actualsum_body_steps)))))) /\\ ((((exists pfa_gap_constant_coefficient_actualresiduebound. pfa_gap_constant_coefficient_actualresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_constant_coefficient_actualresiduecongruence pfa_offset_right_constant_coefficient_actualresiduecongruence. (pfc_natural_sum_constant_coefficient_actual) + (p) * pfa_offset_left_constant_coefficient_actualresiduecongruence = (r) + (p) * pfa_offset_right_constant_coefficient_actualresiduecongruence))))))))) -> (((exists pfa_gap_constant_coefficient_resultleft. pfa_gap_constant_coefficient_resultleft + S (k) = (p)) /\\ (((exists pfa_gap_constant_coefficient_resultright. pfa_gap_constant_coefficient_resultright + S (a) = (p)) /\\ ((((exists pfa_gap_constant_coefficient_resultresultbound. pfa_gap_constant_coefficient_resultresultbound + S (r) = (p)) /\\ ((exists pfa_offset_left_constant_coefficient_resultresultcongruence pfa_offset_right_constant_coefficient_resultresultcongruence. ((k) * (a)) + (p) * pfa_offset_left_constant_coefficient_resultresultcongruence = (r) + (p) * pfa_offset_right_constant_coefficient_resultresultcongruence)))))))))", "statement_sha256": "bf432f7d73e4456d819201429cdc562bf8741c8d3eeb64eb7819bead44461e99", "summary": "The actual antidiagonal coefficient with a length-one right factor is its actual scalar product, by triangular append and proved vanished prior support.", "summary_sha256": "3dad3d3729d14b823ed8e3bb68b5ef2e78498b80f7ec1370b8faceafb9a39a89" }, { "alpha_v33_frontier_enrollment": { "body_receipt_sha256": "7cfa9caa47e568cf9de6e18fc18a0c59246b1ad9fcb28577b85a35db4e9a3666", "bundle_campaign": "polynomial-euclidean-division", "bundle_node_id": 290, "bundle_sha256": "6ae667d8518e4dbe722bb08ad1b08715a0d282c2893e533c8133d770fe861dcf", "campaign": "polynomial-euclidean-division", "first_enrolled_version": "v33", "parent_catalog_sha256": "41b9f387d88a5a4f0fe5ee2bd5578f37a27a4657b0a80f1a1a2cb5109f69a623", "rfc_sha256": 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L", "intro hp", "intro hk", "intro hc", "have hcopy : ((forall fom_index_pfp_constant_product_copyleft. (exists fom_gap_pfp_constant_product_copyleft_index_bound. fom_gap_pfp_constant_product_copyleft_index_bound + S (fom_index_pfp_constant_product_copyleft) = L) -> exists fom_value_pfp_constant_product_copyleft. ((((exists fom_beta_height_pfp_constant_product_copyleft_entry. fom_beta_height_pfp_constant_product_copyleft_entry + S (fom_value_pfp_constant_product_copyleft) = S ((S (fom_index_pfp_constant_product_copyleft)) * ac)) /\\ exists fom_beta_quotient_pfp_constant_product_copyleft_entry. ab = fom_beta_quotient_pfp_constant_product_copyleft_entry * S ((S (fom_index_pfp_constant_product_copyleft)) * ac) + (fom_value_pfp_constant_product_copyleft))) /\\ (exists fom_gap_pfp_constant_product_copyleft_value_bound. fom_gap_pfp_constant_product_copyleft_value_bound + S (fom_value_pfp_constant_product_copyleft) = p))) /\\ (((forall fom_index_pfp_constant_product_copyright. (exists fom_gap_pfp_constant_product_copyright_index_bound. fom_gap_pfp_constant_product_copyright_index_bound + S (fom_index_pfp_constant_product_copyright) = 1) -> exists fom_value_pfp_constant_product_copyright. ((((exists fom_beta_height_pfp_constant_product_copyright_entry. fom_beta_height_pfp_constant_product_copyright_entry + S (fom_value_pfp_constant_product_copyright) = S ((S (fom_index_pfp_constant_product_copyright)) * bc)) /\\ exists fom_beta_quotient_pfp_constant_product_copyright_entry. bb = fom_beta_quotient_pfp_constant_product_copyright_entry * S ((S (fom_index_pfp_constant_product_copyright)) * bc) + (fom_value_pfp_constant_product_copyright))) /\\ (exists fom_gap_pfp_constant_product_copyright_value_bound. fom_gap_pfp_constant_product_copyright_value_bound + S (fom_value_pfp_constant_product_copyright) = p))) /\\ (((((((L)=0 \\/ (1)=0) /\\ (((L)=0)))) \\/ (((~((L)=0)) /\\ (((~((1)=0)) /\\ (((L)+(1)=S (L)))))))) /\\ ((forall pfc_index_constant_product_copycoefficients. (exists pfa_gap_constant_product_copycoefficientsbound. pfa_gap_constant_product_copycoefficientsbound + S (pfc_index_constant_product_copycoefficients) = (L)) -> exists pfc_value_constant_product_copycoefficients. ((((exists ff_h_pfp_constant_product_copycoefficientsentry. ff_h_pfp_constant_product_copycoefficientsentry + S (pfc_value_constant_product_copycoefficients) = S ((S (pfc_index_constant_product_copycoefficients)) * cc)) /\\ exists ff_q_pfp_constant_product_copycoefficientsentry. cb = ff_q_pfp_constant_product_copycoefficientsentry * S ((S (pfc_index_constant_product_copycoefficients)) * cc) + (pfc_value_constant_product_copycoefficients))) /\\ ((exists pfc_terms_code_constant_product_copycoefficientscoefficient pfc_terms_scale_constant_product_copycoefficientscoefficient pfc_natural_sum_constant_product_copycoefficientscoefficient. ((forall pfc_index_constant_product_copycoefficientscoefficientdiagonal. (exists pfa_gap_constant_product_copycoefficientscoefficientdiagonalbound. pfa_gap_constant_product_copycoefficientscoefficientdiagonalbound + S (pfc_index_constant_product_copycoefficientscoefficientdiagonal) = (S (pfc_index_constant_product_copycoefficients))) -> exists pfc_value_constant_product_copycoefficientscoefficientdiagonal. ((((exists ff_h_pfp_constant_product_copycoefficientscoefficientdiagonalentry. ff_h_pfp_constant_product_copycoefficientscoefficientdiagonalentry + S (pfc_value_constant_product_copycoefficientscoefficientdiagonal) = S ((S (pfc_index_constant_product_copycoefficientscoefficientdiagonal)) * pfc_terms_scale_constant_product_copycoefficientscoefficient)) /\\ exists ff_q_pfp_constant_product_copycoefficientscoefficientdiagonalentry. pfc_terms_code_constant_product_copycoefficientscoefficient = ff_q_pfp_constant_product_copycoefficientscoefficientdiagonalentry * S ((S (pfc_index_constant_product_copycoefficientscoefficientdiagonal)) * pfc_terms_scale_constant_product_copycoefficientscoefficient) + (pfc_value_constant_product_copycoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_constant_product_copycoefficientscoefficientdiagonalterm pfc_left_constant_product_copycoefficientscoefficientdiagonalterm pfc_right_constant_product_copycoefficientscoefficientdiagonalterm. (((pfc_index_constant_product_copycoefficientscoefficientdiagonal)+pfc_complement_constant_product_copycoefficientscoefficientdiagonalterm=(pfc_index_constant_product_copycoefficients)) /\\ ((((((exists pfa_gap_constant_product_copycoefficientscoefficientdiagonaltermleftinside. pfa_gap_constant_product_copycoefficientscoefficientdiagonaltermleftinside + S (pfc_index_constant_product_copycoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_constant_product_copycoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_constant_product_copycoefficientscoefficientdiagonaltermleftentry + S (pfc_left_constant_product_copycoefficientscoefficientdiagonalterm) = S ((S (pfc_index_constant_product_copycoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_constant_product_copycoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_constant_product_copycoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_constant_product_copycoefficientscoefficientdiagonal)) * ac) + (pfc_left_constant_product_copycoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_constant_product_copycoefficientscoefficientdiagonaltermleftoutside. pfc_gap_constant_product_copycoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_constant_product_copycoefficientscoefficientdiagonal)) /\\ (((pfc_left_constant_product_copycoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_constant_product_copycoefficientscoefficientdiagonaltermrightinside. pfa_gap_constant_product_copycoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_constant_product_copycoefficientscoefficientdiagonalterm) = (1)) /\\ ((((exists ff_h_pfp_constant_product_copycoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_constant_product_copycoefficientscoefficientdiagonaltermrightentry + S (pfc_right_constant_product_copycoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_constant_product_copycoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_constant_product_copycoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_constant_product_copycoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_constant_product_copycoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_constant_product_copycoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_constant_product_copycoefficientscoefficientdiagonaltermrightoutside. pfc_gap_constant_product_copycoefficientscoefficientdiagonaltermrightoutside+(1)=(pfc_complement_constant_product_copycoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_constant_product_copycoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_constant_product_copycoefficientscoefficientdiagonal)=pfc_left_constant_product_copycoefficientscoefficientdiagonalterm*pfc_right_constant_product_copycoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_constant_product_copycoefficientscoefficientsum fs_v_pfc_constant_product_copycoefficientscoefficientsum. ((((exists fs_h_pfc_constant_product_copycoefficientscoefficientsum_body_start. fs_h_pfc_constant_product_copycoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_constant_product_copycoefficientscoefficientsum)) /\\ exists fs_q_pfc_constant_product_copycoefficientscoefficientsum_body_start. fs_u_pfc_constant_product_copycoefficientscoefficientsum = fs_q_pfc_constant_product_copycoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_constant_product_copycoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_constant_product_copycoefficientscoefficientsum_body_terminal. fs_h_pfc_constant_product_copycoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_constant_product_copycoefficientscoefficient) = S ((S (S (pfc_index_constant_product_copycoefficients))) * fs_v_pfc_constant_product_copycoefficientscoefficientsum)) /\\ exists fs_q_pfc_constant_product_copycoefficientscoefficientsum_body_terminal. fs_u_pfc_constant_product_copycoefficientscoefficientsum = fs_q_pfc_constant_product_copycoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_constant_product_copycoefficients))) * fs_v_pfc_constant_product_copycoefficientscoefficientsum) + (pfc_natural_sum_constant_product_copycoefficientscoefficient))) /\\ forall fs_i_pfc_constant_product_copycoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_constant_product_copycoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_constant_product_copycoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_constant_product_copycoefficientscoefficientsum_body_steps = S (pfc_index_constant_product_copycoefficients)) -> exists fs_a_pfc_constant_product_copycoefficientscoefficientsum_body_steps fs_r_pfc_constant_product_copycoefficientscoefficientsum_body_steps fs_s_pfc_constant_product_copycoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_constant_product_copycoefficientscoefficientsum_body_steps_summand. fs_h_pfc_constant_product_copycoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_constant_product_copycoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_constant_product_copycoefficientscoefficientsum_body_steps)) * pfc_terms_scale_constant_product_copycoefficientscoefficient)) /\\ exists fs_q_pfc_constant_product_copycoefficientscoefficientsum_body_steps_summand. pfc_terms_code_constant_product_copycoefficientscoefficient = fs_q_pfc_constant_product_copycoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_constant_product_copycoefficientscoefficientsum_body_steps)) * pfc_terms_scale_constant_product_copycoefficientscoefficient) + (fs_a_pfc_constant_product_copycoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_constant_product_copycoefficientscoefficientsum_body_steps_partial. fs_h_pfc_constant_product_copycoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_constant_product_copycoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_constant_product_copycoefficientscoefficientsum_body_steps)) * fs_v_pfc_constant_product_copycoefficientscoefficientsum)) /\\ exists fs_q_pfc_constant_product_copycoefficientscoefficientsum_body_steps_partial. fs_u_pfc_constant_product_copycoefficientscoefficientsum = fs_q_pfc_constant_product_copycoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_constant_product_copycoefficientscoefficientsum_body_steps)) * fs_v_pfc_constant_product_copycoefficientscoefficientsum) + (fs_r_pfc_constant_product_copycoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_constant_product_copycoefficientscoefficientsum_body_steps_successor. fs_h_pfc_constant_product_copycoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_constant_product_copycoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_constant_product_copycoefficientscoefficientsum_body_steps)) * fs_v_pfc_constant_product_copycoefficientscoefficientsum)) /\\ exists fs_q_pfc_constant_product_copycoefficientscoefficientsum_body_steps_successor. fs_u_pfc_constant_product_copycoefficientscoefficientsum = fs_q_pfc_constant_product_copycoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_constant_product_copycoefficientscoefficientsum_body_steps)) * fs_v_pfc_constant_product_copycoefficientscoefficientsum) + (fs_s_pfc_constant_product_copycoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_constant_product_copycoefficientscoefficientsum_body_steps = fs_r_pfc_constant_product_copycoefficientscoefficientsum_body_steps + fs_a_pfc_constant_product_copycoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_constant_product_copycoefficientscoefficientresiduebound. pfa_gap_constant_product_copycoefficientscoefficientresiduebound + S (pfc_value_constant_product_copycoefficients) = (p)) /\\ ((exists pfa_offset_left_constant_product_copycoefficientscoefficientresiduecongruence pfa_offset_right_constant_product_copycoefficientscoefficientresiduecongruence. (pfc_natural_sum_constant_product_copycoefficientscoefficient) + (p) * pfa_offset_left_constant_product_copycoefficientscoefficientresiduecongruence = (pfc_value_constant_product_copycoefficients) + (p) * pfa_offset_right_constant_product_copycoefficientscoefficientresiduecongruence))))))))))))))))))", "exact hc", "cases hcopy", "cases hcopy_right", "cases hcopy_right_right", "split", "specialize matrix_rank_bounded_prefix_value (bb)", "specialize matrix_rank_bounded_prefix_value (bc)", "specialize matrix_rank_bounded_prefix_value (1)", "specialize matrix_rank_bounded_prefix_value (p)", "specialize matrix_rank_bounded_prefix_value (0)", "specialize matrix_rank_bounded_prefix_value (k)", "apply matrix_rank_bounded_prefix_value", "exact hcopy_right_left", "exists 0", "simp", "exact hk", "intro i", "intro hi", "have ha : exists a. (((exists ff_h_pfp_constant_scale_source. ff_h_pfp_constant_scale_source + S (a) = S ((S (i)) * ac)) /\\ exists ff_q_pfp_constant_scale_source. ab = ff_q_pfp_constant_scale_source * 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_constant_scale_target. ff_h_pfp_constant_scale_target + S (r) = S ((S (i)) * cc)) /\\ exists ff_q_pfp_constant_scale_target. cb = ff_q_pfp_constant_scale_target * S ((S (i)) * cc) + (r)))", "specialize beta_at_exists (cb)", "specialize beta_at_exists (cc)", "specialize beta_at_exists (i)", "apply beta_at_exists", "cases hr", "exists x", "exists x1", "split", "exact ha_witness", "split", "exact hr_witness", "specialize prime_field_polynomial_constant_right_coefficient (p)", "specialize prime_field_polynomial_constant_right_coefficient (ab)", "specialize prime_field_polynomial_constant_right_coefficient (ac)", "specialize prime_field_polynomial_constant_right_coefficient (L)", "specialize prime_field_polynomial_constant_right_coefficient (bb)", "specialize prime_field_polynomial_constant_right_coefficient (bc)", "specialize prime_field_polynomial_constant_right_coefficient (k)", "specialize prime_field_polynomial_constant_right_coefficient (i)", "specialize prime_field_polynomial_constant_right_coefficient (x)", "specialize prime_field_polynomial_constant_right_coefficient (x1)", "apply prime_field_polynomial_constant_right_coefficient", "exact hp", "exact hcopy_left", "exact hcopy_right_left", "exact hk", "exact hi", "exact ha_witness", "specialize prime_field_polynomial_convolution_entry (p)", "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 (bb)", "specialize prime_field_polynomial_convolution_entry (bc)", "specialize prime_field_polynomial_convolution_entry (1)", "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 (x1)", "apply prime_field_polynomial_convolution_entry", "exact hc", "exact hi", "exact hr_witness" ], "script_sha256": "795811bfb0889069ab511c0eee496617ac69e280380e3f69a23688a7e30ac4e5", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_representation_candidate.py", "sha256": "fc3b40a6ec88841b937251bfc2b4c2dcce55ddeec9932c2533e0f74e46fc5c6a" }, "statement": "forall p k ab ac bb bc cb cc L. (~((p) = 1) /\\ forall pfa_factor_left_constant_scale_prime pfa_factor_right_constant_scale_prime. (p) = pfa_factor_left_constant_scale_prime * pfa_factor_right_constant_scale_prime -> pfa_factor_left_constant_scale_prime = 1 \\/ pfa_factor_right_constant_scale_prime = 1) -> (((exists ff_h_pfp_constant_scale_value. ff_h_pfp_constant_scale_value + S (k) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_constant_scale_value. bb = ff_q_pfp_constant_scale_value * S ((S (0)) * bc) + (k))) -> (((forall fom_index_pfp_constant_scale_convolutionleft. (exists fom_gap_pfp_constant_scale_convolutionleft_index_bound. fom_gap_pfp_constant_scale_convolutionleft_index_bound + S (fom_index_pfp_constant_scale_convolutionleft) = L) -> exists fom_value_pfp_constant_scale_convolutionleft. ((((exists fom_beta_height_pfp_constant_scale_convolutionleft_entry. fom_beta_height_pfp_constant_scale_convolutionleft_entry + S (fom_value_pfp_constant_scale_convolutionleft) = S ((S (fom_index_pfp_constant_scale_convolutionleft)) * ac)) /\\ exists fom_beta_quotient_pfp_constant_scale_convolutionleft_entry. ab = fom_beta_quotient_pfp_constant_scale_convolutionleft_entry * S ((S (fom_index_pfp_constant_scale_convolutionleft)) * ac) + (fom_value_pfp_constant_scale_convolutionleft))) /\\ (exists fom_gap_pfp_constant_scale_convolutionleft_value_bound. fom_gap_pfp_constant_scale_convolutionleft_value_bound + S (fom_value_pfp_constant_scale_convolutionleft) = p))) /\\ (((forall fom_index_pfp_constant_scale_convolutionright. (exists fom_gap_pfp_constant_scale_convolutionright_index_bound. fom_gap_pfp_constant_scale_convolutionright_index_bound + S (fom_index_pfp_constant_scale_convolutionright) = 1) -> exists fom_value_pfp_constant_scale_convolutionright. ((((exists fom_beta_height_pfp_constant_scale_convolutionright_entry. fom_beta_height_pfp_constant_scale_convolutionright_entry + S (fom_value_pfp_constant_scale_convolutionright) = S ((S (fom_index_pfp_constant_scale_convolutionright)) * bc)) /\\ exists fom_beta_quotient_pfp_constant_scale_convolutionright_entry. bb = fom_beta_quotient_pfp_constant_scale_convolutionright_entry * S ((S (fom_index_pfp_constant_scale_convolutionright)) * bc) + (fom_value_pfp_constant_scale_convolutionright))) /\\ (exists fom_gap_pfp_constant_scale_convolutionright_value_bound. fom_gap_pfp_constant_scale_convolutionright_value_bound + S (fom_value_pfp_constant_scale_convolutionright) = p))) /\\ (((((((L)=0 \\/ (1)=0) /\\ (((L)=0)))) \\/ (((~((L)=0)) /\\ (((~((1)=0)) /\\ (((L)+(1)=S (L)))))))) /\\ ((forall pfc_index_constant_scale_convolutioncoefficients. (exists pfa_gap_constant_scale_convolutioncoefficientsbound. pfa_gap_constant_scale_convolutioncoefficientsbound + S (pfc_index_constant_scale_convolutioncoefficients) = (L)) -> exists pfc_value_constant_scale_convolutioncoefficients. ((((exists ff_h_pfp_constant_scale_convolutioncoefficientsentry. ff_h_pfp_constant_scale_convolutioncoefficientsentry + S (pfc_value_constant_scale_convolutioncoefficients) = S ((S (pfc_index_constant_scale_convolutioncoefficients)) * cc)) /\\ exists ff_q_pfp_constant_scale_convolutioncoefficientsentry. cb = ff_q_pfp_constant_scale_convolutioncoefficientsentry * S ((S (pfc_index_constant_scale_convolutioncoefficients)) * cc) + (pfc_value_constant_scale_convolutioncoefficients))) /\\ ((exists pfc_terms_code_constant_scale_convolutioncoefficientscoefficient pfc_terms_scale_constant_scale_convolutioncoefficientscoefficient pfc_natural_sum_constant_scale_convolutioncoefficientscoefficient. ((forall pfc_index_constant_scale_convolutioncoefficientscoefficientdiagonal. (exists pfa_gap_constant_scale_convolutioncoefficientscoefficientdiagonalbound. pfa_gap_constant_scale_convolutioncoefficientscoefficientdiagonalbound + S (pfc_index_constant_scale_convolutioncoefficientscoefficientdiagonal) = (S (pfc_index_constant_scale_convolutioncoefficients))) -> exists pfc_value_constant_scale_convolutioncoefficientscoefficientdiagonal. ((((exists ff_h_pfp_constant_scale_convolutioncoefficientscoefficientdiagonalentry. ff_h_pfp_constant_scale_convolutioncoefficientscoefficientdiagonalentry + S (pfc_value_constant_scale_convolutioncoefficientscoefficientdiagonal) = S ((S (pfc_index_constant_scale_convolutioncoefficientscoefficientdiagonal)) * pfc_terms_scale_constant_scale_convolutioncoefficientscoefficient)) /\\ exists ff_q_pfp_constant_scale_convolutioncoefficientscoefficientdiagonalentry. pfc_terms_code_constant_scale_convolutioncoefficientscoefficient = ff_q_pfp_constant_scale_convolutioncoefficientscoefficientdiagonalentry * S ((S (pfc_index_constant_scale_convolutioncoefficientscoefficientdiagonal)) * pfc_terms_scale_constant_scale_convolutioncoefficientscoefficient) + (pfc_value_constant_scale_convolutioncoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_constant_scale_convolutioncoefficientscoefficientdiagonalterm pfc_left_constant_scale_convolutioncoefficientscoefficientdiagonalterm pfc_right_constant_scale_convolutioncoefficientscoefficientdiagonalterm. (((pfc_index_constant_scale_convolutioncoefficientscoefficientdiagonal)+pfc_complement_constant_scale_convolutioncoefficientscoefficientdiagonalterm=(pfc_index_constant_scale_convolutioncoefficients)) /\\ ((((((exists pfa_gap_constant_scale_convolutioncoefficientscoefficientdiagonaltermleftinside. pfa_gap_constant_scale_convolutioncoefficientscoefficientdiagonaltermleftinside + S (pfc_index_constant_scale_convolutioncoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_constant_scale_convolutioncoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_constant_scale_convolutioncoefficientscoefficientdiagonaltermleftentry + S (pfc_left_constant_scale_convolutioncoefficientscoefficientdiagonalterm) = S ((S (pfc_index_constant_scale_convolutioncoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_constant_scale_convolutioncoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_constant_scale_convolutioncoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_constant_scale_convolutioncoefficientscoefficientdiagonal)) * ac) + (pfc_left_constant_scale_convolutioncoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_constant_scale_convolutioncoefficientscoefficientdiagonaltermleftoutside. pfc_gap_constant_scale_convolutioncoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_constant_scale_convolutioncoefficientscoefficientdiagonal)) /\\ (((pfc_left_constant_scale_convolutioncoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_constant_scale_convolutioncoefficientscoefficientdiagonaltermrightinside. pfa_gap_constant_scale_convolutioncoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_constant_scale_convolutioncoefficientscoefficientdiagonalterm) = (1)) /\\ ((((exists ff_h_pfp_constant_scale_convolutioncoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_constant_scale_convolutioncoefficientscoefficientdiagonaltermrightentry + S (pfc_right_constant_scale_convolutioncoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_constant_scale_convolutioncoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_constant_scale_convolutioncoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_constant_scale_convolutioncoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_constant_scale_convolutioncoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_constant_scale_convolutioncoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_constant_scale_convolutioncoefficientscoefficientdiagonaltermrightoutside. pfc_gap_constant_scale_convolutioncoefficientscoefficientdiagonaltermrightoutside+(1)=(pfc_complement_constant_scale_convolutioncoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_constant_scale_convolutioncoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_constant_scale_convolutioncoefficientscoefficientdiagonal)=pfc_left_constant_scale_convolutioncoefficientscoefficientdiagonalterm*pfc_right_constant_scale_convolutioncoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_constant_scale_convolutioncoefficientscoefficientsum fs_v_pfc_constant_scale_convolutioncoefficientscoefficientsum. ((((exists fs_h_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_start. fs_h_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_constant_scale_convolutioncoefficientscoefficientsum)) /\\ exists fs_q_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_start. fs_u_pfc_constant_scale_convolutioncoefficientscoefficientsum = fs_q_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_constant_scale_convolutioncoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_terminal. fs_h_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_constant_scale_convolutioncoefficientscoefficient) = S ((S (S (pfc_index_constant_scale_convolutioncoefficients))) * fs_v_pfc_constant_scale_convolutioncoefficientscoefficientsum)) /\\ exists fs_q_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_terminal. fs_u_pfc_constant_scale_convolutioncoefficientscoefficientsum = fs_q_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_constant_scale_convolutioncoefficients))) * fs_v_pfc_constant_scale_convolutioncoefficientscoefficientsum) + (pfc_natural_sum_constant_scale_convolutioncoefficientscoefficient))) /\\ forall fs_i_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_steps = S (pfc_index_constant_scale_convolutioncoefficients)) -> exists fs_a_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_steps fs_r_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_steps fs_s_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_steps_summand. fs_h_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_steps)) * pfc_terms_scale_constant_scale_convolutioncoefficientscoefficient)) /\\ exists fs_q_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_steps_summand. pfc_terms_code_constant_scale_convolutioncoefficientscoefficient = fs_q_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_steps)) * pfc_terms_scale_constant_scale_convolutioncoefficientscoefficient) + (fs_a_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_steps_partial. fs_h_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_steps)) * fs_v_pfc_constant_scale_convolutioncoefficientscoefficientsum)) /\\ exists fs_q_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_steps_partial. fs_u_pfc_constant_scale_convolutioncoefficientscoefficientsum = fs_q_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_steps)) * fs_v_pfc_constant_scale_convolutioncoefficientscoefficientsum) + (fs_r_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_steps_successor. fs_h_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_steps)) * fs_v_pfc_constant_scale_convolutioncoefficientscoefficientsum)) /\\ exists fs_q_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_steps_successor. fs_u_pfc_constant_scale_convolutioncoefficientscoefficientsum = fs_q_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_steps)) * fs_v_pfc_constant_scale_convolutioncoefficientscoefficientsum) + (fs_s_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_steps = fs_r_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_steps + fs_a_pfc_constant_scale_convolutioncoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_constant_scale_convolutioncoefficientscoefficientresiduebound. pfa_gap_constant_scale_convolutioncoefficientscoefficientresiduebound + S (pfc_value_constant_scale_convolutioncoefficients) = (p)) /\\ ((exists pfa_offset_left_constant_scale_convolutioncoefficientscoefficientresiduecongruence pfa_offset_right_constant_scale_convolutioncoefficientscoefficientresiduecongruence. (pfc_natural_sum_constant_scale_convolutioncoefficientscoefficient) + (p) * pfa_offset_left_constant_scale_convolutioncoefficientscoefficientresiduecongruence = (pfc_value_constant_scale_convolutioncoefficients) + (p) * pfa_offset_right_constant_scale_convolutioncoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((exists pfa_gap_constant_scale_resultscalar. pfa_gap_constant_scale_resultscalar + S (k) = (p)) /\\ ((forall pfp_index_constant_scale_result. (exists pfa_gap_constant_scale_resultindex. pfa_gap_constant_scale_resultindex + S (pfp_index_constant_scale_result) = (L)) -> exists pfp_source_constant_scale_result pfp_value_constant_scale_result. ((((exists ff_h_pfp_constant_scale_resultsource. ff_h_pfp_constant_scale_resultsource + S (pfp_source_constant_scale_result) = S ((S (pfp_index_constant_scale_result)) * ac)) /\\ exists ff_q_pfp_constant_scale_resultsource. ab = ff_q_pfp_constant_scale_resultsource * S ((S (pfp_index_constant_scale_result)) * ac) + (pfp_source_constant_scale_result))) /\\ (((((exists ff_h_pfp_constant_scale_resulttarget. ff_h_pfp_constant_scale_resulttarget + S (pfp_value_constant_scale_result) = S ((S (pfp_index_constant_scale_result)) * cc)) /\\ exists ff_q_pfp_constant_scale_resulttarget. cb = ff_q_pfp_constant_scale_resulttarget * S ((S (pfp_index_constant_scale_result)) * cc) + (pfp_value_constant_scale_result))) /\\ ((((exists pfa_gap_constant_scale_resultoperationleft. pfa_gap_constant_scale_resultoperationleft + S (k) = (p)) /\\ (((exists pfa_gap_constant_scale_resultoperationright. pfa_gap_constant_scale_resultoperationright + S (pfp_source_constant_scale_result) = (p)) /\\ ((((exists pfa_gap_constant_scale_resultoperationresultbound. pfa_gap_constant_scale_resultoperationresultbound + S (pfp_value_constant_scale_result) = (p)) /\\ ((exists pfa_offset_left_constant_scale_resultoperationresultcongruence pfa_offset_right_constant_scale_resultoperationresultcongruence. 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"script": [ "intro p", "intro k", "intro ab", "intro ac", "intro bb", "intro bc", "intro cb", "intro cc", "intro L", "intro hp", "intro hB", "intro hk", "intro hs", "have hb : ((forall fom_index_pfp_constant_recover_A. (exists fom_gap_pfp_constant_recover_A_index_bound. fom_gap_pfp_constant_recover_A_index_bound + S (fom_index_pfp_constant_recover_A) = L) -> exists fom_value_pfp_constant_recover_A. ((((exists fom_beta_height_pfp_constant_recover_A_entry. fom_beta_height_pfp_constant_recover_A_entry + S (fom_value_pfp_constant_recover_A) = S ((S (fom_index_pfp_constant_recover_A)) * ac)) /\\ exists fom_beta_quotient_pfp_constant_recover_A_entry. ab = fom_beta_quotient_pfp_constant_recover_A_entry * S ((S (fom_index_pfp_constant_recover_A)) * ac) + (fom_value_pfp_constant_recover_A))) /\\ (exists fom_gap_pfp_constant_recover_A_value_bound. fom_gap_pfp_constant_recover_A_value_bound + S (fom_value_pfp_constant_recover_A) = p))) /\\ ((forall fom_index_pfp_constant_recover_C. (exists fom_gap_pfp_constant_recover_C_index_bound. fom_gap_pfp_constant_recover_C_index_bound + S (fom_index_pfp_constant_recover_C) = L) -> exists fom_value_pfp_constant_recover_C. ((((exists fom_beta_height_pfp_constant_recover_C_entry. fom_beta_height_pfp_constant_recover_C_entry + S (fom_value_pfp_constant_recover_C) = S ((S (fom_index_pfp_constant_recover_C)) * cc)) /\\ exists fom_beta_quotient_pfp_constant_recover_C_entry. cb = fom_beta_quotient_pfp_constant_recover_C_entry * S ((S (fom_index_pfp_constant_recover_C)) * cc) + (fom_value_pfp_constant_recover_C))) /\\ (exists fom_gap_pfp_constant_recover_C_value_bound. fom_gap_pfp_constant_recover_C_value_bound + S (fom_value_pfp_constant_recover_C) = 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 (cb)", "specialize prime_field_polynomial_scale_bounded (cc)", "specialize prime_field_polynomial_scale_bounded (L)", "apply prime_field_polynomial_scale_bounded", "exact hs", "cases hb", "split", "exact hb_left", "split", "exact hB", "split", "have hz : L=0 \\/ ~(L=0)", "specialize eq_decidable (L)", "specialize eq_decidable (0)", "apply eq_decidable", "cases hz", "left", "split", "left", "exact hz_left", "exact hz_left", "right", "split", "exact hz_right", "split", "intro hbad", "specialize succ_ne_zero (0)", "apply succ_ne_zero", "exact hbad", "simp", "intro i", "intro hi", "have hv : exists a r. (((((exists ff_h_pfp_constant_recover_source. ff_h_pfp_constant_recover_source + S (a) = S ((S (i)) * ac)) /\\ exists ff_q_pfp_constant_recover_source. ab = ff_q_pfp_constant_recover_source * S ((S (i)) * ac) + (a))) /\\ (((((exists ff_h_pfp_constant_recover_target. ff_h_pfp_constant_recover_target + S (r) = S ((S (i)) * cc)) /\\ exists ff_q_pfp_constant_recover_target. cb = ff_q_pfp_constant_recover_target * S ((S (i)) * cc) + (r))) /\\ ((((exists pfa_gap_constant_recover_mulleft. pfa_gap_constant_recover_mulleft + S (k) = (p)) /\\ (((exists pfa_gap_constant_recover_mulright. pfa_gap_constant_recover_mulright + S (a) = (p)) /\\ ((((exists pfa_gap_constant_recover_mulresultbound. pfa_gap_constant_recover_mulresultbound + S (r) = (p)) /\\ ((exists pfa_offset_left_constant_recover_mulresultcongruence pfa_offset_right_constant_recover_mulresultcongruence. ((k) * (a)) + (p) * pfa_offset_left_constant_recover_mulresultcongruence = (r) + (p) * pfa_offset_right_constant_recover_mulresultcongruence))))))))))))))", "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 hc : exists r. (exists pfc_terms_code_constant_recover_actual pfc_terms_scale_constant_recover_actual pfc_natural_sum_constant_recover_actual. ((forall pfc_index_constant_recover_actualdiagonal. (exists pfa_gap_constant_recover_actualdiagonalbound. pfa_gap_constant_recover_actualdiagonalbound + S (pfc_index_constant_recover_actualdiagonal) = (S (i))) -> exists pfc_value_constant_recover_actualdiagonal. ((((exists ff_h_pfp_constant_recover_actualdiagonalentry. ff_h_pfp_constant_recover_actualdiagonalentry + S (pfc_value_constant_recover_actualdiagonal) = S ((S (pfc_index_constant_recover_actualdiagonal)) * pfc_terms_scale_constant_recover_actual)) /\\ exists ff_q_pfp_constant_recover_actualdiagonalentry. pfc_terms_code_constant_recover_actual = ff_q_pfp_constant_recover_actualdiagonalentry * S ((S (pfc_index_constant_recover_actualdiagonal)) * pfc_terms_scale_constant_recover_actual) + (pfc_value_constant_recover_actualdiagonal))) /\\ ((exists pfc_complement_constant_recover_actualdiagonalterm pfc_left_constant_recover_actualdiagonalterm pfc_right_constant_recover_actualdiagonalterm. (((pfc_index_constant_recover_actualdiagonal)+pfc_complement_constant_recover_actualdiagonalterm=(i)) /\\ ((((((exists pfa_gap_constant_recover_actualdiagonaltermleftinside. pfa_gap_constant_recover_actualdiagonaltermleftinside + S (pfc_index_constant_recover_actualdiagonal) = (L)) /\\ ((((exists ff_h_pfp_constant_recover_actualdiagonaltermleftentry. ff_h_pfp_constant_recover_actualdiagonaltermleftentry + S (pfc_left_constant_recover_actualdiagonalterm) = S ((S (pfc_index_constant_recover_actualdiagonal)) * ac)) /\\ exists ff_q_pfp_constant_recover_actualdiagonaltermleftentry. ab = ff_q_pfp_constant_recover_actualdiagonaltermleftentry * S ((S (pfc_index_constant_recover_actualdiagonal)) * ac) + (pfc_left_constant_recover_actualdiagonalterm)))))) \\/ (((exists pfc_gap_constant_recover_actualdiagonaltermleftoutside. pfc_gap_constant_recover_actualdiagonaltermleftoutside+(L)=(pfc_index_constant_recover_actualdiagonal)) /\\ (((pfc_left_constant_recover_actualdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_constant_recover_actualdiagonaltermrightinside. pfa_gap_constant_recover_actualdiagonaltermrightinside + S (pfc_complement_constant_recover_actualdiagonalterm) = (1)) /\\ ((((exists ff_h_pfp_constant_recover_actualdiagonaltermrightentry. ff_h_pfp_constant_recover_actualdiagonaltermrightentry + S (pfc_right_constant_recover_actualdiagonalterm) = S ((S (pfc_complement_constant_recover_actualdiagonalterm)) * bc)) /\\ exists ff_q_pfp_constant_recover_actualdiagonaltermrightentry. bb = ff_q_pfp_constant_recover_actualdiagonaltermrightentry * S ((S (pfc_complement_constant_recover_actualdiagonalterm)) * bc) + (pfc_right_constant_recover_actualdiagonalterm)))))) \\/ (((exists pfc_gap_constant_recover_actualdiagonaltermrightoutside. pfc_gap_constant_recover_actualdiagonaltermrightoutside+(1)=(pfc_complement_constant_recover_actualdiagonalterm)) /\\ (((pfc_right_constant_recover_actualdiagonalterm)=0))))) /\\ (((pfc_value_constant_recover_actualdiagonal)=pfc_left_constant_recover_actualdiagonalterm*pfc_right_constant_recover_actualdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_constant_recover_actualsum fs_v_pfc_constant_recover_actualsum. ((((exists fs_h_pfc_constant_recover_actualsum_body_start. fs_h_pfc_constant_recover_actualsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_constant_recover_actualsum)) /\\ exists fs_q_pfc_constant_recover_actualsum_body_start. fs_u_pfc_constant_recover_actualsum = fs_q_pfc_constant_recover_actualsum_body_start * S ((S (0)) * fs_v_pfc_constant_recover_actualsum) + (0))) /\\ ((((exists fs_h_pfc_constant_recover_actualsum_body_terminal. fs_h_pfc_constant_recover_actualsum_body_terminal + S (pfc_natural_sum_constant_recover_actual) = S ((S (S (i))) * fs_v_pfc_constant_recover_actualsum)) /\\ exists fs_q_pfc_constant_recover_actualsum_body_terminal. fs_u_pfc_constant_recover_actualsum = fs_q_pfc_constant_recover_actualsum_body_terminal * S ((S (S (i))) * fs_v_pfc_constant_recover_actualsum) + (pfc_natural_sum_constant_recover_actual))) /\\ forall fs_i_pfc_constant_recover_actualsum_body_steps. (exists fs_lt_pfc_constant_recover_actualsum_body_steps_bound. fs_lt_pfc_constant_recover_actualsum_body_steps_bound + S fs_i_pfc_constant_recover_actualsum_body_steps = S (i)) -> exists fs_a_pfc_constant_recover_actualsum_body_steps fs_r_pfc_constant_recover_actualsum_body_steps fs_s_pfc_constant_recover_actualsum_body_steps. ((((exists fs_h_pfc_constant_recover_actualsum_body_steps_summand. fs_h_pfc_constant_recover_actualsum_body_steps_summand + S (fs_a_pfc_constant_recover_actualsum_body_steps) = S ((S (fs_i_pfc_constant_recover_actualsum_body_steps)) * pfc_terms_scale_constant_recover_actual)) /\\ exists fs_q_pfc_constant_recover_actualsum_body_steps_summand. pfc_terms_code_constant_recover_actual = fs_q_pfc_constant_recover_actualsum_body_steps_summand * S ((S (fs_i_pfc_constant_recover_actualsum_body_steps)) * pfc_terms_scale_constant_recover_actual) + (fs_a_pfc_constant_recover_actualsum_body_steps))) /\\ ((((exists fs_h_pfc_constant_recover_actualsum_body_steps_partial. fs_h_pfc_constant_recover_actualsum_body_steps_partial + S (fs_r_pfc_constant_recover_actualsum_body_steps) = S ((S (fs_i_pfc_constant_recover_actualsum_body_steps)) * fs_v_pfc_constant_recover_actualsum)) /\\ exists fs_q_pfc_constant_recover_actualsum_body_steps_partial. fs_u_pfc_constant_recover_actualsum = fs_q_pfc_constant_recover_actualsum_body_steps_partial * S ((S (fs_i_pfc_constant_recover_actualsum_body_steps)) * fs_v_pfc_constant_recover_actualsum) + (fs_r_pfc_constant_recover_actualsum_body_steps))) /\\ ((((exists fs_h_pfc_constant_recover_actualsum_body_steps_successor. fs_h_pfc_constant_recover_actualsum_body_steps_successor + S (fs_s_pfc_constant_recover_actualsum_body_steps) = S ((S (S fs_i_pfc_constant_recover_actualsum_body_steps)) * fs_v_pfc_constant_recover_actualsum)) /\\ exists fs_q_pfc_constant_recover_actualsum_body_steps_successor. fs_u_pfc_constant_recover_actualsum = fs_q_pfc_constant_recover_actualsum_body_steps_successor * S ((S (S fs_i_pfc_constant_recover_actualsum_body_steps)) * fs_v_pfc_constant_recover_actualsum) + (fs_s_pfc_constant_recover_actualsum_body_steps))) /\\ fs_s_pfc_constant_recover_actualsum_body_steps = fs_r_pfc_constant_recover_actualsum_body_steps + fs_a_pfc_constant_recover_actualsum_body_steps)))))) /\\ ((((exists pfa_gap_constant_recover_actualresiduebound. pfa_gap_constant_recover_actualresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_constant_recover_actualresiduecongruence pfa_offset_right_constant_recover_actualresiduecongruence. (pfc_natural_sum_constant_recover_actual) + (p) * pfa_offset_left_constant_recover_actualresiduecongruence = (r) + (p) * pfa_offset_right_constant_recover_actualresiduecongruence)))))))))", "specialize prime_field_convolution_coefficient_exists (p)", "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 (bb)", "specialize prime_field_convolution_coefficient_exists (bc)", "specialize prime_field_convolution_coefficient_exists (1)", "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 hc", "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_polynomial_constant_right_coefficient (p)", "specialize prime_field_polynomial_constant_right_coefficient (ab)", "specialize prime_field_polynomial_constant_right_coefficient (ac)", "specialize prime_field_polynomial_constant_right_coefficient (L)", "specialize prime_field_polynomial_constant_right_coefficient (bb)", "specialize prime_field_polynomial_constant_right_coefficient (bc)", "specialize prime_field_polynomial_constant_right_coefficient (k)", "specialize prime_field_polynomial_constant_right_coefficient (i)", "specialize prime_field_polynomial_constant_right_coefficient (x)", "specialize prime_field_polynomial_constant_right_coefficient (x2)", "apply prime_field_polynomial_constant_right_coefficient", "exact hp", "exact hb_left", "exact hB", "exact hk", "exact hi", "exact hv_witness_witness_left", "exact hc_witness", "exact hv_witness_witness_right_right", "exists x1", "split", "exact hv_witness_witness_right_left", "rewrite heq at hc_witness", "rewrite heq at hc_witness", "exact hc_witness" ], "script_sha256": "efd5da9e6f569ab444c99fc503c2d0a85052eb668f33614aa254dff69cbedba7", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_representation_candidate.py", "sha256": "fc3b40a6ec88841b937251bfc2b4c2dcce55ddeec9932c2533e0f74e46fc5c6a" }, "statement": "forall p k ab ac bb bc cb cc L. (~((p) = 1) /\\ forall pfa_factor_left_constant_recover_prime pfa_factor_right_constant_recover_prime. (p) = pfa_factor_left_constant_recover_prime * pfa_factor_right_constant_recover_prime -> pfa_factor_left_constant_recover_prime = 1 \\/ pfa_factor_right_constant_recover_prime = 1) -> (forall fom_index_pfp_constant_recover_constant. (exists fom_gap_pfp_constant_recover_constant_index_bound. fom_gap_pfp_constant_recover_constant_index_bound + S (fom_index_pfp_constant_recover_constant) = 1) -> exists fom_value_pfp_constant_recover_constant. ((((exists fom_beta_height_pfp_constant_recover_constant_entry. fom_beta_height_pfp_constant_recover_constant_entry + S (fom_value_pfp_constant_recover_constant) = S ((S (fom_index_pfp_constant_recover_constant)) * bc)) /\\ exists fom_beta_quotient_pfp_constant_recover_constant_entry. bb = fom_beta_quotient_pfp_constant_recover_constant_entry * S ((S (fom_index_pfp_constant_recover_constant)) * bc) + (fom_value_pfp_constant_recover_constant))) /\\ (exists fom_gap_pfp_constant_recover_constant_value_bound. fom_gap_pfp_constant_recover_constant_value_bound + S (fom_value_pfp_constant_recover_constant) = p))) -> (((exists ff_h_pfp_constant_recover_value. ff_h_pfp_constant_recover_value + S (k) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_constant_recover_value. bb = ff_q_pfp_constant_recover_value * S ((S (0)) * bc) + (k))) -> (((exists pfa_gap_constant_recover_scalescalar. pfa_gap_constant_recover_scalescalar + S (k) = (p)) /\\ ((forall pfp_index_constant_recover_scale. (exists pfa_gap_constant_recover_scaleindex. pfa_gap_constant_recover_scaleindex + S (pfp_index_constant_recover_scale) = (L)) -> exists pfp_source_constant_recover_scale pfp_value_constant_recover_scale. ((((exists ff_h_pfp_constant_recover_scalesource. ff_h_pfp_constant_recover_scalesource + S (pfp_source_constant_recover_scale) = S ((S (pfp_index_constant_recover_scale)) * ac)) /\\ exists ff_q_pfp_constant_recover_scalesource. ab = ff_q_pfp_constant_recover_scalesource * S ((S (pfp_index_constant_recover_scale)) * ac) + (pfp_source_constant_recover_scale))) /\\ (((((exists ff_h_pfp_constant_recover_scaletarget. ff_h_pfp_constant_recover_scaletarget + S (pfp_value_constant_recover_scale) = S ((S (pfp_index_constant_recover_scale)) * cc)) /\\ exists ff_q_pfp_constant_recover_scaletarget. cb = ff_q_pfp_constant_recover_scaletarget * S ((S (pfp_index_constant_recover_scale)) * cc) + (pfp_value_constant_recover_scale))) /\\ ((((exists pfa_gap_constant_recover_scaleoperationleft. pfa_gap_constant_recover_scaleoperationleft + S (k) = (p)) /\\ (((exists pfa_gap_constant_recover_scaleoperationright. pfa_gap_constant_recover_scaleoperationright + S (pfp_source_constant_recover_scale) = (p)) /\\ ((((exists pfa_gap_constant_recover_scaleoperationresultbound. pfa_gap_constant_recover_scaleoperationresultbound + S (pfp_value_constant_recover_scale) = (p)) /\\ ((exists pfa_offset_left_constant_recover_scaleoperationresultcongruence pfa_offset_right_constant_recover_scaleoperationresultcongruence. ((k) * (pfp_source_constant_recover_scale)) + (p) * pfa_offset_left_constant_recover_scaleoperationresultcongruence = (pfp_value_constant_recover_scale) + (p) * pfa_offset_right_constant_recover_scaleoperationresultcongruence))))))))))))))))) -> (((forall fom_index_pfp_constant_recover_productleft. (exists fom_gap_pfp_constant_recover_productleft_index_bound. fom_gap_pfp_constant_recover_productleft_index_bound + S (fom_index_pfp_constant_recover_productleft) = L) -> exists fom_value_pfp_constant_recover_productleft. ((((exists fom_beta_height_pfp_constant_recover_productleft_entry. fom_beta_height_pfp_constant_recover_productleft_entry + S (fom_value_pfp_constant_recover_productleft) = S ((S (fom_index_pfp_constant_recover_productleft)) * ac)) /\\ exists fom_beta_quotient_pfp_constant_recover_productleft_entry. ab = fom_beta_quotient_pfp_constant_recover_productleft_entry * S ((S (fom_index_pfp_constant_recover_productleft)) * ac) + (fom_value_pfp_constant_recover_productleft))) /\\ (exists fom_gap_pfp_constant_recover_productleft_value_bound. fom_gap_pfp_constant_recover_productleft_value_bound + S (fom_value_pfp_constant_recover_productleft) = p))) /\\ (((forall fom_index_pfp_constant_recover_productright. (exists fom_gap_pfp_constant_recover_productright_index_bound. fom_gap_pfp_constant_recover_productright_index_bound + S (fom_index_pfp_constant_recover_productright) = 1) -> exists fom_value_pfp_constant_recover_productright. ((((exists fom_beta_height_pfp_constant_recover_productright_entry. fom_beta_height_pfp_constant_recover_productright_entry + S (fom_value_pfp_constant_recover_productright) = S ((S (fom_index_pfp_constant_recover_productright)) * bc)) /\\ exists fom_beta_quotient_pfp_constant_recover_productright_entry. bb = fom_beta_quotient_pfp_constant_recover_productright_entry * S ((S (fom_index_pfp_constant_recover_productright)) * bc) + (fom_value_pfp_constant_recover_productright))) /\\ (exists fom_gap_pfp_constant_recover_productright_value_bound. fom_gap_pfp_constant_recover_productright_value_bound + S (fom_value_pfp_constant_recover_productright) = p))) /\\ (((((((L)=0 \\/ (1)=0) /\\ (((L)=0)))) \\/ (((~((L)=0)) /\\ (((~((1)=0)) /\\ (((L)+(1)=S (L)))))))) /\\ ((forall pfc_index_constant_recover_productcoefficients. (exists pfa_gap_constant_recover_productcoefficientsbound. pfa_gap_constant_recover_productcoefficientsbound + S (pfc_index_constant_recover_productcoefficients) = (L)) -> exists pfc_value_constant_recover_productcoefficients. ((((exists ff_h_pfp_constant_recover_productcoefficientsentry. ff_h_pfp_constant_recover_productcoefficientsentry + S (pfc_value_constant_recover_productcoefficients) = S ((S (pfc_index_constant_recover_productcoefficients)) * cc)) /\\ exists ff_q_pfp_constant_recover_productcoefficientsentry. cb = ff_q_pfp_constant_recover_productcoefficientsentry * S ((S (pfc_index_constant_recover_productcoefficients)) * cc) + (pfc_value_constant_recover_productcoefficients))) /\\ ((exists pfc_terms_code_constant_recover_productcoefficientscoefficient pfc_terms_scale_constant_recover_productcoefficientscoefficient pfc_natural_sum_constant_recover_productcoefficientscoefficient. ((forall pfc_index_constant_recover_productcoefficientscoefficientdiagonal. (exists pfa_gap_constant_recover_productcoefficientscoefficientdiagonalbound. pfa_gap_constant_recover_productcoefficientscoefficientdiagonalbound + S (pfc_index_constant_recover_productcoefficientscoefficientdiagonal) = (S (pfc_index_constant_recover_productcoefficients))) -> exists pfc_value_constant_recover_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_constant_recover_productcoefficientscoefficientdiagonalentry. ff_h_pfp_constant_recover_productcoefficientscoefficientdiagonalentry + S (pfc_value_constant_recover_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_constant_recover_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_constant_recover_productcoefficientscoefficient)) /\\ exists ff_q_pfp_constant_recover_productcoefficientscoefficientdiagonalentry. pfc_terms_code_constant_recover_productcoefficientscoefficient = ff_q_pfp_constant_recover_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_constant_recover_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_constant_recover_productcoefficientscoefficient) + (pfc_value_constant_recover_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_constant_recover_productcoefficientscoefficientdiagonalterm pfc_left_constant_recover_productcoefficientscoefficientdiagonalterm pfc_right_constant_recover_productcoefficientscoefficientdiagonalterm. (((pfc_index_constant_recover_productcoefficientscoefficientdiagonal)+pfc_complement_constant_recover_productcoefficientscoefficientdiagonalterm=(pfc_index_constant_recover_productcoefficients)) /\\ ((((((exists pfa_gap_constant_recover_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_constant_recover_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_constant_recover_productcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_constant_recover_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_constant_recover_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_constant_recover_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_constant_recover_productcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_constant_recover_productcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_constant_recover_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_constant_recover_productcoefficientscoefficientdiagonal)) * ac) + (pfc_left_constant_recover_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_constant_recover_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_constant_recover_productcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_constant_recover_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_constant_recover_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_constant_recover_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_constant_recover_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_constant_recover_productcoefficientscoefficientdiagonalterm) = (1)) /\\ ((((exists ff_h_pfp_constant_recover_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_constant_recover_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_constant_recover_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_constant_recover_productcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_constant_recover_productcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_constant_recover_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_constant_recover_productcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_constant_recover_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_constant_recover_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_constant_recover_productcoefficientscoefficientdiagonaltermrightoutside+(1)=(pfc_complement_constant_recover_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_constant_recover_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_constant_recover_productcoefficientscoefficientdiagonal)=pfc_left_constant_recover_productcoefficientscoefficientdiagonalterm*pfc_right_constant_recover_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_constant_recover_productcoefficientscoefficientsum fs_v_pfc_constant_recover_productcoefficientscoefficientsum. ((((exists fs_h_pfc_constant_recover_productcoefficientscoefficientsum_body_start. fs_h_pfc_constant_recover_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_constant_recover_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_constant_recover_productcoefficientscoefficientsum_body_start. fs_u_pfc_constant_recover_productcoefficientscoefficientsum = fs_q_pfc_constant_recover_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_constant_recover_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_constant_recover_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_constant_recover_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_constant_recover_productcoefficientscoefficient) = S ((S (S (pfc_index_constant_recover_productcoefficients))) * fs_v_pfc_constant_recover_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_constant_recover_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_constant_recover_productcoefficientscoefficientsum = fs_q_pfc_constant_recover_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_constant_recover_productcoefficients))) * fs_v_pfc_constant_recover_productcoefficientscoefficientsum) + (pfc_natural_sum_constant_recover_productcoefficientscoefficient))) /\\ forall fs_i_pfc_constant_recover_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_constant_recover_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_constant_recover_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_constant_recover_productcoefficientscoefficientsum_body_steps = S (pfc_index_constant_recover_productcoefficients)) -> exists fs_a_pfc_constant_recover_productcoefficientscoefficientsum_body_steps fs_r_pfc_constant_recover_productcoefficientscoefficientsum_body_steps fs_s_pfc_constant_recover_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_constant_recover_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_constant_recover_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_constant_recover_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_constant_recover_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_constant_recover_productcoefficientscoefficient)) /\\ exists fs_q_pfc_constant_recover_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_constant_recover_productcoefficientscoefficient = fs_q_pfc_constant_recover_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_constant_recover_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_constant_recover_productcoefficientscoefficient) + (fs_a_pfc_constant_recover_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_constant_recover_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_constant_recover_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_constant_recover_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_constant_recover_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_constant_recover_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_constant_recover_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_constant_recover_productcoefficientscoefficientsum = fs_q_pfc_constant_recover_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_constant_recover_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_constant_recover_productcoefficientscoefficientsum) + (fs_r_pfc_constant_recover_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_constant_recover_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_constant_recover_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_constant_recover_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_constant_recover_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_constant_recover_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_constant_recover_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_constant_recover_productcoefficientscoefficientsum = fs_q_pfc_constant_recover_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_constant_recover_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_constant_recover_productcoefficientscoefficientsum) + (fs_s_pfc_constant_recover_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_constant_recover_productcoefficientscoefficientsum_body_steps = fs_r_pfc_constant_recover_productcoefficientscoefficientsum_body_steps + fs_a_pfc_constant_recover_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_constant_recover_productcoefficientscoefficientresiduebound. pfa_gap_constant_recover_productcoefficientscoefficientresiduebound + S (pfc_value_constant_recover_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_constant_recover_productcoefficientscoefficientresiduecongruence pfa_offset_right_constant_recover_productcoefficientscoefficientresiduecongruence. 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"alpha_only", "name": "prime_field_polynomial_inverse_scale", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro a", "intro k", "intro ab", "intro ac", "intro bb", "intro bc", "intro L", "intro hp", "intro hinv", "intro hs", "have hbound : ((forall fom_index_pfp_inverse_scale_A. (exists fom_gap_pfp_inverse_scale_A_index_bound. fom_gap_pfp_inverse_scale_A_index_bound + S (fom_index_pfp_inverse_scale_A) = L) -> exists fom_value_pfp_inverse_scale_A. ((((exists fom_beta_height_pfp_inverse_scale_A_entry. fom_beta_height_pfp_inverse_scale_A_entry + S (fom_value_pfp_inverse_scale_A) = S ((S (fom_index_pfp_inverse_scale_A)) * ac)) /\\ exists fom_beta_quotient_pfp_inverse_scale_A_entry. ab = fom_beta_quotient_pfp_inverse_scale_A_entry * S ((S (fom_index_pfp_inverse_scale_A)) * ac) + (fom_value_pfp_inverse_scale_A))) /\\ (exists fom_gap_pfp_inverse_scale_A_value_bound. fom_gap_pfp_inverse_scale_A_value_bound + S (fom_value_pfp_inverse_scale_A) = p))) /\\ ((forall fom_index_pfp_inverse_scale_B. (exists fom_gap_pfp_inverse_scale_B_index_bound. fom_gap_pfp_inverse_scale_B_index_bound + S (fom_index_pfp_inverse_scale_B) = L) -> exists fom_value_pfp_inverse_scale_B. ((((exists fom_beta_height_pfp_inverse_scale_B_entry. fom_beta_height_pfp_inverse_scale_B_entry + S (fom_value_pfp_inverse_scale_B) = S ((S (fom_index_pfp_inverse_scale_B)) * bc)) /\\ exists fom_beta_quotient_pfp_inverse_scale_B_entry. bb = fom_beta_quotient_pfp_inverse_scale_B_entry * S ((S (fom_index_pfp_inverse_scale_B)) * bc) + (fom_value_pfp_inverse_scale_B))) /\\ (exists fom_gap_pfp_inverse_scale_B_value_bound. fom_gap_pfp_inverse_scale_B_value_bound + S (fom_value_pfp_inverse_scale_B) = 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 (bb)", "specialize prime_field_polynomial_scale_bounded (bc)", "specialize prime_field_polynomial_scale_bounded (L)", "apply prime_field_polynomial_scale_bounded", "exact hs", "cases hbound", "cases hinv", "have hm : ((exists pfa_gap_inverse_scale_productleft. pfa_gap_inverse_scale_productleft + S (a) = (p)) /\\ (((exists pfa_gap_inverse_scale_productright. pfa_gap_inverse_scale_productright + S (k) = (p)) /\\ ((((exists pfa_gap_inverse_scale_productresultbound. pfa_gap_inverse_scale_productresultbound + S (1) = (p)) /\\ ((exists pfa_offset_left_inverse_scale_productresultcongruence pfa_offset_right_inverse_scale_productresultcongruence. ((a) * (k)) + (p) * pfa_offset_left_inverse_scale_productresultcongruence = (1) + (p) * pfa_offset_right_inverse_scale_productresultcongruence))))))))", "exact hinv_right", "cases hinv_right", "cases hinv_right_right", "cases hinv_right_right_right", "have hr : exists cb cc. (((exists pfa_gap_inverse_scale_actualscalar. pfa_gap_inverse_scale_actualscalar + S (a) = (p)) /\\ ((forall pfp_index_inverse_scale_actual. (exists pfa_gap_inverse_scale_actualindex. pfa_gap_inverse_scale_actualindex + S (pfp_index_inverse_scale_actual) = (L)) -> exists pfp_source_inverse_scale_actual pfp_value_inverse_scale_actual. ((((exists ff_h_pfp_inverse_scale_actualsource. ff_h_pfp_inverse_scale_actualsource + S (pfp_source_inverse_scale_actual) = S ((S (pfp_index_inverse_scale_actual)) * bc)) /\\ exists ff_q_pfp_inverse_scale_actualsource. bb = ff_q_pfp_inverse_scale_actualsource * S ((S (pfp_index_inverse_scale_actual)) * bc) + (pfp_source_inverse_scale_actual))) /\\ (((((exists ff_h_pfp_inverse_scale_actualtarget. ff_h_pfp_inverse_scale_actualtarget + S (pfp_value_inverse_scale_actual) = S ((S (pfp_index_inverse_scale_actual)) * cc)) /\\ exists ff_q_pfp_inverse_scale_actualtarget. cb = ff_q_pfp_inverse_scale_actualtarget * S ((S (pfp_index_inverse_scale_actual)) * cc) + (pfp_value_inverse_scale_actual))) /\\ ((((exists pfa_gap_inverse_scale_actualoperationleft. pfa_gap_inverse_scale_actualoperationleft + S (a) = (p)) /\\ (((exists pfa_gap_inverse_scale_actualoperationright. pfa_gap_inverse_scale_actualoperationright + S (pfp_source_inverse_scale_actual) = (p)) /\\ ((((exists pfa_gap_inverse_scale_actualoperationresultbound. pfa_gap_inverse_scale_actualoperationresultbound + S (pfp_value_inverse_scale_actual) = (p)) /\\ ((exists pfa_offset_left_inverse_scale_actualoperationresultcongruence pfa_offset_right_inverse_scale_actualoperationresultcongruence. ((a) * (pfp_source_inverse_scale_actual)) + (p) * pfa_offset_left_inverse_scale_actualoperationresultcongruence = (pfp_value_inverse_scale_actual) + (p) * pfa_offset_right_inverse_scale_actualoperationresultcongruence)))))))))))))))))", "specialize prime_field_polynomial_scale_exists (p)", "specialize prime_field_polynomial_scale_exists (a)", "specialize prime_field_polynomial_scale_exists (bb)", "specialize prime_field_polynomial_scale_exists (bc)", "specialize prime_field_polynomial_scale_exists (L)", "apply prime_field_polynomial_scale_exists", "intro hpzero", "specialize prime_nonzero (p)", "apply prime_nonzero", "exact hp", "exact hpzero", "exact hinv_right_left", "exact hbound_right", "cases hr", "cases hr_witness", "have he : forall mdr_i_pfp_inverse_scale_result_equal mdr_a_pfp_inverse_scale_result_equal. (exists mdr_gap_pfp_inverse_scale_result_equalb. mdr_gap_pfp_inverse_scale_result_equalb + S (mdr_i_pfp_inverse_scale_result_equal) = (L)) -> (((exists ff_h_mdr_pfp_inverse_scale_result_equalo. ff_h_mdr_pfp_inverse_scale_result_equalo + S (mdr_a_pfp_inverse_scale_result_equal) = S ((S (mdr_i_pfp_inverse_scale_result_equal)) * x1)) /\\ exists ff_q_mdr_pfp_inverse_scale_result_equalo. x = ff_q_mdr_pfp_inverse_scale_result_equalo * S ((S (mdr_i_pfp_inverse_scale_result_equal)) * x1) + (mdr_a_pfp_inverse_scale_result_equal))) -> (((exists ff_h_mdr_pfp_inverse_scale_result_equaln. ff_h_mdr_pfp_inverse_scale_result_equaln + S (mdr_a_pfp_inverse_scale_result_equal) = S ((S (mdr_i_pfp_inverse_scale_result_equal)) * ac)) /\\ exists ff_q_mdr_pfp_inverse_scale_result_equaln. ab = ff_q_mdr_pfp_inverse_scale_result_equaln * S ((S (mdr_i_pfp_inverse_scale_result_equal)) * ac) + (mdr_a_pfp_inverse_scale_result_equal)))", "specialize prime_field_polynomial_scale_associative (p)", "specialize prime_field_polynomial_scale_associative (a)", "specialize prime_field_polynomial_scale_associative (k)", "specialize prime_field_polynomial_scale_associative (1)", "specialize prime_field_polynomial_scale_associative (ab)", "specialize prime_field_polynomial_scale_associative (ac)", "specialize prime_field_polynomial_scale_associative (bb)", "specialize prime_field_polynomial_scale_associative (bc)", "specialize prime_field_polynomial_scale_associative (x)", "specialize prime_field_polynomial_scale_associative (x1)", "specialize prime_field_polynomial_scale_associative (ab)", "specialize prime_field_polynomial_scale_associative (ac)", "specialize prime_field_polynomial_scale_associative (L)", "apply prime_field_polynomial_scale_associative", "exact hm", "exact hs", "exact hr_witness_witness", "specialize prime_field_polynomial_scale_one (p)", "specialize prime_field_polynomial_scale_one (ab)", "specialize prime_field_polynomial_scale_one (ac)", "specialize prime_field_polynomial_scale_one (L)", "apply prime_field_polynomial_scale_one", "exact hp", "exact hbound_left", "specialize prime_field_polynomial_scale_transport (p)", "specialize prime_field_polynomial_scale_transport (a)", "specialize prime_field_polynomial_scale_transport (bb)", "specialize prime_field_polynomial_scale_transport (bc)", "specialize prime_field_polynomial_scale_transport (x)", "specialize prime_field_polynomial_scale_transport (x1)", "specialize prime_field_polynomial_scale_transport (bb)", "specialize prime_field_polynomial_scale_transport (bc)", "specialize prime_field_polynomial_scale_transport (ab)", "specialize prime_field_polynomial_scale_transport (ac)", "specialize prime_field_polynomial_scale_transport (L)", "apply prime_field_polynomial_scale_transport", "intro i", "intro r", "intro hi", "intro hat", "exact hat", "exact he", "exact hr_witness_witness" ], "script_sha256": "9cb9bcfce196ef6cbb7983d45e362b845398c7c4c96e1ade0f75d62d15e7ea43", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_representation_candidate.py", "sha256": "fc3b40a6ec88841b937251bfc2b4c2dcce55ddeec9932c2533e0f74e46fc5c6a" }, "statement": "forall p a k ab ac bb bc L. (~((p) = 1) /\\ forall pfa_factor_left_inverse_scale_prime pfa_factor_right_inverse_scale_prime. (p) = pfa_factor_left_inverse_scale_prime * pfa_factor_right_inverse_scale_prime -> pfa_factor_left_inverse_scale_prime = 1 \\/ pfa_factor_right_inverse_scale_prime = 1) -> (((~((a) = 0)) /\\ ((((exists pfa_gap_inverse_scale_inversemultiplicationleft. pfa_gap_inverse_scale_inversemultiplicationleft + S (a) = (p)) /\\ (((exists pfa_gap_inverse_scale_inversemultiplicationright. pfa_gap_inverse_scale_inversemultiplicationright + S (k) = (p)) /\\ ((((exists pfa_gap_inverse_scale_inversemultiplicationresultbound. pfa_gap_inverse_scale_inversemultiplicationresultbound + S (1) = (p)) /\\ ((exists pfa_offset_left_inverse_scale_inversemultiplicationresultcongruence pfa_offset_right_inverse_scale_inversemultiplicationresultcongruence. ((a) * (k)) + (p) * pfa_offset_left_inverse_scale_inversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_inverse_scale_inversemultiplicationresultcongruence)))))))))))) -> (((exists pfa_gap_inverse_scale_forwardscalar. pfa_gap_inverse_scale_forwardscalar + S (k) = (p)) /\\ ((forall pfp_index_inverse_scale_forward. (exists pfa_gap_inverse_scale_forwardindex. pfa_gap_inverse_scale_forwardindex + S (pfp_index_inverse_scale_forward) = (L)) -> exists pfp_source_inverse_scale_forward pfp_value_inverse_scale_forward. ((((exists ff_h_pfp_inverse_scale_forwardsource. ff_h_pfp_inverse_scale_forwardsource + S (pfp_source_inverse_scale_forward) = S ((S (pfp_index_inverse_scale_forward)) * ac)) /\\ exists ff_q_pfp_inverse_scale_forwardsource. ab = ff_q_pfp_inverse_scale_forwardsource * S ((S (pfp_index_inverse_scale_forward)) * ac) + (pfp_source_inverse_scale_forward))) /\\ (((((exists ff_h_pfp_inverse_scale_forwardtarget. ff_h_pfp_inverse_scale_forwardtarget + S (pfp_value_inverse_scale_forward) = S ((S (pfp_index_inverse_scale_forward)) * bc)) /\\ exists ff_q_pfp_inverse_scale_forwardtarget. bb = ff_q_pfp_inverse_scale_forwardtarget * S ((S (pfp_index_inverse_scale_forward)) * bc) + (pfp_value_inverse_scale_forward))) /\\ ((((exists pfa_gap_inverse_scale_forwardoperationleft. pfa_gap_inverse_scale_forwardoperationleft + S (k) = (p)) /\\ (((exists pfa_gap_inverse_scale_forwardoperationright. pfa_gap_inverse_scale_forwardoperationright + S (pfp_source_inverse_scale_forward) = (p)) /\\ ((((exists pfa_gap_inverse_scale_forwardoperationresultbound. pfa_gap_inverse_scale_forwardoperationresultbound + S (pfp_value_inverse_scale_forward) = (p)) /\\ ((exists pfa_offset_left_inverse_scale_forwardoperationresultcongruence pfa_offset_right_inverse_scale_forwardoperationresultcongruence. ((k) * (pfp_source_inverse_scale_forward)) + (p) * pfa_offset_left_inverse_scale_forwardoperationresultcongruence = (pfp_value_inverse_scale_forward) + (p) * pfa_offset_right_inverse_scale_forwardoperationresultcongruence))))))))))))))))) -> (((exists pfa_gap_inverse_scale_backwardscalar. pfa_gap_inverse_scale_backwardscalar + S (a) = (p)) /\\ ((forall pfp_index_inverse_scale_backward. (exists pfa_gap_inverse_scale_backwardindex. pfa_gap_inverse_scale_backwardindex + S (pfp_index_inverse_scale_backward) = (L)) -> exists pfp_source_inverse_scale_backward pfp_value_inverse_scale_backward. ((((exists ff_h_pfp_inverse_scale_backwardsource. ff_h_pfp_inverse_scale_backwardsource + S (pfp_source_inverse_scale_backward) = S ((S (pfp_index_inverse_scale_backward)) * bc)) /\\ exists ff_q_pfp_inverse_scale_backwardsource. bb = ff_q_pfp_inverse_scale_backwardsource * S ((S (pfp_index_inverse_scale_backward)) * bc) + (pfp_source_inverse_scale_backward))) /\\ (((((exists ff_h_pfp_inverse_scale_backwardtarget. ff_h_pfp_inverse_scale_backwardtarget + S (pfp_value_inverse_scale_backward) = S ((S (pfp_index_inverse_scale_backward)) * ac)) /\\ exists ff_q_pfp_inverse_scale_backwardtarget. ab = ff_q_pfp_inverse_scale_backwardtarget * S ((S (pfp_index_inverse_scale_backward)) * ac) + (pfp_value_inverse_scale_backward))) /\\ ((((exists pfa_gap_inverse_scale_backwardoperationleft. pfa_gap_inverse_scale_backwardoperationleft + S (a) = (p)) /\\ (((exists pfa_gap_inverse_scale_backwardoperationright. pfa_gap_inverse_scale_backwardoperationright + S (pfp_source_inverse_scale_backward) = (p)) /\\ ((((exists pfa_gap_inverse_scale_backwardoperationresultbound. pfa_gap_inverse_scale_backwardoperationresultbound + S (pfp_value_inverse_scale_backward) = (p)) /\\ ((exists pfa_offset_left_inverse_scale_backwardoperationresultcongruence pfa_offset_right_inverse_scale_backwardoperationresultcongruence. 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(~((p) = 1) /\\ forall pfa_factor_left_division_scalar_prime pfa_factor_right_division_scalar_prime. (p) = pfa_factor_left_division_scalar_prime * pfa_factor_right_division_scalar_prime -> pfa_factor_left_division_scalar_prime = 1 \\/ pfa_factor_right_division_scalar_prime = 1) -> (((~((b) = 0)) /\\ ((((exists pfa_gap_division_scalar_inversemultiplicationleft. pfa_gap_division_scalar_inversemultiplicationleft + S (b) = (p)) /\\ (((exists pfa_gap_division_scalar_inversemultiplicationright. pfa_gap_division_scalar_inversemultiplicationright + S (k) = (p)) /\\ ((((exists pfa_gap_division_scalar_inversemultiplicationresultbound. pfa_gap_division_scalar_inversemultiplicationresultbound + S (1) = (p)) /\\ ((exists pfa_offset_left_division_scalar_inversemultiplicationresultcongruence pfa_offset_right_division_scalar_inversemultiplicationresultcongruence. ((b) * (k)) + (p) * pfa_offset_left_division_scalar_inversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_division_scalar_inversemultiplicationresultcongruence)))))))))))) -> (((exists pfa_gap_division_scalar_differenceleft. pfa_gap_division_scalar_differenceleft + S (c) = (p)) /\\ (((exists pfa_gap_division_scalar_differenceright. pfa_gap_division_scalar_differenceright + S (s) = (p)) /\\ ((((exists pfa_gap_division_scalar_differenceresultbound. pfa_gap_division_scalar_differenceresultbound + S (a) = (p)) /\\ ((exists pfa_offset_left_division_scalar_differenceresultcongruence pfa_offset_right_division_scalar_differenceresultcongruence. ((c) + (s)) + (p) * pfa_offset_left_division_scalar_differenceresultcongruence = (a) + (p) * pfa_offset_right_division_scalar_differenceresultcongruence))))))))) -> (((exists pfa_gap_division_scalar_quotientleft. pfa_gap_division_scalar_quotientleft + S (k) = (p)) /\\ (((exists pfa_gap_division_scalar_quotientright. pfa_gap_division_scalar_quotientright + S (s) = (p)) /\\ ((((exists pfa_gap_division_scalar_quotientresultbound. pfa_gap_division_scalar_quotientresultbound + S (q) = (p)) /\\ ((exists pfa_offset_left_division_scalar_quotientresultcongruence pfa_offset_right_division_scalar_quotientresultcongruence. ((k) * (s)) + (p) * pfa_offset_left_division_scalar_quotientresultcongruence = (q) + (p) * pfa_offset_right_division_scalar_quotientresultcongruence))))))))) -> (((exists pfa_gap_division_scalar_productleft. pfa_gap_division_scalar_productleft + S (q) = (p)) /\\ (((exists pfa_gap_division_scalar_productright. pfa_gap_division_scalar_productright + S (b) = (p)) /\\ ((((exists pfa_gap_division_scalar_productresultbound. pfa_gap_division_scalar_productresultbound + S (t) = (p)) /\\ ((exists pfa_offset_left_division_scalar_productresultcongruence pfa_offset_right_division_scalar_productresultcongruence. ((q) * (b)) + (p) * pfa_offset_left_division_scalar_productresultcongruence = (t) + (p) * pfa_offset_right_division_scalar_productresultcongruence))))))))) -> (((exists pfa_gap_division_scalar_sumleft. pfa_gap_division_scalar_sumleft + S (c) = (p)) /\\ (((exists pfa_gap_division_scalar_sumright. pfa_gap_division_scalar_sumright + S (t) = (p)) /\\ ((((exists pfa_gap_division_scalar_sumresultbound. pfa_gap_division_scalar_sumresultbound + S (r) = (p)) /\\ ((exists pfa_offset_left_division_scalar_sumresultcongruence pfa_offset_right_division_scalar_sumresultcongruence. 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(forall mdr_i_pfp_division_step_recode mdr_a_pfp_division_step_recode. (exists mdr_gap_pfp_division_step_recodeb. mdr_gap_pfp_division_step_recodeb + S (mdr_i_pfp_division_step_recode) = (i)) -> (((exists ff_h_mdr_pfp_division_step_recodeo. ff_h_mdr_pfp_division_step_recodeo + S (mdr_a_pfp_division_step_recode) = S ((S (mdr_i_pfp_division_step_recode)) * qc)) /\\ exists ff_q_mdr_pfp_division_step_recodeo. qb = ff_q_mdr_pfp_division_step_recodeo * S ((S (mdr_i_pfp_division_step_recode)) * qc) + (mdr_a_pfp_division_step_recode))) -> (((exists ff_h_mdr_pfp_division_step_recoden. ff_h_mdr_pfp_division_step_recoden + S (mdr_a_pfp_division_step_recode) = S ((S (mdr_i_pfp_division_step_recode)) * QC)) /\\ exists ff_q_mdr_pfp_division_step_recoden. QB = ff_q_mdr_pfp_division_step_recoden * S ((S (mdr_i_pfp_division_step_recode)) * QC) + (mdr_a_pfp_division_step_recode)))) -> (exists pfd_input_division_step_old pfd_previous_division_step_old pfd_difference_division_step_old. ((((exists ff_h_pfp_division_step_oldinput. ff_h_pfp_division_step_oldinput + S (pfd_input_division_step_old) = S ((S (i)) * ac)) /\\ exists ff_q_pfp_division_step_oldinput. ab = ff_q_pfp_division_step_oldinput * S ((S (i)) * ac) + (pfd_input_division_step_old))) /\\ (((exists pfc_terms_code_division_step_oldprevious pfc_terms_scale_division_step_oldprevious pfc_natural_sum_division_step_oldprevious. ((forall pfc_index_division_step_oldpreviousdiagonal. (exists pfa_gap_division_step_oldpreviousdiagonalbound. pfa_gap_division_step_oldpreviousdiagonalbound + S (pfc_index_division_step_oldpreviousdiagonal) = (S (i))) -> exists pfc_value_division_step_oldpreviousdiagonal. ((((exists ff_h_pfp_division_step_oldpreviousdiagonalentry. ff_h_pfp_division_step_oldpreviousdiagonalentry + S (pfc_value_division_step_oldpreviousdiagonal) = S ((S (pfc_index_division_step_oldpreviousdiagonal)) * pfc_terms_scale_division_step_oldprevious)) /\\ exists ff_q_pfp_division_step_oldpreviousdiagonalentry. pfc_terms_code_division_step_oldprevious = ff_q_pfp_division_step_oldpreviousdiagonalentry * S ((S (pfc_index_division_step_oldpreviousdiagonal)) * pfc_terms_scale_division_step_oldprevious) + (pfc_value_division_step_oldpreviousdiagonal))) /\\ ((exists pfc_complement_division_step_oldpreviousdiagonalterm pfc_left_division_step_oldpreviousdiagonalterm pfc_right_division_step_oldpreviousdiagonalterm. (((pfc_index_division_step_oldpreviousdiagonal)+pfc_complement_division_step_oldpreviousdiagonalterm=(i)) /\\ ((((((exists pfa_gap_division_step_oldpreviousdiagonaltermleftinside. pfa_gap_division_step_oldpreviousdiagonaltermleftinside + S (pfc_index_division_step_oldpreviousdiagonal) = (i)) /\\ ((((exists ff_h_pfp_division_step_oldpreviousdiagonaltermleftentry. ff_h_pfp_division_step_oldpreviousdiagonaltermleftentry + S (pfc_left_division_step_oldpreviousdiagonalterm) = S ((S (pfc_index_division_step_oldpreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_division_step_oldpreviousdiagonaltermleftentry. qb = ff_q_pfp_division_step_oldpreviousdiagonaltermleftentry * S ((S (pfc_index_division_step_oldpreviousdiagonal)) * qc) + (pfc_left_division_step_oldpreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_step_oldpreviousdiagonaltermleftoutside. pfc_gap_division_step_oldpreviousdiagonaltermleftoutside+(i)=(pfc_index_division_step_oldpreviousdiagonal)) /\\ (((pfc_left_division_step_oldpreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_step_oldpreviousdiagonaltermrightinside. pfa_gap_division_step_oldpreviousdiagonaltermrightinside + S (pfc_complement_division_step_oldpreviousdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_division_step_oldpreviousdiagonaltermrightentry. ff_h_pfp_division_step_oldpreviousdiagonaltermrightentry + S (pfc_right_division_step_oldpreviousdiagonalterm) = S ((S (pfc_complement_division_step_oldpreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_step_oldpreviousdiagonaltermrightentry. bb = ff_q_pfp_division_step_oldpreviousdiagonaltermrightentry * S ((S (pfc_complement_division_step_oldpreviousdiagonalterm)) * bc) + (pfc_right_division_step_oldpreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_step_oldpreviousdiagonaltermrightoutside. pfc_gap_division_step_oldpreviousdiagonaltermrightoutside+(M)=(pfc_complement_division_step_oldpreviousdiagonalterm)) /\\ (((pfc_right_division_step_oldpreviousdiagonalterm)=0))))) /\\ (((pfc_value_division_step_oldpreviousdiagonal)=pfc_left_division_step_oldpreviousdiagonalterm*pfc_right_division_step_oldpreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_step_oldprevioussum fs_v_pfc_division_step_oldprevioussum. ((((exists fs_h_pfc_division_step_oldprevioussum_body_start. fs_h_pfc_division_step_oldprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_step_oldprevioussum)) /\\ exists fs_q_pfc_division_step_oldprevioussum_body_start. fs_u_pfc_division_step_oldprevioussum = fs_q_pfc_division_step_oldprevioussum_body_start * S ((S (0)) * fs_v_pfc_division_step_oldprevioussum) + (0))) /\\ ((((exists fs_h_pfc_division_step_oldprevioussum_body_terminal. fs_h_pfc_division_step_oldprevioussum_body_terminal + S (pfc_natural_sum_division_step_oldprevious) = S ((S (S (i))) * fs_v_pfc_division_step_oldprevioussum)) /\\ exists fs_q_pfc_division_step_oldprevioussum_body_terminal. fs_u_pfc_division_step_oldprevioussum = fs_q_pfc_division_step_oldprevioussum_body_terminal * S ((S (S (i))) * fs_v_pfc_division_step_oldprevioussum) + (pfc_natural_sum_division_step_oldprevious))) /\\ forall fs_i_pfc_division_step_oldprevioussum_body_steps. (exists fs_lt_pfc_division_step_oldprevioussum_body_steps_bound. fs_lt_pfc_division_step_oldprevioussum_body_steps_bound + S fs_i_pfc_division_step_oldprevioussum_body_steps = S (i)) -> exists fs_a_pfc_division_step_oldprevioussum_body_steps fs_r_pfc_division_step_oldprevioussum_body_steps fs_s_pfc_division_step_oldprevioussum_body_steps. ((((exists fs_h_pfc_division_step_oldprevioussum_body_steps_summand. fs_h_pfc_division_step_oldprevioussum_body_steps_summand + S (fs_a_pfc_division_step_oldprevioussum_body_steps) = S ((S (fs_i_pfc_division_step_oldprevioussum_body_steps)) * pfc_terms_scale_division_step_oldprevious)) /\\ exists fs_q_pfc_division_step_oldprevioussum_body_steps_summand. pfc_terms_code_division_step_oldprevious = fs_q_pfc_division_step_oldprevioussum_body_steps_summand * S ((S (fs_i_pfc_division_step_oldprevioussum_body_steps)) * pfc_terms_scale_division_step_oldprevious) + (fs_a_pfc_division_step_oldprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_step_oldprevioussum_body_steps_partial. fs_h_pfc_division_step_oldprevioussum_body_steps_partial + S (fs_r_pfc_division_step_oldprevioussum_body_steps) = S ((S (fs_i_pfc_division_step_oldprevioussum_body_steps)) * fs_v_pfc_division_step_oldprevioussum)) /\\ exists fs_q_pfc_division_step_oldprevioussum_body_steps_partial. fs_u_pfc_division_step_oldprevioussum = fs_q_pfc_division_step_oldprevioussum_body_steps_partial * S ((S (fs_i_pfc_division_step_oldprevioussum_body_steps)) * fs_v_pfc_division_step_oldprevioussum) + (fs_r_pfc_division_step_oldprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_step_oldprevioussum_body_steps_successor. fs_h_pfc_division_step_oldprevioussum_body_steps_successor + S (fs_s_pfc_division_step_oldprevioussum_body_steps) = S ((S (S fs_i_pfc_division_step_oldprevioussum_body_steps)) * fs_v_pfc_division_step_oldprevioussum)) /\\ exists fs_q_pfc_division_step_oldprevioussum_body_steps_successor. fs_u_pfc_division_step_oldprevioussum = fs_q_pfc_division_step_oldprevioussum_body_steps_successor * S ((S (S fs_i_pfc_division_step_oldprevioussum_body_steps)) * fs_v_pfc_division_step_oldprevioussum) + (fs_s_pfc_division_step_oldprevioussum_body_steps))) /\\ fs_s_pfc_division_step_oldprevioussum_body_steps = fs_r_pfc_division_step_oldprevioussum_body_steps + fs_a_pfc_division_step_oldprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_division_step_oldpreviousresiduebound. pfa_gap_division_step_oldpreviousresiduebound + S (pfd_previous_division_step_old) = (p)) /\\ ((exists pfa_offset_left_division_step_oldpreviousresiduecongruence pfa_offset_right_division_step_oldpreviousresiduecongruence. (pfc_natural_sum_division_step_oldprevious) + (p) * pfa_offset_left_division_step_oldpreviousresiduecongruence = (pfd_previous_division_step_old) + (p) * pfa_offset_right_division_step_oldpreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_division_step_oldsubtractleft. pfa_gap_division_step_oldsubtractleft + S (pfd_previous_division_step_old) = (p)) /\\ (((exists pfa_gap_division_step_oldsubtractright. pfa_gap_division_step_oldsubtractright + S (pfd_difference_division_step_old) = (p)) /\\ ((((exists pfa_gap_division_step_oldsubtractresultbound. pfa_gap_division_step_oldsubtractresultbound + S (pfd_input_division_step_old) = (p)) /\\ ((exists pfa_offset_left_division_step_oldsubtractresultcongruence pfa_offset_right_division_step_oldsubtractresultcongruence. ((pfd_previous_division_step_old) + (pfd_difference_division_step_old)) + (p) * pfa_offset_left_division_step_oldsubtractresultcongruence = (pfd_input_division_step_old) + (p) * pfa_offset_right_division_step_oldsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_division_step_oldmultiplyleft. pfa_gap_division_step_oldmultiplyleft + S (k) = (p)) /\\ (((exists pfa_gap_division_step_oldmultiplyright. pfa_gap_division_step_oldmultiplyright + S (pfd_difference_division_step_old) = (p)) /\\ ((((exists pfa_gap_division_step_oldmultiplyresultbound. pfa_gap_division_step_oldmultiplyresultbound + S (q) = (p)) /\\ ((exists pfa_offset_left_division_step_oldmultiplyresultcongruence pfa_offset_right_division_step_oldmultiplyresultcongruence. ((k) * (pfd_difference_division_step_old)) + (p) * pfa_offset_left_division_step_oldmultiplyresultcongruence = (q) + (p) * pfa_offset_right_division_step_oldmultiplyresultcongruence)))))))))))))))) -> (exists pfd_input_division_step_new pfd_previous_division_step_new pfd_difference_division_step_new. ((((exists ff_h_pfp_division_step_newinput. ff_h_pfp_division_step_newinput + S (pfd_input_division_step_new) = S ((S (i)) * ac)) /\\ exists ff_q_pfp_division_step_newinput. ab = ff_q_pfp_division_step_newinput * S ((S (i)) * ac) + (pfd_input_division_step_new))) /\\ (((exists pfc_terms_code_division_step_newprevious pfc_terms_scale_division_step_newprevious pfc_natural_sum_division_step_newprevious. ((forall pfc_index_division_step_newpreviousdiagonal. (exists pfa_gap_division_step_newpreviousdiagonalbound. pfa_gap_division_step_newpreviousdiagonalbound + S (pfc_index_division_step_newpreviousdiagonal) = (S (i))) -> exists pfc_value_division_step_newpreviousdiagonal. ((((exists ff_h_pfp_division_step_newpreviousdiagonalentry. ff_h_pfp_division_step_newpreviousdiagonalentry + S (pfc_value_division_step_newpreviousdiagonal) = S ((S (pfc_index_division_step_newpreviousdiagonal)) * pfc_terms_scale_division_step_newprevious)) /\\ exists ff_q_pfp_division_step_newpreviousdiagonalentry. pfc_terms_code_division_step_newprevious = ff_q_pfp_division_step_newpreviousdiagonalentry * S ((S (pfc_index_division_step_newpreviousdiagonal)) * pfc_terms_scale_division_step_newprevious) + (pfc_value_division_step_newpreviousdiagonal))) /\\ ((exists pfc_complement_division_step_newpreviousdiagonalterm pfc_left_division_step_newpreviousdiagonalterm pfc_right_division_step_newpreviousdiagonalterm. (((pfc_index_division_step_newpreviousdiagonal)+pfc_complement_division_step_newpreviousdiagonalterm=(i)) /\\ ((((((exists pfa_gap_division_step_newpreviousdiagonaltermleftinside. pfa_gap_division_step_newpreviousdiagonaltermleftinside + S (pfc_index_division_step_newpreviousdiagonal) = (i)) /\\ ((((exists ff_h_pfp_division_step_newpreviousdiagonaltermleftentry. ff_h_pfp_division_step_newpreviousdiagonaltermleftentry + S (pfc_left_division_step_newpreviousdiagonalterm) = S ((S (pfc_index_division_step_newpreviousdiagonal)) * QC)) /\\ exists ff_q_pfp_division_step_newpreviousdiagonaltermleftentry. QB = ff_q_pfp_division_step_newpreviousdiagonaltermleftentry * S ((S (pfc_index_division_step_newpreviousdiagonal)) * QC) + (pfc_left_division_step_newpreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_step_newpreviousdiagonaltermleftoutside. pfc_gap_division_step_newpreviousdiagonaltermleftoutside+(i)=(pfc_index_division_step_newpreviousdiagonal)) /\\ (((pfc_left_division_step_newpreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_step_newpreviousdiagonaltermrightinside. pfa_gap_division_step_newpreviousdiagonaltermrightinside + S (pfc_complement_division_step_newpreviousdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_division_step_newpreviousdiagonaltermrightentry. ff_h_pfp_division_step_newpreviousdiagonaltermrightentry + S (pfc_right_division_step_newpreviousdiagonalterm) = S ((S (pfc_complement_division_step_newpreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_step_newpreviousdiagonaltermrightentry. bb = ff_q_pfp_division_step_newpreviousdiagonaltermrightentry * S ((S (pfc_complement_division_step_newpreviousdiagonalterm)) * bc) + (pfc_right_division_step_newpreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_step_newpreviousdiagonaltermrightoutside. pfc_gap_division_step_newpreviousdiagonaltermrightoutside+(M)=(pfc_complement_division_step_newpreviousdiagonalterm)) /\\ (((pfc_right_division_step_newpreviousdiagonalterm)=0))))) /\\ (((pfc_value_division_step_newpreviousdiagonal)=pfc_left_division_step_newpreviousdiagonalterm*pfc_right_division_step_newpreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_step_newprevioussum fs_v_pfc_division_step_newprevioussum. ((((exists fs_h_pfc_division_step_newprevioussum_body_start. fs_h_pfc_division_step_newprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_step_newprevioussum)) /\\ exists fs_q_pfc_division_step_newprevioussum_body_start. fs_u_pfc_division_step_newprevioussum = fs_q_pfc_division_step_newprevioussum_body_start * S ((S (0)) * fs_v_pfc_division_step_newprevioussum) + (0))) /\\ ((((exists fs_h_pfc_division_step_newprevioussum_body_terminal. fs_h_pfc_division_step_newprevioussum_body_terminal + S (pfc_natural_sum_division_step_newprevious) = S ((S (S (i))) * fs_v_pfc_division_step_newprevioussum)) /\\ exists fs_q_pfc_division_step_newprevioussum_body_terminal. fs_u_pfc_division_step_newprevioussum = fs_q_pfc_division_step_newprevioussum_body_terminal * S ((S (S (i))) * fs_v_pfc_division_step_newprevioussum) + (pfc_natural_sum_division_step_newprevious))) /\\ forall fs_i_pfc_division_step_newprevioussum_body_steps. (exists fs_lt_pfc_division_step_newprevioussum_body_steps_bound. fs_lt_pfc_division_step_newprevioussum_body_steps_bound + S fs_i_pfc_division_step_newprevioussum_body_steps = S (i)) -> exists fs_a_pfc_division_step_newprevioussum_body_steps fs_r_pfc_division_step_newprevioussum_body_steps fs_s_pfc_division_step_newprevioussum_body_steps. ((((exists fs_h_pfc_division_step_newprevioussum_body_steps_summand. fs_h_pfc_division_step_newprevioussum_body_steps_summand + S (fs_a_pfc_division_step_newprevioussum_body_steps) = S ((S (fs_i_pfc_division_step_newprevioussum_body_steps)) * pfc_terms_scale_division_step_newprevious)) /\\ exists fs_q_pfc_division_step_newprevioussum_body_steps_summand. pfc_terms_code_division_step_newprevious = fs_q_pfc_division_step_newprevioussum_body_steps_summand * S ((S (fs_i_pfc_division_step_newprevioussum_body_steps)) * pfc_terms_scale_division_step_newprevious) + (fs_a_pfc_division_step_newprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_step_newprevioussum_body_steps_partial. fs_h_pfc_division_step_newprevioussum_body_steps_partial + S (fs_r_pfc_division_step_newprevioussum_body_steps) = S ((S (fs_i_pfc_division_step_newprevioussum_body_steps)) * fs_v_pfc_division_step_newprevioussum)) /\\ exists fs_q_pfc_division_step_newprevioussum_body_steps_partial. fs_u_pfc_division_step_newprevioussum = fs_q_pfc_division_step_newprevioussum_body_steps_partial * S ((S (fs_i_pfc_division_step_newprevioussum_body_steps)) * fs_v_pfc_division_step_newprevioussum) + (fs_r_pfc_division_step_newprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_step_newprevioussum_body_steps_successor. fs_h_pfc_division_step_newprevioussum_body_steps_successor + S (fs_s_pfc_division_step_newprevioussum_body_steps) = S ((S (S fs_i_pfc_division_step_newprevioussum_body_steps)) * fs_v_pfc_division_step_newprevioussum)) /\\ exists fs_q_pfc_division_step_newprevioussum_body_steps_successor. fs_u_pfc_division_step_newprevioussum = fs_q_pfc_division_step_newprevioussum_body_steps_successor * S ((S (S fs_i_pfc_division_step_newprevioussum_body_steps)) * fs_v_pfc_division_step_newprevioussum) + (fs_s_pfc_division_step_newprevioussum_body_steps))) /\\ fs_s_pfc_division_step_newprevioussum_body_steps = fs_r_pfc_division_step_newprevioussum_body_steps + fs_a_pfc_division_step_newprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_division_step_newpreviousresiduebound. pfa_gap_division_step_newpreviousresiduebound + S (pfd_previous_division_step_new) = (p)) /\\ ((exists pfa_offset_left_division_step_newpreviousresiduecongruence pfa_offset_right_division_step_newpreviousresiduecongruence. (pfc_natural_sum_division_step_newprevious) + (p) * pfa_offset_left_division_step_newpreviousresiduecongruence = (pfd_previous_division_step_new) + (p) * pfa_offset_right_division_step_newpreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_division_step_newsubtractleft. pfa_gap_division_step_newsubtractleft + S (pfd_previous_division_step_new) = (p)) /\\ (((exists pfa_gap_division_step_newsubtractright. pfa_gap_division_step_newsubtractright + S (pfd_difference_division_step_new) = (p)) /\\ ((((exists pfa_gap_division_step_newsubtractresultbound. pfa_gap_division_step_newsubtractresultbound + S (pfd_input_division_step_new) = (p)) /\\ ((exists pfa_offset_left_division_step_newsubtractresultcongruence pfa_offset_right_division_step_newsubtractresultcongruence. ((pfd_previous_division_step_new) + (pfd_difference_division_step_new)) + (p) * pfa_offset_left_division_step_newsubtractresultcongruence = (pfd_input_division_step_new) + (p) * pfa_offset_right_division_step_newsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_division_step_newmultiplyleft. pfa_gap_division_step_newmultiplyleft + S (k) = (p)) /\\ (((exists pfa_gap_division_step_newmultiplyright. pfa_gap_division_step_newmultiplyright + S (pfd_difference_division_step_new) = (p)) /\\ ((((exists pfa_gap_division_step_newmultiplyresultbound. pfa_gap_division_step_newmultiplyresultbound + S (q) = (p)) /\\ ((exists pfa_offset_left_division_step_newmultiplyresultcongruence pfa_offset_right_division_step_newmultiplyresultcongruence. ((k) * (pfd_difference_division_step_new)) + (p) * pfa_offset_left_division_step_newmultiplyresultcongruence = (q) + (p) * pfa_offset_right_division_step_newmultiplyresultcongruence))))))))))))))))", "statement_sha256": "a0e233c89aa726b2a4b8a56454333533cc8791e104434ba47fc5e72afc578776", "summary": "An execution step depends only on the actual previously built quotient prefix, never on unused beta entries.", "summary_sha256": "3e1d917bb692b43f53639f9fdee7104a5307a238ab838d0c9658a4bca7844734" }, { "alpha_v33_frontier_enrollment": { "body_receipt_sha256": "7d37f5de2b2a8b2923f995207dcbdfcbb4eb02f29f89cd3a2b15b2ffa2535750", "bundle_campaign": "polynomial-euclidean-division", "bundle_node_id": 295, "bundle_sha256": "6ae667d8518e4dbe722bb08ad1b08715a0d282c2893e533c8133d770fe861dcf", "campaign": "polynomial-euclidean-division", "first_enrolled_version": "v33", "parent_catalog_sha256": "41b9f387d88a5a4f0fe5ee2bd5578f37a27a4657b0a80f1a1a2cb5109f69a623", "rfc_sha256": 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zero_le (i)", "apply zero_le" ], "script_sha256": "3f7708c57808712719f2e2b835f6ae38aa98b93e15c7db4757024b66255ec9a3", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_division_candidate.py", "sha256": "edfc7806caf7a83b9cb0e3e420bd2c3a8679f2d4d9ee6ca9f8eae53faca8d5b2" }, "statement": "forall p k ab ac bb bc M qb qc. forall pfd_index_division_empty. (exists pfa_gap_division_emptybound. pfa_gap_division_emptybound + S (pfd_index_division_empty) = (0)) -> exists pfd_value_division_empty. ((((exists ff_h_pfp_division_emptyentry. ff_h_pfp_division_emptyentry + S (pfd_value_division_empty) = S ((S (pfd_index_division_empty)) * qc)) /\\ exists ff_q_pfp_division_emptyentry. qb = ff_q_pfp_division_emptyentry * S ((S (pfd_index_division_empty)) * qc) + (pfd_value_division_empty))) /\\ ((exists pfd_input_division_emptystep pfd_previous_division_emptystep pfd_difference_division_emptystep. ((((exists ff_h_pfp_division_emptystepinput. ff_h_pfp_division_emptystepinput + S (pfd_input_division_emptystep) = S ((S (pfd_index_division_empty)) * ac)) /\\ exists ff_q_pfp_division_emptystepinput. ab = ff_q_pfp_division_emptystepinput * S ((S (pfd_index_division_empty)) * ac) + (pfd_input_division_emptystep))) /\\ (((exists pfc_terms_code_division_emptystepprevious pfc_terms_scale_division_emptystepprevious pfc_natural_sum_division_emptystepprevious. ((forall pfc_index_division_emptysteppreviousdiagonal. (exists pfa_gap_division_emptysteppreviousdiagonalbound. pfa_gap_division_emptysteppreviousdiagonalbound + S (pfc_index_division_emptysteppreviousdiagonal) = (S (pfd_index_division_empty))) -> exists pfc_value_division_emptysteppreviousdiagonal. ((((exists ff_h_pfp_division_emptysteppreviousdiagonalentry. ff_h_pfp_division_emptysteppreviousdiagonalentry + S (pfc_value_division_emptysteppreviousdiagonal) = S ((S (pfc_index_division_emptysteppreviousdiagonal)) * pfc_terms_scale_division_emptystepprevious)) /\\ exists ff_q_pfp_division_emptysteppreviousdiagonalentry. pfc_terms_code_division_emptystepprevious = ff_q_pfp_division_emptysteppreviousdiagonalentry * S ((S (pfc_index_division_emptysteppreviousdiagonal)) * pfc_terms_scale_division_emptystepprevious) + (pfc_value_division_emptysteppreviousdiagonal))) /\\ ((exists pfc_complement_division_emptysteppreviousdiagonalterm pfc_left_division_emptysteppreviousdiagonalterm pfc_right_division_emptysteppreviousdiagonalterm. (((pfc_index_division_emptysteppreviousdiagonal)+pfc_complement_division_emptysteppreviousdiagonalterm=(pfd_index_division_empty)) /\\ ((((((exists pfa_gap_division_emptysteppreviousdiagonaltermleftinside. pfa_gap_division_emptysteppreviousdiagonaltermleftinside + S (pfc_index_division_emptysteppreviousdiagonal) = (pfd_index_division_empty)) /\\ ((((exists ff_h_pfp_division_emptysteppreviousdiagonaltermleftentry. ff_h_pfp_division_emptysteppreviousdiagonaltermleftentry + S (pfc_left_division_emptysteppreviousdiagonalterm) = S ((S (pfc_index_division_emptysteppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_division_emptysteppreviousdiagonaltermleftentry. qb = ff_q_pfp_division_emptysteppreviousdiagonaltermleftentry * S ((S (pfc_index_division_emptysteppreviousdiagonal)) * qc) + (pfc_left_division_emptysteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_emptysteppreviousdiagonaltermleftoutside. pfc_gap_division_emptysteppreviousdiagonaltermleftoutside+(pfd_index_division_empty)=(pfc_index_division_emptysteppreviousdiagonal)) /\\ (((pfc_left_division_emptysteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_emptysteppreviousdiagonaltermrightinside. pfa_gap_division_emptysteppreviousdiagonaltermrightinside + S (pfc_complement_division_emptysteppreviousdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_division_emptysteppreviousdiagonaltermrightentry. ff_h_pfp_division_emptysteppreviousdiagonaltermrightentry + S (pfc_right_division_emptysteppreviousdiagonalterm) = S ((S (pfc_complement_division_emptysteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_emptysteppreviousdiagonaltermrightentry. bb = ff_q_pfp_division_emptysteppreviousdiagonaltermrightentry * S ((S (pfc_complement_division_emptysteppreviousdiagonalterm)) * bc) + (pfc_right_division_emptysteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_emptysteppreviousdiagonaltermrightoutside. pfc_gap_division_emptysteppreviousdiagonaltermrightoutside+(M)=(pfc_complement_division_emptysteppreviousdiagonalterm)) /\\ (((pfc_right_division_emptysteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_division_emptysteppreviousdiagonal)=pfc_left_division_emptysteppreviousdiagonalterm*pfc_right_division_emptysteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_emptystepprevioussum fs_v_pfc_division_emptystepprevioussum. ((((exists fs_h_pfc_division_emptystepprevioussum_body_start. fs_h_pfc_division_emptystepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_emptystepprevioussum)) /\\ exists fs_q_pfc_division_emptystepprevioussum_body_start. fs_u_pfc_division_emptystepprevioussum = fs_q_pfc_division_emptystepprevioussum_body_start * S ((S (0)) * fs_v_pfc_division_emptystepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_division_emptystepprevioussum_body_terminal. fs_h_pfc_division_emptystepprevioussum_body_terminal + S (pfc_natural_sum_division_emptystepprevious) = S ((S (S (pfd_index_division_empty))) * fs_v_pfc_division_emptystepprevioussum)) /\\ exists fs_q_pfc_division_emptystepprevioussum_body_terminal. fs_u_pfc_division_emptystepprevioussum = fs_q_pfc_division_emptystepprevioussum_body_terminal * S ((S (S (pfd_index_division_empty))) * fs_v_pfc_division_emptystepprevioussum) + (pfc_natural_sum_division_emptystepprevious))) /\\ forall fs_i_pfc_division_emptystepprevioussum_body_steps. (exists fs_lt_pfc_division_emptystepprevioussum_body_steps_bound. fs_lt_pfc_division_emptystepprevioussum_body_steps_bound + S fs_i_pfc_division_emptystepprevioussum_body_steps = S (pfd_index_division_empty)) -> exists fs_a_pfc_division_emptystepprevioussum_body_steps fs_r_pfc_division_emptystepprevioussum_body_steps fs_s_pfc_division_emptystepprevioussum_body_steps. ((((exists fs_h_pfc_division_emptystepprevioussum_body_steps_summand. fs_h_pfc_division_emptystepprevioussum_body_steps_summand + S (fs_a_pfc_division_emptystepprevioussum_body_steps) = S ((S (fs_i_pfc_division_emptystepprevioussum_body_steps)) * pfc_terms_scale_division_emptystepprevious)) /\\ exists fs_q_pfc_division_emptystepprevioussum_body_steps_summand. pfc_terms_code_division_emptystepprevious = fs_q_pfc_division_emptystepprevioussum_body_steps_summand * S ((S (fs_i_pfc_division_emptystepprevioussum_body_steps)) * pfc_terms_scale_division_emptystepprevious) + (fs_a_pfc_division_emptystepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_emptystepprevioussum_body_steps_partial. fs_h_pfc_division_emptystepprevioussum_body_steps_partial + S (fs_r_pfc_division_emptystepprevioussum_body_steps) = S ((S (fs_i_pfc_division_emptystepprevioussum_body_steps)) * fs_v_pfc_division_emptystepprevioussum)) /\\ exists fs_q_pfc_division_emptystepprevioussum_body_steps_partial. fs_u_pfc_division_emptystepprevioussum = fs_q_pfc_division_emptystepprevioussum_body_steps_partial * S ((S (fs_i_pfc_division_emptystepprevioussum_body_steps)) * fs_v_pfc_division_emptystepprevioussum) + (fs_r_pfc_division_emptystepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_emptystepprevioussum_body_steps_successor. fs_h_pfc_division_emptystepprevioussum_body_steps_successor + S (fs_s_pfc_division_emptystepprevioussum_body_steps) = S ((S (S fs_i_pfc_division_emptystepprevioussum_body_steps)) * fs_v_pfc_division_emptystepprevioussum)) /\\ exists fs_q_pfc_division_emptystepprevioussum_body_steps_successor. fs_u_pfc_division_emptystepprevioussum = fs_q_pfc_division_emptystepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_division_emptystepprevioussum_body_steps)) * fs_v_pfc_division_emptystepprevioussum) + (fs_s_pfc_division_emptystepprevioussum_body_steps))) /\\ fs_s_pfc_division_emptystepprevioussum_body_steps = fs_r_pfc_division_emptystepprevioussum_body_steps + fs_a_pfc_division_emptystepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_division_emptysteppreviousresiduebound. pfa_gap_division_emptysteppreviousresiduebound + S (pfd_previous_division_emptystep) = (p)) /\\ ((exists pfa_offset_left_division_emptysteppreviousresiduecongruence pfa_offset_right_division_emptysteppreviousresiduecongruence. (pfc_natural_sum_division_emptystepprevious) + (p) * pfa_offset_left_division_emptysteppreviousresiduecongruence = (pfd_previous_division_emptystep) + (p) * pfa_offset_right_division_emptysteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_division_emptystepsubtractleft. pfa_gap_division_emptystepsubtractleft + S (pfd_previous_division_emptystep) = (p)) /\\ (((exists pfa_gap_division_emptystepsubtractright. pfa_gap_division_emptystepsubtractright + S (pfd_difference_division_emptystep) = (p)) /\\ ((((exists pfa_gap_division_emptystepsubtractresultbound. pfa_gap_division_emptystepsubtractresultbound + S (pfd_input_division_emptystep) = (p)) /\\ ((exists pfa_offset_left_division_emptystepsubtractresultcongruence pfa_offset_right_division_emptystepsubtractresultcongruence. ((pfd_previous_division_emptystep) + (pfd_difference_division_emptystep)) + (p) * pfa_offset_left_division_emptystepsubtractresultcongruence = (pfd_input_division_emptystep) + (p) * pfa_offset_right_division_emptystepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_division_emptystepmultiplyleft. pfa_gap_division_emptystepmultiplyleft + S (k) = (p)) /\\ (((exists pfa_gap_division_emptystepmultiplyright. pfa_gap_division_emptystepmultiplyright + S (pfd_difference_division_emptystep) = (p)) /\\ ((((exists pfa_gap_division_emptystepmultiplyresultbound. pfa_gap_division_emptystepmultiplyresultbound + S (pfd_value_division_empty) = (p)) /\\ ((exists pfa_offset_left_division_emptystepmultiplyresultcongruence pfa_offset_right_division_emptystepmultiplyresultcongruence. 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(exists pfc_gap_division_restrict_bound. pfc_gap_division_restrict_bound+(K)=(N)) -> (forall pfd_index_division_restrict_old. (exists pfa_gap_division_restrict_oldbound. pfa_gap_division_restrict_oldbound + S (pfd_index_division_restrict_old) = (N)) -> exists pfd_value_division_restrict_old. ((((exists ff_h_pfp_division_restrict_oldentry. ff_h_pfp_division_restrict_oldentry + S (pfd_value_division_restrict_old) = S ((S (pfd_index_division_restrict_old)) * qc)) /\\ exists ff_q_pfp_division_restrict_oldentry. qb = ff_q_pfp_division_restrict_oldentry * S ((S (pfd_index_division_restrict_old)) * qc) + (pfd_value_division_restrict_old))) /\\ ((exists pfd_input_division_restrict_oldstep pfd_previous_division_restrict_oldstep pfd_difference_division_restrict_oldstep. ((((exists ff_h_pfp_division_restrict_oldstepinput. ff_h_pfp_division_restrict_oldstepinput + S (pfd_input_division_restrict_oldstep) = S ((S (pfd_index_division_restrict_old)) * ac)) /\\ exists ff_q_pfp_division_restrict_oldstepinput. ab = ff_q_pfp_division_restrict_oldstepinput * S ((S (pfd_index_division_restrict_old)) * ac) + (pfd_input_division_restrict_oldstep))) /\\ (((exists pfc_terms_code_division_restrict_oldstepprevious pfc_terms_scale_division_restrict_oldstepprevious pfc_natural_sum_division_restrict_oldstepprevious. ((forall pfc_index_division_restrict_oldsteppreviousdiagonal. (exists pfa_gap_division_restrict_oldsteppreviousdiagonalbound. pfa_gap_division_restrict_oldsteppreviousdiagonalbound + S (pfc_index_division_restrict_oldsteppreviousdiagonal) = (S (pfd_index_division_restrict_old))) -> exists pfc_value_division_restrict_oldsteppreviousdiagonal. ((((exists ff_h_pfp_division_restrict_oldsteppreviousdiagonalentry. ff_h_pfp_division_restrict_oldsteppreviousdiagonalentry + S (pfc_value_division_restrict_oldsteppreviousdiagonal) = S ((S (pfc_index_division_restrict_oldsteppreviousdiagonal)) * pfc_terms_scale_division_restrict_oldstepprevious)) /\\ exists ff_q_pfp_division_restrict_oldsteppreviousdiagonalentry. pfc_terms_code_division_restrict_oldstepprevious = ff_q_pfp_division_restrict_oldsteppreviousdiagonalentry * S ((S (pfc_index_division_restrict_oldsteppreviousdiagonal)) * pfc_terms_scale_division_restrict_oldstepprevious) + (pfc_value_division_restrict_oldsteppreviousdiagonal))) /\\ ((exists pfc_complement_division_restrict_oldsteppreviousdiagonalterm pfc_left_division_restrict_oldsteppreviousdiagonalterm pfc_right_division_restrict_oldsteppreviousdiagonalterm. (((pfc_index_division_restrict_oldsteppreviousdiagonal)+pfc_complement_division_restrict_oldsteppreviousdiagonalterm=(pfd_index_division_restrict_old)) /\\ ((((((exists pfa_gap_division_restrict_oldsteppreviousdiagonaltermleftinside. pfa_gap_division_restrict_oldsteppreviousdiagonaltermleftinside + S (pfc_index_division_restrict_oldsteppreviousdiagonal) = (pfd_index_division_restrict_old)) /\\ ((((exists ff_h_pfp_division_restrict_oldsteppreviousdiagonaltermleftentry. ff_h_pfp_division_restrict_oldsteppreviousdiagonaltermleftentry + S (pfc_left_division_restrict_oldsteppreviousdiagonalterm) = S ((S (pfc_index_division_restrict_oldsteppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_division_restrict_oldsteppreviousdiagonaltermleftentry. qb = ff_q_pfp_division_restrict_oldsteppreviousdiagonaltermleftentry * S ((S (pfc_index_division_restrict_oldsteppreviousdiagonal)) * qc) + (pfc_left_division_restrict_oldsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_restrict_oldsteppreviousdiagonaltermleftoutside. pfc_gap_division_restrict_oldsteppreviousdiagonaltermleftoutside+(pfd_index_division_restrict_old)=(pfc_index_division_restrict_oldsteppreviousdiagonal)) /\\ (((pfc_left_division_restrict_oldsteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_restrict_oldsteppreviousdiagonaltermrightinside. pfa_gap_division_restrict_oldsteppreviousdiagonaltermrightinside + S (pfc_complement_division_restrict_oldsteppreviousdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_division_restrict_oldsteppreviousdiagonaltermrightentry. ff_h_pfp_division_restrict_oldsteppreviousdiagonaltermrightentry + S (pfc_right_division_restrict_oldsteppreviousdiagonalterm) = S ((S (pfc_complement_division_restrict_oldsteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_restrict_oldsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_division_restrict_oldsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_division_restrict_oldsteppreviousdiagonalterm)) * bc) + (pfc_right_division_restrict_oldsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_restrict_oldsteppreviousdiagonaltermrightoutside. pfc_gap_division_restrict_oldsteppreviousdiagonaltermrightoutside+(M)=(pfc_complement_division_restrict_oldsteppreviousdiagonalterm)) /\\ (((pfc_right_division_restrict_oldsteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_division_restrict_oldsteppreviousdiagonal)=pfc_left_division_restrict_oldsteppreviousdiagonalterm*pfc_right_division_restrict_oldsteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_restrict_oldstepprevioussum fs_v_pfc_division_restrict_oldstepprevioussum. ((((exists fs_h_pfc_division_restrict_oldstepprevioussum_body_start. fs_h_pfc_division_restrict_oldstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_restrict_oldstepprevioussum)) /\\ exists fs_q_pfc_division_restrict_oldstepprevioussum_body_start. fs_u_pfc_division_restrict_oldstepprevioussum = fs_q_pfc_division_restrict_oldstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_division_restrict_oldstepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_division_restrict_oldstepprevioussum_body_terminal. fs_h_pfc_division_restrict_oldstepprevioussum_body_terminal + S (pfc_natural_sum_division_restrict_oldstepprevious) = S ((S (S (pfd_index_division_restrict_old))) * fs_v_pfc_division_restrict_oldstepprevioussum)) /\\ exists fs_q_pfc_division_restrict_oldstepprevioussum_body_terminal. fs_u_pfc_division_restrict_oldstepprevioussum = fs_q_pfc_division_restrict_oldstepprevioussum_body_terminal * S ((S (S (pfd_index_division_restrict_old))) * fs_v_pfc_division_restrict_oldstepprevioussum) + (pfc_natural_sum_division_restrict_oldstepprevious))) /\\ forall fs_i_pfc_division_restrict_oldstepprevioussum_body_steps. (exists fs_lt_pfc_division_restrict_oldstepprevioussum_body_steps_bound. fs_lt_pfc_division_restrict_oldstepprevioussum_body_steps_bound + S fs_i_pfc_division_restrict_oldstepprevioussum_body_steps = S (pfd_index_division_restrict_old)) -> exists fs_a_pfc_division_restrict_oldstepprevioussum_body_steps fs_r_pfc_division_restrict_oldstepprevioussum_body_steps fs_s_pfc_division_restrict_oldstepprevioussum_body_steps. ((((exists fs_h_pfc_division_restrict_oldstepprevioussum_body_steps_summand. fs_h_pfc_division_restrict_oldstepprevioussum_body_steps_summand + S (fs_a_pfc_division_restrict_oldstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_restrict_oldstepprevioussum_body_steps)) * pfc_terms_scale_division_restrict_oldstepprevious)) /\\ exists fs_q_pfc_division_restrict_oldstepprevioussum_body_steps_summand. pfc_terms_code_division_restrict_oldstepprevious = fs_q_pfc_division_restrict_oldstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_division_restrict_oldstepprevioussum_body_steps)) * pfc_terms_scale_division_restrict_oldstepprevious) + (fs_a_pfc_division_restrict_oldstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_restrict_oldstepprevioussum_body_steps_partial. fs_h_pfc_division_restrict_oldstepprevioussum_body_steps_partial + S (fs_r_pfc_division_restrict_oldstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_restrict_oldstepprevioussum_body_steps)) * fs_v_pfc_division_restrict_oldstepprevioussum)) /\\ exists fs_q_pfc_division_restrict_oldstepprevioussum_body_steps_partial. fs_u_pfc_division_restrict_oldstepprevioussum = fs_q_pfc_division_restrict_oldstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_division_restrict_oldstepprevioussum_body_steps)) * fs_v_pfc_division_restrict_oldstepprevioussum) + (fs_r_pfc_division_restrict_oldstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_restrict_oldstepprevioussum_body_steps_successor. fs_h_pfc_division_restrict_oldstepprevioussum_body_steps_successor + S (fs_s_pfc_division_restrict_oldstepprevioussum_body_steps) = S ((S (S fs_i_pfc_division_restrict_oldstepprevioussum_body_steps)) * fs_v_pfc_division_restrict_oldstepprevioussum)) /\\ exists fs_q_pfc_division_restrict_oldstepprevioussum_body_steps_successor. fs_u_pfc_division_restrict_oldstepprevioussum = fs_q_pfc_division_restrict_oldstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_division_restrict_oldstepprevioussum_body_steps)) * fs_v_pfc_division_restrict_oldstepprevioussum) + (fs_s_pfc_division_restrict_oldstepprevioussum_body_steps))) /\\ fs_s_pfc_division_restrict_oldstepprevioussum_body_steps = fs_r_pfc_division_restrict_oldstepprevioussum_body_steps + fs_a_pfc_division_restrict_oldstepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_division_restrict_oldsteppreviousresiduebound. pfa_gap_division_restrict_oldsteppreviousresiduebound + S (pfd_previous_division_restrict_oldstep) = (p)) /\\ ((exists pfa_offset_left_division_restrict_oldsteppreviousresiduecongruence pfa_offset_right_division_restrict_oldsteppreviousresiduecongruence. (pfc_natural_sum_division_restrict_oldstepprevious) + (p) * pfa_offset_left_division_restrict_oldsteppreviousresiduecongruence = (pfd_previous_division_restrict_oldstep) + (p) * pfa_offset_right_division_restrict_oldsteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_division_restrict_oldstepsubtractleft. pfa_gap_division_restrict_oldstepsubtractleft + S (pfd_previous_division_restrict_oldstep) = (p)) /\\ (((exists pfa_gap_division_restrict_oldstepsubtractright. pfa_gap_division_restrict_oldstepsubtractright + S (pfd_difference_division_restrict_oldstep) = (p)) /\\ ((((exists pfa_gap_division_restrict_oldstepsubtractresultbound. pfa_gap_division_restrict_oldstepsubtractresultbound + S (pfd_input_division_restrict_oldstep) = (p)) /\\ ((exists pfa_offset_left_division_restrict_oldstepsubtractresultcongruence pfa_offset_right_division_restrict_oldstepsubtractresultcongruence. ((pfd_previous_division_restrict_oldstep) + (pfd_difference_division_restrict_oldstep)) + (p) * pfa_offset_left_division_restrict_oldstepsubtractresultcongruence = (pfd_input_division_restrict_oldstep) + (p) * pfa_offset_right_division_restrict_oldstepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_division_restrict_oldstepmultiplyleft. pfa_gap_division_restrict_oldstepmultiplyleft + S (k) = (p)) /\\ (((exists pfa_gap_division_restrict_oldstepmultiplyright. pfa_gap_division_restrict_oldstepmultiplyright + S (pfd_difference_division_restrict_oldstep) = (p)) /\\ ((((exists pfa_gap_division_restrict_oldstepmultiplyresultbound. pfa_gap_division_restrict_oldstepmultiplyresultbound + S (pfd_value_division_restrict_old) = (p)) /\\ ((exists pfa_offset_left_division_restrict_oldstepmultiplyresultcongruence pfa_offset_right_division_restrict_oldstepmultiplyresultcongruence. ((k) * (pfd_difference_division_restrict_oldstep)) + (p) * pfa_offset_left_division_restrict_oldstepmultiplyresultcongruence = (pfd_value_division_restrict_old) + (p) * pfa_offset_right_division_restrict_oldstepmultiplyresultcongruence))))))))))))))))))) -> (forall pfd_index_division_restrict_new. (exists pfa_gap_division_restrict_newbound. pfa_gap_division_restrict_newbound + S (pfd_index_division_restrict_new) = (K)) -> exists pfd_value_division_restrict_new. ((((exists ff_h_pfp_division_restrict_newentry. ff_h_pfp_division_restrict_newentry + S (pfd_value_division_restrict_new) = S ((S (pfd_index_division_restrict_new)) * qc)) /\\ exists ff_q_pfp_division_restrict_newentry. qb = ff_q_pfp_division_restrict_newentry * S ((S (pfd_index_division_restrict_new)) * qc) + (pfd_value_division_restrict_new))) /\\ ((exists pfd_input_division_restrict_newstep pfd_previous_division_restrict_newstep pfd_difference_division_restrict_newstep. ((((exists ff_h_pfp_division_restrict_newstepinput. ff_h_pfp_division_restrict_newstepinput + S (pfd_input_division_restrict_newstep) = S ((S (pfd_index_division_restrict_new)) * ac)) /\\ exists ff_q_pfp_division_restrict_newstepinput. ab = ff_q_pfp_division_restrict_newstepinput * S ((S (pfd_index_division_restrict_new)) * ac) + (pfd_input_division_restrict_newstep))) /\\ (((exists pfc_terms_code_division_restrict_newstepprevious pfc_terms_scale_division_restrict_newstepprevious pfc_natural_sum_division_restrict_newstepprevious. ((forall pfc_index_division_restrict_newsteppreviousdiagonal. (exists pfa_gap_division_restrict_newsteppreviousdiagonalbound. pfa_gap_division_restrict_newsteppreviousdiagonalbound + S (pfc_index_division_restrict_newsteppreviousdiagonal) = (S (pfd_index_division_restrict_new))) -> exists pfc_value_division_restrict_newsteppreviousdiagonal. ((((exists ff_h_pfp_division_restrict_newsteppreviousdiagonalentry. ff_h_pfp_division_restrict_newsteppreviousdiagonalentry + S (pfc_value_division_restrict_newsteppreviousdiagonal) = S ((S (pfc_index_division_restrict_newsteppreviousdiagonal)) * pfc_terms_scale_division_restrict_newstepprevious)) /\\ exists ff_q_pfp_division_restrict_newsteppreviousdiagonalentry. pfc_terms_code_division_restrict_newstepprevious = ff_q_pfp_division_restrict_newsteppreviousdiagonalentry * S ((S (pfc_index_division_restrict_newsteppreviousdiagonal)) * pfc_terms_scale_division_restrict_newstepprevious) + (pfc_value_division_restrict_newsteppreviousdiagonal))) /\\ ((exists pfc_complement_division_restrict_newsteppreviousdiagonalterm pfc_left_division_restrict_newsteppreviousdiagonalterm pfc_right_division_restrict_newsteppreviousdiagonalterm. (((pfc_index_division_restrict_newsteppreviousdiagonal)+pfc_complement_division_restrict_newsteppreviousdiagonalterm=(pfd_index_division_restrict_new)) /\\ ((((((exists pfa_gap_division_restrict_newsteppreviousdiagonaltermleftinside. pfa_gap_division_restrict_newsteppreviousdiagonaltermleftinside + S (pfc_index_division_restrict_newsteppreviousdiagonal) = (pfd_index_division_restrict_new)) /\\ ((((exists ff_h_pfp_division_restrict_newsteppreviousdiagonaltermleftentry. ff_h_pfp_division_restrict_newsteppreviousdiagonaltermleftentry + S (pfc_left_division_restrict_newsteppreviousdiagonalterm) = S ((S (pfc_index_division_restrict_newsteppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_division_restrict_newsteppreviousdiagonaltermleftentry. qb = ff_q_pfp_division_restrict_newsteppreviousdiagonaltermleftentry * S ((S (pfc_index_division_restrict_newsteppreviousdiagonal)) * qc) + (pfc_left_division_restrict_newsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_restrict_newsteppreviousdiagonaltermleftoutside. pfc_gap_division_restrict_newsteppreviousdiagonaltermleftoutside+(pfd_index_division_restrict_new)=(pfc_index_division_restrict_newsteppreviousdiagonal)) /\\ (((pfc_left_division_restrict_newsteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_restrict_newsteppreviousdiagonaltermrightinside. pfa_gap_division_restrict_newsteppreviousdiagonaltermrightinside + S (pfc_complement_division_restrict_newsteppreviousdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_division_restrict_newsteppreviousdiagonaltermrightentry. ff_h_pfp_division_restrict_newsteppreviousdiagonaltermrightentry + S (pfc_right_division_restrict_newsteppreviousdiagonalterm) = S ((S (pfc_complement_division_restrict_newsteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_restrict_newsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_division_restrict_newsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_division_restrict_newsteppreviousdiagonalterm)) * bc) + (pfc_right_division_restrict_newsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_restrict_newsteppreviousdiagonaltermrightoutside. pfc_gap_division_restrict_newsteppreviousdiagonaltermrightoutside+(M)=(pfc_complement_division_restrict_newsteppreviousdiagonalterm)) /\\ (((pfc_right_division_restrict_newsteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_division_restrict_newsteppreviousdiagonal)=pfc_left_division_restrict_newsteppreviousdiagonalterm*pfc_right_division_restrict_newsteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_restrict_newstepprevioussum fs_v_pfc_division_restrict_newstepprevioussum. ((((exists fs_h_pfc_division_restrict_newstepprevioussum_body_start. fs_h_pfc_division_restrict_newstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_restrict_newstepprevioussum)) /\\ exists fs_q_pfc_division_restrict_newstepprevioussum_body_start. fs_u_pfc_division_restrict_newstepprevioussum = fs_q_pfc_division_restrict_newstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_division_restrict_newstepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_division_restrict_newstepprevioussum_body_terminal. fs_h_pfc_division_restrict_newstepprevioussum_body_terminal + S (pfc_natural_sum_division_restrict_newstepprevious) = S ((S (S (pfd_index_division_restrict_new))) * fs_v_pfc_division_restrict_newstepprevioussum)) /\\ exists fs_q_pfc_division_restrict_newstepprevioussum_body_terminal. fs_u_pfc_division_restrict_newstepprevioussum = fs_q_pfc_division_restrict_newstepprevioussum_body_terminal * S ((S (S (pfd_index_division_restrict_new))) * fs_v_pfc_division_restrict_newstepprevioussum) + (pfc_natural_sum_division_restrict_newstepprevious))) /\\ forall fs_i_pfc_division_restrict_newstepprevioussum_body_steps. (exists fs_lt_pfc_division_restrict_newstepprevioussum_body_steps_bound. fs_lt_pfc_division_restrict_newstepprevioussum_body_steps_bound + S fs_i_pfc_division_restrict_newstepprevioussum_body_steps = S (pfd_index_division_restrict_new)) -> exists fs_a_pfc_division_restrict_newstepprevioussum_body_steps fs_r_pfc_division_restrict_newstepprevioussum_body_steps fs_s_pfc_division_restrict_newstepprevioussum_body_steps. ((((exists fs_h_pfc_division_restrict_newstepprevioussum_body_steps_summand. fs_h_pfc_division_restrict_newstepprevioussum_body_steps_summand + S (fs_a_pfc_division_restrict_newstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_restrict_newstepprevioussum_body_steps)) * pfc_terms_scale_division_restrict_newstepprevious)) /\\ exists fs_q_pfc_division_restrict_newstepprevioussum_body_steps_summand. pfc_terms_code_division_restrict_newstepprevious = fs_q_pfc_division_restrict_newstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_division_restrict_newstepprevioussum_body_steps)) * pfc_terms_scale_division_restrict_newstepprevious) + (fs_a_pfc_division_restrict_newstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_restrict_newstepprevioussum_body_steps_partial. fs_h_pfc_division_restrict_newstepprevioussum_body_steps_partial + S (fs_r_pfc_division_restrict_newstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_restrict_newstepprevioussum_body_steps)) * fs_v_pfc_division_restrict_newstepprevioussum)) /\\ exists fs_q_pfc_division_restrict_newstepprevioussum_body_steps_partial. fs_u_pfc_division_restrict_newstepprevioussum = fs_q_pfc_division_restrict_newstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_division_restrict_newstepprevioussum_body_steps)) * fs_v_pfc_division_restrict_newstepprevioussum) + (fs_r_pfc_division_restrict_newstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_restrict_newstepprevioussum_body_steps_successor. fs_h_pfc_division_restrict_newstepprevioussum_body_steps_successor + S (fs_s_pfc_division_restrict_newstepprevioussum_body_steps) = S ((S (S fs_i_pfc_division_restrict_newstepprevioussum_body_steps)) * fs_v_pfc_division_restrict_newstepprevioussum)) /\\ exists fs_q_pfc_division_restrict_newstepprevioussum_body_steps_successor. fs_u_pfc_division_restrict_newstepprevioussum = fs_q_pfc_division_restrict_newstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_division_restrict_newstepprevioussum_body_steps)) * fs_v_pfc_division_restrict_newstepprevioussum) + (fs_s_pfc_division_restrict_newstepprevioussum_body_steps))) /\\ fs_s_pfc_division_restrict_newstepprevioussum_body_steps = fs_r_pfc_division_restrict_newstepprevioussum_body_steps + fs_a_pfc_division_restrict_newstepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_division_restrict_newsteppreviousresiduebound. pfa_gap_division_restrict_newsteppreviousresiduebound + S (pfd_previous_division_restrict_newstep) = (p)) /\\ ((exists pfa_offset_left_division_restrict_newsteppreviousresiduecongruence pfa_offset_right_division_restrict_newsteppreviousresiduecongruence. (pfc_natural_sum_division_restrict_newstepprevious) + (p) * pfa_offset_left_division_restrict_newsteppreviousresiduecongruence = (pfd_previous_division_restrict_newstep) + (p) * pfa_offset_right_division_restrict_newsteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_division_restrict_newstepsubtractleft. pfa_gap_division_restrict_newstepsubtractleft + S (pfd_previous_division_restrict_newstep) = (p)) /\\ (((exists pfa_gap_division_restrict_newstepsubtractright. pfa_gap_division_restrict_newstepsubtractright + S (pfd_difference_division_restrict_newstep) = (p)) /\\ ((((exists pfa_gap_division_restrict_newstepsubtractresultbound. pfa_gap_division_restrict_newstepsubtractresultbound + S (pfd_input_division_restrict_newstep) = (p)) /\\ ((exists pfa_offset_left_division_restrict_newstepsubtractresultcongruence pfa_offset_right_division_restrict_newstepsubtractresultcongruence. ((pfd_previous_division_restrict_newstep) + (pfd_difference_division_restrict_newstep)) + (p) * pfa_offset_left_division_restrict_newstepsubtractresultcongruence = (pfd_input_division_restrict_newstep) + (p) * pfa_offset_right_division_restrict_newstepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_division_restrict_newstepmultiplyleft. pfa_gap_division_restrict_newstepmultiplyleft + S (k) = (p)) /\\ (((exists pfa_gap_division_restrict_newstepmultiplyright. pfa_gap_division_restrict_newstepmultiplyright + S (pfd_difference_division_restrict_newstep) = (p)) /\\ ((((exists pfa_gap_division_restrict_newstepmultiplyresultbound. pfa_gap_division_restrict_newstepmultiplyresultbound + S (pfd_value_division_restrict_new) = (p)) /\\ ((exists pfa_offset_left_division_restrict_newstepmultiplyresultcongruence pfa_offset_right_division_restrict_newstepmultiplyresultcongruence. 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((((exists ff_h_pfp_division_entry_chosen. ff_h_pfp_division_entry_chosen + S (r) = S ((S (i)) * qc)) /\\ exists ff_q_pfp_division_entry_chosen. qb = ff_q_pfp_division_entry_chosen * S ((S (i)) * qc) + (r))) /\\ ((exists pfd_input_division_entry_step pfd_previous_division_entry_step pfd_difference_division_entry_step. ((((exists ff_h_pfp_division_entry_stepinput. ff_h_pfp_division_entry_stepinput + S (pfd_input_division_entry_step) = S ((S (i)) * ac)) /\\ exists ff_q_pfp_division_entry_stepinput. ab = ff_q_pfp_division_entry_stepinput * S ((S (i)) * ac) + (pfd_input_division_entry_step))) /\\ (((exists pfc_terms_code_division_entry_stepprevious pfc_terms_scale_division_entry_stepprevious pfc_natural_sum_division_entry_stepprevious. ((forall pfc_index_division_entry_steppreviousdiagonal. (exists pfa_gap_division_entry_steppreviousdiagonalbound. pfa_gap_division_entry_steppreviousdiagonalbound + S (pfc_index_division_entry_steppreviousdiagonal) = (S (i))) -> exists pfc_value_division_entry_steppreviousdiagonal. ((((exists ff_h_pfp_division_entry_steppreviousdiagonalentry. ff_h_pfp_division_entry_steppreviousdiagonalentry + S (pfc_value_division_entry_steppreviousdiagonal) = S ((S (pfc_index_division_entry_steppreviousdiagonal)) * pfc_terms_scale_division_entry_stepprevious)) /\\ exists ff_q_pfp_division_entry_steppreviousdiagonalentry. pfc_terms_code_division_entry_stepprevious = ff_q_pfp_division_entry_steppreviousdiagonalentry * S ((S (pfc_index_division_entry_steppreviousdiagonal)) * pfc_terms_scale_division_entry_stepprevious) + (pfc_value_division_entry_steppreviousdiagonal))) /\\ ((exists pfc_complement_division_entry_steppreviousdiagonalterm pfc_left_division_entry_steppreviousdiagonalterm pfc_right_division_entry_steppreviousdiagonalterm. (((pfc_index_division_entry_steppreviousdiagonal)+pfc_complement_division_entry_steppreviousdiagonalterm=(i)) /\\ ((((((exists pfa_gap_division_entry_steppreviousdiagonaltermleftinside. pfa_gap_division_entry_steppreviousdiagonaltermleftinside + S (pfc_index_division_entry_steppreviousdiagonal) = (i)) /\\ ((((exists ff_h_pfp_division_entry_steppreviousdiagonaltermleftentry. ff_h_pfp_division_entry_steppreviousdiagonaltermleftentry + S (pfc_left_division_entry_steppreviousdiagonalterm) = S ((S (pfc_index_division_entry_steppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_division_entry_steppreviousdiagonaltermleftentry. qb = ff_q_pfp_division_entry_steppreviousdiagonaltermleftentry * S ((S (pfc_index_division_entry_steppreviousdiagonal)) * qc) + (pfc_left_division_entry_steppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_entry_steppreviousdiagonaltermleftoutside. pfc_gap_division_entry_steppreviousdiagonaltermleftoutside+(i)=(pfc_index_division_entry_steppreviousdiagonal)) /\\ (((pfc_left_division_entry_steppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_entry_steppreviousdiagonaltermrightinside. pfa_gap_division_entry_steppreviousdiagonaltermrightinside + S (pfc_complement_division_entry_steppreviousdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_division_entry_steppreviousdiagonaltermrightentry. ff_h_pfp_division_entry_steppreviousdiagonaltermrightentry + S (pfc_right_division_entry_steppreviousdiagonalterm) = S ((S (pfc_complement_division_entry_steppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_entry_steppreviousdiagonaltermrightentry. bb = ff_q_pfp_division_entry_steppreviousdiagonaltermrightentry * S ((S (pfc_complement_division_entry_steppreviousdiagonalterm)) * bc) + (pfc_right_division_entry_steppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_entry_steppreviousdiagonaltermrightoutside. pfc_gap_division_entry_steppreviousdiagonaltermrightoutside+(M)=(pfc_complement_division_entry_steppreviousdiagonalterm)) /\\ (((pfc_right_division_entry_steppreviousdiagonalterm)=0))))) /\\ (((pfc_value_division_entry_steppreviousdiagonal)=pfc_left_division_entry_steppreviousdiagonalterm*pfc_right_division_entry_steppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_entry_stepprevioussum fs_v_pfc_division_entry_stepprevioussum. ((((exists fs_h_pfc_division_entry_stepprevioussum_body_start. fs_h_pfc_division_entry_stepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_entry_stepprevioussum)) /\\ exists fs_q_pfc_division_entry_stepprevioussum_body_start. fs_u_pfc_division_entry_stepprevioussum = fs_q_pfc_division_entry_stepprevioussum_body_start * S ((S (0)) * fs_v_pfc_division_entry_stepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_division_entry_stepprevioussum_body_terminal. fs_h_pfc_division_entry_stepprevioussum_body_terminal + S (pfc_natural_sum_division_entry_stepprevious) = S ((S (S (i))) * fs_v_pfc_division_entry_stepprevioussum)) /\\ exists fs_q_pfc_division_entry_stepprevioussum_body_terminal. fs_u_pfc_division_entry_stepprevioussum = fs_q_pfc_division_entry_stepprevioussum_body_terminal * S ((S (S (i))) * fs_v_pfc_division_entry_stepprevioussum) + (pfc_natural_sum_division_entry_stepprevious))) /\\ forall fs_i_pfc_division_entry_stepprevioussum_body_steps. (exists fs_lt_pfc_division_entry_stepprevioussum_body_steps_bound. fs_lt_pfc_division_entry_stepprevioussum_body_steps_bound + S fs_i_pfc_division_entry_stepprevioussum_body_steps = S (i)) -> exists fs_a_pfc_division_entry_stepprevioussum_body_steps fs_r_pfc_division_entry_stepprevioussum_body_steps fs_s_pfc_division_entry_stepprevioussum_body_steps. ((((exists fs_h_pfc_division_entry_stepprevioussum_body_steps_summand. fs_h_pfc_division_entry_stepprevioussum_body_steps_summand + S (fs_a_pfc_division_entry_stepprevioussum_body_steps) = S ((S (fs_i_pfc_division_entry_stepprevioussum_body_steps)) * pfc_terms_scale_division_entry_stepprevious)) /\\ exists fs_q_pfc_division_entry_stepprevioussum_body_steps_summand. pfc_terms_code_division_entry_stepprevious = fs_q_pfc_division_entry_stepprevioussum_body_steps_summand * S ((S (fs_i_pfc_division_entry_stepprevioussum_body_steps)) * pfc_terms_scale_division_entry_stepprevious) + (fs_a_pfc_division_entry_stepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_entry_stepprevioussum_body_steps_partial. fs_h_pfc_division_entry_stepprevioussum_body_steps_partial + S (fs_r_pfc_division_entry_stepprevioussum_body_steps) = S ((S (fs_i_pfc_division_entry_stepprevioussum_body_steps)) * fs_v_pfc_division_entry_stepprevioussum)) /\\ exists fs_q_pfc_division_entry_stepprevioussum_body_steps_partial. fs_u_pfc_division_entry_stepprevioussum = fs_q_pfc_division_entry_stepprevioussum_body_steps_partial * S ((S (fs_i_pfc_division_entry_stepprevioussum_body_steps)) * fs_v_pfc_division_entry_stepprevioussum) + (fs_r_pfc_division_entry_stepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_entry_stepprevioussum_body_steps_successor. fs_h_pfc_division_entry_stepprevioussum_body_steps_successor + S (fs_s_pfc_division_entry_stepprevioussum_body_steps) = S ((S (S fs_i_pfc_division_entry_stepprevioussum_body_steps)) * fs_v_pfc_division_entry_stepprevioussum)) /\\ exists fs_q_pfc_division_entry_stepprevioussum_body_steps_successor. fs_u_pfc_division_entry_stepprevioussum = fs_q_pfc_division_entry_stepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_division_entry_stepprevioussum_body_steps)) * fs_v_pfc_division_entry_stepprevioussum) + (fs_s_pfc_division_entry_stepprevioussum_body_steps))) /\\ fs_s_pfc_division_entry_stepprevioussum_body_steps = fs_r_pfc_division_entry_stepprevioussum_body_steps + fs_a_pfc_division_entry_stepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_division_entry_steppreviousresiduebound. pfa_gap_division_entry_steppreviousresiduebound + S (pfd_previous_division_entry_step) = (p)) /\\ ((exists pfa_offset_left_division_entry_steppreviousresiduecongruence pfa_offset_right_division_entry_steppreviousresiduecongruence. (pfc_natural_sum_division_entry_stepprevious) + (p) * pfa_offset_left_division_entry_steppreviousresiduecongruence = (pfd_previous_division_entry_step) + (p) * pfa_offset_right_division_entry_steppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_division_entry_stepsubtractleft. pfa_gap_division_entry_stepsubtractleft + S (pfd_previous_division_entry_step) = (p)) /\\ (((exists pfa_gap_division_entry_stepsubtractright. pfa_gap_division_entry_stepsubtractright + S (pfd_difference_division_entry_step) = (p)) /\\ ((((exists pfa_gap_division_entry_stepsubtractresultbound. pfa_gap_division_entry_stepsubtractresultbound + S (pfd_input_division_entry_step) = (p)) /\\ ((exists pfa_offset_left_division_entry_stepsubtractresultcongruence pfa_offset_right_division_entry_stepsubtractresultcongruence. ((pfd_previous_division_entry_step) + (pfd_difference_division_entry_step)) + (p) * pfa_offset_left_division_entry_stepsubtractresultcongruence = (pfd_input_division_entry_step) + (p) * pfa_offset_right_division_entry_stepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_division_entry_stepmultiplyleft. pfa_gap_division_entry_stepmultiplyleft + S (k) = (p)) /\\ (((exists pfa_gap_division_entry_stepmultiplyright. pfa_gap_division_entry_stepmultiplyright + S (pfd_difference_division_entry_step) = (p)) /\\ ((((exists pfa_gap_division_entry_stepmultiplyresultbound. pfa_gap_division_entry_stepmultiplyresultbound + S (r) = (p)) /\\ ((exists pfa_offset_left_division_entry_stepmultiplyresultcongruence pfa_offset_right_division_entry_stepmultiplyresultcongruence. ((k) * (pfd_difference_division_entry_step)) + (p) * pfa_offset_left_division_entry_stepmultiplyresultcongruence = (r) + (p) * pfa_offset_right_division_entry_stepmultiplyresultcongruence))))))))))))))))))", "specialize h (i)", "apply h", "exact hi", "cases hv", "cases hv_witness", "have heq : x=q", "specialize beta_at_unique (qb)", "specialize beta_at_unique (qc)", "specialize beta_at_unique (i)", "specialize beta_at_unique (x)", "specialize beta_at_unique (q)", "apply beta_at_unique", "exact hv_witness_left", "exact hq", "rewrite heq at hv_witness_right", "rewrite heq at hv_witness_right", "exact hv_witness_right" ], "script_sha256": "1fc5dde42cbbd9b7ca344083dfd66549dff8e95ca89c8110bc675e2160c6daeb", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_division_candidate.py", "sha256": "edfc7806caf7a83b9cb0e3e420bd2c3a8679f2d4d9ee6ca9f8eae53faca8d5b2" }, "statement": "forall p k ab ac bb bc M qb qc N i q. (forall pfd_index_division_entry_source. (exists pfa_gap_division_entry_sourcebound. pfa_gap_division_entry_sourcebound + S (pfd_index_division_entry_source) = (N)) -> exists pfd_value_division_entry_source. ((((exists ff_h_pfp_division_entry_sourceentry. ff_h_pfp_division_entry_sourceentry + S (pfd_value_division_entry_source) = S ((S (pfd_index_division_entry_source)) * qc)) /\\ exists ff_q_pfp_division_entry_sourceentry. qb = ff_q_pfp_division_entry_sourceentry * S ((S (pfd_index_division_entry_source)) * qc) + (pfd_value_division_entry_source))) /\\ ((exists pfd_input_division_entry_sourcestep pfd_previous_division_entry_sourcestep pfd_difference_division_entry_sourcestep. ((((exists ff_h_pfp_division_entry_sourcestepinput. ff_h_pfp_division_entry_sourcestepinput + S (pfd_input_division_entry_sourcestep) = S ((S (pfd_index_division_entry_source)) * ac)) /\\ exists ff_q_pfp_division_entry_sourcestepinput. ab = ff_q_pfp_division_entry_sourcestepinput * S ((S (pfd_index_division_entry_source)) * ac) + (pfd_input_division_entry_sourcestep))) /\\ (((exists pfc_terms_code_division_entry_sourcestepprevious pfc_terms_scale_division_entry_sourcestepprevious pfc_natural_sum_division_entry_sourcestepprevious. ((forall pfc_index_division_entry_sourcesteppreviousdiagonal. (exists pfa_gap_division_entry_sourcesteppreviousdiagonalbound. pfa_gap_division_entry_sourcesteppreviousdiagonalbound + S (pfc_index_division_entry_sourcesteppreviousdiagonal) = (S (pfd_index_division_entry_source))) -> exists pfc_value_division_entry_sourcesteppreviousdiagonal. ((((exists ff_h_pfp_division_entry_sourcesteppreviousdiagonalentry. ff_h_pfp_division_entry_sourcesteppreviousdiagonalentry + S (pfc_value_division_entry_sourcesteppreviousdiagonal) = S ((S (pfc_index_division_entry_sourcesteppreviousdiagonal)) * pfc_terms_scale_division_entry_sourcestepprevious)) /\\ exists ff_q_pfp_division_entry_sourcesteppreviousdiagonalentry. pfc_terms_code_division_entry_sourcestepprevious = ff_q_pfp_division_entry_sourcesteppreviousdiagonalentry * S ((S (pfc_index_division_entry_sourcesteppreviousdiagonal)) * pfc_terms_scale_division_entry_sourcestepprevious) + (pfc_value_division_entry_sourcesteppreviousdiagonal))) /\\ ((exists pfc_complement_division_entry_sourcesteppreviousdiagonalterm pfc_left_division_entry_sourcesteppreviousdiagonalterm pfc_right_division_entry_sourcesteppreviousdiagonalterm. (((pfc_index_division_entry_sourcesteppreviousdiagonal)+pfc_complement_division_entry_sourcesteppreviousdiagonalterm=(pfd_index_division_entry_source)) /\\ ((((((exists pfa_gap_division_entry_sourcesteppreviousdiagonaltermleftinside. pfa_gap_division_entry_sourcesteppreviousdiagonaltermleftinside + S (pfc_index_division_entry_sourcesteppreviousdiagonal) = (pfd_index_division_entry_source)) /\\ ((((exists ff_h_pfp_division_entry_sourcesteppreviousdiagonaltermleftentry. ff_h_pfp_division_entry_sourcesteppreviousdiagonaltermleftentry + S (pfc_left_division_entry_sourcesteppreviousdiagonalterm) = S ((S (pfc_index_division_entry_sourcesteppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_division_entry_sourcesteppreviousdiagonaltermleftentry. qb = ff_q_pfp_division_entry_sourcesteppreviousdiagonaltermleftentry * S ((S (pfc_index_division_entry_sourcesteppreviousdiagonal)) * qc) + (pfc_left_division_entry_sourcesteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_entry_sourcesteppreviousdiagonaltermleftoutside. pfc_gap_division_entry_sourcesteppreviousdiagonaltermleftoutside+(pfd_index_division_entry_source)=(pfc_index_division_entry_sourcesteppreviousdiagonal)) /\\ (((pfc_left_division_entry_sourcesteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_entry_sourcesteppreviousdiagonaltermrightinside. pfa_gap_division_entry_sourcesteppreviousdiagonaltermrightinside + S (pfc_complement_division_entry_sourcesteppreviousdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_division_entry_sourcesteppreviousdiagonaltermrightentry. ff_h_pfp_division_entry_sourcesteppreviousdiagonaltermrightentry + S (pfc_right_division_entry_sourcesteppreviousdiagonalterm) = S ((S (pfc_complement_division_entry_sourcesteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_entry_sourcesteppreviousdiagonaltermrightentry. bb = ff_q_pfp_division_entry_sourcesteppreviousdiagonaltermrightentry * S ((S (pfc_complement_division_entry_sourcesteppreviousdiagonalterm)) * bc) + (pfc_right_division_entry_sourcesteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_entry_sourcesteppreviousdiagonaltermrightoutside. pfc_gap_division_entry_sourcesteppreviousdiagonaltermrightoutside+(M)=(pfc_complement_division_entry_sourcesteppreviousdiagonalterm)) /\\ (((pfc_right_division_entry_sourcesteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_division_entry_sourcesteppreviousdiagonal)=pfc_left_division_entry_sourcesteppreviousdiagonalterm*pfc_right_division_entry_sourcesteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_entry_sourcestepprevioussum fs_v_pfc_division_entry_sourcestepprevioussum. ((((exists fs_h_pfc_division_entry_sourcestepprevioussum_body_start. fs_h_pfc_division_entry_sourcestepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_entry_sourcestepprevioussum)) /\\ exists fs_q_pfc_division_entry_sourcestepprevioussum_body_start. fs_u_pfc_division_entry_sourcestepprevioussum = fs_q_pfc_division_entry_sourcestepprevioussum_body_start * S ((S (0)) * fs_v_pfc_division_entry_sourcestepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_division_entry_sourcestepprevioussum_body_terminal. fs_h_pfc_division_entry_sourcestepprevioussum_body_terminal + S (pfc_natural_sum_division_entry_sourcestepprevious) = S ((S (S (pfd_index_division_entry_source))) * fs_v_pfc_division_entry_sourcestepprevioussum)) /\\ exists fs_q_pfc_division_entry_sourcestepprevioussum_body_terminal. fs_u_pfc_division_entry_sourcestepprevioussum = fs_q_pfc_division_entry_sourcestepprevioussum_body_terminal * S ((S (S (pfd_index_division_entry_source))) * fs_v_pfc_division_entry_sourcestepprevioussum) + (pfc_natural_sum_division_entry_sourcestepprevious))) /\\ forall fs_i_pfc_division_entry_sourcestepprevioussum_body_steps. (exists fs_lt_pfc_division_entry_sourcestepprevioussum_body_steps_bound. fs_lt_pfc_division_entry_sourcestepprevioussum_body_steps_bound + S fs_i_pfc_division_entry_sourcestepprevioussum_body_steps = S (pfd_index_division_entry_source)) -> exists fs_a_pfc_division_entry_sourcestepprevioussum_body_steps fs_r_pfc_division_entry_sourcestepprevioussum_body_steps fs_s_pfc_division_entry_sourcestepprevioussum_body_steps. ((((exists fs_h_pfc_division_entry_sourcestepprevioussum_body_steps_summand. fs_h_pfc_division_entry_sourcestepprevioussum_body_steps_summand + S (fs_a_pfc_division_entry_sourcestepprevioussum_body_steps) = S ((S (fs_i_pfc_division_entry_sourcestepprevioussum_body_steps)) * pfc_terms_scale_division_entry_sourcestepprevious)) /\\ exists fs_q_pfc_division_entry_sourcestepprevioussum_body_steps_summand. pfc_terms_code_division_entry_sourcestepprevious = fs_q_pfc_division_entry_sourcestepprevioussum_body_steps_summand * S ((S (fs_i_pfc_division_entry_sourcestepprevioussum_body_steps)) * pfc_terms_scale_division_entry_sourcestepprevious) + (fs_a_pfc_division_entry_sourcestepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_entry_sourcestepprevioussum_body_steps_partial. fs_h_pfc_division_entry_sourcestepprevioussum_body_steps_partial + S (fs_r_pfc_division_entry_sourcestepprevioussum_body_steps) = S ((S (fs_i_pfc_division_entry_sourcestepprevioussum_body_steps)) * fs_v_pfc_division_entry_sourcestepprevioussum)) /\\ exists fs_q_pfc_division_entry_sourcestepprevioussum_body_steps_partial. fs_u_pfc_division_entry_sourcestepprevioussum = fs_q_pfc_division_entry_sourcestepprevioussum_body_steps_partial * S ((S (fs_i_pfc_division_entry_sourcestepprevioussum_body_steps)) * fs_v_pfc_division_entry_sourcestepprevioussum) + (fs_r_pfc_division_entry_sourcestepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_entry_sourcestepprevioussum_body_steps_successor. fs_h_pfc_division_entry_sourcestepprevioussum_body_steps_successor + S (fs_s_pfc_division_entry_sourcestepprevioussum_body_steps) = S ((S (S fs_i_pfc_division_entry_sourcestepprevioussum_body_steps)) * fs_v_pfc_division_entry_sourcestepprevioussum)) /\\ exists fs_q_pfc_division_entry_sourcestepprevioussum_body_steps_successor. fs_u_pfc_division_entry_sourcestepprevioussum = fs_q_pfc_division_entry_sourcestepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_division_entry_sourcestepprevioussum_body_steps)) * fs_v_pfc_division_entry_sourcestepprevioussum) + (fs_s_pfc_division_entry_sourcestepprevioussum_body_steps))) /\\ fs_s_pfc_division_entry_sourcestepprevioussum_body_steps = fs_r_pfc_division_entry_sourcestepprevioussum_body_steps + fs_a_pfc_division_entry_sourcestepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_division_entry_sourcesteppreviousresiduebound. pfa_gap_division_entry_sourcesteppreviousresiduebound + S (pfd_previous_division_entry_sourcestep) = (p)) /\\ ((exists pfa_offset_left_division_entry_sourcesteppreviousresiduecongruence pfa_offset_right_division_entry_sourcesteppreviousresiduecongruence. (pfc_natural_sum_division_entry_sourcestepprevious) + (p) * pfa_offset_left_division_entry_sourcesteppreviousresiduecongruence = (pfd_previous_division_entry_sourcestep) + (p) * pfa_offset_right_division_entry_sourcesteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_division_entry_sourcestepsubtractleft. pfa_gap_division_entry_sourcestepsubtractleft + S (pfd_previous_division_entry_sourcestep) = (p)) /\\ (((exists pfa_gap_division_entry_sourcestepsubtractright. pfa_gap_division_entry_sourcestepsubtractright + S (pfd_difference_division_entry_sourcestep) = (p)) /\\ ((((exists pfa_gap_division_entry_sourcestepsubtractresultbound. pfa_gap_division_entry_sourcestepsubtractresultbound + S (pfd_input_division_entry_sourcestep) = (p)) /\\ ((exists pfa_offset_left_division_entry_sourcestepsubtractresultcongruence pfa_offset_right_division_entry_sourcestepsubtractresultcongruence. ((pfd_previous_division_entry_sourcestep) + (pfd_difference_division_entry_sourcestep)) + (p) * pfa_offset_left_division_entry_sourcestepsubtractresultcongruence = (pfd_input_division_entry_sourcestep) + (p) * pfa_offset_right_division_entry_sourcestepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_division_entry_sourcestepmultiplyleft. pfa_gap_division_entry_sourcestepmultiplyleft + S (k) = (p)) /\\ (((exists pfa_gap_division_entry_sourcestepmultiplyright. pfa_gap_division_entry_sourcestepmultiplyright + S (pfd_difference_division_entry_sourcestep) = (p)) /\\ ((((exists pfa_gap_division_entry_sourcestepmultiplyresultbound. pfa_gap_division_entry_sourcestepmultiplyresultbound + S (pfd_value_division_entry_source) = (p)) /\\ ((exists pfa_offset_left_division_entry_sourcestepmultiplyresultcongruence pfa_offset_right_division_entry_sourcestepmultiplyresultcongruence. ((k) * (pfd_difference_division_entry_sourcestep)) + (p) * pfa_offset_left_division_entry_sourcestepmultiplyresultcongruence = (pfd_value_division_entry_source) + (p) * pfa_offset_right_division_entry_sourcestepmultiplyresultcongruence))))))))))))))))))) -> (exists pfa_gap_division_entry_bound. pfa_gap_division_entry_bound + S (i) = (N)) -> (((exists ff_h_pfp_division_entry_given. ff_h_pfp_division_entry_given + S (q) = S ((S (i)) * qc)) /\\ exists ff_q_pfp_division_entry_given. qb = ff_q_pfp_division_entry_given * S ((S (i)) * qc) + (q))) -> (exists pfd_input_division_entry_result pfd_previous_division_entry_result pfd_difference_division_entry_result. ((((exists ff_h_pfp_division_entry_resultinput. ff_h_pfp_division_entry_resultinput + S (pfd_input_division_entry_result) = S ((S (i)) * ac)) /\\ exists ff_q_pfp_division_entry_resultinput. ab = ff_q_pfp_division_entry_resultinput * S ((S (i)) * ac) + (pfd_input_division_entry_result))) /\\ (((exists pfc_terms_code_division_entry_resultprevious pfc_terms_scale_division_entry_resultprevious pfc_natural_sum_division_entry_resultprevious. ((forall pfc_index_division_entry_resultpreviousdiagonal. (exists pfa_gap_division_entry_resultpreviousdiagonalbound. pfa_gap_division_entry_resultpreviousdiagonalbound + S (pfc_index_division_entry_resultpreviousdiagonal) = (S (i))) -> exists pfc_value_division_entry_resultpreviousdiagonal. ((((exists ff_h_pfp_division_entry_resultpreviousdiagonalentry. ff_h_pfp_division_entry_resultpreviousdiagonalentry + S (pfc_value_division_entry_resultpreviousdiagonal) = S ((S (pfc_index_division_entry_resultpreviousdiagonal)) * pfc_terms_scale_division_entry_resultprevious)) /\\ exists ff_q_pfp_division_entry_resultpreviousdiagonalentry. pfc_terms_code_division_entry_resultprevious = ff_q_pfp_division_entry_resultpreviousdiagonalentry * S ((S (pfc_index_division_entry_resultpreviousdiagonal)) * pfc_terms_scale_division_entry_resultprevious) + (pfc_value_division_entry_resultpreviousdiagonal))) /\\ ((exists pfc_complement_division_entry_resultpreviousdiagonalterm pfc_left_division_entry_resultpreviousdiagonalterm pfc_right_division_entry_resultpreviousdiagonalterm. (((pfc_index_division_entry_resultpreviousdiagonal)+pfc_complement_division_entry_resultpreviousdiagonalterm=(i)) /\\ ((((((exists pfa_gap_division_entry_resultpreviousdiagonaltermleftinside. pfa_gap_division_entry_resultpreviousdiagonaltermleftinside + S (pfc_index_division_entry_resultpreviousdiagonal) = (i)) /\\ ((((exists ff_h_pfp_division_entry_resultpreviousdiagonaltermleftentry. ff_h_pfp_division_entry_resultpreviousdiagonaltermleftentry + S (pfc_left_division_entry_resultpreviousdiagonalterm) = S ((S (pfc_index_division_entry_resultpreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_division_entry_resultpreviousdiagonaltermleftentry. qb = ff_q_pfp_division_entry_resultpreviousdiagonaltermleftentry * S ((S (pfc_index_division_entry_resultpreviousdiagonal)) * qc) + (pfc_left_division_entry_resultpreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_entry_resultpreviousdiagonaltermleftoutside. pfc_gap_division_entry_resultpreviousdiagonaltermleftoutside+(i)=(pfc_index_division_entry_resultpreviousdiagonal)) /\\ (((pfc_left_division_entry_resultpreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_entry_resultpreviousdiagonaltermrightinside. pfa_gap_division_entry_resultpreviousdiagonaltermrightinside + S (pfc_complement_division_entry_resultpreviousdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_division_entry_resultpreviousdiagonaltermrightentry. ff_h_pfp_division_entry_resultpreviousdiagonaltermrightentry + S (pfc_right_division_entry_resultpreviousdiagonalterm) = S ((S (pfc_complement_division_entry_resultpreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_entry_resultpreviousdiagonaltermrightentry. bb = ff_q_pfp_division_entry_resultpreviousdiagonaltermrightentry * S ((S (pfc_complement_division_entry_resultpreviousdiagonalterm)) * bc) + (pfc_right_division_entry_resultpreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_entry_resultpreviousdiagonaltermrightoutside. pfc_gap_division_entry_resultpreviousdiagonaltermrightoutside+(M)=(pfc_complement_division_entry_resultpreviousdiagonalterm)) /\\ (((pfc_right_division_entry_resultpreviousdiagonalterm)=0))))) /\\ (((pfc_value_division_entry_resultpreviousdiagonal)=pfc_left_division_entry_resultpreviousdiagonalterm*pfc_right_division_entry_resultpreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_entry_resultprevioussum fs_v_pfc_division_entry_resultprevioussum. ((((exists fs_h_pfc_division_entry_resultprevioussum_body_start. fs_h_pfc_division_entry_resultprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_entry_resultprevioussum)) /\\ exists fs_q_pfc_division_entry_resultprevioussum_body_start. fs_u_pfc_division_entry_resultprevioussum = fs_q_pfc_division_entry_resultprevioussum_body_start * S ((S (0)) * fs_v_pfc_division_entry_resultprevioussum) + (0))) /\\ ((((exists fs_h_pfc_division_entry_resultprevioussum_body_terminal. fs_h_pfc_division_entry_resultprevioussum_body_terminal + S (pfc_natural_sum_division_entry_resultprevious) = S ((S (S (i))) * fs_v_pfc_division_entry_resultprevioussum)) /\\ exists fs_q_pfc_division_entry_resultprevioussum_body_terminal. fs_u_pfc_division_entry_resultprevioussum = fs_q_pfc_division_entry_resultprevioussum_body_terminal * S ((S (S (i))) * fs_v_pfc_division_entry_resultprevioussum) + (pfc_natural_sum_division_entry_resultprevious))) /\\ forall fs_i_pfc_division_entry_resultprevioussum_body_steps. (exists fs_lt_pfc_division_entry_resultprevioussum_body_steps_bound. fs_lt_pfc_division_entry_resultprevioussum_body_steps_bound + S fs_i_pfc_division_entry_resultprevioussum_body_steps = S (i)) -> exists fs_a_pfc_division_entry_resultprevioussum_body_steps fs_r_pfc_division_entry_resultprevioussum_body_steps fs_s_pfc_division_entry_resultprevioussum_body_steps. ((((exists fs_h_pfc_division_entry_resultprevioussum_body_steps_summand. fs_h_pfc_division_entry_resultprevioussum_body_steps_summand + S (fs_a_pfc_division_entry_resultprevioussum_body_steps) = S ((S (fs_i_pfc_division_entry_resultprevioussum_body_steps)) * pfc_terms_scale_division_entry_resultprevious)) /\\ exists fs_q_pfc_division_entry_resultprevioussum_body_steps_summand. pfc_terms_code_division_entry_resultprevious = fs_q_pfc_division_entry_resultprevioussum_body_steps_summand * S ((S (fs_i_pfc_division_entry_resultprevioussum_body_steps)) * pfc_terms_scale_division_entry_resultprevious) + (fs_a_pfc_division_entry_resultprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_entry_resultprevioussum_body_steps_partial. fs_h_pfc_division_entry_resultprevioussum_body_steps_partial + S (fs_r_pfc_division_entry_resultprevioussum_body_steps) = S ((S (fs_i_pfc_division_entry_resultprevioussum_body_steps)) * fs_v_pfc_division_entry_resultprevioussum)) /\\ exists fs_q_pfc_division_entry_resultprevioussum_body_steps_partial. fs_u_pfc_division_entry_resultprevioussum = fs_q_pfc_division_entry_resultprevioussum_body_steps_partial * S ((S (fs_i_pfc_division_entry_resultprevioussum_body_steps)) * fs_v_pfc_division_entry_resultprevioussum) + (fs_r_pfc_division_entry_resultprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_entry_resultprevioussum_body_steps_successor. fs_h_pfc_division_entry_resultprevioussum_body_steps_successor + S (fs_s_pfc_division_entry_resultprevioussum_body_steps) = S ((S (S fs_i_pfc_division_entry_resultprevioussum_body_steps)) * fs_v_pfc_division_entry_resultprevioussum)) /\\ exists fs_q_pfc_division_entry_resultprevioussum_body_steps_successor. fs_u_pfc_division_entry_resultprevioussum = fs_q_pfc_division_entry_resultprevioussum_body_steps_successor * S ((S (S fs_i_pfc_division_entry_resultprevioussum_body_steps)) * fs_v_pfc_division_entry_resultprevioussum) + (fs_s_pfc_division_entry_resultprevioussum_body_steps))) /\\ fs_s_pfc_division_entry_resultprevioussum_body_steps = fs_r_pfc_division_entry_resultprevioussum_body_steps + fs_a_pfc_division_entry_resultprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_division_entry_resultpreviousresiduebound. pfa_gap_division_entry_resultpreviousresiduebound + S (pfd_previous_division_entry_result) = (p)) /\\ ((exists pfa_offset_left_division_entry_resultpreviousresiduecongruence pfa_offset_right_division_entry_resultpreviousresiduecongruence. (pfc_natural_sum_division_entry_resultprevious) + (p) * pfa_offset_left_division_entry_resultpreviousresiduecongruence = (pfd_previous_division_entry_result) + (p) * pfa_offset_right_division_entry_resultpreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_division_entry_resultsubtractleft. pfa_gap_division_entry_resultsubtractleft + S (pfd_previous_division_entry_result) = (p)) /\\ (((exists pfa_gap_division_entry_resultsubtractright. pfa_gap_division_entry_resultsubtractright + S (pfd_difference_division_entry_result) = (p)) /\\ ((((exists pfa_gap_division_entry_resultsubtractresultbound. pfa_gap_division_entry_resultsubtractresultbound + S (pfd_input_division_entry_result) = (p)) /\\ ((exists pfa_offset_left_division_entry_resultsubtractresultcongruence pfa_offset_right_division_entry_resultsubtractresultcongruence. ((pfd_previous_division_entry_result) + (pfd_difference_division_entry_result)) + (p) * pfa_offset_left_division_entry_resultsubtractresultcongruence = (pfd_input_division_entry_result) + (p) * pfa_offset_right_division_entry_resultsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_division_entry_resultmultiplyleft. pfa_gap_division_entry_resultmultiplyleft + S (k) = (p)) /\\ (((exists pfa_gap_division_entry_resultmultiplyright. pfa_gap_division_entry_resultmultiplyright + S (pfd_difference_division_entry_result) = (p)) /\\ ((((exists pfa_gap_division_entry_resultmultiplyresultbound. pfa_gap_division_entry_resultmultiplyresultbound + S (q) = (p)) /\\ ((exists pfa_offset_left_division_entry_resultmultiplyresultcongruence pfa_offset_right_division_entry_resultmultiplyresultcongruence. 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((((exists ff_h_pfp_division_bound_entry. ff_h_pfp_division_bound_entry + S (q) = S ((S (i)) * qc)) /\\ exists ff_q_pfp_division_bound_entry. qb = ff_q_pfp_division_bound_entry * S ((S (i)) * qc) + (q))) /\\ ((exists pfd_input_division_bound_step pfd_previous_division_bound_step pfd_difference_division_bound_step. ((((exists ff_h_pfp_division_bound_stepinput. ff_h_pfp_division_bound_stepinput + S (pfd_input_division_bound_step) = S ((S (i)) * ac)) /\\ exists ff_q_pfp_division_bound_stepinput. ab = ff_q_pfp_division_bound_stepinput * S ((S (i)) * ac) + (pfd_input_division_bound_step))) /\\ (((exists pfc_terms_code_division_bound_stepprevious pfc_terms_scale_division_bound_stepprevious pfc_natural_sum_division_bound_stepprevious. ((forall pfc_index_division_bound_steppreviousdiagonal. (exists pfa_gap_division_bound_steppreviousdiagonalbound. pfa_gap_division_bound_steppreviousdiagonalbound + S (pfc_index_division_bound_steppreviousdiagonal) = (S (i))) -> exists pfc_value_division_bound_steppreviousdiagonal. ((((exists ff_h_pfp_division_bound_steppreviousdiagonalentry. ff_h_pfp_division_bound_steppreviousdiagonalentry + S (pfc_value_division_bound_steppreviousdiagonal) = S ((S (pfc_index_division_bound_steppreviousdiagonal)) * pfc_terms_scale_division_bound_stepprevious)) /\\ exists ff_q_pfp_division_bound_steppreviousdiagonalentry. pfc_terms_code_division_bound_stepprevious = ff_q_pfp_division_bound_steppreviousdiagonalentry * S ((S (pfc_index_division_bound_steppreviousdiagonal)) * pfc_terms_scale_division_bound_stepprevious) + (pfc_value_division_bound_steppreviousdiagonal))) /\\ ((exists pfc_complement_division_bound_steppreviousdiagonalterm pfc_left_division_bound_steppreviousdiagonalterm pfc_right_division_bound_steppreviousdiagonalterm. (((pfc_index_division_bound_steppreviousdiagonal)+pfc_complement_division_bound_steppreviousdiagonalterm=(i)) /\\ ((((((exists pfa_gap_division_bound_steppreviousdiagonaltermleftinside. pfa_gap_division_bound_steppreviousdiagonaltermleftinside + S (pfc_index_division_bound_steppreviousdiagonal) = (i)) /\\ ((((exists ff_h_pfp_division_bound_steppreviousdiagonaltermleftentry. ff_h_pfp_division_bound_steppreviousdiagonaltermleftentry + S (pfc_left_division_bound_steppreviousdiagonalterm) = S ((S (pfc_index_division_bound_steppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_division_bound_steppreviousdiagonaltermleftentry. qb = ff_q_pfp_division_bound_steppreviousdiagonaltermleftentry * S ((S (pfc_index_division_bound_steppreviousdiagonal)) * qc) + (pfc_left_division_bound_steppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_bound_steppreviousdiagonaltermleftoutside. pfc_gap_division_bound_steppreviousdiagonaltermleftoutside+(i)=(pfc_index_division_bound_steppreviousdiagonal)) /\\ (((pfc_left_division_bound_steppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_bound_steppreviousdiagonaltermrightinside. pfa_gap_division_bound_steppreviousdiagonaltermrightinside + S (pfc_complement_division_bound_steppreviousdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_division_bound_steppreviousdiagonaltermrightentry. ff_h_pfp_division_bound_steppreviousdiagonaltermrightentry + S (pfc_right_division_bound_steppreviousdiagonalterm) = S ((S (pfc_complement_division_bound_steppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_bound_steppreviousdiagonaltermrightentry. bb = ff_q_pfp_division_bound_steppreviousdiagonaltermrightentry * S ((S (pfc_complement_division_bound_steppreviousdiagonalterm)) * bc) + (pfc_right_division_bound_steppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_bound_steppreviousdiagonaltermrightoutside. pfc_gap_division_bound_steppreviousdiagonaltermrightoutside+(M)=(pfc_complement_division_bound_steppreviousdiagonalterm)) /\\ (((pfc_right_division_bound_steppreviousdiagonalterm)=0))))) /\\ (((pfc_value_division_bound_steppreviousdiagonal)=pfc_left_division_bound_steppreviousdiagonalterm*pfc_right_division_bound_steppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_bound_stepprevioussum fs_v_pfc_division_bound_stepprevioussum. ((((exists fs_h_pfc_division_bound_stepprevioussum_body_start. fs_h_pfc_division_bound_stepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_bound_stepprevioussum)) /\\ exists fs_q_pfc_division_bound_stepprevioussum_body_start. fs_u_pfc_division_bound_stepprevioussum = fs_q_pfc_division_bound_stepprevioussum_body_start * S ((S (0)) * fs_v_pfc_division_bound_stepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_division_bound_stepprevioussum_body_terminal. fs_h_pfc_division_bound_stepprevioussum_body_terminal + S (pfc_natural_sum_division_bound_stepprevious) = S ((S (S (i))) * fs_v_pfc_division_bound_stepprevioussum)) /\\ exists fs_q_pfc_division_bound_stepprevioussum_body_terminal. fs_u_pfc_division_bound_stepprevioussum = fs_q_pfc_division_bound_stepprevioussum_body_terminal * S ((S (S (i))) * fs_v_pfc_division_bound_stepprevioussum) + (pfc_natural_sum_division_bound_stepprevious))) /\\ forall fs_i_pfc_division_bound_stepprevioussum_body_steps. (exists fs_lt_pfc_division_bound_stepprevioussum_body_steps_bound. fs_lt_pfc_division_bound_stepprevioussum_body_steps_bound + S fs_i_pfc_division_bound_stepprevioussum_body_steps = S (i)) -> exists fs_a_pfc_division_bound_stepprevioussum_body_steps fs_r_pfc_division_bound_stepprevioussum_body_steps fs_s_pfc_division_bound_stepprevioussum_body_steps. ((((exists fs_h_pfc_division_bound_stepprevioussum_body_steps_summand. fs_h_pfc_division_bound_stepprevioussum_body_steps_summand + S (fs_a_pfc_division_bound_stepprevioussum_body_steps) = S ((S (fs_i_pfc_division_bound_stepprevioussum_body_steps)) * pfc_terms_scale_division_bound_stepprevious)) /\\ exists fs_q_pfc_division_bound_stepprevioussum_body_steps_summand. pfc_terms_code_division_bound_stepprevious = fs_q_pfc_division_bound_stepprevioussum_body_steps_summand * S ((S (fs_i_pfc_division_bound_stepprevioussum_body_steps)) * pfc_terms_scale_division_bound_stepprevious) + (fs_a_pfc_division_bound_stepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_bound_stepprevioussum_body_steps_partial. fs_h_pfc_division_bound_stepprevioussum_body_steps_partial + S (fs_r_pfc_division_bound_stepprevioussum_body_steps) = S ((S (fs_i_pfc_division_bound_stepprevioussum_body_steps)) * fs_v_pfc_division_bound_stepprevioussum)) /\\ exists fs_q_pfc_division_bound_stepprevioussum_body_steps_partial. fs_u_pfc_division_bound_stepprevioussum = fs_q_pfc_division_bound_stepprevioussum_body_steps_partial * S ((S (fs_i_pfc_division_bound_stepprevioussum_body_steps)) * fs_v_pfc_division_bound_stepprevioussum) + (fs_r_pfc_division_bound_stepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_bound_stepprevioussum_body_steps_successor. fs_h_pfc_division_bound_stepprevioussum_body_steps_successor + S (fs_s_pfc_division_bound_stepprevioussum_body_steps) = S ((S (S fs_i_pfc_division_bound_stepprevioussum_body_steps)) * fs_v_pfc_division_bound_stepprevioussum)) /\\ exists fs_q_pfc_division_bound_stepprevioussum_body_steps_successor. fs_u_pfc_division_bound_stepprevioussum = fs_q_pfc_division_bound_stepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_division_bound_stepprevioussum_body_steps)) * fs_v_pfc_division_bound_stepprevioussum) + (fs_s_pfc_division_bound_stepprevioussum_body_steps))) /\\ fs_s_pfc_division_bound_stepprevioussum_body_steps = fs_r_pfc_division_bound_stepprevioussum_body_steps + fs_a_pfc_division_bound_stepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_division_bound_steppreviousresiduebound. pfa_gap_division_bound_steppreviousresiduebound + S (pfd_previous_division_bound_step) = (p)) /\\ ((exists pfa_offset_left_division_bound_steppreviousresiduecongruence pfa_offset_right_division_bound_steppreviousresiduecongruence. (pfc_natural_sum_division_bound_stepprevious) + (p) * pfa_offset_left_division_bound_steppreviousresiduecongruence = (pfd_previous_division_bound_step) + (p) * pfa_offset_right_division_bound_steppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_division_bound_stepsubtractleft. pfa_gap_division_bound_stepsubtractleft + S (pfd_previous_division_bound_step) = (p)) /\\ (((exists pfa_gap_division_bound_stepsubtractright. pfa_gap_division_bound_stepsubtractright + S (pfd_difference_division_bound_step) = (p)) /\\ ((((exists pfa_gap_division_bound_stepsubtractresultbound. pfa_gap_division_bound_stepsubtractresultbound + S (pfd_input_division_bound_step) = (p)) /\\ ((exists pfa_offset_left_division_bound_stepsubtractresultcongruence pfa_offset_right_division_bound_stepsubtractresultcongruence. ((pfd_previous_division_bound_step) + (pfd_difference_division_bound_step)) + (p) * pfa_offset_left_division_bound_stepsubtractresultcongruence = (pfd_input_division_bound_step) + (p) * pfa_offset_right_division_bound_stepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_division_bound_stepmultiplyleft. pfa_gap_division_bound_stepmultiplyleft + S (k) = (p)) /\\ (((exists pfa_gap_division_bound_stepmultiplyright. pfa_gap_division_bound_stepmultiplyright + S (pfd_difference_division_bound_step) = (p)) /\\ ((((exists pfa_gap_division_bound_stepmultiplyresultbound. pfa_gap_division_bound_stepmultiplyresultbound + S (q) = (p)) /\\ ((exists pfa_offset_left_division_bound_stepmultiplyresultcongruence pfa_offset_right_division_bound_stepmultiplyresultcongruence. ((k) * (pfd_difference_division_bound_step)) + (p) * pfa_offset_left_division_bound_stepmultiplyresultcongruence = (q) + (p) * pfa_offset_right_division_bound_stepmultiplyresultcongruence))))))))))))))))))", "specialize h (i)", "apply h", "exact hi", "cases hv", "cases hv_witness", "exists x", "split", "exact hv_witness_left", "cases hv_witness_right", "cases hv_witness_right_witness", "cases hv_witness_right_witness_witness", "cases hv_witness_right_witness_witness_witness", "cases hv_witness_right_witness_witness_witness_right", "cases hv_witness_right_witness_witness_witness_right_right", "cases hv_witness_right_witness_witness_witness_right_right_right", "cases hv_witness_right_witness_witness_witness_right_right_right_right", "cases hv_witness_right_witness_witness_witness_right_right_right_right_right", "exact hv_witness_right_witness_witness_witness_right_right_right_right_right_left" ], "script_sha256": "1067d652059bfe847043213e887a2d7c906b302971deebe97c6f4715187e6629", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_division_candidate.py", "sha256": "edfc7806caf7a83b9cb0e3e420bd2c3a8679f2d4d9ee6ca9f8eae53faca8d5b2" }, "statement": "forall p k ab ac bb bc M qb qc N. (forall pfd_index_division_bound_source. (exists pfa_gap_division_bound_sourcebound. pfa_gap_division_bound_sourcebound + S (pfd_index_division_bound_source) = (N)) -> exists pfd_value_division_bound_source. ((((exists ff_h_pfp_division_bound_sourceentry. ff_h_pfp_division_bound_sourceentry + S (pfd_value_division_bound_source) = S ((S (pfd_index_division_bound_source)) * qc)) /\\ exists ff_q_pfp_division_bound_sourceentry. qb = ff_q_pfp_division_bound_sourceentry * S ((S (pfd_index_division_bound_source)) * qc) + (pfd_value_division_bound_source))) /\\ ((exists pfd_input_division_bound_sourcestep pfd_previous_division_bound_sourcestep pfd_difference_division_bound_sourcestep. ((((exists ff_h_pfp_division_bound_sourcestepinput. ff_h_pfp_division_bound_sourcestepinput + S (pfd_input_division_bound_sourcestep) = S ((S (pfd_index_division_bound_source)) * ac)) /\\ exists ff_q_pfp_division_bound_sourcestepinput. ab = ff_q_pfp_division_bound_sourcestepinput * S ((S (pfd_index_division_bound_source)) * ac) + (pfd_input_division_bound_sourcestep))) /\\ (((exists pfc_terms_code_division_bound_sourcestepprevious pfc_terms_scale_division_bound_sourcestepprevious pfc_natural_sum_division_bound_sourcestepprevious. ((forall pfc_index_division_bound_sourcesteppreviousdiagonal. (exists pfa_gap_division_bound_sourcesteppreviousdiagonalbound. pfa_gap_division_bound_sourcesteppreviousdiagonalbound + S (pfc_index_division_bound_sourcesteppreviousdiagonal) = (S (pfd_index_division_bound_source))) -> exists pfc_value_division_bound_sourcesteppreviousdiagonal. ((((exists ff_h_pfp_division_bound_sourcesteppreviousdiagonalentry. ff_h_pfp_division_bound_sourcesteppreviousdiagonalentry + S (pfc_value_division_bound_sourcesteppreviousdiagonal) = S ((S (pfc_index_division_bound_sourcesteppreviousdiagonal)) * pfc_terms_scale_division_bound_sourcestepprevious)) /\\ exists ff_q_pfp_division_bound_sourcesteppreviousdiagonalentry. pfc_terms_code_division_bound_sourcestepprevious = ff_q_pfp_division_bound_sourcesteppreviousdiagonalentry * S ((S (pfc_index_division_bound_sourcesteppreviousdiagonal)) * pfc_terms_scale_division_bound_sourcestepprevious) + (pfc_value_division_bound_sourcesteppreviousdiagonal))) /\\ ((exists pfc_complement_division_bound_sourcesteppreviousdiagonalterm pfc_left_division_bound_sourcesteppreviousdiagonalterm pfc_right_division_bound_sourcesteppreviousdiagonalterm. (((pfc_index_division_bound_sourcesteppreviousdiagonal)+pfc_complement_division_bound_sourcesteppreviousdiagonalterm=(pfd_index_division_bound_source)) /\\ ((((((exists pfa_gap_division_bound_sourcesteppreviousdiagonaltermleftinside. pfa_gap_division_bound_sourcesteppreviousdiagonaltermleftinside + S (pfc_index_division_bound_sourcesteppreviousdiagonal) = (pfd_index_division_bound_source)) /\\ ((((exists ff_h_pfp_division_bound_sourcesteppreviousdiagonaltermleftentry. ff_h_pfp_division_bound_sourcesteppreviousdiagonaltermleftentry + S (pfc_left_division_bound_sourcesteppreviousdiagonalterm) = S ((S (pfc_index_division_bound_sourcesteppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_division_bound_sourcesteppreviousdiagonaltermleftentry. qb = ff_q_pfp_division_bound_sourcesteppreviousdiagonaltermleftentry * S ((S (pfc_index_division_bound_sourcesteppreviousdiagonal)) * qc) + (pfc_left_division_bound_sourcesteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_bound_sourcesteppreviousdiagonaltermleftoutside. pfc_gap_division_bound_sourcesteppreviousdiagonaltermleftoutside+(pfd_index_division_bound_source)=(pfc_index_division_bound_sourcesteppreviousdiagonal)) /\\ (((pfc_left_division_bound_sourcesteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_bound_sourcesteppreviousdiagonaltermrightinside. pfa_gap_division_bound_sourcesteppreviousdiagonaltermrightinside + S (pfc_complement_division_bound_sourcesteppreviousdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_division_bound_sourcesteppreviousdiagonaltermrightentry. ff_h_pfp_division_bound_sourcesteppreviousdiagonaltermrightentry + S (pfc_right_division_bound_sourcesteppreviousdiagonalterm) = S ((S (pfc_complement_division_bound_sourcesteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_bound_sourcesteppreviousdiagonaltermrightentry. bb = ff_q_pfp_division_bound_sourcesteppreviousdiagonaltermrightentry * S ((S (pfc_complement_division_bound_sourcesteppreviousdiagonalterm)) * bc) + (pfc_right_division_bound_sourcesteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_bound_sourcesteppreviousdiagonaltermrightoutside. pfc_gap_division_bound_sourcesteppreviousdiagonaltermrightoutside+(M)=(pfc_complement_division_bound_sourcesteppreviousdiagonalterm)) /\\ (((pfc_right_division_bound_sourcesteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_division_bound_sourcesteppreviousdiagonal)=pfc_left_division_bound_sourcesteppreviousdiagonalterm*pfc_right_division_bound_sourcesteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_bound_sourcestepprevioussum fs_v_pfc_division_bound_sourcestepprevioussum. ((((exists fs_h_pfc_division_bound_sourcestepprevioussum_body_start. fs_h_pfc_division_bound_sourcestepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_bound_sourcestepprevioussum)) /\\ exists fs_q_pfc_division_bound_sourcestepprevioussum_body_start. fs_u_pfc_division_bound_sourcestepprevioussum = fs_q_pfc_division_bound_sourcestepprevioussum_body_start * S ((S (0)) * fs_v_pfc_division_bound_sourcestepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_division_bound_sourcestepprevioussum_body_terminal. fs_h_pfc_division_bound_sourcestepprevioussum_body_terminal + S (pfc_natural_sum_division_bound_sourcestepprevious) = S ((S (S (pfd_index_division_bound_source))) * fs_v_pfc_division_bound_sourcestepprevioussum)) /\\ exists fs_q_pfc_division_bound_sourcestepprevioussum_body_terminal. fs_u_pfc_division_bound_sourcestepprevioussum = fs_q_pfc_division_bound_sourcestepprevioussum_body_terminal * S ((S (S (pfd_index_division_bound_source))) * fs_v_pfc_division_bound_sourcestepprevioussum) + (pfc_natural_sum_division_bound_sourcestepprevious))) /\\ forall fs_i_pfc_division_bound_sourcestepprevioussum_body_steps. (exists fs_lt_pfc_division_bound_sourcestepprevioussum_body_steps_bound. fs_lt_pfc_division_bound_sourcestepprevioussum_body_steps_bound + S fs_i_pfc_division_bound_sourcestepprevioussum_body_steps = S (pfd_index_division_bound_source)) -> exists fs_a_pfc_division_bound_sourcestepprevioussum_body_steps fs_r_pfc_division_bound_sourcestepprevioussum_body_steps fs_s_pfc_division_bound_sourcestepprevioussum_body_steps. ((((exists fs_h_pfc_division_bound_sourcestepprevioussum_body_steps_summand. fs_h_pfc_division_bound_sourcestepprevioussum_body_steps_summand + S (fs_a_pfc_division_bound_sourcestepprevioussum_body_steps) = S ((S (fs_i_pfc_division_bound_sourcestepprevioussum_body_steps)) * pfc_terms_scale_division_bound_sourcestepprevious)) /\\ exists fs_q_pfc_division_bound_sourcestepprevioussum_body_steps_summand. pfc_terms_code_division_bound_sourcestepprevious = fs_q_pfc_division_bound_sourcestepprevioussum_body_steps_summand * S ((S (fs_i_pfc_division_bound_sourcestepprevioussum_body_steps)) * pfc_terms_scale_division_bound_sourcestepprevious) + (fs_a_pfc_division_bound_sourcestepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_bound_sourcestepprevioussum_body_steps_partial. fs_h_pfc_division_bound_sourcestepprevioussum_body_steps_partial + S (fs_r_pfc_division_bound_sourcestepprevioussum_body_steps) = S ((S (fs_i_pfc_division_bound_sourcestepprevioussum_body_steps)) * fs_v_pfc_division_bound_sourcestepprevioussum)) /\\ exists fs_q_pfc_division_bound_sourcestepprevioussum_body_steps_partial. fs_u_pfc_division_bound_sourcestepprevioussum = fs_q_pfc_division_bound_sourcestepprevioussum_body_steps_partial * S ((S (fs_i_pfc_division_bound_sourcestepprevioussum_body_steps)) * fs_v_pfc_division_bound_sourcestepprevioussum) + (fs_r_pfc_division_bound_sourcestepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_bound_sourcestepprevioussum_body_steps_successor. fs_h_pfc_division_bound_sourcestepprevioussum_body_steps_successor + S (fs_s_pfc_division_bound_sourcestepprevioussum_body_steps) = S ((S (S fs_i_pfc_division_bound_sourcestepprevioussum_body_steps)) * fs_v_pfc_division_bound_sourcestepprevioussum)) /\\ exists fs_q_pfc_division_bound_sourcestepprevioussum_body_steps_successor. fs_u_pfc_division_bound_sourcestepprevioussum = fs_q_pfc_division_bound_sourcestepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_division_bound_sourcestepprevioussum_body_steps)) * fs_v_pfc_division_bound_sourcestepprevioussum) + (fs_s_pfc_division_bound_sourcestepprevioussum_body_steps))) /\\ fs_s_pfc_division_bound_sourcestepprevioussum_body_steps = fs_r_pfc_division_bound_sourcestepprevioussum_body_steps + fs_a_pfc_division_bound_sourcestepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_division_bound_sourcesteppreviousresiduebound. pfa_gap_division_bound_sourcesteppreviousresiduebound + S (pfd_previous_division_bound_sourcestep) = (p)) /\\ ((exists pfa_offset_left_division_bound_sourcesteppreviousresiduecongruence pfa_offset_right_division_bound_sourcesteppreviousresiduecongruence. (pfc_natural_sum_division_bound_sourcestepprevious) + (p) * pfa_offset_left_division_bound_sourcesteppreviousresiduecongruence = (pfd_previous_division_bound_sourcestep) + (p) * pfa_offset_right_division_bound_sourcesteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_division_bound_sourcestepsubtractleft. pfa_gap_division_bound_sourcestepsubtractleft + S (pfd_previous_division_bound_sourcestep) = (p)) /\\ (((exists pfa_gap_division_bound_sourcestepsubtractright. pfa_gap_division_bound_sourcestepsubtractright + S (pfd_difference_division_bound_sourcestep) = (p)) /\\ ((((exists pfa_gap_division_bound_sourcestepsubtractresultbound. pfa_gap_division_bound_sourcestepsubtractresultbound + S (pfd_input_division_bound_sourcestep) = (p)) /\\ ((exists pfa_offset_left_division_bound_sourcestepsubtractresultcongruence pfa_offset_right_division_bound_sourcestepsubtractresultcongruence. ((pfd_previous_division_bound_sourcestep) + (pfd_difference_division_bound_sourcestep)) + (p) * pfa_offset_left_division_bound_sourcestepsubtractresultcongruence = (pfd_input_division_bound_sourcestep) + (p) * pfa_offset_right_division_bound_sourcestepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_division_bound_sourcestepmultiplyleft. pfa_gap_division_bound_sourcestepmultiplyleft + S (k) = (p)) /\\ (((exists pfa_gap_division_bound_sourcestepmultiplyright. pfa_gap_division_bound_sourcestepmultiplyright + S (pfd_difference_division_bound_sourcestep) = (p)) /\\ ((((exists pfa_gap_division_bound_sourcestepmultiplyresultbound. pfa_gap_division_bound_sourcestepmultiplyresultbound + S (pfd_value_division_bound_source) = (p)) /\\ ((exists pfa_offset_left_division_bound_sourcestepmultiplyresultcongruence pfa_offset_right_division_bound_sourcestepmultiplyresultcongruence. ((k) * (pfd_difference_division_bound_sourcestep)) + (p) * pfa_offset_left_division_bound_sourcestepmultiplyresultcongruence = (pfd_value_division_bound_source) + (p) * pfa_offset_right_division_bound_sourcestepmultiplyresultcongruence))))))))))))))))))) -> (forall fom_index_pfp_division_bound_result. (exists fom_gap_pfp_division_bound_result_index_bound. fom_gap_pfp_division_bound_result_index_bound + S (fom_index_pfp_division_bound_result) = N) -> exists fom_value_pfp_division_bound_result. 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((((exists ff_h_pfp_division_append_old_entry. ff_h_pfp_division_append_old_entry + S (r) = S ((S (i)) * qc)) /\\ exists ff_q_pfp_division_append_old_entry. qb = ff_q_pfp_division_append_old_entry * S ((S (i)) * qc) + (r))) /\\ ((exists pfd_input_division_append_old_step pfd_previous_division_append_old_step pfd_difference_division_append_old_step. ((((exists ff_h_pfp_division_append_old_stepinput. ff_h_pfp_division_append_old_stepinput + S (pfd_input_division_append_old_step) = S ((S (i)) * ac)) /\\ exists ff_q_pfp_division_append_old_stepinput. ab = ff_q_pfp_division_append_old_stepinput * S ((S (i)) * ac) + (pfd_input_division_append_old_step))) /\\ (((exists pfc_terms_code_division_append_old_stepprevious pfc_terms_scale_division_append_old_stepprevious pfc_natural_sum_division_append_old_stepprevious. ((forall pfc_index_division_append_old_steppreviousdiagonal. (exists pfa_gap_division_append_old_steppreviousdiagonalbound. pfa_gap_division_append_old_steppreviousdiagonalbound + S (pfc_index_division_append_old_steppreviousdiagonal) = (S (i))) -> exists pfc_value_division_append_old_steppreviousdiagonal. ((((exists ff_h_pfp_division_append_old_steppreviousdiagonalentry. ff_h_pfp_division_append_old_steppreviousdiagonalentry + S (pfc_value_division_append_old_steppreviousdiagonal) = S ((S (pfc_index_division_append_old_steppreviousdiagonal)) * pfc_terms_scale_division_append_old_stepprevious)) /\\ exists ff_q_pfp_division_append_old_steppreviousdiagonalentry. pfc_terms_code_division_append_old_stepprevious = ff_q_pfp_division_append_old_steppreviousdiagonalentry * S ((S (pfc_index_division_append_old_steppreviousdiagonal)) * pfc_terms_scale_division_append_old_stepprevious) + (pfc_value_division_append_old_steppreviousdiagonal))) /\\ ((exists pfc_complement_division_append_old_steppreviousdiagonalterm pfc_left_division_append_old_steppreviousdiagonalterm pfc_right_division_append_old_steppreviousdiagonalterm. (((pfc_index_division_append_old_steppreviousdiagonal)+pfc_complement_division_append_old_steppreviousdiagonalterm=(i)) /\\ ((((((exists pfa_gap_division_append_old_steppreviousdiagonaltermleftinside. pfa_gap_division_append_old_steppreviousdiagonaltermleftinside + S (pfc_index_division_append_old_steppreviousdiagonal) = (i)) /\\ ((((exists ff_h_pfp_division_append_old_steppreviousdiagonaltermleftentry. ff_h_pfp_division_append_old_steppreviousdiagonaltermleftentry + S (pfc_left_division_append_old_steppreviousdiagonalterm) = S ((S (pfc_index_division_append_old_steppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_division_append_old_steppreviousdiagonaltermleftentry. qb = ff_q_pfp_division_append_old_steppreviousdiagonaltermleftentry * S ((S (pfc_index_division_append_old_steppreviousdiagonal)) * qc) + (pfc_left_division_append_old_steppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_append_old_steppreviousdiagonaltermleftoutside. pfc_gap_division_append_old_steppreviousdiagonaltermleftoutside+(i)=(pfc_index_division_append_old_steppreviousdiagonal)) /\\ (((pfc_left_division_append_old_steppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_append_old_steppreviousdiagonaltermrightinside. pfa_gap_division_append_old_steppreviousdiagonaltermrightinside + S (pfc_complement_division_append_old_steppreviousdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_division_append_old_steppreviousdiagonaltermrightentry. ff_h_pfp_division_append_old_steppreviousdiagonaltermrightentry + S (pfc_right_division_append_old_steppreviousdiagonalterm) = S ((S (pfc_complement_division_append_old_steppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_append_old_steppreviousdiagonaltermrightentry. bb = ff_q_pfp_division_append_old_steppreviousdiagonaltermrightentry * S ((S (pfc_complement_division_append_old_steppreviousdiagonalterm)) * bc) + (pfc_right_division_append_old_steppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_append_old_steppreviousdiagonaltermrightoutside. pfc_gap_division_append_old_steppreviousdiagonaltermrightoutside+(M)=(pfc_complement_division_append_old_steppreviousdiagonalterm)) /\\ (((pfc_right_division_append_old_steppreviousdiagonalterm)=0))))) /\\ (((pfc_value_division_append_old_steppreviousdiagonal)=pfc_left_division_append_old_steppreviousdiagonalterm*pfc_right_division_append_old_steppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_append_old_stepprevioussum fs_v_pfc_division_append_old_stepprevioussum. ((((exists fs_h_pfc_division_append_old_stepprevioussum_body_start. fs_h_pfc_division_append_old_stepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_append_old_stepprevioussum)) /\\ exists fs_q_pfc_division_append_old_stepprevioussum_body_start. fs_u_pfc_division_append_old_stepprevioussum = fs_q_pfc_division_append_old_stepprevioussum_body_start * S ((S (0)) * fs_v_pfc_division_append_old_stepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_division_append_old_stepprevioussum_body_terminal. fs_h_pfc_division_append_old_stepprevioussum_body_terminal + S (pfc_natural_sum_division_append_old_stepprevious) = S ((S (S (i))) * fs_v_pfc_division_append_old_stepprevioussum)) /\\ exists fs_q_pfc_division_append_old_stepprevioussum_body_terminal. fs_u_pfc_division_append_old_stepprevioussum = fs_q_pfc_division_append_old_stepprevioussum_body_terminal * S ((S (S (i))) * fs_v_pfc_division_append_old_stepprevioussum) + (pfc_natural_sum_division_append_old_stepprevious))) /\\ forall fs_i_pfc_division_append_old_stepprevioussum_body_steps. (exists fs_lt_pfc_division_append_old_stepprevioussum_body_steps_bound. fs_lt_pfc_division_append_old_stepprevioussum_body_steps_bound + S fs_i_pfc_division_append_old_stepprevioussum_body_steps = S (i)) -> exists fs_a_pfc_division_append_old_stepprevioussum_body_steps fs_r_pfc_division_append_old_stepprevioussum_body_steps fs_s_pfc_division_append_old_stepprevioussum_body_steps. ((((exists fs_h_pfc_division_append_old_stepprevioussum_body_steps_summand. fs_h_pfc_division_append_old_stepprevioussum_body_steps_summand + S (fs_a_pfc_division_append_old_stepprevioussum_body_steps) = S ((S (fs_i_pfc_division_append_old_stepprevioussum_body_steps)) * pfc_terms_scale_division_append_old_stepprevious)) /\\ exists fs_q_pfc_division_append_old_stepprevioussum_body_steps_summand. pfc_terms_code_division_append_old_stepprevious = fs_q_pfc_division_append_old_stepprevioussum_body_steps_summand * S ((S (fs_i_pfc_division_append_old_stepprevioussum_body_steps)) * pfc_terms_scale_division_append_old_stepprevious) + (fs_a_pfc_division_append_old_stepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_append_old_stepprevioussum_body_steps_partial. fs_h_pfc_division_append_old_stepprevioussum_body_steps_partial + S (fs_r_pfc_division_append_old_stepprevioussum_body_steps) = S ((S (fs_i_pfc_division_append_old_stepprevioussum_body_steps)) * fs_v_pfc_division_append_old_stepprevioussum)) /\\ exists fs_q_pfc_division_append_old_stepprevioussum_body_steps_partial. fs_u_pfc_division_append_old_stepprevioussum = fs_q_pfc_division_append_old_stepprevioussum_body_steps_partial * S ((S (fs_i_pfc_division_append_old_stepprevioussum_body_steps)) * fs_v_pfc_division_append_old_stepprevioussum) + (fs_r_pfc_division_append_old_stepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_append_old_stepprevioussum_body_steps_successor. fs_h_pfc_division_append_old_stepprevioussum_body_steps_successor + S (fs_s_pfc_division_append_old_stepprevioussum_body_steps) = S ((S (S fs_i_pfc_division_append_old_stepprevioussum_body_steps)) * fs_v_pfc_division_append_old_stepprevioussum)) /\\ exists fs_q_pfc_division_append_old_stepprevioussum_body_steps_successor. fs_u_pfc_division_append_old_stepprevioussum = fs_q_pfc_division_append_old_stepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_division_append_old_stepprevioussum_body_steps)) * fs_v_pfc_division_append_old_stepprevioussum) + (fs_s_pfc_division_append_old_stepprevioussum_body_steps))) /\\ fs_s_pfc_division_append_old_stepprevioussum_body_steps = fs_r_pfc_division_append_old_stepprevioussum_body_steps + fs_a_pfc_division_append_old_stepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_division_append_old_steppreviousresiduebound. pfa_gap_division_append_old_steppreviousresiduebound + S (pfd_previous_division_append_old_step) = (p)) /\\ ((exists pfa_offset_left_division_append_old_steppreviousresiduecongruence pfa_offset_right_division_append_old_steppreviousresiduecongruence. (pfc_natural_sum_division_append_old_stepprevious) + (p) * pfa_offset_left_division_append_old_steppreviousresiduecongruence = (pfd_previous_division_append_old_step) + (p) * pfa_offset_right_division_append_old_steppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_division_append_old_stepsubtractleft. pfa_gap_division_append_old_stepsubtractleft + S (pfd_previous_division_append_old_step) = (p)) /\\ (((exists pfa_gap_division_append_old_stepsubtractright. pfa_gap_division_append_old_stepsubtractright + S (pfd_difference_division_append_old_step) = (p)) /\\ ((((exists pfa_gap_division_append_old_stepsubtractresultbound. pfa_gap_division_append_old_stepsubtractresultbound + S (pfd_input_division_append_old_step) = (p)) /\\ ((exists pfa_offset_left_division_append_old_stepsubtractresultcongruence pfa_offset_right_division_append_old_stepsubtractresultcongruence. ((pfd_previous_division_append_old_step) + (pfd_difference_division_append_old_step)) + (p) * pfa_offset_left_division_append_old_stepsubtractresultcongruence = (pfd_input_division_append_old_step) + (p) * pfa_offset_right_division_append_old_stepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_division_append_old_stepmultiplyleft. pfa_gap_division_append_old_stepmultiplyleft + S (k) = (p)) /\\ (((exists pfa_gap_division_append_old_stepmultiplyright. pfa_gap_division_append_old_stepmultiplyright + S (pfd_difference_division_append_old_step) = (p)) /\\ ((((exists pfa_gap_division_append_old_stepmultiplyresultbound. pfa_gap_division_append_old_stepmultiplyresultbound + S (r) = (p)) /\\ ((exists pfa_offset_left_division_append_old_stepmultiplyresultcongruence pfa_offset_right_division_append_old_stepmultiplyresultcongruence. ((k) * (pfd_difference_division_append_old_step)) + (p) * pfa_offset_left_division_append_old_stepmultiplyresultcongruence = (r) + (p) * pfa_offset_right_division_append_old_stepmultiplyresultcongruence))))))))))))))))))", "specialize h (i)", "apply h", "exact hcase_right", "cases hv", "cases hv_witness", "exists x", "split", "specialize he (i)", "specialize he (x)", "apply he", "exact hcase_right", "exact hv_witness_left", "specialize prime_field_polynomial_quotient_step_recode (p)", "specialize prime_field_polynomial_quotient_step_recode (k)", "specialize prime_field_polynomial_quotient_step_recode (ab)", "specialize prime_field_polynomial_quotient_step_recode (ac)", "specialize prime_field_polynomial_quotient_step_recode (bb)", "specialize prime_field_polynomial_quotient_step_recode (bc)", "specialize prime_field_polynomial_quotient_step_recode (M)", "specialize prime_field_polynomial_quotient_step_recode (qb)", "specialize prime_field_polynomial_quotient_step_recode (qc)", "specialize prime_field_polynomial_quotient_step_recode (QB)", "specialize prime_field_polynomial_quotient_step_recode (QC)", "specialize prime_field_polynomial_quotient_step_recode (i)", "specialize prime_field_polynomial_quotient_step_recode (x)", "apply prime_field_polynomial_quotient_step_recode", "intro j", "intro a", "intro hj", "intro ha", "specialize he (j)", "specialize he (a)", "apply he", "specialize lt_of_lt_of_le (j)", "specialize lt_of_lt_of_le (S i)", "specialize lt_of_lt_of_le (N)", "apply lt_of_lt_of_le", "specialize le_succ (S j)", "specialize le_succ (i)", "apply le_succ", "exact hj", "exact hcase_right", "exact ha", "exact hv_witness_right" ], "script_sha256": "b860802b9eb320f604ed85251e730862623a87797c90752317fa97e06747fef7", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_division_candidate.py", "sha256": "edfc7806caf7a83b9cb0e3e420bd2c3a8679f2d4d9ee6ca9f8eae53faca8d5b2" }, "statement": "forall p k ab ac bb bc M qb qc QB QC N q. (forall pfd_index_division_append_old. (exists pfa_gap_division_append_oldbound. pfa_gap_division_append_oldbound + S (pfd_index_division_append_old) = (N)) -> exists pfd_value_division_append_old. ((((exists ff_h_pfp_division_append_oldentry. ff_h_pfp_division_append_oldentry + S (pfd_value_division_append_old) = S ((S (pfd_index_division_append_old)) * qc)) /\\ exists ff_q_pfp_division_append_oldentry. qb = ff_q_pfp_division_append_oldentry * S ((S (pfd_index_division_append_old)) * qc) + (pfd_value_division_append_old))) /\\ ((exists pfd_input_division_append_oldstep pfd_previous_division_append_oldstep pfd_difference_division_append_oldstep. ((((exists ff_h_pfp_division_append_oldstepinput. ff_h_pfp_division_append_oldstepinput + S (pfd_input_division_append_oldstep) = S ((S (pfd_index_division_append_old)) * ac)) /\\ exists ff_q_pfp_division_append_oldstepinput. ab = ff_q_pfp_division_append_oldstepinput * S ((S (pfd_index_division_append_old)) * ac) + (pfd_input_division_append_oldstep))) /\\ (((exists pfc_terms_code_division_append_oldstepprevious pfc_terms_scale_division_append_oldstepprevious pfc_natural_sum_division_append_oldstepprevious. ((forall pfc_index_division_append_oldsteppreviousdiagonal. (exists pfa_gap_division_append_oldsteppreviousdiagonalbound. pfa_gap_division_append_oldsteppreviousdiagonalbound + S (pfc_index_division_append_oldsteppreviousdiagonal) = (S (pfd_index_division_append_old))) -> exists pfc_value_division_append_oldsteppreviousdiagonal. ((((exists ff_h_pfp_division_append_oldsteppreviousdiagonalentry. ff_h_pfp_division_append_oldsteppreviousdiagonalentry + S (pfc_value_division_append_oldsteppreviousdiagonal) = S ((S (pfc_index_division_append_oldsteppreviousdiagonal)) * pfc_terms_scale_division_append_oldstepprevious)) /\\ exists ff_q_pfp_division_append_oldsteppreviousdiagonalentry. pfc_terms_code_division_append_oldstepprevious = ff_q_pfp_division_append_oldsteppreviousdiagonalentry * S ((S (pfc_index_division_append_oldsteppreviousdiagonal)) * pfc_terms_scale_division_append_oldstepprevious) + (pfc_value_division_append_oldsteppreviousdiagonal))) /\\ ((exists pfc_complement_division_append_oldsteppreviousdiagonalterm pfc_left_division_append_oldsteppreviousdiagonalterm pfc_right_division_append_oldsteppreviousdiagonalterm. (((pfc_index_division_append_oldsteppreviousdiagonal)+pfc_complement_division_append_oldsteppreviousdiagonalterm=(pfd_index_division_append_old)) /\\ ((((((exists pfa_gap_division_append_oldsteppreviousdiagonaltermleftinside. pfa_gap_division_append_oldsteppreviousdiagonaltermleftinside + S (pfc_index_division_append_oldsteppreviousdiagonal) = (pfd_index_division_append_old)) /\\ ((((exists ff_h_pfp_division_append_oldsteppreviousdiagonaltermleftentry. ff_h_pfp_division_append_oldsteppreviousdiagonaltermleftentry + S (pfc_left_division_append_oldsteppreviousdiagonalterm) = S ((S (pfc_index_division_append_oldsteppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_division_append_oldsteppreviousdiagonaltermleftentry. qb = ff_q_pfp_division_append_oldsteppreviousdiagonaltermleftentry * S ((S (pfc_index_division_append_oldsteppreviousdiagonal)) * qc) + (pfc_left_division_append_oldsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_append_oldsteppreviousdiagonaltermleftoutside. pfc_gap_division_append_oldsteppreviousdiagonaltermleftoutside+(pfd_index_division_append_old)=(pfc_index_division_append_oldsteppreviousdiagonal)) /\\ (((pfc_left_division_append_oldsteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_append_oldsteppreviousdiagonaltermrightinside. pfa_gap_division_append_oldsteppreviousdiagonaltermrightinside + S (pfc_complement_division_append_oldsteppreviousdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_division_append_oldsteppreviousdiagonaltermrightentry. ff_h_pfp_division_append_oldsteppreviousdiagonaltermrightentry + S (pfc_right_division_append_oldsteppreviousdiagonalterm) = S ((S (pfc_complement_division_append_oldsteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_append_oldsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_division_append_oldsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_division_append_oldsteppreviousdiagonalterm)) * bc) + (pfc_right_division_append_oldsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_append_oldsteppreviousdiagonaltermrightoutside. pfc_gap_division_append_oldsteppreviousdiagonaltermrightoutside+(M)=(pfc_complement_division_append_oldsteppreviousdiagonalterm)) /\\ (((pfc_right_division_append_oldsteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_division_append_oldsteppreviousdiagonal)=pfc_left_division_append_oldsteppreviousdiagonalterm*pfc_right_division_append_oldsteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_append_oldstepprevioussum fs_v_pfc_division_append_oldstepprevioussum. ((((exists fs_h_pfc_division_append_oldstepprevioussum_body_start. fs_h_pfc_division_append_oldstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_append_oldstepprevioussum)) /\\ exists fs_q_pfc_division_append_oldstepprevioussum_body_start. fs_u_pfc_division_append_oldstepprevioussum = fs_q_pfc_division_append_oldstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_division_append_oldstepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_division_append_oldstepprevioussum_body_terminal. fs_h_pfc_division_append_oldstepprevioussum_body_terminal + S (pfc_natural_sum_division_append_oldstepprevious) = S ((S (S (pfd_index_division_append_old))) * fs_v_pfc_division_append_oldstepprevioussum)) /\\ exists fs_q_pfc_division_append_oldstepprevioussum_body_terminal. fs_u_pfc_division_append_oldstepprevioussum = fs_q_pfc_division_append_oldstepprevioussum_body_terminal * S ((S (S (pfd_index_division_append_old))) * fs_v_pfc_division_append_oldstepprevioussum) + (pfc_natural_sum_division_append_oldstepprevious))) /\\ forall fs_i_pfc_division_append_oldstepprevioussum_body_steps. (exists fs_lt_pfc_division_append_oldstepprevioussum_body_steps_bound. fs_lt_pfc_division_append_oldstepprevioussum_body_steps_bound + S fs_i_pfc_division_append_oldstepprevioussum_body_steps = S (pfd_index_division_append_old)) -> exists fs_a_pfc_division_append_oldstepprevioussum_body_steps fs_r_pfc_division_append_oldstepprevioussum_body_steps fs_s_pfc_division_append_oldstepprevioussum_body_steps. ((((exists fs_h_pfc_division_append_oldstepprevioussum_body_steps_summand. fs_h_pfc_division_append_oldstepprevioussum_body_steps_summand + S (fs_a_pfc_division_append_oldstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_append_oldstepprevioussum_body_steps)) * pfc_terms_scale_division_append_oldstepprevious)) /\\ exists fs_q_pfc_division_append_oldstepprevioussum_body_steps_summand. pfc_terms_code_division_append_oldstepprevious = fs_q_pfc_division_append_oldstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_division_append_oldstepprevioussum_body_steps)) * pfc_terms_scale_division_append_oldstepprevious) + (fs_a_pfc_division_append_oldstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_append_oldstepprevioussum_body_steps_partial. fs_h_pfc_division_append_oldstepprevioussum_body_steps_partial + S (fs_r_pfc_division_append_oldstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_append_oldstepprevioussum_body_steps)) * fs_v_pfc_division_append_oldstepprevioussum)) /\\ exists fs_q_pfc_division_append_oldstepprevioussum_body_steps_partial. fs_u_pfc_division_append_oldstepprevioussum = fs_q_pfc_division_append_oldstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_division_append_oldstepprevioussum_body_steps)) * fs_v_pfc_division_append_oldstepprevioussum) + (fs_r_pfc_division_append_oldstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_append_oldstepprevioussum_body_steps_successor. fs_h_pfc_division_append_oldstepprevioussum_body_steps_successor + S (fs_s_pfc_division_append_oldstepprevioussum_body_steps) = S ((S (S fs_i_pfc_division_append_oldstepprevioussum_body_steps)) * fs_v_pfc_division_append_oldstepprevioussum)) /\\ exists fs_q_pfc_division_append_oldstepprevioussum_body_steps_successor. fs_u_pfc_division_append_oldstepprevioussum = fs_q_pfc_division_append_oldstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_division_append_oldstepprevioussum_body_steps)) * fs_v_pfc_division_append_oldstepprevioussum) + (fs_s_pfc_division_append_oldstepprevioussum_body_steps))) /\\ fs_s_pfc_division_append_oldstepprevioussum_body_steps = fs_r_pfc_division_append_oldstepprevioussum_body_steps + fs_a_pfc_division_append_oldstepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_division_append_oldsteppreviousresiduebound. pfa_gap_division_append_oldsteppreviousresiduebound + S (pfd_previous_division_append_oldstep) = (p)) /\\ ((exists pfa_offset_left_division_append_oldsteppreviousresiduecongruence pfa_offset_right_division_append_oldsteppreviousresiduecongruence. (pfc_natural_sum_division_append_oldstepprevious) + (p) * pfa_offset_left_division_append_oldsteppreviousresiduecongruence = (pfd_previous_division_append_oldstep) + (p) * pfa_offset_right_division_append_oldsteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_division_append_oldstepsubtractleft. pfa_gap_division_append_oldstepsubtractleft + S (pfd_previous_division_append_oldstep) = (p)) /\\ (((exists pfa_gap_division_append_oldstepsubtractright. pfa_gap_division_append_oldstepsubtractright + S (pfd_difference_division_append_oldstep) = (p)) /\\ ((((exists pfa_gap_division_append_oldstepsubtractresultbound. pfa_gap_division_append_oldstepsubtractresultbound + S (pfd_input_division_append_oldstep) = (p)) /\\ ((exists pfa_offset_left_division_append_oldstepsubtractresultcongruence pfa_offset_right_division_append_oldstepsubtractresultcongruence. ((pfd_previous_division_append_oldstep) + (pfd_difference_division_append_oldstep)) + (p) * pfa_offset_left_division_append_oldstepsubtractresultcongruence = (pfd_input_division_append_oldstep) + (p) * pfa_offset_right_division_append_oldstepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_division_append_oldstepmultiplyleft. pfa_gap_division_append_oldstepmultiplyleft + S (k) = (p)) /\\ (((exists pfa_gap_division_append_oldstepmultiplyright. pfa_gap_division_append_oldstepmultiplyright + S (pfd_difference_division_append_oldstep) = (p)) /\\ ((((exists pfa_gap_division_append_oldstepmultiplyresultbound. pfa_gap_division_append_oldstepmultiplyresultbound + S (pfd_value_division_append_old) = (p)) /\\ ((exists pfa_offset_left_division_append_oldstepmultiplyresultcongruence pfa_offset_right_division_append_oldstepmultiplyresultcongruence. ((k) * (pfd_difference_division_append_oldstep)) + (p) * pfa_offset_left_division_append_oldstepmultiplyresultcongruence = (pfd_value_division_append_old) + (p) * pfa_offset_right_division_append_oldstepmultiplyresultcongruence))))))))))))))))))) -> (forall mdr_i_pfp_division_append_equal mdr_a_pfp_division_append_equal. (exists mdr_gap_pfp_division_append_equalb. mdr_gap_pfp_division_append_equalb + S (mdr_i_pfp_division_append_equal) = (N)) -> (((exists ff_h_mdr_pfp_division_append_equalo. ff_h_mdr_pfp_division_append_equalo + S (mdr_a_pfp_division_append_equal) = S ((S (mdr_i_pfp_division_append_equal)) * qc)) /\\ exists ff_q_mdr_pfp_division_append_equalo. qb = ff_q_mdr_pfp_division_append_equalo * S ((S (mdr_i_pfp_division_append_equal)) * qc) + (mdr_a_pfp_division_append_equal))) -> (((exists ff_h_mdr_pfp_division_append_equaln. ff_h_mdr_pfp_division_append_equaln + S (mdr_a_pfp_division_append_equal) = S ((S (mdr_i_pfp_division_append_equal)) * QC)) /\\ exists ff_q_mdr_pfp_division_append_equaln. QB = ff_q_mdr_pfp_division_append_equaln * S ((S (mdr_i_pfp_division_append_equal)) * QC) + (mdr_a_pfp_division_append_equal)))) -> (((exists ff_h_pfp_division_append_given_entry. ff_h_pfp_division_append_given_entry + S (q) = S ((S (N)) * QC)) /\\ exists ff_q_pfp_division_append_given_entry. QB = ff_q_pfp_division_append_given_entry * S ((S (N)) * QC) + (q))) -> (exists pfd_input_division_append_given_step pfd_previous_division_append_given_step pfd_difference_division_append_given_step. ((((exists ff_h_pfp_division_append_given_stepinput. ff_h_pfp_division_append_given_stepinput + S (pfd_input_division_append_given_step) = S ((S (N)) * ac)) /\\ exists ff_q_pfp_division_append_given_stepinput. ab = ff_q_pfp_division_append_given_stepinput * S ((S (N)) * ac) + (pfd_input_division_append_given_step))) /\\ (((exists pfc_terms_code_division_append_given_stepprevious pfc_terms_scale_division_append_given_stepprevious pfc_natural_sum_division_append_given_stepprevious. ((forall pfc_index_division_append_given_steppreviousdiagonal. (exists pfa_gap_division_append_given_steppreviousdiagonalbound. pfa_gap_division_append_given_steppreviousdiagonalbound + S (pfc_index_division_append_given_steppreviousdiagonal) = (S (N))) -> exists pfc_value_division_append_given_steppreviousdiagonal. ((((exists ff_h_pfp_division_append_given_steppreviousdiagonalentry. ff_h_pfp_division_append_given_steppreviousdiagonalentry + S (pfc_value_division_append_given_steppreviousdiagonal) = S ((S (pfc_index_division_append_given_steppreviousdiagonal)) * pfc_terms_scale_division_append_given_stepprevious)) /\\ exists ff_q_pfp_division_append_given_steppreviousdiagonalentry. pfc_terms_code_division_append_given_stepprevious = ff_q_pfp_division_append_given_steppreviousdiagonalentry * S ((S (pfc_index_division_append_given_steppreviousdiagonal)) * pfc_terms_scale_division_append_given_stepprevious) + (pfc_value_division_append_given_steppreviousdiagonal))) /\\ ((exists pfc_complement_division_append_given_steppreviousdiagonalterm pfc_left_division_append_given_steppreviousdiagonalterm pfc_right_division_append_given_steppreviousdiagonalterm. (((pfc_index_division_append_given_steppreviousdiagonal)+pfc_complement_division_append_given_steppreviousdiagonalterm=(N)) /\\ ((((((exists pfa_gap_division_append_given_steppreviousdiagonaltermleftinside. pfa_gap_division_append_given_steppreviousdiagonaltermleftinside + S (pfc_index_division_append_given_steppreviousdiagonal) = (N)) /\\ ((((exists ff_h_pfp_division_append_given_steppreviousdiagonaltermleftentry. ff_h_pfp_division_append_given_steppreviousdiagonaltermleftentry + S (pfc_left_division_append_given_steppreviousdiagonalterm) = S ((S (pfc_index_division_append_given_steppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_division_append_given_steppreviousdiagonaltermleftentry. qb = ff_q_pfp_division_append_given_steppreviousdiagonaltermleftentry * S ((S (pfc_index_division_append_given_steppreviousdiagonal)) * qc) + (pfc_left_division_append_given_steppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_append_given_steppreviousdiagonaltermleftoutside. pfc_gap_division_append_given_steppreviousdiagonaltermleftoutside+(N)=(pfc_index_division_append_given_steppreviousdiagonal)) /\\ (((pfc_left_division_append_given_steppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_append_given_steppreviousdiagonaltermrightinside. pfa_gap_division_append_given_steppreviousdiagonaltermrightinside + S (pfc_complement_division_append_given_steppreviousdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_division_append_given_steppreviousdiagonaltermrightentry. ff_h_pfp_division_append_given_steppreviousdiagonaltermrightentry + S (pfc_right_division_append_given_steppreviousdiagonalterm) = S ((S (pfc_complement_division_append_given_steppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_append_given_steppreviousdiagonaltermrightentry. bb = ff_q_pfp_division_append_given_steppreviousdiagonaltermrightentry * S ((S (pfc_complement_division_append_given_steppreviousdiagonalterm)) * bc) + (pfc_right_division_append_given_steppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_append_given_steppreviousdiagonaltermrightoutside. pfc_gap_division_append_given_steppreviousdiagonaltermrightoutside+(M)=(pfc_complement_division_append_given_steppreviousdiagonalterm)) /\\ (((pfc_right_division_append_given_steppreviousdiagonalterm)=0))))) /\\ (((pfc_value_division_append_given_steppreviousdiagonal)=pfc_left_division_append_given_steppreviousdiagonalterm*pfc_right_division_append_given_steppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_append_given_stepprevioussum fs_v_pfc_division_append_given_stepprevioussum. ((((exists fs_h_pfc_division_append_given_stepprevioussum_body_start. fs_h_pfc_division_append_given_stepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_append_given_stepprevioussum)) /\\ exists fs_q_pfc_division_append_given_stepprevioussum_body_start. fs_u_pfc_division_append_given_stepprevioussum = fs_q_pfc_division_append_given_stepprevioussum_body_start * S ((S (0)) * fs_v_pfc_division_append_given_stepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_division_append_given_stepprevioussum_body_terminal. fs_h_pfc_division_append_given_stepprevioussum_body_terminal + S (pfc_natural_sum_division_append_given_stepprevious) = S ((S (S (N))) * fs_v_pfc_division_append_given_stepprevioussum)) /\\ exists fs_q_pfc_division_append_given_stepprevioussum_body_terminal. fs_u_pfc_division_append_given_stepprevioussum = fs_q_pfc_division_append_given_stepprevioussum_body_terminal * S ((S (S (N))) * fs_v_pfc_division_append_given_stepprevioussum) + (pfc_natural_sum_division_append_given_stepprevious))) /\\ forall fs_i_pfc_division_append_given_stepprevioussum_body_steps. (exists fs_lt_pfc_division_append_given_stepprevioussum_body_steps_bound. fs_lt_pfc_division_append_given_stepprevioussum_body_steps_bound + S fs_i_pfc_division_append_given_stepprevioussum_body_steps = S (N)) -> exists fs_a_pfc_division_append_given_stepprevioussum_body_steps fs_r_pfc_division_append_given_stepprevioussum_body_steps fs_s_pfc_division_append_given_stepprevioussum_body_steps. ((((exists fs_h_pfc_division_append_given_stepprevioussum_body_steps_summand. fs_h_pfc_division_append_given_stepprevioussum_body_steps_summand + S (fs_a_pfc_division_append_given_stepprevioussum_body_steps) = S ((S (fs_i_pfc_division_append_given_stepprevioussum_body_steps)) * pfc_terms_scale_division_append_given_stepprevious)) /\\ exists fs_q_pfc_division_append_given_stepprevioussum_body_steps_summand. pfc_terms_code_division_append_given_stepprevious = fs_q_pfc_division_append_given_stepprevioussum_body_steps_summand * S ((S (fs_i_pfc_division_append_given_stepprevioussum_body_steps)) * pfc_terms_scale_division_append_given_stepprevious) + (fs_a_pfc_division_append_given_stepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_append_given_stepprevioussum_body_steps_partial. fs_h_pfc_division_append_given_stepprevioussum_body_steps_partial + S (fs_r_pfc_division_append_given_stepprevioussum_body_steps) = S ((S (fs_i_pfc_division_append_given_stepprevioussum_body_steps)) * fs_v_pfc_division_append_given_stepprevioussum)) /\\ exists fs_q_pfc_division_append_given_stepprevioussum_body_steps_partial. fs_u_pfc_division_append_given_stepprevioussum = fs_q_pfc_division_append_given_stepprevioussum_body_steps_partial * S ((S (fs_i_pfc_division_append_given_stepprevioussum_body_steps)) * fs_v_pfc_division_append_given_stepprevioussum) + (fs_r_pfc_division_append_given_stepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_append_given_stepprevioussum_body_steps_successor. fs_h_pfc_division_append_given_stepprevioussum_body_steps_successor + S (fs_s_pfc_division_append_given_stepprevioussum_body_steps) = S ((S (S fs_i_pfc_division_append_given_stepprevioussum_body_steps)) * fs_v_pfc_division_append_given_stepprevioussum)) /\\ exists fs_q_pfc_division_append_given_stepprevioussum_body_steps_successor. fs_u_pfc_division_append_given_stepprevioussum = fs_q_pfc_division_append_given_stepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_division_append_given_stepprevioussum_body_steps)) * fs_v_pfc_division_append_given_stepprevioussum) + (fs_s_pfc_division_append_given_stepprevioussum_body_steps))) /\\ fs_s_pfc_division_append_given_stepprevioussum_body_steps = fs_r_pfc_division_append_given_stepprevioussum_body_steps + fs_a_pfc_division_append_given_stepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_division_append_given_steppreviousresiduebound. pfa_gap_division_append_given_steppreviousresiduebound + S (pfd_previous_division_append_given_step) = (p)) /\\ ((exists pfa_offset_left_division_append_given_steppreviousresiduecongruence pfa_offset_right_division_append_given_steppreviousresiduecongruence. (pfc_natural_sum_division_append_given_stepprevious) + (p) * pfa_offset_left_division_append_given_steppreviousresiduecongruence = (pfd_previous_division_append_given_step) + (p) * pfa_offset_right_division_append_given_steppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_division_append_given_stepsubtractleft. pfa_gap_division_append_given_stepsubtractleft + S (pfd_previous_division_append_given_step) = (p)) /\\ (((exists pfa_gap_division_append_given_stepsubtractright. pfa_gap_division_append_given_stepsubtractright + S (pfd_difference_division_append_given_step) = (p)) /\\ ((((exists pfa_gap_division_append_given_stepsubtractresultbound. pfa_gap_division_append_given_stepsubtractresultbound + S (pfd_input_division_append_given_step) = (p)) /\\ ((exists pfa_offset_left_division_append_given_stepsubtractresultcongruence pfa_offset_right_division_append_given_stepsubtractresultcongruence. ((pfd_previous_division_append_given_step) + (pfd_difference_division_append_given_step)) + (p) * pfa_offset_left_division_append_given_stepsubtractresultcongruence = (pfd_input_division_append_given_step) + (p) * pfa_offset_right_division_append_given_stepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_division_append_given_stepmultiplyleft. pfa_gap_division_append_given_stepmultiplyleft + S (k) = (p)) /\\ (((exists pfa_gap_division_append_given_stepmultiplyright. pfa_gap_division_append_given_stepmultiplyright + S (pfd_difference_division_append_given_step) = (p)) /\\ ((((exists pfa_gap_division_append_given_stepmultiplyresultbound. pfa_gap_division_append_given_stepmultiplyresultbound + S (q) = (p)) /\\ ((exists pfa_offset_left_division_append_given_stepmultiplyresultcongruence pfa_offset_right_division_append_given_stepmultiplyresultcongruence. ((k) * (pfd_difference_division_append_given_step)) + (p) * pfa_offset_left_division_append_given_stepmultiplyresultcongruence = (q) + (p) * pfa_offset_right_division_append_given_stepmultiplyresultcongruence)))))))))))))))) -> (forall pfd_index_division_append_new. (exists pfa_gap_division_append_newbound. pfa_gap_division_append_newbound + S (pfd_index_division_append_new) = (S N)) -> exists pfd_value_division_append_new. ((((exists ff_h_pfp_division_append_newentry. ff_h_pfp_division_append_newentry + S (pfd_value_division_append_new) = S ((S (pfd_index_division_append_new)) * QC)) /\\ exists ff_q_pfp_division_append_newentry. QB = ff_q_pfp_division_append_newentry * S ((S (pfd_index_division_append_new)) * QC) + (pfd_value_division_append_new))) /\\ ((exists pfd_input_division_append_newstep pfd_previous_division_append_newstep pfd_difference_division_append_newstep. ((((exists ff_h_pfp_division_append_newstepinput. ff_h_pfp_division_append_newstepinput + S (pfd_input_division_append_newstep) = S ((S (pfd_index_division_append_new)) * ac)) /\\ exists ff_q_pfp_division_append_newstepinput. ab = ff_q_pfp_division_append_newstepinput * S ((S (pfd_index_division_append_new)) * ac) + (pfd_input_division_append_newstep))) /\\ (((exists pfc_terms_code_division_append_newstepprevious pfc_terms_scale_division_append_newstepprevious pfc_natural_sum_division_append_newstepprevious. ((forall pfc_index_division_append_newsteppreviousdiagonal. (exists pfa_gap_division_append_newsteppreviousdiagonalbound. pfa_gap_division_append_newsteppreviousdiagonalbound + S (pfc_index_division_append_newsteppreviousdiagonal) = (S (pfd_index_division_append_new))) -> exists pfc_value_division_append_newsteppreviousdiagonal. ((((exists ff_h_pfp_division_append_newsteppreviousdiagonalentry. ff_h_pfp_division_append_newsteppreviousdiagonalentry + S (pfc_value_division_append_newsteppreviousdiagonal) = S ((S (pfc_index_division_append_newsteppreviousdiagonal)) * pfc_terms_scale_division_append_newstepprevious)) /\\ exists ff_q_pfp_division_append_newsteppreviousdiagonalentry. pfc_terms_code_division_append_newstepprevious = ff_q_pfp_division_append_newsteppreviousdiagonalentry * S ((S (pfc_index_division_append_newsteppreviousdiagonal)) * pfc_terms_scale_division_append_newstepprevious) + (pfc_value_division_append_newsteppreviousdiagonal))) /\\ ((exists pfc_complement_division_append_newsteppreviousdiagonalterm pfc_left_division_append_newsteppreviousdiagonalterm pfc_right_division_append_newsteppreviousdiagonalterm. (((pfc_index_division_append_newsteppreviousdiagonal)+pfc_complement_division_append_newsteppreviousdiagonalterm=(pfd_index_division_append_new)) /\\ ((((((exists pfa_gap_division_append_newsteppreviousdiagonaltermleftinside. pfa_gap_division_append_newsteppreviousdiagonaltermleftinside + S (pfc_index_division_append_newsteppreviousdiagonal) = (pfd_index_division_append_new)) /\\ ((((exists ff_h_pfp_division_append_newsteppreviousdiagonaltermleftentry. ff_h_pfp_division_append_newsteppreviousdiagonaltermleftentry + S (pfc_left_division_append_newsteppreviousdiagonalterm) = S ((S (pfc_index_division_append_newsteppreviousdiagonal)) * QC)) /\\ exists ff_q_pfp_division_append_newsteppreviousdiagonaltermleftentry. QB = ff_q_pfp_division_append_newsteppreviousdiagonaltermleftentry * S ((S (pfc_index_division_append_newsteppreviousdiagonal)) * QC) + (pfc_left_division_append_newsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_append_newsteppreviousdiagonaltermleftoutside. pfc_gap_division_append_newsteppreviousdiagonaltermleftoutside+(pfd_index_division_append_new)=(pfc_index_division_append_newsteppreviousdiagonal)) /\\ (((pfc_left_division_append_newsteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_append_newsteppreviousdiagonaltermrightinside. pfa_gap_division_append_newsteppreviousdiagonaltermrightinside + S (pfc_complement_division_append_newsteppreviousdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_division_append_newsteppreviousdiagonaltermrightentry. ff_h_pfp_division_append_newsteppreviousdiagonaltermrightentry + S (pfc_right_division_append_newsteppreviousdiagonalterm) = S ((S (pfc_complement_division_append_newsteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_append_newsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_division_append_newsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_division_append_newsteppreviousdiagonalterm)) * bc) + (pfc_right_division_append_newsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_append_newsteppreviousdiagonaltermrightoutside. pfc_gap_division_append_newsteppreviousdiagonaltermrightoutside+(M)=(pfc_complement_division_append_newsteppreviousdiagonalterm)) /\\ (((pfc_right_division_append_newsteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_division_append_newsteppreviousdiagonal)=pfc_left_division_append_newsteppreviousdiagonalterm*pfc_right_division_append_newsteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_append_newstepprevioussum fs_v_pfc_division_append_newstepprevioussum. ((((exists fs_h_pfc_division_append_newstepprevioussum_body_start. fs_h_pfc_division_append_newstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_append_newstepprevioussum)) /\\ exists fs_q_pfc_division_append_newstepprevioussum_body_start. fs_u_pfc_division_append_newstepprevioussum = fs_q_pfc_division_append_newstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_division_append_newstepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_division_append_newstepprevioussum_body_terminal. fs_h_pfc_division_append_newstepprevioussum_body_terminal + S (pfc_natural_sum_division_append_newstepprevious) = S ((S (S (pfd_index_division_append_new))) * fs_v_pfc_division_append_newstepprevioussum)) /\\ exists fs_q_pfc_division_append_newstepprevioussum_body_terminal. fs_u_pfc_division_append_newstepprevioussum = fs_q_pfc_division_append_newstepprevioussum_body_terminal * S ((S (S (pfd_index_division_append_new))) * fs_v_pfc_division_append_newstepprevioussum) + (pfc_natural_sum_division_append_newstepprevious))) /\\ forall fs_i_pfc_division_append_newstepprevioussum_body_steps. (exists fs_lt_pfc_division_append_newstepprevioussum_body_steps_bound. fs_lt_pfc_division_append_newstepprevioussum_body_steps_bound + S fs_i_pfc_division_append_newstepprevioussum_body_steps = S (pfd_index_division_append_new)) -> exists fs_a_pfc_division_append_newstepprevioussum_body_steps fs_r_pfc_division_append_newstepprevioussum_body_steps fs_s_pfc_division_append_newstepprevioussum_body_steps. ((((exists fs_h_pfc_division_append_newstepprevioussum_body_steps_summand. fs_h_pfc_division_append_newstepprevioussum_body_steps_summand + S (fs_a_pfc_division_append_newstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_append_newstepprevioussum_body_steps)) * pfc_terms_scale_division_append_newstepprevious)) /\\ exists fs_q_pfc_division_append_newstepprevioussum_body_steps_summand. pfc_terms_code_division_append_newstepprevious = fs_q_pfc_division_append_newstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_division_append_newstepprevioussum_body_steps)) * pfc_terms_scale_division_append_newstepprevious) + (fs_a_pfc_division_append_newstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_append_newstepprevioussum_body_steps_partial. fs_h_pfc_division_append_newstepprevioussum_body_steps_partial + S (fs_r_pfc_division_append_newstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_append_newstepprevioussum_body_steps)) * fs_v_pfc_division_append_newstepprevioussum)) /\\ exists fs_q_pfc_division_append_newstepprevioussum_body_steps_partial. fs_u_pfc_division_append_newstepprevioussum = fs_q_pfc_division_append_newstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_division_append_newstepprevioussum_body_steps)) * fs_v_pfc_division_append_newstepprevioussum) + (fs_r_pfc_division_append_newstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_append_newstepprevioussum_body_steps_successor. fs_h_pfc_division_append_newstepprevioussum_body_steps_successor + S (fs_s_pfc_division_append_newstepprevioussum_body_steps) = S ((S (S fs_i_pfc_division_append_newstepprevioussum_body_steps)) * fs_v_pfc_division_append_newstepprevioussum)) /\\ exists fs_q_pfc_division_append_newstepprevioussum_body_steps_successor. fs_u_pfc_division_append_newstepprevioussum = fs_q_pfc_division_append_newstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_division_append_newstepprevioussum_body_steps)) * fs_v_pfc_division_append_newstepprevioussum) + (fs_s_pfc_division_append_newstepprevioussum_body_steps))) /\\ fs_s_pfc_division_append_newstepprevioussum_body_steps = fs_r_pfc_division_append_newstepprevioussum_body_steps + fs_a_pfc_division_append_newstepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_division_append_newsteppreviousresiduebound. pfa_gap_division_append_newsteppreviousresiduebound + S (pfd_previous_division_append_newstep) = (p)) /\\ ((exists pfa_offset_left_division_append_newsteppreviousresiduecongruence pfa_offset_right_division_append_newsteppreviousresiduecongruence. (pfc_natural_sum_division_append_newstepprevious) + (p) * pfa_offset_left_division_append_newsteppreviousresiduecongruence = (pfd_previous_division_append_newstep) + (p) * pfa_offset_right_division_append_newsteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_division_append_newstepsubtractleft. pfa_gap_division_append_newstepsubtractleft + S (pfd_previous_division_append_newstep) = (p)) /\\ (((exists pfa_gap_division_append_newstepsubtractright. pfa_gap_division_append_newstepsubtractright + S (pfd_difference_division_append_newstep) = (p)) /\\ ((((exists pfa_gap_division_append_newstepsubtractresultbound. pfa_gap_division_append_newstepsubtractresultbound + S (pfd_input_division_append_newstep) = (p)) /\\ ((exists pfa_offset_left_division_append_newstepsubtractresultcongruence pfa_offset_right_division_append_newstepsubtractresultcongruence. 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(forall pfd_index_division_exists_old. (exists pfa_gap_division_exists_oldbound. pfa_gap_division_exists_oldbound + S (pfd_index_division_exists_old) = (N)) -> exists pfd_value_division_exists_old. ((((exists ff_h_pfp_division_exists_oldentry. ff_h_pfp_division_exists_oldentry + S (pfd_value_division_exists_old) = S ((S (pfd_index_division_exists_old)) * qc)) /\\ exists ff_q_pfp_division_exists_oldentry. qb = ff_q_pfp_division_exists_oldentry * S ((S (pfd_index_division_exists_old)) * qc) + (pfd_value_division_exists_old))) /\\ ((exists pfd_input_division_exists_oldstep pfd_previous_division_exists_oldstep pfd_difference_division_exists_oldstep. ((((exists ff_h_pfp_division_exists_oldstepinput. ff_h_pfp_division_exists_oldstepinput + S (pfd_input_division_exists_oldstep) = S ((S (pfd_index_division_exists_old)) * ac)) /\\ exists ff_q_pfp_division_exists_oldstepinput. ab = ff_q_pfp_division_exists_oldstepinput * S ((S (pfd_index_division_exists_old)) * ac) + (pfd_input_division_exists_oldstep))) /\\ (((exists pfc_terms_code_division_exists_oldstepprevious pfc_terms_scale_division_exists_oldstepprevious pfc_natural_sum_division_exists_oldstepprevious. ((forall pfc_index_division_exists_oldsteppreviousdiagonal. (exists pfa_gap_division_exists_oldsteppreviousdiagonalbound. pfa_gap_division_exists_oldsteppreviousdiagonalbound + S (pfc_index_division_exists_oldsteppreviousdiagonal) = (S (pfd_index_division_exists_old))) -> exists pfc_value_division_exists_oldsteppreviousdiagonal. ((((exists ff_h_pfp_division_exists_oldsteppreviousdiagonalentry. ff_h_pfp_division_exists_oldsteppreviousdiagonalentry + S (pfc_value_division_exists_oldsteppreviousdiagonal) = S ((S (pfc_index_division_exists_oldsteppreviousdiagonal)) * pfc_terms_scale_division_exists_oldstepprevious)) /\\ exists ff_q_pfp_division_exists_oldsteppreviousdiagonalentry. pfc_terms_code_division_exists_oldstepprevious = ff_q_pfp_division_exists_oldsteppreviousdiagonalentry * S ((S (pfc_index_division_exists_oldsteppreviousdiagonal)) * pfc_terms_scale_division_exists_oldstepprevious) + (pfc_value_division_exists_oldsteppreviousdiagonal))) /\\ ((exists pfc_complement_division_exists_oldsteppreviousdiagonalterm pfc_left_division_exists_oldsteppreviousdiagonalterm pfc_right_division_exists_oldsteppreviousdiagonalterm. (((pfc_index_division_exists_oldsteppreviousdiagonal)+pfc_complement_division_exists_oldsteppreviousdiagonalterm=(pfd_index_division_exists_old)) /\\ ((((((exists pfa_gap_division_exists_oldsteppreviousdiagonaltermleftinside. pfa_gap_division_exists_oldsteppreviousdiagonaltermleftinside + S (pfc_index_division_exists_oldsteppreviousdiagonal) = (pfd_index_division_exists_old)) /\\ ((((exists ff_h_pfp_division_exists_oldsteppreviousdiagonaltermleftentry. ff_h_pfp_division_exists_oldsteppreviousdiagonaltermleftentry + S (pfc_left_division_exists_oldsteppreviousdiagonalterm) = S ((S (pfc_index_division_exists_oldsteppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_division_exists_oldsteppreviousdiagonaltermleftentry. qb = ff_q_pfp_division_exists_oldsteppreviousdiagonaltermleftentry * S ((S (pfc_index_division_exists_oldsteppreviousdiagonal)) * qc) + (pfc_left_division_exists_oldsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_exists_oldsteppreviousdiagonaltermleftoutside. pfc_gap_division_exists_oldsteppreviousdiagonaltermleftoutside+(pfd_index_division_exists_old)=(pfc_index_division_exists_oldsteppreviousdiagonal)) /\\ (((pfc_left_division_exists_oldsteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_exists_oldsteppreviousdiagonaltermrightinside. pfa_gap_division_exists_oldsteppreviousdiagonaltermrightinside + S (pfc_complement_division_exists_oldsteppreviousdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_division_exists_oldsteppreviousdiagonaltermrightentry. ff_h_pfp_division_exists_oldsteppreviousdiagonaltermrightentry + S (pfc_right_division_exists_oldsteppreviousdiagonalterm) = S ((S (pfc_complement_division_exists_oldsteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_exists_oldsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_division_exists_oldsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_division_exists_oldsteppreviousdiagonalterm)) * bc) + (pfc_right_division_exists_oldsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_exists_oldsteppreviousdiagonaltermrightoutside. pfc_gap_division_exists_oldsteppreviousdiagonaltermrightoutside+(M)=(pfc_complement_division_exists_oldsteppreviousdiagonalterm)) /\\ (((pfc_right_division_exists_oldsteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_division_exists_oldsteppreviousdiagonal)=pfc_left_division_exists_oldsteppreviousdiagonalterm*pfc_right_division_exists_oldsteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_exists_oldstepprevioussum fs_v_pfc_division_exists_oldstepprevioussum. ((((exists fs_h_pfc_division_exists_oldstepprevioussum_body_start. fs_h_pfc_division_exists_oldstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_exists_oldstepprevioussum)) /\\ exists fs_q_pfc_division_exists_oldstepprevioussum_body_start. fs_u_pfc_division_exists_oldstepprevioussum = fs_q_pfc_division_exists_oldstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_division_exists_oldstepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_division_exists_oldstepprevioussum_body_terminal. fs_h_pfc_division_exists_oldstepprevioussum_body_terminal + S (pfc_natural_sum_division_exists_oldstepprevious) = S ((S (S (pfd_index_division_exists_old))) * fs_v_pfc_division_exists_oldstepprevioussum)) /\\ exists fs_q_pfc_division_exists_oldstepprevioussum_body_terminal. fs_u_pfc_division_exists_oldstepprevioussum = fs_q_pfc_division_exists_oldstepprevioussum_body_terminal * S ((S (S (pfd_index_division_exists_old))) * fs_v_pfc_division_exists_oldstepprevioussum) + (pfc_natural_sum_division_exists_oldstepprevious))) /\\ forall fs_i_pfc_division_exists_oldstepprevioussum_body_steps. (exists fs_lt_pfc_division_exists_oldstepprevioussum_body_steps_bound. fs_lt_pfc_division_exists_oldstepprevioussum_body_steps_bound + S fs_i_pfc_division_exists_oldstepprevioussum_body_steps = S (pfd_index_division_exists_old)) -> exists fs_a_pfc_division_exists_oldstepprevioussum_body_steps fs_r_pfc_division_exists_oldstepprevioussum_body_steps fs_s_pfc_division_exists_oldstepprevioussum_body_steps. ((((exists fs_h_pfc_division_exists_oldstepprevioussum_body_steps_summand. fs_h_pfc_division_exists_oldstepprevioussum_body_steps_summand + S (fs_a_pfc_division_exists_oldstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_exists_oldstepprevioussum_body_steps)) * pfc_terms_scale_division_exists_oldstepprevious)) /\\ exists fs_q_pfc_division_exists_oldstepprevioussum_body_steps_summand. pfc_terms_code_division_exists_oldstepprevious = fs_q_pfc_division_exists_oldstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_division_exists_oldstepprevioussum_body_steps)) * pfc_terms_scale_division_exists_oldstepprevious) + (fs_a_pfc_division_exists_oldstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_exists_oldstepprevioussum_body_steps_partial. fs_h_pfc_division_exists_oldstepprevioussum_body_steps_partial + S (fs_r_pfc_division_exists_oldstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_exists_oldstepprevioussum_body_steps)) * fs_v_pfc_division_exists_oldstepprevioussum)) /\\ exists fs_q_pfc_division_exists_oldstepprevioussum_body_steps_partial. fs_u_pfc_division_exists_oldstepprevioussum = fs_q_pfc_division_exists_oldstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_division_exists_oldstepprevioussum_body_steps)) * fs_v_pfc_division_exists_oldstepprevioussum) + (fs_r_pfc_division_exists_oldstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_exists_oldstepprevioussum_body_steps_successor. fs_h_pfc_division_exists_oldstepprevioussum_body_steps_successor + S (fs_s_pfc_division_exists_oldstepprevioussum_body_steps) = S ((S (S fs_i_pfc_division_exists_oldstepprevioussum_body_steps)) * fs_v_pfc_division_exists_oldstepprevioussum)) /\\ exists fs_q_pfc_division_exists_oldstepprevioussum_body_steps_successor. fs_u_pfc_division_exists_oldstepprevioussum = fs_q_pfc_division_exists_oldstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_division_exists_oldstepprevioussum_body_steps)) * fs_v_pfc_division_exists_oldstepprevioussum) + (fs_s_pfc_division_exists_oldstepprevioussum_body_steps))) /\\ fs_s_pfc_division_exists_oldstepprevioussum_body_steps = fs_r_pfc_division_exists_oldstepprevioussum_body_steps + fs_a_pfc_division_exists_oldstepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_division_exists_oldsteppreviousresiduebound. pfa_gap_division_exists_oldsteppreviousresiduebound + S (pfd_previous_division_exists_oldstep) = (p)) /\\ ((exists pfa_offset_left_division_exists_oldsteppreviousresiduecongruence pfa_offset_right_division_exists_oldsteppreviousresiduecongruence. (pfc_natural_sum_division_exists_oldstepprevious) + (p) * pfa_offset_left_division_exists_oldsteppreviousresiduecongruence = (pfd_previous_division_exists_oldstep) + (p) * pfa_offset_right_division_exists_oldsteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_division_exists_oldstepsubtractleft. pfa_gap_division_exists_oldstepsubtractleft + S (pfd_previous_division_exists_oldstep) = (p)) /\\ (((exists pfa_gap_division_exists_oldstepsubtractright. pfa_gap_division_exists_oldstepsubtractright + S (pfd_difference_division_exists_oldstep) = (p)) /\\ ((((exists pfa_gap_division_exists_oldstepsubtractresultbound. pfa_gap_division_exists_oldstepsubtractresultbound + S (pfd_input_division_exists_oldstep) = (p)) /\\ ((exists pfa_offset_left_division_exists_oldstepsubtractresultcongruence pfa_offset_right_division_exists_oldstepsubtractresultcongruence. ((pfd_previous_division_exists_oldstep) + (pfd_difference_division_exists_oldstep)) + (p) * pfa_offset_left_division_exists_oldstepsubtractresultcongruence = (pfd_input_division_exists_oldstep) + (p) * pfa_offset_right_division_exists_oldstepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_division_exists_oldstepmultiplyleft. pfa_gap_division_exists_oldstepmultiplyleft + S (k) = (p)) /\\ (((exists pfa_gap_division_exists_oldstepmultiplyright. pfa_gap_division_exists_oldstepmultiplyright + S (pfd_difference_division_exists_oldstep) = (p)) /\\ ((((exists pfa_gap_division_exists_oldstepmultiplyresultbound. pfa_gap_division_exists_oldstepmultiplyresultbound + S (pfd_value_division_exists_old) = (p)) /\\ ((exists pfa_offset_left_division_exists_oldstepmultiplyresultcongruence pfa_offset_right_division_exists_oldstepmultiplyresultcongruence. ((k) * (pfd_difference_division_exists_oldstep)) + (p) * pfa_offset_left_division_exists_oldstepmultiplyresultcongruence = (pfd_value_division_exists_old) + (p) * pfa_offset_right_division_exists_oldstepmultiplyresultcongruence)))))))))))))))))))", "apply IH", "intro i", "intro hi", "specialize ha (i)", "apply ha", "specialize le_succ (S i)", "specialize le_succ (N)", "apply le_succ", "exact hi", "cases hold", "cases hold_witness", "have hinput : exists a. ((((exists ff_h_pfp_division_exists_input. ff_h_pfp_division_exists_input + S (a) = S ((S (N)) * ac)) /\\ exists ff_q_pfp_division_exists_input. ab = ff_q_pfp_division_exists_input * S ((S (N)) * ac) + (a))) /\\ ((exists pfa_gap_division_exists_input_bound. pfa_gap_division_exists_input_bound + S (a) = (p))))", "specialize ha (N)", "apply ha", "specialize le_refl (S N)", "apply le_refl", "cases hinput", "cases hinput_witness", "have hc : exists c. (exists pfc_terms_code_division_exists_previous pfc_terms_scale_division_exists_previous pfc_natural_sum_division_exists_previous. ((forall pfc_index_division_exists_previousdiagonal. (exists pfa_gap_division_exists_previousdiagonalbound. pfa_gap_division_exists_previousdiagonalbound + S (pfc_index_division_exists_previousdiagonal) = (S (N))) -> exists pfc_value_division_exists_previousdiagonal. ((((exists ff_h_pfp_division_exists_previousdiagonalentry. ff_h_pfp_division_exists_previousdiagonalentry + S (pfc_value_division_exists_previousdiagonal) = S ((S (pfc_index_division_exists_previousdiagonal)) * pfc_terms_scale_division_exists_previous)) /\\ exists ff_q_pfp_division_exists_previousdiagonalentry. pfc_terms_code_division_exists_previous = ff_q_pfp_division_exists_previousdiagonalentry * S ((S (pfc_index_division_exists_previousdiagonal)) * pfc_terms_scale_division_exists_previous) + (pfc_value_division_exists_previousdiagonal))) /\\ ((exists pfc_complement_division_exists_previousdiagonalterm pfc_left_division_exists_previousdiagonalterm pfc_right_division_exists_previousdiagonalterm. (((pfc_index_division_exists_previousdiagonal)+pfc_complement_division_exists_previousdiagonalterm=(N)) /\\ ((((((exists pfa_gap_division_exists_previousdiagonaltermleftinside. pfa_gap_division_exists_previousdiagonaltermleftinside + S (pfc_index_division_exists_previousdiagonal) = (N)) /\\ ((((exists ff_h_pfp_division_exists_previousdiagonaltermleftentry. ff_h_pfp_division_exists_previousdiagonaltermleftentry + S (pfc_left_division_exists_previousdiagonalterm) = S ((S (pfc_index_division_exists_previousdiagonal)) * x1)) /\\ exists ff_q_pfp_division_exists_previousdiagonaltermleftentry. x = ff_q_pfp_division_exists_previousdiagonaltermleftentry * S ((S (pfc_index_division_exists_previousdiagonal)) * x1) + (pfc_left_division_exists_previousdiagonalterm)))))) \\/ (((exists pfc_gap_division_exists_previousdiagonaltermleftoutside. pfc_gap_division_exists_previousdiagonaltermleftoutside+(N)=(pfc_index_division_exists_previousdiagonal)) /\\ (((pfc_left_division_exists_previousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_exists_previousdiagonaltermrightinside. pfa_gap_division_exists_previousdiagonaltermrightinside + S (pfc_complement_division_exists_previousdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_division_exists_previousdiagonaltermrightentry. ff_h_pfp_division_exists_previousdiagonaltermrightentry + S (pfc_right_division_exists_previousdiagonalterm) = S ((S (pfc_complement_division_exists_previousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_exists_previousdiagonaltermrightentry. bb = ff_q_pfp_division_exists_previousdiagonaltermrightentry * S ((S (pfc_complement_division_exists_previousdiagonalterm)) * bc) + (pfc_right_division_exists_previousdiagonalterm)))))) \\/ (((exists pfc_gap_division_exists_previousdiagonaltermrightoutside. pfc_gap_division_exists_previousdiagonaltermrightoutside+(M)=(pfc_complement_division_exists_previousdiagonalterm)) /\\ (((pfc_right_division_exists_previousdiagonalterm)=0))))) /\\ (((pfc_value_division_exists_previousdiagonal)=pfc_left_division_exists_previousdiagonalterm*pfc_right_division_exists_previousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_exists_previoussum fs_v_pfc_division_exists_previoussum. ((((exists fs_h_pfc_division_exists_previoussum_body_start. fs_h_pfc_division_exists_previoussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_exists_previoussum)) /\\ exists fs_q_pfc_division_exists_previoussum_body_start. fs_u_pfc_division_exists_previoussum = fs_q_pfc_division_exists_previoussum_body_start * S ((S (0)) * fs_v_pfc_division_exists_previoussum) + (0))) /\\ ((((exists fs_h_pfc_division_exists_previoussum_body_terminal. fs_h_pfc_division_exists_previoussum_body_terminal + S (pfc_natural_sum_division_exists_previous) = S ((S (S (N))) * fs_v_pfc_division_exists_previoussum)) /\\ exists fs_q_pfc_division_exists_previoussum_body_terminal. fs_u_pfc_division_exists_previoussum = fs_q_pfc_division_exists_previoussum_body_terminal * S ((S (S (N))) * fs_v_pfc_division_exists_previoussum) + (pfc_natural_sum_division_exists_previous))) /\\ forall fs_i_pfc_division_exists_previoussum_body_steps. (exists fs_lt_pfc_division_exists_previoussum_body_steps_bound. fs_lt_pfc_division_exists_previoussum_body_steps_bound + S fs_i_pfc_division_exists_previoussum_body_steps = S (N)) -> exists fs_a_pfc_division_exists_previoussum_body_steps fs_r_pfc_division_exists_previoussum_body_steps fs_s_pfc_division_exists_previoussum_body_steps. ((((exists fs_h_pfc_division_exists_previoussum_body_steps_summand. fs_h_pfc_division_exists_previoussum_body_steps_summand + S (fs_a_pfc_division_exists_previoussum_body_steps) = S ((S (fs_i_pfc_division_exists_previoussum_body_steps)) * pfc_terms_scale_division_exists_previous)) /\\ exists fs_q_pfc_division_exists_previoussum_body_steps_summand. pfc_terms_code_division_exists_previous = fs_q_pfc_division_exists_previoussum_body_steps_summand * S ((S (fs_i_pfc_division_exists_previoussum_body_steps)) * pfc_terms_scale_division_exists_previous) + (fs_a_pfc_division_exists_previoussum_body_steps))) /\\ ((((exists fs_h_pfc_division_exists_previoussum_body_steps_partial. fs_h_pfc_division_exists_previoussum_body_steps_partial + S (fs_r_pfc_division_exists_previoussum_body_steps) = S ((S (fs_i_pfc_division_exists_previoussum_body_steps)) * fs_v_pfc_division_exists_previoussum)) /\\ exists fs_q_pfc_division_exists_previoussum_body_steps_partial. fs_u_pfc_division_exists_previoussum = fs_q_pfc_division_exists_previoussum_body_steps_partial * S ((S (fs_i_pfc_division_exists_previoussum_body_steps)) * fs_v_pfc_division_exists_previoussum) + (fs_r_pfc_division_exists_previoussum_body_steps))) /\\ ((((exists fs_h_pfc_division_exists_previoussum_body_steps_successor. fs_h_pfc_division_exists_previoussum_body_steps_successor + S (fs_s_pfc_division_exists_previoussum_body_steps) = S ((S (S fs_i_pfc_division_exists_previoussum_body_steps)) * fs_v_pfc_division_exists_previoussum)) /\\ exists fs_q_pfc_division_exists_previoussum_body_steps_successor. fs_u_pfc_division_exists_previoussum = fs_q_pfc_division_exists_previoussum_body_steps_successor * S ((S (S fs_i_pfc_division_exists_previoussum_body_steps)) * fs_v_pfc_division_exists_previoussum) + (fs_s_pfc_division_exists_previoussum_body_steps))) /\\ fs_s_pfc_division_exists_previoussum_body_steps = fs_r_pfc_division_exists_previoussum_body_steps + fs_a_pfc_division_exists_previoussum_body_steps)))))) /\\ ((((exists pfa_gap_division_exists_previousresiduebound. pfa_gap_division_exists_previousresiduebound + S (c) = (p)) /\\ ((exists pfa_offset_left_division_exists_previousresiduecongruence pfa_offset_right_division_exists_previousresiduecongruence. (pfc_natural_sum_division_exists_previous) + (p) * pfa_offset_left_division_exists_previousresiduecongruence = (c) + (p) * pfa_offset_right_division_exists_previousresiduecongruence)))))))))", "specialize prime_field_convolution_coefficient_exists (p)", "specialize prime_field_convolution_coefficient_exists (x)", "specialize prime_field_convolution_coefficient_exists (x1)", "specialize prime_field_convolution_coefficient_exists (N)", "specialize prime_field_convolution_coefficient_exists (bb)", "specialize prime_field_convolution_coefficient_exists (bc)", "specialize prime_field_convolution_coefficient_exists (M)", "specialize prime_field_convolution_coefficient_exists (N)", "apply prime_field_convolution_coefficient_exists", "intro hz", "specialize prime_nonzero (p)", "apply prime_nonzero", "exact hp", "exact hz", "cases hc", "have hs : exists s. (((exists pfa_gap_division_exists_differenceleft. pfa_gap_division_exists_differenceleft + S (x3) = (p)) /\\ (((exists pfa_gap_division_exists_differenceright. pfa_gap_division_exists_differenceright + S (s) = (p)) /\\ ((((exists pfa_gap_division_exists_differenceresultbound. pfa_gap_division_exists_differenceresultbound + S (x2) = (p)) /\\ ((exists pfa_offset_left_division_exists_differenceresultcongruence pfa_offset_right_division_exists_differenceresultcongruence. ((x3) + (s)) + (p) * pfa_offset_left_division_exists_differenceresultcongruence = (x2) + (p) * pfa_offset_right_division_exists_differenceresultcongruence)))))))))", "specialize prime_field_subtract_exists (p)", "specialize prime_field_subtract_exists (x2)", "specialize prime_field_subtract_exists (x3)", "apply prime_field_subtract_exists", "exact hp", "exact hinput_witness_right", "specialize prime_field_convolution_coefficient_bounded (p)", "specialize prime_field_convolution_coefficient_bounded (x)", "specialize prime_field_convolution_coefficient_bounded (x1)", "specialize prime_field_convolution_coefficient_bounded (N)", "specialize prime_field_convolution_coefficient_bounded (bb)", "specialize prime_field_convolution_coefficient_bounded (bc)", "specialize prime_field_convolution_coefficient_bounded (M)", "specialize prime_field_convolution_coefficient_bounded (N)", "specialize prime_field_convolution_coefficient_bounded (x3)", "apply prime_field_convolution_coefficient_bounded", "exact hc_witness", "cases hs", "have hq : exists q. (((exists pfa_gap_division_exists_quotientleft. pfa_gap_division_exists_quotientleft + S (k) = (p)) /\\ (((exists pfa_gap_division_exists_quotientright. pfa_gap_division_exists_quotientright + S (x4) = (p)) /\\ ((((exists pfa_gap_division_exists_quotientresultbound. pfa_gap_division_exists_quotientresultbound + S (q) = (p)) /\\ ((exists pfa_offset_left_division_exists_quotientresultcongruence pfa_offset_right_division_exists_quotientresultcongruence. ((k) * (x4)) + (p) * pfa_offset_left_division_exists_quotientresultcongruence = (q) + (p) * pfa_offset_right_division_exists_quotientresultcongruence)))))))))", "specialize prime_field_multiply_exists (p)", "specialize prime_field_multiply_exists (k)", "specialize prime_field_multiply_exists (x4)", "apply prime_field_multiply_exists", "exact hp", "exact hk", "cases hs_witness", "cases hs_witness_right", "exact hs_witness_right_left", "cases hq", "have hnew : exists QB QC. ((((exists ff_h_pfp_division_exists_new_entry. ff_h_pfp_division_exists_new_entry + S (x5) = S ((S (N)) * QC)) /\\ exists ff_q_pfp_division_exists_new_entry. QB = ff_q_pfp_division_exists_new_entry * S ((S (N)) * QC) + (x5))) /\\ ((forall mdr_i_pfp_division_exists_preservation mdr_a_pfp_division_exists_preservation. (exists mdr_gap_pfp_division_exists_preservationb. mdr_gap_pfp_division_exists_preservationb + S (mdr_i_pfp_division_exists_preservation) = (N)) -> (((exists ff_h_mdr_pfp_division_exists_preservationo. ff_h_mdr_pfp_division_exists_preservationo + S (mdr_a_pfp_division_exists_preservation) = S ((S (mdr_i_pfp_division_exists_preservation)) * x1)) /\\ exists ff_q_mdr_pfp_division_exists_preservationo. x = ff_q_mdr_pfp_division_exists_preservationo * S ((S (mdr_i_pfp_division_exists_preservation)) * x1) + (mdr_a_pfp_division_exists_preservation))) -> (((exists ff_h_mdr_pfp_division_exists_preservationn. ff_h_mdr_pfp_division_exists_preservationn + S (mdr_a_pfp_division_exists_preservation) = S ((S (mdr_i_pfp_division_exists_preservation)) * QC)) /\\ exists ff_q_mdr_pfp_division_exists_preservationn. QB = ff_q_mdr_pfp_division_exists_preservationn * S ((S (mdr_i_pfp_division_exists_preservation)) * QC) + (mdr_a_pfp_division_exists_preservation))))))", "specialize beta_prefix_extend (N)", "specialize beta_prefix_extend (x)", "specialize beta_prefix_extend (x1)", "specialize beta_prefix_extend (x5)", "apply beta_prefix_extend", "cases hnew", "cases hnew_witness", "cases hnew_witness_witness", "exists x6", "exists x7", "specialize prime_field_polynomial_quotient_prefix_append (p)", "specialize prime_field_polynomial_quotient_prefix_append (k)", "specialize prime_field_polynomial_quotient_prefix_append (ab)", "specialize prime_field_polynomial_quotient_prefix_append (ac)", "specialize prime_field_polynomial_quotient_prefix_append (bb)", "specialize prime_field_polynomial_quotient_prefix_append (bc)", "specialize prime_field_polynomial_quotient_prefix_append (M)", "specialize prime_field_polynomial_quotient_prefix_append (x)", "specialize prime_field_polynomial_quotient_prefix_append (x1)", "specialize prime_field_polynomial_quotient_prefix_append (x6)", "specialize prime_field_polynomial_quotient_prefix_append (x7)", "specialize prime_field_polynomial_quotient_prefix_append (N)", "specialize prime_field_polynomial_quotient_prefix_append (x5)", "apply prime_field_polynomial_quotient_prefix_append", "exact hold_witness_witness", "exact hnew_witness_witness_right", "exact hnew_witness_witness_left", "exists x2", "exists x3", "exists x4", "split", "exact hinput_witness_left", "split", "exact hc_witness", "split", "exact hs_witness", "exact hq_witness" ], "script_sha256": "cc04febf037eaccc257e15075c6575b59b372cdf40e9a27de8d44df70df5015b", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_division_candidate.py", "sha256": "edfc7806caf7a83b9cb0e3e420bd2c3a8679f2d4d9ee6ca9f8eae53faca8d5b2" }, "statement": "forall p k ab ac bb bc M N. (~((p) = 1) /\\ forall pfa_factor_left_division_exists_prime pfa_factor_right_division_exists_prime. (p) = pfa_factor_left_division_exists_prime * pfa_factor_right_division_exists_prime -> pfa_factor_left_division_exists_prime = 1 \\/ pfa_factor_right_division_exists_prime = 1) -> (exists pfa_gap_division_exists_scalar. pfa_gap_division_exists_scalar + S (k) = (p)) -> (forall fom_index_pfp_division_exists_source. (exists fom_gap_pfp_division_exists_source_index_bound. fom_gap_pfp_division_exists_source_index_bound + S (fom_index_pfp_division_exists_source) = N) -> exists fom_value_pfp_division_exists_source. ((((exists fom_beta_height_pfp_division_exists_source_entry. fom_beta_height_pfp_division_exists_source_entry + S (fom_value_pfp_division_exists_source) = S ((S (fom_index_pfp_division_exists_source)) * ac)) /\\ exists fom_beta_quotient_pfp_division_exists_source_entry. ab = fom_beta_quotient_pfp_division_exists_source_entry * S ((S (fom_index_pfp_division_exists_source)) * ac) + (fom_value_pfp_division_exists_source))) /\\ (exists fom_gap_pfp_division_exists_source_value_bound. fom_gap_pfp_division_exists_source_value_bound + S (fom_value_pfp_division_exists_source) = p))) -> exists qb qc. (forall pfd_index_division_exists_result. (exists pfa_gap_division_exists_resultbound. pfa_gap_division_exists_resultbound + S (pfd_index_division_exists_result) = (N)) -> exists pfd_value_division_exists_result. ((((exists ff_h_pfp_division_exists_resultentry. ff_h_pfp_division_exists_resultentry + S (pfd_value_division_exists_result) = S ((S (pfd_index_division_exists_result)) * qc)) /\\ exists ff_q_pfp_division_exists_resultentry. qb = ff_q_pfp_division_exists_resultentry * S ((S (pfd_index_division_exists_result)) * qc) + (pfd_value_division_exists_result))) /\\ ((exists pfd_input_division_exists_resultstep pfd_previous_division_exists_resultstep pfd_difference_division_exists_resultstep. ((((exists ff_h_pfp_division_exists_resultstepinput. ff_h_pfp_division_exists_resultstepinput + S (pfd_input_division_exists_resultstep) = S ((S (pfd_index_division_exists_result)) * ac)) /\\ exists ff_q_pfp_division_exists_resultstepinput. ab = ff_q_pfp_division_exists_resultstepinput * S ((S (pfd_index_division_exists_result)) * ac) + (pfd_input_division_exists_resultstep))) /\\ (((exists pfc_terms_code_division_exists_resultstepprevious pfc_terms_scale_division_exists_resultstepprevious pfc_natural_sum_division_exists_resultstepprevious. ((forall pfc_index_division_exists_resultsteppreviousdiagonal. (exists pfa_gap_division_exists_resultsteppreviousdiagonalbound. pfa_gap_division_exists_resultsteppreviousdiagonalbound + S (pfc_index_division_exists_resultsteppreviousdiagonal) = (S (pfd_index_division_exists_result))) -> exists pfc_value_division_exists_resultsteppreviousdiagonal. ((((exists ff_h_pfp_division_exists_resultsteppreviousdiagonalentry. ff_h_pfp_division_exists_resultsteppreviousdiagonalentry + S (pfc_value_division_exists_resultsteppreviousdiagonal) = S ((S (pfc_index_division_exists_resultsteppreviousdiagonal)) * pfc_terms_scale_division_exists_resultstepprevious)) /\\ exists ff_q_pfp_division_exists_resultsteppreviousdiagonalentry. pfc_terms_code_division_exists_resultstepprevious = ff_q_pfp_division_exists_resultsteppreviousdiagonalentry * S ((S (pfc_index_division_exists_resultsteppreviousdiagonal)) * pfc_terms_scale_division_exists_resultstepprevious) + (pfc_value_division_exists_resultsteppreviousdiagonal))) /\\ ((exists pfc_complement_division_exists_resultsteppreviousdiagonalterm pfc_left_division_exists_resultsteppreviousdiagonalterm pfc_right_division_exists_resultsteppreviousdiagonalterm. (((pfc_index_division_exists_resultsteppreviousdiagonal)+pfc_complement_division_exists_resultsteppreviousdiagonalterm=(pfd_index_division_exists_result)) /\\ ((((((exists pfa_gap_division_exists_resultsteppreviousdiagonaltermleftinside. pfa_gap_division_exists_resultsteppreviousdiagonaltermleftinside + S (pfc_index_division_exists_resultsteppreviousdiagonal) = (pfd_index_division_exists_result)) /\\ ((((exists ff_h_pfp_division_exists_resultsteppreviousdiagonaltermleftentry. ff_h_pfp_division_exists_resultsteppreviousdiagonaltermleftentry + S (pfc_left_division_exists_resultsteppreviousdiagonalterm) = S ((S (pfc_index_division_exists_resultsteppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_division_exists_resultsteppreviousdiagonaltermleftentry. qb = ff_q_pfp_division_exists_resultsteppreviousdiagonaltermleftentry * S ((S (pfc_index_division_exists_resultsteppreviousdiagonal)) * qc) + (pfc_left_division_exists_resultsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_exists_resultsteppreviousdiagonaltermleftoutside. pfc_gap_division_exists_resultsteppreviousdiagonaltermleftoutside+(pfd_index_division_exists_result)=(pfc_index_division_exists_resultsteppreviousdiagonal)) /\\ (((pfc_left_division_exists_resultsteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_exists_resultsteppreviousdiagonaltermrightinside. pfa_gap_division_exists_resultsteppreviousdiagonaltermrightinside + S (pfc_complement_division_exists_resultsteppreviousdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_division_exists_resultsteppreviousdiagonaltermrightentry. ff_h_pfp_division_exists_resultsteppreviousdiagonaltermrightentry + S (pfc_right_division_exists_resultsteppreviousdiagonalterm) = S ((S (pfc_complement_division_exists_resultsteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_exists_resultsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_division_exists_resultsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_division_exists_resultsteppreviousdiagonalterm)) * bc) + (pfc_right_division_exists_resultsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_exists_resultsteppreviousdiagonaltermrightoutside. pfc_gap_division_exists_resultsteppreviousdiagonaltermrightoutside+(M)=(pfc_complement_division_exists_resultsteppreviousdiagonalterm)) /\\ (((pfc_right_division_exists_resultsteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_division_exists_resultsteppreviousdiagonal)=pfc_left_division_exists_resultsteppreviousdiagonalterm*pfc_right_division_exists_resultsteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_exists_resultstepprevioussum fs_v_pfc_division_exists_resultstepprevioussum. ((((exists fs_h_pfc_division_exists_resultstepprevioussum_body_start. fs_h_pfc_division_exists_resultstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_exists_resultstepprevioussum)) /\\ exists fs_q_pfc_division_exists_resultstepprevioussum_body_start. fs_u_pfc_division_exists_resultstepprevioussum = fs_q_pfc_division_exists_resultstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_division_exists_resultstepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_division_exists_resultstepprevioussum_body_terminal. fs_h_pfc_division_exists_resultstepprevioussum_body_terminal + S (pfc_natural_sum_division_exists_resultstepprevious) = S ((S (S (pfd_index_division_exists_result))) * fs_v_pfc_division_exists_resultstepprevioussum)) /\\ exists fs_q_pfc_division_exists_resultstepprevioussum_body_terminal. fs_u_pfc_division_exists_resultstepprevioussum = fs_q_pfc_division_exists_resultstepprevioussum_body_terminal * S ((S (S (pfd_index_division_exists_result))) * fs_v_pfc_division_exists_resultstepprevioussum) + (pfc_natural_sum_division_exists_resultstepprevious))) /\\ forall fs_i_pfc_division_exists_resultstepprevioussum_body_steps. (exists fs_lt_pfc_division_exists_resultstepprevioussum_body_steps_bound. fs_lt_pfc_division_exists_resultstepprevioussum_body_steps_bound + S fs_i_pfc_division_exists_resultstepprevioussum_body_steps = S (pfd_index_division_exists_result)) -> exists fs_a_pfc_division_exists_resultstepprevioussum_body_steps fs_r_pfc_division_exists_resultstepprevioussum_body_steps fs_s_pfc_division_exists_resultstepprevioussum_body_steps. ((((exists fs_h_pfc_division_exists_resultstepprevioussum_body_steps_summand. fs_h_pfc_division_exists_resultstepprevioussum_body_steps_summand + S (fs_a_pfc_division_exists_resultstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_exists_resultstepprevioussum_body_steps)) * pfc_terms_scale_division_exists_resultstepprevious)) /\\ exists fs_q_pfc_division_exists_resultstepprevioussum_body_steps_summand. pfc_terms_code_division_exists_resultstepprevious = fs_q_pfc_division_exists_resultstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_division_exists_resultstepprevioussum_body_steps)) * pfc_terms_scale_division_exists_resultstepprevious) + (fs_a_pfc_division_exists_resultstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_exists_resultstepprevioussum_body_steps_partial. fs_h_pfc_division_exists_resultstepprevioussum_body_steps_partial + S (fs_r_pfc_division_exists_resultstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_exists_resultstepprevioussum_body_steps)) * fs_v_pfc_division_exists_resultstepprevioussum)) /\\ exists fs_q_pfc_division_exists_resultstepprevioussum_body_steps_partial. fs_u_pfc_division_exists_resultstepprevioussum = fs_q_pfc_division_exists_resultstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_division_exists_resultstepprevioussum_body_steps)) * fs_v_pfc_division_exists_resultstepprevioussum) + (fs_r_pfc_division_exists_resultstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_exists_resultstepprevioussum_body_steps_successor. fs_h_pfc_division_exists_resultstepprevioussum_body_steps_successor + S (fs_s_pfc_division_exists_resultstepprevioussum_body_steps) = S ((S (S fs_i_pfc_division_exists_resultstepprevioussum_body_steps)) * fs_v_pfc_division_exists_resultstepprevioussum)) /\\ exists fs_q_pfc_division_exists_resultstepprevioussum_body_steps_successor. fs_u_pfc_division_exists_resultstepprevioussum = fs_q_pfc_division_exists_resultstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_division_exists_resultstepprevioussum_body_steps)) * fs_v_pfc_division_exists_resultstepprevioussum) + (fs_s_pfc_division_exists_resultstepprevioussum_body_steps))) /\\ fs_s_pfc_division_exists_resultstepprevioussum_body_steps = fs_r_pfc_division_exists_resultstepprevioussum_body_steps + fs_a_pfc_division_exists_resultstepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_division_exists_resultsteppreviousresiduebound. pfa_gap_division_exists_resultsteppreviousresiduebound + S (pfd_previous_division_exists_resultstep) = (p)) /\\ ((exists pfa_offset_left_division_exists_resultsteppreviousresiduecongruence pfa_offset_right_division_exists_resultsteppreviousresiduecongruence. (pfc_natural_sum_division_exists_resultstepprevious) + (p) * pfa_offset_left_division_exists_resultsteppreviousresiduecongruence = (pfd_previous_division_exists_resultstep) + (p) * pfa_offset_right_division_exists_resultsteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_division_exists_resultstepsubtractleft. pfa_gap_division_exists_resultstepsubtractleft + S (pfd_previous_division_exists_resultstep) = (p)) /\\ (((exists pfa_gap_division_exists_resultstepsubtractright. pfa_gap_division_exists_resultstepsubtractright + S (pfd_difference_division_exists_resultstep) = (p)) /\\ ((((exists pfa_gap_division_exists_resultstepsubtractresultbound. pfa_gap_division_exists_resultstepsubtractresultbound + S (pfd_input_division_exists_resultstep) = (p)) /\\ ((exists pfa_offset_left_division_exists_resultstepsubtractresultcongruence pfa_offset_right_division_exists_resultstepsubtractresultcongruence. ((pfd_previous_division_exists_resultstep) + (pfd_difference_division_exists_resultstep)) + (p) * pfa_offset_left_division_exists_resultstepsubtractresultcongruence = (pfd_input_division_exists_resultstep) + (p) * pfa_offset_right_division_exists_resultstepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_division_exists_resultstepmultiplyleft. pfa_gap_division_exists_resultstepmultiplyleft + S (k) = (p)) /\\ (((exists pfa_gap_division_exists_resultstepmultiplyright. pfa_gap_division_exists_resultstepmultiplyright + S (pfd_difference_division_exists_resultstep) = (p)) /\\ ((((exists pfa_gap_division_exists_resultstepmultiplyresultbound. pfa_gap_division_exists_resultstepmultiplyresultbound + S (pfd_value_division_exists_result) = (p)) /\\ ((exists pfa_offset_left_division_exists_resultstepmultiplyresultcongruence pfa_offset_right_division_exists_resultstepmultiplyresultcongruence. 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"script": [ "intro p", "intro k", "intro ab", "intro ac", "intro bb", "intro bc", "intro d", "intro qb", "intro qc", "intro N", "intro b", "intro i", "intro r", "intro hp", "intro hb", "intro hk", "intro hq", "intro hi", "intro hr", "have hpoint : exists q. ((((exists ff_h_pfp_division_match_quotient_entry. ff_h_pfp_division_match_quotient_entry + S (q) = S ((S (i)) * qc)) /\\ exists ff_q_pfp_division_match_quotient_entry. qb = ff_q_pfp_division_match_quotient_entry * S ((S (i)) * qc) + (q))) /\\ ((exists pfd_input_division_match_step pfd_previous_division_match_step pfd_difference_division_match_step. ((((exists ff_h_pfp_division_match_stepinput. ff_h_pfp_division_match_stepinput + S (pfd_input_division_match_step) = S ((S (i)) * ac)) /\\ exists ff_q_pfp_division_match_stepinput. ab = ff_q_pfp_division_match_stepinput * S ((S (i)) * ac) + (pfd_input_division_match_step))) /\\ (((exists pfc_terms_code_division_match_stepprevious pfc_terms_scale_division_match_stepprevious pfc_natural_sum_division_match_stepprevious. ((forall pfc_index_division_match_steppreviousdiagonal. (exists pfa_gap_division_match_steppreviousdiagonalbound. pfa_gap_division_match_steppreviousdiagonalbound + S (pfc_index_division_match_steppreviousdiagonal) = (S (i))) -> exists pfc_value_division_match_steppreviousdiagonal. ((((exists ff_h_pfp_division_match_steppreviousdiagonalentry. ff_h_pfp_division_match_steppreviousdiagonalentry + S (pfc_value_division_match_steppreviousdiagonal) = S ((S (pfc_index_division_match_steppreviousdiagonal)) * pfc_terms_scale_division_match_stepprevious)) /\\ exists ff_q_pfp_division_match_steppreviousdiagonalentry. pfc_terms_code_division_match_stepprevious = ff_q_pfp_division_match_steppreviousdiagonalentry * S ((S (pfc_index_division_match_steppreviousdiagonal)) * pfc_terms_scale_division_match_stepprevious) + (pfc_value_division_match_steppreviousdiagonal))) /\\ ((exists pfc_complement_division_match_steppreviousdiagonalterm pfc_left_division_match_steppreviousdiagonalterm pfc_right_division_match_steppreviousdiagonalterm. (((pfc_index_division_match_steppreviousdiagonal)+pfc_complement_division_match_steppreviousdiagonalterm=(i)) /\\ ((((((exists pfa_gap_division_match_steppreviousdiagonaltermleftinside. pfa_gap_division_match_steppreviousdiagonaltermleftinside + S (pfc_index_division_match_steppreviousdiagonal) = (i)) /\\ ((((exists ff_h_pfp_division_match_steppreviousdiagonaltermleftentry. ff_h_pfp_division_match_steppreviousdiagonaltermleftentry + S (pfc_left_division_match_steppreviousdiagonalterm) = S ((S (pfc_index_division_match_steppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_division_match_steppreviousdiagonaltermleftentry. qb = ff_q_pfp_division_match_steppreviousdiagonaltermleftentry * S ((S (pfc_index_division_match_steppreviousdiagonal)) * qc) + (pfc_left_division_match_steppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_match_steppreviousdiagonaltermleftoutside. pfc_gap_division_match_steppreviousdiagonaltermleftoutside+(i)=(pfc_index_division_match_steppreviousdiagonal)) /\\ (((pfc_left_division_match_steppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_match_steppreviousdiagonaltermrightinside. pfa_gap_division_match_steppreviousdiagonaltermrightinside + S (pfc_complement_division_match_steppreviousdiagonalterm) = (S d)) /\\ ((((exists ff_h_pfp_division_match_steppreviousdiagonaltermrightentry. ff_h_pfp_division_match_steppreviousdiagonaltermrightentry + S (pfc_right_division_match_steppreviousdiagonalterm) = S ((S (pfc_complement_division_match_steppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_match_steppreviousdiagonaltermrightentry. bb = ff_q_pfp_division_match_steppreviousdiagonaltermrightentry * S ((S (pfc_complement_division_match_steppreviousdiagonalterm)) * bc) + (pfc_right_division_match_steppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_match_steppreviousdiagonaltermrightoutside. pfc_gap_division_match_steppreviousdiagonaltermrightoutside+(S d)=(pfc_complement_division_match_steppreviousdiagonalterm)) /\\ (((pfc_right_division_match_steppreviousdiagonalterm)=0))))) /\\ (((pfc_value_division_match_steppreviousdiagonal)=pfc_left_division_match_steppreviousdiagonalterm*pfc_right_division_match_steppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_match_stepprevioussum fs_v_pfc_division_match_stepprevioussum. ((((exists fs_h_pfc_division_match_stepprevioussum_body_start. fs_h_pfc_division_match_stepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_match_stepprevioussum)) /\\ exists fs_q_pfc_division_match_stepprevioussum_body_start. fs_u_pfc_division_match_stepprevioussum = fs_q_pfc_division_match_stepprevioussum_body_start * S ((S (0)) * fs_v_pfc_division_match_stepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_division_match_stepprevioussum_body_terminal. fs_h_pfc_division_match_stepprevioussum_body_terminal + S (pfc_natural_sum_division_match_stepprevious) = S ((S (S (i))) * fs_v_pfc_division_match_stepprevioussum)) /\\ exists fs_q_pfc_division_match_stepprevioussum_body_terminal. fs_u_pfc_division_match_stepprevioussum = fs_q_pfc_division_match_stepprevioussum_body_terminal * S ((S (S (i))) * fs_v_pfc_division_match_stepprevioussum) + (pfc_natural_sum_division_match_stepprevious))) /\\ forall fs_i_pfc_division_match_stepprevioussum_body_steps. (exists fs_lt_pfc_division_match_stepprevioussum_body_steps_bound. fs_lt_pfc_division_match_stepprevioussum_body_steps_bound + S fs_i_pfc_division_match_stepprevioussum_body_steps = S (i)) -> exists fs_a_pfc_division_match_stepprevioussum_body_steps fs_r_pfc_division_match_stepprevioussum_body_steps fs_s_pfc_division_match_stepprevioussum_body_steps. ((((exists fs_h_pfc_division_match_stepprevioussum_body_steps_summand. fs_h_pfc_division_match_stepprevioussum_body_steps_summand + S (fs_a_pfc_division_match_stepprevioussum_body_steps) = S ((S (fs_i_pfc_division_match_stepprevioussum_body_steps)) * pfc_terms_scale_division_match_stepprevious)) /\\ exists fs_q_pfc_division_match_stepprevioussum_body_steps_summand. pfc_terms_code_division_match_stepprevious = fs_q_pfc_division_match_stepprevioussum_body_steps_summand * S ((S (fs_i_pfc_division_match_stepprevioussum_body_steps)) * pfc_terms_scale_division_match_stepprevious) + (fs_a_pfc_division_match_stepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_match_stepprevioussum_body_steps_partial. fs_h_pfc_division_match_stepprevioussum_body_steps_partial + S (fs_r_pfc_division_match_stepprevioussum_body_steps) = S ((S (fs_i_pfc_division_match_stepprevioussum_body_steps)) * fs_v_pfc_division_match_stepprevioussum)) /\\ exists fs_q_pfc_division_match_stepprevioussum_body_steps_partial. fs_u_pfc_division_match_stepprevioussum = fs_q_pfc_division_match_stepprevioussum_body_steps_partial * S ((S (fs_i_pfc_division_match_stepprevioussum_body_steps)) * fs_v_pfc_division_match_stepprevioussum) + (fs_r_pfc_division_match_stepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_match_stepprevioussum_body_steps_successor. fs_h_pfc_division_match_stepprevioussum_body_steps_successor + S (fs_s_pfc_division_match_stepprevioussum_body_steps) = S ((S (S fs_i_pfc_division_match_stepprevioussum_body_steps)) * fs_v_pfc_division_match_stepprevioussum)) /\\ exists fs_q_pfc_division_match_stepprevioussum_body_steps_successor. fs_u_pfc_division_match_stepprevioussum = fs_q_pfc_division_match_stepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_division_match_stepprevioussum_body_steps)) * fs_v_pfc_division_match_stepprevioussum) + (fs_s_pfc_division_match_stepprevioussum_body_steps))) /\\ fs_s_pfc_division_match_stepprevioussum_body_steps = fs_r_pfc_division_match_stepprevioussum_body_steps + fs_a_pfc_division_match_stepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_division_match_steppreviousresiduebound. pfa_gap_division_match_steppreviousresiduebound + S (pfd_previous_division_match_step) = (p)) /\\ ((exists pfa_offset_left_division_match_steppreviousresiduecongruence pfa_offset_right_division_match_steppreviousresiduecongruence. (pfc_natural_sum_division_match_stepprevious) + (p) * pfa_offset_left_division_match_steppreviousresiduecongruence = (pfd_previous_division_match_step) + (p) * pfa_offset_right_division_match_steppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_division_match_stepsubtractleft. pfa_gap_division_match_stepsubtractleft + S (pfd_previous_division_match_step) = (p)) /\\ (((exists pfa_gap_division_match_stepsubtractright. pfa_gap_division_match_stepsubtractright + S (pfd_difference_division_match_step) = (p)) /\\ ((((exists pfa_gap_division_match_stepsubtractresultbound. pfa_gap_division_match_stepsubtractresultbound + S (pfd_input_division_match_step) = (p)) /\\ ((exists pfa_offset_left_division_match_stepsubtractresultcongruence pfa_offset_right_division_match_stepsubtractresultcongruence. ((pfd_previous_division_match_step) + (pfd_difference_division_match_step)) + (p) * pfa_offset_left_division_match_stepsubtractresultcongruence = (pfd_input_division_match_step) + (p) * pfa_offset_right_division_match_stepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_division_match_stepmultiplyleft. pfa_gap_division_match_stepmultiplyleft + S (k) = (p)) /\\ (((exists pfa_gap_division_match_stepmultiplyright. pfa_gap_division_match_stepmultiplyright + S (pfd_difference_division_match_step) = (p)) /\\ ((((exists pfa_gap_division_match_stepmultiplyresultbound. pfa_gap_division_match_stepmultiplyresultbound + S (q) = (p)) /\\ ((exists pfa_offset_left_division_match_stepmultiplyresultcongruence pfa_offset_right_division_match_stepmultiplyresultcongruence. ((k) * (pfd_difference_division_match_step)) + (p) * pfa_offset_left_division_match_stepmultiplyresultcongruence = (q) + (p) * pfa_offset_right_division_match_stepmultiplyresultcongruence))))))))))))))))))", "specialize hq (i)", "apply hq", "exact hi", "cases hpoint", "cases hpoint_witness", "cases hpoint_witness_right", "cases hpoint_witness_right_witness", "cases hpoint_witness_right_witness_witness", "cases hpoint_witness_right_witness_witness_witness", "cases hpoint_witness_right_witness_witness_witness_right", "cases hpoint_witness_right_witness_witness_witness_right_right", "have hqbound : exists pfa_gap_division_match_value_bound. pfa_gap_division_match_value_bound + S (x) = (p)", "cases hpoint_witness_right_witness_witness_witness_right_right_right", "cases hpoint_witness_right_witness_witness_witness_right_right_right_right", "cases hpoint_witness_right_witness_witness_witness_right_right_right_right_right", "exact hpoint_witness_right_witness_witness_witness_right_right_right_right_right_left", "have hbnd : exists pfa_gap_division_match_head_bound. pfa_gap_division_match_head_bound + S (b) = (p)", "cases hk", "cases hk_right", "exact hk_right_left", "have hproduct : exists t. (((exists pfa_gap_division_match_productleft. pfa_gap_division_match_productleft + S (x) = (p)) /\\ (((exists pfa_gap_division_match_productright. pfa_gap_division_match_productright + S (b) = (p)) /\\ ((((exists pfa_gap_division_match_productresultbound. pfa_gap_division_match_productresultbound + S (t) = (p)) /\\ ((exists pfa_offset_left_division_match_productresultcongruence pfa_offset_right_division_match_productresultcongruence. ((x) * (b)) + (p) * pfa_offset_left_division_match_productresultcongruence = (t) + (p) * pfa_offset_right_division_match_productresultcongruence)))))))))", "specialize prime_field_multiply_exists (p)", "specialize prime_field_multiply_exists (x)", "specialize prime_field_multiply_exists (b)", "apply prime_field_multiply_exists", "exact hp", "exact hqbound", "exact hbnd", "cases hproduct", "have hshort : exists pfc_terms_code_division_match_shorter pfc_terms_scale_division_match_shorter pfc_natural_sum_division_match_shorter. ((forall pfc_index_division_match_shorterdiagonal. (exists pfa_gap_division_match_shorterdiagonalbound. pfa_gap_division_match_shorterdiagonalbound + S (pfc_index_division_match_shorterdiagonal) = (S (i))) -> exists pfc_value_division_match_shorterdiagonal. ((((exists ff_h_pfp_division_match_shorterdiagonalentry. ff_h_pfp_division_match_shorterdiagonalentry + S (pfc_value_division_match_shorterdiagonal) = S ((S (pfc_index_division_match_shorterdiagonal)) * pfc_terms_scale_division_match_shorter)) /\\ exists ff_q_pfp_division_match_shorterdiagonalentry. pfc_terms_code_division_match_shorter = ff_q_pfp_division_match_shorterdiagonalentry * S ((S (pfc_index_division_match_shorterdiagonal)) * pfc_terms_scale_division_match_shorter) + (pfc_value_division_match_shorterdiagonal))) /\\ ((exists pfc_complement_division_match_shorterdiagonalterm pfc_left_division_match_shorterdiagonalterm pfc_right_division_match_shorterdiagonalterm. (((pfc_index_division_match_shorterdiagonal)+pfc_complement_division_match_shorterdiagonalterm=(i)) /\\ ((((((exists pfa_gap_division_match_shorterdiagonaltermleftinside. pfa_gap_division_match_shorterdiagonaltermleftinside + S (pfc_index_division_match_shorterdiagonal) = (S i)) /\\ ((((exists ff_h_pfp_division_match_shorterdiagonaltermleftentry. ff_h_pfp_division_match_shorterdiagonaltermleftentry + S (pfc_left_division_match_shorterdiagonalterm) = S ((S (pfc_index_division_match_shorterdiagonal)) * qc)) /\\ exists ff_q_pfp_division_match_shorterdiagonaltermleftentry. qb = ff_q_pfp_division_match_shorterdiagonaltermleftentry * S ((S (pfc_index_division_match_shorterdiagonal)) * qc) + (pfc_left_division_match_shorterdiagonalterm)))))) \\/ (((exists pfc_gap_division_match_shorterdiagonaltermleftoutside. pfc_gap_division_match_shorterdiagonaltermleftoutside+(S i)=(pfc_index_division_match_shorterdiagonal)) /\\ (((pfc_left_division_match_shorterdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_match_shorterdiagonaltermrightinside. pfa_gap_division_match_shorterdiagonaltermrightinside + S (pfc_complement_division_match_shorterdiagonalterm) = (S d)) /\\ ((((exists ff_h_pfp_division_match_shorterdiagonaltermrightentry. ff_h_pfp_division_match_shorterdiagonaltermrightentry + S (pfc_right_division_match_shorterdiagonalterm) = S ((S (pfc_complement_division_match_shorterdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_match_shorterdiagonaltermrightentry. bb = ff_q_pfp_division_match_shorterdiagonaltermrightentry * S ((S (pfc_complement_division_match_shorterdiagonalterm)) * bc) + (pfc_right_division_match_shorterdiagonalterm)))))) \\/ (((exists pfc_gap_division_match_shorterdiagonaltermrightoutside. pfc_gap_division_match_shorterdiagonaltermrightoutside+(S d)=(pfc_complement_division_match_shorterdiagonalterm)) /\\ (((pfc_right_division_match_shorterdiagonalterm)=0))))) /\\ (((pfc_value_division_match_shorterdiagonal)=pfc_left_division_match_shorterdiagonalterm*pfc_right_division_match_shorterdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_match_shortersum fs_v_pfc_division_match_shortersum. ((((exists fs_h_pfc_division_match_shortersum_body_start. fs_h_pfc_division_match_shortersum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_match_shortersum)) /\\ exists fs_q_pfc_division_match_shortersum_body_start. fs_u_pfc_division_match_shortersum = fs_q_pfc_division_match_shortersum_body_start * S ((S (0)) * fs_v_pfc_division_match_shortersum) + (0))) /\\ ((((exists fs_h_pfc_division_match_shortersum_body_terminal. fs_h_pfc_division_match_shortersum_body_terminal + S (pfc_natural_sum_division_match_shorter) = S ((S (S (i))) * fs_v_pfc_division_match_shortersum)) /\\ exists fs_q_pfc_division_match_shortersum_body_terminal. fs_u_pfc_division_match_shortersum = fs_q_pfc_division_match_shortersum_body_terminal * S ((S (S (i))) * fs_v_pfc_division_match_shortersum) + (pfc_natural_sum_division_match_shorter))) /\\ forall fs_i_pfc_division_match_shortersum_body_steps. (exists fs_lt_pfc_division_match_shortersum_body_steps_bound. fs_lt_pfc_division_match_shortersum_body_steps_bound + S fs_i_pfc_division_match_shortersum_body_steps = S (i)) -> exists fs_a_pfc_division_match_shortersum_body_steps fs_r_pfc_division_match_shortersum_body_steps fs_s_pfc_division_match_shortersum_body_steps. ((((exists fs_h_pfc_division_match_shortersum_body_steps_summand. fs_h_pfc_division_match_shortersum_body_steps_summand + S (fs_a_pfc_division_match_shortersum_body_steps) = S ((S (fs_i_pfc_division_match_shortersum_body_steps)) * pfc_terms_scale_division_match_shorter)) /\\ exists fs_q_pfc_division_match_shortersum_body_steps_summand. pfc_terms_code_division_match_shorter = fs_q_pfc_division_match_shortersum_body_steps_summand * S ((S (fs_i_pfc_division_match_shortersum_body_steps)) * pfc_terms_scale_division_match_shorter) + (fs_a_pfc_division_match_shortersum_body_steps))) /\\ ((((exists fs_h_pfc_division_match_shortersum_body_steps_partial. fs_h_pfc_division_match_shortersum_body_steps_partial + S (fs_r_pfc_division_match_shortersum_body_steps) = S ((S (fs_i_pfc_division_match_shortersum_body_steps)) * fs_v_pfc_division_match_shortersum)) /\\ exists fs_q_pfc_division_match_shortersum_body_steps_partial. fs_u_pfc_division_match_shortersum = fs_q_pfc_division_match_shortersum_body_steps_partial * S ((S (fs_i_pfc_division_match_shortersum_body_steps)) * fs_v_pfc_division_match_shortersum) + (fs_r_pfc_division_match_shortersum_body_steps))) /\\ ((((exists fs_h_pfc_division_match_shortersum_body_steps_successor. fs_h_pfc_division_match_shortersum_body_steps_successor + S (fs_s_pfc_division_match_shortersum_body_steps) = S ((S (S fs_i_pfc_division_match_shortersum_body_steps)) * fs_v_pfc_division_match_shortersum)) /\\ exists fs_q_pfc_division_match_shortersum_body_steps_successor. fs_u_pfc_division_match_shortersum = fs_q_pfc_division_match_shortersum_body_steps_successor * S ((S (S fs_i_pfc_division_match_shortersum_body_steps)) * fs_v_pfc_division_match_shortersum) + (fs_s_pfc_division_match_shortersum_body_steps))) /\\ fs_s_pfc_division_match_shortersum_body_steps = fs_r_pfc_division_match_shortersum_body_steps + fs_a_pfc_division_match_shortersum_body_steps)))))) /\\ ((((exists pfa_gap_division_match_shorterresiduebound. pfa_gap_division_match_shorterresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_division_match_shorterresiduecongruence pfa_offset_right_division_match_shorterresiduecongruence. (pfc_natural_sum_division_match_shorter) + (p) * pfa_offset_left_division_match_shorterresiduecongruence = (r) + (p) * pfa_offset_right_division_match_shorterresiduecongruence))))))))", "specialize prime_field_convolution_coefficient_prefix_transport (p)", "specialize prime_field_convolution_coefficient_prefix_transport (qb)", "specialize prime_field_convolution_coefficient_prefix_transport (qc)", "specialize prime_field_convolution_coefficient_prefix_transport (N)", "specialize prime_field_convolution_coefficient_prefix_transport (qb)", "specialize prime_field_convolution_coefficient_prefix_transport (qc)", "specialize prime_field_convolution_coefficient_prefix_transport (S i)", "specialize prime_field_convolution_coefficient_prefix_transport (bb)", "specialize prime_field_convolution_coefficient_prefix_transport (bc)", "specialize prime_field_convolution_coefficient_prefix_transport (S d)", "specialize prime_field_convolution_coefficient_prefix_transport (S i)", "specialize prime_field_convolution_coefficient_prefix_transport (i)", "specialize prime_field_convolution_coefficient_prefix_transport (r)", "apply prime_field_convolution_coefficient_prefix_transport", "exact hi", "specialize le_refl (S i)", "apply le_refl", "intro j", "intro v", "intro hj", "intro hv", "exact hv", "specialize le_refl (S i)", "apply le_refl", "exact hr", "have hsum : ((exists pfa_gap_division_match_sumleft. pfa_gap_division_match_sumleft + S (x2) = (p)) /\\ (((exists pfa_gap_division_match_sumright. pfa_gap_division_match_sumright + S (x4) = (p)) /\\ ((((exists pfa_gap_division_match_sumresultbound. pfa_gap_division_match_sumresultbound + S (r) = (p)) /\\ ((exists pfa_offset_left_division_match_sumresultcongruence pfa_offset_right_division_match_sumresultcongruence. ((x2) + (x4)) + (p) * pfa_offset_left_division_match_sumresultcongruence = (r) + (p) * pfa_offset_right_division_match_sumresultcongruence))))))))", "specialize prime_field_convolution_coefficient_append (p)", "specialize prime_field_convolution_coefficient_append (qb)", "specialize prime_field_convolution_coefficient_append (qc)", "specialize prime_field_convolution_coefficient_append (qb)", "specialize prime_field_convolution_coefficient_append (qc)", "specialize prime_field_convolution_coefficient_append (bb)", "specialize prime_field_convolution_coefficient_append (bc)", "specialize prime_field_convolution_coefficient_append (d)", "specialize prime_field_convolution_coefficient_append (i)", "specialize prime_field_convolution_coefficient_append (x)", "specialize prime_field_convolution_coefficient_append (b)", "specialize prime_field_convolution_coefficient_append (x2)", "specialize prime_field_convolution_coefficient_append (x4)", "specialize prime_field_convolution_coefficient_append (r)", "apply prime_field_convolution_coefficient_append", "intro j", "intro v", "intro hj", "intro hv", "exact hv", "exact hpoint_witness_left", "exact hb", "exact hpoint_witness_right_witness_witness_witness_right_left", "exact hshort", "exact hproduct_witness", "have heq : r=x1", "specialize prime_field_polynomial_quotient_scalar_cancellation (p)", "specialize prime_field_polynomial_quotient_scalar_cancellation (b)", "specialize prime_field_polynomial_quotient_scalar_cancellation (k)", "specialize prime_field_polynomial_quotient_scalar_cancellation (x2)", "specialize prime_field_polynomial_quotient_scalar_cancellation (x3)", "specialize prime_field_polynomial_quotient_scalar_cancellation (x1)", "specialize prime_field_polynomial_quotient_scalar_cancellation (x)", "specialize prime_field_polynomial_quotient_scalar_cancellation (x4)", "specialize prime_field_polynomial_quotient_scalar_cancellation (r)", "apply prime_field_polynomial_quotient_scalar_cancellation", "exact hp", "exact hk", "exact hpoint_witness_right_witness_witness_witness_right_right_left", "exact hpoint_witness_right_witness_witness_witness_right_right_right", "exact hproduct_witness", "exact hsum", "rewrite heq", "rewrite heq", "exact hpoint_witness_right_witness_witness_witness_left" ], "script_sha256": "15bf65dd20dc24501811aae86d3d5874418e2635e51831434f9454f46d6a6f48", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_division_candidate.py", "sha256": "edfc7806caf7a83b9cb0e3e420bd2c3a8679f2d4d9ee6ca9f8eae53faca8d5b2" }, "statement": "forall p k ab ac bb bc d qb qc N b i r. (~((p) = 1) /\\ forall pfa_factor_left_division_match_prime pfa_factor_right_division_match_prime. (p) = pfa_factor_left_division_match_prime * pfa_factor_right_division_match_prime -> pfa_factor_left_division_match_prime = 1 \\/ pfa_factor_right_division_match_prime = 1) -> (((exists ff_h_pfp_division_match_head. ff_h_pfp_division_match_head + S (b) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_division_match_head. bb = ff_q_pfp_division_match_head * S ((S (0)) * bc) + (b))) -> (((~((b) = 0)) /\\ ((((exists pfa_gap_division_match_inversemultiplicationleft. pfa_gap_division_match_inversemultiplicationleft + S (b) = (p)) /\\ (((exists pfa_gap_division_match_inversemultiplicationright. pfa_gap_division_match_inversemultiplicationright + S (k) = (p)) /\\ ((((exists pfa_gap_division_match_inversemultiplicationresultbound. pfa_gap_division_match_inversemultiplicationresultbound + S (1) = (p)) /\\ ((exists pfa_offset_left_division_match_inversemultiplicationresultcongruence pfa_offset_right_division_match_inversemultiplicationresultcongruence. ((b) * (k)) + (p) * pfa_offset_left_division_match_inversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_division_match_inversemultiplicationresultcongruence)))))))))))) -> (forall pfd_index_division_match_execution. (exists pfa_gap_division_match_executionbound. pfa_gap_division_match_executionbound + S (pfd_index_division_match_execution) = (N)) -> exists pfd_value_division_match_execution. ((((exists ff_h_pfp_division_match_executionentry. ff_h_pfp_division_match_executionentry + S (pfd_value_division_match_execution) = S ((S (pfd_index_division_match_execution)) * qc)) /\\ exists ff_q_pfp_division_match_executionentry. qb = ff_q_pfp_division_match_executionentry * S ((S (pfd_index_division_match_execution)) * qc) + (pfd_value_division_match_execution))) /\\ ((exists pfd_input_division_match_executionstep pfd_previous_division_match_executionstep pfd_difference_division_match_executionstep. ((((exists ff_h_pfp_division_match_executionstepinput. ff_h_pfp_division_match_executionstepinput + S (pfd_input_division_match_executionstep) = S ((S (pfd_index_division_match_execution)) * ac)) /\\ exists ff_q_pfp_division_match_executionstepinput. ab = ff_q_pfp_division_match_executionstepinput * S ((S (pfd_index_division_match_execution)) * ac) + (pfd_input_division_match_executionstep))) /\\ (((exists pfc_terms_code_division_match_executionstepprevious pfc_terms_scale_division_match_executionstepprevious pfc_natural_sum_division_match_executionstepprevious. ((forall pfc_index_division_match_executionsteppreviousdiagonal. (exists pfa_gap_division_match_executionsteppreviousdiagonalbound. pfa_gap_division_match_executionsteppreviousdiagonalbound + S (pfc_index_division_match_executionsteppreviousdiagonal) = (S (pfd_index_division_match_execution))) -> exists pfc_value_division_match_executionsteppreviousdiagonal. ((((exists ff_h_pfp_division_match_executionsteppreviousdiagonalentry. ff_h_pfp_division_match_executionsteppreviousdiagonalentry + S (pfc_value_division_match_executionsteppreviousdiagonal) = S ((S (pfc_index_division_match_executionsteppreviousdiagonal)) * pfc_terms_scale_division_match_executionstepprevious)) /\\ exists ff_q_pfp_division_match_executionsteppreviousdiagonalentry. pfc_terms_code_division_match_executionstepprevious = ff_q_pfp_division_match_executionsteppreviousdiagonalentry * S ((S (pfc_index_division_match_executionsteppreviousdiagonal)) * pfc_terms_scale_division_match_executionstepprevious) + (pfc_value_division_match_executionsteppreviousdiagonal))) /\\ ((exists pfc_complement_division_match_executionsteppreviousdiagonalterm pfc_left_division_match_executionsteppreviousdiagonalterm pfc_right_division_match_executionsteppreviousdiagonalterm. (((pfc_index_division_match_executionsteppreviousdiagonal)+pfc_complement_division_match_executionsteppreviousdiagonalterm=(pfd_index_division_match_execution)) /\\ ((((((exists pfa_gap_division_match_executionsteppreviousdiagonaltermleftinside. pfa_gap_division_match_executionsteppreviousdiagonaltermleftinside + S (pfc_index_division_match_executionsteppreviousdiagonal) = (pfd_index_division_match_execution)) /\\ ((((exists ff_h_pfp_division_match_executionsteppreviousdiagonaltermleftentry. ff_h_pfp_division_match_executionsteppreviousdiagonaltermleftentry + S (pfc_left_division_match_executionsteppreviousdiagonalterm) = S ((S (pfc_index_division_match_executionsteppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_division_match_executionsteppreviousdiagonaltermleftentry. qb = ff_q_pfp_division_match_executionsteppreviousdiagonaltermleftentry * S ((S (pfc_index_division_match_executionsteppreviousdiagonal)) * qc) + (pfc_left_division_match_executionsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_match_executionsteppreviousdiagonaltermleftoutside. pfc_gap_division_match_executionsteppreviousdiagonaltermleftoutside+(pfd_index_division_match_execution)=(pfc_index_division_match_executionsteppreviousdiagonal)) /\\ (((pfc_left_division_match_executionsteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_match_executionsteppreviousdiagonaltermrightinside. pfa_gap_division_match_executionsteppreviousdiagonaltermrightinside + S (pfc_complement_division_match_executionsteppreviousdiagonalterm) = (S d)) /\\ ((((exists ff_h_pfp_division_match_executionsteppreviousdiagonaltermrightentry. ff_h_pfp_division_match_executionsteppreviousdiagonaltermrightentry + S (pfc_right_division_match_executionsteppreviousdiagonalterm) = S ((S (pfc_complement_division_match_executionsteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_match_executionsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_division_match_executionsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_division_match_executionsteppreviousdiagonalterm)) * bc) + (pfc_right_division_match_executionsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_match_executionsteppreviousdiagonaltermrightoutside. pfc_gap_division_match_executionsteppreviousdiagonaltermrightoutside+(S d)=(pfc_complement_division_match_executionsteppreviousdiagonalterm)) /\\ (((pfc_right_division_match_executionsteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_division_match_executionsteppreviousdiagonal)=pfc_left_division_match_executionsteppreviousdiagonalterm*pfc_right_division_match_executionsteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_match_executionstepprevioussum fs_v_pfc_division_match_executionstepprevioussum. ((((exists fs_h_pfc_division_match_executionstepprevioussum_body_start. fs_h_pfc_division_match_executionstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_match_executionstepprevioussum)) /\\ exists fs_q_pfc_division_match_executionstepprevioussum_body_start. fs_u_pfc_division_match_executionstepprevioussum = fs_q_pfc_division_match_executionstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_division_match_executionstepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_division_match_executionstepprevioussum_body_terminal. fs_h_pfc_division_match_executionstepprevioussum_body_terminal + S (pfc_natural_sum_division_match_executionstepprevious) = S ((S (S (pfd_index_division_match_execution))) * fs_v_pfc_division_match_executionstepprevioussum)) /\\ exists fs_q_pfc_division_match_executionstepprevioussum_body_terminal. fs_u_pfc_division_match_executionstepprevioussum = fs_q_pfc_division_match_executionstepprevioussum_body_terminal * S ((S (S (pfd_index_division_match_execution))) * fs_v_pfc_division_match_executionstepprevioussum) + (pfc_natural_sum_division_match_executionstepprevious))) /\\ forall fs_i_pfc_division_match_executionstepprevioussum_body_steps. (exists fs_lt_pfc_division_match_executionstepprevioussum_body_steps_bound. fs_lt_pfc_division_match_executionstepprevioussum_body_steps_bound + S fs_i_pfc_division_match_executionstepprevioussum_body_steps = S (pfd_index_division_match_execution)) -> exists fs_a_pfc_division_match_executionstepprevioussum_body_steps fs_r_pfc_division_match_executionstepprevioussum_body_steps fs_s_pfc_division_match_executionstepprevioussum_body_steps. ((((exists fs_h_pfc_division_match_executionstepprevioussum_body_steps_summand. fs_h_pfc_division_match_executionstepprevioussum_body_steps_summand + S (fs_a_pfc_division_match_executionstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_match_executionstepprevioussum_body_steps)) * pfc_terms_scale_division_match_executionstepprevious)) /\\ exists fs_q_pfc_division_match_executionstepprevioussum_body_steps_summand. pfc_terms_code_division_match_executionstepprevious = fs_q_pfc_division_match_executionstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_division_match_executionstepprevioussum_body_steps)) * pfc_terms_scale_division_match_executionstepprevious) + (fs_a_pfc_division_match_executionstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_match_executionstepprevioussum_body_steps_partial. fs_h_pfc_division_match_executionstepprevioussum_body_steps_partial + S (fs_r_pfc_division_match_executionstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_match_executionstepprevioussum_body_steps)) * fs_v_pfc_division_match_executionstepprevioussum)) /\\ exists fs_q_pfc_division_match_executionstepprevioussum_body_steps_partial. fs_u_pfc_division_match_executionstepprevioussum = fs_q_pfc_division_match_executionstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_division_match_executionstepprevioussum_body_steps)) * fs_v_pfc_division_match_executionstepprevioussum) + (fs_r_pfc_division_match_executionstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_match_executionstepprevioussum_body_steps_successor. fs_h_pfc_division_match_executionstepprevioussum_body_steps_successor + S (fs_s_pfc_division_match_executionstepprevioussum_body_steps) = S ((S (S fs_i_pfc_division_match_executionstepprevioussum_body_steps)) * fs_v_pfc_division_match_executionstepprevioussum)) /\\ exists fs_q_pfc_division_match_executionstepprevioussum_body_steps_successor. fs_u_pfc_division_match_executionstepprevioussum = fs_q_pfc_division_match_executionstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_division_match_executionstepprevioussum_body_steps)) * fs_v_pfc_division_match_executionstepprevioussum) + (fs_s_pfc_division_match_executionstepprevioussum_body_steps))) /\\ fs_s_pfc_division_match_executionstepprevioussum_body_steps = fs_r_pfc_division_match_executionstepprevioussum_body_steps + fs_a_pfc_division_match_executionstepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_division_match_executionsteppreviousresiduebound. pfa_gap_division_match_executionsteppreviousresiduebound + S (pfd_previous_division_match_executionstep) = (p)) /\\ ((exists pfa_offset_left_division_match_executionsteppreviousresiduecongruence pfa_offset_right_division_match_executionsteppreviousresiduecongruence. (pfc_natural_sum_division_match_executionstepprevious) + (p) * pfa_offset_left_division_match_executionsteppreviousresiduecongruence = (pfd_previous_division_match_executionstep) + (p) * pfa_offset_right_division_match_executionsteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_division_match_executionstepsubtractleft. pfa_gap_division_match_executionstepsubtractleft + S (pfd_previous_division_match_executionstep) = (p)) /\\ (((exists pfa_gap_division_match_executionstepsubtractright. pfa_gap_division_match_executionstepsubtractright + S (pfd_difference_division_match_executionstep) = (p)) /\\ ((((exists pfa_gap_division_match_executionstepsubtractresultbound. pfa_gap_division_match_executionstepsubtractresultbound + S (pfd_input_division_match_executionstep) = (p)) /\\ ((exists pfa_offset_left_division_match_executionstepsubtractresultcongruence pfa_offset_right_division_match_executionstepsubtractresultcongruence. ((pfd_previous_division_match_executionstep) + (pfd_difference_division_match_executionstep)) + (p) * pfa_offset_left_division_match_executionstepsubtractresultcongruence = (pfd_input_division_match_executionstep) + (p) * pfa_offset_right_division_match_executionstepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_division_match_executionstepmultiplyleft. pfa_gap_division_match_executionstepmultiplyleft + S (k) = (p)) /\\ (((exists pfa_gap_division_match_executionstepmultiplyright. pfa_gap_division_match_executionstepmultiplyright + S (pfd_difference_division_match_executionstep) = (p)) /\\ ((((exists pfa_gap_division_match_executionstepmultiplyresultbound. pfa_gap_division_match_executionstepmultiplyresultbound + S (pfd_value_division_match_execution) = (p)) /\\ ((exists pfa_offset_left_division_match_executionstepmultiplyresultcongruence pfa_offset_right_division_match_executionstepmultiplyresultcongruence. ((k) * (pfd_difference_division_match_executionstep)) + (p) * pfa_offset_left_division_match_executionstepmultiplyresultcongruence = (pfd_value_division_match_execution) + (p) * pfa_offset_right_division_match_executionstepmultiplyresultcongruence))))))))))))))))))) -> (exists pfa_gap_division_match_index. pfa_gap_division_match_index + S (i) = (N)) -> (exists pfc_terms_code_division_match_actual_coefficient pfc_terms_scale_division_match_actual_coefficient pfc_natural_sum_division_match_actual_coefficient. ((forall pfc_index_division_match_actual_coefficientdiagonal. (exists pfa_gap_division_match_actual_coefficientdiagonalbound. pfa_gap_division_match_actual_coefficientdiagonalbound + S (pfc_index_division_match_actual_coefficientdiagonal) = (S (i))) -> exists pfc_value_division_match_actual_coefficientdiagonal. ((((exists ff_h_pfp_division_match_actual_coefficientdiagonalentry. ff_h_pfp_division_match_actual_coefficientdiagonalentry + S (pfc_value_division_match_actual_coefficientdiagonal) = S ((S (pfc_index_division_match_actual_coefficientdiagonal)) * pfc_terms_scale_division_match_actual_coefficient)) /\\ exists ff_q_pfp_division_match_actual_coefficientdiagonalentry. pfc_terms_code_division_match_actual_coefficient = ff_q_pfp_division_match_actual_coefficientdiagonalentry * S ((S (pfc_index_division_match_actual_coefficientdiagonal)) * pfc_terms_scale_division_match_actual_coefficient) + (pfc_value_division_match_actual_coefficientdiagonal))) /\\ ((exists pfc_complement_division_match_actual_coefficientdiagonalterm pfc_left_division_match_actual_coefficientdiagonalterm pfc_right_division_match_actual_coefficientdiagonalterm. (((pfc_index_division_match_actual_coefficientdiagonal)+pfc_complement_division_match_actual_coefficientdiagonalterm=(i)) /\\ ((((((exists pfa_gap_division_match_actual_coefficientdiagonaltermleftinside. pfa_gap_division_match_actual_coefficientdiagonaltermleftinside + S (pfc_index_division_match_actual_coefficientdiagonal) = (N)) /\\ ((((exists ff_h_pfp_division_match_actual_coefficientdiagonaltermleftentry. ff_h_pfp_division_match_actual_coefficientdiagonaltermleftentry + S (pfc_left_division_match_actual_coefficientdiagonalterm) = S ((S (pfc_index_division_match_actual_coefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_division_match_actual_coefficientdiagonaltermleftentry. qb = ff_q_pfp_division_match_actual_coefficientdiagonaltermleftentry * S ((S (pfc_index_division_match_actual_coefficientdiagonal)) * qc) + (pfc_left_division_match_actual_coefficientdiagonalterm)))))) \\/ (((exists pfc_gap_division_match_actual_coefficientdiagonaltermleftoutside. pfc_gap_division_match_actual_coefficientdiagonaltermleftoutside+(N)=(pfc_index_division_match_actual_coefficientdiagonal)) /\\ (((pfc_left_division_match_actual_coefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_match_actual_coefficientdiagonaltermrightinside. pfa_gap_division_match_actual_coefficientdiagonaltermrightinside + S (pfc_complement_division_match_actual_coefficientdiagonalterm) = (S d)) /\\ ((((exists ff_h_pfp_division_match_actual_coefficientdiagonaltermrightentry. ff_h_pfp_division_match_actual_coefficientdiagonaltermrightentry + S (pfc_right_division_match_actual_coefficientdiagonalterm) = S ((S (pfc_complement_division_match_actual_coefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_match_actual_coefficientdiagonaltermrightentry. bb = ff_q_pfp_division_match_actual_coefficientdiagonaltermrightentry * S ((S (pfc_complement_division_match_actual_coefficientdiagonalterm)) * bc) + (pfc_right_division_match_actual_coefficientdiagonalterm)))))) \\/ (((exists pfc_gap_division_match_actual_coefficientdiagonaltermrightoutside. pfc_gap_division_match_actual_coefficientdiagonaltermrightoutside+(S d)=(pfc_complement_division_match_actual_coefficientdiagonalterm)) /\\ (((pfc_right_division_match_actual_coefficientdiagonalterm)=0))))) /\\ (((pfc_value_division_match_actual_coefficientdiagonal)=pfc_left_division_match_actual_coefficientdiagonalterm*pfc_right_division_match_actual_coefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_match_actual_coefficientsum fs_v_pfc_division_match_actual_coefficientsum. ((((exists fs_h_pfc_division_match_actual_coefficientsum_body_start. fs_h_pfc_division_match_actual_coefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_match_actual_coefficientsum)) /\\ exists fs_q_pfc_division_match_actual_coefficientsum_body_start. fs_u_pfc_division_match_actual_coefficientsum = fs_q_pfc_division_match_actual_coefficientsum_body_start * S ((S (0)) * fs_v_pfc_division_match_actual_coefficientsum) + (0))) /\\ ((((exists fs_h_pfc_division_match_actual_coefficientsum_body_terminal. fs_h_pfc_division_match_actual_coefficientsum_body_terminal + S (pfc_natural_sum_division_match_actual_coefficient) = S ((S (S (i))) * fs_v_pfc_division_match_actual_coefficientsum)) /\\ exists fs_q_pfc_division_match_actual_coefficientsum_body_terminal. fs_u_pfc_division_match_actual_coefficientsum = fs_q_pfc_division_match_actual_coefficientsum_body_terminal * S ((S (S (i))) * fs_v_pfc_division_match_actual_coefficientsum) + (pfc_natural_sum_division_match_actual_coefficient))) /\\ forall fs_i_pfc_division_match_actual_coefficientsum_body_steps. (exists fs_lt_pfc_division_match_actual_coefficientsum_body_steps_bound. fs_lt_pfc_division_match_actual_coefficientsum_body_steps_bound + S fs_i_pfc_division_match_actual_coefficientsum_body_steps = S (i)) -> exists fs_a_pfc_division_match_actual_coefficientsum_body_steps fs_r_pfc_division_match_actual_coefficientsum_body_steps fs_s_pfc_division_match_actual_coefficientsum_body_steps. ((((exists fs_h_pfc_division_match_actual_coefficientsum_body_steps_summand. fs_h_pfc_division_match_actual_coefficientsum_body_steps_summand + S (fs_a_pfc_division_match_actual_coefficientsum_body_steps) = S ((S (fs_i_pfc_division_match_actual_coefficientsum_body_steps)) * pfc_terms_scale_division_match_actual_coefficient)) /\\ exists fs_q_pfc_division_match_actual_coefficientsum_body_steps_summand. pfc_terms_code_division_match_actual_coefficient = fs_q_pfc_division_match_actual_coefficientsum_body_steps_summand * S ((S (fs_i_pfc_division_match_actual_coefficientsum_body_steps)) * pfc_terms_scale_division_match_actual_coefficient) + (fs_a_pfc_division_match_actual_coefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_division_match_actual_coefficientsum_body_steps_partial. fs_h_pfc_division_match_actual_coefficientsum_body_steps_partial + S (fs_r_pfc_division_match_actual_coefficientsum_body_steps) = S ((S (fs_i_pfc_division_match_actual_coefficientsum_body_steps)) * fs_v_pfc_division_match_actual_coefficientsum)) /\\ exists fs_q_pfc_division_match_actual_coefficientsum_body_steps_partial. fs_u_pfc_division_match_actual_coefficientsum = fs_q_pfc_division_match_actual_coefficientsum_body_steps_partial * S ((S (fs_i_pfc_division_match_actual_coefficientsum_body_steps)) * fs_v_pfc_division_match_actual_coefficientsum) + (fs_r_pfc_division_match_actual_coefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_division_match_actual_coefficientsum_body_steps_successor. fs_h_pfc_division_match_actual_coefficientsum_body_steps_successor + S (fs_s_pfc_division_match_actual_coefficientsum_body_steps) = S ((S (S fs_i_pfc_division_match_actual_coefficientsum_body_steps)) * fs_v_pfc_division_match_actual_coefficientsum)) /\\ exists fs_q_pfc_division_match_actual_coefficientsum_body_steps_successor. fs_u_pfc_division_match_actual_coefficientsum = fs_q_pfc_division_match_actual_coefficientsum_body_steps_successor * S ((S (S fs_i_pfc_division_match_actual_coefficientsum_body_steps)) * fs_v_pfc_division_match_actual_coefficientsum) + (fs_s_pfc_division_match_actual_coefficientsum_body_steps))) /\\ fs_s_pfc_division_match_actual_coefficientsum_body_steps = fs_r_pfc_division_match_actual_coefficientsum_body_steps + fs_a_pfc_division_match_actual_coefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_division_match_actual_coefficientresiduebound. pfa_gap_division_match_actual_coefficientresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_division_match_actual_coefficientresiduecongruence pfa_offset_right_division_match_actual_coefficientresiduecongruence. 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"alpha_only", "name": "prime_field_polynomial_quotient_prefix_product_matches", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro k", "intro ab", "intro ac", "intro bb", "intro bc", "intro d", "intro qb", "intro qc", "intro N", "intro b", "intro pb", "intro pc", "intro L", "intro hp", "intro hb", "intro hk", "intro hq", "intro hlen", "intro hproduct", "intro i", "intro a", "intro hi", "intro ha", "have hv : exists r. ((((exists ff_h_pfp_division_match_table_entry. ff_h_pfp_division_match_table_entry + S (r) = S ((S (i)) * pc)) /\\ exists ff_q_pfp_division_match_table_entry. pb = ff_q_pfp_division_match_table_entry * S ((S (i)) * pc) + (r))) /\\ ((exists pfc_terms_code_division_match_table_coefficient pfc_terms_scale_division_match_table_coefficient pfc_natural_sum_division_match_table_coefficient. ((forall pfc_index_division_match_table_coefficientdiagonal. (exists pfa_gap_division_match_table_coefficientdiagonalbound. pfa_gap_division_match_table_coefficientdiagonalbound + S (pfc_index_division_match_table_coefficientdiagonal) = (S (i))) -> exists pfc_value_division_match_table_coefficientdiagonal. ((((exists ff_h_pfp_division_match_table_coefficientdiagonalentry. ff_h_pfp_division_match_table_coefficientdiagonalentry + S (pfc_value_division_match_table_coefficientdiagonal) = S ((S (pfc_index_division_match_table_coefficientdiagonal)) * pfc_terms_scale_division_match_table_coefficient)) /\\ exists ff_q_pfp_division_match_table_coefficientdiagonalentry. pfc_terms_code_division_match_table_coefficient = ff_q_pfp_division_match_table_coefficientdiagonalentry * S ((S (pfc_index_division_match_table_coefficientdiagonal)) * pfc_terms_scale_division_match_table_coefficient) + (pfc_value_division_match_table_coefficientdiagonal))) /\\ ((exists pfc_complement_division_match_table_coefficientdiagonalterm pfc_left_division_match_table_coefficientdiagonalterm pfc_right_division_match_table_coefficientdiagonalterm. (((pfc_index_division_match_table_coefficientdiagonal)+pfc_complement_division_match_table_coefficientdiagonalterm=(i)) /\\ ((((((exists pfa_gap_division_match_table_coefficientdiagonaltermleftinside. pfa_gap_division_match_table_coefficientdiagonaltermleftinside + S (pfc_index_division_match_table_coefficientdiagonal) = (N)) /\\ ((((exists ff_h_pfp_division_match_table_coefficientdiagonaltermleftentry. ff_h_pfp_division_match_table_coefficientdiagonaltermleftentry + S (pfc_left_division_match_table_coefficientdiagonalterm) = S ((S (pfc_index_division_match_table_coefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_division_match_table_coefficientdiagonaltermleftentry. qb = ff_q_pfp_division_match_table_coefficientdiagonaltermleftentry * S ((S (pfc_index_division_match_table_coefficientdiagonal)) * qc) + (pfc_left_division_match_table_coefficientdiagonalterm)))))) \\/ (((exists pfc_gap_division_match_table_coefficientdiagonaltermleftoutside. pfc_gap_division_match_table_coefficientdiagonaltermleftoutside+(N)=(pfc_index_division_match_table_coefficientdiagonal)) /\\ (((pfc_left_division_match_table_coefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_match_table_coefficientdiagonaltermrightinside. pfa_gap_division_match_table_coefficientdiagonaltermrightinside + S (pfc_complement_division_match_table_coefficientdiagonalterm) = (S d)) /\\ ((((exists ff_h_pfp_division_match_table_coefficientdiagonaltermrightentry. ff_h_pfp_division_match_table_coefficientdiagonaltermrightentry + S (pfc_right_division_match_table_coefficientdiagonalterm) = S ((S (pfc_complement_division_match_table_coefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_match_table_coefficientdiagonaltermrightentry. bb = ff_q_pfp_division_match_table_coefficientdiagonaltermrightentry * S ((S (pfc_complement_division_match_table_coefficientdiagonalterm)) * bc) + (pfc_right_division_match_table_coefficientdiagonalterm)))))) \\/ (((exists pfc_gap_division_match_table_coefficientdiagonaltermrightoutside. pfc_gap_division_match_table_coefficientdiagonaltermrightoutside+(S d)=(pfc_complement_division_match_table_coefficientdiagonalterm)) /\\ (((pfc_right_division_match_table_coefficientdiagonalterm)=0))))) /\\ (((pfc_value_division_match_table_coefficientdiagonal)=pfc_left_division_match_table_coefficientdiagonalterm*pfc_right_division_match_table_coefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_match_table_coefficientsum fs_v_pfc_division_match_table_coefficientsum. ((((exists fs_h_pfc_division_match_table_coefficientsum_body_start. fs_h_pfc_division_match_table_coefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_match_table_coefficientsum)) /\\ exists fs_q_pfc_division_match_table_coefficientsum_body_start. fs_u_pfc_division_match_table_coefficientsum = fs_q_pfc_division_match_table_coefficientsum_body_start * S ((S (0)) * fs_v_pfc_division_match_table_coefficientsum) + (0))) /\\ ((((exists fs_h_pfc_division_match_table_coefficientsum_body_terminal. fs_h_pfc_division_match_table_coefficientsum_body_terminal + S (pfc_natural_sum_division_match_table_coefficient) = S ((S (S (i))) * fs_v_pfc_division_match_table_coefficientsum)) /\\ exists fs_q_pfc_division_match_table_coefficientsum_body_terminal. fs_u_pfc_division_match_table_coefficientsum = fs_q_pfc_division_match_table_coefficientsum_body_terminal * S ((S (S (i))) * fs_v_pfc_division_match_table_coefficientsum) + (pfc_natural_sum_division_match_table_coefficient))) /\\ forall fs_i_pfc_division_match_table_coefficientsum_body_steps. (exists fs_lt_pfc_division_match_table_coefficientsum_body_steps_bound. fs_lt_pfc_division_match_table_coefficientsum_body_steps_bound + S fs_i_pfc_division_match_table_coefficientsum_body_steps = S (i)) -> exists fs_a_pfc_division_match_table_coefficientsum_body_steps fs_r_pfc_division_match_table_coefficientsum_body_steps fs_s_pfc_division_match_table_coefficientsum_body_steps. ((((exists fs_h_pfc_division_match_table_coefficientsum_body_steps_summand. fs_h_pfc_division_match_table_coefficientsum_body_steps_summand + S (fs_a_pfc_division_match_table_coefficientsum_body_steps) = S ((S (fs_i_pfc_division_match_table_coefficientsum_body_steps)) * pfc_terms_scale_division_match_table_coefficient)) /\\ exists fs_q_pfc_division_match_table_coefficientsum_body_steps_summand. pfc_terms_code_division_match_table_coefficient = fs_q_pfc_division_match_table_coefficientsum_body_steps_summand * S ((S (fs_i_pfc_division_match_table_coefficientsum_body_steps)) * pfc_terms_scale_division_match_table_coefficient) + (fs_a_pfc_division_match_table_coefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_division_match_table_coefficientsum_body_steps_partial. fs_h_pfc_division_match_table_coefficientsum_body_steps_partial + S (fs_r_pfc_division_match_table_coefficientsum_body_steps) = S ((S (fs_i_pfc_division_match_table_coefficientsum_body_steps)) * fs_v_pfc_division_match_table_coefficientsum)) /\\ exists fs_q_pfc_division_match_table_coefficientsum_body_steps_partial. fs_u_pfc_division_match_table_coefficientsum = fs_q_pfc_division_match_table_coefficientsum_body_steps_partial * S ((S (fs_i_pfc_division_match_table_coefficientsum_body_steps)) * fs_v_pfc_division_match_table_coefficientsum) + (fs_r_pfc_division_match_table_coefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_division_match_table_coefficientsum_body_steps_successor. fs_h_pfc_division_match_table_coefficientsum_body_steps_successor + S (fs_s_pfc_division_match_table_coefficientsum_body_steps) = S ((S (S fs_i_pfc_division_match_table_coefficientsum_body_steps)) * fs_v_pfc_division_match_table_coefficientsum)) /\\ exists fs_q_pfc_division_match_table_coefficientsum_body_steps_successor. fs_u_pfc_division_match_table_coefficientsum = fs_q_pfc_division_match_table_coefficientsum_body_steps_successor * S ((S (S fs_i_pfc_division_match_table_coefficientsum_body_steps)) * fs_v_pfc_division_match_table_coefficientsum) + (fs_s_pfc_division_match_table_coefficientsum_body_steps))) /\\ fs_s_pfc_division_match_table_coefficientsum_body_steps = fs_r_pfc_division_match_table_coefficientsum_body_steps + fs_a_pfc_division_match_table_coefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_division_match_table_coefficientresiduebound. pfa_gap_division_match_table_coefficientresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_division_match_table_coefficientresiduecongruence pfa_offset_right_division_match_table_coefficientresiduecongruence. (pfc_natural_sum_division_match_table_coefficient) + (p) * pfa_offset_left_division_match_table_coefficientresiduecongruence = (r) + (p) * pfa_offset_right_division_match_table_coefficientresiduecongruence)))))))))))", "specialize hproduct (i)", "apply hproduct", "specialize lt_of_lt_of_le (i)", "specialize lt_of_lt_of_le (N)", "specialize lt_of_lt_of_le (L)", "apply lt_of_lt_of_le", "exact hi", "exact hlen", "cases hv", "cases hv_witness", "have hinput : ((exists ff_h_pfp_division_match_table_input. ff_h_pfp_division_match_table_input + S (x) = S ((S (i)) * ac)) /\\ exists ff_q_pfp_division_match_table_input. ab = ff_q_pfp_division_match_table_input * S ((S (i)) * ac) + (x))", "specialize prime_field_polynomial_quotient_prefix_convolution_entry (p)", "specialize prime_field_polynomial_quotient_prefix_convolution_entry (k)", "specialize prime_field_polynomial_quotient_prefix_convolution_entry (ab)", "specialize prime_field_polynomial_quotient_prefix_convolution_entry (ac)", "specialize prime_field_polynomial_quotient_prefix_convolution_entry (bb)", "specialize prime_field_polynomial_quotient_prefix_convolution_entry (bc)", "specialize prime_field_polynomial_quotient_prefix_convolution_entry (d)", "specialize prime_field_polynomial_quotient_prefix_convolution_entry (qb)", "specialize prime_field_polynomial_quotient_prefix_convolution_entry (qc)", "specialize prime_field_polynomial_quotient_prefix_convolution_entry (N)", "specialize prime_field_polynomial_quotient_prefix_convolution_entry (b)", "specialize prime_field_polynomial_quotient_prefix_convolution_entry (i)", "specialize prime_field_polynomial_quotient_prefix_convolution_entry (x)", "apply prime_field_polynomial_quotient_prefix_convolution_entry", "exact hp", "exact hb", "exact hk", "exact hq", "exact hi", "exact hv_witness_right", "have heq : x=a", "specialize beta_at_unique (ab)", "specialize beta_at_unique (ac)", "specialize beta_at_unique (i)", "specialize beta_at_unique (x)", "specialize beta_at_unique (a)", "apply beta_at_unique", "exact hinput", "exact ha", "rewrite heq at hv_witness_left", "rewrite heq at hv_witness_left", "exact hv_witness_left" ], "script_sha256": "de57ddbd3e3b16286373701d0f6f53c5628167e315bdaaf71bcbe8658e3c71dd", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_division_candidate.py", "sha256": "edfc7806caf7a83b9cb0e3e420bd2c3a8679f2d4d9ee6ca9f8eae53faca8d5b2" }, "statement": "forall p k ab ac bb bc d qb qc N b pb pc L. (~((p) = 1) /\\ forall pfa_factor_left_division_match_table_prime pfa_factor_right_division_match_table_prime. (p) = pfa_factor_left_division_match_table_prime * pfa_factor_right_division_match_table_prime -> pfa_factor_left_division_match_table_prime = 1 \\/ pfa_factor_right_division_match_table_prime = 1) -> (((exists ff_h_pfp_division_match_table_head. ff_h_pfp_division_match_table_head + S (b) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_division_match_table_head. bb = ff_q_pfp_division_match_table_head * S ((S (0)) * bc) + (b))) -> (((~((b) = 0)) /\\ ((((exists pfa_gap_division_match_table_inversemultiplicationleft. pfa_gap_division_match_table_inversemultiplicationleft + S (b) = (p)) /\\ (((exists pfa_gap_division_match_table_inversemultiplicationright. pfa_gap_division_match_table_inversemultiplicationright + S (k) = (p)) /\\ ((((exists pfa_gap_division_match_table_inversemultiplicationresultbound. pfa_gap_division_match_table_inversemultiplicationresultbound + S (1) = (p)) /\\ ((exists pfa_offset_left_division_match_table_inversemultiplicationresultcongruence pfa_offset_right_division_match_table_inversemultiplicationresultcongruence. ((b) * (k)) + (p) * pfa_offset_left_division_match_table_inversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_division_match_table_inversemultiplicationresultcongruence)))))))))))) -> (forall pfd_index_division_match_table_execution. (exists pfa_gap_division_match_table_executionbound. pfa_gap_division_match_table_executionbound + S (pfd_index_division_match_table_execution) = (N)) -> exists pfd_value_division_match_table_execution. ((((exists ff_h_pfp_division_match_table_executionentry. ff_h_pfp_division_match_table_executionentry + S (pfd_value_division_match_table_execution) = S ((S (pfd_index_division_match_table_execution)) * qc)) /\\ exists ff_q_pfp_division_match_table_executionentry. qb = ff_q_pfp_division_match_table_executionentry * S ((S (pfd_index_division_match_table_execution)) * qc) + (pfd_value_division_match_table_execution))) /\\ ((exists pfd_input_division_match_table_executionstep pfd_previous_division_match_table_executionstep pfd_difference_division_match_table_executionstep. ((((exists ff_h_pfp_division_match_table_executionstepinput. ff_h_pfp_division_match_table_executionstepinput + S (pfd_input_division_match_table_executionstep) = S ((S (pfd_index_division_match_table_execution)) * ac)) /\\ exists ff_q_pfp_division_match_table_executionstepinput. ab = ff_q_pfp_division_match_table_executionstepinput * S ((S (pfd_index_division_match_table_execution)) * ac) + (pfd_input_division_match_table_executionstep))) /\\ (((exists pfc_terms_code_division_match_table_executionstepprevious pfc_terms_scale_division_match_table_executionstepprevious pfc_natural_sum_division_match_table_executionstepprevious. ((forall pfc_index_division_match_table_executionsteppreviousdiagonal. (exists pfa_gap_division_match_table_executionsteppreviousdiagonalbound. pfa_gap_division_match_table_executionsteppreviousdiagonalbound + S (pfc_index_division_match_table_executionsteppreviousdiagonal) = (S (pfd_index_division_match_table_execution))) -> exists pfc_value_division_match_table_executionsteppreviousdiagonal. ((((exists ff_h_pfp_division_match_table_executionsteppreviousdiagonalentry. ff_h_pfp_division_match_table_executionsteppreviousdiagonalentry + S (pfc_value_division_match_table_executionsteppreviousdiagonal) = S ((S (pfc_index_division_match_table_executionsteppreviousdiagonal)) * pfc_terms_scale_division_match_table_executionstepprevious)) /\\ exists ff_q_pfp_division_match_table_executionsteppreviousdiagonalentry. pfc_terms_code_division_match_table_executionstepprevious = ff_q_pfp_division_match_table_executionsteppreviousdiagonalentry * S ((S (pfc_index_division_match_table_executionsteppreviousdiagonal)) * pfc_terms_scale_division_match_table_executionstepprevious) + (pfc_value_division_match_table_executionsteppreviousdiagonal))) /\\ ((exists pfc_complement_division_match_table_executionsteppreviousdiagonalterm pfc_left_division_match_table_executionsteppreviousdiagonalterm pfc_right_division_match_table_executionsteppreviousdiagonalterm. (((pfc_index_division_match_table_executionsteppreviousdiagonal)+pfc_complement_division_match_table_executionsteppreviousdiagonalterm=(pfd_index_division_match_table_execution)) /\\ ((((((exists pfa_gap_division_match_table_executionsteppreviousdiagonaltermleftinside. pfa_gap_division_match_table_executionsteppreviousdiagonaltermleftinside + S (pfc_index_division_match_table_executionsteppreviousdiagonal) = (pfd_index_division_match_table_execution)) /\\ ((((exists ff_h_pfp_division_match_table_executionsteppreviousdiagonaltermleftentry. ff_h_pfp_division_match_table_executionsteppreviousdiagonaltermleftentry + S (pfc_left_division_match_table_executionsteppreviousdiagonalterm) = S ((S (pfc_index_division_match_table_executionsteppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_division_match_table_executionsteppreviousdiagonaltermleftentry. qb = ff_q_pfp_division_match_table_executionsteppreviousdiagonaltermleftentry * S ((S (pfc_index_division_match_table_executionsteppreviousdiagonal)) * qc) + (pfc_left_division_match_table_executionsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_match_table_executionsteppreviousdiagonaltermleftoutside. pfc_gap_division_match_table_executionsteppreviousdiagonaltermleftoutside+(pfd_index_division_match_table_execution)=(pfc_index_division_match_table_executionsteppreviousdiagonal)) /\\ (((pfc_left_division_match_table_executionsteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_match_table_executionsteppreviousdiagonaltermrightinside. pfa_gap_division_match_table_executionsteppreviousdiagonaltermrightinside + S (pfc_complement_division_match_table_executionsteppreviousdiagonalterm) = (S d)) /\\ ((((exists ff_h_pfp_division_match_table_executionsteppreviousdiagonaltermrightentry. ff_h_pfp_division_match_table_executionsteppreviousdiagonaltermrightentry + S (pfc_right_division_match_table_executionsteppreviousdiagonalterm) = S ((S (pfc_complement_division_match_table_executionsteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_match_table_executionsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_division_match_table_executionsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_division_match_table_executionsteppreviousdiagonalterm)) * bc) + (pfc_right_division_match_table_executionsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_match_table_executionsteppreviousdiagonaltermrightoutside. pfc_gap_division_match_table_executionsteppreviousdiagonaltermrightoutside+(S d)=(pfc_complement_division_match_table_executionsteppreviousdiagonalterm)) /\\ (((pfc_right_division_match_table_executionsteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_division_match_table_executionsteppreviousdiagonal)=pfc_left_division_match_table_executionsteppreviousdiagonalterm*pfc_right_division_match_table_executionsteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_match_table_executionstepprevioussum fs_v_pfc_division_match_table_executionstepprevioussum. ((((exists fs_h_pfc_division_match_table_executionstepprevioussum_body_start. fs_h_pfc_division_match_table_executionstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_match_table_executionstepprevioussum)) /\\ exists fs_q_pfc_division_match_table_executionstepprevioussum_body_start. fs_u_pfc_division_match_table_executionstepprevioussum = fs_q_pfc_division_match_table_executionstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_division_match_table_executionstepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_division_match_table_executionstepprevioussum_body_terminal. fs_h_pfc_division_match_table_executionstepprevioussum_body_terminal + S (pfc_natural_sum_division_match_table_executionstepprevious) = S ((S (S (pfd_index_division_match_table_execution))) * fs_v_pfc_division_match_table_executionstepprevioussum)) /\\ exists fs_q_pfc_division_match_table_executionstepprevioussum_body_terminal. fs_u_pfc_division_match_table_executionstepprevioussum = fs_q_pfc_division_match_table_executionstepprevioussum_body_terminal * S ((S (S (pfd_index_division_match_table_execution))) * fs_v_pfc_division_match_table_executionstepprevioussum) + (pfc_natural_sum_division_match_table_executionstepprevious))) /\\ forall fs_i_pfc_division_match_table_executionstepprevioussum_body_steps. (exists fs_lt_pfc_division_match_table_executionstepprevioussum_body_steps_bound. fs_lt_pfc_division_match_table_executionstepprevioussum_body_steps_bound + S fs_i_pfc_division_match_table_executionstepprevioussum_body_steps = S (pfd_index_division_match_table_execution)) -> exists fs_a_pfc_division_match_table_executionstepprevioussum_body_steps fs_r_pfc_division_match_table_executionstepprevioussum_body_steps fs_s_pfc_division_match_table_executionstepprevioussum_body_steps. ((((exists fs_h_pfc_division_match_table_executionstepprevioussum_body_steps_summand. fs_h_pfc_division_match_table_executionstepprevioussum_body_steps_summand + S (fs_a_pfc_division_match_table_executionstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_match_table_executionstepprevioussum_body_steps)) * pfc_terms_scale_division_match_table_executionstepprevious)) /\\ exists fs_q_pfc_division_match_table_executionstepprevioussum_body_steps_summand. pfc_terms_code_division_match_table_executionstepprevious = fs_q_pfc_division_match_table_executionstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_division_match_table_executionstepprevioussum_body_steps)) * pfc_terms_scale_division_match_table_executionstepprevious) + (fs_a_pfc_division_match_table_executionstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_match_table_executionstepprevioussum_body_steps_partial. fs_h_pfc_division_match_table_executionstepprevioussum_body_steps_partial + S (fs_r_pfc_division_match_table_executionstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_match_table_executionstepprevioussum_body_steps)) * fs_v_pfc_division_match_table_executionstepprevioussum)) /\\ exists fs_q_pfc_division_match_table_executionstepprevioussum_body_steps_partial. fs_u_pfc_division_match_table_executionstepprevioussum = fs_q_pfc_division_match_table_executionstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_division_match_table_executionstepprevioussum_body_steps)) * fs_v_pfc_division_match_table_executionstepprevioussum) + (fs_r_pfc_division_match_table_executionstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_match_table_executionstepprevioussum_body_steps_successor. fs_h_pfc_division_match_table_executionstepprevioussum_body_steps_successor + S (fs_s_pfc_division_match_table_executionstepprevioussum_body_steps) = S ((S (S fs_i_pfc_division_match_table_executionstepprevioussum_body_steps)) * fs_v_pfc_division_match_table_executionstepprevioussum)) /\\ exists fs_q_pfc_division_match_table_executionstepprevioussum_body_steps_successor. fs_u_pfc_division_match_table_executionstepprevioussum = fs_q_pfc_division_match_table_executionstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_division_match_table_executionstepprevioussum_body_steps)) * fs_v_pfc_division_match_table_executionstepprevioussum) + (fs_s_pfc_division_match_table_executionstepprevioussum_body_steps))) /\\ fs_s_pfc_division_match_table_executionstepprevioussum_body_steps = fs_r_pfc_division_match_table_executionstepprevioussum_body_steps + fs_a_pfc_division_match_table_executionstepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_division_match_table_executionsteppreviousresiduebound. pfa_gap_division_match_table_executionsteppreviousresiduebound + S (pfd_previous_division_match_table_executionstep) = (p)) /\\ ((exists pfa_offset_left_division_match_table_executionsteppreviousresiduecongruence pfa_offset_right_division_match_table_executionsteppreviousresiduecongruence. (pfc_natural_sum_division_match_table_executionstepprevious) + (p) * pfa_offset_left_division_match_table_executionsteppreviousresiduecongruence = (pfd_previous_division_match_table_executionstep) + (p) * pfa_offset_right_division_match_table_executionsteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_division_match_table_executionstepsubtractleft. pfa_gap_division_match_table_executionstepsubtractleft + S (pfd_previous_division_match_table_executionstep) = (p)) /\\ (((exists pfa_gap_division_match_table_executionstepsubtractright. pfa_gap_division_match_table_executionstepsubtractright + S (pfd_difference_division_match_table_executionstep) = (p)) /\\ ((((exists pfa_gap_division_match_table_executionstepsubtractresultbound. pfa_gap_division_match_table_executionstepsubtractresultbound + S (pfd_input_division_match_table_executionstep) = (p)) /\\ ((exists pfa_offset_left_division_match_table_executionstepsubtractresultcongruence pfa_offset_right_division_match_table_executionstepsubtractresultcongruence. ((pfd_previous_division_match_table_executionstep) + (pfd_difference_division_match_table_executionstep)) + (p) * pfa_offset_left_division_match_table_executionstepsubtractresultcongruence = (pfd_input_division_match_table_executionstep) + (p) * pfa_offset_right_division_match_table_executionstepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_division_match_table_executionstepmultiplyleft. pfa_gap_division_match_table_executionstepmultiplyleft + S (k) = (p)) /\\ (((exists pfa_gap_division_match_table_executionstepmultiplyright. pfa_gap_division_match_table_executionstepmultiplyright + S (pfd_difference_division_match_table_executionstep) = (p)) /\\ ((((exists pfa_gap_division_match_table_executionstepmultiplyresultbound. pfa_gap_division_match_table_executionstepmultiplyresultbound + S (pfd_value_division_match_table_execution) = (p)) /\\ ((exists pfa_offset_left_division_match_table_executionstepmultiplyresultcongruence pfa_offset_right_division_match_table_executionstepmultiplyresultcongruence. ((k) * (pfd_difference_division_match_table_executionstep)) + (p) * pfa_offset_left_division_match_table_executionstepmultiplyresultcongruence = (pfd_value_division_match_table_execution) + (p) * pfa_offset_right_division_match_table_executionstepmultiplyresultcongruence))))))))))))))))))) -> (exists pfc_gap_division_match_table_length. pfc_gap_division_match_table_length+(N)=(L)) -> (forall pfc_index_division_match_table_product. (exists pfa_gap_division_match_table_productbound. pfa_gap_division_match_table_productbound + S (pfc_index_division_match_table_product) = (L)) -> exists pfc_value_division_match_table_product. ((((exists ff_h_pfp_division_match_table_productentry. ff_h_pfp_division_match_table_productentry + S (pfc_value_division_match_table_product) = S ((S (pfc_index_division_match_table_product)) * pc)) /\\ exists ff_q_pfp_division_match_table_productentry. pb = ff_q_pfp_division_match_table_productentry * S ((S (pfc_index_division_match_table_product)) * pc) + (pfc_value_division_match_table_product))) /\\ ((exists pfc_terms_code_division_match_table_productcoefficient pfc_terms_scale_division_match_table_productcoefficient pfc_natural_sum_division_match_table_productcoefficient. ((forall pfc_index_division_match_table_productcoefficientdiagonal. (exists pfa_gap_division_match_table_productcoefficientdiagonalbound. pfa_gap_division_match_table_productcoefficientdiagonalbound + S (pfc_index_division_match_table_productcoefficientdiagonal) = (S (pfc_index_division_match_table_product))) -> exists pfc_value_division_match_table_productcoefficientdiagonal. ((((exists ff_h_pfp_division_match_table_productcoefficientdiagonalentry. ff_h_pfp_division_match_table_productcoefficientdiagonalentry + S (pfc_value_division_match_table_productcoefficientdiagonal) = S ((S (pfc_index_division_match_table_productcoefficientdiagonal)) * pfc_terms_scale_division_match_table_productcoefficient)) /\\ exists ff_q_pfp_division_match_table_productcoefficientdiagonalentry. pfc_terms_code_division_match_table_productcoefficient = ff_q_pfp_division_match_table_productcoefficientdiagonalentry * S ((S (pfc_index_division_match_table_productcoefficientdiagonal)) * pfc_terms_scale_division_match_table_productcoefficient) + (pfc_value_division_match_table_productcoefficientdiagonal))) /\\ ((exists pfc_complement_division_match_table_productcoefficientdiagonalterm pfc_left_division_match_table_productcoefficientdiagonalterm pfc_right_division_match_table_productcoefficientdiagonalterm. (((pfc_index_division_match_table_productcoefficientdiagonal)+pfc_complement_division_match_table_productcoefficientdiagonalterm=(pfc_index_division_match_table_product)) /\\ ((((((exists pfa_gap_division_match_table_productcoefficientdiagonaltermleftinside. pfa_gap_division_match_table_productcoefficientdiagonaltermleftinside + S (pfc_index_division_match_table_productcoefficientdiagonal) = (N)) /\\ ((((exists ff_h_pfp_division_match_table_productcoefficientdiagonaltermleftentry. ff_h_pfp_division_match_table_productcoefficientdiagonaltermleftentry + S (pfc_left_division_match_table_productcoefficientdiagonalterm) = S ((S (pfc_index_division_match_table_productcoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_division_match_table_productcoefficientdiagonaltermleftentry. qb = ff_q_pfp_division_match_table_productcoefficientdiagonaltermleftentry * S ((S (pfc_index_division_match_table_productcoefficientdiagonal)) * qc) + (pfc_left_division_match_table_productcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_division_match_table_productcoefficientdiagonaltermleftoutside. pfc_gap_division_match_table_productcoefficientdiagonaltermleftoutside+(N)=(pfc_index_division_match_table_productcoefficientdiagonal)) /\\ (((pfc_left_division_match_table_productcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_match_table_productcoefficientdiagonaltermrightinside. pfa_gap_division_match_table_productcoefficientdiagonaltermrightinside + S (pfc_complement_division_match_table_productcoefficientdiagonalterm) = (S d)) /\\ ((((exists ff_h_pfp_division_match_table_productcoefficientdiagonaltermrightentry. ff_h_pfp_division_match_table_productcoefficientdiagonaltermrightentry + S (pfc_right_division_match_table_productcoefficientdiagonalterm) = S ((S (pfc_complement_division_match_table_productcoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_match_table_productcoefficientdiagonaltermrightentry. bb = ff_q_pfp_division_match_table_productcoefficientdiagonaltermrightentry * S ((S (pfc_complement_division_match_table_productcoefficientdiagonalterm)) * bc) + (pfc_right_division_match_table_productcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_division_match_table_productcoefficientdiagonaltermrightoutside. pfc_gap_division_match_table_productcoefficientdiagonaltermrightoutside+(S d)=(pfc_complement_division_match_table_productcoefficientdiagonalterm)) /\\ (((pfc_right_division_match_table_productcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_division_match_table_productcoefficientdiagonal)=pfc_left_division_match_table_productcoefficientdiagonalterm*pfc_right_division_match_table_productcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_match_table_productcoefficientsum fs_v_pfc_division_match_table_productcoefficientsum. ((((exists fs_h_pfc_division_match_table_productcoefficientsum_body_start. fs_h_pfc_division_match_table_productcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_match_table_productcoefficientsum)) /\\ exists fs_q_pfc_division_match_table_productcoefficientsum_body_start. fs_u_pfc_division_match_table_productcoefficientsum = fs_q_pfc_division_match_table_productcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_division_match_table_productcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_division_match_table_productcoefficientsum_body_terminal. fs_h_pfc_division_match_table_productcoefficientsum_body_terminal + S (pfc_natural_sum_division_match_table_productcoefficient) = S ((S (S (pfc_index_division_match_table_product))) * fs_v_pfc_division_match_table_productcoefficientsum)) /\\ exists fs_q_pfc_division_match_table_productcoefficientsum_body_terminal. fs_u_pfc_division_match_table_productcoefficientsum = fs_q_pfc_division_match_table_productcoefficientsum_body_terminal * S ((S (S (pfc_index_division_match_table_product))) * fs_v_pfc_division_match_table_productcoefficientsum) + (pfc_natural_sum_division_match_table_productcoefficient))) /\\ forall fs_i_pfc_division_match_table_productcoefficientsum_body_steps. (exists fs_lt_pfc_division_match_table_productcoefficientsum_body_steps_bound. fs_lt_pfc_division_match_table_productcoefficientsum_body_steps_bound + S fs_i_pfc_division_match_table_productcoefficientsum_body_steps = S (pfc_index_division_match_table_product)) -> exists fs_a_pfc_division_match_table_productcoefficientsum_body_steps fs_r_pfc_division_match_table_productcoefficientsum_body_steps fs_s_pfc_division_match_table_productcoefficientsum_body_steps. ((((exists fs_h_pfc_division_match_table_productcoefficientsum_body_steps_summand. fs_h_pfc_division_match_table_productcoefficientsum_body_steps_summand + S (fs_a_pfc_division_match_table_productcoefficientsum_body_steps) = S ((S (fs_i_pfc_division_match_table_productcoefficientsum_body_steps)) * pfc_terms_scale_division_match_table_productcoefficient)) /\\ exists fs_q_pfc_division_match_table_productcoefficientsum_body_steps_summand. pfc_terms_code_division_match_table_productcoefficient = fs_q_pfc_division_match_table_productcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_division_match_table_productcoefficientsum_body_steps)) * pfc_terms_scale_division_match_table_productcoefficient) + (fs_a_pfc_division_match_table_productcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_division_match_table_productcoefficientsum_body_steps_partial. fs_h_pfc_division_match_table_productcoefficientsum_body_steps_partial + S (fs_r_pfc_division_match_table_productcoefficientsum_body_steps) = S ((S (fs_i_pfc_division_match_table_productcoefficientsum_body_steps)) * fs_v_pfc_division_match_table_productcoefficientsum)) /\\ exists fs_q_pfc_division_match_table_productcoefficientsum_body_steps_partial. fs_u_pfc_division_match_table_productcoefficientsum = fs_q_pfc_division_match_table_productcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_division_match_table_productcoefficientsum_body_steps)) * fs_v_pfc_division_match_table_productcoefficientsum) + (fs_r_pfc_division_match_table_productcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_division_match_table_productcoefficientsum_body_steps_successor. fs_h_pfc_division_match_table_productcoefficientsum_body_steps_successor + S (fs_s_pfc_division_match_table_productcoefficientsum_body_steps) = S ((S (S fs_i_pfc_division_match_table_productcoefficientsum_body_steps)) * fs_v_pfc_division_match_table_productcoefficientsum)) /\\ exists fs_q_pfc_division_match_table_productcoefficientsum_body_steps_successor. fs_u_pfc_division_match_table_productcoefficientsum = fs_q_pfc_division_match_table_productcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_division_match_table_productcoefficientsum_body_steps)) * fs_v_pfc_division_match_table_productcoefficientsum) + (fs_s_pfc_division_match_table_productcoefficientsum_body_steps))) /\\ fs_s_pfc_division_match_table_productcoefficientsum_body_steps = fs_r_pfc_division_match_table_productcoefficientsum_body_steps + fs_a_pfc_division_match_table_productcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_division_match_table_productcoefficientresiduebound. pfa_gap_division_match_table_productcoefficientresiduebound + S (pfc_value_division_match_table_product) = (p)) /\\ ((exists pfa_offset_left_division_match_table_productcoefficientresiduecongruence pfa_offset_right_division_match_table_productcoefficientresiduecongruence. (pfc_natural_sum_division_match_table_productcoefficient) + (p) * pfa_offset_left_division_match_table_productcoefficientresiduecongruence = (pfc_value_division_match_table_product) + (p) * pfa_offset_right_division_match_table_productcoefficientresiduecongruence)))))))))))) -> (forall mdr_i_pfp_division_match_table_result mdr_a_pfp_division_match_table_result. 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(~((p) = 1) /\\ forall pfa_factor_left_division_zero_prime pfa_factor_right_division_zero_prime. (p) = pfa_factor_left_division_zero_prime * pfa_factor_right_division_zero_prime -> pfa_factor_left_division_zero_prime = 1 \\/ pfa_factor_right_division_zero_prime = 1) -> (((exists ff_h_pfp_division_zero_head. ff_h_pfp_division_zero_head + S (b) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_division_zero_head. bb = ff_q_pfp_division_zero_head * S ((S (0)) * bc) + (b))) -> (((~((b) = 0)) /\\ ((((exists pfa_gap_division_zero_inversemultiplicationleft. pfa_gap_division_zero_inversemultiplicationleft + S (b) = (p)) /\\ (((exists pfa_gap_division_zero_inversemultiplicationright. pfa_gap_division_zero_inversemultiplicationright + S (k) = (p)) /\\ ((((exists pfa_gap_division_zero_inversemultiplicationresultbound. pfa_gap_division_zero_inversemultiplicationresultbound + S (1) = (p)) /\\ ((exists pfa_offset_left_division_zero_inversemultiplicationresultcongruence pfa_offset_right_division_zero_inversemultiplicationresultcongruence. ((b) * (k)) + (p) * pfa_offset_left_division_zero_inversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_division_zero_inversemultiplicationresultcongruence)))))))))))) -> (forall pfd_index_division_zero_execution. (exists pfa_gap_division_zero_executionbound. pfa_gap_division_zero_executionbound + S (pfd_index_division_zero_execution) = (N)) -> exists pfd_value_division_zero_execution. ((((exists ff_h_pfp_division_zero_executionentry. ff_h_pfp_division_zero_executionentry + S (pfd_value_division_zero_execution) = S ((S (pfd_index_division_zero_execution)) * qc)) /\\ exists ff_q_pfp_division_zero_executionentry. qb = ff_q_pfp_division_zero_executionentry * S ((S (pfd_index_division_zero_execution)) * qc) + (pfd_value_division_zero_execution))) /\\ ((exists pfd_input_division_zero_executionstep pfd_previous_division_zero_executionstep pfd_difference_division_zero_executionstep. ((((exists ff_h_pfp_division_zero_executionstepinput. ff_h_pfp_division_zero_executionstepinput + S (pfd_input_division_zero_executionstep) = S ((S (pfd_index_division_zero_execution)) * ac)) /\\ exists ff_q_pfp_division_zero_executionstepinput. ab = ff_q_pfp_division_zero_executionstepinput * S ((S (pfd_index_division_zero_execution)) * ac) + (pfd_input_division_zero_executionstep))) /\\ (((exists pfc_terms_code_division_zero_executionstepprevious pfc_terms_scale_division_zero_executionstepprevious pfc_natural_sum_division_zero_executionstepprevious. ((forall pfc_index_division_zero_executionsteppreviousdiagonal. (exists pfa_gap_division_zero_executionsteppreviousdiagonalbound. pfa_gap_division_zero_executionsteppreviousdiagonalbound + S (pfc_index_division_zero_executionsteppreviousdiagonal) = (S (pfd_index_division_zero_execution))) -> exists pfc_value_division_zero_executionsteppreviousdiagonal. ((((exists ff_h_pfp_division_zero_executionsteppreviousdiagonalentry. ff_h_pfp_division_zero_executionsteppreviousdiagonalentry + S (pfc_value_division_zero_executionsteppreviousdiagonal) = S ((S (pfc_index_division_zero_executionsteppreviousdiagonal)) * pfc_terms_scale_division_zero_executionstepprevious)) /\\ exists ff_q_pfp_division_zero_executionsteppreviousdiagonalentry. pfc_terms_code_division_zero_executionstepprevious = ff_q_pfp_division_zero_executionsteppreviousdiagonalentry * S ((S (pfc_index_division_zero_executionsteppreviousdiagonal)) * pfc_terms_scale_division_zero_executionstepprevious) + (pfc_value_division_zero_executionsteppreviousdiagonal))) /\\ ((exists pfc_complement_division_zero_executionsteppreviousdiagonalterm pfc_left_division_zero_executionsteppreviousdiagonalterm pfc_right_division_zero_executionsteppreviousdiagonalterm. (((pfc_index_division_zero_executionsteppreviousdiagonal)+pfc_complement_division_zero_executionsteppreviousdiagonalterm=(pfd_index_division_zero_execution)) /\\ ((((((exists pfa_gap_division_zero_executionsteppreviousdiagonaltermleftinside. pfa_gap_division_zero_executionsteppreviousdiagonaltermleftinside + S (pfc_index_division_zero_executionsteppreviousdiagonal) = (pfd_index_division_zero_execution)) /\\ ((((exists ff_h_pfp_division_zero_executionsteppreviousdiagonaltermleftentry. ff_h_pfp_division_zero_executionsteppreviousdiagonaltermleftentry + S (pfc_left_division_zero_executionsteppreviousdiagonalterm) = S ((S (pfc_index_division_zero_executionsteppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_division_zero_executionsteppreviousdiagonaltermleftentry. qb = ff_q_pfp_division_zero_executionsteppreviousdiagonaltermleftentry * S ((S (pfc_index_division_zero_executionsteppreviousdiagonal)) * qc) + (pfc_left_division_zero_executionsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_zero_executionsteppreviousdiagonaltermleftoutside. pfc_gap_division_zero_executionsteppreviousdiagonaltermleftoutside+(pfd_index_division_zero_execution)=(pfc_index_division_zero_executionsteppreviousdiagonal)) /\\ (((pfc_left_division_zero_executionsteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_zero_executionsteppreviousdiagonaltermrightinside. pfa_gap_division_zero_executionsteppreviousdiagonaltermrightinside + S (pfc_complement_division_zero_executionsteppreviousdiagonalterm) = (S d)) /\\ ((((exists ff_h_pfp_division_zero_executionsteppreviousdiagonaltermrightentry. ff_h_pfp_division_zero_executionsteppreviousdiagonaltermrightentry + S (pfc_right_division_zero_executionsteppreviousdiagonalterm) = S ((S (pfc_complement_division_zero_executionsteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_zero_executionsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_division_zero_executionsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_division_zero_executionsteppreviousdiagonalterm)) * bc) + (pfc_right_division_zero_executionsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_zero_executionsteppreviousdiagonaltermrightoutside. pfc_gap_division_zero_executionsteppreviousdiagonaltermrightoutside+(S d)=(pfc_complement_division_zero_executionsteppreviousdiagonalterm)) /\\ (((pfc_right_division_zero_executionsteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_division_zero_executionsteppreviousdiagonal)=pfc_left_division_zero_executionsteppreviousdiagonalterm*pfc_right_division_zero_executionsteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_zero_executionstepprevioussum fs_v_pfc_division_zero_executionstepprevioussum. ((((exists fs_h_pfc_division_zero_executionstepprevioussum_body_start. fs_h_pfc_division_zero_executionstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_zero_executionstepprevioussum)) /\\ exists fs_q_pfc_division_zero_executionstepprevioussum_body_start. fs_u_pfc_division_zero_executionstepprevioussum = fs_q_pfc_division_zero_executionstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_division_zero_executionstepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_division_zero_executionstepprevioussum_body_terminal. fs_h_pfc_division_zero_executionstepprevioussum_body_terminal + S (pfc_natural_sum_division_zero_executionstepprevious) = S ((S (S (pfd_index_division_zero_execution))) * fs_v_pfc_division_zero_executionstepprevioussum)) /\\ exists fs_q_pfc_division_zero_executionstepprevioussum_body_terminal. fs_u_pfc_division_zero_executionstepprevioussum = fs_q_pfc_division_zero_executionstepprevioussum_body_terminal * S ((S (S (pfd_index_division_zero_execution))) * fs_v_pfc_division_zero_executionstepprevioussum) + (pfc_natural_sum_division_zero_executionstepprevious))) /\\ forall fs_i_pfc_division_zero_executionstepprevioussum_body_steps. (exists fs_lt_pfc_division_zero_executionstepprevioussum_body_steps_bound. fs_lt_pfc_division_zero_executionstepprevioussum_body_steps_bound + S fs_i_pfc_division_zero_executionstepprevioussum_body_steps = S (pfd_index_division_zero_execution)) -> exists fs_a_pfc_division_zero_executionstepprevioussum_body_steps fs_r_pfc_division_zero_executionstepprevioussum_body_steps fs_s_pfc_division_zero_executionstepprevioussum_body_steps. ((((exists fs_h_pfc_division_zero_executionstepprevioussum_body_steps_summand. fs_h_pfc_division_zero_executionstepprevioussum_body_steps_summand + S (fs_a_pfc_division_zero_executionstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_zero_executionstepprevioussum_body_steps)) * pfc_terms_scale_division_zero_executionstepprevious)) /\\ exists fs_q_pfc_division_zero_executionstepprevioussum_body_steps_summand. pfc_terms_code_division_zero_executionstepprevious = fs_q_pfc_division_zero_executionstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_division_zero_executionstepprevioussum_body_steps)) * pfc_terms_scale_division_zero_executionstepprevious) + (fs_a_pfc_division_zero_executionstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_zero_executionstepprevioussum_body_steps_partial. fs_h_pfc_division_zero_executionstepprevioussum_body_steps_partial + S (fs_r_pfc_division_zero_executionstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_zero_executionstepprevioussum_body_steps)) * fs_v_pfc_division_zero_executionstepprevioussum)) /\\ exists fs_q_pfc_division_zero_executionstepprevioussum_body_steps_partial. fs_u_pfc_division_zero_executionstepprevioussum = fs_q_pfc_division_zero_executionstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_division_zero_executionstepprevioussum_body_steps)) * fs_v_pfc_division_zero_executionstepprevioussum) + (fs_r_pfc_division_zero_executionstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_zero_executionstepprevioussum_body_steps_successor. fs_h_pfc_division_zero_executionstepprevioussum_body_steps_successor + S (fs_s_pfc_division_zero_executionstepprevioussum_body_steps) = S ((S (S fs_i_pfc_division_zero_executionstepprevioussum_body_steps)) * fs_v_pfc_division_zero_executionstepprevioussum)) /\\ exists fs_q_pfc_division_zero_executionstepprevioussum_body_steps_successor. fs_u_pfc_division_zero_executionstepprevioussum = fs_q_pfc_division_zero_executionstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_division_zero_executionstepprevioussum_body_steps)) * fs_v_pfc_division_zero_executionstepprevioussum) + (fs_s_pfc_division_zero_executionstepprevioussum_body_steps))) /\\ fs_s_pfc_division_zero_executionstepprevioussum_body_steps = fs_r_pfc_division_zero_executionstepprevioussum_body_steps + fs_a_pfc_division_zero_executionstepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_division_zero_executionsteppreviousresiduebound. pfa_gap_division_zero_executionsteppreviousresiduebound + S (pfd_previous_division_zero_executionstep) = (p)) /\\ ((exists pfa_offset_left_division_zero_executionsteppreviousresiduecongruence pfa_offset_right_division_zero_executionsteppreviousresiduecongruence. (pfc_natural_sum_division_zero_executionstepprevious) + (p) * pfa_offset_left_division_zero_executionsteppreviousresiduecongruence = (pfd_previous_division_zero_executionstep) + (p) * pfa_offset_right_division_zero_executionsteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_division_zero_executionstepsubtractleft. pfa_gap_division_zero_executionstepsubtractleft + S (pfd_previous_division_zero_executionstep) = (p)) /\\ (((exists pfa_gap_division_zero_executionstepsubtractright. pfa_gap_division_zero_executionstepsubtractright + S (pfd_difference_division_zero_executionstep) = (p)) /\\ ((((exists pfa_gap_division_zero_executionstepsubtractresultbound. pfa_gap_division_zero_executionstepsubtractresultbound + S (pfd_input_division_zero_executionstep) = (p)) /\\ ((exists pfa_offset_left_division_zero_executionstepsubtractresultcongruence pfa_offset_right_division_zero_executionstepsubtractresultcongruence. ((pfd_previous_division_zero_executionstep) + (pfd_difference_division_zero_executionstep)) + (p) * pfa_offset_left_division_zero_executionstepsubtractresultcongruence = (pfd_input_division_zero_executionstep) + (p) * pfa_offset_right_division_zero_executionstepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_division_zero_executionstepmultiplyleft. pfa_gap_division_zero_executionstepmultiplyleft + S (k) = (p)) /\\ (((exists pfa_gap_division_zero_executionstepmultiplyright. pfa_gap_division_zero_executionstepmultiplyright + S (pfd_difference_division_zero_executionstep) = (p)) /\\ ((((exists pfa_gap_division_zero_executionstepmultiplyresultbound. pfa_gap_division_zero_executionstepmultiplyresultbound + S (pfd_value_division_zero_execution) = (p)) /\\ ((exists pfa_offset_left_division_zero_executionstepmultiplyresultcongruence pfa_offset_right_division_zero_executionstepmultiplyresultcongruence. ((k) * (pfd_difference_division_zero_executionstep)) + (p) * pfa_offset_left_division_zero_executionstepmultiplyresultcongruence = (pfd_value_division_zero_execution) + (p) * pfa_offset_right_division_zero_executionstepmultiplyresultcongruence))))))))))))))))))) -> (exists pfc_gap_division_zero_length. pfc_gap_division_zero_length+(N)=(L)) -> (forall pfc_index_division_zero_product. (exists pfa_gap_division_zero_productbound. pfa_gap_division_zero_productbound + S (pfc_index_division_zero_product) = (L)) -> exists pfc_value_division_zero_product. ((((exists ff_h_pfp_division_zero_productentry. ff_h_pfp_division_zero_productentry + S (pfc_value_division_zero_product) = S ((S (pfc_index_division_zero_product)) * pc)) /\\ exists ff_q_pfp_division_zero_productentry. pb = ff_q_pfp_division_zero_productentry * S ((S (pfc_index_division_zero_product)) * pc) + (pfc_value_division_zero_product))) /\\ ((exists pfc_terms_code_division_zero_productcoefficient pfc_terms_scale_division_zero_productcoefficient pfc_natural_sum_division_zero_productcoefficient. ((forall pfc_index_division_zero_productcoefficientdiagonal. (exists pfa_gap_division_zero_productcoefficientdiagonalbound. pfa_gap_division_zero_productcoefficientdiagonalbound + S (pfc_index_division_zero_productcoefficientdiagonal) = (S (pfc_index_division_zero_product))) -> exists pfc_value_division_zero_productcoefficientdiagonal. ((((exists ff_h_pfp_division_zero_productcoefficientdiagonalentry. ff_h_pfp_division_zero_productcoefficientdiagonalentry + S (pfc_value_division_zero_productcoefficientdiagonal) = S ((S (pfc_index_division_zero_productcoefficientdiagonal)) * pfc_terms_scale_division_zero_productcoefficient)) /\\ exists ff_q_pfp_division_zero_productcoefficientdiagonalentry. pfc_terms_code_division_zero_productcoefficient = ff_q_pfp_division_zero_productcoefficientdiagonalentry * S ((S (pfc_index_division_zero_productcoefficientdiagonal)) * pfc_terms_scale_division_zero_productcoefficient) + (pfc_value_division_zero_productcoefficientdiagonal))) /\\ ((exists pfc_complement_division_zero_productcoefficientdiagonalterm pfc_left_division_zero_productcoefficientdiagonalterm pfc_right_division_zero_productcoefficientdiagonalterm. (((pfc_index_division_zero_productcoefficientdiagonal)+pfc_complement_division_zero_productcoefficientdiagonalterm=(pfc_index_division_zero_product)) /\\ ((((((exists pfa_gap_division_zero_productcoefficientdiagonaltermleftinside. pfa_gap_division_zero_productcoefficientdiagonaltermleftinside + S (pfc_index_division_zero_productcoefficientdiagonal) = (N)) /\\ ((((exists ff_h_pfp_division_zero_productcoefficientdiagonaltermleftentry. ff_h_pfp_division_zero_productcoefficientdiagonaltermleftentry + S (pfc_left_division_zero_productcoefficientdiagonalterm) = S ((S (pfc_index_division_zero_productcoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_division_zero_productcoefficientdiagonaltermleftentry. qb = ff_q_pfp_division_zero_productcoefficientdiagonaltermleftentry * S ((S (pfc_index_division_zero_productcoefficientdiagonal)) * qc) + (pfc_left_division_zero_productcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_division_zero_productcoefficientdiagonaltermleftoutside. pfc_gap_division_zero_productcoefficientdiagonaltermleftoutside+(N)=(pfc_index_division_zero_productcoefficientdiagonal)) /\\ (((pfc_left_division_zero_productcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_zero_productcoefficientdiagonaltermrightinside. pfa_gap_division_zero_productcoefficientdiagonaltermrightinside + S (pfc_complement_division_zero_productcoefficientdiagonalterm) = (S d)) /\\ ((((exists ff_h_pfp_division_zero_productcoefficientdiagonaltermrightentry. ff_h_pfp_division_zero_productcoefficientdiagonaltermrightentry + S (pfc_right_division_zero_productcoefficientdiagonalterm) = S ((S (pfc_complement_division_zero_productcoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_zero_productcoefficientdiagonaltermrightentry. bb = ff_q_pfp_division_zero_productcoefficientdiagonaltermrightentry * S ((S (pfc_complement_division_zero_productcoefficientdiagonalterm)) * bc) + (pfc_right_division_zero_productcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_division_zero_productcoefficientdiagonaltermrightoutside. pfc_gap_division_zero_productcoefficientdiagonaltermrightoutside+(S d)=(pfc_complement_division_zero_productcoefficientdiagonalterm)) /\\ (((pfc_right_division_zero_productcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_division_zero_productcoefficientdiagonal)=pfc_left_division_zero_productcoefficientdiagonalterm*pfc_right_division_zero_productcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_zero_productcoefficientsum fs_v_pfc_division_zero_productcoefficientsum. ((((exists fs_h_pfc_division_zero_productcoefficientsum_body_start. fs_h_pfc_division_zero_productcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_zero_productcoefficientsum)) /\\ exists fs_q_pfc_division_zero_productcoefficientsum_body_start. fs_u_pfc_division_zero_productcoefficientsum = fs_q_pfc_division_zero_productcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_division_zero_productcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_division_zero_productcoefficientsum_body_terminal. fs_h_pfc_division_zero_productcoefficientsum_body_terminal + S (pfc_natural_sum_division_zero_productcoefficient) = S ((S (S (pfc_index_division_zero_product))) * fs_v_pfc_division_zero_productcoefficientsum)) /\\ exists fs_q_pfc_division_zero_productcoefficientsum_body_terminal. fs_u_pfc_division_zero_productcoefficientsum = fs_q_pfc_division_zero_productcoefficientsum_body_terminal * S ((S (S (pfc_index_division_zero_product))) * fs_v_pfc_division_zero_productcoefficientsum) + (pfc_natural_sum_division_zero_productcoefficient))) /\\ forall fs_i_pfc_division_zero_productcoefficientsum_body_steps. (exists fs_lt_pfc_division_zero_productcoefficientsum_body_steps_bound. fs_lt_pfc_division_zero_productcoefficientsum_body_steps_bound + S fs_i_pfc_division_zero_productcoefficientsum_body_steps = S (pfc_index_division_zero_product)) -> exists fs_a_pfc_division_zero_productcoefficientsum_body_steps fs_r_pfc_division_zero_productcoefficientsum_body_steps fs_s_pfc_division_zero_productcoefficientsum_body_steps. ((((exists fs_h_pfc_division_zero_productcoefficientsum_body_steps_summand. fs_h_pfc_division_zero_productcoefficientsum_body_steps_summand + S (fs_a_pfc_division_zero_productcoefficientsum_body_steps) = S ((S (fs_i_pfc_division_zero_productcoefficientsum_body_steps)) * pfc_terms_scale_division_zero_productcoefficient)) /\\ exists fs_q_pfc_division_zero_productcoefficientsum_body_steps_summand. pfc_terms_code_division_zero_productcoefficient = fs_q_pfc_division_zero_productcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_division_zero_productcoefficientsum_body_steps)) * pfc_terms_scale_division_zero_productcoefficient) + (fs_a_pfc_division_zero_productcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_division_zero_productcoefficientsum_body_steps_partial. fs_h_pfc_division_zero_productcoefficientsum_body_steps_partial + S (fs_r_pfc_division_zero_productcoefficientsum_body_steps) = S ((S (fs_i_pfc_division_zero_productcoefficientsum_body_steps)) * fs_v_pfc_division_zero_productcoefficientsum)) /\\ exists fs_q_pfc_division_zero_productcoefficientsum_body_steps_partial. fs_u_pfc_division_zero_productcoefficientsum = fs_q_pfc_division_zero_productcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_division_zero_productcoefficientsum_body_steps)) * fs_v_pfc_division_zero_productcoefficientsum) + (fs_r_pfc_division_zero_productcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_division_zero_productcoefficientsum_body_steps_successor. fs_h_pfc_division_zero_productcoefficientsum_body_steps_successor + S (fs_s_pfc_division_zero_productcoefficientsum_body_steps) = S ((S (S fs_i_pfc_division_zero_productcoefficientsum_body_steps)) * fs_v_pfc_division_zero_productcoefficientsum)) /\\ exists fs_q_pfc_division_zero_productcoefficientsum_body_steps_successor. fs_u_pfc_division_zero_productcoefficientsum = fs_q_pfc_division_zero_productcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_division_zero_productcoefficientsum_body_steps)) * fs_v_pfc_division_zero_productcoefficientsum) + (fs_s_pfc_division_zero_productcoefficientsum_body_steps))) /\\ fs_s_pfc_division_zero_productcoefficientsum_body_steps = fs_r_pfc_division_zero_productcoefficientsum_body_steps + fs_a_pfc_division_zero_productcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_division_zero_productcoefficientresiduebound. pfa_gap_division_zero_productcoefficientresiduebound + S (pfc_value_division_zero_product) = (p)) /\\ ((exists pfa_offset_left_division_zero_productcoefficientresiduecongruence pfa_offset_right_division_zero_productcoefficientresiduecongruence. (pfc_natural_sum_division_zero_productcoefficient) + (p) * pfa_offset_left_division_zero_productcoefficientresiduecongruence = (pfc_value_division_zero_product) + (p) * pfa_offset_right_division_zero_productcoefficientresiduecongruence)))))))))))) -> (forall pfs_index_division_zero_subtraction. (exists pfa_gap_division_zero_subtractionindex. pfa_gap_division_zero_subtractionindex + S (pfs_index_division_zero_subtraction) = (L)) -> exists pfs_left_division_zero_subtraction pfs_right_division_zero_subtraction pfs_result_division_zero_subtraction. ((((exists ff_h_pfp_division_zero_subtractionleft. ff_h_pfp_division_zero_subtractionleft + S (pfs_left_division_zero_subtraction) = S ((S (pfs_index_division_zero_subtraction)) * ac)) /\\ exists ff_q_pfp_division_zero_subtractionleft. ab = ff_q_pfp_division_zero_subtractionleft * S ((S (pfs_index_division_zero_subtraction)) * ac) + (pfs_left_division_zero_subtraction))) /\\ (((((exists ff_h_pfp_division_zero_subtractionright. ff_h_pfp_division_zero_subtractionright + S (pfs_right_division_zero_subtraction) = S ((S (pfs_index_division_zero_subtraction)) * pc)) /\\ exists ff_q_pfp_division_zero_subtractionright. pb = ff_q_pfp_division_zero_subtractionright * S ((S (pfs_index_division_zero_subtraction)) * pc) + (pfs_right_division_zero_subtraction))) /\\ (((((exists ff_h_pfp_division_zero_subtractionresult. ff_h_pfp_division_zero_subtractionresult + S (pfs_result_division_zero_subtraction) = S ((S (pfs_index_division_zero_subtraction)) * uc)) /\\ exists ff_q_pfp_division_zero_subtractionresult. ub = ff_q_pfp_division_zero_subtractionresult * S ((S (pfs_index_division_zero_subtraction)) * uc) + (pfs_result_division_zero_subtraction))) /\\ ((((exists pfa_gap_division_zero_subtractionoperationleft. pfa_gap_division_zero_subtractionoperationleft + S (pfs_right_division_zero_subtraction) = (p)) /\\ (((exists pfa_gap_division_zero_subtractionoperationright. pfa_gap_division_zero_subtractionoperationright + S (pfs_result_division_zero_subtraction) = (p)) /\\ ((((exists pfa_gap_division_zero_subtractionoperationresultbound. pfa_gap_division_zero_subtractionoperationresultbound + S (pfs_left_division_zero_subtraction) = (p)) /\\ ((exists pfa_offset_left_division_zero_subtractionoperationresultcongruence pfa_offset_right_division_zero_subtractionoperationresultcongruence. ((pfs_right_division_zero_subtraction) + (pfs_result_division_zero_subtraction)) + (p) * pfa_offset_left_division_zero_subtractionoperationresultcongruence = (pfs_left_division_zero_subtraction) + (p) * pfa_offset_right_division_zero_subtractionoperationresultcongruence)))))))))))))))) -> (forall pfp_repeat_index_division_zero_result. 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(exists pfa_gap_division_cut_copyremovedindex. pfa_gap_division_cut_copyremovedindex + S (pfp_repeat_index_division_cut_copyremoved) = (t)) -> (((exists ff_h_pfp_division_cut_copyremovedentry. ff_h_pfp_division_cut_copyremovedentry + S (0) = S ((S (pfp_repeat_index_division_cut_copyremoved)) * uc)) /\\ exists ff_q_pfp_division_cut_copyremovedentry. ub = ff_q_pfp_division_cut_copyremovedentry * S ((S (pfp_repeat_index_division_cut_copyremoved)) * uc) + (0)))) /\\ (((forall pftrim_index_division_cut_copysuffix pftrim_value_division_cut_copysuffix. (exists pfa_gap_division_cut_copysuffixbound. pfa_gap_division_cut_copysuffixbound + S (pftrim_index_division_cut_copysuffix) = (R)) -> (((exists ff_h_pfp_division_cut_copysuffixsource. ff_h_pfp_division_cut_copysuffixsource + S (pftrim_value_division_cut_copysuffix) = S ((S ((t)+pftrim_index_division_cut_copysuffix)) * uc)) /\\ exists ff_q_pfp_division_cut_copysuffixsource. ub = ff_q_pfp_division_cut_copysuffixsource * S ((S ((t)+pftrim_index_division_cut_copysuffix)) * uc) + (pftrim_value_division_cut_copysuffix))) -> (((exists ff_h_pfp_division_cut_copysuffixoutput. ff_h_pfp_division_cut_copysuffixoutput + S (pftrim_value_division_cut_copysuffix) = S ((S (pftrim_index_division_cut_copysuffix)) * rc)) /\\ exists ff_q_pfp_division_cut_copysuffixoutput. rb = ff_q_pfp_division_cut_copysuffixoutput * S ((S (pftrim_index_division_cut_copysuffix)) * rc) + (pftrim_value_division_cut_copysuffix)))) /\\ (((R)=0 \\/ (exists pftrim_leading_division_cut_copynormal. 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(exists pfc_gap_division_cut_length. pfc_gap_division_cut_length+(q)=(L)) -> (forall pfp_repeat_index_division_cut_zero. (exists pfa_gap_division_cut_zeroindex. pfa_gap_division_cut_zeroindex + S (pfp_repeat_index_division_cut_zero) = (q)) -> (((exists ff_h_pfp_division_cut_zeroentry. ff_h_pfp_division_cut_zeroentry + S (0) = S ((S (pfp_repeat_index_division_cut_zero)) * uc)) /\\ exists ff_q_pfp_division_cut_zeroentry. ub = ff_q_pfp_division_cut_zeroentry * S ((S (pfp_repeat_index_division_cut_zero)) * uc) + (0)))) -> ((((L)=(t)+(R)) /\\ (((forall fom_index_pfp_division_cut_triminput. (exists fom_gap_pfp_division_cut_triminput_index_bound. fom_gap_pfp_division_cut_triminput_index_bound + S (fom_index_pfp_division_cut_triminput) = L) -> exists fom_value_pfp_division_cut_triminput. ((((exists fom_beta_height_pfp_division_cut_triminput_entry. fom_beta_height_pfp_division_cut_triminput_entry + S (fom_value_pfp_division_cut_triminput) = S ((S (fom_index_pfp_division_cut_triminput)) * uc)) /\\ exists fom_beta_quotient_pfp_division_cut_triminput_entry. ub = fom_beta_quotient_pfp_division_cut_triminput_entry * S ((S (fom_index_pfp_division_cut_triminput)) * uc) + (fom_value_pfp_division_cut_triminput))) /\\ (exists fom_gap_pfp_division_cut_triminput_value_bound. fom_gap_pfp_division_cut_triminput_value_bound + S (fom_value_pfp_division_cut_triminput) = p))) /\\ (((forall pfp_repeat_index_division_cut_trimremoved. (exists pfa_gap_division_cut_trimremovedindex. pfa_gap_division_cut_trimremovedindex + S (pfp_repeat_index_division_cut_trimremoved) = (t)) -> (((exists ff_h_pfp_division_cut_trimremovedentry. ff_h_pfp_division_cut_trimremovedentry + S (0) = S ((S (pfp_repeat_index_division_cut_trimremoved)) * uc)) /\\ exists ff_q_pfp_division_cut_trimremovedentry. ub = ff_q_pfp_division_cut_trimremovedentry * S ((S (pfp_repeat_index_division_cut_trimremoved)) * uc) + (0)))) /\\ (((forall pftrim_index_division_cut_trimsuffix pftrim_value_division_cut_trimsuffix. (exists pfa_gap_division_cut_trimsuffixbound. pfa_gap_division_cut_trimsuffixbound + S (pftrim_index_division_cut_trimsuffix) = (R)) -> (((exists ff_h_pfp_division_cut_trimsuffixsource. ff_h_pfp_division_cut_trimsuffixsource + S (pftrim_value_division_cut_trimsuffix) = S ((S ((t)+pftrim_index_division_cut_trimsuffix)) * uc)) /\\ exists ff_q_pfp_division_cut_trimsuffixsource. ub = ff_q_pfp_division_cut_trimsuffixsource * S ((S ((t)+pftrim_index_division_cut_trimsuffix)) * uc) + (pftrim_value_division_cut_trimsuffix))) -> (((exists ff_h_pfp_division_cut_trimsuffixoutput. ff_h_pfp_division_cut_trimsuffixoutput + S (pftrim_value_division_cut_trimsuffix) = S ((S (pftrim_index_division_cut_trimsuffix)) * rc)) /\\ exists ff_q_pfp_division_cut_trimsuffixoutput. rb = ff_q_pfp_division_cut_trimsuffixoutput * S ((S (pftrim_index_division_cut_trimsuffix)) * rc) + (pftrim_value_division_cut_trimsuffix)))) /\\ (((R)=0 \\/ (exists pftrim_leading_division_cut_trimnormal. 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"alpha_only", "name": "prime_field_polynomial_trim_zero_prefix_remainder_bound", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro ub", "intro uc", "intro L", "intro t", "intro rb", "intro rc", "intro R", "intro q", "intro d", "intro hqL", "intro hLd", "intro hz", "intro ht", "have hqt : exists pfc_gap_division_remainder_cut. pfc_gap_division_remainder_cut+(q)=(t)", "specialize prime_field_polynomial_trim_zero_prefix_cut_bound (p)", "specialize prime_field_polynomial_trim_zero_prefix_cut_bound (ub)", "specialize prime_field_polynomial_trim_zero_prefix_cut_bound (uc)", "specialize prime_field_polynomial_trim_zero_prefix_cut_bound (L)", "specialize prime_field_polynomial_trim_zero_prefix_cut_bound (t)", "specialize prime_field_polynomial_trim_zero_prefix_cut_bound (rb)", "specialize prime_field_polynomial_trim_zero_prefix_cut_bound (rc)", "specialize prime_field_polynomial_trim_zero_prefix_cut_bound (R)", "specialize prime_field_polynomial_trim_zero_prefix_cut_bound (q)", "apply prime_field_polynomial_trim_zero_prefix_cut_bound", "exact hqL", "exact hz", "exact ht", "have hcopy : (((L)=(t)+(R)) /\\ (((forall fom_index_pfp_division_remainder_copyinput. (exists fom_gap_pfp_division_remainder_copyinput_index_bound. fom_gap_pfp_division_remainder_copyinput_index_bound + S (fom_index_pfp_division_remainder_copyinput) = L) -> exists fom_value_pfp_division_remainder_copyinput. ((((exists fom_beta_height_pfp_division_remainder_copyinput_entry. fom_beta_height_pfp_division_remainder_copyinput_entry + S (fom_value_pfp_division_remainder_copyinput) = S ((S (fom_index_pfp_division_remainder_copyinput)) * uc)) /\\ exists fom_beta_quotient_pfp_division_remainder_copyinput_entry. ub = fom_beta_quotient_pfp_division_remainder_copyinput_entry * S ((S (fom_index_pfp_division_remainder_copyinput)) * uc) + (fom_value_pfp_division_remainder_copyinput))) /\\ (exists fom_gap_pfp_division_remainder_copyinput_value_bound. fom_gap_pfp_division_remainder_copyinput_value_bound + S (fom_value_pfp_division_remainder_copyinput) = p))) /\\ (((forall pfp_repeat_index_division_remainder_copyremoved. (exists pfa_gap_division_remainder_copyremovedindex. pfa_gap_division_remainder_copyremovedindex + S (pfp_repeat_index_division_remainder_copyremoved) = (t)) -> (((exists ff_h_pfp_division_remainder_copyremovedentry. ff_h_pfp_division_remainder_copyremovedentry + S (0) = S ((S (pfp_repeat_index_division_remainder_copyremoved)) * uc)) /\\ exists ff_q_pfp_division_remainder_copyremovedentry. ub = ff_q_pfp_division_remainder_copyremovedentry * S ((S (pfp_repeat_index_division_remainder_copyremoved)) * uc) + (0)))) /\\ (((forall pftrim_index_division_remainder_copysuffix pftrim_value_division_remainder_copysuffix. (exists pfa_gap_division_remainder_copysuffixbound. pfa_gap_division_remainder_copysuffixbound + S (pftrim_index_division_remainder_copysuffix) = (R)) -> (((exists ff_h_pfp_division_remainder_copysuffixsource. ff_h_pfp_division_remainder_copysuffixsource + S (pftrim_value_division_remainder_copysuffix) = S ((S ((t)+pftrim_index_division_remainder_copysuffix)) * uc)) /\\ exists ff_q_pfp_division_remainder_copysuffixsource. ub = ff_q_pfp_division_remainder_copysuffixsource * S ((S ((t)+pftrim_index_division_remainder_copysuffix)) * uc) + (pftrim_value_division_remainder_copysuffix))) -> (((exists ff_h_pfp_division_remainder_copysuffixoutput. ff_h_pfp_division_remainder_copysuffixoutput + S (pftrim_value_division_remainder_copysuffix) = S ((S (pftrim_index_division_remainder_copysuffix)) * rc)) /\\ exists ff_q_pfp_division_remainder_copysuffixoutput. rb = ff_q_pfp_division_remainder_copysuffixoutput * S ((S (pftrim_index_division_remainder_copysuffix)) * rc) + (pftrim_value_division_remainder_copysuffix)))) /\\ (((R)=0 \\/ (exists pftrim_leading_division_remainder_copynormal. ((((exists ff_h_pfp_division_remainder_copynormalentry. ff_h_pfp_division_remainder_copynormalentry + S (pftrim_leading_division_remainder_copynormal) = S ((S (0)) * rc)) /\\ exists ff_q_pfp_division_remainder_copynormalentry. rb = ff_q_pfp_division_remainder_copynormalentry * S ((S (0)) * rc) + (pftrim_leading_division_remainder_copynormal))) /\\ ((~(pftrim_leading_division_remainder_copynormal=0))))))))))))))", "exact ht", "cases hcopy", "cases hcopy_right", "cases hcopy_right_right", "cases hcopy_right_right_right", "have htotal : R+t=L", "trans t+R", "specialize add_comm (R)", "specialize add_comm (t)", "apply add_comm", "symm", "exact hcopy_left", "have hpartial : exists pfc_gap_division_remainder_partial. pfc_gap_division_remainder_partial+(R+q)=(R+t)", "specialize add_le_add_left (q)", "specialize add_le_add_left (t)", "specialize add_le_add_left (R)", "apply add_le_add_left", "exact hqt", "rewrite htotal at hpartial", "have hfull : exists pfc_gap_division_remainder_full. pfc_gap_division_remainder_full+(R+q)=(q+d)", "specialize le_trans (R+q)", "specialize le_trans (L)", "specialize le_trans (q+d)", "apply le_trans", "exact hpartial", "exact hLd", "have hcomm : q+d=d+q", "specialize add_comm (q)", "specialize add_comm (d)", "apply add_comm", "rewrite hcomm at hfull", "specialize add_le_cancel_right (R)", "specialize add_le_cancel_right (d)", "specialize add_le_cancel_right (q)", "apply add_le_cancel_right", "exact hfull" ], "script_sha256": "7bf178d084aade743fdf7901c19b3c0f640013067b2f8e2b587d66fe06df1ae6", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_division_candidate.py", "sha256": "edfc7806caf7a83b9cb0e3e420bd2c3a8679f2d4d9ee6ca9f8eae53faca8d5b2" }, "statement": "forall p ub uc L t rb rc R q d. (exists pfc_gap_division_remainder_quotient_length. pfc_gap_division_remainder_quotient_length+(q)=(L)) -> (exists pfc_gap_division_remainder_cover_length. pfc_gap_division_remainder_cover_length+(L)=(q+d)) -> (forall pfp_repeat_index_division_remainder_zero. (exists pfa_gap_division_remainder_zeroindex. pfa_gap_division_remainder_zeroindex + S (pfp_repeat_index_division_remainder_zero) = (q)) -> (((exists ff_h_pfp_division_remainder_zeroentry. ff_h_pfp_division_remainder_zeroentry + S (0) = S ((S (pfp_repeat_index_division_remainder_zero)) * uc)) /\\ exists ff_q_pfp_division_remainder_zeroentry. ub = ff_q_pfp_division_remainder_zeroentry * S ((S (pfp_repeat_index_division_remainder_zero)) * uc) + (0)))) -> ((((L)=(t)+(R)) /\\ (((forall fom_index_pfp_division_remainder_triminput. (exists fom_gap_pfp_division_remainder_triminput_index_bound. fom_gap_pfp_division_remainder_triminput_index_bound + S (fom_index_pfp_division_remainder_triminput) = L) -> exists fom_value_pfp_division_remainder_triminput. ((((exists fom_beta_height_pfp_division_remainder_triminput_entry. fom_beta_height_pfp_division_remainder_triminput_entry + S (fom_value_pfp_division_remainder_triminput) = S ((S (fom_index_pfp_division_remainder_triminput)) * uc)) /\\ exists fom_beta_quotient_pfp_division_remainder_triminput_entry. ub = fom_beta_quotient_pfp_division_remainder_triminput_entry * S ((S (fom_index_pfp_division_remainder_triminput)) * uc) + (fom_value_pfp_division_remainder_triminput))) /\\ (exists fom_gap_pfp_division_remainder_triminput_value_bound. fom_gap_pfp_division_remainder_triminput_value_bound + S (fom_value_pfp_division_remainder_triminput) = p))) /\\ (((forall pfp_repeat_index_division_remainder_trimremoved. (exists pfa_gap_division_remainder_trimremovedindex. pfa_gap_division_remainder_trimremovedindex + S (pfp_repeat_index_division_remainder_trimremoved) = (t)) -> (((exists ff_h_pfp_division_remainder_trimremovedentry. ff_h_pfp_division_remainder_trimremovedentry + S (0) = S ((S (pfp_repeat_index_division_remainder_trimremoved)) * uc)) /\\ exists ff_q_pfp_division_remainder_trimremovedentry. ub = ff_q_pfp_division_remainder_trimremovedentry * S ((S (pfp_repeat_index_division_remainder_trimremoved)) * uc) + (0)))) /\\ (((forall pftrim_index_division_remainder_trimsuffix pftrim_value_division_remainder_trimsuffix. (exists pfa_gap_division_remainder_trimsuffixbound. pfa_gap_division_remainder_trimsuffixbound + S (pftrim_index_division_remainder_trimsuffix) = (R)) -> (((exists ff_h_pfp_division_remainder_trimsuffixsource. ff_h_pfp_division_remainder_trimsuffixsource + S (pftrim_value_division_remainder_trimsuffix) = S ((S ((t)+pftrim_index_division_remainder_trimsuffix)) * uc)) /\\ exists ff_q_pfp_division_remainder_trimsuffixsource. ub = ff_q_pfp_division_remainder_trimsuffixsource * S ((S ((t)+pftrim_index_division_remainder_trimsuffix)) * uc) + (pftrim_value_division_remainder_trimsuffix))) -> (((exists ff_h_pfp_division_remainder_trimsuffixoutput. ff_h_pfp_division_remainder_trimsuffixoutput + S (pftrim_value_division_remainder_trimsuffix) = S ((S (pftrim_index_division_remainder_trimsuffix)) * rc)) /\\ exists ff_q_pfp_division_remainder_trimsuffixoutput. rb = ff_q_pfp_division_remainder_trimsuffixoutput * S ((S (pftrim_index_division_remainder_trimsuffix)) * rc) + (pftrim_value_division_remainder_trimsuffix)))) /\\ (((R)=0 \\/ (exists pftrim_leading_division_remainder_trimnormal. ((((exists ff_h_pfp_division_remainder_trimnormalentry. ff_h_pfp_division_remainder_trimnormalentry + S (pftrim_leading_division_remainder_trimnormal) = S ((S (0)) * rc)) /\\ exists ff_q_pfp_division_remainder_trimnormalentry. rb = ff_q_pfp_division_remainder_trimnormalentry * S ((S (0)) * rc) + (pftrim_leading_division_remainder_trimnormal))) /\\ ((~(pftrim_leading_division_remainder_trimnormal=0))))))))))))))) -> (exists pfc_gap_division_remainder_bound. pfc_gap_division_remainder_bound+(R)=(d))", "statement_sha256": "571879b4702fda1715323ff293569437148f13b2910e0aee75886dc33bec73fe", "summary": "The actual trimmed residual has at most d coefficients after a length-q zero prefix is proved; no degree is assigned to the empty case.", "summary_sha256": "597c7cdfde7a3c3087b3c64049f9bbd9afb3511fc1c33e6a3032d35a6f352cdd" }, { "alpha_v33_frontier_enrollment": { "body_receipt_sha256": "6e47d5abee407a8fe05bdf0014dedec44c06f569e8cf6e9b79df1a99573ffede", "bundle_campaign": 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((((L)=(t)+(R)) /\\ (((forall fom_index_pfp_division_remainder_degree_triminput. (exists fom_gap_pfp_division_remainder_degree_triminput_index_bound. fom_gap_pfp_division_remainder_degree_triminput_index_bound + S (fom_index_pfp_division_remainder_degree_triminput) = L) -> exists fom_value_pfp_division_remainder_degree_triminput. ((((exists fom_beta_height_pfp_division_remainder_degree_triminput_entry. fom_beta_height_pfp_division_remainder_degree_triminput_entry + S (fom_value_pfp_division_remainder_degree_triminput) = S ((S (fom_index_pfp_division_remainder_degree_triminput)) * uc)) /\\ exists fom_beta_quotient_pfp_division_remainder_degree_triminput_entry. ub = fom_beta_quotient_pfp_division_remainder_degree_triminput_entry * S ((S (fom_index_pfp_division_remainder_degree_triminput)) * uc) + (fom_value_pfp_division_remainder_degree_triminput))) /\\ (exists fom_gap_pfp_division_remainder_degree_triminput_value_bound. fom_gap_pfp_division_remainder_degree_triminput_value_bound + S (fom_value_pfp_division_remainder_degree_triminput) = p))) /\\ (((forall pfp_repeat_index_division_remainder_degree_trimremoved. (exists pfa_gap_division_remainder_degree_trimremovedindex. pfa_gap_division_remainder_degree_trimremovedindex + S (pfp_repeat_index_division_remainder_degree_trimremoved) = (t)) -> (((exists ff_h_pfp_division_remainder_degree_trimremovedentry. ff_h_pfp_division_remainder_degree_trimremovedentry + S (0) = S ((S (pfp_repeat_index_division_remainder_degree_trimremoved)) * uc)) /\\ exists ff_q_pfp_division_remainder_degree_trimremovedentry. ub = ff_q_pfp_division_remainder_degree_trimremovedentry * S ((S (pfp_repeat_index_division_remainder_degree_trimremoved)) * uc) + (0)))) /\\ (((forall pftrim_index_division_remainder_degree_trimsuffix pftrim_value_division_remainder_degree_trimsuffix. (exists pfa_gap_division_remainder_degree_trimsuffixbound. pfa_gap_division_remainder_degree_trimsuffixbound + S (pftrim_index_division_remainder_degree_trimsuffix) = (R)) -> (((exists ff_h_pfp_division_remainder_degree_trimsuffixsource. ff_h_pfp_division_remainder_degree_trimsuffixsource + S (pftrim_value_division_remainder_degree_trimsuffix) = S ((S ((t)+pftrim_index_division_remainder_degree_trimsuffix)) * uc)) /\\ exists ff_q_pfp_division_remainder_degree_trimsuffixsource. ub = ff_q_pfp_division_remainder_degree_trimsuffixsource * S ((S ((t)+pftrim_index_division_remainder_degree_trimsuffix)) * uc) + (pftrim_value_division_remainder_degree_trimsuffix))) -> (((exists ff_h_pfp_division_remainder_degree_trimsuffixoutput. ff_h_pfp_division_remainder_degree_trimsuffixoutput + S (pftrim_value_division_remainder_degree_trimsuffix) = S ((S (pftrim_index_division_remainder_degree_trimsuffix)) * rc)) /\\ exists ff_q_pfp_division_remainder_degree_trimsuffixoutput. rb = ff_q_pfp_division_remainder_degree_trimsuffixoutput * S ((S (pftrim_index_division_remainder_degree_trimsuffix)) * rc) + (pftrim_value_division_remainder_degree_trimsuffix)))) /\\ (((R)=0 \\/ (exists pftrim_leading_division_remainder_degree_trimnormal. ((((exists ff_h_pfp_division_remainder_degree_trimnormalentry. ff_h_pfp_division_remainder_degree_trimnormalentry + S (pftrim_leading_division_remainder_degree_trimnormal) = S ((S (0)) * rc)) /\\ exists ff_q_pfp_division_remainder_degree_trimnormalentry. rb = ff_q_pfp_division_remainder_degree_trimnormalentry * S ((S (0)) * rc) + (pftrim_leading_division_remainder_degree_trimnormal))) /\\ ((~(pftrim_leading_division_remainder_degree_trimnormal=0))))))))))))))) -> (exists pfc_gap_division_remainder_degree_bound. pfc_gap_division_remainder_degree_bound+(R)=(d)) -> ((R)=0 \\/ (exists pfd_remainder_degree_division_remainder_degree. (((((R)=S (pfd_remainder_degree_division_remainder_degree)) /\\ (((forall fom_index_pfp_division_remainder_degreerepresentedcoefficients. (exists fom_gap_pfp_division_remainder_degreerepresentedcoefficients_index_bound. fom_gap_pfp_division_remainder_degreerepresentedcoefficients_index_bound + S (fom_index_pfp_division_remainder_degreerepresentedcoefficients) = R) -> exists fom_value_pfp_division_remainder_degreerepresentedcoefficients. ((((exists fom_beta_height_pfp_division_remainder_degreerepresentedcoefficients_entry. fom_beta_height_pfp_division_remainder_degreerepresentedcoefficients_entry + S (fom_value_pfp_division_remainder_degreerepresentedcoefficients) = S ((S (fom_index_pfp_division_remainder_degreerepresentedcoefficients)) * rc)) /\\ exists fom_beta_quotient_pfp_division_remainder_degreerepresentedcoefficients_entry. rb = fom_beta_quotient_pfp_division_remainder_degreerepresentedcoefficients_entry * S ((S (fom_index_pfp_division_remainder_degreerepresentedcoefficients)) * rc) + (fom_value_pfp_division_remainder_degreerepresentedcoefficients))) /\\ (exists fom_gap_pfp_division_remainder_degreerepresentedcoefficients_value_bound. fom_gap_pfp_division_remainder_degreerepresentedcoefficients_value_bound + S (fom_value_pfp_division_remainder_degreerepresentedcoefficients) = p))) /\\ ((exists pfd_leading_division_remainder_degreerepresented. ((((exists ff_h_pfp_division_remainder_degreerepresentedentry. ff_h_pfp_division_remainder_degreerepresentedentry + S (pfd_leading_division_remainder_degreerepresented) = S ((S (0)) * rc)) /\\ exists ff_q_pfp_division_remainder_degreerepresentedentry. rb = ff_q_pfp_division_remainder_degreerepresentedentry * S ((S (0)) * rc) + (pfd_leading_division_remainder_degreerepresented))) /\\ ((~(pfd_leading_division_remainder_degreerepresented=0)))))))))) /\\ ((exists pfa_gap_division_remainder_degreestrict. pfa_gap_division_remainder_degreestrict + S (pfd_remainder_degree_division_remainder_degree) = (d))))))", "statement_sha256": "e8bbda08de19f4ae16aef1c541704be826f459d26e0c482bde0d27d260e7566c", "summary": "An actual normalized remainder of length at most d is empty or has an actual represented degree strictly below d, even when d=0.", "summary_sha256": "56df34acc3885faf08569a9d41134553a453677899dcd2423e8193a82f9724ff" }, { "alpha_v33_frontier_enrollment": { "body_receipt_sha256": 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((x) * (k)) + (p) * pfa_offset_left_division_total_inversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_division_total_inversemultiplicationresultcongruence))))))))))))", "specialize prime_field_inverse_exists (p)", "specialize prime_field_inverse_exists (x)", "apply prime_field_inverse_exists", "exact hp", "specialize matrix_rank_bounded_prefix_value (bb)", "specialize matrix_rank_bounded_prefix_value (bc)", "specialize matrix_rank_bounded_prefix_value (S d)", "specialize matrix_rank_bounded_prefix_value (p)", "specialize matrix_rank_bounded_prefix_value (0)", "specialize matrix_rank_bounded_prefix_value (x)", "apply matrix_rank_bounded_prefix_value", "exact hb_right_left", "exists d", "simp", "exact hb_right_right_witness_left", "exact hb_right_right_witness_right", "cases hi", "have hk : exists pfa_gap_division_total_scalar_bound. pfa_gap_division_total_scalar_bound + S (x1) = (p)", "cases hi_witness", "cases hi_witness_right", "cases hi_witness_right_right", "exact hi_witness_right_right_left", "have hlength : exists q. (((((q)=0) /\\ ((exists pfc_gap_division_total_lengthshort. pfc_gap_division_total_lengthshort+(L)=(d))))) \\/ (((~((q)=0)) /\\ (((q)+(d)=(L))))))", "specialize polynomial_quotient_length_exists (L)", "specialize polynomial_quotient_length_exists (d)", "apply polynomial_quotient_length_exists", "cases hlength", "have hbounds : ((exists pfc_gap_division_total_q_bound. pfc_gap_division_total_q_bound+(x2)=(L)) /\\ ((exists pfc_gap_division_total_cover. pfc_gap_division_total_cover+(L)=(x2+d))))", "specialize polynomial_quotient_length_bounds (L)", "specialize polynomial_quotient_length_bounds (d)", "specialize polynomial_quotient_length_bounds (x2)", "apply polynomial_quotient_length_bounds", "exact hlength_witness", "cases hbounds", "have hquotient : exists qb qc. (forall pfd_index_division_total_quotient. (exists pfa_gap_division_total_quotientbound. pfa_gap_division_total_quotientbound + S (pfd_index_division_total_quotient) = (x2)) -> exists pfd_value_division_total_quotient. ((((exists ff_h_pfp_division_total_quotiententry. ff_h_pfp_division_total_quotiententry + S (pfd_value_division_total_quotient) = S ((S (pfd_index_division_total_quotient)) * qc)) /\\ exists ff_q_pfp_division_total_quotiententry. qb = ff_q_pfp_division_total_quotiententry * S ((S (pfd_index_division_total_quotient)) * qc) + (pfd_value_division_total_quotient))) /\\ ((exists pfd_input_division_total_quotientstep pfd_previous_division_total_quotientstep pfd_difference_division_total_quotientstep. ((((exists ff_h_pfp_division_total_quotientstepinput. ff_h_pfp_division_total_quotientstepinput + S (pfd_input_division_total_quotientstep) = S ((S (pfd_index_division_total_quotient)) * ac)) /\\ exists ff_q_pfp_division_total_quotientstepinput. ab = ff_q_pfp_division_total_quotientstepinput * S ((S (pfd_index_division_total_quotient)) * ac) + (pfd_input_division_total_quotientstep))) /\\ (((exists pfc_terms_code_division_total_quotientstepprevious pfc_terms_scale_division_total_quotientstepprevious pfc_natural_sum_division_total_quotientstepprevious. ((forall pfc_index_division_total_quotientsteppreviousdiagonal. (exists pfa_gap_division_total_quotientsteppreviousdiagonalbound. pfa_gap_division_total_quotientsteppreviousdiagonalbound + S (pfc_index_division_total_quotientsteppreviousdiagonal) = (S (pfd_index_division_total_quotient))) -> exists pfc_value_division_total_quotientsteppreviousdiagonal. ((((exists ff_h_pfp_division_total_quotientsteppreviousdiagonalentry. ff_h_pfp_division_total_quotientsteppreviousdiagonalentry + S (pfc_value_division_total_quotientsteppreviousdiagonal) = S ((S (pfc_index_division_total_quotientsteppreviousdiagonal)) * pfc_terms_scale_division_total_quotientstepprevious)) /\\ exists ff_q_pfp_division_total_quotientsteppreviousdiagonalentry. pfc_terms_code_division_total_quotientstepprevious = ff_q_pfp_division_total_quotientsteppreviousdiagonalentry * S ((S (pfc_index_division_total_quotientsteppreviousdiagonal)) * pfc_terms_scale_division_total_quotientstepprevious) + (pfc_value_division_total_quotientsteppreviousdiagonal))) /\\ ((exists pfc_complement_division_total_quotientsteppreviousdiagonalterm pfc_left_division_total_quotientsteppreviousdiagonalterm pfc_right_division_total_quotientsteppreviousdiagonalterm. (((pfc_index_division_total_quotientsteppreviousdiagonal)+pfc_complement_division_total_quotientsteppreviousdiagonalterm=(pfd_index_division_total_quotient)) /\\ ((((((exists pfa_gap_division_total_quotientsteppreviousdiagonaltermleftinside. pfa_gap_division_total_quotientsteppreviousdiagonaltermleftinside + S (pfc_index_division_total_quotientsteppreviousdiagonal) = (pfd_index_division_total_quotient)) /\\ ((((exists ff_h_pfp_division_total_quotientsteppreviousdiagonaltermleftentry. ff_h_pfp_division_total_quotientsteppreviousdiagonaltermleftentry + S (pfc_left_division_total_quotientsteppreviousdiagonalterm) = S ((S (pfc_index_division_total_quotientsteppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_division_total_quotientsteppreviousdiagonaltermleftentry. qb = ff_q_pfp_division_total_quotientsteppreviousdiagonaltermleftentry * S ((S (pfc_index_division_total_quotientsteppreviousdiagonal)) * qc) + (pfc_left_division_total_quotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_total_quotientsteppreviousdiagonaltermleftoutside. pfc_gap_division_total_quotientsteppreviousdiagonaltermleftoutside+(pfd_index_division_total_quotient)=(pfc_index_division_total_quotientsteppreviousdiagonal)) /\\ (((pfc_left_division_total_quotientsteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_total_quotientsteppreviousdiagonaltermrightinside. pfa_gap_division_total_quotientsteppreviousdiagonaltermrightinside + S (pfc_complement_division_total_quotientsteppreviousdiagonalterm) = (S d)) /\\ ((((exists ff_h_pfp_division_total_quotientsteppreviousdiagonaltermrightentry. ff_h_pfp_division_total_quotientsteppreviousdiagonaltermrightentry + S (pfc_right_division_total_quotientsteppreviousdiagonalterm) = S ((S (pfc_complement_division_total_quotientsteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_total_quotientsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_division_total_quotientsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_division_total_quotientsteppreviousdiagonalterm)) * bc) + (pfc_right_division_total_quotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_total_quotientsteppreviousdiagonaltermrightoutside. pfc_gap_division_total_quotientsteppreviousdiagonaltermrightoutside+(S d)=(pfc_complement_division_total_quotientsteppreviousdiagonalterm)) /\\ (((pfc_right_division_total_quotientsteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_division_total_quotientsteppreviousdiagonal)=pfc_left_division_total_quotientsteppreviousdiagonalterm*pfc_right_division_total_quotientsteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_total_quotientstepprevioussum fs_v_pfc_division_total_quotientstepprevioussum. ((((exists fs_h_pfc_division_total_quotientstepprevioussum_body_start. fs_h_pfc_division_total_quotientstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_total_quotientstepprevioussum)) /\\ exists fs_q_pfc_division_total_quotientstepprevioussum_body_start. fs_u_pfc_division_total_quotientstepprevioussum = fs_q_pfc_division_total_quotientstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_division_total_quotientstepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_division_total_quotientstepprevioussum_body_terminal. fs_h_pfc_division_total_quotientstepprevioussum_body_terminal + S (pfc_natural_sum_division_total_quotientstepprevious) = S ((S (S (pfd_index_division_total_quotient))) * fs_v_pfc_division_total_quotientstepprevioussum)) /\\ exists fs_q_pfc_division_total_quotientstepprevioussum_body_terminal. fs_u_pfc_division_total_quotientstepprevioussum = fs_q_pfc_division_total_quotientstepprevioussum_body_terminal * S ((S (S (pfd_index_division_total_quotient))) * fs_v_pfc_division_total_quotientstepprevioussum) + (pfc_natural_sum_division_total_quotientstepprevious))) /\\ forall fs_i_pfc_division_total_quotientstepprevioussum_body_steps. (exists fs_lt_pfc_division_total_quotientstepprevioussum_body_steps_bound. fs_lt_pfc_division_total_quotientstepprevioussum_body_steps_bound + S fs_i_pfc_division_total_quotientstepprevioussum_body_steps = S (pfd_index_division_total_quotient)) -> exists fs_a_pfc_division_total_quotientstepprevioussum_body_steps fs_r_pfc_division_total_quotientstepprevioussum_body_steps fs_s_pfc_division_total_quotientstepprevioussum_body_steps. ((((exists fs_h_pfc_division_total_quotientstepprevioussum_body_steps_summand. fs_h_pfc_division_total_quotientstepprevioussum_body_steps_summand + S (fs_a_pfc_division_total_quotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_total_quotientstepprevioussum_body_steps)) * pfc_terms_scale_division_total_quotientstepprevious)) /\\ exists fs_q_pfc_division_total_quotientstepprevioussum_body_steps_summand. pfc_terms_code_division_total_quotientstepprevious = fs_q_pfc_division_total_quotientstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_division_total_quotientstepprevioussum_body_steps)) * pfc_terms_scale_division_total_quotientstepprevious) + (fs_a_pfc_division_total_quotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_total_quotientstepprevioussum_body_steps_partial. fs_h_pfc_division_total_quotientstepprevioussum_body_steps_partial + S (fs_r_pfc_division_total_quotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_total_quotientstepprevioussum_body_steps)) * fs_v_pfc_division_total_quotientstepprevioussum)) /\\ exists fs_q_pfc_division_total_quotientstepprevioussum_body_steps_partial. fs_u_pfc_division_total_quotientstepprevioussum = fs_q_pfc_division_total_quotientstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_division_total_quotientstepprevioussum_body_steps)) * fs_v_pfc_division_total_quotientstepprevioussum) + (fs_r_pfc_division_total_quotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_total_quotientstepprevioussum_body_steps_successor. fs_h_pfc_division_total_quotientstepprevioussum_body_steps_successor + S (fs_s_pfc_division_total_quotientstepprevioussum_body_steps) = S ((S (S fs_i_pfc_division_total_quotientstepprevioussum_body_steps)) * fs_v_pfc_division_total_quotientstepprevioussum)) /\\ exists fs_q_pfc_division_total_quotientstepprevioussum_body_steps_successor. fs_u_pfc_division_total_quotientstepprevioussum = fs_q_pfc_division_total_quotientstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_division_total_quotientstepprevioussum_body_steps)) * fs_v_pfc_division_total_quotientstepprevioussum) + (fs_s_pfc_division_total_quotientstepprevioussum_body_steps))) /\\ fs_s_pfc_division_total_quotientstepprevioussum_body_steps = fs_r_pfc_division_total_quotientstepprevioussum_body_steps + fs_a_pfc_division_total_quotientstepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_division_total_quotientsteppreviousresiduebound. pfa_gap_division_total_quotientsteppreviousresiduebound + S (pfd_previous_division_total_quotientstep) = (p)) /\\ ((exists pfa_offset_left_division_total_quotientsteppreviousresiduecongruence pfa_offset_right_division_total_quotientsteppreviousresiduecongruence. (pfc_natural_sum_division_total_quotientstepprevious) + (p) * pfa_offset_left_division_total_quotientsteppreviousresiduecongruence = (pfd_previous_division_total_quotientstep) + (p) * pfa_offset_right_division_total_quotientsteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_division_total_quotientstepsubtractleft. pfa_gap_division_total_quotientstepsubtractleft + S (pfd_previous_division_total_quotientstep) = (p)) /\\ (((exists pfa_gap_division_total_quotientstepsubtractright. pfa_gap_division_total_quotientstepsubtractright + S (pfd_difference_division_total_quotientstep) = (p)) /\\ ((((exists pfa_gap_division_total_quotientstepsubtractresultbound. pfa_gap_division_total_quotientstepsubtractresultbound + S (pfd_input_division_total_quotientstep) = (p)) /\\ ((exists pfa_offset_left_division_total_quotientstepsubtractresultcongruence pfa_offset_right_division_total_quotientstepsubtractresultcongruence. ((pfd_previous_division_total_quotientstep) + (pfd_difference_division_total_quotientstep)) + (p) * pfa_offset_left_division_total_quotientstepsubtractresultcongruence = (pfd_input_division_total_quotientstep) + (p) * pfa_offset_right_division_total_quotientstepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_division_total_quotientstepmultiplyleft. pfa_gap_division_total_quotientstepmultiplyleft + S (x1) = (p)) /\\ (((exists pfa_gap_division_total_quotientstepmultiplyright. pfa_gap_division_total_quotientstepmultiplyright + S (pfd_difference_division_total_quotientstep) = (p)) /\\ ((((exists pfa_gap_division_total_quotientstepmultiplyresultbound. pfa_gap_division_total_quotientstepmultiplyresultbound + S (pfd_value_division_total_quotient) = (p)) /\\ ((exists pfa_offset_left_division_total_quotientstepmultiplyresultcongruence pfa_offset_right_division_total_quotientstepmultiplyresultcongruence. ((x1) * (pfd_difference_division_total_quotientstep)) + (p) * pfa_offset_left_division_total_quotientstepmultiplyresultcongruence = (pfd_value_division_total_quotient) + (p) * pfa_offset_right_division_total_quotientstepmultiplyresultcongruence)))))))))))))))))))", "specialize prime_field_polynomial_quotient_prefix_exists (p)", "specialize prime_field_polynomial_quotient_prefix_exists (x1)", "specialize prime_field_polynomial_quotient_prefix_exists (ab)", "specialize prime_field_polynomial_quotient_prefix_exists (ac)", "specialize prime_field_polynomial_quotient_prefix_exists (bb)", "specialize prime_field_polynomial_quotient_prefix_exists (bc)", "specialize prime_field_polynomial_quotient_prefix_exists (S d)", "specialize prime_field_polynomial_quotient_prefix_exists (x2)", "apply prime_field_polynomial_quotient_prefix_exists", "exact hp", "exact hk", "intro i", "intro hindex", "specialize ha (i)", "apply ha", "specialize lt_of_lt_of_le (i)", "specialize lt_of_lt_of_le (x2)", "specialize lt_of_lt_of_le (L)", "apply lt_of_lt_of_le", "exact hindex", "exact hbounds_left", "cases hquotient", "cases hquotient_witness", "exists x", "exists x1", "exists x2", "exists x3", "exists x4", "split", "exact hb_right_right_witness_left", "split", "exact hi_witness", "split", "exact hlength_witness", "exact hquotient_witness_witness" ], "script_sha256": "1197cfd43d7d39410e6f282871c7151d7cd0b2975cc6e8c5b48b1820248e09b0", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_division_candidate.py", "sha256": "edfc7806caf7a83b9cb0e3e420bd2c3a8679f2d4d9ee6ca9f8eae53faca8d5b2" }, "statement": "forall p ab ac L bb bc d. (~((p) = 1) /\\ forall pfa_factor_left_division_quotient_data_prime pfa_factor_right_division_quotient_data_prime. (p) = pfa_factor_left_division_quotient_data_prime * pfa_factor_right_division_quotient_data_prime -> pfa_factor_left_division_quotient_data_prime = 1 \\/ pfa_factor_right_division_quotient_data_prime = 1) -> (forall fom_index_pfp_division_quotient_data_input. (exists fom_gap_pfp_division_quotient_data_input_index_bound. fom_gap_pfp_division_quotient_data_input_index_bound + S (fom_index_pfp_division_quotient_data_input) = L) -> exists fom_value_pfp_division_quotient_data_input. ((((exists fom_beta_height_pfp_division_quotient_data_input_entry. fom_beta_height_pfp_division_quotient_data_input_entry + S (fom_value_pfp_division_quotient_data_input) = S ((S (fom_index_pfp_division_quotient_data_input)) * ac)) /\\ exists fom_beta_quotient_pfp_division_quotient_data_input_entry. ab = fom_beta_quotient_pfp_division_quotient_data_input_entry * S ((S (fom_index_pfp_division_quotient_data_input)) * ac) + (fom_value_pfp_division_quotient_data_input))) /\\ (exists fom_gap_pfp_division_quotient_data_input_value_bound. fom_gap_pfp_division_quotient_data_input_value_bound + S (fom_value_pfp_division_quotient_data_input) = p))) -> ((((S d)=S (d)) /\\ (((forall fom_index_pfp_division_quotient_data_divisorcoefficients. (exists fom_gap_pfp_division_quotient_data_divisorcoefficients_index_bound. fom_gap_pfp_division_quotient_data_divisorcoefficients_index_bound + S (fom_index_pfp_division_quotient_data_divisorcoefficients) = S d) -> exists fom_value_pfp_division_quotient_data_divisorcoefficients. ((((exists fom_beta_height_pfp_division_quotient_data_divisorcoefficients_entry. fom_beta_height_pfp_division_quotient_data_divisorcoefficients_entry + S (fom_value_pfp_division_quotient_data_divisorcoefficients) = S ((S (fom_index_pfp_division_quotient_data_divisorcoefficients)) * bc)) /\\ exists fom_beta_quotient_pfp_division_quotient_data_divisorcoefficients_entry. bb = fom_beta_quotient_pfp_division_quotient_data_divisorcoefficients_entry * S ((S (fom_index_pfp_division_quotient_data_divisorcoefficients)) * bc) + (fom_value_pfp_division_quotient_data_divisorcoefficients))) /\\ (exists fom_gap_pfp_division_quotient_data_divisorcoefficients_value_bound. fom_gap_pfp_division_quotient_data_divisorcoefficients_value_bound + S (fom_value_pfp_division_quotient_data_divisorcoefficients) = p))) /\\ ((exists pfd_leading_division_quotient_data_divisor. ((((exists ff_h_pfp_division_quotient_data_divisorentry. ff_h_pfp_division_quotient_data_divisorentry + S (pfd_leading_division_quotient_data_divisor) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_division_quotient_data_divisorentry. bb = ff_q_pfp_division_quotient_data_divisorentry * S ((S (0)) * bc) + (pfd_leading_division_quotient_data_divisor))) /\\ ((~(pfd_leading_division_quotient_data_divisor=0)))))))))) -> exists b k q qb qc. (((((exists ff_h_pfp_division_quotient_data_resulthead. ff_h_pfp_division_quotient_data_resulthead + S (b) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_division_quotient_data_resulthead. bb = ff_q_pfp_division_quotient_data_resulthead * S ((S (0)) * bc) + (b))) /\\ (((((~((b) = 0)) /\\ ((((exists pfa_gap_division_quotient_data_resultinversemultiplicationleft. pfa_gap_division_quotient_data_resultinversemultiplicationleft + S (b) = (p)) /\\ (((exists pfa_gap_division_quotient_data_resultinversemultiplicationright. pfa_gap_division_quotient_data_resultinversemultiplicationright + S (k) = (p)) /\\ ((((exists pfa_gap_division_quotient_data_resultinversemultiplicationresultbound. pfa_gap_division_quotient_data_resultinversemultiplicationresultbound + S (1) = (p)) /\\ ((exists pfa_offset_left_division_quotient_data_resultinversemultiplicationresultcongruence pfa_offset_right_division_quotient_data_resultinversemultiplicationresultcongruence. ((b) * (k)) + (p) * pfa_offset_left_division_quotient_data_resultinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_division_quotient_data_resultinversemultiplicationresultcongruence)))))))))))) /\\ (((((((q)=0) /\\ ((exists pfc_gap_division_quotient_data_resultlengthshort. pfc_gap_division_quotient_data_resultlengthshort+(L)=(d))))) \\/ (((~((q)=0)) /\\ (((q)+(d)=(L)))))) /\\ ((forall pfd_index_division_quotient_data_resultexecution. (exists pfa_gap_division_quotient_data_resultexecutionbound. pfa_gap_division_quotient_data_resultexecutionbound + S (pfd_index_division_quotient_data_resultexecution) = (q)) -> exists pfd_value_division_quotient_data_resultexecution. ((((exists ff_h_pfp_division_quotient_data_resultexecutionentry. ff_h_pfp_division_quotient_data_resultexecutionentry + S (pfd_value_division_quotient_data_resultexecution) = S ((S (pfd_index_division_quotient_data_resultexecution)) * qc)) /\\ exists ff_q_pfp_division_quotient_data_resultexecutionentry. qb = ff_q_pfp_division_quotient_data_resultexecutionentry * S ((S (pfd_index_division_quotient_data_resultexecution)) * qc) + (pfd_value_division_quotient_data_resultexecution))) /\\ ((exists pfd_input_division_quotient_data_resultexecutionstep pfd_previous_division_quotient_data_resultexecutionstep pfd_difference_division_quotient_data_resultexecutionstep. ((((exists ff_h_pfp_division_quotient_data_resultexecutionstepinput. ff_h_pfp_division_quotient_data_resultexecutionstepinput + S (pfd_input_division_quotient_data_resultexecutionstep) = S ((S (pfd_index_division_quotient_data_resultexecution)) * ac)) /\\ exists ff_q_pfp_division_quotient_data_resultexecutionstepinput. ab = ff_q_pfp_division_quotient_data_resultexecutionstepinput * S ((S (pfd_index_division_quotient_data_resultexecution)) * ac) + (pfd_input_division_quotient_data_resultexecutionstep))) /\\ (((exists pfc_terms_code_division_quotient_data_resultexecutionstepprevious pfc_terms_scale_division_quotient_data_resultexecutionstepprevious pfc_natural_sum_division_quotient_data_resultexecutionstepprevious. ((forall pfc_index_division_quotient_data_resultexecutionsteppreviousdiagonal. (exists pfa_gap_division_quotient_data_resultexecutionsteppreviousdiagonalbound. pfa_gap_division_quotient_data_resultexecutionsteppreviousdiagonalbound + S (pfc_index_division_quotient_data_resultexecutionsteppreviousdiagonal) = (S (pfd_index_division_quotient_data_resultexecution))) -> exists pfc_value_division_quotient_data_resultexecutionsteppreviousdiagonal. ((((exists ff_h_pfp_division_quotient_data_resultexecutionsteppreviousdiagonalentry. ff_h_pfp_division_quotient_data_resultexecutionsteppreviousdiagonalentry + S (pfc_value_division_quotient_data_resultexecutionsteppreviousdiagonal) = S ((S (pfc_index_division_quotient_data_resultexecutionsteppreviousdiagonal)) * pfc_terms_scale_division_quotient_data_resultexecutionstepprevious)) /\\ exists ff_q_pfp_division_quotient_data_resultexecutionsteppreviousdiagonalentry. pfc_terms_code_division_quotient_data_resultexecutionstepprevious = ff_q_pfp_division_quotient_data_resultexecutionsteppreviousdiagonalentry * S ((S (pfc_index_division_quotient_data_resultexecutionsteppreviousdiagonal)) * pfc_terms_scale_division_quotient_data_resultexecutionstepprevious) + (pfc_value_division_quotient_data_resultexecutionsteppreviousdiagonal))) /\\ ((exists pfc_complement_division_quotient_data_resultexecutionsteppreviousdiagonalterm pfc_left_division_quotient_data_resultexecutionsteppreviousdiagonalterm pfc_right_division_quotient_data_resultexecutionsteppreviousdiagonalterm. (((pfc_index_division_quotient_data_resultexecutionsteppreviousdiagonal)+pfc_complement_division_quotient_data_resultexecutionsteppreviousdiagonalterm=(pfd_index_division_quotient_data_resultexecution)) /\\ ((((((exists pfa_gap_division_quotient_data_resultexecutionsteppreviousdiagonaltermleftinside. pfa_gap_division_quotient_data_resultexecutionsteppreviousdiagonaltermleftinside + S (pfc_index_division_quotient_data_resultexecutionsteppreviousdiagonal) = (pfd_index_division_quotient_data_resultexecution)) /\\ ((((exists ff_h_pfp_division_quotient_data_resultexecutionsteppreviousdiagonaltermleftentry. ff_h_pfp_division_quotient_data_resultexecutionsteppreviousdiagonaltermleftentry + S (pfc_left_division_quotient_data_resultexecutionsteppreviousdiagonalterm) = S ((S (pfc_index_division_quotient_data_resultexecutionsteppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_division_quotient_data_resultexecutionsteppreviousdiagonaltermleftentry. qb = ff_q_pfp_division_quotient_data_resultexecutionsteppreviousdiagonaltermleftentry * S ((S (pfc_index_division_quotient_data_resultexecutionsteppreviousdiagonal)) * qc) + (pfc_left_division_quotient_data_resultexecutionsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_quotient_data_resultexecutionsteppreviousdiagonaltermleftoutside. pfc_gap_division_quotient_data_resultexecutionsteppreviousdiagonaltermleftoutside+(pfd_index_division_quotient_data_resultexecution)=(pfc_index_division_quotient_data_resultexecutionsteppreviousdiagonal)) /\\ (((pfc_left_division_quotient_data_resultexecutionsteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_quotient_data_resultexecutionsteppreviousdiagonaltermrightinside. pfa_gap_division_quotient_data_resultexecutionsteppreviousdiagonaltermrightinside + S (pfc_complement_division_quotient_data_resultexecutionsteppreviousdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_division_quotient_data_resultexecutionsteppreviousdiagonaltermrightentry. ff_h_pfp_division_quotient_data_resultexecutionsteppreviousdiagonaltermrightentry + S (pfc_right_division_quotient_data_resultexecutionsteppreviousdiagonalterm) = S ((S (pfc_complement_division_quotient_data_resultexecutionsteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_quotient_data_resultexecutionsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_division_quotient_data_resultexecutionsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_division_quotient_data_resultexecutionsteppreviousdiagonalterm)) * bc) + (pfc_right_division_quotient_data_resultexecutionsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_quotient_data_resultexecutionsteppreviousdiagonaltermrightoutside. pfc_gap_division_quotient_data_resultexecutionsteppreviousdiagonaltermrightoutside+(S (d))=(pfc_complement_division_quotient_data_resultexecutionsteppreviousdiagonalterm)) /\\ (((pfc_right_division_quotient_data_resultexecutionsteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_division_quotient_data_resultexecutionsteppreviousdiagonal)=pfc_left_division_quotient_data_resultexecutionsteppreviousdiagonalterm*pfc_right_division_quotient_data_resultexecutionsteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_quotient_data_resultexecutionstepprevioussum fs_v_pfc_division_quotient_data_resultexecutionstepprevioussum. ((((exists fs_h_pfc_division_quotient_data_resultexecutionstepprevioussum_body_start. fs_h_pfc_division_quotient_data_resultexecutionstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_quotient_data_resultexecutionstepprevioussum)) /\\ exists fs_q_pfc_division_quotient_data_resultexecutionstepprevioussum_body_start. fs_u_pfc_division_quotient_data_resultexecutionstepprevioussum = fs_q_pfc_division_quotient_data_resultexecutionstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_division_quotient_data_resultexecutionstepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_division_quotient_data_resultexecutionstepprevioussum_body_terminal. fs_h_pfc_division_quotient_data_resultexecutionstepprevioussum_body_terminal + S (pfc_natural_sum_division_quotient_data_resultexecutionstepprevious) = S ((S (S (pfd_index_division_quotient_data_resultexecution))) * fs_v_pfc_division_quotient_data_resultexecutionstepprevioussum)) /\\ exists fs_q_pfc_division_quotient_data_resultexecutionstepprevioussum_body_terminal. fs_u_pfc_division_quotient_data_resultexecutionstepprevioussum = fs_q_pfc_division_quotient_data_resultexecutionstepprevioussum_body_terminal * S ((S (S (pfd_index_division_quotient_data_resultexecution))) * fs_v_pfc_division_quotient_data_resultexecutionstepprevioussum) + (pfc_natural_sum_division_quotient_data_resultexecutionstepprevious))) /\\ forall fs_i_pfc_division_quotient_data_resultexecutionstepprevioussum_body_steps. (exists fs_lt_pfc_division_quotient_data_resultexecutionstepprevioussum_body_steps_bound. fs_lt_pfc_division_quotient_data_resultexecutionstepprevioussum_body_steps_bound + S fs_i_pfc_division_quotient_data_resultexecutionstepprevioussum_body_steps = S (pfd_index_division_quotient_data_resultexecution)) -> exists fs_a_pfc_division_quotient_data_resultexecutionstepprevioussum_body_steps fs_r_pfc_division_quotient_data_resultexecutionstepprevioussum_body_steps fs_s_pfc_division_quotient_data_resultexecutionstepprevioussum_body_steps. ((((exists fs_h_pfc_division_quotient_data_resultexecutionstepprevioussum_body_steps_summand. fs_h_pfc_division_quotient_data_resultexecutionstepprevioussum_body_steps_summand + S (fs_a_pfc_division_quotient_data_resultexecutionstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_quotient_data_resultexecutionstepprevioussum_body_steps)) * pfc_terms_scale_division_quotient_data_resultexecutionstepprevious)) /\\ exists fs_q_pfc_division_quotient_data_resultexecutionstepprevioussum_body_steps_summand. pfc_terms_code_division_quotient_data_resultexecutionstepprevious = fs_q_pfc_division_quotient_data_resultexecutionstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_division_quotient_data_resultexecutionstepprevioussum_body_steps)) * pfc_terms_scale_division_quotient_data_resultexecutionstepprevious) + (fs_a_pfc_division_quotient_data_resultexecutionstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_quotient_data_resultexecutionstepprevioussum_body_steps_partial. fs_h_pfc_division_quotient_data_resultexecutionstepprevioussum_body_steps_partial + S (fs_r_pfc_division_quotient_data_resultexecutionstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_quotient_data_resultexecutionstepprevioussum_body_steps)) * fs_v_pfc_division_quotient_data_resultexecutionstepprevioussum)) /\\ exists fs_q_pfc_division_quotient_data_resultexecutionstepprevioussum_body_steps_partial. fs_u_pfc_division_quotient_data_resultexecutionstepprevioussum = fs_q_pfc_division_quotient_data_resultexecutionstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_division_quotient_data_resultexecutionstepprevioussum_body_steps)) * fs_v_pfc_division_quotient_data_resultexecutionstepprevioussum) + (fs_r_pfc_division_quotient_data_resultexecutionstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_quotient_data_resultexecutionstepprevioussum_body_steps_successor. fs_h_pfc_division_quotient_data_resultexecutionstepprevioussum_body_steps_successor + S (fs_s_pfc_division_quotient_data_resultexecutionstepprevioussum_body_steps) = S ((S (S fs_i_pfc_division_quotient_data_resultexecutionstepprevioussum_body_steps)) * fs_v_pfc_division_quotient_data_resultexecutionstepprevioussum)) /\\ exists fs_q_pfc_division_quotient_data_resultexecutionstepprevioussum_body_steps_successor. fs_u_pfc_division_quotient_data_resultexecutionstepprevioussum = fs_q_pfc_division_quotient_data_resultexecutionstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_division_quotient_data_resultexecutionstepprevioussum_body_steps)) * fs_v_pfc_division_quotient_data_resultexecutionstepprevioussum) + (fs_s_pfc_division_quotient_data_resultexecutionstepprevioussum_body_steps))) /\\ fs_s_pfc_division_quotient_data_resultexecutionstepprevioussum_body_steps = fs_r_pfc_division_quotient_data_resultexecutionstepprevioussum_body_steps + fs_a_pfc_division_quotient_data_resultexecutionstepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_division_quotient_data_resultexecutionsteppreviousresiduebound. pfa_gap_division_quotient_data_resultexecutionsteppreviousresiduebound + S (pfd_previous_division_quotient_data_resultexecutionstep) = (p)) /\\ ((exists pfa_offset_left_division_quotient_data_resultexecutionsteppreviousresiduecongruence pfa_offset_right_division_quotient_data_resultexecutionsteppreviousresiduecongruence. (pfc_natural_sum_division_quotient_data_resultexecutionstepprevious) + (p) * pfa_offset_left_division_quotient_data_resultexecutionsteppreviousresiduecongruence = (pfd_previous_division_quotient_data_resultexecutionstep) + (p) * pfa_offset_right_division_quotient_data_resultexecutionsteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_division_quotient_data_resultexecutionstepsubtractleft. pfa_gap_division_quotient_data_resultexecutionstepsubtractleft + S (pfd_previous_division_quotient_data_resultexecutionstep) = (p)) /\\ (((exists pfa_gap_division_quotient_data_resultexecutionstepsubtractright. pfa_gap_division_quotient_data_resultexecutionstepsubtractright + S (pfd_difference_division_quotient_data_resultexecutionstep) = (p)) /\\ ((((exists pfa_gap_division_quotient_data_resultexecutionstepsubtractresultbound. pfa_gap_division_quotient_data_resultexecutionstepsubtractresultbound + S (pfd_input_division_quotient_data_resultexecutionstep) = (p)) /\\ ((exists pfa_offset_left_division_quotient_data_resultexecutionstepsubtractresultcongruence pfa_offset_right_division_quotient_data_resultexecutionstepsubtractresultcongruence. ((pfd_previous_division_quotient_data_resultexecutionstep) + (pfd_difference_division_quotient_data_resultexecutionstep)) + (p) * pfa_offset_left_division_quotient_data_resultexecutionstepsubtractresultcongruence = (pfd_input_division_quotient_data_resultexecutionstep) + (p) * pfa_offset_right_division_quotient_data_resultexecutionstepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_division_quotient_data_resultexecutionstepmultiplyleft. pfa_gap_division_quotient_data_resultexecutionstepmultiplyleft + S (k) = (p)) /\\ (((exists pfa_gap_division_quotient_data_resultexecutionstepmultiplyright. pfa_gap_division_quotient_data_resultexecutionstepmultiplyright + S (pfd_difference_division_quotient_data_resultexecutionstep) = (p)) /\\ ((((exists pfa_gap_division_quotient_data_resultexecutionstepmultiplyresultbound. pfa_gap_division_quotient_data_resultexecutionstepmultiplyresultbound + S (pfd_value_division_quotient_data_resultexecution) = (p)) /\\ ((exists pfa_offset_left_division_quotient_data_resultexecutionstepmultiplyresultcongruence pfa_offset_right_division_quotient_data_resultexecutionstepmultiplyresultcongruence. 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"alpha_only", "name": "prime_field_polynomial_division_residual_data_exists", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro ab", "intro ac", "intro L", "intro bb", "intro bc", "intro d", "intro qb", "intro qc", "intro q", "intro hp", "intro ha", "have hproduct : exists pb pc. (forall pfc_index_division_residual_product. (exists pfa_gap_division_residual_productbound. pfa_gap_division_residual_productbound + S (pfc_index_division_residual_product) = (L)) -> exists pfc_value_division_residual_product. ((((exists ff_h_pfp_division_residual_productentry. ff_h_pfp_division_residual_productentry + S (pfc_value_division_residual_product) = S ((S (pfc_index_division_residual_product)) * pc)) /\\ exists ff_q_pfp_division_residual_productentry. pb = ff_q_pfp_division_residual_productentry * S ((S (pfc_index_division_residual_product)) * pc) + (pfc_value_division_residual_product))) /\\ ((exists pfc_terms_code_division_residual_productcoefficient pfc_terms_scale_division_residual_productcoefficient pfc_natural_sum_division_residual_productcoefficient. ((forall pfc_index_division_residual_productcoefficientdiagonal. (exists pfa_gap_division_residual_productcoefficientdiagonalbound. pfa_gap_division_residual_productcoefficientdiagonalbound + S (pfc_index_division_residual_productcoefficientdiagonal) = (S (pfc_index_division_residual_product))) -> exists pfc_value_division_residual_productcoefficientdiagonal. ((((exists ff_h_pfp_division_residual_productcoefficientdiagonalentry. ff_h_pfp_division_residual_productcoefficientdiagonalentry + S (pfc_value_division_residual_productcoefficientdiagonal) = S ((S (pfc_index_division_residual_productcoefficientdiagonal)) * pfc_terms_scale_division_residual_productcoefficient)) /\\ exists ff_q_pfp_division_residual_productcoefficientdiagonalentry. pfc_terms_code_division_residual_productcoefficient = ff_q_pfp_division_residual_productcoefficientdiagonalentry * S ((S (pfc_index_division_residual_productcoefficientdiagonal)) * pfc_terms_scale_division_residual_productcoefficient) + (pfc_value_division_residual_productcoefficientdiagonal))) /\\ ((exists pfc_complement_division_residual_productcoefficientdiagonalterm pfc_left_division_residual_productcoefficientdiagonalterm pfc_right_division_residual_productcoefficientdiagonalterm. (((pfc_index_division_residual_productcoefficientdiagonal)+pfc_complement_division_residual_productcoefficientdiagonalterm=(pfc_index_division_residual_product)) /\\ ((((((exists pfa_gap_division_residual_productcoefficientdiagonaltermleftinside. pfa_gap_division_residual_productcoefficientdiagonaltermleftinside + S (pfc_index_division_residual_productcoefficientdiagonal) = (q)) /\\ ((((exists ff_h_pfp_division_residual_productcoefficientdiagonaltermleftentry. ff_h_pfp_division_residual_productcoefficientdiagonaltermleftentry + S (pfc_left_division_residual_productcoefficientdiagonalterm) = S ((S (pfc_index_division_residual_productcoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_division_residual_productcoefficientdiagonaltermleftentry. qb = ff_q_pfp_division_residual_productcoefficientdiagonaltermleftentry * S ((S (pfc_index_division_residual_productcoefficientdiagonal)) * qc) + (pfc_left_division_residual_productcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_division_residual_productcoefficientdiagonaltermleftoutside. pfc_gap_division_residual_productcoefficientdiagonaltermleftoutside+(q)=(pfc_index_division_residual_productcoefficientdiagonal)) /\\ (((pfc_left_division_residual_productcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_residual_productcoefficientdiagonaltermrightinside. pfa_gap_division_residual_productcoefficientdiagonaltermrightinside + S (pfc_complement_division_residual_productcoefficientdiagonalterm) = (S d)) /\\ ((((exists ff_h_pfp_division_residual_productcoefficientdiagonaltermrightentry. ff_h_pfp_division_residual_productcoefficientdiagonaltermrightentry + S (pfc_right_division_residual_productcoefficientdiagonalterm) = S ((S (pfc_complement_division_residual_productcoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_residual_productcoefficientdiagonaltermrightentry. bb = ff_q_pfp_division_residual_productcoefficientdiagonaltermrightentry * S ((S (pfc_complement_division_residual_productcoefficientdiagonalterm)) * bc) + (pfc_right_division_residual_productcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_division_residual_productcoefficientdiagonaltermrightoutside. pfc_gap_division_residual_productcoefficientdiagonaltermrightoutside+(S d)=(pfc_complement_division_residual_productcoefficientdiagonalterm)) /\\ (((pfc_right_division_residual_productcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_division_residual_productcoefficientdiagonal)=pfc_left_division_residual_productcoefficientdiagonalterm*pfc_right_division_residual_productcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_residual_productcoefficientsum fs_v_pfc_division_residual_productcoefficientsum. ((((exists fs_h_pfc_division_residual_productcoefficientsum_body_start. fs_h_pfc_division_residual_productcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_residual_productcoefficientsum)) /\\ exists fs_q_pfc_division_residual_productcoefficientsum_body_start. fs_u_pfc_division_residual_productcoefficientsum = fs_q_pfc_division_residual_productcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_division_residual_productcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_division_residual_productcoefficientsum_body_terminal. fs_h_pfc_division_residual_productcoefficientsum_body_terminal + S (pfc_natural_sum_division_residual_productcoefficient) = S ((S (S (pfc_index_division_residual_product))) * fs_v_pfc_division_residual_productcoefficientsum)) /\\ exists fs_q_pfc_division_residual_productcoefficientsum_body_terminal. fs_u_pfc_division_residual_productcoefficientsum = fs_q_pfc_division_residual_productcoefficientsum_body_terminal * S ((S (S (pfc_index_division_residual_product))) * fs_v_pfc_division_residual_productcoefficientsum) + (pfc_natural_sum_division_residual_productcoefficient))) /\\ forall fs_i_pfc_division_residual_productcoefficientsum_body_steps. (exists fs_lt_pfc_division_residual_productcoefficientsum_body_steps_bound. fs_lt_pfc_division_residual_productcoefficientsum_body_steps_bound + S fs_i_pfc_division_residual_productcoefficientsum_body_steps = S (pfc_index_division_residual_product)) -> exists fs_a_pfc_division_residual_productcoefficientsum_body_steps fs_r_pfc_division_residual_productcoefficientsum_body_steps fs_s_pfc_division_residual_productcoefficientsum_body_steps. ((((exists fs_h_pfc_division_residual_productcoefficientsum_body_steps_summand. fs_h_pfc_division_residual_productcoefficientsum_body_steps_summand + S (fs_a_pfc_division_residual_productcoefficientsum_body_steps) = S ((S (fs_i_pfc_division_residual_productcoefficientsum_body_steps)) * pfc_terms_scale_division_residual_productcoefficient)) /\\ exists fs_q_pfc_division_residual_productcoefficientsum_body_steps_summand. pfc_terms_code_division_residual_productcoefficient = fs_q_pfc_division_residual_productcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_division_residual_productcoefficientsum_body_steps)) * pfc_terms_scale_division_residual_productcoefficient) + (fs_a_pfc_division_residual_productcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_division_residual_productcoefficientsum_body_steps_partial. fs_h_pfc_division_residual_productcoefficientsum_body_steps_partial + S (fs_r_pfc_division_residual_productcoefficientsum_body_steps) = S ((S (fs_i_pfc_division_residual_productcoefficientsum_body_steps)) * fs_v_pfc_division_residual_productcoefficientsum)) /\\ exists fs_q_pfc_division_residual_productcoefficientsum_body_steps_partial. fs_u_pfc_division_residual_productcoefficientsum = fs_q_pfc_division_residual_productcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_division_residual_productcoefficientsum_body_steps)) * fs_v_pfc_division_residual_productcoefficientsum) + (fs_r_pfc_division_residual_productcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_division_residual_productcoefficientsum_body_steps_successor. fs_h_pfc_division_residual_productcoefficientsum_body_steps_successor + S (fs_s_pfc_division_residual_productcoefficientsum_body_steps) = S ((S (S fs_i_pfc_division_residual_productcoefficientsum_body_steps)) * fs_v_pfc_division_residual_productcoefficientsum)) /\\ exists fs_q_pfc_division_residual_productcoefficientsum_body_steps_successor. fs_u_pfc_division_residual_productcoefficientsum = fs_q_pfc_division_residual_productcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_division_residual_productcoefficientsum_body_steps)) * fs_v_pfc_division_residual_productcoefficientsum) + (fs_s_pfc_division_residual_productcoefficientsum_body_steps))) /\\ fs_s_pfc_division_residual_productcoefficientsum_body_steps = fs_r_pfc_division_residual_productcoefficientsum_body_steps + fs_a_pfc_division_residual_productcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_division_residual_productcoefficientresiduebound. pfa_gap_division_residual_productcoefficientresiduebound + S (pfc_value_division_residual_product) = (p)) /\\ ((exists pfa_offset_left_division_residual_productcoefficientresiduecongruence pfa_offset_right_division_residual_productcoefficientresiduecongruence. (pfc_natural_sum_division_residual_productcoefficient) + (p) * pfa_offset_left_division_residual_productcoefficientresiduecongruence = (pfc_value_division_residual_product) + (p) * pfa_offset_right_division_residual_productcoefficientresiduecongruence))))))))))))", "specialize prime_field_convolution_prefix_exists (p)", "specialize prime_field_convolution_prefix_exists (qb)", "specialize prime_field_convolution_prefix_exists (qc)", "specialize prime_field_convolution_prefix_exists (q)", "specialize prime_field_convolution_prefix_exists (bb)", "specialize prime_field_convolution_prefix_exists (bc)", "specialize prime_field_convolution_prefix_exists (S d)", "specialize prime_field_convolution_prefix_exists (L)", "apply prime_field_convolution_prefix_exists", "intro hz", "specialize prime_nonzero (p)", "apply prime_nonzero", "exact hp", "exact hz", "cases hproduct", "cases hproduct_witness", "have hresidual : exists ub uc. (forall pfs_index_division_residual_difference. (exists pfa_gap_division_residual_differenceindex. pfa_gap_division_residual_differenceindex + S (pfs_index_division_residual_difference) = (L)) -> exists pfs_left_division_residual_difference pfs_right_division_residual_difference pfs_result_division_residual_difference. ((((exists ff_h_pfp_division_residual_differenceleft. ff_h_pfp_division_residual_differenceleft + S (pfs_left_division_residual_difference) = S ((S (pfs_index_division_residual_difference)) * ac)) /\\ exists ff_q_pfp_division_residual_differenceleft. ab = ff_q_pfp_division_residual_differenceleft * S ((S (pfs_index_division_residual_difference)) * ac) + (pfs_left_division_residual_difference))) /\\ (((((exists ff_h_pfp_division_residual_differenceright. ff_h_pfp_division_residual_differenceright + S (pfs_right_division_residual_difference) = S ((S (pfs_index_division_residual_difference)) * x1)) /\\ exists ff_q_pfp_division_residual_differenceright. x = ff_q_pfp_division_residual_differenceright * S ((S (pfs_index_division_residual_difference)) * x1) + (pfs_right_division_residual_difference))) /\\ (((((exists ff_h_pfp_division_residual_differenceresult. ff_h_pfp_division_residual_differenceresult + S (pfs_result_division_residual_difference) = S ((S (pfs_index_division_residual_difference)) * uc)) /\\ exists ff_q_pfp_division_residual_differenceresult. ub = ff_q_pfp_division_residual_differenceresult * S ((S (pfs_index_division_residual_difference)) * uc) + (pfs_result_division_residual_difference))) /\\ ((((exists pfa_gap_division_residual_differenceoperationleft. pfa_gap_division_residual_differenceoperationleft + S (pfs_right_division_residual_difference) = (p)) /\\ (((exists pfa_gap_division_residual_differenceoperationright. pfa_gap_division_residual_differenceoperationright + S (pfs_result_division_residual_difference) = (p)) /\\ ((((exists pfa_gap_division_residual_differenceoperationresultbound. pfa_gap_division_residual_differenceoperationresultbound + S (pfs_left_division_residual_difference) = (p)) /\\ ((exists pfa_offset_left_division_residual_differenceoperationresultcongruence pfa_offset_right_division_residual_differenceoperationresultcongruence. ((pfs_right_division_residual_difference) + (pfs_result_division_residual_difference)) + (p) * pfa_offset_left_division_residual_differenceoperationresultcongruence = (pfs_left_division_residual_difference) + (p) * pfa_offset_right_division_residual_differenceoperationresultcongruence))))))))))))))))", "specialize prime_field_polynomial_subtract_exists (p)", "specialize prime_field_polynomial_subtract_exists (ab)", "specialize prime_field_polynomial_subtract_exists (ac)", "specialize prime_field_polynomial_subtract_exists (x)", "specialize prime_field_polynomial_subtract_exists (x1)", "specialize prime_field_polynomial_subtract_exists (L)", "apply prime_field_polynomial_subtract_exists", "exact hp", "exact ha", "specialize prime_field_convolution_prefix_bounded (p)", "specialize prime_field_convolution_prefix_bounded (qb)", "specialize prime_field_convolution_prefix_bounded (qc)", "specialize prime_field_convolution_prefix_bounded (q)", "specialize prime_field_convolution_prefix_bounded (bb)", "specialize prime_field_convolution_prefix_bounded (bc)", "specialize prime_field_convolution_prefix_bounded (S d)", "specialize prime_field_convolution_prefix_bounded (x)", "specialize prime_field_convolution_prefix_bounded (x1)", "specialize prime_field_convolution_prefix_bounded (L)", "apply prime_field_convolution_prefix_bounded", "exact hproduct_witness_witness", "cases hresidual", "cases hresidual_witness", "have htrim : exists t rb rc R. ((((L)=(t)+(R)) /\\ (((forall fom_index_pfp_division_residual_triminput. (exists fom_gap_pfp_division_residual_triminput_index_bound. fom_gap_pfp_division_residual_triminput_index_bound + S (fom_index_pfp_division_residual_triminput) = L) -> exists fom_value_pfp_division_residual_triminput. ((((exists fom_beta_height_pfp_division_residual_triminput_entry. fom_beta_height_pfp_division_residual_triminput_entry + S (fom_value_pfp_division_residual_triminput) = S ((S (fom_index_pfp_division_residual_triminput)) * x3)) /\\ exists fom_beta_quotient_pfp_division_residual_triminput_entry. x2 = fom_beta_quotient_pfp_division_residual_triminput_entry * S ((S (fom_index_pfp_division_residual_triminput)) * x3) + (fom_value_pfp_division_residual_triminput))) /\\ (exists fom_gap_pfp_division_residual_triminput_value_bound. fom_gap_pfp_division_residual_triminput_value_bound + S (fom_value_pfp_division_residual_triminput) = p))) /\\ (((forall pfp_repeat_index_division_residual_trimremoved. (exists pfa_gap_division_residual_trimremovedindex. pfa_gap_division_residual_trimremovedindex + S (pfp_repeat_index_division_residual_trimremoved) = (t)) -> (((exists ff_h_pfp_division_residual_trimremovedentry. ff_h_pfp_division_residual_trimremovedentry + S (0) = S ((S (pfp_repeat_index_division_residual_trimremoved)) * x3)) /\\ exists ff_q_pfp_division_residual_trimremovedentry. x2 = ff_q_pfp_division_residual_trimremovedentry * S ((S (pfp_repeat_index_division_residual_trimremoved)) * x3) + (0)))) /\\ (((forall pftrim_index_division_residual_trimsuffix pftrim_value_division_residual_trimsuffix. (exists pfa_gap_division_residual_trimsuffixbound. pfa_gap_division_residual_trimsuffixbound + S (pftrim_index_division_residual_trimsuffix) = (R)) -> (((exists ff_h_pfp_division_residual_trimsuffixsource. ff_h_pfp_division_residual_trimsuffixsource + S (pftrim_value_division_residual_trimsuffix) = S ((S ((t)+pftrim_index_division_residual_trimsuffix)) * x3)) /\\ exists ff_q_pfp_division_residual_trimsuffixsource. x2 = ff_q_pfp_division_residual_trimsuffixsource * S ((S ((t)+pftrim_index_division_residual_trimsuffix)) * x3) + (pftrim_value_division_residual_trimsuffix))) -> (((exists ff_h_pfp_division_residual_trimsuffixoutput. ff_h_pfp_division_residual_trimsuffixoutput + S (pftrim_value_division_residual_trimsuffix) = S ((S (pftrim_index_division_residual_trimsuffix)) * rc)) /\\ exists ff_q_pfp_division_residual_trimsuffixoutput. rb = ff_q_pfp_division_residual_trimsuffixoutput * S ((S (pftrim_index_division_residual_trimsuffix)) * rc) + (pftrim_value_division_residual_trimsuffix)))) /\\ (((R)=0 \\/ (exists pftrim_leading_division_residual_trimnormal. ((((exists ff_h_pfp_division_residual_trimnormalentry. ff_h_pfp_division_residual_trimnormalentry + S (pftrim_leading_division_residual_trimnormal) = S ((S (0)) * rc)) /\\ exists ff_q_pfp_division_residual_trimnormalentry. rb = ff_q_pfp_division_residual_trimnormalentry * S ((S (0)) * rc) + (pftrim_leading_division_residual_trimnormal))) /\\ ((~(pftrim_leading_division_residual_trimnormal=0)))))))))))))))", "specialize prime_field_polynomial_trim_exists (p)", "specialize prime_field_polynomial_trim_exists (x2)", "specialize prime_field_polynomial_trim_exists (x3)", "specialize prime_field_polynomial_trim_exists (L)", "apply prime_field_polynomial_trim_exists", "have hcanonical : ((forall fom_index_pfp_division_residual_source_bound. (exists fom_gap_pfp_division_residual_source_bound_index_bound. fom_gap_pfp_division_residual_source_bound_index_bound + S (fom_index_pfp_division_residual_source_bound) = L) -> exists fom_value_pfp_division_residual_source_bound. ((((exists fom_beta_height_pfp_division_residual_source_bound_entry. fom_beta_height_pfp_division_residual_source_bound_entry + S (fom_value_pfp_division_residual_source_bound) = S ((S (fom_index_pfp_division_residual_source_bound)) * ac)) /\\ exists fom_beta_quotient_pfp_division_residual_source_bound_entry. ab = fom_beta_quotient_pfp_division_residual_source_bound_entry * S ((S (fom_index_pfp_division_residual_source_bound)) * ac) + (fom_value_pfp_division_residual_source_bound))) /\\ (exists fom_gap_pfp_division_residual_source_bound_value_bound. fom_gap_pfp_division_residual_source_bound_value_bound + S (fom_value_pfp_division_residual_source_bound) = p))) /\\ (((forall fom_index_pfp_division_residual_product_bound. (exists fom_gap_pfp_division_residual_product_bound_index_bound. fom_gap_pfp_division_residual_product_bound_index_bound + S (fom_index_pfp_division_residual_product_bound) = L) -> exists fom_value_pfp_division_residual_product_bound. ((((exists fom_beta_height_pfp_division_residual_product_bound_entry. fom_beta_height_pfp_division_residual_product_bound_entry + S (fom_value_pfp_division_residual_product_bound) = S ((S (fom_index_pfp_division_residual_product_bound)) * x1)) /\\ exists fom_beta_quotient_pfp_division_residual_product_bound_entry. x = fom_beta_quotient_pfp_division_residual_product_bound_entry * S ((S (fom_index_pfp_division_residual_product_bound)) * x1) + (fom_value_pfp_division_residual_product_bound))) /\\ (exists fom_gap_pfp_division_residual_product_bound_value_bound. fom_gap_pfp_division_residual_product_bound_value_bound + S (fom_value_pfp_division_residual_product_bound) = p))) /\\ ((forall fom_index_pfp_division_residual_result_bound. (exists fom_gap_pfp_division_residual_result_bound_index_bound. fom_gap_pfp_division_residual_result_bound_index_bound + S (fom_index_pfp_division_residual_result_bound) = L) -> exists fom_value_pfp_division_residual_result_bound. ((((exists fom_beta_height_pfp_division_residual_result_bound_entry. fom_beta_height_pfp_division_residual_result_bound_entry + S (fom_value_pfp_division_residual_result_bound) = S ((S (fom_index_pfp_division_residual_result_bound)) * x3)) /\\ exists fom_beta_quotient_pfp_division_residual_result_bound_entry. x2 = fom_beta_quotient_pfp_division_residual_result_bound_entry * S ((S (fom_index_pfp_division_residual_result_bound)) * x3) + (fom_value_pfp_division_residual_result_bound))) /\\ (exists fom_gap_pfp_division_residual_result_bound_value_bound. fom_gap_pfp_division_residual_result_bound_value_bound + S (fom_value_pfp_division_residual_result_bound) = p)))))))", "specialize prime_field_polynomial_subtract_bounded (p)", "specialize prime_field_polynomial_subtract_bounded (ab)", "specialize prime_field_polynomial_subtract_bounded (ac)", "specialize prime_field_polynomial_subtract_bounded (x)", "specialize prime_field_polynomial_subtract_bounded (x1)", "specialize prime_field_polynomial_subtract_bounded (x2)", "specialize prime_field_polynomial_subtract_bounded (x3)", "specialize prime_field_polynomial_subtract_bounded (L)", "apply prime_field_polynomial_subtract_bounded", "exact hresidual_witness_witness", "cases hcanonical", "cases hcanonical_right", "exact hcanonical_right_right", "cases htrim", "cases htrim_witness", "cases htrim_witness_witness", "cases htrim_witness_witness_witness", "exists x", "exists x1", "exists x2", "exists x3", "exists x4", "exists x5", "exists x6", "exists x7", "split", "exact hproduct_witness_witness", "split", "exact hresidual_witness_witness", "exact htrim_witness_witness_witness_witness" ], "script_sha256": "eb2410aa4f09419c671c22f48911a07ec26fce68823795a8a6d9ccc819ef6974", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_division_candidate.py", "sha256": "edfc7806caf7a83b9cb0e3e420bd2c3a8679f2d4d9ee6ca9f8eae53faca8d5b2" }, "statement": "forall p ab ac L bb bc d qb qc q. (~((p) = 1) /\\ forall pfa_factor_left_division_residual_data_prime pfa_factor_right_division_residual_data_prime. (p) = pfa_factor_left_division_residual_data_prime * pfa_factor_right_division_residual_data_prime -> pfa_factor_left_division_residual_data_prime = 1 \\/ pfa_factor_right_division_residual_data_prime = 1) -> (forall fom_index_pfp_division_residual_data_input. (exists fom_gap_pfp_division_residual_data_input_index_bound. fom_gap_pfp_division_residual_data_input_index_bound + S (fom_index_pfp_division_residual_data_input) = L) -> exists fom_value_pfp_division_residual_data_input. ((((exists fom_beta_height_pfp_division_residual_data_input_entry. fom_beta_height_pfp_division_residual_data_input_entry + S (fom_value_pfp_division_residual_data_input) = S ((S (fom_index_pfp_division_residual_data_input)) * ac)) /\\ exists fom_beta_quotient_pfp_division_residual_data_input_entry. ab = fom_beta_quotient_pfp_division_residual_data_input_entry * S ((S (fom_index_pfp_division_residual_data_input)) * ac) + (fom_value_pfp_division_residual_data_input))) /\\ (exists fom_gap_pfp_division_residual_data_input_value_bound. fom_gap_pfp_division_residual_data_input_value_bound + S (fom_value_pfp_division_residual_data_input) = p))) -> exists pb pc ub uc t rb rc R. (((forall pfc_index_division_residual_data_resultproduct. (exists pfa_gap_division_residual_data_resultproductbound. pfa_gap_division_residual_data_resultproductbound + S (pfc_index_division_residual_data_resultproduct) = (L)) -> exists pfc_value_division_residual_data_resultproduct. ((((exists ff_h_pfp_division_residual_data_resultproductentry. ff_h_pfp_division_residual_data_resultproductentry + S (pfc_value_division_residual_data_resultproduct) = S ((S (pfc_index_division_residual_data_resultproduct)) * pc)) /\\ exists ff_q_pfp_division_residual_data_resultproductentry. pb = ff_q_pfp_division_residual_data_resultproductentry * S ((S (pfc_index_division_residual_data_resultproduct)) * pc) + (pfc_value_division_residual_data_resultproduct))) /\\ ((exists pfc_terms_code_division_residual_data_resultproductcoefficient pfc_terms_scale_division_residual_data_resultproductcoefficient pfc_natural_sum_division_residual_data_resultproductcoefficient. ((forall pfc_index_division_residual_data_resultproductcoefficientdiagonal. (exists pfa_gap_division_residual_data_resultproductcoefficientdiagonalbound. pfa_gap_division_residual_data_resultproductcoefficientdiagonalbound + S (pfc_index_division_residual_data_resultproductcoefficientdiagonal) = (S (pfc_index_division_residual_data_resultproduct))) -> exists pfc_value_division_residual_data_resultproductcoefficientdiagonal. ((((exists ff_h_pfp_division_residual_data_resultproductcoefficientdiagonalentry. ff_h_pfp_division_residual_data_resultproductcoefficientdiagonalentry + S (pfc_value_division_residual_data_resultproductcoefficientdiagonal) = S ((S (pfc_index_division_residual_data_resultproductcoefficientdiagonal)) * pfc_terms_scale_division_residual_data_resultproductcoefficient)) /\\ exists ff_q_pfp_division_residual_data_resultproductcoefficientdiagonalentry. pfc_terms_code_division_residual_data_resultproductcoefficient = ff_q_pfp_division_residual_data_resultproductcoefficientdiagonalentry * S ((S (pfc_index_division_residual_data_resultproductcoefficientdiagonal)) * pfc_terms_scale_division_residual_data_resultproductcoefficient) + (pfc_value_division_residual_data_resultproductcoefficientdiagonal))) /\\ ((exists pfc_complement_division_residual_data_resultproductcoefficientdiagonalterm pfc_left_division_residual_data_resultproductcoefficientdiagonalterm pfc_right_division_residual_data_resultproductcoefficientdiagonalterm. (((pfc_index_division_residual_data_resultproductcoefficientdiagonal)+pfc_complement_division_residual_data_resultproductcoefficientdiagonalterm=(pfc_index_division_residual_data_resultproduct)) /\\ ((((((exists pfa_gap_division_residual_data_resultproductcoefficientdiagonaltermleftinside. pfa_gap_division_residual_data_resultproductcoefficientdiagonaltermleftinside + S (pfc_index_division_residual_data_resultproductcoefficientdiagonal) = (q)) /\\ ((((exists ff_h_pfp_division_residual_data_resultproductcoefficientdiagonaltermleftentry. ff_h_pfp_division_residual_data_resultproductcoefficientdiagonaltermleftentry + S (pfc_left_division_residual_data_resultproductcoefficientdiagonalterm) = S ((S (pfc_index_division_residual_data_resultproductcoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_division_residual_data_resultproductcoefficientdiagonaltermleftentry. qb = ff_q_pfp_division_residual_data_resultproductcoefficientdiagonaltermleftentry * S ((S (pfc_index_division_residual_data_resultproductcoefficientdiagonal)) * qc) + (pfc_left_division_residual_data_resultproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_division_residual_data_resultproductcoefficientdiagonaltermleftoutside. pfc_gap_division_residual_data_resultproductcoefficientdiagonaltermleftoutside+(q)=(pfc_index_division_residual_data_resultproductcoefficientdiagonal)) /\\ (((pfc_left_division_residual_data_resultproductcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_residual_data_resultproductcoefficientdiagonaltermrightinside. pfa_gap_division_residual_data_resultproductcoefficientdiagonaltermrightinside + S (pfc_complement_division_residual_data_resultproductcoefficientdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_division_residual_data_resultproductcoefficientdiagonaltermrightentry. ff_h_pfp_division_residual_data_resultproductcoefficientdiagonaltermrightentry + S (pfc_right_division_residual_data_resultproductcoefficientdiagonalterm) = S ((S (pfc_complement_division_residual_data_resultproductcoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_residual_data_resultproductcoefficientdiagonaltermrightentry. bb = ff_q_pfp_division_residual_data_resultproductcoefficientdiagonaltermrightentry * S ((S (pfc_complement_division_residual_data_resultproductcoefficientdiagonalterm)) * bc) + (pfc_right_division_residual_data_resultproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_division_residual_data_resultproductcoefficientdiagonaltermrightoutside. pfc_gap_division_residual_data_resultproductcoefficientdiagonaltermrightoutside+(S (d))=(pfc_complement_division_residual_data_resultproductcoefficientdiagonalterm)) /\\ (((pfc_right_division_residual_data_resultproductcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_division_residual_data_resultproductcoefficientdiagonal)=pfc_left_division_residual_data_resultproductcoefficientdiagonalterm*pfc_right_division_residual_data_resultproductcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_residual_data_resultproductcoefficientsum fs_v_pfc_division_residual_data_resultproductcoefficientsum. ((((exists fs_h_pfc_division_residual_data_resultproductcoefficientsum_body_start. fs_h_pfc_division_residual_data_resultproductcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_residual_data_resultproductcoefficientsum)) /\\ exists fs_q_pfc_division_residual_data_resultproductcoefficientsum_body_start. fs_u_pfc_division_residual_data_resultproductcoefficientsum = fs_q_pfc_division_residual_data_resultproductcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_division_residual_data_resultproductcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_division_residual_data_resultproductcoefficientsum_body_terminal. fs_h_pfc_division_residual_data_resultproductcoefficientsum_body_terminal + S (pfc_natural_sum_division_residual_data_resultproductcoefficient) = S ((S (S (pfc_index_division_residual_data_resultproduct))) * fs_v_pfc_division_residual_data_resultproductcoefficientsum)) /\\ exists fs_q_pfc_division_residual_data_resultproductcoefficientsum_body_terminal. fs_u_pfc_division_residual_data_resultproductcoefficientsum = fs_q_pfc_division_residual_data_resultproductcoefficientsum_body_terminal * S ((S (S (pfc_index_division_residual_data_resultproduct))) * fs_v_pfc_division_residual_data_resultproductcoefficientsum) + (pfc_natural_sum_division_residual_data_resultproductcoefficient))) /\\ forall fs_i_pfc_division_residual_data_resultproductcoefficientsum_body_steps. (exists fs_lt_pfc_division_residual_data_resultproductcoefficientsum_body_steps_bound. fs_lt_pfc_division_residual_data_resultproductcoefficientsum_body_steps_bound + S fs_i_pfc_division_residual_data_resultproductcoefficientsum_body_steps = S (pfc_index_division_residual_data_resultproduct)) -> exists fs_a_pfc_division_residual_data_resultproductcoefficientsum_body_steps fs_r_pfc_division_residual_data_resultproductcoefficientsum_body_steps fs_s_pfc_division_residual_data_resultproductcoefficientsum_body_steps. ((((exists fs_h_pfc_division_residual_data_resultproductcoefficientsum_body_steps_summand. fs_h_pfc_division_residual_data_resultproductcoefficientsum_body_steps_summand + S (fs_a_pfc_division_residual_data_resultproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_division_residual_data_resultproductcoefficientsum_body_steps)) * pfc_terms_scale_division_residual_data_resultproductcoefficient)) /\\ exists fs_q_pfc_division_residual_data_resultproductcoefficientsum_body_steps_summand. pfc_terms_code_division_residual_data_resultproductcoefficient = fs_q_pfc_division_residual_data_resultproductcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_division_residual_data_resultproductcoefficientsum_body_steps)) * pfc_terms_scale_division_residual_data_resultproductcoefficient) + (fs_a_pfc_division_residual_data_resultproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_division_residual_data_resultproductcoefficientsum_body_steps_partial. fs_h_pfc_division_residual_data_resultproductcoefficientsum_body_steps_partial + S (fs_r_pfc_division_residual_data_resultproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_division_residual_data_resultproductcoefficientsum_body_steps)) * fs_v_pfc_division_residual_data_resultproductcoefficientsum)) /\\ exists fs_q_pfc_division_residual_data_resultproductcoefficientsum_body_steps_partial. fs_u_pfc_division_residual_data_resultproductcoefficientsum = fs_q_pfc_division_residual_data_resultproductcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_division_residual_data_resultproductcoefficientsum_body_steps)) * fs_v_pfc_division_residual_data_resultproductcoefficientsum) + (fs_r_pfc_division_residual_data_resultproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_division_residual_data_resultproductcoefficientsum_body_steps_successor. fs_h_pfc_division_residual_data_resultproductcoefficientsum_body_steps_successor + S (fs_s_pfc_division_residual_data_resultproductcoefficientsum_body_steps) = S ((S (S fs_i_pfc_division_residual_data_resultproductcoefficientsum_body_steps)) * fs_v_pfc_division_residual_data_resultproductcoefficientsum)) /\\ exists fs_q_pfc_division_residual_data_resultproductcoefficientsum_body_steps_successor. fs_u_pfc_division_residual_data_resultproductcoefficientsum = fs_q_pfc_division_residual_data_resultproductcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_division_residual_data_resultproductcoefficientsum_body_steps)) * fs_v_pfc_division_residual_data_resultproductcoefficientsum) + (fs_s_pfc_division_residual_data_resultproductcoefficientsum_body_steps))) /\\ fs_s_pfc_division_residual_data_resultproductcoefficientsum_body_steps = fs_r_pfc_division_residual_data_resultproductcoefficientsum_body_steps + fs_a_pfc_division_residual_data_resultproductcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_division_residual_data_resultproductcoefficientresiduebound. pfa_gap_division_residual_data_resultproductcoefficientresiduebound + S (pfc_value_division_residual_data_resultproduct) = (p)) /\\ ((exists pfa_offset_left_division_residual_data_resultproductcoefficientresiduecongruence pfa_offset_right_division_residual_data_resultproductcoefficientresiduecongruence. (pfc_natural_sum_division_residual_data_resultproductcoefficient) + (p) * pfa_offset_left_division_residual_data_resultproductcoefficientresiduecongruence = (pfc_value_division_residual_data_resultproduct) + (p) * pfa_offset_right_division_residual_data_resultproductcoefficientresiduecongruence)))))))))))) /\\ (((forall pfs_index_division_residual_data_resultdifference. (exists pfa_gap_division_residual_data_resultdifferenceindex. pfa_gap_division_residual_data_resultdifferenceindex + S (pfs_index_division_residual_data_resultdifference) = (L)) -> exists pfs_left_division_residual_data_resultdifference pfs_right_division_residual_data_resultdifference pfs_result_division_residual_data_resultdifference. ((((exists ff_h_pfp_division_residual_data_resultdifferenceleft. ff_h_pfp_division_residual_data_resultdifferenceleft + S (pfs_left_division_residual_data_resultdifference) = S ((S (pfs_index_division_residual_data_resultdifference)) * ac)) /\\ exists ff_q_pfp_division_residual_data_resultdifferenceleft. ab = ff_q_pfp_division_residual_data_resultdifferenceleft * S ((S (pfs_index_division_residual_data_resultdifference)) * ac) + (pfs_left_division_residual_data_resultdifference))) /\\ (((((exists ff_h_pfp_division_residual_data_resultdifferenceright. ff_h_pfp_division_residual_data_resultdifferenceright + S (pfs_right_division_residual_data_resultdifference) = S ((S (pfs_index_division_residual_data_resultdifference)) * pc)) /\\ exists ff_q_pfp_division_residual_data_resultdifferenceright. pb = ff_q_pfp_division_residual_data_resultdifferenceright * S ((S (pfs_index_division_residual_data_resultdifference)) * pc) + (pfs_right_division_residual_data_resultdifference))) /\\ (((((exists ff_h_pfp_division_residual_data_resultdifferenceresult. ff_h_pfp_division_residual_data_resultdifferenceresult + S (pfs_result_division_residual_data_resultdifference) = S ((S (pfs_index_division_residual_data_resultdifference)) * uc)) /\\ exists ff_q_pfp_division_residual_data_resultdifferenceresult. ub = ff_q_pfp_division_residual_data_resultdifferenceresult * S ((S (pfs_index_division_residual_data_resultdifference)) * uc) + (pfs_result_division_residual_data_resultdifference))) /\\ ((((exists pfa_gap_division_residual_data_resultdifferenceoperationleft. pfa_gap_division_residual_data_resultdifferenceoperationleft + S (pfs_right_division_residual_data_resultdifference) = (p)) /\\ (((exists pfa_gap_division_residual_data_resultdifferenceoperationright. pfa_gap_division_residual_data_resultdifferenceoperationright + S (pfs_result_division_residual_data_resultdifference) = (p)) /\\ ((((exists pfa_gap_division_residual_data_resultdifferenceoperationresultbound. pfa_gap_division_residual_data_resultdifferenceoperationresultbound + S (pfs_left_division_residual_data_resultdifference) = (p)) /\\ ((exists pfa_offset_left_division_residual_data_resultdifferenceoperationresultcongruence pfa_offset_right_division_residual_data_resultdifferenceoperationresultcongruence. ((pfs_right_division_residual_data_resultdifference) + (pfs_result_division_residual_data_resultdifference)) + (p) * pfa_offset_left_division_residual_data_resultdifferenceoperationresultcongruence = (pfs_left_division_residual_data_resultdifference) + (p) * pfa_offset_right_division_residual_data_resultdifferenceoperationresultcongruence)))))))))))))))) /\\ (((((L)=(t)+(R)) /\\ (((forall fom_index_pfp_division_residual_data_resulttriminput. (exists fom_gap_pfp_division_residual_data_resulttriminput_index_bound. fom_gap_pfp_division_residual_data_resulttriminput_index_bound + S (fom_index_pfp_division_residual_data_resulttriminput) = L) -> exists fom_value_pfp_division_residual_data_resulttriminput. ((((exists fom_beta_height_pfp_division_residual_data_resulttriminput_entry. fom_beta_height_pfp_division_residual_data_resulttriminput_entry + S (fom_value_pfp_division_residual_data_resulttriminput) = S ((S (fom_index_pfp_division_residual_data_resulttriminput)) * uc)) /\\ exists fom_beta_quotient_pfp_division_residual_data_resulttriminput_entry. ub = fom_beta_quotient_pfp_division_residual_data_resulttriminput_entry * S ((S (fom_index_pfp_division_residual_data_resulttriminput)) * uc) + (fom_value_pfp_division_residual_data_resulttriminput))) /\\ (exists fom_gap_pfp_division_residual_data_resulttriminput_value_bound. fom_gap_pfp_division_residual_data_resulttriminput_value_bound + S (fom_value_pfp_division_residual_data_resulttriminput) = p))) /\\ (((forall pfp_repeat_index_division_residual_data_resulttrimremoved. (exists pfa_gap_division_residual_data_resulttrimremovedindex. pfa_gap_division_residual_data_resulttrimremovedindex + S (pfp_repeat_index_division_residual_data_resulttrimremoved) = (t)) -> (((exists ff_h_pfp_division_residual_data_resulttrimremovedentry. ff_h_pfp_division_residual_data_resulttrimremovedentry + S (0) = S ((S (pfp_repeat_index_division_residual_data_resulttrimremoved)) * uc)) /\\ exists ff_q_pfp_division_residual_data_resulttrimremovedentry. ub = ff_q_pfp_division_residual_data_resulttrimremovedentry * S ((S (pfp_repeat_index_division_residual_data_resulttrimremoved)) * uc) + (0)))) /\\ (((forall pftrim_index_division_residual_data_resulttrimsuffix pftrim_value_division_residual_data_resulttrimsuffix. (exists pfa_gap_division_residual_data_resulttrimsuffixbound. pfa_gap_division_residual_data_resulttrimsuffixbound + S (pftrim_index_division_residual_data_resulttrimsuffix) = (R)) -> (((exists ff_h_pfp_division_residual_data_resulttrimsuffixsource. ff_h_pfp_division_residual_data_resulttrimsuffixsource + S (pftrim_value_division_residual_data_resulttrimsuffix) = S ((S ((t)+pftrim_index_division_residual_data_resulttrimsuffix)) * uc)) /\\ exists ff_q_pfp_division_residual_data_resulttrimsuffixsource. ub = ff_q_pfp_division_residual_data_resulttrimsuffixsource * S ((S ((t)+pftrim_index_division_residual_data_resulttrimsuffix)) * uc) + (pftrim_value_division_residual_data_resulttrimsuffix))) -> (((exists ff_h_pfp_division_residual_data_resulttrimsuffixoutput. ff_h_pfp_division_residual_data_resulttrimsuffixoutput + S (pftrim_value_division_residual_data_resulttrimsuffix) = S ((S (pftrim_index_division_residual_data_resulttrimsuffix)) * rc)) /\\ exists ff_q_pfp_division_residual_data_resulttrimsuffixoutput. rb = ff_q_pfp_division_residual_data_resulttrimsuffixoutput * S ((S (pftrim_index_division_residual_data_resulttrimsuffix)) * rc) + (pftrim_value_division_residual_data_resulttrimsuffix)))) /\\ (((R)=0 \\/ (exists pftrim_leading_division_residual_data_resulttrimnormal. 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(((((exists ff_h_pfp_division_total_quotient_datahead. ff_h_pfp_division_total_quotient_datahead + S (b) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_division_total_quotient_datahead. bb = ff_q_pfp_division_total_quotient_datahead * S ((S (0)) * bc) + (b))) /\\ (((((~((b) = 0)) /\\ ((((exists pfa_gap_division_total_quotient_datainversemultiplicationleft. pfa_gap_division_total_quotient_datainversemultiplicationleft + S (b) = (p)) /\\ (((exists pfa_gap_division_total_quotient_datainversemultiplicationright. pfa_gap_division_total_quotient_datainversemultiplicationright + S (k) = (p)) /\\ ((((exists pfa_gap_division_total_quotient_datainversemultiplicationresultbound. pfa_gap_division_total_quotient_datainversemultiplicationresultbound + S (1) = (p)) /\\ ((exists pfa_offset_left_division_total_quotient_datainversemultiplicationresultcongruence pfa_offset_right_division_total_quotient_datainversemultiplicationresultcongruence. ((b) * (k)) + (p) * pfa_offset_left_division_total_quotient_datainversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_division_total_quotient_datainversemultiplicationresultcongruence)))))))))))) /\\ (((((((q)=0) /\\ ((exists pfc_gap_division_total_quotient_datalengthshort. pfc_gap_division_total_quotient_datalengthshort+(L)=(d))))) \\/ (((~((q)=0)) /\\ (((q)+(d)=(L)))))) /\\ ((forall pfd_index_division_total_quotient_dataexecution. (exists pfa_gap_division_total_quotient_dataexecutionbound. pfa_gap_division_total_quotient_dataexecutionbound + S (pfd_index_division_total_quotient_dataexecution) = (q)) -> exists pfd_value_division_total_quotient_dataexecution. ((((exists ff_h_pfp_division_total_quotient_dataexecutionentry. ff_h_pfp_division_total_quotient_dataexecutionentry + S (pfd_value_division_total_quotient_dataexecution) = S ((S (pfd_index_division_total_quotient_dataexecution)) * qc)) /\\ exists ff_q_pfp_division_total_quotient_dataexecutionentry. qb = ff_q_pfp_division_total_quotient_dataexecutionentry * S ((S (pfd_index_division_total_quotient_dataexecution)) * qc) + (pfd_value_division_total_quotient_dataexecution))) /\\ ((exists pfd_input_division_total_quotient_dataexecutionstep pfd_previous_division_total_quotient_dataexecutionstep pfd_difference_division_total_quotient_dataexecutionstep. ((((exists ff_h_pfp_division_total_quotient_dataexecutionstepinput. ff_h_pfp_division_total_quotient_dataexecutionstepinput + S (pfd_input_division_total_quotient_dataexecutionstep) = S ((S (pfd_index_division_total_quotient_dataexecution)) * ac)) /\\ exists ff_q_pfp_division_total_quotient_dataexecutionstepinput. ab = ff_q_pfp_division_total_quotient_dataexecutionstepinput * S ((S (pfd_index_division_total_quotient_dataexecution)) * ac) + (pfd_input_division_total_quotient_dataexecutionstep))) /\\ (((exists pfc_terms_code_division_total_quotient_dataexecutionstepprevious pfc_terms_scale_division_total_quotient_dataexecutionstepprevious pfc_natural_sum_division_total_quotient_dataexecutionstepprevious. ((forall pfc_index_division_total_quotient_dataexecutionsteppreviousdiagonal. (exists pfa_gap_division_total_quotient_dataexecutionsteppreviousdiagonalbound. pfa_gap_division_total_quotient_dataexecutionsteppreviousdiagonalbound + S (pfc_index_division_total_quotient_dataexecutionsteppreviousdiagonal) = (S (pfd_index_division_total_quotient_dataexecution))) -> exists pfc_value_division_total_quotient_dataexecutionsteppreviousdiagonal. ((((exists ff_h_pfp_division_total_quotient_dataexecutionsteppreviousdiagonalentry. ff_h_pfp_division_total_quotient_dataexecutionsteppreviousdiagonalentry + S (pfc_value_division_total_quotient_dataexecutionsteppreviousdiagonal) = S ((S (pfc_index_division_total_quotient_dataexecutionsteppreviousdiagonal)) * pfc_terms_scale_division_total_quotient_dataexecutionstepprevious)) /\\ exists ff_q_pfp_division_total_quotient_dataexecutionsteppreviousdiagonalentry. pfc_terms_code_division_total_quotient_dataexecutionstepprevious = ff_q_pfp_division_total_quotient_dataexecutionsteppreviousdiagonalentry * S ((S (pfc_index_division_total_quotient_dataexecutionsteppreviousdiagonal)) * pfc_terms_scale_division_total_quotient_dataexecutionstepprevious) + (pfc_value_division_total_quotient_dataexecutionsteppreviousdiagonal))) /\\ ((exists pfc_complement_division_total_quotient_dataexecutionsteppreviousdiagonalterm pfc_left_division_total_quotient_dataexecutionsteppreviousdiagonalterm pfc_right_division_total_quotient_dataexecutionsteppreviousdiagonalterm. (((pfc_index_division_total_quotient_dataexecutionsteppreviousdiagonal)+pfc_complement_division_total_quotient_dataexecutionsteppreviousdiagonalterm=(pfd_index_division_total_quotient_dataexecution)) /\\ ((((((exists pfa_gap_division_total_quotient_dataexecutionsteppreviousdiagonaltermleftinside. pfa_gap_division_total_quotient_dataexecutionsteppreviousdiagonaltermleftinside + S (pfc_index_division_total_quotient_dataexecutionsteppreviousdiagonal) = (pfd_index_division_total_quotient_dataexecution)) /\\ ((((exists ff_h_pfp_division_total_quotient_dataexecutionsteppreviousdiagonaltermleftentry. ff_h_pfp_division_total_quotient_dataexecutionsteppreviousdiagonaltermleftentry + S (pfc_left_division_total_quotient_dataexecutionsteppreviousdiagonalterm) = S ((S (pfc_index_division_total_quotient_dataexecutionsteppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_division_total_quotient_dataexecutionsteppreviousdiagonaltermleftentry. qb = ff_q_pfp_division_total_quotient_dataexecutionsteppreviousdiagonaltermleftentry * S ((S (pfc_index_division_total_quotient_dataexecutionsteppreviousdiagonal)) * qc) + (pfc_left_division_total_quotient_dataexecutionsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_total_quotient_dataexecutionsteppreviousdiagonaltermleftoutside. pfc_gap_division_total_quotient_dataexecutionsteppreviousdiagonaltermleftoutside+(pfd_index_division_total_quotient_dataexecution)=(pfc_index_division_total_quotient_dataexecutionsteppreviousdiagonal)) /\\ (((pfc_left_division_total_quotient_dataexecutionsteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_total_quotient_dataexecutionsteppreviousdiagonaltermrightinside. pfa_gap_division_total_quotient_dataexecutionsteppreviousdiagonaltermrightinside + S (pfc_complement_division_total_quotient_dataexecutionsteppreviousdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_division_total_quotient_dataexecutionsteppreviousdiagonaltermrightentry. ff_h_pfp_division_total_quotient_dataexecutionsteppreviousdiagonaltermrightentry + S (pfc_right_division_total_quotient_dataexecutionsteppreviousdiagonalterm) = S ((S (pfc_complement_division_total_quotient_dataexecutionsteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_total_quotient_dataexecutionsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_division_total_quotient_dataexecutionsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_division_total_quotient_dataexecutionsteppreviousdiagonalterm)) * bc) + (pfc_right_division_total_quotient_dataexecutionsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_total_quotient_dataexecutionsteppreviousdiagonaltermrightoutside. pfc_gap_division_total_quotient_dataexecutionsteppreviousdiagonaltermrightoutside+(S (d))=(pfc_complement_division_total_quotient_dataexecutionsteppreviousdiagonalterm)) /\\ (((pfc_right_division_total_quotient_dataexecutionsteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_division_total_quotient_dataexecutionsteppreviousdiagonal)=pfc_left_division_total_quotient_dataexecutionsteppreviousdiagonalterm*pfc_right_division_total_quotient_dataexecutionsteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_total_quotient_dataexecutionstepprevioussum fs_v_pfc_division_total_quotient_dataexecutionstepprevioussum. ((((exists fs_h_pfc_division_total_quotient_dataexecutionstepprevioussum_body_start. fs_h_pfc_division_total_quotient_dataexecutionstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_total_quotient_dataexecutionstepprevioussum)) /\\ exists fs_q_pfc_division_total_quotient_dataexecutionstepprevioussum_body_start. fs_u_pfc_division_total_quotient_dataexecutionstepprevioussum = fs_q_pfc_division_total_quotient_dataexecutionstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_division_total_quotient_dataexecutionstepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_division_total_quotient_dataexecutionstepprevioussum_body_terminal. fs_h_pfc_division_total_quotient_dataexecutionstepprevioussum_body_terminal + S (pfc_natural_sum_division_total_quotient_dataexecutionstepprevious) = S ((S (S (pfd_index_division_total_quotient_dataexecution))) * fs_v_pfc_division_total_quotient_dataexecutionstepprevioussum)) /\\ exists fs_q_pfc_division_total_quotient_dataexecutionstepprevioussum_body_terminal. fs_u_pfc_division_total_quotient_dataexecutionstepprevioussum = fs_q_pfc_division_total_quotient_dataexecutionstepprevioussum_body_terminal * S ((S (S (pfd_index_division_total_quotient_dataexecution))) * fs_v_pfc_division_total_quotient_dataexecutionstepprevioussum) + (pfc_natural_sum_division_total_quotient_dataexecutionstepprevious))) /\\ forall fs_i_pfc_division_total_quotient_dataexecutionstepprevioussum_body_steps. (exists fs_lt_pfc_division_total_quotient_dataexecutionstepprevioussum_body_steps_bound. fs_lt_pfc_division_total_quotient_dataexecutionstepprevioussum_body_steps_bound + S fs_i_pfc_division_total_quotient_dataexecutionstepprevioussum_body_steps = S (pfd_index_division_total_quotient_dataexecution)) -> exists fs_a_pfc_division_total_quotient_dataexecutionstepprevioussum_body_steps fs_r_pfc_division_total_quotient_dataexecutionstepprevioussum_body_steps fs_s_pfc_division_total_quotient_dataexecutionstepprevioussum_body_steps. ((((exists fs_h_pfc_division_total_quotient_dataexecutionstepprevioussum_body_steps_summand. fs_h_pfc_division_total_quotient_dataexecutionstepprevioussum_body_steps_summand + S (fs_a_pfc_division_total_quotient_dataexecutionstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_total_quotient_dataexecutionstepprevioussum_body_steps)) * pfc_terms_scale_division_total_quotient_dataexecutionstepprevious)) /\\ exists fs_q_pfc_division_total_quotient_dataexecutionstepprevioussum_body_steps_summand. pfc_terms_code_division_total_quotient_dataexecutionstepprevious = fs_q_pfc_division_total_quotient_dataexecutionstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_division_total_quotient_dataexecutionstepprevioussum_body_steps)) * pfc_terms_scale_division_total_quotient_dataexecutionstepprevious) + (fs_a_pfc_division_total_quotient_dataexecutionstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_total_quotient_dataexecutionstepprevioussum_body_steps_partial. fs_h_pfc_division_total_quotient_dataexecutionstepprevioussum_body_steps_partial + S (fs_r_pfc_division_total_quotient_dataexecutionstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_total_quotient_dataexecutionstepprevioussum_body_steps)) * fs_v_pfc_division_total_quotient_dataexecutionstepprevioussum)) /\\ exists fs_q_pfc_division_total_quotient_dataexecutionstepprevioussum_body_steps_partial. fs_u_pfc_division_total_quotient_dataexecutionstepprevioussum = fs_q_pfc_division_total_quotient_dataexecutionstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_division_total_quotient_dataexecutionstepprevioussum_body_steps)) * fs_v_pfc_division_total_quotient_dataexecutionstepprevioussum) + (fs_r_pfc_division_total_quotient_dataexecutionstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_total_quotient_dataexecutionstepprevioussum_body_steps_successor. fs_h_pfc_division_total_quotient_dataexecutionstepprevioussum_body_steps_successor + S (fs_s_pfc_division_total_quotient_dataexecutionstepprevioussum_body_steps) = S ((S (S fs_i_pfc_division_total_quotient_dataexecutionstepprevioussum_body_steps)) * fs_v_pfc_division_total_quotient_dataexecutionstepprevioussum)) /\\ exists fs_q_pfc_division_total_quotient_dataexecutionstepprevioussum_body_steps_successor. fs_u_pfc_division_total_quotient_dataexecutionstepprevioussum = fs_q_pfc_division_total_quotient_dataexecutionstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_division_total_quotient_dataexecutionstepprevioussum_body_steps)) * fs_v_pfc_division_total_quotient_dataexecutionstepprevioussum) + (fs_s_pfc_division_total_quotient_dataexecutionstepprevioussum_body_steps))) /\\ fs_s_pfc_division_total_quotient_dataexecutionstepprevioussum_body_steps = fs_r_pfc_division_total_quotient_dataexecutionstepprevioussum_body_steps + fs_a_pfc_division_total_quotient_dataexecutionstepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_division_total_quotient_dataexecutionsteppreviousresiduebound. pfa_gap_division_total_quotient_dataexecutionsteppreviousresiduebound + S (pfd_previous_division_total_quotient_dataexecutionstep) = (p)) /\\ ((exists pfa_offset_left_division_total_quotient_dataexecutionsteppreviousresiduecongruence pfa_offset_right_division_total_quotient_dataexecutionsteppreviousresiduecongruence. (pfc_natural_sum_division_total_quotient_dataexecutionstepprevious) + (p) * pfa_offset_left_division_total_quotient_dataexecutionsteppreviousresiduecongruence = (pfd_previous_division_total_quotient_dataexecutionstep) + (p) * pfa_offset_right_division_total_quotient_dataexecutionsteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_division_total_quotient_dataexecutionstepsubtractleft. pfa_gap_division_total_quotient_dataexecutionstepsubtractleft + S (pfd_previous_division_total_quotient_dataexecutionstep) = (p)) /\\ (((exists pfa_gap_division_total_quotient_dataexecutionstepsubtractright. pfa_gap_division_total_quotient_dataexecutionstepsubtractright + S (pfd_difference_division_total_quotient_dataexecutionstep) = (p)) /\\ ((((exists pfa_gap_division_total_quotient_dataexecutionstepsubtractresultbound. pfa_gap_division_total_quotient_dataexecutionstepsubtractresultbound + S (pfd_input_division_total_quotient_dataexecutionstep) = (p)) /\\ ((exists pfa_offset_left_division_total_quotient_dataexecutionstepsubtractresultcongruence pfa_offset_right_division_total_quotient_dataexecutionstepsubtractresultcongruence. ((pfd_previous_division_total_quotient_dataexecutionstep) + (pfd_difference_division_total_quotient_dataexecutionstep)) + (p) * pfa_offset_left_division_total_quotient_dataexecutionstepsubtractresultcongruence = (pfd_input_division_total_quotient_dataexecutionstep) + (p) * pfa_offset_right_division_total_quotient_dataexecutionstepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_division_total_quotient_dataexecutionstepmultiplyleft. pfa_gap_division_total_quotient_dataexecutionstepmultiplyleft + S (k) = (p)) /\\ (((exists pfa_gap_division_total_quotient_dataexecutionstepmultiplyright. pfa_gap_division_total_quotient_dataexecutionstepmultiplyright + S (pfd_difference_division_total_quotient_dataexecutionstep) = (p)) /\\ ((((exists pfa_gap_division_total_quotient_dataexecutionstepmultiplyresultbound. pfa_gap_division_total_quotient_dataexecutionstepmultiplyresultbound + S (pfd_value_division_total_quotient_dataexecution) = (p)) /\\ ((exists pfa_offset_left_division_total_quotient_dataexecutionstepmultiplyresultcongruence pfa_offset_right_division_total_quotient_dataexecutionstepmultiplyresultcongruence. ((k) * (pfd_difference_division_total_quotient_dataexecutionstep)) + (p) * pfa_offset_left_division_total_quotient_dataexecutionstepmultiplyresultcongruence = (pfd_value_division_total_quotient_dataexecution) + (p) * pfa_offset_right_division_total_quotient_dataexecutionstepmultiplyresultcongruence))))))))))))))))))))))))))", "specialize prime_field_polynomial_division_quotient_data_exists (p)", "specialize prime_field_polynomial_division_quotient_data_exists (ab)", "specialize prime_field_polynomial_division_quotient_data_exists (ac)", "specialize prime_field_polynomial_division_quotient_data_exists (L)", "specialize prime_field_polynomial_division_quotient_data_exists (bb)", "specialize prime_field_polynomial_division_quotient_data_exists (bc)", "specialize prime_field_polynomial_division_quotient_data_exists (d)", "apply prime_field_polynomial_division_quotient_data_exists", "exact hp", "exact ha", "exact hb", "cases hquotient", "cases hquotient_witness", "cases hquotient_witness_witness", "cases hquotient_witness_witness_witness", "cases hquotient_witness_witness_witness_witness", "cases hquotient_witness_witness_witness_witness_witness", "cases hquotient_witness_witness_witness_witness_witness_right", "cases hquotient_witness_witness_witness_witness_witness_right_right", "have hresidual : exists pb pc ub uc t rb rc R. (((forall pfc_index_division_total_residual_dataproduct. (exists pfa_gap_division_total_residual_dataproductbound. pfa_gap_division_total_residual_dataproductbound + S (pfc_index_division_total_residual_dataproduct) = (L)) -> exists pfc_value_division_total_residual_dataproduct. ((((exists ff_h_pfp_division_total_residual_dataproductentry. ff_h_pfp_division_total_residual_dataproductentry + S (pfc_value_division_total_residual_dataproduct) = S ((S (pfc_index_division_total_residual_dataproduct)) * pc)) /\\ exists ff_q_pfp_division_total_residual_dataproductentry. pb = ff_q_pfp_division_total_residual_dataproductentry * S ((S (pfc_index_division_total_residual_dataproduct)) * pc) + (pfc_value_division_total_residual_dataproduct))) /\\ ((exists pfc_terms_code_division_total_residual_dataproductcoefficient pfc_terms_scale_division_total_residual_dataproductcoefficient pfc_natural_sum_division_total_residual_dataproductcoefficient. ((forall pfc_index_division_total_residual_dataproductcoefficientdiagonal. (exists pfa_gap_division_total_residual_dataproductcoefficientdiagonalbound. pfa_gap_division_total_residual_dataproductcoefficientdiagonalbound + S (pfc_index_division_total_residual_dataproductcoefficientdiagonal) = (S (pfc_index_division_total_residual_dataproduct))) -> exists pfc_value_division_total_residual_dataproductcoefficientdiagonal. ((((exists ff_h_pfp_division_total_residual_dataproductcoefficientdiagonalentry. ff_h_pfp_division_total_residual_dataproductcoefficientdiagonalentry + S (pfc_value_division_total_residual_dataproductcoefficientdiagonal) = S ((S (pfc_index_division_total_residual_dataproductcoefficientdiagonal)) * pfc_terms_scale_division_total_residual_dataproductcoefficient)) /\\ exists ff_q_pfp_division_total_residual_dataproductcoefficientdiagonalentry. pfc_terms_code_division_total_residual_dataproductcoefficient = ff_q_pfp_division_total_residual_dataproductcoefficientdiagonalentry * S ((S (pfc_index_division_total_residual_dataproductcoefficientdiagonal)) * pfc_terms_scale_division_total_residual_dataproductcoefficient) + (pfc_value_division_total_residual_dataproductcoefficientdiagonal))) /\\ ((exists pfc_complement_division_total_residual_dataproductcoefficientdiagonalterm pfc_left_division_total_residual_dataproductcoefficientdiagonalterm pfc_right_division_total_residual_dataproductcoefficientdiagonalterm. (((pfc_index_division_total_residual_dataproductcoefficientdiagonal)+pfc_complement_division_total_residual_dataproductcoefficientdiagonalterm=(pfc_index_division_total_residual_dataproduct)) /\\ ((((((exists pfa_gap_division_total_residual_dataproductcoefficientdiagonaltermleftinside. pfa_gap_division_total_residual_dataproductcoefficientdiagonaltermleftinside + S (pfc_index_division_total_residual_dataproductcoefficientdiagonal) = (x2)) /\\ ((((exists ff_h_pfp_division_total_residual_dataproductcoefficientdiagonaltermleftentry. ff_h_pfp_division_total_residual_dataproductcoefficientdiagonaltermleftentry + S (pfc_left_division_total_residual_dataproductcoefficientdiagonalterm) = S ((S (pfc_index_division_total_residual_dataproductcoefficientdiagonal)) * x4)) /\\ exists ff_q_pfp_division_total_residual_dataproductcoefficientdiagonaltermleftentry. x3 = ff_q_pfp_division_total_residual_dataproductcoefficientdiagonaltermleftentry * S ((S (pfc_index_division_total_residual_dataproductcoefficientdiagonal)) * x4) + (pfc_left_division_total_residual_dataproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_division_total_residual_dataproductcoefficientdiagonaltermleftoutside. pfc_gap_division_total_residual_dataproductcoefficientdiagonaltermleftoutside+(x2)=(pfc_index_division_total_residual_dataproductcoefficientdiagonal)) /\\ (((pfc_left_division_total_residual_dataproductcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_total_residual_dataproductcoefficientdiagonaltermrightinside. pfa_gap_division_total_residual_dataproductcoefficientdiagonaltermrightinside + S (pfc_complement_division_total_residual_dataproductcoefficientdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_division_total_residual_dataproductcoefficientdiagonaltermrightentry. ff_h_pfp_division_total_residual_dataproductcoefficientdiagonaltermrightentry + S (pfc_right_division_total_residual_dataproductcoefficientdiagonalterm) = S ((S (pfc_complement_division_total_residual_dataproductcoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_total_residual_dataproductcoefficientdiagonaltermrightentry. bb = ff_q_pfp_division_total_residual_dataproductcoefficientdiagonaltermrightentry * S ((S (pfc_complement_division_total_residual_dataproductcoefficientdiagonalterm)) * bc) + (pfc_right_division_total_residual_dataproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_division_total_residual_dataproductcoefficientdiagonaltermrightoutside. pfc_gap_division_total_residual_dataproductcoefficientdiagonaltermrightoutside+(S (d))=(pfc_complement_division_total_residual_dataproductcoefficientdiagonalterm)) /\\ (((pfc_right_division_total_residual_dataproductcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_division_total_residual_dataproductcoefficientdiagonal)=pfc_left_division_total_residual_dataproductcoefficientdiagonalterm*pfc_right_division_total_residual_dataproductcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_total_residual_dataproductcoefficientsum fs_v_pfc_division_total_residual_dataproductcoefficientsum. ((((exists fs_h_pfc_division_total_residual_dataproductcoefficientsum_body_start. fs_h_pfc_division_total_residual_dataproductcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_total_residual_dataproductcoefficientsum)) /\\ exists fs_q_pfc_division_total_residual_dataproductcoefficientsum_body_start. fs_u_pfc_division_total_residual_dataproductcoefficientsum = fs_q_pfc_division_total_residual_dataproductcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_division_total_residual_dataproductcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_division_total_residual_dataproductcoefficientsum_body_terminal. fs_h_pfc_division_total_residual_dataproductcoefficientsum_body_terminal + S (pfc_natural_sum_division_total_residual_dataproductcoefficient) = S ((S (S (pfc_index_division_total_residual_dataproduct))) * fs_v_pfc_division_total_residual_dataproductcoefficientsum)) /\\ exists fs_q_pfc_division_total_residual_dataproductcoefficientsum_body_terminal. fs_u_pfc_division_total_residual_dataproductcoefficientsum = fs_q_pfc_division_total_residual_dataproductcoefficientsum_body_terminal * S ((S (S (pfc_index_division_total_residual_dataproduct))) * fs_v_pfc_division_total_residual_dataproductcoefficientsum) + (pfc_natural_sum_division_total_residual_dataproductcoefficient))) /\\ forall fs_i_pfc_division_total_residual_dataproductcoefficientsum_body_steps. (exists fs_lt_pfc_division_total_residual_dataproductcoefficientsum_body_steps_bound. fs_lt_pfc_division_total_residual_dataproductcoefficientsum_body_steps_bound + S fs_i_pfc_division_total_residual_dataproductcoefficientsum_body_steps = S (pfc_index_division_total_residual_dataproduct)) -> exists fs_a_pfc_division_total_residual_dataproductcoefficientsum_body_steps fs_r_pfc_division_total_residual_dataproductcoefficientsum_body_steps fs_s_pfc_division_total_residual_dataproductcoefficientsum_body_steps. ((((exists fs_h_pfc_division_total_residual_dataproductcoefficientsum_body_steps_summand. fs_h_pfc_division_total_residual_dataproductcoefficientsum_body_steps_summand + S (fs_a_pfc_division_total_residual_dataproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_division_total_residual_dataproductcoefficientsum_body_steps)) * pfc_terms_scale_division_total_residual_dataproductcoefficient)) /\\ exists fs_q_pfc_division_total_residual_dataproductcoefficientsum_body_steps_summand. pfc_terms_code_division_total_residual_dataproductcoefficient = fs_q_pfc_division_total_residual_dataproductcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_division_total_residual_dataproductcoefficientsum_body_steps)) * pfc_terms_scale_division_total_residual_dataproductcoefficient) + (fs_a_pfc_division_total_residual_dataproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_division_total_residual_dataproductcoefficientsum_body_steps_partial. fs_h_pfc_division_total_residual_dataproductcoefficientsum_body_steps_partial + S (fs_r_pfc_division_total_residual_dataproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_division_total_residual_dataproductcoefficientsum_body_steps)) * fs_v_pfc_division_total_residual_dataproductcoefficientsum)) /\\ exists fs_q_pfc_division_total_residual_dataproductcoefficientsum_body_steps_partial. fs_u_pfc_division_total_residual_dataproductcoefficientsum = fs_q_pfc_division_total_residual_dataproductcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_division_total_residual_dataproductcoefficientsum_body_steps)) * fs_v_pfc_division_total_residual_dataproductcoefficientsum) + (fs_r_pfc_division_total_residual_dataproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_division_total_residual_dataproductcoefficientsum_body_steps_successor. fs_h_pfc_division_total_residual_dataproductcoefficientsum_body_steps_successor + S (fs_s_pfc_division_total_residual_dataproductcoefficientsum_body_steps) = S ((S (S fs_i_pfc_division_total_residual_dataproductcoefficientsum_body_steps)) * fs_v_pfc_division_total_residual_dataproductcoefficientsum)) /\\ exists fs_q_pfc_division_total_residual_dataproductcoefficientsum_body_steps_successor. fs_u_pfc_division_total_residual_dataproductcoefficientsum = fs_q_pfc_division_total_residual_dataproductcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_division_total_residual_dataproductcoefficientsum_body_steps)) * fs_v_pfc_division_total_residual_dataproductcoefficientsum) + (fs_s_pfc_division_total_residual_dataproductcoefficientsum_body_steps))) /\\ fs_s_pfc_division_total_residual_dataproductcoefficientsum_body_steps = fs_r_pfc_division_total_residual_dataproductcoefficientsum_body_steps + fs_a_pfc_division_total_residual_dataproductcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_division_total_residual_dataproductcoefficientresiduebound. pfa_gap_division_total_residual_dataproductcoefficientresiduebound + S (pfc_value_division_total_residual_dataproduct) = (p)) /\\ ((exists pfa_offset_left_division_total_residual_dataproductcoefficientresiduecongruence pfa_offset_right_division_total_residual_dataproductcoefficientresiduecongruence. (pfc_natural_sum_division_total_residual_dataproductcoefficient) + (p) * pfa_offset_left_division_total_residual_dataproductcoefficientresiduecongruence = (pfc_value_division_total_residual_dataproduct) + (p) * pfa_offset_right_division_total_residual_dataproductcoefficientresiduecongruence)))))))))))) /\\ (((forall pfs_index_division_total_residual_datadifference. (exists pfa_gap_division_total_residual_datadifferenceindex. pfa_gap_division_total_residual_datadifferenceindex + S (pfs_index_division_total_residual_datadifference) = (L)) -> exists pfs_left_division_total_residual_datadifference pfs_right_division_total_residual_datadifference pfs_result_division_total_residual_datadifference. ((((exists ff_h_pfp_division_total_residual_datadifferenceleft. ff_h_pfp_division_total_residual_datadifferenceleft + S (pfs_left_division_total_residual_datadifference) = S ((S (pfs_index_division_total_residual_datadifference)) * ac)) /\\ exists ff_q_pfp_division_total_residual_datadifferenceleft. ab = ff_q_pfp_division_total_residual_datadifferenceleft * S ((S (pfs_index_division_total_residual_datadifference)) * ac) + (pfs_left_division_total_residual_datadifference))) /\\ (((((exists ff_h_pfp_division_total_residual_datadifferenceright. ff_h_pfp_division_total_residual_datadifferenceright + S (pfs_right_division_total_residual_datadifference) = S ((S (pfs_index_division_total_residual_datadifference)) * pc)) /\\ exists ff_q_pfp_division_total_residual_datadifferenceright. pb = ff_q_pfp_division_total_residual_datadifferenceright * S ((S (pfs_index_division_total_residual_datadifference)) * pc) + (pfs_right_division_total_residual_datadifference))) /\\ (((((exists ff_h_pfp_division_total_residual_datadifferenceresult. ff_h_pfp_division_total_residual_datadifferenceresult + S (pfs_result_division_total_residual_datadifference) = S ((S (pfs_index_division_total_residual_datadifference)) * uc)) /\\ exists ff_q_pfp_division_total_residual_datadifferenceresult. ub = ff_q_pfp_division_total_residual_datadifferenceresult * S ((S (pfs_index_division_total_residual_datadifference)) * uc) + (pfs_result_division_total_residual_datadifference))) /\\ ((((exists pfa_gap_division_total_residual_datadifferenceoperationleft. pfa_gap_division_total_residual_datadifferenceoperationleft + S (pfs_right_division_total_residual_datadifference) = (p)) /\\ (((exists pfa_gap_division_total_residual_datadifferenceoperationright. pfa_gap_division_total_residual_datadifferenceoperationright + S (pfs_result_division_total_residual_datadifference) = (p)) /\\ ((((exists pfa_gap_division_total_residual_datadifferenceoperationresultbound. pfa_gap_division_total_residual_datadifferenceoperationresultbound + S (pfs_left_division_total_residual_datadifference) = (p)) /\\ ((exists pfa_offset_left_division_total_residual_datadifferenceoperationresultcongruence pfa_offset_right_division_total_residual_datadifferenceoperationresultcongruence. ((pfs_right_division_total_residual_datadifference) + (pfs_result_division_total_residual_datadifference)) + (p) * pfa_offset_left_division_total_residual_datadifferenceoperationresultcongruence = (pfs_left_division_total_residual_datadifference) + (p) * pfa_offset_right_division_total_residual_datadifferenceoperationresultcongruence)))))))))))))))) /\\ (((((L)=(t)+(R)) /\\ (((forall fom_index_pfp_division_total_residual_datatriminput. (exists fom_gap_pfp_division_total_residual_datatriminput_index_bound. fom_gap_pfp_division_total_residual_datatriminput_index_bound + S (fom_index_pfp_division_total_residual_datatriminput) = L) -> exists fom_value_pfp_division_total_residual_datatriminput. ((((exists fom_beta_height_pfp_division_total_residual_datatriminput_entry. fom_beta_height_pfp_division_total_residual_datatriminput_entry + S (fom_value_pfp_division_total_residual_datatriminput) = S ((S (fom_index_pfp_division_total_residual_datatriminput)) * uc)) /\\ exists fom_beta_quotient_pfp_division_total_residual_datatriminput_entry. ub = fom_beta_quotient_pfp_division_total_residual_datatriminput_entry * S ((S (fom_index_pfp_division_total_residual_datatriminput)) * uc) + (fom_value_pfp_division_total_residual_datatriminput))) /\\ (exists fom_gap_pfp_division_total_residual_datatriminput_value_bound. fom_gap_pfp_division_total_residual_datatriminput_value_bound + S (fom_value_pfp_division_total_residual_datatriminput) = p))) /\\ (((forall pfp_repeat_index_division_total_residual_datatrimremoved. (exists pfa_gap_division_total_residual_datatrimremovedindex. pfa_gap_division_total_residual_datatrimremovedindex + S (pfp_repeat_index_division_total_residual_datatrimremoved) = (t)) -> (((exists ff_h_pfp_division_total_residual_datatrimremovedentry. ff_h_pfp_division_total_residual_datatrimremovedentry + S (0) = S ((S (pfp_repeat_index_division_total_residual_datatrimremoved)) * uc)) /\\ exists ff_q_pfp_division_total_residual_datatrimremovedentry. ub = ff_q_pfp_division_total_residual_datatrimremovedentry * S ((S (pfp_repeat_index_division_total_residual_datatrimremoved)) * uc) + (0)))) /\\ (((forall pftrim_index_division_total_residual_datatrimsuffix pftrim_value_division_total_residual_datatrimsuffix. (exists pfa_gap_division_total_residual_datatrimsuffixbound. pfa_gap_division_total_residual_datatrimsuffixbound + S (pftrim_index_division_total_residual_datatrimsuffix) = (R)) -> (((exists ff_h_pfp_division_total_residual_datatrimsuffixsource. ff_h_pfp_division_total_residual_datatrimsuffixsource + S (pftrim_value_division_total_residual_datatrimsuffix) = S ((S ((t)+pftrim_index_division_total_residual_datatrimsuffix)) * uc)) /\\ exists ff_q_pfp_division_total_residual_datatrimsuffixsource. ub = ff_q_pfp_division_total_residual_datatrimsuffixsource * S ((S ((t)+pftrim_index_division_total_residual_datatrimsuffix)) * uc) + (pftrim_value_division_total_residual_datatrimsuffix))) -> (((exists ff_h_pfp_division_total_residual_datatrimsuffixoutput. ff_h_pfp_division_total_residual_datatrimsuffixoutput + S (pftrim_value_division_total_residual_datatrimsuffix) = S ((S (pftrim_index_division_total_residual_datatrimsuffix)) * rc)) /\\ exists ff_q_pfp_division_total_residual_datatrimsuffixoutput. rb = ff_q_pfp_division_total_residual_datatrimsuffixoutput * S ((S (pftrim_index_division_total_residual_datatrimsuffix)) * rc) + (pftrim_value_division_total_residual_datatrimsuffix)))) /\\ (((R)=0 \\/ (exists pftrim_leading_division_total_residual_datatrimnormal. ((((exists ff_h_pfp_division_total_residual_datatrimnormalentry. ff_h_pfp_division_total_residual_datatrimnormalentry + S (pftrim_leading_division_total_residual_datatrimnormal) = S ((S (0)) * rc)) /\\ exists ff_q_pfp_division_total_residual_datatrimnormalentry. rb = ff_q_pfp_division_total_residual_datatrimnormalentry * S ((S (0)) * rc) + (pftrim_leading_division_total_residual_datatrimnormal))) /\\ ((~(pftrim_leading_division_total_residual_datatrimnormal=0))))))))))))))))))))", "specialize prime_field_polynomial_division_residual_data_exists (p)", "specialize prime_field_polynomial_division_residual_data_exists (ab)", "specialize prime_field_polynomial_division_residual_data_exists (ac)", "specialize prime_field_polynomial_division_residual_data_exists (L)", "specialize prime_field_polynomial_division_residual_data_exists (bb)", "specialize prime_field_polynomial_division_residual_data_exists (bc)", "specialize prime_field_polynomial_division_residual_data_exists (d)", "specialize prime_field_polynomial_division_residual_data_exists (x3)", "specialize prime_field_polynomial_division_residual_data_exists (x4)", "specialize prime_field_polynomial_division_residual_data_exists (x2)", "apply prime_field_polynomial_division_residual_data_exists", "exact hp", "exact ha", "cases hresidual", "cases hresidual_witness", "cases hresidual_witness_witness", "cases hresidual_witness_witness_witness", "cases hresidual_witness_witness_witness_witness", "cases hresidual_witness_witness_witness_witness_witness", "cases hresidual_witness_witness_witness_witness_witness_witness", "cases hresidual_witness_witness_witness_witness_witness_witness_witness", "cases hresidual_witness_witness_witness_witness_witness_witness_witness_witness", "cases hresidual_witness_witness_witness_witness_witness_witness_witness_witness_right", "cases hb", "cases hb_right", "exists x3", "exists x4", "exists x2", "exists x10", "exists x11", "exists x12", "split", "exact ha", "split", "exact hb_right_left", "split", "exact hquotient_witness_witness_witness_witness_witness_right_right_left", "exists x", "exists x1", "exists x5", "exists x6", "exists x7", "exists x8", "exists x9", "split", "exact hquotient_witness_witness_witness_witness_witness_left", "split", "exact hquotient_witness_witness_witness_witness_witness_right_left", "split", "exact hquotient_witness_witness_witness_witness_witness_right_right_right", "split", "exact hresidual_witness_witness_witness_witness_witness_witness_witness_witness_left", "split", "exact hresidual_witness_witness_witness_witness_witness_witness_witness_witness_right_left", "exact hresidual_witness_witness_witness_witness_witness_witness_witness_witness_right_right" ], "script_sha256": "0e8ac384ef4dcd6cde8312fd99baa720f5505cf5081e6d7c5e3d028de37cfeef", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_division_candidate.py", "sha256": "edfc7806caf7a83b9cb0e3e420bd2c3a8679f2d4d9ee6ca9f8eae53faca8d5b2" }, "statement": "forall p ab ac L bb bc d. (~((p) = 1) /\\ forall pfa_factor_left_division_total_prime pfa_factor_right_division_total_prime. (p) = pfa_factor_left_division_total_prime * pfa_factor_right_division_total_prime -> pfa_factor_left_division_total_prime = 1 \\/ pfa_factor_right_division_total_prime = 1) -> (forall fom_index_pfp_division_total_input. (exists fom_gap_pfp_division_total_input_index_bound. fom_gap_pfp_division_total_input_index_bound + S (fom_index_pfp_division_total_input) = L) -> exists fom_value_pfp_division_total_input. ((((exists fom_beta_height_pfp_division_total_input_entry. fom_beta_height_pfp_division_total_input_entry + S (fom_value_pfp_division_total_input) = S ((S (fom_index_pfp_division_total_input)) * ac)) /\\ exists fom_beta_quotient_pfp_division_total_input_entry. ab = fom_beta_quotient_pfp_division_total_input_entry * S ((S (fom_index_pfp_division_total_input)) * ac) + (fom_value_pfp_division_total_input))) /\\ (exists fom_gap_pfp_division_total_input_value_bound. fom_gap_pfp_division_total_input_value_bound + S (fom_value_pfp_division_total_input) = p))) -> ((((S d)=S (d)) /\\ (((forall fom_index_pfp_division_total_divisorcoefficients. (exists fom_gap_pfp_division_total_divisorcoefficients_index_bound. fom_gap_pfp_division_total_divisorcoefficients_index_bound + S (fom_index_pfp_division_total_divisorcoefficients) = S d) -> exists fom_value_pfp_division_total_divisorcoefficients. ((((exists fom_beta_height_pfp_division_total_divisorcoefficients_entry. fom_beta_height_pfp_division_total_divisorcoefficients_entry + S (fom_value_pfp_division_total_divisorcoefficients) = S ((S (fom_index_pfp_division_total_divisorcoefficients)) * bc)) /\\ exists fom_beta_quotient_pfp_division_total_divisorcoefficients_entry. bb = fom_beta_quotient_pfp_division_total_divisorcoefficients_entry * S ((S (fom_index_pfp_division_total_divisorcoefficients)) * bc) + (fom_value_pfp_division_total_divisorcoefficients))) /\\ (exists fom_gap_pfp_division_total_divisorcoefficients_value_bound. fom_gap_pfp_division_total_divisorcoefficients_value_bound + S (fom_value_pfp_division_total_divisorcoefficients) = p))) /\\ ((exists pfd_leading_division_total_divisor. ((((exists ff_h_pfp_division_total_divisorentry. ff_h_pfp_division_total_divisorentry + S (pfd_leading_division_total_divisor) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_division_total_divisorentry. bb = ff_q_pfp_division_total_divisorentry * S ((S (0)) * bc) + (pfd_leading_division_total_divisor))) /\\ ((~(pfd_leading_division_total_divisor=0)))))))))) -> exists qb qc q rb rc R. (((forall fom_index_pfp_division_total_resultinput. (exists fom_gap_pfp_division_total_resultinput_index_bound. fom_gap_pfp_division_total_resultinput_index_bound + S (fom_index_pfp_division_total_resultinput) = L) -> exists fom_value_pfp_division_total_resultinput. ((((exists fom_beta_height_pfp_division_total_resultinput_entry. fom_beta_height_pfp_division_total_resultinput_entry + S (fom_value_pfp_division_total_resultinput) = S ((S (fom_index_pfp_division_total_resultinput)) * ac)) /\\ exists fom_beta_quotient_pfp_division_total_resultinput_entry. ab = fom_beta_quotient_pfp_division_total_resultinput_entry * S ((S (fom_index_pfp_division_total_resultinput)) * ac) + (fom_value_pfp_division_total_resultinput))) /\\ (exists fom_gap_pfp_division_total_resultinput_value_bound. fom_gap_pfp_division_total_resultinput_value_bound + S (fom_value_pfp_division_total_resultinput) = p))) /\\ (((forall fom_index_pfp_division_total_resultdivisor. (exists fom_gap_pfp_division_total_resultdivisor_index_bound. fom_gap_pfp_division_total_resultdivisor_index_bound + S (fom_index_pfp_division_total_resultdivisor) = S (d)) -> exists fom_value_pfp_division_total_resultdivisor. ((((exists fom_beta_height_pfp_division_total_resultdivisor_entry. fom_beta_height_pfp_division_total_resultdivisor_entry + S (fom_value_pfp_division_total_resultdivisor) = S ((S (fom_index_pfp_division_total_resultdivisor)) * bc)) /\\ exists fom_beta_quotient_pfp_division_total_resultdivisor_entry. bb = fom_beta_quotient_pfp_division_total_resultdivisor_entry * S ((S (fom_index_pfp_division_total_resultdivisor)) * bc) + (fom_value_pfp_division_total_resultdivisor))) /\\ (exists fom_gap_pfp_division_total_resultdivisor_value_bound. fom_gap_pfp_division_total_resultdivisor_value_bound + S (fom_value_pfp_division_total_resultdivisor) = p))) /\\ (((((((q)=0) /\\ ((exists pfc_gap_division_total_resultlengthshort. pfc_gap_division_total_resultlengthshort+(L)=(d))))) \\/ (((~((q)=0)) /\\ (((q)+(d)=(L)))))) /\\ ((exists pfd_head_division_total_result pfd_inverse_division_total_result pfd_product_code_division_total_result pfd_product_scale_division_total_result pfd_residual_code_division_total_result pfd_residual_scale_division_total_result pfd_cut_division_total_result. ((((exists ff_h_pfp_division_total_resulthead. ff_h_pfp_division_total_resulthead + S (pfd_head_division_total_result) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_division_total_resulthead. bb = ff_q_pfp_division_total_resulthead * S ((S (0)) * bc) + (pfd_head_division_total_result))) /\\ (((((~((pfd_head_division_total_result) = 0)) /\\ ((((exists pfa_gap_division_total_resultinversemultiplicationleft. pfa_gap_division_total_resultinversemultiplicationleft + S (pfd_head_division_total_result) = (p)) /\\ (((exists pfa_gap_division_total_resultinversemultiplicationright. pfa_gap_division_total_resultinversemultiplicationright + S (pfd_inverse_division_total_result) = (p)) /\\ ((((exists pfa_gap_division_total_resultinversemultiplicationresultbound. pfa_gap_division_total_resultinversemultiplicationresultbound + S (1) = (p)) /\\ ((exists pfa_offset_left_division_total_resultinversemultiplicationresultcongruence pfa_offset_right_division_total_resultinversemultiplicationresultcongruence. ((pfd_head_division_total_result) * (pfd_inverse_division_total_result)) + (p) * pfa_offset_left_division_total_resultinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_division_total_resultinversemultiplicationresultcongruence)))))))))))) /\\ (((forall pfd_index_division_total_resultquotient. (exists pfa_gap_division_total_resultquotientbound. pfa_gap_division_total_resultquotientbound + S (pfd_index_division_total_resultquotient) = (q)) -> exists pfd_value_division_total_resultquotient. ((((exists ff_h_pfp_division_total_resultquotiententry. ff_h_pfp_division_total_resultquotiententry + S (pfd_value_division_total_resultquotient) = S ((S (pfd_index_division_total_resultquotient)) * qc)) /\\ exists ff_q_pfp_division_total_resultquotiententry. qb = ff_q_pfp_division_total_resultquotiententry * S ((S (pfd_index_division_total_resultquotient)) * qc) + (pfd_value_division_total_resultquotient))) /\\ ((exists pfd_input_division_total_resultquotientstep pfd_previous_division_total_resultquotientstep pfd_difference_division_total_resultquotientstep. ((((exists ff_h_pfp_division_total_resultquotientstepinput. ff_h_pfp_division_total_resultquotientstepinput + S (pfd_input_division_total_resultquotientstep) = S ((S (pfd_index_division_total_resultquotient)) * ac)) /\\ exists ff_q_pfp_division_total_resultquotientstepinput. ab = ff_q_pfp_division_total_resultquotientstepinput * S ((S (pfd_index_division_total_resultquotient)) * ac) + (pfd_input_division_total_resultquotientstep))) /\\ (((exists pfc_terms_code_division_total_resultquotientstepprevious pfc_terms_scale_division_total_resultquotientstepprevious pfc_natural_sum_division_total_resultquotientstepprevious. ((forall pfc_index_division_total_resultquotientsteppreviousdiagonal. (exists pfa_gap_division_total_resultquotientsteppreviousdiagonalbound. pfa_gap_division_total_resultquotientsteppreviousdiagonalbound + S (pfc_index_division_total_resultquotientsteppreviousdiagonal) = (S (pfd_index_division_total_resultquotient))) -> exists pfc_value_division_total_resultquotientsteppreviousdiagonal. ((((exists ff_h_pfp_division_total_resultquotientsteppreviousdiagonalentry. ff_h_pfp_division_total_resultquotientsteppreviousdiagonalentry + S (pfc_value_division_total_resultquotientsteppreviousdiagonal) = S ((S (pfc_index_division_total_resultquotientsteppreviousdiagonal)) * pfc_terms_scale_division_total_resultquotientstepprevious)) /\\ exists ff_q_pfp_division_total_resultquotientsteppreviousdiagonalentry. pfc_terms_code_division_total_resultquotientstepprevious = ff_q_pfp_division_total_resultquotientsteppreviousdiagonalentry * S ((S (pfc_index_division_total_resultquotientsteppreviousdiagonal)) * pfc_terms_scale_division_total_resultquotientstepprevious) + (pfc_value_division_total_resultquotientsteppreviousdiagonal))) /\\ ((exists pfc_complement_division_total_resultquotientsteppreviousdiagonalterm pfc_left_division_total_resultquotientsteppreviousdiagonalterm pfc_right_division_total_resultquotientsteppreviousdiagonalterm. (((pfc_index_division_total_resultquotientsteppreviousdiagonal)+pfc_complement_division_total_resultquotientsteppreviousdiagonalterm=(pfd_index_division_total_resultquotient)) /\\ ((((((exists pfa_gap_division_total_resultquotientsteppreviousdiagonaltermleftinside. pfa_gap_division_total_resultquotientsteppreviousdiagonaltermleftinside + S (pfc_index_division_total_resultquotientsteppreviousdiagonal) = (pfd_index_division_total_resultquotient)) /\\ ((((exists ff_h_pfp_division_total_resultquotientsteppreviousdiagonaltermleftentry. ff_h_pfp_division_total_resultquotientsteppreviousdiagonaltermleftentry + S (pfc_left_division_total_resultquotientsteppreviousdiagonalterm) = S ((S (pfc_index_division_total_resultquotientsteppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_division_total_resultquotientsteppreviousdiagonaltermleftentry. qb = ff_q_pfp_division_total_resultquotientsteppreviousdiagonaltermleftentry * S ((S (pfc_index_division_total_resultquotientsteppreviousdiagonal)) * qc) + (pfc_left_division_total_resultquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_total_resultquotientsteppreviousdiagonaltermleftoutside. pfc_gap_division_total_resultquotientsteppreviousdiagonaltermleftoutside+(pfd_index_division_total_resultquotient)=(pfc_index_division_total_resultquotientsteppreviousdiagonal)) /\\ (((pfc_left_division_total_resultquotientsteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_total_resultquotientsteppreviousdiagonaltermrightinside. pfa_gap_division_total_resultquotientsteppreviousdiagonaltermrightinside + S (pfc_complement_division_total_resultquotientsteppreviousdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_division_total_resultquotientsteppreviousdiagonaltermrightentry. ff_h_pfp_division_total_resultquotientsteppreviousdiagonaltermrightentry + S (pfc_right_division_total_resultquotientsteppreviousdiagonalterm) = S ((S (pfc_complement_division_total_resultquotientsteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_total_resultquotientsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_division_total_resultquotientsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_division_total_resultquotientsteppreviousdiagonalterm)) * bc) + (pfc_right_division_total_resultquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_total_resultquotientsteppreviousdiagonaltermrightoutside. pfc_gap_division_total_resultquotientsteppreviousdiagonaltermrightoutside+(S (d))=(pfc_complement_division_total_resultquotientsteppreviousdiagonalterm)) /\\ (((pfc_right_division_total_resultquotientsteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_division_total_resultquotientsteppreviousdiagonal)=pfc_left_division_total_resultquotientsteppreviousdiagonalterm*pfc_right_division_total_resultquotientsteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_total_resultquotientstepprevioussum fs_v_pfc_division_total_resultquotientstepprevioussum. ((((exists fs_h_pfc_division_total_resultquotientstepprevioussum_body_start. fs_h_pfc_division_total_resultquotientstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_total_resultquotientstepprevioussum)) /\\ exists fs_q_pfc_division_total_resultquotientstepprevioussum_body_start. fs_u_pfc_division_total_resultquotientstepprevioussum = fs_q_pfc_division_total_resultquotientstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_division_total_resultquotientstepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_division_total_resultquotientstepprevioussum_body_terminal. fs_h_pfc_division_total_resultquotientstepprevioussum_body_terminal + S (pfc_natural_sum_division_total_resultquotientstepprevious) = S ((S (S (pfd_index_division_total_resultquotient))) * fs_v_pfc_division_total_resultquotientstepprevioussum)) /\\ exists fs_q_pfc_division_total_resultquotientstepprevioussum_body_terminal. fs_u_pfc_division_total_resultquotientstepprevioussum = fs_q_pfc_division_total_resultquotientstepprevioussum_body_terminal * S ((S (S (pfd_index_division_total_resultquotient))) * fs_v_pfc_division_total_resultquotientstepprevioussum) + (pfc_natural_sum_division_total_resultquotientstepprevious))) /\\ forall fs_i_pfc_division_total_resultquotientstepprevioussum_body_steps. (exists fs_lt_pfc_division_total_resultquotientstepprevioussum_body_steps_bound. fs_lt_pfc_division_total_resultquotientstepprevioussum_body_steps_bound + S fs_i_pfc_division_total_resultquotientstepprevioussum_body_steps = S (pfd_index_division_total_resultquotient)) -> exists fs_a_pfc_division_total_resultquotientstepprevioussum_body_steps fs_r_pfc_division_total_resultquotientstepprevioussum_body_steps fs_s_pfc_division_total_resultquotientstepprevioussum_body_steps. ((((exists fs_h_pfc_division_total_resultquotientstepprevioussum_body_steps_summand. fs_h_pfc_division_total_resultquotientstepprevioussum_body_steps_summand + S (fs_a_pfc_division_total_resultquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_total_resultquotientstepprevioussum_body_steps)) * pfc_terms_scale_division_total_resultquotientstepprevious)) /\\ exists fs_q_pfc_division_total_resultquotientstepprevioussum_body_steps_summand. pfc_terms_code_division_total_resultquotientstepprevious = fs_q_pfc_division_total_resultquotientstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_division_total_resultquotientstepprevioussum_body_steps)) * pfc_terms_scale_division_total_resultquotientstepprevious) + (fs_a_pfc_division_total_resultquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_total_resultquotientstepprevioussum_body_steps_partial. fs_h_pfc_division_total_resultquotientstepprevioussum_body_steps_partial + S (fs_r_pfc_division_total_resultquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_total_resultquotientstepprevioussum_body_steps)) * fs_v_pfc_division_total_resultquotientstepprevioussum)) /\\ exists fs_q_pfc_division_total_resultquotientstepprevioussum_body_steps_partial. fs_u_pfc_division_total_resultquotientstepprevioussum = fs_q_pfc_division_total_resultquotientstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_division_total_resultquotientstepprevioussum_body_steps)) * fs_v_pfc_division_total_resultquotientstepprevioussum) + (fs_r_pfc_division_total_resultquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_total_resultquotientstepprevioussum_body_steps_successor. fs_h_pfc_division_total_resultquotientstepprevioussum_body_steps_successor + S (fs_s_pfc_division_total_resultquotientstepprevioussum_body_steps) = S ((S (S fs_i_pfc_division_total_resultquotientstepprevioussum_body_steps)) * fs_v_pfc_division_total_resultquotientstepprevioussum)) /\\ exists fs_q_pfc_division_total_resultquotientstepprevioussum_body_steps_successor. fs_u_pfc_division_total_resultquotientstepprevioussum = fs_q_pfc_division_total_resultquotientstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_division_total_resultquotientstepprevioussum_body_steps)) * fs_v_pfc_division_total_resultquotientstepprevioussum) + (fs_s_pfc_division_total_resultquotientstepprevioussum_body_steps))) /\\ fs_s_pfc_division_total_resultquotientstepprevioussum_body_steps = fs_r_pfc_division_total_resultquotientstepprevioussum_body_steps + fs_a_pfc_division_total_resultquotientstepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_division_total_resultquotientsteppreviousresiduebound. pfa_gap_division_total_resultquotientsteppreviousresiduebound + S (pfd_previous_division_total_resultquotientstep) = (p)) /\\ ((exists pfa_offset_left_division_total_resultquotientsteppreviousresiduecongruence pfa_offset_right_division_total_resultquotientsteppreviousresiduecongruence. (pfc_natural_sum_division_total_resultquotientstepprevious) + (p) * pfa_offset_left_division_total_resultquotientsteppreviousresiduecongruence = (pfd_previous_division_total_resultquotientstep) + (p) * pfa_offset_right_division_total_resultquotientsteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_division_total_resultquotientstepsubtractleft. pfa_gap_division_total_resultquotientstepsubtractleft + S (pfd_previous_division_total_resultquotientstep) = (p)) /\\ (((exists pfa_gap_division_total_resultquotientstepsubtractright. pfa_gap_division_total_resultquotientstepsubtractright + S (pfd_difference_division_total_resultquotientstep) = (p)) /\\ ((((exists pfa_gap_division_total_resultquotientstepsubtractresultbound. pfa_gap_division_total_resultquotientstepsubtractresultbound + S (pfd_input_division_total_resultquotientstep) = (p)) /\\ ((exists pfa_offset_left_division_total_resultquotientstepsubtractresultcongruence pfa_offset_right_division_total_resultquotientstepsubtractresultcongruence. ((pfd_previous_division_total_resultquotientstep) + (pfd_difference_division_total_resultquotientstep)) + (p) * pfa_offset_left_division_total_resultquotientstepsubtractresultcongruence = (pfd_input_division_total_resultquotientstep) + (p) * pfa_offset_right_division_total_resultquotientstepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_division_total_resultquotientstepmultiplyleft. pfa_gap_division_total_resultquotientstepmultiplyleft + S (pfd_inverse_division_total_result) = (p)) /\\ (((exists pfa_gap_division_total_resultquotientstepmultiplyright. pfa_gap_division_total_resultquotientstepmultiplyright + S (pfd_difference_division_total_resultquotientstep) = (p)) /\\ ((((exists pfa_gap_division_total_resultquotientstepmultiplyresultbound. pfa_gap_division_total_resultquotientstepmultiplyresultbound + S (pfd_value_division_total_resultquotient) = (p)) /\\ ((exists pfa_offset_left_division_total_resultquotientstepmultiplyresultcongruence pfa_offset_right_division_total_resultquotientstepmultiplyresultcongruence. ((pfd_inverse_division_total_result) * (pfd_difference_division_total_resultquotientstep)) + (p) * pfa_offset_left_division_total_resultquotientstepmultiplyresultcongruence = (pfd_value_division_total_resultquotient) + (p) * pfa_offset_right_division_total_resultquotientstepmultiplyresultcongruence))))))))))))))))))) /\\ (((forall pfc_index_division_total_resultproduct. (exists pfa_gap_division_total_resultproductbound. pfa_gap_division_total_resultproductbound + S (pfc_index_division_total_resultproduct) = (L)) -> exists pfc_value_division_total_resultproduct. ((((exists ff_h_pfp_division_total_resultproductentry. ff_h_pfp_division_total_resultproductentry + S (pfc_value_division_total_resultproduct) = S ((S (pfc_index_division_total_resultproduct)) * pfd_product_scale_division_total_result)) /\\ exists ff_q_pfp_division_total_resultproductentry. pfd_product_code_division_total_result = ff_q_pfp_division_total_resultproductentry * S ((S (pfc_index_division_total_resultproduct)) * pfd_product_scale_division_total_result) + (pfc_value_division_total_resultproduct))) /\\ ((exists pfc_terms_code_division_total_resultproductcoefficient pfc_terms_scale_division_total_resultproductcoefficient pfc_natural_sum_division_total_resultproductcoefficient. ((forall pfc_index_division_total_resultproductcoefficientdiagonal. (exists pfa_gap_division_total_resultproductcoefficientdiagonalbound. pfa_gap_division_total_resultproductcoefficientdiagonalbound + S (pfc_index_division_total_resultproductcoefficientdiagonal) = (S (pfc_index_division_total_resultproduct))) -> exists pfc_value_division_total_resultproductcoefficientdiagonal. ((((exists ff_h_pfp_division_total_resultproductcoefficientdiagonalentry. ff_h_pfp_division_total_resultproductcoefficientdiagonalentry + S (pfc_value_division_total_resultproductcoefficientdiagonal) = S ((S (pfc_index_division_total_resultproductcoefficientdiagonal)) * pfc_terms_scale_division_total_resultproductcoefficient)) /\\ exists ff_q_pfp_division_total_resultproductcoefficientdiagonalentry. pfc_terms_code_division_total_resultproductcoefficient = ff_q_pfp_division_total_resultproductcoefficientdiagonalentry * S ((S (pfc_index_division_total_resultproductcoefficientdiagonal)) * pfc_terms_scale_division_total_resultproductcoefficient) + (pfc_value_division_total_resultproductcoefficientdiagonal))) /\\ ((exists pfc_complement_division_total_resultproductcoefficientdiagonalterm pfc_left_division_total_resultproductcoefficientdiagonalterm pfc_right_division_total_resultproductcoefficientdiagonalterm. (((pfc_index_division_total_resultproductcoefficientdiagonal)+pfc_complement_division_total_resultproductcoefficientdiagonalterm=(pfc_index_division_total_resultproduct)) /\\ ((((((exists pfa_gap_division_total_resultproductcoefficientdiagonaltermleftinside. pfa_gap_division_total_resultproductcoefficientdiagonaltermleftinside + S (pfc_index_division_total_resultproductcoefficientdiagonal) = (q)) /\\ ((((exists ff_h_pfp_division_total_resultproductcoefficientdiagonaltermleftentry. ff_h_pfp_division_total_resultproductcoefficientdiagonaltermleftentry + S (pfc_left_division_total_resultproductcoefficientdiagonalterm) = S ((S (pfc_index_division_total_resultproductcoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_division_total_resultproductcoefficientdiagonaltermleftentry. qb = ff_q_pfp_division_total_resultproductcoefficientdiagonaltermleftentry * S ((S (pfc_index_division_total_resultproductcoefficientdiagonal)) * qc) + (pfc_left_division_total_resultproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_division_total_resultproductcoefficientdiagonaltermleftoutside. pfc_gap_division_total_resultproductcoefficientdiagonaltermleftoutside+(q)=(pfc_index_division_total_resultproductcoefficientdiagonal)) /\\ (((pfc_left_division_total_resultproductcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_total_resultproductcoefficientdiagonaltermrightinside. pfa_gap_division_total_resultproductcoefficientdiagonaltermrightinside + S (pfc_complement_division_total_resultproductcoefficientdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_division_total_resultproductcoefficientdiagonaltermrightentry. ff_h_pfp_division_total_resultproductcoefficientdiagonaltermrightentry + S (pfc_right_division_total_resultproductcoefficientdiagonalterm) = S ((S (pfc_complement_division_total_resultproductcoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_total_resultproductcoefficientdiagonaltermrightentry. bb = ff_q_pfp_division_total_resultproductcoefficientdiagonaltermrightentry * S ((S (pfc_complement_division_total_resultproductcoefficientdiagonalterm)) * bc) + (pfc_right_division_total_resultproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_division_total_resultproductcoefficientdiagonaltermrightoutside. pfc_gap_division_total_resultproductcoefficientdiagonaltermrightoutside+(S (d))=(pfc_complement_division_total_resultproductcoefficientdiagonalterm)) /\\ (((pfc_right_division_total_resultproductcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_division_total_resultproductcoefficientdiagonal)=pfc_left_division_total_resultproductcoefficientdiagonalterm*pfc_right_division_total_resultproductcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_total_resultproductcoefficientsum fs_v_pfc_division_total_resultproductcoefficientsum. ((((exists fs_h_pfc_division_total_resultproductcoefficientsum_body_start. fs_h_pfc_division_total_resultproductcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_total_resultproductcoefficientsum)) /\\ exists fs_q_pfc_division_total_resultproductcoefficientsum_body_start. fs_u_pfc_division_total_resultproductcoefficientsum = fs_q_pfc_division_total_resultproductcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_division_total_resultproductcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_division_total_resultproductcoefficientsum_body_terminal. fs_h_pfc_division_total_resultproductcoefficientsum_body_terminal + S (pfc_natural_sum_division_total_resultproductcoefficient) = S ((S (S (pfc_index_division_total_resultproduct))) * fs_v_pfc_division_total_resultproductcoefficientsum)) /\\ exists fs_q_pfc_division_total_resultproductcoefficientsum_body_terminal. fs_u_pfc_division_total_resultproductcoefficientsum = fs_q_pfc_division_total_resultproductcoefficientsum_body_terminal * S ((S (S (pfc_index_division_total_resultproduct))) * fs_v_pfc_division_total_resultproductcoefficientsum) + (pfc_natural_sum_division_total_resultproductcoefficient))) /\\ forall fs_i_pfc_division_total_resultproductcoefficientsum_body_steps. (exists fs_lt_pfc_division_total_resultproductcoefficientsum_body_steps_bound. fs_lt_pfc_division_total_resultproductcoefficientsum_body_steps_bound + S fs_i_pfc_division_total_resultproductcoefficientsum_body_steps = S (pfc_index_division_total_resultproduct)) -> exists fs_a_pfc_division_total_resultproductcoefficientsum_body_steps fs_r_pfc_division_total_resultproductcoefficientsum_body_steps fs_s_pfc_division_total_resultproductcoefficientsum_body_steps. ((((exists fs_h_pfc_division_total_resultproductcoefficientsum_body_steps_summand. fs_h_pfc_division_total_resultproductcoefficientsum_body_steps_summand + S (fs_a_pfc_division_total_resultproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_division_total_resultproductcoefficientsum_body_steps)) * pfc_terms_scale_division_total_resultproductcoefficient)) /\\ exists fs_q_pfc_division_total_resultproductcoefficientsum_body_steps_summand. pfc_terms_code_division_total_resultproductcoefficient = fs_q_pfc_division_total_resultproductcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_division_total_resultproductcoefficientsum_body_steps)) * pfc_terms_scale_division_total_resultproductcoefficient) + (fs_a_pfc_division_total_resultproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_division_total_resultproductcoefficientsum_body_steps_partial. fs_h_pfc_division_total_resultproductcoefficientsum_body_steps_partial + S (fs_r_pfc_division_total_resultproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_division_total_resultproductcoefficientsum_body_steps)) * fs_v_pfc_division_total_resultproductcoefficientsum)) /\\ exists fs_q_pfc_division_total_resultproductcoefficientsum_body_steps_partial. fs_u_pfc_division_total_resultproductcoefficientsum = fs_q_pfc_division_total_resultproductcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_division_total_resultproductcoefficientsum_body_steps)) * fs_v_pfc_division_total_resultproductcoefficientsum) + (fs_r_pfc_division_total_resultproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_division_total_resultproductcoefficientsum_body_steps_successor. fs_h_pfc_division_total_resultproductcoefficientsum_body_steps_successor + S (fs_s_pfc_division_total_resultproductcoefficientsum_body_steps) = S ((S (S fs_i_pfc_division_total_resultproductcoefficientsum_body_steps)) * fs_v_pfc_division_total_resultproductcoefficientsum)) /\\ exists fs_q_pfc_division_total_resultproductcoefficientsum_body_steps_successor. fs_u_pfc_division_total_resultproductcoefficientsum = fs_q_pfc_division_total_resultproductcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_division_total_resultproductcoefficientsum_body_steps)) * fs_v_pfc_division_total_resultproductcoefficientsum) + (fs_s_pfc_division_total_resultproductcoefficientsum_body_steps))) /\\ fs_s_pfc_division_total_resultproductcoefficientsum_body_steps = fs_r_pfc_division_total_resultproductcoefficientsum_body_steps + fs_a_pfc_division_total_resultproductcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_division_total_resultproductcoefficientresiduebound. pfa_gap_division_total_resultproductcoefficientresiduebound + S (pfc_value_division_total_resultproduct) = (p)) /\\ ((exists pfa_offset_left_division_total_resultproductcoefficientresiduecongruence pfa_offset_right_division_total_resultproductcoefficientresiduecongruence. (pfc_natural_sum_division_total_resultproductcoefficient) + (p) * pfa_offset_left_division_total_resultproductcoefficientresiduecongruence = (pfc_value_division_total_resultproduct) + (p) * pfa_offset_right_division_total_resultproductcoefficientresiduecongruence)))))))))))) /\\ (((forall pfs_index_division_total_resultdifference. (exists pfa_gap_division_total_resultdifferenceindex. pfa_gap_division_total_resultdifferenceindex + S (pfs_index_division_total_resultdifference) = (L)) -> exists pfs_left_division_total_resultdifference pfs_right_division_total_resultdifference pfs_result_division_total_resultdifference. ((((exists ff_h_pfp_division_total_resultdifferenceleft. ff_h_pfp_division_total_resultdifferenceleft + S (pfs_left_division_total_resultdifference) = S ((S (pfs_index_division_total_resultdifference)) * ac)) /\\ exists ff_q_pfp_division_total_resultdifferenceleft. ab = ff_q_pfp_division_total_resultdifferenceleft * S ((S (pfs_index_division_total_resultdifference)) * ac) + (pfs_left_division_total_resultdifference))) /\\ (((((exists ff_h_pfp_division_total_resultdifferenceright. ff_h_pfp_division_total_resultdifferenceright + S (pfs_right_division_total_resultdifference) = S ((S (pfs_index_division_total_resultdifference)) * pfd_product_scale_division_total_result)) /\\ exists ff_q_pfp_division_total_resultdifferenceright. pfd_product_code_division_total_result = ff_q_pfp_division_total_resultdifferenceright * S ((S (pfs_index_division_total_resultdifference)) * pfd_product_scale_division_total_result) + (pfs_right_division_total_resultdifference))) /\\ (((((exists ff_h_pfp_division_total_resultdifferenceresult. ff_h_pfp_division_total_resultdifferenceresult + S (pfs_result_division_total_resultdifference) = S ((S (pfs_index_division_total_resultdifference)) * pfd_residual_scale_division_total_result)) /\\ exists ff_q_pfp_division_total_resultdifferenceresult. pfd_residual_code_division_total_result = ff_q_pfp_division_total_resultdifferenceresult * S ((S (pfs_index_division_total_resultdifference)) * pfd_residual_scale_division_total_result) + (pfs_result_division_total_resultdifference))) /\\ ((((exists pfa_gap_division_total_resultdifferenceoperationleft. pfa_gap_division_total_resultdifferenceoperationleft + S (pfs_right_division_total_resultdifference) = (p)) /\\ (((exists pfa_gap_division_total_resultdifferenceoperationright. pfa_gap_division_total_resultdifferenceoperationright + S (pfs_result_division_total_resultdifference) = (p)) /\\ ((((exists pfa_gap_division_total_resultdifferenceoperationresultbound. pfa_gap_division_total_resultdifferenceoperationresultbound + S (pfs_left_division_total_resultdifference) = (p)) /\\ ((exists pfa_offset_left_division_total_resultdifferenceoperationresultcongruence pfa_offset_right_division_total_resultdifferenceoperationresultcongruence. ((pfs_right_division_total_resultdifference) + (pfs_result_division_total_resultdifference)) + (p) * pfa_offset_left_division_total_resultdifferenceoperationresultcongruence = (pfs_left_division_total_resultdifference) + (p) * pfa_offset_right_division_total_resultdifferenceoperationresultcongruence)))))))))))))))) /\\ (((((L)=(pfd_cut_division_total_result)+(R)) /\\ (((forall fom_index_pfp_division_total_resulttriminput. (exists fom_gap_pfp_division_total_resulttriminput_index_bound. fom_gap_pfp_division_total_resulttriminput_index_bound + S (fom_index_pfp_division_total_resulttriminput) = L) -> exists fom_value_pfp_division_total_resulttriminput. ((((exists fom_beta_height_pfp_division_total_resulttriminput_entry. fom_beta_height_pfp_division_total_resulttriminput_entry + S (fom_value_pfp_division_total_resulttriminput) = S ((S (fom_index_pfp_division_total_resulttriminput)) * pfd_residual_scale_division_total_result)) /\\ exists fom_beta_quotient_pfp_division_total_resulttriminput_entry. pfd_residual_code_division_total_result = fom_beta_quotient_pfp_division_total_resulttriminput_entry * S ((S (fom_index_pfp_division_total_resulttriminput)) * pfd_residual_scale_division_total_result) + (fom_value_pfp_division_total_resulttriminput))) /\\ (exists fom_gap_pfp_division_total_resulttriminput_value_bound. fom_gap_pfp_division_total_resulttriminput_value_bound + S (fom_value_pfp_division_total_resulttriminput) = p))) /\\ (((forall pfp_repeat_index_division_total_resulttrimremoved. (exists pfa_gap_division_total_resulttrimremovedindex. pfa_gap_division_total_resulttrimremovedindex + S (pfp_repeat_index_division_total_resulttrimremoved) = (pfd_cut_division_total_result)) -> (((exists ff_h_pfp_division_total_resulttrimremovedentry. ff_h_pfp_division_total_resulttrimremovedentry + S (0) = S ((S (pfp_repeat_index_division_total_resulttrimremoved)) * pfd_residual_scale_division_total_result)) /\\ exists ff_q_pfp_division_total_resulttrimremovedentry. pfd_residual_code_division_total_result = ff_q_pfp_division_total_resulttrimremovedentry * S ((S (pfp_repeat_index_division_total_resulttrimremoved)) * pfd_residual_scale_division_total_result) + (0)))) /\\ (((forall pftrim_index_division_total_resulttrimsuffix pftrim_value_division_total_resulttrimsuffix. (exists pfa_gap_division_total_resulttrimsuffixbound. pfa_gap_division_total_resulttrimsuffixbound + S (pftrim_index_division_total_resulttrimsuffix) = (R)) -> (((exists ff_h_pfp_division_total_resulttrimsuffixsource. ff_h_pfp_division_total_resulttrimsuffixsource + S (pftrim_value_division_total_resulttrimsuffix) = S ((S ((pfd_cut_division_total_result)+pftrim_index_division_total_resulttrimsuffix)) * pfd_residual_scale_division_total_result)) /\\ exists ff_q_pfp_division_total_resulttrimsuffixsource. pfd_residual_code_division_total_result = ff_q_pfp_division_total_resulttrimsuffixsource * S ((S ((pfd_cut_division_total_result)+pftrim_index_division_total_resulttrimsuffix)) * pfd_residual_scale_division_total_result) + (pftrim_value_division_total_resulttrimsuffix))) -> (((exists ff_h_pfp_division_total_resulttrimsuffixoutput. ff_h_pfp_division_total_resulttrimsuffixoutput + S (pftrim_value_division_total_resulttrimsuffix) = S ((S (pftrim_index_division_total_resulttrimsuffix)) * rc)) /\\ exists ff_q_pfp_division_total_resulttrimsuffixoutput. rb = ff_q_pfp_division_total_resulttrimsuffixoutput * S ((S (pftrim_index_division_total_resulttrimsuffix)) * rc) + (pftrim_value_division_total_resulttrimsuffix)))) /\\ (((R)=0 \\/ (exists pftrim_leading_division_total_resulttrimnormal. ((((exists ff_h_pfp_division_total_resulttrimnormalentry. ff_h_pfp_division_total_resulttrimnormalentry + S (pftrim_leading_division_total_resulttrimnormal) = S ((S (0)) * rc)) /\\ exists ff_q_pfp_division_total_resulttrimnormalentry. rb = ff_q_pfp_division_total_resulttrimnormalentry * S ((S (0)) * rc) + (pftrim_leading_division_total_resulttrimnormal))) /\\ ((~(pftrim_leading_division_total_resulttrimnormal=0)))))))))))))))))))))))))))))))))", "statement_sha256": "2284a51ec603ab07a511343bd12a1d73d9414e266a808b8129d9ca306b54f622", "summary": "Construct general quotient and normalized remainder codes from any canonical input and actual nonzero divisor, without assuming any output identity or degree bound.", "summary_sha256": "edc50a2edafc1dcf57e400d65fb3d56e5336873bf4595e4edf8b07b7fb1db5a9" }, { "alpha_v33_frontier_enrollment": { "body_receipt_sha256": "d08021b2783328fc4508147c77a24e2db3912d0d0f856f9a154af1b66624b40c", "bundle_campaign": "polynomial-euclidean-division", 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"polynomial-euclidean-division", "logical_spec_sha256": "a376a72a7d2a273c4fab91a74fb3c9a679296d5679f1355c0c2a667eed6a3811", "membership": "alpha_only", "name": "prime_field_polynomial_division_remainder_degree", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro ab", "intro ac", "intro L", "intro bb", "intro bc", "intro d", "intro qb", "intro qc", "intro q", "intro rb", "intro rc", "intro R", "intro hp", "intro h", "cases h", "cases h_right", "cases h_right_right", "cases h_right_right_right", "cases h_right_right_right_witness", "cases h_right_right_right_witness_witness", "cases h_right_right_right_witness_witness_witness", "cases h_right_right_right_witness_witness_witness_witness", "cases h_right_right_right_witness_witness_witness_witness_witness", "cases h_right_right_right_witness_witness_witness_witness_witness_witness", "cases h_right_right_right_witness_witness_witness_witness_witness_witness_witness", "cases h_right_right_right_witness_witness_witness_witness_witness_witness_witness_right", "cases h_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right", "cases h_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right_right", "cases h_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right_right_right", "have hbounds : ((exists pfc_gap_division_degree_q_bound. pfc_gap_division_degree_q_bound+(q)=(L)) /\\ ((exists pfc_gap_division_degree_cover. pfc_gap_division_degree_cover+(L)=(q+d))))", "specialize polynomial_quotient_length_bounds (L)", "specialize polynomial_quotient_length_bounds (d)", "specialize polynomial_quotient_length_bounds (q)", "apply polynomial_quotient_length_bounds", "exact h_right_right_left", "cases hbounds", "specialize prime_field_polynomial_trim_bounded_degree (p)", "specialize prime_field_polynomial_trim_bounded_degree (x4)", "specialize prime_field_polynomial_trim_bounded_degree (x5)", "specialize prime_field_polynomial_trim_bounded_degree (L)", "specialize prime_field_polynomial_trim_bounded_degree (x6)", "specialize prime_field_polynomial_trim_bounded_degree (rb)", "specialize prime_field_polynomial_trim_bounded_degree (rc)", "specialize prime_field_polynomial_trim_bounded_degree (R)", "specialize prime_field_polynomial_trim_bounded_degree (d)", "apply prime_field_polynomial_trim_bounded_degree", "exact h_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right", "specialize prime_field_polynomial_trim_zero_prefix_remainder_bound (p)", "specialize prime_field_polynomial_trim_zero_prefix_remainder_bound (x4)", "specialize prime_field_polynomial_trim_zero_prefix_remainder_bound (x5)", "specialize prime_field_polynomial_trim_zero_prefix_remainder_bound (L)", "specialize prime_field_polynomial_trim_zero_prefix_remainder_bound (x6)", "specialize prime_field_polynomial_trim_zero_prefix_remainder_bound (rb)", "specialize prime_field_polynomial_trim_zero_prefix_remainder_bound (rc)", "specialize prime_field_polynomial_trim_zero_prefix_remainder_bound (R)", "specialize prime_field_polynomial_trim_zero_prefix_remainder_bound (q)", "specialize prime_field_polynomial_trim_zero_prefix_remainder_bound (d)", "apply prime_field_polynomial_trim_zero_prefix_remainder_bound", "exact hbounds_left", "exact hbounds_right", "specialize prime_field_polynomial_quotient_prefix_remainder_zero (p)", "specialize prime_field_polynomial_quotient_prefix_remainder_zero (x1)", "specialize prime_field_polynomial_quotient_prefix_remainder_zero (ab)", "specialize prime_field_polynomial_quotient_prefix_remainder_zero (ac)", "specialize prime_field_polynomial_quotient_prefix_remainder_zero (bb)", "specialize prime_field_polynomial_quotient_prefix_remainder_zero (bc)", "specialize prime_field_polynomial_quotient_prefix_remainder_zero (d)", "specialize prime_field_polynomial_quotient_prefix_remainder_zero (qb)", "specialize prime_field_polynomial_quotient_prefix_remainder_zero (qc)", "specialize prime_field_polynomial_quotient_prefix_remainder_zero (q)", "specialize prime_field_polynomial_quotient_prefix_remainder_zero (x)", "specialize prime_field_polynomial_quotient_prefix_remainder_zero (x2)", "specialize prime_field_polynomial_quotient_prefix_remainder_zero (x3)", "specialize prime_field_polynomial_quotient_prefix_remainder_zero (L)", "specialize prime_field_polynomial_quotient_prefix_remainder_zero (x4)", "specialize prime_field_polynomial_quotient_prefix_remainder_zero (x5)", "apply prime_field_polynomial_quotient_prefix_remainder_zero", "exact hp", "exact h_right_right_right_witness_witness_witness_witness_witness_witness_witness_left", "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_left", "exact hbounds_left", "exact h_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right_right_left", "exact h_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_left", "exact h_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right" ], "script_sha256": "37ac4efeafb03dbaf01260ce3a46e37ae2f17bf5a93c192fc4baad8461baf43f", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_division_candidate.py", "sha256": "edfc7806caf7a83b9cb0e3e420bd2c3a8679f2d4d9ee6ca9f8eae53faca8d5b2" }, "statement": "forall p ab ac L bb bc d qb qc q rb rc R. (~((p) = 1) /\\ forall pfa_factor_left_division_degree_prime pfa_factor_right_division_degree_prime. (p) = pfa_factor_left_division_degree_prime * pfa_factor_right_division_degree_prime -> pfa_factor_left_division_degree_prime = 1 \\/ pfa_factor_right_division_degree_prime = 1) -> (((forall fom_index_pfp_division_degree_executioninput. (exists fom_gap_pfp_division_degree_executioninput_index_bound. fom_gap_pfp_division_degree_executioninput_index_bound + S (fom_index_pfp_division_degree_executioninput) = L) -> exists fom_value_pfp_division_degree_executioninput. ((((exists fom_beta_height_pfp_division_degree_executioninput_entry. fom_beta_height_pfp_division_degree_executioninput_entry + S (fom_value_pfp_division_degree_executioninput) = S ((S (fom_index_pfp_division_degree_executioninput)) * ac)) /\\ exists fom_beta_quotient_pfp_division_degree_executioninput_entry. ab = fom_beta_quotient_pfp_division_degree_executioninput_entry * S ((S (fom_index_pfp_division_degree_executioninput)) * ac) + (fom_value_pfp_division_degree_executioninput))) /\\ (exists fom_gap_pfp_division_degree_executioninput_value_bound. fom_gap_pfp_division_degree_executioninput_value_bound + S (fom_value_pfp_division_degree_executioninput) = p))) /\\ (((forall fom_index_pfp_division_degree_executiondivisor. (exists fom_gap_pfp_division_degree_executiondivisor_index_bound. fom_gap_pfp_division_degree_executiondivisor_index_bound + S (fom_index_pfp_division_degree_executiondivisor) = S (d)) -> exists fom_value_pfp_division_degree_executiondivisor. ((((exists fom_beta_height_pfp_division_degree_executiondivisor_entry. fom_beta_height_pfp_division_degree_executiondivisor_entry + S (fom_value_pfp_division_degree_executiondivisor) = S ((S (fom_index_pfp_division_degree_executiondivisor)) * bc)) /\\ exists fom_beta_quotient_pfp_division_degree_executiondivisor_entry. bb = fom_beta_quotient_pfp_division_degree_executiondivisor_entry * S ((S (fom_index_pfp_division_degree_executiondivisor)) * bc) + (fom_value_pfp_division_degree_executiondivisor))) /\\ (exists fom_gap_pfp_division_degree_executiondivisor_value_bound. fom_gap_pfp_division_degree_executiondivisor_value_bound + S (fom_value_pfp_division_degree_executiondivisor) = p))) /\\ (((((((q)=0) /\\ ((exists pfc_gap_division_degree_executionlengthshort. pfc_gap_division_degree_executionlengthshort+(L)=(d))))) \\/ (((~((q)=0)) /\\ (((q)+(d)=(L)))))) /\\ ((exists pfd_head_division_degree_execution pfd_inverse_division_degree_execution pfd_product_code_division_degree_execution pfd_product_scale_division_degree_execution pfd_residual_code_division_degree_execution pfd_residual_scale_division_degree_execution pfd_cut_division_degree_execution. ((((exists ff_h_pfp_division_degree_executionhead. ff_h_pfp_division_degree_executionhead + S (pfd_head_division_degree_execution) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_division_degree_executionhead. bb = ff_q_pfp_division_degree_executionhead * S ((S (0)) * bc) + (pfd_head_division_degree_execution))) /\\ (((((~((pfd_head_division_degree_execution) = 0)) /\\ ((((exists pfa_gap_division_degree_executioninversemultiplicationleft. pfa_gap_division_degree_executioninversemultiplicationleft + S (pfd_head_division_degree_execution) = (p)) /\\ (((exists pfa_gap_division_degree_executioninversemultiplicationright. pfa_gap_division_degree_executioninversemultiplicationright + S (pfd_inverse_division_degree_execution) = (p)) /\\ ((((exists pfa_gap_division_degree_executioninversemultiplicationresultbound. pfa_gap_division_degree_executioninversemultiplicationresultbound + S (1) = (p)) /\\ ((exists pfa_offset_left_division_degree_executioninversemultiplicationresultcongruence pfa_offset_right_division_degree_executioninversemultiplicationresultcongruence. ((pfd_head_division_degree_execution) * (pfd_inverse_division_degree_execution)) + (p) * pfa_offset_left_division_degree_executioninversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_division_degree_executioninversemultiplicationresultcongruence)))))))))))) /\\ (((forall pfd_index_division_degree_executionquotient. (exists pfa_gap_division_degree_executionquotientbound. pfa_gap_division_degree_executionquotientbound + S (pfd_index_division_degree_executionquotient) = (q)) -> exists pfd_value_division_degree_executionquotient. ((((exists ff_h_pfp_division_degree_executionquotiententry. ff_h_pfp_division_degree_executionquotiententry + S (pfd_value_division_degree_executionquotient) = S ((S (pfd_index_division_degree_executionquotient)) * qc)) /\\ exists ff_q_pfp_division_degree_executionquotiententry. qb = ff_q_pfp_division_degree_executionquotiententry * S ((S (pfd_index_division_degree_executionquotient)) * qc) + (pfd_value_division_degree_executionquotient))) /\\ ((exists pfd_input_division_degree_executionquotientstep pfd_previous_division_degree_executionquotientstep pfd_difference_division_degree_executionquotientstep. ((((exists ff_h_pfp_division_degree_executionquotientstepinput. ff_h_pfp_division_degree_executionquotientstepinput + S (pfd_input_division_degree_executionquotientstep) = S ((S (pfd_index_division_degree_executionquotient)) * ac)) /\\ exists ff_q_pfp_division_degree_executionquotientstepinput. ab = ff_q_pfp_division_degree_executionquotientstepinput * S ((S (pfd_index_division_degree_executionquotient)) * ac) + (pfd_input_division_degree_executionquotientstep))) /\\ (((exists pfc_terms_code_division_degree_executionquotientstepprevious pfc_terms_scale_division_degree_executionquotientstepprevious pfc_natural_sum_division_degree_executionquotientstepprevious. ((forall pfc_index_division_degree_executionquotientsteppreviousdiagonal. (exists pfa_gap_division_degree_executionquotientsteppreviousdiagonalbound. pfa_gap_division_degree_executionquotientsteppreviousdiagonalbound + S (pfc_index_division_degree_executionquotientsteppreviousdiagonal) = (S (pfd_index_division_degree_executionquotient))) -> exists pfc_value_division_degree_executionquotientsteppreviousdiagonal. ((((exists ff_h_pfp_division_degree_executionquotientsteppreviousdiagonalentry. ff_h_pfp_division_degree_executionquotientsteppreviousdiagonalentry + S (pfc_value_division_degree_executionquotientsteppreviousdiagonal) = S ((S (pfc_index_division_degree_executionquotientsteppreviousdiagonal)) * pfc_terms_scale_division_degree_executionquotientstepprevious)) /\\ exists ff_q_pfp_division_degree_executionquotientsteppreviousdiagonalentry. pfc_terms_code_division_degree_executionquotientstepprevious = ff_q_pfp_division_degree_executionquotientsteppreviousdiagonalentry * S ((S (pfc_index_division_degree_executionquotientsteppreviousdiagonal)) * pfc_terms_scale_division_degree_executionquotientstepprevious) + (pfc_value_division_degree_executionquotientsteppreviousdiagonal))) /\\ ((exists pfc_complement_division_degree_executionquotientsteppreviousdiagonalterm pfc_left_division_degree_executionquotientsteppreviousdiagonalterm pfc_right_division_degree_executionquotientsteppreviousdiagonalterm. (((pfc_index_division_degree_executionquotientsteppreviousdiagonal)+pfc_complement_division_degree_executionquotientsteppreviousdiagonalterm=(pfd_index_division_degree_executionquotient)) /\\ ((((((exists pfa_gap_division_degree_executionquotientsteppreviousdiagonaltermleftinside. pfa_gap_division_degree_executionquotientsteppreviousdiagonaltermleftinside + S (pfc_index_division_degree_executionquotientsteppreviousdiagonal) = (pfd_index_division_degree_executionquotient)) /\\ ((((exists ff_h_pfp_division_degree_executionquotientsteppreviousdiagonaltermleftentry. ff_h_pfp_division_degree_executionquotientsteppreviousdiagonaltermleftentry + S (pfc_left_division_degree_executionquotientsteppreviousdiagonalterm) = S ((S (pfc_index_division_degree_executionquotientsteppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_division_degree_executionquotientsteppreviousdiagonaltermleftentry. qb = ff_q_pfp_division_degree_executionquotientsteppreviousdiagonaltermleftentry * S ((S (pfc_index_division_degree_executionquotientsteppreviousdiagonal)) * qc) + (pfc_left_division_degree_executionquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_degree_executionquotientsteppreviousdiagonaltermleftoutside. pfc_gap_division_degree_executionquotientsteppreviousdiagonaltermleftoutside+(pfd_index_division_degree_executionquotient)=(pfc_index_division_degree_executionquotientsteppreviousdiagonal)) /\\ (((pfc_left_division_degree_executionquotientsteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_degree_executionquotientsteppreviousdiagonaltermrightinside. pfa_gap_division_degree_executionquotientsteppreviousdiagonaltermrightinside + S (pfc_complement_division_degree_executionquotientsteppreviousdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_division_degree_executionquotientsteppreviousdiagonaltermrightentry. ff_h_pfp_division_degree_executionquotientsteppreviousdiagonaltermrightentry + S (pfc_right_division_degree_executionquotientsteppreviousdiagonalterm) = S ((S (pfc_complement_division_degree_executionquotientsteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_degree_executionquotientsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_division_degree_executionquotientsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_division_degree_executionquotientsteppreviousdiagonalterm)) * bc) + (pfc_right_division_degree_executionquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_degree_executionquotientsteppreviousdiagonaltermrightoutside. pfc_gap_division_degree_executionquotientsteppreviousdiagonaltermrightoutside+(S (d))=(pfc_complement_division_degree_executionquotientsteppreviousdiagonalterm)) /\\ (((pfc_right_division_degree_executionquotientsteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_division_degree_executionquotientsteppreviousdiagonal)=pfc_left_division_degree_executionquotientsteppreviousdiagonalterm*pfc_right_division_degree_executionquotientsteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_degree_executionquotientstepprevioussum fs_v_pfc_division_degree_executionquotientstepprevioussum. ((((exists fs_h_pfc_division_degree_executionquotientstepprevioussum_body_start. fs_h_pfc_division_degree_executionquotientstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_degree_executionquotientstepprevioussum)) /\\ exists fs_q_pfc_division_degree_executionquotientstepprevioussum_body_start. fs_u_pfc_division_degree_executionquotientstepprevioussum = fs_q_pfc_division_degree_executionquotientstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_division_degree_executionquotientstepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_division_degree_executionquotientstepprevioussum_body_terminal. fs_h_pfc_division_degree_executionquotientstepprevioussum_body_terminal + S (pfc_natural_sum_division_degree_executionquotientstepprevious) = S ((S (S (pfd_index_division_degree_executionquotient))) * fs_v_pfc_division_degree_executionquotientstepprevioussum)) /\\ exists fs_q_pfc_division_degree_executionquotientstepprevioussum_body_terminal. fs_u_pfc_division_degree_executionquotientstepprevioussum = fs_q_pfc_division_degree_executionquotientstepprevioussum_body_terminal * S ((S (S (pfd_index_division_degree_executionquotient))) * fs_v_pfc_division_degree_executionquotientstepprevioussum) + (pfc_natural_sum_division_degree_executionquotientstepprevious))) /\\ forall fs_i_pfc_division_degree_executionquotientstepprevioussum_body_steps. (exists fs_lt_pfc_division_degree_executionquotientstepprevioussum_body_steps_bound. fs_lt_pfc_division_degree_executionquotientstepprevioussum_body_steps_bound + S fs_i_pfc_division_degree_executionquotientstepprevioussum_body_steps = S (pfd_index_division_degree_executionquotient)) -> exists fs_a_pfc_division_degree_executionquotientstepprevioussum_body_steps fs_r_pfc_division_degree_executionquotientstepprevioussum_body_steps fs_s_pfc_division_degree_executionquotientstepprevioussum_body_steps. ((((exists fs_h_pfc_division_degree_executionquotientstepprevioussum_body_steps_summand. fs_h_pfc_division_degree_executionquotientstepprevioussum_body_steps_summand + S (fs_a_pfc_division_degree_executionquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_degree_executionquotientstepprevioussum_body_steps)) * pfc_terms_scale_division_degree_executionquotientstepprevious)) /\\ exists fs_q_pfc_division_degree_executionquotientstepprevioussum_body_steps_summand. pfc_terms_code_division_degree_executionquotientstepprevious = fs_q_pfc_division_degree_executionquotientstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_division_degree_executionquotientstepprevioussum_body_steps)) * pfc_terms_scale_division_degree_executionquotientstepprevious) + (fs_a_pfc_division_degree_executionquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_degree_executionquotientstepprevioussum_body_steps_partial. fs_h_pfc_division_degree_executionquotientstepprevioussum_body_steps_partial + S (fs_r_pfc_division_degree_executionquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_degree_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_division_degree_executionquotientstepprevioussum)) /\\ exists fs_q_pfc_division_degree_executionquotientstepprevioussum_body_steps_partial. fs_u_pfc_division_degree_executionquotientstepprevioussum = fs_q_pfc_division_degree_executionquotientstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_division_degree_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_division_degree_executionquotientstepprevioussum) + (fs_r_pfc_division_degree_executionquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_degree_executionquotientstepprevioussum_body_steps_successor. fs_h_pfc_division_degree_executionquotientstepprevioussum_body_steps_successor + S (fs_s_pfc_division_degree_executionquotientstepprevioussum_body_steps) = S ((S (S fs_i_pfc_division_degree_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_division_degree_executionquotientstepprevioussum)) /\\ exists fs_q_pfc_division_degree_executionquotientstepprevioussum_body_steps_successor. fs_u_pfc_division_degree_executionquotientstepprevioussum = fs_q_pfc_division_degree_executionquotientstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_division_degree_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_division_degree_executionquotientstepprevioussum) + (fs_s_pfc_division_degree_executionquotientstepprevioussum_body_steps))) /\\ fs_s_pfc_division_degree_executionquotientstepprevioussum_body_steps = fs_r_pfc_division_degree_executionquotientstepprevioussum_body_steps + fs_a_pfc_division_degree_executionquotientstepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_division_degree_executionquotientsteppreviousresiduebound. pfa_gap_division_degree_executionquotientsteppreviousresiduebound + S (pfd_previous_division_degree_executionquotientstep) = (p)) /\\ ((exists pfa_offset_left_division_degree_executionquotientsteppreviousresiduecongruence pfa_offset_right_division_degree_executionquotientsteppreviousresiduecongruence. (pfc_natural_sum_division_degree_executionquotientstepprevious) + (p) * pfa_offset_left_division_degree_executionquotientsteppreviousresiduecongruence = (pfd_previous_division_degree_executionquotientstep) + (p) * pfa_offset_right_division_degree_executionquotientsteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_division_degree_executionquotientstepsubtractleft. pfa_gap_division_degree_executionquotientstepsubtractleft + S (pfd_previous_division_degree_executionquotientstep) = (p)) /\\ (((exists pfa_gap_division_degree_executionquotientstepsubtractright. pfa_gap_division_degree_executionquotientstepsubtractright + S (pfd_difference_division_degree_executionquotientstep) = (p)) /\\ ((((exists pfa_gap_division_degree_executionquotientstepsubtractresultbound. pfa_gap_division_degree_executionquotientstepsubtractresultbound + S (pfd_input_division_degree_executionquotientstep) = (p)) /\\ ((exists pfa_offset_left_division_degree_executionquotientstepsubtractresultcongruence pfa_offset_right_division_degree_executionquotientstepsubtractresultcongruence. ((pfd_previous_division_degree_executionquotientstep) + (pfd_difference_division_degree_executionquotientstep)) + (p) * pfa_offset_left_division_degree_executionquotientstepsubtractresultcongruence = (pfd_input_division_degree_executionquotientstep) + (p) * pfa_offset_right_division_degree_executionquotientstepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_division_degree_executionquotientstepmultiplyleft. pfa_gap_division_degree_executionquotientstepmultiplyleft + S (pfd_inverse_division_degree_execution) = (p)) /\\ (((exists pfa_gap_division_degree_executionquotientstepmultiplyright. pfa_gap_division_degree_executionquotientstepmultiplyright + S (pfd_difference_division_degree_executionquotientstep) = (p)) /\\ ((((exists pfa_gap_division_degree_executionquotientstepmultiplyresultbound. pfa_gap_division_degree_executionquotientstepmultiplyresultbound + S (pfd_value_division_degree_executionquotient) = (p)) /\\ ((exists pfa_offset_left_division_degree_executionquotientstepmultiplyresultcongruence pfa_offset_right_division_degree_executionquotientstepmultiplyresultcongruence. ((pfd_inverse_division_degree_execution) * (pfd_difference_division_degree_executionquotientstep)) + (p) * pfa_offset_left_division_degree_executionquotientstepmultiplyresultcongruence = (pfd_value_division_degree_executionquotient) + (p) * pfa_offset_right_division_degree_executionquotientstepmultiplyresultcongruence))))))))))))))))))) /\\ (((forall pfc_index_division_degree_executionproduct. (exists pfa_gap_division_degree_executionproductbound. pfa_gap_division_degree_executionproductbound + S (pfc_index_division_degree_executionproduct) = (L)) -> exists pfc_value_division_degree_executionproduct. ((((exists ff_h_pfp_division_degree_executionproductentry. ff_h_pfp_division_degree_executionproductentry + S (pfc_value_division_degree_executionproduct) = S ((S (pfc_index_division_degree_executionproduct)) * pfd_product_scale_division_degree_execution)) /\\ exists ff_q_pfp_division_degree_executionproductentry. pfd_product_code_division_degree_execution = ff_q_pfp_division_degree_executionproductentry * S ((S (pfc_index_division_degree_executionproduct)) * pfd_product_scale_division_degree_execution) + (pfc_value_division_degree_executionproduct))) /\\ ((exists pfc_terms_code_division_degree_executionproductcoefficient pfc_terms_scale_division_degree_executionproductcoefficient pfc_natural_sum_division_degree_executionproductcoefficient. ((forall pfc_index_division_degree_executionproductcoefficientdiagonal. (exists pfa_gap_division_degree_executionproductcoefficientdiagonalbound. pfa_gap_division_degree_executionproductcoefficientdiagonalbound + S (pfc_index_division_degree_executionproductcoefficientdiagonal) = (S (pfc_index_division_degree_executionproduct))) -> exists pfc_value_division_degree_executionproductcoefficientdiagonal. ((((exists ff_h_pfp_division_degree_executionproductcoefficientdiagonalentry. ff_h_pfp_division_degree_executionproductcoefficientdiagonalentry + S (pfc_value_division_degree_executionproductcoefficientdiagonal) = S ((S (pfc_index_division_degree_executionproductcoefficientdiagonal)) * pfc_terms_scale_division_degree_executionproductcoefficient)) /\\ exists ff_q_pfp_division_degree_executionproductcoefficientdiagonalentry. pfc_terms_code_division_degree_executionproductcoefficient = ff_q_pfp_division_degree_executionproductcoefficientdiagonalentry * S ((S (pfc_index_division_degree_executionproductcoefficientdiagonal)) * pfc_terms_scale_division_degree_executionproductcoefficient) + (pfc_value_division_degree_executionproductcoefficientdiagonal))) /\\ ((exists pfc_complement_division_degree_executionproductcoefficientdiagonalterm pfc_left_division_degree_executionproductcoefficientdiagonalterm pfc_right_division_degree_executionproductcoefficientdiagonalterm. (((pfc_index_division_degree_executionproductcoefficientdiagonal)+pfc_complement_division_degree_executionproductcoefficientdiagonalterm=(pfc_index_division_degree_executionproduct)) /\\ ((((((exists pfa_gap_division_degree_executionproductcoefficientdiagonaltermleftinside. pfa_gap_division_degree_executionproductcoefficientdiagonaltermleftinside + S (pfc_index_division_degree_executionproductcoefficientdiagonal) = (q)) /\\ ((((exists ff_h_pfp_division_degree_executionproductcoefficientdiagonaltermleftentry. ff_h_pfp_division_degree_executionproductcoefficientdiagonaltermleftentry + S (pfc_left_division_degree_executionproductcoefficientdiagonalterm) = S ((S (pfc_index_division_degree_executionproductcoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_division_degree_executionproductcoefficientdiagonaltermleftentry. qb = ff_q_pfp_division_degree_executionproductcoefficientdiagonaltermleftentry * S ((S (pfc_index_division_degree_executionproductcoefficientdiagonal)) * qc) + (pfc_left_division_degree_executionproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_division_degree_executionproductcoefficientdiagonaltermleftoutside. pfc_gap_division_degree_executionproductcoefficientdiagonaltermleftoutside+(q)=(pfc_index_division_degree_executionproductcoefficientdiagonal)) /\\ (((pfc_left_division_degree_executionproductcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_degree_executionproductcoefficientdiagonaltermrightinside. pfa_gap_division_degree_executionproductcoefficientdiagonaltermrightinside + S (pfc_complement_division_degree_executionproductcoefficientdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_division_degree_executionproductcoefficientdiagonaltermrightentry. ff_h_pfp_division_degree_executionproductcoefficientdiagonaltermrightentry + S (pfc_right_division_degree_executionproductcoefficientdiagonalterm) = S ((S (pfc_complement_division_degree_executionproductcoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_degree_executionproductcoefficientdiagonaltermrightentry. bb = ff_q_pfp_division_degree_executionproductcoefficientdiagonaltermrightentry * S ((S (pfc_complement_division_degree_executionproductcoefficientdiagonalterm)) * bc) + (pfc_right_division_degree_executionproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_division_degree_executionproductcoefficientdiagonaltermrightoutside. pfc_gap_division_degree_executionproductcoefficientdiagonaltermrightoutside+(S (d))=(pfc_complement_division_degree_executionproductcoefficientdiagonalterm)) /\\ (((pfc_right_division_degree_executionproductcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_division_degree_executionproductcoefficientdiagonal)=pfc_left_division_degree_executionproductcoefficientdiagonalterm*pfc_right_division_degree_executionproductcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_degree_executionproductcoefficientsum fs_v_pfc_division_degree_executionproductcoefficientsum. ((((exists fs_h_pfc_division_degree_executionproductcoefficientsum_body_start. fs_h_pfc_division_degree_executionproductcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_degree_executionproductcoefficientsum)) /\\ exists fs_q_pfc_division_degree_executionproductcoefficientsum_body_start. fs_u_pfc_division_degree_executionproductcoefficientsum = fs_q_pfc_division_degree_executionproductcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_division_degree_executionproductcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_division_degree_executionproductcoefficientsum_body_terminal. fs_h_pfc_division_degree_executionproductcoefficientsum_body_terminal + S (pfc_natural_sum_division_degree_executionproductcoefficient) = S ((S (S (pfc_index_division_degree_executionproduct))) * fs_v_pfc_division_degree_executionproductcoefficientsum)) /\\ exists fs_q_pfc_division_degree_executionproductcoefficientsum_body_terminal. fs_u_pfc_division_degree_executionproductcoefficientsum = fs_q_pfc_division_degree_executionproductcoefficientsum_body_terminal * S ((S (S (pfc_index_division_degree_executionproduct))) * fs_v_pfc_division_degree_executionproductcoefficientsum) + (pfc_natural_sum_division_degree_executionproductcoefficient))) /\\ forall fs_i_pfc_division_degree_executionproductcoefficientsum_body_steps. (exists fs_lt_pfc_division_degree_executionproductcoefficientsum_body_steps_bound. fs_lt_pfc_division_degree_executionproductcoefficientsum_body_steps_bound + S fs_i_pfc_division_degree_executionproductcoefficientsum_body_steps = S (pfc_index_division_degree_executionproduct)) -> exists fs_a_pfc_division_degree_executionproductcoefficientsum_body_steps fs_r_pfc_division_degree_executionproductcoefficientsum_body_steps fs_s_pfc_division_degree_executionproductcoefficientsum_body_steps. ((((exists fs_h_pfc_division_degree_executionproductcoefficientsum_body_steps_summand. fs_h_pfc_division_degree_executionproductcoefficientsum_body_steps_summand + S (fs_a_pfc_division_degree_executionproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_division_degree_executionproductcoefficientsum_body_steps)) * pfc_terms_scale_division_degree_executionproductcoefficient)) /\\ exists fs_q_pfc_division_degree_executionproductcoefficientsum_body_steps_summand. pfc_terms_code_division_degree_executionproductcoefficient = fs_q_pfc_division_degree_executionproductcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_division_degree_executionproductcoefficientsum_body_steps)) * pfc_terms_scale_division_degree_executionproductcoefficient) + (fs_a_pfc_division_degree_executionproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_division_degree_executionproductcoefficientsum_body_steps_partial. fs_h_pfc_division_degree_executionproductcoefficientsum_body_steps_partial + S (fs_r_pfc_division_degree_executionproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_division_degree_executionproductcoefficientsum_body_steps)) * fs_v_pfc_division_degree_executionproductcoefficientsum)) /\\ exists fs_q_pfc_division_degree_executionproductcoefficientsum_body_steps_partial. fs_u_pfc_division_degree_executionproductcoefficientsum = fs_q_pfc_division_degree_executionproductcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_division_degree_executionproductcoefficientsum_body_steps)) * fs_v_pfc_division_degree_executionproductcoefficientsum) + (fs_r_pfc_division_degree_executionproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_division_degree_executionproductcoefficientsum_body_steps_successor. fs_h_pfc_division_degree_executionproductcoefficientsum_body_steps_successor + S (fs_s_pfc_division_degree_executionproductcoefficientsum_body_steps) = S ((S (S fs_i_pfc_division_degree_executionproductcoefficientsum_body_steps)) * fs_v_pfc_division_degree_executionproductcoefficientsum)) /\\ exists fs_q_pfc_division_degree_executionproductcoefficientsum_body_steps_successor. fs_u_pfc_division_degree_executionproductcoefficientsum = fs_q_pfc_division_degree_executionproductcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_division_degree_executionproductcoefficientsum_body_steps)) * fs_v_pfc_division_degree_executionproductcoefficientsum) + (fs_s_pfc_division_degree_executionproductcoefficientsum_body_steps))) /\\ fs_s_pfc_division_degree_executionproductcoefficientsum_body_steps = fs_r_pfc_division_degree_executionproductcoefficientsum_body_steps + fs_a_pfc_division_degree_executionproductcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_division_degree_executionproductcoefficientresiduebound. pfa_gap_division_degree_executionproductcoefficientresiduebound + S (pfc_value_division_degree_executionproduct) = (p)) /\\ ((exists pfa_offset_left_division_degree_executionproductcoefficientresiduecongruence pfa_offset_right_division_degree_executionproductcoefficientresiduecongruence. (pfc_natural_sum_division_degree_executionproductcoefficient) + (p) * pfa_offset_left_division_degree_executionproductcoefficientresiduecongruence = (pfc_value_division_degree_executionproduct) + (p) * pfa_offset_right_division_degree_executionproductcoefficientresiduecongruence)))))))))))) /\\ (((forall pfs_index_division_degree_executiondifference. (exists pfa_gap_division_degree_executiondifferenceindex. pfa_gap_division_degree_executiondifferenceindex + S (pfs_index_division_degree_executiondifference) = (L)) -> exists pfs_left_division_degree_executiondifference pfs_right_division_degree_executiondifference pfs_result_division_degree_executiondifference. ((((exists ff_h_pfp_division_degree_executiondifferenceleft. ff_h_pfp_division_degree_executiondifferenceleft + S (pfs_left_division_degree_executiondifference) = S ((S (pfs_index_division_degree_executiondifference)) * ac)) /\\ exists ff_q_pfp_division_degree_executiondifferenceleft. ab = ff_q_pfp_division_degree_executiondifferenceleft * S ((S (pfs_index_division_degree_executiondifference)) * ac) + (pfs_left_division_degree_executiondifference))) /\\ (((((exists ff_h_pfp_division_degree_executiondifferenceright. ff_h_pfp_division_degree_executiondifferenceright + S (pfs_right_division_degree_executiondifference) = S ((S (pfs_index_division_degree_executiondifference)) * pfd_product_scale_division_degree_execution)) /\\ exists ff_q_pfp_division_degree_executiondifferenceright. pfd_product_code_division_degree_execution = ff_q_pfp_division_degree_executiondifferenceright * S ((S (pfs_index_division_degree_executiondifference)) * pfd_product_scale_division_degree_execution) + (pfs_right_division_degree_executiondifference))) /\\ (((((exists ff_h_pfp_division_degree_executiondifferenceresult. ff_h_pfp_division_degree_executiondifferenceresult + S (pfs_result_division_degree_executiondifference) = S ((S (pfs_index_division_degree_executiondifference)) * pfd_residual_scale_division_degree_execution)) /\\ exists ff_q_pfp_division_degree_executiondifferenceresult. pfd_residual_code_division_degree_execution = ff_q_pfp_division_degree_executiondifferenceresult * S ((S (pfs_index_division_degree_executiondifference)) * pfd_residual_scale_division_degree_execution) + (pfs_result_division_degree_executiondifference))) /\\ ((((exists pfa_gap_division_degree_executiondifferenceoperationleft. pfa_gap_division_degree_executiondifferenceoperationleft + S (pfs_right_division_degree_executiondifference) = (p)) /\\ (((exists pfa_gap_division_degree_executiondifferenceoperationright. pfa_gap_division_degree_executiondifferenceoperationright + S (pfs_result_division_degree_executiondifference) = (p)) /\\ ((((exists pfa_gap_division_degree_executiondifferenceoperationresultbound. pfa_gap_division_degree_executiondifferenceoperationresultbound + S (pfs_left_division_degree_executiondifference) = (p)) /\\ ((exists pfa_offset_left_division_degree_executiondifferenceoperationresultcongruence pfa_offset_right_division_degree_executiondifferenceoperationresultcongruence. ((pfs_right_division_degree_executiondifference) + (pfs_result_division_degree_executiondifference)) + (p) * pfa_offset_left_division_degree_executiondifferenceoperationresultcongruence = (pfs_left_division_degree_executiondifference) + (p) * pfa_offset_right_division_degree_executiondifferenceoperationresultcongruence)))))))))))))))) /\\ (((((L)=(pfd_cut_division_degree_execution)+(R)) /\\ (((forall fom_index_pfp_division_degree_executiontriminput. (exists fom_gap_pfp_division_degree_executiontriminput_index_bound. fom_gap_pfp_division_degree_executiontriminput_index_bound + S (fom_index_pfp_division_degree_executiontriminput) = L) -> exists fom_value_pfp_division_degree_executiontriminput. ((((exists fom_beta_height_pfp_division_degree_executiontriminput_entry. fom_beta_height_pfp_division_degree_executiontriminput_entry + S (fom_value_pfp_division_degree_executiontriminput) = S ((S (fom_index_pfp_division_degree_executiontriminput)) * pfd_residual_scale_division_degree_execution)) /\\ exists fom_beta_quotient_pfp_division_degree_executiontriminput_entry. pfd_residual_code_division_degree_execution = fom_beta_quotient_pfp_division_degree_executiontriminput_entry * S ((S (fom_index_pfp_division_degree_executiontriminput)) * pfd_residual_scale_division_degree_execution) + (fom_value_pfp_division_degree_executiontriminput))) /\\ (exists fom_gap_pfp_division_degree_executiontriminput_value_bound. fom_gap_pfp_division_degree_executiontriminput_value_bound + S (fom_value_pfp_division_degree_executiontriminput) = p))) /\\ (((forall pfp_repeat_index_division_degree_executiontrimremoved. (exists pfa_gap_division_degree_executiontrimremovedindex. pfa_gap_division_degree_executiontrimremovedindex + S (pfp_repeat_index_division_degree_executiontrimremoved) = (pfd_cut_division_degree_execution)) -> (((exists ff_h_pfp_division_degree_executiontrimremovedentry. ff_h_pfp_division_degree_executiontrimremovedentry + S (0) = S ((S (pfp_repeat_index_division_degree_executiontrimremoved)) * pfd_residual_scale_division_degree_execution)) /\\ exists ff_q_pfp_division_degree_executiontrimremovedentry. pfd_residual_code_division_degree_execution = ff_q_pfp_division_degree_executiontrimremovedentry * S ((S (pfp_repeat_index_division_degree_executiontrimremoved)) * pfd_residual_scale_division_degree_execution) + (0)))) /\\ (((forall pftrim_index_division_degree_executiontrimsuffix pftrim_value_division_degree_executiontrimsuffix. (exists pfa_gap_division_degree_executiontrimsuffixbound. pfa_gap_division_degree_executiontrimsuffixbound + S (pftrim_index_division_degree_executiontrimsuffix) = (R)) -> (((exists ff_h_pfp_division_degree_executiontrimsuffixsource. ff_h_pfp_division_degree_executiontrimsuffixsource + S (pftrim_value_division_degree_executiontrimsuffix) = S ((S ((pfd_cut_division_degree_execution)+pftrim_index_division_degree_executiontrimsuffix)) * pfd_residual_scale_division_degree_execution)) /\\ exists ff_q_pfp_division_degree_executiontrimsuffixsource. pfd_residual_code_division_degree_execution = ff_q_pfp_division_degree_executiontrimsuffixsource * S ((S ((pfd_cut_division_degree_execution)+pftrim_index_division_degree_executiontrimsuffix)) * pfd_residual_scale_division_degree_execution) + (pftrim_value_division_degree_executiontrimsuffix))) -> (((exists ff_h_pfp_division_degree_executiontrimsuffixoutput. ff_h_pfp_division_degree_executiontrimsuffixoutput + S (pftrim_value_division_degree_executiontrimsuffix) = S ((S (pftrim_index_division_degree_executiontrimsuffix)) * rc)) /\\ exists ff_q_pfp_division_degree_executiontrimsuffixoutput. rb = ff_q_pfp_division_degree_executiontrimsuffixoutput * S ((S (pftrim_index_division_degree_executiontrimsuffix)) * rc) + (pftrim_value_division_degree_executiontrimsuffix)))) /\\ (((R)=0 \\/ (exists pftrim_leading_division_degree_executiontrimnormal. ((((exists ff_h_pfp_division_degree_executiontrimnormalentry. ff_h_pfp_division_degree_executiontrimnormalentry + S (pftrim_leading_division_degree_executiontrimnormal) = S ((S (0)) * rc)) /\\ exists ff_q_pfp_division_degree_executiontrimnormalentry. rb = ff_q_pfp_division_degree_executiontrimnormalentry * S ((S (0)) * rc) + (pftrim_leading_division_degree_executiontrimnormal))) /\\ ((~(pftrim_leading_division_degree_executiontrimnormal=0))))))))))))))))))))))))))))))))) -> ((R)=0 \\/ (exists pfd_remainder_degree_division_degree_result. (((((R)=S (pfd_remainder_degree_division_degree_result)) /\\ (((forall fom_index_pfp_division_degree_resultrepresentedcoefficients. (exists fom_gap_pfp_division_degree_resultrepresentedcoefficients_index_bound. fom_gap_pfp_division_degree_resultrepresentedcoefficients_index_bound + S (fom_index_pfp_division_degree_resultrepresentedcoefficients) = R) -> exists fom_value_pfp_division_degree_resultrepresentedcoefficients. ((((exists fom_beta_height_pfp_division_degree_resultrepresentedcoefficients_entry. fom_beta_height_pfp_division_degree_resultrepresentedcoefficients_entry + S (fom_value_pfp_division_degree_resultrepresentedcoefficients) = S ((S (fom_index_pfp_division_degree_resultrepresentedcoefficients)) * rc)) /\\ exists fom_beta_quotient_pfp_division_degree_resultrepresentedcoefficients_entry. rb = fom_beta_quotient_pfp_division_degree_resultrepresentedcoefficients_entry * S ((S (fom_index_pfp_division_degree_resultrepresentedcoefficients)) * rc) + (fom_value_pfp_division_degree_resultrepresentedcoefficients))) /\\ (exists fom_gap_pfp_division_degree_resultrepresentedcoefficients_value_bound. fom_gap_pfp_division_degree_resultrepresentedcoefficients_value_bound + S (fom_value_pfp_division_degree_resultrepresentedcoefficients) = p))) /\\ ((exists pfd_leading_division_degree_resultrepresented. 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(((forall fom_index_pfp_division_bounded_choseninput. (exists fom_gap_pfp_division_bounded_choseninput_index_bound. fom_gap_pfp_division_bounded_choseninput_index_bound + S (fom_index_pfp_division_bounded_choseninput) = L) -> exists fom_value_pfp_division_bounded_choseninput. ((((exists fom_beta_height_pfp_division_bounded_choseninput_entry. fom_beta_height_pfp_division_bounded_choseninput_entry + S (fom_value_pfp_division_bounded_choseninput) = S ((S (fom_index_pfp_division_bounded_choseninput)) * ac)) /\\ exists fom_beta_quotient_pfp_division_bounded_choseninput_entry. ab = fom_beta_quotient_pfp_division_bounded_choseninput_entry * S ((S (fom_index_pfp_division_bounded_choseninput)) * ac) + (fom_value_pfp_division_bounded_choseninput))) /\\ (exists fom_gap_pfp_division_bounded_choseninput_value_bound. fom_gap_pfp_division_bounded_choseninput_value_bound + S (fom_value_pfp_division_bounded_choseninput) = p))) /\\ (((forall fom_index_pfp_division_bounded_chosendivisor. (exists fom_gap_pfp_division_bounded_chosendivisor_index_bound. fom_gap_pfp_division_bounded_chosendivisor_index_bound + S (fom_index_pfp_division_bounded_chosendivisor) = S (d)) -> exists fom_value_pfp_division_bounded_chosendivisor. ((((exists fom_beta_height_pfp_division_bounded_chosendivisor_entry. fom_beta_height_pfp_division_bounded_chosendivisor_entry + S (fom_value_pfp_division_bounded_chosendivisor) = S ((S (fom_index_pfp_division_bounded_chosendivisor)) * bc)) /\\ exists fom_beta_quotient_pfp_division_bounded_chosendivisor_entry. bb = fom_beta_quotient_pfp_division_bounded_chosendivisor_entry * S ((S (fom_index_pfp_division_bounded_chosendivisor)) * bc) + (fom_value_pfp_division_bounded_chosendivisor))) /\\ (exists fom_gap_pfp_division_bounded_chosendivisor_value_bound. fom_gap_pfp_division_bounded_chosendivisor_value_bound + S (fom_value_pfp_division_bounded_chosendivisor) = p))) /\\ (((((((q)=0) /\\ ((exists pfc_gap_division_bounded_chosenlengthshort. pfc_gap_division_bounded_chosenlengthshort+(L)=(d))))) \\/ (((~((q)=0)) /\\ (((q)+(d)=(L)))))) /\\ ((exists pfd_head_division_bounded_chosen pfd_inverse_division_bounded_chosen pfd_product_code_division_bounded_chosen pfd_product_scale_division_bounded_chosen pfd_residual_code_division_bounded_chosen pfd_residual_scale_division_bounded_chosen pfd_cut_division_bounded_chosen. ((((exists ff_h_pfp_division_bounded_chosenhead. ff_h_pfp_division_bounded_chosenhead + S (pfd_head_division_bounded_chosen) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_division_bounded_chosenhead. bb = ff_q_pfp_division_bounded_chosenhead * S ((S (0)) * bc) + (pfd_head_division_bounded_chosen))) /\\ (((((~((pfd_head_division_bounded_chosen) = 0)) /\\ ((((exists pfa_gap_division_bounded_choseninversemultiplicationleft. pfa_gap_division_bounded_choseninversemultiplicationleft + S (pfd_head_division_bounded_chosen) = (p)) /\\ (((exists pfa_gap_division_bounded_choseninversemultiplicationright. pfa_gap_division_bounded_choseninversemultiplicationright + S (pfd_inverse_division_bounded_chosen) = (p)) /\\ ((((exists pfa_gap_division_bounded_choseninversemultiplicationresultbound. pfa_gap_division_bounded_choseninversemultiplicationresultbound + S (1) = (p)) /\\ ((exists pfa_offset_left_division_bounded_choseninversemultiplicationresultcongruence pfa_offset_right_division_bounded_choseninversemultiplicationresultcongruence. ((pfd_head_division_bounded_chosen) * (pfd_inverse_division_bounded_chosen)) + (p) * pfa_offset_left_division_bounded_choseninversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_division_bounded_choseninversemultiplicationresultcongruence)))))))))))) /\\ (((forall pfd_index_division_bounded_chosenquotient. (exists pfa_gap_division_bounded_chosenquotientbound. pfa_gap_division_bounded_chosenquotientbound + S (pfd_index_division_bounded_chosenquotient) = (q)) -> exists pfd_value_division_bounded_chosenquotient. ((((exists ff_h_pfp_division_bounded_chosenquotiententry. ff_h_pfp_division_bounded_chosenquotiententry + S (pfd_value_division_bounded_chosenquotient) = S ((S (pfd_index_division_bounded_chosenquotient)) * qc)) /\\ exists ff_q_pfp_division_bounded_chosenquotiententry. qb = ff_q_pfp_division_bounded_chosenquotiententry * S ((S (pfd_index_division_bounded_chosenquotient)) * qc) + (pfd_value_division_bounded_chosenquotient))) /\\ ((exists pfd_input_division_bounded_chosenquotientstep pfd_previous_division_bounded_chosenquotientstep pfd_difference_division_bounded_chosenquotientstep. ((((exists ff_h_pfp_division_bounded_chosenquotientstepinput. ff_h_pfp_division_bounded_chosenquotientstepinput + S (pfd_input_division_bounded_chosenquotientstep) = S ((S (pfd_index_division_bounded_chosenquotient)) * ac)) /\\ exists ff_q_pfp_division_bounded_chosenquotientstepinput. ab = ff_q_pfp_division_bounded_chosenquotientstepinput * S ((S (pfd_index_division_bounded_chosenquotient)) * ac) + (pfd_input_division_bounded_chosenquotientstep))) /\\ (((exists pfc_terms_code_division_bounded_chosenquotientstepprevious pfc_terms_scale_division_bounded_chosenquotientstepprevious pfc_natural_sum_division_bounded_chosenquotientstepprevious. ((forall pfc_index_division_bounded_chosenquotientsteppreviousdiagonal. (exists pfa_gap_division_bounded_chosenquotientsteppreviousdiagonalbound. pfa_gap_division_bounded_chosenquotientsteppreviousdiagonalbound + S (pfc_index_division_bounded_chosenquotientsteppreviousdiagonal) = (S (pfd_index_division_bounded_chosenquotient))) -> exists pfc_value_division_bounded_chosenquotientsteppreviousdiagonal. ((((exists ff_h_pfp_division_bounded_chosenquotientsteppreviousdiagonalentry. ff_h_pfp_division_bounded_chosenquotientsteppreviousdiagonalentry + S (pfc_value_division_bounded_chosenquotientsteppreviousdiagonal) = S ((S (pfc_index_division_bounded_chosenquotientsteppreviousdiagonal)) * pfc_terms_scale_division_bounded_chosenquotientstepprevious)) /\\ exists ff_q_pfp_division_bounded_chosenquotientsteppreviousdiagonalentry. pfc_terms_code_division_bounded_chosenquotientstepprevious = ff_q_pfp_division_bounded_chosenquotientsteppreviousdiagonalentry * S ((S (pfc_index_division_bounded_chosenquotientsteppreviousdiagonal)) * pfc_terms_scale_division_bounded_chosenquotientstepprevious) + (pfc_value_division_bounded_chosenquotientsteppreviousdiagonal))) /\\ ((exists pfc_complement_division_bounded_chosenquotientsteppreviousdiagonalterm pfc_left_division_bounded_chosenquotientsteppreviousdiagonalterm pfc_right_division_bounded_chosenquotientsteppreviousdiagonalterm. (((pfc_index_division_bounded_chosenquotientsteppreviousdiagonal)+pfc_complement_division_bounded_chosenquotientsteppreviousdiagonalterm=(pfd_index_division_bounded_chosenquotient)) /\\ ((((((exists pfa_gap_division_bounded_chosenquotientsteppreviousdiagonaltermleftinside. pfa_gap_division_bounded_chosenquotientsteppreviousdiagonaltermleftinside + S (pfc_index_division_bounded_chosenquotientsteppreviousdiagonal) = (pfd_index_division_bounded_chosenquotient)) /\\ ((((exists ff_h_pfp_division_bounded_chosenquotientsteppreviousdiagonaltermleftentry. ff_h_pfp_division_bounded_chosenquotientsteppreviousdiagonaltermleftentry + S (pfc_left_division_bounded_chosenquotientsteppreviousdiagonalterm) = S ((S (pfc_index_division_bounded_chosenquotientsteppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_division_bounded_chosenquotientsteppreviousdiagonaltermleftentry. qb = ff_q_pfp_division_bounded_chosenquotientsteppreviousdiagonaltermleftentry * S ((S (pfc_index_division_bounded_chosenquotientsteppreviousdiagonal)) * qc) + (pfc_left_division_bounded_chosenquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_bounded_chosenquotientsteppreviousdiagonaltermleftoutside. pfc_gap_division_bounded_chosenquotientsteppreviousdiagonaltermleftoutside+(pfd_index_division_bounded_chosenquotient)=(pfc_index_division_bounded_chosenquotientsteppreviousdiagonal)) /\\ (((pfc_left_division_bounded_chosenquotientsteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_bounded_chosenquotientsteppreviousdiagonaltermrightinside. pfa_gap_division_bounded_chosenquotientsteppreviousdiagonaltermrightinside + S (pfc_complement_division_bounded_chosenquotientsteppreviousdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_division_bounded_chosenquotientsteppreviousdiagonaltermrightentry. ff_h_pfp_division_bounded_chosenquotientsteppreviousdiagonaltermrightentry + S (pfc_right_division_bounded_chosenquotientsteppreviousdiagonalterm) = S ((S (pfc_complement_division_bounded_chosenquotientsteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_bounded_chosenquotientsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_division_bounded_chosenquotientsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_division_bounded_chosenquotientsteppreviousdiagonalterm)) * bc) + (pfc_right_division_bounded_chosenquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_bounded_chosenquotientsteppreviousdiagonaltermrightoutside. pfc_gap_division_bounded_chosenquotientsteppreviousdiagonaltermrightoutside+(S (d))=(pfc_complement_division_bounded_chosenquotientsteppreviousdiagonalterm)) /\\ (((pfc_right_division_bounded_chosenquotientsteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_division_bounded_chosenquotientsteppreviousdiagonal)=pfc_left_division_bounded_chosenquotientsteppreviousdiagonalterm*pfc_right_division_bounded_chosenquotientsteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_bounded_chosenquotientstepprevioussum fs_v_pfc_division_bounded_chosenquotientstepprevioussum. ((((exists fs_h_pfc_division_bounded_chosenquotientstepprevioussum_body_start. fs_h_pfc_division_bounded_chosenquotientstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_bounded_chosenquotientstepprevioussum)) /\\ exists fs_q_pfc_division_bounded_chosenquotientstepprevioussum_body_start. fs_u_pfc_division_bounded_chosenquotientstepprevioussum = fs_q_pfc_division_bounded_chosenquotientstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_division_bounded_chosenquotientstepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_division_bounded_chosenquotientstepprevioussum_body_terminal. fs_h_pfc_division_bounded_chosenquotientstepprevioussum_body_terminal + S (pfc_natural_sum_division_bounded_chosenquotientstepprevious) = S ((S (S (pfd_index_division_bounded_chosenquotient))) * fs_v_pfc_division_bounded_chosenquotientstepprevioussum)) /\\ exists fs_q_pfc_division_bounded_chosenquotientstepprevioussum_body_terminal. fs_u_pfc_division_bounded_chosenquotientstepprevioussum = fs_q_pfc_division_bounded_chosenquotientstepprevioussum_body_terminal * S ((S (S (pfd_index_division_bounded_chosenquotient))) * fs_v_pfc_division_bounded_chosenquotientstepprevioussum) + (pfc_natural_sum_division_bounded_chosenquotientstepprevious))) /\\ forall fs_i_pfc_division_bounded_chosenquotientstepprevioussum_body_steps. (exists fs_lt_pfc_division_bounded_chosenquotientstepprevioussum_body_steps_bound. fs_lt_pfc_division_bounded_chosenquotientstepprevioussum_body_steps_bound + S fs_i_pfc_division_bounded_chosenquotientstepprevioussum_body_steps = S (pfd_index_division_bounded_chosenquotient)) -> exists fs_a_pfc_division_bounded_chosenquotientstepprevioussum_body_steps fs_r_pfc_division_bounded_chosenquotientstepprevioussum_body_steps fs_s_pfc_division_bounded_chosenquotientstepprevioussum_body_steps. ((((exists fs_h_pfc_division_bounded_chosenquotientstepprevioussum_body_steps_summand. fs_h_pfc_division_bounded_chosenquotientstepprevioussum_body_steps_summand + S (fs_a_pfc_division_bounded_chosenquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_bounded_chosenquotientstepprevioussum_body_steps)) * pfc_terms_scale_division_bounded_chosenquotientstepprevious)) /\\ exists fs_q_pfc_division_bounded_chosenquotientstepprevioussum_body_steps_summand. pfc_terms_code_division_bounded_chosenquotientstepprevious = fs_q_pfc_division_bounded_chosenquotientstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_division_bounded_chosenquotientstepprevioussum_body_steps)) * pfc_terms_scale_division_bounded_chosenquotientstepprevious) + (fs_a_pfc_division_bounded_chosenquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_bounded_chosenquotientstepprevioussum_body_steps_partial. fs_h_pfc_division_bounded_chosenquotientstepprevioussum_body_steps_partial + S (fs_r_pfc_division_bounded_chosenquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_bounded_chosenquotientstepprevioussum_body_steps)) * fs_v_pfc_division_bounded_chosenquotientstepprevioussum)) /\\ exists fs_q_pfc_division_bounded_chosenquotientstepprevioussum_body_steps_partial. fs_u_pfc_division_bounded_chosenquotientstepprevioussum = fs_q_pfc_division_bounded_chosenquotientstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_division_bounded_chosenquotientstepprevioussum_body_steps)) * fs_v_pfc_division_bounded_chosenquotientstepprevioussum) + (fs_r_pfc_division_bounded_chosenquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_bounded_chosenquotientstepprevioussum_body_steps_successor. fs_h_pfc_division_bounded_chosenquotientstepprevioussum_body_steps_successor + S (fs_s_pfc_division_bounded_chosenquotientstepprevioussum_body_steps) = S ((S (S fs_i_pfc_division_bounded_chosenquotientstepprevioussum_body_steps)) * fs_v_pfc_division_bounded_chosenquotientstepprevioussum)) /\\ exists fs_q_pfc_division_bounded_chosenquotientstepprevioussum_body_steps_successor. fs_u_pfc_division_bounded_chosenquotientstepprevioussum = fs_q_pfc_division_bounded_chosenquotientstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_division_bounded_chosenquotientstepprevioussum_body_steps)) * fs_v_pfc_division_bounded_chosenquotientstepprevioussum) + (fs_s_pfc_division_bounded_chosenquotientstepprevioussum_body_steps))) /\\ fs_s_pfc_division_bounded_chosenquotientstepprevioussum_body_steps = fs_r_pfc_division_bounded_chosenquotientstepprevioussum_body_steps + fs_a_pfc_division_bounded_chosenquotientstepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_division_bounded_chosenquotientsteppreviousresiduebound. pfa_gap_division_bounded_chosenquotientsteppreviousresiduebound + S (pfd_previous_division_bounded_chosenquotientstep) = (p)) /\\ ((exists pfa_offset_left_division_bounded_chosenquotientsteppreviousresiduecongruence pfa_offset_right_division_bounded_chosenquotientsteppreviousresiduecongruence. (pfc_natural_sum_division_bounded_chosenquotientstepprevious) + (p) * pfa_offset_left_division_bounded_chosenquotientsteppreviousresiduecongruence = (pfd_previous_division_bounded_chosenquotientstep) + (p) * pfa_offset_right_division_bounded_chosenquotientsteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_division_bounded_chosenquotientstepsubtractleft. pfa_gap_division_bounded_chosenquotientstepsubtractleft + S (pfd_previous_division_bounded_chosenquotientstep) = (p)) /\\ (((exists pfa_gap_division_bounded_chosenquotientstepsubtractright. pfa_gap_division_bounded_chosenquotientstepsubtractright + S (pfd_difference_division_bounded_chosenquotientstep) = (p)) /\\ ((((exists pfa_gap_division_bounded_chosenquotientstepsubtractresultbound. pfa_gap_division_bounded_chosenquotientstepsubtractresultbound + S (pfd_input_division_bounded_chosenquotientstep) = (p)) /\\ ((exists pfa_offset_left_division_bounded_chosenquotientstepsubtractresultcongruence pfa_offset_right_division_bounded_chosenquotientstepsubtractresultcongruence. ((pfd_previous_division_bounded_chosenquotientstep) + (pfd_difference_division_bounded_chosenquotientstep)) + (p) * pfa_offset_left_division_bounded_chosenquotientstepsubtractresultcongruence = (pfd_input_division_bounded_chosenquotientstep) + (p) * pfa_offset_right_division_bounded_chosenquotientstepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_division_bounded_chosenquotientstepmultiplyleft. pfa_gap_division_bounded_chosenquotientstepmultiplyleft + S (pfd_inverse_division_bounded_chosen) = (p)) /\\ (((exists pfa_gap_division_bounded_chosenquotientstepmultiplyright. pfa_gap_division_bounded_chosenquotientstepmultiplyright + S (pfd_difference_division_bounded_chosenquotientstep) = (p)) /\\ ((((exists pfa_gap_division_bounded_chosenquotientstepmultiplyresultbound. pfa_gap_division_bounded_chosenquotientstepmultiplyresultbound + S (pfd_value_division_bounded_chosenquotient) = (p)) /\\ ((exists pfa_offset_left_division_bounded_chosenquotientstepmultiplyresultcongruence pfa_offset_right_division_bounded_chosenquotientstepmultiplyresultcongruence. ((pfd_inverse_division_bounded_chosen) * (pfd_difference_division_bounded_chosenquotientstep)) + (p) * pfa_offset_left_division_bounded_chosenquotientstepmultiplyresultcongruence = (pfd_value_division_bounded_chosenquotient) + (p) * pfa_offset_right_division_bounded_chosenquotientstepmultiplyresultcongruence))))))))))))))))))) /\\ (((forall pfc_index_division_bounded_chosenproduct. (exists pfa_gap_division_bounded_chosenproductbound. pfa_gap_division_bounded_chosenproductbound + S (pfc_index_division_bounded_chosenproduct) = (L)) -> exists pfc_value_division_bounded_chosenproduct. ((((exists ff_h_pfp_division_bounded_chosenproductentry. ff_h_pfp_division_bounded_chosenproductentry + S (pfc_value_division_bounded_chosenproduct) = S ((S (pfc_index_division_bounded_chosenproduct)) * pfd_product_scale_division_bounded_chosen)) /\\ exists ff_q_pfp_division_bounded_chosenproductentry. pfd_product_code_division_bounded_chosen = ff_q_pfp_division_bounded_chosenproductentry * S ((S (pfc_index_division_bounded_chosenproduct)) * pfd_product_scale_division_bounded_chosen) + (pfc_value_division_bounded_chosenproduct))) /\\ ((exists pfc_terms_code_division_bounded_chosenproductcoefficient pfc_terms_scale_division_bounded_chosenproductcoefficient pfc_natural_sum_division_bounded_chosenproductcoefficient. ((forall pfc_index_division_bounded_chosenproductcoefficientdiagonal. (exists pfa_gap_division_bounded_chosenproductcoefficientdiagonalbound. pfa_gap_division_bounded_chosenproductcoefficientdiagonalbound + S (pfc_index_division_bounded_chosenproductcoefficientdiagonal) = (S (pfc_index_division_bounded_chosenproduct))) -> exists pfc_value_division_bounded_chosenproductcoefficientdiagonal. ((((exists ff_h_pfp_division_bounded_chosenproductcoefficientdiagonalentry. ff_h_pfp_division_bounded_chosenproductcoefficientdiagonalentry + S (pfc_value_division_bounded_chosenproductcoefficientdiagonal) = S ((S (pfc_index_division_bounded_chosenproductcoefficientdiagonal)) * pfc_terms_scale_division_bounded_chosenproductcoefficient)) /\\ exists ff_q_pfp_division_bounded_chosenproductcoefficientdiagonalentry. pfc_terms_code_division_bounded_chosenproductcoefficient = ff_q_pfp_division_bounded_chosenproductcoefficientdiagonalentry * S ((S (pfc_index_division_bounded_chosenproductcoefficientdiagonal)) * pfc_terms_scale_division_bounded_chosenproductcoefficient) + (pfc_value_division_bounded_chosenproductcoefficientdiagonal))) /\\ ((exists pfc_complement_division_bounded_chosenproductcoefficientdiagonalterm pfc_left_division_bounded_chosenproductcoefficientdiagonalterm pfc_right_division_bounded_chosenproductcoefficientdiagonalterm. (((pfc_index_division_bounded_chosenproductcoefficientdiagonal)+pfc_complement_division_bounded_chosenproductcoefficientdiagonalterm=(pfc_index_division_bounded_chosenproduct)) /\\ ((((((exists pfa_gap_division_bounded_chosenproductcoefficientdiagonaltermleftinside. pfa_gap_division_bounded_chosenproductcoefficientdiagonaltermleftinside + S (pfc_index_division_bounded_chosenproductcoefficientdiagonal) = (q)) /\\ ((((exists ff_h_pfp_division_bounded_chosenproductcoefficientdiagonaltermleftentry. ff_h_pfp_division_bounded_chosenproductcoefficientdiagonaltermleftentry + S (pfc_left_division_bounded_chosenproductcoefficientdiagonalterm) = S ((S (pfc_index_division_bounded_chosenproductcoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_division_bounded_chosenproductcoefficientdiagonaltermleftentry. qb = ff_q_pfp_division_bounded_chosenproductcoefficientdiagonaltermleftentry * S ((S (pfc_index_division_bounded_chosenproductcoefficientdiagonal)) * qc) + (pfc_left_division_bounded_chosenproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_division_bounded_chosenproductcoefficientdiagonaltermleftoutside. pfc_gap_division_bounded_chosenproductcoefficientdiagonaltermleftoutside+(q)=(pfc_index_division_bounded_chosenproductcoefficientdiagonal)) /\\ (((pfc_left_division_bounded_chosenproductcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_bounded_chosenproductcoefficientdiagonaltermrightinside. pfa_gap_division_bounded_chosenproductcoefficientdiagonaltermrightinside + S (pfc_complement_division_bounded_chosenproductcoefficientdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_division_bounded_chosenproductcoefficientdiagonaltermrightentry. ff_h_pfp_division_bounded_chosenproductcoefficientdiagonaltermrightentry + S (pfc_right_division_bounded_chosenproductcoefficientdiagonalterm) = S ((S (pfc_complement_division_bounded_chosenproductcoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_bounded_chosenproductcoefficientdiagonaltermrightentry. bb = ff_q_pfp_division_bounded_chosenproductcoefficientdiagonaltermrightentry * S ((S (pfc_complement_division_bounded_chosenproductcoefficientdiagonalterm)) * bc) + (pfc_right_division_bounded_chosenproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_division_bounded_chosenproductcoefficientdiagonaltermrightoutside. pfc_gap_division_bounded_chosenproductcoefficientdiagonaltermrightoutside+(S (d))=(pfc_complement_division_bounded_chosenproductcoefficientdiagonalterm)) /\\ (((pfc_right_division_bounded_chosenproductcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_division_bounded_chosenproductcoefficientdiagonal)=pfc_left_division_bounded_chosenproductcoefficientdiagonalterm*pfc_right_division_bounded_chosenproductcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_bounded_chosenproductcoefficientsum fs_v_pfc_division_bounded_chosenproductcoefficientsum. ((((exists fs_h_pfc_division_bounded_chosenproductcoefficientsum_body_start. fs_h_pfc_division_bounded_chosenproductcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_bounded_chosenproductcoefficientsum)) /\\ exists fs_q_pfc_division_bounded_chosenproductcoefficientsum_body_start. fs_u_pfc_division_bounded_chosenproductcoefficientsum = fs_q_pfc_division_bounded_chosenproductcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_division_bounded_chosenproductcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_division_bounded_chosenproductcoefficientsum_body_terminal. fs_h_pfc_division_bounded_chosenproductcoefficientsum_body_terminal + S (pfc_natural_sum_division_bounded_chosenproductcoefficient) = S ((S (S (pfc_index_division_bounded_chosenproduct))) * fs_v_pfc_division_bounded_chosenproductcoefficientsum)) /\\ exists fs_q_pfc_division_bounded_chosenproductcoefficientsum_body_terminal. fs_u_pfc_division_bounded_chosenproductcoefficientsum = fs_q_pfc_division_bounded_chosenproductcoefficientsum_body_terminal * S ((S (S (pfc_index_division_bounded_chosenproduct))) * fs_v_pfc_division_bounded_chosenproductcoefficientsum) + (pfc_natural_sum_division_bounded_chosenproductcoefficient))) /\\ forall fs_i_pfc_division_bounded_chosenproductcoefficientsum_body_steps. (exists fs_lt_pfc_division_bounded_chosenproductcoefficientsum_body_steps_bound. fs_lt_pfc_division_bounded_chosenproductcoefficientsum_body_steps_bound + S fs_i_pfc_division_bounded_chosenproductcoefficientsum_body_steps = S (pfc_index_division_bounded_chosenproduct)) -> exists fs_a_pfc_division_bounded_chosenproductcoefficientsum_body_steps fs_r_pfc_division_bounded_chosenproductcoefficientsum_body_steps fs_s_pfc_division_bounded_chosenproductcoefficientsum_body_steps. ((((exists fs_h_pfc_division_bounded_chosenproductcoefficientsum_body_steps_summand. fs_h_pfc_division_bounded_chosenproductcoefficientsum_body_steps_summand + S (fs_a_pfc_division_bounded_chosenproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_division_bounded_chosenproductcoefficientsum_body_steps)) * pfc_terms_scale_division_bounded_chosenproductcoefficient)) /\\ exists fs_q_pfc_division_bounded_chosenproductcoefficientsum_body_steps_summand. pfc_terms_code_division_bounded_chosenproductcoefficient = fs_q_pfc_division_bounded_chosenproductcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_division_bounded_chosenproductcoefficientsum_body_steps)) * pfc_terms_scale_division_bounded_chosenproductcoefficient) + (fs_a_pfc_division_bounded_chosenproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_division_bounded_chosenproductcoefficientsum_body_steps_partial. fs_h_pfc_division_bounded_chosenproductcoefficientsum_body_steps_partial + S (fs_r_pfc_division_bounded_chosenproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_division_bounded_chosenproductcoefficientsum_body_steps)) * fs_v_pfc_division_bounded_chosenproductcoefficientsum)) /\\ exists fs_q_pfc_division_bounded_chosenproductcoefficientsum_body_steps_partial. fs_u_pfc_division_bounded_chosenproductcoefficientsum = fs_q_pfc_division_bounded_chosenproductcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_division_bounded_chosenproductcoefficientsum_body_steps)) * fs_v_pfc_division_bounded_chosenproductcoefficientsum) + (fs_r_pfc_division_bounded_chosenproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_division_bounded_chosenproductcoefficientsum_body_steps_successor. fs_h_pfc_division_bounded_chosenproductcoefficientsum_body_steps_successor + S (fs_s_pfc_division_bounded_chosenproductcoefficientsum_body_steps) = S ((S (S fs_i_pfc_division_bounded_chosenproductcoefficientsum_body_steps)) * fs_v_pfc_division_bounded_chosenproductcoefficientsum)) /\\ exists fs_q_pfc_division_bounded_chosenproductcoefficientsum_body_steps_successor. fs_u_pfc_division_bounded_chosenproductcoefficientsum = fs_q_pfc_division_bounded_chosenproductcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_division_bounded_chosenproductcoefficientsum_body_steps)) * fs_v_pfc_division_bounded_chosenproductcoefficientsum) + (fs_s_pfc_division_bounded_chosenproductcoefficientsum_body_steps))) /\\ fs_s_pfc_division_bounded_chosenproductcoefficientsum_body_steps = fs_r_pfc_division_bounded_chosenproductcoefficientsum_body_steps + fs_a_pfc_division_bounded_chosenproductcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_division_bounded_chosenproductcoefficientresiduebound. pfa_gap_division_bounded_chosenproductcoefficientresiduebound + S (pfc_value_division_bounded_chosenproduct) = (p)) /\\ ((exists pfa_offset_left_division_bounded_chosenproductcoefficientresiduecongruence pfa_offset_right_division_bounded_chosenproductcoefficientresiduecongruence. (pfc_natural_sum_division_bounded_chosenproductcoefficient) + (p) * pfa_offset_left_division_bounded_chosenproductcoefficientresiduecongruence = (pfc_value_division_bounded_chosenproduct) + (p) * pfa_offset_right_division_bounded_chosenproductcoefficientresiduecongruence)))))))))))) /\\ (((forall pfs_index_division_bounded_chosendifference. (exists pfa_gap_division_bounded_chosendifferenceindex. pfa_gap_division_bounded_chosendifferenceindex + S (pfs_index_division_bounded_chosendifference) = (L)) -> exists pfs_left_division_bounded_chosendifference pfs_right_division_bounded_chosendifference pfs_result_division_bounded_chosendifference. ((((exists ff_h_pfp_division_bounded_chosendifferenceleft. ff_h_pfp_division_bounded_chosendifferenceleft + S (pfs_left_division_bounded_chosendifference) = S ((S (pfs_index_division_bounded_chosendifference)) * ac)) /\\ exists ff_q_pfp_division_bounded_chosendifferenceleft. ab = ff_q_pfp_division_bounded_chosendifferenceleft * S ((S (pfs_index_division_bounded_chosendifference)) * ac) + (pfs_left_division_bounded_chosendifference))) /\\ (((((exists ff_h_pfp_division_bounded_chosendifferenceright. ff_h_pfp_division_bounded_chosendifferenceright + S (pfs_right_division_bounded_chosendifference) = S ((S (pfs_index_division_bounded_chosendifference)) * pfd_product_scale_division_bounded_chosen)) /\\ exists ff_q_pfp_division_bounded_chosendifferenceright. pfd_product_code_division_bounded_chosen = ff_q_pfp_division_bounded_chosendifferenceright * S ((S (pfs_index_division_bounded_chosendifference)) * pfd_product_scale_division_bounded_chosen) + (pfs_right_division_bounded_chosendifference))) /\\ (((((exists ff_h_pfp_division_bounded_chosendifferenceresult. ff_h_pfp_division_bounded_chosendifferenceresult + S (pfs_result_division_bounded_chosendifference) = S ((S (pfs_index_division_bounded_chosendifference)) * pfd_residual_scale_division_bounded_chosen)) /\\ exists ff_q_pfp_division_bounded_chosendifferenceresult. pfd_residual_code_division_bounded_chosen = ff_q_pfp_division_bounded_chosendifferenceresult * S ((S (pfs_index_division_bounded_chosendifference)) * pfd_residual_scale_division_bounded_chosen) + (pfs_result_division_bounded_chosendifference))) /\\ ((((exists pfa_gap_division_bounded_chosendifferenceoperationleft. pfa_gap_division_bounded_chosendifferenceoperationleft + S (pfs_right_division_bounded_chosendifference) = (p)) /\\ (((exists pfa_gap_division_bounded_chosendifferenceoperationright. pfa_gap_division_bounded_chosendifferenceoperationright + S (pfs_result_division_bounded_chosendifference) = (p)) /\\ ((((exists pfa_gap_division_bounded_chosendifferenceoperationresultbound. pfa_gap_division_bounded_chosendifferenceoperationresultbound + S (pfs_left_division_bounded_chosendifference) = (p)) /\\ ((exists pfa_offset_left_division_bounded_chosendifferenceoperationresultcongruence pfa_offset_right_division_bounded_chosendifferenceoperationresultcongruence. ((pfs_right_division_bounded_chosendifference) + (pfs_result_division_bounded_chosendifference)) + (p) * pfa_offset_left_division_bounded_chosendifferenceoperationresultcongruence = (pfs_left_division_bounded_chosendifference) + (p) * pfa_offset_right_division_bounded_chosendifferenceoperationresultcongruence)))))))))))))))) /\\ (((((L)=(pfd_cut_division_bounded_chosen)+(R)) /\\ (((forall fom_index_pfp_division_bounded_chosentriminput. (exists fom_gap_pfp_division_bounded_chosentriminput_index_bound. fom_gap_pfp_division_bounded_chosentriminput_index_bound + S (fom_index_pfp_division_bounded_chosentriminput) = L) -> exists fom_value_pfp_division_bounded_chosentriminput. ((((exists fom_beta_height_pfp_division_bounded_chosentriminput_entry. fom_beta_height_pfp_division_bounded_chosentriminput_entry + S (fom_value_pfp_division_bounded_chosentriminput) = S ((S (fom_index_pfp_division_bounded_chosentriminput)) * pfd_residual_scale_division_bounded_chosen)) /\\ exists fom_beta_quotient_pfp_division_bounded_chosentriminput_entry. pfd_residual_code_division_bounded_chosen = fom_beta_quotient_pfp_division_bounded_chosentriminput_entry * S ((S (fom_index_pfp_division_bounded_chosentriminput)) * pfd_residual_scale_division_bounded_chosen) + (fom_value_pfp_division_bounded_chosentriminput))) /\\ (exists fom_gap_pfp_division_bounded_chosentriminput_value_bound. fom_gap_pfp_division_bounded_chosentriminput_value_bound + S (fom_value_pfp_division_bounded_chosentriminput) = p))) /\\ (((forall pfp_repeat_index_division_bounded_chosentrimremoved. (exists pfa_gap_division_bounded_chosentrimremovedindex. pfa_gap_division_bounded_chosentrimremovedindex + S (pfp_repeat_index_division_bounded_chosentrimremoved) = (pfd_cut_division_bounded_chosen)) -> (((exists ff_h_pfp_division_bounded_chosentrimremovedentry. ff_h_pfp_division_bounded_chosentrimremovedentry + S (0) = S ((S (pfp_repeat_index_division_bounded_chosentrimremoved)) * pfd_residual_scale_division_bounded_chosen)) /\\ exists ff_q_pfp_division_bounded_chosentrimremovedentry. pfd_residual_code_division_bounded_chosen = ff_q_pfp_division_bounded_chosentrimremovedentry * S ((S (pfp_repeat_index_division_bounded_chosentrimremoved)) * pfd_residual_scale_division_bounded_chosen) + (0)))) /\\ (((forall pftrim_index_division_bounded_chosentrimsuffix pftrim_value_division_bounded_chosentrimsuffix. (exists pfa_gap_division_bounded_chosentrimsuffixbound. pfa_gap_division_bounded_chosentrimsuffixbound + S (pftrim_index_division_bounded_chosentrimsuffix) = (R)) -> (((exists ff_h_pfp_division_bounded_chosentrimsuffixsource. ff_h_pfp_division_bounded_chosentrimsuffixsource + S (pftrim_value_division_bounded_chosentrimsuffix) = S ((S ((pfd_cut_division_bounded_chosen)+pftrim_index_division_bounded_chosentrimsuffix)) * pfd_residual_scale_division_bounded_chosen)) /\\ exists ff_q_pfp_division_bounded_chosentrimsuffixsource. pfd_residual_code_division_bounded_chosen = ff_q_pfp_division_bounded_chosentrimsuffixsource * S ((S ((pfd_cut_division_bounded_chosen)+pftrim_index_division_bounded_chosentrimsuffix)) * pfd_residual_scale_division_bounded_chosen) + (pftrim_value_division_bounded_chosentrimsuffix))) -> (((exists ff_h_pfp_division_bounded_chosentrimsuffixoutput. ff_h_pfp_division_bounded_chosentrimsuffixoutput + S (pftrim_value_division_bounded_chosentrimsuffix) = S ((S (pftrim_index_division_bounded_chosentrimsuffix)) * rc)) /\\ exists ff_q_pfp_division_bounded_chosentrimsuffixoutput. rb = ff_q_pfp_division_bounded_chosentrimsuffixoutput * S ((S (pftrim_index_division_bounded_chosentrimsuffix)) * rc) + (pftrim_value_division_bounded_chosentrimsuffix)))) /\\ (((R)=0 \\/ (exists pftrim_leading_division_bounded_chosentrimnormal. ((((exists ff_h_pfp_division_bounded_chosentrimnormalentry. ff_h_pfp_division_bounded_chosentrimnormalentry + S (pftrim_leading_division_bounded_chosentrimnormal) = S ((S (0)) * rc)) /\\ exists ff_q_pfp_division_bounded_chosentrimnormalentry. rb = ff_q_pfp_division_bounded_chosentrimnormalentry * S ((S (0)) * rc) + (pftrim_leading_division_bounded_chosentrimnormal))) /\\ ((~(pftrim_leading_division_bounded_chosentrimnormal=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 (bb)", "specialize prime_field_polynomial_division_execution_exists (bc)", "specialize prime_field_polynomial_division_execution_exists (d)", "apply prime_field_polynomial_division_execution_exists", "exact hp", "exact ha", "exact hb", "cases h", "cases h_witness", "cases h_witness_witness", "cases h_witness_witness_witness", "cases h_witness_witness_witness_witness", "cases h_witness_witness_witness_witness_witness", "exists x", "exists x1", "exists x2", "exists x3", "exists x4", "exists x5", "split", "exact h_witness_witness_witness_witness_witness_witness", "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 (x)", "specialize prime_field_polynomial_division_remainder_degree (x1)", "specialize prime_field_polynomial_division_remainder_degree (x2)", "specialize prime_field_polynomial_division_remainder_degree (x3)", "specialize prime_field_polynomial_division_remainder_degree (x4)", "specialize prime_field_polynomial_division_remainder_degree (x5)", "apply prime_field_polynomial_division_remainder_degree", "exact hp", "exact h_witness_witness_witness_witness_witness_witness" ], "script_sha256": "3ef0dfd32cff8e3b8c93a4c3bbc2d99911403290f41d5018a71840e08a76fd64", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_division_candidate.py", "sha256": "edfc7806caf7a83b9cb0e3e420bd2c3a8679f2d4d9ee6ca9f8eae53faca8d5b2" }, "statement": "forall p ab ac L bb bc d. (~((p) = 1) /\\ forall pfa_factor_left_division_bounded_prime pfa_factor_right_division_bounded_prime. (p) = pfa_factor_left_division_bounded_prime * pfa_factor_right_division_bounded_prime -> pfa_factor_left_division_bounded_prime = 1 \\/ pfa_factor_right_division_bounded_prime = 1) -> (forall fom_index_pfp_division_bounded_input. (exists fom_gap_pfp_division_bounded_input_index_bound. fom_gap_pfp_division_bounded_input_index_bound + S (fom_index_pfp_division_bounded_input) = L) -> exists fom_value_pfp_division_bounded_input. ((((exists fom_beta_height_pfp_division_bounded_input_entry. fom_beta_height_pfp_division_bounded_input_entry + S (fom_value_pfp_division_bounded_input) = S ((S (fom_index_pfp_division_bounded_input)) * ac)) /\\ exists fom_beta_quotient_pfp_division_bounded_input_entry. ab = fom_beta_quotient_pfp_division_bounded_input_entry * S ((S (fom_index_pfp_division_bounded_input)) * ac) + (fom_value_pfp_division_bounded_input))) /\\ (exists fom_gap_pfp_division_bounded_input_value_bound. fom_gap_pfp_division_bounded_input_value_bound + S (fom_value_pfp_division_bounded_input) = p))) -> ((((S d)=S (d)) /\\ (((forall fom_index_pfp_division_bounded_divisorcoefficients. (exists fom_gap_pfp_division_bounded_divisorcoefficients_index_bound. fom_gap_pfp_division_bounded_divisorcoefficients_index_bound + S (fom_index_pfp_division_bounded_divisorcoefficients) = S d) -> exists fom_value_pfp_division_bounded_divisorcoefficients. ((((exists fom_beta_height_pfp_division_bounded_divisorcoefficients_entry. fom_beta_height_pfp_division_bounded_divisorcoefficients_entry + S (fom_value_pfp_division_bounded_divisorcoefficients) = S ((S (fom_index_pfp_division_bounded_divisorcoefficients)) * bc)) /\\ exists fom_beta_quotient_pfp_division_bounded_divisorcoefficients_entry. bb = fom_beta_quotient_pfp_division_bounded_divisorcoefficients_entry * S ((S (fom_index_pfp_division_bounded_divisorcoefficients)) * bc) + (fom_value_pfp_division_bounded_divisorcoefficients))) /\\ (exists fom_gap_pfp_division_bounded_divisorcoefficients_value_bound. fom_gap_pfp_division_bounded_divisorcoefficients_value_bound + S (fom_value_pfp_division_bounded_divisorcoefficients) = p))) /\\ ((exists pfd_leading_division_bounded_divisor. ((((exists ff_h_pfp_division_bounded_divisorentry. ff_h_pfp_division_bounded_divisorentry + S (pfd_leading_division_bounded_divisor) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_division_bounded_divisorentry. bb = ff_q_pfp_division_bounded_divisorentry * S ((S (0)) * bc) + (pfd_leading_division_bounded_divisor))) /\\ ((~(pfd_leading_division_bounded_divisor=0)))))))))) -> exists qb qc q rb rc R. ((((forall fom_index_pfp_division_bounded_executioninput. (exists fom_gap_pfp_division_bounded_executioninput_index_bound. fom_gap_pfp_division_bounded_executioninput_index_bound + S (fom_index_pfp_division_bounded_executioninput) = L) -> exists fom_value_pfp_division_bounded_executioninput. ((((exists fom_beta_height_pfp_division_bounded_executioninput_entry. fom_beta_height_pfp_division_bounded_executioninput_entry + S (fom_value_pfp_division_bounded_executioninput) = S ((S (fom_index_pfp_division_bounded_executioninput)) * ac)) /\\ exists fom_beta_quotient_pfp_division_bounded_executioninput_entry. ab = fom_beta_quotient_pfp_division_bounded_executioninput_entry * S ((S (fom_index_pfp_division_bounded_executioninput)) * ac) + (fom_value_pfp_division_bounded_executioninput))) /\\ (exists fom_gap_pfp_division_bounded_executioninput_value_bound. fom_gap_pfp_division_bounded_executioninput_value_bound + S (fom_value_pfp_division_bounded_executioninput) = p))) /\\ (((forall fom_index_pfp_division_bounded_executiondivisor. (exists fom_gap_pfp_division_bounded_executiondivisor_index_bound. fom_gap_pfp_division_bounded_executiondivisor_index_bound + S (fom_index_pfp_division_bounded_executiondivisor) = S (d)) -> exists fom_value_pfp_division_bounded_executiondivisor. ((((exists fom_beta_height_pfp_division_bounded_executiondivisor_entry. fom_beta_height_pfp_division_bounded_executiondivisor_entry + S (fom_value_pfp_division_bounded_executiondivisor) = S ((S (fom_index_pfp_division_bounded_executiondivisor)) * bc)) /\\ exists fom_beta_quotient_pfp_division_bounded_executiondivisor_entry. bb = fom_beta_quotient_pfp_division_bounded_executiondivisor_entry * S ((S (fom_index_pfp_division_bounded_executiondivisor)) * bc) + (fom_value_pfp_division_bounded_executiondivisor))) /\\ (exists fom_gap_pfp_division_bounded_executiondivisor_value_bound. fom_gap_pfp_division_bounded_executiondivisor_value_bound + S (fom_value_pfp_division_bounded_executiondivisor) = p))) /\\ (((((((q)=0) /\\ ((exists pfc_gap_division_bounded_executionlengthshort. pfc_gap_division_bounded_executionlengthshort+(L)=(d))))) \\/ (((~((q)=0)) /\\ (((q)+(d)=(L)))))) /\\ ((exists pfd_head_division_bounded_execution pfd_inverse_division_bounded_execution pfd_product_code_division_bounded_execution pfd_product_scale_division_bounded_execution pfd_residual_code_division_bounded_execution pfd_residual_scale_division_bounded_execution pfd_cut_division_bounded_execution. ((((exists ff_h_pfp_division_bounded_executionhead. ff_h_pfp_division_bounded_executionhead + S (pfd_head_division_bounded_execution) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_division_bounded_executionhead. bb = ff_q_pfp_division_bounded_executionhead * S ((S (0)) * bc) + (pfd_head_division_bounded_execution))) /\\ (((((~((pfd_head_division_bounded_execution) = 0)) /\\ ((((exists pfa_gap_division_bounded_executioninversemultiplicationleft. pfa_gap_division_bounded_executioninversemultiplicationleft + S (pfd_head_division_bounded_execution) = (p)) /\\ (((exists pfa_gap_division_bounded_executioninversemultiplicationright. pfa_gap_division_bounded_executioninversemultiplicationright + S (pfd_inverse_division_bounded_execution) = (p)) /\\ ((((exists pfa_gap_division_bounded_executioninversemultiplicationresultbound. pfa_gap_division_bounded_executioninversemultiplicationresultbound + S (1) = (p)) /\\ ((exists pfa_offset_left_division_bounded_executioninversemultiplicationresultcongruence pfa_offset_right_division_bounded_executioninversemultiplicationresultcongruence. ((pfd_head_division_bounded_execution) * (pfd_inverse_division_bounded_execution)) + (p) * pfa_offset_left_division_bounded_executioninversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_division_bounded_executioninversemultiplicationresultcongruence)))))))))))) /\\ (((forall pfd_index_division_bounded_executionquotient. (exists pfa_gap_division_bounded_executionquotientbound. pfa_gap_division_bounded_executionquotientbound + S (pfd_index_division_bounded_executionquotient) = (q)) -> exists pfd_value_division_bounded_executionquotient. ((((exists ff_h_pfp_division_bounded_executionquotiententry. ff_h_pfp_division_bounded_executionquotiententry + S (pfd_value_division_bounded_executionquotient) = S ((S (pfd_index_division_bounded_executionquotient)) * qc)) /\\ exists ff_q_pfp_division_bounded_executionquotiententry. qb = ff_q_pfp_division_bounded_executionquotiententry * S ((S (pfd_index_division_bounded_executionquotient)) * qc) + (pfd_value_division_bounded_executionquotient))) /\\ ((exists pfd_input_division_bounded_executionquotientstep pfd_previous_division_bounded_executionquotientstep pfd_difference_division_bounded_executionquotientstep. ((((exists ff_h_pfp_division_bounded_executionquotientstepinput. ff_h_pfp_division_bounded_executionquotientstepinput + S (pfd_input_division_bounded_executionquotientstep) = S ((S (pfd_index_division_bounded_executionquotient)) * ac)) /\\ exists ff_q_pfp_division_bounded_executionquotientstepinput. ab = ff_q_pfp_division_bounded_executionquotientstepinput * S ((S (pfd_index_division_bounded_executionquotient)) * ac) + (pfd_input_division_bounded_executionquotientstep))) /\\ (((exists pfc_terms_code_division_bounded_executionquotientstepprevious pfc_terms_scale_division_bounded_executionquotientstepprevious pfc_natural_sum_division_bounded_executionquotientstepprevious. ((forall pfc_index_division_bounded_executionquotientsteppreviousdiagonal. (exists pfa_gap_division_bounded_executionquotientsteppreviousdiagonalbound. pfa_gap_division_bounded_executionquotientsteppreviousdiagonalbound + S (pfc_index_division_bounded_executionquotientsteppreviousdiagonal) = (S (pfd_index_division_bounded_executionquotient))) -> exists pfc_value_division_bounded_executionquotientsteppreviousdiagonal. ((((exists ff_h_pfp_division_bounded_executionquotientsteppreviousdiagonalentry. ff_h_pfp_division_bounded_executionquotientsteppreviousdiagonalentry + S (pfc_value_division_bounded_executionquotientsteppreviousdiagonal) = S ((S (pfc_index_division_bounded_executionquotientsteppreviousdiagonal)) * pfc_terms_scale_division_bounded_executionquotientstepprevious)) /\\ exists ff_q_pfp_division_bounded_executionquotientsteppreviousdiagonalentry. pfc_terms_code_division_bounded_executionquotientstepprevious = ff_q_pfp_division_bounded_executionquotientsteppreviousdiagonalentry * S ((S (pfc_index_division_bounded_executionquotientsteppreviousdiagonal)) * pfc_terms_scale_division_bounded_executionquotientstepprevious) + (pfc_value_division_bounded_executionquotientsteppreviousdiagonal))) /\\ ((exists pfc_complement_division_bounded_executionquotientsteppreviousdiagonalterm pfc_left_division_bounded_executionquotientsteppreviousdiagonalterm pfc_right_division_bounded_executionquotientsteppreviousdiagonalterm. (((pfc_index_division_bounded_executionquotientsteppreviousdiagonal)+pfc_complement_division_bounded_executionquotientsteppreviousdiagonalterm=(pfd_index_division_bounded_executionquotient)) /\\ ((((((exists pfa_gap_division_bounded_executionquotientsteppreviousdiagonaltermleftinside. pfa_gap_division_bounded_executionquotientsteppreviousdiagonaltermleftinside + S (pfc_index_division_bounded_executionquotientsteppreviousdiagonal) = (pfd_index_division_bounded_executionquotient)) /\\ ((((exists ff_h_pfp_division_bounded_executionquotientsteppreviousdiagonaltermleftentry. ff_h_pfp_division_bounded_executionquotientsteppreviousdiagonaltermleftentry + S (pfc_left_division_bounded_executionquotientsteppreviousdiagonalterm) = S ((S (pfc_index_division_bounded_executionquotientsteppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_division_bounded_executionquotientsteppreviousdiagonaltermleftentry. qb = ff_q_pfp_division_bounded_executionquotientsteppreviousdiagonaltermleftentry * S ((S (pfc_index_division_bounded_executionquotientsteppreviousdiagonal)) * qc) + (pfc_left_division_bounded_executionquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_bounded_executionquotientsteppreviousdiagonaltermleftoutside. pfc_gap_division_bounded_executionquotientsteppreviousdiagonaltermleftoutside+(pfd_index_division_bounded_executionquotient)=(pfc_index_division_bounded_executionquotientsteppreviousdiagonal)) /\\ (((pfc_left_division_bounded_executionquotientsteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_bounded_executionquotientsteppreviousdiagonaltermrightinside. pfa_gap_division_bounded_executionquotientsteppreviousdiagonaltermrightinside + S (pfc_complement_division_bounded_executionquotientsteppreviousdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_division_bounded_executionquotientsteppreviousdiagonaltermrightentry. ff_h_pfp_division_bounded_executionquotientsteppreviousdiagonaltermrightentry + S (pfc_right_division_bounded_executionquotientsteppreviousdiagonalterm) = S ((S (pfc_complement_division_bounded_executionquotientsteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_bounded_executionquotientsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_division_bounded_executionquotientsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_division_bounded_executionquotientsteppreviousdiagonalterm)) * bc) + (pfc_right_division_bounded_executionquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_bounded_executionquotientsteppreviousdiagonaltermrightoutside. pfc_gap_division_bounded_executionquotientsteppreviousdiagonaltermrightoutside+(S (d))=(pfc_complement_division_bounded_executionquotientsteppreviousdiagonalterm)) /\\ (((pfc_right_division_bounded_executionquotientsteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_division_bounded_executionquotientsteppreviousdiagonal)=pfc_left_division_bounded_executionquotientsteppreviousdiagonalterm*pfc_right_division_bounded_executionquotientsteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_bounded_executionquotientstepprevioussum fs_v_pfc_division_bounded_executionquotientstepprevioussum. ((((exists fs_h_pfc_division_bounded_executionquotientstepprevioussum_body_start. fs_h_pfc_division_bounded_executionquotientstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_bounded_executionquotientstepprevioussum)) /\\ exists fs_q_pfc_division_bounded_executionquotientstepprevioussum_body_start. fs_u_pfc_division_bounded_executionquotientstepprevioussum = fs_q_pfc_division_bounded_executionquotientstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_division_bounded_executionquotientstepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_division_bounded_executionquotientstepprevioussum_body_terminal. fs_h_pfc_division_bounded_executionquotientstepprevioussum_body_terminal + S (pfc_natural_sum_division_bounded_executionquotientstepprevious) = S ((S (S (pfd_index_division_bounded_executionquotient))) * fs_v_pfc_division_bounded_executionquotientstepprevioussum)) /\\ exists fs_q_pfc_division_bounded_executionquotientstepprevioussum_body_terminal. fs_u_pfc_division_bounded_executionquotientstepprevioussum = fs_q_pfc_division_bounded_executionquotientstepprevioussum_body_terminal * S ((S (S (pfd_index_division_bounded_executionquotient))) * fs_v_pfc_division_bounded_executionquotientstepprevioussum) + (pfc_natural_sum_division_bounded_executionquotientstepprevious))) /\\ forall fs_i_pfc_division_bounded_executionquotientstepprevioussum_body_steps. (exists fs_lt_pfc_division_bounded_executionquotientstepprevioussum_body_steps_bound. fs_lt_pfc_division_bounded_executionquotientstepprevioussum_body_steps_bound + S fs_i_pfc_division_bounded_executionquotientstepprevioussum_body_steps = S (pfd_index_division_bounded_executionquotient)) -> exists fs_a_pfc_division_bounded_executionquotientstepprevioussum_body_steps fs_r_pfc_division_bounded_executionquotientstepprevioussum_body_steps fs_s_pfc_division_bounded_executionquotientstepprevioussum_body_steps. ((((exists fs_h_pfc_division_bounded_executionquotientstepprevioussum_body_steps_summand. fs_h_pfc_division_bounded_executionquotientstepprevioussum_body_steps_summand + S (fs_a_pfc_division_bounded_executionquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_bounded_executionquotientstepprevioussum_body_steps)) * pfc_terms_scale_division_bounded_executionquotientstepprevious)) /\\ exists fs_q_pfc_division_bounded_executionquotientstepprevioussum_body_steps_summand. pfc_terms_code_division_bounded_executionquotientstepprevious = fs_q_pfc_division_bounded_executionquotientstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_division_bounded_executionquotientstepprevioussum_body_steps)) * pfc_terms_scale_division_bounded_executionquotientstepprevious) + (fs_a_pfc_division_bounded_executionquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_bounded_executionquotientstepprevioussum_body_steps_partial. fs_h_pfc_division_bounded_executionquotientstepprevioussum_body_steps_partial + S (fs_r_pfc_division_bounded_executionquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_bounded_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_division_bounded_executionquotientstepprevioussum)) /\\ exists fs_q_pfc_division_bounded_executionquotientstepprevioussum_body_steps_partial. fs_u_pfc_division_bounded_executionquotientstepprevioussum = fs_q_pfc_division_bounded_executionquotientstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_division_bounded_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_division_bounded_executionquotientstepprevioussum) + (fs_r_pfc_division_bounded_executionquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_bounded_executionquotientstepprevioussum_body_steps_successor. fs_h_pfc_division_bounded_executionquotientstepprevioussum_body_steps_successor + S (fs_s_pfc_division_bounded_executionquotientstepprevioussum_body_steps) = S ((S (S fs_i_pfc_division_bounded_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_division_bounded_executionquotientstepprevioussum)) /\\ exists fs_q_pfc_division_bounded_executionquotientstepprevioussum_body_steps_successor. fs_u_pfc_division_bounded_executionquotientstepprevioussum = fs_q_pfc_division_bounded_executionquotientstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_division_bounded_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_division_bounded_executionquotientstepprevioussum) + (fs_s_pfc_division_bounded_executionquotientstepprevioussum_body_steps))) /\\ fs_s_pfc_division_bounded_executionquotientstepprevioussum_body_steps = fs_r_pfc_division_bounded_executionquotientstepprevioussum_body_steps + fs_a_pfc_division_bounded_executionquotientstepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_division_bounded_executionquotientsteppreviousresiduebound. pfa_gap_division_bounded_executionquotientsteppreviousresiduebound + S (pfd_previous_division_bounded_executionquotientstep) = (p)) /\\ ((exists pfa_offset_left_division_bounded_executionquotientsteppreviousresiduecongruence pfa_offset_right_division_bounded_executionquotientsteppreviousresiduecongruence. (pfc_natural_sum_division_bounded_executionquotientstepprevious) + (p) * pfa_offset_left_division_bounded_executionquotientsteppreviousresiduecongruence = (pfd_previous_division_bounded_executionquotientstep) + (p) * pfa_offset_right_division_bounded_executionquotientsteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_division_bounded_executionquotientstepsubtractleft. pfa_gap_division_bounded_executionquotientstepsubtractleft + S (pfd_previous_division_bounded_executionquotientstep) = (p)) /\\ (((exists pfa_gap_division_bounded_executionquotientstepsubtractright. pfa_gap_division_bounded_executionquotientstepsubtractright + S (pfd_difference_division_bounded_executionquotientstep) = (p)) /\\ ((((exists pfa_gap_division_bounded_executionquotientstepsubtractresultbound. pfa_gap_division_bounded_executionquotientstepsubtractresultbound + S (pfd_input_division_bounded_executionquotientstep) = (p)) /\\ ((exists pfa_offset_left_division_bounded_executionquotientstepsubtractresultcongruence pfa_offset_right_division_bounded_executionquotientstepsubtractresultcongruence. ((pfd_previous_division_bounded_executionquotientstep) + (pfd_difference_division_bounded_executionquotientstep)) + (p) * pfa_offset_left_division_bounded_executionquotientstepsubtractresultcongruence = (pfd_input_division_bounded_executionquotientstep) + (p) * pfa_offset_right_division_bounded_executionquotientstepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_division_bounded_executionquotientstepmultiplyleft. pfa_gap_division_bounded_executionquotientstepmultiplyleft + S (pfd_inverse_division_bounded_execution) = (p)) /\\ (((exists pfa_gap_division_bounded_executionquotientstepmultiplyright. pfa_gap_division_bounded_executionquotientstepmultiplyright + S (pfd_difference_division_bounded_executionquotientstep) = (p)) /\\ ((((exists pfa_gap_division_bounded_executionquotientstepmultiplyresultbound. pfa_gap_division_bounded_executionquotientstepmultiplyresultbound + S (pfd_value_division_bounded_executionquotient) = (p)) /\\ ((exists pfa_offset_left_division_bounded_executionquotientstepmultiplyresultcongruence pfa_offset_right_division_bounded_executionquotientstepmultiplyresultcongruence. ((pfd_inverse_division_bounded_execution) * (pfd_difference_division_bounded_executionquotientstep)) + (p) * pfa_offset_left_division_bounded_executionquotientstepmultiplyresultcongruence = (pfd_value_division_bounded_executionquotient) + (p) * pfa_offset_right_division_bounded_executionquotientstepmultiplyresultcongruence))))))))))))))))))) /\\ (((forall pfc_index_division_bounded_executionproduct. (exists pfa_gap_division_bounded_executionproductbound. pfa_gap_division_bounded_executionproductbound + S (pfc_index_division_bounded_executionproduct) = (L)) -> exists pfc_value_division_bounded_executionproduct. ((((exists ff_h_pfp_division_bounded_executionproductentry. ff_h_pfp_division_bounded_executionproductentry + S (pfc_value_division_bounded_executionproduct) = S ((S (pfc_index_division_bounded_executionproduct)) * pfd_product_scale_division_bounded_execution)) /\\ exists ff_q_pfp_division_bounded_executionproductentry. pfd_product_code_division_bounded_execution = ff_q_pfp_division_bounded_executionproductentry * S ((S (pfc_index_division_bounded_executionproduct)) * pfd_product_scale_division_bounded_execution) + (pfc_value_division_bounded_executionproduct))) /\\ ((exists pfc_terms_code_division_bounded_executionproductcoefficient pfc_terms_scale_division_bounded_executionproductcoefficient pfc_natural_sum_division_bounded_executionproductcoefficient. ((forall pfc_index_division_bounded_executionproductcoefficientdiagonal. (exists pfa_gap_division_bounded_executionproductcoefficientdiagonalbound. pfa_gap_division_bounded_executionproductcoefficientdiagonalbound + S (pfc_index_division_bounded_executionproductcoefficientdiagonal) = (S (pfc_index_division_bounded_executionproduct))) -> exists pfc_value_division_bounded_executionproductcoefficientdiagonal. ((((exists ff_h_pfp_division_bounded_executionproductcoefficientdiagonalentry. ff_h_pfp_division_bounded_executionproductcoefficientdiagonalentry + S (pfc_value_division_bounded_executionproductcoefficientdiagonal) = S ((S (pfc_index_division_bounded_executionproductcoefficientdiagonal)) * pfc_terms_scale_division_bounded_executionproductcoefficient)) /\\ exists ff_q_pfp_division_bounded_executionproductcoefficientdiagonalentry. pfc_terms_code_division_bounded_executionproductcoefficient = ff_q_pfp_division_bounded_executionproductcoefficientdiagonalentry * S ((S (pfc_index_division_bounded_executionproductcoefficientdiagonal)) * pfc_terms_scale_division_bounded_executionproductcoefficient) + (pfc_value_division_bounded_executionproductcoefficientdiagonal))) /\\ ((exists pfc_complement_division_bounded_executionproductcoefficientdiagonalterm pfc_left_division_bounded_executionproductcoefficientdiagonalterm pfc_right_division_bounded_executionproductcoefficientdiagonalterm. (((pfc_index_division_bounded_executionproductcoefficientdiagonal)+pfc_complement_division_bounded_executionproductcoefficientdiagonalterm=(pfc_index_division_bounded_executionproduct)) /\\ ((((((exists pfa_gap_division_bounded_executionproductcoefficientdiagonaltermleftinside. pfa_gap_division_bounded_executionproductcoefficientdiagonaltermleftinside + S (pfc_index_division_bounded_executionproductcoefficientdiagonal) = (q)) /\\ ((((exists ff_h_pfp_division_bounded_executionproductcoefficientdiagonaltermleftentry. ff_h_pfp_division_bounded_executionproductcoefficientdiagonaltermleftentry + S (pfc_left_division_bounded_executionproductcoefficientdiagonalterm) = S ((S (pfc_index_division_bounded_executionproductcoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_division_bounded_executionproductcoefficientdiagonaltermleftentry. qb = ff_q_pfp_division_bounded_executionproductcoefficientdiagonaltermleftentry * S ((S (pfc_index_division_bounded_executionproductcoefficientdiagonal)) * qc) + (pfc_left_division_bounded_executionproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_division_bounded_executionproductcoefficientdiagonaltermleftoutside. pfc_gap_division_bounded_executionproductcoefficientdiagonaltermleftoutside+(q)=(pfc_index_division_bounded_executionproductcoefficientdiagonal)) /\\ (((pfc_left_division_bounded_executionproductcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_bounded_executionproductcoefficientdiagonaltermrightinside. pfa_gap_division_bounded_executionproductcoefficientdiagonaltermrightinside + S (pfc_complement_division_bounded_executionproductcoefficientdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_division_bounded_executionproductcoefficientdiagonaltermrightentry. ff_h_pfp_division_bounded_executionproductcoefficientdiagonaltermrightentry + S (pfc_right_division_bounded_executionproductcoefficientdiagonalterm) = S ((S (pfc_complement_division_bounded_executionproductcoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_bounded_executionproductcoefficientdiagonaltermrightentry. bb = ff_q_pfp_division_bounded_executionproductcoefficientdiagonaltermrightentry * S ((S (pfc_complement_division_bounded_executionproductcoefficientdiagonalterm)) * bc) + (pfc_right_division_bounded_executionproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_division_bounded_executionproductcoefficientdiagonaltermrightoutside. pfc_gap_division_bounded_executionproductcoefficientdiagonaltermrightoutside+(S (d))=(pfc_complement_division_bounded_executionproductcoefficientdiagonalterm)) /\\ (((pfc_right_division_bounded_executionproductcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_division_bounded_executionproductcoefficientdiagonal)=pfc_left_division_bounded_executionproductcoefficientdiagonalterm*pfc_right_division_bounded_executionproductcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_bounded_executionproductcoefficientsum fs_v_pfc_division_bounded_executionproductcoefficientsum. ((((exists fs_h_pfc_division_bounded_executionproductcoefficientsum_body_start. fs_h_pfc_division_bounded_executionproductcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_bounded_executionproductcoefficientsum)) /\\ exists fs_q_pfc_division_bounded_executionproductcoefficientsum_body_start. fs_u_pfc_division_bounded_executionproductcoefficientsum = fs_q_pfc_division_bounded_executionproductcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_division_bounded_executionproductcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_division_bounded_executionproductcoefficientsum_body_terminal. fs_h_pfc_division_bounded_executionproductcoefficientsum_body_terminal + S (pfc_natural_sum_division_bounded_executionproductcoefficient) = S ((S (S (pfc_index_division_bounded_executionproduct))) * fs_v_pfc_division_bounded_executionproductcoefficientsum)) /\\ exists fs_q_pfc_division_bounded_executionproductcoefficientsum_body_terminal. fs_u_pfc_division_bounded_executionproductcoefficientsum = fs_q_pfc_division_bounded_executionproductcoefficientsum_body_terminal * S ((S (S (pfc_index_division_bounded_executionproduct))) * fs_v_pfc_division_bounded_executionproductcoefficientsum) + (pfc_natural_sum_division_bounded_executionproductcoefficient))) /\\ forall fs_i_pfc_division_bounded_executionproductcoefficientsum_body_steps. (exists fs_lt_pfc_division_bounded_executionproductcoefficientsum_body_steps_bound. fs_lt_pfc_division_bounded_executionproductcoefficientsum_body_steps_bound + S fs_i_pfc_division_bounded_executionproductcoefficientsum_body_steps = S (pfc_index_division_bounded_executionproduct)) -> exists fs_a_pfc_division_bounded_executionproductcoefficientsum_body_steps fs_r_pfc_division_bounded_executionproductcoefficientsum_body_steps fs_s_pfc_division_bounded_executionproductcoefficientsum_body_steps. ((((exists fs_h_pfc_division_bounded_executionproductcoefficientsum_body_steps_summand. fs_h_pfc_division_bounded_executionproductcoefficientsum_body_steps_summand + S (fs_a_pfc_division_bounded_executionproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_division_bounded_executionproductcoefficientsum_body_steps)) * pfc_terms_scale_division_bounded_executionproductcoefficient)) /\\ exists fs_q_pfc_division_bounded_executionproductcoefficientsum_body_steps_summand. pfc_terms_code_division_bounded_executionproductcoefficient = fs_q_pfc_division_bounded_executionproductcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_division_bounded_executionproductcoefficientsum_body_steps)) * pfc_terms_scale_division_bounded_executionproductcoefficient) + (fs_a_pfc_division_bounded_executionproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_division_bounded_executionproductcoefficientsum_body_steps_partial. fs_h_pfc_division_bounded_executionproductcoefficientsum_body_steps_partial + S (fs_r_pfc_division_bounded_executionproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_division_bounded_executionproductcoefficientsum_body_steps)) * fs_v_pfc_division_bounded_executionproductcoefficientsum)) /\\ exists fs_q_pfc_division_bounded_executionproductcoefficientsum_body_steps_partial. fs_u_pfc_division_bounded_executionproductcoefficientsum = fs_q_pfc_division_bounded_executionproductcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_division_bounded_executionproductcoefficientsum_body_steps)) * fs_v_pfc_division_bounded_executionproductcoefficientsum) + (fs_r_pfc_division_bounded_executionproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_division_bounded_executionproductcoefficientsum_body_steps_successor. fs_h_pfc_division_bounded_executionproductcoefficientsum_body_steps_successor + S (fs_s_pfc_division_bounded_executionproductcoefficientsum_body_steps) = S ((S (S fs_i_pfc_division_bounded_executionproductcoefficientsum_body_steps)) * fs_v_pfc_division_bounded_executionproductcoefficientsum)) /\\ exists fs_q_pfc_division_bounded_executionproductcoefficientsum_body_steps_successor. fs_u_pfc_division_bounded_executionproductcoefficientsum = fs_q_pfc_division_bounded_executionproductcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_division_bounded_executionproductcoefficientsum_body_steps)) * fs_v_pfc_division_bounded_executionproductcoefficientsum) + (fs_s_pfc_division_bounded_executionproductcoefficientsum_body_steps))) /\\ fs_s_pfc_division_bounded_executionproductcoefficientsum_body_steps = fs_r_pfc_division_bounded_executionproductcoefficientsum_body_steps + fs_a_pfc_division_bounded_executionproductcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_division_bounded_executionproductcoefficientresiduebound. pfa_gap_division_bounded_executionproductcoefficientresiduebound + S (pfc_value_division_bounded_executionproduct) = (p)) /\\ ((exists pfa_offset_left_division_bounded_executionproductcoefficientresiduecongruence pfa_offset_right_division_bounded_executionproductcoefficientresiduecongruence. (pfc_natural_sum_division_bounded_executionproductcoefficient) + (p) * pfa_offset_left_division_bounded_executionproductcoefficientresiduecongruence = (pfc_value_division_bounded_executionproduct) + (p) * pfa_offset_right_division_bounded_executionproductcoefficientresiduecongruence)))))))))))) /\\ (((forall pfs_index_division_bounded_executiondifference. (exists pfa_gap_division_bounded_executiondifferenceindex. pfa_gap_division_bounded_executiondifferenceindex + S (pfs_index_division_bounded_executiondifference) = (L)) -> exists pfs_left_division_bounded_executiondifference pfs_right_division_bounded_executiondifference pfs_result_division_bounded_executiondifference. ((((exists ff_h_pfp_division_bounded_executiondifferenceleft. ff_h_pfp_division_bounded_executiondifferenceleft + S (pfs_left_division_bounded_executiondifference) = S ((S (pfs_index_division_bounded_executiondifference)) * ac)) /\\ exists ff_q_pfp_division_bounded_executiondifferenceleft. ab = ff_q_pfp_division_bounded_executiondifferenceleft * S ((S (pfs_index_division_bounded_executiondifference)) * ac) + (pfs_left_division_bounded_executiondifference))) /\\ (((((exists ff_h_pfp_division_bounded_executiondifferenceright. ff_h_pfp_division_bounded_executiondifferenceright + S (pfs_right_division_bounded_executiondifference) = S ((S (pfs_index_division_bounded_executiondifference)) * pfd_product_scale_division_bounded_execution)) /\\ exists ff_q_pfp_division_bounded_executiondifferenceright. pfd_product_code_division_bounded_execution = ff_q_pfp_division_bounded_executiondifferenceright * S ((S (pfs_index_division_bounded_executiondifference)) * pfd_product_scale_division_bounded_execution) + (pfs_right_division_bounded_executiondifference))) /\\ (((((exists ff_h_pfp_division_bounded_executiondifferenceresult. ff_h_pfp_division_bounded_executiondifferenceresult + S (pfs_result_division_bounded_executiondifference) = S ((S (pfs_index_division_bounded_executiondifference)) * pfd_residual_scale_division_bounded_execution)) /\\ exists ff_q_pfp_division_bounded_executiondifferenceresult. pfd_residual_code_division_bounded_execution = ff_q_pfp_division_bounded_executiondifferenceresult * S ((S (pfs_index_division_bounded_executiondifference)) * pfd_residual_scale_division_bounded_execution) + (pfs_result_division_bounded_executiondifference))) /\\ ((((exists pfa_gap_division_bounded_executiondifferenceoperationleft. pfa_gap_division_bounded_executiondifferenceoperationleft + S (pfs_right_division_bounded_executiondifference) = (p)) /\\ (((exists pfa_gap_division_bounded_executiondifferenceoperationright. pfa_gap_division_bounded_executiondifferenceoperationright + S (pfs_result_division_bounded_executiondifference) = (p)) /\\ ((((exists pfa_gap_division_bounded_executiondifferenceoperationresultbound. pfa_gap_division_bounded_executiondifferenceoperationresultbound + S (pfs_left_division_bounded_executiondifference) = (p)) /\\ ((exists pfa_offset_left_division_bounded_executiondifferenceoperationresultcongruence pfa_offset_right_division_bounded_executiondifferenceoperationresultcongruence. ((pfs_right_division_bounded_executiondifference) + (pfs_result_division_bounded_executiondifference)) + (p) * pfa_offset_left_division_bounded_executiondifferenceoperationresultcongruence = (pfs_left_division_bounded_executiondifference) + (p) * pfa_offset_right_division_bounded_executiondifferenceoperationresultcongruence)))))))))))))))) /\\ (((((L)=(pfd_cut_division_bounded_execution)+(R)) /\\ (((forall fom_index_pfp_division_bounded_executiontriminput. (exists fom_gap_pfp_division_bounded_executiontriminput_index_bound. fom_gap_pfp_division_bounded_executiontriminput_index_bound + S (fom_index_pfp_division_bounded_executiontriminput) = L) -> exists fom_value_pfp_division_bounded_executiontriminput. ((((exists fom_beta_height_pfp_division_bounded_executiontriminput_entry. fom_beta_height_pfp_division_bounded_executiontriminput_entry + S (fom_value_pfp_division_bounded_executiontriminput) = S ((S (fom_index_pfp_division_bounded_executiontriminput)) * pfd_residual_scale_division_bounded_execution)) /\\ exists fom_beta_quotient_pfp_division_bounded_executiontriminput_entry. pfd_residual_code_division_bounded_execution = fom_beta_quotient_pfp_division_bounded_executiontriminput_entry * S ((S (fom_index_pfp_division_bounded_executiontriminput)) * pfd_residual_scale_division_bounded_execution) + (fom_value_pfp_division_bounded_executiontriminput))) /\\ (exists fom_gap_pfp_division_bounded_executiontriminput_value_bound. fom_gap_pfp_division_bounded_executiontriminput_value_bound + S (fom_value_pfp_division_bounded_executiontriminput) = p))) /\\ (((forall pfp_repeat_index_division_bounded_executiontrimremoved. (exists pfa_gap_division_bounded_executiontrimremovedindex. pfa_gap_division_bounded_executiontrimremovedindex + S (pfp_repeat_index_division_bounded_executiontrimremoved) = (pfd_cut_division_bounded_execution)) -> (((exists ff_h_pfp_division_bounded_executiontrimremovedentry. ff_h_pfp_division_bounded_executiontrimremovedentry + S (0) = S ((S (pfp_repeat_index_division_bounded_executiontrimremoved)) * pfd_residual_scale_division_bounded_execution)) /\\ exists ff_q_pfp_division_bounded_executiontrimremovedentry. pfd_residual_code_division_bounded_execution = ff_q_pfp_division_bounded_executiontrimremovedentry * S ((S (pfp_repeat_index_division_bounded_executiontrimremoved)) * pfd_residual_scale_division_bounded_execution) + (0)))) /\\ (((forall pftrim_index_division_bounded_executiontrimsuffix pftrim_value_division_bounded_executiontrimsuffix. (exists pfa_gap_division_bounded_executiontrimsuffixbound. pfa_gap_division_bounded_executiontrimsuffixbound + S (pftrim_index_division_bounded_executiontrimsuffix) = (R)) -> (((exists ff_h_pfp_division_bounded_executiontrimsuffixsource. ff_h_pfp_division_bounded_executiontrimsuffixsource + S (pftrim_value_division_bounded_executiontrimsuffix) = S ((S ((pfd_cut_division_bounded_execution)+pftrim_index_division_bounded_executiontrimsuffix)) * pfd_residual_scale_division_bounded_execution)) /\\ exists ff_q_pfp_division_bounded_executiontrimsuffixsource. pfd_residual_code_division_bounded_execution = ff_q_pfp_division_bounded_executiontrimsuffixsource * S ((S ((pfd_cut_division_bounded_execution)+pftrim_index_division_bounded_executiontrimsuffix)) * pfd_residual_scale_division_bounded_execution) + (pftrim_value_division_bounded_executiontrimsuffix))) -> (((exists ff_h_pfp_division_bounded_executiontrimsuffixoutput. ff_h_pfp_division_bounded_executiontrimsuffixoutput + S (pftrim_value_division_bounded_executiontrimsuffix) = S ((S (pftrim_index_division_bounded_executiontrimsuffix)) * rc)) /\\ exists ff_q_pfp_division_bounded_executiontrimsuffixoutput. rb = ff_q_pfp_division_bounded_executiontrimsuffixoutput * S ((S (pftrim_index_division_bounded_executiontrimsuffix)) * rc) + (pftrim_value_division_bounded_executiontrimsuffix)))) /\\ (((R)=0 \\/ (exists pftrim_leading_division_bounded_executiontrimnormal. ((((exists ff_h_pfp_division_bounded_executiontrimnormalentry. ff_h_pfp_division_bounded_executiontrimnormalentry + S (pftrim_leading_division_bounded_executiontrimnormal) = S ((S (0)) * rc)) /\\ exists ff_q_pfp_division_bounded_executiontrimnormalentry. rb = ff_q_pfp_division_bounded_executiontrimnormalentry * S ((S (0)) * rc) + (pftrim_leading_division_bounded_executiontrimnormal))) /\\ ((~(pftrim_leading_division_bounded_executiontrimnormal=0))))))))))))))))))))))))))))))))) /\\ (((R)=0 \\/ (exists pfd_remainder_degree_division_bounded_degree. (((((R)=S (pfd_remainder_degree_division_bounded_degree)) /\\ (((forall fom_index_pfp_division_bounded_degreerepresentedcoefficients. (exists fom_gap_pfp_division_bounded_degreerepresentedcoefficients_index_bound. fom_gap_pfp_division_bounded_degreerepresentedcoefficients_index_bound + S (fom_index_pfp_division_bounded_degreerepresentedcoefficients) = R) -> exists fom_value_pfp_division_bounded_degreerepresentedcoefficients. 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"intro q", "intro pb", "intro pc", "intro L", "intro hlen", "intro hpositive", "intro hb", "intro hq", "intro hproduct", "split", "specialize prime_field_polynomial_quotient_prefix_bounded (p)", "specialize prime_field_polynomial_quotient_prefix_bounded (k)", "specialize prime_field_polynomial_quotient_prefix_bounded (ab)", "specialize prime_field_polynomial_quotient_prefix_bounded (ac)", "specialize prime_field_polynomial_quotient_prefix_bounded (bb)", "specialize prime_field_polynomial_quotient_prefix_bounded (bc)", "specialize prime_field_polynomial_quotient_prefix_bounded (S d)", "specialize prime_field_polynomial_quotient_prefix_bounded (qb)", "specialize prime_field_polynomial_quotient_prefix_bounded (qc)", "specialize prime_field_polynomial_quotient_prefix_bounded (q)", "apply prime_field_polynomial_quotient_prefix_bounded", "exact hq", "split", "exact hb", "split", "specialize polynomial_quotient_length_product (L)", "specialize polynomial_quotient_length_product (d)", "specialize polynomial_quotient_length_product (q)", "apply polynomial_quotient_length_product", "exact hlen", "exact hpositive", "exact hproduct" ], "script_sha256": "241467404e1842cdec1fe638ed668b103568931ac7d1c73ae24b6cc855f9dd22", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_division_candidate.py", "sha256": "edfc7806caf7a83b9cb0e3e420bd2c3a8679f2d4d9ee6ca9f8eae53faca8d5b2" }, "statement": "forall p k ab ac bb bc d qb qc q pb pc L. (((((q)=0) /\\ ((exists pfc_gap_division_product_chosen_lengthshort. pfc_gap_division_product_chosen_lengthshort+(L)=(d))))) \\/ (((~((q)=0)) /\\ (((q)+(d)=(L)))))) -> ~(q=0) -> (forall fom_index_pfp_division_product_divisor. (exists fom_gap_pfp_division_product_divisor_index_bound. fom_gap_pfp_division_product_divisor_index_bound + S (fom_index_pfp_division_product_divisor) = S d) -> exists fom_value_pfp_division_product_divisor. ((((exists fom_beta_height_pfp_division_product_divisor_entry. fom_beta_height_pfp_division_product_divisor_entry + S (fom_value_pfp_division_product_divisor) = S ((S (fom_index_pfp_division_product_divisor)) * bc)) /\\ exists fom_beta_quotient_pfp_division_product_divisor_entry. bb = fom_beta_quotient_pfp_division_product_divisor_entry * S ((S (fom_index_pfp_division_product_divisor)) * bc) + (fom_value_pfp_division_product_divisor))) /\\ (exists fom_gap_pfp_division_product_divisor_value_bound. fom_gap_pfp_division_product_divisor_value_bound + S (fom_value_pfp_division_product_divisor) = p))) -> (forall pfd_index_division_product_quotient. (exists pfa_gap_division_product_quotientbound. pfa_gap_division_product_quotientbound + S (pfd_index_division_product_quotient) = (q)) -> exists pfd_value_division_product_quotient. ((((exists ff_h_pfp_division_product_quotiententry. ff_h_pfp_division_product_quotiententry + S (pfd_value_division_product_quotient) = S ((S (pfd_index_division_product_quotient)) * qc)) /\\ exists ff_q_pfp_division_product_quotiententry. qb = ff_q_pfp_division_product_quotiententry * S ((S (pfd_index_division_product_quotient)) * qc) + (pfd_value_division_product_quotient))) /\\ ((exists pfd_input_division_product_quotientstep pfd_previous_division_product_quotientstep pfd_difference_division_product_quotientstep. ((((exists ff_h_pfp_division_product_quotientstepinput. ff_h_pfp_division_product_quotientstepinput + S (pfd_input_division_product_quotientstep) = S ((S (pfd_index_division_product_quotient)) * ac)) /\\ exists ff_q_pfp_division_product_quotientstepinput. ab = ff_q_pfp_division_product_quotientstepinput * S ((S (pfd_index_division_product_quotient)) * ac) + (pfd_input_division_product_quotientstep))) /\\ (((exists pfc_terms_code_division_product_quotientstepprevious pfc_terms_scale_division_product_quotientstepprevious pfc_natural_sum_division_product_quotientstepprevious. ((forall pfc_index_division_product_quotientsteppreviousdiagonal. (exists pfa_gap_division_product_quotientsteppreviousdiagonalbound. pfa_gap_division_product_quotientsteppreviousdiagonalbound + S (pfc_index_division_product_quotientsteppreviousdiagonal) = (S (pfd_index_division_product_quotient))) -> exists pfc_value_division_product_quotientsteppreviousdiagonal. ((((exists ff_h_pfp_division_product_quotientsteppreviousdiagonalentry. ff_h_pfp_division_product_quotientsteppreviousdiagonalentry + S (pfc_value_division_product_quotientsteppreviousdiagonal) = S ((S (pfc_index_division_product_quotientsteppreviousdiagonal)) * pfc_terms_scale_division_product_quotientstepprevious)) /\\ exists ff_q_pfp_division_product_quotientsteppreviousdiagonalentry. pfc_terms_code_division_product_quotientstepprevious = ff_q_pfp_division_product_quotientsteppreviousdiagonalentry * S ((S (pfc_index_division_product_quotientsteppreviousdiagonal)) * pfc_terms_scale_division_product_quotientstepprevious) + (pfc_value_division_product_quotientsteppreviousdiagonal))) /\\ ((exists pfc_complement_division_product_quotientsteppreviousdiagonalterm pfc_left_division_product_quotientsteppreviousdiagonalterm pfc_right_division_product_quotientsteppreviousdiagonalterm. (((pfc_index_division_product_quotientsteppreviousdiagonal)+pfc_complement_division_product_quotientsteppreviousdiagonalterm=(pfd_index_division_product_quotient)) /\\ ((((((exists pfa_gap_division_product_quotientsteppreviousdiagonaltermleftinside. pfa_gap_division_product_quotientsteppreviousdiagonaltermleftinside + S (pfc_index_division_product_quotientsteppreviousdiagonal) = (pfd_index_division_product_quotient)) /\\ ((((exists ff_h_pfp_division_product_quotientsteppreviousdiagonaltermleftentry. ff_h_pfp_division_product_quotientsteppreviousdiagonaltermleftentry + S (pfc_left_division_product_quotientsteppreviousdiagonalterm) = S ((S (pfc_index_division_product_quotientsteppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_division_product_quotientsteppreviousdiagonaltermleftentry. qb = ff_q_pfp_division_product_quotientsteppreviousdiagonaltermleftentry * S ((S (pfc_index_division_product_quotientsteppreviousdiagonal)) * qc) + (pfc_left_division_product_quotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_product_quotientsteppreviousdiagonaltermleftoutside. pfc_gap_division_product_quotientsteppreviousdiagonaltermleftoutside+(pfd_index_division_product_quotient)=(pfc_index_division_product_quotientsteppreviousdiagonal)) /\\ (((pfc_left_division_product_quotientsteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_product_quotientsteppreviousdiagonaltermrightinside. pfa_gap_division_product_quotientsteppreviousdiagonaltermrightinside + S (pfc_complement_division_product_quotientsteppreviousdiagonalterm) = (S d)) /\\ ((((exists ff_h_pfp_division_product_quotientsteppreviousdiagonaltermrightentry. ff_h_pfp_division_product_quotientsteppreviousdiagonaltermrightentry + S (pfc_right_division_product_quotientsteppreviousdiagonalterm) = S ((S (pfc_complement_division_product_quotientsteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_product_quotientsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_division_product_quotientsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_division_product_quotientsteppreviousdiagonalterm)) * bc) + (pfc_right_division_product_quotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_product_quotientsteppreviousdiagonaltermrightoutside. pfc_gap_division_product_quotientsteppreviousdiagonaltermrightoutside+(S d)=(pfc_complement_division_product_quotientsteppreviousdiagonalterm)) /\\ (((pfc_right_division_product_quotientsteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_division_product_quotientsteppreviousdiagonal)=pfc_left_division_product_quotientsteppreviousdiagonalterm*pfc_right_division_product_quotientsteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_product_quotientstepprevioussum fs_v_pfc_division_product_quotientstepprevioussum. ((((exists fs_h_pfc_division_product_quotientstepprevioussum_body_start. fs_h_pfc_division_product_quotientstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_product_quotientstepprevioussum)) /\\ exists fs_q_pfc_division_product_quotientstepprevioussum_body_start. fs_u_pfc_division_product_quotientstepprevioussum = fs_q_pfc_division_product_quotientstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_division_product_quotientstepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_division_product_quotientstepprevioussum_body_terminal. fs_h_pfc_division_product_quotientstepprevioussum_body_terminal + S (pfc_natural_sum_division_product_quotientstepprevious) = S ((S (S (pfd_index_division_product_quotient))) * fs_v_pfc_division_product_quotientstepprevioussum)) /\\ exists fs_q_pfc_division_product_quotientstepprevioussum_body_terminal. fs_u_pfc_division_product_quotientstepprevioussum = fs_q_pfc_division_product_quotientstepprevioussum_body_terminal * S ((S (S (pfd_index_division_product_quotient))) * fs_v_pfc_division_product_quotientstepprevioussum) + (pfc_natural_sum_division_product_quotientstepprevious))) /\\ forall fs_i_pfc_division_product_quotientstepprevioussum_body_steps. (exists fs_lt_pfc_division_product_quotientstepprevioussum_body_steps_bound. fs_lt_pfc_division_product_quotientstepprevioussum_body_steps_bound + S fs_i_pfc_division_product_quotientstepprevioussum_body_steps = S (pfd_index_division_product_quotient)) -> exists fs_a_pfc_division_product_quotientstepprevioussum_body_steps fs_r_pfc_division_product_quotientstepprevioussum_body_steps fs_s_pfc_division_product_quotientstepprevioussum_body_steps. ((((exists fs_h_pfc_division_product_quotientstepprevioussum_body_steps_summand. fs_h_pfc_division_product_quotientstepprevioussum_body_steps_summand + S (fs_a_pfc_division_product_quotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_product_quotientstepprevioussum_body_steps)) * pfc_terms_scale_division_product_quotientstepprevious)) /\\ exists fs_q_pfc_division_product_quotientstepprevioussum_body_steps_summand. pfc_terms_code_division_product_quotientstepprevious = fs_q_pfc_division_product_quotientstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_division_product_quotientstepprevioussum_body_steps)) * pfc_terms_scale_division_product_quotientstepprevious) + (fs_a_pfc_division_product_quotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_product_quotientstepprevioussum_body_steps_partial. fs_h_pfc_division_product_quotientstepprevioussum_body_steps_partial + S (fs_r_pfc_division_product_quotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_product_quotientstepprevioussum_body_steps)) * fs_v_pfc_division_product_quotientstepprevioussum)) /\\ exists fs_q_pfc_division_product_quotientstepprevioussum_body_steps_partial. fs_u_pfc_division_product_quotientstepprevioussum = fs_q_pfc_division_product_quotientstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_division_product_quotientstepprevioussum_body_steps)) * fs_v_pfc_division_product_quotientstepprevioussum) + (fs_r_pfc_division_product_quotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_product_quotientstepprevioussum_body_steps_successor. fs_h_pfc_division_product_quotientstepprevioussum_body_steps_successor + S (fs_s_pfc_division_product_quotientstepprevioussum_body_steps) = S ((S (S fs_i_pfc_division_product_quotientstepprevioussum_body_steps)) * fs_v_pfc_division_product_quotientstepprevioussum)) /\\ exists fs_q_pfc_division_product_quotientstepprevioussum_body_steps_successor. fs_u_pfc_division_product_quotientstepprevioussum = fs_q_pfc_division_product_quotientstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_division_product_quotientstepprevioussum_body_steps)) * fs_v_pfc_division_product_quotientstepprevioussum) + (fs_s_pfc_division_product_quotientstepprevioussum_body_steps))) /\\ fs_s_pfc_division_product_quotientstepprevioussum_body_steps = fs_r_pfc_division_product_quotientstepprevioussum_body_steps + fs_a_pfc_division_product_quotientstepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_division_product_quotientsteppreviousresiduebound. pfa_gap_division_product_quotientsteppreviousresiduebound + S (pfd_previous_division_product_quotientstep) = (p)) /\\ ((exists pfa_offset_left_division_product_quotientsteppreviousresiduecongruence pfa_offset_right_division_product_quotientsteppreviousresiduecongruence. (pfc_natural_sum_division_product_quotientstepprevious) + (p) * pfa_offset_left_division_product_quotientsteppreviousresiduecongruence = (pfd_previous_division_product_quotientstep) + (p) * pfa_offset_right_division_product_quotientsteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_division_product_quotientstepsubtractleft. pfa_gap_division_product_quotientstepsubtractleft + S (pfd_previous_division_product_quotientstep) = (p)) /\\ (((exists pfa_gap_division_product_quotientstepsubtractright. pfa_gap_division_product_quotientstepsubtractright + S (pfd_difference_division_product_quotientstep) = (p)) /\\ ((((exists pfa_gap_division_product_quotientstepsubtractresultbound. pfa_gap_division_product_quotientstepsubtractresultbound + S (pfd_input_division_product_quotientstep) = (p)) /\\ ((exists pfa_offset_left_division_product_quotientstepsubtractresultcongruence pfa_offset_right_division_product_quotientstepsubtractresultcongruence. ((pfd_previous_division_product_quotientstep) + (pfd_difference_division_product_quotientstep)) + (p) * pfa_offset_left_division_product_quotientstepsubtractresultcongruence = (pfd_input_division_product_quotientstep) + (p) * pfa_offset_right_division_product_quotientstepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_division_product_quotientstepmultiplyleft. pfa_gap_division_product_quotientstepmultiplyleft + S (k) = (p)) /\\ (((exists pfa_gap_division_product_quotientstepmultiplyright. pfa_gap_division_product_quotientstepmultiplyright + S (pfd_difference_division_product_quotientstep) = (p)) /\\ ((((exists pfa_gap_division_product_quotientstepmultiplyresultbound. pfa_gap_division_product_quotientstepmultiplyresultbound + S (pfd_value_division_product_quotient) = (p)) /\\ ((exists pfa_offset_left_division_product_quotientstepmultiplyresultcongruence pfa_offset_right_division_product_quotientstepmultiplyresultcongruence. ((k) * (pfd_difference_division_product_quotientstep)) + (p) * pfa_offset_left_division_product_quotientstepmultiplyresultcongruence = (pfd_value_division_product_quotient) + (p) * pfa_offset_right_division_product_quotientstepmultiplyresultcongruence))))))))))))))))))) -> (forall pfc_index_division_product_ambient. (exists pfa_gap_division_product_ambientbound. pfa_gap_division_product_ambientbound + S (pfc_index_division_product_ambient) = (L)) -> exists pfc_value_division_product_ambient. ((((exists ff_h_pfp_division_product_ambiententry. ff_h_pfp_division_product_ambiententry + S (pfc_value_division_product_ambient) = S ((S (pfc_index_division_product_ambient)) * pc)) /\\ exists ff_q_pfp_division_product_ambiententry. pb = ff_q_pfp_division_product_ambiententry * S ((S (pfc_index_division_product_ambient)) * pc) + (pfc_value_division_product_ambient))) /\\ ((exists pfc_terms_code_division_product_ambientcoefficient pfc_terms_scale_division_product_ambientcoefficient pfc_natural_sum_division_product_ambientcoefficient. ((forall pfc_index_division_product_ambientcoefficientdiagonal. (exists pfa_gap_division_product_ambientcoefficientdiagonalbound. pfa_gap_division_product_ambientcoefficientdiagonalbound + S (pfc_index_division_product_ambientcoefficientdiagonal) = (S (pfc_index_division_product_ambient))) -> exists pfc_value_division_product_ambientcoefficientdiagonal. ((((exists ff_h_pfp_division_product_ambientcoefficientdiagonalentry. ff_h_pfp_division_product_ambientcoefficientdiagonalentry + S (pfc_value_division_product_ambientcoefficientdiagonal) = S ((S (pfc_index_division_product_ambientcoefficientdiagonal)) * pfc_terms_scale_division_product_ambientcoefficient)) /\\ exists ff_q_pfp_division_product_ambientcoefficientdiagonalentry. pfc_terms_code_division_product_ambientcoefficient = ff_q_pfp_division_product_ambientcoefficientdiagonalentry * S ((S (pfc_index_division_product_ambientcoefficientdiagonal)) * pfc_terms_scale_division_product_ambientcoefficient) + (pfc_value_division_product_ambientcoefficientdiagonal))) /\\ ((exists pfc_complement_division_product_ambientcoefficientdiagonalterm pfc_left_division_product_ambientcoefficientdiagonalterm pfc_right_division_product_ambientcoefficientdiagonalterm. (((pfc_index_division_product_ambientcoefficientdiagonal)+pfc_complement_division_product_ambientcoefficientdiagonalterm=(pfc_index_division_product_ambient)) /\\ ((((((exists pfa_gap_division_product_ambientcoefficientdiagonaltermleftinside. pfa_gap_division_product_ambientcoefficientdiagonaltermleftinside + S (pfc_index_division_product_ambientcoefficientdiagonal) = (q)) /\\ ((((exists ff_h_pfp_division_product_ambientcoefficientdiagonaltermleftentry. ff_h_pfp_division_product_ambientcoefficientdiagonaltermleftentry + S (pfc_left_division_product_ambientcoefficientdiagonalterm) = S ((S (pfc_index_division_product_ambientcoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_division_product_ambientcoefficientdiagonaltermleftentry. qb = ff_q_pfp_division_product_ambientcoefficientdiagonaltermleftentry * S ((S (pfc_index_division_product_ambientcoefficientdiagonal)) * qc) + (pfc_left_division_product_ambientcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_division_product_ambientcoefficientdiagonaltermleftoutside. pfc_gap_division_product_ambientcoefficientdiagonaltermleftoutside+(q)=(pfc_index_division_product_ambientcoefficientdiagonal)) /\\ (((pfc_left_division_product_ambientcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_product_ambientcoefficientdiagonaltermrightinside. pfa_gap_division_product_ambientcoefficientdiagonaltermrightinside + S (pfc_complement_division_product_ambientcoefficientdiagonalterm) = (S d)) /\\ ((((exists ff_h_pfp_division_product_ambientcoefficientdiagonaltermrightentry. ff_h_pfp_division_product_ambientcoefficientdiagonaltermrightentry + S (pfc_right_division_product_ambientcoefficientdiagonalterm) = S ((S (pfc_complement_division_product_ambientcoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_product_ambientcoefficientdiagonaltermrightentry. bb = ff_q_pfp_division_product_ambientcoefficientdiagonaltermrightentry * S ((S (pfc_complement_division_product_ambientcoefficientdiagonalterm)) * bc) + (pfc_right_division_product_ambientcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_division_product_ambientcoefficientdiagonaltermrightoutside. pfc_gap_division_product_ambientcoefficientdiagonaltermrightoutside+(S d)=(pfc_complement_division_product_ambientcoefficientdiagonalterm)) /\\ (((pfc_right_division_product_ambientcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_division_product_ambientcoefficientdiagonal)=pfc_left_division_product_ambientcoefficientdiagonalterm*pfc_right_division_product_ambientcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_product_ambientcoefficientsum fs_v_pfc_division_product_ambientcoefficientsum. ((((exists fs_h_pfc_division_product_ambientcoefficientsum_body_start. fs_h_pfc_division_product_ambientcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_product_ambientcoefficientsum)) /\\ exists fs_q_pfc_division_product_ambientcoefficientsum_body_start. fs_u_pfc_division_product_ambientcoefficientsum = fs_q_pfc_division_product_ambientcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_division_product_ambientcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_division_product_ambientcoefficientsum_body_terminal. fs_h_pfc_division_product_ambientcoefficientsum_body_terminal + S (pfc_natural_sum_division_product_ambientcoefficient) = S ((S (S (pfc_index_division_product_ambient))) * fs_v_pfc_division_product_ambientcoefficientsum)) /\\ exists fs_q_pfc_division_product_ambientcoefficientsum_body_terminal. fs_u_pfc_division_product_ambientcoefficientsum = fs_q_pfc_division_product_ambientcoefficientsum_body_terminal * S ((S (S (pfc_index_division_product_ambient))) * fs_v_pfc_division_product_ambientcoefficientsum) + (pfc_natural_sum_division_product_ambientcoefficient))) /\\ forall fs_i_pfc_division_product_ambientcoefficientsum_body_steps. (exists fs_lt_pfc_division_product_ambientcoefficientsum_body_steps_bound. fs_lt_pfc_division_product_ambientcoefficientsum_body_steps_bound + S fs_i_pfc_division_product_ambientcoefficientsum_body_steps = S (pfc_index_division_product_ambient)) -> exists fs_a_pfc_division_product_ambientcoefficientsum_body_steps fs_r_pfc_division_product_ambientcoefficientsum_body_steps fs_s_pfc_division_product_ambientcoefficientsum_body_steps. ((((exists fs_h_pfc_division_product_ambientcoefficientsum_body_steps_summand. fs_h_pfc_division_product_ambientcoefficientsum_body_steps_summand + S (fs_a_pfc_division_product_ambientcoefficientsum_body_steps) = S ((S (fs_i_pfc_division_product_ambientcoefficientsum_body_steps)) * pfc_terms_scale_division_product_ambientcoefficient)) /\\ exists fs_q_pfc_division_product_ambientcoefficientsum_body_steps_summand. pfc_terms_code_division_product_ambientcoefficient = fs_q_pfc_division_product_ambientcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_division_product_ambientcoefficientsum_body_steps)) * pfc_terms_scale_division_product_ambientcoefficient) + (fs_a_pfc_division_product_ambientcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_division_product_ambientcoefficientsum_body_steps_partial. fs_h_pfc_division_product_ambientcoefficientsum_body_steps_partial + S (fs_r_pfc_division_product_ambientcoefficientsum_body_steps) = S ((S (fs_i_pfc_division_product_ambientcoefficientsum_body_steps)) * fs_v_pfc_division_product_ambientcoefficientsum)) /\\ exists fs_q_pfc_division_product_ambientcoefficientsum_body_steps_partial. fs_u_pfc_division_product_ambientcoefficientsum = fs_q_pfc_division_product_ambientcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_division_product_ambientcoefficientsum_body_steps)) * fs_v_pfc_division_product_ambientcoefficientsum) + (fs_r_pfc_division_product_ambientcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_division_product_ambientcoefficientsum_body_steps_successor. fs_h_pfc_division_product_ambientcoefficientsum_body_steps_successor + S (fs_s_pfc_division_product_ambientcoefficientsum_body_steps) = S ((S (S fs_i_pfc_division_product_ambientcoefficientsum_body_steps)) * fs_v_pfc_division_product_ambientcoefficientsum)) /\\ exists fs_q_pfc_division_product_ambientcoefficientsum_body_steps_successor. fs_u_pfc_division_product_ambientcoefficientsum = fs_q_pfc_division_product_ambientcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_division_product_ambientcoefficientsum_body_steps)) * fs_v_pfc_division_product_ambientcoefficientsum) + (fs_s_pfc_division_product_ambientcoefficientsum_body_steps))) /\\ fs_s_pfc_division_product_ambientcoefficientsum_body_steps = fs_r_pfc_division_product_ambientcoefficientsum_body_steps + fs_a_pfc_division_product_ambientcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_division_product_ambientcoefficientresiduebound. pfa_gap_division_product_ambientcoefficientresiduebound + S (pfc_value_division_product_ambient) = (p)) /\\ ((exists pfa_offset_left_division_product_ambientcoefficientresiduecongruence pfa_offset_right_division_product_ambientcoefficientresiduecongruence. (pfc_natural_sum_division_product_ambientcoefficient) + (p) * pfa_offset_left_division_product_ambientcoefficientresiduecongruence = (pfc_value_division_product_ambient) + (p) * pfa_offset_right_division_product_ambientcoefficientresiduecongruence)))))))))))) -> (((forall fom_index_pfp_division_product_actualleft. (exists fom_gap_pfp_division_product_actualleft_index_bound. fom_gap_pfp_division_product_actualleft_index_bound + S (fom_index_pfp_division_product_actualleft) = q) -> exists fom_value_pfp_division_product_actualleft. ((((exists fom_beta_height_pfp_division_product_actualleft_entry. fom_beta_height_pfp_division_product_actualleft_entry + S (fom_value_pfp_division_product_actualleft) = S ((S (fom_index_pfp_division_product_actualleft)) * qc)) /\\ exists fom_beta_quotient_pfp_division_product_actualleft_entry. qb = fom_beta_quotient_pfp_division_product_actualleft_entry * S ((S (fom_index_pfp_division_product_actualleft)) * qc) + (fom_value_pfp_division_product_actualleft))) /\\ (exists fom_gap_pfp_division_product_actualleft_value_bound. fom_gap_pfp_division_product_actualleft_value_bound + S (fom_value_pfp_division_product_actualleft) = p))) /\\ (((forall fom_index_pfp_division_product_actualright. (exists fom_gap_pfp_division_product_actualright_index_bound. fom_gap_pfp_division_product_actualright_index_bound + S (fom_index_pfp_division_product_actualright) = S d) -> exists fom_value_pfp_division_product_actualright. ((((exists fom_beta_height_pfp_division_product_actualright_entry. fom_beta_height_pfp_division_product_actualright_entry + S (fom_value_pfp_division_product_actualright) = S ((S (fom_index_pfp_division_product_actualright)) * bc)) /\\ exists fom_beta_quotient_pfp_division_product_actualright_entry. bb = fom_beta_quotient_pfp_division_product_actualright_entry * S ((S (fom_index_pfp_division_product_actualright)) * bc) + (fom_value_pfp_division_product_actualright))) /\\ (exists fom_gap_pfp_division_product_actualright_value_bound. fom_gap_pfp_division_product_actualright_value_bound + S (fom_value_pfp_division_product_actualright) = p))) /\\ (((((((q)=0 \\/ (S d)=0) /\\ (((L)=0)))) \\/ (((~((q)=0)) /\\ (((~((S d)=0)) /\\ (((q)+(S d)=S (L)))))))) /\\ ((forall pfc_index_division_product_actualcoefficients. (exists pfa_gap_division_product_actualcoefficientsbound. pfa_gap_division_product_actualcoefficientsbound + S (pfc_index_division_product_actualcoefficients) = (L)) -> exists pfc_value_division_product_actualcoefficients. ((((exists ff_h_pfp_division_product_actualcoefficientsentry. ff_h_pfp_division_product_actualcoefficientsentry + S (pfc_value_division_product_actualcoefficients) = S ((S (pfc_index_division_product_actualcoefficients)) * pc)) /\\ exists ff_q_pfp_division_product_actualcoefficientsentry. pb = ff_q_pfp_division_product_actualcoefficientsentry * S ((S (pfc_index_division_product_actualcoefficients)) * pc) + (pfc_value_division_product_actualcoefficients))) /\\ ((exists pfc_terms_code_division_product_actualcoefficientscoefficient pfc_terms_scale_division_product_actualcoefficientscoefficient pfc_natural_sum_division_product_actualcoefficientscoefficient. ((forall pfc_index_division_product_actualcoefficientscoefficientdiagonal. (exists pfa_gap_division_product_actualcoefficientscoefficientdiagonalbound. pfa_gap_division_product_actualcoefficientscoefficientdiagonalbound + S (pfc_index_division_product_actualcoefficientscoefficientdiagonal) = (S (pfc_index_division_product_actualcoefficients))) -> exists pfc_value_division_product_actualcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_division_product_actualcoefficientscoefficientdiagonalentry. ff_h_pfp_division_product_actualcoefficientscoefficientdiagonalentry + S (pfc_value_division_product_actualcoefficientscoefficientdiagonal) = S ((S (pfc_index_division_product_actualcoefficientscoefficientdiagonal)) * pfc_terms_scale_division_product_actualcoefficientscoefficient)) /\\ exists ff_q_pfp_division_product_actualcoefficientscoefficientdiagonalentry. pfc_terms_code_division_product_actualcoefficientscoefficient = ff_q_pfp_division_product_actualcoefficientscoefficientdiagonalentry * S ((S (pfc_index_division_product_actualcoefficientscoefficientdiagonal)) * pfc_terms_scale_division_product_actualcoefficientscoefficient) + (pfc_value_division_product_actualcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_division_product_actualcoefficientscoefficientdiagonalterm pfc_left_division_product_actualcoefficientscoefficientdiagonalterm pfc_right_division_product_actualcoefficientscoefficientdiagonalterm. (((pfc_index_division_product_actualcoefficientscoefficientdiagonal)+pfc_complement_division_product_actualcoefficientscoefficientdiagonalterm=(pfc_index_division_product_actualcoefficients)) /\\ ((((((exists pfa_gap_division_product_actualcoefficientscoefficientdiagonaltermleftinside. pfa_gap_division_product_actualcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_division_product_actualcoefficientscoefficientdiagonal) = (q)) /\\ ((((exists ff_h_pfp_division_product_actualcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_division_product_actualcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_division_product_actualcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_division_product_actualcoefficientscoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_division_product_actualcoefficientscoefficientdiagonaltermleftentry. qb = ff_q_pfp_division_product_actualcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_division_product_actualcoefficientscoefficientdiagonal)) * qc) + (pfc_left_division_product_actualcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_division_product_actualcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_division_product_actualcoefficientscoefficientdiagonaltermleftoutside+(q)=(pfc_index_division_product_actualcoefficientscoefficientdiagonal)) /\\ (((pfc_left_division_product_actualcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_product_actualcoefficientscoefficientdiagonaltermrightinside. pfa_gap_division_product_actualcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_division_product_actualcoefficientscoefficientdiagonalterm) = (S d)) /\\ ((((exists ff_h_pfp_division_product_actualcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_division_product_actualcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_division_product_actualcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_division_product_actualcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_product_actualcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_division_product_actualcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_division_product_actualcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_division_product_actualcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_division_product_actualcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_division_product_actualcoefficientscoefficientdiagonaltermrightoutside+(S d)=(pfc_complement_division_product_actualcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_division_product_actualcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_division_product_actualcoefficientscoefficientdiagonal)=pfc_left_division_product_actualcoefficientscoefficientdiagonalterm*pfc_right_division_product_actualcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_product_actualcoefficientscoefficientsum fs_v_pfc_division_product_actualcoefficientscoefficientsum. ((((exists fs_h_pfc_division_product_actualcoefficientscoefficientsum_body_start. fs_h_pfc_division_product_actualcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_product_actualcoefficientscoefficientsum)) /\\ exists fs_q_pfc_division_product_actualcoefficientscoefficientsum_body_start. fs_u_pfc_division_product_actualcoefficientscoefficientsum = fs_q_pfc_division_product_actualcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_division_product_actualcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_division_product_actualcoefficientscoefficientsum_body_terminal. fs_h_pfc_division_product_actualcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_division_product_actualcoefficientscoefficient) = S ((S (S (pfc_index_division_product_actualcoefficients))) * fs_v_pfc_division_product_actualcoefficientscoefficientsum)) /\\ exists fs_q_pfc_division_product_actualcoefficientscoefficientsum_body_terminal. fs_u_pfc_division_product_actualcoefficientscoefficientsum = fs_q_pfc_division_product_actualcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_division_product_actualcoefficients))) * fs_v_pfc_division_product_actualcoefficientscoefficientsum) + (pfc_natural_sum_division_product_actualcoefficientscoefficient))) /\\ forall fs_i_pfc_division_product_actualcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_division_product_actualcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_division_product_actualcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_division_product_actualcoefficientscoefficientsum_body_steps = S (pfc_index_division_product_actualcoefficients)) -> exists fs_a_pfc_division_product_actualcoefficientscoefficientsum_body_steps fs_r_pfc_division_product_actualcoefficientscoefficientsum_body_steps fs_s_pfc_division_product_actualcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_division_product_actualcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_division_product_actualcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_division_product_actualcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_division_product_actualcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_division_product_actualcoefficientscoefficient)) /\\ exists fs_q_pfc_division_product_actualcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_division_product_actualcoefficientscoefficient = fs_q_pfc_division_product_actualcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_division_product_actualcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_division_product_actualcoefficientscoefficient) + (fs_a_pfc_division_product_actualcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_division_product_actualcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_division_product_actualcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_division_product_actualcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_division_product_actualcoefficientscoefficientsum_body_steps)) * fs_v_pfc_division_product_actualcoefficientscoefficientsum)) /\\ exists fs_q_pfc_division_product_actualcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_division_product_actualcoefficientscoefficientsum = fs_q_pfc_division_product_actualcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_division_product_actualcoefficientscoefficientsum_body_steps)) * fs_v_pfc_division_product_actualcoefficientscoefficientsum) + (fs_r_pfc_division_product_actualcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_division_product_actualcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_division_product_actualcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_division_product_actualcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_division_product_actualcoefficientscoefficientsum_body_steps)) * fs_v_pfc_division_product_actualcoefficientscoefficientsum)) /\\ exists fs_q_pfc_division_product_actualcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_division_product_actualcoefficientscoefficientsum = fs_q_pfc_division_product_actualcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_division_product_actualcoefficientscoefficientsum_body_steps)) * fs_v_pfc_division_product_actualcoefficientscoefficientsum) + (fs_s_pfc_division_product_actualcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_division_product_actualcoefficientscoefficientsum_body_steps = fs_r_pfc_division_product_actualcoefficientscoefficientsum_body_steps + fs_a_pfc_division_product_actualcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_division_product_actualcoefficientscoefficientresiduebound. pfa_gap_division_product_actualcoefficientscoefficientresiduebound + S (pfc_value_division_product_actualcoefficients) = (p)) /\\ ((exists pfa_offset_left_division_product_actualcoefficientscoefficientresiduecongruence pfa_offset_right_division_product_actualcoefficientscoefficientresiduecongruence. 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"intro hp", "intro h", "intro i", "intro hi", "have hv : exists r. ((((exists ff_h_pfp_division_empty_product_entry. ff_h_pfp_division_empty_product_entry + S (r) = S ((S (i)) * pc)) /\\ exists ff_q_pfp_division_empty_product_entry. pb = ff_q_pfp_division_empty_product_entry * S ((S (i)) * pc) + (r))) /\\ ((exists pfc_terms_code_division_empty_product_coefficient pfc_terms_scale_division_empty_product_coefficient pfc_natural_sum_division_empty_product_coefficient. ((forall pfc_index_division_empty_product_coefficientdiagonal. (exists pfa_gap_division_empty_product_coefficientdiagonalbound. pfa_gap_division_empty_product_coefficientdiagonalbound + S (pfc_index_division_empty_product_coefficientdiagonal) = (S (i))) -> exists pfc_value_division_empty_product_coefficientdiagonal. ((((exists ff_h_pfp_division_empty_product_coefficientdiagonalentry. ff_h_pfp_division_empty_product_coefficientdiagonalentry + S (pfc_value_division_empty_product_coefficientdiagonal) = S ((S (pfc_index_division_empty_product_coefficientdiagonal)) * pfc_terms_scale_division_empty_product_coefficient)) /\\ exists ff_q_pfp_division_empty_product_coefficientdiagonalentry. pfc_terms_code_division_empty_product_coefficient = ff_q_pfp_division_empty_product_coefficientdiagonalentry * S ((S (pfc_index_division_empty_product_coefficientdiagonal)) * pfc_terms_scale_division_empty_product_coefficient) + (pfc_value_division_empty_product_coefficientdiagonal))) /\\ ((exists pfc_complement_division_empty_product_coefficientdiagonalterm pfc_left_division_empty_product_coefficientdiagonalterm pfc_right_division_empty_product_coefficientdiagonalterm. (((pfc_index_division_empty_product_coefficientdiagonal)+pfc_complement_division_empty_product_coefficientdiagonalterm=(i)) /\\ ((((((exists pfa_gap_division_empty_product_coefficientdiagonaltermleftinside. pfa_gap_division_empty_product_coefficientdiagonaltermleftinside + S (pfc_index_division_empty_product_coefficientdiagonal) = (0)) /\\ ((((exists ff_h_pfp_division_empty_product_coefficientdiagonaltermleftentry. ff_h_pfp_division_empty_product_coefficientdiagonaltermleftentry + S (pfc_left_division_empty_product_coefficientdiagonalterm) = S ((S (pfc_index_division_empty_product_coefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_division_empty_product_coefficientdiagonaltermleftentry. qb = ff_q_pfp_division_empty_product_coefficientdiagonaltermleftentry * S ((S (pfc_index_division_empty_product_coefficientdiagonal)) * qc) + (pfc_left_division_empty_product_coefficientdiagonalterm)))))) \\/ (((exists pfc_gap_division_empty_product_coefficientdiagonaltermleftoutside. pfc_gap_division_empty_product_coefficientdiagonaltermleftoutside+(0)=(pfc_index_division_empty_product_coefficientdiagonal)) /\\ (((pfc_left_division_empty_product_coefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_empty_product_coefficientdiagonaltermrightinside. pfa_gap_division_empty_product_coefficientdiagonaltermrightinside + S (pfc_complement_division_empty_product_coefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_division_empty_product_coefficientdiagonaltermrightentry. ff_h_pfp_division_empty_product_coefficientdiagonaltermrightentry + S (pfc_right_division_empty_product_coefficientdiagonalterm) = S ((S (pfc_complement_division_empty_product_coefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_empty_product_coefficientdiagonaltermrightentry. bb = ff_q_pfp_division_empty_product_coefficientdiagonaltermrightentry * S ((S (pfc_complement_division_empty_product_coefficientdiagonalterm)) * bc) + (pfc_right_division_empty_product_coefficientdiagonalterm)))))) \\/ (((exists pfc_gap_division_empty_product_coefficientdiagonaltermrightoutside. pfc_gap_division_empty_product_coefficientdiagonaltermrightoutside+(M)=(pfc_complement_division_empty_product_coefficientdiagonalterm)) /\\ (((pfc_right_division_empty_product_coefficientdiagonalterm)=0))))) /\\ (((pfc_value_division_empty_product_coefficientdiagonal)=pfc_left_division_empty_product_coefficientdiagonalterm*pfc_right_division_empty_product_coefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_empty_product_coefficientsum fs_v_pfc_division_empty_product_coefficientsum. ((((exists fs_h_pfc_division_empty_product_coefficientsum_body_start. fs_h_pfc_division_empty_product_coefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_empty_product_coefficientsum)) /\\ exists fs_q_pfc_division_empty_product_coefficientsum_body_start. fs_u_pfc_division_empty_product_coefficientsum = fs_q_pfc_division_empty_product_coefficientsum_body_start * S ((S (0)) * fs_v_pfc_division_empty_product_coefficientsum) + (0))) /\\ ((((exists fs_h_pfc_division_empty_product_coefficientsum_body_terminal. fs_h_pfc_division_empty_product_coefficientsum_body_terminal + S (pfc_natural_sum_division_empty_product_coefficient) = S ((S (S (i))) * fs_v_pfc_division_empty_product_coefficientsum)) /\\ exists fs_q_pfc_division_empty_product_coefficientsum_body_terminal. fs_u_pfc_division_empty_product_coefficientsum = fs_q_pfc_division_empty_product_coefficientsum_body_terminal * S ((S (S (i))) * fs_v_pfc_division_empty_product_coefficientsum) + (pfc_natural_sum_division_empty_product_coefficient))) /\\ forall fs_i_pfc_division_empty_product_coefficientsum_body_steps. (exists fs_lt_pfc_division_empty_product_coefficientsum_body_steps_bound. fs_lt_pfc_division_empty_product_coefficientsum_body_steps_bound + S fs_i_pfc_division_empty_product_coefficientsum_body_steps = S (i)) -> exists fs_a_pfc_division_empty_product_coefficientsum_body_steps fs_r_pfc_division_empty_product_coefficientsum_body_steps fs_s_pfc_division_empty_product_coefficientsum_body_steps. ((((exists fs_h_pfc_division_empty_product_coefficientsum_body_steps_summand. fs_h_pfc_division_empty_product_coefficientsum_body_steps_summand + S (fs_a_pfc_division_empty_product_coefficientsum_body_steps) = S ((S (fs_i_pfc_division_empty_product_coefficientsum_body_steps)) * pfc_terms_scale_division_empty_product_coefficient)) /\\ exists fs_q_pfc_division_empty_product_coefficientsum_body_steps_summand. pfc_terms_code_division_empty_product_coefficient = fs_q_pfc_division_empty_product_coefficientsum_body_steps_summand * S ((S (fs_i_pfc_division_empty_product_coefficientsum_body_steps)) * pfc_terms_scale_division_empty_product_coefficient) + (fs_a_pfc_division_empty_product_coefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_division_empty_product_coefficientsum_body_steps_partial. fs_h_pfc_division_empty_product_coefficientsum_body_steps_partial + S (fs_r_pfc_division_empty_product_coefficientsum_body_steps) = S ((S (fs_i_pfc_division_empty_product_coefficientsum_body_steps)) * fs_v_pfc_division_empty_product_coefficientsum)) /\\ exists fs_q_pfc_division_empty_product_coefficientsum_body_steps_partial. fs_u_pfc_division_empty_product_coefficientsum = fs_q_pfc_division_empty_product_coefficientsum_body_steps_partial * S ((S (fs_i_pfc_division_empty_product_coefficientsum_body_steps)) * fs_v_pfc_division_empty_product_coefficientsum) + (fs_r_pfc_division_empty_product_coefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_division_empty_product_coefficientsum_body_steps_successor. fs_h_pfc_division_empty_product_coefficientsum_body_steps_successor + S (fs_s_pfc_division_empty_product_coefficientsum_body_steps) = S ((S (S fs_i_pfc_division_empty_product_coefficientsum_body_steps)) * fs_v_pfc_division_empty_product_coefficientsum)) /\\ exists fs_q_pfc_division_empty_product_coefficientsum_body_steps_successor. fs_u_pfc_division_empty_product_coefficientsum = fs_q_pfc_division_empty_product_coefficientsum_body_steps_successor * S ((S (S fs_i_pfc_division_empty_product_coefficientsum_body_steps)) * fs_v_pfc_division_empty_product_coefficientsum) + (fs_s_pfc_division_empty_product_coefficientsum_body_steps))) /\\ fs_s_pfc_division_empty_product_coefficientsum_body_steps = fs_r_pfc_division_empty_product_coefficientsum_body_steps + fs_a_pfc_division_empty_product_coefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_division_empty_product_coefficientresiduebound. pfa_gap_division_empty_product_coefficientresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_division_empty_product_coefficientresiduecongruence pfa_offset_right_division_empty_product_coefficientresiduecongruence. (pfc_natural_sum_division_empty_product_coefficient) + (p) * pfa_offset_left_division_empty_product_coefficientresiduecongruence = (r) + (p) * pfa_offset_right_division_empty_product_coefficientresiduecongruence)))))))))))", "specialize h (i)", "apply h", "exact hi", "cases hv", "cases hv_witness", "have hz : x=0", "specialize prime_field_convolution_coefficient_zero_left (p)", "specialize prime_field_convolution_coefficient_zero_left (qb)", "specialize prime_field_convolution_coefficient_zero_left (qc)", "specialize prime_field_convolution_coefficient_zero_left (0)", "specialize prime_field_convolution_coefficient_zero_left (bb)", "specialize prime_field_convolution_coefficient_zero_left (bc)", "specialize prime_field_convolution_coefficient_zero_left (M)", "specialize prime_field_convolution_coefficient_zero_left (i)", "specialize prime_field_convolution_coefficient_zero_left (x)", "apply prime_field_convolution_coefficient_zero_left", "exact hp", "intro j", "intro hj", "exfalso", "specialize lt_not_le (j)", "specialize lt_not_le (0)", "apply lt_not_le", "exact hj", "specialize zero_le (j)", "apply zero_le", "exact hv_witness_right", "rewrite hz at hv_witness_left", "rewrite hz at hv_witness_left", "exact hv_witness_left" ], "script_sha256": "4d3779a12210cd4faad3d5e6b96949b944a7956b9ab84b7c7fa2d70c85aad77f", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_division_candidate.py", "sha256": "edfc7806caf7a83b9cb0e3e420bd2c3a8679f2d4d9ee6ca9f8eae53faca8d5b2" }, "statement": "forall p qb qc bb bc M pb pc L. ~(p=0) -> (forall pfc_index_division_empty_product_source. (exists pfa_gap_division_empty_product_sourcebound. pfa_gap_division_empty_product_sourcebound + S (pfc_index_division_empty_product_source) = (L)) -> exists pfc_value_division_empty_product_source. ((((exists ff_h_pfp_division_empty_product_sourceentry. ff_h_pfp_division_empty_product_sourceentry + S (pfc_value_division_empty_product_source) = S ((S (pfc_index_division_empty_product_source)) * pc)) /\\ exists ff_q_pfp_division_empty_product_sourceentry. pb = ff_q_pfp_division_empty_product_sourceentry * S ((S (pfc_index_division_empty_product_source)) * pc) + (pfc_value_division_empty_product_source))) /\\ ((exists pfc_terms_code_division_empty_product_sourcecoefficient pfc_terms_scale_division_empty_product_sourcecoefficient pfc_natural_sum_division_empty_product_sourcecoefficient. ((forall pfc_index_division_empty_product_sourcecoefficientdiagonal. (exists pfa_gap_division_empty_product_sourcecoefficientdiagonalbound. pfa_gap_division_empty_product_sourcecoefficientdiagonalbound + S (pfc_index_division_empty_product_sourcecoefficientdiagonal) = (S (pfc_index_division_empty_product_source))) -> exists pfc_value_division_empty_product_sourcecoefficientdiagonal. ((((exists ff_h_pfp_division_empty_product_sourcecoefficientdiagonalentry. ff_h_pfp_division_empty_product_sourcecoefficientdiagonalentry + S (pfc_value_division_empty_product_sourcecoefficientdiagonal) = S ((S (pfc_index_division_empty_product_sourcecoefficientdiagonal)) * pfc_terms_scale_division_empty_product_sourcecoefficient)) /\\ exists ff_q_pfp_division_empty_product_sourcecoefficientdiagonalentry. pfc_terms_code_division_empty_product_sourcecoefficient = ff_q_pfp_division_empty_product_sourcecoefficientdiagonalentry * S ((S (pfc_index_division_empty_product_sourcecoefficientdiagonal)) * pfc_terms_scale_division_empty_product_sourcecoefficient) + (pfc_value_division_empty_product_sourcecoefficientdiagonal))) /\\ ((exists pfc_complement_division_empty_product_sourcecoefficientdiagonalterm pfc_left_division_empty_product_sourcecoefficientdiagonalterm pfc_right_division_empty_product_sourcecoefficientdiagonalterm. (((pfc_index_division_empty_product_sourcecoefficientdiagonal)+pfc_complement_division_empty_product_sourcecoefficientdiagonalterm=(pfc_index_division_empty_product_source)) /\\ ((((((exists pfa_gap_division_empty_product_sourcecoefficientdiagonaltermleftinside. pfa_gap_division_empty_product_sourcecoefficientdiagonaltermleftinside + S (pfc_index_division_empty_product_sourcecoefficientdiagonal) = (0)) /\\ ((((exists ff_h_pfp_division_empty_product_sourcecoefficientdiagonaltermleftentry. ff_h_pfp_division_empty_product_sourcecoefficientdiagonaltermleftentry + S (pfc_left_division_empty_product_sourcecoefficientdiagonalterm) = S ((S (pfc_index_division_empty_product_sourcecoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_division_empty_product_sourcecoefficientdiagonaltermleftentry. qb = ff_q_pfp_division_empty_product_sourcecoefficientdiagonaltermleftentry * S ((S (pfc_index_division_empty_product_sourcecoefficientdiagonal)) * qc) + (pfc_left_division_empty_product_sourcecoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_division_empty_product_sourcecoefficientdiagonaltermleftoutside. pfc_gap_division_empty_product_sourcecoefficientdiagonaltermleftoutside+(0)=(pfc_index_division_empty_product_sourcecoefficientdiagonal)) /\\ (((pfc_left_division_empty_product_sourcecoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_empty_product_sourcecoefficientdiagonaltermrightinside. pfa_gap_division_empty_product_sourcecoefficientdiagonaltermrightinside + S (pfc_complement_division_empty_product_sourcecoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_division_empty_product_sourcecoefficientdiagonaltermrightentry. ff_h_pfp_division_empty_product_sourcecoefficientdiagonaltermrightentry + S (pfc_right_division_empty_product_sourcecoefficientdiagonalterm) = S ((S (pfc_complement_division_empty_product_sourcecoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_empty_product_sourcecoefficientdiagonaltermrightentry. bb = ff_q_pfp_division_empty_product_sourcecoefficientdiagonaltermrightentry * S ((S (pfc_complement_division_empty_product_sourcecoefficientdiagonalterm)) * bc) + (pfc_right_division_empty_product_sourcecoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_division_empty_product_sourcecoefficientdiagonaltermrightoutside. pfc_gap_division_empty_product_sourcecoefficientdiagonaltermrightoutside+(M)=(pfc_complement_division_empty_product_sourcecoefficientdiagonalterm)) /\\ (((pfc_right_division_empty_product_sourcecoefficientdiagonalterm)=0))))) /\\ (((pfc_value_division_empty_product_sourcecoefficientdiagonal)=pfc_left_division_empty_product_sourcecoefficientdiagonalterm*pfc_right_division_empty_product_sourcecoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_empty_product_sourcecoefficientsum fs_v_pfc_division_empty_product_sourcecoefficientsum. ((((exists fs_h_pfc_division_empty_product_sourcecoefficientsum_body_start. fs_h_pfc_division_empty_product_sourcecoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_empty_product_sourcecoefficientsum)) /\\ exists fs_q_pfc_division_empty_product_sourcecoefficientsum_body_start. fs_u_pfc_division_empty_product_sourcecoefficientsum = fs_q_pfc_division_empty_product_sourcecoefficientsum_body_start * S ((S (0)) * fs_v_pfc_division_empty_product_sourcecoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_division_empty_product_sourcecoefficientsum_body_terminal. fs_h_pfc_division_empty_product_sourcecoefficientsum_body_terminal + S (pfc_natural_sum_division_empty_product_sourcecoefficient) = S ((S (S (pfc_index_division_empty_product_source))) * fs_v_pfc_division_empty_product_sourcecoefficientsum)) /\\ exists fs_q_pfc_division_empty_product_sourcecoefficientsum_body_terminal. fs_u_pfc_division_empty_product_sourcecoefficientsum = fs_q_pfc_division_empty_product_sourcecoefficientsum_body_terminal * S ((S (S (pfc_index_division_empty_product_source))) * fs_v_pfc_division_empty_product_sourcecoefficientsum) + (pfc_natural_sum_division_empty_product_sourcecoefficient))) /\\ forall fs_i_pfc_division_empty_product_sourcecoefficientsum_body_steps. (exists fs_lt_pfc_division_empty_product_sourcecoefficientsum_body_steps_bound. fs_lt_pfc_division_empty_product_sourcecoefficientsum_body_steps_bound + S fs_i_pfc_division_empty_product_sourcecoefficientsum_body_steps = S (pfc_index_division_empty_product_source)) -> exists fs_a_pfc_division_empty_product_sourcecoefficientsum_body_steps fs_r_pfc_division_empty_product_sourcecoefficientsum_body_steps fs_s_pfc_division_empty_product_sourcecoefficientsum_body_steps. ((((exists fs_h_pfc_division_empty_product_sourcecoefficientsum_body_steps_summand. fs_h_pfc_division_empty_product_sourcecoefficientsum_body_steps_summand + S (fs_a_pfc_division_empty_product_sourcecoefficientsum_body_steps) = S ((S (fs_i_pfc_division_empty_product_sourcecoefficientsum_body_steps)) * pfc_terms_scale_division_empty_product_sourcecoefficient)) /\\ exists fs_q_pfc_division_empty_product_sourcecoefficientsum_body_steps_summand. pfc_terms_code_division_empty_product_sourcecoefficient = fs_q_pfc_division_empty_product_sourcecoefficientsum_body_steps_summand * S ((S (fs_i_pfc_division_empty_product_sourcecoefficientsum_body_steps)) * pfc_terms_scale_division_empty_product_sourcecoefficient) + (fs_a_pfc_division_empty_product_sourcecoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_division_empty_product_sourcecoefficientsum_body_steps_partial. fs_h_pfc_division_empty_product_sourcecoefficientsum_body_steps_partial + S (fs_r_pfc_division_empty_product_sourcecoefficientsum_body_steps) = S ((S (fs_i_pfc_division_empty_product_sourcecoefficientsum_body_steps)) * fs_v_pfc_division_empty_product_sourcecoefficientsum)) /\\ exists fs_q_pfc_division_empty_product_sourcecoefficientsum_body_steps_partial. fs_u_pfc_division_empty_product_sourcecoefficientsum = fs_q_pfc_division_empty_product_sourcecoefficientsum_body_steps_partial * S ((S (fs_i_pfc_division_empty_product_sourcecoefficientsum_body_steps)) * fs_v_pfc_division_empty_product_sourcecoefficientsum) + (fs_r_pfc_division_empty_product_sourcecoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_division_empty_product_sourcecoefficientsum_body_steps_successor. fs_h_pfc_division_empty_product_sourcecoefficientsum_body_steps_successor + S (fs_s_pfc_division_empty_product_sourcecoefficientsum_body_steps) = S ((S (S fs_i_pfc_division_empty_product_sourcecoefficientsum_body_steps)) * fs_v_pfc_division_empty_product_sourcecoefficientsum)) /\\ exists fs_q_pfc_division_empty_product_sourcecoefficientsum_body_steps_successor. fs_u_pfc_division_empty_product_sourcecoefficientsum = fs_q_pfc_division_empty_product_sourcecoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_division_empty_product_sourcecoefficientsum_body_steps)) * fs_v_pfc_division_empty_product_sourcecoefficientsum) + (fs_s_pfc_division_empty_product_sourcecoefficientsum_body_steps))) /\\ fs_s_pfc_division_empty_product_sourcecoefficientsum_body_steps = fs_r_pfc_division_empty_product_sourcecoefficientsum_body_steps + fs_a_pfc_division_empty_product_sourcecoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_division_empty_product_sourcecoefficientresiduebound. pfa_gap_division_empty_product_sourcecoefficientresiduebound + S (pfc_value_division_empty_product_source) = (p)) /\\ ((exists pfa_offset_left_division_empty_product_sourcecoefficientresiduecongruence pfa_offset_right_division_empty_product_sourcecoefficientresiduecongruence. 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(~((p) = 1) /\\ forall pfa_factor_left_division_identity_prime pfa_factor_right_division_identity_prime. (p) = pfa_factor_left_division_identity_prime * pfa_factor_right_division_identity_prime -> pfa_factor_left_division_identity_prime = 1 \\/ pfa_factor_right_division_identity_prime = 1) -> (((forall fom_index_pfp_division_identity_executioninput. (exists fom_gap_pfp_division_identity_executioninput_index_bound. fom_gap_pfp_division_identity_executioninput_index_bound + S (fom_index_pfp_division_identity_executioninput) = L) -> exists fom_value_pfp_division_identity_executioninput. ((((exists fom_beta_height_pfp_division_identity_executioninput_entry. fom_beta_height_pfp_division_identity_executioninput_entry + S (fom_value_pfp_division_identity_executioninput) = S ((S (fom_index_pfp_division_identity_executioninput)) * ac)) /\\ exists fom_beta_quotient_pfp_division_identity_executioninput_entry. ab = fom_beta_quotient_pfp_division_identity_executioninput_entry * S ((S (fom_index_pfp_division_identity_executioninput)) * ac) + (fom_value_pfp_division_identity_executioninput))) /\\ (exists fom_gap_pfp_division_identity_executioninput_value_bound. fom_gap_pfp_division_identity_executioninput_value_bound + S (fom_value_pfp_division_identity_executioninput) = p))) /\\ (((forall fom_index_pfp_division_identity_executiondivisor. (exists fom_gap_pfp_division_identity_executiondivisor_index_bound. fom_gap_pfp_division_identity_executiondivisor_index_bound + S (fom_index_pfp_division_identity_executiondivisor) = S (d)) -> exists fom_value_pfp_division_identity_executiondivisor. ((((exists fom_beta_height_pfp_division_identity_executiondivisor_entry. fom_beta_height_pfp_division_identity_executiondivisor_entry + S (fom_value_pfp_division_identity_executiondivisor) = S ((S (fom_index_pfp_division_identity_executiondivisor)) * bc)) /\\ exists fom_beta_quotient_pfp_division_identity_executiondivisor_entry. bb = fom_beta_quotient_pfp_division_identity_executiondivisor_entry * S ((S (fom_index_pfp_division_identity_executiondivisor)) * bc) + (fom_value_pfp_division_identity_executiondivisor))) /\\ (exists fom_gap_pfp_division_identity_executiondivisor_value_bound. fom_gap_pfp_division_identity_executiondivisor_value_bound + S (fom_value_pfp_division_identity_executiondivisor) = p))) /\\ (((((((q)=0) /\\ ((exists pfc_gap_division_identity_executionlengthshort. pfc_gap_division_identity_executionlengthshort+(L)=(d))))) \\/ (((~((q)=0)) /\\ (((q)+(d)=(L)))))) /\\ ((exists pfd_head_division_identity_execution pfd_inverse_division_identity_execution pfd_product_code_division_identity_execution pfd_product_scale_division_identity_execution pfd_residual_code_division_identity_execution pfd_residual_scale_division_identity_execution pfd_cut_division_identity_execution. ((((exists ff_h_pfp_division_identity_executionhead. ff_h_pfp_division_identity_executionhead + S (pfd_head_division_identity_execution) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_division_identity_executionhead. bb = ff_q_pfp_division_identity_executionhead * S ((S (0)) * bc) + (pfd_head_division_identity_execution))) /\\ (((((~((pfd_head_division_identity_execution) = 0)) /\\ ((((exists pfa_gap_division_identity_executioninversemultiplicationleft. pfa_gap_division_identity_executioninversemultiplicationleft + S (pfd_head_division_identity_execution) = (p)) /\\ (((exists pfa_gap_division_identity_executioninversemultiplicationright. pfa_gap_division_identity_executioninversemultiplicationright + S (pfd_inverse_division_identity_execution) = (p)) /\\ ((((exists pfa_gap_division_identity_executioninversemultiplicationresultbound. pfa_gap_division_identity_executioninversemultiplicationresultbound + S (1) = (p)) /\\ ((exists pfa_offset_left_division_identity_executioninversemultiplicationresultcongruence pfa_offset_right_division_identity_executioninversemultiplicationresultcongruence. ((pfd_head_division_identity_execution) * (pfd_inverse_division_identity_execution)) + (p) * pfa_offset_left_division_identity_executioninversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_division_identity_executioninversemultiplicationresultcongruence)))))))))))) /\\ (((forall pfd_index_division_identity_executionquotient. (exists pfa_gap_division_identity_executionquotientbound. pfa_gap_division_identity_executionquotientbound + S (pfd_index_division_identity_executionquotient) = (q)) -> exists pfd_value_division_identity_executionquotient. ((((exists ff_h_pfp_division_identity_executionquotiententry. ff_h_pfp_division_identity_executionquotiententry + S (pfd_value_division_identity_executionquotient) = S ((S (pfd_index_division_identity_executionquotient)) * qc)) /\\ exists ff_q_pfp_division_identity_executionquotiententry. qb = ff_q_pfp_division_identity_executionquotiententry * S ((S (pfd_index_division_identity_executionquotient)) * qc) + (pfd_value_division_identity_executionquotient))) /\\ ((exists pfd_input_division_identity_executionquotientstep pfd_previous_division_identity_executionquotientstep pfd_difference_division_identity_executionquotientstep. ((((exists ff_h_pfp_division_identity_executionquotientstepinput. ff_h_pfp_division_identity_executionquotientstepinput + S (pfd_input_division_identity_executionquotientstep) = S ((S (pfd_index_division_identity_executionquotient)) * ac)) /\\ exists ff_q_pfp_division_identity_executionquotientstepinput. ab = ff_q_pfp_division_identity_executionquotientstepinput * S ((S (pfd_index_division_identity_executionquotient)) * ac) + (pfd_input_division_identity_executionquotientstep))) /\\ (((exists pfc_terms_code_division_identity_executionquotientstepprevious pfc_terms_scale_division_identity_executionquotientstepprevious pfc_natural_sum_division_identity_executionquotientstepprevious. ((forall pfc_index_division_identity_executionquotientsteppreviousdiagonal. (exists pfa_gap_division_identity_executionquotientsteppreviousdiagonalbound. pfa_gap_division_identity_executionquotientsteppreviousdiagonalbound + S (pfc_index_division_identity_executionquotientsteppreviousdiagonal) = (S (pfd_index_division_identity_executionquotient))) -> exists pfc_value_division_identity_executionquotientsteppreviousdiagonal. ((((exists ff_h_pfp_division_identity_executionquotientsteppreviousdiagonalentry. ff_h_pfp_division_identity_executionquotientsteppreviousdiagonalentry + S (pfc_value_division_identity_executionquotientsteppreviousdiagonal) = S ((S (pfc_index_division_identity_executionquotientsteppreviousdiagonal)) * pfc_terms_scale_division_identity_executionquotientstepprevious)) /\\ exists ff_q_pfp_division_identity_executionquotientsteppreviousdiagonalentry. pfc_terms_code_division_identity_executionquotientstepprevious = ff_q_pfp_division_identity_executionquotientsteppreviousdiagonalentry * S ((S (pfc_index_division_identity_executionquotientsteppreviousdiagonal)) * pfc_terms_scale_division_identity_executionquotientstepprevious) + (pfc_value_division_identity_executionquotientsteppreviousdiagonal))) /\\ ((exists pfc_complement_division_identity_executionquotientsteppreviousdiagonalterm pfc_left_division_identity_executionquotientsteppreviousdiagonalterm pfc_right_division_identity_executionquotientsteppreviousdiagonalterm. (((pfc_index_division_identity_executionquotientsteppreviousdiagonal)+pfc_complement_division_identity_executionquotientsteppreviousdiagonalterm=(pfd_index_division_identity_executionquotient)) /\\ ((((((exists pfa_gap_division_identity_executionquotientsteppreviousdiagonaltermleftinside. pfa_gap_division_identity_executionquotientsteppreviousdiagonaltermleftinside + S (pfc_index_division_identity_executionquotientsteppreviousdiagonal) = (pfd_index_division_identity_executionquotient)) /\\ ((((exists ff_h_pfp_division_identity_executionquotientsteppreviousdiagonaltermleftentry. ff_h_pfp_division_identity_executionquotientsteppreviousdiagonaltermleftentry + S (pfc_left_division_identity_executionquotientsteppreviousdiagonalterm) = S ((S (pfc_index_division_identity_executionquotientsteppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_division_identity_executionquotientsteppreviousdiagonaltermleftentry. qb = ff_q_pfp_division_identity_executionquotientsteppreviousdiagonaltermleftentry * S ((S (pfc_index_division_identity_executionquotientsteppreviousdiagonal)) * qc) + (pfc_left_division_identity_executionquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_identity_executionquotientsteppreviousdiagonaltermleftoutside. pfc_gap_division_identity_executionquotientsteppreviousdiagonaltermleftoutside+(pfd_index_division_identity_executionquotient)=(pfc_index_division_identity_executionquotientsteppreviousdiagonal)) /\\ (((pfc_left_division_identity_executionquotientsteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_identity_executionquotientsteppreviousdiagonaltermrightinside. pfa_gap_division_identity_executionquotientsteppreviousdiagonaltermrightinside + S (pfc_complement_division_identity_executionquotientsteppreviousdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_division_identity_executionquotientsteppreviousdiagonaltermrightentry. ff_h_pfp_division_identity_executionquotientsteppreviousdiagonaltermrightentry + S (pfc_right_division_identity_executionquotientsteppreviousdiagonalterm) = S ((S (pfc_complement_division_identity_executionquotientsteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_identity_executionquotientsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_division_identity_executionquotientsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_division_identity_executionquotientsteppreviousdiagonalterm)) * bc) + (pfc_right_division_identity_executionquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_division_identity_executionquotientsteppreviousdiagonaltermrightoutside. pfc_gap_division_identity_executionquotientsteppreviousdiagonaltermrightoutside+(S (d))=(pfc_complement_division_identity_executionquotientsteppreviousdiagonalterm)) /\\ (((pfc_right_division_identity_executionquotientsteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_division_identity_executionquotientsteppreviousdiagonal)=pfc_left_division_identity_executionquotientsteppreviousdiagonalterm*pfc_right_division_identity_executionquotientsteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_identity_executionquotientstepprevioussum fs_v_pfc_division_identity_executionquotientstepprevioussum. ((((exists fs_h_pfc_division_identity_executionquotientstepprevioussum_body_start. fs_h_pfc_division_identity_executionquotientstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_identity_executionquotientstepprevioussum)) /\\ exists fs_q_pfc_division_identity_executionquotientstepprevioussum_body_start. fs_u_pfc_division_identity_executionquotientstepprevioussum = fs_q_pfc_division_identity_executionquotientstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_division_identity_executionquotientstepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_division_identity_executionquotientstepprevioussum_body_terminal. fs_h_pfc_division_identity_executionquotientstepprevioussum_body_terminal + S (pfc_natural_sum_division_identity_executionquotientstepprevious) = S ((S (S (pfd_index_division_identity_executionquotient))) * fs_v_pfc_division_identity_executionquotientstepprevioussum)) /\\ exists fs_q_pfc_division_identity_executionquotientstepprevioussum_body_terminal. fs_u_pfc_division_identity_executionquotientstepprevioussum = fs_q_pfc_division_identity_executionquotientstepprevioussum_body_terminal * S ((S (S (pfd_index_division_identity_executionquotient))) * fs_v_pfc_division_identity_executionquotientstepprevioussum) + (pfc_natural_sum_division_identity_executionquotientstepprevious))) /\\ forall fs_i_pfc_division_identity_executionquotientstepprevioussum_body_steps. (exists fs_lt_pfc_division_identity_executionquotientstepprevioussum_body_steps_bound. fs_lt_pfc_division_identity_executionquotientstepprevioussum_body_steps_bound + S fs_i_pfc_division_identity_executionquotientstepprevioussum_body_steps = S (pfd_index_division_identity_executionquotient)) -> exists fs_a_pfc_division_identity_executionquotientstepprevioussum_body_steps fs_r_pfc_division_identity_executionquotientstepprevioussum_body_steps fs_s_pfc_division_identity_executionquotientstepprevioussum_body_steps. ((((exists fs_h_pfc_division_identity_executionquotientstepprevioussum_body_steps_summand. fs_h_pfc_division_identity_executionquotientstepprevioussum_body_steps_summand + S (fs_a_pfc_division_identity_executionquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_identity_executionquotientstepprevioussum_body_steps)) * pfc_terms_scale_division_identity_executionquotientstepprevious)) /\\ exists fs_q_pfc_division_identity_executionquotientstepprevioussum_body_steps_summand. pfc_terms_code_division_identity_executionquotientstepprevious = fs_q_pfc_division_identity_executionquotientstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_division_identity_executionquotientstepprevioussum_body_steps)) * pfc_terms_scale_division_identity_executionquotientstepprevious) + (fs_a_pfc_division_identity_executionquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_identity_executionquotientstepprevioussum_body_steps_partial. fs_h_pfc_division_identity_executionquotientstepprevioussum_body_steps_partial + S (fs_r_pfc_division_identity_executionquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_identity_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_division_identity_executionquotientstepprevioussum)) /\\ exists fs_q_pfc_division_identity_executionquotientstepprevioussum_body_steps_partial. fs_u_pfc_division_identity_executionquotientstepprevioussum = fs_q_pfc_division_identity_executionquotientstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_division_identity_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_division_identity_executionquotientstepprevioussum) + (fs_r_pfc_division_identity_executionquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_division_identity_executionquotientstepprevioussum_body_steps_successor. fs_h_pfc_division_identity_executionquotientstepprevioussum_body_steps_successor + S (fs_s_pfc_division_identity_executionquotientstepprevioussum_body_steps) = S ((S (S fs_i_pfc_division_identity_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_division_identity_executionquotientstepprevioussum)) /\\ exists fs_q_pfc_division_identity_executionquotientstepprevioussum_body_steps_successor. fs_u_pfc_division_identity_executionquotientstepprevioussum = fs_q_pfc_division_identity_executionquotientstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_division_identity_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_division_identity_executionquotientstepprevioussum) + (fs_s_pfc_division_identity_executionquotientstepprevioussum_body_steps))) /\\ fs_s_pfc_division_identity_executionquotientstepprevioussum_body_steps = fs_r_pfc_division_identity_executionquotientstepprevioussum_body_steps + fs_a_pfc_division_identity_executionquotientstepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_division_identity_executionquotientsteppreviousresiduebound. pfa_gap_division_identity_executionquotientsteppreviousresiduebound + S (pfd_previous_division_identity_executionquotientstep) = (p)) /\\ ((exists pfa_offset_left_division_identity_executionquotientsteppreviousresiduecongruence pfa_offset_right_division_identity_executionquotientsteppreviousresiduecongruence. (pfc_natural_sum_division_identity_executionquotientstepprevious) + (p) * pfa_offset_left_division_identity_executionquotientsteppreviousresiduecongruence = (pfd_previous_division_identity_executionquotientstep) + (p) * pfa_offset_right_division_identity_executionquotientsteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_division_identity_executionquotientstepsubtractleft. pfa_gap_division_identity_executionquotientstepsubtractleft + S (pfd_previous_division_identity_executionquotientstep) = (p)) /\\ (((exists pfa_gap_division_identity_executionquotientstepsubtractright. pfa_gap_division_identity_executionquotientstepsubtractright + S (pfd_difference_division_identity_executionquotientstep) = (p)) /\\ ((((exists pfa_gap_division_identity_executionquotientstepsubtractresultbound. pfa_gap_division_identity_executionquotientstepsubtractresultbound + S (pfd_input_division_identity_executionquotientstep) = (p)) /\\ ((exists pfa_offset_left_division_identity_executionquotientstepsubtractresultcongruence pfa_offset_right_division_identity_executionquotientstepsubtractresultcongruence. ((pfd_previous_division_identity_executionquotientstep) + (pfd_difference_division_identity_executionquotientstep)) + (p) * pfa_offset_left_division_identity_executionquotientstepsubtractresultcongruence = (pfd_input_division_identity_executionquotientstep) + (p) * pfa_offset_right_division_identity_executionquotientstepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_division_identity_executionquotientstepmultiplyleft. pfa_gap_division_identity_executionquotientstepmultiplyleft + S (pfd_inverse_division_identity_execution) = (p)) /\\ (((exists pfa_gap_division_identity_executionquotientstepmultiplyright. pfa_gap_division_identity_executionquotientstepmultiplyright + S (pfd_difference_division_identity_executionquotientstep) = (p)) /\\ ((((exists pfa_gap_division_identity_executionquotientstepmultiplyresultbound. pfa_gap_division_identity_executionquotientstepmultiplyresultbound + S (pfd_value_division_identity_executionquotient) = (p)) /\\ ((exists pfa_offset_left_division_identity_executionquotientstepmultiplyresultcongruence pfa_offset_right_division_identity_executionquotientstepmultiplyresultcongruence. ((pfd_inverse_division_identity_execution) * (pfd_difference_division_identity_executionquotientstep)) + (p) * pfa_offset_left_division_identity_executionquotientstepmultiplyresultcongruence = (pfd_value_division_identity_executionquotient) + (p) * pfa_offset_right_division_identity_executionquotientstepmultiplyresultcongruence))))))))))))))))))) /\\ (((forall pfc_index_division_identity_executionproduct. (exists pfa_gap_division_identity_executionproductbound. pfa_gap_division_identity_executionproductbound + S (pfc_index_division_identity_executionproduct) = (L)) -> exists pfc_value_division_identity_executionproduct. ((((exists ff_h_pfp_division_identity_executionproductentry. ff_h_pfp_division_identity_executionproductentry + S (pfc_value_division_identity_executionproduct) = S ((S (pfc_index_division_identity_executionproduct)) * pfd_product_scale_division_identity_execution)) /\\ exists ff_q_pfp_division_identity_executionproductentry. pfd_product_code_division_identity_execution = ff_q_pfp_division_identity_executionproductentry * S ((S (pfc_index_division_identity_executionproduct)) * pfd_product_scale_division_identity_execution) + (pfc_value_division_identity_executionproduct))) /\\ ((exists pfc_terms_code_division_identity_executionproductcoefficient pfc_terms_scale_division_identity_executionproductcoefficient pfc_natural_sum_division_identity_executionproductcoefficient. ((forall pfc_index_division_identity_executionproductcoefficientdiagonal. (exists pfa_gap_division_identity_executionproductcoefficientdiagonalbound. pfa_gap_division_identity_executionproductcoefficientdiagonalbound + S (pfc_index_division_identity_executionproductcoefficientdiagonal) = (S (pfc_index_division_identity_executionproduct))) -> exists pfc_value_division_identity_executionproductcoefficientdiagonal. ((((exists ff_h_pfp_division_identity_executionproductcoefficientdiagonalentry. ff_h_pfp_division_identity_executionproductcoefficientdiagonalentry + S (pfc_value_division_identity_executionproductcoefficientdiagonal) = S ((S (pfc_index_division_identity_executionproductcoefficientdiagonal)) * pfc_terms_scale_division_identity_executionproductcoefficient)) /\\ exists ff_q_pfp_division_identity_executionproductcoefficientdiagonalentry. pfc_terms_code_division_identity_executionproductcoefficient = ff_q_pfp_division_identity_executionproductcoefficientdiagonalentry * S ((S (pfc_index_division_identity_executionproductcoefficientdiagonal)) * pfc_terms_scale_division_identity_executionproductcoefficient) + (pfc_value_division_identity_executionproductcoefficientdiagonal))) /\\ ((exists pfc_complement_division_identity_executionproductcoefficientdiagonalterm pfc_left_division_identity_executionproductcoefficientdiagonalterm pfc_right_division_identity_executionproductcoefficientdiagonalterm. (((pfc_index_division_identity_executionproductcoefficientdiagonal)+pfc_complement_division_identity_executionproductcoefficientdiagonalterm=(pfc_index_division_identity_executionproduct)) /\\ ((((((exists pfa_gap_division_identity_executionproductcoefficientdiagonaltermleftinside. pfa_gap_division_identity_executionproductcoefficientdiagonaltermleftinside + S (pfc_index_division_identity_executionproductcoefficientdiagonal) = (q)) /\\ ((((exists ff_h_pfp_division_identity_executionproductcoefficientdiagonaltermleftentry. ff_h_pfp_division_identity_executionproductcoefficientdiagonaltermleftentry + S (pfc_left_division_identity_executionproductcoefficientdiagonalterm) = S ((S (pfc_index_division_identity_executionproductcoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_division_identity_executionproductcoefficientdiagonaltermleftentry. qb = ff_q_pfp_division_identity_executionproductcoefficientdiagonaltermleftentry * S ((S (pfc_index_division_identity_executionproductcoefficientdiagonal)) * qc) + (pfc_left_division_identity_executionproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_division_identity_executionproductcoefficientdiagonaltermleftoutside. pfc_gap_division_identity_executionproductcoefficientdiagonaltermleftoutside+(q)=(pfc_index_division_identity_executionproductcoefficientdiagonal)) /\\ (((pfc_left_division_identity_executionproductcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_identity_executionproductcoefficientdiagonaltermrightinside. pfa_gap_division_identity_executionproductcoefficientdiagonaltermrightinside + S (pfc_complement_division_identity_executionproductcoefficientdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_division_identity_executionproductcoefficientdiagonaltermrightentry. ff_h_pfp_division_identity_executionproductcoefficientdiagonaltermrightentry + S (pfc_right_division_identity_executionproductcoefficientdiagonalterm) = S ((S (pfc_complement_division_identity_executionproductcoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_identity_executionproductcoefficientdiagonaltermrightentry. bb = ff_q_pfp_division_identity_executionproductcoefficientdiagonaltermrightentry * S ((S (pfc_complement_division_identity_executionproductcoefficientdiagonalterm)) * bc) + (pfc_right_division_identity_executionproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_division_identity_executionproductcoefficientdiagonaltermrightoutside. pfc_gap_division_identity_executionproductcoefficientdiagonaltermrightoutside+(S (d))=(pfc_complement_division_identity_executionproductcoefficientdiagonalterm)) /\\ (((pfc_right_division_identity_executionproductcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_division_identity_executionproductcoefficientdiagonal)=pfc_left_division_identity_executionproductcoefficientdiagonalterm*pfc_right_division_identity_executionproductcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_identity_executionproductcoefficientsum fs_v_pfc_division_identity_executionproductcoefficientsum. ((((exists fs_h_pfc_division_identity_executionproductcoefficientsum_body_start. fs_h_pfc_division_identity_executionproductcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_identity_executionproductcoefficientsum)) /\\ exists fs_q_pfc_division_identity_executionproductcoefficientsum_body_start. fs_u_pfc_division_identity_executionproductcoefficientsum = fs_q_pfc_division_identity_executionproductcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_division_identity_executionproductcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_division_identity_executionproductcoefficientsum_body_terminal. fs_h_pfc_division_identity_executionproductcoefficientsum_body_terminal + S (pfc_natural_sum_division_identity_executionproductcoefficient) = S ((S (S (pfc_index_division_identity_executionproduct))) * fs_v_pfc_division_identity_executionproductcoefficientsum)) /\\ exists fs_q_pfc_division_identity_executionproductcoefficientsum_body_terminal. fs_u_pfc_division_identity_executionproductcoefficientsum = fs_q_pfc_division_identity_executionproductcoefficientsum_body_terminal * S ((S (S (pfc_index_division_identity_executionproduct))) * fs_v_pfc_division_identity_executionproductcoefficientsum) + (pfc_natural_sum_division_identity_executionproductcoefficient))) /\\ forall fs_i_pfc_division_identity_executionproductcoefficientsum_body_steps. (exists fs_lt_pfc_division_identity_executionproductcoefficientsum_body_steps_bound. fs_lt_pfc_division_identity_executionproductcoefficientsum_body_steps_bound + S fs_i_pfc_division_identity_executionproductcoefficientsum_body_steps = S (pfc_index_division_identity_executionproduct)) -> exists fs_a_pfc_division_identity_executionproductcoefficientsum_body_steps fs_r_pfc_division_identity_executionproductcoefficientsum_body_steps fs_s_pfc_division_identity_executionproductcoefficientsum_body_steps. ((((exists fs_h_pfc_division_identity_executionproductcoefficientsum_body_steps_summand. fs_h_pfc_division_identity_executionproductcoefficientsum_body_steps_summand + S (fs_a_pfc_division_identity_executionproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_division_identity_executionproductcoefficientsum_body_steps)) * pfc_terms_scale_division_identity_executionproductcoefficient)) /\\ exists fs_q_pfc_division_identity_executionproductcoefficientsum_body_steps_summand. pfc_terms_code_division_identity_executionproductcoefficient = fs_q_pfc_division_identity_executionproductcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_division_identity_executionproductcoefficientsum_body_steps)) * pfc_terms_scale_division_identity_executionproductcoefficient) + (fs_a_pfc_division_identity_executionproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_division_identity_executionproductcoefficientsum_body_steps_partial. fs_h_pfc_division_identity_executionproductcoefficientsum_body_steps_partial + S (fs_r_pfc_division_identity_executionproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_division_identity_executionproductcoefficientsum_body_steps)) * fs_v_pfc_division_identity_executionproductcoefficientsum)) /\\ exists fs_q_pfc_division_identity_executionproductcoefficientsum_body_steps_partial. fs_u_pfc_division_identity_executionproductcoefficientsum = fs_q_pfc_division_identity_executionproductcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_division_identity_executionproductcoefficientsum_body_steps)) * fs_v_pfc_division_identity_executionproductcoefficientsum) + (fs_r_pfc_division_identity_executionproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_division_identity_executionproductcoefficientsum_body_steps_successor. fs_h_pfc_division_identity_executionproductcoefficientsum_body_steps_successor + S (fs_s_pfc_division_identity_executionproductcoefficientsum_body_steps) = S ((S (S fs_i_pfc_division_identity_executionproductcoefficientsum_body_steps)) * fs_v_pfc_division_identity_executionproductcoefficientsum)) /\\ exists fs_q_pfc_division_identity_executionproductcoefficientsum_body_steps_successor. fs_u_pfc_division_identity_executionproductcoefficientsum = fs_q_pfc_division_identity_executionproductcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_division_identity_executionproductcoefficientsum_body_steps)) * fs_v_pfc_division_identity_executionproductcoefficientsum) + (fs_s_pfc_division_identity_executionproductcoefficientsum_body_steps))) /\\ fs_s_pfc_division_identity_executionproductcoefficientsum_body_steps = fs_r_pfc_division_identity_executionproductcoefficientsum_body_steps + fs_a_pfc_division_identity_executionproductcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_division_identity_executionproductcoefficientresiduebound. pfa_gap_division_identity_executionproductcoefficientresiduebound + S (pfc_value_division_identity_executionproduct) = (p)) /\\ ((exists pfa_offset_left_division_identity_executionproductcoefficientresiduecongruence pfa_offset_right_division_identity_executionproductcoefficientresiduecongruence. (pfc_natural_sum_division_identity_executionproductcoefficient) + (p) * pfa_offset_left_division_identity_executionproductcoefficientresiduecongruence = (pfc_value_division_identity_executionproduct) + (p) * pfa_offset_right_division_identity_executionproductcoefficientresiduecongruence)))))))))))) /\\ (((forall pfs_index_division_identity_executiondifference. (exists pfa_gap_division_identity_executiondifferenceindex. pfa_gap_division_identity_executiondifferenceindex + S (pfs_index_division_identity_executiondifference) = (L)) -> exists pfs_left_division_identity_executiondifference pfs_right_division_identity_executiondifference pfs_result_division_identity_executiondifference. ((((exists ff_h_pfp_division_identity_executiondifferenceleft. ff_h_pfp_division_identity_executiondifferenceleft + S (pfs_left_division_identity_executiondifference) = S ((S (pfs_index_division_identity_executiondifference)) * ac)) /\\ exists ff_q_pfp_division_identity_executiondifferenceleft. ab = ff_q_pfp_division_identity_executiondifferenceleft * S ((S (pfs_index_division_identity_executiondifference)) * ac) + (pfs_left_division_identity_executiondifference))) /\\ (((((exists ff_h_pfp_division_identity_executiondifferenceright. ff_h_pfp_division_identity_executiondifferenceright + S (pfs_right_division_identity_executiondifference) = S ((S (pfs_index_division_identity_executiondifference)) * pfd_product_scale_division_identity_execution)) /\\ exists ff_q_pfp_division_identity_executiondifferenceright. pfd_product_code_division_identity_execution = ff_q_pfp_division_identity_executiondifferenceright * S ((S (pfs_index_division_identity_executiondifference)) * pfd_product_scale_division_identity_execution) + (pfs_right_division_identity_executiondifference))) /\\ (((((exists ff_h_pfp_division_identity_executiondifferenceresult. ff_h_pfp_division_identity_executiondifferenceresult + S (pfs_result_division_identity_executiondifference) = S ((S (pfs_index_division_identity_executiondifference)) * pfd_residual_scale_division_identity_execution)) /\\ exists ff_q_pfp_division_identity_executiondifferenceresult. pfd_residual_code_division_identity_execution = ff_q_pfp_division_identity_executiondifferenceresult * S ((S (pfs_index_division_identity_executiondifference)) * pfd_residual_scale_division_identity_execution) + (pfs_result_division_identity_executiondifference))) /\\ ((((exists pfa_gap_division_identity_executiondifferenceoperationleft. pfa_gap_division_identity_executiondifferenceoperationleft + S (pfs_right_division_identity_executiondifference) = (p)) /\\ (((exists pfa_gap_division_identity_executiondifferenceoperationright. pfa_gap_division_identity_executiondifferenceoperationright + S (pfs_result_division_identity_executiondifference) = (p)) /\\ ((((exists pfa_gap_division_identity_executiondifferenceoperationresultbound. pfa_gap_division_identity_executiondifferenceoperationresultbound + S (pfs_left_division_identity_executiondifference) = (p)) /\\ ((exists pfa_offset_left_division_identity_executiondifferenceoperationresultcongruence pfa_offset_right_division_identity_executiondifferenceoperationresultcongruence. ((pfs_right_division_identity_executiondifference) + (pfs_result_division_identity_executiondifference)) + (p) * pfa_offset_left_division_identity_executiondifferenceoperationresultcongruence = (pfs_left_division_identity_executiondifference) + (p) * pfa_offset_right_division_identity_executiondifferenceoperationresultcongruence)))))))))))))))) /\\ (((((L)=(pfd_cut_division_identity_execution)+(R)) /\\ (((forall fom_index_pfp_division_identity_executiontriminput. (exists fom_gap_pfp_division_identity_executiontriminput_index_bound. fom_gap_pfp_division_identity_executiontriminput_index_bound + S (fom_index_pfp_division_identity_executiontriminput) = L) -> exists fom_value_pfp_division_identity_executiontriminput. ((((exists fom_beta_height_pfp_division_identity_executiontriminput_entry. fom_beta_height_pfp_division_identity_executiontriminput_entry + S (fom_value_pfp_division_identity_executiontriminput) = S ((S (fom_index_pfp_division_identity_executiontriminput)) * pfd_residual_scale_division_identity_execution)) /\\ exists fom_beta_quotient_pfp_division_identity_executiontriminput_entry. pfd_residual_code_division_identity_execution = fom_beta_quotient_pfp_division_identity_executiontriminput_entry * S ((S (fom_index_pfp_division_identity_executiontriminput)) * pfd_residual_scale_division_identity_execution) + (fom_value_pfp_division_identity_executiontriminput))) /\\ (exists fom_gap_pfp_division_identity_executiontriminput_value_bound. fom_gap_pfp_division_identity_executiontriminput_value_bound + S (fom_value_pfp_division_identity_executiontriminput) = p))) /\\ (((forall pfp_repeat_index_division_identity_executiontrimremoved. (exists pfa_gap_division_identity_executiontrimremovedindex. pfa_gap_division_identity_executiontrimremovedindex + S (pfp_repeat_index_division_identity_executiontrimremoved) = (pfd_cut_division_identity_execution)) -> (((exists ff_h_pfp_division_identity_executiontrimremovedentry. ff_h_pfp_division_identity_executiontrimremovedentry + S (0) = S ((S (pfp_repeat_index_division_identity_executiontrimremoved)) * pfd_residual_scale_division_identity_execution)) /\\ exists ff_q_pfp_division_identity_executiontrimremovedentry. pfd_residual_code_division_identity_execution = ff_q_pfp_division_identity_executiontrimremovedentry * S ((S (pfp_repeat_index_division_identity_executiontrimremoved)) * pfd_residual_scale_division_identity_execution) + (0)))) /\\ (((forall pftrim_index_division_identity_executiontrimsuffix pftrim_value_division_identity_executiontrimsuffix. (exists pfa_gap_division_identity_executiontrimsuffixbound. pfa_gap_division_identity_executiontrimsuffixbound + S (pftrim_index_division_identity_executiontrimsuffix) = (R)) -> (((exists ff_h_pfp_division_identity_executiontrimsuffixsource. ff_h_pfp_division_identity_executiontrimsuffixsource + S (pftrim_value_division_identity_executiontrimsuffix) = S ((S ((pfd_cut_division_identity_execution)+pftrim_index_division_identity_executiontrimsuffix)) * pfd_residual_scale_division_identity_execution)) /\\ exists ff_q_pfp_division_identity_executiontrimsuffixsource. pfd_residual_code_division_identity_execution = ff_q_pfp_division_identity_executiontrimsuffixsource * S ((S ((pfd_cut_division_identity_execution)+pftrim_index_division_identity_executiontrimsuffix)) * pfd_residual_scale_division_identity_execution) + (pftrim_value_division_identity_executiontrimsuffix))) -> (((exists ff_h_pfp_division_identity_executiontrimsuffixoutput. ff_h_pfp_division_identity_executiontrimsuffixoutput + S (pftrim_value_division_identity_executiontrimsuffix) = S ((S (pftrim_index_division_identity_executiontrimsuffix)) * rc)) /\\ exists ff_q_pfp_division_identity_executiontrimsuffixoutput. rb = ff_q_pfp_division_identity_executiontrimsuffixoutput * S ((S (pftrim_index_division_identity_executiontrimsuffix)) * rc) + (pftrim_value_division_identity_executiontrimsuffix)))) /\\ (((R)=0 \\/ (exists pftrim_leading_division_identity_executiontrimnormal. ((((exists ff_h_pfp_division_identity_executiontrimnormalentry. ff_h_pfp_division_identity_executiontrimnormalentry + S (pftrim_leading_division_identity_executiontrimnormal) = S ((S (0)) * rc)) /\\ exists ff_q_pfp_division_identity_executiontrimnormalentry. rb = ff_q_pfp_division_identity_executiontrimnormalentry * S ((S (0)) * rc) + (pftrim_leading_division_identity_executiontrimnormal))) /\\ ((~(pftrim_leading_division_identity_executiontrimnormal=0))))))))))))))))))))))))))))))))) -> (exists pfd_identity_pb_division_identity_result pfd_identity_pc_division_identity_result pfd_identity_ub_division_identity_result pfd_identity_uc_division_identity_result pfd_identity_t_division_identity_result. ((((((q)=0) /\\ ((forall pfp_repeat_index_division_identity_resultempty. (exists pfa_gap_division_identity_resultemptyindex. pfa_gap_division_identity_resultemptyindex + S (pfp_repeat_index_division_identity_resultempty) = (L)) -> (((exists ff_h_pfp_division_identity_resultemptyentry. ff_h_pfp_division_identity_resultemptyentry + S (0) = S ((S (pfp_repeat_index_division_identity_resultempty)) * pfd_identity_pc_division_identity_result)) /\\ exists ff_q_pfp_division_identity_resultemptyentry. pfd_identity_pb_division_identity_result = ff_q_pfp_division_identity_resultemptyentry * S ((S (pfp_repeat_index_division_identity_resultempty)) * pfd_identity_pc_division_identity_result) + (0))))))) \\/ (((~((q)=0)) /\\ ((((forall fom_index_pfp_division_identity_resultproductleft. (exists fom_gap_pfp_division_identity_resultproductleft_index_bound. fom_gap_pfp_division_identity_resultproductleft_index_bound + S (fom_index_pfp_division_identity_resultproductleft) = q) -> exists fom_value_pfp_division_identity_resultproductleft. ((((exists fom_beta_height_pfp_division_identity_resultproductleft_entry. fom_beta_height_pfp_division_identity_resultproductleft_entry + S (fom_value_pfp_division_identity_resultproductleft) = S ((S (fom_index_pfp_division_identity_resultproductleft)) * qc)) /\\ exists fom_beta_quotient_pfp_division_identity_resultproductleft_entry. qb = fom_beta_quotient_pfp_division_identity_resultproductleft_entry * S ((S (fom_index_pfp_division_identity_resultproductleft)) * qc) + (fom_value_pfp_division_identity_resultproductleft))) /\\ (exists fom_gap_pfp_division_identity_resultproductleft_value_bound. fom_gap_pfp_division_identity_resultproductleft_value_bound + S (fom_value_pfp_division_identity_resultproductleft) = p))) /\\ (((forall fom_index_pfp_division_identity_resultproductright. (exists fom_gap_pfp_division_identity_resultproductright_index_bound. fom_gap_pfp_division_identity_resultproductright_index_bound + S (fom_index_pfp_division_identity_resultproductright) = S (d)) -> exists fom_value_pfp_division_identity_resultproductright. ((((exists fom_beta_height_pfp_division_identity_resultproductright_entry. fom_beta_height_pfp_division_identity_resultproductright_entry + S (fom_value_pfp_division_identity_resultproductright) = S ((S (fom_index_pfp_division_identity_resultproductright)) * bc)) /\\ exists fom_beta_quotient_pfp_division_identity_resultproductright_entry. bb = fom_beta_quotient_pfp_division_identity_resultproductright_entry * S ((S (fom_index_pfp_division_identity_resultproductright)) * bc) + (fom_value_pfp_division_identity_resultproductright))) /\\ (exists fom_gap_pfp_division_identity_resultproductright_value_bound. fom_gap_pfp_division_identity_resultproductright_value_bound + S (fom_value_pfp_division_identity_resultproductright) = p))) /\\ (((((((q)=0 \\/ (S (d))=0) /\\ (((L)=0)))) \\/ (((~((q)=0)) /\\ (((~((S (d))=0)) /\\ (((q)+(S (d))=S (L)))))))) /\\ ((forall pfc_index_division_identity_resultproductcoefficients. (exists pfa_gap_division_identity_resultproductcoefficientsbound. pfa_gap_division_identity_resultproductcoefficientsbound + S (pfc_index_division_identity_resultproductcoefficients) = (L)) -> exists pfc_value_division_identity_resultproductcoefficients. ((((exists ff_h_pfp_division_identity_resultproductcoefficientsentry. ff_h_pfp_division_identity_resultproductcoefficientsentry + S (pfc_value_division_identity_resultproductcoefficients) = S ((S (pfc_index_division_identity_resultproductcoefficients)) * pfd_identity_pc_division_identity_result)) /\\ exists ff_q_pfp_division_identity_resultproductcoefficientsentry. pfd_identity_pb_division_identity_result = ff_q_pfp_division_identity_resultproductcoefficientsentry * S ((S (pfc_index_division_identity_resultproductcoefficients)) * pfd_identity_pc_division_identity_result) + (pfc_value_division_identity_resultproductcoefficients))) /\\ ((exists pfc_terms_code_division_identity_resultproductcoefficientscoefficient pfc_terms_scale_division_identity_resultproductcoefficientscoefficient pfc_natural_sum_division_identity_resultproductcoefficientscoefficient. ((forall pfc_index_division_identity_resultproductcoefficientscoefficientdiagonal. (exists pfa_gap_division_identity_resultproductcoefficientscoefficientdiagonalbound. pfa_gap_division_identity_resultproductcoefficientscoefficientdiagonalbound + S (pfc_index_division_identity_resultproductcoefficientscoefficientdiagonal) = (S (pfc_index_division_identity_resultproductcoefficients))) -> exists pfc_value_division_identity_resultproductcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_division_identity_resultproductcoefficientscoefficientdiagonalentry. ff_h_pfp_division_identity_resultproductcoefficientscoefficientdiagonalentry + S (pfc_value_division_identity_resultproductcoefficientscoefficientdiagonal) = S ((S (pfc_index_division_identity_resultproductcoefficientscoefficientdiagonal)) * pfc_terms_scale_division_identity_resultproductcoefficientscoefficient)) /\\ exists ff_q_pfp_division_identity_resultproductcoefficientscoefficientdiagonalentry. pfc_terms_code_division_identity_resultproductcoefficientscoefficient = ff_q_pfp_division_identity_resultproductcoefficientscoefficientdiagonalentry * S ((S (pfc_index_division_identity_resultproductcoefficientscoefficientdiagonal)) * pfc_terms_scale_division_identity_resultproductcoefficientscoefficient) + (pfc_value_division_identity_resultproductcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_division_identity_resultproductcoefficientscoefficientdiagonalterm pfc_left_division_identity_resultproductcoefficientscoefficientdiagonalterm pfc_right_division_identity_resultproductcoefficientscoefficientdiagonalterm. (((pfc_index_division_identity_resultproductcoefficientscoefficientdiagonal)+pfc_complement_division_identity_resultproductcoefficientscoefficientdiagonalterm=(pfc_index_division_identity_resultproductcoefficients)) /\\ ((((((exists pfa_gap_division_identity_resultproductcoefficientscoefficientdiagonaltermleftinside. pfa_gap_division_identity_resultproductcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_division_identity_resultproductcoefficientscoefficientdiagonal) = (q)) /\\ ((((exists ff_h_pfp_division_identity_resultproductcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_division_identity_resultproductcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_division_identity_resultproductcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_division_identity_resultproductcoefficientscoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_division_identity_resultproductcoefficientscoefficientdiagonaltermleftentry. qb = ff_q_pfp_division_identity_resultproductcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_division_identity_resultproductcoefficientscoefficientdiagonal)) * qc) + (pfc_left_division_identity_resultproductcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_division_identity_resultproductcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_division_identity_resultproductcoefficientscoefficientdiagonaltermleftoutside+(q)=(pfc_index_division_identity_resultproductcoefficientscoefficientdiagonal)) /\\ (((pfc_left_division_identity_resultproductcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_division_identity_resultproductcoefficientscoefficientdiagonaltermrightinside. pfa_gap_division_identity_resultproductcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_division_identity_resultproductcoefficientscoefficientdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_division_identity_resultproductcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_division_identity_resultproductcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_division_identity_resultproductcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_division_identity_resultproductcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_division_identity_resultproductcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_division_identity_resultproductcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_division_identity_resultproductcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_division_identity_resultproductcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_division_identity_resultproductcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_division_identity_resultproductcoefficientscoefficientdiagonaltermrightoutside+(S (d))=(pfc_complement_division_identity_resultproductcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_division_identity_resultproductcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_division_identity_resultproductcoefficientscoefficientdiagonal)=pfc_left_division_identity_resultproductcoefficientscoefficientdiagonalterm*pfc_right_division_identity_resultproductcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_division_identity_resultproductcoefficientscoefficientsum fs_v_pfc_division_identity_resultproductcoefficientscoefficientsum. ((((exists fs_h_pfc_division_identity_resultproductcoefficientscoefficientsum_body_start. fs_h_pfc_division_identity_resultproductcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_identity_resultproductcoefficientscoefficientsum)) /\\ exists fs_q_pfc_division_identity_resultproductcoefficientscoefficientsum_body_start. fs_u_pfc_division_identity_resultproductcoefficientscoefficientsum = fs_q_pfc_division_identity_resultproductcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_division_identity_resultproductcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_division_identity_resultproductcoefficientscoefficientsum_body_terminal. fs_h_pfc_division_identity_resultproductcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_division_identity_resultproductcoefficientscoefficient) = S ((S (S (pfc_index_division_identity_resultproductcoefficients))) * fs_v_pfc_division_identity_resultproductcoefficientscoefficientsum)) /\\ exists fs_q_pfc_division_identity_resultproductcoefficientscoefficientsum_body_terminal. fs_u_pfc_division_identity_resultproductcoefficientscoefficientsum = fs_q_pfc_division_identity_resultproductcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_division_identity_resultproductcoefficients))) * fs_v_pfc_division_identity_resultproductcoefficientscoefficientsum) + (pfc_natural_sum_division_identity_resultproductcoefficientscoefficient))) /\\ forall fs_i_pfc_division_identity_resultproductcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_division_identity_resultproductcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_division_identity_resultproductcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_division_identity_resultproductcoefficientscoefficientsum_body_steps = S (pfc_index_division_identity_resultproductcoefficients)) -> exists fs_a_pfc_division_identity_resultproductcoefficientscoefficientsum_body_steps fs_r_pfc_division_identity_resultproductcoefficientscoefficientsum_body_steps fs_s_pfc_division_identity_resultproductcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_division_identity_resultproductcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_division_identity_resultproductcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_division_identity_resultproductcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_division_identity_resultproductcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_division_identity_resultproductcoefficientscoefficient)) /\\ exists fs_q_pfc_division_identity_resultproductcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_division_identity_resultproductcoefficientscoefficient = fs_q_pfc_division_identity_resultproductcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_division_identity_resultproductcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_division_identity_resultproductcoefficientscoefficient) + (fs_a_pfc_division_identity_resultproductcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_division_identity_resultproductcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_division_identity_resultproductcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_division_identity_resultproductcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_division_identity_resultproductcoefficientscoefficientsum_body_steps)) * fs_v_pfc_division_identity_resultproductcoefficientscoefficientsum)) /\\ exists fs_q_pfc_division_identity_resultproductcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_division_identity_resultproductcoefficientscoefficientsum = fs_q_pfc_division_identity_resultproductcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_division_identity_resultproductcoefficientscoefficientsum_body_steps)) * fs_v_pfc_division_identity_resultproductcoefficientscoefficientsum) + (fs_r_pfc_division_identity_resultproductcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_division_identity_resultproductcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_division_identity_resultproductcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_division_identity_resultproductcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_division_identity_resultproductcoefficientscoefficientsum_body_steps)) * fs_v_pfc_division_identity_resultproductcoefficientscoefficientsum)) /\\ exists fs_q_pfc_division_identity_resultproductcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_division_identity_resultproductcoefficientscoefficientsum = fs_q_pfc_division_identity_resultproductcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_division_identity_resultproductcoefficientscoefficientsum_body_steps)) * fs_v_pfc_division_identity_resultproductcoefficientscoefficientsum) + (fs_s_pfc_division_identity_resultproductcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_division_identity_resultproductcoefficientscoefficientsum_body_steps = fs_r_pfc_division_identity_resultproductcoefficientscoefficientsum_body_steps + fs_a_pfc_division_identity_resultproductcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_division_identity_resultproductcoefficientscoefficientresiduebound. pfa_gap_division_identity_resultproductcoefficientscoefficientresiduebound + S (pfc_value_division_identity_resultproductcoefficients) = (p)) /\\ ((exists pfa_offset_left_division_identity_resultproductcoefficientscoefficientresiduecongruence pfa_offset_right_division_identity_resultproductcoefficientscoefficientresiduecongruence. (pfc_natural_sum_division_identity_resultproductcoefficientscoefficient) + (p) * pfa_offset_left_division_identity_resultproductcoefficientscoefficientresiduecongruence = (pfc_value_division_identity_resultproductcoefficients) + (p) * pfa_offset_right_division_identity_resultproductcoefficientscoefficientresiduecongruence))))))))))))))))))))))) /\\ (((forall pfp_index_division_identity_resultaddition. (exists pfa_gap_division_identity_resultadditionindex. pfa_gap_division_identity_resultadditionindex + S (pfp_index_division_identity_resultaddition) = (L)) -> exists pfp_left_division_identity_resultaddition pfp_right_division_identity_resultaddition pfp_value_division_identity_resultaddition. ((((exists ff_h_pfp_division_identity_resultadditionleft. ff_h_pfp_division_identity_resultadditionleft + S (pfp_left_division_identity_resultaddition) = S ((S (pfp_index_division_identity_resultaddition)) * pfd_identity_pc_division_identity_result)) /\\ exists ff_q_pfp_division_identity_resultadditionleft. pfd_identity_pb_division_identity_result = ff_q_pfp_division_identity_resultadditionleft * S ((S (pfp_index_division_identity_resultaddition)) * pfd_identity_pc_division_identity_result) + (pfp_left_division_identity_resultaddition))) /\\ (((((exists ff_h_pfp_division_identity_resultadditionright. ff_h_pfp_division_identity_resultadditionright + S (pfp_right_division_identity_resultaddition) = S ((S (pfp_index_division_identity_resultaddition)) * pfd_identity_uc_division_identity_result)) /\\ exists ff_q_pfp_division_identity_resultadditionright. pfd_identity_ub_division_identity_result = ff_q_pfp_division_identity_resultadditionright * S ((S (pfp_index_division_identity_resultaddition)) * pfd_identity_uc_division_identity_result) + (pfp_right_division_identity_resultaddition))) /\\ (((((exists ff_h_pfp_division_identity_resultadditiontarget. ff_h_pfp_division_identity_resultadditiontarget + S (pfp_value_division_identity_resultaddition) = S ((S (pfp_index_division_identity_resultaddition)) * ac)) /\\ exists ff_q_pfp_division_identity_resultadditiontarget. ab = ff_q_pfp_division_identity_resultadditiontarget * S ((S (pfp_index_division_identity_resultaddition)) * ac) + (pfp_value_division_identity_resultaddition))) /\\ ((((exists pfa_gap_division_identity_resultadditionoperationleft. pfa_gap_division_identity_resultadditionoperationleft + S (pfp_left_division_identity_resultaddition) = (p)) /\\ (((exists pfa_gap_division_identity_resultadditionoperationright. pfa_gap_division_identity_resultadditionoperationright + S (pfp_right_division_identity_resultaddition) = (p)) /\\ ((((exists pfa_gap_division_identity_resultadditionoperationresultbound. pfa_gap_division_identity_resultadditionoperationresultbound + S (pfp_value_division_identity_resultaddition) = (p)) /\\ ((exists pfa_offset_left_division_identity_resultadditionoperationresultcongruence pfa_offset_right_division_identity_resultadditionoperationresultcongruence. ((pfp_left_division_identity_resultaddition) + (pfp_right_division_identity_resultaddition)) + (p) * pfa_offset_left_division_identity_resultadditionoperationresultcongruence = (pfp_value_division_identity_resultaddition) + (p) * pfa_offset_right_division_identity_resultadditionoperationresultcongruence)))))))))))))))) /\\ (((((L)=(pfd_identity_t_division_identity_result)+(R)) /\\ (((forall fom_index_pfp_division_identity_resultremainderinput. (exists fom_gap_pfp_division_identity_resultremainderinput_index_bound. fom_gap_pfp_division_identity_resultremainderinput_index_bound + S (fom_index_pfp_division_identity_resultremainderinput) = L) -> exists fom_value_pfp_division_identity_resultremainderinput. ((((exists fom_beta_height_pfp_division_identity_resultremainderinput_entry. fom_beta_height_pfp_division_identity_resultremainderinput_entry + S (fom_value_pfp_division_identity_resultremainderinput) = S ((S (fom_index_pfp_division_identity_resultremainderinput)) * pfd_identity_uc_division_identity_result)) /\\ exists fom_beta_quotient_pfp_division_identity_resultremainderinput_entry. pfd_identity_ub_division_identity_result = fom_beta_quotient_pfp_division_identity_resultremainderinput_entry * S ((S (fom_index_pfp_division_identity_resultremainderinput)) * pfd_identity_uc_division_identity_result) + (fom_value_pfp_division_identity_resultremainderinput))) /\\ (exists fom_gap_pfp_division_identity_resultremainderinput_value_bound. fom_gap_pfp_division_identity_resultremainderinput_value_bound + S (fom_value_pfp_division_identity_resultremainderinput) = p))) /\\ (((forall pfp_repeat_index_division_identity_resultremainderremoved. (exists pfa_gap_division_identity_resultremainderremovedindex. pfa_gap_division_identity_resultremainderremovedindex + S (pfp_repeat_index_division_identity_resultremainderremoved) = (pfd_identity_t_division_identity_result)) -> (((exists ff_h_pfp_division_identity_resultremainderremovedentry. ff_h_pfp_division_identity_resultremainderremovedentry + S (0) = S ((S (pfp_repeat_index_division_identity_resultremainderremoved)) * pfd_identity_uc_division_identity_result)) /\\ exists ff_q_pfp_division_identity_resultremainderremovedentry. pfd_identity_ub_division_identity_result = ff_q_pfp_division_identity_resultremainderremovedentry * S ((S (pfp_repeat_index_division_identity_resultremainderremoved)) * pfd_identity_uc_division_identity_result) + (0)))) /\\ (((forall pftrim_index_division_identity_resultremaindersuffix pftrim_value_division_identity_resultremaindersuffix. (exists pfa_gap_division_identity_resultremaindersuffixbound. pfa_gap_division_identity_resultremaindersuffixbound + S (pftrim_index_division_identity_resultremaindersuffix) = (R)) -> (((exists ff_h_pfp_division_identity_resultremaindersuffixsource. ff_h_pfp_division_identity_resultremaindersuffixsource + S (pftrim_value_division_identity_resultremaindersuffix) = S ((S ((pfd_identity_t_division_identity_result)+pftrim_index_division_identity_resultremaindersuffix)) * pfd_identity_uc_division_identity_result)) /\\ exists ff_q_pfp_division_identity_resultremaindersuffixsource. pfd_identity_ub_division_identity_result = ff_q_pfp_division_identity_resultremaindersuffixsource * S ((S ((pfd_identity_t_division_identity_result)+pftrim_index_division_identity_resultremaindersuffix)) * pfd_identity_uc_division_identity_result) + (pftrim_value_division_identity_resultremaindersuffix))) -> (((exists ff_h_pfp_division_identity_resultremaindersuffixoutput. ff_h_pfp_division_identity_resultremaindersuffixoutput + S (pftrim_value_division_identity_resultremaindersuffix) = S ((S (pftrim_index_division_identity_resultremaindersuffix)) * rc)) /\\ exists ff_q_pfp_division_identity_resultremaindersuffixoutput. rb = ff_q_pfp_division_identity_resultremaindersuffixoutput * S ((S (pftrim_index_division_identity_resultremaindersuffix)) * rc) + (pftrim_value_division_identity_resultremaindersuffix)))) /\\ (((R)=0 \\/ (exists pftrim_leading_division_identity_resultremaindernormal. ((((exists ff_h_pfp_division_identity_resultremaindernormalentry. ff_h_pfp_division_identity_resultremaindernormalentry + S (pftrim_leading_division_identity_resultremaindernormal) = S ((S (0)) * rc)) /\\ exists ff_q_pfp_division_identity_resultremaindernormalentry. rb = ff_q_pfp_division_identity_resultremaindernormalentry * S ((S (0)) * rc) + (pftrim_leading_division_identity_resultremaindernormal))) /\\ ((~(pftrim_leading_division_identity_resultremaindernormal=0))))))))))))))))))))", "statement_sha256": "8cc8fa1862d8912600d53092565c286758eb9796e6d8cc047df55dc0245ab195", "summary": "Derive the actual coefficient identity A=P+U, where P is the proper product Q*B (or padded empty product), and the actual trim makes U precisely a leading-zero representation of the normalized remainder.", "summary_sha256": "6270586f00deb7173d15c6aade1199a6418fc84259f82fbc1726d2f89b39e3f0" }, { "alpha_v33_frontier_enrollment": { "body_receipt_sha256": 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"2f18bde4f9b9419c420316fd232690ab7dd79b6702b558bac20986308870f7d5", "membership": "alpha_only", "name": "beta_sum_pointwise_mod_add", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro ab", "intro ac", "intro bb", "intro bc", "intro cb", "intro cc", "induction L", "intro A", "intro B", "intro C", "intro ha", "intro hb", "intro hc", "intro hpw", "have hA : A=0", "specialize beta_sum_zero (ab)", "specialize beta_sum_zero (ac)", "specialize beta_sum_zero (A)", "apply beta_sum_zero", "exact ha", "have hB : B=0", "specialize beta_sum_zero (bb)", "specialize beta_sum_zero (bc)", "specialize beta_sum_zero (B)", "apply beta_sum_zero", "exact hb", "have hC : C=0", "specialize beta_sum_zero (cb)", "specialize beta_sum_zero (cc)", "specialize beta_sum_zero (C)", "apply beta_sum_zero", "exact hc", "rewrite hA", "rewrite hB", "rewrite hC", "exists 0", "exists 0", "simp", "intro A", "intro B", "intro C", "intro ha", "intro hb", "intro hc", "intro hpw", "have hda : exists a r. ((((exists ff_h_pfp_hda_entry. ff_h_pfp_hda_entry + S (a) = S ((S (L)) * ac)) /\\ exists ff_q_pfp_hda_entry. ab = ff_q_pfp_hda_entry * S ((S (L)) * ac) + (a))) /\\ (((exists fs_u_pfc_hda_prefix fs_v_pfc_hda_prefix. ((((exists fs_h_pfc_hda_prefix_body_start. fs_h_pfc_hda_prefix_body_start + S (0) = S ((S (0)) * fs_v_pfc_hda_prefix)) /\\ exists fs_q_pfc_hda_prefix_body_start. fs_u_pfc_hda_prefix = fs_q_pfc_hda_prefix_body_start * S ((S (0)) * fs_v_pfc_hda_prefix) + (0))) /\\ ((((exists fs_h_pfc_hda_prefix_body_terminal. fs_h_pfc_hda_prefix_body_terminal + S (r) = S ((S (L)) * fs_v_pfc_hda_prefix)) /\\ exists fs_q_pfc_hda_prefix_body_terminal. fs_u_pfc_hda_prefix = fs_q_pfc_hda_prefix_body_terminal * S ((S (L)) * fs_v_pfc_hda_prefix) + (r))) /\\ forall fs_i_pfc_hda_prefix_body_steps. (exists fs_lt_pfc_hda_prefix_body_steps_bound. fs_lt_pfc_hda_prefix_body_steps_bound + S fs_i_pfc_hda_prefix_body_steps = L) -> exists fs_a_pfc_hda_prefix_body_steps fs_r_pfc_hda_prefix_body_steps fs_s_pfc_hda_prefix_body_steps. ((((exists fs_h_pfc_hda_prefix_body_steps_summand. fs_h_pfc_hda_prefix_body_steps_summand + S (fs_a_pfc_hda_prefix_body_steps) = S ((S (fs_i_pfc_hda_prefix_body_steps)) * ac)) /\\ exists fs_q_pfc_hda_prefix_body_steps_summand. ab = fs_q_pfc_hda_prefix_body_steps_summand * S ((S (fs_i_pfc_hda_prefix_body_steps)) * ac) + (fs_a_pfc_hda_prefix_body_steps))) /\\ ((((exists fs_h_pfc_hda_prefix_body_steps_partial. fs_h_pfc_hda_prefix_body_steps_partial + S (fs_r_pfc_hda_prefix_body_steps) = S ((S (fs_i_pfc_hda_prefix_body_steps)) * fs_v_pfc_hda_prefix)) /\\ exists fs_q_pfc_hda_prefix_body_steps_partial. fs_u_pfc_hda_prefix = fs_q_pfc_hda_prefix_body_steps_partial * S ((S (fs_i_pfc_hda_prefix_body_steps)) * fs_v_pfc_hda_prefix) + (fs_r_pfc_hda_prefix_body_steps))) /\\ ((((exists fs_h_pfc_hda_prefix_body_steps_successor. fs_h_pfc_hda_prefix_body_steps_successor + S (fs_s_pfc_hda_prefix_body_steps) = S ((S (S fs_i_pfc_hda_prefix_body_steps)) * fs_v_pfc_hda_prefix)) /\\ exists fs_q_pfc_hda_prefix_body_steps_successor. fs_u_pfc_hda_prefix = fs_q_pfc_hda_prefix_body_steps_successor * S ((S (S fs_i_pfc_hda_prefix_body_steps)) * fs_v_pfc_hda_prefix) + (fs_s_pfc_hda_prefix_body_steps))) /\\ fs_s_pfc_hda_prefix_body_steps = fs_r_pfc_hda_prefix_body_steps + fs_a_pfc_hda_prefix_body_steps)))))) /\\ ((A=r+a)))))", "specialize beta_sum_succ_decompose (ab)", "specialize beta_sum_succ_decompose (ac)", "specialize beta_sum_succ_decompose (L)", "specialize beta_sum_succ_decompose (A)", "apply beta_sum_succ_decompose", "exact ha", "cases hda", "cases hda_witness", "cases hda_witness_witness", "cases hda_witness_witness_right", "have hdb : exists a r. ((((exists ff_h_pfp_hdb_entry. ff_h_pfp_hdb_entry + S (a) = S ((S (L)) * bc)) /\\ exists ff_q_pfp_hdb_entry. bb = ff_q_pfp_hdb_entry * S ((S (L)) * bc) + (a))) /\\ (((exists fs_u_pfc_hdb_prefix fs_v_pfc_hdb_prefix. ((((exists fs_h_pfc_hdb_prefix_body_start. fs_h_pfc_hdb_prefix_body_start + S (0) = S ((S (0)) * fs_v_pfc_hdb_prefix)) /\\ exists fs_q_pfc_hdb_prefix_body_start. fs_u_pfc_hdb_prefix = fs_q_pfc_hdb_prefix_body_start * S ((S (0)) * fs_v_pfc_hdb_prefix) + (0))) /\\ ((((exists fs_h_pfc_hdb_prefix_body_terminal. fs_h_pfc_hdb_prefix_body_terminal + S (r) = S ((S (L)) * fs_v_pfc_hdb_prefix)) /\\ exists fs_q_pfc_hdb_prefix_body_terminal. fs_u_pfc_hdb_prefix = fs_q_pfc_hdb_prefix_body_terminal * S ((S (L)) * fs_v_pfc_hdb_prefix) + (r))) /\\ forall fs_i_pfc_hdb_prefix_body_steps. (exists fs_lt_pfc_hdb_prefix_body_steps_bound. fs_lt_pfc_hdb_prefix_body_steps_bound + S fs_i_pfc_hdb_prefix_body_steps = L) -> exists fs_a_pfc_hdb_prefix_body_steps fs_r_pfc_hdb_prefix_body_steps fs_s_pfc_hdb_prefix_body_steps. ((((exists fs_h_pfc_hdb_prefix_body_steps_summand. fs_h_pfc_hdb_prefix_body_steps_summand + S (fs_a_pfc_hdb_prefix_body_steps) = S ((S (fs_i_pfc_hdb_prefix_body_steps)) * bc)) /\\ exists fs_q_pfc_hdb_prefix_body_steps_summand. bb = fs_q_pfc_hdb_prefix_body_steps_summand * S ((S (fs_i_pfc_hdb_prefix_body_steps)) * bc) + (fs_a_pfc_hdb_prefix_body_steps))) /\\ ((((exists fs_h_pfc_hdb_prefix_body_steps_partial. fs_h_pfc_hdb_prefix_body_steps_partial + S (fs_r_pfc_hdb_prefix_body_steps) = S ((S (fs_i_pfc_hdb_prefix_body_steps)) * fs_v_pfc_hdb_prefix)) /\\ exists fs_q_pfc_hdb_prefix_body_steps_partial. fs_u_pfc_hdb_prefix = fs_q_pfc_hdb_prefix_body_steps_partial * S ((S (fs_i_pfc_hdb_prefix_body_steps)) * fs_v_pfc_hdb_prefix) + (fs_r_pfc_hdb_prefix_body_steps))) /\\ ((((exists fs_h_pfc_hdb_prefix_body_steps_successor. fs_h_pfc_hdb_prefix_body_steps_successor + S (fs_s_pfc_hdb_prefix_body_steps) = S ((S (S fs_i_pfc_hdb_prefix_body_steps)) * fs_v_pfc_hdb_prefix)) /\\ exists fs_q_pfc_hdb_prefix_body_steps_successor. fs_u_pfc_hdb_prefix = fs_q_pfc_hdb_prefix_body_steps_successor * S ((S (S fs_i_pfc_hdb_prefix_body_steps)) * fs_v_pfc_hdb_prefix) + (fs_s_pfc_hdb_prefix_body_steps))) /\\ fs_s_pfc_hdb_prefix_body_steps = fs_r_pfc_hdb_prefix_body_steps + fs_a_pfc_hdb_prefix_body_steps)))))) /\\ ((B=r+a)))))", "specialize beta_sum_succ_decompose (bb)", "specialize beta_sum_succ_decompose (bc)", "specialize beta_sum_succ_decompose (L)", "specialize beta_sum_succ_decompose (B)", "apply beta_sum_succ_decompose", "exact hb", "cases hdb", "cases hdb_witness", "cases hdb_witness_witness", "cases hdb_witness_witness_right", "have hdc : exists a r. ((((exists ff_h_pfp_hdc_entry. ff_h_pfp_hdc_entry + S (a) = S ((S (L)) * cc)) /\\ exists ff_q_pfp_hdc_entry. cb = ff_q_pfp_hdc_entry * S ((S (L)) * cc) + (a))) /\\ (((exists fs_u_pfc_hdc_prefix fs_v_pfc_hdc_prefix. ((((exists fs_h_pfc_hdc_prefix_body_start. fs_h_pfc_hdc_prefix_body_start + S (0) = S ((S (0)) * fs_v_pfc_hdc_prefix)) /\\ exists fs_q_pfc_hdc_prefix_body_start. fs_u_pfc_hdc_prefix = fs_q_pfc_hdc_prefix_body_start * S ((S (0)) * fs_v_pfc_hdc_prefix) + (0))) /\\ ((((exists fs_h_pfc_hdc_prefix_body_terminal. fs_h_pfc_hdc_prefix_body_terminal + S (r) = S ((S (L)) * fs_v_pfc_hdc_prefix)) /\\ exists fs_q_pfc_hdc_prefix_body_terminal. fs_u_pfc_hdc_prefix = fs_q_pfc_hdc_prefix_body_terminal * S ((S (L)) * fs_v_pfc_hdc_prefix) + (r))) /\\ forall fs_i_pfc_hdc_prefix_body_steps. (exists fs_lt_pfc_hdc_prefix_body_steps_bound. fs_lt_pfc_hdc_prefix_body_steps_bound + S fs_i_pfc_hdc_prefix_body_steps = L) -> exists fs_a_pfc_hdc_prefix_body_steps fs_r_pfc_hdc_prefix_body_steps fs_s_pfc_hdc_prefix_body_steps. ((((exists fs_h_pfc_hdc_prefix_body_steps_summand. fs_h_pfc_hdc_prefix_body_steps_summand + S (fs_a_pfc_hdc_prefix_body_steps) = S ((S (fs_i_pfc_hdc_prefix_body_steps)) * cc)) /\\ exists fs_q_pfc_hdc_prefix_body_steps_summand. cb = fs_q_pfc_hdc_prefix_body_steps_summand * S ((S (fs_i_pfc_hdc_prefix_body_steps)) * cc) + (fs_a_pfc_hdc_prefix_body_steps))) /\\ ((((exists fs_h_pfc_hdc_prefix_body_steps_partial. fs_h_pfc_hdc_prefix_body_steps_partial + S (fs_r_pfc_hdc_prefix_body_steps) = S ((S (fs_i_pfc_hdc_prefix_body_steps)) * fs_v_pfc_hdc_prefix)) /\\ exists fs_q_pfc_hdc_prefix_body_steps_partial. fs_u_pfc_hdc_prefix = fs_q_pfc_hdc_prefix_body_steps_partial * S ((S (fs_i_pfc_hdc_prefix_body_steps)) * fs_v_pfc_hdc_prefix) + (fs_r_pfc_hdc_prefix_body_steps))) /\\ ((((exists fs_h_pfc_hdc_prefix_body_steps_successor. fs_h_pfc_hdc_prefix_body_steps_successor + S (fs_s_pfc_hdc_prefix_body_steps) = S ((S (S fs_i_pfc_hdc_prefix_body_steps)) * fs_v_pfc_hdc_prefix)) /\\ exists fs_q_pfc_hdc_prefix_body_steps_successor. fs_u_pfc_hdc_prefix = fs_q_pfc_hdc_prefix_body_steps_successor * S ((S (S fs_i_pfc_hdc_prefix_body_steps)) * fs_v_pfc_hdc_prefix) + (fs_s_pfc_hdc_prefix_body_steps))) /\\ fs_s_pfc_hdc_prefix_body_steps = fs_r_pfc_hdc_prefix_body_steps + fs_a_pfc_hdc_prefix_body_steps)))))) /\\ ((C=r+a)))))", "specialize beta_sum_succ_decompose (cb)", "specialize beta_sum_succ_decompose (cc)", "specialize beta_sum_succ_decompose (L)", "specialize beta_sum_succ_decompose (C)", "apply beta_sum_succ_decompose", "exact hc", "cases hdc", "cases hdc_witness", "cases hdc_witness_witness", "cases hdc_witness_witness_right", "have hprefix : exists pfa_offset_left_distribution_sum_prefix pfa_offset_right_distribution_sum_prefix. (x1+x3) + (p) * pfa_offset_left_distribution_sum_prefix = (x5) + (p) * pfa_offset_right_distribution_sum_prefix", "specialize IH (x1)", "specialize IH (x3)", "specialize IH (x5)", "apply IH", "exact hda_witness_witness_right_left", "exact hdb_witness_witness_right_left", "exact hdc_witness_witness_right_left", "intro i", "intro a", "intro b", "intro c", "intro hi", "intro hea", "intro heb", "intro hec", "specialize hpw (i)", "specialize hpw (a)", "specialize hpw (b)", "specialize hpw (c)", "apply hpw", "specialize le_succ (S i)", "specialize le_succ (L)", "apply le_succ", "exact hi", "exact hea", "exact heb", "exact hec", "have hlast : exists pfa_offset_left_distribution_sum_last pfa_offset_right_distribution_sum_last. (x+x2) + (p) * pfa_offset_left_distribution_sum_last = (x4) + (p) * pfa_offset_right_distribution_sum_last", "specialize hpw (L)", "specialize hpw (x)", "specialize hpw (x2)", "specialize hpw (x4)", "apply hpw", "specialize le_refl (S L)", "apply le_refl", "exact hda_witness_witness_left", "exact hdb_witness_witness_left", "exact hdc_witness_witness_left", "have hcombined : exists pfa_offset_left_distribution_sum_combined pfa_offset_right_distribution_sum_combined. ((x1+x3)+(x+x2)) + (p) * pfa_offset_left_distribution_sum_combined = (x5+x4) + (p) * pfa_offset_right_distribution_sum_combined", "specialize mod_eq_add (p)", "specialize mod_eq_add (x1+x3)", "specialize mod_eq_add (x5)", "specialize mod_eq_add (x+x2)", "specialize mod_eq_add (x4)", "apply mod_eq_add", "exact hprefix", "exact hlast", "have hshuffle : (x1+x)+(x3+x2)=(x1+x3)+(x+x2)", "trans x1+(x+(x3+x2))", "apply add_assoc", "trans x1+((x+x3)+x2)", "congr", "refl", "symm", "apply add_assoc", "trans x1+((x3+x)+x2)", "congr", "refl", "congr", "apply add_comm", "refl", "trans x1+(x3+(x+x2))", "congr", "refl", "apply add_assoc", "symm", "apply add_assoc", "rewrite hda_witness_witness_right_right", "rewrite hdb_witness_witness_right_right", "rewrite hdc_witness_witness_right_right", "rewrite hshuffle", "exact hcombined" ], "script_sha256": "acede9ac21115a8c4c17ae5c80c196744d5b1ae55c50e77be04a1e822ed7bb1b", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_distributivity_candidate.py", "sha256": "a959962d631759cd1fc773dd7eef2fadf4f3f95361d6d7bc8c6a9e82d0d4ab86" }, "statement": "forall p ab ac bb bc cb cc L A B C. (exists fs_u_pfc_distribution_sum_a fs_v_pfc_distribution_sum_a. ((((exists fs_h_pfc_distribution_sum_a_body_start. fs_h_pfc_distribution_sum_a_body_start + S (0) = S ((S (0)) * fs_v_pfc_distribution_sum_a)) /\\ exists fs_q_pfc_distribution_sum_a_body_start. fs_u_pfc_distribution_sum_a = fs_q_pfc_distribution_sum_a_body_start * S ((S (0)) * fs_v_pfc_distribution_sum_a) + (0))) /\\ ((((exists fs_h_pfc_distribution_sum_a_body_terminal. fs_h_pfc_distribution_sum_a_body_terminal + S (A) = S ((S (L)) * fs_v_pfc_distribution_sum_a)) /\\ exists fs_q_pfc_distribution_sum_a_body_terminal. fs_u_pfc_distribution_sum_a = fs_q_pfc_distribution_sum_a_body_terminal * S ((S (L)) * fs_v_pfc_distribution_sum_a) + (A))) /\\ forall fs_i_pfc_distribution_sum_a_body_steps. (exists fs_lt_pfc_distribution_sum_a_body_steps_bound. fs_lt_pfc_distribution_sum_a_body_steps_bound + S fs_i_pfc_distribution_sum_a_body_steps = L) -> exists fs_a_pfc_distribution_sum_a_body_steps fs_r_pfc_distribution_sum_a_body_steps fs_s_pfc_distribution_sum_a_body_steps. ((((exists fs_h_pfc_distribution_sum_a_body_steps_summand. fs_h_pfc_distribution_sum_a_body_steps_summand + S (fs_a_pfc_distribution_sum_a_body_steps) = S ((S (fs_i_pfc_distribution_sum_a_body_steps)) * ac)) /\\ exists fs_q_pfc_distribution_sum_a_body_steps_summand. ab = fs_q_pfc_distribution_sum_a_body_steps_summand * S ((S (fs_i_pfc_distribution_sum_a_body_steps)) * ac) + (fs_a_pfc_distribution_sum_a_body_steps))) /\\ ((((exists fs_h_pfc_distribution_sum_a_body_steps_partial. fs_h_pfc_distribution_sum_a_body_steps_partial + S (fs_r_pfc_distribution_sum_a_body_steps) = S ((S (fs_i_pfc_distribution_sum_a_body_steps)) * fs_v_pfc_distribution_sum_a)) /\\ exists fs_q_pfc_distribution_sum_a_body_steps_partial. fs_u_pfc_distribution_sum_a = fs_q_pfc_distribution_sum_a_body_steps_partial * S ((S (fs_i_pfc_distribution_sum_a_body_steps)) * fs_v_pfc_distribution_sum_a) + (fs_r_pfc_distribution_sum_a_body_steps))) /\\ ((((exists fs_h_pfc_distribution_sum_a_body_steps_successor. fs_h_pfc_distribution_sum_a_body_steps_successor + S (fs_s_pfc_distribution_sum_a_body_steps) = S ((S (S fs_i_pfc_distribution_sum_a_body_steps)) * fs_v_pfc_distribution_sum_a)) /\\ exists fs_q_pfc_distribution_sum_a_body_steps_successor. fs_u_pfc_distribution_sum_a = fs_q_pfc_distribution_sum_a_body_steps_successor * S ((S (S fs_i_pfc_distribution_sum_a_body_steps)) * fs_v_pfc_distribution_sum_a) + (fs_s_pfc_distribution_sum_a_body_steps))) /\\ fs_s_pfc_distribution_sum_a_body_steps = fs_r_pfc_distribution_sum_a_body_steps + fs_a_pfc_distribution_sum_a_body_steps)))))) -> (exists fs_u_pfc_distribution_sum_b fs_v_pfc_distribution_sum_b. ((((exists fs_h_pfc_distribution_sum_b_body_start. fs_h_pfc_distribution_sum_b_body_start + S (0) = S ((S (0)) * fs_v_pfc_distribution_sum_b)) /\\ exists fs_q_pfc_distribution_sum_b_body_start. fs_u_pfc_distribution_sum_b = fs_q_pfc_distribution_sum_b_body_start * S ((S (0)) * fs_v_pfc_distribution_sum_b) + (0))) /\\ ((((exists fs_h_pfc_distribution_sum_b_body_terminal. fs_h_pfc_distribution_sum_b_body_terminal + S (B) = S ((S (L)) * fs_v_pfc_distribution_sum_b)) /\\ exists fs_q_pfc_distribution_sum_b_body_terminal. fs_u_pfc_distribution_sum_b = fs_q_pfc_distribution_sum_b_body_terminal * S ((S (L)) * fs_v_pfc_distribution_sum_b) + (B))) /\\ forall fs_i_pfc_distribution_sum_b_body_steps. (exists fs_lt_pfc_distribution_sum_b_body_steps_bound. fs_lt_pfc_distribution_sum_b_body_steps_bound + S fs_i_pfc_distribution_sum_b_body_steps = L) -> exists fs_a_pfc_distribution_sum_b_body_steps fs_r_pfc_distribution_sum_b_body_steps fs_s_pfc_distribution_sum_b_body_steps. ((((exists fs_h_pfc_distribution_sum_b_body_steps_summand. fs_h_pfc_distribution_sum_b_body_steps_summand + S (fs_a_pfc_distribution_sum_b_body_steps) = S ((S (fs_i_pfc_distribution_sum_b_body_steps)) * bc)) /\\ exists fs_q_pfc_distribution_sum_b_body_steps_summand. bb = fs_q_pfc_distribution_sum_b_body_steps_summand * S ((S (fs_i_pfc_distribution_sum_b_body_steps)) * bc) + (fs_a_pfc_distribution_sum_b_body_steps))) /\\ ((((exists fs_h_pfc_distribution_sum_b_body_steps_partial. fs_h_pfc_distribution_sum_b_body_steps_partial + S (fs_r_pfc_distribution_sum_b_body_steps) = S ((S (fs_i_pfc_distribution_sum_b_body_steps)) * fs_v_pfc_distribution_sum_b)) /\\ exists fs_q_pfc_distribution_sum_b_body_steps_partial. fs_u_pfc_distribution_sum_b = fs_q_pfc_distribution_sum_b_body_steps_partial * S ((S (fs_i_pfc_distribution_sum_b_body_steps)) * fs_v_pfc_distribution_sum_b) + (fs_r_pfc_distribution_sum_b_body_steps))) /\\ ((((exists fs_h_pfc_distribution_sum_b_body_steps_successor. fs_h_pfc_distribution_sum_b_body_steps_successor + S (fs_s_pfc_distribution_sum_b_body_steps) = S ((S (S fs_i_pfc_distribution_sum_b_body_steps)) * fs_v_pfc_distribution_sum_b)) /\\ exists fs_q_pfc_distribution_sum_b_body_steps_successor. fs_u_pfc_distribution_sum_b = fs_q_pfc_distribution_sum_b_body_steps_successor * S ((S (S fs_i_pfc_distribution_sum_b_body_steps)) * fs_v_pfc_distribution_sum_b) + (fs_s_pfc_distribution_sum_b_body_steps))) /\\ fs_s_pfc_distribution_sum_b_body_steps = fs_r_pfc_distribution_sum_b_body_steps + fs_a_pfc_distribution_sum_b_body_steps)))))) -> (exists fs_u_pfc_distribution_sum_c fs_v_pfc_distribution_sum_c. ((((exists fs_h_pfc_distribution_sum_c_body_start. fs_h_pfc_distribution_sum_c_body_start + S (0) = S ((S (0)) * fs_v_pfc_distribution_sum_c)) /\\ exists fs_q_pfc_distribution_sum_c_body_start. fs_u_pfc_distribution_sum_c = fs_q_pfc_distribution_sum_c_body_start * S ((S (0)) * fs_v_pfc_distribution_sum_c) + (0))) /\\ ((((exists fs_h_pfc_distribution_sum_c_body_terminal. fs_h_pfc_distribution_sum_c_body_terminal + S (C) = S ((S (L)) * fs_v_pfc_distribution_sum_c)) /\\ exists fs_q_pfc_distribution_sum_c_body_terminal. fs_u_pfc_distribution_sum_c = fs_q_pfc_distribution_sum_c_body_terminal * S ((S (L)) * fs_v_pfc_distribution_sum_c) + (C))) /\\ forall fs_i_pfc_distribution_sum_c_body_steps. (exists fs_lt_pfc_distribution_sum_c_body_steps_bound. fs_lt_pfc_distribution_sum_c_body_steps_bound + S fs_i_pfc_distribution_sum_c_body_steps = L) -> exists fs_a_pfc_distribution_sum_c_body_steps fs_r_pfc_distribution_sum_c_body_steps fs_s_pfc_distribution_sum_c_body_steps. ((((exists fs_h_pfc_distribution_sum_c_body_steps_summand. fs_h_pfc_distribution_sum_c_body_steps_summand + S (fs_a_pfc_distribution_sum_c_body_steps) = S ((S (fs_i_pfc_distribution_sum_c_body_steps)) * cc)) /\\ exists fs_q_pfc_distribution_sum_c_body_steps_summand. cb = fs_q_pfc_distribution_sum_c_body_steps_summand * S ((S (fs_i_pfc_distribution_sum_c_body_steps)) * cc) + (fs_a_pfc_distribution_sum_c_body_steps))) /\\ ((((exists fs_h_pfc_distribution_sum_c_body_steps_partial. fs_h_pfc_distribution_sum_c_body_steps_partial + S (fs_r_pfc_distribution_sum_c_body_steps) = S ((S (fs_i_pfc_distribution_sum_c_body_steps)) * fs_v_pfc_distribution_sum_c)) /\\ exists fs_q_pfc_distribution_sum_c_body_steps_partial. fs_u_pfc_distribution_sum_c = fs_q_pfc_distribution_sum_c_body_steps_partial * S ((S (fs_i_pfc_distribution_sum_c_body_steps)) * fs_v_pfc_distribution_sum_c) + (fs_r_pfc_distribution_sum_c_body_steps))) /\\ ((((exists fs_h_pfc_distribution_sum_c_body_steps_successor. fs_h_pfc_distribution_sum_c_body_steps_successor + S (fs_s_pfc_distribution_sum_c_body_steps) = S ((S (S fs_i_pfc_distribution_sum_c_body_steps)) * fs_v_pfc_distribution_sum_c)) /\\ exists fs_q_pfc_distribution_sum_c_body_steps_successor. fs_u_pfc_distribution_sum_c = fs_q_pfc_distribution_sum_c_body_steps_successor * S ((S (S fs_i_pfc_distribution_sum_c_body_steps)) * fs_v_pfc_distribution_sum_c) + (fs_s_pfc_distribution_sum_c_body_steps))) /\\ fs_s_pfc_distribution_sum_c_body_steps = fs_r_pfc_distribution_sum_c_body_steps + fs_a_pfc_distribution_sum_c_body_steps)))))) -> (forall i a b c. (exists pfa_gap_distribution_sum_pointwise_bound. pfa_gap_distribution_sum_pointwise_bound + S (i) = (L)) -> (((exists ff_h_pfp_distribution_sum_pointwise_a. ff_h_pfp_distribution_sum_pointwise_a + S (a) = S ((S (i)) * ac)) /\\ exists ff_q_pfp_distribution_sum_pointwise_a. ab = ff_q_pfp_distribution_sum_pointwise_a * S ((S (i)) * ac) + (a))) -> (((exists ff_h_pfp_distribution_sum_pointwise_b. ff_h_pfp_distribution_sum_pointwise_b + S (b) = S ((S (i)) * bc)) /\\ exists ff_q_pfp_distribution_sum_pointwise_b. bb = ff_q_pfp_distribution_sum_pointwise_b * S ((S (i)) * bc) + (b))) -> (((exists ff_h_pfp_distribution_sum_pointwise_c. ff_h_pfp_distribution_sum_pointwise_c + S (c) = S ((S (i)) * cc)) /\\ exists ff_q_pfp_distribution_sum_pointwise_c. cb = ff_q_pfp_distribution_sum_pointwise_c * S ((S (i)) * cc) + (c))) -> (exists pfa_offset_left_distribution_sum_pointwise_value pfa_offset_right_distribution_sum_pointwise_value. 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(forall pfp_index_padded_add_source. (exists pfa_gap_padded_add_sourceindex. pfa_gap_padded_add_sourceindex + S (pfp_index_padded_add_source) = (L)) -> exists pfp_left_padded_add_source pfp_right_padded_add_source pfp_value_padded_add_source. ((((exists ff_h_pfp_padded_add_sourceleft. ff_h_pfp_padded_add_sourceleft + S (pfp_left_padded_add_source) = S ((S (pfp_index_padded_add_source)) * ac)) /\\ exists ff_q_pfp_padded_add_sourceleft. ab = ff_q_pfp_padded_add_sourceleft * S ((S (pfp_index_padded_add_source)) * ac) + (pfp_left_padded_add_source))) /\\ (((((exists ff_h_pfp_padded_add_sourceright. ff_h_pfp_padded_add_sourceright + S (pfp_right_padded_add_source) = S ((S (pfp_index_padded_add_source)) * bc)) /\\ exists ff_q_pfp_padded_add_sourceright. bb = ff_q_pfp_padded_add_sourceright * S ((S (pfp_index_padded_add_source)) * bc) + (pfp_right_padded_add_source))) /\\ (((((exists ff_h_pfp_padded_add_sourcetarget. ff_h_pfp_padded_add_sourcetarget + S (pfp_value_padded_add_source) = S ((S (pfp_index_padded_add_source)) * cc)) /\\ exists ff_q_pfp_padded_add_sourcetarget. cb = ff_q_pfp_padded_add_sourcetarget * S ((S (pfp_index_padded_add_source)) * cc) + (pfp_value_padded_add_source))) /\\ ((((exists pfa_gap_padded_add_sourceoperationleft. pfa_gap_padded_add_sourceoperationleft + S (pfp_left_padded_add_source) = (p)) /\\ (((exists pfa_gap_padded_add_sourceoperationright. pfa_gap_padded_add_sourceoperationright + S (pfp_right_padded_add_source) = (p)) /\\ ((((exists pfa_gap_padded_add_sourceoperationresultbound. pfa_gap_padded_add_sourceoperationresultbound + S (pfp_value_padded_add_source) = (p)) /\\ ((exists pfa_offset_left_padded_add_sourceoperationresultcongruence pfa_offset_right_padded_add_sourceoperationresultcongruence. ((pfp_left_padded_add_source) + (pfp_right_padded_add_source)) + (p) * pfa_offset_left_padded_add_sourceoperationresultcongruence = (pfp_value_padded_add_source) + (p) * pfa_offset_right_padded_add_sourceoperationresultcongruence)))))))))))))))) -> ((((exists pfa_gap_padded_add_ainside. pfa_gap_padded_add_ainside + S (i) = (L)) /\\ ((((exists ff_h_pfp_padded_add_aentry. ff_h_pfp_padded_add_aentry + S (a) = S ((S (i)) * ac)) /\\ exists ff_q_pfp_padded_add_aentry. ab = ff_q_pfp_padded_add_aentry * S ((S (i)) * ac) + (a)))))) \\/ (((exists pfc_gap_padded_add_aoutside. pfc_gap_padded_add_aoutside+(L)=(i)) /\\ (((a)=0))))) -> ((((exists pfa_gap_padded_add_binside. pfa_gap_padded_add_binside + S (i) = (L)) /\\ ((((exists ff_h_pfp_padded_add_bentry. ff_h_pfp_padded_add_bentry + S (b) = S ((S (i)) * bc)) /\\ exists ff_q_pfp_padded_add_bentry. bb = ff_q_pfp_padded_add_bentry * S ((S (i)) * bc) + (b)))))) \\/ (((exists pfc_gap_padded_add_boutside. pfc_gap_padded_add_boutside+(L)=(i)) /\\ (((b)=0))))) -> ((((exists pfa_gap_padded_add_rinside. pfa_gap_padded_add_rinside + S (i) = (L)) /\\ ((((exists ff_h_pfp_padded_add_rentry. ff_h_pfp_padded_add_rentry + S (r) = S ((S (i)) * cc)) /\\ exists ff_q_pfp_padded_add_rentry. cb = ff_q_pfp_padded_add_rentry * S ((S (i)) * cc) + (r)))))) \\/ (((exists pfc_gap_padded_add_routside. pfc_gap_padded_add_routside+(L)=(i)) /\\ (((r)=0))))) -> (exists pfa_offset_left_padded_add_result pfa_offset_right_padded_add_result. (a+b) + (p) * pfa_offset_left_padded_add_result = (r) + (p) * pfa_offset_right_padded_add_result)", "statement_sha256": "892f91ef9172e8ae7a7b2ac6669bb0fdcaf2f20a6c7ffa900e7b302566fb0e46", "summary": "An actual coefficient sum extends by actual zeros to an additive congruence at every index; no claim is made about arbitrary decoded entries outside the original prefixes.", "summary_sha256": "d7c59f4d7cddaef84b486ed61442bc8e3ffd13b7e1664d0212f2dbc2a1de62c3" }, { "alpha_v33_frontier_enrollment": { "body_receipt_sha256": "4fec0f1e15bd94b73338a29c5913a188d12abbf445d8ecf1c1e04a8a8daa4dc4", "bundle_campaign": "polynomial-euclidean-division", "bundle_node_id": 320, "bundle_sha256": "6ae667d8518e4dbe722bb08ad1b08715a0d282c2893e533c8133d770fe861dcf", "campaign": "polynomial-euclidean-division", "first_enrolled_version": "v33", "parent_catalog_sha256": "41b9f387d88a5a4f0fe5ee2bd5578f37a27a4657b0a80f1a1a2cb5109f69a623", "rfc_sha256": 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"intro dc", "intro M", "intro i", "intro j", "intro u", "intro v", "intro w", "intro hs", "intro hu", "intro hv", "intro hw", "cases hu", "cases hu_witness", "cases hu_witness_witness", "cases hu_witness_witness_witness", "cases hu_witness_witness_witness_right", "cases hu_witness_witness_witness_right_right", "cases hv", "cases hv_witness", "cases hv_witness_witness", "cases hv_witness_witness_witness", "cases hv_witness_witness_witness_right", "cases hv_witness_witness_witness_right_right", "cases hw", "cases hw_witness", "cases hw_witness_witness", "cases hw_witness_witness_witness", "cases hw_witness_witness_witness_right", "cases hw_witness_witness_witness_right_right", "have hku : x=x6", "specialize add_left_cancel (j)", "specialize add_left_cancel (x)", "specialize add_left_cancel (x6)", "apply add_left_cancel", "trans i", "exact hu_witness_witness_witness_left", "symm", "exact hw_witness_witness_witness_left", "rewrite hku at hu_witness_witness_witness_right_right_left", "rewrite hku at hu_witness_witness_witness_right_right_left", "rewrite hku at hu_witness_witness_witness_right_right_left", "rewrite hku at hu_witness_witness_witness_right_right_left", "have hkv : x3=x6", "specialize add_left_cancel (j)", "specialize add_left_cancel (x3)", "specialize add_left_cancel (x6)", "apply add_left_cancel", "trans i", "exact hv_witness_witness_witness_left", "symm", "exact hw_witness_witness_witness_left", "rewrite hkv at hv_witness_witness_witness_right_right_left", "rewrite hkv at hv_witness_witness_witness_right_right_left", "rewrite hkv at hv_witness_witness_witness_right_right_left", "rewrite hkv at hv_witness_witness_witness_right_right_left", "have hfu : x1=x7", "specialize polynomial_zero_extended_entry_functional (db)", "specialize polynomial_zero_extended_entry_functional (dc)", "specialize polynomial_zero_extended_entry_functional (M)", "specialize polynomial_zero_extended_entry_functional (j)", "specialize polynomial_zero_extended_entry_functional (x1)", "specialize polynomial_zero_extended_entry_functional (x7)", "apply polynomial_zero_extended_entry_functional", "exact hu_witness_witness_witness_right_left", "exact hw_witness_witness_witness_right_left", "have hfv : x4=x7", "specialize polynomial_zero_extended_entry_functional (db)", "specialize polynomial_zero_extended_entry_functional (dc)", "specialize polynomial_zero_extended_entry_functional (M)", "specialize polynomial_zero_extended_entry_functional (j)", "specialize polynomial_zero_extended_entry_functional (x4)", "specialize polynomial_zero_extended_entry_functional (x7)", "apply polynomial_zero_extended_entry_functional", "exact hv_witness_witness_witness_right_left", "exact hw_witness_witness_witness_right_left", "have hsum : exists pfa_offset_left_term_add_left_sum pfa_offset_right_term_add_left_sum. (x2+x5) + (p) * pfa_offset_left_term_add_left_sum = (x8) + (p) * pfa_offset_right_term_add_left_sum", "specialize polynomial_zero_extended_add_congruent (p)", "specialize polynomial_zero_extended_add_congruent (ab)", "specialize polynomial_zero_extended_add_congruent (ac)", "specialize polynomial_zero_extended_add_congruent (bb)", "specialize polynomial_zero_extended_add_congruent (bc)", "specialize polynomial_zero_extended_add_congruent (cb)", "specialize polynomial_zero_extended_add_congruent (cc)", "specialize polynomial_zero_extended_add_congruent (L)", "specialize polynomial_zero_extended_add_congruent (x6)", "specialize polynomial_zero_extended_add_congruent (x2)", "specialize polynomial_zero_extended_add_congruent (x5)", "specialize polynomial_zero_extended_add_congruent (x8)", "apply polynomial_zero_extended_add_congruent", "exact hs", "exact hu_witness_witness_witness_right_right_left", "exact hv_witness_witness_witness_right_right_left", "exact hw_witness_witness_witness_right_right_left", "rewrite hu_witness_witness_witness_right_right_right", "rewrite hv_witness_witness_witness_right_right_right", "rewrite hw_witness_witness_witness_right_right_right", "rewrite hfu", "rewrite hfv", "have hfactor : x7*x2+x7*x5=x7*(x2+x5)", "symm", "specialize mul_add (x7)", "specialize mul_add (x2)", "specialize mul_add (x5)", "apply mul_add", "rewrite hfactor", "specialize mod_eq_mul_left (p)", "specialize mod_eq_mul_left (x2+x5)", "specialize mod_eq_mul_left (x8)", "specialize mod_eq_mul_left (x7)", "apply mod_eq_mul_left", "exact hsum" ], "script_sha256": "a57af0844c3c253640151f87c80bc08fadf2b4b00e9259b71f8254a109f5d885", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_distributivity_candidate.py", "sha256": "a959962d631759cd1fc773dd7eef2fadf4f3f95361d6d7bc8c6a9e82d0d4ab86" }, "statement": "forall p ab ac bb bc cb cc L db dc M i j u v w. (forall pfp_index_term_add_left_source. (exists pfa_gap_term_add_left_sourceindex. pfa_gap_term_add_left_sourceindex + S (pfp_index_term_add_left_source) = (L)) -> exists pfp_left_term_add_left_source pfp_right_term_add_left_source pfp_value_term_add_left_source. ((((exists ff_h_pfp_term_add_left_sourceleft. ff_h_pfp_term_add_left_sourceleft + S (pfp_left_term_add_left_source) = S ((S (pfp_index_term_add_left_source)) * ac)) /\\ exists ff_q_pfp_term_add_left_sourceleft. ab = ff_q_pfp_term_add_left_sourceleft * S ((S (pfp_index_term_add_left_source)) * ac) + (pfp_left_term_add_left_source))) /\\ (((((exists ff_h_pfp_term_add_left_sourceright. ff_h_pfp_term_add_left_sourceright + S (pfp_right_term_add_left_source) = S ((S (pfp_index_term_add_left_source)) * bc)) /\\ exists ff_q_pfp_term_add_left_sourceright. bb = ff_q_pfp_term_add_left_sourceright * S ((S (pfp_index_term_add_left_source)) * bc) + (pfp_right_term_add_left_source))) /\\ (((((exists ff_h_pfp_term_add_left_sourcetarget. ff_h_pfp_term_add_left_sourcetarget + S (pfp_value_term_add_left_source) = S ((S (pfp_index_term_add_left_source)) * cc)) /\\ exists ff_q_pfp_term_add_left_sourcetarget. cb = ff_q_pfp_term_add_left_sourcetarget * S ((S (pfp_index_term_add_left_source)) * cc) + (pfp_value_term_add_left_source))) /\\ ((((exists pfa_gap_term_add_left_sourceoperationleft. pfa_gap_term_add_left_sourceoperationleft + S (pfp_left_term_add_left_source) = (p)) /\\ (((exists pfa_gap_term_add_left_sourceoperationright. pfa_gap_term_add_left_sourceoperationright + S (pfp_right_term_add_left_source) = (p)) /\\ ((((exists pfa_gap_term_add_left_sourceoperationresultbound. pfa_gap_term_add_left_sourceoperationresultbound + S (pfp_value_term_add_left_source) = (p)) /\\ ((exists pfa_offset_left_term_add_left_sourceoperationresultcongruence pfa_offset_right_term_add_left_sourceoperationresultcongruence. ((pfp_left_term_add_left_source) + (pfp_right_term_add_left_source)) + (p) * pfa_offset_left_term_add_left_sourceoperationresultcongruence = (pfp_value_term_add_left_source) + (p) * pfa_offset_right_term_add_left_sourceoperationresultcongruence)))))))))))))))) -> (exists pfc_complement_term_add_left_u pfc_left_term_add_left_u pfc_right_term_add_left_u. (((j)+pfc_complement_term_add_left_u=(i)) /\\ ((((((exists pfa_gap_term_add_left_uleftinside. pfa_gap_term_add_left_uleftinside + S (j) = (M)) /\\ ((((exists ff_h_pfp_term_add_left_uleftentry. ff_h_pfp_term_add_left_uleftentry + S (pfc_left_term_add_left_u) = S ((S (j)) * dc)) /\\ exists ff_q_pfp_term_add_left_uleftentry. db = ff_q_pfp_term_add_left_uleftentry * S ((S (j)) * dc) + (pfc_left_term_add_left_u)))))) \\/ (((exists pfc_gap_term_add_left_uleftoutside. pfc_gap_term_add_left_uleftoutside+(M)=(j)) /\\ (((pfc_left_term_add_left_u)=0))))) /\\ ((((((exists pfa_gap_term_add_left_urightinside. pfa_gap_term_add_left_urightinside + S (pfc_complement_term_add_left_u) = (L)) /\\ ((((exists ff_h_pfp_term_add_left_urightentry. ff_h_pfp_term_add_left_urightentry + S (pfc_right_term_add_left_u) = S ((S (pfc_complement_term_add_left_u)) * ac)) /\\ exists ff_q_pfp_term_add_left_urightentry. ab = ff_q_pfp_term_add_left_urightentry * S ((S (pfc_complement_term_add_left_u)) * ac) + (pfc_right_term_add_left_u)))))) \\/ (((exists pfc_gap_term_add_left_urightoutside. pfc_gap_term_add_left_urightoutside+(L)=(pfc_complement_term_add_left_u)) /\\ (((pfc_right_term_add_left_u)=0))))) /\\ (((u)=pfc_left_term_add_left_u*pfc_right_term_add_left_u)))))))) -> (exists pfc_complement_term_add_left_v pfc_left_term_add_left_v pfc_right_term_add_left_v. (((j)+pfc_complement_term_add_left_v=(i)) /\\ ((((((exists pfa_gap_term_add_left_vleftinside. pfa_gap_term_add_left_vleftinside + S (j) = (M)) /\\ ((((exists ff_h_pfp_term_add_left_vleftentry. ff_h_pfp_term_add_left_vleftentry + S (pfc_left_term_add_left_v) = S ((S (j)) * dc)) /\\ exists ff_q_pfp_term_add_left_vleftentry. db = ff_q_pfp_term_add_left_vleftentry * S ((S (j)) * dc) + (pfc_left_term_add_left_v)))))) \\/ (((exists pfc_gap_term_add_left_vleftoutside. pfc_gap_term_add_left_vleftoutside+(M)=(j)) /\\ (((pfc_left_term_add_left_v)=0))))) /\\ ((((((exists pfa_gap_term_add_left_vrightinside. pfa_gap_term_add_left_vrightinside + S (pfc_complement_term_add_left_v) = (L)) /\\ ((((exists ff_h_pfp_term_add_left_vrightentry. ff_h_pfp_term_add_left_vrightentry + S (pfc_right_term_add_left_v) = S ((S (pfc_complement_term_add_left_v)) * bc)) /\\ exists ff_q_pfp_term_add_left_vrightentry. bb = ff_q_pfp_term_add_left_vrightentry * S ((S (pfc_complement_term_add_left_v)) * bc) + (pfc_right_term_add_left_v)))))) \\/ (((exists pfc_gap_term_add_left_vrightoutside. pfc_gap_term_add_left_vrightoutside+(L)=(pfc_complement_term_add_left_v)) /\\ (((pfc_right_term_add_left_v)=0))))) /\\ (((v)=pfc_left_term_add_left_v*pfc_right_term_add_left_v)))))))) -> (exists pfc_complement_term_add_left_w pfc_left_term_add_left_w pfc_right_term_add_left_w. (((j)+pfc_complement_term_add_left_w=(i)) /\\ ((((((exists pfa_gap_term_add_left_wleftinside. pfa_gap_term_add_left_wleftinside + S (j) = (M)) /\\ ((((exists ff_h_pfp_term_add_left_wleftentry. ff_h_pfp_term_add_left_wleftentry + S (pfc_left_term_add_left_w) = S ((S (j)) * dc)) /\\ exists ff_q_pfp_term_add_left_wleftentry. db = ff_q_pfp_term_add_left_wleftentry * S ((S (j)) * dc) + (pfc_left_term_add_left_w)))))) \\/ (((exists pfc_gap_term_add_left_wleftoutside. pfc_gap_term_add_left_wleftoutside+(M)=(j)) /\\ (((pfc_left_term_add_left_w)=0))))) /\\ ((((((exists pfa_gap_term_add_left_wrightinside. pfa_gap_term_add_left_wrightinside + S (pfc_complement_term_add_left_w) = (L)) /\\ ((((exists ff_h_pfp_term_add_left_wrightentry. ff_h_pfp_term_add_left_wrightentry + S (pfc_right_term_add_left_w) = S ((S (pfc_complement_term_add_left_w)) * cc)) /\\ exists ff_q_pfp_term_add_left_wrightentry. cb = ff_q_pfp_term_add_left_wrightentry * S ((S (pfc_complement_term_add_left_w)) * cc) + (pfc_right_term_add_left_w)))))) \\/ (((exists pfc_gap_term_add_left_wrightoutside. pfc_gap_term_add_left_wrightoutside+(L)=(pfc_complement_term_add_left_w)) /\\ (((pfc_right_term_add_left_w)=0))))) /\\ (((w)=pfc_left_term_add_left_w*pfc_right_term_add_left_w)))))))) -> (exists pfa_offset_left_term_add_left_result pfa_offset_right_term_add_left_result. (u+v) + (p) * pfa_offset_left_term_add_left_result = (w) + (p) * pfa_offset_right_term_add_left_result)", "statement_sha256": "7e8af4a9214224c102e5ccf065264620b099ed58c8996e279302854baef746f9", "summary": "At one actual antidiagonal position, left multiplication carries the genuine padded coefficient sum to the sum of the two genuine multiplication terms modulo the same modulus.", "summary_sha256": "60f0583732461ba8a99213d2a9f5953fd61a99325951cd8a87aaaf67bf279986" }, { "alpha_v33_frontier_enrollment": { "body_receipt_sha256": "f21aab17ec0525843c94ea8e8bd00886ed3a280d906d3b90735cb662c3d6f252", "bundle_campaign": "polynomial-euclidean-division", "bundle_node_id": 321, "bundle_sha256": "6ae667d8518e4dbe722bb08ad1b08715a0d282c2893e533c8133d770fe861dcf", "campaign": "polynomial-euclidean-division", "first_enrolled_version": "v33", "parent_catalog_sha256": "41b9f387d88a5a4f0fe5ee2bd5578f37a27a4657b0a80f1a1a2cb5109f69a623", "rfc_sha256": 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"cea85e5c595a021061fb50997df7ad489c8905ab563695209dadc8235f6762cb", "kind": "alpha_v33_live_original_kernel_and_lean_receipt", "path": "research/arithmetic-library/artifacts/alpha-v33-research-receipt-v1.json", "role": "fresh_release_proof_verification", "selector": "families[slug=polynomial-euclidean-division]" }, { "document_sha256": "41b9f387d88a5a4f0fe5ee2bd5578f37a27a4657b0a80f1a1a2cb5109f69a623", "kind": "sealed_alpha_v32_parent", "path": "artifacts/peano-library/alpha/catalog-v32.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "polynomial-euclidean-division", "logical_spec_sha256": "b6c4d5ac05a7df1a45111642e8dc62d54a5cc72bfbd7570a2f3b94fe58bb5bb8", "membership": "alpha_only", "name": "polynomial_diagonal_term_right_add_congruent", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro ab", "intro ac", "intro bb", "intro bc", "intro cb", "intro cc", "intro L", "intro db", "intro dc", "intro M", "intro i", "intro j", "intro u", "intro v", "intro w", "intro hs", "intro hu", "intro hv", "intro hw", "cases hu", "cases hu_witness", "cases hu_witness_witness", "cases hu_witness_witness_witness", "cases hu_witness_witness_witness_right", "cases hu_witness_witness_witness_right_right", "cases hv", "cases hv_witness", "cases hv_witness_witness", "cases hv_witness_witness_witness", "cases hv_witness_witness_witness_right", "cases hv_witness_witness_witness_right_right", "cases hw", "cases hw_witness", "cases hw_witness_witness", "cases hw_witness_witness_witness", "cases hw_witness_witness_witness_right", "cases hw_witness_witness_witness_right_right", "have hku : x=x6", "specialize add_left_cancel (j)", "specialize add_left_cancel (x)", "specialize add_left_cancel (x6)", "apply add_left_cancel", "trans i", "exact hu_witness_witness_witness_left", "symm", "exact hw_witness_witness_witness_left", "rewrite hku at hu_witness_witness_witness_right_right_left", "rewrite hku at hu_witness_witness_witness_right_right_left", "rewrite hku at hu_witness_witness_witness_right_right_left", "rewrite hku at hu_witness_witness_witness_right_right_left", "have hkv : x3=x6", "specialize add_left_cancel (j)", "specialize add_left_cancel (x3)", "specialize add_left_cancel (x6)", "apply add_left_cancel", "trans i", "exact hv_witness_witness_witness_left", "symm", "exact hw_witness_witness_witness_left", "rewrite hkv at hv_witness_witness_witness_right_right_left", "rewrite hkv at hv_witness_witness_witness_right_right_left", "rewrite hkv at hv_witness_witness_witness_right_right_left", "rewrite hkv at hv_witness_witness_witness_right_right_left", "have hfu : x2=x8", "specialize polynomial_zero_extended_entry_functional (db)", "specialize polynomial_zero_extended_entry_functional (dc)", "specialize polynomial_zero_extended_entry_functional (M)", "specialize polynomial_zero_extended_entry_functional (x6)", "specialize polynomial_zero_extended_entry_functional (x2)", "specialize polynomial_zero_extended_entry_functional (x8)", "apply polynomial_zero_extended_entry_functional", "exact hu_witness_witness_witness_right_right_left", "exact hw_witness_witness_witness_right_right_left", "have hfv : x5=x8", "specialize polynomial_zero_extended_entry_functional (db)", "specialize polynomial_zero_extended_entry_functional (dc)", "specialize polynomial_zero_extended_entry_functional (M)", "specialize polynomial_zero_extended_entry_functional (x6)", "specialize polynomial_zero_extended_entry_functional (x5)", "specialize polynomial_zero_extended_entry_functional (x8)", "apply polynomial_zero_extended_entry_functional", "exact hv_witness_witness_witness_right_right_left", "exact hw_witness_witness_witness_right_right_left", "have hsum : exists pfa_offset_left_term_add_right_sum pfa_offset_right_term_add_right_sum. (x1+x4) + (p) * pfa_offset_left_term_add_right_sum = (x7) + (p) * pfa_offset_right_term_add_right_sum", "specialize polynomial_zero_extended_add_congruent (p)", "specialize polynomial_zero_extended_add_congruent (ab)", "specialize polynomial_zero_extended_add_congruent (ac)", "specialize polynomial_zero_extended_add_congruent (bb)", "specialize polynomial_zero_extended_add_congruent (bc)", "specialize polynomial_zero_extended_add_congruent (cb)", "specialize polynomial_zero_extended_add_congruent (cc)", "specialize polynomial_zero_extended_add_congruent (L)", "specialize polynomial_zero_extended_add_congruent (j)", "specialize polynomial_zero_extended_add_congruent (x1)", "specialize polynomial_zero_extended_add_congruent (x4)", "specialize polynomial_zero_extended_add_congruent (x7)", "apply polynomial_zero_extended_add_congruent", "exact hs", "exact hu_witness_witness_witness_right_left", "exact hv_witness_witness_witness_right_left", "exact hw_witness_witness_witness_right_left", "rewrite hu_witness_witness_witness_right_right_right", "rewrite hv_witness_witness_witness_right_right_right", "rewrite hw_witness_witness_witness_right_right_right", "rewrite hfu", "rewrite hfv", "have hfactor : x1*x8+x4*x8=(x1+x4)*x8", "symm", "specialize add_mul (x1)", "specialize add_mul (x4)", "specialize add_mul (x8)", "apply add_mul", "rewrite hfactor", "specialize mod_eq_mul_right (p)", "specialize mod_eq_mul_right (x1+x4)", "specialize mod_eq_mul_right (x7)", "specialize mod_eq_mul_right (x8)", "apply mod_eq_mul_right", "exact hsum" ], "script_sha256": "7f4bb8650c2637bceff315914dee3b93b1359e81a42d7c31fd1296e087d6e299", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_distributivity_candidate.py", "sha256": "a959962d631759cd1fc773dd7eef2fadf4f3f95361d6d7bc8c6a9e82d0d4ab86" }, "statement": "forall p ab ac bb bc cb cc L db dc M i j u v w. (forall pfp_index_term_add_right_source. (exists pfa_gap_term_add_right_sourceindex. pfa_gap_term_add_right_sourceindex + S (pfp_index_term_add_right_source) = (L)) -> exists pfp_left_term_add_right_source pfp_right_term_add_right_source pfp_value_term_add_right_source. ((((exists ff_h_pfp_term_add_right_sourceleft. ff_h_pfp_term_add_right_sourceleft + S (pfp_left_term_add_right_source) = S ((S (pfp_index_term_add_right_source)) * ac)) /\\ exists ff_q_pfp_term_add_right_sourceleft. ab = ff_q_pfp_term_add_right_sourceleft * S ((S (pfp_index_term_add_right_source)) * ac) + (pfp_left_term_add_right_source))) /\\ (((((exists ff_h_pfp_term_add_right_sourceright. ff_h_pfp_term_add_right_sourceright + S (pfp_right_term_add_right_source) = S ((S (pfp_index_term_add_right_source)) * bc)) /\\ exists ff_q_pfp_term_add_right_sourceright. bb = ff_q_pfp_term_add_right_sourceright * S ((S (pfp_index_term_add_right_source)) * bc) + (pfp_right_term_add_right_source))) /\\ (((((exists ff_h_pfp_term_add_right_sourcetarget. ff_h_pfp_term_add_right_sourcetarget + S (pfp_value_term_add_right_source) = S ((S (pfp_index_term_add_right_source)) * cc)) /\\ exists ff_q_pfp_term_add_right_sourcetarget. cb = ff_q_pfp_term_add_right_sourcetarget * S ((S (pfp_index_term_add_right_source)) * cc) + (pfp_value_term_add_right_source))) /\\ ((((exists pfa_gap_term_add_right_sourceoperationleft. pfa_gap_term_add_right_sourceoperationleft + S (pfp_left_term_add_right_source) = (p)) /\\ (((exists pfa_gap_term_add_right_sourceoperationright. pfa_gap_term_add_right_sourceoperationright + S (pfp_right_term_add_right_source) = (p)) /\\ ((((exists pfa_gap_term_add_right_sourceoperationresultbound. pfa_gap_term_add_right_sourceoperationresultbound + S (pfp_value_term_add_right_source) = (p)) /\\ ((exists pfa_offset_left_term_add_right_sourceoperationresultcongruence pfa_offset_right_term_add_right_sourceoperationresultcongruence. ((pfp_left_term_add_right_source) + (pfp_right_term_add_right_source)) + (p) * pfa_offset_left_term_add_right_sourceoperationresultcongruence = (pfp_value_term_add_right_source) + (p) * pfa_offset_right_term_add_right_sourceoperationresultcongruence)))))))))))))))) -> (exists pfc_complement_term_add_right_u pfc_left_term_add_right_u pfc_right_term_add_right_u. (((j)+pfc_complement_term_add_right_u=(i)) /\\ ((((((exists pfa_gap_term_add_right_uleftinside. pfa_gap_term_add_right_uleftinside + S (j) = (L)) /\\ ((((exists ff_h_pfp_term_add_right_uleftentry. ff_h_pfp_term_add_right_uleftentry + S (pfc_left_term_add_right_u) = S ((S (j)) * ac)) /\\ exists ff_q_pfp_term_add_right_uleftentry. ab = ff_q_pfp_term_add_right_uleftentry * S ((S (j)) * ac) + (pfc_left_term_add_right_u)))))) \\/ (((exists pfc_gap_term_add_right_uleftoutside. pfc_gap_term_add_right_uleftoutside+(L)=(j)) /\\ (((pfc_left_term_add_right_u)=0))))) /\\ ((((((exists pfa_gap_term_add_right_urightinside. pfa_gap_term_add_right_urightinside + S (pfc_complement_term_add_right_u) = (M)) /\\ ((((exists ff_h_pfp_term_add_right_urightentry. ff_h_pfp_term_add_right_urightentry + S (pfc_right_term_add_right_u) = S ((S (pfc_complement_term_add_right_u)) * dc)) /\\ exists ff_q_pfp_term_add_right_urightentry. db = ff_q_pfp_term_add_right_urightentry * S ((S (pfc_complement_term_add_right_u)) * dc) + (pfc_right_term_add_right_u)))))) \\/ (((exists pfc_gap_term_add_right_urightoutside. pfc_gap_term_add_right_urightoutside+(M)=(pfc_complement_term_add_right_u)) /\\ (((pfc_right_term_add_right_u)=0))))) /\\ (((u)=pfc_left_term_add_right_u*pfc_right_term_add_right_u)))))))) -> (exists pfc_complement_term_add_right_v pfc_left_term_add_right_v pfc_right_term_add_right_v. (((j)+pfc_complement_term_add_right_v=(i)) /\\ ((((((exists pfa_gap_term_add_right_vleftinside. pfa_gap_term_add_right_vleftinside + S (j) = (L)) /\\ ((((exists ff_h_pfp_term_add_right_vleftentry. ff_h_pfp_term_add_right_vleftentry + S (pfc_left_term_add_right_v) = S ((S (j)) * bc)) /\\ exists ff_q_pfp_term_add_right_vleftentry. bb = ff_q_pfp_term_add_right_vleftentry * S ((S (j)) * bc) + (pfc_left_term_add_right_v)))))) \\/ (((exists pfc_gap_term_add_right_vleftoutside. pfc_gap_term_add_right_vleftoutside+(L)=(j)) /\\ (((pfc_left_term_add_right_v)=0))))) /\\ ((((((exists pfa_gap_term_add_right_vrightinside. pfa_gap_term_add_right_vrightinside + S (pfc_complement_term_add_right_v) = (M)) /\\ ((((exists ff_h_pfp_term_add_right_vrightentry. ff_h_pfp_term_add_right_vrightentry + S (pfc_right_term_add_right_v) = S ((S (pfc_complement_term_add_right_v)) * dc)) /\\ exists ff_q_pfp_term_add_right_vrightentry. db = ff_q_pfp_term_add_right_vrightentry * S ((S (pfc_complement_term_add_right_v)) * dc) + (pfc_right_term_add_right_v)))))) \\/ (((exists pfc_gap_term_add_right_vrightoutside. pfc_gap_term_add_right_vrightoutside+(M)=(pfc_complement_term_add_right_v)) /\\ (((pfc_right_term_add_right_v)=0))))) /\\ (((v)=pfc_left_term_add_right_v*pfc_right_term_add_right_v)))))))) -> (exists pfc_complement_term_add_right_w pfc_left_term_add_right_w pfc_right_term_add_right_w. (((j)+pfc_complement_term_add_right_w=(i)) /\\ ((((((exists pfa_gap_term_add_right_wleftinside. pfa_gap_term_add_right_wleftinside + S (j) = (L)) /\\ ((((exists ff_h_pfp_term_add_right_wleftentry. ff_h_pfp_term_add_right_wleftentry + S (pfc_left_term_add_right_w) = S ((S (j)) * cc)) /\\ exists ff_q_pfp_term_add_right_wleftentry. cb = ff_q_pfp_term_add_right_wleftentry * S ((S (j)) * cc) + (pfc_left_term_add_right_w)))))) \\/ (((exists pfc_gap_term_add_right_wleftoutside. pfc_gap_term_add_right_wleftoutside+(L)=(j)) /\\ (((pfc_left_term_add_right_w)=0))))) /\\ ((((((exists pfa_gap_term_add_right_wrightinside. pfa_gap_term_add_right_wrightinside + S (pfc_complement_term_add_right_w) = (M)) /\\ ((((exists ff_h_pfp_term_add_right_wrightentry. ff_h_pfp_term_add_right_wrightentry + S (pfc_right_term_add_right_w) = S ((S (pfc_complement_term_add_right_w)) * dc)) /\\ exists ff_q_pfp_term_add_right_wrightentry. db = ff_q_pfp_term_add_right_wrightentry * S ((S (pfc_complement_term_add_right_w)) * dc) + (pfc_right_term_add_right_w)))))) \\/ (((exists pfc_gap_term_add_right_wrightoutside. pfc_gap_term_add_right_wrightoutside+(M)=(pfc_complement_term_add_right_w)) /\\ (((pfc_right_term_add_right_w)=0))))) /\\ (((w)=pfc_left_term_add_right_w*pfc_right_term_add_right_w)))))))) -> (exists pfa_offset_left_term_add_right_result pfa_offset_right_term_add_right_result. (u+v) + (p) * pfa_offset_left_term_add_right_result = (w) + (p) * pfa_offset_right_term_add_right_result)", "statement_sha256": "2c017040b8c25306b8cc784c3212ba45956044f67a6a6211a597054f466f681e", "summary": "At one actual antidiagonal position, right multiplication carries the genuine padded coefficient sum to the sum of the two genuine multiplication terms modulo the same modulus.", "summary_sha256": "0beb641f7f230f2ebb4f910c7d2cb1fc42f441107cbbf56aea8d700d444f3479" }, { "alpha_v33_frontier_enrollment": { "body_receipt_sha256": "770fd392c4236b68d4b91fac020fbfdfc6984fab46f6bc6e6c28c1f4b3300251", "bundle_campaign": "polynomial-euclidean-division", "bundle_node_id": 322, "bundle_sha256": "6ae667d8518e4dbe722bb08ad1b08715a0d282c2893e533c8133d770fe861dcf", "campaign": "polynomial-euclidean-division", "first_enrolled_version": "v33", "parent_catalog_sha256": "41b9f387d88a5a4f0fe5ee2bd5578f37a27a4657b0a80f1a1a2cb5109f69a623", "rfc_sha256": 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(forall pfp_index_diagonal_add_left_source. (exists pfa_gap_diagonal_add_left_sourceindex. pfa_gap_diagonal_add_left_sourceindex + S (pfp_index_diagonal_add_left_source) = (L)) -> exists pfp_left_diagonal_add_left_source pfp_right_diagonal_add_left_source pfp_value_diagonal_add_left_source. ((((exists ff_h_pfp_diagonal_add_left_sourceleft. ff_h_pfp_diagonal_add_left_sourceleft + S (pfp_left_diagonal_add_left_source) = S ((S (pfp_index_diagonal_add_left_source)) * ac)) /\\ exists ff_q_pfp_diagonal_add_left_sourceleft. ab = ff_q_pfp_diagonal_add_left_sourceleft * S ((S (pfp_index_diagonal_add_left_source)) * ac) + (pfp_left_diagonal_add_left_source))) /\\ (((((exists ff_h_pfp_diagonal_add_left_sourceright. ff_h_pfp_diagonal_add_left_sourceright + S (pfp_right_diagonal_add_left_source) = S ((S (pfp_index_diagonal_add_left_source)) * bc)) /\\ exists ff_q_pfp_diagonal_add_left_sourceright. bb = ff_q_pfp_diagonal_add_left_sourceright * S ((S (pfp_index_diagonal_add_left_source)) * bc) + (pfp_right_diagonal_add_left_source))) /\\ (((((exists ff_h_pfp_diagonal_add_left_sourcetarget. ff_h_pfp_diagonal_add_left_sourcetarget + S (pfp_value_diagonal_add_left_source) = S ((S (pfp_index_diagonal_add_left_source)) * cc)) /\\ exists ff_q_pfp_diagonal_add_left_sourcetarget. cb = ff_q_pfp_diagonal_add_left_sourcetarget * S ((S (pfp_index_diagonal_add_left_source)) * cc) + (pfp_value_diagonal_add_left_source))) /\\ ((((exists pfa_gap_diagonal_add_left_sourceoperationleft. pfa_gap_diagonal_add_left_sourceoperationleft + S (pfp_left_diagonal_add_left_source) = (p)) /\\ (((exists pfa_gap_diagonal_add_left_sourceoperationright. pfa_gap_diagonal_add_left_sourceoperationright + S (pfp_right_diagonal_add_left_source) = (p)) /\\ ((((exists pfa_gap_diagonal_add_left_sourceoperationresultbound. pfa_gap_diagonal_add_left_sourceoperationresultbound + S (pfp_value_diagonal_add_left_source) = (p)) /\\ ((exists pfa_offset_left_diagonal_add_left_sourceoperationresultcongruence pfa_offset_right_diagonal_add_left_sourceoperationresultcongruence. ((pfp_left_diagonal_add_left_source) + (pfp_right_diagonal_add_left_source)) + (p) * pfa_offset_left_diagonal_add_left_sourceoperationresultcongruence = (pfp_value_diagonal_add_left_source) + (p) * pfa_offset_right_diagonal_add_left_sourceoperationresultcongruence)))))))))))))))) -> (forall pfc_index_diagonal_add_left_u. (exists pfa_gap_diagonal_add_left_ubound. pfa_gap_diagonal_add_left_ubound + S (pfc_index_diagonal_add_left_u) = (N)) -> exists pfc_value_diagonal_add_left_u. ((((exists ff_h_pfp_diagonal_add_left_uentry. ff_h_pfp_diagonal_add_left_uentry + S (pfc_value_diagonal_add_left_u) = S ((S (pfc_index_diagonal_add_left_u)) * uc)) /\\ exists ff_q_pfp_diagonal_add_left_uentry. ub = ff_q_pfp_diagonal_add_left_uentry * S ((S (pfc_index_diagonal_add_left_u)) * uc) + (pfc_value_diagonal_add_left_u))) /\\ ((exists pfc_complement_diagonal_add_left_uterm pfc_left_diagonal_add_left_uterm pfc_right_diagonal_add_left_uterm. (((pfc_index_diagonal_add_left_u)+pfc_complement_diagonal_add_left_uterm=(i)) /\\ ((((((exists pfa_gap_diagonal_add_left_utermleftinside. pfa_gap_diagonal_add_left_utermleftinside + S (pfc_index_diagonal_add_left_u) = (M)) /\\ ((((exists ff_h_pfp_diagonal_add_left_utermleftentry. ff_h_pfp_diagonal_add_left_utermleftentry + S (pfc_left_diagonal_add_left_uterm) = S ((S (pfc_index_diagonal_add_left_u)) * dc)) /\\ exists ff_q_pfp_diagonal_add_left_utermleftentry. db = ff_q_pfp_diagonal_add_left_utermleftentry * S ((S (pfc_index_diagonal_add_left_u)) * dc) + (pfc_left_diagonal_add_left_uterm)))))) \\/ (((exists pfc_gap_diagonal_add_left_utermleftoutside. pfc_gap_diagonal_add_left_utermleftoutside+(M)=(pfc_index_diagonal_add_left_u)) /\\ (((pfc_left_diagonal_add_left_uterm)=0))))) /\\ ((((((exists pfa_gap_diagonal_add_left_utermrightinside. pfa_gap_diagonal_add_left_utermrightinside + S (pfc_complement_diagonal_add_left_uterm) = (L)) /\\ ((((exists ff_h_pfp_diagonal_add_left_utermrightentry. ff_h_pfp_diagonal_add_left_utermrightentry + S (pfc_right_diagonal_add_left_uterm) = S ((S (pfc_complement_diagonal_add_left_uterm)) * ac)) /\\ exists ff_q_pfp_diagonal_add_left_utermrightentry. ab = ff_q_pfp_diagonal_add_left_utermrightentry * S ((S (pfc_complement_diagonal_add_left_uterm)) * ac) + (pfc_right_diagonal_add_left_uterm)))))) \\/ (((exists pfc_gap_diagonal_add_left_utermrightoutside. pfc_gap_diagonal_add_left_utermrightoutside+(L)=(pfc_complement_diagonal_add_left_uterm)) /\\ (((pfc_right_diagonal_add_left_uterm)=0))))) /\\ (((pfc_value_diagonal_add_left_u)=pfc_left_diagonal_add_left_uterm*pfc_right_diagonal_add_left_uterm))))))))))) -> (exists fs_u_pfc_diagonal_add_left_sum_u fs_v_pfc_diagonal_add_left_sum_u. ((((exists fs_h_pfc_diagonal_add_left_sum_u_body_start. fs_h_pfc_diagonal_add_left_sum_u_body_start + S (0) = S ((S (0)) * fs_v_pfc_diagonal_add_left_sum_u)) /\\ exists fs_q_pfc_diagonal_add_left_sum_u_body_start. fs_u_pfc_diagonal_add_left_sum_u = fs_q_pfc_diagonal_add_left_sum_u_body_start * S ((S (0)) * fs_v_pfc_diagonal_add_left_sum_u) + (0))) /\\ ((((exists fs_h_pfc_diagonal_add_left_sum_u_body_terminal. fs_h_pfc_diagonal_add_left_sum_u_body_terminal + S (u) = S ((S (N)) * fs_v_pfc_diagonal_add_left_sum_u)) /\\ exists fs_q_pfc_diagonal_add_left_sum_u_body_terminal. fs_u_pfc_diagonal_add_left_sum_u = fs_q_pfc_diagonal_add_left_sum_u_body_terminal * S ((S (N)) * fs_v_pfc_diagonal_add_left_sum_u) + (u))) /\\ forall fs_i_pfc_diagonal_add_left_sum_u_body_steps. (exists fs_lt_pfc_diagonal_add_left_sum_u_body_steps_bound. fs_lt_pfc_diagonal_add_left_sum_u_body_steps_bound + S fs_i_pfc_diagonal_add_left_sum_u_body_steps = N) -> exists fs_a_pfc_diagonal_add_left_sum_u_body_steps fs_r_pfc_diagonal_add_left_sum_u_body_steps fs_s_pfc_diagonal_add_left_sum_u_body_steps. ((((exists fs_h_pfc_diagonal_add_left_sum_u_body_steps_summand. fs_h_pfc_diagonal_add_left_sum_u_body_steps_summand + S (fs_a_pfc_diagonal_add_left_sum_u_body_steps) = S ((S (fs_i_pfc_diagonal_add_left_sum_u_body_steps)) * uc)) /\\ exists fs_q_pfc_diagonal_add_left_sum_u_body_steps_summand. ub = fs_q_pfc_diagonal_add_left_sum_u_body_steps_summand * S ((S (fs_i_pfc_diagonal_add_left_sum_u_body_steps)) * uc) + (fs_a_pfc_diagonal_add_left_sum_u_body_steps))) /\\ ((((exists fs_h_pfc_diagonal_add_left_sum_u_body_steps_partial. fs_h_pfc_diagonal_add_left_sum_u_body_steps_partial + S (fs_r_pfc_diagonal_add_left_sum_u_body_steps) = S ((S (fs_i_pfc_diagonal_add_left_sum_u_body_steps)) * fs_v_pfc_diagonal_add_left_sum_u)) /\\ exists fs_q_pfc_diagonal_add_left_sum_u_body_steps_partial. fs_u_pfc_diagonal_add_left_sum_u = fs_q_pfc_diagonal_add_left_sum_u_body_steps_partial * S ((S (fs_i_pfc_diagonal_add_left_sum_u_body_steps)) * fs_v_pfc_diagonal_add_left_sum_u) + (fs_r_pfc_diagonal_add_left_sum_u_body_steps))) /\\ ((((exists fs_h_pfc_diagonal_add_left_sum_u_body_steps_successor. fs_h_pfc_diagonal_add_left_sum_u_body_steps_successor + S (fs_s_pfc_diagonal_add_left_sum_u_body_steps) = S ((S (S fs_i_pfc_diagonal_add_left_sum_u_body_steps)) * fs_v_pfc_diagonal_add_left_sum_u)) /\\ exists fs_q_pfc_diagonal_add_left_sum_u_body_steps_successor. fs_u_pfc_diagonal_add_left_sum_u = fs_q_pfc_diagonal_add_left_sum_u_body_steps_successor * S ((S (S fs_i_pfc_diagonal_add_left_sum_u_body_steps)) * fs_v_pfc_diagonal_add_left_sum_u) + (fs_s_pfc_diagonal_add_left_sum_u_body_steps))) /\\ fs_s_pfc_diagonal_add_left_sum_u_body_steps = fs_r_pfc_diagonal_add_left_sum_u_body_steps + fs_a_pfc_diagonal_add_left_sum_u_body_steps)))))) -> (forall pfc_index_diagonal_add_left_v. (exists pfa_gap_diagonal_add_left_vbound. pfa_gap_diagonal_add_left_vbound + S (pfc_index_diagonal_add_left_v) = (N)) -> exists pfc_value_diagonal_add_left_v. ((((exists ff_h_pfp_diagonal_add_left_ventry. ff_h_pfp_diagonal_add_left_ventry + S (pfc_value_diagonal_add_left_v) = S ((S (pfc_index_diagonal_add_left_v)) * vc)) /\\ exists ff_q_pfp_diagonal_add_left_ventry. vb = ff_q_pfp_diagonal_add_left_ventry * S ((S (pfc_index_diagonal_add_left_v)) * vc) + (pfc_value_diagonal_add_left_v))) /\\ ((exists pfc_complement_diagonal_add_left_vterm pfc_left_diagonal_add_left_vterm pfc_right_diagonal_add_left_vterm. (((pfc_index_diagonal_add_left_v)+pfc_complement_diagonal_add_left_vterm=(i)) /\\ ((((((exists pfa_gap_diagonal_add_left_vtermleftinside. pfa_gap_diagonal_add_left_vtermleftinside + S (pfc_index_diagonal_add_left_v) = (M)) /\\ ((((exists ff_h_pfp_diagonal_add_left_vtermleftentry. ff_h_pfp_diagonal_add_left_vtermleftentry + S (pfc_left_diagonal_add_left_vterm) = S ((S (pfc_index_diagonal_add_left_v)) * dc)) /\\ exists ff_q_pfp_diagonal_add_left_vtermleftentry. db = ff_q_pfp_diagonal_add_left_vtermleftentry * S ((S (pfc_index_diagonal_add_left_v)) * dc) + (pfc_left_diagonal_add_left_vterm)))))) \\/ (((exists pfc_gap_diagonal_add_left_vtermleftoutside. pfc_gap_diagonal_add_left_vtermleftoutside+(M)=(pfc_index_diagonal_add_left_v)) /\\ (((pfc_left_diagonal_add_left_vterm)=0))))) /\\ ((((((exists pfa_gap_diagonal_add_left_vtermrightinside. pfa_gap_diagonal_add_left_vtermrightinside + S (pfc_complement_diagonal_add_left_vterm) = (L)) /\\ ((((exists ff_h_pfp_diagonal_add_left_vtermrightentry. ff_h_pfp_diagonal_add_left_vtermrightentry + S (pfc_right_diagonal_add_left_vterm) = S ((S (pfc_complement_diagonal_add_left_vterm)) * bc)) /\\ exists ff_q_pfp_diagonal_add_left_vtermrightentry. bb = ff_q_pfp_diagonal_add_left_vtermrightentry * S ((S (pfc_complement_diagonal_add_left_vterm)) * bc) + (pfc_right_diagonal_add_left_vterm)))))) \\/ (((exists pfc_gap_diagonal_add_left_vtermrightoutside. pfc_gap_diagonal_add_left_vtermrightoutside+(L)=(pfc_complement_diagonal_add_left_vterm)) /\\ (((pfc_right_diagonal_add_left_vterm)=0))))) /\\ (((pfc_value_diagonal_add_left_v)=pfc_left_diagonal_add_left_vterm*pfc_right_diagonal_add_left_vterm))))))))))) -> (exists fs_u_pfc_diagonal_add_left_sum_v fs_v_pfc_diagonal_add_left_sum_v. ((((exists fs_h_pfc_diagonal_add_left_sum_v_body_start. fs_h_pfc_diagonal_add_left_sum_v_body_start + S (0) = S ((S (0)) * fs_v_pfc_diagonal_add_left_sum_v)) /\\ exists fs_q_pfc_diagonal_add_left_sum_v_body_start. fs_u_pfc_diagonal_add_left_sum_v = fs_q_pfc_diagonal_add_left_sum_v_body_start * S ((S (0)) * fs_v_pfc_diagonal_add_left_sum_v) + (0))) /\\ ((((exists fs_h_pfc_diagonal_add_left_sum_v_body_terminal. fs_h_pfc_diagonal_add_left_sum_v_body_terminal + S (v) = S ((S (N)) * fs_v_pfc_diagonal_add_left_sum_v)) /\\ exists fs_q_pfc_diagonal_add_left_sum_v_body_terminal. fs_u_pfc_diagonal_add_left_sum_v = fs_q_pfc_diagonal_add_left_sum_v_body_terminal * S ((S (N)) * fs_v_pfc_diagonal_add_left_sum_v) + (v))) /\\ forall fs_i_pfc_diagonal_add_left_sum_v_body_steps. (exists fs_lt_pfc_diagonal_add_left_sum_v_body_steps_bound. fs_lt_pfc_diagonal_add_left_sum_v_body_steps_bound + S fs_i_pfc_diagonal_add_left_sum_v_body_steps = N) -> exists fs_a_pfc_diagonal_add_left_sum_v_body_steps fs_r_pfc_diagonal_add_left_sum_v_body_steps fs_s_pfc_diagonal_add_left_sum_v_body_steps. ((((exists fs_h_pfc_diagonal_add_left_sum_v_body_steps_summand. fs_h_pfc_diagonal_add_left_sum_v_body_steps_summand + S (fs_a_pfc_diagonal_add_left_sum_v_body_steps) = S ((S (fs_i_pfc_diagonal_add_left_sum_v_body_steps)) * vc)) /\\ exists fs_q_pfc_diagonal_add_left_sum_v_body_steps_summand. vb = fs_q_pfc_diagonal_add_left_sum_v_body_steps_summand * S ((S (fs_i_pfc_diagonal_add_left_sum_v_body_steps)) * vc) + (fs_a_pfc_diagonal_add_left_sum_v_body_steps))) /\\ ((((exists fs_h_pfc_diagonal_add_left_sum_v_body_steps_partial. fs_h_pfc_diagonal_add_left_sum_v_body_steps_partial + S (fs_r_pfc_diagonal_add_left_sum_v_body_steps) = S ((S (fs_i_pfc_diagonal_add_left_sum_v_body_steps)) * fs_v_pfc_diagonal_add_left_sum_v)) /\\ exists fs_q_pfc_diagonal_add_left_sum_v_body_steps_partial. fs_u_pfc_diagonal_add_left_sum_v = fs_q_pfc_diagonal_add_left_sum_v_body_steps_partial * S ((S (fs_i_pfc_diagonal_add_left_sum_v_body_steps)) * fs_v_pfc_diagonal_add_left_sum_v) + (fs_r_pfc_diagonal_add_left_sum_v_body_steps))) /\\ ((((exists fs_h_pfc_diagonal_add_left_sum_v_body_steps_successor. fs_h_pfc_diagonal_add_left_sum_v_body_steps_successor + S (fs_s_pfc_diagonal_add_left_sum_v_body_steps) = S ((S (S fs_i_pfc_diagonal_add_left_sum_v_body_steps)) * fs_v_pfc_diagonal_add_left_sum_v)) /\\ exists fs_q_pfc_diagonal_add_left_sum_v_body_steps_successor. fs_u_pfc_diagonal_add_left_sum_v = fs_q_pfc_diagonal_add_left_sum_v_body_steps_successor * S ((S (S fs_i_pfc_diagonal_add_left_sum_v_body_steps)) * fs_v_pfc_diagonal_add_left_sum_v) + (fs_s_pfc_diagonal_add_left_sum_v_body_steps))) /\\ fs_s_pfc_diagonal_add_left_sum_v_body_steps = fs_r_pfc_diagonal_add_left_sum_v_body_steps + fs_a_pfc_diagonal_add_left_sum_v_body_steps)))))) -> (forall pfc_index_diagonal_add_left_w. (exists pfa_gap_diagonal_add_left_wbound. pfa_gap_diagonal_add_left_wbound + S (pfc_index_diagonal_add_left_w) = (N)) -> exists pfc_value_diagonal_add_left_w. ((((exists ff_h_pfp_diagonal_add_left_wentry. ff_h_pfp_diagonal_add_left_wentry + S (pfc_value_diagonal_add_left_w) = S ((S (pfc_index_diagonal_add_left_w)) * wc)) /\\ exists ff_q_pfp_diagonal_add_left_wentry. wb = ff_q_pfp_diagonal_add_left_wentry * S ((S (pfc_index_diagonal_add_left_w)) * wc) + (pfc_value_diagonal_add_left_w))) /\\ ((exists pfc_complement_diagonal_add_left_wterm pfc_left_diagonal_add_left_wterm pfc_right_diagonal_add_left_wterm. (((pfc_index_diagonal_add_left_w)+pfc_complement_diagonal_add_left_wterm=(i)) /\\ ((((((exists pfa_gap_diagonal_add_left_wtermleftinside. pfa_gap_diagonal_add_left_wtermleftinside + S (pfc_index_diagonal_add_left_w) = (M)) /\\ ((((exists ff_h_pfp_diagonal_add_left_wtermleftentry. ff_h_pfp_diagonal_add_left_wtermleftentry + S (pfc_left_diagonal_add_left_wterm) = S ((S (pfc_index_diagonal_add_left_w)) * dc)) /\\ exists ff_q_pfp_diagonal_add_left_wtermleftentry. db = ff_q_pfp_diagonal_add_left_wtermleftentry * S ((S (pfc_index_diagonal_add_left_w)) * dc) + (pfc_left_diagonal_add_left_wterm)))))) \\/ (((exists pfc_gap_diagonal_add_left_wtermleftoutside. pfc_gap_diagonal_add_left_wtermleftoutside+(M)=(pfc_index_diagonal_add_left_w)) /\\ (((pfc_left_diagonal_add_left_wterm)=0))))) /\\ ((((((exists pfa_gap_diagonal_add_left_wtermrightinside. pfa_gap_diagonal_add_left_wtermrightinside + S (pfc_complement_diagonal_add_left_wterm) = (L)) /\\ ((((exists ff_h_pfp_diagonal_add_left_wtermrightentry. ff_h_pfp_diagonal_add_left_wtermrightentry + S (pfc_right_diagonal_add_left_wterm) = S ((S (pfc_complement_diagonal_add_left_wterm)) * cc)) /\\ exists ff_q_pfp_diagonal_add_left_wtermrightentry. cb = ff_q_pfp_diagonal_add_left_wtermrightentry * S ((S (pfc_complement_diagonal_add_left_wterm)) * cc) + (pfc_right_diagonal_add_left_wterm)))))) \\/ (((exists pfc_gap_diagonal_add_left_wtermrightoutside. pfc_gap_diagonal_add_left_wtermrightoutside+(L)=(pfc_complement_diagonal_add_left_wterm)) /\\ (((pfc_right_diagonal_add_left_wterm)=0))))) /\\ (((pfc_value_diagonal_add_left_w)=pfc_left_diagonal_add_left_wterm*pfc_right_diagonal_add_left_wterm))))))))))) -> (exists fs_u_pfc_diagonal_add_left_sum_w fs_v_pfc_diagonal_add_left_sum_w. ((((exists fs_h_pfc_diagonal_add_left_sum_w_body_start. fs_h_pfc_diagonal_add_left_sum_w_body_start + S (0) = S ((S (0)) * fs_v_pfc_diagonal_add_left_sum_w)) /\\ exists fs_q_pfc_diagonal_add_left_sum_w_body_start. fs_u_pfc_diagonal_add_left_sum_w = fs_q_pfc_diagonal_add_left_sum_w_body_start * S ((S (0)) * fs_v_pfc_diagonal_add_left_sum_w) + (0))) /\\ ((((exists fs_h_pfc_diagonal_add_left_sum_w_body_terminal. fs_h_pfc_diagonal_add_left_sum_w_body_terminal + S (w) = S ((S (N)) * fs_v_pfc_diagonal_add_left_sum_w)) /\\ exists fs_q_pfc_diagonal_add_left_sum_w_body_terminal. fs_u_pfc_diagonal_add_left_sum_w = fs_q_pfc_diagonal_add_left_sum_w_body_terminal * S ((S (N)) * fs_v_pfc_diagonal_add_left_sum_w) + (w))) /\\ forall fs_i_pfc_diagonal_add_left_sum_w_body_steps. (exists fs_lt_pfc_diagonal_add_left_sum_w_body_steps_bound. fs_lt_pfc_diagonal_add_left_sum_w_body_steps_bound + S fs_i_pfc_diagonal_add_left_sum_w_body_steps = N) -> exists fs_a_pfc_diagonal_add_left_sum_w_body_steps fs_r_pfc_diagonal_add_left_sum_w_body_steps fs_s_pfc_diagonal_add_left_sum_w_body_steps. ((((exists fs_h_pfc_diagonal_add_left_sum_w_body_steps_summand. fs_h_pfc_diagonal_add_left_sum_w_body_steps_summand + S (fs_a_pfc_diagonal_add_left_sum_w_body_steps) = S ((S (fs_i_pfc_diagonal_add_left_sum_w_body_steps)) * wc)) /\\ exists fs_q_pfc_diagonal_add_left_sum_w_body_steps_summand. wb = fs_q_pfc_diagonal_add_left_sum_w_body_steps_summand * S ((S (fs_i_pfc_diagonal_add_left_sum_w_body_steps)) * wc) + (fs_a_pfc_diagonal_add_left_sum_w_body_steps))) /\\ ((((exists fs_h_pfc_diagonal_add_left_sum_w_body_steps_partial. fs_h_pfc_diagonal_add_left_sum_w_body_steps_partial + S (fs_r_pfc_diagonal_add_left_sum_w_body_steps) = S ((S (fs_i_pfc_diagonal_add_left_sum_w_body_steps)) * fs_v_pfc_diagonal_add_left_sum_w)) /\\ exists fs_q_pfc_diagonal_add_left_sum_w_body_steps_partial. fs_u_pfc_diagonal_add_left_sum_w = fs_q_pfc_diagonal_add_left_sum_w_body_steps_partial * S ((S (fs_i_pfc_diagonal_add_left_sum_w_body_steps)) * fs_v_pfc_diagonal_add_left_sum_w) + (fs_r_pfc_diagonal_add_left_sum_w_body_steps))) /\\ ((((exists fs_h_pfc_diagonal_add_left_sum_w_body_steps_successor. fs_h_pfc_diagonal_add_left_sum_w_body_steps_successor + S (fs_s_pfc_diagonal_add_left_sum_w_body_steps) = S ((S (S fs_i_pfc_diagonal_add_left_sum_w_body_steps)) * fs_v_pfc_diagonal_add_left_sum_w)) /\\ exists fs_q_pfc_diagonal_add_left_sum_w_body_steps_successor. fs_u_pfc_diagonal_add_left_sum_w = fs_q_pfc_diagonal_add_left_sum_w_body_steps_successor * S ((S (S fs_i_pfc_diagonal_add_left_sum_w_body_steps)) * fs_v_pfc_diagonal_add_left_sum_w) + (fs_s_pfc_diagonal_add_left_sum_w_body_steps))) /\\ fs_s_pfc_diagonal_add_left_sum_w_body_steps = fs_r_pfc_diagonal_add_left_sum_w_body_steps + fs_a_pfc_diagonal_add_left_sum_w_body_steps)))))) -> (exists pfa_offset_left_diagonal_add_left_result pfa_offset_right_diagonal_add_left_result. 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(forall pfp_index_diagonal_add_right_source. (exists pfa_gap_diagonal_add_right_sourceindex. pfa_gap_diagonal_add_right_sourceindex + S (pfp_index_diagonal_add_right_source) = (L)) -> exists pfp_left_diagonal_add_right_source pfp_right_diagonal_add_right_source pfp_value_diagonal_add_right_source. ((((exists ff_h_pfp_diagonal_add_right_sourceleft. ff_h_pfp_diagonal_add_right_sourceleft + S (pfp_left_diagonal_add_right_source) = S ((S (pfp_index_diagonal_add_right_source)) * ac)) /\\ exists ff_q_pfp_diagonal_add_right_sourceleft. ab = ff_q_pfp_diagonal_add_right_sourceleft * S ((S (pfp_index_diagonal_add_right_source)) * ac) + (pfp_left_diagonal_add_right_source))) /\\ (((((exists ff_h_pfp_diagonal_add_right_sourceright. ff_h_pfp_diagonal_add_right_sourceright + S (pfp_right_diagonal_add_right_source) = S ((S (pfp_index_diagonal_add_right_source)) * bc)) /\\ exists ff_q_pfp_diagonal_add_right_sourceright. bb = ff_q_pfp_diagonal_add_right_sourceright * S ((S (pfp_index_diagonal_add_right_source)) * bc) + (pfp_right_diagonal_add_right_source))) /\\ (((((exists ff_h_pfp_diagonal_add_right_sourcetarget. ff_h_pfp_diagonal_add_right_sourcetarget + S (pfp_value_diagonal_add_right_source) = S ((S (pfp_index_diagonal_add_right_source)) * cc)) /\\ exists ff_q_pfp_diagonal_add_right_sourcetarget. cb = ff_q_pfp_diagonal_add_right_sourcetarget * S ((S (pfp_index_diagonal_add_right_source)) * cc) + (pfp_value_diagonal_add_right_source))) /\\ ((((exists pfa_gap_diagonal_add_right_sourceoperationleft. pfa_gap_diagonal_add_right_sourceoperationleft + S (pfp_left_diagonal_add_right_source) = (p)) /\\ (((exists pfa_gap_diagonal_add_right_sourceoperationright. pfa_gap_diagonal_add_right_sourceoperationright + S (pfp_right_diagonal_add_right_source) = (p)) /\\ ((((exists pfa_gap_diagonal_add_right_sourceoperationresultbound. pfa_gap_diagonal_add_right_sourceoperationresultbound + S (pfp_value_diagonal_add_right_source) = (p)) /\\ ((exists pfa_offset_left_diagonal_add_right_sourceoperationresultcongruence pfa_offset_right_diagonal_add_right_sourceoperationresultcongruence. ((pfp_left_diagonal_add_right_source) + (pfp_right_diagonal_add_right_source)) + (p) * pfa_offset_left_diagonal_add_right_sourceoperationresultcongruence = (pfp_value_diagonal_add_right_source) + (p) * pfa_offset_right_diagonal_add_right_sourceoperationresultcongruence)))))))))))))))) -> (forall pfc_index_diagonal_add_right_u. (exists pfa_gap_diagonal_add_right_ubound. pfa_gap_diagonal_add_right_ubound + S (pfc_index_diagonal_add_right_u) = (N)) -> exists pfc_value_diagonal_add_right_u. ((((exists ff_h_pfp_diagonal_add_right_uentry. ff_h_pfp_diagonal_add_right_uentry + S (pfc_value_diagonal_add_right_u) = S ((S (pfc_index_diagonal_add_right_u)) * uc)) /\\ exists ff_q_pfp_diagonal_add_right_uentry. ub = ff_q_pfp_diagonal_add_right_uentry * S ((S (pfc_index_diagonal_add_right_u)) * uc) + (pfc_value_diagonal_add_right_u))) /\\ ((exists pfc_complement_diagonal_add_right_uterm pfc_left_diagonal_add_right_uterm pfc_right_diagonal_add_right_uterm. (((pfc_index_diagonal_add_right_u)+pfc_complement_diagonal_add_right_uterm=(i)) /\\ ((((((exists pfa_gap_diagonal_add_right_utermleftinside. pfa_gap_diagonal_add_right_utermleftinside + S (pfc_index_diagonal_add_right_u) = (L)) /\\ ((((exists ff_h_pfp_diagonal_add_right_utermleftentry. ff_h_pfp_diagonal_add_right_utermleftentry + S (pfc_left_diagonal_add_right_uterm) = S ((S (pfc_index_diagonal_add_right_u)) * ac)) /\\ exists ff_q_pfp_diagonal_add_right_utermleftentry. ab = ff_q_pfp_diagonal_add_right_utermleftentry * S ((S (pfc_index_diagonal_add_right_u)) * ac) + (pfc_left_diagonal_add_right_uterm)))))) \\/ (((exists pfc_gap_diagonal_add_right_utermleftoutside. pfc_gap_diagonal_add_right_utermleftoutside+(L)=(pfc_index_diagonal_add_right_u)) /\\ (((pfc_left_diagonal_add_right_uterm)=0))))) /\\ ((((((exists pfa_gap_diagonal_add_right_utermrightinside. pfa_gap_diagonal_add_right_utermrightinside + S (pfc_complement_diagonal_add_right_uterm) = (M)) /\\ ((((exists ff_h_pfp_diagonal_add_right_utermrightentry. ff_h_pfp_diagonal_add_right_utermrightentry + S (pfc_right_diagonal_add_right_uterm) = S ((S (pfc_complement_diagonal_add_right_uterm)) * dc)) /\\ exists ff_q_pfp_diagonal_add_right_utermrightentry. db = ff_q_pfp_diagonal_add_right_utermrightentry * S ((S (pfc_complement_diagonal_add_right_uterm)) * dc) + (pfc_right_diagonal_add_right_uterm)))))) \\/ (((exists pfc_gap_diagonal_add_right_utermrightoutside. pfc_gap_diagonal_add_right_utermrightoutside+(M)=(pfc_complement_diagonal_add_right_uterm)) /\\ (((pfc_right_diagonal_add_right_uterm)=0))))) /\\ (((pfc_value_diagonal_add_right_u)=pfc_left_diagonal_add_right_uterm*pfc_right_diagonal_add_right_uterm))))))))))) -> (exists fs_u_pfc_diagonal_add_right_sum_u fs_v_pfc_diagonal_add_right_sum_u. ((((exists fs_h_pfc_diagonal_add_right_sum_u_body_start. fs_h_pfc_diagonal_add_right_sum_u_body_start + S (0) = S ((S (0)) * fs_v_pfc_diagonal_add_right_sum_u)) /\\ exists fs_q_pfc_diagonal_add_right_sum_u_body_start. fs_u_pfc_diagonal_add_right_sum_u = fs_q_pfc_diagonal_add_right_sum_u_body_start * S ((S (0)) * fs_v_pfc_diagonal_add_right_sum_u) + (0))) /\\ ((((exists fs_h_pfc_diagonal_add_right_sum_u_body_terminal. fs_h_pfc_diagonal_add_right_sum_u_body_terminal + S (u) = S ((S (N)) * fs_v_pfc_diagonal_add_right_sum_u)) /\\ exists fs_q_pfc_diagonal_add_right_sum_u_body_terminal. fs_u_pfc_diagonal_add_right_sum_u = fs_q_pfc_diagonal_add_right_sum_u_body_terminal * S ((S (N)) * fs_v_pfc_diagonal_add_right_sum_u) + (u))) /\\ forall fs_i_pfc_diagonal_add_right_sum_u_body_steps. (exists fs_lt_pfc_diagonal_add_right_sum_u_body_steps_bound. fs_lt_pfc_diagonal_add_right_sum_u_body_steps_bound + S fs_i_pfc_diagonal_add_right_sum_u_body_steps = N) -> exists fs_a_pfc_diagonal_add_right_sum_u_body_steps fs_r_pfc_diagonal_add_right_sum_u_body_steps fs_s_pfc_diagonal_add_right_sum_u_body_steps. ((((exists fs_h_pfc_diagonal_add_right_sum_u_body_steps_summand. fs_h_pfc_diagonal_add_right_sum_u_body_steps_summand + S (fs_a_pfc_diagonal_add_right_sum_u_body_steps) = S ((S (fs_i_pfc_diagonal_add_right_sum_u_body_steps)) * uc)) /\\ exists fs_q_pfc_diagonal_add_right_sum_u_body_steps_summand. ub = fs_q_pfc_diagonal_add_right_sum_u_body_steps_summand * S ((S (fs_i_pfc_diagonal_add_right_sum_u_body_steps)) * uc) + (fs_a_pfc_diagonal_add_right_sum_u_body_steps))) /\\ ((((exists fs_h_pfc_diagonal_add_right_sum_u_body_steps_partial. fs_h_pfc_diagonal_add_right_sum_u_body_steps_partial + S (fs_r_pfc_diagonal_add_right_sum_u_body_steps) = S ((S (fs_i_pfc_diagonal_add_right_sum_u_body_steps)) * fs_v_pfc_diagonal_add_right_sum_u)) /\\ exists fs_q_pfc_diagonal_add_right_sum_u_body_steps_partial. fs_u_pfc_diagonal_add_right_sum_u = fs_q_pfc_diagonal_add_right_sum_u_body_steps_partial * S ((S (fs_i_pfc_diagonal_add_right_sum_u_body_steps)) * fs_v_pfc_diagonal_add_right_sum_u) + (fs_r_pfc_diagonal_add_right_sum_u_body_steps))) /\\ ((((exists fs_h_pfc_diagonal_add_right_sum_u_body_steps_successor. fs_h_pfc_diagonal_add_right_sum_u_body_steps_successor + S (fs_s_pfc_diagonal_add_right_sum_u_body_steps) = S ((S (S fs_i_pfc_diagonal_add_right_sum_u_body_steps)) * fs_v_pfc_diagonal_add_right_sum_u)) /\\ exists fs_q_pfc_diagonal_add_right_sum_u_body_steps_successor. fs_u_pfc_diagonal_add_right_sum_u = fs_q_pfc_diagonal_add_right_sum_u_body_steps_successor * S ((S (S fs_i_pfc_diagonal_add_right_sum_u_body_steps)) * fs_v_pfc_diagonal_add_right_sum_u) + (fs_s_pfc_diagonal_add_right_sum_u_body_steps))) /\\ fs_s_pfc_diagonal_add_right_sum_u_body_steps = fs_r_pfc_diagonal_add_right_sum_u_body_steps + fs_a_pfc_diagonal_add_right_sum_u_body_steps)))))) -> (forall pfc_index_diagonal_add_right_v. (exists pfa_gap_diagonal_add_right_vbound. pfa_gap_diagonal_add_right_vbound + S (pfc_index_diagonal_add_right_v) = (N)) -> exists pfc_value_diagonal_add_right_v. ((((exists ff_h_pfp_diagonal_add_right_ventry. ff_h_pfp_diagonal_add_right_ventry + S (pfc_value_diagonal_add_right_v) = S ((S (pfc_index_diagonal_add_right_v)) * vc)) /\\ exists ff_q_pfp_diagonal_add_right_ventry. vb = ff_q_pfp_diagonal_add_right_ventry * S ((S (pfc_index_diagonal_add_right_v)) * vc) + (pfc_value_diagonal_add_right_v))) /\\ ((exists pfc_complement_diagonal_add_right_vterm pfc_left_diagonal_add_right_vterm pfc_right_diagonal_add_right_vterm. (((pfc_index_diagonal_add_right_v)+pfc_complement_diagonal_add_right_vterm=(i)) /\\ ((((((exists pfa_gap_diagonal_add_right_vtermleftinside. pfa_gap_diagonal_add_right_vtermleftinside + S (pfc_index_diagonal_add_right_v) = (L)) /\\ ((((exists ff_h_pfp_diagonal_add_right_vtermleftentry. ff_h_pfp_diagonal_add_right_vtermleftentry + S (pfc_left_diagonal_add_right_vterm) = S ((S (pfc_index_diagonal_add_right_v)) * bc)) /\\ exists ff_q_pfp_diagonal_add_right_vtermleftentry. bb = ff_q_pfp_diagonal_add_right_vtermleftentry * S ((S (pfc_index_diagonal_add_right_v)) * bc) + (pfc_left_diagonal_add_right_vterm)))))) \\/ (((exists pfc_gap_diagonal_add_right_vtermleftoutside. pfc_gap_diagonal_add_right_vtermleftoutside+(L)=(pfc_index_diagonal_add_right_v)) /\\ (((pfc_left_diagonal_add_right_vterm)=0))))) /\\ ((((((exists pfa_gap_diagonal_add_right_vtermrightinside. pfa_gap_diagonal_add_right_vtermrightinside + S (pfc_complement_diagonal_add_right_vterm) = (M)) /\\ ((((exists ff_h_pfp_diagonal_add_right_vtermrightentry. ff_h_pfp_diagonal_add_right_vtermrightentry + S (pfc_right_diagonal_add_right_vterm) = S ((S (pfc_complement_diagonal_add_right_vterm)) * dc)) /\\ exists ff_q_pfp_diagonal_add_right_vtermrightentry. db = ff_q_pfp_diagonal_add_right_vtermrightentry * S ((S (pfc_complement_diagonal_add_right_vterm)) * dc) + (pfc_right_diagonal_add_right_vterm)))))) \\/ (((exists pfc_gap_diagonal_add_right_vtermrightoutside. pfc_gap_diagonal_add_right_vtermrightoutside+(M)=(pfc_complement_diagonal_add_right_vterm)) /\\ (((pfc_right_diagonal_add_right_vterm)=0))))) /\\ (((pfc_value_diagonal_add_right_v)=pfc_left_diagonal_add_right_vterm*pfc_right_diagonal_add_right_vterm))))))))))) -> (exists fs_u_pfc_diagonal_add_right_sum_v fs_v_pfc_diagonal_add_right_sum_v. ((((exists fs_h_pfc_diagonal_add_right_sum_v_body_start. fs_h_pfc_diagonal_add_right_sum_v_body_start + S (0) = S ((S (0)) * fs_v_pfc_diagonal_add_right_sum_v)) /\\ exists fs_q_pfc_diagonal_add_right_sum_v_body_start. fs_u_pfc_diagonal_add_right_sum_v = fs_q_pfc_diagonal_add_right_sum_v_body_start * S ((S (0)) * fs_v_pfc_diagonal_add_right_sum_v) + (0))) /\\ ((((exists fs_h_pfc_diagonal_add_right_sum_v_body_terminal. fs_h_pfc_diagonal_add_right_sum_v_body_terminal + S (v) = S ((S (N)) * fs_v_pfc_diagonal_add_right_sum_v)) /\\ exists fs_q_pfc_diagonal_add_right_sum_v_body_terminal. fs_u_pfc_diagonal_add_right_sum_v = fs_q_pfc_diagonal_add_right_sum_v_body_terminal * S ((S (N)) * fs_v_pfc_diagonal_add_right_sum_v) + (v))) /\\ forall fs_i_pfc_diagonal_add_right_sum_v_body_steps. (exists fs_lt_pfc_diagonal_add_right_sum_v_body_steps_bound. fs_lt_pfc_diagonal_add_right_sum_v_body_steps_bound + S fs_i_pfc_diagonal_add_right_sum_v_body_steps = N) -> exists fs_a_pfc_diagonal_add_right_sum_v_body_steps fs_r_pfc_diagonal_add_right_sum_v_body_steps fs_s_pfc_diagonal_add_right_sum_v_body_steps. ((((exists fs_h_pfc_diagonal_add_right_sum_v_body_steps_summand. fs_h_pfc_diagonal_add_right_sum_v_body_steps_summand + S (fs_a_pfc_diagonal_add_right_sum_v_body_steps) = S ((S (fs_i_pfc_diagonal_add_right_sum_v_body_steps)) * vc)) /\\ exists fs_q_pfc_diagonal_add_right_sum_v_body_steps_summand. vb = fs_q_pfc_diagonal_add_right_sum_v_body_steps_summand * S ((S (fs_i_pfc_diagonal_add_right_sum_v_body_steps)) * vc) + (fs_a_pfc_diagonal_add_right_sum_v_body_steps))) /\\ ((((exists fs_h_pfc_diagonal_add_right_sum_v_body_steps_partial. fs_h_pfc_diagonal_add_right_sum_v_body_steps_partial + S (fs_r_pfc_diagonal_add_right_sum_v_body_steps) = S ((S (fs_i_pfc_diagonal_add_right_sum_v_body_steps)) * fs_v_pfc_diagonal_add_right_sum_v)) /\\ exists fs_q_pfc_diagonal_add_right_sum_v_body_steps_partial. fs_u_pfc_diagonal_add_right_sum_v = fs_q_pfc_diagonal_add_right_sum_v_body_steps_partial * S ((S (fs_i_pfc_diagonal_add_right_sum_v_body_steps)) * fs_v_pfc_diagonal_add_right_sum_v) + (fs_r_pfc_diagonal_add_right_sum_v_body_steps))) /\\ ((((exists fs_h_pfc_diagonal_add_right_sum_v_body_steps_successor. fs_h_pfc_diagonal_add_right_sum_v_body_steps_successor + S (fs_s_pfc_diagonal_add_right_sum_v_body_steps) = S ((S (S fs_i_pfc_diagonal_add_right_sum_v_body_steps)) * fs_v_pfc_diagonal_add_right_sum_v)) /\\ exists fs_q_pfc_diagonal_add_right_sum_v_body_steps_successor. fs_u_pfc_diagonal_add_right_sum_v = fs_q_pfc_diagonal_add_right_sum_v_body_steps_successor * S ((S (S fs_i_pfc_diagonal_add_right_sum_v_body_steps)) * fs_v_pfc_diagonal_add_right_sum_v) + (fs_s_pfc_diagonal_add_right_sum_v_body_steps))) /\\ fs_s_pfc_diagonal_add_right_sum_v_body_steps = fs_r_pfc_diagonal_add_right_sum_v_body_steps + fs_a_pfc_diagonal_add_right_sum_v_body_steps)))))) -> (forall pfc_index_diagonal_add_right_w. (exists pfa_gap_diagonal_add_right_wbound. pfa_gap_diagonal_add_right_wbound + S (pfc_index_diagonal_add_right_w) = (N)) -> exists pfc_value_diagonal_add_right_w. ((((exists ff_h_pfp_diagonal_add_right_wentry. ff_h_pfp_diagonal_add_right_wentry + S (pfc_value_diagonal_add_right_w) = S ((S (pfc_index_diagonal_add_right_w)) * wc)) /\\ exists ff_q_pfp_diagonal_add_right_wentry. wb = ff_q_pfp_diagonal_add_right_wentry * S ((S (pfc_index_diagonal_add_right_w)) * wc) + (pfc_value_diagonal_add_right_w))) /\\ ((exists pfc_complement_diagonal_add_right_wterm pfc_left_diagonal_add_right_wterm pfc_right_diagonal_add_right_wterm. (((pfc_index_diagonal_add_right_w)+pfc_complement_diagonal_add_right_wterm=(i)) /\\ ((((((exists pfa_gap_diagonal_add_right_wtermleftinside. pfa_gap_diagonal_add_right_wtermleftinside + S (pfc_index_diagonal_add_right_w) = (L)) /\\ ((((exists ff_h_pfp_diagonal_add_right_wtermleftentry. ff_h_pfp_diagonal_add_right_wtermleftentry + S (pfc_left_diagonal_add_right_wterm) = S ((S (pfc_index_diagonal_add_right_w)) * cc)) /\\ exists ff_q_pfp_diagonal_add_right_wtermleftentry. cb = ff_q_pfp_diagonal_add_right_wtermleftentry * S ((S (pfc_index_diagonal_add_right_w)) * cc) + (pfc_left_diagonal_add_right_wterm)))))) \\/ (((exists pfc_gap_diagonal_add_right_wtermleftoutside. pfc_gap_diagonal_add_right_wtermleftoutside+(L)=(pfc_index_diagonal_add_right_w)) /\\ (((pfc_left_diagonal_add_right_wterm)=0))))) /\\ ((((((exists pfa_gap_diagonal_add_right_wtermrightinside. pfa_gap_diagonal_add_right_wtermrightinside + S (pfc_complement_diagonal_add_right_wterm) = (M)) /\\ ((((exists ff_h_pfp_diagonal_add_right_wtermrightentry. ff_h_pfp_diagonal_add_right_wtermrightentry + S (pfc_right_diagonal_add_right_wterm) = S ((S (pfc_complement_diagonal_add_right_wterm)) * dc)) /\\ exists ff_q_pfp_diagonal_add_right_wtermrightentry. db = ff_q_pfp_diagonal_add_right_wtermrightentry * S ((S (pfc_complement_diagonal_add_right_wterm)) * dc) + (pfc_right_diagonal_add_right_wterm)))))) \\/ (((exists pfc_gap_diagonal_add_right_wtermrightoutside. pfc_gap_diagonal_add_right_wtermrightoutside+(M)=(pfc_complement_diagonal_add_right_wterm)) /\\ (((pfc_right_diagonal_add_right_wterm)=0))))) /\\ (((pfc_value_diagonal_add_right_w)=pfc_left_diagonal_add_right_wterm*pfc_right_diagonal_add_right_wterm))))))))))) -> (exists fs_u_pfc_diagonal_add_right_sum_w fs_v_pfc_diagonal_add_right_sum_w. ((((exists fs_h_pfc_diagonal_add_right_sum_w_body_start. fs_h_pfc_diagonal_add_right_sum_w_body_start + S (0) = S ((S (0)) * fs_v_pfc_diagonal_add_right_sum_w)) /\\ exists fs_q_pfc_diagonal_add_right_sum_w_body_start. fs_u_pfc_diagonal_add_right_sum_w = fs_q_pfc_diagonal_add_right_sum_w_body_start * S ((S (0)) * fs_v_pfc_diagonal_add_right_sum_w) + (0))) /\\ ((((exists fs_h_pfc_diagonal_add_right_sum_w_body_terminal. fs_h_pfc_diagonal_add_right_sum_w_body_terminal + S (w) = S ((S (N)) * fs_v_pfc_diagonal_add_right_sum_w)) /\\ exists fs_q_pfc_diagonal_add_right_sum_w_body_terminal. fs_u_pfc_diagonal_add_right_sum_w = fs_q_pfc_diagonal_add_right_sum_w_body_terminal * S ((S (N)) * fs_v_pfc_diagonal_add_right_sum_w) + (w))) /\\ forall fs_i_pfc_diagonal_add_right_sum_w_body_steps. (exists fs_lt_pfc_diagonal_add_right_sum_w_body_steps_bound. fs_lt_pfc_diagonal_add_right_sum_w_body_steps_bound + S fs_i_pfc_diagonal_add_right_sum_w_body_steps = N) -> exists fs_a_pfc_diagonal_add_right_sum_w_body_steps fs_r_pfc_diagonal_add_right_sum_w_body_steps fs_s_pfc_diagonal_add_right_sum_w_body_steps. ((((exists fs_h_pfc_diagonal_add_right_sum_w_body_steps_summand. fs_h_pfc_diagonal_add_right_sum_w_body_steps_summand + S (fs_a_pfc_diagonal_add_right_sum_w_body_steps) = S ((S (fs_i_pfc_diagonal_add_right_sum_w_body_steps)) * wc)) /\\ exists fs_q_pfc_diagonal_add_right_sum_w_body_steps_summand. wb = fs_q_pfc_diagonal_add_right_sum_w_body_steps_summand * S ((S (fs_i_pfc_diagonal_add_right_sum_w_body_steps)) * wc) + (fs_a_pfc_diagonal_add_right_sum_w_body_steps))) /\\ ((((exists fs_h_pfc_diagonal_add_right_sum_w_body_steps_partial. fs_h_pfc_diagonal_add_right_sum_w_body_steps_partial + S (fs_r_pfc_diagonal_add_right_sum_w_body_steps) = S ((S (fs_i_pfc_diagonal_add_right_sum_w_body_steps)) * fs_v_pfc_diagonal_add_right_sum_w)) /\\ exists fs_q_pfc_diagonal_add_right_sum_w_body_steps_partial. fs_u_pfc_diagonal_add_right_sum_w = fs_q_pfc_diagonal_add_right_sum_w_body_steps_partial * S ((S (fs_i_pfc_diagonal_add_right_sum_w_body_steps)) * fs_v_pfc_diagonal_add_right_sum_w) + (fs_r_pfc_diagonal_add_right_sum_w_body_steps))) /\\ ((((exists fs_h_pfc_diagonal_add_right_sum_w_body_steps_successor. fs_h_pfc_diagonal_add_right_sum_w_body_steps_successor + S (fs_s_pfc_diagonal_add_right_sum_w_body_steps) = S ((S (S fs_i_pfc_diagonal_add_right_sum_w_body_steps)) * fs_v_pfc_diagonal_add_right_sum_w)) /\\ exists fs_q_pfc_diagonal_add_right_sum_w_body_steps_successor. fs_u_pfc_diagonal_add_right_sum_w = fs_q_pfc_diagonal_add_right_sum_w_body_steps_successor * S ((S (S fs_i_pfc_diagonal_add_right_sum_w_body_steps)) * fs_v_pfc_diagonal_add_right_sum_w) + (fs_s_pfc_diagonal_add_right_sum_w_body_steps))) /\\ fs_s_pfc_diagonal_add_right_sum_w_body_steps = fs_r_pfc_diagonal_add_right_sum_w_body_steps + fs_a_pfc_diagonal_add_right_sum_w_body_steps)))))) -> (exists pfa_offset_left_diagonal_add_right_result pfa_offset_right_diagonal_add_right_result. (u+v) + (p) * pfa_offset_left_diagonal_add_right_result = (w) + (p) * pfa_offset_right_diagonal_add_right_result)", "statement_sha256": "9c53c047069e8473a706c3e957dd75ba6b90ac4523f41ed1564f5c05412e0288", "summary": "The three independently beta-coded actual antidiagonal sums obey right additive congruence, including empty sum prefixes and with no raw-code equality.", "summary_sha256": "6cbc348540dbd3bce2d31e2ef4e0d9460fa43c7b41ec61dae26dae6efaab41d3" }, { "alpha_v33_frontier_enrollment": { "body_receipt_sha256": "76cb611d0562ca84742dd486bcafee8bf9007cb91bba09e809ad424ca6613f46", "bundle_campaign": "polynomial-euclidean-division", "bundle_node_id": 324, "bundle_sha256": "6ae667d8518e4dbe722bb08ad1b08715a0d282c2893e533c8133d770fe861dcf", "campaign": "polynomial-euclidean-division", "first_enrolled_version": "v33", "parent_catalog_sha256": "41b9f387d88a5a4f0fe5ee2bd5578f37a27a4657b0a80f1a1a2cb5109f69a623", "rfc_sha256": "38d609ffe5dbf7c36cd0e5112dcfb219934bac3c8af09da5d96f30844ae44895", 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hu_witness", "cases hu_witness_witness", "cases hu_witness_witness_witness", "cases hu_witness_witness_witness_right", "cases hv", "cases hv_witness", "cases hv_witness_witness", "cases hv_witness_witness_witness", "cases hv_witness_witness_witness_right", "cases hw", "cases hw_witness", "cases hw_witness_witness", "cases hw_witness_witness_witness", "cases hw_witness_witness_witness_right", "have hsum : exists pfa_offset_left_coefficient_add_left_sum pfa_offset_right_coefficient_add_left_sum. (x2+x5) + (p) * pfa_offset_left_coefficient_add_left_sum = (x8) + (p) * pfa_offset_right_coefficient_add_left_sum", "specialize polynomial_diagonal_sum_left_add_congruent (p)", "specialize polynomial_diagonal_sum_left_add_congruent (ab)", "specialize polynomial_diagonal_sum_left_add_congruent (ac)", "specialize polynomial_diagonal_sum_left_add_congruent (bb)", "specialize polynomial_diagonal_sum_left_add_congruent (bc)", "specialize polynomial_diagonal_sum_left_add_congruent (cb)", "specialize polynomial_diagonal_sum_left_add_congruent (cc)", "specialize polynomial_diagonal_sum_left_add_congruent (L)", "specialize polynomial_diagonal_sum_left_add_congruent (db)", "specialize polynomial_diagonal_sum_left_add_congruent (dc)", "specialize polynomial_diagonal_sum_left_add_congruent (M)", "specialize polynomial_diagonal_sum_left_add_congruent (i)", "specialize polynomial_diagonal_sum_left_add_congruent (S i)", "specialize polynomial_diagonal_sum_left_add_congruent (x)", "specialize polynomial_diagonal_sum_left_add_congruent (x1)", "specialize polynomial_diagonal_sum_left_add_congruent (x3)", "specialize polynomial_diagonal_sum_left_add_congruent (x4)", "specialize polynomial_diagonal_sum_left_add_congruent (x6)", "specialize polynomial_diagonal_sum_left_add_congruent (x7)", "specialize polynomial_diagonal_sum_left_add_congruent (x2)", "specialize polynomial_diagonal_sum_left_add_congruent (x5)", "specialize polynomial_diagonal_sum_left_add_congruent (x8)", "apply polynomial_diagonal_sum_left_add_congruent", "exact hs", "exact hu_witness_witness_witness_left", "exact hu_witness_witness_witness_right_left", "exact hv_witness_witness_witness_left", "exact hv_witness_witness_witness_right_left", "exact hw_witness_witness_witness_left", "exact hw_witness_witness_witness_right_left", "cases hu_witness_witness_witness_right_right", "cases hv_witness_witness_witness_right_right", "cases hw_witness_witness_witness_right_right", "have hraw : exists pfa_offset_left_coefficient_add_left_raw pfa_offset_right_coefficient_add_left_raw. (x2+x5) + (p) * pfa_offset_left_coefficient_add_left_raw = (w) + (p) * pfa_offset_right_coefficient_add_left_raw", "specialize mod_eq_trans (p)", "specialize mod_eq_trans (x2+x5)", "specialize mod_eq_trans (x8)", "specialize mod_eq_trans (w)", "apply mod_eq_trans", "exact hsum", "exact hw_witness_witness_witness_right_right_right", "split", "exact hu_witness_witness_witness_right_right_left", "split", "exact hv_witness_witness_witness_right_right_left", "split", "exact hw_witness_witness_witness_right_right_left", "specialize mod_eq_trans (p)", "specialize mod_eq_trans (u+v)", "specialize mod_eq_trans (x2+x5)", "specialize mod_eq_trans (w)", "apply mod_eq_trans", "specialize mod_eq_symm (p)", "specialize mod_eq_symm (x2+x5)", "specialize mod_eq_symm (u+v)", "apply mod_eq_symm", "specialize mod_eq_add (p)", "specialize mod_eq_add (x2)", "specialize mod_eq_add (u)", "specialize mod_eq_add (x5)", "specialize mod_eq_add (v)", "apply mod_eq_add", "exact hu_witness_witness_witness_right_right_right", "exact hv_witness_witness_witness_right_right_right", "exact hraw" ], "script_sha256": "9e01eb601ef3f9ddec621b4e76b0f8f8813e16de780dc6567d8e09c6acc42892", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_distributivity_candidate.py", "sha256": "a959962d631759cd1fc773dd7eef2fadf4f3f95361d6d7bc8c6a9e82d0d4ab86" }, "statement": "forall p ab ac bb bc cb cc L db dc M i u v w. (forall pfp_index_coefficient_add_left_source. (exists pfa_gap_coefficient_add_left_sourceindex. pfa_gap_coefficient_add_left_sourceindex + S (pfp_index_coefficient_add_left_source) = (L)) -> exists pfp_left_coefficient_add_left_source pfp_right_coefficient_add_left_source pfp_value_coefficient_add_left_source. ((((exists ff_h_pfp_coefficient_add_left_sourceleft. ff_h_pfp_coefficient_add_left_sourceleft + S (pfp_left_coefficient_add_left_source) = S ((S (pfp_index_coefficient_add_left_source)) * ac)) /\\ exists ff_q_pfp_coefficient_add_left_sourceleft. ab = ff_q_pfp_coefficient_add_left_sourceleft * S ((S (pfp_index_coefficient_add_left_source)) * ac) + (pfp_left_coefficient_add_left_source))) /\\ (((((exists ff_h_pfp_coefficient_add_left_sourceright. ff_h_pfp_coefficient_add_left_sourceright + S (pfp_right_coefficient_add_left_source) = S ((S (pfp_index_coefficient_add_left_source)) * bc)) /\\ exists ff_q_pfp_coefficient_add_left_sourceright. bb = ff_q_pfp_coefficient_add_left_sourceright * S ((S (pfp_index_coefficient_add_left_source)) * bc) + (pfp_right_coefficient_add_left_source))) /\\ (((((exists ff_h_pfp_coefficient_add_left_sourcetarget. ff_h_pfp_coefficient_add_left_sourcetarget + S (pfp_value_coefficient_add_left_source) = S ((S (pfp_index_coefficient_add_left_source)) * cc)) /\\ exists ff_q_pfp_coefficient_add_left_sourcetarget. cb = ff_q_pfp_coefficient_add_left_sourcetarget * S ((S (pfp_index_coefficient_add_left_source)) * cc) + (pfp_value_coefficient_add_left_source))) /\\ ((((exists pfa_gap_coefficient_add_left_sourceoperationleft. pfa_gap_coefficient_add_left_sourceoperationleft + S (pfp_left_coefficient_add_left_source) = (p)) /\\ (((exists pfa_gap_coefficient_add_left_sourceoperationright. pfa_gap_coefficient_add_left_sourceoperationright + S (pfp_right_coefficient_add_left_source) = (p)) /\\ ((((exists pfa_gap_coefficient_add_left_sourceoperationresultbound. pfa_gap_coefficient_add_left_sourceoperationresultbound + S (pfp_value_coefficient_add_left_source) = (p)) /\\ ((exists pfa_offset_left_coefficient_add_left_sourceoperationresultcongruence pfa_offset_right_coefficient_add_left_sourceoperationresultcongruence. ((pfp_left_coefficient_add_left_source) + (pfp_right_coefficient_add_left_source)) + (p) * pfa_offset_left_coefficient_add_left_sourceoperationresultcongruence = (pfp_value_coefficient_add_left_source) + (p) * pfa_offset_right_coefficient_add_left_sourceoperationresultcongruence)))))))))))))))) -> (exists pfc_terms_code_coefficient_add_left_u pfc_terms_scale_coefficient_add_left_u pfc_natural_sum_coefficient_add_left_u. ((forall pfc_index_coefficient_add_left_udiagonal. (exists pfa_gap_coefficient_add_left_udiagonalbound. pfa_gap_coefficient_add_left_udiagonalbound + S (pfc_index_coefficient_add_left_udiagonal) = (S (i))) -> exists pfc_value_coefficient_add_left_udiagonal. ((((exists ff_h_pfp_coefficient_add_left_udiagonalentry. ff_h_pfp_coefficient_add_left_udiagonalentry + S (pfc_value_coefficient_add_left_udiagonal) = S ((S (pfc_index_coefficient_add_left_udiagonal)) * pfc_terms_scale_coefficient_add_left_u)) /\\ exists ff_q_pfp_coefficient_add_left_udiagonalentry. pfc_terms_code_coefficient_add_left_u = ff_q_pfp_coefficient_add_left_udiagonalentry * S ((S (pfc_index_coefficient_add_left_udiagonal)) * pfc_terms_scale_coefficient_add_left_u) + (pfc_value_coefficient_add_left_udiagonal))) /\\ ((exists pfc_complement_coefficient_add_left_udiagonalterm pfc_left_coefficient_add_left_udiagonalterm pfc_right_coefficient_add_left_udiagonalterm. (((pfc_index_coefficient_add_left_udiagonal)+pfc_complement_coefficient_add_left_udiagonalterm=(i)) /\\ ((((((exists pfa_gap_coefficient_add_left_udiagonaltermleftinside. pfa_gap_coefficient_add_left_udiagonaltermleftinside + S (pfc_index_coefficient_add_left_udiagonal) = (M)) /\\ ((((exists ff_h_pfp_coefficient_add_left_udiagonaltermleftentry. ff_h_pfp_coefficient_add_left_udiagonaltermleftentry + S (pfc_left_coefficient_add_left_udiagonalterm) = S ((S (pfc_index_coefficient_add_left_udiagonal)) * dc)) /\\ exists ff_q_pfp_coefficient_add_left_udiagonaltermleftentry. db = ff_q_pfp_coefficient_add_left_udiagonaltermleftentry * S ((S (pfc_index_coefficient_add_left_udiagonal)) * dc) + (pfc_left_coefficient_add_left_udiagonalterm)))))) \\/ (((exists pfc_gap_coefficient_add_left_udiagonaltermleftoutside. pfc_gap_coefficient_add_left_udiagonaltermleftoutside+(M)=(pfc_index_coefficient_add_left_udiagonal)) /\\ (((pfc_left_coefficient_add_left_udiagonalterm)=0))))) /\\ ((((((exists pfa_gap_coefficient_add_left_udiagonaltermrightinside. pfa_gap_coefficient_add_left_udiagonaltermrightinside + S (pfc_complement_coefficient_add_left_udiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_coefficient_add_left_udiagonaltermrightentry. ff_h_pfp_coefficient_add_left_udiagonaltermrightentry + S (pfc_right_coefficient_add_left_udiagonalterm) = S ((S (pfc_complement_coefficient_add_left_udiagonalterm)) * ac)) /\\ exists ff_q_pfp_coefficient_add_left_udiagonaltermrightentry. ab = ff_q_pfp_coefficient_add_left_udiagonaltermrightentry * S ((S (pfc_complement_coefficient_add_left_udiagonalterm)) * ac) + (pfc_right_coefficient_add_left_udiagonalterm)))))) \\/ (((exists pfc_gap_coefficient_add_left_udiagonaltermrightoutside. pfc_gap_coefficient_add_left_udiagonaltermrightoutside+(L)=(pfc_complement_coefficient_add_left_udiagonalterm)) /\\ (((pfc_right_coefficient_add_left_udiagonalterm)=0))))) /\\ (((pfc_value_coefficient_add_left_udiagonal)=pfc_left_coefficient_add_left_udiagonalterm*pfc_right_coefficient_add_left_udiagonalterm))))))))))) /\\ (((exists fs_u_pfc_coefficient_add_left_usum fs_v_pfc_coefficient_add_left_usum. ((((exists fs_h_pfc_coefficient_add_left_usum_body_start. fs_h_pfc_coefficient_add_left_usum_body_start + S (0) = S ((S (0)) * fs_v_pfc_coefficient_add_left_usum)) /\\ exists fs_q_pfc_coefficient_add_left_usum_body_start. fs_u_pfc_coefficient_add_left_usum = fs_q_pfc_coefficient_add_left_usum_body_start * S ((S (0)) * fs_v_pfc_coefficient_add_left_usum) + (0))) /\\ ((((exists fs_h_pfc_coefficient_add_left_usum_body_terminal. fs_h_pfc_coefficient_add_left_usum_body_terminal + S (pfc_natural_sum_coefficient_add_left_u) = S ((S (S (i))) * fs_v_pfc_coefficient_add_left_usum)) /\\ exists fs_q_pfc_coefficient_add_left_usum_body_terminal. fs_u_pfc_coefficient_add_left_usum = fs_q_pfc_coefficient_add_left_usum_body_terminal * S ((S (S (i))) * fs_v_pfc_coefficient_add_left_usum) + (pfc_natural_sum_coefficient_add_left_u))) /\\ forall fs_i_pfc_coefficient_add_left_usum_body_steps. (exists fs_lt_pfc_coefficient_add_left_usum_body_steps_bound. fs_lt_pfc_coefficient_add_left_usum_body_steps_bound + S fs_i_pfc_coefficient_add_left_usum_body_steps = S (i)) -> exists fs_a_pfc_coefficient_add_left_usum_body_steps fs_r_pfc_coefficient_add_left_usum_body_steps fs_s_pfc_coefficient_add_left_usum_body_steps. ((((exists fs_h_pfc_coefficient_add_left_usum_body_steps_summand. fs_h_pfc_coefficient_add_left_usum_body_steps_summand + S (fs_a_pfc_coefficient_add_left_usum_body_steps) = S ((S (fs_i_pfc_coefficient_add_left_usum_body_steps)) * pfc_terms_scale_coefficient_add_left_u)) /\\ exists fs_q_pfc_coefficient_add_left_usum_body_steps_summand. pfc_terms_code_coefficient_add_left_u = fs_q_pfc_coefficient_add_left_usum_body_steps_summand * S ((S (fs_i_pfc_coefficient_add_left_usum_body_steps)) * pfc_terms_scale_coefficient_add_left_u) + (fs_a_pfc_coefficient_add_left_usum_body_steps))) /\\ ((((exists fs_h_pfc_coefficient_add_left_usum_body_steps_partial. fs_h_pfc_coefficient_add_left_usum_body_steps_partial + S (fs_r_pfc_coefficient_add_left_usum_body_steps) = S ((S (fs_i_pfc_coefficient_add_left_usum_body_steps)) * fs_v_pfc_coefficient_add_left_usum)) /\\ exists fs_q_pfc_coefficient_add_left_usum_body_steps_partial. fs_u_pfc_coefficient_add_left_usum = fs_q_pfc_coefficient_add_left_usum_body_steps_partial * S ((S (fs_i_pfc_coefficient_add_left_usum_body_steps)) * fs_v_pfc_coefficient_add_left_usum) + (fs_r_pfc_coefficient_add_left_usum_body_steps))) /\\ ((((exists fs_h_pfc_coefficient_add_left_usum_body_steps_successor. fs_h_pfc_coefficient_add_left_usum_body_steps_successor + S (fs_s_pfc_coefficient_add_left_usum_body_steps) = S ((S (S fs_i_pfc_coefficient_add_left_usum_body_steps)) * fs_v_pfc_coefficient_add_left_usum)) /\\ exists fs_q_pfc_coefficient_add_left_usum_body_steps_successor. fs_u_pfc_coefficient_add_left_usum = fs_q_pfc_coefficient_add_left_usum_body_steps_successor * S ((S (S fs_i_pfc_coefficient_add_left_usum_body_steps)) * fs_v_pfc_coefficient_add_left_usum) + (fs_s_pfc_coefficient_add_left_usum_body_steps))) /\\ fs_s_pfc_coefficient_add_left_usum_body_steps = fs_r_pfc_coefficient_add_left_usum_body_steps + fs_a_pfc_coefficient_add_left_usum_body_steps)))))) /\\ ((((exists pfa_gap_coefficient_add_left_uresiduebound. pfa_gap_coefficient_add_left_uresiduebound + S (u) = (p)) /\\ ((exists pfa_offset_left_coefficient_add_left_uresiduecongruence pfa_offset_right_coefficient_add_left_uresiduecongruence. (pfc_natural_sum_coefficient_add_left_u) + (p) * pfa_offset_left_coefficient_add_left_uresiduecongruence = (u) + (p) * pfa_offset_right_coefficient_add_left_uresiduecongruence))))))))) -> (exists pfc_terms_code_coefficient_add_left_v pfc_terms_scale_coefficient_add_left_v pfc_natural_sum_coefficient_add_left_v. ((forall pfc_index_coefficient_add_left_vdiagonal. (exists pfa_gap_coefficient_add_left_vdiagonalbound. pfa_gap_coefficient_add_left_vdiagonalbound + S (pfc_index_coefficient_add_left_vdiagonal) = (S (i))) -> exists pfc_value_coefficient_add_left_vdiagonal. ((((exists ff_h_pfp_coefficient_add_left_vdiagonalentry. ff_h_pfp_coefficient_add_left_vdiagonalentry + S (pfc_value_coefficient_add_left_vdiagonal) = S ((S (pfc_index_coefficient_add_left_vdiagonal)) * pfc_terms_scale_coefficient_add_left_v)) /\\ exists ff_q_pfp_coefficient_add_left_vdiagonalentry. pfc_terms_code_coefficient_add_left_v = ff_q_pfp_coefficient_add_left_vdiagonalentry * S ((S (pfc_index_coefficient_add_left_vdiagonal)) * pfc_terms_scale_coefficient_add_left_v) + (pfc_value_coefficient_add_left_vdiagonal))) /\\ ((exists pfc_complement_coefficient_add_left_vdiagonalterm pfc_left_coefficient_add_left_vdiagonalterm pfc_right_coefficient_add_left_vdiagonalterm. (((pfc_index_coefficient_add_left_vdiagonal)+pfc_complement_coefficient_add_left_vdiagonalterm=(i)) /\\ ((((((exists pfa_gap_coefficient_add_left_vdiagonaltermleftinside. pfa_gap_coefficient_add_left_vdiagonaltermleftinside + S (pfc_index_coefficient_add_left_vdiagonal) = (M)) /\\ ((((exists ff_h_pfp_coefficient_add_left_vdiagonaltermleftentry. ff_h_pfp_coefficient_add_left_vdiagonaltermleftentry + S (pfc_left_coefficient_add_left_vdiagonalterm) = S ((S (pfc_index_coefficient_add_left_vdiagonal)) * dc)) /\\ exists ff_q_pfp_coefficient_add_left_vdiagonaltermleftentry. db = ff_q_pfp_coefficient_add_left_vdiagonaltermleftentry * S ((S (pfc_index_coefficient_add_left_vdiagonal)) * dc) + (pfc_left_coefficient_add_left_vdiagonalterm)))))) \\/ (((exists pfc_gap_coefficient_add_left_vdiagonaltermleftoutside. pfc_gap_coefficient_add_left_vdiagonaltermleftoutside+(M)=(pfc_index_coefficient_add_left_vdiagonal)) /\\ (((pfc_left_coefficient_add_left_vdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_coefficient_add_left_vdiagonaltermrightinside. pfa_gap_coefficient_add_left_vdiagonaltermrightinside + S (pfc_complement_coefficient_add_left_vdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_coefficient_add_left_vdiagonaltermrightentry. ff_h_pfp_coefficient_add_left_vdiagonaltermrightentry + S (pfc_right_coefficient_add_left_vdiagonalterm) = S ((S (pfc_complement_coefficient_add_left_vdiagonalterm)) * bc)) /\\ exists ff_q_pfp_coefficient_add_left_vdiagonaltermrightentry. bb = ff_q_pfp_coefficient_add_left_vdiagonaltermrightentry * S ((S (pfc_complement_coefficient_add_left_vdiagonalterm)) * bc) + (pfc_right_coefficient_add_left_vdiagonalterm)))))) \\/ (((exists pfc_gap_coefficient_add_left_vdiagonaltermrightoutside. pfc_gap_coefficient_add_left_vdiagonaltermrightoutside+(L)=(pfc_complement_coefficient_add_left_vdiagonalterm)) /\\ (((pfc_right_coefficient_add_left_vdiagonalterm)=0))))) /\\ (((pfc_value_coefficient_add_left_vdiagonal)=pfc_left_coefficient_add_left_vdiagonalterm*pfc_right_coefficient_add_left_vdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_coefficient_add_left_vsum fs_v_pfc_coefficient_add_left_vsum. ((((exists fs_h_pfc_coefficient_add_left_vsum_body_start. fs_h_pfc_coefficient_add_left_vsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_coefficient_add_left_vsum)) /\\ exists fs_q_pfc_coefficient_add_left_vsum_body_start. fs_u_pfc_coefficient_add_left_vsum = fs_q_pfc_coefficient_add_left_vsum_body_start * S ((S (0)) * fs_v_pfc_coefficient_add_left_vsum) + (0))) /\\ ((((exists fs_h_pfc_coefficient_add_left_vsum_body_terminal. fs_h_pfc_coefficient_add_left_vsum_body_terminal + S (pfc_natural_sum_coefficient_add_left_v) = S ((S (S (i))) * fs_v_pfc_coefficient_add_left_vsum)) /\\ exists fs_q_pfc_coefficient_add_left_vsum_body_terminal. fs_u_pfc_coefficient_add_left_vsum = fs_q_pfc_coefficient_add_left_vsum_body_terminal * S ((S (S (i))) * fs_v_pfc_coefficient_add_left_vsum) + (pfc_natural_sum_coefficient_add_left_v))) /\\ forall fs_i_pfc_coefficient_add_left_vsum_body_steps. (exists fs_lt_pfc_coefficient_add_left_vsum_body_steps_bound. fs_lt_pfc_coefficient_add_left_vsum_body_steps_bound + S fs_i_pfc_coefficient_add_left_vsum_body_steps = S (i)) -> exists fs_a_pfc_coefficient_add_left_vsum_body_steps fs_r_pfc_coefficient_add_left_vsum_body_steps fs_s_pfc_coefficient_add_left_vsum_body_steps. ((((exists fs_h_pfc_coefficient_add_left_vsum_body_steps_summand. fs_h_pfc_coefficient_add_left_vsum_body_steps_summand + S (fs_a_pfc_coefficient_add_left_vsum_body_steps) = S ((S (fs_i_pfc_coefficient_add_left_vsum_body_steps)) * pfc_terms_scale_coefficient_add_left_v)) /\\ exists fs_q_pfc_coefficient_add_left_vsum_body_steps_summand. pfc_terms_code_coefficient_add_left_v = fs_q_pfc_coefficient_add_left_vsum_body_steps_summand * S ((S (fs_i_pfc_coefficient_add_left_vsum_body_steps)) * pfc_terms_scale_coefficient_add_left_v) + (fs_a_pfc_coefficient_add_left_vsum_body_steps))) /\\ ((((exists fs_h_pfc_coefficient_add_left_vsum_body_steps_partial. fs_h_pfc_coefficient_add_left_vsum_body_steps_partial + S (fs_r_pfc_coefficient_add_left_vsum_body_steps) = S ((S (fs_i_pfc_coefficient_add_left_vsum_body_steps)) * fs_v_pfc_coefficient_add_left_vsum)) /\\ exists fs_q_pfc_coefficient_add_left_vsum_body_steps_partial. fs_u_pfc_coefficient_add_left_vsum = fs_q_pfc_coefficient_add_left_vsum_body_steps_partial * S ((S (fs_i_pfc_coefficient_add_left_vsum_body_steps)) * fs_v_pfc_coefficient_add_left_vsum) + (fs_r_pfc_coefficient_add_left_vsum_body_steps))) /\\ ((((exists fs_h_pfc_coefficient_add_left_vsum_body_steps_successor. fs_h_pfc_coefficient_add_left_vsum_body_steps_successor + S (fs_s_pfc_coefficient_add_left_vsum_body_steps) = S ((S (S fs_i_pfc_coefficient_add_left_vsum_body_steps)) * fs_v_pfc_coefficient_add_left_vsum)) /\\ exists fs_q_pfc_coefficient_add_left_vsum_body_steps_successor. fs_u_pfc_coefficient_add_left_vsum = fs_q_pfc_coefficient_add_left_vsum_body_steps_successor * S ((S (S fs_i_pfc_coefficient_add_left_vsum_body_steps)) * fs_v_pfc_coefficient_add_left_vsum) + (fs_s_pfc_coefficient_add_left_vsum_body_steps))) /\\ fs_s_pfc_coefficient_add_left_vsum_body_steps = fs_r_pfc_coefficient_add_left_vsum_body_steps + fs_a_pfc_coefficient_add_left_vsum_body_steps)))))) /\\ ((((exists pfa_gap_coefficient_add_left_vresiduebound. pfa_gap_coefficient_add_left_vresiduebound + S (v) = (p)) /\\ ((exists pfa_offset_left_coefficient_add_left_vresiduecongruence pfa_offset_right_coefficient_add_left_vresiduecongruence. (pfc_natural_sum_coefficient_add_left_v) + (p) * pfa_offset_left_coefficient_add_left_vresiduecongruence = (v) + (p) * pfa_offset_right_coefficient_add_left_vresiduecongruence))))))))) -> (exists pfc_terms_code_coefficient_add_left_w pfc_terms_scale_coefficient_add_left_w pfc_natural_sum_coefficient_add_left_w. ((forall pfc_index_coefficient_add_left_wdiagonal. (exists pfa_gap_coefficient_add_left_wdiagonalbound. pfa_gap_coefficient_add_left_wdiagonalbound + S (pfc_index_coefficient_add_left_wdiagonal) = (S (i))) -> exists pfc_value_coefficient_add_left_wdiagonal. ((((exists ff_h_pfp_coefficient_add_left_wdiagonalentry. ff_h_pfp_coefficient_add_left_wdiagonalentry + S (pfc_value_coefficient_add_left_wdiagonal) = S ((S (pfc_index_coefficient_add_left_wdiagonal)) * pfc_terms_scale_coefficient_add_left_w)) /\\ exists ff_q_pfp_coefficient_add_left_wdiagonalentry. pfc_terms_code_coefficient_add_left_w = ff_q_pfp_coefficient_add_left_wdiagonalentry * S ((S (pfc_index_coefficient_add_left_wdiagonal)) * pfc_terms_scale_coefficient_add_left_w) + (pfc_value_coefficient_add_left_wdiagonal))) /\\ ((exists pfc_complement_coefficient_add_left_wdiagonalterm pfc_left_coefficient_add_left_wdiagonalterm pfc_right_coefficient_add_left_wdiagonalterm. (((pfc_index_coefficient_add_left_wdiagonal)+pfc_complement_coefficient_add_left_wdiagonalterm=(i)) /\\ ((((((exists pfa_gap_coefficient_add_left_wdiagonaltermleftinside. pfa_gap_coefficient_add_left_wdiagonaltermleftinside + S (pfc_index_coefficient_add_left_wdiagonal) = (M)) /\\ ((((exists ff_h_pfp_coefficient_add_left_wdiagonaltermleftentry. ff_h_pfp_coefficient_add_left_wdiagonaltermleftentry + S (pfc_left_coefficient_add_left_wdiagonalterm) = S ((S (pfc_index_coefficient_add_left_wdiagonal)) * dc)) /\\ exists ff_q_pfp_coefficient_add_left_wdiagonaltermleftentry. db = ff_q_pfp_coefficient_add_left_wdiagonaltermleftentry * S ((S (pfc_index_coefficient_add_left_wdiagonal)) * dc) + (pfc_left_coefficient_add_left_wdiagonalterm)))))) \\/ (((exists pfc_gap_coefficient_add_left_wdiagonaltermleftoutside. pfc_gap_coefficient_add_left_wdiagonaltermleftoutside+(M)=(pfc_index_coefficient_add_left_wdiagonal)) /\\ (((pfc_left_coefficient_add_left_wdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_coefficient_add_left_wdiagonaltermrightinside. pfa_gap_coefficient_add_left_wdiagonaltermrightinside + S (pfc_complement_coefficient_add_left_wdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_coefficient_add_left_wdiagonaltermrightentry. ff_h_pfp_coefficient_add_left_wdiagonaltermrightentry + S (pfc_right_coefficient_add_left_wdiagonalterm) = S ((S (pfc_complement_coefficient_add_left_wdiagonalterm)) * cc)) /\\ exists ff_q_pfp_coefficient_add_left_wdiagonaltermrightentry. cb = ff_q_pfp_coefficient_add_left_wdiagonaltermrightentry * S ((S (pfc_complement_coefficient_add_left_wdiagonalterm)) * cc) + (pfc_right_coefficient_add_left_wdiagonalterm)))))) \\/ (((exists pfc_gap_coefficient_add_left_wdiagonaltermrightoutside. pfc_gap_coefficient_add_left_wdiagonaltermrightoutside+(L)=(pfc_complement_coefficient_add_left_wdiagonalterm)) /\\ (((pfc_right_coefficient_add_left_wdiagonalterm)=0))))) /\\ (((pfc_value_coefficient_add_left_wdiagonal)=pfc_left_coefficient_add_left_wdiagonalterm*pfc_right_coefficient_add_left_wdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_coefficient_add_left_wsum fs_v_pfc_coefficient_add_left_wsum. ((((exists fs_h_pfc_coefficient_add_left_wsum_body_start. fs_h_pfc_coefficient_add_left_wsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_coefficient_add_left_wsum)) /\\ exists fs_q_pfc_coefficient_add_left_wsum_body_start. fs_u_pfc_coefficient_add_left_wsum = fs_q_pfc_coefficient_add_left_wsum_body_start * S ((S (0)) * fs_v_pfc_coefficient_add_left_wsum) + (0))) /\\ ((((exists fs_h_pfc_coefficient_add_left_wsum_body_terminal. fs_h_pfc_coefficient_add_left_wsum_body_terminal + S (pfc_natural_sum_coefficient_add_left_w) = S ((S (S (i))) * fs_v_pfc_coefficient_add_left_wsum)) /\\ exists fs_q_pfc_coefficient_add_left_wsum_body_terminal. fs_u_pfc_coefficient_add_left_wsum = fs_q_pfc_coefficient_add_left_wsum_body_terminal * S ((S (S (i))) * fs_v_pfc_coefficient_add_left_wsum) + (pfc_natural_sum_coefficient_add_left_w))) /\\ forall fs_i_pfc_coefficient_add_left_wsum_body_steps. (exists fs_lt_pfc_coefficient_add_left_wsum_body_steps_bound. fs_lt_pfc_coefficient_add_left_wsum_body_steps_bound + S fs_i_pfc_coefficient_add_left_wsum_body_steps = S (i)) -> exists fs_a_pfc_coefficient_add_left_wsum_body_steps fs_r_pfc_coefficient_add_left_wsum_body_steps fs_s_pfc_coefficient_add_left_wsum_body_steps. ((((exists fs_h_pfc_coefficient_add_left_wsum_body_steps_summand. fs_h_pfc_coefficient_add_left_wsum_body_steps_summand + S (fs_a_pfc_coefficient_add_left_wsum_body_steps) = S ((S (fs_i_pfc_coefficient_add_left_wsum_body_steps)) * pfc_terms_scale_coefficient_add_left_w)) /\\ exists fs_q_pfc_coefficient_add_left_wsum_body_steps_summand. pfc_terms_code_coefficient_add_left_w = fs_q_pfc_coefficient_add_left_wsum_body_steps_summand * S ((S (fs_i_pfc_coefficient_add_left_wsum_body_steps)) * pfc_terms_scale_coefficient_add_left_w) + (fs_a_pfc_coefficient_add_left_wsum_body_steps))) /\\ ((((exists fs_h_pfc_coefficient_add_left_wsum_body_steps_partial. fs_h_pfc_coefficient_add_left_wsum_body_steps_partial + S (fs_r_pfc_coefficient_add_left_wsum_body_steps) = S ((S (fs_i_pfc_coefficient_add_left_wsum_body_steps)) * fs_v_pfc_coefficient_add_left_wsum)) /\\ exists fs_q_pfc_coefficient_add_left_wsum_body_steps_partial. fs_u_pfc_coefficient_add_left_wsum = fs_q_pfc_coefficient_add_left_wsum_body_steps_partial * S ((S (fs_i_pfc_coefficient_add_left_wsum_body_steps)) * fs_v_pfc_coefficient_add_left_wsum) + (fs_r_pfc_coefficient_add_left_wsum_body_steps))) /\\ ((((exists fs_h_pfc_coefficient_add_left_wsum_body_steps_successor. fs_h_pfc_coefficient_add_left_wsum_body_steps_successor + S (fs_s_pfc_coefficient_add_left_wsum_body_steps) = S ((S (S fs_i_pfc_coefficient_add_left_wsum_body_steps)) * fs_v_pfc_coefficient_add_left_wsum)) /\\ exists fs_q_pfc_coefficient_add_left_wsum_body_steps_successor. fs_u_pfc_coefficient_add_left_wsum = fs_q_pfc_coefficient_add_left_wsum_body_steps_successor * S ((S (S fs_i_pfc_coefficient_add_left_wsum_body_steps)) * fs_v_pfc_coefficient_add_left_wsum) + (fs_s_pfc_coefficient_add_left_wsum_body_steps))) /\\ fs_s_pfc_coefficient_add_left_wsum_body_steps = fs_r_pfc_coefficient_add_left_wsum_body_steps + fs_a_pfc_coefficient_add_left_wsum_body_steps)))))) /\\ ((((exists pfa_gap_coefficient_add_left_wresiduebound. pfa_gap_coefficient_add_left_wresiduebound + S (w) = (p)) /\\ ((exists pfa_offset_left_coefficient_add_left_wresiduecongruence pfa_offset_right_coefficient_add_left_wresiduecongruence. (pfc_natural_sum_coefficient_add_left_w) + (p) * pfa_offset_left_coefficient_add_left_wresiduecongruence = (w) + (p) * pfa_offset_right_coefficient_add_left_wresiduecongruence))))))))) -> (((exists pfa_gap_coefficient_add_left_resultleft. pfa_gap_coefficient_add_left_resultleft + S (u) = (p)) /\\ (((exists pfa_gap_coefficient_add_left_resultright. pfa_gap_coefficient_add_left_resultright + S (v) = (p)) /\\ ((((exists pfa_gap_coefficient_add_left_resultresultbound. pfa_gap_coefficient_add_left_resultresultbound + S (w) = (p)) /\\ ((exists pfa_offset_left_coefficient_add_left_resultresultcongruence pfa_offset_right_coefficient_add_left_resultresultcongruence. 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"cea85e5c595a021061fb50997df7ad489c8905ab563695209dadc8235f6762cb", "kind": "alpha_v33_live_original_kernel_and_lean_receipt", "path": "research/arithmetic-library/artifacts/alpha-v33-research-receipt-v1.json", "role": "fresh_release_proof_verification", "selector": "families[slug=polynomial-euclidean-division]" }, { "document_sha256": "41b9f387d88a5a4f0fe5ee2bd5578f37a27a4657b0a80f1a1a2cb5109f69a623", "kind": "sealed_alpha_v32_parent", "path": "artifacts/peano-library/alpha/catalog-v32.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "polynomial-euclidean-division", "logical_spec_sha256": "f52dbe4a10c0280fdef872cc0967eadb04d0127f3d209c00144dee5a112c7ec4", "membership": "alpha_only", "name": "prime_field_convolution_coefficient_right_add", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro ab", "intro ac", "intro bb", "intro bc", "intro cb", "intro cc", "intro L", "intro db", "intro dc", "intro M", "intro i", "intro u", "intro v", "intro w", "intro hs", "intro hu", "intro hv", "intro hw", "cases hu", "cases hu_witness", "cases hu_witness_witness", "cases hu_witness_witness_witness", "cases hu_witness_witness_witness_right", "cases hv", "cases hv_witness", "cases hv_witness_witness", "cases hv_witness_witness_witness", "cases hv_witness_witness_witness_right", "cases hw", "cases hw_witness", "cases hw_witness_witness", "cases hw_witness_witness_witness", "cases hw_witness_witness_witness_right", "have hsum : exists pfa_offset_left_coefficient_add_right_sum pfa_offset_right_coefficient_add_right_sum. (x2+x5) + (p) * pfa_offset_left_coefficient_add_right_sum = (x8) + (p) * pfa_offset_right_coefficient_add_right_sum", "specialize polynomial_diagonal_sum_right_add_congruent (p)", "specialize polynomial_diagonal_sum_right_add_congruent (ab)", "specialize polynomial_diagonal_sum_right_add_congruent (ac)", "specialize polynomial_diagonal_sum_right_add_congruent (bb)", "specialize polynomial_diagonal_sum_right_add_congruent (bc)", "specialize polynomial_diagonal_sum_right_add_congruent (cb)", "specialize polynomial_diagonal_sum_right_add_congruent (cc)", "specialize polynomial_diagonal_sum_right_add_congruent (L)", "specialize polynomial_diagonal_sum_right_add_congruent (db)", "specialize polynomial_diagonal_sum_right_add_congruent (dc)", "specialize polynomial_diagonal_sum_right_add_congruent (M)", "specialize polynomial_diagonal_sum_right_add_congruent (i)", "specialize polynomial_diagonal_sum_right_add_congruent (S i)", "specialize polynomial_diagonal_sum_right_add_congruent (x)", "specialize polynomial_diagonal_sum_right_add_congruent (x1)", "specialize polynomial_diagonal_sum_right_add_congruent (x3)", "specialize polynomial_diagonal_sum_right_add_congruent (x4)", "specialize polynomial_diagonal_sum_right_add_congruent (x6)", "specialize polynomial_diagonal_sum_right_add_congruent (x7)", "specialize polynomial_diagonal_sum_right_add_congruent (x2)", "specialize polynomial_diagonal_sum_right_add_congruent (x5)", "specialize polynomial_diagonal_sum_right_add_congruent (x8)", "apply polynomial_diagonal_sum_right_add_congruent", "exact hs", "exact hu_witness_witness_witness_left", "exact hu_witness_witness_witness_right_left", "exact hv_witness_witness_witness_left", "exact hv_witness_witness_witness_right_left", "exact hw_witness_witness_witness_left", "exact hw_witness_witness_witness_right_left", "cases hu_witness_witness_witness_right_right", "cases hv_witness_witness_witness_right_right", "cases hw_witness_witness_witness_right_right", "have hraw : exists pfa_offset_left_coefficient_add_right_raw pfa_offset_right_coefficient_add_right_raw. (x2+x5) + (p) * pfa_offset_left_coefficient_add_right_raw = (w) + (p) * pfa_offset_right_coefficient_add_right_raw", "specialize mod_eq_trans (p)", "specialize mod_eq_trans (x2+x5)", "specialize mod_eq_trans (x8)", "specialize mod_eq_trans (w)", "apply mod_eq_trans", "exact hsum", "exact hw_witness_witness_witness_right_right_right", "split", "exact hu_witness_witness_witness_right_right_left", "split", "exact hv_witness_witness_witness_right_right_left", "split", "exact hw_witness_witness_witness_right_right_left", "specialize mod_eq_trans (p)", "specialize mod_eq_trans (u+v)", "specialize mod_eq_trans (x2+x5)", "specialize mod_eq_trans (w)", "apply mod_eq_trans", "specialize mod_eq_symm (p)", "specialize mod_eq_symm (x2+x5)", "specialize mod_eq_symm (u+v)", "apply mod_eq_symm", "specialize mod_eq_add (p)", "specialize mod_eq_add (x2)", "specialize mod_eq_add (u)", "specialize mod_eq_add (x5)", "specialize mod_eq_add (v)", "apply mod_eq_add", "exact hu_witness_witness_witness_right_right_right", "exact hv_witness_witness_witness_right_right_right", "exact hraw" ], "script_sha256": "071a19087e0e7b513348d396212448d7afaaf0bb51c2f51f9d168b3856a049f6", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_distributivity_candidate.py", "sha256": "a959962d631759cd1fc773dd7eef2fadf4f3f95361d6d7bc8c6a9e82d0d4ab86" }, "statement": "forall p ab ac bb bc cb cc L db dc M i u v w. (forall pfp_index_coefficient_add_right_source. (exists pfa_gap_coefficient_add_right_sourceindex. pfa_gap_coefficient_add_right_sourceindex + S (pfp_index_coefficient_add_right_source) = (L)) -> exists pfp_left_coefficient_add_right_source pfp_right_coefficient_add_right_source pfp_value_coefficient_add_right_source. ((((exists ff_h_pfp_coefficient_add_right_sourceleft. ff_h_pfp_coefficient_add_right_sourceleft + S (pfp_left_coefficient_add_right_source) = S ((S (pfp_index_coefficient_add_right_source)) * ac)) /\\ exists ff_q_pfp_coefficient_add_right_sourceleft. ab = ff_q_pfp_coefficient_add_right_sourceleft * S ((S (pfp_index_coefficient_add_right_source)) * ac) + (pfp_left_coefficient_add_right_source))) /\\ (((((exists ff_h_pfp_coefficient_add_right_sourceright. ff_h_pfp_coefficient_add_right_sourceright + S (pfp_right_coefficient_add_right_source) = S ((S (pfp_index_coefficient_add_right_source)) * bc)) /\\ exists ff_q_pfp_coefficient_add_right_sourceright. bb = ff_q_pfp_coefficient_add_right_sourceright * S ((S (pfp_index_coefficient_add_right_source)) * bc) + (pfp_right_coefficient_add_right_source))) /\\ (((((exists ff_h_pfp_coefficient_add_right_sourcetarget. ff_h_pfp_coefficient_add_right_sourcetarget + S (pfp_value_coefficient_add_right_source) = S ((S (pfp_index_coefficient_add_right_source)) * cc)) /\\ exists ff_q_pfp_coefficient_add_right_sourcetarget. cb = ff_q_pfp_coefficient_add_right_sourcetarget * S ((S (pfp_index_coefficient_add_right_source)) * cc) + (pfp_value_coefficient_add_right_source))) /\\ ((((exists pfa_gap_coefficient_add_right_sourceoperationleft. pfa_gap_coefficient_add_right_sourceoperationleft + S (pfp_left_coefficient_add_right_source) = (p)) /\\ (((exists pfa_gap_coefficient_add_right_sourceoperationright. pfa_gap_coefficient_add_right_sourceoperationright + S (pfp_right_coefficient_add_right_source) = (p)) /\\ ((((exists pfa_gap_coefficient_add_right_sourceoperationresultbound. pfa_gap_coefficient_add_right_sourceoperationresultbound + S (pfp_value_coefficient_add_right_source) = (p)) /\\ ((exists pfa_offset_left_coefficient_add_right_sourceoperationresultcongruence pfa_offset_right_coefficient_add_right_sourceoperationresultcongruence. ((pfp_left_coefficient_add_right_source) + (pfp_right_coefficient_add_right_source)) + (p) * pfa_offset_left_coefficient_add_right_sourceoperationresultcongruence = (pfp_value_coefficient_add_right_source) + (p) * pfa_offset_right_coefficient_add_right_sourceoperationresultcongruence)))))))))))))))) -> (exists pfc_terms_code_coefficient_add_right_u pfc_terms_scale_coefficient_add_right_u pfc_natural_sum_coefficient_add_right_u. ((forall pfc_index_coefficient_add_right_udiagonal. (exists pfa_gap_coefficient_add_right_udiagonalbound. pfa_gap_coefficient_add_right_udiagonalbound + S (pfc_index_coefficient_add_right_udiagonal) = (S (i))) -> exists pfc_value_coefficient_add_right_udiagonal. ((((exists ff_h_pfp_coefficient_add_right_udiagonalentry. ff_h_pfp_coefficient_add_right_udiagonalentry + S (pfc_value_coefficient_add_right_udiagonal) = S ((S (pfc_index_coefficient_add_right_udiagonal)) * pfc_terms_scale_coefficient_add_right_u)) /\\ exists ff_q_pfp_coefficient_add_right_udiagonalentry. pfc_terms_code_coefficient_add_right_u = ff_q_pfp_coefficient_add_right_udiagonalentry * S ((S (pfc_index_coefficient_add_right_udiagonal)) * pfc_terms_scale_coefficient_add_right_u) + (pfc_value_coefficient_add_right_udiagonal))) /\\ ((exists pfc_complement_coefficient_add_right_udiagonalterm pfc_left_coefficient_add_right_udiagonalterm pfc_right_coefficient_add_right_udiagonalterm. (((pfc_index_coefficient_add_right_udiagonal)+pfc_complement_coefficient_add_right_udiagonalterm=(i)) /\\ ((((((exists pfa_gap_coefficient_add_right_udiagonaltermleftinside. pfa_gap_coefficient_add_right_udiagonaltermleftinside + S (pfc_index_coefficient_add_right_udiagonal) = (L)) /\\ ((((exists ff_h_pfp_coefficient_add_right_udiagonaltermleftentry. ff_h_pfp_coefficient_add_right_udiagonaltermleftentry + S (pfc_left_coefficient_add_right_udiagonalterm) = S ((S (pfc_index_coefficient_add_right_udiagonal)) * ac)) /\\ exists ff_q_pfp_coefficient_add_right_udiagonaltermleftentry. ab = ff_q_pfp_coefficient_add_right_udiagonaltermleftentry * S ((S (pfc_index_coefficient_add_right_udiagonal)) * ac) + (pfc_left_coefficient_add_right_udiagonalterm)))))) \\/ (((exists pfc_gap_coefficient_add_right_udiagonaltermleftoutside. pfc_gap_coefficient_add_right_udiagonaltermleftoutside+(L)=(pfc_index_coefficient_add_right_udiagonal)) /\\ (((pfc_left_coefficient_add_right_udiagonalterm)=0))))) /\\ ((((((exists pfa_gap_coefficient_add_right_udiagonaltermrightinside. pfa_gap_coefficient_add_right_udiagonaltermrightinside + S (pfc_complement_coefficient_add_right_udiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_coefficient_add_right_udiagonaltermrightentry. ff_h_pfp_coefficient_add_right_udiagonaltermrightentry + S (pfc_right_coefficient_add_right_udiagonalterm) = S ((S (pfc_complement_coefficient_add_right_udiagonalterm)) * dc)) /\\ exists ff_q_pfp_coefficient_add_right_udiagonaltermrightentry. db = ff_q_pfp_coefficient_add_right_udiagonaltermrightentry * S ((S (pfc_complement_coefficient_add_right_udiagonalterm)) * dc) + (pfc_right_coefficient_add_right_udiagonalterm)))))) \\/ (((exists pfc_gap_coefficient_add_right_udiagonaltermrightoutside. pfc_gap_coefficient_add_right_udiagonaltermrightoutside+(M)=(pfc_complement_coefficient_add_right_udiagonalterm)) /\\ (((pfc_right_coefficient_add_right_udiagonalterm)=0))))) /\\ (((pfc_value_coefficient_add_right_udiagonal)=pfc_left_coefficient_add_right_udiagonalterm*pfc_right_coefficient_add_right_udiagonalterm))))))))))) /\\ (((exists fs_u_pfc_coefficient_add_right_usum fs_v_pfc_coefficient_add_right_usum. ((((exists fs_h_pfc_coefficient_add_right_usum_body_start. fs_h_pfc_coefficient_add_right_usum_body_start + S (0) = S ((S (0)) * fs_v_pfc_coefficient_add_right_usum)) /\\ exists fs_q_pfc_coefficient_add_right_usum_body_start. fs_u_pfc_coefficient_add_right_usum = fs_q_pfc_coefficient_add_right_usum_body_start * S ((S (0)) * fs_v_pfc_coefficient_add_right_usum) + (0))) /\\ ((((exists fs_h_pfc_coefficient_add_right_usum_body_terminal. fs_h_pfc_coefficient_add_right_usum_body_terminal + S (pfc_natural_sum_coefficient_add_right_u) = S ((S (S (i))) * fs_v_pfc_coefficient_add_right_usum)) /\\ exists fs_q_pfc_coefficient_add_right_usum_body_terminal. fs_u_pfc_coefficient_add_right_usum = fs_q_pfc_coefficient_add_right_usum_body_terminal * S ((S (S (i))) * fs_v_pfc_coefficient_add_right_usum) + (pfc_natural_sum_coefficient_add_right_u))) /\\ forall fs_i_pfc_coefficient_add_right_usum_body_steps. (exists fs_lt_pfc_coefficient_add_right_usum_body_steps_bound. fs_lt_pfc_coefficient_add_right_usum_body_steps_bound + S fs_i_pfc_coefficient_add_right_usum_body_steps = S (i)) -> exists fs_a_pfc_coefficient_add_right_usum_body_steps fs_r_pfc_coefficient_add_right_usum_body_steps fs_s_pfc_coefficient_add_right_usum_body_steps. ((((exists fs_h_pfc_coefficient_add_right_usum_body_steps_summand. fs_h_pfc_coefficient_add_right_usum_body_steps_summand + S (fs_a_pfc_coefficient_add_right_usum_body_steps) = S ((S (fs_i_pfc_coefficient_add_right_usum_body_steps)) * pfc_terms_scale_coefficient_add_right_u)) /\\ exists fs_q_pfc_coefficient_add_right_usum_body_steps_summand. pfc_terms_code_coefficient_add_right_u = fs_q_pfc_coefficient_add_right_usum_body_steps_summand * S ((S (fs_i_pfc_coefficient_add_right_usum_body_steps)) * pfc_terms_scale_coefficient_add_right_u) + (fs_a_pfc_coefficient_add_right_usum_body_steps))) /\\ ((((exists fs_h_pfc_coefficient_add_right_usum_body_steps_partial. fs_h_pfc_coefficient_add_right_usum_body_steps_partial + S (fs_r_pfc_coefficient_add_right_usum_body_steps) = S ((S (fs_i_pfc_coefficient_add_right_usum_body_steps)) * fs_v_pfc_coefficient_add_right_usum)) /\\ exists fs_q_pfc_coefficient_add_right_usum_body_steps_partial. fs_u_pfc_coefficient_add_right_usum = fs_q_pfc_coefficient_add_right_usum_body_steps_partial * S ((S (fs_i_pfc_coefficient_add_right_usum_body_steps)) * fs_v_pfc_coefficient_add_right_usum) + (fs_r_pfc_coefficient_add_right_usum_body_steps))) /\\ ((((exists fs_h_pfc_coefficient_add_right_usum_body_steps_successor. fs_h_pfc_coefficient_add_right_usum_body_steps_successor + S (fs_s_pfc_coefficient_add_right_usum_body_steps) = S ((S (S fs_i_pfc_coefficient_add_right_usum_body_steps)) * fs_v_pfc_coefficient_add_right_usum)) /\\ exists fs_q_pfc_coefficient_add_right_usum_body_steps_successor. fs_u_pfc_coefficient_add_right_usum = fs_q_pfc_coefficient_add_right_usum_body_steps_successor * S ((S (S fs_i_pfc_coefficient_add_right_usum_body_steps)) * fs_v_pfc_coefficient_add_right_usum) + (fs_s_pfc_coefficient_add_right_usum_body_steps))) /\\ fs_s_pfc_coefficient_add_right_usum_body_steps = fs_r_pfc_coefficient_add_right_usum_body_steps + fs_a_pfc_coefficient_add_right_usum_body_steps)))))) /\\ ((((exists pfa_gap_coefficient_add_right_uresiduebound. pfa_gap_coefficient_add_right_uresiduebound + S (u) = (p)) /\\ ((exists pfa_offset_left_coefficient_add_right_uresiduecongruence pfa_offset_right_coefficient_add_right_uresiduecongruence. (pfc_natural_sum_coefficient_add_right_u) + (p) * pfa_offset_left_coefficient_add_right_uresiduecongruence = (u) + (p) * pfa_offset_right_coefficient_add_right_uresiduecongruence))))))))) -> (exists pfc_terms_code_coefficient_add_right_v pfc_terms_scale_coefficient_add_right_v pfc_natural_sum_coefficient_add_right_v. ((forall pfc_index_coefficient_add_right_vdiagonal. (exists pfa_gap_coefficient_add_right_vdiagonalbound. pfa_gap_coefficient_add_right_vdiagonalbound + S (pfc_index_coefficient_add_right_vdiagonal) = (S (i))) -> exists pfc_value_coefficient_add_right_vdiagonal. ((((exists ff_h_pfp_coefficient_add_right_vdiagonalentry. ff_h_pfp_coefficient_add_right_vdiagonalentry + S (pfc_value_coefficient_add_right_vdiagonal) = S ((S (pfc_index_coefficient_add_right_vdiagonal)) * pfc_terms_scale_coefficient_add_right_v)) /\\ exists ff_q_pfp_coefficient_add_right_vdiagonalentry. pfc_terms_code_coefficient_add_right_v = ff_q_pfp_coefficient_add_right_vdiagonalentry * S ((S (pfc_index_coefficient_add_right_vdiagonal)) * pfc_terms_scale_coefficient_add_right_v) + (pfc_value_coefficient_add_right_vdiagonal))) /\\ ((exists pfc_complement_coefficient_add_right_vdiagonalterm pfc_left_coefficient_add_right_vdiagonalterm pfc_right_coefficient_add_right_vdiagonalterm. (((pfc_index_coefficient_add_right_vdiagonal)+pfc_complement_coefficient_add_right_vdiagonalterm=(i)) /\\ ((((((exists pfa_gap_coefficient_add_right_vdiagonaltermleftinside. pfa_gap_coefficient_add_right_vdiagonaltermleftinside + S (pfc_index_coefficient_add_right_vdiagonal) = (L)) /\\ ((((exists ff_h_pfp_coefficient_add_right_vdiagonaltermleftentry. ff_h_pfp_coefficient_add_right_vdiagonaltermleftentry + S (pfc_left_coefficient_add_right_vdiagonalterm) = S ((S (pfc_index_coefficient_add_right_vdiagonal)) * bc)) /\\ exists ff_q_pfp_coefficient_add_right_vdiagonaltermleftentry. bb = ff_q_pfp_coefficient_add_right_vdiagonaltermleftentry * S ((S (pfc_index_coefficient_add_right_vdiagonal)) * bc) + (pfc_left_coefficient_add_right_vdiagonalterm)))))) \\/ (((exists pfc_gap_coefficient_add_right_vdiagonaltermleftoutside. pfc_gap_coefficient_add_right_vdiagonaltermleftoutside+(L)=(pfc_index_coefficient_add_right_vdiagonal)) /\\ (((pfc_left_coefficient_add_right_vdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_coefficient_add_right_vdiagonaltermrightinside. pfa_gap_coefficient_add_right_vdiagonaltermrightinside + S (pfc_complement_coefficient_add_right_vdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_coefficient_add_right_vdiagonaltermrightentry. ff_h_pfp_coefficient_add_right_vdiagonaltermrightentry + S (pfc_right_coefficient_add_right_vdiagonalterm) = S ((S (pfc_complement_coefficient_add_right_vdiagonalterm)) * dc)) /\\ exists ff_q_pfp_coefficient_add_right_vdiagonaltermrightentry. db = ff_q_pfp_coefficient_add_right_vdiagonaltermrightentry * S ((S (pfc_complement_coefficient_add_right_vdiagonalterm)) * dc) + (pfc_right_coefficient_add_right_vdiagonalterm)))))) \\/ (((exists pfc_gap_coefficient_add_right_vdiagonaltermrightoutside. pfc_gap_coefficient_add_right_vdiagonaltermrightoutside+(M)=(pfc_complement_coefficient_add_right_vdiagonalterm)) /\\ (((pfc_right_coefficient_add_right_vdiagonalterm)=0))))) /\\ (((pfc_value_coefficient_add_right_vdiagonal)=pfc_left_coefficient_add_right_vdiagonalterm*pfc_right_coefficient_add_right_vdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_coefficient_add_right_vsum fs_v_pfc_coefficient_add_right_vsum. ((((exists fs_h_pfc_coefficient_add_right_vsum_body_start. fs_h_pfc_coefficient_add_right_vsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_coefficient_add_right_vsum)) /\\ exists fs_q_pfc_coefficient_add_right_vsum_body_start. fs_u_pfc_coefficient_add_right_vsum = fs_q_pfc_coefficient_add_right_vsum_body_start * S ((S (0)) * fs_v_pfc_coefficient_add_right_vsum) + (0))) /\\ ((((exists fs_h_pfc_coefficient_add_right_vsum_body_terminal. fs_h_pfc_coefficient_add_right_vsum_body_terminal + S (pfc_natural_sum_coefficient_add_right_v) = S ((S (S (i))) * fs_v_pfc_coefficient_add_right_vsum)) /\\ exists fs_q_pfc_coefficient_add_right_vsum_body_terminal. fs_u_pfc_coefficient_add_right_vsum = fs_q_pfc_coefficient_add_right_vsum_body_terminal * S ((S (S (i))) * fs_v_pfc_coefficient_add_right_vsum) + (pfc_natural_sum_coefficient_add_right_v))) /\\ forall fs_i_pfc_coefficient_add_right_vsum_body_steps. (exists fs_lt_pfc_coefficient_add_right_vsum_body_steps_bound. fs_lt_pfc_coefficient_add_right_vsum_body_steps_bound + S fs_i_pfc_coefficient_add_right_vsum_body_steps = S (i)) -> exists fs_a_pfc_coefficient_add_right_vsum_body_steps fs_r_pfc_coefficient_add_right_vsum_body_steps fs_s_pfc_coefficient_add_right_vsum_body_steps. ((((exists fs_h_pfc_coefficient_add_right_vsum_body_steps_summand. fs_h_pfc_coefficient_add_right_vsum_body_steps_summand + S (fs_a_pfc_coefficient_add_right_vsum_body_steps) = S ((S (fs_i_pfc_coefficient_add_right_vsum_body_steps)) * pfc_terms_scale_coefficient_add_right_v)) /\\ exists fs_q_pfc_coefficient_add_right_vsum_body_steps_summand. pfc_terms_code_coefficient_add_right_v = fs_q_pfc_coefficient_add_right_vsum_body_steps_summand * S ((S (fs_i_pfc_coefficient_add_right_vsum_body_steps)) * pfc_terms_scale_coefficient_add_right_v) + (fs_a_pfc_coefficient_add_right_vsum_body_steps))) /\\ ((((exists fs_h_pfc_coefficient_add_right_vsum_body_steps_partial. fs_h_pfc_coefficient_add_right_vsum_body_steps_partial + S (fs_r_pfc_coefficient_add_right_vsum_body_steps) = S ((S (fs_i_pfc_coefficient_add_right_vsum_body_steps)) * fs_v_pfc_coefficient_add_right_vsum)) /\\ exists fs_q_pfc_coefficient_add_right_vsum_body_steps_partial. fs_u_pfc_coefficient_add_right_vsum = fs_q_pfc_coefficient_add_right_vsum_body_steps_partial * S ((S (fs_i_pfc_coefficient_add_right_vsum_body_steps)) * fs_v_pfc_coefficient_add_right_vsum) + (fs_r_pfc_coefficient_add_right_vsum_body_steps))) /\\ ((((exists fs_h_pfc_coefficient_add_right_vsum_body_steps_successor. fs_h_pfc_coefficient_add_right_vsum_body_steps_successor + S (fs_s_pfc_coefficient_add_right_vsum_body_steps) = S ((S (S fs_i_pfc_coefficient_add_right_vsum_body_steps)) * fs_v_pfc_coefficient_add_right_vsum)) /\\ exists fs_q_pfc_coefficient_add_right_vsum_body_steps_successor. fs_u_pfc_coefficient_add_right_vsum = fs_q_pfc_coefficient_add_right_vsum_body_steps_successor * S ((S (S fs_i_pfc_coefficient_add_right_vsum_body_steps)) * fs_v_pfc_coefficient_add_right_vsum) + (fs_s_pfc_coefficient_add_right_vsum_body_steps))) /\\ fs_s_pfc_coefficient_add_right_vsum_body_steps = fs_r_pfc_coefficient_add_right_vsum_body_steps + fs_a_pfc_coefficient_add_right_vsum_body_steps)))))) /\\ ((((exists pfa_gap_coefficient_add_right_vresiduebound. pfa_gap_coefficient_add_right_vresiduebound + S (v) = (p)) /\\ ((exists pfa_offset_left_coefficient_add_right_vresiduecongruence pfa_offset_right_coefficient_add_right_vresiduecongruence. (pfc_natural_sum_coefficient_add_right_v) + (p) * pfa_offset_left_coefficient_add_right_vresiduecongruence = (v) + (p) * pfa_offset_right_coefficient_add_right_vresiduecongruence))))))))) -> (exists pfc_terms_code_coefficient_add_right_w pfc_terms_scale_coefficient_add_right_w pfc_natural_sum_coefficient_add_right_w. ((forall pfc_index_coefficient_add_right_wdiagonal. (exists pfa_gap_coefficient_add_right_wdiagonalbound. pfa_gap_coefficient_add_right_wdiagonalbound + S (pfc_index_coefficient_add_right_wdiagonal) = (S (i))) -> exists pfc_value_coefficient_add_right_wdiagonal. ((((exists ff_h_pfp_coefficient_add_right_wdiagonalentry. ff_h_pfp_coefficient_add_right_wdiagonalentry + S (pfc_value_coefficient_add_right_wdiagonal) = S ((S (pfc_index_coefficient_add_right_wdiagonal)) * pfc_terms_scale_coefficient_add_right_w)) /\\ exists ff_q_pfp_coefficient_add_right_wdiagonalentry. pfc_terms_code_coefficient_add_right_w = ff_q_pfp_coefficient_add_right_wdiagonalentry * S ((S (pfc_index_coefficient_add_right_wdiagonal)) * pfc_terms_scale_coefficient_add_right_w) + (pfc_value_coefficient_add_right_wdiagonal))) /\\ ((exists pfc_complement_coefficient_add_right_wdiagonalterm pfc_left_coefficient_add_right_wdiagonalterm pfc_right_coefficient_add_right_wdiagonalterm. (((pfc_index_coefficient_add_right_wdiagonal)+pfc_complement_coefficient_add_right_wdiagonalterm=(i)) /\\ ((((((exists pfa_gap_coefficient_add_right_wdiagonaltermleftinside. pfa_gap_coefficient_add_right_wdiagonaltermleftinside + S (pfc_index_coefficient_add_right_wdiagonal) = (L)) /\\ ((((exists ff_h_pfp_coefficient_add_right_wdiagonaltermleftentry. ff_h_pfp_coefficient_add_right_wdiagonaltermleftentry + S (pfc_left_coefficient_add_right_wdiagonalterm) = S ((S (pfc_index_coefficient_add_right_wdiagonal)) * cc)) /\\ exists ff_q_pfp_coefficient_add_right_wdiagonaltermleftentry. cb = ff_q_pfp_coefficient_add_right_wdiagonaltermleftentry * S ((S (pfc_index_coefficient_add_right_wdiagonal)) * cc) + (pfc_left_coefficient_add_right_wdiagonalterm)))))) \\/ (((exists pfc_gap_coefficient_add_right_wdiagonaltermleftoutside. pfc_gap_coefficient_add_right_wdiagonaltermleftoutside+(L)=(pfc_index_coefficient_add_right_wdiagonal)) /\\ (((pfc_left_coefficient_add_right_wdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_coefficient_add_right_wdiagonaltermrightinside. pfa_gap_coefficient_add_right_wdiagonaltermrightinside + S (pfc_complement_coefficient_add_right_wdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_coefficient_add_right_wdiagonaltermrightentry. ff_h_pfp_coefficient_add_right_wdiagonaltermrightentry + S (pfc_right_coefficient_add_right_wdiagonalterm) = S ((S (pfc_complement_coefficient_add_right_wdiagonalterm)) * dc)) /\\ exists ff_q_pfp_coefficient_add_right_wdiagonaltermrightentry. db = ff_q_pfp_coefficient_add_right_wdiagonaltermrightentry * S ((S (pfc_complement_coefficient_add_right_wdiagonalterm)) * dc) + (pfc_right_coefficient_add_right_wdiagonalterm)))))) \\/ (((exists pfc_gap_coefficient_add_right_wdiagonaltermrightoutside. pfc_gap_coefficient_add_right_wdiagonaltermrightoutside+(M)=(pfc_complement_coefficient_add_right_wdiagonalterm)) /\\ (((pfc_right_coefficient_add_right_wdiagonalterm)=0))))) /\\ (((pfc_value_coefficient_add_right_wdiagonal)=pfc_left_coefficient_add_right_wdiagonalterm*pfc_right_coefficient_add_right_wdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_coefficient_add_right_wsum fs_v_pfc_coefficient_add_right_wsum. ((((exists fs_h_pfc_coefficient_add_right_wsum_body_start. fs_h_pfc_coefficient_add_right_wsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_coefficient_add_right_wsum)) /\\ exists fs_q_pfc_coefficient_add_right_wsum_body_start. fs_u_pfc_coefficient_add_right_wsum = fs_q_pfc_coefficient_add_right_wsum_body_start * S ((S (0)) * fs_v_pfc_coefficient_add_right_wsum) + (0))) /\\ ((((exists fs_h_pfc_coefficient_add_right_wsum_body_terminal. fs_h_pfc_coefficient_add_right_wsum_body_terminal + S (pfc_natural_sum_coefficient_add_right_w) = S ((S (S (i))) * fs_v_pfc_coefficient_add_right_wsum)) /\\ exists fs_q_pfc_coefficient_add_right_wsum_body_terminal. fs_u_pfc_coefficient_add_right_wsum = fs_q_pfc_coefficient_add_right_wsum_body_terminal * S ((S (S (i))) * fs_v_pfc_coefficient_add_right_wsum) + (pfc_natural_sum_coefficient_add_right_w))) /\\ forall fs_i_pfc_coefficient_add_right_wsum_body_steps. (exists fs_lt_pfc_coefficient_add_right_wsum_body_steps_bound. fs_lt_pfc_coefficient_add_right_wsum_body_steps_bound + S fs_i_pfc_coefficient_add_right_wsum_body_steps = S (i)) -> exists fs_a_pfc_coefficient_add_right_wsum_body_steps fs_r_pfc_coefficient_add_right_wsum_body_steps fs_s_pfc_coefficient_add_right_wsum_body_steps. ((((exists fs_h_pfc_coefficient_add_right_wsum_body_steps_summand. fs_h_pfc_coefficient_add_right_wsum_body_steps_summand + S (fs_a_pfc_coefficient_add_right_wsum_body_steps) = S ((S (fs_i_pfc_coefficient_add_right_wsum_body_steps)) * pfc_terms_scale_coefficient_add_right_w)) /\\ exists fs_q_pfc_coefficient_add_right_wsum_body_steps_summand. pfc_terms_code_coefficient_add_right_w = fs_q_pfc_coefficient_add_right_wsum_body_steps_summand * S ((S (fs_i_pfc_coefficient_add_right_wsum_body_steps)) * pfc_terms_scale_coefficient_add_right_w) + (fs_a_pfc_coefficient_add_right_wsum_body_steps))) /\\ ((((exists fs_h_pfc_coefficient_add_right_wsum_body_steps_partial. fs_h_pfc_coefficient_add_right_wsum_body_steps_partial + S (fs_r_pfc_coefficient_add_right_wsum_body_steps) = S ((S (fs_i_pfc_coefficient_add_right_wsum_body_steps)) * fs_v_pfc_coefficient_add_right_wsum)) /\\ exists fs_q_pfc_coefficient_add_right_wsum_body_steps_partial. fs_u_pfc_coefficient_add_right_wsum = fs_q_pfc_coefficient_add_right_wsum_body_steps_partial * S ((S (fs_i_pfc_coefficient_add_right_wsum_body_steps)) * fs_v_pfc_coefficient_add_right_wsum) + (fs_r_pfc_coefficient_add_right_wsum_body_steps))) /\\ ((((exists fs_h_pfc_coefficient_add_right_wsum_body_steps_successor. fs_h_pfc_coefficient_add_right_wsum_body_steps_successor + S (fs_s_pfc_coefficient_add_right_wsum_body_steps) = S ((S (S fs_i_pfc_coefficient_add_right_wsum_body_steps)) * fs_v_pfc_coefficient_add_right_wsum)) /\\ exists fs_q_pfc_coefficient_add_right_wsum_body_steps_successor. fs_u_pfc_coefficient_add_right_wsum = fs_q_pfc_coefficient_add_right_wsum_body_steps_successor * S ((S (S fs_i_pfc_coefficient_add_right_wsum_body_steps)) * fs_v_pfc_coefficient_add_right_wsum) + (fs_s_pfc_coefficient_add_right_wsum_body_steps))) /\\ fs_s_pfc_coefficient_add_right_wsum_body_steps = fs_r_pfc_coefficient_add_right_wsum_body_steps + fs_a_pfc_coefficient_add_right_wsum_body_steps)))))) /\\ ((((exists pfa_gap_coefficient_add_right_wresiduebound. pfa_gap_coefficient_add_right_wresiduebound + S (w) = (p)) /\\ ((exists pfa_offset_left_coefficient_add_right_wresiduecongruence pfa_offset_right_coefficient_add_right_wresiduecongruence. (pfc_natural_sum_coefficient_add_right_w) + (p) * pfa_offset_left_coefficient_add_right_wresiduecongruence = (w) + (p) * pfa_offset_right_coefficient_add_right_wresiduecongruence))))))))) -> (((exists pfa_gap_coefficient_add_right_resultleft. pfa_gap_coefficient_add_right_resultleft + S (u) = (p)) /\\ (((exists pfa_gap_coefficient_add_right_resultright. pfa_gap_coefficient_add_right_resultright + S (v) = (p)) /\\ ((((exists pfa_gap_coefficient_add_right_resultresultbound. pfa_gap_coefficient_add_right_resultresultbound + S (w) = (p)) /\\ ((exists pfa_offset_left_coefficient_add_right_resultresultcongruence pfa_offset_right_coefficient_add_right_resultresultcongruence. ((u) + (v)) + (p) * pfa_offset_left_coefficient_add_right_resultresultcongruence = (w) + (p) * pfa_offset_right_coefficient_add_right_resultresultcongruence)))))))))", "statement_sha256": "a3da8edf3f7d91b305f66b51e3c252429358e2b7e85e41a8251fb38ecd991acb", "summary": "Three genuine convolution coefficients satisfy actual canonical field addition under right distributivity, proved from their independently witnessed natural sums and residues.", "summary_sha256": "d8674363d1bfcd33a1b57a030e227ce1616c08c2024e766a22df72d1aa239b75" }, { "alpha_v33_frontier_enrollment": { "body_receipt_sha256": "2b0c526b3fc45c11d73c9d6979bdb45e5d1c5856ec61cfdc35461b8e54227f72", "bundle_campaign": "polynomial-euclidean-division", "bundle_node_id": 326, "bundle_sha256": "6ae667d8518e4dbe722bb08ad1b08715a0d282c2893e533c8133d770fe861dcf", "campaign": "polynomial-euclidean-division", "first_enrolled_version": "v33", "parent_catalog_sha256": "41b9f387d88a5a4f0fe5ee2bd5578f37a27a4657b0a80f1a1a2cb5109f69a623", 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"intro hw", "intro i", "intro hi", "have hcu : exists r. ((((exists ff_h_pfp_prefix_add_left_hcu. ff_h_pfp_prefix_add_left_hcu + S (r) = S ((S (i)) * uc)) /\\ exists ff_q_pfp_prefix_add_left_hcu. ub = ff_q_pfp_prefix_add_left_hcu * S ((S (i)) * uc) + (r))) /\\ ((exists pfc_terms_code_prefix_add_left_value_hcu pfc_terms_scale_prefix_add_left_value_hcu pfc_natural_sum_prefix_add_left_value_hcu. ((forall pfc_index_prefix_add_left_value_hcudiagonal. (exists pfa_gap_prefix_add_left_value_hcudiagonalbound. pfa_gap_prefix_add_left_value_hcudiagonalbound + S (pfc_index_prefix_add_left_value_hcudiagonal) = (S (i))) -> exists pfc_value_prefix_add_left_value_hcudiagonal. ((((exists ff_h_pfp_prefix_add_left_value_hcudiagonalentry. ff_h_pfp_prefix_add_left_value_hcudiagonalentry + S (pfc_value_prefix_add_left_value_hcudiagonal) = S ((S (pfc_index_prefix_add_left_value_hcudiagonal)) * pfc_terms_scale_prefix_add_left_value_hcu)) /\\ exists ff_q_pfp_prefix_add_left_value_hcudiagonalentry. pfc_terms_code_prefix_add_left_value_hcu = ff_q_pfp_prefix_add_left_value_hcudiagonalentry * S ((S (pfc_index_prefix_add_left_value_hcudiagonal)) * pfc_terms_scale_prefix_add_left_value_hcu) + (pfc_value_prefix_add_left_value_hcudiagonal))) /\\ ((exists pfc_complement_prefix_add_left_value_hcudiagonalterm pfc_left_prefix_add_left_value_hcudiagonalterm pfc_right_prefix_add_left_value_hcudiagonalterm. (((pfc_index_prefix_add_left_value_hcudiagonal)+pfc_complement_prefix_add_left_value_hcudiagonalterm=(i)) /\\ ((((((exists pfa_gap_prefix_add_left_value_hcudiagonaltermleftinside. pfa_gap_prefix_add_left_value_hcudiagonaltermleftinside + S (pfc_index_prefix_add_left_value_hcudiagonal) = (M)) /\\ ((((exists ff_h_pfp_prefix_add_left_value_hcudiagonaltermleftentry. ff_h_pfp_prefix_add_left_value_hcudiagonaltermleftentry + S (pfc_left_prefix_add_left_value_hcudiagonalterm) = S ((S (pfc_index_prefix_add_left_value_hcudiagonal)) * dc)) /\\ exists ff_q_pfp_prefix_add_left_value_hcudiagonaltermleftentry. db = ff_q_pfp_prefix_add_left_value_hcudiagonaltermleftentry * S ((S (pfc_index_prefix_add_left_value_hcudiagonal)) * dc) + (pfc_left_prefix_add_left_value_hcudiagonalterm)))))) \\/ (((exists pfc_gap_prefix_add_left_value_hcudiagonaltermleftoutside. pfc_gap_prefix_add_left_value_hcudiagonaltermleftoutside+(M)=(pfc_index_prefix_add_left_value_hcudiagonal)) /\\ (((pfc_left_prefix_add_left_value_hcudiagonalterm)=0))))) /\\ ((((((exists pfa_gap_prefix_add_left_value_hcudiagonaltermrightinside. pfa_gap_prefix_add_left_value_hcudiagonaltermrightinside + S (pfc_complement_prefix_add_left_value_hcudiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_prefix_add_left_value_hcudiagonaltermrightentry. ff_h_pfp_prefix_add_left_value_hcudiagonaltermrightentry + S (pfc_right_prefix_add_left_value_hcudiagonalterm) = S ((S (pfc_complement_prefix_add_left_value_hcudiagonalterm)) * ac)) /\\ exists ff_q_pfp_prefix_add_left_value_hcudiagonaltermrightentry. ab = ff_q_pfp_prefix_add_left_value_hcudiagonaltermrightentry * S ((S (pfc_complement_prefix_add_left_value_hcudiagonalterm)) * ac) + (pfc_right_prefix_add_left_value_hcudiagonalterm)))))) \\/ (((exists pfc_gap_prefix_add_left_value_hcudiagonaltermrightoutside. pfc_gap_prefix_add_left_value_hcudiagonaltermrightoutside+(L)=(pfc_complement_prefix_add_left_value_hcudiagonalterm)) /\\ (((pfc_right_prefix_add_left_value_hcudiagonalterm)=0))))) /\\ (((pfc_value_prefix_add_left_value_hcudiagonal)=pfc_left_prefix_add_left_value_hcudiagonalterm*pfc_right_prefix_add_left_value_hcudiagonalterm))))))))))) /\\ (((exists fs_u_pfc_prefix_add_left_value_hcusum fs_v_pfc_prefix_add_left_value_hcusum. ((((exists fs_h_pfc_prefix_add_left_value_hcusum_body_start. fs_h_pfc_prefix_add_left_value_hcusum_body_start + S (0) = S ((S (0)) * fs_v_pfc_prefix_add_left_value_hcusum)) /\\ exists fs_q_pfc_prefix_add_left_value_hcusum_body_start. fs_u_pfc_prefix_add_left_value_hcusum = fs_q_pfc_prefix_add_left_value_hcusum_body_start * S ((S (0)) * fs_v_pfc_prefix_add_left_value_hcusum) + (0))) /\\ ((((exists fs_h_pfc_prefix_add_left_value_hcusum_body_terminal. fs_h_pfc_prefix_add_left_value_hcusum_body_terminal + S (pfc_natural_sum_prefix_add_left_value_hcu) = S ((S (S (i))) * fs_v_pfc_prefix_add_left_value_hcusum)) /\\ exists fs_q_pfc_prefix_add_left_value_hcusum_body_terminal. fs_u_pfc_prefix_add_left_value_hcusum = fs_q_pfc_prefix_add_left_value_hcusum_body_terminal * S ((S (S (i))) * fs_v_pfc_prefix_add_left_value_hcusum) + (pfc_natural_sum_prefix_add_left_value_hcu))) /\\ forall fs_i_pfc_prefix_add_left_value_hcusum_body_steps. (exists fs_lt_pfc_prefix_add_left_value_hcusum_body_steps_bound. fs_lt_pfc_prefix_add_left_value_hcusum_body_steps_bound + S fs_i_pfc_prefix_add_left_value_hcusum_body_steps = S (i)) -> exists fs_a_pfc_prefix_add_left_value_hcusum_body_steps fs_r_pfc_prefix_add_left_value_hcusum_body_steps fs_s_pfc_prefix_add_left_value_hcusum_body_steps. ((((exists fs_h_pfc_prefix_add_left_value_hcusum_body_steps_summand. fs_h_pfc_prefix_add_left_value_hcusum_body_steps_summand + S (fs_a_pfc_prefix_add_left_value_hcusum_body_steps) = S ((S (fs_i_pfc_prefix_add_left_value_hcusum_body_steps)) * pfc_terms_scale_prefix_add_left_value_hcu)) /\\ exists fs_q_pfc_prefix_add_left_value_hcusum_body_steps_summand. pfc_terms_code_prefix_add_left_value_hcu = fs_q_pfc_prefix_add_left_value_hcusum_body_steps_summand * S ((S (fs_i_pfc_prefix_add_left_value_hcusum_body_steps)) * pfc_terms_scale_prefix_add_left_value_hcu) + (fs_a_pfc_prefix_add_left_value_hcusum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_add_left_value_hcusum_body_steps_partial. fs_h_pfc_prefix_add_left_value_hcusum_body_steps_partial + S (fs_r_pfc_prefix_add_left_value_hcusum_body_steps) = S ((S (fs_i_pfc_prefix_add_left_value_hcusum_body_steps)) * fs_v_pfc_prefix_add_left_value_hcusum)) /\\ exists fs_q_pfc_prefix_add_left_value_hcusum_body_steps_partial. fs_u_pfc_prefix_add_left_value_hcusum = fs_q_pfc_prefix_add_left_value_hcusum_body_steps_partial * S ((S (fs_i_pfc_prefix_add_left_value_hcusum_body_steps)) * fs_v_pfc_prefix_add_left_value_hcusum) + (fs_r_pfc_prefix_add_left_value_hcusum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_add_left_value_hcusum_body_steps_successor. fs_h_pfc_prefix_add_left_value_hcusum_body_steps_successor + S (fs_s_pfc_prefix_add_left_value_hcusum_body_steps) = S ((S (S fs_i_pfc_prefix_add_left_value_hcusum_body_steps)) * fs_v_pfc_prefix_add_left_value_hcusum)) /\\ exists fs_q_pfc_prefix_add_left_value_hcusum_body_steps_successor. fs_u_pfc_prefix_add_left_value_hcusum = fs_q_pfc_prefix_add_left_value_hcusum_body_steps_successor * S ((S (S fs_i_pfc_prefix_add_left_value_hcusum_body_steps)) * fs_v_pfc_prefix_add_left_value_hcusum) + (fs_s_pfc_prefix_add_left_value_hcusum_body_steps))) /\\ fs_s_pfc_prefix_add_left_value_hcusum_body_steps = fs_r_pfc_prefix_add_left_value_hcusum_body_steps + fs_a_pfc_prefix_add_left_value_hcusum_body_steps)))))) /\\ ((((exists pfa_gap_prefix_add_left_value_hcuresiduebound. pfa_gap_prefix_add_left_value_hcuresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_prefix_add_left_value_hcuresiduecongruence pfa_offset_right_prefix_add_left_value_hcuresiduecongruence. (pfc_natural_sum_prefix_add_left_value_hcu) + (p) * pfa_offset_left_prefix_add_left_value_hcuresiduecongruence = (r) + (p) * pfa_offset_right_prefix_add_left_value_hcuresiduecongruence)))))))))))", "specialize hu (i)", "apply hu", "exact hi", "cases hcu", "cases hcu_witness", "have hcv : exists r. ((((exists ff_h_pfp_prefix_add_left_hcv. ff_h_pfp_prefix_add_left_hcv + S (r) = S ((S (i)) * vc)) /\\ exists ff_q_pfp_prefix_add_left_hcv. vb = ff_q_pfp_prefix_add_left_hcv * S ((S (i)) * vc) + (r))) /\\ ((exists pfc_terms_code_prefix_add_left_value_hcv pfc_terms_scale_prefix_add_left_value_hcv pfc_natural_sum_prefix_add_left_value_hcv. ((forall pfc_index_prefix_add_left_value_hcvdiagonal. (exists pfa_gap_prefix_add_left_value_hcvdiagonalbound. pfa_gap_prefix_add_left_value_hcvdiagonalbound + S (pfc_index_prefix_add_left_value_hcvdiagonal) = (S (i))) -> exists pfc_value_prefix_add_left_value_hcvdiagonal. ((((exists ff_h_pfp_prefix_add_left_value_hcvdiagonalentry. ff_h_pfp_prefix_add_left_value_hcvdiagonalentry + S (pfc_value_prefix_add_left_value_hcvdiagonal) = S ((S (pfc_index_prefix_add_left_value_hcvdiagonal)) * pfc_terms_scale_prefix_add_left_value_hcv)) /\\ exists ff_q_pfp_prefix_add_left_value_hcvdiagonalentry. pfc_terms_code_prefix_add_left_value_hcv = ff_q_pfp_prefix_add_left_value_hcvdiagonalentry * S ((S (pfc_index_prefix_add_left_value_hcvdiagonal)) * pfc_terms_scale_prefix_add_left_value_hcv) + (pfc_value_prefix_add_left_value_hcvdiagonal))) /\\ ((exists pfc_complement_prefix_add_left_value_hcvdiagonalterm pfc_left_prefix_add_left_value_hcvdiagonalterm pfc_right_prefix_add_left_value_hcvdiagonalterm. (((pfc_index_prefix_add_left_value_hcvdiagonal)+pfc_complement_prefix_add_left_value_hcvdiagonalterm=(i)) /\\ ((((((exists pfa_gap_prefix_add_left_value_hcvdiagonaltermleftinside. pfa_gap_prefix_add_left_value_hcvdiagonaltermleftinside + S (pfc_index_prefix_add_left_value_hcvdiagonal) = (M)) /\\ ((((exists ff_h_pfp_prefix_add_left_value_hcvdiagonaltermleftentry. ff_h_pfp_prefix_add_left_value_hcvdiagonaltermleftentry + S (pfc_left_prefix_add_left_value_hcvdiagonalterm) = S ((S (pfc_index_prefix_add_left_value_hcvdiagonal)) * dc)) /\\ exists ff_q_pfp_prefix_add_left_value_hcvdiagonaltermleftentry. db = ff_q_pfp_prefix_add_left_value_hcvdiagonaltermleftentry * S ((S (pfc_index_prefix_add_left_value_hcvdiagonal)) * dc) + (pfc_left_prefix_add_left_value_hcvdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_add_left_value_hcvdiagonaltermleftoutside. pfc_gap_prefix_add_left_value_hcvdiagonaltermleftoutside+(M)=(pfc_index_prefix_add_left_value_hcvdiagonal)) /\\ (((pfc_left_prefix_add_left_value_hcvdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_prefix_add_left_value_hcvdiagonaltermrightinside. pfa_gap_prefix_add_left_value_hcvdiagonaltermrightinside + S (pfc_complement_prefix_add_left_value_hcvdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_prefix_add_left_value_hcvdiagonaltermrightentry. ff_h_pfp_prefix_add_left_value_hcvdiagonaltermrightentry + S (pfc_right_prefix_add_left_value_hcvdiagonalterm) = S ((S (pfc_complement_prefix_add_left_value_hcvdiagonalterm)) * bc)) /\\ exists ff_q_pfp_prefix_add_left_value_hcvdiagonaltermrightentry. bb = ff_q_pfp_prefix_add_left_value_hcvdiagonaltermrightentry * S ((S (pfc_complement_prefix_add_left_value_hcvdiagonalterm)) * bc) + (pfc_right_prefix_add_left_value_hcvdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_add_left_value_hcvdiagonaltermrightoutside. pfc_gap_prefix_add_left_value_hcvdiagonaltermrightoutside+(L)=(pfc_complement_prefix_add_left_value_hcvdiagonalterm)) /\\ (((pfc_right_prefix_add_left_value_hcvdiagonalterm)=0))))) /\\ (((pfc_value_prefix_add_left_value_hcvdiagonal)=pfc_left_prefix_add_left_value_hcvdiagonalterm*pfc_right_prefix_add_left_value_hcvdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_prefix_add_left_value_hcvsum fs_v_pfc_prefix_add_left_value_hcvsum. ((((exists fs_h_pfc_prefix_add_left_value_hcvsum_body_start. fs_h_pfc_prefix_add_left_value_hcvsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_prefix_add_left_value_hcvsum)) /\\ exists fs_q_pfc_prefix_add_left_value_hcvsum_body_start. fs_u_pfc_prefix_add_left_value_hcvsum = fs_q_pfc_prefix_add_left_value_hcvsum_body_start * S ((S (0)) * fs_v_pfc_prefix_add_left_value_hcvsum) + (0))) /\\ ((((exists fs_h_pfc_prefix_add_left_value_hcvsum_body_terminal. fs_h_pfc_prefix_add_left_value_hcvsum_body_terminal + S (pfc_natural_sum_prefix_add_left_value_hcv) = S ((S (S (i))) * fs_v_pfc_prefix_add_left_value_hcvsum)) /\\ exists fs_q_pfc_prefix_add_left_value_hcvsum_body_terminal. fs_u_pfc_prefix_add_left_value_hcvsum = fs_q_pfc_prefix_add_left_value_hcvsum_body_terminal * S ((S (S (i))) * fs_v_pfc_prefix_add_left_value_hcvsum) + (pfc_natural_sum_prefix_add_left_value_hcv))) /\\ forall fs_i_pfc_prefix_add_left_value_hcvsum_body_steps. (exists fs_lt_pfc_prefix_add_left_value_hcvsum_body_steps_bound. fs_lt_pfc_prefix_add_left_value_hcvsum_body_steps_bound + S fs_i_pfc_prefix_add_left_value_hcvsum_body_steps = S (i)) -> exists fs_a_pfc_prefix_add_left_value_hcvsum_body_steps fs_r_pfc_prefix_add_left_value_hcvsum_body_steps fs_s_pfc_prefix_add_left_value_hcvsum_body_steps. ((((exists fs_h_pfc_prefix_add_left_value_hcvsum_body_steps_summand. fs_h_pfc_prefix_add_left_value_hcvsum_body_steps_summand + S (fs_a_pfc_prefix_add_left_value_hcvsum_body_steps) = S ((S (fs_i_pfc_prefix_add_left_value_hcvsum_body_steps)) * pfc_terms_scale_prefix_add_left_value_hcv)) /\\ exists fs_q_pfc_prefix_add_left_value_hcvsum_body_steps_summand. pfc_terms_code_prefix_add_left_value_hcv = fs_q_pfc_prefix_add_left_value_hcvsum_body_steps_summand * S ((S (fs_i_pfc_prefix_add_left_value_hcvsum_body_steps)) * pfc_terms_scale_prefix_add_left_value_hcv) + (fs_a_pfc_prefix_add_left_value_hcvsum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_add_left_value_hcvsum_body_steps_partial. fs_h_pfc_prefix_add_left_value_hcvsum_body_steps_partial + S (fs_r_pfc_prefix_add_left_value_hcvsum_body_steps) = S ((S (fs_i_pfc_prefix_add_left_value_hcvsum_body_steps)) * fs_v_pfc_prefix_add_left_value_hcvsum)) /\\ exists fs_q_pfc_prefix_add_left_value_hcvsum_body_steps_partial. fs_u_pfc_prefix_add_left_value_hcvsum = fs_q_pfc_prefix_add_left_value_hcvsum_body_steps_partial * S ((S (fs_i_pfc_prefix_add_left_value_hcvsum_body_steps)) * fs_v_pfc_prefix_add_left_value_hcvsum) + (fs_r_pfc_prefix_add_left_value_hcvsum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_add_left_value_hcvsum_body_steps_successor. fs_h_pfc_prefix_add_left_value_hcvsum_body_steps_successor + S (fs_s_pfc_prefix_add_left_value_hcvsum_body_steps) = S ((S (S fs_i_pfc_prefix_add_left_value_hcvsum_body_steps)) * fs_v_pfc_prefix_add_left_value_hcvsum)) /\\ exists fs_q_pfc_prefix_add_left_value_hcvsum_body_steps_successor. fs_u_pfc_prefix_add_left_value_hcvsum = fs_q_pfc_prefix_add_left_value_hcvsum_body_steps_successor * S ((S (S fs_i_pfc_prefix_add_left_value_hcvsum_body_steps)) * fs_v_pfc_prefix_add_left_value_hcvsum) + (fs_s_pfc_prefix_add_left_value_hcvsum_body_steps))) /\\ fs_s_pfc_prefix_add_left_value_hcvsum_body_steps = fs_r_pfc_prefix_add_left_value_hcvsum_body_steps + fs_a_pfc_prefix_add_left_value_hcvsum_body_steps)))))) /\\ ((((exists pfa_gap_prefix_add_left_value_hcvresiduebound. pfa_gap_prefix_add_left_value_hcvresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_prefix_add_left_value_hcvresiduecongruence pfa_offset_right_prefix_add_left_value_hcvresiduecongruence. (pfc_natural_sum_prefix_add_left_value_hcv) + (p) * pfa_offset_left_prefix_add_left_value_hcvresiduecongruence = (r) + (p) * pfa_offset_right_prefix_add_left_value_hcvresiduecongruence)))))))))))", "specialize hv (i)", "apply hv", "exact hi", "cases hcv", "cases hcv_witness", "have hcw : exists r. ((((exists ff_h_pfp_prefix_add_left_hcw. ff_h_pfp_prefix_add_left_hcw + S (r) = S ((S (i)) * wc)) /\\ exists ff_q_pfp_prefix_add_left_hcw. wb = ff_q_pfp_prefix_add_left_hcw * S ((S (i)) * wc) + (r))) /\\ ((exists pfc_terms_code_prefix_add_left_value_hcw pfc_terms_scale_prefix_add_left_value_hcw pfc_natural_sum_prefix_add_left_value_hcw. ((forall pfc_index_prefix_add_left_value_hcwdiagonal. (exists pfa_gap_prefix_add_left_value_hcwdiagonalbound. pfa_gap_prefix_add_left_value_hcwdiagonalbound + S (pfc_index_prefix_add_left_value_hcwdiagonal) = (S (i))) -> exists pfc_value_prefix_add_left_value_hcwdiagonal. ((((exists ff_h_pfp_prefix_add_left_value_hcwdiagonalentry. ff_h_pfp_prefix_add_left_value_hcwdiagonalentry + S (pfc_value_prefix_add_left_value_hcwdiagonal) = S ((S (pfc_index_prefix_add_left_value_hcwdiagonal)) * pfc_terms_scale_prefix_add_left_value_hcw)) /\\ exists ff_q_pfp_prefix_add_left_value_hcwdiagonalentry. pfc_terms_code_prefix_add_left_value_hcw = ff_q_pfp_prefix_add_left_value_hcwdiagonalentry * S ((S (pfc_index_prefix_add_left_value_hcwdiagonal)) * pfc_terms_scale_prefix_add_left_value_hcw) + (pfc_value_prefix_add_left_value_hcwdiagonal))) /\\ ((exists pfc_complement_prefix_add_left_value_hcwdiagonalterm pfc_left_prefix_add_left_value_hcwdiagonalterm pfc_right_prefix_add_left_value_hcwdiagonalterm. (((pfc_index_prefix_add_left_value_hcwdiagonal)+pfc_complement_prefix_add_left_value_hcwdiagonalterm=(i)) /\\ ((((((exists pfa_gap_prefix_add_left_value_hcwdiagonaltermleftinside. pfa_gap_prefix_add_left_value_hcwdiagonaltermleftinside + S (pfc_index_prefix_add_left_value_hcwdiagonal) = (M)) /\\ ((((exists ff_h_pfp_prefix_add_left_value_hcwdiagonaltermleftentry. ff_h_pfp_prefix_add_left_value_hcwdiagonaltermleftentry + S (pfc_left_prefix_add_left_value_hcwdiagonalterm) = S ((S (pfc_index_prefix_add_left_value_hcwdiagonal)) * dc)) /\\ exists ff_q_pfp_prefix_add_left_value_hcwdiagonaltermleftentry. db = ff_q_pfp_prefix_add_left_value_hcwdiagonaltermleftentry * S ((S (pfc_index_prefix_add_left_value_hcwdiagonal)) * dc) + (pfc_left_prefix_add_left_value_hcwdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_add_left_value_hcwdiagonaltermleftoutside. pfc_gap_prefix_add_left_value_hcwdiagonaltermleftoutside+(M)=(pfc_index_prefix_add_left_value_hcwdiagonal)) /\\ (((pfc_left_prefix_add_left_value_hcwdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_prefix_add_left_value_hcwdiagonaltermrightinside. pfa_gap_prefix_add_left_value_hcwdiagonaltermrightinside + S (pfc_complement_prefix_add_left_value_hcwdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_prefix_add_left_value_hcwdiagonaltermrightentry. ff_h_pfp_prefix_add_left_value_hcwdiagonaltermrightentry + S (pfc_right_prefix_add_left_value_hcwdiagonalterm) = S ((S (pfc_complement_prefix_add_left_value_hcwdiagonalterm)) * cc)) /\\ exists ff_q_pfp_prefix_add_left_value_hcwdiagonaltermrightentry. cb = ff_q_pfp_prefix_add_left_value_hcwdiagonaltermrightentry * S ((S (pfc_complement_prefix_add_left_value_hcwdiagonalterm)) * cc) + (pfc_right_prefix_add_left_value_hcwdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_add_left_value_hcwdiagonaltermrightoutside. pfc_gap_prefix_add_left_value_hcwdiagonaltermrightoutside+(L)=(pfc_complement_prefix_add_left_value_hcwdiagonalterm)) /\\ (((pfc_right_prefix_add_left_value_hcwdiagonalterm)=0))))) /\\ (((pfc_value_prefix_add_left_value_hcwdiagonal)=pfc_left_prefix_add_left_value_hcwdiagonalterm*pfc_right_prefix_add_left_value_hcwdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_prefix_add_left_value_hcwsum fs_v_pfc_prefix_add_left_value_hcwsum. ((((exists fs_h_pfc_prefix_add_left_value_hcwsum_body_start. fs_h_pfc_prefix_add_left_value_hcwsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_prefix_add_left_value_hcwsum)) /\\ exists fs_q_pfc_prefix_add_left_value_hcwsum_body_start. fs_u_pfc_prefix_add_left_value_hcwsum = fs_q_pfc_prefix_add_left_value_hcwsum_body_start * S ((S (0)) * fs_v_pfc_prefix_add_left_value_hcwsum) + (0))) /\\ ((((exists fs_h_pfc_prefix_add_left_value_hcwsum_body_terminal. fs_h_pfc_prefix_add_left_value_hcwsum_body_terminal + S (pfc_natural_sum_prefix_add_left_value_hcw) = S ((S (S (i))) * fs_v_pfc_prefix_add_left_value_hcwsum)) /\\ exists fs_q_pfc_prefix_add_left_value_hcwsum_body_terminal. fs_u_pfc_prefix_add_left_value_hcwsum = fs_q_pfc_prefix_add_left_value_hcwsum_body_terminal * S ((S (S (i))) * fs_v_pfc_prefix_add_left_value_hcwsum) + (pfc_natural_sum_prefix_add_left_value_hcw))) /\\ forall fs_i_pfc_prefix_add_left_value_hcwsum_body_steps. (exists fs_lt_pfc_prefix_add_left_value_hcwsum_body_steps_bound. fs_lt_pfc_prefix_add_left_value_hcwsum_body_steps_bound + S fs_i_pfc_prefix_add_left_value_hcwsum_body_steps = S (i)) -> exists fs_a_pfc_prefix_add_left_value_hcwsum_body_steps fs_r_pfc_prefix_add_left_value_hcwsum_body_steps fs_s_pfc_prefix_add_left_value_hcwsum_body_steps. ((((exists fs_h_pfc_prefix_add_left_value_hcwsum_body_steps_summand. fs_h_pfc_prefix_add_left_value_hcwsum_body_steps_summand + S (fs_a_pfc_prefix_add_left_value_hcwsum_body_steps) = S ((S (fs_i_pfc_prefix_add_left_value_hcwsum_body_steps)) * pfc_terms_scale_prefix_add_left_value_hcw)) /\\ exists fs_q_pfc_prefix_add_left_value_hcwsum_body_steps_summand. pfc_terms_code_prefix_add_left_value_hcw = fs_q_pfc_prefix_add_left_value_hcwsum_body_steps_summand * S ((S (fs_i_pfc_prefix_add_left_value_hcwsum_body_steps)) * pfc_terms_scale_prefix_add_left_value_hcw) + (fs_a_pfc_prefix_add_left_value_hcwsum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_add_left_value_hcwsum_body_steps_partial. fs_h_pfc_prefix_add_left_value_hcwsum_body_steps_partial + S (fs_r_pfc_prefix_add_left_value_hcwsum_body_steps) = S ((S (fs_i_pfc_prefix_add_left_value_hcwsum_body_steps)) * fs_v_pfc_prefix_add_left_value_hcwsum)) /\\ exists fs_q_pfc_prefix_add_left_value_hcwsum_body_steps_partial. fs_u_pfc_prefix_add_left_value_hcwsum = fs_q_pfc_prefix_add_left_value_hcwsum_body_steps_partial * S ((S (fs_i_pfc_prefix_add_left_value_hcwsum_body_steps)) * fs_v_pfc_prefix_add_left_value_hcwsum) + (fs_r_pfc_prefix_add_left_value_hcwsum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_add_left_value_hcwsum_body_steps_successor. fs_h_pfc_prefix_add_left_value_hcwsum_body_steps_successor + S (fs_s_pfc_prefix_add_left_value_hcwsum_body_steps) = S ((S (S fs_i_pfc_prefix_add_left_value_hcwsum_body_steps)) * fs_v_pfc_prefix_add_left_value_hcwsum)) /\\ exists fs_q_pfc_prefix_add_left_value_hcwsum_body_steps_successor. fs_u_pfc_prefix_add_left_value_hcwsum = fs_q_pfc_prefix_add_left_value_hcwsum_body_steps_successor * S ((S (S fs_i_pfc_prefix_add_left_value_hcwsum_body_steps)) * fs_v_pfc_prefix_add_left_value_hcwsum) + (fs_s_pfc_prefix_add_left_value_hcwsum_body_steps))) /\\ fs_s_pfc_prefix_add_left_value_hcwsum_body_steps = fs_r_pfc_prefix_add_left_value_hcwsum_body_steps + fs_a_pfc_prefix_add_left_value_hcwsum_body_steps)))))) /\\ ((((exists pfa_gap_prefix_add_left_value_hcwresiduebound. pfa_gap_prefix_add_left_value_hcwresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_prefix_add_left_value_hcwresiduecongruence pfa_offset_right_prefix_add_left_value_hcwresiduecongruence. (pfc_natural_sum_prefix_add_left_value_hcw) + (p) * pfa_offset_left_prefix_add_left_value_hcwresiduecongruence = (r) + (p) * pfa_offset_right_prefix_add_left_value_hcwresiduecongruence)))))))))))", "specialize hw (i)", "apply hw", "exact hi", "cases hcw", "cases hcw_witness", "exists x", "exists x1", "exists x2", "split", "exact hcu_witness_left", "split", "exact hcv_witness_left", "split", "exact hcw_witness_left", "specialize prime_field_convolution_coefficient_left_add (p)", "specialize prime_field_convolution_coefficient_left_add (ab)", "specialize prime_field_convolution_coefficient_left_add (ac)", "specialize prime_field_convolution_coefficient_left_add (bb)", "specialize prime_field_convolution_coefficient_left_add (bc)", "specialize prime_field_convolution_coefficient_left_add (cb)", "specialize prime_field_convolution_coefficient_left_add (cc)", "specialize prime_field_convolution_coefficient_left_add (L)", "specialize prime_field_convolution_coefficient_left_add (db)", "specialize prime_field_convolution_coefficient_left_add (dc)", "specialize prime_field_convolution_coefficient_left_add (M)", "specialize prime_field_convolution_coefficient_left_add (i)", "specialize prime_field_convolution_coefficient_left_add (x)", "specialize prime_field_convolution_coefficient_left_add (x1)", "specialize prime_field_convolution_coefficient_left_add (x2)", "apply prime_field_convolution_coefficient_left_add", "exact hs", "exact hcu_witness_right", "exact hcv_witness_right", "exact hcw_witness_right" ], "script_sha256": "9274601b8053dcc36e384c1682b917607c384160b5071e63c154e1a23459c249", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_distributivity_candidate.py", "sha256": "a959962d631759cd1fc773dd7eef2fadf4f3f95361d6d7bc8c6a9e82d0d4ab86" }, "statement": "forall p ab ac bb bc cb cc L db dc M ub uc vb vc wb wc N. (forall pfp_index_prefix_add_left_source. (exists pfa_gap_prefix_add_left_sourceindex. pfa_gap_prefix_add_left_sourceindex + S (pfp_index_prefix_add_left_source) = (L)) -> exists pfp_left_prefix_add_left_source pfp_right_prefix_add_left_source pfp_value_prefix_add_left_source. ((((exists ff_h_pfp_prefix_add_left_sourceleft. ff_h_pfp_prefix_add_left_sourceleft + S (pfp_left_prefix_add_left_source) = S ((S (pfp_index_prefix_add_left_source)) * ac)) /\\ exists ff_q_pfp_prefix_add_left_sourceleft. ab = ff_q_pfp_prefix_add_left_sourceleft * S ((S (pfp_index_prefix_add_left_source)) * ac) + (pfp_left_prefix_add_left_source))) /\\ (((((exists ff_h_pfp_prefix_add_left_sourceright. ff_h_pfp_prefix_add_left_sourceright + S (pfp_right_prefix_add_left_source) = S ((S (pfp_index_prefix_add_left_source)) * bc)) /\\ exists ff_q_pfp_prefix_add_left_sourceright. bb = ff_q_pfp_prefix_add_left_sourceright * S ((S (pfp_index_prefix_add_left_source)) * bc) + (pfp_right_prefix_add_left_source))) /\\ (((((exists ff_h_pfp_prefix_add_left_sourcetarget. ff_h_pfp_prefix_add_left_sourcetarget + S (pfp_value_prefix_add_left_source) = S ((S (pfp_index_prefix_add_left_source)) * cc)) /\\ exists ff_q_pfp_prefix_add_left_sourcetarget. cb = ff_q_pfp_prefix_add_left_sourcetarget * S ((S (pfp_index_prefix_add_left_source)) * cc) + (pfp_value_prefix_add_left_source))) /\\ ((((exists pfa_gap_prefix_add_left_sourceoperationleft. pfa_gap_prefix_add_left_sourceoperationleft + S (pfp_left_prefix_add_left_source) = (p)) /\\ (((exists pfa_gap_prefix_add_left_sourceoperationright. pfa_gap_prefix_add_left_sourceoperationright + S (pfp_right_prefix_add_left_source) = (p)) /\\ ((((exists pfa_gap_prefix_add_left_sourceoperationresultbound. pfa_gap_prefix_add_left_sourceoperationresultbound + S (pfp_value_prefix_add_left_source) = (p)) /\\ ((exists pfa_offset_left_prefix_add_left_sourceoperationresultcongruence pfa_offset_right_prefix_add_left_sourceoperationresultcongruence. ((pfp_left_prefix_add_left_source) + (pfp_right_prefix_add_left_source)) + (p) * pfa_offset_left_prefix_add_left_sourceoperationresultcongruence = (pfp_value_prefix_add_left_source) + (p) * pfa_offset_right_prefix_add_left_sourceoperationresultcongruence)))))))))))))))) -> (forall pfc_index_prefix_add_left_ub. (exists pfa_gap_prefix_add_left_ubbound. pfa_gap_prefix_add_left_ubbound + S (pfc_index_prefix_add_left_ub) = (N)) -> exists pfc_value_prefix_add_left_ub. ((((exists ff_h_pfp_prefix_add_left_ubentry. ff_h_pfp_prefix_add_left_ubentry + S (pfc_value_prefix_add_left_ub) = S ((S (pfc_index_prefix_add_left_ub)) * uc)) /\\ exists ff_q_pfp_prefix_add_left_ubentry. ub = ff_q_pfp_prefix_add_left_ubentry * S ((S (pfc_index_prefix_add_left_ub)) * uc) + (pfc_value_prefix_add_left_ub))) /\\ ((exists pfc_terms_code_prefix_add_left_ubcoefficient pfc_terms_scale_prefix_add_left_ubcoefficient pfc_natural_sum_prefix_add_left_ubcoefficient. ((forall pfc_index_prefix_add_left_ubcoefficientdiagonal. (exists pfa_gap_prefix_add_left_ubcoefficientdiagonalbound. pfa_gap_prefix_add_left_ubcoefficientdiagonalbound + S (pfc_index_prefix_add_left_ubcoefficientdiagonal) = (S (pfc_index_prefix_add_left_ub))) -> exists pfc_value_prefix_add_left_ubcoefficientdiagonal. ((((exists ff_h_pfp_prefix_add_left_ubcoefficientdiagonalentry. ff_h_pfp_prefix_add_left_ubcoefficientdiagonalentry + S (pfc_value_prefix_add_left_ubcoefficientdiagonal) = S ((S (pfc_index_prefix_add_left_ubcoefficientdiagonal)) * pfc_terms_scale_prefix_add_left_ubcoefficient)) /\\ exists ff_q_pfp_prefix_add_left_ubcoefficientdiagonalentry. pfc_terms_code_prefix_add_left_ubcoefficient = ff_q_pfp_prefix_add_left_ubcoefficientdiagonalentry * S ((S (pfc_index_prefix_add_left_ubcoefficientdiagonal)) * pfc_terms_scale_prefix_add_left_ubcoefficient) + (pfc_value_prefix_add_left_ubcoefficientdiagonal))) /\\ ((exists pfc_complement_prefix_add_left_ubcoefficientdiagonalterm pfc_left_prefix_add_left_ubcoefficientdiagonalterm pfc_right_prefix_add_left_ubcoefficientdiagonalterm. (((pfc_index_prefix_add_left_ubcoefficientdiagonal)+pfc_complement_prefix_add_left_ubcoefficientdiagonalterm=(pfc_index_prefix_add_left_ub)) /\\ ((((((exists pfa_gap_prefix_add_left_ubcoefficientdiagonaltermleftinside. pfa_gap_prefix_add_left_ubcoefficientdiagonaltermleftinside + S (pfc_index_prefix_add_left_ubcoefficientdiagonal) = (M)) /\\ ((((exists ff_h_pfp_prefix_add_left_ubcoefficientdiagonaltermleftentry. ff_h_pfp_prefix_add_left_ubcoefficientdiagonaltermleftentry + S (pfc_left_prefix_add_left_ubcoefficientdiagonalterm) = S ((S (pfc_index_prefix_add_left_ubcoefficientdiagonal)) * dc)) /\\ exists ff_q_pfp_prefix_add_left_ubcoefficientdiagonaltermleftentry. db = ff_q_pfp_prefix_add_left_ubcoefficientdiagonaltermleftentry * S ((S (pfc_index_prefix_add_left_ubcoefficientdiagonal)) * dc) + (pfc_left_prefix_add_left_ubcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_add_left_ubcoefficientdiagonaltermleftoutside. pfc_gap_prefix_add_left_ubcoefficientdiagonaltermleftoutside+(M)=(pfc_index_prefix_add_left_ubcoefficientdiagonal)) /\\ (((pfc_left_prefix_add_left_ubcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_prefix_add_left_ubcoefficientdiagonaltermrightinside. pfa_gap_prefix_add_left_ubcoefficientdiagonaltermrightinside + S (pfc_complement_prefix_add_left_ubcoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_prefix_add_left_ubcoefficientdiagonaltermrightentry. ff_h_pfp_prefix_add_left_ubcoefficientdiagonaltermrightentry + S (pfc_right_prefix_add_left_ubcoefficientdiagonalterm) = S ((S (pfc_complement_prefix_add_left_ubcoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_prefix_add_left_ubcoefficientdiagonaltermrightentry. ab = ff_q_pfp_prefix_add_left_ubcoefficientdiagonaltermrightentry * S ((S (pfc_complement_prefix_add_left_ubcoefficientdiagonalterm)) * ac) + (pfc_right_prefix_add_left_ubcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_add_left_ubcoefficientdiagonaltermrightoutside. pfc_gap_prefix_add_left_ubcoefficientdiagonaltermrightoutside+(L)=(pfc_complement_prefix_add_left_ubcoefficientdiagonalterm)) /\\ (((pfc_right_prefix_add_left_ubcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_prefix_add_left_ubcoefficientdiagonal)=pfc_left_prefix_add_left_ubcoefficientdiagonalterm*pfc_right_prefix_add_left_ubcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_prefix_add_left_ubcoefficientsum fs_v_pfc_prefix_add_left_ubcoefficientsum. ((((exists fs_h_pfc_prefix_add_left_ubcoefficientsum_body_start. fs_h_pfc_prefix_add_left_ubcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_prefix_add_left_ubcoefficientsum)) /\\ exists fs_q_pfc_prefix_add_left_ubcoefficientsum_body_start. fs_u_pfc_prefix_add_left_ubcoefficientsum = fs_q_pfc_prefix_add_left_ubcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_prefix_add_left_ubcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_prefix_add_left_ubcoefficientsum_body_terminal. fs_h_pfc_prefix_add_left_ubcoefficientsum_body_terminal + S (pfc_natural_sum_prefix_add_left_ubcoefficient) = S ((S (S (pfc_index_prefix_add_left_ub))) * fs_v_pfc_prefix_add_left_ubcoefficientsum)) /\\ exists fs_q_pfc_prefix_add_left_ubcoefficientsum_body_terminal. fs_u_pfc_prefix_add_left_ubcoefficientsum = fs_q_pfc_prefix_add_left_ubcoefficientsum_body_terminal * S ((S (S (pfc_index_prefix_add_left_ub))) * fs_v_pfc_prefix_add_left_ubcoefficientsum) + (pfc_natural_sum_prefix_add_left_ubcoefficient))) /\\ forall fs_i_pfc_prefix_add_left_ubcoefficientsum_body_steps. (exists fs_lt_pfc_prefix_add_left_ubcoefficientsum_body_steps_bound. fs_lt_pfc_prefix_add_left_ubcoefficientsum_body_steps_bound + S fs_i_pfc_prefix_add_left_ubcoefficientsum_body_steps = S (pfc_index_prefix_add_left_ub)) -> exists fs_a_pfc_prefix_add_left_ubcoefficientsum_body_steps fs_r_pfc_prefix_add_left_ubcoefficientsum_body_steps fs_s_pfc_prefix_add_left_ubcoefficientsum_body_steps. ((((exists fs_h_pfc_prefix_add_left_ubcoefficientsum_body_steps_summand. fs_h_pfc_prefix_add_left_ubcoefficientsum_body_steps_summand + S (fs_a_pfc_prefix_add_left_ubcoefficientsum_body_steps) = S ((S (fs_i_pfc_prefix_add_left_ubcoefficientsum_body_steps)) * pfc_terms_scale_prefix_add_left_ubcoefficient)) /\\ exists fs_q_pfc_prefix_add_left_ubcoefficientsum_body_steps_summand. pfc_terms_code_prefix_add_left_ubcoefficient = fs_q_pfc_prefix_add_left_ubcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_prefix_add_left_ubcoefficientsum_body_steps)) * pfc_terms_scale_prefix_add_left_ubcoefficient) + (fs_a_pfc_prefix_add_left_ubcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_add_left_ubcoefficientsum_body_steps_partial. fs_h_pfc_prefix_add_left_ubcoefficientsum_body_steps_partial + S (fs_r_pfc_prefix_add_left_ubcoefficientsum_body_steps) = S ((S (fs_i_pfc_prefix_add_left_ubcoefficientsum_body_steps)) * fs_v_pfc_prefix_add_left_ubcoefficientsum)) /\\ exists fs_q_pfc_prefix_add_left_ubcoefficientsum_body_steps_partial. fs_u_pfc_prefix_add_left_ubcoefficientsum = fs_q_pfc_prefix_add_left_ubcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_prefix_add_left_ubcoefficientsum_body_steps)) * fs_v_pfc_prefix_add_left_ubcoefficientsum) + (fs_r_pfc_prefix_add_left_ubcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_add_left_ubcoefficientsum_body_steps_successor. fs_h_pfc_prefix_add_left_ubcoefficientsum_body_steps_successor + S (fs_s_pfc_prefix_add_left_ubcoefficientsum_body_steps) = S ((S (S fs_i_pfc_prefix_add_left_ubcoefficientsum_body_steps)) * fs_v_pfc_prefix_add_left_ubcoefficientsum)) /\\ exists fs_q_pfc_prefix_add_left_ubcoefficientsum_body_steps_successor. fs_u_pfc_prefix_add_left_ubcoefficientsum = fs_q_pfc_prefix_add_left_ubcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_prefix_add_left_ubcoefficientsum_body_steps)) * fs_v_pfc_prefix_add_left_ubcoefficientsum) + (fs_s_pfc_prefix_add_left_ubcoefficientsum_body_steps))) /\\ fs_s_pfc_prefix_add_left_ubcoefficientsum_body_steps = fs_r_pfc_prefix_add_left_ubcoefficientsum_body_steps + fs_a_pfc_prefix_add_left_ubcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_prefix_add_left_ubcoefficientresiduebound. pfa_gap_prefix_add_left_ubcoefficientresiduebound + S (pfc_value_prefix_add_left_ub) = (p)) /\\ ((exists pfa_offset_left_prefix_add_left_ubcoefficientresiduecongruence pfa_offset_right_prefix_add_left_ubcoefficientresiduecongruence. (pfc_natural_sum_prefix_add_left_ubcoefficient) + (p) * pfa_offset_left_prefix_add_left_ubcoefficientresiduecongruence = (pfc_value_prefix_add_left_ub) + (p) * pfa_offset_right_prefix_add_left_ubcoefficientresiduecongruence)))))))))))) -> (forall pfc_index_prefix_add_left_vb. (exists pfa_gap_prefix_add_left_vbbound. pfa_gap_prefix_add_left_vbbound + S (pfc_index_prefix_add_left_vb) = (N)) -> exists pfc_value_prefix_add_left_vb. ((((exists ff_h_pfp_prefix_add_left_vbentry. ff_h_pfp_prefix_add_left_vbentry + S (pfc_value_prefix_add_left_vb) = S ((S (pfc_index_prefix_add_left_vb)) * vc)) /\\ exists ff_q_pfp_prefix_add_left_vbentry. vb = ff_q_pfp_prefix_add_left_vbentry * S ((S (pfc_index_prefix_add_left_vb)) * vc) + (pfc_value_prefix_add_left_vb))) /\\ ((exists pfc_terms_code_prefix_add_left_vbcoefficient pfc_terms_scale_prefix_add_left_vbcoefficient pfc_natural_sum_prefix_add_left_vbcoefficient. ((forall pfc_index_prefix_add_left_vbcoefficientdiagonal. (exists pfa_gap_prefix_add_left_vbcoefficientdiagonalbound. pfa_gap_prefix_add_left_vbcoefficientdiagonalbound + S (pfc_index_prefix_add_left_vbcoefficientdiagonal) = (S (pfc_index_prefix_add_left_vb))) -> exists pfc_value_prefix_add_left_vbcoefficientdiagonal. ((((exists ff_h_pfp_prefix_add_left_vbcoefficientdiagonalentry. ff_h_pfp_prefix_add_left_vbcoefficientdiagonalentry + S (pfc_value_prefix_add_left_vbcoefficientdiagonal) = S ((S (pfc_index_prefix_add_left_vbcoefficientdiagonal)) * pfc_terms_scale_prefix_add_left_vbcoefficient)) /\\ exists ff_q_pfp_prefix_add_left_vbcoefficientdiagonalentry. pfc_terms_code_prefix_add_left_vbcoefficient = ff_q_pfp_prefix_add_left_vbcoefficientdiagonalentry * S ((S (pfc_index_prefix_add_left_vbcoefficientdiagonal)) * pfc_terms_scale_prefix_add_left_vbcoefficient) + (pfc_value_prefix_add_left_vbcoefficientdiagonal))) /\\ ((exists pfc_complement_prefix_add_left_vbcoefficientdiagonalterm pfc_left_prefix_add_left_vbcoefficientdiagonalterm pfc_right_prefix_add_left_vbcoefficientdiagonalterm. (((pfc_index_prefix_add_left_vbcoefficientdiagonal)+pfc_complement_prefix_add_left_vbcoefficientdiagonalterm=(pfc_index_prefix_add_left_vb)) /\\ ((((((exists pfa_gap_prefix_add_left_vbcoefficientdiagonaltermleftinside. pfa_gap_prefix_add_left_vbcoefficientdiagonaltermleftinside + S (pfc_index_prefix_add_left_vbcoefficientdiagonal) = (M)) /\\ ((((exists ff_h_pfp_prefix_add_left_vbcoefficientdiagonaltermleftentry. ff_h_pfp_prefix_add_left_vbcoefficientdiagonaltermleftentry + S (pfc_left_prefix_add_left_vbcoefficientdiagonalterm) = S ((S (pfc_index_prefix_add_left_vbcoefficientdiagonal)) * dc)) /\\ exists ff_q_pfp_prefix_add_left_vbcoefficientdiagonaltermleftentry. db = ff_q_pfp_prefix_add_left_vbcoefficientdiagonaltermleftentry * S ((S (pfc_index_prefix_add_left_vbcoefficientdiagonal)) * dc) + (pfc_left_prefix_add_left_vbcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_add_left_vbcoefficientdiagonaltermleftoutside. pfc_gap_prefix_add_left_vbcoefficientdiagonaltermleftoutside+(M)=(pfc_index_prefix_add_left_vbcoefficientdiagonal)) /\\ (((pfc_left_prefix_add_left_vbcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_prefix_add_left_vbcoefficientdiagonaltermrightinside. pfa_gap_prefix_add_left_vbcoefficientdiagonaltermrightinside + S (pfc_complement_prefix_add_left_vbcoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_prefix_add_left_vbcoefficientdiagonaltermrightentry. ff_h_pfp_prefix_add_left_vbcoefficientdiagonaltermrightentry + S (pfc_right_prefix_add_left_vbcoefficientdiagonalterm) = S ((S (pfc_complement_prefix_add_left_vbcoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_prefix_add_left_vbcoefficientdiagonaltermrightentry. bb = ff_q_pfp_prefix_add_left_vbcoefficientdiagonaltermrightentry * S ((S (pfc_complement_prefix_add_left_vbcoefficientdiagonalterm)) * bc) + (pfc_right_prefix_add_left_vbcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_add_left_vbcoefficientdiagonaltermrightoutside. pfc_gap_prefix_add_left_vbcoefficientdiagonaltermrightoutside+(L)=(pfc_complement_prefix_add_left_vbcoefficientdiagonalterm)) /\\ (((pfc_right_prefix_add_left_vbcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_prefix_add_left_vbcoefficientdiagonal)=pfc_left_prefix_add_left_vbcoefficientdiagonalterm*pfc_right_prefix_add_left_vbcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_prefix_add_left_vbcoefficientsum fs_v_pfc_prefix_add_left_vbcoefficientsum. ((((exists fs_h_pfc_prefix_add_left_vbcoefficientsum_body_start. fs_h_pfc_prefix_add_left_vbcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_prefix_add_left_vbcoefficientsum)) /\\ exists fs_q_pfc_prefix_add_left_vbcoefficientsum_body_start. fs_u_pfc_prefix_add_left_vbcoefficientsum = fs_q_pfc_prefix_add_left_vbcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_prefix_add_left_vbcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_prefix_add_left_vbcoefficientsum_body_terminal. fs_h_pfc_prefix_add_left_vbcoefficientsum_body_terminal + S (pfc_natural_sum_prefix_add_left_vbcoefficient) = S ((S (S (pfc_index_prefix_add_left_vb))) * fs_v_pfc_prefix_add_left_vbcoefficientsum)) /\\ exists fs_q_pfc_prefix_add_left_vbcoefficientsum_body_terminal. fs_u_pfc_prefix_add_left_vbcoefficientsum = fs_q_pfc_prefix_add_left_vbcoefficientsum_body_terminal * S ((S (S (pfc_index_prefix_add_left_vb))) * fs_v_pfc_prefix_add_left_vbcoefficientsum) + (pfc_natural_sum_prefix_add_left_vbcoefficient))) /\\ forall fs_i_pfc_prefix_add_left_vbcoefficientsum_body_steps. (exists fs_lt_pfc_prefix_add_left_vbcoefficientsum_body_steps_bound. fs_lt_pfc_prefix_add_left_vbcoefficientsum_body_steps_bound + S fs_i_pfc_prefix_add_left_vbcoefficientsum_body_steps = S (pfc_index_prefix_add_left_vb)) -> exists fs_a_pfc_prefix_add_left_vbcoefficientsum_body_steps fs_r_pfc_prefix_add_left_vbcoefficientsum_body_steps fs_s_pfc_prefix_add_left_vbcoefficientsum_body_steps. ((((exists fs_h_pfc_prefix_add_left_vbcoefficientsum_body_steps_summand. fs_h_pfc_prefix_add_left_vbcoefficientsum_body_steps_summand + S (fs_a_pfc_prefix_add_left_vbcoefficientsum_body_steps) = S ((S (fs_i_pfc_prefix_add_left_vbcoefficientsum_body_steps)) * pfc_terms_scale_prefix_add_left_vbcoefficient)) /\\ exists fs_q_pfc_prefix_add_left_vbcoefficientsum_body_steps_summand. pfc_terms_code_prefix_add_left_vbcoefficient = fs_q_pfc_prefix_add_left_vbcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_prefix_add_left_vbcoefficientsum_body_steps)) * pfc_terms_scale_prefix_add_left_vbcoefficient) + (fs_a_pfc_prefix_add_left_vbcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_add_left_vbcoefficientsum_body_steps_partial. fs_h_pfc_prefix_add_left_vbcoefficientsum_body_steps_partial + S (fs_r_pfc_prefix_add_left_vbcoefficientsum_body_steps) = S ((S (fs_i_pfc_prefix_add_left_vbcoefficientsum_body_steps)) * fs_v_pfc_prefix_add_left_vbcoefficientsum)) /\\ exists fs_q_pfc_prefix_add_left_vbcoefficientsum_body_steps_partial. fs_u_pfc_prefix_add_left_vbcoefficientsum = fs_q_pfc_prefix_add_left_vbcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_prefix_add_left_vbcoefficientsum_body_steps)) * fs_v_pfc_prefix_add_left_vbcoefficientsum) + (fs_r_pfc_prefix_add_left_vbcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_add_left_vbcoefficientsum_body_steps_successor. fs_h_pfc_prefix_add_left_vbcoefficientsum_body_steps_successor + S (fs_s_pfc_prefix_add_left_vbcoefficientsum_body_steps) = S ((S (S fs_i_pfc_prefix_add_left_vbcoefficientsum_body_steps)) * fs_v_pfc_prefix_add_left_vbcoefficientsum)) /\\ exists fs_q_pfc_prefix_add_left_vbcoefficientsum_body_steps_successor. fs_u_pfc_prefix_add_left_vbcoefficientsum = fs_q_pfc_prefix_add_left_vbcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_prefix_add_left_vbcoefficientsum_body_steps)) * fs_v_pfc_prefix_add_left_vbcoefficientsum) + (fs_s_pfc_prefix_add_left_vbcoefficientsum_body_steps))) /\\ fs_s_pfc_prefix_add_left_vbcoefficientsum_body_steps = fs_r_pfc_prefix_add_left_vbcoefficientsum_body_steps + fs_a_pfc_prefix_add_left_vbcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_prefix_add_left_vbcoefficientresiduebound. pfa_gap_prefix_add_left_vbcoefficientresiduebound + S (pfc_value_prefix_add_left_vb) = (p)) /\\ ((exists pfa_offset_left_prefix_add_left_vbcoefficientresiduecongruence pfa_offset_right_prefix_add_left_vbcoefficientresiduecongruence. (pfc_natural_sum_prefix_add_left_vbcoefficient) + (p) * pfa_offset_left_prefix_add_left_vbcoefficientresiduecongruence = (pfc_value_prefix_add_left_vb) + (p) * pfa_offset_right_prefix_add_left_vbcoefficientresiduecongruence)))))))))))) -> (forall pfc_index_prefix_add_left_wb. (exists pfa_gap_prefix_add_left_wbbound. pfa_gap_prefix_add_left_wbbound + S (pfc_index_prefix_add_left_wb) = (N)) -> exists pfc_value_prefix_add_left_wb. ((((exists ff_h_pfp_prefix_add_left_wbentry. ff_h_pfp_prefix_add_left_wbentry + S (pfc_value_prefix_add_left_wb) = S ((S (pfc_index_prefix_add_left_wb)) * wc)) /\\ exists ff_q_pfp_prefix_add_left_wbentry. wb = ff_q_pfp_prefix_add_left_wbentry * S ((S (pfc_index_prefix_add_left_wb)) * wc) + (pfc_value_prefix_add_left_wb))) /\\ ((exists pfc_terms_code_prefix_add_left_wbcoefficient pfc_terms_scale_prefix_add_left_wbcoefficient pfc_natural_sum_prefix_add_left_wbcoefficient. ((forall pfc_index_prefix_add_left_wbcoefficientdiagonal. (exists pfa_gap_prefix_add_left_wbcoefficientdiagonalbound. pfa_gap_prefix_add_left_wbcoefficientdiagonalbound + S (pfc_index_prefix_add_left_wbcoefficientdiagonal) = (S (pfc_index_prefix_add_left_wb))) -> exists pfc_value_prefix_add_left_wbcoefficientdiagonal. ((((exists ff_h_pfp_prefix_add_left_wbcoefficientdiagonalentry. ff_h_pfp_prefix_add_left_wbcoefficientdiagonalentry + S (pfc_value_prefix_add_left_wbcoefficientdiagonal) = S ((S (pfc_index_prefix_add_left_wbcoefficientdiagonal)) * pfc_terms_scale_prefix_add_left_wbcoefficient)) /\\ exists ff_q_pfp_prefix_add_left_wbcoefficientdiagonalentry. pfc_terms_code_prefix_add_left_wbcoefficient = ff_q_pfp_prefix_add_left_wbcoefficientdiagonalentry * S ((S (pfc_index_prefix_add_left_wbcoefficientdiagonal)) * pfc_terms_scale_prefix_add_left_wbcoefficient) + (pfc_value_prefix_add_left_wbcoefficientdiagonal))) /\\ ((exists pfc_complement_prefix_add_left_wbcoefficientdiagonalterm pfc_left_prefix_add_left_wbcoefficientdiagonalterm pfc_right_prefix_add_left_wbcoefficientdiagonalterm. (((pfc_index_prefix_add_left_wbcoefficientdiagonal)+pfc_complement_prefix_add_left_wbcoefficientdiagonalterm=(pfc_index_prefix_add_left_wb)) /\\ ((((((exists pfa_gap_prefix_add_left_wbcoefficientdiagonaltermleftinside. pfa_gap_prefix_add_left_wbcoefficientdiagonaltermleftinside + S (pfc_index_prefix_add_left_wbcoefficientdiagonal) = (M)) /\\ ((((exists ff_h_pfp_prefix_add_left_wbcoefficientdiagonaltermleftentry. ff_h_pfp_prefix_add_left_wbcoefficientdiagonaltermleftentry + S (pfc_left_prefix_add_left_wbcoefficientdiagonalterm) = S ((S (pfc_index_prefix_add_left_wbcoefficientdiagonal)) * dc)) /\\ exists ff_q_pfp_prefix_add_left_wbcoefficientdiagonaltermleftentry. db = ff_q_pfp_prefix_add_left_wbcoefficientdiagonaltermleftentry * S ((S (pfc_index_prefix_add_left_wbcoefficientdiagonal)) * dc) + (pfc_left_prefix_add_left_wbcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_add_left_wbcoefficientdiagonaltermleftoutside. pfc_gap_prefix_add_left_wbcoefficientdiagonaltermleftoutside+(M)=(pfc_index_prefix_add_left_wbcoefficientdiagonal)) /\\ (((pfc_left_prefix_add_left_wbcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_prefix_add_left_wbcoefficientdiagonaltermrightinside. pfa_gap_prefix_add_left_wbcoefficientdiagonaltermrightinside + S (pfc_complement_prefix_add_left_wbcoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_prefix_add_left_wbcoefficientdiagonaltermrightentry. ff_h_pfp_prefix_add_left_wbcoefficientdiagonaltermrightentry + S (pfc_right_prefix_add_left_wbcoefficientdiagonalterm) = S ((S (pfc_complement_prefix_add_left_wbcoefficientdiagonalterm)) * cc)) /\\ exists ff_q_pfp_prefix_add_left_wbcoefficientdiagonaltermrightentry. cb = ff_q_pfp_prefix_add_left_wbcoefficientdiagonaltermrightentry * S ((S (pfc_complement_prefix_add_left_wbcoefficientdiagonalterm)) * cc) + (pfc_right_prefix_add_left_wbcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_add_left_wbcoefficientdiagonaltermrightoutside. pfc_gap_prefix_add_left_wbcoefficientdiagonaltermrightoutside+(L)=(pfc_complement_prefix_add_left_wbcoefficientdiagonalterm)) /\\ (((pfc_right_prefix_add_left_wbcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_prefix_add_left_wbcoefficientdiagonal)=pfc_left_prefix_add_left_wbcoefficientdiagonalterm*pfc_right_prefix_add_left_wbcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_prefix_add_left_wbcoefficientsum fs_v_pfc_prefix_add_left_wbcoefficientsum. ((((exists fs_h_pfc_prefix_add_left_wbcoefficientsum_body_start. fs_h_pfc_prefix_add_left_wbcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_prefix_add_left_wbcoefficientsum)) /\\ exists fs_q_pfc_prefix_add_left_wbcoefficientsum_body_start. fs_u_pfc_prefix_add_left_wbcoefficientsum = fs_q_pfc_prefix_add_left_wbcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_prefix_add_left_wbcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_prefix_add_left_wbcoefficientsum_body_terminal. fs_h_pfc_prefix_add_left_wbcoefficientsum_body_terminal + S (pfc_natural_sum_prefix_add_left_wbcoefficient) = S ((S (S (pfc_index_prefix_add_left_wb))) * fs_v_pfc_prefix_add_left_wbcoefficientsum)) /\\ exists fs_q_pfc_prefix_add_left_wbcoefficientsum_body_terminal. fs_u_pfc_prefix_add_left_wbcoefficientsum = fs_q_pfc_prefix_add_left_wbcoefficientsum_body_terminal * S ((S (S (pfc_index_prefix_add_left_wb))) * fs_v_pfc_prefix_add_left_wbcoefficientsum) + (pfc_natural_sum_prefix_add_left_wbcoefficient))) /\\ forall fs_i_pfc_prefix_add_left_wbcoefficientsum_body_steps. (exists fs_lt_pfc_prefix_add_left_wbcoefficientsum_body_steps_bound. fs_lt_pfc_prefix_add_left_wbcoefficientsum_body_steps_bound + S fs_i_pfc_prefix_add_left_wbcoefficientsum_body_steps = S (pfc_index_prefix_add_left_wb)) -> exists fs_a_pfc_prefix_add_left_wbcoefficientsum_body_steps fs_r_pfc_prefix_add_left_wbcoefficientsum_body_steps fs_s_pfc_prefix_add_left_wbcoefficientsum_body_steps. ((((exists fs_h_pfc_prefix_add_left_wbcoefficientsum_body_steps_summand. fs_h_pfc_prefix_add_left_wbcoefficientsum_body_steps_summand + S (fs_a_pfc_prefix_add_left_wbcoefficientsum_body_steps) = S ((S (fs_i_pfc_prefix_add_left_wbcoefficientsum_body_steps)) * pfc_terms_scale_prefix_add_left_wbcoefficient)) /\\ exists fs_q_pfc_prefix_add_left_wbcoefficientsum_body_steps_summand. pfc_terms_code_prefix_add_left_wbcoefficient = fs_q_pfc_prefix_add_left_wbcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_prefix_add_left_wbcoefficientsum_body_steps)) * pfc_terms_scale_prefix_add_left_wbcoefficient) + (fs_a_pfc_prefix_add_left_wbcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_add_left_wbcoefficientsum_body_steps_partial. fs_h_pfc_prefix_add_left_wbcoefficientsum_body_steps_partial + S (fs_r_pfc_prefix_add_left_wbcoefficientsum_body_steps) = S ((S (fs_i_pfc_prefix_add_left_wbcoefficientsum_body_steps)) * fs_v_pfc_prefix_add_left_wbcoefficientsum)) /\\ exists fs_q_pfc_prefix_add_left_wbcoefficientsum_body_steps_partial. fs_u_pfc_prefix_add_left_wbcoefficientsum = fs_q_pfc_prefix_add_left_wbcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_prefix_add_left_wbcoefficientsum_body_steps)) * fs_v_pfc_prefix_add_left_wbcoefficientsum) + (fs_r_pfc_prefix_add_left_wbcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_add_left_wbcoefficientsum_body_steps_successor. fs_h_pfc_prefix_add_left_wbcoefficientsum_body_steps_successor + S (fs_s_pfc_prefix_add_left_wbcoefficientsum_body_steps) = S ((S (S fs_i_pfc_prefix_add_left_wbcoefficientsum_body_steps)) * fs_v_pfc_prefix_add_left_wbcoefficientsum)) /\\ exists fs_q_pfc_prefix_add_left_wbcoefficientsum_body_steps_successor. fs_u_pfc_prefix_add_left_wbcoefficientsum = fs_q_pfc_prefix_add_left_wbcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_prefix_add_left_wbcoefficientsum_body_steps)) * fs_v_pfc_prefix_add_left_wbcoefficientsum) + (fs_s_pfc_prefix_add_left_wbcoefficientsum_body_steps))) /\\ fs_s_pfc_prefix_add_left_wbcoefficientsum_body_steps = fs_r_pfc_prefix_add_left_wbcoefficientsum_body_steps + fs_a_pfc_prefix_add_left_wbcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_prefix_add_left_wbcoefficientresiduebound. pfa_gap_prefix_add_left_wbcoefficientresiduebound + S (pfc_value_prefix_add_left_wb) = (p)) /\\ ((exists pfa_offset_left_prefix_add_left_wbcoefficientresiduecongruence pfa_offset_right_prefix_add_left_wbcoefficientresiduecongruence. (pfc_natural_sum_prefix_add_left_wbcoefficient) + (p) * pfa_offset_left_prefix_add_left_wbcoefficientresiduecongruence = (pfc_value_prefix_add_left_wb) + (p) * pfa_offset_right_prefix_add_left_wbcoefficientresiduecongruence)))))))))))) -> (forall pfp_index_prefix_add_left_result. (exists pfa_gap_prefix_add_left_resultindex. pfa_gap_prefix_add_left_resultindex + S (pfp_index_prefix_add_left_result) = (N)) -> exists pfp_left_prefix_add_left_result pfp_right_prefix_add_left_result pfp_value_prefix_add_left_result. ((((exists ff_h_pfp_prefix_add_left_resultleft. ff_h_pfp_prefix_add_left_resultleft + S (pfp_left_prefix_add_left_result) = S ((S (pfp_index_prefix_add_left_result)) * uc)) /\\ exists ff_q_pfp_prefix_add_left_resultleft. ub = ff_q_pfp_prefix_add_left_resultleft * S ((S (pfp_index_prefix_add_left_result)) * uc) + (pfp_left_prefix_add_left_result))) /\\ (((((exists ff_h_pfp_prefix_add_left_resultright. ff_h_pfp_prefix_add_left_resultright + S (pfp_right_prefix_add_left_result) = S ((S (pfp_index_prefix_add_left_result)) * vc)) /\\ exists ff_q_pfp_prefix_add_left_resultright. vb = ff_q_pfp_prefix_add_left_resultright * S ((S (pfp_index_prefix_add_left_result)) * vc) + (pfp_right_prefix_add_left_result))) /\\ (((((exists ff_h_pfp_prefix_add_left_resulttarget. ff_h_pfp_prefix_add_left_resulttarget + S (pfp_value_prefix_add_left_result) = S ((S (pfp_index_prefix_add_left_result)) * wc)) /\\ exists ff_q_pfp_prefix_add_left_resulttarget. wb = ff_q_pfp_prefix_add_left_resulttarget * S ((S (pfp_index_prefix_add_left_result)) * wc) + (pfp_value_prefix_add_left_result))) /\\ ((((exists pfa_gap_prefix_add_left_resultoperationleft. pfa_gap_prefix_add_left_resultoperationleft + S (pfp_left_prefix_add_left_result) = (p)) /\\ (((exists pfa_gap_prefix_add_left_resultoperationright. pfa_gap_prefix_add_left_resultoperationright + S (pfp_right_prefix_add_left_result) = (p)) /\\ ((((exists pfa_gap_prefix_add_left_resultoperationresultbound. pfa_gap_prefix_add_left_resultoperationresultbound + S (pfp_value_prefix_add_left_result) = (p)) /\\ ((exists pfa_offset_left_prefix_add_left_resultoperationresultcongruence pfa_offset_right_prefix_add_left_resultoperationresultcongruence. ((pfp_left_prefix_add_left_result) + (pfp_right_prefix_add_left_result)) + (p) * pfa_offset_left_prefix_add_left_resultoperationresultcongruence = (pfp_value_prefix_add_left_result) + (p) * pfa_offset_right_prefix_add_left_resultoperationresultcongruence))))))))))))))))", "statement_sha256": "eac9d40997c063f48f846e141fefac76a4e120f96e354e08e57264e532a3b411", "summary": "Every requested ambient output prefix of the three actual left products satisfies actual coefficientwise addition, including N=0 and prefixes extending past product support.", "summary_sha256": "9f8e4e859f7afac21675ae43614af59674501e4225f6a35aa8eb591b688734cb" }, { "alpha_v33_frontier_enrollment": { "body_receipt_sha256": "341f148cb126b41f963b9aee6dc5f9942bacb9e127d69fb7315d14aa8608d926", "bundle_campaign": "polynomial-euclidean-division", "bundle_node_id": 327, "bundle_sha256": "6ae667d8518e4dbe722bb08ad1b08715a0d282c2893e533c8133d770fe861dcf", "campaign": "polynomial-euclidean-division", "first_enrolled_version": 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"intro dc", "intro M", "intro ub", "intro uc", "intro vb", "intro vc", "intro wb", "intro wc", "intro N", "intro hs", "intro hu", "intro hv", "intro hw", "intro i", "intro hi", "have hcu : exists r. ((((exists ff_h_pfp_prefix_add_right_hcu. ff_h_pfp_prefix_add_right_hcu + S (r) = S ((S (i)) * uc)) /\\ exists ff_q_pfp_prefix_add_right_hcu. ub = ff_q_pfp_prefix_add_right_hcu * S ((S (i)) * uc) + (r))) /\\ ((exists pfc_terms_code_prefix_add_right_value_hcu pfc_terms_scale_prefix_add_right_value_hcu pfc_natural_sum_prefix_add_right_value_hcu. ((forall pfc_index_prefix_add_right_value_hcudiagonal. (exists pfa_gap_prefix_add_right_value_hcudiagonalbound. pfa_gap_prefix_add_right_value_hcudiagonalbound + S (pfc_index_prefix_add_right_value_hcudiagonal) = (S (i))) -> exists pfc_value_prefix_add_right_value_hcudiagonal. ((((exists ff_h_pfp_prefix_add_right_value_hcudiagonalentry. ff_h_pfp_prefix_add_right_value_hcudiagonalentry + S (pfc_value_prefix_add_right_value_hcudiagonal) = S ((S (pfc_index_prefix_add_right_value_hcudiagonal)) * pfc_terms_scale_prefix_add_right_value_hcu)) /\\ exists ff_q_pfp_prefix_add_right_value_hcudiagonalentry. pfc_terms_code_prefix_add_right_value_hcu = ff_q_pfp_prefix_add_right_value_hcudiagonalentry * S ((S (pfc_index_prefix_add_right_value_hcudiagonal)) * pfc_terms_scale_prefix_add_right_value_hcu) + (pfc_value_prefix_add_right_value_hcudiagonal))) /\\ ((exists pfc_complement_prefix_add_right_value_hcudiagonalterm pfc_left_prefix_add_right_value_hcudiagonalterm pfc_right_prefix_add_right_value_hcudiagonalterm. (((pfc_index_prefix_add_right_value_hcudiagonal)+pfc_complement_prefix_add_right_value_hcudiagonalterm=(i)) /\\ ((((((exists pfa_gap_prefix_add_right_value_hcudiagonaltermleftinside. pfa_gap_prefix_add_right_value_hcudiagonaltermleftinside + S (pfc_index_prefix_add_right_value_hcudiagonal) = (L)) /\\ ((((exists ff_h_pfp_prefix_add_right_value_hcudiagonaltermleftentry. ff_h_pfp_prefix_add_right_value_hcudiagonaltermleftentry + S (pfc_left_prefix_add_right_value_hcudiagonalterm) = S ((S (pfc_index_prefix_add_right_value_hcudiagonal)) * ac)) /\\ exists ff_q_pfp_prefix_add_right_value_hcudiagonaltermleftentry. ab = ff_q_pfp_prefix_add_right_value_hcudiagonaltermleftentry * S ((S (pfc_index_prefix_add_right_value_hcudiagonal)) * ac) + (pfc_left_prefix_add_right_value_hcudiagonalterm)))))) \\/ (((exists pfc_gap_prefix_add_right_value_hcudiagonaltermleftoutside. pfc_gap_prefix_add_right_value_hcudiagonaltermleftoutside+(L)=(pfc_index_prefix_add_right_value_hcudiagonal)) /\\ (((pfc_left_prefix_add_right_value_hcudiagonalterm)=0))))) /\\ ((((((exists pfa_gap_prefix_add_right_value_hcudiagonaltermrightinside. pfa_gap_prefix_add_right_value_hcudiagonaltermrightinside + S (pfc_complement_prefix_add_right_value_hcudiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_prefix_add_right_value_hcudiagonaltermrightentry. ff_h_pfp_prefix_add_right_value_hcudiagonaltermrightentry + S (pfc_right_prefix_add_right_value_hcudiagonalterm) = S ((S (pfc_complement_prefix_add_right_value_hcudiagonalterm)) * dc)) /\\ exists ff_q_pfp_prefix_add_right_value_hcudiagonaltermrightentry. db = ff_q_pfp_prefix_add_right_value_hcudiagonaltermrightentry * S ((S (pfc_complement_prefix_add_right_value_hcudiagonalterm)) * dc) + (pfc_right_prefix_add_right_value_hcudiagonalterm)))))) \\/ (((exists pfc_gap_prefix_add_right_value_hcudiagonaltermrightoutside. pfc_gap_prefix_add_right_value_hcudiagonaltermrightoutside+(M)=(pfc_complement_prefix_add_right_value_hcudiagonalterm)) /\\ (((pfc_right_prefix_add_right_value_hcudiagonalterm)=0))))) /\\ (((pfc_value_prefix_add_right_value_hcudiagonal)=pfc_left_prefix_add_right_value_hcudiagonalterm*pfc_right_prefix_add_right_value_hcudiagonalterm))))))))))) /\\ (((exists fs_u_pfc_prefix_add_right_value_hcusum fs_v_pfc_prefix_add_right_value_hcusum. ((((exists fs_h_pfc_prefix_add_right_value_hcusum_body_start. fs_h_pfc_prefix_add_right_value_hcusum_body_start + S (0) = S ((S (0)) * fs_v_pfc_prefix_add_right_value_hcusum)) /\\ exists fs_q_pfc_prefix_add_right_value_hcusum_body_start. fs_u_pfc_prefix_add_right_value_hcusum = fs_q_pfc_prefix_add_right_value_hcusum_body_start * S ((S (0)) * fs_v_pfc_prefix_add_right_value_hcusum) + (0))) /\\ ((((exists fs_h_pfc_prefix_add_right_value_hcusum_body_terminal. fs_h_pfc_prefix_add_right_value_hcusum_body_terminal + S (pfc_natural_sum_prefix_add_right_value_hcu) = S ((S (S (i))) * fs_v_pfc_prefix_add_right_value_hcusum)) /\\ exists fs_q_pfc_prefix_add_right_value_hcusum_body_terminal. fs_u_pfc_prefix_add_right_value_hcusum = fs_q_pfc_prefix_add_right_value_hcusum_body_terminal * S ((S (S (i))) * fs_v_pfc_prefix_add_right_value_hcusum) + (pfc_natural_sum_prefix_add_right_value_hcu))) /\\ forall fs_i_pfc_prefix_add_right_value_hcusum_body_steps. (exists fs_lt_pfc_prefix_add_right_value_hcusum_body_steps_bound. fs_lt_pfc_prefix_add_right_value_hcusum_body_steps_bound + S fs_i_pfc_prefix_add_right_value_hcusum_body_steps = S (i)) -> exists fs_a_pfc_prefix_add_right_value_hcusum_body_steps fs_r_pfc_prefix_add_right_value_hcusum_body_steps fs_s_pfc_prefix_add_right_value_hcusum_body_steps. ((((exists fs_h_pfc_prefix_add_right_value_hcusum_body_steps_summand. fs_h_pfc_prefix_add_right_value_hcusum_body_steps_summand + S (fs_a_pfc_prefix_add_right_value_hcusum_body_steps) = S ((S (fs_i_pfc_prefix_add_right_value_hcusum_body_steps)) * pfc_terms_scale_prefix_add_right_value_hcu)) /\\ exists fs_q_pfc_prefix_add_right_value_hcusum_body_steps_summand. pfc_terms_code_prefix_add_right_value_hcu = fs_q_pfc_prefix_add_right_value_hcusum_body_steps_summand * S ((S (fs_i_pfc_prefix_add_right_value_hcusum_body_steps)) * pfc_terms_scale_prefix_add_right_value_hcu) + (fs_a_pfc_prefix_add_right_value_hcusum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_add_right_value_hcusum_body_steps_partial. fs_h_pfc_prefix_add_right_value_hcusum_body_steps_partial + S (fs_r_pfc_prefix_add_right_value_hcusum_body_steps) = S ((S (fs_i_pfc_prefix_add_right_value_hcusum_body_steps)) * fs_v_pfc_prefix_add_right_value_hcusum)) /\\ exists fs_q_pfc_prefix_add_right_value_hcusum_body_steps_partial. fs_u_pfc_prefix_add_right_value_hcusum = fs_q_pfc_prefix_add_right_value_hcusum_body_steps_partial * S ((S (fs_i_pfc_prefix_add_right_value_hcusum_body_steps)) * fs_v_pfc_prefix_add_right_value_hcusum) + (fs_r_pfc_prefix_add_right_value_hcusum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_add_right_value_hcusum_body_steps_successor. fs_h_pfc_prefix_add_right_value_hcusum_body_steps_successor + S (fs_s_pfc_prefix_add_right_value_hcusum_body_steps) = S ((S (S fs_i_pfc_prefix_add_right_value_hcusum_body_steps)) * fs_v_pfc_prefix_add_right_value_hcusum)) /\\ exists fs_q_pfc_prefix_add_right_value_hcusum_body_steps_successor. fs_u_pfc_prefix_add_right_value_hcusum = fs_q_pfc_prefix_add_right_value_hcusum_body_steps_successor * S ((S (S fs_i_pfc_prefix_add_right_value_hcusum_body_steps)) * fs_v_pfc_prefix_add_right_value_hcusum) + (fs_s_pfc_prefix_add_right_value_hcusum_body_steps))) /\\ fs_s_pfc_prefix_add_right_value_hcusum_body_steps = fs_r_pfc_prefix_add_right_value_hcusum_body_steps + fs_a_pfc_prefix_add_right_value_hcusum_body_steps)))))) /\\ ((((exists pfa_gap_prefix_add_right_value_hcuresiduebound. pfa_gap_prefix_add_right_value_hcuresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_prefix_add_right_value_hcuresiduecongruence pfa_offset_right_prefix_add_right_value_hcuresiduecongruence. (pfc_natural_sum_prefix_add_right_value_hcu) + (p) * pfa_offset_left_prefix_add_right_value_hcuresiduecongruence = (r) + (p) * pfa_offset_right_prefix_add_right_value_hcuresiduecongruence)))))))))))", "specialize hu (i)", "apply hu", "exact hi", "cases hcu", "cases hcu_witness", "have hcv : exists r. ((((exists ff_h_pfp_prefix_add_right_hcv. ff_h_pfp_prefix_add_right_hcv + S (r) = S ((S (i)) * vc)) /\\ exists ff_q_pfp_prefix_add_right_hcv. vb = ff_q_pfp_prefix_add_right_hcv * S ((S (i)) * vc) + (r))) /\\ ((exists pfc_terms_code_prefix_add_right_value_hcv pfc_terms_scale_prefix_add_right_value_hcv pfc_natural_sum_prefix_add_right_value_hcv. ((forall pfc_index_prefix_add_right_value_hcvdiagonal. (exists pfa_gap_prefix_add_right_value_hcvdiagonalbound. pfa_gap_prefix_add_right_value_hcvdiagonalbound + S (pfc_index_prefix_add_right_value_hcvdiagonal) = (S (i))) -> exists pfc_value_prefix_add_right_value_hcvdiagonal. ((((exists ff_h_pfp_prefix_add_right_value_hcvdiagonalentry. ff_h_pfp_prefix_add_right_value_hcvdiagonalentry + S (pfc_value_prefix_add_right_value_hcvdiagonal) = S ((S (pfc_index_prefix_add_right_value_hcvdiagonal)) * pfc_terms_scale_prefix_add_right_value_hcv)) /\\ exists ff_q_pfp_prefix_add_right_value_hcvdiagonalentry. pfc_terms_code_prefix_add_right_value_hcv = ff_q_pfp_prefix_add_right_value_hcvdiagonalentry * S ((S (pfc_index_prefix_add_right_value_hcvdiagonal)) * pfc_terms_scale_prefix_add_right_value_hcv) + (pfc_value_prefix_add_right_value_hcvdiagonal))) /\\ ((exists pfc_complement_prefix_add_right_value_hcvdiagonalterm pfc_left_prefix_add_right_value_hcvdiagonalterm pfc_right_prefix_add_right_value_hcvdiagonalterm. (((pfc_index_prefix_add_right_value_hcvdiagonal)+pfc_complement_prefix_add_right_value_hcvdiagonalterm=(i)) /\\ ((((((exists pfa_gap_prefix_add_right_value_hcvdiagonaltermleftinside. pfa_gap_prefix_add_right_value_hcvdiagonaltermleftinside + S (pfc_index_prefix_add_right_value_hcvdiagonal) = (L)) /\\ ((((exists ff_h_pfp_prefix_add_right_value_hcvdiagonaltermleftentry. ff_h_pfp_prefix_add_right_value_hcvdiagonaltermleftentry + S (pfc_left_prefix_add_right_value_hcvdiagonalterm) = S ((S (pfc_index_prefix_add_right_value_hcvdiagonal)) * bc)) /\\ exists ff_q_pfp_prefix_add_right_value_hcvdiagonaltermleftentry. bb = ff_q_pfp_prefix_add_right_value_hcvdiagonaltermleftentry * S ((S (pfc_index_prefix_add_right_value_hcvdiagonal)) * bc) + (pfc_left_prefix_add_right_value_hcvdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_add_right_value_hcvdiagonaltermleftoutside. pfc_gap_prefix_add_right_value_hcvdiagonaltermleftoutside+(L)=(pfc_index_prefix_add_right_value_hcvdiagonal)) /\\ (((pfc_left_prefix_add_right_value_hcvdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_prefix_add_right_value_hcvdiagonaltermrightinside. pfa_gap_prefix_add_right_value_hcvdiagonaltermrightinside + S (pfc_complement_prefix_add_right_value_hcvdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_prefix_add_right_value_hcvdiagonaltermrightentry. ff_h_pfp_prefix_add_right_value_hcvdiagonaltermrightentry + S (pfc_right_prefix_add_right_value_hcvdiagonalterm) = S ((S (pfc_complement_prefix_add_right_value_hcvdiagonalterm)) * dc)) /\\ exists ff_q_pfp_prefix_add_right_value_hcvdiagonaltermrightentry. db = ff_q_pfp_prefix_add_right_value_hcvdiagonaltermrightentry * S ((S (pfc_complement_prefix_add_right_value_hcvdiagonalterm)) * dc) + (pfc_right_prefix_add_right_value_hcvdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_add_right_value_hcvdiagonaltermrightoutside. pfc_gap_prefix_add_right_value_hcvdiagonaltermrightoutside+(M)=(pfc_complement_prefix_add_right_value_hcvdiagonalterm)) /\\ (((pfc_right_prefix_add_right_value_hcvdiagonalterm)=0))))) /\\ (((pfc_value_prefix_add_right_value_hcvdiagonal)=pfc_left_prefix_add_right_value_hcvdiagonalterm*pfc_right_prefix_add_right_value_hcvdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_prefix_add_right_value_hcvsum fs_v_pfc_prefix_add_right_value_hcvsum. ((((exists fs_h_pfc_prefix_add_right_value_hcvsum_body_start. fs_h_pfc_prefix_add_right_value_hcvsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_prefix_add_right_value_hcvsum)) /\\ exists fs_q_pfc_prefix_add_right_value_hcvsum_body_start. fs_u_pfc_prefix_add_right_value_hcvsum = fs_q_pfc_prefix_add_right_value_hcvsum_body_start * S ((S (0)) * fs_v_pfc_prefix_add_right_value_hcvsum) + (0))) /\\ ((((exists fs_h_pfc_prefix_add_right_value_hcvsum_body_terminal. fs_h_pfc_prefix_add_right_value_hcvsum_body_terminal + S (pfc_natural_sum_prefix_add_right_value_hcv) = S ((S (S (i))) * fs_v_pfc_prefix_add_right_value_hcvsum)) /\\ exists fs_q_pfc_prefix_add_right_value_hcvsum_body_terminal. fs_u_pfc_prefix_add_right_value_hcvsum = fs_q_pfc_prefix_add_right_value_hcvsum_body_terminal * S ((S (S (i))) * fs_v_pfc_prefix_add_right_value_hcvsum) + (pfc_natural_sum_prefix_add_right_value_hcv))) /\\ forall fs_i_pfc_prefix_add_right_value_hcvsum_body_steps. (exists fs_lt_pfc_prefix_add_right_value_hcvsum_body_steps_bound. fs_lt_pfc_prefix_add_right_value_hcvsum_body_steps_bound + S fs_i_pfc_prefix_add_right_value_hcvsum_body_steps = S (i)) -> exists fs_a_pfc_prefix_add_right_value_hcvsum_body_steps fs_r_pfc_prefix_add_right_value_hcvsum_body_steps fs_s_pfc_prefix_add_right_value_hcvsum_body_steps. ((((exists fs_h_pfc_prefix_add_right_value_hcvsum_body_steps_summand. fs_h_pfc_prefix_add_right_value_hcvsum_body_steps_summand + S (fs_a_pfc_prefix_add_right_value_hcvsum_body_steps) = S ((S (fs_i_pfc_prefix_add_right_value_hcvsum_body_steps)) * pfc_terms_scale_prefix_add_right_value_hcv)) /\\ exists fs_q_pfc_prefix_add_right_value_hcvsum_body_steps_summand. pfc_terms_code_prefix_add_right_value_hcv = fs_q_pfc_prefix_add_right_value_hcvsum_body_steps_summand * S ((S (fs_i_pfc_prefix_add_right_value_hcvsum_body_steps)) * pfc_terms_scale_prefix_add_right_value_hcv) + (fs_a_pfc_prefix_add_right_value_hcvsum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_add_right_value_hcvsum_body_steps_partial. fs_h_pfc_prefix_add_right_value_hcvsum_body_steps_partial + S (fs_r_pfc_prefix_add_right_value_hcvsum_body_steps) = S ((S (fs_i_pfc_prefix_add_right_value_hcvsum_body_steps)) * fs_v_pfc_prefix_add_right_value_hcvsum)) /\\ exists fs_q_pfc_prefix_add_right_value_hcvsum_body_steps_partial. fs_u_pfc_prefix_add_right_value_hcvsum = fs_q_pfc_prefix_add_right_value_hcvsum_body_steps_partial * S ((S (fs_i_pfc_prefix_add_right_value_hcvsum_body_steps)) * fs_v_pfc_prefix_add_right_value_hcvsum) + (fs_r_pfc_prefix_add_right_value_hcvsum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_add_right_value_hcvsum_body_steps_successor. fs_h_pfc_prefix_add_right_value_hcvsum_body_steps_successor + S (fs_s_pfc_prefix_add_right_value_hcvsum_body_steps) = S ((S (S fs_i_pfc_prefix_add_right_value_hcvsum_body_steps)) * fs_v_pfc_prefix_add_right_value_hcvsum)) /\\ exists fs_q_pfc_prefix_add_right_value_hcvsum_body_steps_successor. fs_u_pfc_prefix_add_right_value_hcvsum = fs_q_pfc_prefix_add_right_value_hcvsum_body_steps_successor * S ((S (S fs_i_pfc_prefix_add_right_value_hcvsum_body_steps)) * fs_v_pfc_prefix_add_right_value_hcvsum) + (fs_s_pfc_prefix_add_right_value_hcvsum_body_steps))) /\\ fs_s_pfc_prefix_add_right_value_hcvsum_body_steps = fs_r_pfc_prefix_add_right_value_hcvsum_body_steps + fs_a_pfc_prefix_add_right_value_hcvsum_body_steps)))))) /\\ ((((exists pfa_gap_prefix_add_right_value_hcvresiduebound. pfa_gap_prefix_add_right_value_hcvresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_prefix_add_right_value_hcvresiduecongruence pfa_offset_right_prefix_add_right_value_hcvresiduecongruence. (pfc_natural_sum_prefix_add_right_value_hcv) + (p) * pfa_offset_left_prefix_add_right_value_hcvresiduecongruence = (r) + (p) * pfa_offset_right_prefix_add_right_value_hcvresiduecongruence)))))))))))", "specialize hv (i)", "apply hv", "exact hi", "cases hcv", "cases hcv_witness", "have hcw : exists r. ((((exists ff_h_pfp_prefix_add_right_hcw. ff_h_pfp_prefix_add_right_hcw + S (r) = S ((S (i)) * wc)) /\\ exists ff_q_pfp_prefix_add_right_hcw. wb = ff_q_pfp_prefix_add_right_hcw * S ((S (i)) * wc) + (r))) /\\ ((exists pfc_terms_code_prefix_add_right_value_hcw pfc_terms_scale_prefix_add_right_value_hcw pfc_natural_sum_prefix_add_right_value_hcw. ((forall pfc_index_prefix_add_right_value_hcwdiagonal. (exists pfa_gap_prefix_add_right_value_hcwdiagonalbound. pfa_gap_prefix_add_right_value_hcwdiagonalbound + S (pfc_index_prefix_add_right_value_hcwdiagonal) = (S (i))) -> exists pfc_value_prefix_add_right_value_hcwdiagonal. ((((exists ff_h_pfp_prefix_add_right_value_hcwdiagonalentry. ff_h_pfp_prefix_add_right_value_hcwdiagonalentry + S (pfc_value_prefix_add_right_value_hcwdiagonal) = S ((S (pfc_index_prefix_add_right_value_hcwdiagonal)) * pfc_terms_scale_prefix_add_right_value_hcw)) /\\ exists ff_q_pfp_prefix_add_right_value_hcwdiagonalentry. pfc_terms_code_prefix_add_right_value_hcw = ff_q_pfp_prefix_add_right_value_hcwdiagonalentry * S ((S (pfc_index_prefix_add_right_value_hcwdiagonal)) * pfc_terms_scale_prefix_add_right_value_hcw) + (pfc_value_prefix_add_right_value_hcwdiagonal))) /\\ ((exists pfc_complement_prefix_add_right_value_hcwdiagonalterm pfc_left_prefix_add_right_value_hcwdiagonalterm pfc_right_prefix_add_right_value_hcwdiagonalterm. (((pfc_index_prefix_add_right_value_hcwdiagonal)+pfc_complement_prefix_add_right_value_hcwdiagonalterm=(i)) /\\ ((((((exists pfa_gap_prefix_add_right_value_hcwdiagonaltermleftinside. pfa_gap_prefix_add_right_value_hcwdiagonaltermleftinside + S (pfc_index_prefix_add_right_value_hcwdiagonal) = (L)) /\\ ((((exists ff_h_pfp_prefix_add_right_value_hcwdiagonaltermleftentry. ff_h_pfp_prefix_add_right_value_hcwdiagonaltermleftentry + S (pfc_left_prefix_add_right_value_hcwdiagonalterm) = S ((S (pfc_index_prefix_add_right_value_hcwdiagonal)) * cc)) /\\ exists ff_q_pfp_prefix_add_right_value_hcwdiagonaltermleftentry. cb = ff_q_pfp_prefix_add_right_value_hcwdiagonaltermleftentry * S ((S (pfc_index_prefix_add_right_value_hcwdiagonal)) * cc) + (pfc_left_prefix_add_right_value_hcwdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_add_right_value_hcwdiagonaltermleftoutside. pfc_gap_prefix_add_right_value_hcwdiagonaltermleftoutside+(L)=(pfc_index_prefix_add_right_value_hcwdiagonal)) /\\ (((pfc_left_prefix_add_right_value_hcwdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_prefix_add_right_value_hcwdiagonaltermrightinside. pfa_gap_prefix_add_right_value_hcwdiagonaltermrightinside + S (pfc_complement_prefix_add_right_value_hcwdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_prefix_add_right_value_hcwdiagonaltermrightentry. ff_h_pfp_prefix_add_right_value_hcwdiagonaltermrightentry + S (pfc_right_prefix_add_right_value_hcwdiagonalterm) = S ((S (pfc_complement_prefix_add_right_value_hcwdiagonalterm)) * dc)) /\\ exists ff_q_pfp_prefix_add_right_value_hcwdiagonaltermrightentry. db = ff_q_pfp_prefix_add_right_value_hcwdiagonaltermrightentry * S ((S (pfc_complement_prefix_add_right_value_hcwdiagonalterm)) * dc) + (pfc_right_prefix_add_right_value_hcwdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_add_right_value_hcwdiagonaltermrightoutside. pfc_gap_prefix_add_right_value_hcwdiagonaltermrightoutside+(M)=(pfc_complement_prefix_add_right_value_hcwdiagonalterm)) /\\ (((pfc_right_prefix_add_right_value_hcwdiagonalterm)=0))))) /\\ (((pfc_value_prefix_add_right_value_hcwdiagonal)=pfc_left_prefix_add_right_value_hcwdiagonalterm*pfc_right_prefix_add_right_value_hcwdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_prefix_add_right_value_hcwsum fs_v_pfc_prefix_add_right_value_hcwsum. ((((exists fs_h_pfc_prefix_add_right_value_hcwsum_body_start. fs_h_pfc_prefix_add_right_value_hcwsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_prefix_add_right_value_hcwsum)) /\\ exists fs_q_pfc_prefix_add_right_value_hcwsum_body_start. fs_u_pfc_prefix_add_right_value_hcwsum = fs_q_pfc_prefix_add_right_value_hcwsum_body_start * S ((S (0)) * fs_v_pfc_prefix_add_right_value_hcwsum) + (0))) /\\ ((((exists fs_h_pfc_prefix_add_right_value_hcwsum_body_terminal. fs_h_pfc_prefix_add_right_value_hcwsum_body_terminal + S (pfc_natural_sum_prefix_add_right_value_hcw) = S ((S (S (i))) * fs_v_pfc_prefix_add_right_value_hcwsum)) /\\ exists fs_q_pfc_prefix_add_right_value_hcwsum_body_terminal. fs_u_pfc_prefix_add_right_value_hcwsum = fs_q_pfc_prefix_add_right_value_hcwsum_body_terminal * S ((S (S (i))) * fs_v_pfc_prefix_add_right_value_hcwsum) + (pfc_natural_sum_prefix_add_right_value_hcw))) /\\ forall fs_i_pfc_prefix_add_right_value_hcwsum_body_steps. (exists fs_lt_pfc_prefix_add_right_value_hcwsum_body_steps_bound. fs_lt_pfc_prefix_add_right_value_hcwsum_body_steps_bound + S fs_i_pfc_prefix_add_right_value_hcwsum_body_steps = S (i)) -> exists fs_a_pfc_prefix_add_right_value_hcwsum_body_steps fs_r_pfc_prefix_add_right_value_hcwsum_body_steps fs_s_pfc_prefix_add_right_value_hcwsum_body_steps. ((((exists fs_h_pfc_prefix_add_right_value_hcwsum_body_steps_summand. fs_h_pfc_prefix_add_right_value_hcwsum_body_steps_summand + S (fs_a_pfc_prefix_add_right_value_hcwsum_body_steps) = S ((S (fs_i_pfc_prefix_add_right_value_hcwsum_body_steps)) * pfc_terms_scale_prefix_add_right_value_hcw)) /\\ exists fs_q_pfc_prefix_add_right_value_hcwsum_body_steps_summand. pfc_terms_code_prefix_add_right_value_hcw = fs_q_pfc_prefix_add_right_value_hcwsum_body_steps_summand * S ((S (fs_i_pfc_prefix_add_right_value_hcwsum_body_steps)) * pfc_terms_scale_prefix_add_right_value_hcw) + (fs_a_pfc_prefix_add_right_value_hcwsum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_add_right_value_hcwsum_body_steps_partial. fs_h_pfc_prefix_add_right_value_hcwsum_body_steps_partial + S (fs_r_pfc_prefix_add_right_value_hcwsum_body_steps) = S ((S (fs_i_pfc_prefix_add_right_value_hcwsum_body_steps)) * fs_v_pfc_prefix_add_right_value_hcwsum)) /\\ exists fs_q_pfc_prefix_add_right_value_hcwsum_body_steps_partial. fs_u_pfc_prefix_add_right_value_hcwsum = fs_q_pfc_prefix_add_right_value_hcwsum_body_steps_partial * S ((S (fs_i_pfc_prefix_add_right_value_hcwsum_body_steps)) * fs_v_pfc_prefix_add_right_value_hcwsum) + (fs_r_pfc_prefix_add_right_value_hcwsum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_add_right_value_hcwsum_body_steps_successor. fs_h_pfc_prefix_add_right_value_hcwsum_body_steps_successor + S (fs_s_pfc_prefix_add_right_value_hcwsum_body_steps) = S ((S (S fs_i_pfc_prefix_add_right_value_hcwsum_body_steps)) * fs_v_pfc_prefix_add_right_value_hcwsum)) /\\ exists fs_q_pfc_prefix_add_right_value_hcwsum_body_steps_successor. fs_u_pfc_prefix_add_right_value_hcwsum = fs_q_pfc_prefix_add_right_value_hcwsum_body_steps_successor * S ((S (S fs_i_pfc_prefix_add_right_value_hcwsum_body_steps)) * fs_v_pfc_prefix_add_right_value_hcwsum) + (fs_s_pfc_prefix_add_right_value_hcwsum_body_steps))) /\\ fs_s_pfc_prefix_add_right_value_hcwsum_body_steps = fs_r_pfc_prefix_add_right_value_hcwsum_body_steps + fs_a_pfc_prefix_add_right_value_hcwsum_body_steps)))))) /\\ ((((exists pfa_gap_prefix_add_right_value_hcwresiduebound. pfa_gap_prefix_add_right_value_hcwresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_prefix_add_right_value_hcwresiduecongruence pfa_offset_right_prefix_add_right_value_hcwresiduecongruence. (pfc_natural_sum_prefix_add_right_value_hcw) + (p) * pfa_offset_left_prefix_add_right_value_hcwresiduecongruence = (r) + (p) * pfa_offset_right_prefix_add_right_value_hcwresiduecongruence)))))))))))", "specialize hw (i)", "apply hw", "exact hi", "cases hcw", "cases hcw_witness", "exists x", "exists x1", "exists x2", "split", "exact hcu_witness_left", "split", "exact hcv_witness_left", "split", "exact hcw_witness_left", "specialize prime_field_convolution_coefficient_right_add (p)", "specialize prime_field_convolution_coefficient_right_add (ab)", "specialize prime_field_convolution_coefficient_right_add (ac)", "specialize prime_field_convolution_coefficient_right_add (bb)", "specialize prime_field_convolution_coefficient_right_add (bc)", "specialize prime_field_convolution_coefficient_right_add (cb)", "specialize prime_field_convolution_coefficient_right_add (cc)", "specialize prime_field_convolution_coefficient_right_add (L)", "specialize prime_field_convolution_coefficient_right_add (db)", "specialize prime_field_convolution_coefficient_right_add (dc)", "specialize prime_field_convolution_coefficient_right_add (M)", "specialize prime_field_convolution_coefficient_right_add (i)", "specialize prime_field_convolution_coefficient_right_add (x)", "specialize prime_field_convolution_coefficient_right_add (x1)", "specialize prime_field_convolution_coefficient_right_add (x2)", "apply prime_field_convolution_coefficient_right_add", "exact hs", "exact hcu_witness_right", "exact hcv_witness_right", "exact hcw_witness_right" ], "script_sha256": "a48426ed1c60416d5d7a5cef87ab90d0be3cef00665d27676f082e358e2cc556", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_distributivity_candidate.py", "sha256": "a959962d631759cd1fc773dd7eef2fadf4f3f95361d6d7bc8c6a9e82d0d4ab86" }, "statement": "forall p ab ac bb bc cb cc L db dc M ub uc vb vc wb wc N. (forall pfp_index_prefix_add_right_source. (exists pfa_gap_prefix_add_right_sourceindex. pfa_gap_prefix_add_right_sourceindex + S (pfp_index_prefix_add_right_source) = (L)) -> exists pfp_left_prefix_add_right_source pfp_right_prefix_add_right_source pfp_value_prefix_add_right_source. ((((exists ff_h_pfp_prefix_add_right_sourceleft. ff_h_pfp_prefix_add_right_sourceleft + S (pfp_left_prefix_add_right_source) = S ((S (pfp_index_prefix_add_right_source)) * ac)) /\\ exists ff_q_pfp_prefix_add_right_sourceleft. ab = ff_q_pfp_prefix_add_right_sourceleft * S ((S (pfp_index_prefix_add_right_source)) * ac) + (pfp_left_prefix_add_right_source))) /\\ (((((exists ff_h_pfp_prefix_add_right_sourceright. ff_h_pfp_prefix_add_right_sourceright + S (pfp_right_prefix_add_right_source) = S ((S (pfp_index_prefix_add_right_source)) * bc)) /\\ exists ff_q_pfp_prefix_add_right_sourceright. bb = ff_q_pfp_prefix_add_right_sourceright * S ((S (pfp_index_prefix_add_right_source)) * bc) + (pfp_right_prefix_add_right_source))) /\\ (((((exists ff_h_pfp_prefix_add_right_sourcetarget. ff_h_pfp_prefix_add_right_sourcetarget + S (pfp_value_prefix_add_right_source) = S ((S (pfp_index_prefix_add_right_source)) * cc)) /\\ exists ff_q_pfp_prefix_add_right_sourcetarget. cb = ff_q_pfp_prefix_add_right_sourcetarget * S ((S (pfp_index_prefix_add_right_source)) * cc) + (pfp_value_prefix_add_right_source))) /\\ ((((exists pfa_gap_prefix_add_right_sourceoperationleft. pfa_gap_prefix_add_right_sourceoperationleft + S (pfp_left_prefix_add_right_source) = (p)) /\\ (((exists pfa_gap_prefix_add_right_sourceoperationright. pfa_gap_prefix_add_right_sourceoperationright + S (pfp_right_prefix_add_right_source) = (p)) /\\ ((((exists pfa_gap_prefix_add_right_sourceoperationresultbound. pfa_gap_prefix_add_right_sourceoperationresultbound + S (pfp_value_prefix_add_right_source) = (p)) /\\ ((exists pfa_offset_left_prefix_add_right_sourceoperationresultcongruence pfa_offset_right_prefix_add_right_sourceoperationresultcongruence. ((pfp_left_prefix_add_right_source) + (pfp_right_prefix_add_right_source)) + (p) * pfa_offset_left_prefix_add_right_sourceoperationresultcongruence = (pfp_value_prefix_add_right_source) + (p) * pfa_offset_right_prefix_add_right_sourceoperationresultcongruence)))))))))))))))) -> (forall pfc_index_prefix_add_right_ub. (exists pfa_gap_prefix_add_right_ubbound. pfa_gap_prefix_add_right_ubbound + S (pfc_index_prefix_add_right_ub) = (N)) -> exists pfc_value_prefix_add_right_ub. ((((exists ff_h_pfp_prefix_add_right_ubentry. ff_h_pfp_prefix_add_right_ubentry + S (pfc_value_prefix_add_right_ub) = S ((S (pfc_index_prefix_add_right_ub)) * uc)) /\\ exists ff_q_pfp_prefix_add_right_ubentry. ub = ff_q_pfp_prefix_add_right_ubentry * S ((S (pfc_index_prefix_add_right_ub)) * uc) + (pfc_value_prefix_add_right_ub))) /\\ ((exists pfc_terms_code_prefix_add_right_ubcoefficient pfc_terms_scale_prefix_add_right_ubcoefficient pfc_natural_sum_prefix_add_right_ubcoefficient. ((forall pfc_index_prefix_add_right_ubcoefficientdiagonal. (exists pfa_gap_prefix_add_right_ubcoefficientdiagonalbound. pfa_gap_prefix_add_right_ubcoefficientdiagonalbound + S (pfc_index_prefix_add_right_ubcoefficientdiagonal) = (S (pfc_index_prefix_add_right_ub))) -> exists pfc_value_prefix_add_right_ubcoefficientdiagonal. ((((exists ff_h_pfp_prefix_add_right_ubcoefficientdiagonalentry. ff_h_pfp_prefix_add_right_ubcoefficientdiagonalentry + S (pfc_value_prefix_add_right_ubcoefficientdiagonal) = S ((S (pfc_index_prefix_add_right_ubcoefficientdiagonal)) * pfc_terms_scale_prefix_add_right_ubcoefficient)) /\\ exists ff_q_pfp_prefix_add_right_ubcoefficientdiagonalentry. pfc_terms_code_prefix_add_right_ubcoefficient = ff_q_pfp_prefix_add_right_ubcoefficientdiagonalentry * S ((S (pfc_index_prefix_add_right_ubcoefficientdiagonal)) * pfc_terms_scale_prefix_add_right_ubcoefficient) + (pfc_value_prefix_add_right_ubcoefficientdiagonal))) /\\ ((exists pfc_complement_prefix_add_right_ubcoefficientdiagonalterm pfc_left_prefix_add_right_ubcoefficientdiagonalterm pfc_right_prefix_add_right_ubcoefficientdiagonalterm. (((pfc_index_prefix_add_right_ubcoefficientdiagonal)+pfc_complement_prefix_add_right_ubcoefficientdiagonalterm=(pfc_index_prefix_add_right_ub)) /\\ ((((((exists pfa_gap_prefix_add_right_ubcoefficientdiagonaltermleftinside. pfa_gap_prefix_add_right_ubcoefficientdiagonaltermleftinside + S (pfc_index_prefix_add_right_ubcoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_prefix_add_right_ubcoefficientdiagonaltermleftentry. ff_h_pfp_prefix_add_right_ubcoefficientdiagonaltermleftentry + S (pfc_left_prefix_add_right_ubcoefficientdiagonalterm) = S ((S (pfc_index_prefix_add_right_ubcoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_prefix_add_right_ubcoefficientdiagonaltermleftentry. ab = ff_q_pfp_prefix_add_right_ubcoefficientdiagonaltermleftentry * S ((S (pfc_index_prefix_add_right_ubcoefficientdiagonal)) * ac) + (pfc_left_prefix_add_right_ubcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_add_right_ubcoefficientdiagonaltermleftoutside. pfc_gap_prefix_add_right_ubcoefficientdiagonaltermleftoutside+(L)=(pfc_index_prefix_add_right_ubcoefficientdiagonal)) /\\ (((pfc_left_prefix_add_right_ubcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_prefix_add_right_ubcoefficientdiagonaltermrightinside. pfa_gap_prefix_add_right_ubcoefficientdiagonaltermrightinside + S (pfc_complement_prefix_add_right_ubcoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_prefix_add_right_ubcoefficientdiagonaltermrightentry. ff_h_pfp_prefix_add_right_ubcoefficientdiagonaltermrightentry + S (pfc_right_prefix_add_right_ubcoefficientdiagonalterm) = S ((S (pfc_complement_prefix_add_right_ubcoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_prefix_add_right_ubcoefficientdiagonaltermrightentry. db = ff_q_pfp_prefix_add_right_ubcoefficientdiagonaltermrightentry * S ((S (pfc_complement_prefix_add_right_ubcoefficientdiagonalterm)) * dc) + (pfc_right_prefix_add_right_ubcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_add_right_ubcoefficientdiagonaltermrightoutside. pfc_gap_prefix_add_right_ubcoefficientdiagonaltermrightoutside+(M)=(pfc_complement_prefix_add_right_ubcoefficientdiagonalterm)) /\\ (((pfc_right_prefix_add_right_ubcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_prefix_add_right_ubcoefficientdiagonal)=pfc_left_prefix_add_right_ubcoefficientdiagonalterm*pfc_right_prefix_add_right_ubcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_prefix_add_right_ubcoefficientsum fs_v_pfc_prefix_add_right_ubcoefficientsum. ((((exists fs_h_pfc_prefix_add_right_ubcoefficientsum_body_start. fs_h_pfc_prefix_add_right_ubcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_prefix_add_right_ubcoefficientsum)) /\\ exists fs_q_pfc_prefix_add_right_ubcoefficientsum_body_start. fs_u_pfc_prefix_add_right_ubcoefficientsum = fs_q_pfc_prefix_add_right_ubcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_prefix_add_right_ubcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_prefix_add_right_ubcoefficientsum_body_terminal. fs_h_pfc_prefix_add_right_ubcoefficientsum_body_terminal + S (pfc_natural_sum_prefix_add_right_ubcoefficient) = S ((S (S (pfc_index_prefix_add_right_ub))) * fs_v_pfc_prefix_add_right_ubcoefficientsum)) /\\ exists fs_q_pfc_prefix_add_right_ubcoefficientsum_body_terminal. fs_u_pfc_prefix_add_right_ubcoefficientsum = fs_q_pfc_prefix_add_right_ubcoefficientsum_body_terminal * S ((S (S (pfc_index_prefix_add_right_ub))) * fs_v_pfc_prefix_add_right_ubcoefficientsum) + (pfc_natural_sum_prefix_add_right_ubcoefficient))) /\\ forall fs_i_pfc_prefix_add_right_ubcoefficientsum_body_steps. (exists fs_lt_pfc_prefix_add_right_ubcoefficientsum_body_steps_bound. fs_lt_pfc_prefix_add_right_ubcoefficientsum_body_steps_bound + S fs_i_pfc_prefix_add_right_ubcoefficientsum_body_steps = S (pfc_index_prefix_add_right_ub)) -> exists fs_a_pfc_prefix_add_right_ubcoefficientsum_body_steps fs_r_pfc_prefix_add_right_ubcoefficientsum_body_steps fs_s_pfc_prefix_add_right_ubcoefficientsum_body_steps. ((((exists fs_h_pfc_prefix_add_right_ubcoefficientsum_body_steps_summand. fs_h_pfc_prefix_add_right_ubcoefficientsum_body_steps_summand + S (fs_a_pfc_prefix_add_right_ubcoefficientsum_body_steps) = S ((S (fs_i_pfc_prefix_add_right_ubcoefficientsum_body_steps)) * pfc_terms_scale_prefix_add_right_ubcoefficient)) /\\ exists fs_q_pfc_prefix_add_right_ubcoefficientsum_body_steps_summand. pfc_terms_code_prefix_add_right_ubcoefficient = fs_q_pfc_prefix_add_right_ubcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_prefix_add_right_ubcoefficientsum_body_steps)) * pfc_terms_scale_prefix_add_right_ubcoefficient) + (fs_a_pfc_prefix_add_right_ubcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_add_right_ubcoefficientsum_body_steps_partial. fs_h_pfc_prefix_add_right_ubcoefficientsum_body_steps_partial + S (fs_r_pfc_prefix_add_right_ubcoefficientsum_body_steps) = S ((S (fs_i_pfc_prefix_add_right_ubcoefficientsum_body_steps)) * fs_v_pfc_prefix_add_right_ubcoefficientsum)) /\\ exists fs_q_pfc_prefix_add_right_ubcoefficientsum_body_steps_partial. fs_u_pfc_prefix_add_right_ubcoefficientsum = fs_q_pfc_prefix_add_right_ubcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_prefix_add_right_ubcoefficientsum_body_steps)) * fs_v_pfc_prefix_add_right_ubcoefficientsum) + (fs_r_pfc_prefix_add_right_ubcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_add_right_ubcoefficientsum_body_steps_successor. fs_h_pfc_prefix_add_right_ubcoefficientsum_body_steps_successor + S (fs_s_pfc_prefix_add_right_ubcoefficientsum_body_steps) = S ((S (S fs_i_pfc_prefix_add_right_ubcoefficientsum_body_steps)) * fs_v_pfc_prefix_add_right_ubcoefficientsum)) /\\ exists fs_q_pfc_prefix_add_right_ubcoefficientsum_body_steps_successor. fs_u_pfc_prefix_add_right_ubcoefficientsum = fs_q_pfc_prefix_add_right_ubcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_prefix_add_right_ubcoefficientsum_body_steps)) * fs_v_pfc_prefix_add_right_ubcoefficientsum) + (fs_s_pfc_prefix_add_right_ubcoefficientsum_body_steps))) /\\ fs_s_pfc_prefix_add_right_ubcoefficientsum_body_steps = fs_r_pfc_prefix_add_right_ubcoefficientsum_body_steps + fs_a_pfc_prefix_add_right_ubcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_prefix_add_right_ubcoefficientresiduebound. pfa_gap_prefix_add_right_ubcoefficientresiduebound + S (pfc_value_prefix_add_right_ub) = (p)) /\\ ((exists pfa_offset_left_prefix_add_right_ubcoefficientresiduecongruence pfa_offset_right_prefix_add_right_ubcoefficientresiduecongruence. (pfc_natural_sum_prefix_add_right_ubcoefficient) + (p) * pfa_offset_left_prefix_add_right_ubcoefficientresiduecongruence = (pfc_value_prefix_add_right_ub) + (p) * pfa_offset_right_prefix_add_right_ubcoefficientresiduecongruence)))))))))))) -> (forall pfc_index_prefix_add_right_vb. (exists pfa_gap_prefix_add_right_vbbound. pfa_gap_prefix_add_right_vbbound + S (pfc_index_prefix_add_right_vb) = (N)) -> exists pfc_value_prefix_add_right_vb. ((((exists ff_h_pfp_prefix_add_right_vbentry. ff_h_pfp_prefix_add_right_vbentry + S (pfc_value_prefix_add_right_vb) = S ((S (pfc_index_prefix_add_right_vb)) * vc)) /\\ exists ff_q_pfp_prefix_add_right_vbentry. vb = ff_q_pfp_prefix_add_right_vbentry * S ((S (pfc_index_prefix_add_right_vb)) * vc) + (pfc_value_prefix_add_right_vb))) /\\ ((exists pfc_terms_code_prefix_add_right_vbcoefficient pfc_terms_scale_prefix_add_right_vbcoefficient pfc_natural_sum_prefix_add_right_vbcoefficient. ((forall pfc_index_prefix_add_right_vbcoefficientdiagonal. (exists pfa_gap_prefix_add_right_vbcoefficientdiagonalbound. pfa_gap_prefix_add_right_vbcoefficientdiagonalbound + S (pfc_index_prefix_add_right_vbcoefficientdiagonal) = (S (pfc_index_prefix_add_right_vb))) -> exists pfc_value_prefix_add_right_vbcoefficientdiagonal. ((((exists ff_h_pfp_prefix_add_right_vbcoefficientdiagonalentry. ff_h_pfp_prefix_add_right_vbcoefficientdiagonalentry + S (pfc_value_prefix_add_right_vbcoefficientdiagonal) = S ((S (pfc_index_prefix_add_right_vbcoefficientdiagonal)) * pfc_terms_scale_prefix_add_right_vbcoefficient)) /\\ exists ff_q_pfp_prefix_add_right_vbcoefficientdiagonalentry. pfc_terms_code_prefix_add_right_vbcoefficient = ff_q_pfp_prefix_add_right_vbcoefficientdiagonalentry * S ((S (pfc_index_prefix_add_right_vbcoefficientdiagonal)) * pfc_terms_scale_prefix_add_right_vbcoefficient) + (pfc_value_prefix_add_right_vbcoefficientdiagonal))) /\\ ((exists pfc_complement_prefix_add_right_vbcoefficientdiagonalterm pfc_left_prefix_add_right_vbcoefficientdiagonalterm pfc_right_prefix_add_right_vbcoefficientdiagonalterm. (((pfc_index_prefix_add_right_vbcoefficientdiagonal)+pfc_complement_prefix_add_right_vbcoefficientdiagonalterm=(pfc_index_prefix_add_right_vb)) /\\ ((((((exists pfa_gap_prefix_add_right_vbcoefficientdiagonaltermleftinside. pfa_gap_prefix_add_right_vbcoefficientdiagonaltermleftinside + S (pfc_index_prefix_add_right_vbcoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_prefix_add_right_vbcoefficientdiagonaltermleftentry. ff_h_pfp_prefix_add_right_vbcoefficientdiagonaltermleftentry + S (pfc_left_prefix_add_right_vbcoefficientdiagonalterm) = S ((S (pfc_index_prefix_add_right_vbcoefficientdiagonal)) * bc)) /\\ exists ff_q_pfp_prefix_add_right_vbcoefficientdiagonaltermleftentry. bb = ff_q_pfp_prefix_add_right_vbcoefficientdiagonaltermleftentry * S ((S (pfc_index_prefix_add_right_vbcoefficientdiagonal)) * bc) + (pfc_left_prefix_add_right_vbcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_add_right_vbcoefficientdiagonaltermleftoutside. pfc_gap_prefix_add_right_vbcoefficientdiagonaltermleftoutside+(L)=(pfc_index_prefix_add_right_vbcoefficientdiagonal)) /\\ (((pfc_left_prefix_add_right_vbcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_prefix_add_right_vbcoefficientdiagonaltermrightinside. pfa_gap_prefix_add_right_vbcoefficientdiagonaltermrightinside + S (pfc_complement_prefix_add_right_vbcoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_prefix_add_right_vbcoefficientdiagonaltermrightentry. ff_h_pfp_prefix_add_right_vbcoefficientdiagonaltermrightentry + S (pfc_right_prefix_add_right_vbcoefficientdiagonalterm) = S ((S (pfc_complement_prefix_add_right_vbcoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_prefix_add_right_vbcoefficientdiagonaltermrightentry. db = ff_q_pfp_prefix_add_right_vbcoefficientdiagonaltermrightentry * S ((S (pfc_complement_prefix_add_right_vbcoefficientdiagonalterm)) * dc) + (pfc_right_prefix_add_right_vbcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_add_right_vbcoefficientdiagonaltermrightoutside. pfc_gap_prefix_add_right_vbcoefficientdiagonaltermrightoutside+(M)=(pfc_complement_prefix_add_right_vbcoefficientdiagonalterm)) /\\ (((pfc_right_prefix_add_right_vbcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_prefix_add_right_vbcoefficientdiagonal)=pfc_left_prefix_add_right_vbcoefficientdiagonalterm*pfc_right_prefix_add_right_vbcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_prefix_add_right_vbcoefficientsum fs_v_pfc_prefix_add_right_vbcoefficientsum. ((((exists fs_h_pfc_prefix_add_right_vbcoefficientsum_body_start. fs_h_pfc_prefix_add_right_vbcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_prefix_add_right_vbcoefficientsum)) /\\ exists fs_q_pfc_prefix_add_right_vbcoefficientsum_body_start. fs_u_pfc_prefix_add_right_vbcoefficientsum = fs_q_pfc_prefix_add_right_vbcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_prefix_add_right_vbcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_prefix_add_right_vbcoefficientsum_body_terminal. fs_h_pfc_prefix_add_right_vbcoefficientsum_body_terminal + S (pfc_natural_sum_prefix_add_right_vbcoefficient) = S ((S (S (pfc_index_prefix_add_right_vb))) * fs_v_pfc_prefix_add_right_vbcoefficientsum)) /\\ exists fs_q_pfc_prefix_add_right_vbcoefficientsum_body_terminal. fs_u_pfc_prefix_add_right_vbcoefficientsum = fs_q_pfc_prefix_add_right_vbcoefficientsum_body_terminal * S ((S (S (pfc_index_prefix_add_right_vb))) * fs_v_pfc_prefix_add_right_vbcoefficientsum) + (pfc_natural_sum_prefix_add_right_vbcoefficient))) /\\ forall fs_i_pfc_prefix_add_right_vbcoefficientsum_body_steps. (exists fs_lt_pfc_prefix_add_right_vbcoefficientsum_body_steps_bound. fs_lt_pfc_prefix_add_right_vbcoefficientsum_body_steps_bound + S fs_i_pfc_prefix_add_right_vbcoefficientsum_body_steps = S (pfc_index_prefix_add_right_vb)) -> exists fs_a_pfc_prefix_add_right_vbcoefficientsum_body_steps fs_r_pfc_prefix_add_right_vbcoefficientsum_body_steps fs_s_pfc_prefix_add_right_vbcoefficientsum_body_steps. ((((exists fs_h_pfc_prefix_add_right_vbcoefficientsum_body_steps_summand. fs_h_pfc_prefix_add_right_vbcoefficientsum_body_steps_summand + S (fs_a_pfc_prefix_add_right_vbcoefficientsum_body_steps) = S ((S (fs_i_pfc_prefix_add_right_vbcoefficientsum_body_steps)) * pfc_terms_scale_prefix_add_right_vbcoefficient)) /\\ exists fs_q_pfc_prefix_add_right_vbcoefficientsum_body_steps_summand. pfc_terms_code_prefix_add_right_vbcoefficient = fs_q_pfc_prefix_add_right_vbcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_prefix_add_right_vbcoefficientsum_body_steps)) * pfc_terms_scale_prefix_add_right_vbcoefficient) + (fs_a_pfc_prefix_add_right_vbcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_add_right_vbcoefficientsum_body_steps_partial. fs_h_pfc_prefix_add_right_vbcoefficientsum_body_steps_partial + S (fs_r_pfc_prefix_add_right_vbcoefficientsum_body_steps) = S ((S (fs_i_pfc_prefix_add_right_vbcoefficientsum_body_steps)) * fs_v_pfc_prefix_add_right_vbcoefficientsum)) /\\ exists fs_q_pfc_prefix_add_right_vbcoefficientsum_body_steps_partial. fs_u_pfc_prefix_add_right_vbcoefficientsum = fs_q_pfc_prefix_add_right_vbcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_prefix_add_right_vbcoefficientsum_body_steps)) * fs_v_pfc_prefix_add_right_vbcoefficientsum) + (fs_r_pfc_prefix_add_right_vbcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_add_right_vbcoefficientsum_body_steps_successor. fs_h_pfc_prefix_add_right_vbcoefficientsum_body_steps_successor + S (fs_s_pfc_prefix_add_right_vbcoefficientsum_body_steps) = S ((S (S fs_i_pfc_prefix_add_right_vbcoefficientsum_body_steps)) * fs_v_pfc_prefix_add_right_vbcoefficientsum)) /\\ exists fs_q_pfc_prefix_add_right_vbcoefficientsum_body_steps_successor. fs_u_pfc_prefix_add_right_vbcoefficientsum = fs_q_pfc_prefix_add_right_vbcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_prefix_add_right_vbcoefficientsum_body_steps)) * fs_v_pfc_prefix_add_right_vbcoefficientsum) + (fs_s_pfc_prefix_add_right_vbcoefficientsum_body_steps))) /\\ fs_s_pfc_prefix_add_right_vbcoefficientsum_body_steps = fs_r_pfc_prefix_add_right_vbcoefficientsum_body_steps + fs_a_pfc_prefix_add_right_vbcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_prefix_add_right_vbcoefficientresiduebound. pfa_gap_prefix_add_right_vbcoefficientresiduebound + S (pfc_value_prefix_add_right_vb) = (p)) /\\ ((exists pfa_offset_left_prefix_add_right_vbcoefficientresiduecongruence pfa_offset_right_prefix_add_right_vbcoefficientresiduecongruence. (pfc_natural_sum_prefix_add_right_vbcoefficient) + (p) * pfa_offset_left_prefix_add_right_vbcoefficientresiduecongruence = (pfc_value_prefix_add_right_vb) + (p) * pfa_offset_right_prefix_add_right_vbcoefficientresiduecongruence)))))))))))) -> (forall pfc_index_prefix_add_right_wb. (exists pfa_gap_prefix_add_right_wbbound. pfa_gap_prefix_add_right_wbbound + S (pfc_index_prefix_add_right_wb) = (N)) -> exists pfc_value_prefix_add_right_wb. ((((exists ff_h_pfp_prefix_add_right_wbentry. ff_h_pfp_prefix_add_right_wbentry + S (pfc_value_prefix_add_right_wb) = S ((S (pfc_index_prefix_add_right_wb)) * wc)) /\\ exists ff_q_pfp_prefix_add_right_wbentry. wb = ff_q_pfp_prefix_add_right_wbentry * S ((S (pfc_index_prefix_add_right_wb)) * wc) + (pfc_value_prefix_add_right_wb))) /\\ ((exists pfc_terms_code_prefix_add_right_wbcoefficient pfc_terms_scale_prefix_add_right_wbcoefficient pfc_natural_sum_prefix_add_right_wbcoefficient. ((forall pfc_index_prefix_add_right_wbcoefficientdiagonal. (exists pfa_gap_prefix_add_right_wbcoefficientdiagonalbound. pfa_gap_prefix_add_right_wbcoefficientdiagonalbound + S (pfc_index_prefix_add_right_wbcoefficientdiagonal) = (S (pfc_index_prefix_add_right_wb))) -> exists pfc_value_prefix_add_right_wbcoefficientdiagonal. ((((exists ff_h_pfp_prefix_add_right_wbcoefficientdiagonalentry. ff_h_pfp_prefix_add_right_wbcoefficientdiagonalentry + S (pfc_value_prefix_add_right_wbcoefficientdiagonal) = S ((S (pfc_index_prefix_add_right_wbcoefficientdiagonal)) * pfc_terms_scale_prefix_add_right_wbcoefficient)) /\\ exists ff_q_pfp_prefix_add_right_wbcoefficientdiagonalentry. pfc_terms_code_prefix_add_right_wbcoefficient = ff_q_pfp_prefix_add_right_wbcoefficientdiagonalentry * S ((S (pfc_index_prefix_add_right_wbcoefficientdiagonal)) * pfc_terms_scale_prefix_add_right_wbcoefficient) + (pfc_value_prefix_add_right_wbcoefficientdiagonal))) /\\ ((exists pfc_complement_prefix_add_right_wbcoefficientdiagonalterm pfc_left_prefix_add_right_wbcoefficientdiagonalterm pfc_right_prefix_add_right_wbcoefficientdiagonalterm. (((pfc_index_prefix_add_right_wbcoefficientdiagonal)+pfc_complement_prefix_add_right_wbcoefficientdiagonalterm=(pfc_index_prefix_add_right_wb)) /\\ ((((((exists pfa_gap_prefix_add_right_wbcoefficientdiagonaltermleftinside. pfa_gap_prefix_add_right_wbcoefficientdiagonaltermleftinside + S (pfc_index_prefix_add_right_wbcoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_prefix_add_right_wbcoefficientdiagonaltermleftentry. ff_h_pfp_prefix_add_right_wbcoefficientdiagonaltermleftentry + S (pfc_left_prefix_add_right_wbcoefficientdiagonalterm) = S ((S (pfc_index_prefix_add_right_wbcoefficientdiagonal)) * cc)) /\\ exists ff_q_pfp_prefix_add_right_wbcoefficientdiagonaltermleftentry. cb = ff_q_pfp_prefix_add_right_wbcoefficientdiagonaltermleftentry * S ((S (pfc_index_prefix_add_right_wbcoefficientdiagonal)) * cc) + (pfc_left_prefix_add_right_wbcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_add_right_wbcoefficientdiagonaltermleftoutside. pfc_gap_prefix_add_right_wbcoefficientdiagonaltermleftoutside+(L)=(pfc_index_prefix_add_right_wbcoefficientdiagonal)) /\\ (((pfc_left_prefix_add_right_wbcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_prefix_add_right_wbcoefficientdiagonaltermrightinside. pfa_gap_prefix_add_right_wbcoefficientdiagonaltermrightinside + S (pfc_complement_prefix_add_right_wbcoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_prefix_add_right_wbcoefficientdiagonaltermrightentry. ff_h_pfp_prefix_add_right_wbcoefficientdiagonaltermrightentry + S (pfc_right_prefix_add_right_wbcoefficientdiagonalterm) = S ((S (pfc_complement_prefix_add_right_wbcoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_prefix_add_right_wbcoefficientdiagonaltermrightentry. db = ff_q_pfp_prefix_add_right_wbcoefficientdiagonaltermrightentry * S ((S (pfc_complement_prefix_add_right_wbcoefficientdiagonalterm)) * dc) + (pfc_right_prefix_add_right_wbcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_add_right_wbcoefficientdiagonaltermrightoutside. pfc_gap_prefix_add_right_wbcoefficientdiagonaltermrightoutside+(M)=(pfc_complement_prefix_add_right_wbcoefficientdiagonalterm)) /\\ (((pfc_right_prefix_add_right_wbcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_prefix_add_right_wbcoefficientdiagonal)=pfc_left_prefix_add_right_wbcoefficientdiagonalterm*pfc_right_prefix_add_right_wbcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_prefix_add_right_wbcoefficientsum fs_v_pfc_prefix_add_right_wbcoefficientsum. ((((exists fs_h_pfc_prefix_add_right_wbcoefficientsum_body_start. fs_h_pfc_prefix_add_right_wbcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_prefix_add_right_wbcoefficientsum)) /\\ exists fs_q_pfc_prefix_add_right_wbcoefficientsum_body_start. fs_u_pfc_prefix_add_right_wbcoefficientsum = fs_q_pfc_prefix_add_right_wbcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_prefix_add_right_wbcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_prefix_add_right_wbcoefficientsum_body_terminal. fs_h_pfc_prefix_add_right_wbcoefficientsum_body_terminal + S (pfc_natural_sum_prefix_add_right_wbcoefficient) = S ((S (S (pfc_index_prefix_add_right_wb))) * fs_v_pfc_prefix_add_right_wbcoefficientsum)) /\\ exists fs_q_pfc_prefix_add_right_wbcoefficientsum_body_terminal. fs_u_pfc_prefix_add_right_wbcoefficientsum = fs_q_pfc_prefix_add_right_wbcoefficientsum_body_terminal * S ((S (S (pfc_index_prefix_add_right_wb))) * fs_v_pfc_prefix_add_right_wbcoefficientsum) + (pfc_natural_sum_prefix_add_right_wbcoefficient))) /\\ forall fs_i_pfc_prefix_add_right_wbcoefficientsum_body_steps. (exists fs_lt_pfc_prefix_add_right_wbcoefficientsum_body_steps_bound. fs_lt_pfc_prefix_add_right_wbcoefficientsum_body_steps_bound + S fs_i_pfc_prefix_add_right_wbcoefficientsum_body_steps = S (pfc_index_prefix_add_right_wb)) -> exists fs_a_pfc_prefix_add_right_wbcoefficientsum_body_steps fs_r_pfc_prefix_add_right_wbcoefficientsum_body_steps fs_s_pfc_prefix_add_right_wbcoefficientsum_body_steps. ((((exists fs_h_pfc_prefix_add_right_wbcoefficientsum_body_steps_summand. fs_h_pfc_prefix_add_right_wbcoefficientsum_body_steps_summand + S (fs_a_pfc_prefix_add_right_wbcoefficientsum_body_steps) = S ((S (fs_i_pfc_prefix_add_right_wbcoefficientsum_body_steps)) * pfc_terms_scale_prefix_add_right_wbcoefficient)) /\\ exists fs_q_pfc_prefix_add_right_wbcoefficientsum_body_steps_summand. pfc_terms_code_prefix_add_right_wbcoefficient = fs_q_pfc_prefix_add_right_wbcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_prefix_add_right_wbcoefficientsum_body_steps)) * pfc_terms_scale_prefix_add_right_wbcoefficient) + (fs_a_pfc_prefix_add_right_wbcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_add_right_wbcoefficientsum_body_steps_partial. fs_h_pfc_prefix_add_right_wbcoefficientsum_body_steps_partial + S (fs_r_pfc_prefix_add_right_wbcoefficientsum_body_steps) = S ((S (fs_i_pfc_prefix_add_right_wbcoefficientsum_body_steps)) * fs_v_pfc_prefix_add_right_wbcoefficientsum)) /\\ exists fs_q_pfc_prefix_add_right_wbcoefficientsum_body_steps_partial. fs_u_pfc_prefix_add_right_wbcoefficientsum = fs_q_pfc_prefix_add_right_wbcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_prefix_add_right_wbcoefficientsum_body_steps)) * fs_v_pfc_prefix_add_right_wbcoefficientsum) + (fs_r_pfc_prefix_add_right_wbcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_add_right_wbcoefficientsum_body_steps_successor. fs_h_pfc_prefix_add_right_wbcoefficientsum_body_steps_successor + S (fs_s_pfc_prefix_add_right_wbcoefficientsum_body_steps) = S ((S (S fs_i_pfc_prefix_add_right_wbcoefficientsum_body_steps)) * fs_v_pfc_prefix_add_right_wbcoefficientsum)) /\\ exists fs_q_pfc_prefix_add_right_wbcoefficientsum_body_steps_successor. fs_u_pfc_prefix_add_right_wbcoefficientsum = fs_q_pfc_prefix_add_right_wbcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_prefix_add_right_wbcoefficientsum_body_steps)) * fs_v_pfc_prefix_add_right_wbcoefficientsum) + (fs_s_pfc_prefix_add_right_wbcoefficientsum_body_steps))) /\\ fs_s_pfc_prefix_add_right_wbcoefficientsum_body_steps = fs_r_pfc_prefix_add_right_wbcoefficientsum_body_steps + fs_a_pfc_prefix_add_right_wbcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_prefix_add_right_wbcoefficientresiduebound. pfa_gap_prefix_add_right_wbcoefficientresiduebound + S (pfc_value_prefix_add_right_wb) = (p)) /\\ ((exists pfa_offset_left_prefix_add_right_wbcoefficientresiduecongruence pfa_offset_right_prefix_add_right_wbcoefficientresiduecongruence. (pfc_natural_sum_prefix_add_right_wbcoefficient) + (p) * pfa_offset_left_prefix_add_right_wbcoefficientresiduecongruence = (pfc_value_prefix_add_right_wb) + (p) * pfa_offset_right_prefix_add_right_wbcoefficientresiduecongruence)))))))))))) -> (forall pfp_index_prefix_add_right_result. (exists pfa_gap_prefix_add_right_resultindex. pfa_gap_prefix_add_right_resultindex + S (pfp_index_prefix_add_right_result) = (N)) -> exists pfp_left_prefix_add_right_result pfp_right_prefix_add_right_result pfp_value_prefix_add_right_result. ((((exists ff_h_pfp_prefix_add_right_resultleft. ff_h_pfp_prefix_add_right_resultleft + S (pfp_left_prefix_add_right_result) = S ((S (pfp_index_prefix_add_right_result)) * uc)) /\\ exists ff_q_pfp_prefix_add_right_resultleft. ub = ff_q_pfp_prefix_add_right_resultleft * S ((S (pfp_index_prefix_add_right_result)) * uc) + (pfp_left_prefix_add_right_result))) /\\ (((((exists ff_h_pfp_prefix_add_right_resultright. ff_h_pfp_prefix_add_right_resultright + S (pfp_right_prefix_add_right_result) = S ((S (pfp_index_prefix_add_right_result)) * vc)) /\\ exists ff_q_pfp_prefix_add_right_resultright. vb = ff_q_pfp_prefix_add_right_resultright * S ((S (pfp_index_prefix_add_right_result)) * vc) + (pfp_right_prefix_add_right_result))) /\\ (((((exists ff_h_pfp_prefix_add_right_resulttarget. ff_h_pfp_prefix_add_right_resulttarget + S (pfp_value_prefix_add_right_result) = S ((S (pfp_index_prefix_add_right_result)) * wc)) /\\ exists ff_q_pfp_prefix_add_right_resulttarget. wb = ff_q_pfp_prefix_add_right_resulttarget * S ((S (pfp_index_prefix_add_right_result)) * wc) + (pfp_value_prefix_add_right_result))) /\\ ((((exists pfa_gap_prefix_add_right_resultoperationleft. pfa_gap_prefix_add_right_resultoperationleft + S (pfp_left_prefix_add_right_result) = (p)) /\\ (((exists pfa_gap_prefix_add_right_resultoperationright. pfa_gap_prefix_add_right_resultoperationright + S (pfp_right_prefix_add_right_result) = (p)) /\\ ((((exists pfa_gap_prefix_add_right_resultoperationresultbound. pfa_gap_prefix_add_right_resultoperationresultbound + S (pfp_value_prefix_add_right_result) = (p)) /\\ ((exists pfa_offset_left_prefix_add_right_resultoperationresultcongruence pfa_offset_right_prefix_add_right_resultoperationresultcongruence. 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prime_field_polynomial_subtract_recover_add (ab)", "specialize prime_field_polynomial_subtract_recover_add (ac)", "specialize prime_field_polynomial_subtract_recover_add (bb)", "specialize prime_field_polynomial_subtract_recover_add (bc)", "specialize prime_field_polynomial_subtract_recover_add (cb)", "specialize prime_field_polynomial_subtract_recover_add (cc)", "specialize prime_field_polynomial_subtract_recover_add (L)", "apply prime_field_polynomial_subtract_recover_add", "exact hs", "exact hv", "exact hw", "exact hu" ], "script_sha256": "1c3b612e0289e4b4a28607eb6de1252c040b58b56bba4299a514970b8f2f53d7", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_distributivity_candidate.py", "sha256": "a959962d631759cd1fc773dd7eef2fadf4f3f95361d6d7bc8c6a9e82d0d4ab86" }, "statement": "forall p ab ac bb bc cb cc L db dc M ub uc vb vc wb wc N. (forall pfs_index_prefix_subtract_left_source. (exists pfa_gap_prefix_subtract_left_sourceindex. pfa_gap_prefix_subtract_left_sourceindex + S (pfs_index_prefix_subtract_left_source) = (L)) -> exists pfs_left_prefix_subtract_left_source pfs_right_prefix_subtract_left_source pfs_result_prefix_subtract_left_source. ((((exists ff_h_pfp_prefix_subtract_left_sourceleft. ff_h_pfp_prefix_subtract_left_sourceleft + S (pfs_left_prefix_subtract_left_source) = S ((S (pfs_index_prefix_subtract_left_source)) * ac)) /\\ exists ff_q_pfp_prefix_subtract_left_sourceleft. ab = ff_q_pfp_prefix_subtract_left_sourceleft * S ((S (pfs_index_prefix_subtract_left_source)) * ac) + (pfs_left_prefix_subtract_left_source))) /\\ (((((exists ff_h_pfp_prefix_subtract_left_sourceright. ff_h_pfp_prefix_subtract_left_sourceright + S (pfs_right_prefix_subtract_left_source) = S ((S (pfs_index_prefix_subtract_left_source)) * bc)) /\\ exists ff_q_pfp_prefix_subtract_left_sourceright. bb = ff_q_pfp_prefix_subtract_left_sourceright * S ((S (pfs_index_prefix_subtract_left_source)) * bc) + (pfs_right_prefix_subtract_left_source))) /\\ (((((exists ff_h_pfp_prefix_subtract_left_sourceresult. ff_h_pfp_prefix_subtract_left_sourceresult + S (pfs_result_prefix_subtract_left_source) = S ((S (pfs_index_prefix_subtract_left_source)) * cc)) /\\ exists ff_q_pfp_prefix_subtract_left_sourceresult. cb = ff_q_pfp_prefix_subtract_left_sourceresult * S ((S (pfs_index_prefix_subtract_left_source)) * cc) + (pfs_result_prefix_subtract_left_source))) /\\ ((((exists pfa_gap_prefix_subtract_left_sourceoperationleft. pfa_gap_prefix_subtract_left_sourceoperationleft + S (pfs_right_prefix_subtract_left_source) = (p)) /\\ (((exists pfa_gap_prefix_subtract_left_sourceoperationright. pfa_gap_prefix_subtract_left_sourceoperationright + S (pfs_result_prefix_subtract_left_source) = (p)) /\\ ((((exists pfa_gap_prefix_subtract_left_sourceoperationresultbound. pfa_gap_prefix_subtract_left_sourceoperationresultbound + S (pfs_left_prefix_subtract_left_source) = (p)) /\\ ((exists pfa_offset_left_prefix_subtract_left_sourceoperationresultcongruence pfa_offset_right_prefix_subtract_left_sourceoperationresultcongruence. ((pfs_right_prefix_subtract_left_source) + (pfs_result_prefix_subtract_left_source)) + (p) * pfa_offset_left_prefix_subtract_left_sourceoperationresultcongruence = (pfs_left_prefix_subtract_left_source) + (p) * pfa_offset_right_prefix_subtract_left_sourceoperationresultcongruence)))))))))))))))) -> (forall pfc_index_prefix_subtract_left_ub. (exists pfa_gap_prefix_subtract_left_ubbound. pfa_gap_prefix_subtract_left_ubbound + S (pfc_index_prefix_subtract_left_ub) = (N)) -> exists pfc_value_prefix_subtract_left_ub. ((((exists ff_h_pfp_prefix_subtract_left_ubentry. ff_h_pfp_prefix_subtract_left_ubentry + S (pfc_value_prefix_subtract_left_ub) = S ((S (pfc_index_prefix_subtract_left_ub)) * uc)) /\\ exists ff_q_pfp_prefix_subtract_left_ubentry. ub = ff_q_pfp_prefix_subtract_left_ubentry * S ((S (pfc_index_prefix_subtract_left_ub)) * uc) + (pfc_value_prefix_subtract_left_ub))) /\\ ((exists pfc_terms_code_prefix_subtract_left_ubcoefficient pfc_terms_scale_prefix_subtract_left_ubcoefficient pfc_natural_sum_prefix_subtract_left_ubcoefficient. ((forall pfc_index_prefix_subtract_left_ubcoefficientdiagonal. (exists pfa_gap_prefix_subtract_left_ubcoefficientdiagonalbound. pfa_gap_prefix_subtract_left_ubcoefficientdiagonalbound + S (pfc_index_prefix_subtract_left_ubcoefficientdiagonal) = (S (pfc_index_prefix_subtract_left_ub))) -> exists pfc_value_prefix_subtract_left_ubcoefficientdiagonal. ((((exists ff_h_pfp_prefix_subtract_left_ubcoefficientdiagonalentry. ff_h_pfp_prefix_subtract_left_ubcoefficientdiagonalentry + S (pfc_value_prefix_subtract_left_ubcoefficientdiagonal) = S ((S (pfc_index_prefix_subtract_left_ubcoefficientdiagonal)) * pfc_terms_scale_prefix_subtract_left_ubcoefficient)) /\\ exists ff_q_pfp_prefix_subtract_left_ubcoefficientdiagonalentry. pfc_terms_code_prefix_subtract_left_ubcoefficient = ff_q_pfp_prefix_subtract_left_ubcoefficientdiagonalentry * S ((S (pfc_index_prefix_subtract_left_ubcoefficientdiagonal)) * pfc_terms_scale_prefix_subtract_left_ubcoefficient) + (pfc_value_prefix_subtract_left_ubcoefficientdiagonal))) /\\ ((exists pfc_complement_prefix_subtract_left_ubcoefficientdiagonalterm pfc_left_prefix_subtract_left_ubcoefficientdiagonalterm pfc_right_prefix_subtract_left_ubcoefficientdiagonalterm. (((pfc_index_prefix_subtract_left_ubcoefficientdiagonal)+pfc_complement_prefix_subtract_left_ubcoefficientdiagonalterm=(pfc_index_prefix_subtract_left_ub)) /\\ ((((((exists pfa_gap_prefix_subtract_left_ubcoefficientdiagonaltermleftinside. pfa_gap_prefix_subtract_left_ubcoefficientdiagonaltermleftinside + S (pfc_index_prefix_subtract_left_ubcoefficientdiagonal) = (M)) /\\ ((((exists ff_h_pfp_prefix_subtract_left_ubcoefficientdiagonaltermleftentry. ff_h_pfp_prefix_subtract_left_ubcoefficientdiagonaltermleftentry + S (pfc_left_prefix_subtract_left_ubcoefficientdiagonalterm) = S ((S (pfc_index_prefix_subtract_left_ubcoefficientdiagonal)) * dc)) /\\ exists ff_q_pfp_prefix_subtract_left_ubcoefficientdiagonaltermleftentry. db = ff_q_pfp_prefix_subtract_left_ubcoefficientdiagonaltermleftentry * S ((S (pfc_index_prefix_subtract_left_ubcoefficientdiagonal)) * dc) + (pfc_left_prefix_subtract_left_ubcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_subtract_left_ubcoefficientdiagonaltermleftoutside. pfc_gap_prefix_subtract_left_ubcoefficientdiagonaltermleftoutside+(M)=(pfc_index_prefix_subtract_left_ubcoefficientdiagonal)) /\\ (((pfc_left_prefix_subtract_left_ubcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_prefix_subtract_left_ubcoefficientdiagonaltermrightinside. pfa_gap_prefix_subtract_left_ubcoefficientdiagonaltermrightinside + S (pfc_complement_prefix_subtract_left_ubcoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_prefix_subtract_left_ubcoefficientdiagonaltermrightentry. ff_h_pfp_prefix_subtract_left_ubcoefficientdiagonaltermrightentry + S (pfc_right_prefix_subtract_left_ubcoefficientdiagonalterm) = S ((S (pfc_complement_prefix_subtract_left_ubcoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_prefix_subtract_left_ubcoefficientdiagonaltermrightentry. ab = ff_q_pfp_prefix_subtract_left_ubcoefficientdiagonaltermrightentry * S ((S (pfc_complement_prefix_subtract_left_ubcoefficientdiagonalterm)) * ac) + (pfc_right_prefix_subtract_left_ubcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_subtract_left_ubcoefficientdiagonaltermrightoutside. pfc_gap_prefix_subtract_left_ubcoefficientdiagonaltermrightoutside+(L)=(pfc_complement_prefix_subtract_left_ubcoefficientdiagonalterm)) /\\ (((pfc_right_prefix_subtract_left_ubcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_prefix_subtract_left_ubcoefficientdiagonal)=pfc_left_prefix_subtract_left_ubcoefficientdiagonalterm*pfc_right_prefix_subtract_left_ubcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_prefix_subtract_left_ubcoefficientsum fs_v_pfc_prefix_subtract_left_ubcoefficientsum. ((((exists fs_h_pfc_prefix_subtract_left_ubcoefficientsum_body_start. fs_h_pfc_prefix_subtract_left_ubcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_prefix_subtract_left_ubcoefficientsum)) /\\ exists fs_q_pfc_prefix_subtract_left_ubcoefficientsum_body_start. fs_u_pfc_prefix_subtract_left_ubcoefficientsum = fs_q_pfc_prefix_subtract_left_ubcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_prefix_subtract_left_ubcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_prefix_subtract_left_ubcoefficientsum_body_terminal. fs_h_pfc_prefix_subtract_left_ubcoefficientsum_body_terminal + S (pfc_natural_sum_prefix_subtract_left_ubcoefficient) = S ((S (S (pfc_index_prefix_subtract_left_ub))) * fs_v_pfc_prefix_subtract_left_ubcoefficientsum)) /\\ exists fs_q_pfc_prefix_subtract_left_ubcoefficientsum_body_terminal. fs_u_pfc_prefix_subtract_left_ubcoefficientsum = fs_q_pfc_prefix_subtract_left_ubcoefficientsum_body_terminal * S ((S (S (pfc_index_prefix_subtract_left_ub))) * fs_v_pfc_prefix_subtract_left_ubcoefficientsum) + (pfc_natural_sum_prefix_subtract_left_ubcoefficient))) /\\ forall fs_i_pfc_prefix_subtract_left_ubcoefficientsum_body_steps. (exists fs_lt_pfc_prefix_subtract_left_ubcoefficientsum_body_steps_bound. fs_lt_pfc_prefix_subtract_left_ubcoefficientsum_body_steps_bound + S fs_i_pfc_prefix_subtract_left_ubcoefficientsum_body_steps = S (pfc_index_prefix_subtract_left_ub)) -> exists fs_a_pfc_prefix_subtract_left_ubcoefficientsum_body_steps fs_r_pfc_prefix_subtract_left_ubcoefficientsum_body_steps fs_s_pfc_prefix_subtract_left_ubcoefficientsum_body_steps. ((((exists fs_h_pfc_prefix_subtract_left_ubcoefficientsum_body_steps_summand. fs_h_pfc_prefix_subtract_left_ubcoefficientsum_body_steps_summand + S (fs_a_pfc_prefix_subtract_left_ubcoefficientsum_body_steps) = S ((S (fs_i_pfc_prefix_subtract_left_ubcoefficientsum_body_steps)) * pfc_terms_scale_prefix_subtract_left_ubcoefficient)) /\\ exists fs_q_pfc_prefix_subtract_left_ubcoefficientsum_body_steps_summand. pfc_terms_code_prefix_subtract_left_ubcoefficient = fs_q_pfc_prefix_subtract_left_ubcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_prefix_subtract_left_ubcoefficientsum_body_steps)) * pfc_terms_scale_prefix_subtract_left_ubcoefficient) + (fs_a_pfc_prefix_subtract_left_ubcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_subtract_left_ubcoefficientsum_body_steps_partial. fs_h_pfc_prefix_subtract_left_ubcoefficientsum_body_steps_partial + S (fs_r_pfc_prefix_subtract_left_ubcoefficientsum_body_steps) = S ((S (fs_i_pfc_prefix_subtract_left_ubcoefficientsum_body_steps)) * fs_v_pfc_prefix_subtract_left_ubcoefficientsum)) /\\ exists fs_q_pfc_prefix_subtract_left_ubcoefficientsum_body_steps_partial. fs_u_pfc_prefix_subtract_left_ubcoefficientsum = fs_q_pfc_prefix_subtract_left_ubcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_prefix_subtract_left_ubcoefficientsum_body_steps)) * fs_v_pfc_prefix_subtract_left_ubcoefficientsum) + (fs_r_pfc_prefix_subtract_left_ubcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_subtract_left_ubcoefficientsum_body_steps_successor. fs_h_pfc_prefix_subtract_left_ubcoefficientsum_body_steps_successor + S (fs_s_pfc_prefix_subtract_left_ubcoefficientsum_body_steps) = S ((S (S fs_i_pfc_prefix_subtract_left_ubcoefficientsum_body_steps)) * fs_v_pfc_prefix_subtract_left_ubcoefficientsum)) /\\ exists fs_q_pfc_prefix_subtract_left_ubcoefficientsum_body_steps_successor. fs_u_pfc_prefix_subtract_left_ubcoefficientsum = fs_q_pfc_prefix_subtract_left_ubcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_prefix_subtract_left_ubcoefficientsum_body_steps)) * fs_v_pfc_prefix_subtract_left_ubcoefficientsum) + (fs_s_pfc_prefix_subtract_left_ubcoefficientsum_body_steps))) /\\ fs_s_pfc_prefix_subtract_left_ubcoefficientsum_body_steps = fs_r_pfc_prefix_subtract_left_ubcoefficientsum_body_steps + fs_a_pfc_prefix_subtract_left_ubcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_prefix_subtract_left_ubcoefficientresiduebound. pfa_gap_prefix_subtract_left_ubcoefficientresiduebound + S (pfc_value_prefix_subtract_left_ub) = (p)) /\\ ((exists pfa_offset_left_prefix_subtract_left_ubcoefficientresiduecongruence pfa_offset_right_prefix_subtract_left_ubcoefficientresiduecongruence. (pfc_natural_sum_prefix_subtract_left_ubcoefficient) + (p) * pfa_offset_left_prefix_subtract_left_ubcoefficientresiduecongruence = (pfc_value_prefix_subtract_left_ub) + (p) * pfa_offset_right_prefix_subtract_left_ubcoefficientresiduecongruence)))))))))))) -> (forall pfc_index_prefix_subtract_left_vb. (exists pfa_gap_prefix_subtract_left_vbbound. pfa_gap_prefix_subtract_left_vbbound + S (pfc_index_prefix_subtract_left_vb) = (N)) -> exists pfc_value_prefix_subtract_left_vb. ((((exists ff_h_pfp_prefix_subtract_left_vbentry. ff_h_pfp_prefix_subtract_left_vbentry + S (pfc_value_prefix_subtract_left_vb) = S ((S (pfc_index_prefix_subtract_left_vb)) * vc)) /\\ exists ff_q_pfp_prefix_subtract_left_vbentry. vb = ff_q_pfp_prefix_subtract_left_vbentry * S ((S (pfc_index_prefix_subtract_left_vb)) * vc) + (pfc_value_prefix_subtract_left_vb))) /\\ ((exists pfc_terms_code_prefix_subtract_left_vbcoefficient pfc_terms_scale_prefix_subtract_left_vbcoefficient pfc_natural_sum_prefix_subtract_left_vbcoefficient. ((forall pfc_index_prefix_subtract_left_vbcoefficientdiagonal. (exists pfa_gap_prefix_subtract_left_vbcoefficientdiagonalbound. pfa_gap_prefix_subtract_left_vbcoefficientdiagonalbound + S (pfc_index_prefix_subtract_left_vbcoefficientdiagonal) = (S (pfc_index_prefix_subtract_left_vb))) -> exists pfc_value_prefix_subtract_left_vbcoefficientdiagonal. ((((exists ff_h_pfp_prefix_subtract_left_vbcoefficientdiagonalentry. ff_h_pfp_prefix_subtract_left_vbcoefficientdiagonalentry + S (pfc_value_prefix_subtract_left_vbcoefficientdiagonal) = S ((S (pfc_index_prefix_subtract_left_vbcoefficientdiagonal)) * pfc_terms_scale_prefix_subtract_left_vbcoefficient)) /\\ exists ff_q_pfp_prefix_subtract_left_vbcoefficientdiagonalentry. pfc_terms_code_prefix_subtract_left_vbcoefficient = ff_q_pfp_prefix_subtract_left_vbcoefficientdiagonalentry * S ((S (pfc_index_prefix_subtract_left_vbcoefficientdiagonal)) * pfc_terms_scale_prefix_subtract_left_vbcoefficient) + (pfc_value_prefix_subtract_left_vbcoefficientdiagonal))) /\\ ((exists pfc_complement_prefix_subtract_left_vbcoefficientdiagonalterm pfc_left_prefix_subtract_left_vbcoefficientdiagonalterm pfc_right_prefix_subtract_left_vbcoefficientdiagonalterm. (((pfc_index_prefix_subtract_left_vbcoefficientdiagonal)+pfc_complement_prefix_subtract_left_vbcoefficientdiagonalterm=(pfc_index_prefix_subtract_left_vb)) /\\ ((((((exists pfa_gap_prefix_subtract_left_vbcoefficientdiagonaltermleftinside. pfa_gap_prefix_subtract_left_vbcoefficientdiagonaltermleftinside + S (pfc_index_prefix_subtract_left_vbcoefficientdiagonal) = (M)) /\\ ((((exists ff_h_pfp_prefix_subtract_left_vbcoefficientdiagonaltermleftentry. ff_h_pfp_prefix_subtract_left_vbcoefficientdiagonaltermleftentry + S (pfc_left_prefix_subtract_left_vbcoefficientdiagonalterm) = S ((S (pfc_index_prefix_subtract_left_vbcoefficientdiagonal)) * dc)) /\\ exists ff_q_pfp_prefix_subtract_left_vbcoefficientdiagonaltermleftentry. db = ff_q_pfp_prefix_subtract_left_vbcoefficientdiagonaltermleftentry * S ((S (pfc_index_prefix_subtract_left_vbcoefficientdiagonal)) * dc) + (pfc_left_prefix_subtract_left_vbcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_subtract_left_vbcoefficientdiagonaltermleftoutside. pfc_gap_prefix_subtract_left_vbcoefficientdiagonaltermleftoutside+(M)=(pfc_index_prefix_subtract_left_vbcoefficientdiagonal)) /\\ (((pfc_left_prefix_subtract_left_vbcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_prefix_subtract_left_vbcoefficientdiagonaltermrightinside. pfa_gap_prefix_subtract_left_vbcoefficientdiagonaltermrightinside + S (pfc_complement_prefix_subtract_left_vbcoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_prefix_subtract_left_vbcoefficientdiagonaltermrightentry. ff_h_pfp_prefix_subtract_left_vbcoefficientdiagonaltermrightentry + S (pfc_right_prefix_subtract_left_vbcoefficientdiagonalterm) = S ((S (pfc_complement_prefix_subtract_left_vbcoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_prefix_subtract_left_vbcoefficientdiagonaltermrightentry. bb = ff_q_pfp_prefix_subtract_left_vbcoefficientdiagonaltermrightentry * S ((S (pfc_complement_prefix_subtract_left_vbcoefficientdiagonalterm)) * bc) + (pfc_right_prefix_subtract_left_vbcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_subtract_left_vbcoefficientdiagonaltermrightoutside. pfc_gap_prefix_subtract_left_vbcoefficientdiagonaltermrightoutside+(L)=(pfc_complement_prefix_subtract_left_vbcoefficientdiagonalterm)) /\\ (((pfc_right_prefix_subtract_left_vbcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_prefix_subtract_left_vbcoefficientdiagonal)=pfc_left_prefix_subtract_left_vbcoefficientdiagonalterm*pfc_right_prefix_subtract_left_vbcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_prefix_subtract_left_vbcoefficientsum fs_v_pfc_prefix_subtract_left_vbcoefficientsum. ((((exists fs_h_pfc_prefix_subtract_left_vbcoefficientsum_body_start. fs_h_pfc_prefix_subtract_left_vbcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_prefix_subtract_left_vbcoefficientsum)) /\\ exists fs_q_pfc_prefix_subtract_left_vbcoefficientsum_body_start. fs_u_pfc_prefix_subtract_left_vbcoefficientsum = fs_q_pfc_prefix_subtract_left_vbcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_prefix_subtract_left_vbcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_prefix_subtract_left_vbcoefficientsum_body_terminal. fs_h_pfc_prefix_subtract_left_vbcoefficientsum_body_terminal + S (pfc_natural_sum_prefix_subtract_left_vbcoefficient) = S ((S (S (pfc_index_prefix_subtract_left_vb))) * fs_v_pfc_prefix_subtract_left_vbcoefficientsum)) /\\ exists fs_q_pfc_prefix_subtract_left_vbcoefficientsum_body_terminal. fs_u_pfc_prefix_subtract_left_vbcoefficientsum = fs_q_pfc_prefix_subtract_left_vbcoefficientsum_body_terminal * S ((S (S (pfc_index_prefix_subtract_left_vb))) * fs_v_pfc_prefix_subtract_left_vbcoefficientsum) + (pfc_natural_sum_prefix_subtract_left_vbcoefficient))) /\\ forall fs_i_pfc_prefix_subtract_left_vbcoefficientsum_body_steps. (exists fs_lt_pfc_prefix_subtract_left_vbcoefficientsum_body_steps_bound. fs_lt_pfc_prefix_subtract_left_vbcoefficientsum_body_steps_bound + S fs_i_pfc_prefix_subtract_left_vbcoefficientsum_body_steps = S (pfc_index_prefix_subtract_left_vb)) -> exists fs_a_pfc_prefix_subtract_left_vbcoefficientsum_body_steps fs_r_pfc_prefix_subtract_left_vbcoefficientsum_body_steps fs_s_pfc_prefix_subtract_left_vbcoefficientsum_body_steps. ((((exists fs_h_pfc_prefix_subtract_left_vbcoefficientsum_body_steps_summand. fs_h_pfc_prefix_subtract_left_vbcoefficientsum_body_steps_summand + S (fs_a_pfc_prefix_subtract_left_vbcoefficientsum_body_steps) = S ((S (fs_i_pfc_prefix_subtract_left_vbcoefficientsum_body_steps)) * pfc_terms_scale_prefix_subtract_left_vbcoefficient)) /\\ exists fs_q_pfc_prefix_subtract_left_vbcoefficientsum_body_steps_summand. pfc_terms_code_prefix_subtract_left_vbcoefficient = fs_q_pfc_prefix_subtract_left_vbcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_prefix_subtract_left_vbcoefficientsum_body_steps)) * pfc_terms_scale_prefix_subtract_left_vbcoefficient) + (fs_a_pfc_prefix_subtract_left_vbcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_subtract_left_vbcoefficientsum_body_steps_partial. fs_h_pfc_prefix_subtract_left_vbcoefficientsum_body_steps_partial + S (fs_r_pfc_prefix_subtract_left_vbcoefficientsum_body_steps) = S ((S (fs_i_pfc_prefix_subtract_left_vbcoefficientsum_body_steps)) * fs_v_pfc_prefix_subtract_left_vbcoefficientsum)) /\\ exists fs_q_pfc_prefix_subtract_left_vbcoefficientsum_body_steps_partial. fs_u_pfc_prefix_subtract_left_vbcoefficientsum = fs_q_pfc_prefix_subtract_left_vbcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_prefix_subtract_left_vbcoefficientsum_body_steps)) * fs_v_pfc_prefix_subtract_left_vbcoefficientsum) + (fs_r_pfc_prefix_subtract_left_vbcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_subtract_left_vbcoefficientsum_body_steps_successor. fs_h_pfc_prefix_subtract_left_vbcoefficientsum_body_steps_successor + S (fs_s_pfc_prefix_subtract_left_vbcoefficientsum_body_steps) = S ((S (S fs_i_pfc_prefix_subtract_left_vbcoefficientsum_body_steps)) * fs_v_pfc_prefix_subtract_left_vbcoefficientsum)) /\\ exists fs_q_pfc_prefix_subtract_left_vbcoefficientsum_body_steps_successor. fs_u_pfc_prefix_subtract_left_vbcoefficientsum = fs_q_pfc_prefix_subtract_left_vbcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_prefix_subtract_left_vbcoefficientsum_body_steps)) * fs_v_pfc_prefix_subtract_left_vbcoefficientsum) + (fs_s_pfc_prefix_subtract_left_vbcoefficientsum_body_steps))) /\\ fs_s_pfc_prefix_subtract_left_vbcoefficientsum_body_steps = fs_r_pfc_prefix_subtract_left_vbcoefficientsum_body_steps + fs_a_pfc_prefix_subtract_left_vbcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_prefix_subtract_left_vbcoefficientresiduebound. pfa_gap_prefix_subtract_left_vbcoefficientresiduebound + S (pfc_value_prefix_subtract_left_vb) = (p)) /\\ ((exists pfa_offset_left_prefix_subtract_left_vbcoefficientresiduecongruence pfa_offset_right_prefix_subtract_left_vbcoefficientresiduecongruence. (pfc_natural_sum_prefix_subtract_left_vbcoefficient) + (p) * pfa_offset_left_prefix_subtract_left_vbcoefficientresiduecongruence = (pfc_value_prefix_subtract_left_vb) + (p) * pfa_offset_right_prefix_subtract_left_vbcoefficientresiduecongruence)))))))))))) -> (forall pfc_index_prefix_subtract_left_wb. (exists pfa_gap_prefix_subtract_left_wbbound. pfa_gap_prefix_subtract_left_wbbound + S (pfc_index_prefix_subtract_left_wb) = (N)) -> exists pfc_value_prefix_subtract_left_wb. ((((exists ff_h_pfp_prefix_subtract_left_wbentry. ff_h_pfp_prefix_subtract_left_wbentry + S (pfc_value_prefix_subtract_left_wb) = S ((S (pfc_index_prefix_subtract_left_wb)) * wc)) /\\ exists ff_q_pfp_prefix_subtract_left_wbentry. wb = ff_q_pfp_prefix_subtract_left_wbentry * S ((S (pfc_index_prefix_subtract_left_wb)) * wc) + (pfc_value_prefix_subtract_left_wb))) /\\ ((exists pfc_terms_code_prefix_subtract_left_wbcoefficient pfc_terms_scale_prefix_subtract_left_wbcoefficient pfc_natural_sum_prefix_subtract_left_wbcoefficient. ((forall pfc_index_prefix_subtract_left_wbcoefficientdiagonal. (exists pfa_gap_prefix_subtract_left_wbcoefficientdiagonalbound. pfa_gap_prefix_subtract_left_wbcoefficientdiagonalbound + S (pfc_index_prefix_subtract_left_wbcoefficientdiagonal) = (S (pfc_index_prefix_subtract_left_wb))) -> exists pfc_value_prefix_subtract_left_wbcoefficientdiagonal. ((((exists ff_h_pfp_prefix_subtract_left_wbcoefficientdiagonalentry. ff_h_pfp_prefix_subtract_left_wbcoefficientdiagonalentry + S (pfc_value_prefix_subtract_left_wbcoefficientdiagonal) = S ((S (pfc_index_prefix_subtract_left_wbcoefficientdiagonal)) * pfc_terms_scale_prefix_subtract_left_wbcoefficient)) /\\ exists ff_q_pfp_prefix_subtract_left_wbcoefficientdiagonalentry. pfc_terms_code_prefix_subtract_left_wbcoefficient = ff_q_pfp_prefix_subtract_left_wbcoefficientdiagonalentry * S ((S (pfc_index_prefix_subtract_left_wbcoefficientdiagonal)) * pfc_terms_scale_prefix_subtract_left_wbcoefficient) + (pfc_value_prefix_subtract_left_wbcoefficientdiagonal))) /\\ ((exists pfc_complement_prefix_subtract_left_wbcoefficientdiagonalterm pfc_left_prefix_subtract_left_wbcoefficientdiagonalterm pfc_right_prefix_subtract_left_wbcoefficientdiagonalterm. (((pfc_index_prefix_subtract_left_wbcoefficientdiagonal)+pfc_complement_prefix_subtract_left_wbcoefficientdiagonalterm=(pfc_index_prefix_subtract_left_wb)) /\\ ((((((exists pfa_gap_prefix_subtract_left_wbcoefficientdiagonaltermleftinside. pfa_gap_prefix_subtract_left_wbcoefficientdiagonaltermleftinside + S (pfc_index_prefix_subtract_left_wbcoefficientdiagonal) = (M)) /\\ ((((exists ff_h_pfp_prefix_subtract_left_wbcoefficientdiagonaltermleftentry. ff_h_pfp_prefix_subtract_left_wbcoefficientdiagonaltermleftentry + S (pfc_left_prefix_subtract_left_wbcoefficientdiagonalterm) = S ((S (pfc_index_prefix_subtract_left_wbcoefficientdiagonal)) * dc)) /\\ exists ff_q_pfp_prefix_subtract_left_wbcoefficientdiagonaltermleftentry. db = ff_q_pfp_prefix_subtract_left_wbcoefficientdiagonaltermleftentry * S ((S (pfc_index_prefix_subtract_left_wbcoefficientdiagonal)) * dc) + (pfc_left_prefix_subtract_left_wbcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_subtract_left_wbcoefficientdiagonaltermleftoutside. pfc_gap_prefix_subtract_left_wbcoefficientdiagonaltermleftoutside+(M)=(pfc_index_prefix_subtract_left_wbcoefficientdiagonal)) /\\ (((pfc_left_prefix_subtract_left_wbcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_prefix_subtract_left_wbcoefficientdiagonaltermrightinside. pfa_gap_prefix_subtract_left_wbcoefficientdiagonaltermrightinside + S (pfc_complement_prefix_subtract_left_wbcoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_prefix_subtract_left_wbcoefficientdiagonaltermrightentry. ff_h_pfp_prefix_subtract_left_wbcoefficientdiagonaltermrightentry + S (pfc_right_prefix_subtract_left_wbcoefficientdiagonalterm) = S ((S (pfc_complement_prefix_subtract_left_wbcoefficientdiagonalterm)) * cc)) /\\ exists ff_q_pfp_prefix_subtract_left_wbcoefficientdiagonaltermrightentry. cb = ff_q_pfp_prefix_subtract_left_wbcoefficientdiagonaltermrightentry * S ((S (pfc_complement_prefix_subtract_left_wbcoefficientdiagonalterm)) * cc) + (pfc_right_prefix_subtract_left_wbcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_subtract_left_wbcoefficientdiagonaltermrightoutside. pfc_gap_prefix_subtract_left_wbcoefficientdiagonaltermrightoutside+(L)=(pfc_complement_prefix_subtract_left_wbcoefficientdiagonalterm)) /\\ (((pfc_right_prefix_subtract_left_wbcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_prefix_subtract_left_wbcoefficientdiagonal)=pfc_left_prefix_subtract_left_wbcoefficientdiagonalterm*pfc_right_prefix_subtract_left_wbcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_prefix_subtract_left_wbcoefficientsum fs_v_pfc_prefix_subtract_left_wbcoefficientsum. ((((exists fs_h_pfc_prefix_subtract_left_wbcoefficientsum_body_start. fs_h_pfc_prefix_subtract_left_wbcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_prefix_subtract_left_wbcoefficientsum)) /\\ exists fs_q_pfc_prefix_subtract_left_wbcoefficientsum_body_start. fs_u_pfc_prefix_subtract_left_wbcoefficientsum = fs_q_pfc_prefix_subtract_left_wbcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_prefix_subtract_left_wbcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_prefix_subtract_left_wbcoefficientsum_body_terminal. fs_h_pfc_prefix_subtract_left_wbcoefficientsum_body_terminal + S (pfc_natural_sum_prefix_subtract_left_wbcoefficient) = S ((S (S (pfc_index_prefix_subtract_left_wb))) * fs_v_pfc_prefix_subtract_left_wbcoefficientsum)) /\\ exists fs_q_pfc_prefix_subtract_left_wbcoefficientsum_body_terminal. fs_u_pfc_prefix_subtract_left_wbcoefficientsum = fs_q_pfc_prefix_subtract_left_wbcoefficientsum_body_terminal * S ((S (S (pfc_index_prefix_subtract_left_wb))) * fs_v_pfc_prefix_subtract_left_wbcoefficientsum) + (pfc_natural_sum_prefix_subtract_left_wbcoefficient))) /\\ forall fs_i_pfc_prefix_subtract_left_wbcoefficientsum_body_steps. (exists fs_lt_pfc_prefix_subtract_left_wbcoefficientsum_body_steps_bound. fs_lt_pfc_prefix_subtract_left_wbcoefficientsum_body_steps_bound + S fs_i_pfc_prefix_subtract_left_wbcoefficientsum_body_steps = S (pfc_index_prefix_subtract_left_wb)) -> exists fs_a_pfc_prefix_subtract_left_wbcoefficientsum_body_steps fs_r_pfc_prefix_subtract_left_wbcoefficientsum_body_steps fs_s_pfc_prefix_subtract_left_wbcoefficientsum_body_steps. ((((exists fs_h_pfc_prefix_subtract_left_wbcoefficientsum_body_steps_summand. fs_h_pfc_prefix_subtract_left_wbcoefficientsum_body_steps_summand + S (fs_a_pfc_prefix_subtract_left_wbcoefficientsum_body_steps) = S ((S (fs_i_pfc_prefix_subtract_left_wbcoefficientsum_body_steps)) * pfc_terms_scale_prefix_subtract_left_wbcoefficient)) /\\ exists fs_q_pfc_prefix_subtract_left_wbcoefficientsum_body_steps_summand. pfc_terms_code_prefix_subtract_left_wbcoefficient = fs_q_pfc_prefix_subtract_left_wbcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_prefix_subtract_left_wbcoefficientsum_body_steps)) * pfc_terms_scale_prefix_subtract_left_wbcoefficient) + (fs_a_pfc_prefix_subtract_left_wbcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_subtract_left_wbcoefficientsum_body_steps_partial. fs_h_pfc_prefix_subtract_left_wbcoefficientsum_body_steps_partial + S (fs_r_pfc_prefix_subtract_left_wbcoefficientsum_body_steps) = S ((S (fs_i_pfc_prefix_subtract_left_wbcoefficientsum_body_steps)) * fs_v_pfc_prefix_subtract_left_wbcoefficientsum)) /\\ exists fs_q_pfc_prefix_subtract_left_wbcoefficientsum_body_steps_partial. fs_u_pfc_prefix_subtract_left_wbcoefficientsum = fs_q_pfc_prefix_subtract_left_wbcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_prefix_subtract_left_wbcoefficientsum_body_steps)) * fs_v_pfc_prefix_subtract_left_wbcoefficientsum) + (fs_r_pfc_prefix_subtract_left_wbcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_subtract_left_wbcoefficientsum_body_steps_successor. fs_h_pfc_prefix_subtract_left_wbcoefficientsum_body_steps_successor + S (fs_s_pfc_prefix_subtract_left_wbcoefficientsum_body_steps) = S ((S (S fs_i_pfc_prefix_subtract_left_wbcoefficientsum_body_steps)) * fs_v_pfc_prefix_subtract_left_wbcoefficientsum)) /\\ exists fs_q_pfc_prefix_subtract_left_wbcoefficientsum_body_steps_successor. fs_u_pfc_prefix_subtract_left_wbcoefficientsum = fs_q_pfc_prefix_subtract_left_wbcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_prefix_subtract_left_wbcoefficientsum_body_steps)) * fs_v_pfc_prefix_subtract_left_wbcoefficientsum) + (fs_s_pfc_prefix_subtract_left_wbcoefficientsum_body_steps))) /\\ fs_s_pfc_prefix_subtract_left_wbcoefficientsum_body_steps = fs_r_pfc_prefix_subtract_left_wbcoefficientsum_body_steps + fs_a_pfc_prefix_subtract_left_wbcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_prefix_subtract_left_wbcoefficientresiduebound. pfa_gap_prefix_subtract_left_wbcoefficientresiduebound + S (pfc_value_prefix_subtract_left_wb) = (p)) /\\ ((exists pfa_offset_left_prefix_subtract_left_wbcoefficientresiduecongruence pfa_offset_right_prefix_subtract_left_wbcoefficientresiduecongruence. (pfc_natural_sum_prefix_subtract_left_wbcoefficient) + (p) * pfa_offset_left_prefix_subtract_left_wbcoefficientresiduecongruence = (pfc_value_prefix_subtract_left_wb) + (p) * pfa_offset_right_prefix_subtract_left_wbcoefficientresiduecongruence)))))))))))) -> (forall pfs_index_prefix_subtract_left_result. (exists pfa_gap_prefix_subtract_left_resultindex. pfa_gap_prefix_subtract_left_resultindex + S (pfs_index_prefix_subtract_left_result) = (N)) -> exists pfs_left_prefix_subtract_left_result pfs_right_prefix_subtract_left_result pfs_result_prefix_subtract_left_result. ((((exists ff_h_pfp_prefix_subtract_left_resultleft. ff_h_pfp_prefix_subtract_left_resultleft + S (pfs_left_prefix_subtract_left_result) = S ((S (pfs_index_prefix_subtract_left_result)) * uc)) /\\ exists ff_q_pfp_prefix_subtract_left_resultleft. ub = ff_q_pfp_prefix_subtract_left_resultleft * S ((S (pfs_index_prefix_subtract_left_result)) * uc) + (pfs_left_prefix_subtract_left_result))) /\\ (((((exists ff_h_pfp_prefix_subtract_left_resultright. ff_h_pfp_prefix_subtract_left_resultright + S (pfs_right_prefix_subtract_left_result) = S ((S (pfs_index_prefix_subtract_left_result)) * vc)) /\\ exists ff_q_pfp_prefix_subtract_left_resultright. vb = ff_q_pfp_prefix_subtract_left_resultright * S ((S (pfs_index_prefix_subtract_left_result)) * vc) + (pfs_right_prefix_subtract_left_result))) /\\ (((((exists ff_h_pfp_prefix_subtract_left_resultresult. ff_h_pfp_prefix_subtract_left_resultresult + S (pfs_result_prefix_subtract_left_result) = S ((S (pfs_index_prefix_subtract_left_result)) * wc)) /\\ exists ff_q_pfp_prefix_subtract_left_resultresult. wb = ff_q_pfp_prefix_subtract_left_resultresult * S ((S (pfs_index_prefix_subtract_left_result)) * wc) + (pfs_result_prefix_subtract_left_result))) /\\ ((((exists pfa_gap_prefix_subtract_left_resultoperationleft. pfa_gap_prefix_subtract_left_resultoperationleft + S (pfs_right_prefix_subtract_left_result) = (p)) /\\ (((exists pfa_gap_prefix_subtract_left_resultoperationright. pfa_gap_prefix_subtract_left_resultoperationright + S (pfs_result_prefix_subtract_left_result) = (p)) /\\ ((((exists pfa_gap_prefix_subtract_left_resultoperationresultbound. pfa_gap_prefix_subtract_left_resultoperationresultbound + S (pfs_left_prefix_subtract_left_result) = (p)) /\\ ((exists pfa_offset_left_prefix_subtract_left_resultoperationresultcongruence pfa_offset_right_prefix_subtract_left_resultoperationresultcongruence. 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[ "intro p", "intro ab", "intro ac", "intro bb", "intro bc", "intro cb", "intro cc", "intro L", "intro db", "intro dc", "intro M", "intro ub", "intro uc", "intro vb", "intro vc", "intro wb", "intro wc", "intro N", "intro hs", "intro hu", "intro hv", "intro hw", "specialize prime_field_polynomial_subtract_from_add (p)", "specialize prime_field_polynomial_subtract_from_add (ub)", "specialize prime_field_polynomial_subtract_from_add (uc)", "specialize prime_field_polynomial_subtract_from_add (vb)", "specialize prime_field_polynomial_subtract_from_add (vc)", "specialize prime_field_polynomial_subtract_from_add (wb)", "specialize prime_field_polynomial_subtract_from_add (wc)", "specialize prime_field_polynomial_subtract_from_add (N)", "apply prime_field_polynomial_subtract_from_add", "specialize prime_field_convolution_prefix_right_add (p)", "specialize prime_field_convolution_prefix_right_add (bb)", "specialize prime_field_convolution_prefix_right_add (bc)", "specialize prime_field_convolution_prefix_right_add (cb)", "specialize prime_field_convolution_prefix_right_add (cc)", "specialize prime_field_convolution_prefix_right_add (ab)", "specialize prime_field_convolution_prefix_right_add (ac)", "specialize prime_field_convolution_prefix_right_add (L)", "specialize prime_field_convolution_prefix_right_add (db)", "specialize prime_field_convolution_prefix_right_add (dc)", "specialize prime_field_convolution_prefix_right_add (M)", "specialize prime_field_convolution_prefix_right_add (vb)", "specialize prime_field_convolution_prefix_right_add (vc)", "specialize prime_field_convolution_prefix_right_add (wb)", "specialize prime_field_convolution_prefix_right_add (wc)", "specialize prime_field_convolution_prefix_right_add (ub)", "specialize prime_field_convolution_prefix_right_add (uc)", "specialize prime_field_convolution_prefix_right_add (N)", "apply prime_field_convolution_prefix_right_add", "specialize prime_field_polynomial_subtract_recover_add (p)", "specialize prime_field_polynomial_subtract_recover_add (ab)", "specialize prime_field_polynomial_subtract_recover_add (ac)", "specialize prime_field_polynomial_subtract_recover_add (bb)", "specialize prime_field_polynomial_subtract_recover_add (bc)", "specialize prime_field_polynomial_subtract_recover_add (cb)", "specialize prime_field_polynomial_subtract_recover_add (cc)", "specialize prime_field_polynomial_subtract_recover_add (L)", "apply prime_field_polynomial_subtract_recover_add", "exact hs", "exact hv", "exact hw", "exact hu" ], "script_sha256": "ddd70dce6dcc31799906da15b64813fbcc167eb5a9012cf6f921349ecc37b822", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_distributivity_candidate.py", "sha256": "a959962d631759cd1fc773dd7eef2fadf4f3f95361d6d7bc8c6a9e82d0d4ab86" }, "statement": "forall p ab ac bb bc cb cc L db dc M ub uc vb vc wb wc N. (forall pfs_index_prefix_subtract_right_source. (exists pfa_gap_prefix_subtract_right_sourceindex. pfa_gap_prefix_subtract_right_sourceindex + S (pfs_index_prefix_subtract_right_source) = (L)) -> exists pfs_left_prefix_subtract_right_source pfs_right_prefix_subtract_right_source pfs_result_prefix_subtract_right_source. ((((exists ff_h_pfp_prefix_subtract_right_sourceleft. ff_h_pfp_prefix_subtract_right_sourceleft + S (pfs_left_prefix_subtract_right_source) = S ((S (pfs_index_prefix_subtract_right_source)) * ac)) /\\ exists ff_q_pfp_prefix_subtract_right_sourceleft. ab = ff_q_pfp_prefix_subtract_right_sourceleft * S ((S (pfs_index_prefix_subtract_right_source)) * ac) + (pfs_left_prefix_subtract_right_source))) /\\ (((((exists ff_h_pfp_prefix_subtract_right_sourceright. ff_h_pfp_prefix_subtract_right_sourceright + S (pfs_right_prefix_subtract_right_source) = S ((S (pfs_index_prefix_subtract_right_source)) * bc)) /\\ exists ff_q_pfp_prefix_subtract_right_sourceright. bb = ff_q_pfp_prefix_subtract_right_sourceright * S ((S (pfs_index_prefix_subtract_right_source)) * bc) + (pfs_right_prefix_subtract_right_source))) /\\ (((((exists ff_h_pfp_prefix_subtract_right_sourceresult. ff_h_pfp_prefix_subtract_right_sourceresult + S (pfs_result_prefix_subtract_right_source) = S ((S (pfs_index_prefix_subtract_right_source)) * cc)) /\\ exists ff_q_pfp_prefix_subtract_right_sourceresult. cb = ff_q_pfp_prefix_subtract_right_sourceresult * S ((S (pfs_index_prefix_subtract_right_source)) * cc) + (pfs_result_prefix_subtract_right_source))) /\\ ((((exists pfa_gap_prefix_subtract_right_sourceoperationleft. pfa_gap_prefix_subtract_right_sourceoperationleft + S (pfs_right_prefix_subtract_right_source) = (p)) /\\ (((exists pfa_gap_prefix_subtract_right_sourceoperationright. pfa_gap_prefix_subtract_right_sourceoperationright + S (pfs_result_prefix_subtract_right_source) = (p)) /\\ ((((exists pfa_gap_prefix_subtract_right_sourceoperationresultbound. pfa_gap_prefix_subtract_right_sourceoperationresultbound + S (pfs_left_prefix_subtract_right_source) = (p)) /\\ ((exists pfa_offset_left_prefix_subtract_right_sourceoperationresultcongruence pfa_offset_right_prefix_subtract_right_sourceoperationresultcongruence. ((pfs_right_prefix_subtract_right_source) + (pfs_result_prefix_subtract_right_source)) + (p) * pfa_offset_left_prefix_subtract_right_sourceoperationresultcongruence = (pfs_left_prefix_subtract_right_source) + (p) * pfa_offset_right_prefix_subtract_right_sourceoperationresultcongruence)))))))))))))))) -> (forall pfc_index_prefix_subtract_right_ub. (exists pfa_gap_prefix_subtract_right_ubbound. pfa_gap_prefix_subtract_right_ubbound + S (pfc_index_prefix_subtract_right_ub) = (N)) -> exists pfc_value_prefix_subtract_right_ub. ((((exists ff_h_pfp_prefix_subtract_right_ubentry. ff_h_pfp_prefix_subtract_right_ubentry + S (pfc_value_prefix_subtract_right_ub) = S ((S (pfc_index_prefix_subtract_right_ub)) * uc)) /\\ exists ff_q_pfp_prefix_subtract_right_ubentry. ub = ff_q_pfp_prefix_subtract_right_ubentry * S ((S (pfc_index_prefix_subtract_right_ub)) * uc) + (pfc_value_prefix_subtract_right_ub))) /\\ ((exists pfc_terms_code_prefix_subtract_right_ubcoefficient pfc_terms_scale_prefix_subtract_right_ubcoefficient pfc_natural_sum_prefix_subtract_right_ubcoefficient. ((forall pfc_index_prefix_subtract_right_ubcoefficientdiagonal. (exists pfa_gap_prefix_subtract_right_ubcoefficientdiagonalbound. pfa_gap_prefix_subtract_right_ubcoefficientdiagonalbound + S (pfc_index_prefix_subtract_right_ubcoefficientdiagonal) = (S (pfc_index_prefix_subtract_right_ub))) -> exists pfc_value_prefix_subtract_right_ubcoefficientdiagonal. ((((exists ff_h_pfp_prefix_subtract_right_ubcoefficientdiagonalentry. ff_h_pfp_prefix_subtract_right_ubcoefficientdiagonalentry + S (pfc_value_prefix_subtract_right_ubcoefficientdiagonal) = S ((S (pfc_index_prefix_subtract_right_ubcoefficientdiagonal)) * pfc_terms_scale_prefix_subtract_right_ubcoefficient)) /\\ exists ff_q_pfp_prefix_subtract_right_ubcoefficientdiagonalentry. pfc_terms_code_prefix_subtract_right_ubcoefficient = ff_q_pfp_prefix_subtract_right_ubcoefficientdiagonalentry * S ((S (pfc_index_prefix_subtract_right_ubcoefficientdiagonal)) * pfc_terms_scale_prefix_subtract_right_ubcoefficient) + (pfc_value_prefix_subtract_right_ubcoefficientdiagonal))) /\\ ((exists pfc_complement_prefix_subtract_right_ubcoefficientdiagonalterm pfc_left_prefix_subtract_right_ubcoefficientdiagonalterm pfc_right_prefix_subtract_right_ubcoefficientdiagonalterm. (((pfc_index_prefix_subtract_right_ubcoefficientdiagonal)+pfc_complement_prefix_subtract_right_ubcoefficientdiagonalterm=(pfc_index_prefix_subtract_right_ub)) /\\ ((((((exists pfa_gap_prefix_subtract_right_ubcoefficientdiagonaltermleftinside. pfa_gap_prefix_subtract_right_ubcoefficientdiagonaltermleftinside + S (pfc_index_prefix_subtract_right_ubcoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_prefix_subtract_right_ubcoefficientdiagonaltermleftentry. ff_h_pfp_prefix_subtract_right_ubcoefficientdiagonaltermleftentry + S (pfc_left_prefix_subtract_right_ubcoefficientdiagonalterm) = S ((S (pfc_index_prefix_subtract_right_ubcoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_prefix_subtract_right_ubcoefficientdiagonaltermleftentry. ab = ff_q_pfp_prefix_subtract_right_ubcoefficientdiagonaltermleftentry * S ((S (pfc_index_prefix_subtract_right_ubcoefficientdiagonal)) * ac) + (pfc_left_prefix_subtract_right_ubcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_subtract_right_ubcoefficientdiagonaltermleftoutside. pfc_gap_prefix_subtract_right_ubcoefficientdiagonaltermleftoutside+(L)=(pfc_index_prefix_subtract_right_ubcoefficientdiagonal)) /\\ (((pfc_left_prefix_subtract_right_ubcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_prefix_subtract_right_ubcoefficientdiagonaltermrightinside. pfa_gap_prefix_subtract_right_ubcoefficientdiagonaltermrightinside + S (pfc_complement_prefix_subtract_right_ubcoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_prefix_subtract_right_ubcoefficientdiagonaltermrightentry. ff_h_pfp_prefix_subtract_right_ubcoefficientdiagonaltermrightentry + S (pfc_right_prefix_subtract_right_ubcoefficientdiagonalterm) = S ((S (pfc_complement_prefix_subtract_right_ubcoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_prefix_subtract_right_ubcoefficientdiagonaltermrightentry. db = ff_q_pfp_prefix_subtract_right_ubcoefficientdiagonaltermrightentry * S ((S (pfc_complement_prefix_subtract_right_ubcoefficientdiagonalterm)) * dc) + (pfc_right_prefix_subtract_right_ubcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_subtract_right_ubcoefficientdiagonaltermrightoutside. pfc_gap_prefix_subtract_right_ubcoefficientdiagonaltermrightoutside+(M)=(pfc_complement_prefix_subtract_right_ubcoefficientdiagonalterm)) /\\ (((pfc_right_prefix_subtract_right_ubcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_prefix_subtract_right_ubcoefficientdiagonal)=pfc_left_prefix_subtract_right_ubcoefficientdiagonalterm*pfc_right_prefix_subtract_right_ubcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_prefix_subtract_right_ubcoefficientsum fs_v_pfc_prefix_subtract_right_ubcoefficientsum. ((((exists fs_h_pfc_prefix_subtract_right_ubcoefficientsum_body_start. fs_h_pfc_prefix_subtract_right_ubcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_prefix_subtract_right_ubcoefficientsum)) /\\ exists fs_q_pfc_prefix_subtract_right_ubcoefficientsum_body_start. fs_u_pfc_prefix_subtract_right_ubcoefficientsum = fs_q_pfc_prefix_subtract_right_ubcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_prefix_subtract_right_ubcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_prefix_subtract_right_ubcoefficientsum_body_terminal. fs_h_pfc_prefix_subtract_right_ubcoefficientsum_body_terminal + S (pfc_natural_sum_prefix_subtract_right_ubcoefficient) = S ((S (S (pfc_index_prefix_subtract_right_ub))) * fs_v_pfc_prefix_subtract_right_ubcoefficientsum)) /\\ exists fs_q_pfc_prefix_subtract_right_ubcoefficientsum_body_terminal. fs_u_pfc_prefix_subtract_right_ubcoefficientsum = fs_q_pfc_prefix_subtract_right_ubcoefficientsum_body_terminal * S ((S (S (pfc_index_prefix_subtract_right_ub))) * fs_v_pfc_prefix_subtract_right_ubcoefficientsum) + (pfc_natural_sum_prefix_subtract_right_ubcoefficient))) /\\ forall fs_i_pfc_prefix_subtract_right_ubcoefficientsum_body_steps. (exists fs_lt_pfc_prefix_subtract_right_ubcoefficientsum_body_steps_bound. fs_lt_pfc_prefix_subtract_right_ubcoefficientsum_body_steps_bound + S fs_i_pfc_prefix_subtract_right_ubcoefficientsum_body_steps = S (pfc_index_prefix_subtract_right_ub)) -> exists fs_a_pfc_prefix_subtract_right_ubcoefficientsum_body_steps fs_r_pfc_prefix_subtract_right_ubcoefficientsum_body_steps fs_s_pfc_prefix_subtract_right_ubcoefficientsum_body_steps. ((((exists fs_h_pfc_prefix_subtract_right_ubcoefficientsum_body_steps_summand. fs_h_pfc_prefix_subtract_right_ubcoefficientsum_body_steps_summand + S (fs_a_pfc_prefix_subtract_right_ubcoefficientsum_body_steps) = S ((S (fs_i_pfc_prefix_subtract_right_ubcoefficientsum_body_steps)) * pfc_terms_scale_prefix_subtract_right_ubcoefficient)) /\\ exists fs_q_pfc_prefix_subtract_right_ubcoefficientsum_body_steps_summand. pfc_terms_code_prefix_subtract_right_ubcoefficient = fs_q_pfc_prefix_subtract_right_ubcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_prefix_subtract_right_ubcoefficientsum_body_steps)) * pfc_terms_scale_prefix_subtract_right_ubcoefficient) + (fs_a_pfc_prefix_subtract_right_ubcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_subtract_right_ubcoefficientsum_body_steps_partial. fs_h_pfc_prefix_subtract_right_ubcoefficientsum_body_steps_partial + S (fs_r_pfc_prefix_subtract_right_ubcoefficientsum_body_steps) = S ((S (fs_i_pfc_prefix_subtract_right_ubcoefficientsum_body_steps)) * fs_v_pfc_prefix_subtract_right_ubcoefficientsum)) /\\ exists fs_q_pfc_prefix_subtract_right_ubcoefficientsum_body_steps_partial. fs_u_pfc_prefix_subtract_right_ubcoefficientsum = fs_q_pfc_prefix_subtract_right_ubcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_prefix_subtract_right_ubcoefficientsum_body_steps)) * fs_v_pfc_prefix_subtract_right_ubcoefficientsum) + (fs_r_pfc_prefix_subtract_right_ubcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_subtract_right_ubcoefficientsum_body_steps_successor. fs_h_pfc_prefix_subtract_right_ubcoefficientsum_body_steps_successor + S (fs_s_pfc_prefix_subtract_right_ubcoefficientsum_body_steps) = S ((S (S fs_i_pfc_prefix_subtract_right_ubcoefficientsum_body_steps)) * fs_v_pfc_prefix_subtract_right_ubcoefficientsum)) /\\ exists fs_q_pfc_prefix_subtract_right_ubcoefficientsum_body_steps_successor. fs_u_pfc_prefix_subtract_right_ubcoefficientsum = fs_q_pfc_prefix_subtract_right_ubcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_prefix_subtract_right_ubcoefficientsum_body_steps)) * fs_v_pfc_prefix_subtract_right_ubcoefficientsum) + (fs_s_pfc_prefix_subtract_right_ubcoefficientsum_body_steps))) /\\ fs_s_pfc_prefix_subtract_right_ubcoefficientsum_body_steps = fs_r_pfc_prefix_subtract_right_ubcoefficientsum_body_steps + fs_a_pfc_prefix_subtract_right_ubcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_prefix_subtract_right_ubcoefficientresiduebound. pfa_gap_prefix_subtract_right_ubcoefficientresiduebound + S (pfc_value_prefix_subtract_right_ub) = (p)) /\\ ((exists pfa_offset_left_prefix_subtract_right_ubcoefficientresiduecongruence pfa_offset_right_prefix_subtract_right_ubcoefficientresiduecongruence. (pfc_natural_sum_prefix_subtract_right_ubcoefficient) + (p) * pfa_offset_left_prefix_subtract_right_ubcoefficientresiduecongruence = (pfc_value_prefix_subtract_right_ub) + (p) * pfa_offset_right_prefix_subtract_right_ubcoefficientresiduecongruence)))))))))))) -> (forall pfc_index_prefix_subtract_right_vb. (exists pfa_gap_prefix_subtract_right_vbbound. pfa_gap_prefix_subtract_right_vbbound + S (pfc_index_prefix_subtract_right_vb) = (N)) -> exists pfc_value_prefix_subtract_right_vb. ((((exists ff_h_pfp_prefix_subtract_right_vbentry. ff_h_pfp_prefix_subtract_right_vbentry + S (pfc_value_prefix_subtract_right_vb) = S ((S (pfc_index_prefix_subtract_right_vb)) * vc)) /\\ exists ff_q_pfp_prefix_subtract_right_vbentry. vb = ff_q_pfp_prefix_subtract_right_vbentry * S ((S (pfc_index_prefix_subtract_right_vb)) * vc) + (pfc_value_prefix_subtract_right_vb))) /\\ ((exists pfc_terms_code_prefix_subtract_right_vbcoefficient pfc_terms_scale_prefix_subtract_right_vbcoefficient pfc_natural_sum_prefix_subtract_right_vbcoefficient. ((forall pfc_index_prefix_subtract_right_vbcoefficientdiagonal. (exists pfa_gap_prefix_subtract_right_vbcoefficientdiagonalbound. pfa_gap_prefix_subtract_right_vbcoefficientdiagonalbound + S (pfc_index_prefix_subtract_right_vbcoefficientdiagonal) = (S (pfc_index_prefix_subtract_right_vb))) -> exists pfc_value_prefix_subtract_right_vbcoefficientdiagonal. ((((exists ff_h_pfp_prefix_subtract_right_vbcoefficientdiagonalentry. ff_h_pfp_prefix_subtract_right_vbcoefficientdiagonalentry + S (pfc_value_prefix_subtract_right_vbcoefficientdiagonal) = S ((S (pfc_index_prefix_subtract_right_vbcoefficientdiagonal)) * pfc_terms_scale_prefix_subtract_right_vbcoefficient)) /\\ exists ff_q_pfp_prefix_subtract_right_vbcoefficientdiagonalentry. pfc_terms_code_prefix_subtract_right_vbcoefficient = ff_q_pfp_prefix_subtract_right_vbcoefficientdiagonalentry * S ((S (pfc_index_prefix_subtract_right_vbcoefficientdiagonal)) * pfc_terms_scale_prefix_subtract_right_vbcoefficient) + (pfc_value_prefix_subtract_right_vbcoefficientdiagonal))) /\\ ((exists pfc_complement_prefix_subtract_right_vbcoefficientdiagonalterm pfc_left_prefix_subtract_right_vbcoefficientdiagonalterm pfc_right_prefix_subtract_right_vbcoefficientdiagonalterm. (((pfc_index_prefix_subtract_right_vbcoefficientdiagonal)+pfc_complement_prefix_subtract_right_vbcoefficientdiagonalterm=(pfc_index_prefix_subtract_right_vb)) /\\ ((((((exists pfa_gap_prefix_subtract_right_vbcoefficientdiagonaltermleftinside. pfa_gap_prefix_subtract_right_vbcoefficientdiagonaltermleftinside + S (pfc_index_prefix_subtract_right_vbcoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_prefix_subtract_right_vbcoefficientdiagonaltermleftentry. ff_h_pfp_prefix_subtract_right_vbcoefficientdiagonaltermleftentry + S (pfc_left_prefix_subtract_right_vbcoefficientdiagonalterm) = S ((S (pfc_index_prefix_subtract_right_vbcoefficientdiagonal)) * bc)) /\\ exists ff_q_pfp_prefix_subtract_right_vbcoefficientdiagonaltermleftentry. bb = ff_q_pfp_prefix_subtract_right_vbcoefficientdiagonaltermleftentry * S ((S (pfc_index_prefix_subtract_right_vbcoefficientdiagonal)) * bc) + (pfc_left_prefix_subtract_right_vbcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_subtract_right_vbcoefficientdiagonaltermleftoutside. pfc_gap_prefix_subtract_right_vbcoefficientdiagonaltermleftoutside+(L)=(pfc_index_prefix_subtract_right_vbcoefficientdiagonal)) /\\ (((pfc_left_prefix_subtract_right_vbcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_prefix_subtract_right_vbcoefficientdiagonaltermrightinside. pfa_gap_prefix_subtract_right_vbcoefficientdiagonaltermrightinside + S (pfc_complement_prefix_subtract_right_vbcoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_prefix_subtract_right_vbcoefficientdiagonaltermrightentry. ff_h_pfp_prefix_subtract_right_vbcoefficientdiagonaltermrightentry + S (pfc_right_prefix_subtract_right_vbcoefficientdiagonalterm) = S ((S (pfc_complement_prefix_subtract_right_vbcoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_prefix_subtract_right_vbcoefficientdiagonaltermrightentry. db = ff_q_pfp_prefix_subtract_right_vbcoefficientdiagonaltermrightentry * S ((S (pfc_complement_prefix_subtract_right_vbcoefficientdiagonalterm)) * dc) + (pfc_right_prefix_subtract_right_vbcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_subtract_right_vbcoefficientdiagonaltermrightoutside. pfc_gap_prefix_subtract_right_vbcoefficientdiagonaltermrightoutside+(M)=(pfc_complement_prefix_subtract_right_vbcoefficientdiagonalterm)) /\\ (((pfc_right_prefix_subtract_right_vbcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_prefix_subtract_right_vbcoefficientdiagonal)=pfc_left_prefix_subtract_right_vbcoefficientdiagonalterm*pfc_right_prefix_subtract_right_vbcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_prefix_subtract_right_vbcoefficientsum fs_v_pfc_prefix_subtract_right_vbcoefficientsum. ((((exists fs_h_pfc_prefix_subtract_right_vbcoefficientsum_body_start. fs_h_pfc_prefix_subtract_right_vbcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_prefix_subtract_right_vbcoefficientsum)) /\\ exists fs_q_pfc_prefix_subtract_right_vbcoefficientsum_body_start. fs_u_pfc_prefix_subtract_right_vbcoefficientsum = fs_q_pfc_prefix_subtract_right_vbcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_prefix_subtract_right_vbcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_prefix_subtract_right_vbcoefficientsum_body_terminal. fs_h_pfc_prefix_subtract_right_vbcoefficientsum_body_terminal + S (pfc_natural_sum_prefix_subtract_right_vbcoefficient) = S ((S (S (pfc_index_prefix_subtract_right_vb))) * fs_v_pfc_prefix_subtract_right_vbcoefficientsum)) /\\ exists fs_q_pfc_prefix_subtract_right_vbcoefficientsum_body_terminal. fs_u_pfc_prefix_subtract_right_vbcoefficientsum = fs_q_pfc_prefix_subtract_right_vbcoefficientsum_body_terminal * S ((S (S (pfc_index_prefix_subtract_right_vb))) * fs_v_pfc_prefix_subtract_right_vbcoefficientsum) + (pfc_natural_sum_prefix_subtract_right_vbcoefficient))) /\\ forall fs_i_pfc_prefix_subtract_right_vbcoefficientsum_body_steps. (exists fs_lt_pfc_prefix_subtract_right_vbcoefficientsum_body_steps_bound. fs_lt_pfc_prefix_subtract_right_vbcoefficientsum_body_steps_bound + S fs_i_pfc_prefix_subtract_right_vbcoefficientsum_body_steps = S (pfc_index_prefix_subtract_right_vb)) -> exists fs_a_pfc_prefix_subtract_right_vbcoefficientsum_body_steps fs_r_pfc_prefix_subtract_right_vbcoefficientsum_body_steps fs_s_pfc_prefix_subtract_right_vbcoefficientsum_body_steps. ((((exists fs_h_pfc_prefix_subtract_right_vbcoefficientsum_body_steps_summand. fs_h_pfc_prefix_subtract_right_vbcoefficientsum_body_steps_summand + S (fs_a_pfc_prefix_subtract_right_vbcoefficientsum_body_steps) = S ((S (fs_i_pfc_prefix_subtract_right_vbcoefficientsum_body_steps)) * pfc_terms_scale_prefix_subtract_right_vbcoefficient)) /\\ exists fs_q_pfc_prefix_subtract_right_vbcoefficientsum_body_steps_summand. pfc_terms_code_prefix_subtract_right_vbcoefficient = fs_q_pfc_prefix_subtract_right_vbcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_prefix_subtract_right_vbcoefficientsum_body_steps)) * pfc_terms_scale_prefix_subtract_right_vbcoefficient) + (fs_a_pfc_prefix_subtract_right_vbcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_subtract_right_vbcoefficientsum_body_steps_partial. fs_h_pfc_prefix_subtract_right_vbcoefficientsum_body_steps_partial + S (fs_r_pfc_prefix_subtract_right_vbcoefficientsum_body_steps) = S ((S (fs_i_pfc_prefix_subtract_right_vbcoefficientsum_body_steps)) * fs_v_pfc_prefix_subtract_right_vbcoefficientsum)) /\\ exists fs_q_pfc_prefix_subtract_right_vbcoefficientsum_body_steps_partial. fs_u_pfc_prefix_subtract_right_vbcoefficientsum = fs_q_pfc_prefix_subtract_right_vbcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_prefix_subtract_right_vbcoefficientsum_body_steps)) * fs_v_pfc_prefix_subtract_right_vbcoefficientsum) + (fs_r_pfc_prefix_subtract_right_vbcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_subtract_right_vbcoefficientsum_body_steps_successor. fs_h_pfc_prefix_subtract_right_vbcoefficientsum_body_steps_successor + S (fs_s_pfc_prefix_subtract_right_vbcoefficientsum_body_steps) = S ((S (S fs_i_pfc_prefix_subtract_right_vbcoefficientsum_body_steps)) * fs_v_pfc_prefix_subtract_right_vbcoefficientsum)) /\\ exists fs_q_pfc_prefix_subtract_right_vbcoefficientsum_body_steps_successor. fs_u_pfc_prefix_subtract_right_vbcoefficientsum = fs_q_pfc_prefix_subtract_right_vbcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_prefix_subtract_right_vbcoefficientsum_body_steps)) * fs_v_pfc_prefix_subtract_right_vbcoefficientsum) + (fs_s_pfc_prefix_subtract_right_vbcoefficientsum_body_steps))) /\\ fs_s_pfc_prefix_subtract_right_vbcoefficientsum_body_steps = fs_r_pfc_prefix_subtract_right_vbcoefficientsum_body_steps + fs_a_pfc_prefix_subtract_right_vbcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_prefix_subtract_right_vbcoefficientresiduebound. pfa_gap_prefix_subtract_right_vbcoefficientresiduebound + S (pfc_value_prefix_subtract_right_vb) = (p)) /\\ ((exists pfa_offset_left_prefix_subtract_right_vbcoefficientresiduecongruence pfa_offset_right_prefix_subtract_right_vbcoefficientresiduecongruence. (pfc_natural_sum_prefix_subtract_right_vbcoefficient) + (p) * pfa_offset_left_prefix_subtract_right_vbcoefficientresiduecongruence = (pfc_value_prefix_subtract_right_vb) + (p) * pfa_offset_right_prefix_subtract_right_vbcoefficientresiduecongruence)))))))))))) -> (forall pfc_index_prefix_subtract_right_wb. (exists pfa_gap_prefix_subtract_right_wbbound. pfa_gap_prefix_subtract_right_wbbound + S (pfc_index_prefix_subtract_right_wb) = (N)) -> exists pfc_value_prefix_subtract_right_wb. ((((exists ff_h_pfp_prefix_subtract_right_wbentry. ff_h_pfp_prefix_subtract_right_wbentry + S (pfc_value_prefix_subtract_right_wb) = S ((S (pfc_index_prefix_subtract_right_wb)) * wc)) /\\ exists ff_q_pfp_prefix_subtract_right_wbentry. wb = ff_q_pfp_prefix_subtract_right_wbentry * S ((S (pfc_index_prefix_subtract_right_wb)) * wc) + (pfc_value_prefix_subtract_right_wb))) /\\ ((exists pfc_terms_code_prefix_subtract_right_wbcoefficient pfc_terms_scale_prefix_subtract_right_wbcoefficient pfc_natural_sum_prefix_subtract_right_wbcoefficient. ((forall pfc_index_prefix_subtract_right_wbcoefficientdiagonal. (exists pfa_gap_prefix_subtract_right_wbcoefficientdiagonalbound. pfa_gap_prefix_subtract_right_wbcoefficientdiagonalbound + S (pfc_index_prefix_subtract_right_wbcoefficientdiagonal) = (S (pfc_index_prefix_subtract_right_wb))) -> exists pfc_value_prefix_subtract_right_wbcoefficientdiagonal. ((((exists ff_h_pfp_prefix_subtract_right_wbcoefficientdiagonalentry. ff_h_pfp_prefix_subtract_right_wbcoefficientdiagonalentry + S (pfc_value_prefix_subtract_right_wbcoefficientdiagonal) = S ((S (pfc_index_prefix_subtract_right_wbcoefficientdiagonal)) * pfc_terms_scale_prefix_subtract_right_wbcoefficient)) /\\ exists ff_q_pfp_prefix_subtract_right_wbcoefficientdiagonalentry. pfc_terms_code_prefix_subtract_right_wbcoefficient = ff_q_pfp_prefix_subtract_right_wbcoefficientdiagonalentry * S ((S (pfc_index_prefix_subtract_right_wbcoefficientdiagonal)) * pfc_terms_scale_prefix_subtract_right_wbcoefficient) + (pfc_value_prefix_subtract_right_wbcoefficientdiagonal))) /\\ ((exists pfc_complement_prefix_subtract_right_wbcoefficientdiagonalterm pfc_left_prefix_subtract_right_wbcoefficientdiagonalterm pfc_right_prefix_subtract_right_wbcoefficientdiagonalterm. (((pfc_index_prefix_subtract_right_wbcoefficientdiagonal)+pfc_complement_prefix_subtract_right_wbcoefficientdiagonalterm=(pfc_index_prefix_subtract_right_wb)) /\\ ((((((exists pfa_gap_prefix_subtract_right_wbcoefficientdiagonaltermleftinside. pfa_gap_prefix_subtract_right_wbcoefficientdiagonaltermleftinside + S (pfc_index_prefix_subtract_right_wbcoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_prefix_subtract_right_wbcoefficientdiagonaltermleftentry. ff_h_pfp_prefix_subtract_right_wbcoefficientdiagonaltermleftentry + S (pfc_left_prefix_subtract_right_wbcoefficientdiagonalterm) = S ((S (pfc_index_prefix_subtract_right_wbcoefficientdiagonal)) * cc)) /\\ exists ff_q_pfp_prefix_subtract_right_wbcoefficientdiagonaltermleftentry. cb = ff_q_pfp_prefix_subtract_right_wbcoefficientdiagonaltermleftentry * S ((S (pfc_index_prefix_subtract_right_wbcoefficientdiagonal)) * cc) + (pfc_left_prefix_subtract_right_wbcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_subtract_right_wbcoefficientdiagonaltermleftoutside. pfc_gap_prefix_subtract_right_wbcoefficientdiagonaltermleftoutside+(L)=(pfc_index_prefix_subtract_right_wbcoefficientdiagonal)) /\\ (((pfc_left_prefix_subtract_right_wbcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_prefix_subtract_right_wbcoefficientdiagonaltermrightinside. pfa_gap_prefix_subtract_right_wbcoefficientdiagonaltermrightinside + S (pfc_complement_prefix_subtract_right_wbcoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_prefix_subtract_right_wbcoefficientdiagonaltermrightentry. ff_h_pfp_prefix_subtract_right_wbcoefficientdiagonaltermrightentry + S (pfc_right_prefix_subtract_right_wbcoefficientdiagonalterm) = S ((S (pfc_complement_prefix_subtract_right_wbcoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_prefix_subtract_right_wbcoefficientdiagonaltermrightentry. db = ff_q_pfp_prefix_subtract_right_wbcoefficientdiagonaltermrightentry * S ((S (pfc_complement_prefix_subtract_right_wbcoefficientdiagonalterm)) * dc) + (pfc_right_prefix_subtract_right_wbcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_subtract_right_wbcoefficientdiagonaltermrightoutside. pfc_gap_prefix_subtract_right_wbcoefficientdiagonaltermrightoutside+(M)=(pfc_complement_prefix_subtract_right_wbcoefficientdiagonalterm)) /\\ (((pfc_right_prefix_subtract_right_wbcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_prefix_subtract_right_wbcoefficientdiagonal)=pfc_left_prefix_subtract_right_wbcoefficientdiagonalterm*pfc_right_prefix_subtract_right_wbcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_prefix_subtract_right_wbcoefficientsum fs_v_pfc_prefix_subtract_right_wbcoefficientsum. ((((exists fs_h_pfc_prefix_subtract_right_wbcoefficientsum_body_start. fs_h_pfc_prefix_subtract_right_wbcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_prefix_subtract_right_wbcoefficientsum)) /\\ exists fs_q_pfc_prefix_subtract_right_wbcoefficientsum_body_start. fs_u_pfc_prefix_subtract_right_wbcoefficientsum = fs_q_pfc_prefix_subtract_right_wbcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_prefix_subtract_right_wbcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_prefix_subtract_right_wbcoefficientsum_body_terminal. fs_h_pfc_prefix_subtract_right_wbcoefficientsum_body_terminal + S (pfc_natural_sum_prefix_subtract_right_wbcoefficient) = S ((S (S (pfc_index_prefix_subtract_right_wb))) * fs_v_pfc_prefix_subtract_right_wbcoefficientsum)) /\\ exists fs_q_pfc_prefix_subtract_right_wbcoefficientsum_body_terminal. fs_u_pfc_prefix_subtract_right_wbcoefficientsum = fs_q_pfc_prefix_subtract_right_wbcoefficientsum_body_terminal * S ((S (S (pfc_index_prefix_subtract_right_wb))) * fs_v_pfc_prefix_subtract_right_wbcoefficientsum) + (pfc_natural_sum_prefix_subtract_right_wbcoefficient))) /\\ forall fs_i_pfc_prefix_subtract_right_wbcoefficientsum_body_steps. (exists fs_lt_pfc_prefix_subtract_right_wbcoefficientsum_body_steps_bound. fs_lt_pfc_prefix_subtract_right_wbcoefficientsum_body_steps_bound + S fs_i_pfc_prefix_subtract_right_wbcoefficientsum_body_steps = S (pfc_index_prefix_subtract_right_wb)) -> exists fs_a_pfc_prefix_subtract_right_wbcoefficientsum_body_steps fs_r_pfc_prefix_subtract_right_wbcoefficientsum_body_steps fs_s_pfc_prefix_subtract_right_wbcoefficientsum_body_steps. ((((exists fs_h_pfc_prefix_subtract_right_wbcoefficientsum_body_steps_summand. fs_h_pfc_prefix_subtract_right_wbcoefficientsum_body_steps_summand + S (fs_a_pfc_prefix_subtract_right_wbcoefficientsum_body_steps) = S ((S (fs_i_pfc_prefix_subtract_right_wbcoefficientsum_body_steps)) * pfc_terms_scale_prefix_subtract_right_wbcoefficient)) /\\ exists fs_q_pfc_prefix_subtract_right_wbcoefficientsum_body_steps_summand. pfc_terms_code_prefix_subtract_right_wbcoefficient = fs_q_pfc_prefix_subtract_right_wbcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_prefix_subtract_right_wbcoefficientsum_body_steps)) * pfc_terms_scale_prefix_subtract_right_wbcoefficient) + (fs_a_pfc_prefix_subtract_right_wbcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_subtract_right_wbcoefficientsum_body_steps_partial. fs_h_pfc_prefix_subtract_right_wbcoefficientsum_body_steps_partial + S (fs_r_pfc_prefix_subtract_right_wbcoefficientsum_body_steps) = S ((S (fs_i_pfc_prefix_subtract_right_wbcoefficientsum_body_steps)) * fs_v_pfc_prefix_subtract_right_wbcoefficientsum)) /\\ exists fs_q_pfc_prefix_subtract_right_wbcoefficientsum_body_steps_partial. fs_u_pfc_prefix_subtract_right_wbcoefficientsum = fs_q_pfc_prefix_subtract_right_wbcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_prefix_subtract_right_wbcoefficientsum_body_steps)) * fs_v_pfc_prefix_subtract_right_wbcoefficientsum) + (fs_r_pfc_prefix_subtract_right_wbcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_subtract_right_wbcoefficientsum_body_steps_successor. fs_h_pfc_prefix_subtract_right_wbcoefficientsum_body_steps_successor + S (fs_s_pfc_prefix_subtract_right_wbcoefficientsum_body_steps) = S ((S (S fs_i_pfc_prefix_subtract_right_wbcoefficientsum_body_steps)) * fs_v_pfc_prefix_subtract_right_wbcoefficientsum)) /\\ exists fs_q_pfc_prefix_subtract_right_wbcoefficientsum_body_steps_successor. fs_u_pfc_prefix_subtract_right_wbcoefficientsum = fs_q_pfc_prefix_subtract_right_wbcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_prefix_subtract_right_wbcoefficientsum_body_steps)) * fs_v_pfc_prefix_subtract_right_wbcoefficientsum) + (fs_s_pfc_prefix_subtract_right_wbcoefficientsum_body_steps))) /\\ fs_s_pfc_prefix_subtract_right_wbcoefficientsum_body_steps = fs_r_pfc_prefix_subtract_right_wbcoefficientsum_body_steps + fs_a_pfc_prefix_subtract_right_wbcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_prefix_subtract_right_wbcoefficientresiduebound. pfa_gap_prefix_subtract_right_wbcoefficientresiduebound + S (pfc_value_prefix_subtract_right_wb) = (p)) /\\ ((exists pfa_offset_left_prefix_subtract_right_wbcoefficientresiduecongruence pfa_offset_right_prefix_subtract_right_wbcoefficientresiduecongruence. (pfc_natural_sum_prefix_subtract_right_wbcoefficient) + (p) * pfa_offset_left_prefix_subtract_right_wbcoefficientresiduecongruence = (pfc_value_prefix_subtract_right_wb) + (p) * pfa_offset_right_prefix_subtract_right_wbcoefficientresiduecongruence)))))))))))) -> (forall pfs_index_prefix_subtract_right_result. (exists pfa_gap_prefix_subtract_right_resultindex. pfa_gap_prefix_subtract_right_resultindex + S (pfs_index_prefix_subtract_right_result) = (N)) -> exists pfs_left_prefix_subtract_right_result pfs_right_prefix_subtract_right_result pfs_result_prefix_subtract_right_result. ((((exists ff_h_pfp_prefix_subtract_right_resultleft. ff_h_pfp_prefix_subtract_right_resultleft + S (pfs_left_prefix_subtract_right_result) = S ((S (pfs_index_prefix_subtract_right_result)) * uc)) /\\ exists ff_q_pfp_prefix_subtract_right_resultleft. ub = ff_q_pfp_prefix_subtract_right_resultleft * S ((S (pfs_index_prefix_subtract_right_result)) * uc) + (pfs_left_prefix_subtract_right_result))) /\\ (((((exists ff_h_pfp_prefix_subtract_right_resultright. ff_h_pfp_prefix_subtract_right_resultright + S (pfs_right_prefix_subtract_right_result) = S ((S (pfs_index_prefix_subtract_right_result)) * vc)) /\\ exists ff_q_pfp_prefix_subtract_right_resultright. vb = ff_q_pfp_prefix_subtract_right_resultright * S ((S (pfs_index_prefix_subtract_right_result)) * vc) + (pfs_right_prefix_subtract_right_result))) /\\ (((((exists ff_h_pfp_prefix_subtract_right_resultresult. ff_h_pfp_prefix_subtract_right_resultresult + S (pfs_result_prefix_subtract_right_result) = S ((S (pfs_index_prefix_subtract_right_result)) * wc)) /\\ exists ff_q_pfp_prefix_subtract_right_resultresult. wb = ff_q_pfp_prefix_subtract_right_resultresult * S ((S (pfs_index_prefix_subtract_right_result)) * wc) + (pfs_result_prefix_subtract_right_result))) /\\ ((((exists pfa_gap_prefix_subtract_right_resultoperationleft. pfa_gap_prefix_subtract_right_resultoperationleft + S (pfs_right_prefix_subtract_right_result) = (p)) /\\ (((exists pfa_gap_prefix_subtract_right_resultoperationright. pfa_gap_prefix_subtract_right_resultoperationright + S (pfs_result_prefix_subtract_right_result) = (p)) /\\ ((((exists pfa_gap_prefix_subtract_right_resultoperationresultbound. pfa_gap_prefix_subtract_right_resultoperationresultbound + S (pfs_left_prefix_subtract_right_result) = (p)) /\\ ((exists pfa_offset_left_prefix_subtract_right_resultoperationresultcongruence pfa_offset_right_prefix_subtract_right_resultoperationresultcongruence. 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(forall pfp_index_proper_left_add_source. (exists pfa_gap_proper_left_add_sourceindex. pfa_gap_proper_left_add_sourceindex + S (pfp_index_proper_left_add_source) = (L)) -> exists pfp_left_proper_left_add_source pfp_right_proper_left_add_source pfp_value_proper_left_add_source. ((((exists ff_h_pfp_proper_left_add_sourceleft. ff_h_pfp_proper_left_add_sourceleft + S (pfp_left_proper_left_add_source) = S ((S (pfp_index_proper_left_add_source)) * ac)) /\\ exists ff_q_pfp_proper_left_add_sourceleft. ab = ff_q_pfp_proper_left_add_sourceleft * S ((S (pfp_index_proper_left_add_source)) * ac) + (pfp_left_proper_left_add_source))) /\\ (((((exists ff_h_pfp_proper_left_add_sourceright. ff_h_pfp_proper_left_add_sourceright + S (pfp_right_proper_left_add_source) = S ((S (pfp_index_proper_left_add_source)) * bc)) /\\ exists ff_q_pfp_proper_left_add_sourceright. bb = ff_q_pfp_proper_left_add_sourceright * S ((S (pfp_index_proper_left_add_source)) * bc) + (pfp_right_proper_left_add_source))) /\\ (((((exists ff_h_pfp_proper_left_add_sourcetarget. ff_h_pfp_proper_left_add_sourcetarget + S (pfp_value_proper_left_add_source) = S ((S (pfp_index_proper_left_add_source)) * cc)) /\\ exists ff_q_pfp_proper_left_add_sourcetarget. cb = ff_q_pfp_proper_left_add_sourcetarget * S ((S (pfp_index_proper_left_add_source)) * cc) + (pfp_value_proper_left_add_source))) /\\ ((((exists pfa_gap_proper_left_add_sourceoperationleft. pfa_gap_proper_left_add_sourceoperationleft + S (pfp_left_proper_left_add_source) = (p)) /\\ (((exists pfa_gap_proper_left_add_sourceoperationright. pfa_gap_proper_left_add_sourceoperationright + S (pfp_right_proper_left_add_source) = (p)) /\\ ((((exists pfa_gap_proper_left_add_sourceoperationresultbound. pfa_gap_proper_left_add_sourceoperationresultbound + S (pfp_value_proper_left_add_source) = (p)) /\\ ((exists pfa_offset_left_proper_left_add_sourceoperationresultcongruence pfa_offset_right_proper_left_add_sourceoperationresultcongruence. ((pfp_left_proper_left_add_source) + (pfp_right_proper_left_add_source)) + (p) * pfa_offset_left_proper_left_add_sourceoperationresultcongruence = (pfp_value_proper_left_add_source) + (p) * pfa_offset_right_proper_left_add_sourceoperationresultcongruence)))))))))))))))) -> (((forall fom_index_pfp_proper_left_add_ubleft. (exists fom_gap_pfp_proper_left_add_ubleft_index_bound. fom_gap_pfp_proper_left_add_ubleft_index_bound + S (fom_index_pfp_proper_left_add_ubleft) = M) -> exists fom_value_pfp_proper_left_add_ubleft. ((((exists fom_beta_height_pfp_proper_left_add_ubleft_entry. fom_beta_height_pfp_proper_left_add_ubleft_entry + S (fom_value_pfp_proper_left_add_ubleft) = S ((S (fom_index_pfp_proper_left_add_ubleft)) * dc)) /\\ exists fom_beta_quotient_pfp_proper_left_add_ubleft_entry. db = fom_beta_quotient_pfp_proper_left_add_ubleft_entry * S ((S (fom_index_pfp_proper_left_add_ubleft)) * dc) + (fom_value_pfp_proper_left_add_ubleft))) /\\ (exists fom_gap_pfp_proper_left_add_ubleft_value_bound. fom_gap_pfp_proper_left_add_ubleft_value_bound + S (fom_value_pfp_proper_left_add_ubleft) = p))) /\\ (((forall fom_index_pfp_proper_left_add_ubright. (exists fom_gap_pfp_proper_left_add_ubright_index_bound. fom_gap_pfp_proper_left_add_ubright_index_bound + S (fom_index_pfp_proper_left_add_ubright) = L) -> exists fom_value_pfp_proper_left_add_ubright. ((((exists fom_beta_height_pfp_proper_left_add_ubright_entry. fom_beta_height_pfp_proper_left_add_ubright_entry + S (fom_value_pfp_proper_left_add_ubright) = S ((S (fom_index_pfp_proper_left_add_ubright)) * ac)) /\\ exists fom_beta_quotient_pfp_proper_left_add_ubright_entry. ab = fom_beta_quotient_pfp_proper_left_add_ubright_entry * S ((S (fom_index_pfp_proper_left_add_ubright)) * ac) + (fom_value_pfp_proper_left_add_ubright))) /\\ (exists fom_gap_pfp_proper_left_add_ubright_value_bound. fom_gap_pfp_proper_left_add_ubright_value_bound + S (fom_value_pfp_proper_left_add_ubright) = p))) /\\ (((((((M)=0 \\/ (L)=0) /\\ (((N)=0)))) \\/ (((~((M)=0)) /\\ (((~((L)=0)) /\\ (((M)+(L)=S (N)))))))) /\\ ((forall pfc_index_proper_left_add_ubcoefficients. (exists pfa_gap_proper_left_add_ubcoefficientsbound. pfa_gap_proper_left_add_ubcoefficientsbound + S (pfc_index_proper_left_add_ubcoefficients) = (N)) -> exists pfc_value_proper_left_add_ubcoefficients. ((((exists ff_h_pfp_proper_left_add_ubcoefficientsentry. ff_h_pfp_proper_left_add_ubcoefficientsentry + S (pfc_value_proper_left_add_ubcoefficients) = S ((S (pfc_index_proper_left_add_ubcoefficients)) * uc)) /\\ exists ff_q_pfp_proper_left_add_ubcoefficientsentry. ub = ff_q_pfp_proper_left_add_ubcoefficientsentry * S ((S (pfc_index_proper_left_add_ubcoefficients)) * uc) + (pfc_value_proper_left_add_ubcoefficients))) /\\ ((exists pfc_terms_code_proper_left_add_ubcoefficientscoefficient pfc_terms_scale_proper_left_add_ubcoefficientscoefficient pfc_natural_sum_proper_left_add_ubcoefficientscoefficient. ((forall pfc_index_proper_left_add_ubcoefficientscoefficientdiagonal. (exists pfa_gap_proper_left_add_ubcoefficientscoefficientdiagonalbound. pfa_gap_proper_left_add_ubcoefficientscoefficientdiagonalbound + S (pfc_index_proper_left_add_ubcoefficientscoefficientdiagonal) = (S (pfc_index_proper_left_add_ubcoefficients))) -> exists pfc_value_proper_left_add_ubcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_proper_left_add_ubcoefficientscoefficientdiagonalentry. ff_h_pfp_proper_left_add_ubcoefficientscoefficientdiagonalentry + S (pfc_value_proper_left_add_ubcoefficientscoefficientdiagonal) = S ((S (pfc_index_proper_left_add_ubcoefficientscoefficientdiagonal)) * pfc_terms_scale_proper_left_add_ubcoefficientscoefficient)) /\\ exists ff_q_pfp_proper_left_add_ubcoefficientscoefficientdiagonalentry. pfc_terms_code_proper_left_add_ubcoefficientscoefficient = ff_q_pfp_proper_left_add_ubcoefficientscoefficientdiagonalentry * S ((S (pfc_index_proper_left_add_ubcoefficientscoefficientdiagonal)) * pfc_terms_scale_proper_left_add_ubcoefficientscoefficient) + (pfc_value_proper_left_add_ubcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_proper_left_add_ubcoefficientscoefficientdiagonalterm pfc_left_proper_left_add_ubcoefficientscoefficientdiagonalterm pfc_right_proper_left_add_ubcoefficientscoefficientdiagonalterm. (((pfc_index_proper_left_add_ubcoefficientscoefficientdiagonal)+pfc_complement_proper_left_add_ubcoefficientscoefficientdiagonalterm=(pfc_index_proper_left_add_ubcoefficients)) /\\ ((((((exists pfa_gap_proper_left_add_ubcoefficientscoefficientdiagonaltermleftinside. pfa_gap_proper_left_add_ubcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_proper_left_add_ubcoefficientscoefficientdiagonal) = (M)) /\\ ((((exists ff_h_pfp_proper_left_add_ubcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_proper_left_add_ubcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_proper_left_add_ubcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_proper_left_add_ubcoefficientscoefficientdiagonal)) * dc)) /\\ exists ff_q_pfp_proper_left_add_ubcoefficientscoefficientdiagonaltermleftentry. db = ff_q_pfp_proper_left_add_ubcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_proper_left_add_ubcoefficientscoefficientdiagonal)) * dc) + (pfc_left_proper_left_add_ubcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_proper_left_add_ubcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_proper_left_add_ubcoefficientscoefficientdiagonaltermleftoutside+(M)=(pfc_index_proper_left_add_ubcoefficientscoefficientdiagonal)) /\\ (((pfc_left_proper_left_add_ubcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_proper_left_add_ubcoefficientscoefficientdiagonaltermrightinside. pfa_gap_proper_left_add_ubcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_proper_left_add_ubcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_proper_left_add_ubcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_proper_left_add_ubcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_proper_left_add_ubcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_proper_left_add_ubcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_proper_left_add_ubcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_proper_left_add_ubcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_proper_left_add_ubcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_proper_left_add_ubcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_proper_left_add_ubcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_proper_left_add_ubcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_proper_left_add_ubcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_proper_left_add_ubcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_proper_left_add_ubcoefficientscoefficientdiagonal)=pfc_left_proper_left_add_ubcoefficientscoefficientdiagonalterm*pfc_right_proper_left_add_ubcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_proper_left_add_ubcoefficientscoefficientsum fs_v_pfc_proper_left_add_ubcoefficientscoefficientsum. ((((exists fs_h_pfc_proper_left_add_ubcoefficientscoefficientsum_body_start. fs_h_pfc_proper_left_add_ubcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_proper_left_add_ubcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_left_add_ubcoefficientscoefficientsum_body_start. fs_u_pfc_proper_left_add_ubcoefficientscoefficientsum = fs_q_pfc_proper_left_add_ubcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_proper_left_add_ubcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_proper_left_add_ubcoefficientscoefficientsum_body_terminal. fs_h_pfc_proper_left_add_ubcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_proper_left_add_ubcoefficientscoefficient) = S ((S (S (pfc_index_proper_left_add_ubcoefficients))) * fs_v_pfc_proper_left_add_ubcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_left_add_ubcoefficientscoefficientsum_body_terminal. fs_u_pfc_proper_left_add_ubcoefficientscoefficientsum = fs_q_pfc_proper_left_add_ubcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_proper_left_add_ubcoefficients))) * fs_v_pfc_proper_left_add_ubcoefficientscoefficientsum) + (pfc_natural_sum_proper_left_add_ubcoefficientscoefficient))) /\\ forall fs_i_pfc_proper_left_add_ubcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_proper_left_add_ubcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_proper_left_add_ubcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_proper_left_add_ubcoefficientscoefficientsum_body_steps = S (pfc_index_proper_left_add_ubcoefficients)) -> exists fs_a_pfc_proper_left_add_ubcoefficientscoefficientsum_body_steps fs_r_pfc_proper_left_add_ubcoefficientscoefficientsum_body_steps fs_s_pfc_proper_left_add_ubcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_proper_left_add_ubcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_proper_left_add_ubcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_proper_left_add_ubcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_proper_left_add_ubcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_proper_left_add_ubcoefficientscoefficient)) /\\ exists fs_q_pfc_proper_left_add_ubcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_proper_left_add_ubcoefficientscoefficient = fs_q_pfc_proper_left_add_ubcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_proper_left_add_ubcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_proper_left_add_ubcoefficientscoefficient) + (fs_a_pfc_proper_left_add_ubcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_proper_left_add_ubcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_proper_left_add_ubcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_proper_left_add_ubcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_proper_left_add_ubcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_left_add_ubcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_left_add_ubcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_proper_left_add_ubcoefficientscoefficientsum = fs_q_pfc_proper_left_add_ubcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_proper_left_add_ubcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_left_add_ubcoefficientscoefficientsum) + (fs_r_pfc_proper_left_add_ubcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_proper_left_add_ubcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_proper_left_add_ubcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_proper_left_add_ubcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_proper_left_add_ubcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_left_add_ubcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_left_add_ubcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_proper_left_add_ubcoefficientscoefficientsum = fs_q_pfc_proper_left_add_ubcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_proper_left_add_ubcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_left_add_ubcoefficientscoefficientsum) + (fs_s_pfc_proper_left_add_ubcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_proper_left_add_ubcoefficientscoefficientsum_body_steps = fs_r_pfc_proper_left_add_ubcoefficientscoefficientsum_body_steps + fs_a_pfc_proper_left_add_ubcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_proper_left_add_ubcoefficientscoefficientresiduebound. pfa_gap_proper_left_add_ubcoefficientscoefficientresiduebound + S (pfc_value_proper_left_add_ubcoefficients) = (p)) /\\ ((exists pfa_offset_left_proper_left_add_ubcoefficientscoefficientresiduecongruence pfa_offset_right_proper_left_add_ubcoefficientscoefficientresiduecongruence. (pfc_natural_sum_proper_left_add_ubcoefficientscoefficient) + (p) * pfa_offset_left_proper_left_add_ubcoefficientscoefficientresiduecongruence = (pfc_value_proper_left_add_ubcoefficients) + (p) * pfa_offset_right_proper_left_add_ubcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_proper_left_add_vbleft. (exists fom_gap_pfp_proper_left_add_vbleft_index_bound. fom_gap_pfp_proper_left_add_vbleft_index_bound + S (fom_index_pfp_proper_left_add_vbleft) = M) -> exists fom_value_pfp_proper_left_add_vbleft. ((((exists fom_beta_height_pfp_proper_left_add_vbleft_entry. fom_beta_height_pfp_proper_left_add_vbleft_entry + S (fom_value_pfp_proper_left_add_vbleft) = S ((S (fom_index_pfp_proper_left_add_vbleft)) * dc)) /\\ exists fom_beta_quotient_pfp_proper_left_add_vbleft_entry. db = fom_beta_quotient_pfp_proper_left_add_vbleft_entry * S ((S (fom_index_pfp_proper_left_add_vbleft)) * dc) + (fom_value_pfp_proper_left_add_vbleft))) /\\ (exists fom_gap_pfp_proper_left_add_vbleft_value_bound. fom_gap_pfp_proper_left_add_vbleft_value_bound + S (fom_value_pfp_proper_left_add_vbleft) = p))) /\\ (((forall fom_index_pfp_proper_left_add_vbright. (exists fom_gap_pfp_proper_left_add_vbright_index_bound. fom_gap_pfp_proper_left_add_vbright_index_bound + S (fom_index_pfp_proper_left_add_vbright) = L) -> exists fom_value_pfp_proper_left_add_vbright. ((((exists fom_beta_height_pfp_proper_left_add_vbright_entry. fom_beta_height_pfp_proper_left_add_vbright_entry + S (fom_value_pfp_proper_left_add_vbright) = S ((S (fom_index_pfp_proper_left_add_vbright)) * bc)) /\\ exists fom_beta_quotient_pfp_proper_left_add_vbright_entry. bb = fom_beta_quotient_pfp_proper_left_add_vbright_entry * S ((S (fom_index_pfp_proper_left_add_vbright)) * bc) + (fom_value_pfp_proper_left_add_vbright))) /\\ (exists fom_gap_pfp_proper_left_add_vbright_value_bound. fom_gap_pfp_proper_left_add_vbright_value_bound + S (fom_value_pfp_proper_left_add_vbright) = p))) /\\ (((((((M)=0 \\/ (L)=0) /\\ (((N)=0)))) \\/ (((~((M)=0)) /\\ (((~((L)=0)) /\\ (((M)+(L)=S (N)))))))) /\\ ((forall pfc_index_proper_left_add_vbcoefficients. (exists pfa_gap_proper_left_add_vbcoefficientsbound. pfa_gap_proper_left_add_vbcoefficientsbound + S (pfc_index_proper_left_add_vbcoefficients) = (N)) -> exists pfc_value_proper_left_add_vbcoefficients. ((((exists ff_h_pfp_proper_left_add_vbcoefficientsentry. ff_h_pfp_proper_left_add_vbcoefficientsentry + S (pfc_value_proper_left_add_vbcoefficients) = S ((S (pfc_index_proper_left_add_vbcoefficients)) * vc)) /\\ exists ff_q_pfp_proper_left_add_vbcoefficientsentry. vb = ff_q_pfp_proper_left_add_vbcoefficientsentry * S ((S (pfc_index_proper_left_add_vbcoefficients)) * vc) + (pfc_value_proper_left_add_vbcoefficients))) /\\ ((exists pfc_terms_code_proper_left_add_vbcoefficientscoefficient pfc_terms_scale_proper_left_add_vbcoefficientscoefficient pfc_natural_sum_proper_left_add_vbcoefficientscoefficient. ((forall pfc_index_proper_left_add_vbcoefficientscoefficientdiagonal. (exists pfa_gap_proper_left_add_vbcoefficientscoefficientdiagonalbound. pfa_gap_proper_left_add_vbcoefficientscoefficientdiagonalbound + S (pfc_index_proper_left_add_vbcoefficientscoefficientdiagonal) = (S (pfc_index_proper_left_add_vbcoefficients))) -> exists pfc_value_proper_left_add_vbcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_proper_left_add_vbcoefficientscoefficientdiagonalentry. ff_h_pfp_proper_left_add_vbcoefficientscoefficientdiagonalentry + S (pfc_value_proper_left_add_vbcoefficientscoefficientdiagonal) = S ((S (pfc_index_proper_left_add_vbcoefficientscoefficientdiagonal)) * pfc_terms_scale_proper_left_add_vbcoefficientscoefficient)) /\\ exists ff_q_pfp_proper_left_add_vbcoefficientscoefficientdiagonalentry. pfc_terms_code_proper_left_add_vbcoefficientscoefficient = ff_q_pfp_proper_left_add_vbcoefficientscoefficientdiagonalentry * S ((S (pfc_index_proper_left_add_vbcoefficientscoefficientdiagonal)) * pfc_terms_scale_proper_left_add_vbcoefficientscoefficient) + (pfc_value_proper_left_add_vbcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_proper_left_add_vbcoefficientscoefficientdiagonalterm pfc_left_proper_left_add_vbcoefficientscoefficientdiagonalterm pfc_right_proper_left_add_vbcoefficientscoefficientdiagonalterm. (((pfc_index_proper_left_add_vbcoefficientscoefficientdiagonal)+pfc_complement_proper_left_add_vbcoefficientscoefficientdiagonalterm=(pfc_index_proper_left_add_vbcoefficients)) /\\ ((((((exists pfa_gap_proper_left_add_vbcoefficientscoefficientdiagonaltermleftinside. pfa_gap_proper_left_add_vbcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_proper_left_add_vbcoefficientscoefficientdiagonal) = (M)) /\\ ((((exists ff_h_pfp_proper_left_add_vbcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_proper_left_add_vbcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_proper_left_add_vbcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_proper_left_add_vbcoefficientscoefficientdiagonal)) * dc)) /\\ exists ff_q_pfp_proper_left_add_vbcoefficientscoefficientdiagonaltermleftentry. db = ff_q_pfp_proper_left_add_vbcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_proper_left_add_vbcoefficientscoefficientdiagonal)) * dc) + (pfc_left_proper_left_add_vbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_proper_left_add_vbcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_proper_left_add_vbcoefficientscoefficientdiagonaltermleftoutside+(M)=(pfc_index_proper_left_add_vbcoefficientscoefficientdiagonal)) /\\ (((pfc_left_proper_left_add_vbcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_proper_left_add_vbcoefficientscoefficientdiagonaltermrightinside. pfa_gap_proper_left_add_vbcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_proper_left_add_vbcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_proper_left_add_vbcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_proper_left_add_vbcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_proper_left_add_vbcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_proper_left_add_vbcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_proper_left_add_vbcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_proper_left_add_vbcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_proper_left_add_vbcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_proper_left_add_vbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_proper_left_add_vbcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_proper_left_add_vbcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_proper_left_add_vbcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_proper_left_add_vbcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_proper_left_add_vbcoefficientscoefficientdiagonal)=pfc_left_proper_left_add_vbcoefficientscoefficientdiagonalterm*pfc_right_proper_left_add_vbcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_proper_left_add_vbcoefficientscoefficientsum fs_v_pfc_proper_left_add_vbcoefficientscoefficientsum. ((((exists fs_h_pfc_proper_left_add_vbcoefficientscoefficientsum_body_start. fs_h_pfc_proper_left_add_vbcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_proper_left_add_vbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_left_add_vbcoefficientscoefficientsum_body_start. fs_u_pfc_proper_left_add_vbcoefficientscoefficientsum = fs_q_pfc_proper_left_add_vbcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_proper_left_add_vbcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_proper_left_add_vbcoefficientscoefficientsum_body_terminal. fs_h_pfc_proper_left_add_vbcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_proper_left_add_vbcoefficientscoefficient) = S ((S (S (pfc_index_proper_left_add_vbcoefficients))) * fs_v_pfc_proper_left_add_vbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_left_add_vbcoefficientscoefficientsum_body_terminal. fs_u_pfc_proper_left_add_vbcoefficientscoefficientsum = fs_q_pfc_proper_left_add_vbcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_proper_left_add_vbcoefficients))) * fs_v_pfc_proper_left_add_vbcoefficientscoefficientsum) + (pfc_natural_sum_proper_left_add_vbcoefficientscoefficient))) /\\ forall fs_i_pfc_proper_left_add_vbcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_proper_left_add_vbcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_proper_left_add_vbcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_proper_left_add_vbcoefficientscoefficientsum_body_steps = S (pfc_index_proper_left_add_vbcoefficients)) -> exists fs_a_pfc_proper_left_add_vbcoefficientscoefficientsum_body_steps fs_r_pfc_proper_left_add_vbcoefficientscoefficientsum_body_steps fs_s_pfc_proper_left_add_vbcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_proper_left_add_vbcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_proper_left_add_vbcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_proper_left_add_vbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_proper_left_add_vbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_proper_left_add_vbcoefficientscoefficient)) /\\ exists fs_q_pfc_proper_left_add_vbcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_proper_left_add_vbcoefficientscoefficient = fs_q_pfc_proper_left_add_vbcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_proper_left_add_vbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_proper_left_add_vbcoefficientscoefficient) + (fs_a_pfc_proper_left_add_vbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_proper_left_add_vbcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_proper_left_add_vbcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_proper_left_add_vbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_proper_left_add_vbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_left_add_vbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_left_add_vbcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_proper_left_add_vbcoefficientscoefficientsum = fs_q_pfc_proper_left_add_vbcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_proper_left_add_vbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_left_add_vbcoefficientscoefficientsum) + (fs_r_pfc_proper_left_add_vbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_proper_left_add_vbcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_proper_left_add_vbcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_proper_left_add_vbcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_proper_left_add_vbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_left_add_vbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_left_add_vbcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_proper_left_add_vbcoefficientscoefficientsum = fs_q_pfc_proper_left_add_vbcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_proper_left_add_vbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_left_add_vbcoefficientscoefficientsum) + (fs_s_pfc_proper_left_add_vbcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_proper_left_add_vbcoefficientscoefficientsum_body_steps = fs_r_pfc_proper_left_add_vbcoefficientscoefficientsum_body_steps + fs_a_pfc_proper_left_add_vbcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_proper_left_add_vbcoefficientscoefficientresiduebound. pfa_gap_proper_left_add_vbcoefficientscoefficientresiduebound + S (pfc_value_proper_left_add_vbcoefficients) = (p)) /\\ ((exists pfa_offset_left_proper_left_add_vbcoefficientscoefficientresiduecongruence pfa_offset_right_proper_left_add_vbcoefficientscoefficientresiduecongruence. (pfc_natural_sum_proper_left_add_vbcoefficientscoefficient) + (p) * pfa_offset_left_proper_left_add_vbcoefficientscoefficientresiduecongruence = (pfc_value_proper_left_add_vbcoefficients) + (p) * pfa_offset_right_proper_left_add_vbcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_proper_left_add_wbleft. (exists fom_gap_pfp_proper_left_add_wbleft_index_bound. fom_gap_pfp_proper_left_add_wbleft_index_bound + S (fom_index_pfp_proper_left_add_wbleft) = M) -> exists fom_value_pfp_proper_left_add_wbleft. ((((exists fom_beta_height_pfp_proper_left_add_wbleft_entry. fom_beta_height_pfp_proper_left_add_wbleft_entry + S (fom_value_pfp_proper_left_add_wbleft) = S ((S (fom_index_pfp_proper_left_add_wbleft)) * dc)) /\\ exists fom_beta_quotient_pfp_proper_left_add_wbleft_entry. db = fom_beta_quotient_pfp_proper_left_add_wbleft_entry * S ((S (fom_index_pfp_proper_left_add_wbleft)) * dc) + (fom_value_pfp_proper_left_add_wbleft))) /\\ (exists fom_gap_pfp_proper_left_add_wbleft_value_bound. fom_gap_pfp_proper_left_add_wbleft_value_bound + S (fom_value_pfp_proper_left_add_wbleft) = p))) /\\ (((forall fom_index_pfp_proper_left_add_wbright. (exists fom_gap_pfp_proper_left_add_wbright_index_bound. fom_gap_pfp_proper_left_add_wbright_index_bound + S (fom_index_pfp_proper_left_add_wbright) = L) -> exists fom_value_pfp_proper_left_add_wbright. ((((exists fom_beta_height_pfp_proper_left_add_wbright_entry. fom_beta_height_pfp_proper_left_add_wbright_entry + S (fom_value_pfp_proper_left_add_wbright) = S ((S (fom_index_pfp_proper_left_add_wbright)) * cc)) /\\ exists fom_beta_quotient_pfp_proper_left_add_wbright_entry. cb = fom_beta_quotient_pfp_proper_left_add_wbright_entry * S ((S (fom_index_pfp_proper_left_add_wbright)) * cc) + (fom_value_pfp_proper_left_add_wbright))) /\\ (exists fom_gap_pfp_proper_left_add_wbright_value_bound. fom_gap_pfp_proper_left_add_wbright_value_bound + S (fom_value_pfp_proper_left_add_wbright) = p))) /\\ (((((((M)=0 \\/ (L)=0) /\\ (((N)=0)))) \\/ (((~((M)=0)) /\\ (((~((L)=0)) /\\ (((M)+(L)=S (N)))))))) /\\ ((forall pfc_index_proper_left_add_wbcoefficients. (exists pfa_gap_proper_left_add_wbcoefficientsbound. pfa_gap_proper_left_add_wbcoefficientsbound + S (pfc_index_proper_left_add_wbcoefficients) = (N)) -> exists pfc_value_proper_left_add_wbcoefficients. ((((exists ff_h_pfp_proper_left_add_wbcoefficientsentry. ff_h_pfp_proper_left_add_wbcoefficientsentry + S (pfc_value_proper_left_add_wbcoefficients) = S ((S (pfc_index_proper_left_add_wbcoefficients)) * wc)) /\\ exists ff_q_pfp_proper_left_add_wbcoefficientsentry. wb = ff_q_pfp_proper_left_add_wbcoefficientsentry * S ((S (pfc_index_proper_left_add_wbcoefficients)) * wc) + (pfc_value_proper_left_add_wbcoefficients))) /\\ ((exists pfc_terms_code_proper_left_add_wbcoefficientscoefficient pfc_terms_scale_proper_left_add_wbcoefficientscoefficient pfc_natural_sum_proper_left_add_wbcoefficientscoefficient. ((forall pfc_index_proper_left_add_wbcoefficientscoefficientdiagonal. (exists pfa_gap_proper_left_add_wbcoefficientscoefficientdiagonalbound. pfa_gap_proper_left_add_wbcoefficientscoefficientdiagonalbound + S (pfc_index_proper_left_add_wbcoefficientscoefficientdiagonal) = (S (pfc_index_proper_left_add_wbcoefficients))) -> exists pfc_value_proper_left_add_wbcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_proper_left_add_wbcoefficientscoefficientdiagonalentry. ff_h_pfp_proper_left_add_wbcoefficientscoefficientdiagonalentry + S (pfc_value_proper_left_add_wbcoefficientscoefficientdiagonal) = S ((S (pfc_index_proper_left_add_wbcoefficientscoefficientdiagonal)) * pfc_terms_scale_proper_left_add_wbcoefficientscoefficient)) /\\ exists ff_q_pfp_proper_left_add_wbcoefficientscoefficientdiagonalentry. pfc_terms_code_proper_left_add_wbcoefficientscoefficient = ff_q_pfp_proper_left_add_wbcoefficientscoefficientdiagonalentry * S ((S (pfc_index_proper_left_add_wbcoefficientscoefficientdiagonal)) * pfc_terms_scale_proper_left_add_wbcoefficientscoefficient) + (pfc_value_proper_left_add_wbcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_proper_left_add_wbcoefficientscoefficientdiagonalterm pfc_left_proper_left_add_wbcoefficientscoefficientdiagonalterm pfc_right_proper_left_add_wbcoefficientscoefficientdiagonalterm. (((pfc_index_proper_left_add_wbcoefficientscoefficientdiagonal)+pfc_complement_proper_left_add_wbcoefficientscoefficientdiagonalterm=(pfc_index_proper_left_add_wbcoefficients)) /\\ ((((((exists pfa_gap_proper_left_add_wbcoefficientscoefficientdiagonaltermleftinside. pfa_gap_proper_left_add_wbcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_proper_left_add_wbcoefficientscoefficientdiagonal) = (M)) /\\ ((((exists ff_h_pfp_proper_left_add_wbcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_proper_left_add_wbcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_proper_left_add_wbcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_proper_left_add_wbcoefficientscoefficientdiagonal)) * dc)) /\\ exists ff_q_pfp_proper_left_add_wbcoefficientscoefficientdiagonaltermleftentry. db = ff_q_pfp_proper_left_add_wbcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_proper_left_add_wbcoefficientscoefficientdiagonal)) * dc) + (pfc_left_proper_left_add_wbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_proper_left_add_wbcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_proper_left_add_wbcoefficientscoefficientdiagonaltermleftoutside+(M)=(pfc_index_proper_left_add_wbcoefficientscoefficientdiagonal)) /\\ (((pfc_left_proper_left_add_wbcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_proper_left_add_wbcoefficientscoefficientdiagonaltermrightinside. pfa_gap_proper_left_add_wbcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_proper_left_add_wbcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_proper_left_add_wbcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_proper_left_add_wbcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_proper_left_add_wbcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_proper_left_add_wbcoefficientscoefficientdiagonalterm)) * cc)) /\\ exists ff_q_pfp_proper_left_add_wbcoefficientscoefficientdiagonaltermrightentry. cb = ff_q_pfp_proper_left_add_wbcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_proper_left_add_wbcoefficientscoefficientdiagonalterm)) * cc) + (pfc_right_proper_left_add_wbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_proper_left_add_wbcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_proper_left_add_wbcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_proper_left_add_wbcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_proper_left_add_wbcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_proper_left_add_wbcoefficientscoefficientdiagonal)=pfc_left_proper_left_add_wbcoefficientscoefficientdiagonalterm*pfc_right_proper_left_add_wbcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_proper_left_add_wbcoefficientscoefficientsum fs_v_pfc_proper_left_add_wbcoefficientscoefficientsum. ((((exists fs_h_pfc_proper_left_add_wbcoefficientscoefficientsum_body_start. fs_h_pfc_proper_left_add_wbcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_proper_left_add_wbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_left_add_wbcoefficientscoefficientsum_body_start. fs_u_pfc_proper_left_add_wbcoefficientscoefficientsum = fs_q_pfc_proper_left_add_wbcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_proper_left_add_wbcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_proper_left_add_wbcoefficientscoefficientsum_body_terminal. fs_h_pfc_proper_left_add_wbcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_proper_left_add_wbcoefficientscoefficient) = S ((S (S (pfc_index_proper_left_add_wbcoefficients))) * fs_v_pfc_proper_left_add_wbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_left_add_wbcoefficientscoefficientsum_body_terminal. fs_u_pfc_proper_left_add_wbcoefficientscoefficientsum = fs_q_pfc_proper_left_add_wbcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_proper_left_add_wbcoefficients))) * fs_v_pfc_proper_left_add_wbcoefficientscoefficientsum) + (pfc_natural_sum_proper_left_add_wbcoefficientscoefficient))) /\\ forall fs_i_pfc_proper_left_add_wbcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_proper_left_add_wbcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_proper_left_add_wbcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_proper_left_add_wbcoefficientscoefficientsum_body_steps = S (pfc_index_proper_left_add_wbcoefficients)) -> exists fs_a_pfc_proper_left_add_wbcoefficientscoefficientsum_body_steps fs_r_pfc_proper_left_add_wbcoefficientscoefficientsum_body_steps fs_s_pfc_proper_left_add_wbcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_proper_left_add_wbcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_proper_left_add_wbcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_proper_left_add_wbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_proper_left_add_wbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_proper_left_add_wbcoefficientscoefficient)) /\\ exists fs_q_pfc_proper_left_add_wbcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_proper_left_add_wbcoefficientscoefficient = fs_q_pfc_proper_left_add_wbcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_proper_left_add_wbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_proper_left_add_wbcoefficientscoefficient) + (fs_a_pfc_proper_left_add_wbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_proper_left_add_wbcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_proper_left_add_wbcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_proper_left_add_wbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_proper_left_add_wbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_left_add_wbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_left_add_wbcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_proper_left_add_wbcoefficientscoefficientsum = fs_q_pfc_proper_left_add_wbcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_proper_left_add_wbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_left_add_wbcoefficientscoefficientsum) + (fs_r_pfc_proper_left_add_wbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_proper_left_add_wbcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_proper_left_add_wbcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_proper_left_add_wbcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_proper_left_add_wbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_left_add_wbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_left_add_wbcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_proper_left_add_wbcoefficientscoefficientsum = fs_q_pfc_proper_left_add_wbcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_proper_left_add_wbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_left_add_wbcoefficientscoefficientsum) + (fs_s_pfc_proper_left_add_wbcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_proper_left_add_wbcoefficientscoefficientsum_body_steps = fs_r_pfc_proper_left_add_wbcoefficientscoefficientsum_body_steps + fs_a_pfc_proper_left_add_wbcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_proper_left_add_wbcoefficientscoefficientresiduebound. pfa_gap_proper_left_add_wbcoefficientscoefficientresiduebound + S (pfc_value_proper_left_add_wbcoefficients) = (p)) /\\ ((exists pfa_offset_left_proper_left_add_wbcoefficientscoefficientresiduecongruence pfa_offset_right_proper_left_add_wbcoefficientscoefficientresiduecongruence. (pfc_natural_sum_proper_left_add_wbcoefficientscoefficient) + (p) * pfa_offset_left_proper_left_add_wbcoefficientscoefficientresiduecongruence = (pfc_value_proper_left_add_wbcoefficients) + (p) * pfa_offset_right_proper_left_add_wbcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (forall pfp_index_proper_left_add_result. (exists pfa_gap_proper_left_add_resultindex. pfa_gap_proper_left_add_resultindex + S (pfp_index_proper_left_add_result) = (N)) -> exists pfp_left_proper_left_add_result pfp_right_proper_left_add_result pfp_value_proper_left_add_result. ((((exists ff_h_pfp_proper_left_add_resultleft. ff_h_pfp_proper_left_add_resultleft + S (pfp_left_proper_left_add_result) = S ((S (pfp_index_proper_left_add_result)) * uc)) /\\ exists ff_q_pfp_proper_left_add_resultleft. ub = ff_q_pfp_proper_left_add_resultleft * S ((S (pfp_index_proper_left_add_result)) * uc) + (pfp_left_proper_left_add_result))) /\\ (((((exists ff_h_pfp_proper_left_add_resultright. ff_h_pfp_proper_left_add_resultright + S (pfp_right_proper_left_add_result) = S ((S (pfp_index_proper_left_add_result)) * vc)) /\\ exists ff_q_pfp_proper_left_add_resultright. vb = ff_q_pfp_proper_left_add_resultright * S ((S (pfp_index_proper_left_add_result)) * vc) + (pfp_right_proper_left_add_result))) /\\ (((((exists ff_h_pfp_proper_left_add_resulttarget. ff_h_pfp_proper_left_add_resulttarget + S (pfp_value_proper_left_add_result) = S ((S (pfp_index_proper_left_add_result)) * wc)) /\\ exists ff_q_pfp_proper_left_add_resulttarget. wb = ff_q_pfp_proper_left_add_resulttarget * S ((S (pfp_index_proper_left_add_result)) * wc) + (pfp_value_proper_left_add_result))) /\\ ((((exists pfa_gap_proper_left_add_resultoperationleft. pfa_gap_proper_left_add_resultoperationleft + S (pfp_left_proper_left_add_result) = (p)) /\\ (((exists pfa_gap_proper_left_add_resultoperationright. pfa_gap_proper_left_add_resultoperationright + S (pfp_right_proper_left_add_result) = (p)) /\\ ((((exists pfa_gap_proper_left_add_resultoperationresultbound. pfa_gap_proper_left_add_resultoperationresultbound + S (pfp_value_proper_left_add_result) = (p)) /\\ ((exists pfa_offset_left_proper_left_add_resultoperationresultcongruence pfa_offset_right_proper_left_add_resultoperationresultcongruence. 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(forall pfp_index_proper_right_add_source. (exists pfa_gap_proper_right_add_sourceindex. pfa_gap_proper_right_add_sourceindex + S (pfp_index_proper_right_add_source) = (L)) -> exists pfp_left_proper_right_add_source pfp_right_proper_right_add_source pfp_value_proper_right_add_source. ((((exists ff_h_pfp_proper_right_add_sourceleft. ff_h_pfp_proper_right_add_sourceleft + S (pfp_left_proper_right_add_source) = S ((S (pfp_index_proper_right_add_source)) * ac)) /\\ exists ff_q_pfp_proper_right_add_sourceleft. ab = ff_q_pfp_proper_right_add_sourceleft * S ((S (pfp_index_proper_right_add_source)) * ac) + (pfp_left_proper_right_add_source))) /\\ (((((exists ff_h_pfp_proper_right_add_sourceright. ff_h_pfp_proper_right_add_sourceright + S (pfp_right_proper_right_add_source) = S ((S (pfp_index_proper_right_add_source)) * bc)) /\\ exists ff_q_pfp_proper_right_add_sourceright. bb = ff_q_pfp_proper_right_add_sourceright * S ((S (pfp_index_proper_right_add_source)) * bc) + (pfp_right_proper_right_add_source))) /\\ (((((exists ff_h_pfp_proper_right_add_sourcetarget. ff_h_pfp_proper_right_add_sourcetarget + S (pfp_value_proper_right_add_source) = S ((S (pfp_index_proper_right_add_source)) * cc)) /\\ exists ff_q_pfp_proper_right_add_sourcetarget. cb = ff_q_pfp_proper_right_add_sourcetarget * S ((S (pfp_index_proper_right_add_source)) * cc) + (pfp_value_proper_right_add_source))) /\\ ((((exists pfa_gap_proper_right_add_sourceoperationleft. pfa_gap_proper_right_add_sourceoperationleft + S (pfp_left_proper_right_add_source) = (p)) /\\ (((exists pfa_gap_proper_right_add_sourceoperationright. pfa_gap_proper_right_add_sourceoperationright + S (pfp_right_proper_right_add_source) = (p)) /\\ ((((exists pfa_gap_proper_right_add_sourceoperationresultbound. pfa_gap_proper_right_add_sourceoperationresultbound + S (pfp_value_proper_right_add_source) = (p)) /\\ ((exists pfa_offset_left_proper_right_add_sourceoperationresultcongruence pfa_offset_right_proper_right_add_sourceoperationresultcongruence. ((pfp_left_proper_right_add_source) + (pfp_right_proper_right_add_source)) + (p) * pfa_offset_left_proper_right_add_sourceoperationresultcongruence = (pfp_value_proper_right_add_source) + (p) * pfa_offset_right_proper_right_add_sourceoperationresultcongruence)))))))))))))))) -> (((forall fom_index_pfp_proper_right_add_ubleft. (exists fom_gap_pfp_proper_right_add_ubleft_index_bound. fom_gap_pfp_proper_right_add_ubleft_index_bound + S (fom_index_pfp_proper_right_add_ubleft) = L) -> exists fom_value_pfp_proper_right_add_ubleft. ((((exists fom_beta_height_pfp_proper_right_add_ubleft_entry. fom_beta_height_pfp_proper_right_add_ubleft_entry + S (fom_value_pfp_proper_right_add_ubleft) = S ((S (fom_index_pfp_proper_right_add_ubleft)) * ac)) /\\ exists fom_beta_quotient_pfp_proper_right_add_ubleft_entry. ab = fom_beta_quotient_pfp_proper_right_add_ubleft_entry * S ((S (fom_index_pfp_proper_right_add_ubleft)) * ac) + (fom_value_pfp_proper_right_add_ubleft))) /\\ (exists fom_gap_pfp_proper_right_add_ubleft_value_bound. fom_gap_pfp_proper_right_add_ubleft_value_bound + S (fom_value_pfp_proper_right_add_ubleft) = p))) /\\ (((forall fom_index_pfp_proper_right_add_ubright. (exists fom_gap_pfp_proper_right_add_ubright_index_bound. fom_gap_pfp_proper_right_add_ubright_index_bound + S (fom_index_pfp_proper_right_add_ubright) = M) -> exists fom_value_pfp_proper_right_add_ubright. ((((exists fom_beta_height_pfp_proper_right_add_ubright_entry. fom_beta_height_pfp_proper_right_add_ubright_entry + S (fom_value_pfp_proper_right_add_ubright) = S ((S (fom_index_pfp_proper_right_add_ubright)) * dc)) /\\ exists fom_beta_quotient_pfp_proper_right_add_ubright_entry. db = fom_beta_quotient_pfp_proper_right_add_ubright_entry * S ((S (fom_index_pfp_proper_right_add_ubright)) * dc) + (fom_value_pfp_proper_right_add_ubright))) /\\ (exists fom_gap_pfp_proper_right_add_ubright_value_bound. fom_gap_pfp_proper_right_add_ubright_value_bound + S (fom_value_pfp_proper_right_add_ubright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_proper_right_add_ubcoefficients. (exists pfa_gap_proper_right_add_ubcoefficientsbound. pfa_gap_proper_right_add_ubcoefficientsbound + S (pfc_index_proper_right_add_ubcoefficients) = (N)) -> exists pfc_value_proper_right_add_ubcoefficients. ((((exists ff_h_pfp_proper_right_add_ubcoefficientsentry. ff_h_pfp_proper_right_add_ubcoefficientsentry + S (pfc_value_proper_right_add_ubcoefficients) = S ((S (pfc_index_proper_right_add_ubcoefficients)) * uc)) /\\ exists ff_q_pfp_proper_right_add_ubcoefficientsentry. ub = ff_q_pfp_proper_right_add_ubcoefficientsentry * S ((S (pfc_index_proper_right_add_ubcoefficients)) * uc) + (pfc_value_proper_right_add_ubcoefficients))) /\\ ((exists pfc_terms_code_proper_right_add_ubcoefficientscoefficient pfc_terms_scale_proper_right_add_ubcoefficientscoefficient pfc_natural_sum_proper_right_add_ubcoefficientscoefficient. ((forall pfc_index_proper_right_add_ubcoefficientscoefficientdiagonal. (exists pfa_gap_proper_right_add_ubcoefficientscoefficientdiagonalbound. pfa_gap_proper_right_add_ubcoefficientscoefficientdiagonalbound + S (pfc_index_proper_right_add_ubcoefficientscoefficientdiagonal) = (S (pfc_index_proper_right_add_ubcoefficients))) -> exists pfc_value_proper_right_add_ubcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_proper_right_add_ubcoefficientscoefficientdiagonalentry. ff_h_pfp_proper_right_add_ubcoefficientscoefficientdiagonalentry + S (pfc_value_proper_right_add_ubcoefficientscoefficientdiagonal) = S ((S (pfc_index_proper_right_add_ubcoefficientscoefficientdiagonal)) * pfc_terms_scale_proper_right_add_ubcoefficientscoefficient)) /\\ exists ff_q_pfp_proper_right_add_ubcoefficientscoefficientdiagonalentry. pfc_terms_code_proper_right_add_ubcoefficientscoefficient = ff_q_pfp_proper_right_add_ubcoefficientscoefficientdiagonalentry * S ((S (pfc_index_proper_right_add_ubcoefficientscoefficientdiagonal)) * pfc_terms_scale_proper_right_add_ubcoefficientscoefficient) + (pfc_value_proper_right_add_ubcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_proper_right_add_ubcoefficientscoefficientdiagonalterm pfc_left_proper_right_add_ubcoefficientscoefficientdiagonalterm pfc_right_proper_right_add_ubcoefficientscoefficientdiagonalterm. (((pfc_index_proper_right_add_ubcoefficientscoefficientdiagonal)+pfc_complement_proper_right_add_ubcoefficientscoefficientdiagonalterm=(pfc_index_proper_right_add_ubcoefficients)) /\\ ((((((exists pfa_gap_proper_right_add_ubcoefficientscoefficientdiagonaltermleftinside. pfa_gap_proper_right_add_ubcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_proper_right_add_ubcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_proper_right_add_ubcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_proper_right_add_ubcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_proper_right_add_ubcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_proper_right_add_ubcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_proper_right_add_ubcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_proper_right_add_ubcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_proper_right_add_ubcoefficientscoefficientdiagonal)) * ac) + (pfc_left_proper_right_add_ubcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_proper_right_add_ubcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_proper_right_add_ubcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_proper_right_add_ubcoefficientscoefficientdiagonal)) /\\ (((pfc_left_proper_right_add_ubcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_proper_right_add_ubcoefficientscoefficientdiagonaltermrightinside. pfa_gap_proper_right_add_ubcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_proper_right_add_ubcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_proper_right_add_ubcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_proper_right_add_ubcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_proper_right_add_ubcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_proper_right_add_ubcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_proper_right_add_ubcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_proper_right_add_ubcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_proper_right_add_ubcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_proper_right_add_ubcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_proper_right_add_ubcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_proper_right_add_ubcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_proper_right_add_ubcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_proper_right_add_ubcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_proper_right_add_ubcoefficientscoefficientdiagonal)=pfc_left_proper_right_add_ubcoefficientscoefficientdiagonalterm*pfc_right_proper_right_add_ubcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_proper_right_add_ubcoefficientscoefficientsum fs_v_pfc_proper_right_add_ubcoefficientscoefficientsum. ((((exists fs_h_pfc_proper_right_add_ubcoefficientscoefficientsum_body_start. fs_h_pfc_proper_right_add_ubcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_proper_right_add_ubcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_right_add_ubcoefficientscoefficientsum_body_start. fs_u_pfc_proper_right_add_ubcoefficientscoefficientsum = fs_q_pfc_proper_right_add_ubcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_proper_right_add_ubcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_proper_right_add_ubcoefficientscoefficientsum_body_terminal. fs_h_pfc_proper_right_add_ubcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_proper_right_add_ubcoefficientscoefficient) = S ((S (S (pfc_index_proper_right_add_ubcoefficients))) * fs_v_pfc_proper_right_add_ubcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_right_add_ubcoefficientscoefficientsum_body_terminal. fs_u_pfc_proper_right_add_ubcoefficientscoefficientsum = fs_q_pfc_proper_right_add_ubcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_proper_right_add_ubcoefficients))) * fs_v_pfc_proper_right_add_ubcoefficientscoefficientsum) + (pfc_natural_sum_proper_right_add_ubcoefficientscoefficient))) /\\ forall fs_i_pfc_proper_right_add_ubcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_proper_right_add_ubcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_proper_right_add_ubcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_proper_right_add_ubcoefficientscoefficientsum_body_steps = S (pfc_index_proper_right_add_ubcoefficients)) -> exists fs_a_pfc_proper_right_add_ubcoefficientscoefficientsum_body_steps fs_r_pfc_proper_right_add_ubcoefficientscoefficientsum_body_steps fs_s_pfc_proper_right_add_ubcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_proper_right_add_ubcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_proper_right_add_ubcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_proper_right_add_ubcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_proper_right_add_ubcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_proper_right_add_ubcoefficientscoefficient)) /\\ exists fs_q_pfc_proper_right_add_ubcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_proper_right_add_ubcoefficientscoefficient = fs_q_pfc_proper_right_add_ubcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_proper_right_add_ubcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_proper_right_add_ubcoefficientscoefficient) + (fs_a_pfc_proper_right_add_ubcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_proper_right_add_ubcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_proper_right_add_ubcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_proper_right_add_ubcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_proper_right_add_ubcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_right_add_ubcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_right_add_ubcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_proper_right_add_ubcoefficientscoefficientsum = fs_q_pfc_proper_right_add_ubcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_proper_right_add_ubcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_right_add_ubcoefficientscoefficientsum) + (fs_r_pfc_proper_right_add_ubcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_proper_right_add_ubcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_proper_right_add_ubcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_proper_right_add_ubcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_proper_right_add_ubcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_right_add_ubcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_right_add_ubcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_proper_right_add_ubcoefficientscoefficientsum = fs_q_pfc_proper_right_add_ubcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_proper_right_add_ubcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_right_add_ubcoefficientscoefficientsum) + (fs_s_pfc_proper_right_add_ubcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_proper_right_add_ubcoefficientscoefficientsum_body_steps = fs_r_pfc_proper_right_add_ubcoefficientscoefficientsum_body_steps + fs_a_pfc_proper_right_add_ubcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_proper_right_add_ubcoefficientscoefficientresiduebound. pfa_gap_proper_right_add_ubcoefficientscoefficientresiduebound + S (pfc_value_proper_right_add_ubcoefficients) = (p)) /\\ ((exists pfa_offset_left_proper_right_add_ubcoefficientscoefficientresiduecongruence pfa_offset_right_proper_right_add_ubcoefficientscoefficientresiduecongruence. (pfc_natural_sum_proper_right_add_ubcoefficientscoefficient) + (p) * pfa_offset_left_proper_right_add_ubcoefficientscoefficientresiduecongruence = (pfc_value_proper_right_add_ubcoefficients) + (p) * pfa_offset_right_proper_right_add_ubcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_proper_right_add_vbleft. (exists fom_gap_pfp_proper_right_add_vbleft_index_bound. fom_gap_pfp_proper_right_add_vbleft_index_bound + S (fom_index_pfp_proper_right_add_vbleft) = L) -> exists fom_value_pfp_proper_right_add_vbleft. ((((exists fom_beta_height_pfp_proper_right_add_vbleft_entry. fom_beta_height_pfp_proper_right_add_vbleft_entry + S (fom_value_pfp_proper_right_add_vbleft) = S ((S (fom_index_pfp_proper_right_add_vbleft)) * bc)) /\\ exists fom_beta_quotient_pfp_proper_right_add_vbleft_entry. bb = fom_beta_quotient_pfp_proper_right_add_vbleft_entry * S ((S (fom_index_pfp_proper_right_add_vbleft)) * bc) + (fom_value_pfp_proper_right_add_vbleft))) /\\ (exists fom_gap_pfp_proper_right_add_vbleft_value_bound. fom_gap_pfp_proper_right_add_vbleft_value_bound + S (fom_value_pfp_proper_right_add_vbleft) = p))) /\\ (((forall fom_index_pfp_proper_right_add_vbright. (exists fom_gap_pfp_proper_right_add_vbright_index_bound. fom_gap_pfp_proper_right_add_vbright_index_bound + S (fom_index_pfp_proper_right_add_vbright) = M) -> exists fom_value_pfp_proper_right_add_vbright. ((((exists fom_beta_height_pfp_proper_right_add_vbright_entry. fom_beta_height_pfp_proper_right_add_vbright_entry + S (fom_value_pfp_proper_right_add_vbright) = S ((S (fom_index_pfp_proper_right_add_vbright)) * dc)) /\\ exists fom_beta_quotient_pfp_proper_right_add_vbright_entry. db = fom_beta_quotient_pfp_proper_right_add_vbright_entry * S ((S (fom_index_pfp_proper_right_add_vbright)) * dc) + (fom_value_pfp_proper_right_add_vbright))) /\\ (exists fom_gap_pfp_proper_right_add_vbright_value_bound. fom_gap_pfp_proper_right_add_vbright_value_bound + S (fom_value_pfp_proper_right_add_vbright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_proper_right_add_vbcoefficients. (exists pfa_gap_proper_right_add_vbcoefficientsbound. pfa_gap_proper_right_add_vbcoefficientsbound + S (pfc_index_proper_right_add_vbcoefficients) = (N)) -> exists pfc_value_proper_right_add_vbcoefficients. ((((exists ff_h_pfp_proper_right_add_vbcoefficientsentry. ff_h_pfp_proper_right_add_vbcoefficientsentry + S (pfc_value_proper_right_add_vbcoefficients) = S ((S (pfc_index_proper_right_add_vbcoefficients)) * vc)) /\\ exists ff_q_pfp_proper_right_add_vbcoefficientsentry. vb = ff_q_pfp_proper_right_add_vbcoefficientsentry * S ((S (pfc_index_proper_right_add_vbcoefficients)) * vc) + (pfc_value_proper_right_add_vbcoefficients))) /\\ ((exists pfc_terms_code_proper_right_add_vbcoefficientscoefficient pfc_terms_scale_proper_right_add_vbcoefficientscoefficient pfc_natural_sum_proper_right_add_vbcoefficientscoefficient. ((forall pfc_index_proper_right_add_vbcoefficientscoefficientdiagonal. (exists pfa_gap_proper_right_add_vbcoefficientscoefficientdiagonalbound. pfa_gap_proper_right_add_vbcoefficientscoefficientdiagonalbound + S (pfc_index_proper_right_add_vbcoefficientscoefficientdiagonal) = (S (pfc_index_proper_right_add_vbcoefficients))) -> exists pfc_value_proper_right_add_vbcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_proper_right_add_vbcoefficientscoefficientdiagonalentry. ff_h_pfp_proper_right_add_vbcoefficientscoefficientdiagonalentry + S (pfc_value_proper_right_add_vbcoefficientscoefficientdiagonal) = S ((S (pfc_index_proper_right_add_vbcoefficientscoefficientdiagonal)) * pfc_terms_scale_proper_right_add_vbcoefficientscoefficient)) /\\ exists ff_q_pfp_proper_right_add_vbcoefficientscoefficientdiagonalentry. pfc_terms_code_proper_right_add_vbcoefficientscoefficient = ff_q_pfp_proper_right_add_vbcoefficientscoefficientdiagonalentry * S ((S (pfc_index_proper_right_add_vbcoefficientscoefficientdiagonal)) * pfc_terms_scale_proper_right_add_vbcoefficientscoefficient) + (pfc_value_proper_right_add_vbcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_proper_right_add_vbcoefficientscoefficientdiagonalterm pfc_left_proper_right_add_vbcoefficientscoefficientdiagonalterm pfc_right_proper_right_add_vbcoefficientscoefficientdiagonalterm. (((pfc_index_proper_right_add_vbcoefficientscoefficientdiagonal)+pfc_complement_proper_right_add_vbcoefficientscoefficientdiagonalterm=(pfc_index_proper_right_add_vbcoefficients)) /\\ ((((((exists pfa_gap_proper_right_add_vbcoefficientscoefficientdiagonaltermleftinside. pfa_gap_proper_right_add_vbcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_proper_right_add_vbcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_proper_right_add_vbcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_proper_right_add_vbcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_proper_right_add_vbcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_proper_right_add_vbcoefficientscoefficientdiagonal)) * bc)) /\\ exists ff_q_pfp_proper_right_add_vbcoefficientscoefficientdiagonaltermleftentry. bb = ff_q_pfp_proper_right_add_vbcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_proper_right_add_vbcoefficientscoefficientdiagonal)) * bc) + (pfc_left_proper_right_add_vbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_proper_right_add_vbcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_proper_right_add_vbcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_proper_right_add_vbcoefficientscoefficientdiagonal)) /\\ (((pfc_left_proper_right_add_vbcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_proper_right_add_vbcoefficientscoefficientdiagonaltermrightinside. pfa_gap_proper_right_add_vbcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_proper_right_add_vbcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_proper_right_add_vbcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_proper_right_add_vbcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_proper_right_add_vbcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_proper_right_add_vbcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_proper_right_add_vbcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_proper_right_add_vbcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_proper_right_add_vbcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_proper_right_add_vbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_proper_right_add_vbcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_proper_right_add_vbcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_proper_right_add_vbcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_proper_right_add_vbcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_proper_right_add_vbcoefficientscoefficientdiagonal)=pfc_left_proper_right_add_vbcoefficientscoefficientdiagonalterm*pfc_right_proper_right_add_vbcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_proper_right_add_vbcoefficientscoefficientsum fs_v_pfc_proper_right_add_vbcoefficientscoefficientsum. ((((exists fs_h_pfc_proper_right_add_vbcoefficientscoefficientsum_body_start. fs_h_pfc_proper_right_add_vbcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_proper_right_add_vbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_right_add_vbcoefficientscoefficientsum_body_start. fs_u_pfc_proper_right_add_vbcoefficientscoefficientsum = fs_q_pfc_proper_right_add_vbcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_proper_right_add_vbcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_proper_right_add_vbcoefficientscoefficientsum_body_terminal. fs_h_pfc_proper_right_add_vbcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_proper_right_add_vbcoefficientscoefficient) = S ((S (S (pfc_index_proper_right_add_vbcoefficients))) * fs_v_pfc_proper_right_add_vbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_right_add_vbcoefficientscoefficientsum_body_terminal. fs_u_pfc_proper_right_add_vbcoefficientscoefficientsum = fs_q_pfc_proper_right_add_vbcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_proper_right_add_vbcoefficients))) * fs_v_pfc_proper_right_add_vbcoefficientscoefficientsum) + (pfc_natural_sum_proper_right_add_vbcoefficientscoefficient))) /\\ forall fs_i_pfc_proper_right_add_vbcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_proper_right_add_vbcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_proper_right_add_vbcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_proper_right_add_vbcoefficientscoefficientsum_body_steps = S (pfc_index_proper_right_add_vbcoefficients)) -> exists fs_a_pfc_proper_right_add_vbcoefficientscoefficientsum_body_steps fs_r_pfc_proper_right_add_vbcoefficientscoefficientsum_body_steps fs_s_pfc_proper_right_add_vbcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_proper_right_add_vbcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_proper_right_add_vbcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_proper_right_add_vbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_proper_right_add_vbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_proper_right_add_vbcoefficientscoefficient)) /\\ exists fs_q_pfc_proper_right_add_vbcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_proper_right_add_vbcoefficientscoefficient = fs_q_pfc_proper_right_add_vbcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_proper_right_add_vbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_proper_right_add_vbcoefficientscoefficient) + (fs_a_pfc_proper_right_add_vbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_proper_right_add_vbcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_proper_right_add_vbcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_proper_right_add_vbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_proper_right_add_vbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_right_add_vbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_right_add_vbcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_proper_right_add_vbcoefficientscoefficientsum = fs_q_pfc_proper_right_add_vbcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_proper_right_add_vbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_right_add_vbcoefficientscoefficientsum) + (fs_r_pfc_proper_right_add_vbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_proper_right_add_vbcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_proper_right_add_vbcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_proper_right_add_vbcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_proper_right_add_vbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_right_add_vbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_right_add_vbcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_proper_right_add_vbcoefficientscoefficientsum = fs_q_pfc_proper_right_add_vbcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_proper_right_add_vbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_right_add_vbcoefficientscoefficientsum) + (fs_s_pfc_proper_right_add_vbcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_proper_right_add_vbcoefficientscoefficientsum_body_steps = fs_r_pfc_proper_right_add_vbcoefficientscoefficientsum_body_steps + fs_a_pfc_proper_right_add_vbcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_proper_right_add_vbcoefficientscoefficientresiduebound. pfa_gap_proper_right_add_vbcoefficientscoefficientresiduebound + S (pfc_value_proper_right_add_vbcoefficients) = (p)) /\\ ((exists pfa_offset_left_proper_right_add_vbcoefficientscoefficientresiduecongruence pfa_offset_right_proper_right_add_vbcoefficientscoefficientresiduecongruence. (pfc_natural_sum_proper_right_add_vbcoefficientscoefficient) + (p) * pfa_offset_left_proper_right_add_vbcoefficientscoefficientresiduecongruence = (pfc_value_proper_right_add_vbcoefficients) + (p) * pfa_offset_right_proper_right_add_vbcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_proper_right_add_wbleft. (exists fom_gap_pfp_proper_right_add_wbleft_index_bound. fom_gap_pfp_proper_right_add_wbleft_index_bound + S (fom_index_pfp_proper_right_add_wbleft) = L) -> exists fom_value_pfp_proper_right_add_wbleft. ((((exists fom_beta_height_pfp_proper_right_add_wbleft_entry. fom_beta_height_pfp_proper_right_add_wbleft_entry + S (fom_value_pfp_proper_right_add_wbleft) = S ((S (fom_index_pfp_proper_right_add_wbleft)) * cc)) /\\ exists fom_beta_quotient_pfp_proper_right_add_wbleft_entry. cb = fom_beta_quotient_pfp_proper_right_add_wbleft_entry * S ((S (fom_index_pfp_proper_right_add_wbleft)) * cc) + (fom_value_pfp_proper_right_add_wbleft))) /\\ (exists fom_gap_pfp_proper_right_add_wbleft_value_bound. fom_gap_pfp_proper_right_add_wbleft_value_bound + S (fom_value_pfp_proper_right_add_wbleft) = p))) /\\ (((forall fom_index_pfp_proper_right_add_wbright. (exists fom_gap_pfp_proper_right_add_wbright_index_bound. fom_gap_pfp_proper_right_add_wbright_index_bound + S (fom_index_pfp_proper_right_add_wbright) = M) -> exists fom_value_pfp_proper_right_add_wbright. ((((exists fom_beta_height_pfp_proper_right_add_wbright_entry. fom_beta_height_pfp_proper_right_add_wbright_entry + S (fom_value_pfp_proper_right_add_wbright) = S ((S (fom_index_pfp_proper_right_add_wbright)) * dc)) /\\ exists fom_beta_quotient_pfp_proper_right_add_wbright_entry. db = fom_beta_quotient_pfp_proper_right_add_wbright_entry * S ((S (fom_index_pfp_proper_right_add_wbright)) * dc) + (fom_value_pfp_proper_right_add_wbright))) /\\ (exists fom_gap_pfp_proper_right_add_wbright_value_bound. fom_gap_pfp_proper_right_add_wbright_value_bound + S (fom_value_pfp_proper_right_add_wbright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_proper_right_add_wbcoefficients. (exists pfa_gap_proper_right_add_wbcoefficientsbound. pfa_gap_proper_right_add_wbcoefficientsbound + S (pfc_index_proper_right_add_wbcoefficients) = (N)) -> exists pfc_value_proper_right_add_wbcoefficients. ((((exists ff_h_pfp_proper_right_add_wbcoefficientsentry. ff_h_pfp_proper_right_add_wbcoefficientsentry + S (pfc_value_proper_right_add_wbcoefficients) = S ((S (pfc_index_proper_right_add_wbcoefficients)) * wc)) /\\ exists ff_q_pfp_proper_right_add_wbcoefficientsentry. wb = ff_q_pfp_proper_right_add_wbcoefficientsentry * S ((S (pfc_index_proper_right_add_wbcoefficients)) * wc) + (pfc_value_proper_right_add_wbcoefficients))) /\\ ((exists pfc_terms_code_proper_right_add_wbcoefficientscoefficient pfc_terms_scale_proper_right_add_wbcoefficientscoefficient pfc_natural_sum_proper_right_add_wbcoefficientscoefficient. ((forall pfc_index_proper_right_add_wbcoefficientscoefficientdiagonal. (exists pfa_gap_proper_right_add_wbcoefficientscoefficientdiagonalbound. pfa_gap_proper_right_add_wbcoefficientscoefficientdiagonalbound + S (pfc_index_proper_right_add_wbcoefficientscoefficientdiagonal) = (S (pfc_index_proper_right_add_wbcoefficients))) -> exists pfc_value_proper_right_add_wbcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_proper_right_add_wbcoefficientscoefficientdiagonalentry. ff_h_pfp_proper_right_add_wbcoefficientscoefficientdiagonalentry + S (pfc_value_proper_right_add_wbcoefficientscoefficientdiagonal) = S ((S (pfc_index_proper_right_add_wbcoefficientscoefficientdiagonal)) * pfc_terms_scale_proper_right_add_wbcoefficientscoefficient)) /\\ exists ff_q_pfp_proper_right_add_wbcoefficientscoefficientdiagonalentry. pfc_terms_code_proper_right_add_wbcoefficientscoefficient = ff_q_pfp_proper_right_add_wbcoefficientscoefficientdiagonalentry * S ((S (pfc_index_proper_right_add_wbcoefficientscoefficientdiagonal)) * pfc_terms_scale_proper_right_add_wbcoefficientscoefficient) + (pfc_value_proper_right_add_wbcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_proper_right_add_wbcoefficientscoefficientdiagonalterm pfc_left_proper_right_add_wbcoefficientscoefficientdiagonalterm pfc_right_proper_right_add_wbcoefficientscoefficientdiagonalterm. (((pfc_index_proper_right_add_wbcoefficientscoefficientdiagonal)+pfc_complement_proper_right_add_wbcoefficientscoefficientdiagonalterm=(pfc_index_proper_right_add_wbcoefficients)) /\\ ((((((exists pfa_gap_proper_right_add_wbcoefficientscoefficientdiagonaltermleftinside. pfa_gap_proper_right_add_wbcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_proper_right_add_wbcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_proper_right_add_wbcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_proper_right_add_wbcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_proper_right_add_wbcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_proper_right_add_wbcoefficientscoefficientdiagonal)) * cc)) /\\ exists ff_q_pfp_proper_right_add_wbcoefficientscoefficientdiagonaltermleftentry. cb = ff_q_pfp_proper_right_add_wbcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_proper_right_add_wbcoefficientscoefficientdiagonal)) * cc) + (pfc_left_proper_right_add_wbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_proper_right_add_wbcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_proper_right_add_wbcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_proper_right_add_wbcoefficientscoefficientdiagonal)) /\\ (((pfc_left_proper_right_add_wbcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_proper_right_add_wbcoefficientscoefficientdiagonaltermrightinside. pfa_gap_proper_right_add_wbcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_proper_right_add_wbcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_proper_right_add_wbcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_proper_right_add_wbcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_proper_right_add_wbcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_proper_right_add_wbcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_proper_right_add_wbcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_proper_right_add_wbcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_proper_right_add_wbcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_proper_right_add_wbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_proper_right_add_wbcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_proper_right_add_wbcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_proper_right_add_wbcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_proper_right_add_wbcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_proper_right_add_wbcoefficientscoefficientdiagonal)=pfc_left_proper_right_add_wbcoefficientscoefficientdiagonalterm*pfc_right_proper_right_add_wbcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_proper_right_add_wbcoefficientscoefficientsum fs_v_pfc_proper_right_add_wbcoefficientscoefficientsum. ((((exists fs_h_pfc_proper_right_add_wbcoefficientscoefficientsum_body_start. fs_h_pfc_proper_right_add_wbcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_proper_right_add_wbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_right_add_wbcoefficientscoefficientsum_body_start. fs_u_pfc_proper_right_add_wbcoefficientscoefficientsum = fs_q_pfc_proper_right_add_wbcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_proper_right_add_wbcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_proper_right_add_wbcoefficientscoefficientsum_body_terminal. fs_h_pfc_proper_right_add_wbcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_proper_right_add_wbcoefficientscoefficient) = S ((S (S (pfc_index_proper_right_add_wbcoefficients))) * fs_v_pfc_proper_right_add_wbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_right_add_wbcoefficientscoefficientsum_body_terminal. fs_u_pfc_proper_right_add_wbcoefficientscoefficientsum = fs_q_pfc_proper_right_add_wbcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_proper_right_add_wbcoefficients))) * fs_v_pfc_proper_right_add_wbcoefficientscoefficientsum) + (pfc_natural_sum_proper_right_add_wbcoefficientscoefficient))) /\\ forall fs_i_pfc_proper_right_add_wbcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_proper_right_add_wbcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_proper_right_add_wbcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_proper_right_add_wbcoefficientscoefficientsum_body_steps = S (pfc_index_proper_right_add_wbcoefficients)) -> exists fs_a_pfc_proper_right_add_wbcoefficientscoefficientsum_body_steps fs_r_pfc_proper_right_add_wbcoefficientscoefficientsum_body_steps fs_s_pfc_proper_right_add_wbcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_proper_right_add_wbcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_proper_right_add_wbcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_proper_right_add_wbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_proper_right_add_wbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_proper_right_add_wbcoefficientscoefficient)) /\\ exists fs_q_pfc_proper_right_add_wbcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_proper_right_add_wbcoefficientscoefficient = fs_q_pfc_proper_right_add_wbcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_proper_right_add_wbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_proper_right_add_wbcoefficientscoefficient) + (fs_a_pfc_proper_right_add_wbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_proper_right_add_wbcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_proper_right_add_wbcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_proper_right_add_wbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_proper_right_add_wbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_right_add_wbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_right_add_wbcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_proper_right_add_wbcoefficientscoefficientsum = fs_q_pfc_proper_right_add_wbcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_proper_right_add_wbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_right_add_wbcoefficientscoefficientsum) + (fs_r_pfc_proper_right_add_wbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_proper_right_add_wbcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_proper_right_add_wbcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_proper_right_add_wbcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_proper_right_add_wbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_right_add_wbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_right_add_wbcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_proper_right_add_wbcoefficientscoefficientsum = fs_q_pfc_proper_right_add_wbcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_proper_right_add_wbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_right_add_wbcoefficientscoefficientsum) + (fs_s_pfc_proper_right_add_wbcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_proper_right_add_wbcoefficientscoefficientsum_body_steps = fs_r_pfc_proper_right_add_wbcoefficientscoefficientsum_body_steps + fs_a_pfc_proper_right_add_wbcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_proper_right_add_wbcoefficientscoefficientresiduebound. pfa_gap_proper_right_add_wbcoefficientscoefficientresiduebound + S (pfc_value_proper_right_add_wbcoefficients) = (p)) /\\ ((exists pfa_offset_left_proper_right_add_wbcoefficientscoefficientresiduecongruence pfa_offset_right_proper_right_add_wbcoefficientscoefficientresiduecongruence. (pfc_natural_sum_proper_right_add_wbcoefficientscoefficient) + (p) * pfa_offset_left_proper_right_add_wbcoefficientscoefficientresiduecongruence = (pfc_value_proper_right_add_wbcoefficients) + (p) * pfa_offset_right_proper_right_add_wbcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (forall pfp_index_proper_right_add_result. (exists pfa_gap_proper_right_add_resultindex. pfa_gap_proper_right_add_resultindex + S (pfp_index_proper_right_add_result) = (N)) -> exists pfp_left_proper_right_add_result pfp_right_proper_right_add_result pfp_value_proper_right_add_result. ((((exists ff_h_pfp_proper_right_add_resultleft. ff_h_pfp_proper_right_add_resultleft + S (pfp_left_proper_right_add_result) = S ((S (pfp_index_proper_right_add_result)) * uc)) /\\ exists ff_q_pfp_proper_right_add_resultleft. ub = ff_q_pfp_proper_right_add_resultleft * S ((S (pfp_index_proper_right_add_result)) * uc) + (pfp_left_proper_right_add_result))) /\\ (((((exists ff_h_pfp_proper_right_add_resultright. ff_h_pfp_proper_right_add_resultright + S (pfp_right_proper_right_add_result) = S ((S (pfp_index_proper_right_add_result)) * vc)) /\\ exists ff_q_pfp_proper_right_add_resultright. vb = ff_q_pfp_proper_right_add_resultright * S ((S (pfp_index_proper_right_add_result)) * vc) + (pfp_right_proper_right_add_result))) /\\ (((((exists ff_h_pfp_proper_right_add_resulttarget. ff_h_pfp_proper_right_add_resulttarget + S (pfp_value_proper_right_add_result) = S ((S (pfp_index_proper_right_add_result)) * wc)) /\\ exists ff_q_pfp_proper_right_add_resulttarget. wb = ff_q_pfp_proper_right_add_resulttarget * S ((S (pfp_index_proper_right_add_result)) * wc) + (pfp_value_proper_right_add_result))) /\\ ((((exists pfa_gap_proper_right_add_resultoperationleft. pfa_gap_proper_right_add_resultoperationleft + S (pfp_left_proper_right_add_result) = (p)) /\\ (((exists pfa_gap_proper_right_add_resultoperationright. pfa_gap_proper_right_add_resultoperationright + S (pfp_right_proper_right_add_result) = (p)) /\\ ((((exists pfa_gap_proper_right_add_resultoperationresultbound. pfa_gap_proper_right_add_resultoperationresultbound + S (pfp_value_proper_right_add_result) = (p)) /\\ ((exists pfa_offset_left_proper_right_add_resultoperationresultcongruence pfa_offset_right_proper_right_add_resultoperationresultcongruence. 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dc M ub uc vb vc wb wc N. (forall pfs_index_proper_left_subtract_source. (exists pfa_gap_proper_left_subtract_sourceindex. pfa_gap_proper_left_subtract_sourceindex + S (pfs_index_proper_left_subtract_source) = (L)) -> exists pfs_left_proper_left_subtract_source pfs_right_proper_left_subtract_source pfs_result_proper_left_subtract_source. ((((exists ff_h_pfp_proper_left_subtract_sourceleft. ff_h_pfp_proper_left_subtract_sourceleft + S (pfs_left_proper_left_subtract_source) = S ((S (pfs_index_proper_left_subtract_source)) * ac)) /\\ exists ff_q_pfp_proper_left_subtract_sourceleft. ab = ff_q_pfp_proper_left_subtract_sourceleft * S ((S (pfs_index_proper_left_subtract_source)) * ac) + (pfs_left_proper_left_subtract_source))) /\\ (((((exists ff_h_pfp_proper_left_subtract_sourceright. ff_h_pfp_proper_left_subtract_sourceright + S (pfs_right_proper_left_subtract_source) = S ((S (pfs_index_proper_left_subtract_source)) * bc)) /\\ exists ff_q_pfp_proper_left_subtract_sourceright. bb = ff_q_pfp_proper_left_subtract_sourceright * S ((S (pfs_index_proper_left_subtract_source)) * bc) + (pfs_right_proper_left_subtract_source))) /\\ (((((exists ff_h_pfp_proper_left_subtract_sourceresult. ff_h_pfp_proper_left_subtract_sourceresult + S (pfs_result_proper_left_subtract_source) = S ((S (pfs_index_proper_left_subtract_source)) * cc)) /\\ exists ff_q_pfp_proper_left_subtract_sourceresult. cb = ff_q_pfp_proper_left_subtract_sourceresult * S ((S (pfs_index_proper_left_subtract_source)) * cc) + (pfs_result_proper_left_subtract_source))) /\\ ((((exists pfa_gap_proper_left_subtract_sourceoperationleft. pfa_gap_proper_left_subtract_sourceoperationleft + S (pfs_right_proper_left_subtract_source) = (p)) /\\ (((exists pfa_gap_proper_left_subtract_sourceoperationright. pfa_gap_proper_left_subtract_sourceoperationright + S (pfs_result_proper_left_subtract_source) = (p)) /\\ ((((exists pfa_gap_proper_left_subtract_sourceoperationresultbound. pfa_gap_proper_left_subtract_sourceoperationresultbound + S (pfs_left_proper_left_subtract_source) = (p)) /\\ ((exists pfa_offset_left_proper_left_subtract_sourceoperationresultcongruence pfa_offset_right_proper_left_subtract_sourceoperationresultcongruence. ((pfs_right_proper_left_subtract_source) + (pfs_result_proper_left_subtract_source)) + (p) * pfa_offset_left_proper_left_subtract_sourceoperationresultcongruence = (pfs_left_proper_left_subtract_source) + (p) * pfa_offset_right_proper_left_subtract_sourceoperationresultcongruence)))))))))))))))) -> (((forall fom_index_pfp_proper_left_subtract_ubleft. (exists fom_gap_pfp_proper_left_subtract_ubleft_index_bound. fom_gap_pfp_proper_left_subtract_ubleft_index_bound + S (fom_index_pfp_proper_left_subtract_ubleft) = M) -> exists fom_value_pfp_proper_left_subtract_ubleft. ((((exists fom_beta_height_pfp_proper_left_subtract_ubleft_entry. fom_beta_height_pfp_proper_left_subtract_ubleft_entry + S (fom_value_pfp_proper_left_subtract_ubleft) = S ((S (fom_index_pfp_proper_left_subtract_ubleft)) * dc)) /\\ exists fom_beta_quotient_pfp_proper_left_subtract_ubleft_entry. db = fom_beta_quotient_pfp_proper_left_subtract_ubleft_entry * S ((S (fom_index_pfp_proper_left_subtract_ubleft)) * dc) + (fom_value_pfp_proper_left_subtract_ubleft))) /\\ (exists fom_gap_pfp_proper_left_subtract_ubleft_value_bound. fom_gap_pfp_proper_left_subtract_ubleft_value_bound + S (fom_value_pfp_proper_left_subtract_ubleft) = p))) /\\ (((forall fom_index_pfp_proper_left_subtract_ubright. (exists fom_gap_pfp_proper_left_subtract_ubright_index_bound. fom_gap_pfp_proper_left_subtract_ubright_index_bound + S (fom_index_pfp_proper_left_subtract_ubright) = L) -> exists fom_value_pfp_proper_left_subtract_ubright. ((((exists fom_beta_height_pfp_proper_left_subtract_ubright_entry. fom_beta_height_pfp_proper_left_subtract_ubright_entry + S (fom_value_pfp_proper_left_subtract_ubright) = S ((S (fom_index_pfp_proper_left_subtract_ubright)) * ac)) /\\ exists fom_beta_quotient_pfp_proper_left_subtract_ubright_entry. ab = fom_beta_quotient_pfp_proper_left_subtract_ubright_entry * S ((S (fom_index_pfp_proper_left_subtract_ubright)) * ac) + (fom_value_pfp_proper_left_subtract_ubright))) /\\ (exists fom_gap_pfp_proper_left_subtract_ubright_value_bound. fom_gap_pfp_proper_left_subtract_ubright_value_bound + S (fom_value_pfp_proper_left_subtract_ubright) = p))) /\\ (((((((M)=0 \\/ (L)=0) /\\ (((N)=0)))) \\/ (((~((M)=0)) /\\ (((~((L)=0)) /\\ (((M)+(L)=S (N)))))))) /\\ ((forall pfc_index_proper_left_subtract_ubcoefficients. (exists pfa_gap_proper_left_subtract_ubcoefficientsbound. pfa_gap_proper_left_subtract_ubcoefficientsbound + S (pfc_index_proper_left_subtract_ubcoefficients) = (N)) -> exists pfc_value_proper_left_subtract_ubcoefficients. ((((exists ff_h_pfp_proper_left_subtract_ubcoefficientsentry. ff_h_pfp_proper_left_subtract_ubcoefficientsentry + S (pfc_value_proper_left_subtract_ubcoefficients) = S ((S (pfc_index_proper_left_subtract_ubcoefficients)) * uc)) /\\ exists ff_q_pfp_proper_left_subtract_ubcoefficientsentry. ub = ff_q_pfp_proper_left_subtract_ubcoefficientsentry * S ((S (pfc_index_proper_left_subtract_ubcoefficients)) * uc) + (pfc_value_proper_left_subtract_ubcoefficients))) /\\ ((exists pfc_terms_code_proper_left_subtract_ubcoefficientscoefficient pfc_terms_scale_proper_left_subtract_ubcoefficientscoefficient pfc_natural_sum_proper_left_subtract_ubcoefficientscoefficient. ((forall pfc_index_proper_left_subtract_ubcoefficientscoefficientdiagonal. (exists pfa_gap_proper_left_subtract_ubcoefficientscoefficientdiagonalbound. pfa_gap_proper_left_subtract_ubcoefficientscoefficientdiagonalbound + S (pfc_index_proper_left_subtract_ubcoefficientscoefficientdiagonal) = (S (pfc_index_proper_left_subtract_ubcoefficients))) -> exists pfc_value_proper_left_subtract_ubcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_proper_left_subtract_ubcoefficientscoefficientdiagonalentry. ff_h_pfp_proper_left_subtract_ubcoefficientscoefficientdiagonalentry + S (pfc_value_proper_left_subtract_ubcoefficientscoefficientdiagonal) = S ((S (pfc_index_proper_left_subtract_ubcoefficientscoefficientdiagonal)) * pfc_terms_scale_proper_left_subtract_ubcoefficientscoefficient)) /\\ exists ff_q_pfp_proper_left_subtract_ubcoefficientscoefficientdiagonalentry. pfc_terms_code_proper_left_subtract_ubcoefficientscoefficient = ff_q_pfp_proper_left_subtract_ubcoefficientscoefficientdiagonalentry * S ((S (pfc_index_proper_left_subtract_ubcoefficientscoefficientdiagonal)) * pfc_terms_scale_proper_left_subtract_ubcoefficientscoefficient) + (pfc_value_proper_left_subtract_ubcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_proper_left_subtract_ubcoefficientscoefficientdiagonalterm pfc_left_proper_left_subtract_ubcoefficientscoefficientdiagonalterm pfc_right_proper_left_subtract_ubcoefficientscoefficientdiagonalterm. (((pfc_index_proper_left_subtract_ubcoefficientscoefficientdiagonal)+pfc_complement_proper_left_subtract_ubcoefficientscoefficientdiagonalterm=(pfc_index_proper_left_subtract_ubcoefficients)) /\\ ((((((exists pfa_gap_proper_left_subtract_ubcoefficientscoefficientdiagonaltermleftinside. pfa_gap_proper_left_subtract_ubcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_proper_left_subtract_ubcoefficientscoefficientdiagonal) = (M)) /\\ ((((exists ff_h_pfp_proper_left_subtract_ubcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_proper_left_subtract_ubcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_proper_left_subtract_ubcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_proper_left_subtract_ubcoefficientscoefficientdiagonal)) * dc)) /\\ exists ff_q_pfp_proper_left_subtract_ubcoefficientscoefficientdiagonaltermleftentry. db = ff_q_pfp_proper_left_subtract_ubcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_proper_left_subtract_ubcoefficientscoefficientdiagonal)) * dc) + (pfc_left_proper_left_subtract_ubcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_proper_left_subtract_ubcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_proper_left_subtract_ubcoefficientscoefficientdiagonaltermleftoutside+(M)=(pfc_index_proper_left_subtract_ubcoefficientscoefficientdiagonal)) /\\ (((pfc_left_proper_left_subtract_ubcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_proper_left_subtract_ubcoefficientscoefficientdiagonaltermrightinside. pfa_gap_proper_left_subtract_ubcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_proper_left_subtract_ubcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_proper_left_subtract_ubcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_proper_left_subtract_ubcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_proper_left_subtract_ubcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_proper_left_subtract_ubcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_proper_left_subtract_ubcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_proper_left_subtract_ubcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_proper_left_subtract_ubcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_proper_left_subtract_ubcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_proper_left_subtract_ubcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_proper_left_subtract_ubcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_proper_left_subtract_ubcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_proper_left_subtract_ubcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_proper_left_subtract_ubcoefficientscoefficientdiagonal)=pfc_left_proper_left_subtract_ubcoefficientscoefficientdiagonalterm*pfc_right_proper_left_subtract_ubcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_proper_left_subtract_ubcoefficientscoefficientsum fs_v_pfc_proper_left_subtract_ubcoefficientscoefficientsum. ((((exists fs_h_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_start. fs_h_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_proper_left_subtract_ubcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_start. fs_u_pfc_proper_left_subtract_ubcoefficientscoefficientsum = fs_q_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_proper_left_subtract_ubcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_terminal. fs_h_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_proper_left_subtract_ubcoefficientscoefficient) = S ((S (S (pfc_index_proper_left_subtract_ubcoefficients))) * fs_v_pfc_proper_left_subtract_ubcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_terminal. fs_u_pfc_proper_left_subtract_ubcoefficientscoefficientsum = fs_q_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_proper_left_subtract_ubcoefficients))) * fs_v_pfc_proper_left_subtract_ubcoefficientscoefficientsum) + (pfc_natural_sum_proper_left_subtract_ubcoefficientscoefficient))) /\\ forall fs_i_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_steps = S (pfc_index_proper_left_subtract_ubcoefficients)) -> exists fs_a_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_steps fs_r_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_steps fs_s_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_proper_left_subtract_ubcoefficientscoefficient)) /\\ exists fs_q_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_proper_left_subtract_ubcoefficientscoefficient = fs_q_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_proper_left_subtract_ubcoefficientscoefficient) + (fs_a_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_left_subtract_ubcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_proper_left_subtract_ubcoefficientscoefficientsum = fs_q_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_left_subtract_ubcoefficientscoefficientsum) + (fs_r_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_left_subtract_ubcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_proper_left_subtract_ubcoefficientscoefficientsum = fs_q_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_left_subtract_ubcoefficientscoefficientsum) + (fs_s_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_steps = fs_r_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_steps + fs_a_pfc_proper_left_subtract_ubcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_proper_left_subtract_ubcoefficientscoefficientresiduebound. pfa_gap_proper_left_subtract_ubcoefficientscoefficientresiduebound + S (pfc_value_proper_left_subtract_ubcoefficients) = (p)) /\\ ((exists pfa_offset_left_proper_left_subtract_ubcoefficientscoefficientresiduecongruence pfa_offset_right_proper_left_subtract_ubcoefficientscoefficientresiduecongruence. (pfc_natural_sum_proper_left_subtract_ubcoefficientscoefficient) + (p) * pfa_offset_left_proper_left_subtract_ubcoefficientscoefficientresiduecongruence = (pfc_value_proper_left_subtract_ubcoefficients) + (p) * pfa_offset_right_proper_left_subtract_ubcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_proper_left_subtract_vbleft. (exists fom_gap_pfp_proper_left_subtract_vbleft_index_bound. fom_gap_pfp_proper_left_subtract_vbleft_index_bound + S (fom_index_pfp_proper_left_subtract_vbleft) = M) -> exists fom_value_pfp_proper_left_subtract_vbleft. ((((exists fom_beta_height_pfp_proper_left_subtract_vbleft_entry. fom_beta_height_pfp_proper_left_subtract_vbleft_entry + S (fom_value_pfp_proper_left_subtract_vbleft) = S ((S (fom_index_pfp_proper_left_subtract_vbleft)) * dc)) /\\ exists fom_beta_quotient_pfp_proper_left_subtract_vbleft_entry. db = fom_beta_quotient_pfp_proper_left_subtract_vbleft_entry * S ((S (fom_index_pfp_proper_left_subtract_vbleft)) * dc) + (fom_value_pfp_proper_left_subtract_vbleft))) /\\ (exists fom_gap_pfp_proper_left_subtract_vbleft_value_bound. fom_gap_pfp_proper_left_subtract_vbleft_value_bound + S (fom_value_pfp_proper_left_subtract_vbleft) = p))) /\\ (((forall fom_index_pfp_proper_left_subtract_vbright. (exists fom_gap_pfp_proper_left_subtract_vbright_index_bound. fom_gap_pfp_proper_left_subtract_vbright_index_bound + S (fom_index_pfp_proper_left_subtract_vbright) = L) -> exists fom_value_pfp_proper_left_subtract_vbright. ((((exists fom_beta_height_pfp_proper_left_subtract_vbright_entry. fom_beta_height_pfp_proper_left_subtract_vbright_entry + S (fom_value_pfp_proper_left_subtract_vbright) = S ((S (fom_index_pfp_proper_left_subtract_vbright)) * bc)) /\\ exists fom_beta_quotient_pfp_proper_left_subtract_vbright_entry. bb = fom_beta_quotient_pfp_proper_left_subtract_vbright_entry * S ((S (fom_index_pfp_proper_left_subtract_vbright)) * bc) + (fom_value_pfp_proper_left_subtract_vbright))) /\\ (exists fom_gap_pfp_proper_left_subtract_vbright_value_bound. fom_gap_pfp_proper_left_subtract_vbright_value_bound + S (fom_value_pfp_proper_left_subtract_vbright) = p))) /\\ (((((((M)=0 \\/ (L)=0) /\\ (((N)=0)))) \\/ (((~((M)=0)) /\\ (((~((L)=0)) /\\ (((M)+(L)=S (N)))))))) /\\ ((forall pfc_index_proper_left_subtract_vbcoefficients. (exists pfa_gap_proper_left_subtract_vbcoefficientsbound. pfa_gap_proper_left_subtract_vbcoefficientsbound + S (pfc_index_proper_left_subtract_vbcoefficients) = (N)) -> exists pfc_value_proper_left_subtract_vbcoefficients. ((((exists ff_h_pfp_proper_left_subtract_vbcoefficientsentry. ff_h_pfp_proper_left_subtract_vbcoefficientsentry + S (pfc_value_proper_left_subtract_vbcoefficients) = S ((S (pfc_index_proper_left_subtract_vbcoefficients)) * vc)) /\\ exists ff_q_pfp_proper_left_subtract_vbcoefficientsentry. vb = ff_q_pfp_proper_left_subtract_vbcoefficientsentry * S ((S (pfc_index_proper_left_subtract_vbcoefficients)) * vc) + (pfc_value_proper_left_subtract_vbcoefficients))) /\\ ((exists pfc_terms_code_proper_left_subtract_vbcoefficientscoefficient pfc_terms_scale_proper_left_subtract_vbcoefficientscoefficient pfc_natural_sum_proper_left_subtract_vbcoefficientscoefficient. ((forall pfc_index_proper_left_subtract_vbcoefficientscoefficientdiagonal. (exists pfa_gap_proper_left_subtract_vbcoefficientscoefficientdiagonalbound. pfa_gap_proper_left_subtract_vbcoefficientscoefficientdiagonalbound + S (pfc_index_proper_left_subtract_vbcoefficientscoefficientdiagonal) = (S (pfc_index_proper_left_subtract_vbcoefficients))) -> exists pfc_value_proper_left_subtract_vbcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_proper_left_subtract_vbcoefficientscoefficientdiagonalentry. ff_h_pfp_proper_left_subtract_vbcoefficientscoefficientdiagonalentry + S (pfc_value_proper_left_subtract_vbcoefficientscoefficientdiagonal) = S ((S (pfc_index_proper_left_subtract_vbcoefficientscoefficientdiagonal)) * pfc_terms_scale_proper_left_subtract_vbcoefficientscoefficient)) /\\ exists ff_q_pfp_proper_left_subtract_vbcoefficientscoefficientdiagonalentry. pfc_terms_code_proper_left_subtract_vbcoefficientscoefficient = ff_q_pfp_proper_left_subtract_vbcoefficientscoefficientdiagonalentry * S ((S (pfc_index_proper_left_subtract_vbcoefficientscoefficientdiagonal)) * pfc_terms_scale_proper_left_subtract_vbcoefficientscoefficient) + (pfc_value_proper_left_subtract_vbcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_proper_left_subtract_vbcoefficientscoefficientdiagonalterm pfc_left_proper_left_subtract_vbcoefficientscoefficientdiagonalterm pfc_right_proper_left_subtract_vbcoefficientscoefficientdiagonalterm. (((pfc_index_proper_left_subtract_vbcoefficientscoefficientdiagonal)+pfc_complement_proper_left_subtract_vbcoefficientscoefficientdiagonalterm=(pfc_index_proper_left_subtract_vbcoefficients)) /\\ ((((((exists pfa_gap_proper_left_subtract_vbcoefficientscoefficientdiagonaltermleftinside. pfa_gap_proper_left_subtract_vbcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_proper_left_subtract_vbcoefficientscoefficientdiagonal) = (M)) /\\ ((((exists ff_h_pfp_proper_left_subtract_vbcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_proper_left_subtract_vbcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_proper_left_subtract_vbcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_proper_left_subtract_vbcoefficientscoefficientdiagonal)) * dc)) /\\ exists ff_q_pfp_proper_left_subtract_vbcoefficientscoefficientdiagonaltermleftentry. db = ff_q_pfp_proper_left_subtract_vbcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_proper_left_subtract_vbcoefficientscoefficientdiagonal)) * dc) + (pfc_left_proper_left_subtract_vbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_proper_left_subtract_vbcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_proper_left_subtract_vbcoefficientscoefficientdiagonaltermleftoutside+(M)=(pfc_index_proper_left_subtract_vbcoefficientscoefficientdiagonal)) /\\ (((pfc_left_proper_left_subtract_vbcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_proper_left_subtract_vbcoefficientscoefficientdiagonaltermrightinside. pfa_gap_proper_left_subtract_vbcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_proper_left_subtract_vbcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_proper_left_subtract_vbcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_proper_left_subtract_vbcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_proper_left_subtract_vbcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_proper_left_subtract_vbcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_proper_left_subtract_vbcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_proper_left_subtract_vbcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_proper_left_subtract_vbcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_proper_left_subtract_vbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_proper_left_subtract_vbcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_proper_left_subtract_vbcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_proper_left_subtract_vbcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_proper_left_subtract_vbcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_proper_left_subtract_vbcoefficientscoefficientdiagonal)=pfc_left_proper_left_subtract_vbcoefficientscoefficientdiagonalterm*pfc_right_proper_left_subtract_vbcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_proper_left_subtract_vbcoefficientscoefficientsum fs_v_pfc_proper_left_subtract_vbcoefficientscoefficientsum. ((((exists fs_h_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_start. fs_h_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_proper_left_subtract_vbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_start. fs_u_pfc_proper_left_subtract_vbcoefficientscoefficientsum = fs_q_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_proper_left_subtract_vbcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_terminal. fs_h_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_proper_left_subtract_vbcoefficientscoefficient) = S ((S (S (pfc_index_proper_left_subtract_vbcoefficients))) * fs_v_pfc_proper_left_subtract_vbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_terminal. fs_u_pfc_proper_left_subtract_vbcoefficientscoefficientsum = fs_q_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_proper_left_subtract_vbcoefficients))) * fs_v_pfc_proper_left_subtract_vbcoefficientscoefficientsum) + (pfc_natural_sum_proper_left_subtract_vbcoefficientscoefficient))) /\\ forall fs_i_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_steps = S (pfc_index_proper_left_subtract_vbcoefficients)) -> exists fs_a_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_steps fs_r_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_steps fs_s_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_proper_left_subtract_vbcoefficientscoefficient)) /\\ exists fs_q_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_proper_left_subtract_vbcoefficientscoefficient = fs_q_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_proper_left_subtract_vbcoefficientscoefficient) + (fs_a_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_left_subtract_vbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_proper_left_subtract_vbcoefficientscoefficientsum = fs_q_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_left_subtract_vbcoefficientscoefficientsum) + (fs_r_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_left_subtract_vbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_proper_left_subtract_vbcoefficientscoefficientsum = fs_q_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_left_subtract_vbcoefficientscoefficientsum) + (fs_s_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_steps = fs_r_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_steps + fs_a_pfc_proper_left_subtract_vbcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_proper_left_subtract_vbcoefficientscoefficientresiduebound. pfa_gap_proper_left_subtract_vbcoefficientscoefficientresiduebound + S (pfc_value_proper_left_subtract_vbcoefficients) = (p)) /\\ ((exists pfa_offset_left_proper_left_subtract_vbcoefficientscoefficientresiduecongruence pfa_offset_right_proper_left_subtract_vbcoefficientscoefficientresiduecongruence. (pfc_natural_sum_proper_left_subtract_vbcoefficientscoefficient) + (p) * pfa_offset_left_proper_left_subtract_vbcoefficientscoefficientresiduecongruence = (pfc_value_proper_left_subtract_vbcoefficients) + (p) * pfa_offset_right_proper_left_subtract_vbcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_proper_left_subtract_wbleft. (exists fom_gap_pfp_proper_left_subtract_wbleft_index_bound. fom_gap_pfp_proper_left_subtract_wbleft_index_bound + S (fom_index_pfp_proper_left_subtract_wbleft) = M) -> exists fom_value_pfp_proper_left_subtract_wbleft. ((((exists fom_beta_height_pfp_proper_left_subtract_wbleft_entry. fom_beta_height_pfp_proper_left_subtract_wbleft_entry + S (fom_value_pfp_proper_left_subtract_wbleft) = S ((S (fom_index_pfp_proper_left_subtract_wbleft)) * dc)) /\\ exists fom_beta_quotient_pfp_proper_left_subtract_wbleft_entry. db = fom_beta_quotient_pfp_proper_left_subtract_wbleft_entry * S ((S (fom_index_pfp_proper_left_subtract_wbleft)) * dc) + (fom_value_pfp_proper_left_subtract_wbleft))) /\\ (exists fom_gap_pfp_proper_left_subtract_wbleft_value_bound. fom_gap_pfp_proper_left_subtract_wbleft_value_bound + S (fom_value_pfp_proper_left_subtract_wbleft) = p))) /\\ (((forall fom_index_pfp_proper_left_subtract_wbright. (exists fom_gap_pfp_proper_left_subtract_wbright_index_bound. fom_gap_pfp_proper_left_subtract_wbright_index_bound + S (fom_index_pfp_proper_left_subtract_wbright) = L) -> exists fom_value_pfp_proper_left_subtract_wbright. ((((exists fom_beta_height_pfp_proper_left_subtract_wbright_entry. fom_beta_height_pfp_proper_left_subtract_wbright_entry + S (fom_value_pfp_proper_left_subtract_wbright) = S ((S (fom_index_pfp_proper_left_subtract_wbright)) * cc)) /\\ exists fom_beta_quotient_pfp_proper_left_subtract_wbright_entry. cb = fom_beta_quotient_pfp_proper_left_subtract_wbright_entry * S ((S (fom_index_pfp_proper_left_subtract_wbright)) * cc) + (fom_value_pfp_proper_left_subtract_wbright))) /\\ (exists fom_gap_pfp_proper_left_subtract_wbright_value_bound. fom_gap_pfp_proper_left_subtract_wbright_value_bound + S (fom_value_pfp_proper_left_subtract_wbright) = p))) /\\ (((((((M)=0 \\/ (L)=0) /\\ (((N)=0)))) \\/ (((~((M)=0)) /\\ (((~((L)=0)) /\\ (((M)+(L)=S (N)))))))) /\\ ((forall pfc_index_proper_left_subtract_wbcoefficients. (exists pfa_gap_proper_left_subtract_wbcoefficientsbound. pfa_gap_proper_left_subtract_wbcoefficientsbound + S (pfc_index_proper_left_subtract_wbcoefficients) = (N)) -> exists pfc_value_proper_left_subtract_wbcoefficients. ((((exists ff_h_pfp_proper_left_subtract_wbcoefficientsentry. ff_h_pfp_proper_left_subtract_wbcoefficientsentry + S (pfc_value_proper_left_subtract_wbcoefficients) = S ((S (pfc_index_proper_left_subtract_wbcoefficients)) * wc)) /\\ exists ff_q_pfp_proper_left_subtract_wbcoefficientsentry. wb = ff_q_pfp_proper_left_subtract_wbcoefficientsentry * S ((S (pfc_index_proper_left_subtract_wbcoefficients)) * wc) + (pfc_value_proper_left_subtract_wbcoefficients))) /\\ ((exists pfc_terms_code_proper_left_subtract_wbcoefficientscoefficient pfc_terms_scale_proper_left_subtract_wbcoefficientscoefficient pfc_natural_sum_proper_left_subtract_wbcoefficientscoefficient. ((forall pfc_index_proper_left_subtract_wbcoefficientscoefficientdiagonal. (exists pfa_gap_proper_left_subtract_wbcoefficientscoefficientdiagonalbound. pfa_gap_proper_left_subtract_wbcoefficientscoefficientdiagonalbound + S (pfc_index_proper_left_subtract_wbcoefficientscoefficientdiagonal) = (S (pfc_index_proper_left_subtract_wbcoefficients))) -> exists pfc_value_proper_left_subtract_wbcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_proper_left_subtract_wbcoefficientscoefficientdiagonalentry. ff_h_pfp_proper_left_subtract_wbcoefficientscoefficientdiagonalentry + S (pfc_value_proper_left_subtract_wbcoefficientscoefficientdiagonal) = S ((S (pfc_index_proper_left_subtract_wbcoefficientscoefficientdiagonal)) * pfc_terms_scale_proper_left_subtract_wbcoefficientscoefficient)) /\\ exists ff_q_pfp_proper_left_subtract_wbcoefficientscoefficientdiagonalentry. pfc_terms_code_proper_left_subtract_wbcoefficientscoefficient = ff_q_pfp_proper_left_subtract_wbcoefficientscoefficientdiagonalentry * S ((S (pfc_index_proper_left_subtract_wbcoefficientscoefficientdiagonal)) * pfc_terms_scale_proper_left_subtract_wbcoefficientscoefficient) + (pfc_value_proper_left_subtract_wbcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_proper_left_subtract_wbcoefficientscoefficientdiagonalterm pfc_left_proper_left_subtract_wbcoefficientscoefficientdiagonalterm pfc_right_proper_left_subtract_wbcoefficientscoefficientdiagonalterm. (((pfc_index_proper_left_subtract_wbcoefficientscoefficientdiagonal)+pfc_complement_proper_left_subtract_wbcoefficientscoefficientdiagonalterm=(pfc_index_proper_left_subtract_wbcoefficients)) /\\ ((((((exists pfa_gap_proper_left_subtract_wbcoefficientscoefficientdiagonaltermleftinside. pfa_gap_proper_left_subtract_wbcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_proper_left_subtract_wbcoefficientscoefficientdiagonal) = (M)) /\\ ((((exists ff_h_pfp_proper_left_subtract_wbcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_proper_left_subtract_wbcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_proper_left_subtract_wbcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_proper_left_subtract_wbcoefficientscoefficientdiagonal)) * dc)) /\\ exists ff_q_pfp_proper_left_subtract_wbcoefficientscoefficientdiagonaltermleftentry. db = ff_q_pfp_proper_left_subtract_wbcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_proper_left_subtract_wbcoefficientscoefficientdiagonal)) * dc) + (pfc_left_proper_left_subtract_wbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_proper_left_subtract_wbcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_proper_left_subtract_wbcoefficientscoefficientdiagonaltermleftoutside+(M)=(pfc_index_proper_left_subtract_wbcoefficientscoefficientdiagonal)) /\\ (((pfc_left_proper_left_subtract_wbcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_proper_left_subtract_wbcoefficientscoefficientdiagonaltermrightinside. pfa_gap_proper_left_subtract_wbcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_proper_left_subtract_wbcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_proper_left_subtract_wbcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_proper_left_subtract_wbcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_proper_left_subtract_wbcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_proper_left_subtract_wbcoefficientscoefficientdiagonalterm)) * cc)) /\\ exists ff_q_pfp_proper_left_subtract_wbcoefficientscoefficientdiagonaltermrightentry. cb = ff_q_pfp_proper_left_subtract_wbcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_proper_left_subtract_wbcoefficientscoefficientdiagonalterm)) * cc) + (pfc_right_proper_left_subtract_wbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_proper_left_subtract_wbcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_proper_left_subtract_wbcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_proper_left_subtract_wbcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_proper_left_subtract_wbcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_proper_left_subtract_wbcoefficientscoefficientdiagonal)=pfc_left_proper_left_subtract_wbcoefficientscoefficientdiagonalterm*pfc_right_proper_left_subtract_wbcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_proper_left_subtract_wbcoefficientscoefficientsum fs_v_pfc_proper_left_subtract_wbcoefficientscoefficientsum. ((((exists fs_h_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_start. fs_h_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_proper_left_subtract_wbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_start. fs_u_pfc_proper_left_subtract_wbcoefficientscoefficientsum = fs_q_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_proper_left_subtract_wbcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_terminal. fs_h_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_proper_left_subtract_wbcoefficientscoefficient) = S ((S (S (pfc_index_proper_left_subtract_wbcoefficients))) * fs_v_pfc_proper_left_subtract_wbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_terminal. fs_u_pfc_proper_left_subtract_wbcoefficientscoefficientsum = fs_q_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_proper_left_subtract_wbcoefficients))) * fs_v_pfc_proper_left_subtract_wbcoefficientscoefficientsum) + (pfc_natural_sum_proper_left_subtract_wbcoefficientscoefficient))) /\\ forall fs_i_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_steps = S (pfc_index_proper_left_subtract_wbcoefficients)) -> exists fs_a_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_steps fs_r_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_steps fs_s_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_proper_left_subtract_wbcoefficientscoefficient)) /\\ exists fs_q_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_proper_left_subtract_wbcoefficientscoefficient = fs_q_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_proper_left_subtract_wbcoefficientscoefficient) + (fs_a_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_left_subtract_wbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_proper_left_subtract_wbcoefficientscoefficientsum = fs_q_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_left_subtract_wbcoefficientscoefficientsum) + (fs_r_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_left_subtract_wbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_proper_left_subtract_wbcoefficientscoefficientsum = fs_q_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_left_subtract_wbcoefficientscoefficientsum) + (fs_s_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_steps = fs_r_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_steps + fs_a_pfc_proper_left_subtract_wbcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_proper_left_subtract_wbcoefficientscoefficientresiduebound. pfa_gap_proper_left_subtract_wbcoefficientscoefficientresiduebound + S (pfc_value_proper_left_subtract_wbcoefficients) = (p)) /\\ ((exists pfa_offset_left_proper_left_subtract_wbcoefficientscoefficientresiduecongruence pfa_offset_right_proper_left_subtract_wbcoefficientscoefficientresiduecongruence. (pfc_natural_sum_proper_left_subtract_wbcoefficientscoefficient) + (p) * pfa_offset_left_proper_left_subtract_wbcoefficientscoefficientresiduecongruence = (pfc_value_proper_left_subtract_wbcoefficients) + (p) * pfa_offset_right_proper_left_subtract_wbcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (forall pfs_index_proper_left_subtract_result. (exists pfa_gap_proper_left_subtract_resultindex. pfa_gap_proper_left_subtract_resultindex + S (pfs_index_proper_left_subtract_result) = (N)) -> exists pfs_left_proper_left_subtract_result pfs_right_proper_left_subtract_result pfs_result_proper_left_subtract_result. ((((exists ff_h_pfp_proper_left_subtract_resultleft. ff_h_pfp_proper_left_subtract_resultleft + S (pfs_left_proper_left_subtract_result) = S ((S (pfs_index_proper_left_subtract_result)) * uc)) /\\ exists ff_q_pfp_proper_left_subtract_resultleft. ub = ff_q_pfp_proper_left_subtract_resultleft * S ((S (pfs_index_proper_left_subtract_result)) * uc) + (pfs_left_proper_left_subtract_result))) /\\ (((((exists ff_h_pfp_proper_left_subtract_resultright. ff_h_pfp_proper_left_subtract_resultright + S (pfs_right_proper_left_subtract_result) = S ((S (pfs_index_proper_left_subtract_result)) * vc)) /\\ exists ff_q_pfp_proper_left_subtract_resultright. vb = ff_q_pfp_proper_left_subtract_resultright * S ((S (pfs_index_proper_left_subtract_result)) * vc) + (pfs_right_proper_left_subtract_result))) /\\ (((((exists ff_h_pfp_proper_left_subtract_resultresult. ff_h_pfp_proper_left_subtract_resultresult + S (pfs_result_proper_left_subtract_result) = S ((S (pfs_index_proper_left_subtract_result)) * wc)) /\\ exists ff_q_pfp_proper_left_subtract_resultresult. wb = ff_q_pfp_proper_left_subtract_resultresult * S ((S (pfs_index_proper_left_subtract_result)) * wc) + (pfs_result_proper_left_subtract_result))) /\\ ((((exists pfa_gap_proper_left_subtract_resultoperationleft. pfa_gap_proper_left_subtract_resultoperationleft + S (pfs_right_proper_left_subtract_result) = (p)) /\\ (((exists pfa_gap_proper_left_subtract_resultoperationright. pfa_gap_proper_left_subtract_resultoperationright + S (pfs_result_proper_left_subtract_result) = (p)) /\\ ((((exists pfa_gap_proper_left_subtract_resultoperationresultbound. pfa_gap_proper_left_subtract_resultoperationresultbound + S (pfs_left_proper_left_subtract_result) = (p)) /\\ ((exists pfa_offset_left_proper_left_subtract_resultoperationresultcongruence pfa_offset_right_proper_left_subtract_resultoperationresultcongruence. 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(forall pfs_index_proper_right_subtract_source. (exists pfa_gap_proper_right_subtract_sourceindex. pfa_gap_proper_right_subtract_sourceindex + S (pfs_index_proper_right_subtract_source) = (L)) -> exists pfs_left_proper_right_subtract_source pfs_right_proper_right_subtract_source pfs_result_proper_right_subtract_source. ((((exists ff_h_pfp_proper_right_subtract_sourceleft. ff_h_pfp_proper_right_subtract_sourceleft + S (pfs_left_proper_right_subtract_source) = S ((S (pfs_index_proper_right_subtract_source)) * ac)) /\\ exists ff_q_pfp_proper_right_subtract_sourceleft. ab = ff_q_pfp_proper_right_subtract_sourceleft * S ((S (pfs_index_proper_right_subtract_source)) * ac) + (pfs_left_proper_right_subtract_source))) /\\ (((((exists ff_h_pfp_proper_right_subtract_sourceright. ff_h_pfp_proper_right_subtract_sourceright + S (pfs_right_proper_right_subtract_source) = S ((S (pfs_index_proper_right_subtract_source)) * bc)) /\\ exists ff_q_pfp_proper_right_subtract_sourceright. bb = ff_q_pfp_proper_right_subtract_sourceright * S ((S (pfs_index_proper_right_subtract_source)) * bc) + (pfs_right_proper_right_subtract_source))) /\\ (((((exists ff_h_pfp_proper_right_subtract_sourceresult. ff_h_pfp_proper_right_subtract_sourceresult + S (pfs_result_proper_right_subtract_source) = S ((S (pfs_index_proper_right_subtract_source)) * cc)) /\\ exists ff_q_pfp_proper_right_subtract_sourceresult. cb = ff_q_pfp_proper_right_subtract_sourceresult * S ((S (pfs_index_proper_right_subtract_source)) * cc) + (pfs_result_proper_right_subtract_source))) /\\ ((((exists pfa_gap_proper_right_subtract_sourceoperationleft. pfa_gap_proper_right_subtract_sourceoperationleft + S (pfs_right_proper_right_subtract_source) = (p)) /\\ (((exists pfa_gap_proper_right_subtract_sourceoperationright. pfa_gap_proper_right_subtract_sourceoperationright + S (pfs_result_proper_right_subtract_source) = (p)) /\\ ((((exists pfa_gap_proper_right_subtract_sourceoperationresultbound. pfa_gap_proper_right_subtract_sourceoperationresultbound + S (pfs_left_proper_right_subtract_source) = (p)) /\\ ((exists pfa_offset_left_proper_right_subtract_sourceoperationresultcongruence pfa_offset_right_proper_right_subtract_sourceoperationresultcongruence. ((pfs_right_proper_right_subtract_source) + (pfs_result_proper_right_subtract_source)) + (p) * pfa_offset_left_proper_right_subtract_sourceoperationresultcongruence = (pfs_left_proper_right_subtract_source) + (p) * pfa_offset_right_proper_right_subtract_sourceoperationresultcongruence)))))))))))))))) -> (((forall fom_index_pfp_proper_right_subtract_ubleft. (exists fom_gap_pfp_proper_right_subtract_ubleft_index_bound. fom_gap_pfp_proper_right_subtract_ubleft_index_bound + S (fom_index_pfp_proper_right_subtract_ubleft) = L) -> exists fom_value_pfp_proper_right_subtract_ubleft. ((((exists fom_beta_height_pfp_proper_right_subtract_ubleft_entry. fom_beta_height_pfp_proper_right_subtract_ubleft_entry + S (fom_value_pfp_proper_right_subtract_ubleft) = S ((S (fom_index_pfp_proper_right_subtract_ubleft)) * ac)) /\\ exists fom_beta_quotient_pfp_proper_right_subtract_ubleft_entry. ab = fom_beta_quotient_pfp_proper_right_subtract_ubleft_entry * S ((S (fom_index_pfp_proper_right_subtract_ubleft)) * ac) + (fom_value_pfp_proper_right_subtract_ubleft))) /\\ (exists fom_gap_pfp_proper_right_subtract_ubleft_value_bound. fom_gap_pfp_proper_right_subtract_ubleft_value_bound + S (fom_value_pfp_proper_right_subtract_ubleft) = p))) /\\ (((forall fom_index_pfp_proper_right_subtract_ubright. (exists fom_gap_pfp_proper_right_subtract_ubright_index_bound. fom_gap_pfp_proper_right_subtract_ubright_index_bound + S (fom_index_pfp_proper_right_subtract_ubright) = M) -> exists fom_value_pfp_proper_right_subtract_ubright. ((((exists fom_beta_height_pfp_proper_right_subtract_ubright_entry. fom_beta_height_pfp_proper_right_subtract_ubright_entry + S (fom_value_pfp_proper_right_subtract_ubright) = S ((S (fom_index_pfp_proper_right_subtract_ubright)) * dc)) /\\ exists fom_beta_quotient_pfp_proper_right_subtract_ubright_entry. db = fom_beta_quotient_pfp_proper_right_subtract_ubright_entry * S ((S (fom_index_pfp_proper_right_subtract_ubright)) * dc) + (fom_value_pfp_proper_right_subtract_ubright))) /\\ (exists fom_gap_pfp_proper_right_subtract_ubright_value_bound. fom_gap_pfp_proper_right_subtract_ubright_value_bound + S (fom_value_pfp_proper_right_subtract_ubright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_proper_right_subtract_ubcoefficients. (exists pfa_gap_proper_right_subtract_ubcoefficientsbound. pfa_gap_proper_right_subtract_ubcoefficientsbound + S (pfc_index_proper_right_subtract_ubcoefficients) = (N)) -> exists pfc_value_proper_right_subtract_ubcoefficients. ((((exists ff_h_pfp_proper_right_subtract_ubcoefficientsentry. ff_h_pfp_proper_right_subtract_ubcoefficientsentry + S (pfc_value_proper_right_subtract_ubcoefficients) = S ((S (pfc_index_proper_right_subtract_ubcoefficients)) * uc)) /\\ exists ff_q_pfp_proper_right_subtract_ubcoefficientsentry. ub = ff_q_pfp_proper_right_subtract_ubcoefficientsentry * S ((S (pfc_index_proper_right_subtract_ubcoefficients)) * uc) + (pfc_value_proper_right_subtract_ubcoefficients))) /\\ ((exists pfc_terms_code_proper_right_subtract_ubcoefficientscoefficient pfc_terms_scale_proper_right_subtract_ubcoefficientscoefficient pfc_natural_sum_proper_right_subtract_ubcoefficientscoefficient. ((forall pfc_index_proper_right_subtract_ubcoefficientscoefficientdiagonal. (exists pfa_gap_proper_right_subtract_ubcoefficientscoefficientdiagonalbound. pfa_gap_proper_right_subtract_ubcoefficientscoefficientdiagonalbound + S (pfc_index_proper_right_subtract_ubcoefficientscoefficientdiagonal) = (S (pfc_index_proper_right_subtract_ubcoefficients))) -> exists pfc_value_proper_right_subtract_ubcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_proper_right_subtract_ubcoefficientscoefficientdiagonalentry. ff_h_pfp_proper_right_subtract_ubcoefficientscoefficientdiagonalentry + S (pfc_value_proper_right_subtract_ubcoefficientscoefficientdiagonal) = S ((S (pfc_index_proper_right_subtract_ubcoefficientscoefficientdiagonal)) * pfc_terms_scale_proper_right_subtract_ubcoefficientscoefficient)) /\\ exists ff_q_pfp_proper_right_subtract_ubcoefficientscoefficientdiagonalentry. pfc_terms_code_proper_right_subtract_ubcoefficientscoefficient = ff_q_pfp_proper_right_subtract_ubcoefficientscoefficientdiagonalentry * S ((S (pfc_index_proper_right_subtract_ubcoefficientscoefficientdiagonal)) * pfc_terms_scale_proper_right_subtract_ubcoefficientscoefficient) + (pfc_value_proper_right_subtract_ubcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_proper_right_subtract_ubcoefficientscoefficientdiagonalterm pfc_left_proper_right_subtract_ubcoefficientscoefficientdiagonalterm pfc_right_proper_right_subtract_ubcoefficientscoefficientdiagonalterm. (((pfc_index_proper_right_subtract_ubcoefficientscoefficientdiagonal)+pfc_complement_proper_right_subtract_ubcoefficientscoefficientdiagonalterm=(pfc_index_proper_right_subtract_ubcoefficients)) /\\ ((((((exists pfa_gap_proper_right_subtract_ubcoefficientscoefficientdiagonaltermleftinside. pfa_gap_proper_right_subtract_ubcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_proper_right_subtract_ubcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_proper_right_subtract_ubcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_proper_right_subtract_ubcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_proper_right_subtract_ubcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_proper_right_subtract_ubcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_proper_right_subtract_ubcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_proper_right_subtract_ubcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_proper_right_subtract_ubcoefficientscoefficientdiagonal)) * ac) + (pfc_left_proper_right_subtract_ubcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_proper_right_subtract_ubcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_proper_right_subtract_ubcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_proper_right_subtract_ubcoefficientscoefficientdiagonal)) /\\ (((pfc_left_proper_right_subtract_ubcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_proper_right_subtract_ubcoefficientscoefficientdiagonaltermrightinside. pfa_gap_proper_right_subtract_ubcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_proper_right_subtract_ubcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_proper_right_subtract_ubcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_proper_right_subtract_ubcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_proper_right_subtract_ubcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_proper_right_subtract_ubcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_proper_right_subtract_ubcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_proper_right_subtract_ubcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_proper_right_subtract_ubcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_proper_right_subtract_ubcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_proper_right_subtract_ubcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_proper_right_subtract_ubcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_proper_right_subtract_ubcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_proper_right_subtract_ubcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_proper_right_subtract_ubcoefficientscoefficientdiagonal)=pfc_left_proper_right_subtract_ubcoefficientscoefficientdiagonalterm*pfc_right_proper_right_subtract_ubcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_proper_right_subtract_ubcoefficientscoefficientsum fs_v_pfc_proper_right_subtract_ubcoefficientscoefficientsum. ((((exists fs_h_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_start. fs_h_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_proper_right_subtract_ubcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_start. fs_u_pfc_proper_right_subtract_ubcoefficientscoefficientsum = fs_q_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_proper_right_subtract_ubcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_terminal. fs_h_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_proper_right_subtract_ubcoefficientscoefficient) = S ((S (S (pfc_index_proper_right_subtract_ubcoefficients))) * fs_v_pfc_proper_right_subtract_ubcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_terminal. fs_u_pfc_proper_right_subtract_ubcoefficientscoefficientsum = fs_q_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_proper_right_subtract_ubcoefficients))) * fs_v_pfc_proper_right_subtract_ubcoefficientscoefficientsum) + (pfc_natural_sum_proper_right_subtract_ubcoefficientscoefficient))) /\\ forall fs_i_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_steps = S (pfc_index_proper_right_subtract_ubcoefficients)) -> exists fs_a_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_steps fs_r_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_steps fs_s_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_proper_right_subtract_ubcoefficientscoefficient)) /\\ exists fs_q_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_proper_right_subtract_ubcoefficientscoefficient = fs_q_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_proper_right_subtract_ubcoefficientscoefficient) + (fs_a_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_right_subtract_ubcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_proper_right_subtract_ubcoefficientscoefficientsum = fs_q_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_right_subtract_ubcoefficientscoefficientsum) + (fs_r_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_right_subtract_ubcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_proper_right_subtract_ubcoefficientscoefficientsum = fs_q_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_right_subtract_ubcoefficientscoefficientsum) + (fs_s_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_steps = fs_r_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_steps + fs_a_pfc_proper_right_subtract_ubcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_proper_right_subtract_ubcoefficientscoefficientresiduebound. pfa_gap_proper_right_subtract_ubcoefficientscoefficientresiduebound + S (pfc_value_proper_right_subtract_ubcoefficients) = (p)) /\\ ((exists pfa_offset_left_proper_right_subtract_ubcoefficientscoefficientresiduecongruence pfa_offset_right_proper_right_subtract_ubcoefficientscoefficientresiduecongruence. (pfc_natural_sum_proper_right_subtract_ubcoefficientscoefficient) + (p) * pfa_offset_left_proper_right_subtract_ubcoefficientscoefficientresiduecongruence = (pfc_value_proper_right_subtract_ubcoefficients) + (p) * pfa_offset_right_proper_right_subtract_ubcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_proper_right_subtract_vbleft. (exists fom_gap_pfp_proper_right_subtract_vbleft_index_bound. fom_gap_pfp_proper_right_subtract_vbleft_index_bound + S (fom_index_pfp_proper_right_subtract_vbleft) = L) -> exists fom_value_pfp_proper_right_subtract_vbleft. ((((exists fom_beta_height_pfp_proper_right_subtract_vbleft_entry. fom_beta_height_pfp_proper_right_subtract_vbleft_entry + S (fom_value_pfp_proper_right_subtract_vbleft) = S ((S (fom_index_pfp_proper_right_subtract_vbleft)) * bc)) /\\ exists fom_beta_quotient_pfp_proper_right_subtract_vbleft_entry. bb = fom_beta_quotient_pfp_proper_right_subtract_vbleft_entry * S ((S (fom_index_pfp_proper_right_subtract_vbleft)) * bc) + (fom_value_pfp_proper_right_subtract_vbleft))) /\\ (exists fom_gap_pfp_proper_right_subtract_vbleft_value_bound. fom_gap_pfp_proper_right_subtract_vbleft_value_bound + S (fom_value_pfp_proper_right_subtract_vbleft) = p))) /\\ (((forall fom_index_pfp_proper_right_subtract_vbright. (exists fom_gap_pfp_proper_right_subtract_vbright_index_bound. fom_gap_pfp_proper_right_subtract_vbright_index_bound + S (fom_index_pfp_proper_right_subtract_vbright) = M) -> exists fom_value_pfp_proper_right_subtract_vbright. ((((exists fom_beta_height_pfp_proper_right_subtract_vbright_entry. fom_beta_height_pfp_proper_right_subtract_vbright_entry + S (fom_value_pfp_proper_right_subtract_vbright) = S ((S (fom_index_pfp_proper_right_subtract_vbright)) * dc)) /\\ exists fom_beta_quotient_pfp_proper_right_subtract_vbright_entry. db = fom_beta_quotient_pfp_proper_right_subtract_vbright_entry * S ((S (fom_index_pfp_proper_right_subtract_vbright)) * dc) + (fom_value_pfp_proper_right_subtract_vbright))) /\\ (exists fom_gap_pfp_proper_right_subtract_vbright_value_bound. fom_gap_pfp_proper_right_subtract_vbright_value_bound + S (fom_value_pfp_proper_right_subtract_vbright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_proper_right_subtract_vbcoefficients. (exists pfa_gap_proper_right_subtract_vbcoefficientsbound. pfa_gap_proper_right_subtract_vbcoefficientsbound + S (pfc_index_proper_right_subtract_vbcoefficients) = (N)) -> exists pfc_value_proper_right_subtract_vbcoefficients. ((((exists ff_h_pfp_proper_right_subtract_vbcoefficientsentry. ff_h_pfp_proper_right_subtract_vbcoefficientsentry + S (pfc_value_proper_right_subtract_vbcoefficients) = S ((S (pfc_index_proper_right_subtract_vbcoefficients)) * vc)) /\\ exists ff_q_pfp_proper_right_subtract_vbcoefficientsentry. vb = ff_q_pfp_proper_right_subtract_vbcoefficientsentry * S ((S (pfc_index_proper_right_subtract_vbcoefficients)) * vc) + (pfc_value_proper_right_subtract_vbcoefficients))) /\\ ((exists pfc_terms_code_proper_right_subtract_vbcoefficientscoefficient pfc_terms_scale_proper_right_subtract_vbcoefficientscoefficient pfc_natural_sum_proper_right_subtract_vbcoefficientscoefficient. ((forall pfc_index_proper_right_subtract_vbcoefficientscoefficientdiagonal. (exists pfa_gap_proper_right_subtract_vbcoefficientscoefficientdiagonalbound. pfa_gap_proper_right_subtract_vbcoefficientscoefficientdiagonalbound + S (pfc_index_proper_right_subtract_vbcoefficientscoefficientdiagonal) = (S (pfc_index_proper_right_subtract_vbcoefficients))) -> exists pfc_value_proper_right_subtract_vbcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_proper_right_subtract_vbcoefficientscoefficientdiagonalentry. ff_h_pfp_proper_right_subtract_vbcoefficientscoefficientdiagonalentry + S (pfc_value_proper_right_subtract_vbcoefficientscoefficientdiagonal) = S ((S (pfc_index_proper_right_subtract_vbcoefficientscoefficientdiagonal)) * pfc_terms_scale_proper_right_subtract_vbcoefficientscoefficient)) /\\ exists ff_q_pfp_proper_right_subtract_vbcoefficientscoefficientdiagonalentry. pfc_terms_code_proper_right_subtract_vbcoefficientscoefficient = ff_q_pfp_proper_right_subtract_vbcoefficientscoefficientdiagonalentry * S ((S (pfc_index_proper_right_subtract_vbcoefficientscoefficientdiagonal)) * pfc_terms_scale_proper_right_subtract_vbcoefficientscoefficient) + (pfc_value_proper_right_subtract_vbcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_proper_right_subtract_vbcoefficientscoefficientdiagonalterm pfc_left_proper_right_subtract_vbcoefficientscoefficientdiagonalterm pfc_right_proper_right_subtract_vbcoefficientscoefficientdiagonalterm. (((pfc_index_proper_right_subtract_vbcoefficientscoefficientdiagonal)+pfc_complement_proper_right_subtract_vbcoefficientscoefficientdiagonalterm=(pfc_index_proper_right_subtract_vbcoefficients)) /\\ ((((((exists pfa_gap_proper_right_subtract_vbcoefficientscoefficientdiagonaltermleftinside. pfa_gap_proper_right_subtract_vbcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_proper_right_subtract_vbcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_proper_right_subtract_vbcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_proper_right_subtract_vbcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_proper_right_subtract_vbcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_proper_right_subtract_vbcoefficientscoefficientdiagonal)) * bc)) /\\ exists ff_q_pfp_proper_right_subtract_vbcoefficientscoefficientdiagonaltermleftentry. bb = ff_q_pfp_proper_right_subtract_vbcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_proper_right_subtract_vbcoefficientscoefficientdiagonal)) * bc) + (pfc_left_proper_right_subtract_vbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_proper_right_subtract_vbcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_proper_right_subtract_vbcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_proper_right_subtract_vbcoefficientscoefficientdiagonal)) /\\ (((pfc_left_proper_right_subtract_vbcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_proper_right_subtract_vbcoefficientscoefficientdiagonaltermrightinside. pfa_gap_proper_right_subtract_vbcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_proper_right_subtract_vbcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_proper_right_subtract_vbcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_proper_right_subtract_vbcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_proper_right_subtract_vbcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_proper_right_subtract_vbcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_proper_right_subtract_vbcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_proper_right_subtract_vbcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_proper_right_subtract_vbcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_proper_right_subtract_vbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_proper_right_subtract_vbcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_proper_right_subtract_vbcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_proper_right_subtract_vbcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_proper_right_subtract_vbcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_proper_right_subtract_vbcoefficientscoefficientdiagonal)=pfc_left_proper_right_subtract_vbcoefficientscoefficientdiagonalterm*pfc_right_proper_right_subtract_vbcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_proper_right_subtract_vbcoefficientscoefficientsum fs_v_pfc_proper_right_subtract_vbcoefficientscoefficientsum. ((((exists fs_h_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_start. fs_h_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_proper_right_subtract_vbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_start. fs_u_pfc_proper_right_subtract_vbcoefficientscoefficientsum = fs_q_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_proper_right_subtract_vbcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_terminal. fs_h_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_proper_right_subtract_vbcoefficientscoefficient) = S ((S (S (pfc_index_proper_right_subtract_vbcoefficients))) * fs_v_pfc_proper_right_subtract_vbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_terminal. fs_u_pfc_proper_right_subtract_vbcoefficientscoefficientsum = fs_q_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_proper_right_subtract_vbcoefficients))) * fs_v_pfc_proper_right_subtract_vbcoefficientscoefficientsum) + (pfc_natural_sum_proper_right_subtract_vbcoefficientscoefficient))) /\\ forall fs_i_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_steps = S (pfc_index_proper_right_subtract_vbcoefficients)) -> exists fs_a_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_steps fs_r_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_steps fs_s_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_proper_right_subtract_vbcoefficientscoefficient)) /\\ exists fs_q_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_proper_right_subtract_vbcoefficientscoefficient = fs_q_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_proper_right_subtract_vbcoefficientscoefficient) + (fs_a_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_right_subtract_vbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_proper_right_subtract_vbcoefficientscoefficientsum = fs_q_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_right_subtract_vbcoefficientscoefficientsum) + (fs_r_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_right_subtract_vbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_proper_right_subtract_vbcoefficientscoefficientsum = fs_q_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_right_subtract_vbcoefficientscoefficientsum) + (fs_s_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_steps = fs_r_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_steps + fs_a_pfc_proper_right_subtract_vbcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_proper_right_subtract_vbcoefficientscoefficientresiduebound. pfa_gap_proper_right_subtract_vbcoefficientscoefficientresiduebound + S (pfc_value_proper_right_subtract_vbcoefficients) = (p)) /\\ ((exists pfa_offset_left_proper_right_subtract_vbcoefficientscoefficientresiduecongruence pfa_offset_right_proper_right_subtract_vbcoefficientscoefficientresiduecongruence. (pfc_natural_sum_proper_right_subtract_vbcoefficientscoefficient) + (p) * pfa_offset_left_proper_right_subtract_vbcoefficientscoefficientresiduecongruence = (pfc_value_proper_right_subtract_vbcoefficients) + (p) * pfa_offset_right_proper_right_subtract_vbcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_proper_right_subtract_wbleft. (exists fom_gap_pfp_proper_right_subtract_wbleft_index_bound. fom_gap_pfp_proper_right_subtract_wbleft_index_bound + S (fom_index_pfp_proper_right_subtract_wbleft) = L) -> exists fom_value_pfp_proper_right_subtract_wbleft. ((((exists fom_beta_height_pfp_proper_right_subtract_wbleft_entry. fom_beta_height_pfp_proper_right_subtract_wbleft_entry + S (fom_value_pfp_proper_right_subtract_wbleft) = S ((S (fom_index_pfp_proper_right_subtract_wbleft)) * cc)) /\\ exists fom_beta_quotient_pfp_proper_right_subtract_wbleft_entry. cb = fom_beta_quotient_pfp_proper_right_subtract_wbleft_entry * S ((S (fom_index_pfp_proper_right_subtract_wbleft)) * cc) + (fom_value_pfp_proper_right_subtract_wbleft))) /\\ (exists fom_gap_pfp_proper_right_subtract_wbleft_value_bound. fom_gap_pfp_proper_right_subtract_wbleft_value_bound + S (fom_value_pfp_proper_right_subtract_wbleft) = p))) /\\ (((forall fom_index_pfp_proper_right_subtract_wbright. (exists fom_gap_pfp_proper_right_subtract_wbright_index_bound. fom_gap_pfp_proper_right_subtract_wbright_index_bound + S (fom_index_pfp_proper_right_subtract_wbright) = M) -> exists fom_value_pfp_proper_right_subtract_wbright. ((((exists fom_beta_height_pfp_proper_right_subtract_wbright_entry. fom_beta_height_pfp_proper_right_subtract_wbright_entry + S (fom_value_pfp_proper_right_subtract_wbright) = S ((S (fom_index_pfp_proper_right_subtract_wbright)) * dc)) /\\ exists fom_beta_quotient_pfp_proper_right_subtract_wbright_entry. db = fom_beta_quotient_pfp_proper_right_subtract_wbright_entry * S ((S (fom_index_pfp_proper_right_subtract_wbright)) * dc) + (fom_value_pfp_proper_right_subtract_wbright))) /\\ (exists fom_gap_pfp_proper_right_subtract_wbright_value_bound. fom_gap_pfp_proper_right_subtract_wbright_value_bound + S (fom_value_pfp_proper_right_subtract_wbright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_proper_right_subtract_wbcoefficients. (exists pfa_gap_proper_right_subtract_wbcoefficientsbound. pfa_gap_proper_right_subtract_wbcoefficientsbound + S (pfc_index_proper_right_subtract_wbcoefficients) = (N)) -> exists pfc_value_proper_right_subtract_wbcoefficients. ((((exists ff_h_pfp_proper_right_subtract_wbcoefficientsentry. ff_h_pfp_proper_right_subtract_wbcoefficientsentry + S (pfc_value_proper_right_subtract_wbcoefficients) = S ((S (pfc_index_proper_right_subtract_wbcoefficients)) * wc)) /\\ exists ff_q_pfp_proper_right_subtract_wbcoefficientsentry. wb = ff_q_pfp_proper_right_subtract_wbcoefficientsentry * S ((S (pfc_index_proper_right_subtract_wbcoefficients)) * wc) + (pfc_value_proper_right_subtract_wbcoefficients))) /\\ ((exists pfc_terms_code_proper_right_subtract_wbcoefficientscoefficient pfc_terms_scale_proper_right_subtract_wbcoefficientscoefficient pfc_natural_sum_proper_right_subtract_wbcoefficientscoefficient. ((forall pfc_index_proper_right_subtract_wbcoefficientscoefficientdiagonal. (exists pfa_gap_proper_right_subtract_wbcoefficientscoefficientdiagonalbound. pfa_gap_proper_right_subtract_wbcoefficientscoefficientdiagonalbound + S (pfc_index_proper_right_subtract_wbcoefficientscoefficientdiagonal) = (S (pfc_index_proper_right_subtract_wbcoefficients))) -> exists pfc_value_proper_right_subtract_wbcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_proper_right_subtract_wbcoefficientscoefficientdiagonalentry. ff_h_pfp_proper_right_subtract_wbcoefficientscoefficientdiagonalentry + S (pfc_value_proper_right_subtract_wbcoefficientscoefficientdiagonal) = S ((S (pfc_index_proper_right_subtract_wbcoefficientscoefficientdiagonal)) * pfc_terms_scale_proper_right_subtract_wbcoefficientscoefficient)) /\\ exists ff_q_pfp_proper_right_subtract_wbcoefficientscoefficientdiagonalentry. pfc_terms_code_proper_right_subtract_wbcoefficientscoefficient = ff_q_pfp_proper_right_subtract_wbcoefficientscoefficientdiagonalentry * S ((S (pfc_index_proper_right_subtract_wbcoefficientscoefficientdiagonal)) * pfc_terms_scale_proper_right_subtract_wbcoefficientscoefficient) + (pfc_value_proper_right_subtract_wbcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_proper_right_subtract_wbcoefficientscoefficientdiagonalterm pfc_left_proper_right_subtract_wbcoefficientscoefficientdiagonalterm pfc_right_proper_right_subtract_wbcoefficientscoefficientdiagonalterm. (((pfc_index_proper_right_subtract_wbcoefficientscoefficientdiagonal)+pfc_complement_proper_right_subtract_wbcoefficientscoefficientdiagonalterm=(pfc_index_proper_right_subtract_wbcoefficients)) /\\ ((((((exists pfa_gap_proper_right_subtract_wbcoefficientscoefficientdiagonaltermleftinside. pfa_gap_proper_right_subtract_wbcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_proper_right_subtract_wbcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_proper_right_subtract_wbcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_proper_right_subtract_wbcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_proper_right_subtract_wbcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_proper_right_subtract_wbcoefficientscoefficientdiagonal)) * cc)) /\\ exists ff_q_pfp_proper_right_subtract_wbcoefficientscoefficientdiagonaltermleftentry. cb = ff_q_pfp_proper_right_subtract_wbcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_proper_right_subtract_wbcoefficientscoefficientdiagonal)) * cc) + (pfc_left_proper_right_subtract_wbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_proper_right_subtract_wbcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_proper_right_subtract_wbcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_proper_right_subtract_wbcoefficientscoefficientdiagonal)) /\\ (((pfc_left_proper_right_subtract_wbcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_proper_right_subtract_wbcoefficientscoefficientdiagonaltermrightinside. pfa_gap_proper_right_subtract_wbcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_proper_right_subtract_wbcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_proper_right_subtract_wbcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_proper_right_subtract_wbcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_proper_right_subtract_wbcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_proper_right_subtract_wbcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_proper_right_subtract_wbcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_proper_right_subtract_wbcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_proper_right_subtract_wbcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_proper_right_subtract_wbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_proper_right_subtract_wbcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_proper_right_subtract_wbcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_proper_right_subtract_wbcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_proper_right_subtract_wbcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_proper_right_subtract_wbcoefficientscoefficientdiagonal)=pfc_left_proper_right_subtract_wbcoefficientscoefficientdiagonalterm*pfc_right_proper_right_subtract_wbcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_proper_right_subtract_wbcoefficientscoefficientsum fs_v_pfc_proper_right_subtract_wbcoefficientscoefficientsum. ((((exists fs_h_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_start. fs_h_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_proper_right_subtract_wbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_start. fs_u_pfc_proper_right_subtract_wbcoefficientscoefficientsum = fs_q_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_proper_right_subtract_wbcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_terminal. fs_h_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_proper_right_subtract_wbcoefficientscoefficient) = S ((S (S (pfc_index_proper_right_subtract_wbcoefficients))) * fs_v_pfc_proper_right_subtract_wbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_terminal. fs_u_pfc_proper_right_subtract_wbcoefficientscoefficientsum = fs_q_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_proper_right_subtract_wbcoefficients))) * fs_v_pfc_proper_right_subtract_wbcoefficientscoefficientsum) + (pfc_natural_sum_proper_right_subtract_wbcoefficientscoefficient))) /\\ forall fs_i_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_steps = S (pfc_index_proper_right_subtract_wbcoefficients)) -> exists fs_a_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_steps fs_r_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_steps fs_s_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_proper_right_subtract_wbcoefficientscoefficient)) /\\ exists fs_q_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_proper_right_subtract_wbcoefficientscoefficient = fs_q_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_proper_right_subtract_wbcoefficientscoefficient) + (fs_a_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_right_subtract_wbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_proper_right_subtract_wbcoefficientscoefficientsum = fs_q_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_right_subtract_wbcoefficientscoefficientsum) + (fs_r_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_right_subtract_wbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_proper_right_subtract_wbcoefficientscoefficientsum = fs_q_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_proper_right_subtract_wbcoefficientscoefficientsum) + (fs_s_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_steps = fs_r_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_steps + fs_a_pfc_proper_right_subtract_wbcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_proper_right_subtract_wbcoefficientscoefficientresiduebound. pfa_gap_proper_right_subtract_wbcoefficientscoefficientresiduebound + S (pfc_value_proper_right_subtract_wbcoefficients) = (p)) /\\ ((exists pfa_offset_left_proper_right_subtract_wbcoefficientscoefficientresiduecongruence pfa_offset_right_proper_right_subtract_wbcoefficientscoefficientresiduecongruence. (pfc_natural_sum_proper_right_subtract_wbcoefficientscoefficient) + (p) * pfa_offset_left_proper_right_subtract_wbcoefficientscoefficientresiduecongruence = (pfc_value_proper_right_subtract_wbcoefficients) + (p) * pfa_offset_right_proper_right_subtract_wbcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (forall pfs_index_proper_right_subtract_result. (exists pfa_gap_proper_right_subtract_resultindex. pfa_gap_proper_right_subtract_resultindex + S (pfs_index_proper_right_subtract_result) = (N)) -> exists pfs_left_proper_right_subtract_result pfs_right_proper_right_subtract_result pfs_result_proper_right_subtract_result. ((((exists ff_h_pfp_proper_right_subtract_resultleft. ff_h_pfp_proper_right_subtract_resultleft + S (pfs_left_proper_right_subtract_result) = S ((S (pfs_index_proper_right_subtract_result)) * uc)) /\\ exists ff_q_pfp_proper_right_subtract_resultleft. ub = ff_q_pfp_proper_right_subtract_resultleft * S ((S (pfs_index_proper_right_subtract_result)) * uc) + (pfs_left_proper_right_subtract_result))) /\\ (((((exists ff_h_pfp_proper_right_subtract_resultright. ff_h_pfp_proper_right_subtract_resultright + S (pfs_right_proper_right_subtract_result) = S ((S (pfs_index_proper_right_subtract_result)) * vc)) /\\ exists ff_q_pfp_proper_right_subtract_resultright. vb = ff_q_pfp_proper_right_subtract_resultright * S ((S (pfs_index_proper_right_subtract_result)) * vc) + (pfs_right_proper_right_subtract_result))) /\\ (((((exists ff_h_pfp_proper_right_subtract_resultresult. ff_h_pfp_proper_right_subtract_resultresult + S (pfs_result_proper_right_subtract_result) = S ((S (pfs_index_proper_right_subtract_result)) * wc)) /\\ exists ff_q_pfp_proper_right_subtract_resultresult. wb = ff_q_pfp_proper_right_subtract_resultresult * S ((S (pfs_index_proper_right_subtract_result)) * wc) + (pfs_result_proper_right_subtract_result))) /\\ ((((exists pfa_gap_proper_right_subtract_resultoperationleft. pfa_gap_proper_right_subtract_resultoperationleft + S (pfs_right_proper_right_subtract_result) = (p)) /\\ (((exists pfa_gap_proper_right_subtract_resultoperationright. pfa_gap_proper_right_subtract_resultoperationright + S (pfs_result_proper_right_subtract_result) = (p)) /\\ ((((exists pfa_gap_proper_right_subtract_resultoperationresultbound. pfa_gap_proper_right_subtract_resultoperationresultbound + S (pfs_left_proper_right_subtract_result) = (p)) /\\ ((exists pfa_offset_left_proper_right_subtract_resultoperationresultcongruence pfa_offset_right_proper_right_subtract_resultoperationresultcongruence. ((pfs_right_proper_right_subtract_result) + (pfs_result_proper_right_subtract_result)) + (p) * pfa_offset_left_proper_right_subtract_resultoperationresultcongruence = (pfs_left_proper_right_subtract_result) + (p) * pfa_offset_right_proper_right_subtract_resultoperationresultcongruence))))))))))))))))", "statement_sha256": "1ad4bbbd98be6fd361b4dcf76d989068ada8061e9fdbe73bea8a84cfee0cdf25", "summary": "The existing proper-length convolution graphs obey actual right subtract distributivity; this is a formal coefficient law, not an evaluation test.", "summary_sha256": "0f8bdfbf41f9433a0cb5d254e8dc15d11869991adc09c62cbbc061de86275032" }, { "alpha_v33_frontier_enrollment": { "body_receipt_sha256": "13164720b7fe2019609109a2cf8b00effd4a6fa26b3e8f168c6a9c6070fd275e", "bundle_campaign": "polynomial-euclidean-division", "bundle_node_id": 334, "bundle_sha256": "6ae667d8518e4dbe722bb08ad1b08715a0d282c2893e533c8133d770fe861dcf", "campaign": "polynomial-euclidean-division", "first_enrolled_version": 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"script": [ "intro p", "intro ab", "intro ac", "intro bb", "intro bc", "intro cb", "intro cc", "intro L", "intro db", "intro dc", "intro M", "intro hp", "intro hd", "intro hs", "have hlength : exists N. ((((M)=0 \\/ (L)=0) /\\ (((N)=0)))) \\/ (((~((M)=0)) /\\ (((~((L)=0)) /\\ (((M)+(L)=S (N)))))))", "specialize polynomial_product_length_exists (M)", "specialize polynomial_product_length_exists (L)", "apply polynomial_product_length_exists", "cases hlength", "have hbounds : ((forall fom_index_pfp_products_left_bounded_ab. (exists fom_gap_pfp_products_left_bounded_ab_index_bound. fom_gap_pfp_products_left_bounded_ab_index_bound + S (fom_index_pfp_products_left_bounded_ab) = L) -> exists fom_value_pfp_products_left_bounded_ab. ((((exists fom_beta_height_pfp_products_left_bounded_ab_entry. fom_beta_height_pfp_products_left_bounded_ab_entry + S (fom_value_pfp_products_left_bounded_ab) = S ((S (fom_index_pfp_products_left_bounded_ab)) * ac)) /\\ exists fom_beta_quotient_pfp_products_left_bounded_ab_entry. ab = fom_beta_quotient_pfp_products_left_bounded_ab_entry * S ((S (fom_index_pfp_products_left_bounded_ab)) * ac) + (fom_value_pfp_products_left_bounded_ab))) /\\ (exists fom_gap_pfp_products_left_bounded_ab_value_bound. fom_gap_pfp_products_left_bounded_ab_value_bound + S (fom_value_pfp_products_left_bounded_ab) = p))) /\\ (((forall fom_index_pfp_products_left_bounded_bb. (exists fom_gap_pfp_products_left_bounded_bb_index_bound. fom_gap_pfp_products_left_bounded_bb_index_bound + S (fom_index_pfp_products_left_bounded_bb) = L) -> exists fom_value_pfp_products_left_bounded_bb. ((((exists fom_beta_height_pfp_products_left_bounded_bb_entry. fom_beta_height_pfp_products_left_bounded_bb_entry + S (fom_value_pfp_products_left_bounded_bb) = S ((S (fom_index_pfp_products_left_bounded_bb)) * bc)) /\\ exists fom_beta_quotient_pfp_products_left_bounded_bb_entry. bb = fom_beta_quotient_pfp_products_left_bounded_bb_entry * S ((S (fom_index_pfp_products_left_bounded_bb)) * bc) + (fom_value_pfp_products_left_bounded_bb))) /\\ (exists fom_gap_pfp_products_left_bounded_bb_value_bound. fom_gap_pfp_products_left_bounded_bb_value_bound + S (fom_value_pfp_products_left_bounded_bb) = p))) /\\ ((forall fom_index_pfp_products_left_bounded_cb. (exists fom_gap_pfp_products_left_bounded_cb_index_bound. fom_gap_pfp_products_left_bounded_cb_index_bound + S (fom_index_pfp_products_left_bounded_cb) = L) -> exists fom_value_pfp_products_left_bounded_cb. ((((exists fom_beta_height_pfp_products_left_bounded_cb_entry. fom_beta_height_pfp_products_left_bounded_cb_entry + S (fom_value_pfp_products_left_bounded_cb) = S ((S (fom_index_pfp_products_left_bounded_cb)) * cc)) /\\ exists fom_beta_quotient_pfp_products_left_bounded_cb_entry. cb = fom_beta_quotient_pfp_products_left_bounded_cb_entry * S ((S (fom_index_pfp_products_left_bounded_cb)) * cc) + (fom_value_pfp_products_left_bounded_cb))) /\\ (exists fom_gap_pfp_products_left_bounded_cb_value_bound. fom_gap_pfp_products_left_bounded_cb_value_bound + S (fom_value_pfp_products_left_bounded_cb) = 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 (cb)", "specialize prime_field_polynomial_add_bounded (cc)", "specialize prime_field_polynomial_add_bounded (L)", "apply prime_field_polynomial_add_bounded", "exact hs", "cases hbounds", "cases hbounds_right", "have hu : exists ub uc. ((forall fom_index_pfp_products_left_huleft. (exists fom_gap_pfp_products_left_huleft_index_bound. fom_gap_pfp_products_left_huleft_index_bound + S (fom_index_pfp_products_left_huleft) = M) -> exists fom_value_pfp_products_left_huleft. ((((exists fom_beta_height_pfp_products_left_huleft_entry. fom_beta_height_pfp_products_left_huleft_entry + S (fom_value_pfp_products_left_huleft) = S ((S (fom_index_pfp_products_left_huleft)) * dc)) /\\ exists fom_beta_quotient_pfp_products_left_huleft_entry. db = fom_beta_quotient_pfp_products_left_huleft_entry * S ((S (fom_index_pfp_products_left_huleft)) * dc) + (fom_value_pfp_products_left_huleft))) /\\ (exists fom_gap_pfp_products_left_huleft_value_bound. fom_gap_pfp_products_left_huleft_value_bound + S (fom_value_pfp_products_left_huleft) = p))) /\\ (((forall fom_index_pfp_products_left_huright. (exists fom_gap_pfp_products_left_huright_index_bound. fom_gap_pfp_products_left_huright_index_bound + S (fom_index_pfp_products_left_huright) = L) -> exists fom_value_pfp_products_left_huright. ((((exists fom_beta_height_pfp_products_left_huright_entry. fom_beta_height_pfp_products_left_huright_entry + S (fom_value_pfp_products_left_huright) = S ((S (fom_index_pfp_products_left_huright)) * ac)) /\\ exists fom_beta_quotient_pfp_products_left_huright_entry. ab = fom_beta_quotient_pfp_products_left_huright_entry * S ((S (fom_index_pfp_products_left_huright)) * ac) + (fom_value_pfp_products_left_huright))) /\\ (exists fom_gap_pfp_products_left_huright_value_bound. fom_gap_pfp_products_left_huright_value_bound + S (fom_value_pfp_products_left_huright) = p))) /\\ (((((((M)=0 \\/ (L)=0) /\\ (((x)=0)))) \\/ (((~((M)=0)) /\\ (((~((L)=0)) /\\ (((M)+(L)=S (x)))))))) /\\ ((forall pfc_index_products_left_hucoefficients. (exists pfa_gap_products_left_hucoefficientsbound. pfa_gap_products_left_hucoefficientsbound + S (pfc_index_products_left_hucoefficients) = (x)) -> exists pfc_value_products_left_hucoefficients. ((((exists ff_h_pfp_products_left_hucoefficientsentry. ff_h_pfp_products_left_hucoefficientsentry + S (pfc_value_products_left_hucoefficients) = S ((S (pfc_index_products_left_hucoefficients)) * uc)) /\\ exists ff_q_pfp_products_left_hucoefficientsentry. ub = ff_q_pfp_products_left_hucoefficientsentry * S ((S (pfc_index_products_left_hucoefficients)) * uc) + (pfc_value_products_left_hucoefficients))) /\\ ((exists pfc_terms_code_products_left_hucoefficientscoefficient pfc_terms_scale_products_left_hucoefficientscoefficient pfc_natural_sum_products_left_hucoefficientscoefficient. ((forall pfc_index_products_left_hucoefficientscoefficientdiagonal. (exists pfa_gap_products_left_hucoefficientscoefficientdiagonalbound. pfa_gap_products_left_hucoefficientscoefficientdiagonalbound + S (pfc_index_products_left_hucoefficientscoefficientdiagonal) = (S (pfc_index_products_left_hucoefficients))) -> exists pfc_value_products_left_hucoefficientscoefficientdiagonal. ((((exists ff_h_pfp_products_left_hucoefficientscoefficientdiagonalentry. ff_h_pfp_products_left_hucoefficientscoefficientdiagonalentry + S (pfc_value_products_left_hucoefficientscoefficientdiagonal) = S ((S (pfc_index_products_left_hucoefficientscoefficientdiagonal)) * pfc_terms_scale_products_left_hucoefficientscoefficient)) /\\ exists ff_q_pfp_products_left_hucoefficientscoefficientdiagonalentry. pfc_terms_code_products_left_hucoefficientscoefficient = ff_q_pfp_products_left_hucoefficientscoefficientdiagonalentry * S ((S (pfc_index_products_left_hucoefficientscoefficientdiagonal)) * pfc_terms_scale_products_left_hucoefficientscoefficient) + (pfc_value_products_left_hucoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_products_left_hucoefficientscoefficientdiagonalterm pfc_left_products_left_hucoefficientscoefficientdiagonalterm pfc_right_products_left_hucoefficientscoefficientdiagonalterm. (((pfc_index_products_left_hucoefficientscoefficientdiagonal)+pfc_complement_products_left_hucoefficientscoefficientdiagonalterm=(pfc_index_products_left_hucoefficients)) /\\ ((((((exists pfa_gap_products_left_hucoefficientscoefficientdiagonaltermleftinside. pfa_gap_products_left_hucoefficientscoefficientdiagonaltermleftinside + S (pfc_index_products_left_hucoefficientscoefficientdiagonal) = (M)) /\\ ((((exists ff_h_pfp_products_left_hucoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_products_left_hucoefficientscoefficientdiagonaltermleftentry + S (pfc_left_products_left_hucoefficientscoefficientdiagonalterm) = S ((S (pfc_index_products_left_hucoefficientscoefficientdiagonal)) * dc)) /\\ exists ff_q_pfp_products_left_hucoefficientscoefficientdiagonaltermleftentry. db = ff_q_pfp_products_left_hucoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_products_left_hucoefficientscoefficientdiagonal)) * dc) + (pfc_left_products_left_hucoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_products_left_hucoefficientscoefficientdiagonaltermleftoutside. pfc_gap_products_left_hucoefficientscoefficientdiagonaltermleftoutside+(M)=(pfc_index_products_left_hucoefficientscoefficientdiagonal)) /\\ (((pfc_left_products_left_hucoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_products_left_hucoefficientscoefficientdiagonaltermrightinside. pfa_gap_products_left_hucoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_products_left_hucoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_products_left_hucoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_products_left_hucoefficientscoefficientdiagonaltermrightentry + S (pfc_right_products_left_hucoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_products_left_hucoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_products_left_hucoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_products_left_hucoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_products_left_hucoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_products_left_hucoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_products_left_hucoefficientscoefficientdiagonaltermrightoutside. pfc_gap_products_left_hucoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_products_left_hucoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_products_left_hucoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_products_left_hucoefficientscoefficientdiagonal)=pfc_left_products_left_hucoefficientscoefficientdiagonalterm*pfc_right_products_left_hucoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_products_left_hucoefficientscoefficientsum fs_v_pfc_products_left_hucoefficientscoefficientsum. ((((exists fs_h_pfc_products_left_hucoefficientscoefficientsum_body_start. fs_h_pfc_products_left_hucoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_products_left_hucoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_left_hucoefficientscoefficientsum_body_start. fs_u_pfc_products_left_hucoefficientscoefficientsum = fs_q_pfc_products_left_hucoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_products_left_hucoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_products_left_hucoefficientscoefficientsum_body_terminal. fs_h_pfc_products_left_hucoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_products_left_hucoefficientscoefficient) = S ((S (S (pfc_index_products_left_hucoefficients))) * fs_v_pfc_products_left_hucoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_left_hucoefficientscoefficientsum_body_terminal. fs_u_pfc_products_left_hucoefficientscoefficientsum = fs_q_pfc_products_left_hucoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_products_left_hucoefficients))) * fs_v_pfc_products_left_hucoefficientscoefficientsum) + (pfc_natural_sum_products_left_hucoefficientscoefficient))) /\\ forall fs_i_pfc_products_left_hucoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_products_left_hucoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_products_left_hucoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_products_left_hucoefficientscoefficientsum_body_steps = S (pfc_index_products_left_hucoefficients)) -> exists fs_a_pfc_products_left_hucoefficientscoefficientsum_body_steps fs_r_pfc_products_left_hucoefficientscoefficientsum_body_steps fs_s_pfc_products_left_hucoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_products_left_hucoefficientscoefficientsum_body_steps_summand. fs_h_pfc_products_left_hucoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_products_left_hucoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_products_left_hucoefficientscoefficientsum_body_steps)) * pfc_terms_scale_products_left_hucoefficientscoefficient)) /\\ exists fs_q_pfc_products_left_hucoefficientscoefficientsum_body_steps_summand. pfc_terms_code_products_left_hucoefficientscoefficient = fs_q_pfc_products_left_hucoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_products_left_hucoefficientscoefficientsum_body_steps)) * pfc_terms_scale_products_left_hucoefficientscoefficient) + (fs_a_pfc_products_left_hucoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_products_left_hucoefficientscoefficientsum_body_steps_partial. fs_h_pfc_products_left_hucoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_products_left_hucoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_products_left_hucoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_hucoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_left_hucoefficientscoefficientsum_body_steps_partial. fs_u_pfc_products_left_hucoefficientscoefficientsum = fs_q_pfc_products_left_hucoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_products_left_hucoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_hucoefficientscoefficientsum) + (fs_r_pfc_products_left_hucoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_products_left_hucoefficientscoefficientsum_body_steps_successor. fs_h_pfc_products_left_hucoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_products_left_hucoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_products_left_hucoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_hucoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_left_hucoefficientscoefficientsum_body_steps_successor. fs_u_pfc_products_left_hucoefficientscoefficientsum = fs_q_pfc_products_left_hucoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_products_left_hucoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_hucoefficientscoefficientsum) + (fs_s_pfc_products_left_hucoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_products_left_hucoefficientscoefficientsum_body_steps = fs_r_pfc_products_left_hucoefficientscoefficientsum_body_steps + fs_a_pfc_products_left_hucoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_products_left_hucoefficientscoefficientresiduebound. pfa_gap_products_left_hucoefficientscoefficientresiduebound + S (pfc_value_products_left_hucoefficients) = (p)) /\\ ((exists pfa_offset_left_products_left_hucoefficientscoefficientresiduecongruence pfa_offset_right_products_left_hucoefficientscoefficientresiduecongruence. (pfc_natural_sum_products_left_hucoefficientscoefficient) + (p) * pfa_offset_left_products_left_hucoefficientscoefficientresiduecongruence = (pfc_value_products_left_hucoefficients) + (p) * pfa_offset_right_products_left_hucoefficientscoefficientresiduecongruence))))))))))))))))))", "specialize prime_field_polynomial_convolution_at_length_exists (p)", "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 (M)", "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)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hp", "exact hd", "exact hbounds_left", "exact hlength_witness", "cases hu", "cases hu_witness", "have hv : exists ub uc. ((forall fom_index_pfp_products_left_hvleft. (exists fom_gap_pfp_products_left_hvleft_index_bound. fom_gap_pfp_products_left_hvleft_index_bound + S (fom_index_pfp_products_left_hvleft) = M) -> exists fom_value_pfp_products_left_hvleft. ((((exists fom_beta_height_pfp_products_left_hvleft_entry. fom_beta_height_pfp_products_left_hvleft_entry + S (fom_value_pfp_products_left_hvleft) = S ((S (fom_index_pfp_products_left_hvleft)) * dc)) /\\ exists fom_beta_quotient_pfp_products_left_hvleft_entry. db = fom_beta_quotient_pfp_products_left_hvleft_entry * S ((S (fom_index_pfp_products_left_hvleft)) * dc) + (fom_value_pfp_products_left_hvleft))) /\\ (exists fom_gap_pfp_products_left_hvleft_value_bound. fom_gap_pfp_products_left_hvleft_value_bound + S (fom_value_pfp_products_left_hvleft) = p))) /\\ (((forall fom_index_pfp_products_left_hvright. (exists fom_gap_pfp_products_left_hvright_index_bound. fom_gap_pfp_products_left_hvright_index_bound + S (fom_index_pfp_products_left_hvright) = L) -> exists fom_value_pfp_products_left_hvright. ((((exists fom_beta_height_pfp_products_left_hvright_entry. fom_beta_height_pfp_products_left_hvright_entry + S (fom_value_pfp_products_left_hvright) = S ((S (fom_index_pfp_products_left_hvright)) * bc)) /\\ exists fom_beta_quotient_pfp_products_left_hvright_entry. bb = fom_beta_quotient_pfp_products_left_hvright_entry * S ((S (fom_index_pfp_products_left_hvright)) * bc) + (fom_value_pfp_products_left_hvright))) /\\ (exists fom_gap_pfp_products_left_hvright_value_bound. fom_gap_pfp_products_left_hvright_value_bound + S (fom_value_pfp_products_left_hvright) = p))) /\\ (((((((M)=0 \\/ (L)=0) /\\ (((x)=0)))) \\/ (((~((M)=0)) /\\ (((~((L)=0)) /\\ (((M)+(L)=S (x)))))))) /\\ ((forall pfc_index_products_left_hvcoefficients. (exists pfa_gap_products_left_hvcoefficientsbound. pfa_gap_products_left_hvcoefficientsbound + S (pfc_index_products_left_hvcoefficients) = (x)) -> exists pfc_value_products_left_hvcoefficients. ((((exists ff_h_pfp_products_left_hvcoefficientsentry. ff_h_pfp_products_left_hvcoefficientsentry + S (pfc_value_products_left_hvcoefficients) = S ((S (pfc_index_products_left_hvcoefficients)) * uc)) /\\ exists ff_q_pfp_products_left_hvcoefficientsentry. ub = ff_q_pfp_products_left_hvcoefficientsentry * S ((S (pfc_index_products_left_hvcoefficients)) * uc) + (pfc_value_products_left_hvcoefficients))) /\\ ((exists pfc_terms_code_products_left_hvcoefficientscoefficient pfc_terms_scale_products_left_hvcoefficientscoefficient pfc_natural_sum_products_left_hvcoefficientscoefficient. ((forall pfc_index_products_left_hvcoefficientscoefficientdiagonal. (exists pfa_gap_products_left_hvcoefficientscoefficientdiagonalbound. pfa_gap_products_left_hvcoefficientscoefficientdiagonalbound + S (pfc_index_products_left_hvcoefficientscoefficientdiagonal) = (S (pfc_index_products_left_hvcoefficients))) -> exists pfc_value_products_left_hvcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_products_left_hvcoefficientscoefficientdiagonalentry. ff_h_pfp_products_left_hvcoefficientscoefficientdiagonalentry + S (pfc_value_products_left_hvcoefficientscoefficientdiagonal) = S ((S (pfc_index_products_left_hvcoefficientscoefficientdiagonal)) * pfc_terms_scale_products_left_hvcoefficientscoefficient)) /\\ exists ff_q_pfp_products_left_hvcoefficientscoefficientdiagonalentry. pfc_terms_code_products_left_hvcoefficientscoefficient = ff_q_pfp_products_left_hvcoefficientscoefficientdiagonalentry * S ((S (pfc_index_products_left_hvcoefficientscoefficientdiagonal)) * pfc_terms_scale_products_left_hvcoefficientscoefficient) + (pfc_value_products_left_hvcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_products_left_hvcoefficientscoefficientdiagonalterm pfc_left_products_left_hvcoefficientscoefficientdiagonalterm pfc_right_products_left_hvcoefficientscoefficientdiagonalterm. (((pfc_index_products_left_hvcoefficientscoefficientdiagonal)+pfc_complement_products_left_hvcoefficientscoefficientdiagonalterm=(pfc_index_products_left_hvcoefficients)) /\\ ((((((exists pfa_gap_products_left_hvcoefficientscoefficientdiagonaltermleftinside. pfa_gap_products_left_hvcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_products_left_hvcoefficientscoefficientdiagonal) = (M)) /\\ ((((exists ff_h_pfp_products_left_hvcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_products_left_hvcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_products_left_hvcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_products_left_hvcoefficientscoefficientdiagonal)) * dc)) /\\ exists ff_q_pfp_products_left_hvcoefficientscoefficientdiagonaltermleftentry. db = ff_q_pfp_products_left_hvcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_products_left_hvcoefficientscoefficientdiagonal)) * dc) + (pfc_left_products_left_hvcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_products_left_hvcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_products_left_hvcoefficientscoefficientdiagonaltermleftoutside+(M)=(pfc_index_products_left_hvcoefficientscoefficientdiagonal)) /\\ (((pfc_left_products_left_hvcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_products_left_hvcoefficientscoefficientdiagonaltermrightinside. pfa_gap_products_left_hvcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_products_left_hvcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_products_left_hvcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_products_left_hvcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_products_left_hvcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_products_left_hvcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_products_left_hvcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_products_left_hvcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_products_left_hvcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_products_left_hvcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_products_left_hvcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_products_left_hvcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_products_left_hvcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_products_left_hvcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_products_left_hvcoefficientscoefficientdiagonal)=pfc_left_products_left_hvcoefficientscoefficientdiagonalterm*pfc_right_products_left_hvcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_products_left_hvcoefficientscoefficientsum fs_v_pfc_products_left_hvcoefficientscoefficientsum. ((((exists fs_h_pfc_products_left_hvcoefficientscoefficientsum_body_start. fs_h_pfc_products_left_hvcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_products_left_hvcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_left_hvcoefficientscoefficientsum_body_start. fs_u_pfc_products_left_hvcoefficientscoefficientsum = fs_q_pfc_products_left_hvcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_products_left_hvcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_products_left_hvcoefficientscoefficientsum_body_terminal. fs_h_pfc_products_left_hvcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_products_left_hvcoefficientscoefficient) = S ((S (S (pfc_index_products_left_hvcoefficients))) * fs_v_pfc_products_left_hvcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_left_hvcoefficientscoefficientsum_body_terminal. fs_u_pfc_products_left_hvcoefficientscoefficientsum = fs_q_pfc_products_left_hvcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_products_left_hvcoefficients))) * fs_v_pfc_products_left_hvcoefficientscoefficientsum) + (pfc_natural_sum_products_left_hvcoefficientscoefficient))) /\\ forall fs_i_pfc_products_left_hvcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_products_left_hvcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_products_left_hvcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_products_left_hvcoefficientscoefficientsum_body_steps = S (pfc_index_products_left_hvcoefficients)) -> exists fs_a_pfc_products_left_hvcoefficientscoefficientsum_body_steps fs_r_pfc_products_left_hvcoefficientscoefficientsum_body_steps fs_s_pfc_products_left_hvcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_products_left_hvcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_products_left_hvcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_products_left_hvcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_products_left_hvcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_products_left_hvcoefficientscoefficient)) /\\ exists fs_q_pfc_products_left_hvcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_products_left_hvcoefficientscoefficient = fs_q_pfc_products_left_hvcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_products_left_hvcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_products_left_hvcoefficientscoefficient) + (fs_a_pfc_products_left_hvcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_products_left_hvcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_products_left_hvcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_products_left_hvcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_products_left_hvcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_hvcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_left_hvcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_products_left_hvcoefficientscoefficientsum = fs_q_pfc_products_left_hvcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_products_left_hvcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_hvcoefficientscoefficientsum) + (fs_r_pfc_products_left_hvcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_products_left_hvcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_products_left_hvcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_products_left_hvcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_products_left_hvcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_hvcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_left_hvcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_products_left_hvcoefficientscoefficientsum = fs_q_pfc_products_left_hvcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_products_left_hvcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_hvcoefficientscoefficientsum) + (fs_s_pfc_products_left_hvcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_products_left_hvcoefficientscoefficientsum_body_steps = fs_r_pfc_products_left_hvcoefficientscoefficientsum_body_steps + fs_a_pfc_products_left_hvcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_products_left_hvcoefficientscoefficientresiduebound. pfa_gap_products_left_hvcoefficientscoefficientresiduebound + S (pfc_value_products_left_hvcoefficients) = (p)) /\\ ((exists pfa_offset_left_products_left_hvcoefficientscoefficientresiduecongruence pfa_offset_right_products_left_hvcoefficientscoefficientresiduecongruence. (pfc_natural_sum_products_left_hvcoefficientscoefficient) + (p) * pfa_offset_left_products_left_hvcoefficientscoefficientresiduecongruence = (pfc_value_products_left_hvcoefficients) + (p) * pfa_offset_right_products_left_hvcoefficientscoefficientresiduecongruence))))))))))))))))))", "specialize prime_field_polynomial_convolution_at_length_exists (p)", "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 (M)", "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 (L)", "specialize prime_field_polynomial_convolution_at_length_exists (x)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hp", "exact hd", "exact hbounds_right_left", "exact hlength_witness", "cases hv", "cases hv_witness", "have hw : exists ub uc. ((forall fom_index_pfp_products_left_hwleft. (exists fom_gap_pfp_products_left_hwleft_index_bound. fom_gap_pfp_products_left_hwleft_index_bound + S (fom_index_pfp_products_left_hwleft) = M) -> exists fom_value_pfp_products_left_hwleft. ((((exists fom_beta_height_pfp_products_left_hwleft_entry. fom_beta_height_pfp_products_left_hwleft_entry + S (fom_value_pfp_products_left_hwleft) = S ((S (fom_index_pfp_products_left_hwleft)) * dc)) /\\ exists fom_beta_quotient_pfp_products_left_hwleft_entry. db = fom_beta_quotient_pfp_products_left_hwleft_entry * S ((S (fom_index_pfp_products_left_hwleft)) * dc) + (fom_value_pfp_products_left_hwleft))) /\\ (exists fom_gap_pfp_products_left_hwleft_value_bound. fom_gap_pfp_products_left_hwleft_value_bound + S (fom_value_pfp_products_left_hwleft) = p))) /\\ (((forall fom_index_pfp_products_left_hwright. (exists fom_gap_pfp_products_left_hwright_index_bound. fom_gap_pfp_products_left_hwright_index_bound + S (fom_index_pfp_products_left_hwright) = L) -> exists fom_value_pfp_products_left_hwright. ((((exists fom_beta_height_pfp_products_left_hwright_entry. fom_beta_height_pfp_products_left_hwright_entry + S (fom_value_pfp_products_left_hwright) = S ((S (fom_index_pfp_products_left_hwright)) * cc)) /\\ exists fom_beta_quotient_pfp_products_left_hwright_entry. cb = fom_beta_quotient_pfp_products_left_hwright_entry * S ((S (fom_index_pfp_products_left_hwright)) * cc) + (fom_value_pfp_products_left_hwright))) /\\ (exists fom_gap_pfp_products_left_hwright_value_bound. fom_gap_pfp_products_left_hwright_value_bound + S (fom_value_pfp_products_left_hwright) = p))) /\\ (((((((M)=0 \\/ (L)=0) /\\ (((x)=0)))) \\/ (((~((M)=0)) /\\ (((~((L)=0)) /\\ (((M)+(L)=S (x)))))))) /\\ ((forall pfc_index_products_left_hwcoefficients. (exists pfa_gap_products_left_hwcoefficientsbound. pfa_gap_products_left_hwcoefficientsbound + S (pfc_index_products_left_hwcoefficients) = (x)) -> exists pfc_value_products_left_hwcoefficients. ((((exists ff_h_pfp_products_left_hwcoefficientsentry. ff_h_pfp_products_left_hwcoefficientsentry + S (pfc_value_products_left_hwcoefficients) = S ((S (pfc_index_products_left_hwcoefficients)) * uc)) /\\ exists ff_q_pfp_products_left_hwcoefficientsentry. ub = ff_q_pfp_products_left_hwcoefficientsentry * S ((S (pfc_index_products_left_hwcoefficients)) * uc) + (pfc_value_products_left_hwcoefficients))) /\\ ((exists pfc_terms_code_products_left_hwcoefficientscoefficient pfc_terms_scale_products_left_hwcoefficientscoefficient pfc_natural_sum_products_left_hwcoefficientscoefficient. ((forall pfc_index_products_left_hwcoefficientscoefficientdiagonal. (exists pfa_gap_products_left_hwcoefficientscoefficientdiagonalbound. pfa_gap_products_left_hwcoefficientscoefficientdiagonalbound + S (pfc_index_products_left_hwcoefficientscoefficientdiagonal) = (S (pfc_index_products_left_hwcoefficients))) -> exists pfc_value_products_left_hwcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_products_left_hwcoefficientscoefficientdiagonalentry. ff_h_pfp_products_left_hwcoefficientscoefficientdiagonalentry + S (pfc_value_products_left_hwcoefficientscoefficientdiagonal) = S ((S (pfc_index_products_left_hwcoefficientscoefficientdiagonal)) * pfc_terms_scale_products_left_hwcoefficientscoefficient)) /\\ exists ff_q_pfp_products_left_hwcoefficientscoefficientdiagonalentry. pfc_terms_code_products_left_hwcoefficientscoefficient = ff_q_pfp_products_left_hwcoefficientscoefficientdiagonalentry * S ((S (pfc_index_products_left_hwcoefficientscoefficientdiagonal)) * pfc_terms_scale_products_left_hwcoefficientscoefficient) + (pfc_value_products_left_hwcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_products_left_hwcoefficientscoefficientdiagonalterm pfc_left_products_left_hwcoefficientscoefficientdiagonalterm pfc_right_products_left_hwcoefficientscoefficientdiagonalterm. (((pfc_index_products_left_hwcoefficientscoefficientdiagonal)+pfc_complement_products_left_hwcoefficientscoefficientdiagonalterm=(pfc_index_products_left_hwcoefficients)) /\\ ((((((exists pfa_gap_products_left_hwcoefficientscoefficientdiagonaltermleftinside. pfa_gap_products_left_hwcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_products_left_hwcoefficientscoefficientdiagonal) = (M)) /\\ ((((exists ff_h_pfp_products_left_hwcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_products_left_hwcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_products_left_hwcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_products_left_hwcoefficientscoefficientdiagonal)) * dc)) /\\ exists ff_q_pfp_products_left_hwcoefficientscoefficientdiagonaltermleftentry. db = ff_q_pfp_products_left_hwcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_products_left_hwcoefficientscoefficientdiagonal)) * dc) + (pfc_left_products_left_hwcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_products_left_hwcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_products_left_hwcoefficientscoefficientdiagonaltermleftoutside+(M)=(pfc_index_products_left_hwcoefficientscoefficientdiagonal)) /\\ (((pfc_left_products_left_hwcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_products_left_hwcoefficientscoefficientdiagonaltermrightinside. pfa_gap_products_left_hwcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_products_left_hwcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_products_left_hwcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_products_left_hwcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_products_left_hwcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_products_left_hwcoefficientscoefficientdiagonalterm)) * cc)) /\\ exists ff_q_pfp_products_left_hwcoefficientscoefficientdiagonaltermrightentry. cb = ff_q_pfp_products_left_hwcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_products_left_hwcoefficientscoefficientdiagonalterm)) * cc) + (pfc_right_products_left_hwcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_products_left_hwcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_products_left_hwcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_products_left_hwcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_products_left_hwcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_products_left_hwcoefficientscoefficientdiagonal)=pfc_left_products_left_hwcoefficientscoefficientdiagonalterm*pfc_right_products_left_hwcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_products_left_hwcoefficientscoefficientsum fs_v_pfc_products_left_hwcoefficientscoefficientsum. ((((exists fs_h_pfc_products_left_hwcoefficientscoefficientsum_body_start. fs_h_pfc_products_left_hwcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_products_left_hwcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_left_hwcoefficientscoefficientsum_body_start. fs_u_pfc_products_left_hwcoefficientscoefficientsum = fs_q_pfc_products_left_hwcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_products_left_hwcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_products_left_hwcoefficientscoefficientsum_body_terminal. fs_h_pfc_products_left_hwcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_products_left_hwcoefficientscoefficient) = S ((S (S (pfc_index_products_left_hwcoefficients))) * fs_v_pfc_products_left_hwcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_left_hwcoefficientscoefficientsum_body_terminal. fs_u_pfc_products_left_hwcoefficientscoefficientsum = fs_q_pfc_products_left_hwcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_products_left_hwcoefficients))) * fs_v_pfc_products_left_hwcoefficientscoefficientsum) + (pfc_natural_sum_products_left_hwcoefficientscoefficient))) /\\ forall fs_i_pfc_products_left_hwcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_products_left_hwcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_products_left_hwcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_products_left_hwcoefficientscoefficientsum_body_steps = S (pfc_index_products_left_hwcoefficients)) -> exists fs_a_pfc_products_left_hwcoefficientscoefficientsum_body_steps fs_r_pfc_products_left_hwcoefficientscoefficientsum_body_steps fs_s_pfc_products_left_hwcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_products_left_hwcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_products_left_hwcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_products_left_hwcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_products_left_hwcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_products_left_hwcoefficientscoefficient)) /\\ exists fs_q_pfc_products_left_hwcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_products_left_hwcoefficientscoefficient = fs_q_pfc_products_left_hwcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_products_left_hwcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_products_left_hwcoefficientscoefficient) + (fs_a_pfc_products_left_hwcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_products_left_hwcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_products_left_hwcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_products_left_hwcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_products_left_hwcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_hwcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_left_hwcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_products_left_hwcoefficientscoefficientsum = fs_q_pfc_products_left_hwcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_products_left_hwcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_hwcoefficientscoefficientsum) + (fs_r_pfc_products_left_hwcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_products_left_hwcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_products_left_hwcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_products_left_hwcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_products_left_hwcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_hwcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_left_hwcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_products_left_hwcoefficientscoefficientsum = fs_q_pfc_products_left_hwcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_products_left_hwcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_hwcoefficientscoefficientsum) + (fs_s_pfc_products_left_hwcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_products_left_hwcoefficientscoefficientsum_body_steps = fs_r_pfc_products_left_hwcoefficientscoefficientsum_body_steps + fs_a_pfc_products_left_hwcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_products_left_hwcoefficientscoefficientresiduebound. pfa_gap_products_left_hwcoefficientscoefficientresiduebound + S (pfc_value_products_left_hwcoefficients) = (p)) /\\ ((exists pfa_offset_left_products_left_hwcoefficientscoefficientresiduecongruence pfa_offset_right_products_left_hwcoefficientscoefficientresiduecongruence. (pfc_natural_sum_products_left_hwcoefficientscoefficient) + (p) * pfa_offset_left_products_left_hwcoefficientscoefficientresiduecongruence = (pfc_value_products_left_hwcoefficients) + (p) * pfa_offset_right_products_left_hwcoefficientscoefficientresiduecongruence))))))))))))))))))", "specialize prime_field_polynomial_convolution_at_length_exists (p)", "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 (M)", "specialize prime_field_polynomial_convolution_at_length_exists (cb)", "specialize prime_field_polynomial_convolution_at_length_exists (cc)", "specialize prime_field_polynomial_convolution_at_length_exists (L)", "specialize prime_field_polynomial_convolution_at_length_exists (x)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hp", "exact hd", "exact hbounds_right_right", "exact hlength_witness", "cases hw", "cases hw_witness", "exists x", "exists x1", "exists x2", "exists x3", "exists x4", "exists x5", "exists x6", "split", "exact hu_witness_witness", "split", "exact hv_witness_witness", "split", "exact hw_witness_witness", "specialize prime_field_polynomial_convolution_left_add (p)", "specialize prime_field_polynomial_convolution_left_add (ab)", "specialize prime_field_polynomial_convolution_left_add (ac)", "specialize prime_field_polynomial_convolution_left_add (bb)", "specialize prime_field_polynomial_convolution_left_add (bc)", "specialize prime_field_polynomial_convolution_left_add (cb)", "specialize prime_field_polynomial_convolution_left_add (cc)", "specialize prime_field_polynomial_convolution_left_add (L)", "specialize prime_field_polynomial_convolution_left_add (db)", "specialize prime_field_polynomial_convolution_left_add (dc)", "specialize prime_field_polynomial_convolution_left_add (M)", "specialize prime_field_polynomial_convolution_left_add (x1)", "specialize prime_field_polynomial_convolution_left_add (x2)", "specialize prime_field_polynomial_convolution_left_add (x3)", "specialize prime_field_polynomial_convolution_left_add (x4)", "specialize prime_field_polynomial_convolution_left_add (x5)", "specialize prime_field_polynomial_convolution_left_add (x6)", "specialize prime_field_polynomial_convolution_left_add (x)", "apply prime_field_polynomial_convolution_left_add", "exact hs", "exact hu_witness_witness", "exact hv_witness_witness", "exact hw_witness_witness" ], "script_sha256": "cfde9b382b214f324990879ea1680bf6ab8fef3468fecf4a7e609410299cc2e9", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_distributivity_candidate.py", "sha256": "a959962d631759cd1fc773dd7eef2fadf4f3f95361d6d7bc8c6a9e82d0d4ab86" }, "statement": "forall p ab ac bb bc cb cc L db dc M. (~(p=0)) -> (forall fom_index_pfp_products_left_fixed_input. (exists fom_gap_pfp_products_left_fixed_input_index_bound. fom_gap_pfp_products_left_fixed_input_index_bound + S (fom_index_pfp_products_left_fixed_input) = M) -> exists fom_value_pfp_products_left_fixed_input. ((((exists fom_beta_height_pfp_products_left_fixed_input_entry. fom_beta_height_pfp_products_left_fixed_input_entry + S (fom_value_pfp_products_left_fixed_input) = S ((S (fom_index_pfp_products_left_fixed_input)) * dc)) /\\ exists fom_beta_quotient_pfp_products_left_fixed_input_entry. db = fom_beta_quotient_pfp_products_left_fixed_input_entry * S ((S (fom_index_pfp_products_left_fixed_input)) * dc) + (fom_value_pfp_products_left_fixed_input))) /\\ (exists fom_gap_pfp_products_left_fixed_input_value_bound. fom_gap_pfp_products_left_fixed_input_value_bound + S (fom_value_pfp_products_left_fixed_input) = p))) -> (forall pfp_index_products_left_input_addition. (exists pfa_gap_products_left_input_additionindex. pfa_gap_products_left_input_additionindex + S (pfp_index_products_left_input_addition) = (L)) -> exists pfp_left_products_left_input_addition pfp_right_products_left_input_addition pfp_value_products_left_input_addition. ((((exists ff_h_pfp_products_left_input_additionleft. ff_h_pfp_products_left_input_additionleft + S (pfp_left_products_left_input_addition) = S ((S (pfp_index_products_left_input_addition)) * ac)) /\\ exists ff_q_pfp_products_left_input_additionleft. ab = ff_q_pfp_products_left_input_additionleft * S ((S (pfp_index_products_left_input_addition)) * ac) + (pfp_left_products_left_input_addition))) /\\ (((((exists ff_h_pfp_products_left_input_additionright. ff_h_pfp_products_left_input_additionright + S (pfp_right_products_left_input_addition) = S ((S (pfp_index_products_left_input_addition)) * bc)) /\\ exists ff_q_pfp_products_left_input_additionright. bb = ff_q_pfp_products_left_input_additionright * S ((S (pfp_index_products_left_input_addition)) * bc) + (pfp_right_products_left_input_addition))) /\\ (((((exists ff_h_pfp_products_left_input_additiontarget. ff_h_pfp_products_left_input_additiontarget + S (pfp_value_products_left_input_addition) = S ((S (pfp_index_products_left_input_addition)) * cc)) /\\ exists ff_q_pfp_products_left_input_additiontarget. cb = ff_q_pfp_products_left_input_additiontarget * S ((S (pfp_index_products_left_input_addition)) * cc) + (pfp_value_products_left_input_addition))) /\\ ((((exists pfa_gap_products_left_input_additionoperationleft. pfa_gap_products_left_input_additionoperationleft + S (pfp_left_products_left_input_addition) = (p)) /\\ (((exists pfa_gap_products_left_input_additionoperationright. pfa_gap_products_left_input_additionoperationright + S (pfp_right_products_left_input_addition) = (p)) /\\ ((((exists pfa_gap_products_left_input_additionoperationresultbound. pfa_gap_products_left_input_additionoperationresultbound + S (pfp_value_products_left_input_addition) = (p)) /\\ ((exists pfa_offset_left_products_left_input_additionoperationresultcongruence pfa_offset_right_products_left_input_additionoperationresultcongruence. ((pfp_left_products_left_input_addition) + (pfp_right_products_left_input_addition)) + (p) * pfa_offset_left_products_left_input_additionoperationresultcongruence = (pfp_value_products_left_input_addition) + (p) * pfa_offset_right_products_left_input_additionoperationresultcongruence)))))))))))))))) -> (exists N ub uc vb vc wb wc. ((((forall fom_index_pfp_products_left_result_ubleft. (exists fom_gap_pfp_products_left_result_ubleft_index_bound. fom_gap_pfp_products_left_result_ubleft_index_bound + S (fom_index_pfp_products_left_result_ubleft) = M) -> exists fom_value_pfp_products_left_result_ubleft. ((((exists fom_beta_height_pfp_products_left_result_ubleft_entry. fom_beta_height_pfp_products_left_result_ubleft_entry + S (fom_value_pfp_products_left_result_ubleft) = S ((S (fom_index_pfp_products_left_result_ubleft)) * dc)) /\\ exists fom_beta_quotient_pfp_products_left_result_ubleft_entry. db = fom_beta_quotient_pfp_products_left_result_ubleft_entry * S ((S (fom_index_pfp_products_left_result_ubleft)) * dc) + (fom_value_pfp_products_left_result_ubleft))) /\\ (exists fom_gap_pfp_products_left_result_ubleft_value_bound. fom_gap_pfp_products_left_result_ubleft_value_bound + S (fom_value_pfp_products_left_result_ubleft) = p))) /\\ (((forall fom_index_pfp_products_left_result_ubright. (exists fom_gap_pfp_products_left_result_ubright_index_bound. fom_gap_pfp_products_left_result_ubright_index_bound + S (fom_index_pfp_products_left_result_ubright) = L) -> exists fom_value_pfp_products_left_result_ubright. ((((exists fom_beta_height_pfp_products_left_result_ubright_entry. fom_beta_height_pfp_products_left_result_ubright_entry + S (fom_value_pfp_products_left_result_ubright) = S ((S (fom_index_pfp_products_left_result_ubright)) * ac)) /\\ exists fom_beta_quotient_pfp_products_left_result_ubright_entry. ab = fom_beta_quotient_pfp_products_left_result_ubright_entry * S ((S (fom_index_pfp_products_left_result_ubright)) * ac) + (fom_value_pfp_products_left_result_ubright))) /\\ (exists fom_gap_pfp_products_left_result_ubright_value_bound. fom_gap_pfp_products_left_result_ubright_value_bound + S (fom_value_pfp_products_left_result_ubright) = p))) /\\ (((((((M)=0 \\/ (L)=0) /\\ (((N)=0)))) \\/ (((~((M)=0)) /\\ (((~((L)=0)) /\\ (((M)+(L)=S (N)))))))) /\\ ((forall pfc_index_products_left_result_ubcoefficients. (exists pfa_gap_products_left_result_ubcoefficientsbound. pfa_gap_products_left_result_ubcoefficientsbound + S (pfc_index_products_left_result_ubcoefficients) = (N)) -> exists pfc_value_products_left_result_ubcoefficients. ((((exists ff_h_pfp_products_left_result_ubcoefficientsentry. ff_h_pfp_products_left_result_ubcoefficientsentry + S (pfc_value_products_left_result_ubcoefficients) = S ((S (pfc_index_products_left_result_ubcoefficients)) * uc)) /\\ exists ff_q_pfp_products_left_result_ubcoefficientsentry. ub = ff_q_pfp_products_left_result_ubcoefficientsentry * S ((S (pfc_index_products_left_result_ubcoefficients)) * uc) + (pfc_value_products_left_result_ubcoefficients))) /\\ ((exists pfc_terms_code_products_left_result_ubcoefficientscoefficient pfc_terms_scale_products_left_result_ubcoefficientscoefficient pfc_natural_sum_products_left_result_ubcoefficientscoefficient. ((forall pfc_index_products_left_result_ubcoefficientscoefficientdiagonal. (exists pfa_gap_products_left_result_ubcoefficientscoefficientdiagonalbound. pfa_gap_products_left_result_ubcoefficientscoefficientdiagonalbound + S (pfc_index_products_left_result_ubcoefficientscoefficientdiagonal) = (S (pfc_index_products_left_result_ubcoefficients))) -> exists pfc_value_products_left_result_ubcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_products_left_result_ubcoefficientscoefficientdiagonalentry. ff_h_pfp_products_left_result_ubcoefficientscoefficientdiagonalentry + S (pfc_value_products_left_result_ubcoefficientscoefficientdiagonal) = S ((S (pfc_index_products_left_result_ubcoefficientscoefficientdiagonal)) * pfc_terms_scale_products_left_result_ubcoefficientscoefficient)) /\\ exists ff_q_pfp_products_left_result_ubcoefficientscoefficientdiagonalentry. pfc_terms_code_products_left_result_ubcoefficientscoefficient = ff_q_pfp_products_left_result_ubcoefficientscoefficientdiagonalentry * S ((S (pfc_index_products_left_result_ubcoefficientscoefficientdiagonal)) * pfc_terms_scale_products_left_result_ubcoefficientscoefficient) + (pfc_value_products_left_result_ubcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_products_left_result_ubcoefficientscoefficientdiagonalterm pfc_left_products_left_result_ubcoefficientscoefficientdiagonalterm pfc_right_products_left_result_ubcoefficientscoefficientdiagonalterm. (((pfc_index_products_left_result_ubcoefficientscoefficientdiagonal)+pfc_complement_products_left_result_ubcoefficientscoefficientdiagonalterm=(pfc_index_products_left_result_ubcoefficients)) /\\ ((((((exists pfa_gap_products_left_result_ubcoefficientscoefficientdiagonaltermleftinside. pfa_gap_products_left_result_ubcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_products_left_result_ubcoefficientscoefficientdiagonal) = (M)) /\\ ((((exists ff_h_pfp_products_left_result_ubcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_products_left_result_ubcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_products_left_result_ubcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_products_left_result_ubcoefficientscoefficientdiagonal)) * dc)) /\\ exists ff_q_pfp_products_left_result_ubcoefficientscoefficientdiagonaltermleftentry. db = ff_q_pfp_products_left_result_ubcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_products_left_result_ubcoefficientscoefficientdiagonal)) * dc) + (pfc_left_products_left_result_ubcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_products_left_result_ubcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_products_left_result_ubcoefficientscoefficientdiagonaltermleftoutside+(M)=(pfc_index_products_left_result_ubcoefficientscoefficientdiagonal)) /\\ (((pfc_left_products_left_result_ubcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_products_left_result_ubcoefficientscoefficientdiagonaltermrightinside. pfa_gap_products_left_result_ubcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_products_left_result_ubcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_products_left_result_ubcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_products_left_result_ubcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_products_left_result_ubcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_products_left_result_ubcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_products_left_result_ubcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_products_left_result_ubcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_products_left_result_ubcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_products_left_result_ubcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_products_left_result_ubcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_products_left_result_ubcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_products_left_result_ubcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_products_left_result_ubcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_products_left_result_ubcoefficientscoefficientdiagonal)=pfc_left_products_left_result_ubcoefficientscoefficientdiagonalterm*pfc_right_products_left_result_ubcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_products_left_result_ubcoefficientscoefficientsum fs_v_pfc_products_left_result_ubcoefficientscoefficientsum. ((((exists fs_h_pfc_products_left_result_ubcoefficientscoefficientsum_body_start. fs_h_pfc_products_left_result_ubcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_products_left_result_ubcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_left_result_ubcoefficientscoefficientsum_body_start. fs_u_pfc_products_left_result_ubcoefficientscoefficientsum = fs_q_pfc_products_left_result_ubcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_products_left_result_ubcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_products_left_result_ubcoefficientscoefficientsum_body_terminal. fs_h_pfc_products_left_result_ubcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_products_left_result_ubcoefficientscoefficient) = S ((S (S (pfc_index_products_left_result_ubcoefficients))) * fs_v_pfc_products_left_result_ubcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_left_result_ubcoefficientscoefficientsum_body_terminal. fs_u_pfc_products_left_result_ubcoefficientscoefficientsum = fs_q_pfc_products_left_result_ubcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_products_left_result_ubcoefficients))) * fs_v_pfc_products_left_result_ubcoefficientscoefficientsum) + (pfc_natural_sum_products_left_result_ubcoefficientscoefficient))) /\\ forall fs_i_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps = S (pfc_index_products_left_result_ubcoefficients)) -> exists fs_a_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps fs_r_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps fs_s_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_products_left_result_ubcoefficientscoefficient)) /\\ exists fs_q_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_products_left_result_ubcoefficientscoefficient = fs_q_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_products_left_result_ubcoefficientscoefficient) + (fs_a_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_result_ubcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_products_left_result_ubcoefficientscoefficientsum = fs_q_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_result_ubcoefficientscoefficientsum) + (fs_r_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_result_ubcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_products_left_result_ubcoefficientscoefficientsum = fs_q_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_result_ubcoefficientscoefficientsum) + (fs_s_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps = fs_r_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps + fs_a_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_products_left_result_ubcoefficientscoefficientresiduebound. pfa_gap_products_left_result_ubcoefficientscoefficientresiduebound + S (pfc_value_products_left_result_ubcoefficients) = (p)) /\\ ((exists pfa_offset_left_products_left_result_ubcoefficientscoefficientresiduecongruence pfa_offset_right_products_left_result_ubcoefficientscoefficientresiduecongruence. (pfc_natural_sum_products_left_result_ubcoefficientscoefficient) + (p) * pfa_offset_left_products_left_result_ubcoefficientscoefficientresiduecongruence = (pfc_value_products_left_result_ubcoefficients) + (p) * pfa_offset_right_products_left_result_ubcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall fom_index_pfp_products_left_result_vbleft. (exists fom_gap_pfp_products_left_result_vbleft_index_bound. fom_gap_pfp_products_left_result_vbleft_index_bound + S (fom_index_pfp_products_left_result_vbleft) = M) -> exists fom_value_pfp_products_left_result_vbleft. ((((exists fom_beta_height_pfp_products_left_result_vbleft_entry. fom_beta_height_pfp_products_left_result_vbleft_entry + S (fom_value_pfp_products_left_result_vbleft) = S ((S (fom_index_pfp_products_left_result_vbleft)) * dc)) /\\ exists fom_beta_quotient_pfp_products_left_result_vbleft_entry. db = fom_beta_quotient_pfp_products_left_result_vbleft_entry * S ((S (fom_index_pfp_products_left_result_vbleft)) * dc) + (fom_value_pfp_products_left_result_vbleft))) /\\ (exists fom_gap_pfp_products_left_result_vbleft_value_bound. fom_gap_pfp_products_left_result_vbleft_value_bound + S (fom_value_pfp_products_left_result_vbleft) = p))) /\\ (((forall fom_index_pfp_products_left_result_vbright. (exists fom_gap_pfp_products_left_result_vbright_index_bound. fom_gap_pfp_products_left_result_vbright_index_bound + S (fom_index_pfp_products_left_result_vbright) = L) -> exists fom_value_pfp_products_left_result_vbright. ((((exists fom_beta_height_pfp_products_left_result_vbright_entry. fom_beta_height_pfp_products_left_result_vbright_entry + S (fom_value_pfp_products_left_result_vbright) = S ((S (fom_index_pfp_products_left_result_vbright)) * bc)) /\\ exists fom_beta_quotient_pfp_products_left_result_vbright_entry. bb = fom_beta_quotient_pfp_products_left_result_vbright_entry * S ((S (fom_index_pfp_products_left_result_vbright)) * bc) + (fom_value_pfp_products_left_result_vbright))) /\\ (exists fom_gap_pfp_products_left_result_vbright_value_bound. fom_gap_pfp_products_left_result_vbright_value_bound + S (fom_value_pfp_products_left_result_vbright) = p))) /\\ (((((((M)=0 \\/ (L)=0) /\\ (((N)=0)))) \\/ (((~((M)=0)) /\\ (((~((L)=0)) /\\ (((M)+(L)=S (N)))))))) /\\ ((forall pfc_index_products_left_result_vbcoefficients. (exists pfa_gap_products_left_result_vbcoefficientsbound. pfa_gap_products_left_result_vbcoefficientsbound + S (pfc_index_products_left_result_vbcoefficients) = (N)) -> exists pfc_value_products_left_result_vbcoefficients. ((((exists ff_h_pfp_products_left_result_vbcoefficientsentry. ff_h_pfp_products_left_result_vbcoefficientsentry + S (pfc_value_products_left_result_vbcoefficients) = S ((S (pfc_index_products_left_result_vbcoefficients)) * vc)) /\\ exists ff_q_pfp_products_left_result_vbcoefficientsentry. vb = ff_q_pfp_products_left_result_vbcoefficientsentry * S ((S (pfc_index_products_left_result_vbcoefficients)) * vc) + (pfc_value_products_left_result_vbcoefficients))) /\\ ((exists pfc_terms_code_products_left_result_vbcoefficientscoefficient pfc_terms_scale_products_left_result_vbcoefficientscoefficient pfc_natural_sum_products_left_result_vbcoefficientscoefficient. ((forall pfc_index_products_left_result_vbcoefficientscoefficientdiagonal. (exists pfa_gap_products_left_result_vbcoefficientscoefficientdiagonalbound. pfa_gap_products_left_result_vbcoefficientscoefficientdiagonalbound + S (pfc_index_products_left_result_vbcoefficientscoefficientdiagonal) = (S (pfc_index_products_left_result_vbcoefficients))) -> exists pfc_value_products_left_result_vbcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_products_left_result_vbcoefficientscoefficientdiagonalentry. ff_h_pfp_products_left_result_vbcoefficientscoefficientdiagonalentry + S (pfc_value_products_left_result_vbcoefficientscoefficientdiagonal) = S ((S (pfc_index_products_left_result_vbcoefficientscoefficientdiagonal)) * pfc_terms_scale_products_left_result_vbcoefficientscoefficient)) /\\ exists ff_q_pfp_products_left_result_vbcoefficientscoefficientdiagonalentry. pfc_terms_code_products_left_result_vbcoefficientscoefficient = ff_q_pfp_products_left_result_vbcoefficientscoefficientdiagonalentry * S ((S (pfc_index_products_left_result_vbcoefficientscoefficientdiagonal)) * pfc_terms_scale_products_left_result_vbcoefficientscoefficient) + (pfc_value_products_left_result_vbcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_products_left_result_vbcoefficientscoefficientdiagonalterm pfc_left_products_left_result_vbcoefficientscoefficientdiagonalterm pfc_right_products_left_result_vbcoefficientscoefficientdiagonalterm. (((pfc_index_products_left_result_vbcoefficientscoefficientdiagonal)+pfc_complement_products_left_result_vbcoefficientscoefficientdiagonalterm=(pfc_index_products_left_result_vbcoefficients)) /\\ ((((((exists pfa_gap_products_left_result_vbcoefficientscoefficientdiagonaltermleftinside. pfa_gap_products_left_result_vbcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_products_left_result_vbcoefficientscoefficientdiagonal) = (M)) /\\ ((((exists ff_h_pfp_products_left_result_vbcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_products_left_result_vbcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_products_left_result_vbcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_products_left_result_vbcoefficientscoefficientdiagonal)) * dc)) /\\ exists ff_q_pfp_products_left_result_vbcoefficientscoefficientdiagonaltermleftentry. db = ff_q_pfp_products_left_result_vbcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_products_left_result_vbcoefficientscoefficientdiagonal)) * dc) + (pfc_left_products_left_result_vbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_products_left_result_vbcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_products_left_result_vbcoefficientscoefficientdiagonaltermleftoutside+(M)=(pfc_index_products_left_result_vbcoefficientscoefficientdiagonal)) /\\ (((pfc_left_products_left_result_vbcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_products_left_result_vbcoefficientscoefficientdiagonaltermrightinside. pfa_gap_products_left_result_vbcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_products_left_result_vbcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_products_left_result_vbcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_products_left_result_vbcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_products_left_result_vbcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_products_left_result_vbcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_products_left_result_vbcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_products_left_result_vbcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_products_left_result_vbcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_products_left_result_vbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_products_left_result_vbcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_products_left_result_vbcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_products_left_result_vbcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_products_left_result_vbcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_products_left_result_vbcoefficientscoefficientdiagonal)=pfc_left_products_left_result_vbcoefficientscoefficientdiagonalterm*pfc_right_products_left_result_vbcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_products_left_result_vbcoefficientscoefficientsum fs_v_pfc_products_left_result_vbcoefficientscoefficientsum. ((((exists fs_h_pfc_products_left_result_vbcoefficientscoefficientsum_body_start. fs_h_pfc_products_left_result_vbcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_products_left_result_vbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_left_result_vbcoefficientscoefficientsum_body_start. fs_u_pfc_products_left_result_vbcoefficientscoefficientsum = fs_q_pfc_products_left_result_vbcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_products_left_result_vbcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_products_left_result_vbcoefficientscoefficientsum_body_terminal. fs_h_pfc_products_left_result_vbcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_products_left_result_vbcoefficientscoefficient) = S ((S (S (pfc_index_products_left_result_vbcoefficients))) * fs_v_pfc_products_left_result_vbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_left_result_vbcoefficientscoefficientsum_body_terminal. fs_u_pfc_products_left_result_vbcoefficientscoefficientsum = fs_q_pfc_products_left_result_vbcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_products_left_result_vbcoefficients))) * fs_v_pfc_products_left_result_vbcoefficientscoefficientsum) + (pfc_natural_sum_products_left_result_vbcoefficientscoefficient))) /\\ forall fs_i_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps = S (pfc_index_products_left_result_vbcoefficients)) -> exists fs_a_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps fs_r_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps fs_s_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_products_left_result_vbcoefficientscoefficient)) /\\ exists fs_q_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_products_left_result_vbcoefficientscoefficient = fs_q_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_products_left_result_vbcoefficientscoefficient) + (fs_a_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_result_vbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_products_left_result_vbcoefficientscoefficientsum = fs_q_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_result_vbcoefficientscoefficientsum) + (fs_r_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_result_vbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_products_left_result_vbcoefficientscoefficientsum = fs_q_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_result_vbcoefficientscoefficientsum) + (fs_s_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps = fs_r_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps + fs_a_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_products_left_result_vbcoefficientscoefficientresiduebound. pfa_gap_products_left_result_vbcoefficientscoefficientresiduebound + S (pfc_value_products_left_result_vbcoefficients) = (p)) /\\ ((exists pfa_offset_left_products_left_result_vbcoefficientscoefficientresiduecongruence pfa_offset_right_products_left_result_vbcoefficientscoefficientresiduecongruence. (pfc_natural_sum_products_left_result_vbcoefficientscoefficient) + (p) * pfa_offset_left_products_left_result_vbcoefficientscoefficientresiduecongruence = (pfc_value_products_left_result_vbcoefficients) + (p) * pfa_offset_right_products_left_result_vbcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall fom_index_pfp_products_left_result_wbleft. (exists fom_gap_pfp_products_left_result_wbleft_index_bound. fom_gap_pfp_products_left_result_wbleft_index_bound + S (fom_index_pfp_products_left_result_wbleft) = M) -> exists fom_value_pfp_products_left_result_wbleft. ((((exists fom_beta_height_pfp_products_left_result_wbleft_entry. fom_beta_height_pfp_products_left_result_wbleft_entry + S (fom_value_pfp_products_left_result_wbleft) = S ((S (fom_index_pfp_products_left_result_wbleft)) * dc)) /\\ exists fom_beta_quotient_pfp_products_left_result_wbleft_entry. db = fom_beta_quotient_pfp_products_left_result_wbleft_entry * S ((S (fom_index_pfp_products_left_result_wbleft)) * dc) + (fom_value_pfp_products_left_result_wbleft))) /\\ (exists fom_gap_pfp_products_left_result_wbleft_value_bound. fom_gap_pfp_products_left_result_wbleft_value_bound + S (fom_value_pfp_products_left_result_wbleft) = p))) /\\ (((forall fom_index_pfp_products_left_result_wbright. (exists fom_gap_pfp_products_left_result_wbright_index_bound. fom_gap_pfp_products_left_result_wbright_index_bound + S (fom_index_pfp_products_left_result_wbright) = L) -> exists fom_value_pfp_products_left_result_wbright. ((((exists fom_beta_height_pfp_products_left_result_wbright_entry. fom_beta_height_pfp_products_left_result_wbright_entry + S (fom_value_pfp_products_left_result_wbright) = S ((S (fom_index_pfp_products_left_result_wbright)) * cc)) /\\ exists fom_beta_quotient_pfp_products_left_result_wbright_entry. cb = fom_beta_quotient_pfp_products_left_result_wbright_entry * S ((S (fom_index_pfp_products_left_result_wbright)) * cc) + (fom_value_pfp_products_left_result_wbright))) /\\ (exists fom_gap_pfp_products_left_result_wbright_value_bound. fom_gap_pfp_products_left_result_wbright_value_bound + S (fom_value_pfp_products_left_result_wbright) = p))) /\\ (((((((M)=0 \\/ (L)=0) /\\ (((N)=0)))) \\/ (((~((M)=0)) /\\ (((~((L)=0)) /\\ (((M)+(L)=S (N)))))))) /\\ ((forall pfc_index_products_left_result_wbcoefficients. (exists pfa_gap_products_left_result_wbcoefficientsbound. pfa_gap_products_left_result_wbcoefficientsbound + S (pfc_index_products_left_result_wbcoefficients) = (N)) -> exists pfc_value_products_left_result_wbcoefficients. ((((exists ff_h_pfp_products_left_result_wbcoefficientsentry. ff_h_pfp_products_left_result_wbcoefficientsentry + S (pfc_value_products_left_result_wbcoefficients) = S ((S (pfc_index_products_left_result_wbcoefficients)) * wc)) /\\ exists ff_q_pfp_products_left_result_wbcoefficientsentry. wb = ff_q_pfp_products_left_result_wbcoefficientsentry * S ((S (pfc_index_products_left_result_wbcoefficients)) * wc) + (pfc_value_products_left_result_wbcoefficients))) /\\ ((exists pfc_terms_code_products_left_result_wbcoefficientscoefficient pfc_terms_scale_products_left_result_wbcoefficientscoefficient pfc_natural_sum_products_left_result_wbcoefficientscoefficient. ((forall pfc_index_products_left_result_wbcoefficientscoefficientdiagonal. (exists pfa_gap_products_left_result_wbcoefficientscoefficientdiagonalbound. pfa_gap_products_left_result_wbcoefficientscoefficientdiagonalbound + S (pfc_index_products_left_result_wbcoefficientscoefficientdiagonal) = (S (pfc_index_products_left_result_wbcoefficients))) -> exists pfc_value_products_left_result_wbcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_products_left_result_wbcoefficientscoefficientdiagonalentry. ff_h_pfp_products_left_result_wbcoefficientscoefficientdiagonalentry + S (pfc_value_products_left_result_wbcoefficientscoefficientdiagonal) = S ((S (pfc_index_products_left_result_wbcoefficientscoefficientdiagonal)) * pfc_terms_scale_products_left_result_wbcoefficientscoefficient)) /\\ exists ff_q_pfp_products_left_result_wbcoefficientscoefficientdiagonalentry. pfc_terms_code_products_left_result_wbcoefficientscoefficient = ff_q_pfp_products_left_result_wbcoefficientscoefficientdiagonalentry * S ((S (pfc_index_products_left_result_wbcoefficientscoefficientdiagonal)) * pfc_terms_scale_products_left_result_wbcoefficientscoefficient) + (pfc_value_products_left_result_wbcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_products_left_result_wbcoefficientscoefficientdiagonalterm pfc_left_products_left_result_wbcoefficientscoefficientdiagonalterm pfc_right_products_left_result_wbcoefficientscoefficientdiagonalterm. (((pfc_index_products_left_result_wbcoefficientscoefficientdiagonal)+pfc_complement_products_left_result_wbcoefficientscoefficientdiagonalterm=(pfc_index_products_left_result_wbcoefficients)) /\\ ((((((exists pfa_gap_products_left_result_wbcoefficientscoefficientdiagonaltermleftinside. pfa_gap_products_left_result_wbcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_products_left_result_wbcoefficientscoefficientdiagonal) = (M)) /\\ ((((exists ff_h_pfp_products_left_result_wbcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_products_left_result_wbcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_products_left_result_wbcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_products_left_result_wbcoefficientscoefficientdiagonal)) * dc)) /\\ exists ff_q_pfp_products_left_result_wbcoefficientscoefficientdiagonaltermleftentry. db = ff_q_pfp_products_left_result_wbcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_products_left_result_wbcoefficientscoefficientdiagonal)) * dc) + (pfc_left_products_left_result_wbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_products_left_result_wbcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_products_left_result_wbcoefficientscoefficientdiagonaltermleftoutside+(M)=(pfc_index_products_left_result_wbcoefficientscoefficientdiagonal)) /\\ (((pfc_left_products_left_result_wbcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_products_left_result_wbcoefficientscoefficientdiagonaltermrightinside. pfa_gap_products_left_result_wbcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_products_left_result_wbcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_products_left_result_wbcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_products_left_result_wbcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_products_left_result_wbcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_products_left_result_wbcoefficientscoefficientdiagonalterm)) * cc)) /\\ exists ff_q_pfp_products_left_result_wbcoefficientscoefficientdiagonaltermrightentry. cb = ff_q_pfp_products_left_result_wbcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_products_left_result_wbcoefficientscoefficientdiagonalterm)) * cc) + (pfc_right_products_left_result_wbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_products_left_result_wbcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_products_left_result_wbcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_products_left_result_wbcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_products_left_result_wbcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_products_left_result_wbcoefficientscoefficientdiagonal)=pfc_left_products_left_result_wbcoefficientscoefficientdiagonalterm*pfc_right_products_left_result_wbcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_products_left_result_wbcoefficientscoefficientsum fs_v_pfc_products_left_result_wbcoefficientscoefficientsum. ((((exists fs_h_pfc_products_left_result_wbcoefficientscoefficientsum_body_start. fs_h_pfc_products_left_result_wbcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_products_left_result_wbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_left_result_wbcoefficientscoefficientsum_body_start. fs_u_pfc_products_left_result_wbcoefficientscoefficientsum = fs_q_pfc_products_left_result_wbcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_products_left_result_wbcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_products_left_result_wbcoefficientscoefficientsum_body_terminal. fs_h_pfc_products_left_result_wbcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_products_left_result_wbcoefficientscoefficient) = S ((S (S (pfc_index_products_left_result_wbcoefficients))) * fs_v_pfc_products_left_result_wbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_left_result_wbcoefficientscoefficientsum_body_terminal. fs_u_pfc_products_left_result_wbcoefficientscoefficientsum = fs_q_pfc_products_left_result_wbcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_products_left_result_wbcoefficients))) * fs_v_pfc_products_left_result_wbcoefficientscoefficientsum) + (pfc_natural_sum_products_left_result_wbcoefficientscoefficient))) /\\ forall fs_i_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps = S (pfc_index_products_left_result_wbcoefficients)) -> exists fs_a_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps fs_r_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps fs_s_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_products_left_result_wbcoefficientscoefficient)) /\\ exists fs_q_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_products_left_result_wbcoefficientscoefficient = fs_q_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_products_left_result_wbcoefficientscoefficient) + (fs_a_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_result_wbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_products_left_result_wbcoefficientscoefficientsum = fs_q_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_result_wbcoefficientscoefficientsum) + (fs_r_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_result_wbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_products_left_result_wbcoefficientscoefficientsum = fs_q_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_result_wbcoefficientscoefficientsum) + (fs_s_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps = fs_r_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps + fs_a_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_products_left_result_wbcoefficientscoefficientresiduebound. pfa_gap_products_left_result_wbcoefficientscoefficientresiduebound + S (pfc_value_products_left_result_wbcoefficients) = (p)) /\\ ((exists pfa_offset_left_products_left_result_wbcoefficientscoefficientresiduecongruence pfa_offset_right_products_left_result_wbcoefficientscoefficientresiduecongruence. (pfc_natural_sum_products_left_result_wbcoefficientscoefficient) + (p) * pfa_offset_left_products_left_result_wbcoefficientscoefficientresiduecongruence = (pfc_value_products_left_result_wbcoefficients) + (p) * pfa_offset_right_products_left_result_wbcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfp_index_products_left_addition. (exists pfa_gap_products_left_additionindex. pfa_gap_products_left_additionindex + S (pfp_index_products_left_addition) = (N)) -> exists pfp_left_products_left_addition pfp_right_products_left_addition pfp_value_products_left_addition. ((((exists ff_h_pfp_products_left_additionleft. ff_h_pfp_products_left_additionleft + S (pfp_left_products_left_addition) = S ((S (pfp_index_products_left_addition)) * uc)) /\\ exists ff_q_pfp_products_left_additionleft. ub = ff_q_pfp_products_left_additionleft * S ((S (pfp_index_products_left_addition)) * uc) + (pfp_left_products_left_addition))) /\\ (((((exists ff_h_pfp_products_left_additionright. ff_h_pfp_products_left_additionright + S (pfp_right_products_left_addition) = S ((S (pfp_index_products_left_addition)) * vc)) /\\ exists ff_q_pfp_products_left_additionright. vb = ff_q_pfp_products_left_additionright * S ((S (pfp_index_products_left_addition)) * vc) + (pfp_right_products_left_addition))) /\\ (((((exists ff_h_pfp_products_left_additiontarget. ff_h_pfp_products_left_additiontarget + S (pfp_value_products_left_addition) = S ((S (pfp_index_products_left_addition)) * wc)) /\\ exists ff_q_pfp_products_left_additiontarget. wb = ff_q_pfp_products_left_additiontarget * S ((S (pfp_index_products_left_addition)) * wc) + (pfp_value_products_left_addition))) /\\ ((((exists pfa_gap_products_left_additionoperationleft. pfa_gap_products_left_additionoperationleft + S (pfp_left_products_left_addition) = (p)) /\\ (((exists pfa_gap_products_left_additionoperationright. pfa_gap_products_left_additionoperationright + S (pfp_right_products_left_addition) = (p)) /\\ ((((exists pfa_gap_products_left_additionoperationresultbound. pfa_gap_products_left_additionoperationresultbound + S (pfp_value_products_left_addition) = (p)) /\\ ((exists pfa_offset_left_products_left_additionoperationresultcongruence pfa_offset_right_products_left_additionoperationresultcongruence. ((pfp_left_products_left_addition) + (pfp_right_products_left_addition)) + (p) * pfa_offset_left_products_left_additionoperationresultcongruence = (pfp_value_products_left_addition) + (p) * pfa_offset_right_products_left_additionoperationresultcongruence)))))))))))))))))))))))", "statement_sha256": "9df1dde91e1e57221d120a6aff8a84019d6f860ed274037ecab2e9281c7a00c9", "summary": "Construct all three genuine proper-length left products and then prove their coefficient-addition identity; the product witnesses and the distributive conclusion are outputs, never input assumptions.", "summary_sha256": "704132d6652d6b3450dabac7375a6cc116ee5dae861f5336a26419d86ee4b5d9" }, { "alpha_v33_frontier_enrollment": { "body_receipt_sha256": "9a9829705d7c805d1be33cec2364018906b86a9790315f6af46ca09dfd84059c", "bundle_campaign": "polynomial-euclidean-division", "bundle_node_id": 335, "bundle_sha256": "6ae667d8518e4dbe722bb08ad1b08715a0d282c2893e533c8133d770fe861dcf", "campaign": 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"alpha_only", "name": "prime_field_polynomial_right_distributive_products_exists", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro ab", "intro ac", "intro bb", "intro bc", "intro cb", "intro cc", "intro L", "intro db", "intro dc", "intro M", "intro hp", "intro hd", "intro hs", "have hlength : exists N. ((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))", "specialize polynomial_product_length_exists (L)", "specialize polynomial_product_length_exists (M)", "apply polynomial_product_length_exists", "cases hlength", "have hbounds : ((forall fom_index_pfp_products_right_bounded_ab. (exists fom_gap_pfp_products_right_bounded_ab_index_bound. fom_gap_pfp_products_right_bounded_ab_index_bound + S (fom_index_pfp_products_right_bounded_ab) = L) -> exists fom_value_pfp_products_right_bounded_ab. ((((exists fom_beta_height_pfp_products_right_bounded_ab_entry. fom_beta_height_pfp_products_right_bounded_ab_entry + S (fom_value_pfp_products_right_bounded_ab) = S ((S (fom_index_pfp_products_right_bounded_ab)) * ac)) /\\ exists fom_beta_quotient_pfp_products_right_bounded_ab_entry. ab = fom_beta_quotient_pfp_products_right_bounded_ab_entry * S ((S (fom_index_pfp_products_right_bounded_ab)) * ac) + (fom_value_pfp_products_right_bounded_ab))) /\\ (exists fom_gap_pfp_products_right_bounded_ab_value_bound. fom_gap_pfp_products_right_bounded_ab_value_bound + S (fom_value_pfp_products_right_bounded_ab) = p))) /\\ (((forall fom_index_pfp_products_right_bounded_bb. (exists fom_gap_pfp_products_right_bounded_bb_index_bound. fom_gap_pfp_products_right_bounded_bb_index_bound + S (fom_index_pfp_products_right_bounded_bb) = L) -> exists fom_value_pfp_products_right_bounded_bb. ((((exists fom_beta_height_pfp_products_right_bounded_bb_entry. fom_beta_height_pfp_products_right_bounded_bb_entry + S (fom_value_pfp_products_right_bounded_bb) = S ((S (fom_index_pfp_products_right_bounded_bb)) * bc)) /\\ exists fom_beta_quotient_pfp_products_right_bounded_bb_entry. bb = fom_beta_quotient_pfp_products_right_bounded_bb_entry * S ((S (fom_index_pfp_products_right_bounded_bb)) * bc) + (fom_value_pfp_products_right_bounded_bb))) /\\ (exists fom_gap_pfp_products_right_bounded_bb_value_bound. fom_gap_pfp_products_right_bounded_bb_value_bound + S (fom_value_pfp_products_right_bounded_bb) = p))) /\\ ((forall fom_index_pfp_products_right_bounded_cb. (exists fom_gap_pfp_products_right_bounded_cb_index_bound. fom_gap_pfp_products_right_bounded_cb_index_bound + S (fom_index_pfp_products_right_bounded_cb) = L) -> exists fom_value_pfp_products_right_bounded_cb. ((((exists fom_beta_height_pfp_products_right_bounded_cb_entry. fom_beta_height_pfp_products_right_bounded_cb_entry + S (fom_value_pfp_products_right_bounded_cb) = S ((S (fom_index_pfp_products_right_bounded_cb)) * cc)) /\\ exists fom_beta_quotient_pfp_products_right_bounded_cb_entry. cb = fom_beta_quotient_pfp_products_right_bounded_cb_entry * S ((S (fom_index_pfp_products_right_bounded_cb)) * cc) + (fom_value_pfp_products_right_bounded_cb))) /\\ (exists fom_gap_pfp_products_right_bounded_cb_value_bound. fom_gap_pfp_products_right_bounded_cb_value_bound + S (fom_value_pfp_products_right_bounded_cb) = 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 (cb)", "specialize prime_field_polynomial_add_bounded (cc)", "specialize prime_field_polynomial_add_bounded (L)", "apply prime_field_polynomial_add_bounded", "exact hs", "cases hbounds", "cases hbounds_right", "have hu : exists ub uc. ((forall fom_index_pfp_products_right_huleft. (exists fom_gap_pfp_products_right_huleft_index_bound. fom_gap_pfp_products_right_huleft_index_bound + S (fom_index_pfp_products_right_huleft) = L) -> exists fom_value_pfp_products_right_huleft. ((((exists fom_beta_height_pfp_products_right_huleft_entry. fom_beta_height_pfp_products_right_huleft_entry + S (fom_value_pfp_products_right_huleft) = S ((S (fom_index_pfp_products_right_huleft)) * ac)) /\\ exists fom_beta_quotient_pfp_products_right_huleft_entry. ab = fom_beta_quotient_pfp_products_right_huleft_entry * S ((S (fom_index_pfp_products_right_huleft)) * ac) + (fom_value_pfp_products_right_huleft))) /\\ (exists fom_gap_pfp_products_right_huleft_value_bound. fom_gap_pfp_products_right_huleft_value_bound + S (fom_value_pfp_products_right_huleft) = p))) /\\ (((forall fom_index_pfp_products_right_huright. (exists fom_gap_pfp_products_right_huright_index_bound. fom_gap_pfp_products_right_huright_index_bound + S (fom_index_pfp_products_right_huright) = M) -> exists fom_value_pfp_products_right_huright. ((((exists fom_beta_height_pfp_products_right_huright_entry. fom_beta_height_pfp_products_right_huright_entry + S (fom_value_pfp_products_right_huright) = S ((S (fom_index_pfp_products_right_huright)) * dc)) /\\ exists fom_beta_quotient_pfp_products_right_huright_entry. db = fom_beta_quotient_pfp_products_right_huright_entry * S ((S (fom_index_pfp_products_right_huright)) * dc) + (fom_value_pfp_products_right_huright))) /\\ (exists fom_gap_pfp_products_right_huright_value_bound. fom_gap_pfp_products_right_huright_value_bound + S (fom_value_pfp_products_right_huright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((x)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (x)))))))) /\\ ((forall pfc_index_products_right_hucoefficients. (exists pfa_gap_products_right_hucoefficientsbound. pfa_gap_products_right_hucoefficientsbound + S (pfc_index_products_right_hucoefficients) = (x)) -> exists pfc_value_products_right_hucoefficients. ((((exists ff_h_pfp_products_right_hucoefficientsentry. ff_h_pfp_products_right_hucoefficientsentry + S (pfc_value_products_right_hucoefficients) = S ((S (pfc_index_products_right_hucoefficients)) * uc)) /\\ exists ff_q_pfp_products_right_hucoefficientsentry. ub = ff_q_pfp_products_right_hucoefficientsentry * S ((S (pfc_index_products_right_hucoefficients)) * uc) + (pfc_value_products_right_hucoefficients))) /\\ ((exists pfc_terms_code_products_right_hucoefficientscoefficient pfc_terms_scale_products_right_hucoefficientscoefficient pfc_natural_sum_products_right_hucoefficientscoefficient. ((forall pfc_index_products_right_hucoefficientscoefficientdiagonal. (exists pfa_gap_products_right_hucoefficientscoefficientdiagonalbound. pfa_gap_products_right_hucoefficientscoefficientdiagonalbound + S (pfc_index_products_right_hucoefficientscoefficientdiagonal) = (S (pfc_index_products_right_hucoefficients))) -> exists pfc_value_products_right_hucoefficientscoefficientdiagonal. ((((exists ff_h_pfp_products_right_hucoefficientscoefficientdiagonalentry. ff_h_pfp_products_right_hucoefficientscoefficientdiagonalentry + S (pfc_value_products_right_hucoefficientscoefficientdiagonal) = S ((S (pfc_index_products_right_hucoefficientscoefficientdiagonal)) * pfc_terms_scale_products_right_hucoefficientscoefficient)) /\\ exists ff_q_pfp_products_right_hucoefficientscoefficientdiagonalentry. pfc_terms_code_products_right_hucoefficientscoefficient = ff_q_pfp_products_right_hucoefficientscoefficientdiagonalentry * S ((S (pfc_index_products_right_hucoefficientscoefficientdiagonal)) * pfc_terms_scale_products_right_hucoefficientscoefficient) + (pfc_value_products_right_hucoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_products_right_hucoefficientscoefficientdiagonalterm pfc_left_products_right_hucoefficientscoefficientdiagonalterm pfc_right_products_right_hucoefficientscoefficientdiagonalterm. (((pfc_index_products_right_hucoefficientscoefficientdiagonal)+pfc_complement_products_right_hucoefficientscoefficientdiagonalterm=(pfc_index_products_right_hucoefficients)) /\\ ((((((exists pfa_gap_products_right_hucoefficientscoefficientdiagonaltermleftinside. pfa_gap_products_right_hucoefficientscoefficientdiagonaltermleftinside + S (pfc_index_products_right_hucoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_products_right_hucoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_products_right_hucoefficientscoefficientdiagonaltermleftentry + S (pfc_left_products_right_hucoefficientscoefficientdiagonalterm) = S ((S (pfc_index_products_right_hucoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_products_right_hucoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_products_right_hucoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_products_right_hucoefficientscoefficientdiagonal)) * ac) + (pfc_left_products_right_hucoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_products_right_hucoefficientscoefficientdiagonaltermleftoutside. pfc_gap_products_right_hucoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_products_right_hucoefficientscoefficientdiagonal)) /\\ (((pfc_left_products_right_hucoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_products_right_hucoefficientscoefficientdiagonaltermrightinside. pfa_gap_products_right_hucoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_products_right_hucoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_products_right_hucoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_products_right_hucoefficientscoefficientdiagonaltermrightentry + S (pfc_right_products_right_hucoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_products_right_hucoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_products_right_hucoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_products_right_hucoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_products_right_hucoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_products_right_hucoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_products_right_hucoefficientscoefficientdiagonaltermrightoutside. pfc_gap_products_right_hucoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_products_right_hucoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_products_right_hucoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_products_right_hucoefficientscoefficientdiagonal)=pfc_left_products_right_hucoefficientscoefficientdiagonalterm*pfc_right_products_right_hucoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_products_right_hucoefficientscoefficientsum fs_v_pfc_products_right_hucoefficientscoefficientsum. ((((exists fs_h_pfc_products_right_hucoefficientscoefficientsum_body_start. fs_h_pfc_products_right_hucoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_products_right_hucoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_right_hucoefficientscoefficientsum_body_start. fs_u_pfc_products_right_hucoefficientscoefficientsum = fs_q_pfc_products_right_hucoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_products_right_hucoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_products_right_hucoefficientscoefficientsum_body_terminal. fs_h_pfc_products_right_hucoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_products_right_hucoefficientscoefficient) = S ((S (S (pfc_index_products_right_hucoefficients))) * fs_v_pfc_products_right_hucoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_right_hucoefficientscoefficientsum_body_terminal. fs_u_pfc_products_right_hucoefficientscoefficientsum = fs_q_pfc_products_right_hucoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_products_right_hucoefficients))) * fs_v_pfc_products_right_hucoefficientscoefficientsum) + (pfc_natural_sum_products_right_hucoefficientscoefficient))) /\\ forall fs_i_pfc_products_right_hucoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_products_right_hucoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_products_right_hucoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_products_right_hucoefficientscoefficientsum_body_steps = S (pfc_index_products_right_hucoefficients)) -> exists fs_a_pfc_products_right_hucoefficientscoefficientsum_body_steps fs_r_pfc_products_right_hucoefficientscoefficientsum_body_steps fs_s_pfc_products_right_hucoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_products_right_hucoefficientscoefficientsum_body_steps_summand. fs_h_pfc_products_right_hucoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_products_right_hucoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_products_right_hucoefficientscoefficientsum_body_steps)) * pfc_terms_scale_products_right_hucoefficientscoefficient)) /\\ exists fs_q_pfc_products_right_hucoefficientscoefficientsum_body_steps_summand. pfc_terms_code_products_right_hucoefficientscoefficient = fs_q_pfc_products_right_hucoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_products_right_hucoefficientscoefficientsum_body_steps)) * pfc_terms_scale_products_right_hucoefficientscoefficient) + (fs_a_pfc_products_right_hucoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_products_right_hucoefficientscoefficientsum_body_steps_partial. fs_h_pfc_products_right_hucoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_products_right_hucoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_products_right_hucoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_right_hucoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_right_hucoefficientscoefficientsum_body_steps_partial. fs_u_pfc_products_right_hucoefficientscoefficientsum = fs_q_pfc_products_right_hucoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_products_right_hucoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_right_hucoefficientscoefficientsum) + (fs_r_pfc_products_right_hucoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_products_right_hucoefficientscoefficientsum_body_steps_successor. fs_h_pfc_products_right_hucoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_products_right_hucoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_products_right_hucoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_right_hucoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_right_hucoefficientscoefficientsum_body_steps_successor. fs_u_pfc_products_right_hucoefficientscoefficientsum = fs_q_pfc_products_right_hucoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_products_right_hucoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_right_hucoefficientscoefficientsum) + (fs_s_pfc_products_right_hucoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_products_right_hucoefficientscoefficientsum_body_steps = fs_r_pfc_products_right_hucoefficientscoefficientsum_body_steps + fs_a_pfc_products_right_hucoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_products_right_hucoefficientscoefficientresiduebound. pfa_gap_products_right_hucoefficientscoefficientresiduebound + S (pfc_value_products_right_hucoefficients) = (p)) /\\ ((exists pfa_offset_left_products_right_hucoefficientscoefficientresiduecongruence pfa_offset_right_products_right_hucoefficientscoefficientresiduecongruence. (pfc_natural_sum_products_right_hucoefficientscoefficient) + (p) * pfa_offset_left_products_right_hucoefficientscoefficientresiduecongruence = (pfc_value_products_right_hucoefficients) + (p) * pfa_offset_right_products_right_hucoefficientscoefficientresiduecongruence))))))))))))))))))", "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 (db)", "specialize prime_field_polynomial_convolution_at_length_exists (dc)", "specialize prime_field_polynomial_convolution_at_length_exists (M)", "specialize prime_field_polynomial_convolution_at_length_exists (x)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hp", "exact hbounds_left", "exact hd", "exact hlength_witness", "cases hu", "cases hu_witness", "have hv : exists ub uc. ((forall fom_index_pfp_products_right_hvleft. (exists fom_gap_pfp_products_right_hvleft_index_bound. fom_gap_pfp_products_right_hvleft_index_bound + S (fom_index_pfp_products_right_hvleft) = L) -> exists fom_value_pfp_products_right_hvleft. ((((exists fom_beta_height_pfp_products_right_hvleft_entry. fom_beta_height_pfp_products_right_hvleft_entry + S (fom_value_pfp_products_right_hvleft) = S ((S (fom_index_pfp_products_right_hvleft)) * bc)) /\\ exists fom_beta_quotient_pfp_products_right_hvleft_entry. bb = fom_beta_quotient_pfp_products_right_hvleft_entry * S ((S (fom_index_pfp_products_right_hvleft)) * bc) + (fom_value_pfp_products_right_hvleft))) /\\ (exists fom_gap_pfp_products_right_hvleft_value_bound. fom_gap_pfp_products_right_hvleft_value_bound + S (fom_value_pfp_products_right_hvleft) = p))) /\\ (((forall fom_index_pfp_products_right_hvright. (exists fom_gap_pfp_products_right_hvright_index_bound. fom_gap_pfp_products_right_hvright_index_bound + S (fom_index_pfp_products_right_hvright) = M) -> exists fom_value_pfp_products_right_hvright. ((((exists fom_beta_height_pfp_products_right_hvright_entry. fom_beta_height_pfp_products_right_hvright_entry + S (fom_value_pfp_products_right_hvright) = S ((S (fom_index_pfp_products_right_hvright)) * dc)) /\\ exists fom_beta_quotient_pfp_products_right_hvright_entry. db = fom_beta_quotient_pfp_products_right_hvright_entry * S ((S (fom_index_pfp_products_right_hvright)) * dc) + (fom_value_pfp_products_right_hvright))) /\\ (exists fom_gap_pfp_products_right_hvright_value_bound. fom_gap_pfp_products_right_hvright_value_bound + S (fom_value_pfp_products_right_hvright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((x)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (x)))))))) /\\ ((forall pfc_index_products_right_hvcoefficients. (exists pfa_gap_products_right_hvcoefficientsbound. pfa_gap_products_right_hvcoefficientsbound + S (pfc_index_products_right_hvcoefficients) = (x)) -> exists pfc_value_products_right_hvcoefficients. ((((exists ff_h_pfp_products_right_hvcoefficientsentry. ff_h_pfp_products_right_hvcoefficientsentry + S (pfc_value_products_right_hvcoefficients) = S ((S (pfc_index_products_right_hvcoefficients)) * uc)) /\\ exists ff_q_pfp_products_right_hvcoefficientsentry. ub = ff_q_pfp_products_right_hvcoefficientsentry * S ((S (pfc_index_products_right_hvcoefficients)) * uc) + (pfc_value_products_right_hvcoefficients))) /\\ ((exists pfc_terms_code_products_right_hvcoefficientscoefficient pfc_terms_scale_products_right_hvcoefficientscoefficient pfc_natural_sum_products_right_hvcoefficientscoefficient. ((forall pfc_index_products_right_hvcoefficientscoefficientdiagonal. (exists pfa_gap_products_right_hvcoefficientscoefficientdiagonalbound. pfa_gap_products_right_hvcoefficientscoefficientdiagonalbound + S (pfc_index_products_right_hvcoefficientscoefficientdiagonal) = (S (pfc_index_products_right_hvcoefficients))) -> exists pfc_value_products_right_hvcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_products_right_hvcoefficientscoefficientdiagonalentry. ff_h_pfp_products_right_hvcoefficientscoefficientdiagonalentry + S (pfc_value_products_right_hvcoefficientscoefficientdiagonal) = S ((S (pfc_index_products_right_hvcoefficientscoefficientdiagonal)) * pfc_terms_scale_products_right_hvcoefficientscoefficient)) /\\ exists ff_q_pfp_products_right_hvcoefficientscoefficientdiagonalentry. pfc_terms_code_products_right_hvcoefficientscoefficient = ff_q_pfp_products_right_hvcoefficientscoefficientdiagonalentry * S ((S (pfc_index_products_right_hvcoefficientscoefficientdiagonal)) * pfc_terms_scale_products_right_hvcoefficientscoefficient) + (pfc_value_products_right_hvcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_products_right_hvcoefficientscoefficientdiagonalterm pfc_left_products_right_hvcoefficientscoefficientdiagonalterm pfc_right_products_right_hvcoefficientscoefficientdiagonalterm. (((pfc_index_products_right_hvcoefficientscoefficientdiagonal)+pfc_complement_products_right_hvcoefficientscoefficientdiagonalterm=(pfc_index_products_right_hvcoefficients)) /\\ ((((((exists pfa_gap_products_right_hvcoefficientscoefficientdiagonaltermleftinside. pfa_gap_products_right_hvcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_products_right_hvcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_products_right_hvcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_products_right_hvcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_products_right_hvcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_products_right_hvcoefficientscoefficientdiagonal)) * bc)) /\\ exists ff_q_pfp_products_right_hvcoefficientscoefficientdiagonaltermleftentry. bb = ff_q_pfp_products_right_hvcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_products_right_hvcoefficientscoefficientdiagonal)) * bc) + (pfc_left_products_right_hvcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_products_right_hvcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_products_right_hvcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_products_right_hvcoefficientscoefficientdiagonal)) /\\ (((pfc_left_products_right_hvcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_products_right_hvcoefficientscoefficientdiagonaltermrightinside. pfa_gap_products_right_hvcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_products_right_hvcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_products_right_hvcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_products_right_hvcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_products_right_hvcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_products_right_hvcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_products_right_hvcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_products_right_hvcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_products_right_hvcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_products_right_hvcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_products_right_hvcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_products_right_hvcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_products_right_hvcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_products_right_hvcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_products_right_hvcoefficientscoefficientdiagonal)=pfc_left_products_right_hvcoefficientscoefficientdiagonalterm*pfc_right_products_right_hvcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_products_right_hvcoefficientscoefficientsum fs_v_pfc_products_right_hvcoefficientscoefficientsum. ((((exists fs_h_pfc_products_right_hvcoefficientscoefficientsum_body_start. fs_h_pfc_products_right_hvcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_products_right_hvcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_right_hvcoefficientscoefficientsum_body_start. fs_u_pfc_products_right_hvcoefficientscoefficientsum = fs_q_pfc_products_right_hvcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_products_right_hvcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_products_right_hvcoefficientscoefficientsum_body_terminal. fs_h_pfc_products_right_hvcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_products_right_hvcoefficientscoefficient) = S ((S (S (pfc_index_products_right_hvcoefficients))) * fs_v_pfc_products_right_hvcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_right_hvcoefficientscoefficientsum_body_terminal. fs_u_pfc_products_right_hvcoefficientscoefficientsum = fs_q_pfc_products_right_hvcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_products_right_hvcoefficients))) * fs_v_pfc_products_right_hvcoefficientscoefficientsum) + (pfc_natural_sum_products_right_hvcoefficientscoefficient))) /\\ forall fs_i_pfc_products_right_hvcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_products_right_hvcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_products_right_hvcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_products_right_hvcoefficientscoefficientsum_body_steps = S (pfc_index_products_right_hvcoefficients)) -> exists fs_a_pfc_products_right_hvcoefficientscoefficientsum_body_steps fs_r_pfc_products_right_hvcoefficientscoefficientsum_body_steps fs_s_pfc_products_right_hvcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_products_right_hvcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_products_right_hvcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_products_right_hvcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_products_right_hvcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_products_right_hvcoefficientscoefficient)) /\\ exists fs_q_pfc_products_right_hvcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_products_right_hvcoefficientscoefficient = fs_q_pfc_products_right_hvcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_products_right_hvcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_products_right_hvcoefficientscoefficient) + (fs_a_pfc_products_right_hvcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_products_right_hvcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_products_right_hvcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_products_right_hvcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_products_right_hvcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_right_hvcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_right_hvcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_products_right_hvcoefficientscoefficientsum = fs_q_pfc_products_right_hvcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_products_right_hvcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_right_hvcoefficientscoefficientsum) + (fs_r_pfc_products_right_hvcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_products_right_hvcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_products_right_hvcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_products_right_hvcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_products_right_hvcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_right_hvcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_right_hvcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_products_right_hvcoefficientscoefficientsum = fs_q_pfc_products_right_hvcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_products_right_hvcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_right_hvcoefficientscoefficientsum) + (fs_s_pfc_products_right_hvcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_products_right_hvcoefficientscoefficientsum_body_steps = fs_r_pfc_products_right_hvcoefficientscoefficientsum_body_steps + fs_a_pfc_products_right_hvcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_products_right_hvcoefficientscoefficientresiduebound. pfa_gap_products_right_hvcoefficientscoefficientresiduebound + S (pfc_value_products_right_hvcoefficients) = (p)) /\\ ((exists pfa_offset_left_products_right_hvcoefficientscoefficientresiduecongruence pfa_offset_right_products_right_hvcoefficientscoefficientresiduecongruence. (pfc_natural_sum_products_right_hvcoefficientscoefficient) + (p) * pfa_offset_left_products_right_hvcoefficientscoefficientresiduecongruence = (pfc_value_products_right_hvcoefficients) + (p) * pfa_offset_right_products_right_hvcoefficientscoefficientresiduecongruence))))))))))))))))))", "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 (L)", "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 (M)", "specialize prime_field_polynomial_convolution_at_length_exists (x)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hp", "exact hbounds_right_left", "exact hd", "exact hlength_witness", "cases hv", "cases hv_witness", "have hw : exists ub uc. ((forall fom_index_pfp_products_right_hwleft. (exists fom_gap_pfp_products_right_hwleft_index_bound. fom_gap_pfp_products_right_hwleft_index_bound + S (fom_index_pfp_products_right_hwleft) = L) -> exists fom_value_pfp_products_right_hwleft. ((((exists fom_beta_height_pfp_products_right_hwleft_entry. fom_beta_height_pfp_products_right_hwleft_entry + S (fom_value_pfp_products_right_hwleft) = S ((S (fom_index_pfp_products_right_hwleft)) * cc)) /\\ exists fom_beta_quotient_pfp_products_right_hwleft_entry. cb = fom_beta_quotient_pfp_products_right_hwleft_entry * S ((S (fom_index_pfp_products_right_hwleft)) * cc) + (fom_value_pfp_products_right_hwleft))) /\\ (exists fom_gap_pfp_products_right_hwleft_value_bound. fom_gap_pfp_products_right_hwleft_value_bound + S (fom_value_pfp_products_right_hwleft) = p))) /\\ (((forall fom_index_pfp_products_right_hwright. (exists fom_gap_pfp_products_right_hwright_index_bound. fom_gap_pfp_products_right_hwright_index_bound + S (fom_index_pfp_products_right_hwright) = M) -> exists fom_value_pfp_products_right_hwright. ((((exists fom_beta_height_pfp_products_right_hwright_entry. fom_beta_height_pfp_products_right_hwright_entry + S (fom_value_pfp_products_right_hwright) = S ((S (fom_index_pfp_products_right_hwright)) * dc)) /\\ exists fom_beta_quotient_pfp_products_right_hwright_entry. db = fom_beta_quotient_pfp_products_right_hwright_entry * S ((S (fom_index_pfp_products_right_hwright)) * dc) + (fom_value_pfp_products_right_hwright))) /\\ (exists fom_gap_pfp_products_right_hwright_value_bound. fom_gap_pfp_products_right_hwright_value_bound + S (fom_value_pfp_products_right_hwright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((x)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (x)))))))) /\\ ((forall pfc_index_products_right_hwcoefficients. (exists pfa_gap_products_right_hwcoefficientsbound. pfa_gap_products_right_hwcoefficientsbound + S (pfc_index_products_right_hwcoefficients) = (x)) -> exists pfc_value_products_right_hwcoefficients. ((((exists ff_h_pfp_products_right_hwcoefficientsentry. ff_h_pfp_products_right_hwcoefficientsentry + S (pfc_value_products_right_hwcoefficients) = S ((S (pfc_index_products_right_hwcoefficients)) * uc)) /\\ exists ff_q_pfp_products_right_hwcoefficientsentry. ub = ff_q_pfp_products_right_hwcoefficientsentry * S ((S (pfc_index_products_right_hwcoefficients)) * uc) + (pfc_value_products_right_hwcoefficients))) /\\ ((exists pfc_terms_code_products_right_hwcoefficientscoefficient pfc_terms_scale_products_right_hwcoefficientscoefficient pfc_natural_sum_products_right_hwcoefficientscoefficient. ((forall pfc_index_products_right_hwcoefficientscoefficientdiagonal. (exists pfa_gap_products_right_hwcoefficientscoefficientdiagonalbound. pfa_gap_products_right_hwcoefficientscoefficientdiagonalbound + S (pfc_index_products_right_hwcoefficientscoefficientdiagonal) = (S (pfc_index_products_right_hwcoefficients))) -> exists pfc_value_products_right_hwcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_products_right_hwcoefficientscoefficientdiagonalentry. ff_h_pfp_products_right_hwcoefficientscoefficientdiagonalentry + S (pfc_value_products_right_hwcoefficientscoefficientdiagonal) = S ((S (pfc_index_products_right_hwcoefficientscoefficientdiagonal)) * pfc_terms_scale_products_right_hwcoefficientscoefficient)) /\\ exists ff_q_pfp_products_right_hwcoefficientscoefficientdiagonalentry. pfc_terms_code_products_right_hwcoefficientscoefficient = ff_q_pfp_products_right_hwcoefficientscoefficientdiagonalentry * S ((S (pfc_index_products_right_hwcoefficientscoefficientdiagonal)) * pfc_terms_scale_products_right_hwcoefficientscoefficient) + (pfc_value_products_right_hwcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_products_right_hwcoefficientscoefficientdiagonalterm pfc_left_products_right_hwcoefficientscoefficientdiagonalterm pfc_right_products_right_hwcoefficientscoefficientdiagonalterm. (((pfc_index_products_right_hwcoefficientscoefficientdiagonal)+pfc_complement_products_right_hwcoefficientscoefficientdiagonalterm=(pfc_index_products_right_hwcoefficients)) /\\ ((((((exists pfa_gap_products_right_hwcoefficientscoefficientdiagonaltermleftinside. pfa_gap_products_right_hwcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_products_right_hwcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_products_right_hwcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_products_right_hwcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_products_right_hwcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_products_right_hwcoefficientscoefficientdiagonal)) * cc)) /\\ exists ff_q_pfp_products_right_hwcoefficientscoefficientdiagonaltermleftentry. cb = ff_q_pfp_products_right_hwcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_products_right_hwcoefficientscoefficientdiagonal)) * cc) + (pfc_left_products_right_hwcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_products_right_hwcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_products_right_hwcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_products_right_hwcoefficientscoefficientdiagonal)) /\\ (((pfc_left_products_right_hwcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_products_right_hwcoefficientscoefficientdiagonaltermrightinside. pfa_gap_products_right_hwcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_products_right_hwcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_products_right_hwcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_products_right_hwcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_products_right_hwcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_products_right_hwcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_products_right_hwcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_products_right_hwcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_products_right_hwcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_products_right_hwcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_products_right_hwcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_products_right_hwcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_products_right_hwcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_products_right_hwcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_products_right_hwcoefficientscoefficientdiagonal)=pfc_left_products_right_hwcoefficientscoefficientdiagonalterm*pfc_right_products_right_hwcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_products_right_hwcoefficientscoefficientsum fs_v_pfc_products_right_hwcoefficientscoefficientsum. ((((exists fs_h_pfc_products_right_hwcoefficientscoefficientsum_body_start. fs_h_pfc_products_right_hwcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_products_right_hwcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_right_hwcoefficientscoefficientsum_body_start. fs_u_pfc_products_right_hwcoefficientscoefficientsum = fs_q_pfc_products_right_hwcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_products_right_hwcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_products_right_hwcoefficientscoefficientsum_body_terminal. fs_h_pfc_products_right_hwcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_products_right_hwcoefficientscoefficient) = S ((S (S (pfc_index_products_right_hwcoefficients))) * fs_v_pfc_products_right_hwcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_right_hwcoefficientscoefficientsum_body_terminal. fs_u_pfc_products_right_hwcoefficientscoefficientsum = fs_q_pfc_products_right_hwcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_products_right_hwcoefficients))) * fs_v_pfc_products_right_hwcoefficientscoefficientsum) + (pfc_natural_sum_products_right_hwcoefficientscoefficient))) /\\ forall fs_i_pfc_products_right_hwcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_products_right_hwcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_products_right_hwcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_products_right_hwcoefficientscoefficientsum_body_steps = S (pfc_index_products_right_hwcoefficients)) -> exists fs_a_pfc_products_right_hwcoefficientscoefficientsum_body_steps fs_r_pfc_products_right_hwcoefficientscoefficientsum_body_steps fs_s_pfc_products_right_hwcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_products_right_hwcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_products_right_hwcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_products_right_hwcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_products_right_hwcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_products_right_hwcoefficientscoefficient)) /\\ exists fs_q_pfc_products_right_hwcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_products_right_hwcoefficientscoefficient = fs_q_pfc_products_right_hwcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_products_right_hwcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_products_right_hwcoefficientscoefficient) + (fs_a_pfc_products_right_hwcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_products_right_hwcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_products_right_hwcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_products_right_hwcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_products_right_hwcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_right_hwcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_right_hwcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_products_right_hwcoefficientscoefficientsum = fs_q_pfc_products_right_hwcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_products_right_hwcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_right_hwcoefficientscoefficientsum) + (fs_r_pfc_products_right_hwcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_products_right_hwcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_products_right_hwcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_products_right_hwcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_products_right_hwcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_right_hwcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_right_hwcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_products_right_hwcoefficientscoefficientsum = fs_q_pfc_products_right_hwcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_products_right_hwcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_right_hwcoefficientscoefficientsum) + (fs_s_pfc_products_right_hwcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_products_right_hwcoefficientscoefficientsum_body_steps = fs_r_pfc_products_right_hwcoefficientscoefficientsum_body_steps + fs_a_pfc_products_right_hwcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_products_right_hwcoefficientscoefficientresiduebound. pfa_gap_products_right_hwcoefficientscoefficientresiduebound + S (pfc_value_products_right_hwcoefficients) = (p)) /\\ ((exists pfa_offset_left_products_right_hwcoefficientscoefficientresiduecongruence pfa_offset_right_products_right_hwcoefficientscoefficientresiduecongruence. (pfc_natural_sum_products_right_hwcoefficientscoefficient) + (p) * pfa_offset_left_products_right_hwcoefficientscoefficientresiduecongruence = (pfc_value_products_right_hwcoefficients) + (p) * pfa_offset_right_products_right_hwcoefficientscoefficientresiduecongruence))))))))))))))))))", "specialize prime_field_polynomial_convolution_at_length_exists (p)", "specialize prime_field_polynomial_convolution_at_length_exists (cb)", "specialize prime_field_polynomial_convolution_at_length_exists (cc)", "specialize prime_field_polynomial_convolution_at_length_exists (L)", "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 (M)", "specialize prime_field_polynomial_convolution_at_length_exists (x)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hp", "exact hbounds_right_right", "exact hd", "exact hlength_witness", "cases hw", "cases hw_witness", "exists x", "exists x1", "exists x2", "exists x3", "exists x4", "exists x5", "exists x6", "split", "exact hu_witness_witness", "split", "exact hv_witness_witness", "split", "exact hw_witness_witness", "specialize prime_field_polynomial_convolution_right_add (p)", "specialize prime_field_polynomial_convolution_right_add (ab)", "specialize prime_field_polynomial_convolution_right_add (ac)", "specialize prime_field_polynomial_convolution_right_add (bb)", "specialize prime_field_polynomial_convolution_right_add (bc)", "specialize prime_field_polynomial_convolution_right_add (cb)", "specialize prime_field_polynomial_convolution_right_add (cc)", "specialize prime_field_polynomial_convolution_right_add (L)", "specialize prime_field_polynomial_convolution_right_add (db)", "specialize prime_field_polynomial_convolution_right_add (dc)", "specialize prime_field_polynomial_convolution_right_add (M)", "specialize prime_field_polynomial_convolution_right_add (x1)", "specialize prime_field_polynomial_convolution_right_add (x2)", "specialize prime_field_polynomial_convolution_right_add (x3)", "specialize prime_field_polynomial_convolution_right_add (x4)", "specialize prime_field_polynomial_convolution_right_add (x5)", "specialize prime_field_polynomial_convolution_right_add (x6)", "specialize prime_field_polynomial_convolution_right_add (x)", "apply prime_field_polynomial_convolution_right_add", "exact hs", "exact hu_witness_witness", "exact hv_witness_witness", "exact hw_witness_witness" ], "script_sha256": "c5cc9c8c4f0945bde59d2c5bb5758769bc24e89de80fa22822c2a97a0f53ab41", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_distributivity_candidate.py", "sha256": "a959962d631759cd1fc773dd7eef2fadf4f3f95361d6d7bc8c6a9e82d0d4ab86" }, "statement": "forall p ab ac bb bc cb cc L db dc M. (~(p=0)) -> (forall fom_index_pfp_products_right_fixed_input. (exists fom_gap_pfp_products_right_fixed_input_index_bound. fom_gap_pfp_products_right_fixed_input_index_bound + S (fom_index_pfp_products_right_fixed_input) = M) -> exists fom_value_pfp_products_right_fixed_input. ((((exists fom_beta_height_pfp_products_right_fixed_input_entry. fom_beta_height_pfp_products_right_fixed_input_entry + S (fom_value_pfp_products_right_fixed_input) = S ((S (fom_index_pfp_products_right_fixed_input)) * dc)) /\\ exists fom_beta_quotient_pfp_products_right_fixed_input_entry. db = fom_beta_quotient_pfp_products_right_fixed_input_entry * S ((S (fom_index_pfp_products_right_fixed_input)) * dc) + (fom_value_pfp_products_right_fixed_input))) /\\ (exists fom_gap_pfp_products_right_fixed_input_value_bound. fom_gap_pfp_products_right_fixed_input_value_bound + S (fom_value_pfp_products_right_fixed_input) = p))) -> (forall pfp_index_products_right_input_addition. (exists pfa_gap_products_right_input_additionindex. pfa_gap_products_right_input_additionindex + S (pfp_index_products_right_input_addition) = (L)) -> exists pfp_left_products_right_input_addition pfp_right_products_right_input_addition pfp_value_products_right_input_addition. ((((exists ff_h_pfp_products_right_input_additionleft. ff_h_pfp_products_right_input_additionleft + S (pfp_left_products_right_input_addition) = S ((S (pfp_index_products_right_input_addition)) * ac)) /\\ exists ff_q_pfp_products_right_input_additionleft. ab = ff_q_pfp_products_right_input_additionleft * S ((S (pfp_index_products_right_input_addition)) * ac) + (pfp_left_products_right_input_addition))) /\\ (((((exists ff_h_pfp_products_right_input_additionright. ff_h_pfp_products_right_input_additionright + S (pfp_right_products_right_input_addition) = S ((S (pfp_index_products_right_input_addition)) * bc)) /\\ exists ff_q_pfp_products_right_input_additionright. bb = ff_q_pfp_products_right_input_additionright * S ((S (pfp_index_products_right_input_addition)) * bc) + (pfp_right_products_right_input_addition))) /\\ (((((exists ff_h_pfp_products_right_input_additiontarget. ff_h_pfp_products_right_input_additiontarget + S (pfp_value_products_right_input_addition) = S ((S (pfp_index_products_right_input_addition)) * cc)) /\\ exists ff_q_pfp_products_right_input_additiontarget. cb = ff_q_pfp_products_right_input_additiontarget * S ((S (pfp_index_products_right_input_addition)) * cc) + (pfp_value_products_right_input_addition))) /\\ ((((exists pfa_gap_products_right_input_additionoperationleft. pfa_gap_products_right_input_additionoperationleft + S (pfp_left_products_right_input_addition) = (p)) /\\ (((exists pfa_gap_products_right_input_additionoperationright. pfa_gap_products_right_input_additionoperationright + S (pfp_right_products_right_input_addition) = (p)) /\\ ((((exists pfa_gap_products_right_input_additionoperationresultbound. pfa_gap_products_right_input_additionoperationresultbound + S (pfp_value_products_right_input_addition) = (p)) /\\ ((exists pfa_offset_left_products_right_input_additionoperationresultcongruence pfa_offset_right_products_right_input_additionoperationresultcongruence. ((pfp_left_products_right_input_addition) + (pfp_right_products_right_input_addition)) + (p) * pfa_offset_left_products_right_input_additionoperationresultcongruence = (pfp_value_products_right_input_addition) + (p) * pfa_offset_right_products_right_input_additionoperationresultcongruence)))))))))))))))) -> (exists N ub uc vb vc wb wc. ((((forall fom_index_pfp_products_right_result_ubleft. (exists fom_gap_pfp_products_right_result_ubleft_index_bound. fom_gap_pfp_products_right_result_ubleft_index_bound + S (fom_index_pfp_products_right_result_ubleft) = L) -> exists fom_value_pfp_products_right_result_ubleft. ((((exists fom_beta_height_pfp_products_right_result_ubleft_entry. fom_beta_height_pfp_products_right_result_ubleft_entry + S (fom_value_pfp_products_right_result_ubleft) = S ((S (fom_index_pfp_products_right_result_ubleft)) * ac)) /\\ exists fom_beta_quotient_pfp_products_right_result_ubleft_entry. ab = fom_beta_quotient_pfp_products_right_result_ubleft_entry * S ((S (fom_index_pfp_products_right_result_ubleft)) * ac) + (fom_value_pfp_products_right_result_ubleft))) /\\ (exists fom_gap_pfp_products_right_result_ubleft_value_bound. fom_gap_pfp_products_right_result_ubleft_value_bound + S (fom_value_pfp_products_right_result_ubleft) = p))) /\\ (((forall fom_index_pfp_products_right_result_ubright. (exists fom_gap_pfp_products_right_result_ubright_index_bound. fom_gap_pfp_products_right_result_ubright_index_bound + S (fom_index_pfp_products_right_result_ubright) = M) -> exists fom_value_pfp_products_right_result_ubright. ((((exists fom_beta_height_pfp_products_right_result_ubright_entry. fom_beta_height_pfp_products_right_result_ubright_entry + S (fom_value_pfp_products_right_result_ubright) = S ((S (fom_index_pfp_products_right_result_ubright)) * dc)) /\\ exists fom_beta_quotient_pfp_products_right_result_ubright_entry. db = fom_beta_quotient_pfp_products_right_result_ubright_entry * S ((S (fom_index_pfp_products_right_result_ubright)) * dc) + (fom_value_pfp_products_right_result_ubright))) /\\ (exists fom_gap_pfp_products_right_result_ubright_value_bound. fom_gap_pfp_products_right_result_ubright_value_bound + S (fom_value_pfp_products_right_result_ubright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_products_right_result_ubcoefficients. (exists pfa_gap_products_right_result_ubcoefficientsbound. pfa_gap_products_right_result_ubcoefficientsbound + S (pfc_index_products_right_result_ubcoefficients) = (N)) -> exists pfc_value_products_right_result_ubcoefficients. ((((exists ff_h_pfp_products_right_result_ubcoefficientsentry. ff_h_pfp_products_right_result_ubcoefficientsentry + S (pfc_value_products_right_result_ubcoefficients) = S ((S (pfc_index_products_right_result_ubcoefficients)) * uc)) /\\ exists ff_q_pfp_products_right_result_ubcoefficientsentry. ub = ff_q_pfp_products_right_result_ubcoefficientsentry * S ((S (pfc_index_products_right_result_ubcoefficients)) * uc) + (pfc_value_products_right_result_ubcoefficients))) /\\ ((exists pfc_terms_code_products_right_result_ubcoefficientscoefficient pfc_terms_scale_products_right_result_ubcoefficientscoefficient pfc_natural_sum_products_right_result_ubcoefficientscoefficient. ((forall pfc_index_products_right_result_ubcoefficientscoefficientdiagonal. (exists pfa_gap_products_right_result_ubcoefficientscoefficientdiagonalbound. pfa_gap_products_right_result_ubcoefficientscoefficientdiagonalbound + S (pfc_index_products_right_result_ubcoefficientscoefficientdiagonal) = (S (pfc_index_products_right_result_ubcoefficients))) -> exists pfc_value_products_right_result_ubcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_products_right_result_ubcoefficientscoefficientdiagonalentry. ff_h_pfp_products_right_result_ubcoefficientscoefficientdiagonalentry + S (pfc_value_products_right_result_ubcoefficientscoefficientdiagonal) = S ((S (pfc_index_products_right_result_ubcoefficientscoefficientdiagonal)) * pfc_terms_scale_products_right_result_ubcoefficientscoefficient)) /\\ exists ff_q_pfp_products_right_result_ubcoefficientscoefficientdiagonalentry. pfc_terms_code_products_right_result_ubcoefficientscoefficient = ff_q_pfp_products_right_result_ubcoefficientscoefficientdiagonalentry * S ((S (pfc_index_products_right_result_ubcoefficientscoefficientdiagonal)) * pfc_terms_scale_products_right_result_ubcoefficientscoefficient) + (pfc_value_products_right_result_ubcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_products_right_result_ubcoefficientscoefficientdiagonalterm pfc_left_products_right_result_ubcoefficientscoefficientdiagonalterm pfc_right_products_right_result_ubcoefficientscoefficientdiagonalterm. (((pfc_index_products_right_result_ubcoefficientscoefficientdiagonal)+pfc_complement_products_right_result_ubcoefficientscoefficientdiagonalterm=(pfc_index_products_right_result_ubcoefficients)) /\\ ((((((exists pfa_gap_products_right_result_ubcoefficientscoefficientdiagonaltermleftinside. pfa_gap_products_right_result_ubcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_products_right_result_ubcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_products_right_result_ubcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_products_right_result_ubcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_products_right_result_ubcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_products_right_result_ubcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_products_right_result_ubcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_products_right_result_ubcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_products_right_result_ubcoefficientscoefficientdiagonal)) * ac) + (pfc_left_products_right_result_ubcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_products_right_result_ubcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_products_right_result_ubcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_products_right_result_ubcoefficientscoefficientdiagonal)) /\\ (((pfc_left_products_right_result_ubcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_products_right_result_ubcoefficientscoefficientdiagonaltermrightinside. pfa_gap_products_right_result_ubcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_products_right_result_ubcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_products_right_result_ubcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_products_right_result_ubcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_products_right_result_ubcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_products_right_result_ubcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_products_right_result_ubcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_products_right_result_ubcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_products_right_result_ubcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_products_right_result_ubcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_products_right_result_ubcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_products_right_result_ubcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_products_right_result_ubcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_products_right_result_ubcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_products_right_result_ubcoefficientscoefficientdiagonal)=pfc_left_products_right_result_ubcoefficientscoefficientdiagonalterm*pfc_right_products_right_result_ubcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_products_right_result_ubcoefficientscoefficientsum fs_v_pfc_products_right_result_ubcoefficientscoefficientsum. ((((exists fs_h_pfc_products_right_result_ubcoefficientscoefficientsum_body_start. fs_h_pfc_products_right_result_ubcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_products_right_result_ubcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_right_result_ubcoefficientscoefficientsum_body_start. fs_u_pfc_products_right_result_ubcoefficientscoefficientsum = fs_q_pfc_products_right_result_ubcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_products_right_result_ubcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_products_right_result_ubcoefficientscoefficientsum_body_terminal. fs_h_pfc_products_right_result_ubcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_products_right_result_ubcoefficientscoefficient) = S ((S (S (pfc_index_products_right_result_ubcoefficients))) * fs_v_pfc_products_right_result_ubcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_right_result_ubcoefficientscoefficientsum_body_terminal. fs_u_pfc_products_right_result_ubcoefficientscoefficientsum = fs_q_pfc_products_right_result_ubcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_products_right_result_ubcoefficients))) * fs_v_pfc_products_right_result_ubcoefficientscoefficientsum) + (pfc_natural_sum_products_right_result_ubcoefficientscoefficient))) /\\ forall fs_i_pfc_products_right_result_ubcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_products_right_result_ubcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_products_right_result_ubcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_products_right_result_ubcoefficientscoefficientsum_body_steps = S (pfc_index_products_right_result_ubcoefficients)) -> exists fs_a_pfc_products_right_result_ubcoefficientscoefficientsum_body_steps fs_r_pfc_products_right_result_ubcoefficientscoefficientsum_body_steps fs_s_pfc_products_right_result_ubcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_products_right_result_ubcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_products_right_result_ubcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_products_right_result_ubcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_products_right_result_ubcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_products_right_result_ubcoefficientscoefficient)) /\\ exists fs_q_pfc_products_right_result_ubcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_products_right_result_ubcoefficientscoefficient = fs_q_pfc_products_right_result_ubcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_products_right_result_ubcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_products_right_result_ubcoefficientscoefficient) + (fs_a_pfc_products_right_result_ubcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_products_right_result_ubcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_products_right_result_ubcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_products_right_result_ubcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_products_right_result_ubcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_right_result_ubcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_right_result_ubcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_products_right_result_ubcoefficientscoefficientsum = fs_q_pfc_products_right_result_ubcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_products_right_result_ubcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_right_result_ubcoefficientscoefficientsum) + (fs_r_pfc_products_right_result_ubcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_products_right_result_ubcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_products_right_result_ubcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_products_right_result_ubcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_products_right_result_ubcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_right_result_ubcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_right_result_ubcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_products_right_result_ubcoefficientscoefficientsum = fs_q_pfc_products_right_result_ubcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_products_right_result_ubcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_right_result_ubcoefficientscoefficientsum) + (fs_s_pfc_products_right_result_ubcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_products_right_result_ubcoefficientscoefficientsum_body_steps = fs_r_pfc_products_right_result_ubcoefficientscoefficientsum_body_steps + fs_a_pfc_products_right_result_ubcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_products_right_result_ubcoefficientscoefficientresiduebound. pfa_gap_products_right_result_ubcoefficientscoefficientresiduebound + S (pfc_value_products_right_result_ubcoefficients) = (p)) /\\ ((exists pfa_offset_left_products_right_result_ubcoefficientscoefficientresiduecongruence pfa_offset_right_products_right_result_ubcoefficientscoefficientresiduecongruence. (pfc_natural_sum_products_right_result_ubcoefficientscoefficient) + (p) * pfa_offset_left_products_right_result_ubcoefficientscoefficientresiduecongruence = (pfc_value_products_right_result_ubcoefficients) + (p) * pfa_offset_right_products_right_result_ubcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall fom_index_pfp_products_right_result_vbleft. (exists fom_gap_pfp_products_right_result_vbleft_index_bound. fom_gap_pfp_products_right_result_vbleft_index_bound + S (fom_index_pfp_products_right_result_vbleft) = L) -> exists fom_value_pfp_products_right_result_vbleft. ((((exists fom_beta_height_pfp_products_right_result_vbleft_entry. fom_beta_height_pfp_products_right_result_vbleft_entry + S (fom_value_pfp_products_right_result_vbleft) = S ((S (fom_index_pfp_products_right_result_vbleft)) * bc)) /\\ exists fom_beta_quotient_pfp_products_right_result_vbleft_entry. bb = fom_beta_quotient_pfp_products_right_result_vbleft_entry * S ((S (fom_index_pfp_products_right_result_vbleft)) * bc) + (fom_value_pfp_products_right_result_vbleft))) /\\ (exists fom_gap_pfp_products_right_result_vbleft_value_bound. fom_gap_pfp_products_right_result_vbleft_value_bound + S (fom_value_pfp_products_right_result_vbleft) = p))) /\\ (((forall fom_index_pfp_products_right_result_vbright. (exists fom_gap_pfp_products_right_result_vbright_index_bound. fom_gap_pfp_products_right_result_vbright_index_bound + S (fom_index_pfp_products_right_result_vbright) = M) -> exists fom_value_pfp_products_right_result_vbright. ((((exists fom_beta_height_pfp_products_right_result_vbright_entry. fom_beta_height_pfp_products_right_result_vbright_entry + S (fom_value_pfp_products_right_result_vbright) = S ((S (fom_index_pfp_products_right_result_vbright)) * dc)) /\\ exists fom_beta_quotient_pfp_products_right_result_vbright_entry. db = fom_beta_quotient_pfp_products_right_result_vbright_entry * S ((S (fom_index_pfp_products_right_result_vbright)) * dc) + (fom_value_pfp_products_right_result_vbright))) /\\ (exists fom_gap_pfp_products_right_result_vbright_value_bound. fom_gap_pfp_products_right_result_vbright_value_bound + S (fom_value_pfp_products_right_result_vbright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_products_right_result_vbcoefficients. (exists pfa_gap_products_right_result_vbcoefficientsbound. pfa_gap_products_right_result_vbcoefficientsbound + S (pfc_index_products_right_result_vbcoefficients) = (N)) -> exists pfc_value_products_right_result_vbcoefficients. ((((exists ff_h_pfp_products_right_result_vbcoefficientsentry. ff_h_pfp_products_right_result_vbcoefficientsentry + S (pfc_value_products_right_result_vbcoefficients) = S ((S (pfc_index_products_right_result_vbcoefficients)) * vc)) /\\ exists ff_q_pfp_products_right_result_vbcoefficientsentry. vb = ff_q_pfp_products_right_result_vbcoefficientsentry * S ((S (pfc_index_products_right_result_vbcoefficients)) * vc) + (pfc_value_products_right_result_vbcoefficients))) /\\ ((exists pfc_terms_code_products_right_result_vbcoefficientscoefficient pfc_terms_scale_products_right_result_vbcoefficientscoefficient pfc_natural_sum_products_right_result_vbcoefficientscoefficient. ((forall pfc_index_products_right_result_vbcoefficientscoefficientdiagonal. (exists pfa_gap_products_right_result_vbcoefficientscoefficientdiagonalbound. pfa_gap_products_right_result_vbcoefficientscoefficientdiagonalbound + S (pfc_index_products_right_result_vbcoefficientscoefficientdiagonal) = (S (pfc_index_products_right_result_vbcoefficients))) -> exists pfc_value_products_right_result_vbcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_products_right_result_vbcoefficientscoefficientdiagonalentry. ff_h_pfp_products_right_result_vbcoefficientscoefficientdiagonalentry + S (pfc_value_products_right_result_vbcoefficientscoefficientdiagonal) = S ((S (pfc_index_products_right_result_vbcoefficientscoefficientdiagonal)) * pfc_terms_scale_products_right_result_vbcoefficientscoefficient)) /\\ exists ff_q_pfp_products_right_result_vbcoefficientscoefficientdiagonalentry. pfc_terms_code_products_right_result_vbcoefficientscoefficient = ff_q_pfp_products_right_result_vbcoefficientscoefficientdiagonalentry * S ((S (pfc_index_products_right_result_vbcoefficientscoefficientdiagonal)) * pfc_terms_scale_products_right_result_vbcoefficientscoefficient) + (pfc_value_products_right_result_vbcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_products_right_result_vbcoefficientscoefficientdiagonalterm pfc_left_products_right_result_vbcoefficientscoefficientdiagonalterm pfc_right_products_right_result_vbcoefficientscoefficientdiagonalterm. (((pfc_index_products_right_result_vbcoefficientscoefficientdiagonal)+pfc_complement_products_right_result_vbcoefficientscoefficientdiagonalterm=(pfc_index_products_right_result_vbcoefficients)) /\\ ((((((exists pfa_gap_products_right_result_vbcoefficientscoefficientdiagonaltermleftinside. pfa_gap_products_right_result_vbcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_products_right_result_vbcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_products_right_result_vbcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_products_right_result_vbcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_products_right_result_vbcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_products_right_result_vbcoefficientscoefficientdiagonal)) * bc)) /\\ exists ff_q_pfp_products_right_result_vbcoefficientscoefficientdiagonaltermleftentry. bb = ff_q_pfp_products_right_result_vbcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_products_right_result_vbcoefficientscoefficientdiagonal)) * bc) + (pfc_left_products_right_result_vbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_products_right_result_vbcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_products_right_result_vbcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_products_right_result_vbcoefficientscoefficientdiagonal)) /\\ (((pfc_left_products_right_result_vbcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_products_right_result_vbcoefficientscoefficientdiagonaltermrightinside. pfa_gap_products_right_result_vbcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_products_right_result_vbcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_products_right_result_vbcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_products_right_result_vbcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_products_right_result_vbcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_products_right_result_vbcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_products_right_result_vbcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_products_right_result_vbcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_products_right_result_vbcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_products_right_result_vbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_products_right_result_vbcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_products_right_result_vbcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_products_right_result_vbcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_products_right_result_vbcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_products_right_result_vbcoefficientscoefficientdiagonal)=pfc_left_products_right_result_vbcoefficientscoefficientdiagonalterm*pfc_right_products_right_result_vbcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_products_right_result_vbcoefficientscoefficientsum fs_v_pfc_products_right_result_vbcoefficientscoefficientsum. ((((exists fs_h_pfc_products_right_result_vbcoefficientscoefficientsum_body_start. fs_h_pfc_products_right_result_vbcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_products_right_result_vbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_right_result_vbcoefficientscoefficientsum_body_start. fs_u_pfc_products_right_result_vbcoefficientscoefficientsum = fs_q_pfc_products_right_result_vbcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_products_right_result_vbcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_products_right_result_vbcoefficientscoefficientsum_body_terminal. fs_h_pfc_products_right_result_vbcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_products_right_result_vbcoefficientscoefficient) = S ((S (S (pfc_index_products_right_result_vbcoefficients))) * fs_v_pfc_products_right_result_vbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_right_result_vbcoefficientscoefficientsum_body_terminal. fs_u_pfc_products_right_result_vbcoefficientscoefficientsum = fs_q_pfc_products_right_result_vbcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_products_right_result_vbcoefficients))) * fs_v_pfc_products_right_result_vbcoefficientscoefficientsum) + (pfc_natural_sum_products_right_result_vbcoefficientscoefficient))) /\\ forall fs_i_pfc_products_right_result_vbcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_products_right_result_vbcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_products_right_result_vbcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_products_right_result_vbcoefficientscoefficientsum_body_steps = S (pfc_index_products_right_result_vbcoefficients)) -> exists fs_a_pfc_products_right_result_vbcoefficientscoefficientsum_body_steps fs_r_pfc_products_right_result_vbcoefficientscoefficientsum_body_steps fs_s_pfc_products_right_result_vbcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_products_right_result_vbcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_products_right_result_vbcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_products_right_result_vbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_products_right_result_vbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_products_right_result_vbcoefficientscoefficient)) /\\ exists fs_q_pfc_products_right_result_vbcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_products_right_result_vbcoefficientscoefficient = fs_q_pfc_products_right_result_vbcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_products_right_result_vbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_products_right_result_vbcoefficientscoefficient) + (fs_a_pfc_products_right_result_vbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_products_right_result_vbcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_products_right_result_vbcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_products_right_result_vbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_products_right_result_vbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_right_result_vbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_right_result_vbcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_products_right_result_vbcoefficientscoefficientsum = fs_q_pfc_products_right_result_vbcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_products_right_result_vbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_right_result_vbcoefficientscoefficientsum) + (fs_r_pfc_products_right_result_vbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_products_right_result_vbcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_products_right_result_vbcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_products_right_result_vbcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_products_right_result_vbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_right_result_vbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_right_result_vbcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_products_right_result_vbcoefficientscoefficientsum = fs_q_pfc_products_right_result_vbcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_products_right_result_vbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_right_result_vbcoefficientscoefficientsum) + (fs_s_pfc_products_right_result_vbcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_products_right_result_vbcoefficientscoefficientsum_body_steps = fs_r_pfc_products_right_result_vbcoefficientscoefficientsum_body_steps + fs_a_pfc_products_right_result_vbcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_products_right_result_vbcoefficientscoefficientresiduebound. pfa_gap_products_right_result_vbcoefficientscoefficientresiduebound + S (pfc_value_products_right_result_vbcoefficients) = (p)) /\\ ((exists pfa_offset_left_products_right_result_vbcoefficientscoefficientresiduecongruence pfa_offset_right_products_right_result_vbcoefficientscoefficientresiduecongruence. (pfc_natural_sum_products_right_result_vbcoefficientscoefficient) + (p) * pfa_offset_left_products_right_result_vbcoefficientscoefficientresiduecongruence = (pfc_value_products_right_result_vbcoefficients) + (p) * pfa_offset_right_products_right_result_vbcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall fom_index_pfp_products_right_result_wbleft. (exists fom_gap_pfp_products_right_result_wbleft_index_bound. fom_gap_pfp_products_right_result_wbleft_index_bound + S (fom_index_pfp_products_right_result_wbleft) = L) -> exists fom_value_pfp_products_right_result_wbleft. ((((exists fom_beta_height_pfp_products_right_result_wbleft_entry. fom_beta_height_pfp_products_right_result_wbleft_entry + S (fom_value_pfp_products_right_result_wbleft) = S ((S (fom_index_pfp_products_right_result_wbleft)) * cc)) /\\ exists fom_beta_quotient_pfp_products_right_result_wbleft_entry. cb = fom_beta_quotient_pfp_products_right_result_wbleft_entry * S ((S (fom_index_pfp_products_right_result_wbleft)) * cc) + (fom_value_pfp_products_right_result_wbleft))) /\\ (exists fom_gap_pfp_products_right_result_wbleft_value_bound. fom_gap_pfp_products_right_result_wbleft_value_bound + S (fom_value_pfp_products_right_result_wbleft) = p))) /\\ (((forall fom_index_pfp_products_right_result_wbright. (exists fom_gap_pfp_products_right_result_wbright_index_bound. fom_gap_pfp_products_right_result_wbright_index_bound + S (fom_index_pfp_products_right_result_wbright) = M) -> exists fom_value_pfp_products_right_result_wbright. ((((exists fom_beta_height_pfp_products_right_result_wbright_entry. fom_beta_height_pfp_products_right_result_wbright_entry + S (fom_value_pfp_products_right_result_wbright) = S ((S (fom_index_pfp_products_right_result_wbright)) * dc)) /\\ exists fom_beta_quotient_pfp_products_right_result_wbright_entry. db = fom_beta_quotient_pfp_products_right_result_wbright_entry * S ((S (fom_index_pfp_products_right_result_wbright)) * dc) + (fom_value_pfp_products_right_result_wbright))) /\\ (exists fom_gap_pfp_products_right_result_wbright_value_bound. fom_gap_pfp_products_right_result_wbright_value_bound + S (fom_value_pfp_products_right_result_wbright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_products_right_result_wbcoefficients. (exists pfa_gap_products_right_result_wbcoefficientsbound. pfa_gap_products_right_result_wbcoefficientsbound + S (pfc_index_products_right_result_wbcoefficients) = (N)) -> exists pfc_value_products_right_result_wbcoefficients. ((((exists ff_h_pfp_products_right_result_wbcoefficientsentry. ff_h_pfp_products_right_result_wbcoefficientsentry + S (pfc_value_products_right_result_wbcoefficients) = S ((S (pfc_index_products_right_result_wbcoefficients)) * wc)) /\\ exists ff_q_pfp_products_right_result_wbcoefficientsentry. wb = ff_q_pfp_products_right_result_wbcoefficientsentry * S ((S (pfc_index_products_right_result_wbcoefficients)) * wc) + (pfc_value_products_right_result_wbcoefficients))) /\\ ((exists pfc_terms_code_products_right_result_wbcoefficientscoefficient pfc_terms_scale_products_right_result_wbcoefficientscoefficient pfc_natural_sum_products_right_result_wbcoefficientscoefficient. ((forall pfc_index_products_right_result_wbcoefficientscoefficientdiagonal. (exists pfa_gap_products_right_result_wbcoefficientscoefficientdiagonalbound. pfa_gap_products_right_result_wbcoefficientscoefficientdiagonalbound + S (pfc_index_products_right_result_wbcoefficientscoefficientdiagonal) = (S (pfc_index_products_right_result_wbcoefficients))) -> exists pfc_value_products_right_result_wbcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_products_right_result_wbcoefficientscoefficientdiagonalentry. ff_h_pfp_products_right_result_wbcoefficientscoefficientdiagonalentry + S (pfc_value_products_right_result_wbcoefficientscoefficientdiagonal) = S ((S (pfc_index_products_right_result_wbcoefficientscoefficientdiagonal)) * pfc_terms_scale_products_right_result_wbcoefficientscoefficient)) /\\ exists ff_q_pfp_products_right_result_wbcoefficientscoefficientdiagonalentry. pfc_terms_code_products_right_result_wbcoefficientscoefficient = ff_q_pfp_products_right_result_wbcoefficientscoefficientdiagonalentry * S ((S (pfc_index_products_right_result_wbcoefficientscoefficientdiagonal)) * pfc_terms_scale_products_right_result_wbcoefficientscoefficient) + (pfc_value_products_right_result_wbcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_products_right_result_wbcoefficientscoefficientdiagonalterm pfc_left_products_right_result_wbcoefficientscoefficientdiagonalterm pfc_right_products_right_result_wbcoefficientscoefficientdiagonalterm. (((pfc_index_products_right_result_wbcoefficientscoefficientdiagonal)+pfc_complement_products_right_result_wbcoefficientscoefficientdiagonalterm=(pfc_index_products_right_result_wbcoefficients)) /\\ ((((((exists pfa_gap_products_right_result_wbcoefficientscoefficientdiagonaltermleftinside. pfa_gap_products_right_result_wbcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_products_right_result_wbcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_products_right_result_wbcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_products_right_result_wbcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_products_right_result_wbcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_products_right_result_wbcoefficientscoefficientdiagonal)) * cc)) /\\ exists ff_q_pfp_products_right_result_wbcoefficientscoefficientdiagonaltermleftentry. cb = ff_q_pfp_products_right_result_wbcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_products_right_result_wbcoefficientscoefficientdiagonal)) * cc) + (pfc_left_products_right_result_wbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_products_right_result_wbcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_products_right_result_wbcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_products_right_result_wbcoefficientscoefficientdiagonal)) /\\ (((pfc_left_products_right_result_wbcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_products_right_result_wbcoefficientscoefficientdiagonaltermrightinside. pfa_gap_products_right_result_wbcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_products_right_result_wbcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_products_right_result_wbcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_products_right_result_wbcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_products_right_result_wbcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_products_right_result_wbcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_products_right_result_wbcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_products_right_result_wbcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_products_right_result_wbcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_products_right_result_wbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_products_right_result_wbcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_products_right_result_wbcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_products_right_result_wbcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_products_right_result_wbcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_products_right_result_wbcoefficientscoefficientdiagonal)=pfc_left_products_right_result_wbcoefficientscoefficientdiagonalterm*pfc_right_products_right_result_wbcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_products_right_result_wbcoefficientscoefficientsum fs_v_pfc_products_right_result_wbcoefficientscoefficientsum. ((((exists fs_h_pfc_products_right_result_wbcoefficientscoefficientsum_body_start. fs_h_pfc_products_right_result_wbcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_products_right_result_wbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_right_result_wbcoefficientscoefficientsum_body_start. fs_u_pfc_products_right_result_wbcoefficientscoefficientsum = fs_q_pfc_products_right_result_wbcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_products_right_result_wbcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_products_right_result_wbcoefficientscoefficientsum_body_terminal. fs_h_pfc_products_right_result_wbcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_products_right_result_wbcoefficientscoefficient) = S ((S (S (pfc_index_products_right_result_wbcoefficients))) * fs_v_pfc_products_right_result_wbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_right_result_wbcoefficientscoefficientsum_body_terminal. fs_u_pfc_products_right_result_wbcoefficientscoefficientsum = fs_q_pfc_products_right_result_wbcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_products_right_result_wbcoefficients))) * fs_v_pfc_products_right_result_wbcoefficientscoefficientsum) + (pfc_natural_sum_products_right_result_wbcoefficientscoefficient))) /\\ forall fs_i_pfc_products_right_result_wbcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_products_right_result_wbcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_products_right_result_wbcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_products_right_result_wbcoefficientscoefficientsum_body_steps = S (pfc_index_products_right_result_wbcoefficients)) -> exists fs_a_pfc_products_right_result_wbcoefficientscoefficientsum_body_steps fs_r_pfc_products_right_result_wbcoefficientscoefficientsum_body_steps fs_s_pfc_products_right_result_wbcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_products_right_result_wbcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_products_right_result_wbcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_products_right_result_wbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_products_right_result_wbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_products_right_result_wbcoefficientscoefficient)) /\\ exists fs_q_pfc_products_right_result_wbcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_products_right_result_wbcoefficientscoefficient = fs_q_pfc_products_right_result_wbcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_products_right_result_wbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_products_right_result_wbcoefficientscoefficient) + (fs_a_pfc_products_right_result_wbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_products_right_result_wbcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_products_right_result_wbcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_products_right_result_wbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_products_right_result_wbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_right_result_wbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_right_result_wbcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_products_right_result_wbcoefficientscoefficientsum = fs_q_pfc_products_right_result_wbcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_products_right_result_wbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_right_result_wbcoefficientscoefficientsum) + (fs_r_pfc_products_right_result_wbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_products_right_result_wbcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_products_right_result_wbcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_products_right_result_wbcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_products_right_result_wbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_right_result_wbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_products_right_result_wbcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_products_right_result_wbcoefficientscoefficientsum = fs_q_pfc_products_right_result_wbcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_products_right_result_wbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_right_result_wbcoefficientscoefficientsum) + (fs_s_pfc_products_right_result_wbcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_products_right_result_wbcoefficientscoefficientsum_body_steps = fs_r_pfc_products_right_result_wbcoefficientscoefficientsum_body_steps + fs_a_pfc_products_right_result_wbcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_products_right_result_wbcoefficientscoefficientresiduebound. pfa_gap_products_right_result_wbcoefficientscoefficientresiduebound + S (pfc_value_products_right_result_wbcoefficients) = (p)) /\\ ((exists pfa_offset_left_products_right_result_wbcoefficientscoefficientresiduecongruence pfa_offset_right_products_right_result_wbcoefficientscoefficientresiduecongruence. (pfc_natural_sum_products_right_result_wbcoefficientscoefficient) + (p) * pfa_offset_left_products_right_result_wbcoefficientscoefficientresiduecongruence = (pfc_value_products_right_result_wbcoefficients) + (p) * pfa_offset_right_products_right_result_wbcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfp_index_products_right_addition. (exists pfa_gap_products_right_additionindex. pfa_gap_products_right_additionindex + S (pfp_index_products_right_addition) = (N)) -> exists pfp_left_products_right_addition pfp_right_products_right_addition pfp_value_products_right_addition. ((((exists ff_h_pfp_products_right_additionleft. ff_h_pfp_products_right_additionleft + S (pfp_left_products_right_addition) = S ((S (pfp_index_products_right_addition)) * uc)) /\\ exists ff_q_pfp_products_right_additionleft. ub = ff_q_pfp_products_right_additionleft * S ((S (pfp_index_products_right_addition)) * uc) + (pfp_left_products_right_addition))) /\\ (((((exists ff_h_pfp_products_right_additionright. ff_h_pfp_products_right_additionright + S (pfp_right_products_right_addition) = S ((S (pfp_index_products_right_addition)) * vc)) /\\ exists ff_q_pfp_products_right_additionright. vb = ff_q_pfp_products_right_additionright * S ((S (pfp_index_products_right_addition)) * vc) + (pfp_right_products_right_addition))) /\\ (((((exists ff_h_pfp_products_right_additiontarget. ff_h_pfp_products_right_additiontarget + S (pfp_value_products_right_addition) = S ((S (pfp_index_products_right_addition)) * wc)) /\\ exists ff_q_pfp_products_right_additiontarget. wb = ff_q_pfp_products_right_additiontarget * S ((S (pfp_index_products_right_addition)) * wc) + (pfp_value_products_right_addition))) /\\ ((((exists pfa_gap_products_right_additionoperationleft. pfa_gap_products_right_additionoperationleft + S (pfp_left_products_right_addition) = (p)) /\\ (((exists pfa_gap_products_right_additionoperationright. pfa_gap_products_right_additionoperationright + S (pfp_right_products_right_addition) = (p)) /\\ ((((exists pfa_gap_products_right_additionoperationresultbound. pfa_gap_products_right_additionoperationresultbound + S (pfp_value_products_right_addition) = (p)) /\\ ((exists pfa_offset_left_products_right_additionoperationresultcongruence pfa_offset_right_products_right_additionoperationresultcongruence. 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(exists pfd_input_step_unique_first pfd_previous_step_unique_first pfd_difference_step_unique_first. ((((exists ff_h_pfp_step_unique_firstinput. ff_h_pfp_step_unique_firstinput + S (pfd_input_step_unique_first) = S ((S (i)) * ac)) /\\ exists ff_q_pfp_step_unique_firstinput. ab = ff_q_pfp_step_unique_firstinput * S ((S (i)) * ac) + (pfd_input_step_unique_first))) /\\ (((exists pfc_terms_code_step_unique_firstprevious pfc_terms_scale_step_unique_firstprevious pfc_natural_sum_step_unique_firstprevious. ((forall pfc_index_step_unique_firstpreviousdiagonal. (exists pfa_gap_step_unique_firstpreviousdiagonalbound. pfa_gap_step_unique_firstpreviousdiagonalbound + S (pfc_index_step_unique_firstpreviousdiagonal) = (S (i))) -> exists pfc_value_step_unique_firstpreviousdiagonal. ((((exists ff_h_pfp_step_unique_firstpreviousdiagonalentry. ff_h_pfp_step_unique_firstpreviousdiagonalentry + S (pfc_value_step_unique_firstpreviousdiagonal) = S ((S (pfc_index_step_unique_firstpreviousdiagonal)) * pfc_terms_scale_step_unique_firstprevious)) /\\ exists ff_q_pfp_step_unique_firstpreviousdiagonalentry. pfc_terms_code_step_unique_firstprevious = ff_q_pfp_step_unique_firstpreviousdiagonalentry * S ((S (pfc_index_step_unique_firstpreviousdiagonal)) * pfc_terms_scale_step_unique_firstprevious) + (pfc_value_step_unique_firstpreviousdiagonal))) /\\ ((exists pfc_complement_step_unique_firstpreviousdiagonalterm pfc_left_step_unique_firstpreviousdiagonalterm pfc_right_step_unique_firstpreviousdiagonalterm. (((pfc_index_step_unique_firstpreviousdiagonal)+pfc_complement_step_unique_firstpreviousdiagonalterm=(i)) /\\ ((((((exists pfa_gap_step_unique_firstpreviousdiagonaltermleftinside. pfa_gap_step_unique_firstpreviousdiagonaltermleftinside + S (pfc_index_step_unique_firstpreviousdiagonal) = (i)) /\\ ((((exists ff_h_pfp_step_unique_firstpreviousdiagonaltermleftentry. ff_h_pfp_step_unique_firstpreviousdiagonaltermleftentry + S (pfc_left_step_unique_firstpreviousdiagonalterm) = S ((S (pfc_index_step_unique_firstpreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_step_unique_firstpreviousdiagonaltermleftentry. qb = ff_q_pfp_step_unique_firstpreviousdiagonaltermleftentry * S ((S (pfc_index_step_unique_firstpreviousdiagonal)) * qc) + (pfc_left_step_unique_firstpreviousdiagonalterm)))))) \\/ (((exists pfc_gap_step_unique_firstpreviousdiagonaltermleftoutside. pfc_gap_step_unique_firstpreviousdiagonaltermleftoutside+(i)=(pfc_index_step_unique_firstpreviousdiagonal)) /\\ (((pfc_left_step_unique_firstpreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_step_unique_firstpreviousdiagonaltermrightinside. pfa_gap_step_unique_firstpreviousdiagonaltermrightinside + S (pfc_complement_step_unique_firstpreviousdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_step_unique_firstpreviousdiagonaltermrightentry. ff_h_pfp_step_unique_firstpreviousdiagonaltermrightentry + S (pfc_right_step_unique_firstpreviousdiagonalterm) = S ((S (pfc_complement_step_unique_firstpreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_step_unique_firstpreviousdiagonaltermrightentry. bb = ff_q_pfp_step_unique_firstpreviousdiagonaltermrightentry * S ((S (pfc_complement_step_unique_firstpreviousdiagonalterm)) * bc) + (pfc_right_step_unique_firstpreviousdiagonalterm)))))) \\/ (((exists pfc_gap_step_unique_firstpreviousdiagonaltermrightoutside. pfc_gap_step_unique_firstpreviousdiagonaltermrightoutside+(M)=(pfc_complement_step_unique_firstpreviousdiagonalterm)) /\\ (((pfc_right_step_unique_firstpreviousdiagonalterm)=0))))) /\\ (((pfc_value_step_unique_firstpreviousdiagonal)=pfc_left_step_unique_firstpreviousdiagonalterm*pfc_right_step_unique_firstpreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_step_unique_firstprevioussum fs_v_pfc_step_unique_firstprevioussum. ((((exists fs_h_pfc_step_unique_firstprevioussum_body_start. fs_h_pfc_step_unique_firstprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_step_unique_firstprevioussum)) /\\ exists fs_q_pfc_step_unique_firstprevioussum_body_start. fs_u_pfc_step_unique_firstprevioussum = fs_q_pfc_step_unique_firstprevioussum_body_start * S ((S (0)) * fs_v_pfc_step_unique_firstprevioussum) + (0))) /\\ ((((exists fs_h_pfc_step_unique_firstprevioussum_body_terminal. fs_h_pfc_step_unique_firstprevioussum_body_terminal + S (pfc_natural_sum_step_unique_firstprevious) = S ((S (S (i))) * fs_v_pfc_step_unique_firstprevioussum)) /\\ exists fs_q_pfc_step_unique_firstprevioussum_body_terminal. fs_u_pfc_step_unique_firstprevioussum = fs_q_pfc_step_unique_firstprevioussum_body_terminal * S ((S (S (i))) * fs_v_pfc_step_unique_firstprevioussum) + (pfc_natural_sum_step_unique_firstprevious))) /\\ forall fs_i_pfc_step_unique_firstprevioussum_body_steps. (exists fs_lt_pfc_step_unique_firstprevioussum_body_steps_bound. fs_lt_pfc_step_unique_firstprevioussum_body_steps_bound + S fs_i_pfc_step_unique_firstprevioussum_body_steps = S (i)) -> exists fs_a_pfc_step_unique_firstprevioussum_body_steps fs_r_pfc_step_unique_firstprevioussum_body_steps fs_s_pfc_step_unique_firstprevioussum_body_steps. ((((exists fs_h_pfc_step_unique_firstprevioussum_body_steps_summand. fs_h_pfc_step_unique_firstprevioussum_body_steps_summand + S (fs_a_pfc_step_unique_firstprevioussum_body_steps) = S ((S (fs_i_pfc_step_unique_firstprevioussum_body_steps)) * pfc_terms_scale_step_unique_firstprevious)) /\\ exists fs_q_pfc_step_unique_firstprevioussum_body_steps_summand. pfc_terms_code_step_unique_firstprevious = fs_q_pfc_step_unique_firstprevioussum_body_steps_summand * S ((S (fs_i_pfc_step_unique_firstprevioussum_body_steps)) * pfc_terms_scale_step_unique_firstprevious) + (fs_a_pfc_step_unique_firstprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_step_unique_firstprevioussum_body_steps_partial. fs_h_pfc_step_unique_firstprevioussum_body_steps_partial + S (fs_r_pfc_step_unique_firstprevioussum_body_steps) = S ((S (fs_i_pfc_step_unique_firstprevioussum_body_steps)) * fs_v_pfc_step_unique_firstprevioussum)) /\\ exists fs_q_pfc_step_unique_firstprevioussum_body_steps_partial. fs_u_pfc_step_unique_firstprevioussum = fs_q_pfc_step_unique_firstprevioussum_body_steps_partial * S ((S (fs_i_pfc_step_unique_firstprevioussum_body_steps)) * fs_v_pfc_step_unique_firstprevioussum) + (fs_r_pfc_step_unique_firstprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_step_unique_firstprevioussum_body_steps_successor. fs_h_pfc_step_unique_firstprevioussum_body_steps_successor + S (fs_s_pfc_step_unique_firstprevioussum_body_steps) = S ((S (S fs_i_pfc_step_unique_firstprevioussum_body_steps)) * fs_v_pfc_step_unique_firstprevioussum)) /\\ exists fs_q_pfc_step_unique_firstprevioussum_body_steps_successor. fs_u_pfc_step_unique_firstprevioussum = fs_q_pfc_step_unique_firstprevioussum_body_steps_successor * S ((S (S fs_i_pfc_step_unique_firstprevioussum_body_steps)) * fs_v_pfc_step_unique_firstprevioussum) + (fs_s_pfc_step_unique_firstprevioussum_body_steps))) /\\ fs_s_pfc_step_unique_firstprevioussum_body_steps = fs_r_pfc_step_unique_firstprevioussum_body_steps + fs_a_pfc_step_unique_firstprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_step_unique_firstpreviousresiduebound. pfa_gap_step_unique_firstpreviousresiduebound + S (pfd_previous_step_unique_first) = (p)) /\\ ((exists pfa_offset_left_step_unique_firstpreviousresiduecongruence pfa_offset_right_step_unique_firstpreviousresiduecongruence. (pfc_natural_sum_step_unique_firstprevious) + (p) * pfa_offset_left_step_unique_firstpreviousresiduecongruence = (pfd_previous_step_unique_first) + (p) * pfa_offset_right_step_unique_firstpreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_step_unique_firstsubtractleft. pfa_gap_step_unique_firstsubtractleft + S (pfd_previous_step_unique_first) = (p)) /\\ (((exists pfa_gap_step_unique_firstsubtractright. pfa_gap_step_unique_firstsubtractright + S (pfd_difference_step_unique_first) = (p)) /\\ ((((exists pfa_gap_step_unique_firstsubtractresultbound. pfa_gap_step_unique_firstsubtractresultbound + S (pfd_input_step_unique_first) = (p)) /\\ ((exists pfa_offset_left_step_unique_firstsubtractresultcongruence pfa_offset_right_step_unique_firstsubtractresultcongruence. ((pfd_previous_step_unique_first) + (pfd_difference_step_unique_first)) + (p) * pfa_offset_left_step_unique_firstsubtractresultcongruence = (pfd_input_step_unique_first) + (p) * pfa_offset_right_step_unique_firstsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_step_unique_firstmultiplyleft. pfa_gap_step_unique_firstmultiplyleft + S (k) = (p)) /\\ (((exists pfa_gap_step_unique_firstmultiplyright. pfa_gap_step_unique_firstmultiplyright + S (pfd_difference_step_unique_first) = (p)) /\\ ((((exists pfa_gap_step_unique_firstmultiplyresultbound. pfa_gap_step_unique_firstmultiplyresultbound + S (q) = (p)) /\\ ((exists pfa_offset_left_step_unique_firstmultiplyresultcongruence pfa_offset_right_step_unique_firstmultiplyresultcongruence. ((k) * (pfd_difference_step_unique_first)) + (p) * pfa_offset_left_step_unique_firstmultiplyresultcongruence = (q) + (p) * pfa_offset_right_step_unique_firstmultiplyresultcongruence)))))))))))))))) -> (exists pfd_input_step_unique_second pfd_previous_step_unique_second pfd_difference_step_unique_second. ((((exists ff_h_pfp_step_unique_secondinput. ff_h_pfp_step_unique_secondinput + S (pfd_input_step_unique_second) = S ((S (i)) * ac)) /\\ exists ff_q_pfp_step_unique_secondinput. ab = ff_q_pfp_step_unique_secondinput * S ((S (i)) * ac) + (pfd_input_step_unique_second))) /\\ (((exists pfc_terms_code_step_unique_secondprevious pfc_terms_scale_step_unique_secondprevious pfc_natural_sum_step_unique_secondprevious. ((forall pfc_index_step_unique_secondpreviousdiagonal. (exists pfa_gap_step_unique_secondpreviousdiagonalbound. pfa_gap_step_unique_secondpreviousdiagonalbound + S (pfc_index_step_unique_secondpreviousdiagonal) = (S (i))) -> exists pfc_value_step_unique_secondpreviousdiagonal. ((((exists ff_h_pfp_step_unique_secondpreviousdiagonalentry. ff_h_pfp_step_unique_secondpreviousdiagonalentry + S (pfc_value_step_unique_secondpreviousdiagonal) = S ((S (pfc_index_step_unique_secondpreviousdiagonal)) * pfc_terms_scale_step_unique_secondprevious)) /\\ exists ff_q_pfp_step_unique_secondpreviousdiagonalentry. pfc_terms_code_step_unique_secondprevious = ff_q_pfp_step_unique_secondpreviousdiagonalentry * S ((S (pfc_index_step_unique_secondpreviousdiagonal)) * pfc_terms_scale_step_unique_secondprevious) + (pfc_value_step_unique_secondpreviousdiagonal))) /\\ ((exists pfc_complement_step_unique_secondpreviousdiagonalterm pfc_left_step_unique_secondpreviousdiagonalterm pfc_right_step_unique_secondpreviousdiagonalterm. (((pfc_index_step_unique_secondpreviousdiagonal)+pfc_complement_step_unique_secondpreviousdiagonalterm=(i)) /\\ ((((((exists pfa_gap_step_unique_secondpreviousdiagonaltermleftinside. pfa_gap_step_unique_secondpreviousdiagonaltermleftinside + S (pfc_index_step_unique_secondpreviousdiagonal) = (i)) /\\ ((((exists ff_h_pfp_step_unique_secondpreviousdiagonaltermleftentry. ff_h_pfp_step_unique_secondpreviousdiagonaltermleftentry + S (pfc_left_step_unique_secondpreviousdiagonalterm) = S ((S (pfc_index_step_unique_secondpreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_step_unique_secondpreviousdiagonaltermleftentry. qb = ff_q_pfp_step_unique_secondpreviousdiagonaltermleftentry * S ((S (pfc_index_step_unique_secondpreviousdiagonal)) * qc) + (pfc_left_step_unique_secondpreviousdiagonalterm)))))) \\/ (((exists pfc_gap_step_unique_secondpreviousdiagonaltermleftoutside. pfc_gap_step_unique_secondpreviousdiagonaltermleftoutside+(i)=(pfc_index_step_unique_secondpreviousdiagonal)) /\\ (((pfc_left_step_unique_secondpreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_step_unique_secondpreviousdiagonaltermrightinside. pfa_gap_step_unique_secondpreviousdiagonaltermrightinside + S (pfc_complement_step_unique_secondpreviousdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_step_unique_secondpreviousdiagonaltermrightentry. ff_h_pfp_step_unique_secondpreviousdiagonaltermrightentry + S (pfc_right_step_unique_secondpreviousdiagonalterm) = S ((S (pfc_complement_step_unique_secondpreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_step_unique_secondpreviousdiagonaltermrightentry. bb = ff_q_pfp_step_unique_secondpreviousdiagonaltermrightentry * S ((S (pfc_complement_step_unique_secondpreviousdiagonalterm)) * bc) + (pfc_right_step_unique_secondpreviousdiagonalterm)))))) \\/ (((exists pfc_gap_step_unique_secondpreviousdiagonaltermrightoutside. pfc_gap_step_unique_secondpreviousdiagonaltermrightoutside+(M)=(pfc_complement_step_unique_secondpreviousdiagonalterm)) /\\ (((pfc_right_step_unique_secondpreviousdiagonalterm)=0))))) /\\ (((pfc_value_step_unique_secondpreviousdiagonal)=pfc_left_step_unique_secondpreviousdiagonalterm*pfc_right_step_unique_secondpreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_step_unique_secondprevioussum fs_v_pfc_step_unique_secondprevioussum. ((((exists fs_h_pfc_step_unique_secondprevioussum_body_start. fs_h_pfc_step_unique_secondprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_step_unique_secondprevioussum)) /\\ exists fs_q_pfc_step_unique_secondprevioussum_body_start. fs_u_pfc_step_unique_secondprevioussum = fs_q_pfc_step_unique_secondprevioussum_body_start * S ((S (0)) * fs_v_pfc_step_unique_secondprevioussum) + (0))) /\\ ((((exists fs_h_pfc_step_unique_secondprevioussum_body_terminal. fs_h_pfc_step_unique_secondprevioussum_body_terminal + S (pfc_natural_sum_step_unique_secondprevious) = S ((S (S (i))) * fs_v_pfc_step_unique_secondprevioussum)) /\\ exists fs_q_pfc_step_unique_secondprevioussum_body_terminal. fs_u_pfc_step_unique_secondprevioussum = fs_q_pfc_step_unique_secondprevioussum_body_terminal * S ((S (S (i))) * fs_v_pfc_step_unique_secondprevioussum) + (pfc_natural_sum_step_unique_secondprevious))) /\\ forall fs_i_pfc_step_unique_secondprevioussum_body_steps. (exists fs_lt_pfc_step_unique_secondprevioussum_body_steps_bound. fs_lt_pfc_step_unique_secondprevioussum_body_steps_bound + S fs_i_pfc_step_unique_secondprevioussum_body_steps = S (i)) -> exists fs_a_pfc_step_unique_secondprevioussum_body_steps fs_r_pfc_step_unique_secondprevioussum_body_steps fs_s_pfc_step_unique_secondprevioussum_body_steps. ((((exists fs_h_pfc_step_unique_secondprevioussum_body_steps_summand. fs_h_pfc_step_unique_secondprevioussum_body_steps_summand + S (fs_a_pfc_step_unique_secondprevioussum_body_steps) = S ((S (fs_i_pfc_step_unique_secondprevioussum_body_steps)) * pfc_terms_scale_step_unique_secondprevious)) /\\ exists fs_q_pfc_step_unique_secondprevioussum_body_steps_summand. pfc_terms_code_step_unique_secondprevious = fs_q_pfc_step_unique_secondprevioussum_body_steps_summand * S ((S (fs_i_pfc_step_unique_secondprevioussum_body_steps)) * pfc_terms_scale_step_unique_secondprevious) + (fs_a_pfc_step_unique_secondprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_step_unique_secondprevioussum_body_steps_partial. fs_h_pfc_step_unique_secondprevioussum_body_steps_partial + S (fs_r_pfc_step_unique_secondprevioussum_body_steps) = S ((S (fs_i_pfc_step_unique_secondprevioussum_body_steps)) * fs_v_pfc_step_unique_secondprevioussum)) /\\ exists fs_q_pfc_step_unique_secondprevioussum_body_steps_partial. fs_u_pfc_step_unique_secondprevioussum = fs_q_pfc_step_unique_secondprevioussum_body_steps_partial * S ((S (fs_i_pfc_step_unique_secondprevioussum_body_steps)) * fs_v_pfc_step_unique_secondprevioussum) + (fs_r_pfc_step_unique_secondprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_step_unique_secondprevioussum_body_steps_successor. fs_h_pfc_step_unique_secondprevioussum_body_steps_successor + S (fs_s_pfc_step_unique_secondprevioussum_body_steps) = S ((S (S fs_i_pfc_step_unique_secondprevioussum_body_steps)) * fs_v_pfc_step_unique_secondprevioussum)) /\\ exists fs_q_pfc_step_unique_secondprevioussum_body_steps_successor. fs_u_pfc_step_unique_secondprevioussum = fs_q_pfc_step_unique_secondprevioussum_body_steps_successor * S ((S (S fs_i_pfc_step_unique_secondprevioussum_body_steps)) * fs_v_pfc_step_unique_secondprevioussum) + (fs_s_pfc_step_unique_secondprevioussum_body_steps))) /\\ fs_s_pfc_step_unique_secondprevioussum_body_steps = fs_r_pfc_step_unique_secondprevioussum_body_steps + fs_a_pfc_step_unique_secondprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_step_unique_secondpreviousresiduebound. pfa_gap_step_unique_secondpreviousresiduebound + S (pfd_previous_step_unique_second) = (p)) /\\ ((exists pfa_offset_left_step_unique_secondpreviousresiduecongruence pfa_offset_right_step_unique_secondpreviousresiduecongruence. (pfc_natural_sum_step_unique_secondprevious) + (p) * pfa_offset_left_step_unique_secondpreviousresiduecongruence = (pfd_previous_step_unique_second) + (p) * pfa_offset_right_step_unique_secondpreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_step_unique_secondsubtractleft. pfa_gap_step_unique_secondsubtractleft + S (pfd_previous_step_unique_second) = (p)) /\\ (((exists pfa_gap_step_unique_secondsubtractright. pfa_gap_step_unique_secondsubtractright + S (pfd_difference_step_unique_second) = (p)) /\\ ((((exists pfa_gap_step_unique_secondsubtractresultbound. pfa_gap_step_unique_secondsubtractresultbound + S (pfd_input_step_unique_second) = (p)) /\\ ((exists pfa_offset_left_step_unique_secondsubtractresultcongruence pfa_offset_right_step_unique_secondsubtractresultcongruence. 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prime_field_polynomial_quotient_step_recode (p)", "specialize prime_field_polynomial_quotient_step_recode (k)", "specialize prime_field_polynomial_quotient_step_recode (ab)", "specialize prime_field_polynomial_quotient_step_recode (ac)", "specialize prime_field_polynomial_quotient_step_recode (bb)", "specialize prime_field_polynomial_quotient_step_recode (bc)", "specialize prime_field_polynomial_quotient_step_recode (M)", "specialize prime_field_polynomial_quotient_step_recode (qb)", "specialize prime_field_polynomial_quotient_step_recode (qc)", "specialize prime_field_polynomial_quotient_step_recode (QB)", "specialize prime_field_polynomial_quotient_step_recode (QC)", "specialize prime_field_polynomial_quotient_step_recode (i)", "specialize prime_field_polynomial_quotient_step_recode (q)", "apply prime_field_polynomial_quotient_step_recode", "exact hequal", "exact hfirst", "exact hsecond" ], "script_sha256": "a10a7a9853ee5eb2937aca9819c1b9c7359467e61b3a73ad7a4fc25be2cbd952", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_division_uniqueness_candidate.py", "sha256": "6a9d9ebe1f72202743e5df2c069b9aa367fdb3d61108f1d9354cdc9276ab2d15" }, "statement": "forall p k ab ac bb bc M qb qc QB QC i q r. (forall mdr_i_pfp_step_unique_prefix mdr_a_pfp_step_unique_prefix. (exists mdr_gap_pfp_step_unique_prefixb. mdr_gap_pfp_step_unique_prefixb + S (mdr_i_pfp_step_unique_prefix) = (i)) -> (((exists ff_h_mdr_pfp_step_unique_prefixo. ff_h_mdr_pfp_step_unique_prefixo + S (mdr_a_pfp_step_unique_prefix) = S ((S (mdr_i_pfp_step_unique_prefix)) * qc)) /\\ exists ff_q_mdr_pfp_step_unique_prefixo. qb = ff_q_mdr_pfp_step_unique_prefixo * S ((S (mdr_i_pfp_step_unique_prefix)) * qc) + (mdr_a_pfp_step_unique_prefix))) -> (((exists ff_h_mdr_pfp_step_unique_prefixn. ff_h_mdr_pfp_step_unique_prefixn + S (mdr_a_pfp_step_unique_prefix) = S ((S (mdr_i_pfp_step_unique_prefix)) * QC)) /\\ exists ff_q_mdr_pfp_step_unique_prefixn. QB = ff_q_mdr_pfp_step_unique_prefixn * S ((S (mdr_i_pfp_step_unique_prefix)) * QC) + (mdr_a_pfp_step_unique_prefix)))) -> (exists pfd_input_step_unique_old pfd_previous_step_unique_old pfd_difference_step_unique_old. ((((exists ff_h_pfp_step_unique_oldinput. ff_h_pfp_step_unique_oldinput + S (pfd_input_step_unique_old) = S ((S (i)) * ac)) /\\ exists ff_q_pfp_step_unique_oldinput. ab = ff_q_pfp_step_unique_oldinput * S ((S (i)) * ac) + (pfd_input_step_unique_old))) /\\ (((exists pfc_terms_code_step_unique_oldprevious pfc_terms_scale_step_unique_oldprevious pfc_natural_sum_step_unique_oldprevious. ((forall pfc_index_step_unique_oldpreviousdiagonal. (exists pfa_gap_step_unique_oldpreviousdiagonalbound. pfa_gap_step_unique_oldpreviousdiagonalbound + S (pfc_index_step_unique_oldpreviousdiagonal) = (S (i))) -> exists pfc_value_step_unique_oldpreviousdiagonal. ((((exists ff_h_pfp_step_unique_oldpreviousdiagonalentry. ff_h_pfp_step_unique_oldpreviousdiagonalentry + S (pfc_value_step_unique_oldpreviousdiagonal) = S ((S (pfc_index_step_unique_oldpreviousdiagonal)) * pfc_terms_scale_step_unique_oldprevious)) /\\ exists ff_q_pfp_step_unique_oldpreviousdiagonalentry. pfc_terms_code_step_unique_oldprevious = ff_q_pfp_step_unique_oldpreviousdiagonalentry * S ((S (pfc_index_step_unique_oldpreviousdiagonal)) * pfc_terms_scale_step_unique_oldprevious) + (pfc_value_step_unique_oldpreviousdiagonal))) /\\ ((exists pfc_complement_step_unique_oldpreviousdiagonalterm pfc_left_step_unique_oldpreviousdiagonalterm pfc_right_step_unique_oldpreviousdiagonalterm. (((pfc_index_step_unique_oldpreviousdiagonal)+pfc_complement_step_unique_oldpreviousdiagonalterm=(i)) /\\ ((((((exists pfa_gap_step_unique_oldpreviousdiagonaltermleftinside. pfa_gap_step_unique_oldpreviousdiagonaltermleftinside + S (pfc_index_step_unique_oldpreviousdiagonal) = (i)) /\\ ((((exists ff_h_pfp_step_unique_oldpreviousdiagonaltermleftentry. ff_h_pfp_step_unique_oldpreviousdiagonaltermleftentry + S (pfc_left_step_unique_oldpreviousdiagonalterm) = S ((S (pfc_index_step_unique_oldpreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_step_unique_oldpreviousdiagonaltermleftentry. qb = ff_q_pfp_step_unique_oldpreviousdiagonaltermleftentry * S ((S (pfc_index_step_unique_oldpreviousdiagonal)) * qc) + (pfc_left_step_unique_oldpreviousdiagonalterm)))))) \\/ (((exists pfc_gap_step_unique_oldpreviousdiagonaltermleftoutside. pfc_gap_step_unique_oldpreviousdiagonaltermleftoutside+(i)=(pfc_index_step_unique_oldpreviousdiagonal)) /\\ (((pfc_left_step_unique_oldpreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_step_unique_oldpreviousdiagonaltermrightinside. pfa_gap_step_unique_oldpreviousdiagonaltermrightinside + S (pfc_complement_step_unique_oldpreviousdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_step_unique_oldpreviousdiagonaltermrightentry. ff_h_pfp_step_unique_oldpreviousdiagonaltermrightentry + S (pfc_right_step_unique_oldpreviousdiagonalterm) = S ((S (pfc_complement_step_unique_oldpreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_step_unique_oldpreviousdiagonaltermrightentry. bb = ff_q_pfp_step_unique_oldpreviousdiagonaltermrightentry * S ((S (pfc_complement_step_unique_oldpreviousdiagonalterm)) * bc) + (pfc_right_step_unique_oldpreviousdiagonalterm)))))) \\/ (((exists pfc_gap_step_unique_oldpreviousdiagonaltermrightoutside. pfc_gap_step_unique_oldpreviousdiagonaltermrightoutside+(M)=(pfc_complement_step_unique_oldpreviousdiagonalterm)) /\\ (((pfc_right_step_unique_oldpreviousdiagonalterm)=0))))) /\\ (((pfc_value_step_unique_oldpreviousdiagonal)=pfc_left_step_unique_oldpreviousdiagonalterm*pfc_right_step_unique_oldpreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_step_unique_oldprevioussum fs_v_pfc_step_unique_oldprevioussum. ((((exists fs_h_pfc_step_unique_oldprevioussum_body_start. fs_h_pfc_step_unique_oldprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_step_unique_oldprevioussum)) /\\ exists fs_q_pfc_step_unique_oldprevioussum_body_start. fs_u_pfc_step_unique_oldprevioussum = fs_q_pfc_step_unique_oldprevioussum_body_start * S ((S (0)) * fs_v_pfc_step_unique_oldprevioussum) + (0))) /\\ ((((exists fs_h_pfc_step_unique_oldprevioussum_body_terminal. fs_h_pfc_step_unique_oldprevioussum_body_terminal + S (pfc_natural_sum_step_unique_oldprevious) = S ((S (S (i))) * fs_v_pfc_step_unique_oldprevioussum)) /\\ exists fs_q_pfc_step_unique_oldprevioussum_body_terminal. fs_u_pfc_step_unique_oldprevioussum = fs_q_pfc_step_unique_oldprevioussum_body_terminal * S ((S (S (i))) * fs_v_pfc_step_unique_oldprevioussum) + (pfc_natural_sum_step_unique_oldprevious))) /\\ forall fs_i_pfc_step_unique_oldprevioussum_body_steps. (exists fs_lt_pfc_step_unique_oldprevioussum_body_steps_bound. fs_lt_pfc_step_unique_oldprevioussum_body_steps_bound + S fs_i_pfc_step_unique_oldprevioussum_body_steps = S (i)) -> exists fs_a_pfc_step_unique_oldprevioussum_body_steps fs_r_pfc_step_unique_oldprevioussum_body_steps fs_s_pfc_step_unique_oldprevioussum_body_steps. ((((exists fs_h_pfc_step_unique_oldprevioussum_body_steps_summand. fs_h_pfc_step_unique_oldprevioussum_body_steps_summand + S (fs_a_pfc_step_unique_oldprevioussum_body_steps) = S ((S (fs_i_pfc_step_unique_oldprevioussum_body_steps)) * pfc_terms_scale_step_unique_oldprevious)) /\\ exists fs_q_pfc_step_unique_oldprevioussum_body_steps_summand. pfc_terms_code_step_unique_oldprevious = fs_q_pfc_step_unique_oldprevioussum_body_steps_summand * S ((S (fs_i_pfc_step_unique_oldprevioussum_body_steps)) * pfc_terms_scale_step_unique_oldprevious) + (fs_a_pfc_step_unique_oldprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_step_unique_oldprevioussum_body_steps_partial. fs_h_pfc_step_unique_oldprevioussum_body_steps_partial + S (fs_r_pfc_step_unique_oldprevioussum_body_steps) = S ((S (fs_i_pfc_step_unique_oldprevioussum_body_steps)) * fs_v_pfc_step_unique_oldprevioussum)) /\\ exists fs_q_pfc_step_unique_oldprevioussum_body_steps_partial. fs_u_pfc_step_unique_oldprevioussum = fs_q_pfc_step_unique_oldprevioussum_body_steps_partial * S ((S (fs_i_pfc_step_unique_oldprevioussum_body_steps)) * fs_v_pfc_step_unique_oldprevioussum) + (fs_r_pfc_step_unique_oldprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_step_unique_oldprevioussum_body_steps_successor. fs_h_pfc_step_unique_oldprevioussum_body_steps_successor + S (fs_s_pfc_step_unique_oldprevioussum_body_steps) = S ((S (S fs_i_pfc_step_unique_oldprevioussum_body_steps)) * fs_v_pfc_step_unique_oldprevioussum)) /\\ exists fs_q_pfc_step_unique_oldprevioussum_body_steps_successor. fs_u_pfc_step_unique_oldprevioussum = fs_q_pfc_step_unique_oldprevioussum_body_steps_successor * S ((S (S fs_i_pfc_step_unique_oldprevioussum_body_steps)) * fs_v_pfc_step_unique_oldprevioussum) + (fs_s_pfc_step_unique_oldprevioussum_body_steps))) /\\ fs_s_pfc_step_unique_oldprevioussum_body_steps = fs_r_pfc_step_unique_oldprevioussum_body_steps + fs_a_pfc_step_unique_oldprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_step_unique_oldpreviousresiduebound. pfa_gap_step_unique_oldpreviousresiduebound + S (pfd_previous_step_unique_old) = (p)) /\\ ((exists pfa_offset_left_step_unique_oldpreviousresiduecongruence pfa_offset_right_step_unique_oldpreviousresiduecongruence. (pfc_natural_sum_step_unique_oldprevious) + (p) * pfa_offset_left_step_unique_oldpreviousresiduecongruence = (pfd_previous_step_unique_old) + (p) * pfa_offset_right_step_unique_oldpreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_step_unique_oldsubtractleft. pfa_gap_step_unique_oldsubtractleft + S (pfd_previous_step_unique_old) = (p)) /\\ (((exists pfa_gap_step_unique_oldsubtractright. pfa_gap_step_unique_oldsubtractright + S (pfd_difference_step_unique_old) = (p)) /\\ ((((exists pfa_gap_step_unique_oldsubtractresultbound. pfa_gap_step_unique_oldsubtractresultbound + S (pfd_input_step_unique_old) = (p)) /\\ ((exists pfa_offset_left_step_unique_oldsubtractresultcongruence pfa_offset_right_step_unique_oldsubtractresultcongruence. ((pfd_previous_step_unique_old) + (pfd_difference_step_unique_old)) + (p) * pfa_offset_left_step_unique_oldsubtractresultcongruence = (pfd_input_step_unique_old) + (p) * pfa_offset_right_step_unique_oldsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_step_unique_oldmultiplyleft. pfa_gap_step_unique_oldmultiplyleft + S (k) = (p)) /\\ (((exists pfa_gap_step_unique_oldmultiplyright. pfa_gap_step_unique_oldmultiplyright + S (pfd_difference_step_unique_old) = (p)) /\\ ((((exists pfa_gap_step_unique_oldmultiplyresultbound. pfa_gap_step_unique_oldmultiplyresultbound + S (q) = (p)) /\\ ((exists pfa_offset_left_step_unique_oldmultiplyresultcongruence pfa_offset_right_step_unique_oldmultiplyresultcongruence. ((k) * (pfd_difference_step_unique_old)) + (p) * pfa_offset_left_step_unique_oldmultiplyresultcongruence = (q) + (p) * pfa_offset_right_step_unique_oldmultiplyresultcongruence)))))))))))))))) -> (exists pfd_input_step_unique_new pfd_previous_step_unique_new pfd_difference_step_unique_new. ((((exists ff_h_pfp_step_unique_newinput. ff_h_pfp_step_unique_newinput + S (pfd_input_step_unique_new) = S ((S (i)) * ac)) /\\ exists ff_q_pfp_step_unique_newinput. ab = ff_q_pfp_step_unique_newinput * S ((S (i)) * ac) + (pfd_input_step_unique_new))) /\\ (((exists pfc_terms_code_step_unique_newprevious pfc_terms_scale_step_unique_newprevious pfc_natural_sum_step_unique_newprevious. ((forall pfc_index_step_unique_newpreviousdiagonal. (exists pfa_gap_step_unique_newpreviousdiagonalbound. pfa_gap_step_unique_newpreviousdiagonalbound + S (pfc_index_step_unique_newpreviousdiagonal) = (S (i))) -> exists pfc_value_step_unique_newpreviousdiagonal. ((((exists ff_h_pfp_step_unique_newpreviousdiagonalentry. ff_h_pfp_step_unique_newpreviousdiagonalentry + S (pfc_value_step_unique_newpreviousdiagonal) = S ((S (pfc_index_step_unique_newpreviousdiagonal)) * pfc_terms_scale_step_unique_newprevious)) /\\ exists ff_q_pfp_step_unique_newpreviousdiagonalentry. pfc_terms_code_step_unique_newprevious = ff_q_pfp_step_unique_newpreviousdiagonalentry * S ((S (pfc_index_step_unique_newpreviousdiagonal)) * pfc_terms_scale_step_unique_newprevious) + (pfc_value_step_unique_newpreviousdiagonal))) /\\ ((exists pfc_complement_step_unique_newpreviousdiagonalterm pfc_left_step_unique_newpreviousdiagonalterm pfc_right_step_unique_newpreviousdiagonalterm. (((pfc_index_step_unique_newpreviousdiagonal)+pfc_complement_step_unique_newpreviousdiagonalterm=(i)) /\\ ((((((exists pfa_gap_step_unique_newpreviousdiagonaltermleftinside. pfa_gap_step_unique_newpreviousdiagonaltermleftinside + S (pfc_index_step_unique_newpreviousdiagonal) = (i)) /\\ ((((exists ff_h_pfp_step_unique_newpreviousdiagonaltermleftentry. ff_h_pfp_step_unique_newpreviousdiagonaltermleftentry + S (pfc_left_step_unique_newpreviousdiagonalterm) = S ((S (pfc_index_step_unique_newpreviousdiagonal)) * QC)) /\\ exists ff_q_pfp_step_unique_newpreviousdiagonaltermleftentry. QB = ff_q_pfp_step_unique_newpreviousdiagonaltermleftentry * S ((S (pfc_index_step_unique_newpreviousdiagonal)) * QC) + (pfc_left_step_unique_newpreviousdiagonalterm)))))) \\/ (((exists pfc_gap_step_unique_newpreviousdiagonaltermleftoutside. pfc_gap_step_unique_newpreviousdiagonaltermleftoutside+(i)=(pfc_index_step_unique_newpreviousdiagonal)) /\\ (((pfc_left_step_unique_newpreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_step_unique_newpreviousdiagonaltermrightinside. pfa_gap_step_unique_newpreviousdiagonaltermrightinside + S (pfc_complement_step_unique_newpreviousdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_step_unique_newpreviousdiagonaltermrightentry. ff_h_pfp_step_unique_newpreviousdiagonaltermrightentry + S (pfc_right_step_unique_newpreviousdiagonalterm) = S ((S (pfc_complement_step_unique_newpreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_step_unique_newpreviousdiagonaltermrightentry. bb = ff_q_pfp_step_unique_newpreviousdiagonaltermrightentry * S ((S (pfc_complement_step_unique_newpreviousdiagonalterm)) * bc) + (pfc_right_step_unique_newpreviousdiagonalterm)))))) \\/ (((exists pfc_gap_step_unique_newpreviousdiagonaltermrightoutside. pfc_gap_step_unique_newpreviousdiagonaltermrightoutside+(M)=(pfc_complement_step_unique_newpreviousdiagonalterm)) /\\ (((pfc_right_step_unique_newpreviousdiagonalterm)=0))))) /\\ (((pfc_value_step_unique_newpreviousdiagonal)=pfc_left_step_unique_newpreviousdiagonalterm*pfc_right_step_unique_newpreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_step_unique_newprevioussum fs_v_pfc_step_unique_newprevioussum. ((((exists fs_h_pfc_step_unique_newprevioussum_body_start. fs_h_pfc_step_unique_newprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_step_unique_newprevioussum)) /\\ exists fs_q_pfc_step_unique_newprevioussum_body_start. fs_u_pfc_step_unique_newprevioussum = fs_q_pfc_step_unique_newprevioussum_body_start * S ((S (0)) * fs_v_pfc_step_unique_newprevioussum) + (0))) /\\ ((((exists fs_h_pfc_step_unique_newprevioussum_body_terminal. fs_h_pfc_step_unique_newprevioussum_body_terminal + S (pfc_natural_sum_step_unique_newprevious) = S ((S (S (i))) * fs_v_pfc_step_unique_newprevioussum)) /\\ exists fs_q_pfc_step_unique_newprevioussum_body_terminal. fs_u_pfc_step_unique_newprevioussum = fs_q_pfc_step_unique_newprevioussum_body_terminal * S ((S (S (i))) * fs_v_pfc_step_unique_newprevioussum) + (pfc_natural_sum_step_unique_newprevious))) /\\ forall fs_i_pfc_step_unique_newprevioussum_body_steps. (exists fs_lt_pfc_step_unique_newprevioussum_body_steps_bound. fs_lt_pfc_step_unique_newprevioussum_body_steps_bound + S fs_i_pfc_step_unique_newprevioussum_body_steps = S (i)) -> exists fs_a_pfc_step_unique_newprevioussum_body_steps fs_r_pfc_step_unique_newprevioussum_body_steps fs_s_pfc_step_unique_newprevioussum_body_steps. ((((exists fs_h_pfc_step_unique_newprevioussum_body_steps_summand. fs_h_pfc_step_unique_newprevioussum_body_steps_summand + S (fs_a_pfc_step_unique_newprevioussum_body_steps) = S ((S (fs_i_pfc_step_unique_newprevioussum_body_steps)) * pfc_terms_scale_step_unique_newprevious)) /\\ exists fs_q_pfc_step_unique_newprevioussum_body_steps_summand. pfc_terms_code_step_unique_newprevious = fs_q_pfc_step_unique_newprevioussum_body_steps_summand * S ((S (fs_i_pfc_step_unique_newprevioussum_body_steps)) * pfc_terms_scale_step_unique_newprevious) + (fs_a_pfc_step_unique_newprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_step_unique_newprevioussum_body_steps_partial. fs_h_pfc_step_unique_newprevioussum_body_steps_partial + S (fs_r_pfc_step_unique_newprevioussum_body_steps) = S ((S (fs_i_pfc_step_unique_newprevioussum_body_steps)) * fs_v_pfc_step_unique_newprevioussum)) /\\ exists fs_q_pfc_step_unique_newprevioussum_body_steps_partial. fs_u_pfc_step_unique_newprevioussum = fs_q_pfc_step_unique_newprevioussum_body_steps_partial * S ((S (fs_i_pfc_step_unique_newprevioussum_body_steps)) * fs_v_pfc_step_unique_newprevioussum) + (fs_r_pfc_step_unique_newprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_step_unique_newprevioussum_body_steps_successor. fs_h_pfc_step_unique_newprevioussum_body_steps_successor + S (fs_s_pfc_step_unique_newprevioussum_body_steps) = S ((S (S fs_i_pfc_step_unique_newprevioussum_body_steps)) * fs_v_pfc_step_unique_newprevioussum)) /\\ exists fs_q_pfc_step_unique_newprevioussum_body_steps_successor. fs_u_pfc_step_unique_newprevioussum = fs_q_pfc_step_unique_newprevioussum_body_steps_successor * S ((S (S fs_i_pfc_step_unique_newprevioussum_body_steps)) * fs_v_pfc_step_unique_newprevioussum) + (fs_s_pfc_step_unique_newprevioussum_body_steps))) /\\ fs_s_pfc_step_unique_newprevioussum_body_steps = fs_r_pfc_step_unique_newprevioussum_body_steps + fs_a_pfc_step_unique_newprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_step_unique_newpreviousresiduebound. pfa_gap_step_unique_newpreviousresiduebound + S (pfd_previous_step_unique_new) = (p)) /\\ ((exists pfa_offset_left_step_unique_newpreviousresiduecongruence pfa_offset_right_step_unique_newpreviousresiduecongruence. (pfc_natural_sum_step_unique_newprevious) + (p) * pfa_offset_left_step_unique_newpreviousresiduecongruence = (pfd_previous_step_unique_new) + (p) * pfa_offset_right_step_unique_newpreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_step_unique_newsubtractleft. pfa_gap_step_unique_newsubtractleft + S (pfd_previous_step_unique_new) = (p)) /\\ (((exists pfa_gap_step_unique_newsubtractright. pfa_gap_step_unique_newsubtractright + S (pfd_difference_step_unique_new) = (p)) /\\ ((((exists pfa_gap_step_unique_newsubtractresultbound. pfa_gap_step_unique_newsubtractresultbound + S (pfd_input_step_unique_new) = (p)) /\\ ((exists pfa_offset_left_step_unique_newsubtractresultcongruence pfa_offset_right_step_unique_newsubtractresultcongruence. ((pfd_previous_step_unique_new) + (pfd_difference_step_unique_new)) + (p) * pfa_offset_left_step_unique_newsubtractresultcongruence = (pfd_input_step_unique_new) + (p) * pfa_offset_right_step_unique_newsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_step_unique_newmultiplyleft. pfa_gap_step_unique_newmultiplyleft + S (k) = (p)) /\\ (((exists pfa_gap_step_unique_newmultiplyright. pfa_gap_step_unique_newmultiplyright + S (pfd_difference_step_unique_new) = (p)) /\\ ((((exists pfa_gap_step_unique_newmultiplyresultbound. pfa_gap_step_unique_newmultiplyresultbound + S (r) = (p)) /\\ ((exists pfa_offset_left_step_unique_newmultiplyresultcongruence pfa_offset_right_step_unique_newmultiplyresultcongruence. ((k) * (pfd_difference_step_unique_new)) + (p) * pfa_offset_left_step_unique_newmultiplyresultcongruence = (r) + (p) * pfa_offset_right_step_unique_newmultiplyresultcongruence)))))))))))))))) -> q=r", "statement_sha256": "58805d6e1d85049ca6200d8928f054296e8a019fba83884a36584136d2555517", "summary": "Two genuine steps with equal previously computed coefficients agree, even when their beta encodings and unused entries differ.", "summary_sha256": "362c570472b7a2eb2bc746807d16c256585209041efb694314ec34cf08174797" }, { "alpha_v33_frontier_enrollment": { "body_receipt_sha256": "4a7c6d18b0bf02d8ce7cd7f8bafe06c8e1b886376a22ae4f4232819d1143c410", "bundle_campaign": "polynomial-euclidean-division", "bundle_node_id": 338, "bundle_sha256": "6ae667d8518e4dbe722bb08ad1b08715a0d282c2893e533c8133d770fe861dcf", "campaign": "polynomial-euclidean-division", "first_enrolled_version": "v33", "parent_catalog_sha256": "41b9f387d88a5a4f0fe5ee2bd5578f37a27a4657b0a80f1a1a2cb5109f69a623", "rfc_sha256": 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"intro qb", "intro qc", "intro QB", "intro QC", "intro N", "induction N", "intro hfirst", "intro hsecond", "intro i", "intro a", "intro hindex", "intro hvalue", "exfalso", "specialize lt_not_le (i)", "specialize lt_not_le (0)", "apply lt_not_le", "exact hindex", "specialize zero_le (i)", "apply zero_le", "intro hfirst", "intro hsecond", "have hequal : forall mdr_i_pfp_prefix_unique_induction mdr_a_pfp_prefix_unique_induction. (exists mdr_gap_pfp_prefix_unique_inductionb. mdr_gap_pfp_prefix_unique_inductionb + S (mdr_i_pfp_prefix_unique_induction) = (N)) -> (((exists ff_h_mdr_pfp_prefix_unique_inductiono. ff_h_mdr_pfp_prefix_unique_inductiono + S (mdr_a_pfp_prefix_unique_induction) = S ((S (mdr_i_pfp_prefix_unique_induction)) * qc)) /\\ exists ff_q_mdr_pfp_prefix_unique_inductiono. qb = ff_q_mdr_pfp_prefix_unique_inductiono * S ((S (mdr_i_pfp_prefix_unique_induction)) * qc) + (mdr_a_pfp_prefix_unique_induction))) -> (((exists ff_h_mdr_pfp_prefix_unique_inductionn. ff_h_mdr_pfp_prefix_unique_inductionn + S (mdr_a_pfp_prefix_unique_induction) = S ((S (mdr_i_pfp_prefix_unique_induction)) * QC)) /\\ exists ff_q_mdr_pfp_prefix_unique_inductionn. QB = ff_q_mdr_pfp_prefix_unique_inductionn * S ((S (mdr_i_pfp_prefix_unique_induction)) * QC) + (mdr_a_pfp_prefix_unique_induction)))", "apply IH", "specialize prime_field_polynomial_quotient_prefix_restrict (p)", "specialize prime_field_polynomial_quotient_prefix_restrict (k)", "specialize prime_field_polynomial_quotient_prefix_restrict (ab)", "specialize prime_field_polynomial_quotient_prefix_restrict (ac)", "specialize prime_field_polynomial_quotient_prefix_restrict (bb)", "specialize prime_field_polynomial_quotient_prefix_restrict (bc)", "specialize prime_field_polynomial_quotient_prefix_restrict (M)", "specialize prime_field_polynomial_quotient_prefix_restrict (qb)", "specialize prime_field_polynomial_quotient_prefix_restrict (qc)", "specialize prime_field_polynomial_quotient_prefix_restrict (S N)", "specialize prime_field_polynomial_quotient_prefix_restrict (N)", "apply prime_field_polynomial_quotient_prefix_restrict", "specialize le_succ (N)", "specialize le_succ (N)", "apply le_succ", "specialize le_refl (N)", "apply le_refl", "exact hfirst", "specialize prime_field_polynomial_quotient_prefix_restrict (p)", "specialize prime_field_polynomial_quotient_prefix_restrict (k)", "specialize prime_field_polynomial_quotient_prefix_restrict (ab)", "specialize prime_field_polynomial_quotient_prefix_restrict (ac)", "specialize prime_field_polynomial_quotient_prefix_restrict (bb)", "specialize prime_field_polynomial_quotient_prefix_restrict (bc)", "specialize prime_field_polynomial_quotient_prefix_restrict (M)", "specialize prime_field_polynomial_quotient_prefix_restrict (QB)", "specialize prime_field_polynomial_quotient_prefix_restrict (QC)", "specialize prime_field_polynomial_quotient_prefix_restrict (S N)", "specialize prime_field_polynomial_quotient_prefix_restrict (N)", "apply prime_field_polynomial_quotient_prefix_restrict", "specialize le_succ (N)", "specialize le_succ (N)", "apply le_succ", "specialize le_refl (N)", "apply le_refl", "exact hsecond", "intro i", "intro a", "intro hindex", "intro hvalue", "have hcase : i=N \\/ (exists pfa_gap_prefix_unique_earlier. pfa_gap_prefix_unique_earlier + S (i) = (N))", "specialize finite_lt_succ_eq_or_lt (N)", "specialize finite_lt_succ_eq_or_lt (i)", "apply finite_lt_succ_eq_or_lt", "exact hindex", "cases hcase", "rewrite hcase_left at hvalue", "rewrite hcase_left at hvalue", "rewrite hcase_left", "rewrite hcase_left", "have hchosen : exists r. (((((exists ff_h_pfp_prefix_unique_other_value. ff_h_pfp_prefix_unique_other_value + S (r) = S ((S (N)) * QC)) /\\ exists ff_q_pfp_prefix_unique_other_value. QB = ff_q_pfp_prefix_unique_other_value * S ((S (N)) * QC) + (r))) /\\ ((exists pfd_input_prefix_unique_other_step pfd_previous_prefix_unique_other_step pfd_difference_prefix_unique_other_step. ((((exists ff_h_pfp_prefix_unique_other_stepinput. ff_h_pfp_prefix_unique_other_stepinput + S (pfd_input_prefix_unique_other_step) = S ((S (N)) * ac)) /\\ exists ff_q_pfp_prefix_unique_other_stepinput. ab = ff_q_pfp_prefix_unique_other_stepinput * S ((S (N)) * ac) + (pfd_input_prefix_unique_other_step))) /\\ (((exists pfc_terms_code_prefix_unique_other_stepprevious pfc_terms_scale_prefix_unique_other_stepprevious pfc_natural_sum_prefix_unique_other_stepprevious. ((forall pfc_index_prefix_unique_other_steppreviousdiagonal. (exists pfa_gap_prefix_unique_other_steppreviousdiagonalbound. pfa_gap_prefix_unique_other_steppreviousdiagonalbound + S (pfc_index_prefix_unique_other_steppreviousdiagonal) = (S (N))) -> exists pfc_value_prefix_unique_other_steppreviousdiagonal. ((((exists ff_h_pfp_prefix_unique_other_steppreviousdiagonalentry. ff_h_pfp_prefix_unique_other_steppreviousdiagonalentry + S (pfc_value_prefix_unique_other_steppreviousdiagonal) = S ((S (pfc_index_prefix_unique_other_steppreviousdiagonal)) * pfc_terms_scale_prefix_unique_other_stepprevious)) /\\ exists ff_q_pfp_prefix_unique_other_steppreviousdiagonalentry. pfc_terms_code_prefix_unique_other_stepprevious = ff_q_pfp_prefix_unique_other_steppreviousdiagonalentry * S ((S (pfc_index_prefix_unique_other_steppreviousdiagonal)) * pfc_terms_scale_prefix_unique_other_stepprevious) + (pfc_value_prefix_unique_other_steppreviousdiagonal))) /\\ ((exists pfc_complement_prefix_unique_other_steppreviousdiagonalterm pfc_left_prefix_unique_other_steppreviousdiagonalterm pfc_right_prefix_unique_other_steppreviousdiagonalterm. (((pfc_index_prefix_unique_other_steppreviousdiagonal)+pfc_complement_prefix_unique_other_steppreviousdiagonalterm=(N)) /\\ ((((((exists pfa_gap_prefix_unique_other_steppreviousdiagonaltermleftinside. pfa_gap_prefix_unique_other_steppreviousdiagonaltermleftinside + S (pfc_index_prefix_unique_other_steppreviousdiagonal) = (N)) /\\ ((((exists ff_h_pfp_prefix_unique_other_steppreviousdiagonaltermleftentry. ff_h_pfp_prefix_unique_other_steppreviousdiagonaltermleftentry + S (pfc_left_prefix_unique_other_steppreviousdiagonalterm) = S ((S (pfc_index_prefix_unique_other_steppreviousdiagonal)) * QC)) /\\ exists ff_q_pfp_prefix_unique_other_steppreviousdiagonaltermleftentry. QB = ff_q_pfp_prefix_unique_other_steppreviousdiagonaltermleftentry * S ((S (pfc_index_prefix_unique_other_steppreviousdiagonal)) * QC) + (pfc_left_prefix_unique_other_steppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_unique_other_steppreviousdiagonaltermleftoutside. pfc_gap_prefix_unique_other_steppreviousdiagonaltermleftoutside+(N)=(pfc_index_prefix_unique_other_steppreviousdiagonal)) /\\ (((pfc_left_prefix_unique_other_steppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_prefix_unique_other_steppreviousdiagonaltermrightinside. pfa_gap_prefix_unique_other_steppreviousdiagonaltermrightinside + S (pfc_complement_prefix_unique_other_steppreviousdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_prefix_unique_other_steppreviousdiagonaltermrightentry. ff_h_pfp_prefix_unique_other_steppreviousdiagonaltermrightentry + S (pfc_right_prefix_unique_other_steppreviousdiagonalterm) = S ((S (pfc_complement_prefix_unique_other_steppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_prefix_unique_other_steppreviousdiagonaltermrightentry. bb = ff_q_pfp_prefix_unique_other_steppreviousdiagonaltermrightentry * S ((S (pfc_complement_prefix_unique_other_steppreviousdiagonalterm)) * bc) + (pfc_right_prefix_unique_other_steppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_unique_other_steppreviousdiagonaltermrightoutside. pfc_gap_prefix_unique_other_steppreviousdiagonaltermrightoutside+(M)=(pfc_complement_prefix_unique_other_steppreviousdiagonalterm)) /\\ (((pfc_right_prefix_unique_other_steppreviousdiagonalterm)=0))))) /\\ (((pfc_value_prefix_unique_other_steppreviousdiagonal)=pfc_left_prefix_unique_other_steppreviousdiagonalterm*pfc_right_prefix_unique_other_steppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_prefix_unique_other_stepprevioussum fs_v_pfc_prefix_unique_other_stepprevioussum. ((((exists fs_h_pfc_prefix_unique_other_stepprevioussum_body_start. fs_h_pfc_prefix_unique_other_stepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_prefix_unique_other_stepprevioussum)) /\\ exists fs_q_pfc_prefix_unique_other_stepprevioussum_body_start. fs_u_pfc_prefix_unique_other_stepprevioussum = fs_q_pfc_prefix_unique_other_stepprevioussum_body_start * S ((S (0)) * fs_v_pfc_prefix_unique_other_stepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_prefix_unique_other_stepprevioussum_body_terminal. fs_h_pfc_prefix_unique_other_stepprevioussum_body_terminal + S (pfc_natural_sum_prefix_unique_other_stepprevious) = S ((S (S (N))) * fs_v_pfc_prefix_unique_other_stepprevioussum)) /\\ exists fs_q_pfc_prefix_unique_other_stepprevioussum_body_terminal. fs_u_pfc_prefix_unique_other_stepprevioussum = fs_q_pfc_prefix_unique_other_stepprevioussum_body_terminal * S ((S (S (N))) * fs_v_pfc_prefix_unique_other_stepprevioussum) + (pfc_natural_sum_prefix_unique_other_stepprevious))) /\\ forall fs_i_pfc_prefix_unique_other_stepprevioussum_body_steps. (exists fs_lt_pfc_prefix_unique_other_stepprevioussum_body_steps_bound. fs_lt_pfc_prefix_unique_other_stepprevioussum_body_steps_bound + S fs_i_pfc_prefix_unique_other_stepprevioussum_body_steps = S (N)) -> exists fs_a_pfc_prefix_unique_other_stepprevioussum_body_steps fs_r_pfc_prefix_unique_other_stepprevioussum_body_steps fs_s_pfc_prefix_unique_other_stepprevioussum_body_steps. ((((exists fs_h_pfc_prefix_unique_other_stepprevioussum_body_steps_summand. fs_h_pfc_prefix_unique_other_stepprevioussum_body_steps_summand + S (fs_a_pfc_prefix_unique_other_stepprevioussum_body_steps) = S ((S (fs_i_pfc_prefix_unique_other_stepprevioussum_body_steps)) * pfc_terms_scale_prefix_unique_other_stepprevious)) /\\ exists fs_q_pfc_prefix_unique_other_stepprevioussum_body_steps_summand. pfc_terms_code_prefix_unique_other_stepprevious = fs_q_pfc_prefix_unique_other_stepprevioussum_body_steps_summand * S ((S (fs_i_pfc_prefix_unique_other_stepprevioussum_body_steps)) * pfc_terms_scale_prefix_unique_other_stepprevious) + (fs_a_pfc_prefix_unique_other_stepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_unique_other_stepprevioussum_body_steps_partial. fs_h_pfc_prefix_unique_other_stepprevioussum_body_steps_partial + S (fs_r_pfc_prefix_unique_other_stepprevioussum_body_steps) = S ((S (fs_i_pfc_prefix_unique_other_stepprevioussum_body_steps)) * fs_v_pfc_prefix_unique_other_stepprevioussum)) /\\ exists fs_q_pfc_prefix_unique_other_stepprevioussum_body_steps_partial. fs_u_pfc_prefix_unique_other_stepprevioussum = fs_q_pfc_prefix_unique_other_stepprevioussum_body_steps_partial * S ((S (fs_i_pfc_prefix_unique_other_stepprevioussum_body_steps)) * fs_v_pfc_prefix_unique_other_stepprevioussum) + (fs_r_pfc_prefix_unique_other_stepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_unique_other_stepprevioussum_body_steps_successor. fs_h_pfc_prefix_unique_other_stepprevioussum_body_steps_successor + S (fs_s_pfc_prefix_unique_other_stepprevioussum_body_steps) = S ((S (S fs_i_pfc_prefix_unique_other_stepprevioussum_body_steps)) * fs_v_pfc_prefix_unique_other_stepprevioussum)) /\\ exists fs_q_pfc_prefix_unique_other_stepprevioussum_body_steps_successor. fs_u_pfc_prefix_unique_other_stepprevioussum = fs_q_pfc_prefix_unique_other_stepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_prefix_unique_other_stepprevioussum_body_steps)) * fs_v_pfc_prefix_unique_other_stepprevioussum) + (fs_s_pfc_prefix_unique_other_stepprevioussum_body_steps))) /\\ fs_s_pfc_prefix_unique_other_stepprevioussum_body_steps = fs_r_pfc_prefix_unique_other_stepprevioussum_body_steps + fs_a_pfc_prefix_unique_other_stepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_prefix_unique_other_steppreviousresiduebound. pfa_gap_prefix_unique_other_steppreviousresiduebound + S (pfd_previous_prefix_unique_other_step) = (p)) /\\ ((exists pfa_offset_left_prefix_unique_other_steppreviousresiduecongruence pfa_offset_right_prefix_unique_other_steppreviousresiduecongruence. (pfc_natural_sum_prefix_unique_other_stepprevious) + (p) * pfa_offset_left_prefix_unique_other_steppreviousresiduecongruence = (pfd_previous_prefix_unique_other_step) + (p) * pfa_offset_right_prefix_unique_other_steppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_prefix_unique_other_stepsubtractleft. pfa_gap_prefix_unique_other_stepsubtractleft + S (pfd_previous_prefix_unique_other_step) = (p)) /\\ (((exists pfa_gap_prefix_unique_other_stepsubtractright. pfa_gap_prefix_unique_other_stepsubtractright + S (pfd_difference_prefix_unique_other_step) = (p)) /\\ ((((exists pfa_gap_prefix_unique_other_stepsubtractresultbound. pfa_gap_prefix_unique_other_stepsubtractresultbound + S (pfd_input_prefix_unique_other_step) = (p)) /\\ ((exists pfa_offset_left_prefix_unique_other_stepsubtractresultcongruence pfa_offset_right_prefix_unique_other_stepsubtractresultcongruence. ((pfd_previous_prefix_unique_other_step) + (pfd_difference_prefix_unique_other_step)) + (p) * pfa_offset_left_prefix_unique_other_stepsubtractresultcongruence = (pfd_input_prefix_unique_other_step) + (p) * pfa_offset_right_prefix_unique_other_stepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_prefix_unique_other_stepmultiplyleft. pfa_gap_prefix_unique_other_stepmultiplyleft + S (k) = (p)) /\\ (((exists pfa_gap_prefix_unique_other_stepmultiplyright. pfa_gap_prefix_unique_other_stepmultiplyright + S (pfd_difference_prefix_unique_other_step) = (p)) /\\ ((((exists pfa_gap_prefix_unique_other_stepmultiplyresultbound. pfa_gap_prefix_unique_other_stepmultiplyresultbound + S (r) = (p)) /\\ ((exists pfa_offset_left_prefix_unique_other_stepmultiplyresultcongruence pfa_offset_right_prefix_unique_other_stepmultiplyresultcongruence. ((k) * (pfd_difference_prefix_unique_other_step)) + (p) * pfa_offset_left_prefix_unique_other_stepmultiplyresultcongruence = (r) + (p) * pfa_offset_right_prefix_unique_other_stepmultiplyresultcongruence)))))))))))))))))))", "specialize hsecond (N)", "apply hsecond", "specialize le_refl (S N)", "apply le_refl", "cases hchosen", "cases hchosen_witness", "have hlast : a=x", "specialize prime_field_polynomial_quotient_step_prefix_functional (p)", "specialize prime_field_polynomial_quotient_step_prefix_functional (k)", "specialize prime_field_polynomial_quotient_step_prefix_functional (ab)", "specialize prime_field_polynomial_quotient_step_prefix_functional (ac)", "specialize prime_field_polynomial_quotient_step_prefix_functional (bb)", "specialize prime_field_polynomial_quotient_step_prefix_functional (bc)", "specialize prime_field_polynomial_quotient_step_prefix_functional (M)", "specialize prime_field_polynomial_quotient_step_prefix_functional (qb)", "specialize prime_field_polynomial_quotient_step_prefix_functional (qc)", "specialize prime_field_polynomial_quotient_step_prefix_functional (QB)", "specialize prime_field_polynomial_quotient_step_prefix_functional (QC)", "specialize prime_field_polynomial_quotient_step_prefix_functional (N)", "specialize prime_field_polynomial_quotient_step_prefix_functional (a)", "specialize prime_field_polynomial_quotient_step_prefix_functional (x)", "apply prime_field_polynomial_quotient_step_prefix_functional", "exact hequal", "specialize prime_field_polynomial_quotient_prefix_entry (p)", "specialize prime_field_polynomial_quotient_prefix_entry (k)", "specialize prime_field_polynomial_quotient_prefix_entry (ab)", "specialize prime_field_polynomial_quotient_prefix_entry (ac)", "specialize prime_field_polynomial_quotient_prefix_entry (bb)", "specialize prime_field_polynomial_quotient_prefix_entry (bc)", "specialize prime_field_polynomial_quotient_prefix_entry (M)", "specialize prime_field_polynomial_quotient_prefix_entry (qb)", "specialize prime_field_polynomial_quotient_prefix_entry (qc)", "specialize prime_field_polynomial_quotient_prefix_entry (S N)", "specialize prime_field_polynomial_quotient_prefix_entry (N)", "specialize prime_field_polynomial_quotient_prefix_entry (a)", "apply prime_field_polynomial_quotient_prefix_entry", "exact hfirst", "specialize le_refl (S N)", "apply le_refl", "exact hvalue", "exact hchosen_witness_right", "rewrite hlast", "rewrite hlast", "exact hchosen_witness_left", "specialize hequal (i)", "specialize hequal (a)", "apply hequal", "exact hcase_right", "exact hvalue" ], "script_sha256": "408834edf538a17d317056f23907faa36dd6c55ad6fb1ee8df4b98c54525e802", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_division_uniqueness_candidate.py", "sha256": "6a9d9ebe1f72202743e5df2c069b9aa367fdb3d61108f1d9354cdc9276ab2d15" }, "statement": "forall p k ab ac bb bc M qb qc QB QC N. (forall pfd_index_prefix_unique_first. (exists pfa_gap_prefix_unique_firstbound. pfa_gap_prefix_unique_firstbound + S (pfd_index_prefix_unique_first) = (N)) -> exists pfd_value_prefix_unique_first. ((((exists ff_h_pfp_prefix_unique_firstentry. ff_h_pfp_prefix_unique_firstentry + S (pfd_value_prefix_unique_first) = S ((S (pfd_index_prefix_unique_first)) * qc)) /\\ exists ff_q_pfp_prefix_unique_firstentry. qb = ff_q_pfp_prefix_unique_firstentry * S ((S (pfd_index_prefix_unique_first)) * qc) + (pfd_value_prefix_unique_first))) /\\ ((exists pfd_input_prefix_unique_firststep pfd_previous_prefix_unique_firststep pfd_difference_prefix_unique_firststep. ((((exists ff_h_pfp_prefix_unique_firststepinput. ff_h_pfp_prefix_unique_firststepinput + S (pfd_input_prefix_unique_firststep) = S ((S (pfd_index_prefix_unique_first)) * ac)) /\\ exists ff_q_pfp_prefix_unique_firststepinput. ab = ff_q_pfp_prefix_unique_firststepinput * S ((S (pfd_index_prefix_unique_first)) * ac) + (pfd_input_prefix_unique_firststep))) /\\ (((exists pfc_terms_code_prefix_unique_firststepprevious pfc_terms_scale_prefix_unique_firststepprevious pfc_natural_sum_prefix_unique_firststepprevious. ((forall pfc_index_prefix_unique_firststeppreviousdiagonal. (exists pfa_gap_prefix_unique_firststeppreviousdiagonalbound. pfa_gap_prefix_unique_firststeppreviousdiagonalbound + S (pfc_index_prefix_unique_firststeppreviousdiagonal) = (S (pfd_index_prefix_unique_first))) -> exists pfc_value_prefix_unique_firststeppreviousdiagonal. ((((exists ff_h_pfp_prefix_unique_firststeppreviousdiagonalentry. ff_h_pfp_prefix_unique_firststeppreviousdiagonalentry + S (pfc_value_prefix_unique_firststeppreviousdiagonal) = S ((S (pfc_index_prefix_unique_firststeppreviousdiagonal)) * pfc_terms_scale_prefix_unique_firststepprevious)) /\\ exists ff_q_pfp_prefix_unique_firststeppreviousdiagonalentry. pfc_terms_code_prefix_unique_firststepprevious = ff_q_pfp_prefix_unique_firststeppreviousdiagonalentry * S ((S (pfc_index_prefix_unique_firststeppreviousdiagonal)) * pfc_terms_scale_prefix_unique_firststepprevious) + (pfc_value_prefix_unique_firststeppreviousdiagonal))) /\\ ((exists pfc_complement_prefix_unique_firststeppreviousdiagonalterm pfc_left_prefix_unique_firststeppreviousdiagonalterm pfc_right_prefix_unique_firststeppreviousdiagonalterm. (((pfc_index_prefix_unique_firststeppreviousdiagonal)+pfc_complement_prefix_unique_firststeppreviousdiagonalterm=(pfd_index_prefix_unique_first)) /\\ ((((((exists pfa_gap_prefix_unique_firststeppreviousdiagonaltermleftinside. pfa_gap_prefix_unique_firststeppreviousdiagonaltermleftinside + S (pfc_index_prefix_unique_firststeppreviousdiagonal) = (pfd_index_prefix_unique_first)) /\\ ((((exists ff_h_pfp_prefix_unique_firststeppreviousdiagonaltermleftentry. ff_h_pfp_prefix_unique_firststeppreviousdiagonaltermleftentry + S (pfc_left_prefix_unique_firststeppreviousdiagonalterm) = S ((S (pfc_index_prefix_unique_firststeppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_prefix_unique_firststeppreviousdiagonaltermleftentry. qb = ff_q_pfp_prefix_unique_firststeppreviousdiagonaltermleftentry * S ((S (pfc_index_prefix_unique_firststeppreviousdiagonal)) * qc) + (pfc_left_prefix_unique_firststeppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_unique_firststeppreviousdiagonaltermleftoutside. pfc_gap_prefix_unique_firststeppreviousdiagonaltermleftoutside+(pfd_index_prefix_unique_first)=(pfc_index_prefix_unique_firststeppreviousdiagonal)) /\\ (((pfc_left_prefix_unique_firststeppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_prefix_unique_firststeppreviousdiagonaltermrightinside. pfa_gap_prefix_unique_firststeppreviousdiagonaltermrightinside + S (pfc_complement_prefix_unique_firststeppreviousdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_prefix_unique_firststeppreviousdiagonaltermrightentry. ff_h_pfp_prefix_unique_firststeppreviousdiagonaltermrightentry + S (pfc_right_prefix_unique_firststeppreviousdiagonalterm) = S ((S (pfc_complement_prefix_unique_firststeppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_prefix_unique_firststeppreviousdiagonaltermrightentry. bb = ff_q_pfp_prefix_unique_firststeppreviousdiagonaltermrightentry * S ((S (pfc_complement_prefix_unique_firststeppreviousdiagonalterm)) * bc) + (pfc_right_prefix_unique_firststeppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_unique_firststeppreviousdiagonaltermrightoutside. pfc_gap_prefix_unique_firststeppreviousdiagonaltermrightoutside+(M)=(pfc_complement_prefix_unique_firststeppreviousdiagonalterm)) /\\ (((pfc_right_prefix_unique_firststeppreviousdiagonalterm)=0))))) /\\ (((pfc_value_prefix_unique_firststeppreviousdiagonal)=pfc_left_prefix_unique_firststeppreviousdiagonalterm*pfc_right_prefix_unique_firststeppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_prefix_unique_firststepprevioussum fs_v_pfc_prefix_unique_firststepprevioussum. ((((exists fs_h_pfc_prefix_unique_firststepprevioussum_body_start. fs_h_pfc_prefix_unique_firststepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_prefix_unique_firststepprevioussum)) /\\ exists fs_q_pfc_prefix_unique_firststepprevioussum_body_start. fs_u_pfc_prefix_unique_firststepprevioussum = fs_q_pfc_prefix_unique_firststepprevioussum_body_start * S ((S (0)) * fs_v_pfc_prefix_unique_firststepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_prefix_unique_firststepprevioussum_body_terminal. fs_h_pfc_prefix_unique_firststepprevioussum_body_terminal + S (pfc_natural_sum_prefix_unique_firststepprevious) = S ((S (S (pfd_index_prefix_unique_first))) * fs_v_pfc_prefix_unique_firststepprevioussum)) /\\ exists fs_q_pfc_prefix_unique_firststepprevioussum_body_terminal. fs_u_pfc_prefix_unique_firststepprevioussum = fs_q_pfc_prefix_unique_firststepprevioussum_body_terminal * S ((S (S (pfd_index_prefix_unique_first))) * fs_v_pfc_prefix_unique_firststepprevioussum) + (pfc_natural_sum_prefix_unique_firststepprevious))) /\\ forall fs_i_pfc_prefix_unique_firststepprevioussum_body_steps. (exists fs_lt_pfc_prefix_unique_firststepprevioussum_body_steps_bound. fs_lt_pfc_prefix_unique_firststepprevioussum_body_steps_bound + S fs_i_pfc_prefix_unique_firststepprevioussum_body_steps = S (pfd_index_prefix_unique_first)) -> exists fs_a_pfc_prefix_unique_firststepprevioussum_body_steps fs_r_pfc_prefix_unique_firststepprevioussum_body_steps fs_s_pfc_prefix_unique_firststepprevioussum_body_steps. ((((exists fs_h_pfc_prefix_unique_firststepprevioussum_body_steps_summand. fs_h_pfc_prefix_unique_firststepprevioussum_body_steps_summand + S (fs_a_pfc_prefix_unique_firststepprevioussum_body_steps) = S ((S (fs_i_pfc_prefix_unique_firststepprevioussum_body_steps)) * pfc_terms_scale_prefix_unique_firststepprevious)) /\\ exists fs_q_pfc_prefix_unique_firststepprevioussum_body_steps_summand. pfc_terms_code_prefix_unique_firststepprevious = fs_q_pfc_prefix_unique_firststepprevioussum_body_steps_summand * S ((S (fs_i_pfc_prefix_unique_firststepprevioussum_body_steps)) * pfc_terms_scale_prefix_unique_firststepprevious) + (fs_a_pfc_prefix_unique_firststepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_unique_firststepprevioussum_body_steps_partial. fs_h_pfc_prefix_unique_firststepprevioussum_body_steps_partial + S (fs_r_pfc_prefix_unique_firststepprevioussum_body_steps) = S ((S (fs_i_pfc_prefix_unique_firststepprevioussum_body_steps)) * fs_v_pfc_prefix_unique_firststepprevioussum)) /\\ exists fs_q_pfc_prefix_unique_firststepprevioussum_body_steps_partial. fs_u_pfc_prefix_unique_firststepprevioussum = fs_q_pfc_prefix_unique_firststepprevioussum_body_steps_partial * S ((S (fs_i_pfc_prefix_unique_firststepprevioussum_body_steps)) * fs_v_pfc_prefix_unique_firststepprevioussum) + (fs_r_pfc_prefix_unique_firststepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_unique_firststepprevioussum_body_steps_successor. fs_h_pfc_prefix_unique_firststepprevioussum_body_steps_successor + S (fs_s_pfc_prefix_unique_firststepprevioussum_body_steps) = S ((S (S fs_i_pfc_prefix_unique_firststepprevioussum_body_steps)) * fs_v_pfc_prefix_unique_firststepprevioussum)) /\\ exists fs_q_pfc_prefix_unique_firststepprevioussum_body_steps_successor. fs_u_pfc_prefix_unique_firststepprevioussum = fs_q_pfc_prefix_unique_firststepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_prefix_unique_firststepprevioussum_body_steps)) * fs_v_pfc_prefix_unique_firststepprevioussum) + (fs_s_pfc_prefix_unique_firststepprevioussum_body_steps))) /\\ fs_s_pfc_prefix_unique_firststepprevioussum_body_steps = fs_r_pfc_prefix_unique_firststepprevioussum_body_steps + fs_a_pfc_prefix_unique_firststepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_prefix_unique_firststeppreviousresiduebound. pfa_gap_prefix_unique_firststeppreviousresiduebound + S (pfd_previous_prefix_unique_firststep) = (p)) /\\ ((exists pfa_offset_left_prefix_unique_firststeppreviousresiduecongruence pfa_offset_right_prefix_unique_firststeppreviousresiduecongruence. (pfc_natural_sum_prefix_unique_firststepprevious) + (p) * pfa_offset_left_prefix_unique_firststeppreviousresiduecongruence = (pfd_previous_prefix_unique_firststep) + (p) * pfa_offset_right_prefix_unique_firststeppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_prefix_unique_firststepsubtractleft. pfa_gap_prefix_unique_firststepsubtractleft + S (pfd_previous_prefix_unique_firststep) = (p)) /\\ (((exists pfa_gap_prefix_unique_firststepsubtractright. pfa_gap_prefix_unique_firststepsubtractright + S (pfd_difference_prefix_unique_firststep) = (p)) /\\ ((((exists pfa_gap_prefix_unique_firststepsubtractresultbound. pfa_gap_prefix_unique_firststepsubtractresultbound + S (pfd_input_prefix_unique_firststep) = (p)) /\\ ((exists pfa_offset_left_prefix_unique_firststepsubtractresultcongruence pfa_offset_right_prefix_unique_firststepsubtractresultcongruence. ((pfd_previous_prefix_unique_firststep) + (pfd_difference_prefix_unique_firststep)) + (p) * pfa_offset_left_prefix_unique_firststepsubtractresultcongruence = (pfd_input_prefix_unique_firststep) + (p) * pfa_offset_right_prefix_unique_firststepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_prefix_unique_firststepmultiplyleft. pfa_gap_prefix_unique_firststepmultiplyleft + S (k) = (p)) /\\ (((exists pfa_gap_prefix_unique_firststepmultiplyright. pfa_gap_prefix_unique_firststepmultiplyright + S (pfd_difference_prefix_unique_firststep) = (p)) /\\ ((((exists pfa_gap_prefix_unique_firststepmultiplyresultbound. pfa_gap_prefix_unique_firststepmultiplyresultbound + S (pfd_value_prefix_unique_first) = (p)) /\\ ((exists pfa_offset_left_prefix_unique_firststepmultiplyresultcongruence pfa_offset_right_prefix_unique_firststepmultiplyresultcongruence. ((k) * (pfd_difference_prefix_unique_firststep)) + (p) * pfa_offset_left_prefix_unique_firststepmultiplyresultcongruence = (pfd_value_prefix_unique_first) + (p) * pfa_offset_right_prefix_unique_firststepmultiplyresultcongruence))))))))))))))))))) -> (forall pfd_index_prefix_unique_second. (exists pfa_gap_prefix_unique_secondbound. pfa_gap_prefix_unique_secondbound + S (pfd_index_prefix_unique_second) = (N)) -> exists pfd_value_prefix_unique_second. ((((exists ff_h_pfp_prefix_unique_secondentry. ff_h_pfp_prefix_unique_secondentry + S (pfd_value_prefix_unique_second) = S ((S (pfd_index_prefix_unique_second)) * QC)) /\\ exists ff_q_pfp_prefix_unique_secondentry. QB = ff_q_pfp_prefix_unique_secondentry * S ((S (pfd_index_prefix_unique_second)) * QC) + (pfd_value_prefix_unique_second))) /\\ ((exists pfd_input_prefix_unique_secondstep pfd_previous_prefix_unique_secondstep pfd_difference_prefix_unique_secondstep. ((((exists ff_h_pfp_prefix_unique_secondstepinput. ff_h_pfp_prefix_unique_secondstepinput + S (pfd_input_prefix_unique_secondstep) = S ((S (pfd_index_prefix_unique_second)) * ac)) /\\ exists ff_q_pfp_prefix_unique_secondstepinput. ab = ff_q_pfp_prefix_unique_secondstepinput * S ((S (pfd_index_prefix_unique_second)) * ac) + (pfd_input_prefix_unique_secondstep))) /\\ (((exists pfc_terms_code_prefix_unique_secondstepprevious pfc_terms_scale_prefix_unique_secondstepprevious pfc_natural_sum_prefix_unique_secondstepprevious. ((forall pfc_index_prefix_unique_secondsteppreviousdiagonal. (exists pfa_gap_prefix_unique_secondsteppreviousdiagonalbound. pfa_gap_prefix_unique_secondsteppreviousdiagonalbound + S (pfc_index_prefix_unique_secondsteppreviousdiagonal) = (S (pfd_index_prefix_unique_second))) -> exists pfc_value_prefix_unique_secondsteppreviousdiagonal. ((((exists ff_h_pfp_prefix_unique_secondsteppreviousdiagonalentry. ff_h_pfp_prefix_unique_secondsteppreviousdiagonalentry + S (pfc_value_prefix_unique_secondsteppreviousdiagonal) = S ((S (pfc_index_prefix_unique_secondsteppreviousdiagonal)) * pfc_terms_scale_prefix_unique_secondstepprevious)) /\\ exists ff_q_pfp_prefix_unique_secondsteppreviousdiagonalentry. pfc_terms_code_prefix_unique_secondstepprevious = ff_q_pfp_prefix_unique_secondsteppreviousdiagonalentry * S ((S (pfc_index_prefix_unique_secondsteppreviousdiagonal)) * pfc_terms_scale_prefix_unique_secondstepprevious) + (pfc_value_prefix_unique_secondsteppreviousdiagonal))) /\\ ((exists pfc_complement_prefix_unique_secondsteppreviousdiagonalterm pfc_left_prefix_unique_secondsteppreviousdiagonalterm pfc_right_prefix_unique_secondsteppreviousdiagonalterm. (((pfc_index_prefix_unique_secondsteppreviousdiagonal)+pfc_complement_prefix_unique_secondsteppreviousdiagonalterm=(pfd_index_prefix_unique_second)) /\\ ((((((exists pfa_gap_prefix_unique_secondsteppreviousdiagonaltermleftinside. pfa_gap_prefix_unique_secondsteppreviousdiagonaltermleftinside + S (pfc_index_prefix_unique_secondsteppreviousdiagonal) = (pfd_index_prefix_unique_second)) /\\ ((((exists ff_h_pfp_prefix_unique_secondsteppreviousdiagonaltermleftentry. ff_h_pfp_prefix_unique_secondsteppreviousdiagonaltermleftentry + S (pfc_left_prefix_unique_secondsteppreviousdiagonalterm) = S ((S (pfc_index_prefix_unique_secondsteppreviousdiagonal)) * QC)) /\\ exists ff_q_pfp_prefix_unique_secondsteppreviousdiagonaltermleftentry. QB = ff_q_pfp_prefix_unique_secondsteppreviousdiagonaltermleftentry * S ((S (pfc_index_prefix_unique_secondsteppreviousdiagonal)) * QC) + (pfc_left_prefix_unique_secondsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_unique_secondsteppreviousdiagonaltermleftoutside. pfc_gap_prefix_unique_secondsteppreviousdiagonaltermleftoutside+(pfd_index_prefix_unique_second)=(pfc_index_prefix_unique_secondsteppreviousdiagonal)) /\\ (((pfc_left_prefix_unique_secondsteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_prefix_unique_secondsteppreviousdiagonaltermrightinside. pfa_gap_prefix_unique_secondsteppreviousdiagonaltermrightinside + S (pfc_complement_prefix_unique_secondsteppreviousdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_prefix_unique_secondsteppreviousdiagonaltermrightentry. ff_h_pfp_prefix_unique_secondsteppreviousdiagonaltermrightentry + S (pfc_right_prefix_unique_secondsteppreviousdiagonalterm) = S ((S (pfc_complement_prefix_unique_secondsteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_prefix_unique_secondsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_prefix_unique_secondsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_prefix_unique_secondsteppreviousdiagonalterm)) * bc) + (pfc_right_prefix_unique_secondsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_prefix_unique_secondsteppreviousdiagonaltermrightoutside. pfc_gap_prefix_unique_secondsteppreviousdiagonaltermrightoutside+(M)=(pfc_complement_prefix_unique_secondsteppreviousdiagonalterm)) /\\ (((pfc_right_prefix_unique_secondsteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_prefix_unique_secondsteppreviousdiagonal)=pfc_left_prefix_unique_secondsteppreviousdiagonalterm*pfc_right_prefix_unique_secondsteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_prefix_unique_secondstepprevioussum fs_v_pfc_prefix_unique_secondstepprevioussum. ((((exists fs_h_pfc_prefix_unique_secondstepprevioussum_body_start. fs_h_pfc_prefix_unique_secondstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_prefix_unique_secondstepprevioussum)) /\\ exists fs_q_pfc_prefix_unique_secondstepprevioussum_body_start. fs_u_pfc_prefix_unique_secondstepprevioussum = fs_q_pfc_prefix_unique_secondstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_prefix_unique_secondstepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_prefix_unique_secondstepprevioussum_body_terminal. fs_h_pfc_prefix_unique_secondstepprevioussum_body_terminal + S (pfc_natural_sum_prefix_unique_secondstepprevious) = S ((S (S (pfd_index_prefix_unique_second))) * fs_v_pfc_prefix_unique_secondstepprevioussum)) /\\ exists fs_q_pfc_prefix_unique_secondstepprevioussum_body_terminal. fs_u_pfc_prefix_unique_secondstepprevioussum = fs_q_pfc_prefix_unique_secondstepprevioussum_body_terminal * S ((S (S (pfd_index_prefix_unique_second))) * fs_v_pfc_prefix_unique_secondstepprevioussum) + (pfc_natural_sum_prefix_unique_secondstepprevious))) /\\ forall fs_i_pfc_prefix_unique_secondstepprevioussum_body_steps. (exists fs_lt_pfc_prefix_unique_secondstepprevioussum_body_steps_bound. fs_lt_pfc_prefix_unique_secondstepprevioussum_body_steps_bound + S fs_i_pfc_prefix_unique_secondstepprevioussum_body_steps = S (pfd_index_prefix_unique_second)) -> exists fs_a_pfc_prefix_unique_secondstepprevioussum_body_steps fs_r_pfc_prefix_unique_secondstepprevioussum_body_steps fs_s_pfc_prefix_unique_secondstepprevioussum_body_steps. ((((exists fs_h_pfc_prefix_unique_secondstepprevioussum_body_steps_summand. fs_h_pfc_prefix_unique_secondstepprevioussum_body_steps_summand + S (fs_a_pfc_prefix_unique_secondstepprevioussum_body_steps) = S ((S (fs_i_pfc_prefix_unique_secondstepprevioussum_body_steps)) * pfc_terms_scale_prefix_unique_secondstepprevious)) /\\ exists fs_q_pfc_prefix_unique_secondstepprevioussum_body_steps_summand. pfc_terms_code_prefix_unique_secondstepprevious = fs_q_pfc_prefix_unique_secondstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_prefix_unique_secondstepprevioussum_body_steps)) * pfc_terms_scale_prefix_unique_secondstepprevious) + (fs_a_pfc_prefix_unique_secondstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_unique_secondstepprevioussum_body_steps_partial. fs_h_pfc_prefix_unique_secondstepprevioussum_body_steps_partial + S (fs_r_pfc_prefix_unique_secondstepprevioussum_body_steps) = S ((S (fs_i_pfc_prefix_unique_secondstepprevioussum_body_steps)) * fs_v_pfc_prefix_unique_secondstepprevioussum)) /\\ exists fs_q_pfc_prefix_unique_secondstepprevioussum_body_steps_partial. fs_u_pfc_prefix_unique_secondstepprevioussum = fs_q_pfc_prefix_unique_secondstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_prefix_unique_secondstepprevioussum_body_steps)) * fs_v_pfc_prefix_unique_secondstepprevioussum) + (fs_r_pfc_prefix_unique_secondstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_prefix_unique_secondstepprevioussum_body_steps_successor. fs_h_pfc_prefix_unique_secondstepprevioussum_body_steps_successor + S (fs_s_pfc_prefix_unique_secondstepprevioussum_body_steps) = S ((S (S fs_i_pfc_prefix_unique_secondstepprevioussum_body_steps)) * fs_v_pfc_prefix_unique_secondstepprevioussum)) /\\ exists fs_q_pfc_prefix_unique_secondstepprevioussum_body_steps_successor. fs_u_pfc_prefix_unique_secondstepprevioussum = fs_q_pfc_prefix_unique_secondstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_prefix_unique_secondstepprevioussum_body_steps)) * fs_v_pfc_prefix_unique_secondstepprevioussum) + (fs_s_pfc_prefix_unique_secondstepprevioussum_body_steps))) /\\ fs_s_pfc_prefix_unique_secondstepprevioussum_body_steps = fs_r_pfc_prefix_unique_secondstepprevioussum_body_steps + fs_a_pfc_prefix_unique_secondstepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_prefix_unique_secondsteppreviousresiduebound. pfa_gap_prefix_unique_secondsteppreviousresiduebound + S (pfd_previous_prefix_unique_secondstep) = (p)) /\\ ((exists pfa_offset_left_prefix_unique_secondsteppreviousresiduecongruence pfa_offset_right_prefix_unique_secondsteppreviousresiduecongruence. (pfc_natural_sum_prefix_unique_secondstepprevious) + (p) * pfa_offset_left_prefix_unique_secondsteppreviousresiduecongruence = (pfd_previous_prefix_unique_secondstep) + (p) * pfa_offset_right_prefix_unique_secondsteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_prefix_unique_secondstepsubtractleft. pfa_gap_prefix_unique_secondstepsubtractleft + S (pfd_previous_prefix_unique_secondstep) = (p)) /\\ (((exists pfa_gap_prefix_unique_secondstepsubtractright. pfa_gap_prefix_unique_secondstepsubtractright + S (pfd_difference_prefix_unique_secondstep) = (p)) /\\ ((((exists pfa_gap_prefix_unique_secondstepsubtractresultbound. pfa_gap_prefix_unique_secondstepsubtractresultbound + S (pfd_input_prefix_unique_secondstep) = (p)) /\\ ((exists pfa_offset_left_prefix_unique_secondstepsubtractresultcongruence pfa_offset_right_prefix_unique_secondstepsubtractresultcongruence. ((pfd_previous_prefix_unique_secondstep) + (pfd_difference_prefix_unique_secondstep)) + (p) * pfa_offset_left_prefix_unique_secondstepsubtractresultcongruence = (pfd_input_prefix_unique_secondstep) + (p) * pfa_offset_right_prefix_unique_secondstepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_prefix_unique_secondstepmultiplyleft. pfa_gap_prefix_unique_secondstepmultiplyleft + S (k) = (p)) /\\ (((exists pfa_gap_prefix_unique_secondstepmultiplyright. pfa_gap_prefix_unique_secondstepmultiplyright + S (pfd_difference_prefix_unique_secondstep) = (p)) /\\ ((((exists pfa_gap_prefix_unique_secondstepmultiplyresultbound. pfa_gap_prefix_unique_secondstepmultiplyresultbound + S (pfd_value_prefix_unique_second) = (p)) /\\ ((exists pfa_offset_left_prefix_unique_secondstepmultiplyresultcongruence pfa_offset_right_prefix_unique_secondstepmultiplyresultcongruence. ((k) * (pfd_difference_prefix_unique_secondstep)) + (p) * pfa_offset_left_prefix_unique_secondstepmultiplyresultcongruence = (pfd_value_prefix_unique_second) + (p) * pfa_offset_right_prefix_unique_secondstepmultiplyresultcongruence))))))))))))))))))) -> (forall mdr_i_pfp_prefix_unique_result mdr_a_pfp_prefix_unique_result. 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"intro p", "intro b", "intro c", "intro B", "intro C", "intro L", "intro t", "intro rb", "intro rc", "intro R", "intro hequal", "intro htrim", "have hreverse : forall mdr_i_pfp_trim_transport_reverse mdr_a_pfp_trim_transport_reverse. (exists mdr_gap_pfp_trim_transport_reverseb. mdr_gap_pfp_trim_transport_reverseb + S (mdr_i_pfp_trim_transport_reverse) = (L)) -> (((exists ff_h_mdr_pfp_trim_transport_reverseo. ff_h_mdr_pfp_trim_transport_reverseo + S (mdr_a_pfp_trim_transport_reverse) = S ((S (mdr_i_pfp_trim_transport_reverse)) * C)) /\\ exists ff_q_mdr_pfp_trim_transport_reverseo. B = ff_q_mdr_pfp_trim_transport_reverseo * S ((S (mdr_i_pfp_trim_transport_reverse)) * C) + (mdr_a_pfp_trim_transport_reverse))) -> (((exists ff_h_mdr_pfp_trim_transport_reversen. ff_h_mdr_pfp_trim_transport_reversen + S (mdr_a_pfp_trim_transport_reverse) = S ((S (mdr_i_pfp_trim_transport_reverse)) * c)) /\\ exists ff_q_mdr_pfp_trim_transport_reversen. b = ff_q_mdr_pfp_trim_transport_reversen * S ((S (mdr_i_pfp_trim_transport_reverse)) * c) + (mdr_a_pfp_trim_transport_reverse)))", "specialize matrix_rank_prefix_equality_symmetric (b)", "specialize matrix_rank_prefix_equality_symmetric (c)", "specialize matrix_rank_prefix_equality_symmetric (B)", "specialize matrix_rank_prefix_equality_symmetric (C)", "specialize matrix_rank_prefix_equality_symmetric (L)", "apply matrix_rank_prefix_equality_symmetric", "exact hequal", "have hbounds : ((exists pfc_gap_trim_transport_cut. pfc_gap_trim_transport_cut+(t)=(L)) /\\ ((exists pfc_gap_trim_transport_length. pfc_gap_trim_transport_length+(R)=(L))))", "specialize prime_field_polynomial_trim_length_bounds (p)", "specialize prime_field_polynomial_trim_length_bounds (b)", "specialize prime_field_polynomial_trim_length_bounds (c)", "specialize prime_field_polynomial_trim_length_bounds (L)", "specialize prime_field_polynomial_trim_length_bounds (t)", "specialize prime_field_polynomial_trim_length_bounds (rb)", "specialize prime_field_polynomial_trim_length_bounds (rc)", "specialize prime_field_polynomial_trim_length_bounds (R)", "apply prime_field_polynomial_trim_length_bounds", "exact htrim", "cases hbounds", "cases htrim", "cases htrim_right", "cases htrim_right_right", "cases htrim_right_right_right", "split", "exact htrim_left", "split", "specialize matrix_rank_bounded_prefix_transport (b)", "specialize matrix_rank_bounded_prefix_transport (c)", "specialize matrix_rank_bounded_prefix_transport (B)", "specialize matrix_rank_bounded_prefix_transport (C)", "specialize matrix_rank_bounded_prefix_transport (L)", "specialize matrix_rank_bounded_prefix_transport (p)", "apply matrix_rank_bounded_prefix_transport", "exact hequal", "exact htrim_right_left", "split", "intro i", "intro hindex", "specialize hequal (i)", "specialize hequal (0)", "apply hequal", "specialize lt_of_lt_of_le (i)", "specialize lt_of_lt_of_le (t)", "specialize lt_of_lt_of_le (L)", "apply lt_of_lt_of_le", "exact hindex", "exact hbounds_left", "specialize htrim_right_right_left (i)", "apply htrim_right_right_left", "exact hindex", "split", "intro i", "intro a", "intro hindex", "intro hvalue", "specialize htrim_right_right_right_left (i)", "specialize htrim_right_right_right_left (a)", "apply htrim_right_right_right_left", "exact hindex", "specialize hreverse (t+i)", "specialize hreverse (a)", "apply hreverse", "rewrite htrim_left", "specialize matrix_recursive_lt_add_left (i)", "specialize matrix_recursive_lt_add_left (R)", "specialize matrix_recursive_lt_add_left (t)", "apply matrix_recursive_lt_add_left", "exact hindex", "exact hvalue", "exact htrim_right_right_right_right" ], "script_sha256": "abe9341943324e9c1a3d414a3a0a83a5b3dc0e3954f1c24e2c563a482982e7f6", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_division_uniqueness_candidate.py", "sha256": "6a9d9ebe1f72202743e5df2c069b9aa367fdb3d61108f1d9354cdc9276ab2d15" }, "statement": "forall p b c B C L t rb rc R. (forall mdr_i_pfp_trim_transport_equal mdr_a_pfp_trim_transport_equal. (exists mdr_gap_pfp_trim_transport_equalb. mdr_gap_pfp_trim_transport_equalb + S (mdr_i_pfp_trim_transport_equal) = (L)) -> (((exists ff_h_mdr_pfp_trim_transport_equalo. ff_h_mdr_pfp_trim_transport_equalo + S (mdr_a_pfp_trim_transport_equal) = S ((S (mdr_i_pfp_trim_transport_equal)) * c)) /\\ exists ff_q_mdr_pfp_trim_transport_equalo. b = ff_q_mdr_pfp_trim_transport_equalo * S ((S (mdr_i_pfp_trim_transport_equal)) * c) + (mdr_a_pfp_trim_transport_equal))) -> (((exists ff_h_mdr_pfp_trim_transport_equaln. ff_h_mdr_pfp_trim_transport_equaln + S (mdr_a_pfp_trim_transport_equal) = S ((S (mdr_i_pfp_trim_transport_equal)) * C)) /\\ exists ff_q_mdr_pfp_trim_transport_equaln. B = ff_q_mdr_pfp_trim_transport_equaln * S ((S (mdr_i_pfp_trim_transport_equal)) * C) + (mdr_a_pfp_trim_transport_equal)))) -> ((((L)=(t)+(R)) /\\ (((forall fom_index_pfp_trim_transport_sourceinput. (exists fom_gap_pfp_trim_transport_sourceinput_index_bound. fom_gap_pfp_trim_transport_sourceinput_index_bound + S (fom_index_pfp_trim_transport_sourceinput) = L) -> exists fom_value_pfp_trim_transport_sourceinput. ((((exists fom_beta_height_pfp_trim_transport_sourceinput_entry. fom_beta_height_pfp_trim_transport_sourceinput_entry + S (fom_value_pfp_trim_transport_sourceinput) = S ((S (fom_index_pfp_trim_transport_sourceinput)) * c)) /\\ exists fom_beta_quotient_pfp_trim_transport_sourceinput_entry. b = fom_beta_quotient_pfp_trim_transport_sourceinput_entry * S ((S (fom_index_pfp_trim_transport_sourceinput)) * c) + (fom_value_pfp_trim_transport_sourceinput))) /\\ (exists fom_gap_pfp_trim_transport_sourceinput_value_bound. fom_gap_pfp_trim_transport_sourceinput_value_bound + S (fom_value_pfp_trim_transport_sourceinput) = p))) /\\ (((forall pfp_repeat_index_trim_transport_sourceremoved. (exists pfa_gap_trim_transport_sourceremovedindex. pfa_gap_trim_transport_sourceremovedindex + S (pfp_repeat_index_trim_transport_sourceremoved) = (t)) -> (((exists ff_h_pfp_trim_transport_sourceremovedentry. ff_h_pfp_trim_transport_sourceremovedentry + S (0) = S ((S (pfp_repeat_index_trim_transport_sourceremoved)) * c)) /\\ exists ff_q_pfp_trim_transport_sourceremovedentry. b = ff_q_pfp_trim_transport_sourceremovedentry * S ((S (pfp_repeat_index_trim_transport_sourceremoved)) * c) + (0)))) /\\ (((forall pftrim_index_trim_transport_sourcesuffix pftrim_value_trim_transport_sourcesuffix. (exists pfa_gap_trim_transport_sourcesuffixbound. pfa_gap_trim_transport_sourcesuffixbound + S (pftrim_index_trim_transport_sourcesuffix) = (R)) -> (((exists ff_h_pfp_trim_transport_sourcesuffixsource. ff_h_pfp_trim_transport_sourcesuffixsource + S (pftrim_value_trim_transport_sourcesuffix) = S ((S ((t)+pftrim_index_trim_transport_sourcesuffix)) * c)) /\\ exists ff_q_pfp_trim_transport_sourcesuffixsource. b = ff_q_pfp_trim_transport_sourcesuffixsource * S ((S ((t)+pftrim_index_trim_transport_sourcesuffix)) * c) + (pftrim_value_trim_transport_sourcesuffix))) -> (((exists ff_h_pfp_trim_transport_sourcesuffixoutput. ff_h_pfp_trim_transport_sourcesuffixoutput + S (pftrim_value_trim_transport_sourcesuffix) = S ((S (pftrim_index_trim_transport_sourcesuffix)) * rc)) /\\ exists ff_q_pfp_trim_transport_sourcesuffixoutput. rb = ff_q_pfp_trim_transport_sourcesuffixoutput * S ((S (pftrim_index_trim_transport_sourcesuffix)) * rc) + (pftrim_value_trim_transport_sourcesuffix)))) /\\ (((R)=0 \\/ (exists pftrim_leading_trim_transport_sourcenormal. ((((exists ff_h_pfp_trim_transport_sourcenormalentry. ff_h_pfp_trim_transport_sourcenormalentry + S (pftrim_leading_trim_transport_sourcenormal) = S ((S (0)) * rc)) /\\ exists ff_q_pfp_trim_transport_sourcenormalentry. rb = ff_q_pfp_trim_transport_sourcenormalentry * S ((S (0)) * rc) + (pftrim_leading_trim_transport_sourcenormal))) /\\ ((~(pftrim_leading_trim_transport_sourcenormal=0))))))))))))))) -> ((((L)=(t)+(R)) /\\ (((forall fom_index_pfp_trim_transport_resultinput. (exists fom_gap_pfp_trim_transport_resultinput_index_bound. fom_gap_pfp_trim_transport_resultinput_index_bound + S (fom_index_pfp_trim_transport_resultinput) = L) -> exists fom_value_pfp_trim_transport_resultinput. ((((exists fom_beta_height_pfp_trim_transport_resultinput_entry. fom_beta_height_pfp_trim_transport_resultinput_entry + S (fom_value_pfp_trim_transport_resultinput) = S ((S (fom_index_pfp_trim_transport_resultinput)) * C)) /\\ exists fom_beta_quotient_pfp_trim_transport_resultinput_entry. B = fom_beta_quotient_pfp_trim_transport_resultinput_entry * S ((S (fom_index_pfp_trim_transport_resultinput)) * C) + (fom_value_pfp_trim_transport_resultinput))) /\\ (exists fom_gap_pfp_trim_transport_resultinput_value_bound. fom_gap_pfp_trim_transport_resultinput_value_bound + S (fom_value_pfp_trim_transport_resultinput) = p))) /\\ (((forall pfp_repeat_index_trim_transport_resultremoved. (exists pfa_gap_trim_transport_resultremovedindex. pfa_gap_trim_transport_resultremovedindex + S (pfp_repeat_index_trim_transport_resultremoved) = (t)) -> (((exists ff_h_pfp_trim_transport_resultremovedentry. ff_h_pfp_trim_transport_resultremovedentry + S (0) = S ((S (pfp_repeat_index_trim_transport_resultremoved)) * C)) /\\ exists ff_q_pfp_trim_transport_resultremovedentry. B = ff_q_pfp_trim_transport_resultremovedentry * S ((S (pfp_repeat_index_trim_transport_resultremoved)) * C) + (0)))) /\\ (((forall pftrim_index_trim_transport_resultsuffix pftrim_value_trim_transport_resultsuffix. (exists pfa_gap_trim_transport_resultsuffixbound. pfa_gap_trim_transport_resultsuffixbound + S (pftrim_index_trim_transport_resultsuffix) = (R)) -> (((exists ff_h_pfp_trim_transport_resultsuffixsource. ff_h_pfp_trim_transport_resultsuffixsource + S (pftrim_value_trim_transport_resultsuffix) = S ((S ((t)+pftrim_index_trim_transport_resultsuffix)) * C)) /\\ exists ff_q_pfp_trim_transport_resultsuffixsource. B = ff_q_pfp_trim_transport_resultsuffixsource * S ((S ((t)+pftrim_index_trim_transport_resultsuffix)) * C) + (pftrim_value_trim_transport_resultsuffix))) -> (((exists ff_h_pfp_trim_transport_resultsuffixoutput. ff_h_pfp_trim_transport_resultsuffixoutput + S (pftrim_value_trim_transport_resultsuffix) = S ((S (pftrim_index_trim_transport_resultsuffix)) * rc)) /\\ exists ff_q_pfp_trim_transport_resultsuffixoutput. rb = ff_q_pfp_trim_transport_resultsuffixoutput * S ((S (pftrim_index_trim_transport_resultsuffix)) * rc) + (pftrim_value_trim_transport_resultsuffix)))) /\\ (((R)=0 \\/ (exists pftrim_leading_trim_transport_resultnormal. 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(((((exists ff_h_pfp_data_unique_firsthead. ff_h_pfp_data_unique_firsthead + S (b) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_data_unique_firsthead. bb = ff_q_pfp_data_unique_firsthead * S ((S (0)) * bc) + (b))) /\\ (((((~((b) = 0)) /\\ ((((exists pfa_gap_data_unique_firstinversemultiplicationleft. pfa_gap_data_unique_firstinversemultiplicationleft + S (b) = (p)) /\\ (((exists pfa_gap_data_unique_firstinversemultiplicationright. pfa_gap_data_unique_firstinversemultiplicationright + S (k) = (p)) /\\ ((((exists pfa_gap_data_unique_firstinversemultiplicationresultbound. pfa_gap_data_unique_firstinversemultiplicationresultbound + S (1) = (p)) /\\ ((exists pfa_offset_left_data_unique_firstinversemultiplicationresultcongruence pfa_offset_right_data_unique_firstinversemultiplicationresultcongruence. ((b) * (k)) + (p) * pfa_offset_left_data_unique_firstinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_data_unique_firstinversemultiplicationresultcongruence)))))))))))) /\\ (((((((q)=0) /\\ ((exists pfc_gap_data_unique_firstlengthshort. pfc_gap_data_unique_firstlengthshort+(L)=(d))))) \\/ (((~((q)=0)) /\\ (((q)+(d)=(L)))))) /\\ ((forall pfd_index_data_unique_firstexecution. (exists pfa_gap_data_unique_firstexecutionbound. pfa_gap_data_unique_firstexecutionbound + S (pfd_index_data_unique_firstexecution) = (q)) -> exists pfd_value_data_unique_firstexecution. ((((exists ff_h_pfp_data_unique_firstexecutionentry. ff_h_pfp_data_unique_firstexecutionentry + S (pfd_value_data_unique_firstexecution) = S ((S (pfd_index_data_unique_firstexecution)) * qc)) /\\ exists ff_q_pfp_data_unique_firstexecutionentry. qb = ff_q_pfp_data_unique_firstexecutionentry * S ((S (pfd_index_data_unique_firstexecution)) * qc) + (pfd_value_data_unique_firstexecution))) /\\ ((exists pfd_input_data_unique_firstexecutionstep pfd_previous_data_unique_firstexecutionstep pfd_difference_data_unique_firstexecutionstep. ((((exists ff_h_pfp_data_unique_firstexecutionstepinput. ff_h_pfp_data_unique_firstexecutionstepinput + S (pfd_input_data_unique_firstexecutionstep) = S ((S (pfd_index_data_unique_firstexecution)) * ac)) /\\ exists ff_q_pfp_data_unique_firstexecutionstepinput. ab = ff_q_pfp_data_unique_firstexecutionstepinput * S ((S (pfd_index_data_unique_firstexecution)) * ac) + (pfd_input_data_unique_firstexecutionstep))) /\\ (((exists pfc_terms_code_data_unique_firstexecutionstepprevious pfc_terms_scale_data_unique_firstexecutionstepprevious pfc_natural_sum_data_unique_firstexecutionstepprevious. ((forall pfc_index_data_unique_firstexecutionsteppreviousdiagonal. (exists pfa_gap_data_unique_firstexecutionsteppreviousdiagonalbound. pfa_gap_data_unique_firstexecutionsteppreviousdiagonalbound + S (pfc_index_data_unique_firstexecutionsteppreviousdiagonal) = (S (pfd_index_data_unique_firstexecution))) -> exists pfc_value_data_unique_firstexecutionsteppreviousdiagonal. ((((exists ff_h_pfp_data_unique_firstexecutionsteppreviousdiagonalentry. ff_h_pfp_data_unique_firstexecutionsteppreviousdiagonalentry + S (pfc_value_data_unique_firstexecutionsteppreviousdiagonal) = S ((S (pfc_index_data_unique_firstexecutionsteppreviousdiagonal)) * pfc_terms_scale_data_unique_firstexecutionstepprevious)) /\\ exists ff_q_pfp_data_unique_firstexecutionsteppreviousdiagonalentry. pfc_terms_code_data_unique_firstexecutionstepprevious = ff_q_pfp_data_unique_firstexecutionsteppreviousdiagonalentry * S ((S (pfc_index_data_unique_firstexecutionsteppreviousdiagonal)) * pfc_terms_scale_data_unique_firstexecutionstepprevious) + (pfc_value_data_unique_firstexecutionsteppreviousdiagonal))) /\\ ((exists pfc_complement_data_unique_firstexecutionsteppreviousdiagonalterm pfc_left_data_unique_firstexecutionsteppreviousdiagonalterm pfc_right_data_unique_firstexecutionsteppreviousdiagonalterm. (((pfc_index_data_unique_firstexecutionsteppreviousdiagonal)+pfc_complement_data_unique_firstexecutionsteppreviousdiagonalterm=(pfd_index_data_unique_firstexecution)) /\\ ((((((exists pfa_gap_data_unique_firstexecutionsteppreviousdiagonaltermleftinside. pfa_gap_data_unique_firstexecutionsteppreviousdiagonaltermleftinside + S (pfc_index_data_unique_firstexecutionsteppreviousdiagonal) = (pfd_index_data_unique_firstexecution)) /\\ ((((exists ff_h_pfp_data_unique_firstexecutionsteppreviousdiagonaltermleftentry. ff_h_pfp_data_unique_firstexecutionsteppreviousdiagonaltermleftentry + S (pfc_left_data_unique_firstexecutionsteppreviousdiagonalterm) = S ((S (pfc_index_data_unique_firstexecutionsteppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_data_unique_firstexecutionsteppreviousdiagonaltermleftentry. qb = ff_q_pfp_data_unique_firstexecutionsteppreviousdiagonaltermleftentry * S ((S (pfc_index_data_unique_firstexecutionsteppreviousdiagonal)) * qc) + (pfc_left_data_unique_firstexecutionsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_data_unique_firstexecutionsteppreviousdiagonaltermleftoutside. pfc_gap_data_unique_firstexecutionsteppreviousdiagonaltermleftoutside+(pfd_index_data_unique_firstexecution)=(pfc_index_data_unique_firstexecutionsteppreviousdiagonal)) /\\ (((pfc_left_data_unique_firstexecutionsteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_data_unique_firstexecutionsteppreviousdiagonaltermrightinside. pfa_gap_data_unique_firstexecutionsteppreviousdiagonaltermrightinside + S (pfc_complement_data_unique_firstexecutionsteppreviousdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_data_unique_firstexecutionsteppreviousdiagonaltermrightentry. ff_h_pfp_data_unique_firstexecutionsteppreviousdiagonaltermrightentry + S (pfc_right_data_unique_firstexecutionsteppreviousdiagonalterm) = S ((S (pfc_complement_data_unique_firstexecutionsteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_data_unique_firstexecutionsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_data_unique_firstexecutionsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_data_unique_firstexecutionsteppreviousdiagonalterm)) * bc) + (pfc_right_data_unique_firstexecutionsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_data_unique_firstexecutionsteppreviousdiagonaltermrightoutside. pfc_gap_data_unique_firstexecutionsteppreviousdiagonaltermrightoutside+(S (d))=(pfc_complement_data_unique_firstexecutionsteppreviousdiagonalterm)) /\\ (((pfc_right_data_unique_firstexecutionsteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_data_unique_firstexecutionsteppreviousdiagonal)=pfc_left_data_unique_firstexecutionsteppreviousdiagonalterm*pfc_right_data_unique_firstexecutionsteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_data_unique_firstexecutionstepprevioussum fs_v_pfc_data_unique_firstexecutionstepprevioussum. ((((exists fs_h_pfc_data_unique_firstexecutionstepprevioussum_body_start. fs_h_pfc_data_unique_firstexecutionstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_data_unique_firstexecutionstepprevioussum)) /\\ exists fs_q_pfc_data_unique_firstexecutionstepprevioussum_body_start. fs_u_pfc_data_unique_firstexecutionstepprevioussum = fs_q_pfc_data_unique_firstexecutionstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_data_unique_firstexecutionstepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_data_unique_firstexecutionstepprevioussum_body_terminal. fs_h_pfc_data_unique_firstexecutionstepprevioussum_body_terminal + S (pfc_natural_sum_data_unique_firstexecutionstepprevious) = S ((S (S (pfd_index_data_unique_firstexecution))) * fs_v_pfc_data_unique_firstexecutionstepprevioussum)) /\\ exists fs_q_pfc_data_unique_firstexecutionstepprevioussum_body_terminal. fs_u_pfc_data_unique_firstexecutionstepprevioussum = fs_q_pfc_data_unique_firstexecutionstepprevioussum_body_terminal * S ((S (S (pfd_index_data_unique_firstexecution))) * fs_v_pfc_data_unique_firstexecutionstepprevioussum) + (pfc_natural_sum_data_unique_firstexecutionstepprevious))) /\\ forall fs_i_pfc_data_unique_firstexecutionstepprevioussum_body_steps. (exists fs_lt_pfc_data_unique_firstexecutionstepprevioussum_body_steps_bound. fs_lt_pfc_data_unique_firstexecutionstepprevioussum_body_steps_bound + S fs_i_pfc_data_unique_firstexecutionstepprevioussum_body_steps = S (pfd_index_data_unique_firstexecution)) -> exists fs_a_pfc_data_unique_firstexecutionstepprevioussum_body_steps fs_r_pfc_data_unique_firstexecutionstepprevioussum_body_steps fs_s_pfc_data_unique_firstexecutionstepprevioussum_body_steps. ((((exists fs_h_pfc_data_unique_firstexecutionstepprevioussum_body_steps_summand. fs_h_pfc_data_unique_firstexecutionstepprevioussum_body_steps_summand + S (fs_a_pfc_data_unique_firstexecutionstepprevioussum_body_steps) = S ((S (fs_i_pfc_data_unique_firstexecutionstepprevioussum_body_steps)) * pfc_terms_scale_data_unique_firstexecutionstepprevious)) /\\ exists fs_q_pfc_data_unique_firstexecutionstepprevioussum_body_steps_summand. pfc_terms_code_data_unique_firstexecutionstepprevious = fs_q_pfc_data_unique_firstexecutionstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_data_unique_firstexecutionstepprevioussum_body_steps)) * pfc_terms_scale_data_unique_firstexecutionstepprevious) + (fs_a_pfc_data_unique_firstexecutionstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_data_unique_firstexecutionstepprevioussum_body_steps_partial. fs_h_pfc_data_unique_firstexecutionstepprevioussum_body_steps_partial + S (fs_r_pfc_data_unique_firstexecutionstepprevioussum_body_steps) = S ((S (fs_i_pfc_data_unique_firstexecutionstepprevioussum_body_steps)) * fs_v_pfc_data_unique_firstexecutionstepprevioussum)) /\\ exists fs_q_pfc_data_unique_firstexecutionstepprevioussum_body_steps_partial. fs_u_pfc_data_unique_firstexecutionstepprevioussum = fs_q_pfc_data_unique_firstexecutionstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_data_unique_firstexecutionstepprevioussum_body_steps)) * fs_v_pfc_data_unique_firstexecutionstepprevioussum) + (fs_r_pfc_data_unique_firstexecutionstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_data_unique_firstexecutionstepprevioussum_body_steps_successor. fs_h_pfc_data_unique_firstexecutionstepprevioussum_body_steps_successor + S (fs_s_pfc_data_unique_firstexecutionstepprevioussum_body_steps) = S ((S (S fs_i_pfc_data_unique_firstexecutionstepprevioussum_body_steps)) * fs_v_pfc_data_unique_firstexecutionstepprevioussum)) /\\ exists fs_q_pfc_data_unique_firstexecutionstepprevioussum_body_steps_successor. fs_u_pfc_data_unique_firstexecutionstepprevioussum = fs_q_pfc_data_unique_firstexecutionstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_data_unique_firstexecutionstepprevioussum_body_steps)) * fs_v_pfc_data_unique_firstexecutionstepprevioussum) + (fs_s_pfc_data_unique_firstexecutionstepprevioussum_body_steps))) /\\ fs_s_pfc_data_unique_firstexecutionstepprevioussum_body_steps = fs_r_pfc_data_unique_firstexecutionstepprevioussum_body_steps + fs_a_pfc_data_unique_firstexecutionstepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_data_unique_firstexecutionsteppreviousresiduebound. pfa_gap_data_unique_firstexecutionsteppreviousresiduebound + S (pfd_previous_data_unique_firstexecutionstep) = (p)) /\\ ((exists pfa_offset_left_data_unique_firstexecutionsteppreviousresiduecongruence pfa_offset_right_data_unique_firstexecutionsteppreviousresiduecongruence. (pfc_natural_sum_data_unique_firstexecutionstepprevious) + (p) * pfa_offset_left_data_unique_firstexecutionsteppreviousresiduecongruence = (pfd_previous_data_unique_firstexecutionstep) + (p) * pfa_offset_right_data_unique_firstexecutionsteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_data_unique_firstexecutionstepsubtractleft. pfa_gap_data_unique_firstexecutionstepsubtractleft + S (pfd_previous_data_unique_firstexecutionstep) = (p)) /\\ (((exists pfa_gap_data_unique_firstexecutionstepsubtractright. pfa_gap_data_unique_firstexecutionstepsubtractright + S (pfd_difference_data_unique_firstexecutionstep) = (p)) /\\ ((((exists pfa_gap_data_unique_firstexecutionstepsubtractresultbound. pfa_gap_data_unique_firstexecutionstepsubtractresultbound + S (pfd_input_data_unique_firstexecutionstep) = (p)) /\\ ((exists pfa_offset_left_data_unique_firstexecutionstepsubtractresultcongruence pfa_offset_right_data_unique_firstexecutionstepsubtractresultcongruence. ((pfd_previous_data_unique_firstexecutionstep) + (pfd_difference_data_unique_firstexecutionstep)) + (p) * pfa_offset_left_data_unique_firstexecutionstepsubtractresultcongruence = (pfd_input_data_unique_firstexecutionstep) + (p) * pfa_offset_right_data_unique_firstexecutionstepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_data_unique_firstexecutionstepmultiplyleft. pfa_gap_data_unique_firstexecutionstepmultiplyleft + S (k) = (p)) /\\ (((exists pfa_gap_data_unique_firstexecutionstepmultiplyright. pfa_gap_data_unique_firstexecutionstepmultiplyright + S (pfd_difference_data_unique_firstexecutionstep) = (p)) /\\ ((((exists pfa_gap_data_unique_firstexecutionstepmultiplyresultbound. pfa_gap_data_unique_firstexecutionstepmultiplyresultbound + S (pfd_value_data_unique_firstexecution) = (p)) /\\ ((exists pfa_offset_left_data_unique_firstexecutionstepmultiplyresultcongruence pfa_offset_right_data_unique_firstexecutionstepmultiplyresultcongruence. ((k) * (pfd_difference_data_unique_firstexecutionstep)) + (p) * pfa_offset_left_data_unique_firstexecutionstepmultiplyresultcongruence = (pfd_value_data_unique_firstexecution) + (p) * pfa_offset_right_data_unique_firstexecutionstepmultiplyresultcongruence)))))))))))))))))))))))))) -> (((((exists ff_h_pfp_data_unique_secondhead. ff_h_pfp_data_unique_secondhead + S (B) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_data_unique_secondhead. bb = ff_q_pfp_data_unique_secondhead * S ((S (0)) * bc) + (B))) /\\ (((((~((B) = 0)) /\\ ((((exists pfa_gap_data_unique_secondinversemultiplicationleft. pfa_gap_data_unique_secondinversemultiplicationleft + S (B) = (p)) /\\ (((exists pfa_gap_data_unique_secondinversemultiplicationright. pfa_gap_data_unique_secondinversemultiplicationright + S (K) = (p)) /\\ ((((exists pfa_gap_data_unique_secondinversemultiplicationresultbound. pfa_gap_data_unique_secondinversemultiplicationresultbound + S (1) = (p)) /\\ ((exists pfa_offset_left_data_unique_secondinversemultiplicationresultcongruence pfa_offset_right_data_unique_secondinversemultiplicationresultcongruence. ((B) * (K)) + (p) * pfa_offset_left_data_unique_secondinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_data_unique_secondinversemultiplicationresultcongruence)))))))))))) /\\ (((((((Q)=0) /\\ ((exists pfc_gap_data_unique_secondlengthshort. pfc_gap_data_unique_secondlengthshort+(L)=(d))))) \\/ (((~((Q)=0)) /\\ (((Q)+(d)=(L)))))) /\\ ((forall pfd_index_data_unique_secondexecution. (exists pfa_gap_data_unique_secondexecutionbound. pfa_gap_data_unique_secondexecutionbound + S (pfd_index_data_unique_secondexecution) = (Q)) -> exists pfd_value_data_unique_secondexecution. ((((exists ff_h_pfp_data_unique_secondexecutionentry. ff_h_pfp_data_unique_secondexecutionentry + S (pfd_value_data_unique_secondexecution) = S ((S (pfd_index_data_unique_secondexecution)) * QC)) /\\ exists ff_q_pfp_data_unique_secondexecutionentry. QB = ff_q_pfp_data_unique_secondexecutionentry * S ((S (pfd_index_data_unique_secondexecution)) * QC) + (pfd_value_data_unique_secondexecution))) /\\ ((exists pfd_input_data_unique_secondexecutionstep pfd_previous_data_unique_secondexecutionstep pfd_difference_data_unique_secondexecutionstep. ((((exists ff_h_pfp_data_unique_secondexecutionstepinput. ff_h_pfp_data_unique_secondexecutionstepinput + S (pfd_input_data_unique_secondexecutionstep) = S ((S (pfd_index_data_unique_secondexecution)) * ac)) /\\ exists ff_q_pfp_data_unique_secondexecutionstepinput. ab = ff_q_pfp_data_unique_secondexecutionstepinput * S ((S (pfd_index_data_unique_secondexecution)) * ac) + (pfd_input_data_unique_secondexecutionstep))) /\\ (((exists pfc_terms_code_data_unique_secondexecutionstepprevious pfc_terms_scale_data_unique_secondexecutionstepprevious pfc_natural_sum_data_unique_secondexecutionstepprevious. ((forall pfc_index_data_unique_secondexecutionsteppreviousdiagonal. (exists pfa_gap_data_unique_secondexecutionsteppreviousdiagonalbound. pfa_gap_data_unique_secondexecutionsteppreviousdiagonalbound + S (pfc_index_data_unique_secondexecutionsteppreviousdiagonal) = (S (pfd_index_data_unique_secondexecution))) -> exists pfc_value_data_unique_secondexecutionsteppreviousdiagonal. ((((exists ff_h_pfp_data_unique_secondexecutionsteppreviousdiagonalentry. ff_h_pfp_data_unique_secondexecutionsteppreviousdiagonalentry + S (pfc_value_data_unique_secondexecutionsteppreviousdiagonal) = S ((S (pfc_index_data_unique_secondexecutionsteppreviousdiagonal)) * pfc_terms_scale_data_unique_secondexecutionstepprevious)) /\\ exists ff_q_pfp_data_unique_secondexecutionsteppreviousdiagonalentry. pfc_terms_code_data_unique_secondexecutionstepprevious = ff_q_pfp_data_unique_secondexecutionsteppreviousdiagonalentry * S ((S (pfc_index_data_unique_secondexecutionsteppreviousdiagonal)) * pfc_terms_scale_data_unique_secondexecutionstepprevious) + (pfc_value_data_unique_secondexecutionsteppreviousdiagonal))) /\\ ((exists pfc_complement_data_unique_secondexecutionsteppreviousdiagonalterm pfc_left_data_unique_secondexecutionsteppreviousdiagonalterm pfc_right_data_unique_secondexecutionsteppreviousdiagonalterm. (((pfc_index_data_unique_secondexecutionsteppreviousdiagonal)+pfc_complement_data_unique_secondexecutionsteppreviousdiagonalterm=(pfd_index_data_unique_secondexecution)) /\\ ((((((exists pfa_gap_data_unique_secondexecutionsteppreviousdiagonaltermleftinside. pfa_gap_data_unique_secondexecutionsteppreviousdiagonaltermleftinside + S (pfc_index_data_unique_secondexecutionsteppreviousdiagonal) = (pfd_index_data_unique_secondexecution)) /\\ ((((exists ff_h_pfp_data_unique_secondexecutionsteppreviousdiagonaltermleftentry. ff_h_pfp_data_unique_secondexecutionsteppreviousdiagonaltermleftentry + S (pfc_left_data_unique_secondexecutionsteppreviousdiagonalterm) = S ((S (pfc_index_data_unique_secondexecutionsteppreviousdiagonal)) * QC)) /\\ exists ff_q_pfp_data_unique_secondexecutionsteppreviousdiagonaltermleftentry. QB = ff_q_pfp_data_unique_secondexecutionsteppreviousdiagonaltermleftentry * S ((S (pfc_index_data_unique_secondexecutionsteppreviousdiagonal)) * QC) + (pfc_left_data_unique_secondexecutionsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_data_unique_secondexecutionsteppreviousdiagonaltermleftoutside. pfc_gap_data_unique_secondexecutionsteppreviousdiagonaltermleftoutside+(pfd_index_data_unique_secondexecution)=(pfc_index_data_unique_secondexecutionsteppreviousdiagonal)) /\\ (((pfc_left_data_unique_secondexecutionsteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_data_unique_secondexecutionsteppreviousdiagonaltermrightinside. pfa_gap_data_unique_secondexecutionsteppreviousdiagonaltermrightinside + S (pfc_complement_data_unique_secondexecutionsteppreviousdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_data_unique_secondexecutionsteppreviousdiagonaltermrightentry. ff_h_pfp_data_unique_secondexecutionsteppreviousdiagonaltermrightentry + S (pfc_right_data_unique_secondexecutionsteppreviousdiagonalterm) = S ((S (pfc_complement_data_unique_secondexecutionsteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_data_unique_secondexecutionsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_data_unique_secondexecutionsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_data_unique_secondexecutionsteppreviousdiagonalterm)) * bc) + (pfc_right_data_unique_secondexecutionsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_data_unique_secondexecutionsteppreviousdiagonaltermrightoutside. pfc_gap_data_unique_secondexecutionsteppreviousdiagonaltermrightoutside+(S (d))=(pfc_complement_data_unique_secondexecutionsteppreviousdiagonalterm)) /\\ (((pfc_right_data_unique_secondexecutionsteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_data_unique_secondexecutionsteppreviousdiagonal)=pfc_left_data_unique_secondexecutionsteppreviousdiagonalterm*pfc_right_data_unique_secondexecutionsteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_data_unique_secondexecutionstepprevioussum fs_v_pfc_data_unique_secondexecutionstepprevioussum. ((((exists fs_h_pfc_data_unique_secondexecutionstepprevioussum_body_start. fs_h_pfc_data_unique_secondexecutionstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_data_unique_secondexecutionstepprevioussum)) /\\ exists fs_q_pfc_data_unique_secondexecutionstepprevioussum_body_start. fs_u_pfc_data_unique_secondexecutionstepprevioussum = fs_q_pfc_data_unique_secondexecutionstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_data_unique_secondexecutionstepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_data_unique_secondexecutionstepprevioussum_body_terminal. fs_h_pfc_data_unique_secondexecutionstepprevioussum_body_terminal + S (pfc_natural_sum_data_unique_secondexecutionstepprevious) = S ((S (S (pfd_index_data_unique_secondexecution))) * fs_v_pfc_data_unique_secondexecutionstepprevioussum)) /\\ exists fs_q_pfc_data_unique_secondexecutionstepprevioussum_body_terminal. fs_u_pfc_data_unique_secondexecutionstepprevioussum = fs_q_pfc_data_unique_secondexecutionstepprevioussum_body_terminal * S ((S (S (pfd_index_data_unique_secondexecution))) * fs_v_pfc_data_unique_secondexecutionstepprevioussum) + (pfc_natural_sum_data_unique_secondexecutionstepprevious))) /\\ forall fs_i_pfc_data_unique_secondexecutionstepprevioussum_body_steps. (exists fs_lt_pfc_data_unique_secondexecutionstepprevioussum_body_steps_bound. fs_lt_pfc_data_unique_secondexecutionstepprevioussum_body_steps_bound + S fs_i_pfc_data_unique_secondexecutionstepprevioussum_body_steps = S (pfd_index_data_unique_secondexecution)) -> exists fs_a_pfc_data_unique_secondexecutionstepprevioussum_body_steps fs_r_pfc_data_unique_secondexecutionstepprevioussum_body_steps fs_s_pfc_data_unique_secondexecutionstepprevioussum_body_steps. ((((exists fs_h_pfc_data_unique_secondexecutionstepprevioussum_body_steps_summand. fs_h_pfc_data_unique_secondexecutionstepprevioussum_body_steps_summand + S (fs_a_pfc_data_unique_secondexecutionstepprevioussum_body_steps) = S ((S (fs_i_pfc_data_unique_secondexecutionstepprevioussum_body_steps)) * pfc_terms_scale_data_unique_secondexecutionstepprevious)) /\\ exists fs_q_pfc_data_unique_secondexecutionstepprevioussum_body_steps_summand. pfc_terms_code_data_unique_secondexecutionstepprevious = fs_q_pfc_data_unique_secondexecutionstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_data_unique_secondexecutionstepprevioussum_body_steps)) * pfc_terms_scale_data_unique_secondexecutionstepprevious) + (fs_a_pfc_data_unique_secondexecutionstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_data_unique_secondexecutionstepprevioussum_body_steps_partial. fs_h_pfc_data_unique_secondexecutionstepprevioussum_body_steps_partial + S (fs_r_pfc_data_unique_secondexecutionstepprevioussum_body_steps) = S ((S (fs_i_pfc_data_unique_secondexecutionstepprevioussum_body_steps)) * fs_v_pfc_data_unique_secondexecutionstepprevioussum)) /\\ exists fs_q_pfc_data_unique_secondexecutionstepprevioussum_body_steps_partial. fs_u_pfc_data_unique_secondexecutionstepprevioussum = fs_q_pfc_data_unique_secondexecutionstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_data_unique_secondexecutionstepprevioussum_body_steps)) * fs_v_pfc_data_unique_secondexecutionstepprevioussum) + (fs_r_pfc_data_unique_secondexecutionstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_data_unique_secondexecutionstepprevioussum_body_steps_successor. fs_h_pfc_data_unique_secondexecutionstepprevioussum_body_steps_successor + S (fs_s_pfc_data_unique_secondexecutionstepprevioussum_body_steps) = S ((S (S fs_i_pfc_data_unique_secondexecutionstepprevioussum_body_steps)) * fs_v_pfc_data_unique_secondexecutionstepprevioussum)) /\\ exists fs_q_pfc_data_unique_secondexecutionstepprevioussum_body_steps_successor. fs_u_pfc_data_unique_secondexecutionstepprevioussum = fs_q_pfc_data_unique_secondexecutionstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_data_unique_secondexecutionstepprevioussum_body_steps)) * fs_v_pfc_data_unique_secondexecutionstepprevioussum) + (fs_s_pfc_data_unique_secondexecutionstepprevioussum_body_steps))) /\\ fs_s_pfc_data_unique_secondexecutionstepprevioussum_body_steps = fs_r_pfc_data_unique_secondexecutionstepprevioussum_body_steps + fs_a_pfc_data_unique_secondexecutionstepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_data_unique_secondexecutionsteppreviousresiduebound. pfa_gap_data_unique_secondexecutionsteppreviousresiduebound + S (pfd_previous_data_unique_secondexecutionstep) = (p)) /\\ ((exists pfa_offset_left_data_unique_secondexecutionsteppreviousresiduecongruence pfa_offset_right_data_unique_secondexecutionsteppreviousresiduecongruence. (pfc_natural_sum_data_unique_secondexecutionstepprevious) + (p) * pfa_offset_left_data_unique_secondexecutionsteppreviousresiduecongruence = (pfd_previous_data_unique_secondexecutionstep) + (p) * pfa_offset_right_data_unique_secondexecutionsteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_data_unique_secondexecutionstepsubtractleft. pfa_gap_data_unique_secondexecutionstepsubtractleft + S (pfd_previous_data_unique_secondexecutionstep) = (p)) /\\ (((exists pfa_gap_data_unique_secondexecutionstepsubtractright. pfa_gap_data_unique_secondexecutionstepsubtractright + S (pfd_difference_data_unique_secondexecutionstep) = (p)) /\\ ((((exists pfa_gap_data_unique_secondexecutionstepsubtractresultbound. pfa_gap_data_unique_secondexecutionstepsubtractresultbound + S (pfd_input_data_unique_secondexecutionstep) = (p)) /\\ ((exists pfa_offset_left_data_unique_secondexecutionstepsubtractresultcongruence pfa_offset_right_data_unique_secondexecutionstepsubtractresultcongruence. ((pfd_previous_data_unique_secondexecutionstep) + (pfd_difference_data_unique_secondexecutionstep)) + (p) * pfa_offset_left_data_unique_secondexecutionstepsubtractresultcongruence = (pfd_input_data_unique_secondexecutionstep) + (p) * pfa_offset_right_data_unique_secondexecutionstepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_data_unique_secondexecutionstepmultiplyleft. pfa_gap_data_unique_secondexecutionstepmultiplyleft + S (K) = (p)) /\\ (((exists pfa_gap_data_unique_secondexecutionstepmultiplyright. pfa_gap_data_unique_secondexecutionstepmultiplyright + S (pfd_difference_data_unique_secondexecutionstep) = (p)) /\\ ((((exists pfa_gap_data_unique_secondexecutionstepmultiplyresultbound. pfa_gap_data_unique_secondexecutionstepmultiplyresultbound + S (pfd_value_data_unique_secondexecution) = (p)) /\\ ((exists pfa_offset_left_data_unique_secondexecutionstepmultiplyresultcongruence pfa_offset_right_data_unique_secondexecutionstepmultiplyresultcongruence. ((K) * (pfd_difference_data_unique_secondexecutionstep)) + (p) * pfa_offset_left_data_unique_secondexecutionstepmultiplyresultcongruence = (pfd_value_data_unique_secondexecution) + (p) * pfa_offset_right_data_unique_secondexecutionstepmultiplyresultcongruence)))))))))))))))))))))))))) -> (((b=B) /\\ (((k=K) /\\ (((q=Q) /\\ ((forall mdr_i_pfp_data_unique_quotient mdr_a_pfp_data_unique_quotient. 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"bc1d5686d3af25ced058fc4737540ce8bceb39164c1e4b5a8cb8d6abfc2e9301", "membership": "alpha_only", "name": "prime_field_polynomial_division_residual_data_functional", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro ab", "intro ac", "intro L", "intro bb", "intro bc", "intro d", "intro qb", "intro qc", "intro QB", "intro QC", "intro q", "intro pb", "intro pc", "intro ub", "intro uc", "intro t", "intro rb", "intro rc", "intro R", "intro PB", "intro PC", "intro UB", "intro UC", "intro T", "intro RB", "intro RC", "intro K", "intro hequal", "intro hfirst", "intro hsecond", "cases hfirst", "cases hfirst_right", "cases hsecond", "cases hsecond_right", "have hproduct : forall pfc_index_residual_unique_recode. (exists pfa_gap_residual_unique_recodebound. pfa_gap_residual_unique_recodebound + S (pfc_index_residual_unique_recode) = (L)) -> exists pfc_value_residual_unique_recode. ((((exists ff_h_pfp_residual_unique_recodeentry. ff_h_pfp_residual_unique_recodeentry + S (pfc_value_residual_unique_recode) = S ((S (pfc_index_residual_unique_recode)) * pc)) /\\ exists ff_q_pfp_residual_unique_recodeentry. pb = ff_q_pfp_residual_unique_recodeentry * S ((S (pfc_index_residual_unique_recode)) * pc) + (pfc_value_residual_unique_recode))) /\\ ((exists pfc_terms_code_residual_unique_recodecoefficient pfc_terms_scale_residual_unique_recodecoefficient pfc_natural_sum_residual_unique_recodecoefficient. ((forall pfc_index_residual_unique_recodecoefficientdiagonal. (exists pfa_gap_residual_unique_recodecoefficientdiagonalbound. pfa_gap_residual_unique_recodecoefficientdiagonalbound + S (pfc_index_residual_unique_recodecoefficientdiagonal) = (S (pfc_index_residual_unique_recode))) -> exists pfc_value_residual_unique_recodecoefficientdiagonal. ((((exists ff_h_pfp_residual_unique_recodecoefficientdiagonalentry. ff_h_pfp_residual_unique_recodecoefficientdiagonalentry + S (pfc_value_residual_unique_recodecoefficientdiagonal) = S ((S (pfc_index_residual_unique_recodecoefficientdiagonal)) * pfc_terms_scale_residual_unique_recodecoefficient)) /\\ exists ff_q_pfp_residual_unique_recodecoefficientdiagonalentry. pfc_terms_code_residual_unique_recodecoefficient = ff_q_pfp_residual_unique_recodecoefficientdiagonalentry * S ((S (pfc_index_residual_unique_recodecoefficientdiagonal)) * pfc_terms_scale_residual_unique_recodecoefficient) + (pfc_value_residual_unique_recodecoefficientdiagonal))) /\\ ((exists pfc_complement_residual_unique_recodecoefficientdiagonalterm pfc_left_residual_unique_recodecoefficientdiagonalterm pfc_right_residual_unique_recodecoefficientdiagonalterm. (((pfc_index_residual_unique_recodecoefficientdiagonal)+pfc_complement_residual_unique_recodecoefficientdiagonalterm=(pfc_index_residual_unique_recode)) /\\ ((((((exists pfa_gap_residual_unique_recodecoefficientdiagonaltermleftinside. pfa_gap_residual_unique_recodecoefficientdiagonaltermleftinside + S (pfc_index_residual_unique_recodecoefficientdiagonal) = (q)) /\\ ((((exists ff_h_pfp_residual_unique_recodecoefficientdiagonaltermleftentry. ff_h_pfp_residual_unique_recodecoefficientdiagonaltermleftentry + S (pfc_left_residual_unique_recodecoefficientdiagonalterm) = S ((S (pfc_index_residual_unique_recodecoefficientdiagonal)) * QC)) /\\ exists ff_q_pfp_residual_unique_recodecoefficientdiagonaltermleftentry. QB = ff_q_pfp_residual_unique_recodecoefficientdiagonaltermleftentry * S ((S (pfc_index_residual_unique_recodecoefficientdiagonal)) * QC) + (pfc_left_residual_unique_recodecoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_residual_unique_recodecoefficientdiagonaltermleftoutside. pfc_gap_residual_unique_recodecoefficientdiagonaltermleftoutside+(q)=(pfc_index_residual_unique_recodecoefficientdiagonal)) /\\ (((pfc_left_residual_unique_recodecoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_residual_unique_recodecoefficientdiagonaltermrightinside. pfa_gap_residual_unique_recodecoefficientdiagonaltermrightinside + S (pfc_complement_residual_unique_recodecoefficientdiagonalterm) = (S d)) /\\ ((((exists ff_h_pfp_residual_unique_recodecoefficientdiagonaltermrightentry. ff_h_pfp_residual_unique_recodecoefficientdiagonaltermrightentry + S (pfc_right_residual_unique_recodecoefficientdiagonalterm) = S ((S (pfc_complement_residual_unique_recodecoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_residual_unique_recodecoefficientdiagonaltermrightentry. bb = ff_q_pfp_residual_unique_recodecoefficientdiagonaltermrightentry * S ((S (pfc_complement_residual_unique_recodecoefficientdiagonalterm)) * bc) + (pfc_right_residual_unique_recodecoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_residual_unique_recodecoefficientdiagonaltermrightoutside. pfc_gap_residual_unique_recodecoefficientdiagonaltermrightoutside+(S d)=(pfc_complement_residual_unique_recodecoefficientdiagonalterm)) /\\ (((pfc_right_residual_unique_recodecoefficientdiagonalterm)=0))))) /\\ (((pfc_value_residual_unique_recodecoefficientdiagonal)=pfc_left_residual_unique_recodecoefficientdiagonalterm*pfc_right_residual_unique_recodecoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_residual_unique_recodecoefficientsum fs_v_pfc_residual_unique_recodecoefficientsum. ((((exists fs_h_pfc_residual_unique_recodecoefficientsum_body_start. fs_h_pfc_residual_unique_recodecoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_residual_unique_recodecoefficientsum)) /\\ exists fs_q_pfc_residual_unique_recodecoefficientsum_body_start. fs_u_pfc_residual_unique_recodecoefficientsum = fs_q_pfc_residual_unique_recodecoefficientsum_body_start * S ((S (0)) * fs_v_pfc_residual_unique_recodecoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_residual_unique_recodecoefficientsum_body_terminal. fs_h_pfc_residual_unique_recodecoefficientsum_body_terminal + S (pfc_natural_sum_residual_unique_recodecoefficient) = S ((S (S (pfc_index_residual_unique_recode))) * fs_v_pfc_residual_unique_recodecoefficientsum)) /\\ exists fs_q_pfc_residual_unique_recodecoefficientsum_body_terminal. fs_u_pfc_residual_unique_recodecoefficientsum = fs_q_pfc_residual_unique_recodecoefficientsum_body_terminal * S ((S (S (pfc_index_residual_unique_recode))) * fs_v_pfc_residual_unique_recodecoefficientsum) + (pfc_natural_sum_residual_unique_recodecoefficient))) /\\ forall fs_i_pfc_residual_unique_recodecoefficientsum_body_steps. (exists fs_lt_pfc_residual_unique_recodecoefficientsum_body_steps_bound. fs_lt_pfc_residual_unique_recodecoefficientsum_body_steps_bound + S fs_i_pfc_residual_unique_recodecoefficientsum_body_steps = S (pfc_index_residual_unique_recode)) -> exists fs_a_pfc_residual_unique_recodecoefficientsum_body_steps fs_r_pfc_residual_unique_recodecoefficientsum_body_steps fs_s_pfc_residual_unique_recodecoefficientsum_body_steps. ((((exists fs_h_pfc_residual_unique_recodecoefficientsum_body_steps_summand. fs_h_pfc_residual_unique_recodecoefficientsum_body_steps_summand + S (fs_a_pfc_residual_unique_recodecoefficientsum_body_steps) = S ((S (fs_i_pfc_residual_unique_recodecoefficientsum_body_steps)) * pfc_terms_scale_residual_unique_recodecoefficient)) /\\ exists fs_q_pfc_residual_unique_recodecoefficientsum_body_steps_summand. pfc_terms_code_residual_unique_recodecoefficient = fs_q_pfc_residual_unique_recodecoefficientsum_body_steps_summand * S ((S (fs_i_pfc_residual_unique_recodecoefficientsum_body_steps)) * pfc_terms_scale_residual_unique_recodecoefficient) + (fs_a_pfc_residual_unique_recodecoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_residual_unique_recodecoefficientsum_body_steps_partial. fs_h_pfc_residual_unique_recodecoefficientsum_body_steps_partial + S (fs_r_pfc_residual_unique_recodecoefficientsum_body_steps) = S ((S (fs_i_pfc_residual_unique_recodecoefficientsum_body_steps)) * fs_v_pfc_residual_unique_recodecoefficientsum)) /\\ exists fs_q_pfc_residual_unique_recodecoefficientsum_body_steps_partial. fs_u_pfc_residual_unique_recodecoefficientsum = fs_q_pfc_residual_unique_recodecoefficientsum_body_steps_partial * S ((S (fs_i_pfc_residual_unique_recodecoefficientsum_body_steps)) * fs_v_pfc_residual_unique_recodecoefficientsum) + (fs_r_pfc_residual_unique_recodecoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_residual_unique_recodecoefficientsum_body_steps_successor. fs_h_pfc_residual_unique_recodecoefficientsum_body_steps_successor + S (fs_s_pfc_residual_unique_recodecoefficientsum_body_steps) = S ((S (S fs_i_pfc_residual_unique_recodecoefficientsum_body_steps)) * fs_v_pfc_residual_unique_recodecoefficientsum)) /\\ exists fs_q_pfc_residual_unique_recodecoefficientsum_body_steps_successor. fs_u_pfc_residual_unique_recodecoefficientsum = fs_q_pfc_residual_unique_recodecoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_residual_unique_recodecoefficientsum_body_steps)) * fs_v_pfc_residual_unique_recodecoefficientsum) + (fs_s_pfc_residual_unique_recodecoefficientsum_body_steps))) /\\ fs_s_pfc_residual_unique_recodecoefficientsum_body_steps = fs_r_pfc_residual_unique_recodecoefficientsum_body_steps + fs_a_pfc_residual_unique_recodecoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_residual_unique_recodecoefficientresiduebound. pfa_gap_residual_unique_recodecoefficientresiduebound + S (pfc_value_residual_unique_recode) = (p)) /\\ ((exists pfa_offset_left_residual_unique_recodecoefficientresiduecongruence pfa_offset_right_residual_unique_recodecoefficientresiduecongruence. (pfc_natural_sum_residual_unique_recodecoefficient) + (p) * pfa_offset_left_residual_unique_recodecoefficientresiduecongruence = (pfc_value_residual_unique_recode) + (p) * pfa_offset_right_residual_unique_recodecoefficientresiduecongruence)))))))))))", "specialize prime_field_convolution_prefix_input_transport (p)", "specialize prime_field_convolution_prefix_input_transport (qb)", "specialize prime_field_convolution_prefix_input_transport (qc)", "specialize prime_field_convolution_prefix_input_transport (q)", "specialize prime_field_convolution_prefix_input_transport (bb)", "specialize prime_field_convolution_prefix_input_transport (bc)", "specialize prime_field_convolution_prefix_input_transport (S d)", "specialize prime_field_convolution_prefix_input_transport (QB)", "specialize prime_field_convolution_prefix_input_transport (QC)", "specialize prime_field_convolution_prefix_input_transport (bb)", "specialize prime_field_convolution_prefix_input_transport (bc)", "specialize prime_field_convolution_prefix_input_transport (pb)", "specialize prime_field_convolution_prefix_input_transport (pc)", "specialize prime_field_convolution_prefix_input_transport (L)", "apply prime_field_convolution_prefix_input_transport", "exact hequal", "intro i", "intro a", "intro hindex", "intro hvalue", "exact hvalue", "exact hfirst_left", "have hproducts : forall mdr_i_pfp_residual_unique_products mdr_a_pfp_residual_unique_products. (exists mdr_gap_pfp_residual_unique_productsb. mdr_gap_pfp_residual_unique_productsb + S (mdr_i_pfp_residual_unique_products) = (L)) -> (((exists ff_h_mdr_pfp_residual_unique_productso. ff_h_mdr_pfp_residual_unique_productso + S (mdr_a_pfp_residual_unique_products) = S ((S (mdr_i_pfp_residual_unique_products)) * pc)) /\\ exists ff_q_mdr_pfp_residual_unique_productso. pb = ff_q_mdr_pfp_residual_unique_productso * S ((S (mdr_i_pfp_residual_unique_products)) * pc) + (mdr_a_pfp_residual_unique_products))) -> (((exists ff_h_mdr_pfp_residual_unique_productsn. ff_h_mdr_pfp_residual_unique_productsn + S (mdr_a_pfp_residual_unique_products) = S ((S (mdr_i_pfp_residual_unique_products)) * PC)) /\\ exists ff_q_mdr_pfp_residual_unique_productsn. PB = ff_q_mdr_pfp_residual_unique_productsn * S ((S (mdr_i_pfp_residual_unique_products)) * PC) + (mdr_a_pfp_residual_unique_products)))", "specialize prime_field_convolution_prefix_functional (p)", "specialize prime_field_convolution_prefix_functional (QB)", "specialize prime_field_convolution_prefix_functional (QC)", "specialize prime_field_convolution_prefix_functional (q)", "specialize prime_field_convolution_prefix_functional (bb)", "specialize prime_field_convolution_prefix_functional (bc)", "specialize prime_field_convolution_prefix_functional (S d)", "specialize prime_field_convolution_prefix_functional (pb)", "specialize prime_field_convolution_prefix_functional (pc)", "specialize prime_field_convolution_prefix_functional (PB)", "specialize prime_field_convolution_prefix_functional (PC)", "specialize prime_field_convolution_prefix_functional (L)", "apply prime_field_convolution_prefix_functional", "exact hproduct", "exact hsecond_left", "have hsubtract : forall pfs_index_residual_unique_difference. (exists pfa_gap_residual_unique_differenceindex. pfa_gap_residual_unique_differenceindex + S (pfs_index_residual_unique_difference) = (L)) -> exists pfs_left_residual_unique_difference pfs_right_residual_unique_difference pfs_result_residual_unique_difference. ((((exists ff_h_pfp_residual_unique_differenceleft. ff_h_pfp_residual_unique_differenceleft + S (pfs_left_residual_unique_difference) = S ((S (pfs_index_residual_unique_difference)) * ac)) /\\ exists ff_q_pfp_residual_unique_differenceleft. ab = ff_q_pfp_residual_unique_differenceleft * S ((S (pfs_index_residual_unique_difference)) * ac) + (pfs_left_residual_unique_difference))) /\\ (((((exists ff_h_pfp_residual_unique_differenceright. ff_h_pfp_residual_unique_differenceright + S (pfs_right_residual_unique_difference) = S ((S (pfs_index_residual_unique_difference)) * PC)) /\\ exists ff_q_pfp_residual_unique_differenceright. PB = ff_q_pfp_residual_unique_differenceright * S ((S (pfs_index_residual_unique_difference)) * PC) + (pfs_right_residual_unique_difference))) /\\ (((((exists ff_h_pfp_residual_unique_differenceresult. ff_h_pfp_residual_unique_differenceresult + S (pfs_result_residual_unique_difference) = S ((S (pfs_index_residual_unique_difference)) * uc)) /\\ exists ff_q_pfp_residual_unique_differenceresult. ub = ff_q_pfp_residual_unique_differenceresult * S ((S (pfs_index_residual_unique_difference)) * uc) + (pfs_result_residual_unique_difference))) /\\ ((((exists pfa_gap_residual_unique_differenceoperationleft. pfa_gap_residual_unique_differenceoperationleft + S (pfs_right_residual_unique_difference) = (p)) /\\ (((exists pfa_gap_residual_unique_differenceoperationright. pfa_gap_residual_unique_differenceoperationright + S (pfs_result_residual_unique_difference) = (p)) /\\ ((((exists pfa_gap_residual_unique_differenceoperationresultbound. pfa_gap_residual_unique_differenceoperationresultbound + S (pfs_left_residual_unique_difference) = (p)) /\\ ((exists pfa_offset_left_residual_unique_differenceoperationresultcongruence pfa_offset_right_residual_unique_differenceoperationresultcongruence. ((pfs_right_residual_unique_difference) + (pfs_result_residual_unique_difference)) + (p) * pfa_offset_left_residual_unique_differenceoperationresultcongruence = (pfs_left_residual_unique_difference) + (p) * pfa_offset_right_residual_unique_differenceoperationresultcongruence)))))))))))))))", "specialize prime_field_polynomial_subtract_transport (p)", "specialize prime_field_polynomial_subtract_transport (ab)", "specialize prime_field_polynomial_subtract_transport (ac)", "specialize prime_field_polynomial_subtract_transport (pb)", "specialize prime_field_polynomial_subtract_transport (pc)", "specialize prime_field_polynomial_subtract_transport (ub)", "specialize prime_field_polynomial_subtract_transport (uc)", "specialize prime_field_polynomial_subtract_transport (ab)", "specialize prime_field_polynomial_subtract_transport (ac)", "specialize prime_field_polynomial_subtract_transport (PB)", "specialize prime_field_polynomial_subtract_transport (PC)", "specialize prime_field_polynomial_subtract_transport (ub)", "specialize prime_field_polynomial_subtract_transport (uc)", "specialize prime_field_polynomial_subtract_transport (L)", "apply prime_field_polynomial_subtract_transport", "intro i", "intro a", "intro hindex", "intro hvalue", "exact hvalue", "exact hproducts", "intro i", "intro a", "intro hindex", "intro hvalue", "exact hvalue", "exact hfirst_right_left", "have hresiduals : forall mdr_i_pfp_residual_unique_inputs mdr_a_pfp_residual_unique_inputs. (exists mdr_gap_pfp_residual_unique_inputsb. mdr_gap_pfp_residual_unique_inputsb + S (mdr_i_pfp_residual_unique_inputs) = (L)) -> (((exists ff_h_mdr_pfp_residual_unique_inputso. ff_h_mdr_pfp_residual_unique_inputso + S (mdr_a_pfp_residual_unique_inputs) = S ((S (mdr_i_pfp_residual_unique_inputs)) * uc)) /\\ exists ff_q_mdr_pfp_residual_unique_inputso. ub = ff_q_mdr_pfp_residual_unique_inputso * S ((S (mdr_i_pfp_residual_unique_inputs)) * uc) + (mdr_a_pfp_residual_unique_inputs))) -> (((exists ff_h_mdr_pfp_residual_unique_inputsn. ff_h_mdr_pfp_residual_unique_inputsn + S (mdr_a_pfp_residual_unique_inputs) = S ((S (mdr_i_pfp_residual_unique_inputs)) * UC)) /\\ exists ff_q_mdr_pfp_residual_unique_inputsn. UB = ff_q_mdr_pfp_residual_unique_inputsn * S ((S (mdr_i_pfp_residual_unique_inputs)) * UC) + (mdr_a_pfp_residual_unique_inputs)))", "specialize prime_field_polynomial_subtract_functional (p)", "specialize prime_field_polynomial_subtract_functional (ab)", "specialize prime_field_polynomial_subtract_functional (ac)", "specialize prime_field_polynomial_subtract_functional (PB)", "specialize prime_field_polynomial_subtract_functional (PC)", "specialize prime_field_polynomial_subtract_functional (ub)", "specialize prime_field_polynomial_subtract_functional (uc)", "specialize prime_field_polynomial_subtract_functional (UB)", "specialize prime_field_polynomial_subtract_functional (UC)", "specialize prime_field_polynomial_subtract_functional (L)", "apply prime_field_polynomial_subtract_functional", "exact hsubtract", "exact hsecond_right_left", "have htrim : (((L)=(t)+(R)) /\\ (((forall fom_index_pfp_residual_unique_trim_recodeinput. (exists fom_gap_pfp_residual_unique_trim_recodeinput_index_bound. fom_gap_pfp_residual_unique_trim_recodeinput_index_bound + S (fom_index_pfp_residual_unique_trim_recodeinput) = L) -> exists fom_value_pfp_residual_unique_trim_recodeinput. ((((exists fom_beta_height_pfp_residual_unique_trim_recodeinput_entry. fom_beta_height_pfp_residual_unique_trim_recodeinput_entry + S (fom_value_pfp_residual_unique_trim_recodeinput) = S ((S (fom_index_pfp_residual_unique_trim_recodeinput)) * UC)) /\\ exists fom_beta_quotient_pfp_residual_unique_trim_recodeinput_entry. UB = fom_beta_quotient_pfp_residual_unique_trim_recodeinput_entry * S ((S (fom_index_pfp_residual_unique_trim_recodeinput)) * UC) + (fom_value_pfp_residual_unique_trim_recodeinput))) /\\ (exists fom_gap_pfp_residual_unique_trim_recodeinput_value_bound. fom_gap_pfp_residual_unique_trim_recodeinput_value_bound + S (fom_value_pfp_residual_unique_trim_recodeinput) = p))) /\\ (((forall pfp_repeat_index_residual_unique_trim_recoderemoved. (exists pfa_gap_residual_unique_trim_recoderemovedindex. pfa_gap_residual_unique_trim_recoderemovedindex + S (pfp_repeat_index_residual_unique_trim_recoderemoved) = (t)) -> (((exists ff_h_pfp_residual_unique_trim_recoderemovedentry. ff_h_pfp_residual_unique_trim_recoderemovedentry + S (0) = S ((S (pfp_repeat_index_residual_unique_trim_recoderemoved)) * UC)) /\\ exists ff_q_pfp_residual_unique_trim_recoderemovedentry. UB = ff_q_pfp_residual_unique_trim_recoderemovedentry * S ((S (pfp_repeat_index_residual_unique_trim_recoderemoved)) * UC) + (0)))) /\\ (((forall pftrim_index_residual_unique_trim_recodesuffix pftrim_value_residual_unique_trim_recodesuffix. (exists pfa_gap_residual_unique_trim_recodesuffixbound. pfa_gap_residual_unique_trim_recodesuffixbound + S (pftrim_index_residual_unique_trim_recodesuffix) = (R)) -> (((exists ff_h_pfp_residual_unique_trim_recodesuffixsource. ff_h_pfp_residual_unique_trim_recodesuffixsource + S (pftrim_value_residual_unique_trim_recodesuffix) = S ((S ((t)+pftrim_index_residual_unique_trim_recodesuffix)) * UC)) /\\ exists ff_q_pfp_residual_unique_trim_recodesuffixsource. UB = ff_q_pfp_residual_unique_trim_recodesuffixsource * S ((S ((t)+pftrim_index_residual_unique_trim_recodesuffix)) * UC) + (pftrim_value_residual_unique_trim_recodesuffix))) -> (((exists ff_h_pfp_residual_unique_trim_recodesuffixoutput. ff_h_pfp_residual_unique_trim_recodesuffixoutput + S (pftrim_value_residual_unique_trim_recodesuffix) = S ((S (pftrim_index_residual_unique_trim_recodesuffix)) * rc)) /\\ exists ff_q_pfp_residual_unique_trim_recodesuffixoutput. rb = ff_q_pfp_residual_unique_trim_recodesuffixoutput * S ((S (pftrim_index_residual_unique_trim_recodesuffix)) * rc) + (pftrim_value_residual_unique_trim_recodesuffix)))) /\\ (((R)=0 \\/ (exists pftrim_leading_residual_unique_trim_recodenormal. ((((exists ff_h_pfp_residual_unique_trim_recodenormalentry. ff_h_pfp_residual_unique_trim_recodenormalentry + S (pftrim_leading_residual_unique_trim_recodenormal) = S ((S (0)) * rc)) /\\ exists ff_q_pfp_residual_unique_trim_recodenormalentry. rb = ff_q_pfp_residual_unique_trim_recodenormalentry * S ((S (0)) * rc) + (pftrim_leading_residual_unique_trim_recodenormal))) /\\ ((~(pftrim_leading_residual_unique_trim_recodenormal=0))))))))))))))", "specialize prime_field_polynomial_trim_input_transport (p)", "specialize prime_field_polynomial_trim_input_transport (ub)", "specialize prime_field_polynomial_trim_input_transport (uc)", "specialize prime_field_polynomial_trim_input_transport (UB)", "specialize prime_field_polynomial_trim_input_transport (UC)", "specialize prime_field_polynomial_trim_input_transport (L)", "specialize prime_field_polynomial_trim_input_transport (t)", "specialize prime_field_polynomial_trim_input_transport (rb)", "specialize prime_field_polynomial_trim_input_transport (rc)", "specialize prime_field_polynomial_trim_input_transport (R)", "apply prime_field_polynomial_trim_input_transport", "exact hresiduals", "exact hfirst_right_right", "split", "specialize prime_field_polynomial_trim_removed_count_unique (p)", "specialize prime_field_polynomial_trim_removed_count_unique (UB)", "specialize prime_field_polynomial_trim_removed_count_unique (UC)", "specialize prime_field_polynomial_trim_removed_count_unique (L)", "specialize prime_field_polynomial_trim_removed_count_unique (t)", "specialize prime_field_polynomial_trim_removed_count_unique (rb)", "specialize prime_field_polynomial_trim_removed_count_unique (rc)", "specialize prime_field_polynomial_trim_removed_count_unique (R)", "specialize prime_field_polynomial_trim_removed_count_unique (T)", "specialize prime_field_polynomial_trim_removed_count_unique (RB)", "specialize prime_field_polynomial_trim_removed_count_unique (RC)", "specialize prime_field_polynomial_trim_removed_count_unique (K)", "apply prime_field_polynomial_trim_removed_count_unique", "exact htrim", "exact hsecond_right_right", "split", "specialize prime_field_polynomial_trim_retained_length_unique (p)", "specialize prime_field_polynomial_trim_retained_length_unique (UB)", "specialize prime_field_polynomial_trim_retained_length_unique (UC)", "specialize prime_field_polynomial_trim_retained_length_unique (L)", "specialize prime_field_polynomial_trim_retained_length_unique (t)", "specialize prime_field_polynomial_trim_retained_length_unique (rb)", "specialize prime_field_polynomial_trim_retained_length_unique (rc)", "specialize prime_field_polynomial_trim_retained_length_unique (R)", "specialize prime_field_polynomial_trim_retained_length_unique (T)", "specialize prime_field_polynomial_trim_retained_length_unique (RB)", "specialize prime_field_polynomial_trim_retained_length_unique (RC)", "specialize prime_field_polynomial_trim_retained_length_unique (K)", "apply prime_field_polynomial_trim_retained_length_unique", "exact htrim", "exact hsecond_right_right", "specialize prime_field_polynomial_trim_output_equal (p)", "specialize prime_field_polynomial_trim_output_equal (UB)", "specialize prime_field_polynomial_trim_output_equal (UC)", "specialize prime_field_polynomial_trim_output_equal (L)", "specialize prime_field_polynomial_trim_output_equal (t)", "specialize prime_field_polynomial_trim_output_equal (rb)", "specialize prime_field_polynomial_trim_output_equal (rc)", "specialize prime_field_polynomial_trim_output_equal (R)", "specialize prime_field_polynomial_trim_output_equal (T)", "specialize prime_field_polynomial_trim_output_equal (RB)", "specialize prime_field_polynomial_trim_output_equal (RC)", "specialize prime_field_polynomial_trim_output_equal (K)", "apply prime_field_polynomial_trim_output_equal", "exact htrim", "exact hsecond_right_right" ], "script_sha256": "a8f932d1cdd05af1f2cdbe73ff7662ec7f5986070f17aa354aebfdff8984f6b1", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_division_uniqueness_candidate.py", "sha256": "6a9d9ebe1f72202743e5df2c069b9aa367fdb3d61108f1d9354cdc9276ab2d15" }, "statement": "forall p ab ac L bb bc d qb qc QB QC q pb pc ub uc t rb rc R PB PC UB UC T RB RC K. (forall mdr_i_pfp_residual_unique_quotients mdr_a_pfp_residual_unique_quotients. (exists mdr_gap_pfp_residual_unique_quotientsb. mdr_gap_pfp_residual_unique_quotientsb + S (mdr_i_pfp_residual_unique_quotients) = (q)) -> (((exists ff_h_mdr_pfp_residual_unique_quotientso. ff_h_mdr_pfp_residual_unique_quotientso + S (mdr_a_pfp_residual_unique_quotients) = S ((S (mdr_i_pfp_residual_unique_quotients)) * qc)) /\\ exists ff_q_mdr_pfp_residual_unique_quotientso. qb = ff_q_mdr_pfp_residual_unique_quotientso * S ((S (mdr_i_pfp_residual_unique_quotients)) * qc) + (mdr_a_pfp_residual_unique_quotients))) -> (((exists ff_h_mdr_pfp_residual_unique_quotientsn. ff_h_mdr_pfp_residual_unique_quotientsn + S (mdr_a_pfp_residual_unique_quotients) = S ((S (mdr_i_pfp_residual_unique_quotients)) * QC)) /\\ exists ff_q_mdr_pfp_residual_unique_quotientsn. QB = ff_q_mdr_pfp_residual_unique_quotientsn * S ((S (mdr_i_pfp_residual_unique_quotients)) * QC) + (mdr_a_pfp_residual_unique_quotients)))) -> (((forall pfc_index_residual_unique_firstproduct. (exists pfa_gap_residual_unique_firstproductbound. pfa_gap_residual_unique_firstproductbound + S (pfc_index_residual_unique_firstproduct) = (L)) -> exists pfc_value_residual_unique_firstproduct. ((((exists ff_h_pfp_residual_unique_firstproductentry. ff_h_pfp_residual_unique_firstproductentry + S (pfc_value_residual_unique_firstproduct) = S ((S (pfc_index_residual_unique_firstproduct)) * pc)) /\\ exists ff_q_pfp_residual_unique_firstproductentry. pb = ff_q_pfp_residual_unique_firstproductentry * S ((S (pfc_index_residual_unique_firstproduct)) * pc) + (pfc_value_residual_unique_firstproduct))) /\\ ((exists pfc_terms_code_residual_unique_firstproductcoefficient pfc_terms_scale_residual_unique_firstproductcoefficient pfc_natural_sum_residual_unique_firstproductcoefficient. ((forall pfc_index_residual_unique_firstproductcoefficientdiagonal. (exists pfa_gap_residual_unique_firstproductcoefficientdiagonalbound. pfa_gap_residual_unique_firstproductcoefficientdiagonalbound + S (pfc_index_residual_unique_firstproductcoefficientdiagonal) = (S (pfc_index_residual_unique_firstproduct))) -> exists pfc_value_residual_unique_firstproductcoefficientdiagonal. ((((exists ff_h_pfp_residual_unique_firstproductcoefficientdiagonalentry. ff_h_pfp_residual_unique_firstproductcoefficientdiagonalentry + S (pfc_value_residual_unique_firstproductcoefficientdiagonal) = S ((S (pfc_index_residual_unique_firstproductcoefficientdiagonal)) * pfc_terms_scale_residual_unique_firstproductcoefficient)) /\\ exists ff_q_pfp_residual_unique_firstproductcoefficientdiagonalentry. pfc_terms_code_residual_unique_firstproductcoefficient = ff_q_pfp_residual_unique_firstproductcoefficientdiagonalentry * S ((S (pfc_index_residual_unique_firstproductcoefficientdiagonal)) * pfc_terms_scale_residual_unique_firstproductcoefficient) + (pfc_value_residual_unique_firstproductcoefficientdiagonal))) /\\ ((exists pfc_complement_residual_unique_firstproductcoefficientdiagonalterm pfc_left_residual_unique_firstproductcoefficientdiagonalterm pfc_right_residual_unique_firstproductcoefficientdiagonalterm. (((pfc_index_residual_unique_firstproductcoefficientdiagonal)+pfc_complement_residual_unique_firstproductcoefficientdiagonalterm=(pfc_index_residual_unique_firstproduct)) /\\ ((((((exists pfa_gap_residual_unique_firstproductcoefficientdiagonaltermleftinside. pfa_gap_residual_unique_firstproductcoefficientdiagonaltermleftinside + S (pfc_index_residual_unique_firstproductcoefficientdiagonal) = (q)) /\\ ((((exists ff_h_pfp_residual_unique_firstproductcoefficientdiagonaltermleftentry. ff_h_pfp_residual_unique_firstproductcoefficientdiagonaltermleftentry + S (pfc_left_residual_unique_firstproductcoefficientdiagonalterm) = S ((S (pfc_index_residual_unique_firstproductcoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_residual_unique_firstproductcoefficientdiagonaltermleftentry. qb = ff_q_pfp_residual_unique_firstproductcoefficientdiagonaltermleftentry * S ((S (pfc_index_residual_unique_firstproductcoefficientdiagonal)) * qc) + (pfc_left_residual_unique_firstproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_residual_unique_firstproductcoefficientdiagonaltermleftoutside. pfc_gap_residual_unique_firstproductcoefficientdiagonaltermleftoutside+(q)=(pfc_index_residual_unique_firstproductcoefficientdiagonal)) /\\ (((pfc_left_residual_unique_firstproductcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_residual_unique_firstproductcoefficientdiagonaltermrightinside. pfa_gap_residual_unique_firstproductcoefficientdiagonaltermrightinside + S (pfc_complement_residual_unique_firstproductcoefficientdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_residual_unique_firstproductcoefficientdiagonaltermrightentry. ff_h_pfp_residual_unique_firstproductcoefficientdiagonaltermrightentry + S (pfc_right_residual_unique_firstproductcoefficientdiagonalterm) = S ((S (pfc_complement_residual_unique_firstproductcoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_residual_unique_firstproductcoefficientdiagonaltermrightentry. bb = ff_q_pfp_residual_unique_firstproductcoefficientdiagonaltermrightentry * S ((S (pfc_complement_residual_unique_firstproductcoefficientdiagonalterm)) * bc) + (pfc_right_residual_unique_firstproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_residual_unique_firstproductcoefficientdiagonaltermrightoutside. pfc_gap_residual_unique_firstproductcoefficientdiagonaltermrightoutside+(S (d))=(pfc_complement_residual_unique_firstproductcoefficientdiagonalterm)) /\\ (((pfc_right_residual_unique_firstproductcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_residual_unique_firstproductcoefficientdiagonal)=pfc_left_residual_unique_firstproductcoefficientdiagonalterm*pfc_right_residual_unique_firstproductcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_residual_unique_firstproductcoefficientsum fs_v_pfc_residual_unique_firstproductcoefficientsum. ((((exists fs_h_pfc_residual_unique_firstproductcoefficientsum_body_start. fs_h_pfc_residual_unique_firstproductcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_residual_unique_firstproductcoefficientsum)) /\\ exists fs_q_pfc_residual_unique_firstproductcoefficientsum_body_start. fs_u_pfc_residual_unique_firstproductcoefficientsum = fs_q_pfc_residual_unique_firstproductcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_residual_unique_firstproductcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_residual_unique_firstproductcoefficientsum_body_terminal. fs_h_pfc_residual_unique_firstproductcoefficientsum_body_terminal + S (pfc_natural_sum_residual_unique_firstproductcoefficient) = S ((S (S (pfc_index_residual_unique_firstproduct))) * fs_v_pfc_residual_unique_firstproductcoefficientsum)) /\\ exists fs_q_pfc_residual_unique_firstproductcoefficientsum_body_terminal. fs_u_pfc_residual_unique_firstproductcoefficientsum = fs_q_pfc_residual_unique_firstproductcoefficientsum_body_terminal * S ((S (S (pfc_index_residual_unique_firstproduct))) * fs_v_pfc_residual_unique_firstproductcoefficientsum) + (pfc_natural_sum_residual_unique_firstproductcoefficient))) /\\ forall fs_i_pfc_residual_unique_firstproductcoefficientsum_body_steps. (exists fs_lt_pfc_residual_unique_firstproductcoefficientsum_body_steps_bound. fs_lt_pfc_residual_unique_firstproductcoefficientsum_body_steps_bound + S fs_i_pfc_residual_unique_firstproductcoefficientsum_body_steps = S (pfc_index_residual_unique_firstproduct)) -> exists fs_a_pfc_residual_unique_firstproductcoefficientsum_body_steps fs_r_pfc_residual_unique_firstproductcoefficientsum_body_steps fs_s_pfc_residual_unique_firstproductcoefficientsum_body_steps. ((((exists fs_h_pfc_residual_unique_firstproductcoefficientsum_body_steps_summand. fs_h_pfc_residual_unique_firstproductcoefficientsum_body_steps_summand + S (fs_a_pfc_residual_unique_firstproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_residual_unique_firstproductcoefficientsum_body_steps)) * pfc_terms_scale_residual_unique_firstproductcoefficient)) /\\ exists fs_q_pfc_residual_unique_firstproductcoefficientsum_body_steps_summand. pfc_terms_code_residual_unique_firstproductcoefficient = fs_q_pfc_residual_unique_firstproductcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_residual_unique_firstproductcoefficientsum_body_steps)) * pfc_terms_scale_residual_unique_firstproductcoefficient) + (fs_a_pfc_residual_unique_firstproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_residual_unique_firstproductcoefficientsum_body_steps_partial. fs_h_pfc_residual_unique_firstproductcoefficientsum_body_steps_partial + S (fs_r_pfc_residual_unique_firstproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_residual_unique_firstproductcoefficientsum_body_steps)) * fs_v_pfc_residual_unique_firstproductcoefficientsum)) /\\ exists fs_q_pfc_residual_unique_firstproductcoefficientsum_body_steps_partial. fs_u_pfc_residual_unique_firstproductcoefficientsum = fs_q_pfc_residual_unique_firstproductcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_residual_unique_firstproductcoefficientsum_body_steps)) * fs_v_pfc_residual_unique_firstproductcoefficientsum) + (fs_r_pfc_residual_unique_firstproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_residual_unique_firstproductcoefficientsum_body_steps_successor. fs_h_pfc_residual_unique_firstproductcoefficientsum_body_steps_successor + S (fs_s_pfc_residual_unique_firstproductcoefficientsum_body_steps) = S ((S (S fs_i_pfc_residual_unique_firstproductcoefficientsum_body_steps)) * fs_v_pfc_residual_unique_firstproductcoefficientsum)) /\\ exists fs_q_pfc_residual_unique_firstproductcoefficientsum_body_steps_successor. fs_u_pfc_residual_unique_firstproductcoefficientsum = fs_q_pfc_residual_unique_firstproductcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_residual_unique_firstproductcoefficientsum_body_steps)) * fs_v_pfc_residual_unique_firstproductcoefficientsum) + (fs_s_pfc_residual_unique_firstproductcoefficientsum_body_steps))) /\\ fs_s_pfc_residual_unique_firstproductcoefficientsum_body_steps = fs_r_pfc_residual_unique_firstproductcoefficientsum_body_steps + fs_a_pfc_residual_unique_firstproductcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_residual_unique_firstproductcoefficientresiduebound. pfa_gap_residual_unique_firstproductcoefficientresiduebound + S (pfc_value_residual_unique_firstproduct) = (p)) /\\ ((exists pfa_offset_left_residual_unique_firstproductcoefficientresiduecongruence pfa_offset_right_residual_unique_firstproductcoefficientresiduecongruence. (pfc_natural_sum_residual_unique_firstproductcoefficient) + (p) * pfa_offset_left_residual_unique_firstproductcoefficientresiduecongruence = (pfc_value_residual_unique_firstproduct) + (p) * pfa_offset_right_residual_unique_firstproductcoefficientresiduecongruence)))))))))))) /\\ (((forall pfs_index_residual_unique_firstdifference. (exists pfa_gap_residual_unique_firstdifferenceindex. pfa_gap_residual_unique_firstdifferenceindex + S (pfs_index_residual_unique_firstdifference) = (L)) -> exists pfs_left_residual_unique_firstdifference pfs_right_residual_unique_firstdifference pfs_result_residual_unique_firstdifference. ((((exists ff_h_pfp_residual_unique_firstdifferenceleft. ff_h_pfp_residual_unique_firstdifferenceleft + S (pfs_left_residual_unique_firstdifference) = S ((S (pfs_index_residual_unique_firstdifference)) * ac)) /\\ exists ff_q_pfp_residual_unique_firstdifferenceleft. ab = ff_q_pfp_residual_unique_firstdifferenceleft * S ((S (pfs_index_residual_unique_firstdifference)) * ac) + (pfs_left_residual_unique_firstdifference))) /\\ (((((exists ff_h_pfp_residual_unique_firstdifferenceright. ff_h_pfp_residual_unique_firstdifferenceright + S (pfs_right_residual_unique_firstdifference) = S ((S (pfs_index_residual_unique_firstdifference)) * pc)) /\\ exists ff_q_pfp_residual_unique_firstdifferenceright. pb = ff_q_pfp_residual_unique_firstdifferenceright * S ((S (pfs_index_residual_unique_firstdifference)) * pc) + (pfs_right_residual_unique_firstdifference))) /\\ (((((exists ff_h_pfp_residual_unique_firstdifferenceresult. ff_h_pfp_residual_unique_firstdifferenceresult + S (pfs_result_residual_unique_firstdifference) = S ((S (pfs_index_residual_unique_firstdifference)) * uc)) /\\ exists ff_q_pfp_residual_unique_firstdifferenceresult. ub = ff_q_pfp_residual_unique_firstdifferenceresult * S ((S (pfs_index_residual_unique_firstdifference)) * uc) + (pfs_result_residual_unique_firstdifference))) /\\ ((((exists pfa_gap_residual_unique_firstdifferenceoperationleft. pfa_gap_residual_unique_firstdifferenceoperationleft + S (pfs_right_residual_unique_firstdifference) = (p)) /\\ (((exists pfa_gap_residual_unique_firstdifferenceoperationright. pfa_gap_residual_unique_firstdifferenceoperationright + S (pfs_result_residual_unique_firstdifference) = (p)) /\\ ((((exists pfa_gap_residual_unique_firstdifferenceoperationresultbound. pfa_gap_residual_unique_firstdifferenceoperationresultbound + S (pfs_left_residual_unique_firstdifference) = (p)) /\\ ((exists pfa_offset_left_residual_unique_firstdifferenceoperationresultcongruence pfa_offset_right_residual_unique_firstdifferenceoperationresultcongruence. ((pfs_right_residual_unique_firstdifference) + (pfs_result_residual_unique_firstdifference)) + (p) * pfa_offset_left_residual_unique_firstdifferenceoperationresultcongruence = (pfs_left_residual_unique_firstdifference) + (p) * pfa_offset_right_residual_unique_firstdifferenceoperationresultcongruence)))))))))))))))) /\\ (((((L)=(t)+(R)) /\\ (((forall fom_index_pfp_residual_unique_firsttriminput. (exists fom_gap_pfp_residual_unique_firsttriminput_index_bound. fom_gap_pfp_residual_unique_firsttriminput_index_bound + S (fom_index_pfp_residual_unique_firsttriminput) = L) -> exists fom_value_pfp_residual_unique_firsttriminput. ((((exists fom_beta_height_pfp_residual_unique_firsttriminput_entry. fom_beta_height_pfp_residual_unique_firsttriminput_entry + S (fom_value_pfp_residual_unique_firsttriminput) = S ((S (fom_index_pfp_residual_unique_firsttriminput)) * uc)) /\\ exists fom_beta_quotient_pfp_residual_unique_firsttriminput_entry. ub = fom_beta_quotient_pfp_residual_unique_firsttriminput_entry * S ((S (fom_index_pfp_residual_unique_firsttriminput)) * uc) + (fom_value_pfp_residual_unique_firsttriminput))) /\\ (exists fom_gap_pfp_residual_unique_firsttriminput_value_bound. fom_gap_pfp_residual_unique_firsttriminput_value_bound + S (fom_value_pfp_residual_unique_firsttriminput) = p))) /\\ (((forall pfp_repeat_index_residual_unique_firsttrimremoved. (exists pfa_gap_residual_unique_firsttrimremovedindex. pfa_gap_residual_unique_firsttrimremovedindex + S (pfp_repeat_index_residual_unique_firsttrimremoved) = (t)) -> (((exists ff_h_pfp_residual_unique_firsttrimremovedentry. ff_h_pfp_residual_unique_firsttrimremovedentry + S (0) = S ((S (pfp_repeat_index_residual_unique_firsttrimremoved)) * uc)) /\\ exists ff_q_pfp_residual_unique_firsttrimremovedentry. ub = ff_q_pfp_residual_unique_firsttrimremovedentry * S ((S (pfp_repeat_index_residual_unique_firsttrimremoved)) * uc) + (0)))) /\\ (((forall pftrim_index_residual_unique_firsttrimsuffix pftrim_value_residual_unique_firsttrimsuffix. (exists pfa_gap_residual_unique_firsttrimsuffixbound. pfa_gap_residual_unique_firsttrimsuffixbound + S (pftrim_index_residual_unique_firsttrimsuffix) = (R)) -> (((exists ff_h_pfp_residual_unique_firsttrimsuffixsource. ff_h_pfp_residual_unique_firsttrimsuffixsource + S (pftrim_value_residual_unique_firsttrimsuffix) = S ((S ((t)+pftrim_index_residual_unique_firsttrimsuffix)) * uc)) /\\ exists ff_q_pfp_residual_unique_firsttrimsuffixsource. ub = ff_q_pfp_residual_unique_firsttrimsuffixsource * S ((S ((t)+pftrim_index_residual_unique_firsttrimsuffix)) * uc) + (pftrim_value_residual_unique_firsttrimsuffix))) -> (((exists ff_h_pfp_residual_unique_firsttrimsuffixoutput. ff_h_pfp_residual_unique_firsttrimsuffixoutput + S (pftrim_value_residual_unique_firsttrimsuffix) = S ((S (pftrim_index_residual_unique_firsttrimsuffix)) * rc)) /\\ exists ff_q_pfp_residual_unique_firsttrimsuffixoutput. rb = ff_q_pfp_residual_unique_firsttrimsuffixoutput * S ((S (pftrim_index_residual_unique_firsttrimsuffix)) * rc) + (pftrim_value_residual_unique_firsttrimsuffix)))) /\\ (((R)=0 \\/ (exists pftrim_leading_residual_unique_firsttrimnormal. ((((exists ff_h_pfp_residual_unique_firsttrimnormalentry. ff_h_pfp_residual_unique_firsttrimnormalentry + S (pftrim_leading_residual_unique_firsttrimnormal) = S ((S (0)) * rc)) /\\ exists ff_q_pfp_residual_unique_firsttrimnormalentry. rb = ff_q_pfp_residual_unique_firsttrimnormalentry * S ((S (0)) * rc) + (pftrim_leading_residual_unique_firsttrimnormal))) /\\ ((~(pftrim_leading_residual_unique_firsttrimnormal=0)))))))))))))))))))) -> (((forall pfc_index_residual_unique_secondproduct. (exists pfa_gap_residual_unique_secondproductbound. pfa_gap_residual_unique_secondproductbound + S (pfc_index_residual_unique_secondproduct) = (L)) -> exists pfc_value_residual_unique_secondproduct. ((((exists ff_h_pfp_residual_unique_secondproductentry. ff_h_pfp_residual_unique_secondproductentry + S (pfc_value_residual_unique_secondproduct) = S ((S (pfc_index_residual_unique_secondproduct)) * PC)) /\\ exists ff_q_pfp_residual_unique_secondproductentry. PB = ff_q_pfp_residual_unique_secondproductentry * S ((S (pfc_index_residual_unique_secondproduct)) * PC) + (pfc_value_residual_unique_secondproduct))) /\\ ((exists pfc_terms_code_residual_unique_secondproductcoefficient pfc_terms_scale_residual_unique_secondproductcoefficient pfc_natural_sum_residual_unique_secondproductcoefficient. ((forall pfc_index_residual_unique_secondproductcoefficientdiagonal. (exists pfa_gap_residual_unique_secondproductcoefficientdiagonalbound. pfa_gap_residual_unique_secondproductcoefficientdiagonalbound + S (pfc_index_residual_unique_secondproductcoefficientdiagonal) = (S (pfc_index_residual_unique_secondproduct))) -> exists pfc_value_residual_unique_secondproductcoefficientdiagonal. ((((exists ff_h_pfp_residual_unique_secondproductcoefficientdiagonalentry. ff_h_pfp_residual_unique_secondproductcoefficientdiagonalentry + S (pfc_value_residual_unique_secondproductcoefficientdiagonal) = S ((S (pfc_index_residual_unique_secondproductcoefficientdiagonal)) * pfc_terms_scale_residual_unique_secondproductcoefficient)) /\\ exists ff_q_pfp_residual_unique_secondproductcoefficientdiagonalentry. pfc_terms_code_residual_unique_secondproductcoefficient = ff_q_pfp_residual_unique_secondproductcoefficientdiagonalentry * S ((S (pfc_index_residual_unique_secondproductcoefficientdiagonal)) * pfc_terms_scale_residual_unique_secondproductcoefficient) + (pfc_value_residual_unique_secondproductcoefficientdiagonal))) /\\ ((exists pfc_complement_residual_unique_secondproductcoefficientdiagonalterm pfc_left_residual_unique_secondproductcoefficientdiagonalterm pfc_right_residual_unique_secondproductcoefficientdiagonalterm. (((pfc_index_residual_unique_secondproductcoefficientdiagonal)+pfc_complement_residual_unique_secondproductcoefficientdiagonalterm=(pfc_index_residual_unique_secondproduct)) /\\ ((((((exists pfa_gap_residual_unique_secondproductcoefficientdiagonaltermleftinside. pfa_gap_residual_unique_secondproductcoefficientdiagonaltermleftinside + S (pfc_index_residual_unique_secondproductcoefficientdiagonal) = (q)) /\\ ((((exists ff_h_pfp_residual_unique_secondproductcoefficientdiagonaltermleftentry. ff_h_pfp_residual_unique_secondproductcoefficientdiagonaltermleftentry + S (pfc_left_residual_unique_secondproductcoefficientdiagonalterm) = S ((S (pfc_index_residual_unique_secondproductcoefficientdiagonal)) * QC)) /\\ exists ff_q_pfp_residual_unique_secondproductcoefficientdiagonaltermleftentry. QB = ff_q_pfp_residual_unique_secondproductcoefficientdiagonaltermleftentry * S ((S (pfc_index_residual_unique_secondproductcoefficientdiagonal)) * QC) + (pfc_left_residual_unique_secondproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_residual_unique_secondproductcoefficientdiagonaltermleftoutside. pfc_gap_residual_unique_secondproductcoefficientdiagonaltermleftoutside+(q)=(pfc_index_residual_unique_secondproductcoefficientdiagonal)) /\\ (((pfc_left_residual_unique_secondproductcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_residual_unique_secondproductcoefficientdiagonaltermrightinside. pfa_gap_residual_unique_secondproductcoefficientdiagonaltermrightinside + S (pfc_complement_residual_unique_secondproductcoefficientdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_residual_unique_secondproductcoefficientdiagonaltermrightentry. ff_h_pfp_residual_unique_secondproductcoefficientdiagonaltermrightentry + S (pfc_right_residual_unique_secondproductcoefficientdiagonalterm) = S ((S (pfc_complement_residual_unique_secondproductcoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_residual_unique_secondproductcoefficientdiagonaltermrightentry. bb = ff_q_pfp_residual_unique_secondproductcoefficientdiagonaltermrightentry * S ((S (pfc_complement_residual_unique_secondproductcoefficientdiagonalterm)) * bc) + (pfc_right_residual_unique_secondproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_residual_unique_secondproductcoefficientdiagonaltermrightoutside. pfc_gap_residual_unique_secondproductcoefficientdiagonaltermrightoutside+(S (d))=(pfc_complement_residual_unique_secondproductcoefficientdiagonalterm)) /\\ (((pfc_right_residual_unique_secondproductcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_residual_unique_secondproductcoefficientdiagonal)=pfc_left_residual_unique_secondproductcoefficientdiagonalterm*pfc_right_residual_unique_secondproductcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_residual_unique_secondproductcoefficientsum fs_v_pfc_residual_unique_secondproductcoefficientsum. ((((exists fs_h_pfc_residual_unique_secondproductcoefficientsum_body_start. fs_h_pfc_residual_unique_secondproductcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_residual_unique_secondproductcoefficientsum)) /\\ exists fs_q_pfc_residual_unique_secondproductcoefficientsum_body_start. fs_u_pfc_residual_unique_secondproductcoefficientsum = fs_q_pfc_residual_unique_secondproductcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_residual_unique_secondproductcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_residual_unique_secondproductcoefficientsum_body_terminal. fs_h_pfc_residual_unique_secondproductcoefficientsum_body_terminal + S (pfc_natural_sum_residual_unique_secondproductcoefficient) = S ((S (S (pfc_index_residual_unique_secondproduct))) * fs_v_pfc_residual_unique_secondproductcoefficientsum)) /\\ exists fs_q_pfc_residual_unique_secondproductcoefficientsum_body_terminal. fs_u_pfc_residual_unique_secondproductcoefficientsum = fs_q_pfc_residual_unique_secondproductcoefficientsum_body_terminal * S ((S (S (pfc_index_residual_unique_secondproduct))) * fs_v_pfc_residual_unique_secondproductcoefficientsum) + (pfc_natural_sum_residual_unique_secondproductcoefficient))) /\\ forall fs_i_pfc_residual_unique_secondproductcoefficientsum_body_steps. (exists fs_lt_pfc_residual_unique_secondproductcoefficientsum_body_steps_bound. fs_lt_pfc_residual_unique_secondproductcoefficientsum_body_steps_bound + S fs_i_pfc_residual_unique_secondproductcoefficientsum_body_steps = S (pfc_index_residual_unique_secondproduct)) -> exists fs_a_pfc_residual_unique_secondproductcoefficientsum_body_steps fs_r_pfc_residual_unique_secondproductcoefficientsum_body_steps fs_s_pfc_residual_unique_secondproductcoefficientsum_body_steps. ((((exists fs_h_pfc_residual_unique_secondproductcoefficientsum_body_steps_summand. fs_h_pfc_residual_unique_secondproductcoefficientsum_body_steps_summand + S (fs_a_pfc_residual_unique_secondproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_residual_unique_secondproductcoefficientsum_body_steps)) * pfc_terms_scale_residual_unique_secondproductcoefficient)) /\\ exists fs_q_pfc_residual_unique_secondproductcoefficientsum_body_steps_summand. pfc_terms_code_residual_unique_secondproductcoefficient = fs_q_pfc_residual_unique_secondproductcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_residual_unique_secondproductcoefficientsum_body_steps)) * pfc_terms_scale_residual_unique_secondproductcoefficient) + (fs_a_pfc_residual_unique_secondproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_residual_unique_secondproductcoefficientsum_body_steps_partial. fs_h_pfc_residual_unique_secondproductcoefficientsum_body_steps_partial + S (fs_r_pfc_residual_unique_secondproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_residual_unique_secondproductcoefficientsum_body_steps)) * fs_v_pfc_residual_unique_secondproductcoefficientsum)) /\\ exists fs_q_pfc_residual_unique_secondproductcoefficientsum_body_steps_partial. fs_u_pfc_residual_unique_secondproductcoefficientsum = fs_q_pfc_residual_unique_secondproductcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_residual_unique_secondproductcoefficientsum_body_steps)) * fs_v_pfc_residual_unique_secondproductcoefficientsum) + (fs_r_pfc_residual_unique_secondproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_residual_unique_secondproductcoefficientsum_body_steps_successor. fs_h_pfc_residual_unique_secondproductcoefficientsum_body_steps_successor + S (fs_s_pfc_residual_unique_secondproductcoefficientsum_body_steps) = S ((S (S fs_i_pfc_residual_unique_secondproductcoefficientsum_body_steps)) * fs_v_pfc_residual_unique_secondproductcoefficientsum)) /\\ exists fs_q_pfc_residual_unique_secondproductcoefficientsum_body_steps_successor. fs_u_pfc_residual_unique_secondproductcoefficientsum = fs_q_pfc_residual_unique_secondproductcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_residual_unique_secondproductcoefficientsum_body_steps)) * fs_v_pfc_residual_unique_secondproductcoefficientsum) + (fs_s_pfc_residual_unique_secondproductcoefficientsum_body_steps))) /\\ fs_s_pfc_residual_unique_secondproductcoefficientsum_body_steps = fs_r_pfc_residual_unique_secondproductcoefficientsum_body_steps + fs_a_pfc_residual_unique_secondproductcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_residual_unique_secondproductcoefficientresiduebound. pfa_gap_residual_unique_secondproductcoefficientresiduebound + S (pfc_value_residual_unique_secondproduct) = (p)) /\\ ((exists pfa_offset_left_residual_unique_secondproductcoefficientresiduecongruence pfa_offset_right_residual_unique_secondproductcoefficientresiduecongruence. (pfc_natural_sum_residual_unique_secondproductcoefficient) + (p) * pfa_offset_left_residual_unique_secondproductcoefficientresiduecongruence = (pfc_value_residual_unique_secondproduct) + (p) * pfa_offset_right_residual_unique_secondproductcoefficientresiduecongruence)))))))))))) /\\ (((forall pfs_index_residual_unique_seconddifference. (exists pfa_gap_residual_unique_seconddifferenceindex. pfa_gap_residual_unique_seconddifferenceindex + S (pfs_index_residual_unique_seconddifference) = (L)) -> exists pfs_left_residual_unique_seconddifference pfs_right_residual_unique_seconddifference pfs_result_residual_unique_seconddifference. ((((exists ff_h_pfp_residual_unique_seconddifferenceleft. ff_h_pfp_residual_unique_seconddifferenceleft + S (pfs_left_residual_unique_seconddifference) = S ((S (pfs_index_residual_unique_seconddifference)) * ac)) /\\ exists ff_q_pfp_residual_unique_seconddifferenceleft. ab = ff_q_pfp_residual_unique_seconddifferenceleft * S ((S (pfs_index_residual_unique_seconddifference)) * ac) + (pfs_left_residual_unique_seconddifference))) /\\ (((((exists ff_h_pfp_residual_unique_seconddifferenceright. ff_h_pfp_residual_unique_seconddifferenceright + S (pfs_right_residual_unique_seconddifference) = S ((S (pfs_index_residual_unique_seconddifference)) * PC)) /\\ exists ff_q_pfp_residual_unique_seconddifferenceright. PB = ff_q_pfp_residual_unique_seconddifferenceright * S ((S (pfs_index_residual_unique_seconddifference)) * PC) + (pfs_right_residual_unique_seconddifference))) /\\ (((((exists ff_h_pfp_residual_unique_seconddifferenceresult. ff_h_pfp_residual_unique_seconddifferenceresult + S (pfs_result_residual_unique_seconddifference) = S ((S (pfs_index_residual_unique_seconddifference)) * UC)) /\\ exists ff_q_pfp_residual_unique_seconddifferenceresult. UB = ff_q_pfp_residual_unique_seconddifferenceresult * S ((S (pfs_index_residual_unique_seconddifference)) * UC) + (pfs_result_residual_unique_seconddifference))) /\\ ((((exists pfa_gap_residual_unique_seconddifferenceoperationleft. pfa_gap_residual_unique_seconddifferenceoperationleft + S (pfs_right_residual_unique_seconddifference) = (p)) /\\ (((exists pfa_gap_residual_unique_seconddifferenceoperationright. pfa_gap_residual_unique_seconddifferenceoperationright + S (pfs_result_residual_unique_seconddifference) = (p)) /\\ ((((exists pfa_gap_residual_unique_seconddifferenceoperationresultbound. pfa_gap_residual_unique_seconddifferenceoperationresultbound + S (pfs_left_residual_unique_seconddifference) = (p)) /\\ ((exists pfa_offset_left_residual_unique_seconddifferenceoperationresultcongruence pfa_offset_right_residual_unique_seconddifferenceoperationresultcongruence. ((pfs_right_residual_unique_seconddifference) + (pfs_result_residual_unique_seconddifference)) + (p) * pfa_offset_left_residual_unique_seconddifferenceoperationresultcongruence = (pfs_left_residual_unique_seconddifference) + (p) * pfa_offset_right_residual_unique_seconddifferenceoperationresultcongruence)))))))))))))))) /\\ (((((L)=(T)+(K)) /\\ (((forall fom_index_pfp_residual_unique_secondtriminput. (exists fom_gap_pfp_residual_unique_secondtriminput_index_bound. fom_gap_pfp_residual_unique_secondtriminput_index_bound + S (fom_index_pfp_residual_unique_secondtriminput) = L) -> exists fom_value_pfp_residual_unique_secondtriminput. ((((exists fom_beta_height_pfp_residual_unique_secondtriminput_entry. fom_beta_height_pfp_residual_unique_secondtriminput_entry + S (fom_value_pfp_residual_unique_secondtriminput) = S ((S (fom_index_pfp_residual_unique_secondtriminput)) * UC)) /\\ exists fom_beta_quotient_pfp_residual_unique_secondtriminput_entry. UB = fom_beta_quotient_pfp_residual_unique_secondtriminput_entry * S ((S (fom_index_pfp_residual_unique_secondtriminput)) * UC) + (fom_value_pfp_residual_unique_secondtriminput))) /\\ (exists fom_gap_pfp_residual_unique_secondtriminput_value_bound. fom_gap_pfp_residual_unique_secondtriminput_value_bound + S (fom_value_pfp_residual_unique_secondtriminput) = p))) /\\ (((forall pfp_repeat_index_residual_unique_secondtrimremoved. (exists pfa_gap_residual_unique_secondtrimremovedindex. pfa_gap_residual_unique_secondtrimremovedindex + S (pfp_repeat_index_residual_unique_secondtrimremoved) = (T)) -> (((exists ff_h_pfp_residual_unique_secondtrimremovedentry. ff_h_pfp_residual_unique_secondtrimremovedentry + S (0) = S ((S (pfp_repeat_index_residual_unique_secondtrimremoved)) * UC)) /\\ exists ff_q_pfp_residual_unique_secondtrimremovedentry. UB = ff_q_pfp_residual_unique_secondtrimremovedentry * S ((S (pfp_repeat_index_residual_unique_secondtrimremoved)) * UC) + (0)))) /\\ (((forall pftrim_index_residual_unique_secondtrimsuffix pftrim_value_residual_unique_secondtrimsuffix. (exists pfa_gap_residual_unique_secondtrimsuffixbound. pfa_gap_residual_unique_secondtrimsuffixbound + S (pftrim_index_residual_unique_secondtrimsuffix) = (K)) -> (((exists ff_h_pfp_residual_unique_secondtrimsuffixsource. ff_h_pfp_residual_unique_secondtrimsuffixsource + S (pftrim_value_residual_unique_secondtrimsuffix) = S ((S ((T)+pftrim_index_residual_unique_secondtrimsuffix)) * UC)) /\\ exists ff_q_pfp_residual_unique_secondtrimsuffixsource. UB = ff_q_pfp_residual_unique_secondtrimsuffixsource * S ((S ((T)+pftrim_index_residual_unique_secondtrimsuffix)) * UC) + (pftrim_value_residual_unique_secondtrimsuffix))) -> (((exists ff_h_pfp_residual_unique_secondtrimsuffixoutput. ff_h_pfp_residual_unique_secondtrimsuffixoutput + S (pftrim_value_residual_unique_secondtrimsuffix) = S ((S (pftrim_index_residual_unique_secondtrimsuffix)) * RC)) /\\ exists ff_q_pfp_residual_unique_secondtrimsuffixoutput. RB = ff_q_pfp_residual_unique_secondtrimsuffixoutput * S ((S (pftrim_index_residual_unique_secondtrimsuffix)) * RC) + (pftrim_value_residual_unique_secondtrimsuffix)))) /\\ (((K)=0 \\/ (exists pftrim_leading_residual_unique_secondtrimnormal. ((((exists ff_h_pfp_residual_unique_secondtrimnormalentry. ff_h_pfp_residual_unique_secondtrimnormalentry + S (pftrim_leading_residual_unique_secondtrimnormal) = S ((S (0)) * RC)) /\\ exists ff_q_pfp_residual_unique_secondtrimnormalentry. RB = ff_q_pfp_residual_unique_secondtrimnormalentry * S ((S (0)) * RC) + (pftrim_leading_residual_unique_secondtrimnormal))) /\\ ((~(pftrim_leading_residual_unique_secondtrimnormal=0)))))))))))))))))))) -> (((t=T) /\\ (((R=K) /\\ ((forall mdr_i_pfp_residual_unique_output mdr_a_pfp_residual_unique_output. (exists mdr_gap_pfp_residual_unique_outputb. mdr_gap_pfp_residual_unique_outputb + S (mdr_i_pfp_residual_unique_output) = (R)) -> (((exists ff_h_mdr_pfp_residual_unique_outputo. ff_h_mdr_pfp_residual_unique_outputo + S (mdr_a_pfp_residual_unique_output) = S ((S (mdr_i_pfp_residual_unique_output)) * rc)) /\\ exists ff_q_mdr_pfp_residual_unique_outputo. rb = ff_q_mdr_pfp_residual_unique_outputo * S ((S (mdr_i_pfp_residual_unique_output)) * rc) + (mdr_a_pfp_residual_unique_output))) -> (((exists ff_h_mdr_pfp_residual_unique_outputn. ff_h_mdr_pfp_residual_unique_outputn + S (mdr_a_pfp_residual_unique_output) = S ((S (mdr_i_pfp_residual_unique_output)) * RC)) /\\ exists ff_q_mdr_pfp_residual_unique_outputn. 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qc", "intro q", "intro rb", "intro rc", "intro R", "intro QB", "intro QC", "intro Q", "intro RB", "intro RC", "intro K", "intro hfirst", "intro hsecond", "cases hfirst", "cases hfirst_right", "cases hfirst_right_right", "cases hsecond", "cases hsecond_right", "cases hsecond_right_right", "cases hfirst_right_right_right", "cases hfirst_right_right_right_witness", "cases hfirst_right_right_right_witness_witness", "cases hfirst_right_right_right_witness_witness_witness", "cases hfirst_right_right_right_witness_witness_witness_witness", "cases hfirst_right_right_right_witness_witness_witness_witness_witness", "cases hfirst_right_right_right_witness_witness_witness_witness_witness_witness", "cases hfirst_right_right_right_witness_witness_witness_witness_witness_witness_witness", "cases hfirst_right_right_right_witness_witness_witness_witness_witness_witness_witness_right", "cases hfirst_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right", "cases hfirst_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right_right", "cases hfirst_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right_right_right", "cases hsecond_right_right_right", "cases hsecond_right_right_right_witness", "cases hsecond_right_right_right_witness_witness", "cases hsecond_right_right_right_witness_witness_witness", "cases hsecond_right_right_right_witness_witness_witness_witness", "cases hsecond_right_right_right_witness_witness_witness_witness_witness", "cases hsecond_right_right_right_witness_witness_witness_witness_witness_witness", "cases hsecond_right_right_right_witness_witness_witness_witness_witness_witness_witness", "cases hsecond_right_right_right_witness_witness_witness_witness_witness_witness_witness_right", "cases hsecond_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right", "cases hsecond_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right_right", "cases hsecond_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right_right_right", "have hquotients : ((x=x7) /\\ (((x1=x8) /\\ (((q=Q) /\\ ((forall mdr_i_pfp_execution_unique_quotients mdr_a_pfp_execution_unique_quotients. (exists mdr_gap_pfp_execution_unique_quotientsb. mdr_gap_pfp_execution_unique_quotientsb + S (mdr_i_pfp_execution_unique_quotients) = (q)) -> (((exists ff_h_mdr_pfp_execution_unique_quotientso. ff_h_mdr_pfp_execution_unique_quotientso + S (mdr_a_pfp_execution_unique_quotients) = S ((S (mdr_i_pfp_execution_unique_quotients)) * qc)) /\\ exists ff_q_mdr_pfp_execution_unique_quotientso. qb = ff_q_mdr_pfp_execution_unique_quotientso * S ((S (mdr_i_pfp_execution_unique_quotients)) * qc) + (mdr_a_pfp_execution_unique_quotients))) -> (((exists ff_h_mdr_pfp_execution_unique_quotientsn. ff_h_mdr_pfp_execution_unique_quotientsn + S (mdr_a_pfp_execution_unique_quotients) = S ((S (mdr_i_pfp_execution_unique_quotients)) * QC)) /\\ exists ff_q_mdr_pfp_execution_unique_quotientsn. QB = ff_q_mdr_pfp_execution_unique_quotientsn * S ((S (mdr_i_pfp_execution_unique_quotients)) * QC) + (mdr_a_pfp_execution_unique_quotients))))))))))", "specialize prime_field_polynomial_division_quotient_data_functional (p)", "specialize prime_field_polynomial_division_quotient_data_functional (ab)", "specialize prime_field_polynomial_division_quotient_data_functional (ac)", "specialize prime_field_polynomial_division_quotient_data_functional (L)", "specialize prime_field_polynomial_division_quotient_data_functional (bb)", "specialize prime_field_polynomial_division_quotient_data_functional (bc)", "specialize prime_field_polynomial_division_quotient_data_functional (d)", "specialize prime_field_polynomial_division_quotient_data_functional (x)", "specialize prime_field_polynomial_division_quotient_data_functional (x1)", "specialize prime_field_polynomial_division_quotient_data_functional (q)", "specialize prime_field_polynomial_division_quotient_data_functional (qb)", "specialize prime_field_polynomial_division_quotient_data_functional (qc)", "specialize prime_field_polynomial_division_quotient_data_functional (x7)", "specialize prime_field_polynomial_division_quotient_data_functional (x8)", "specialize prime_field_polynomial_division_quotient_data_functional (Q)", "specialize prime_field_polynomial_division_quotient_data_functional (QB)", "specialize prime_field_polynomial_division_quotient_data_functional (QC)", "apply prime_field_polynomial_division_quotient_data_functional", "split", "exact hfirst_right_right_right_witness_witness_witness_witness_witness_witness_witness_left", "split", "exact hfirst_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_left", "split", "exact hfirst_right_right_left", "exact hfirst_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right_left", "split", "exact hsecond_right_right_right_witness_witness_witness_witness_witness_witness_witness_left", "split", "exact hsecond_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_left", "split", "exact hsecond_right_right_left", "exact hsecond_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right_left", "cases hquotients", "cases hquotients_right", "cases hquotients_right_right", "have hremainder : ((x6=x13) /\\ (((R=K) /\\ ((forall mdr_i_pfp_execution_unique_residuals mdr_a_pfp_execution_unique_residuals. (exists mdr_gap_pfp_execution_unique_residualsb. mdr_gap_pfp_execution_unique_residualsb + S (mdr_i_pfp_execution_unique_residuals) = (R)) -> (((exists ff_h_mdr_pfp_execution_unique_residualso. ff_h_mdr_pfp_execution_unique_residualso + S (mdr_a_pfp_execution_unique_residuals) = S ((S (mdr_i_pfp_execution_unique_residuals)) * rc)) /\\ exists ff_q_mdr_pfp_execution_unique_residualso. rb = ff_q_mdr_pfp_execution_unique_residualso * S ((S (mdr_i_pfp_execution_unique_residuals)) * rc) + (mdr_a_pfp_execution_unique_residuals))) -> (((exists ff_h_mdr_pfp_execution_unique_residualsn. ff_h_mdr_pfp_execution_unique_residualsn + S (mdr_a_pfp_execution_unique_residuals) = S ((S (mdr_i_pfp_execution_unique_residuals)) * RC)) /\\ exists ff_q_mdr_pfp_execution_unique_residualsn. RB = ff_q_mdr_pfp_execution_unique_residualsn * S ((S (mdr_i_pfp_execution_unique_residuals)) * RC) + (mdr_a_pfp_execution_unique_residuals))))))))", "specialize prime_field_polynomial_division_residual_data_functional (p)", "specialize prime_field_polynomial_division_residual_data_functional (ab)", "specialize prime_field_polynomial_division_residual_data_functional (ac)", "specialize prime_field_polynomial_division_residual_data_functional (L)", "specialize prime_field_polynomial_division_residual_data_functional (bb)", "specialize prime_field_polynomial_division_residual_data_functional (bc)", "specialize prime_field_polynomial_division_residual_data_functional (d)", "specialize prime_field_polynomial_division_residual_data_functional (qb)", "specialize prime_field_polynomial_division_residual_data_functional (qc)", "specialize prime_field_polynomial_division_residual_data_functional (QB)", "specialize prime_field_polynomial_division_residual_data_functional (QC)", "specialize prime_field_polynomial_division_residual_data_functional (q)", "specialize prime_field_polynomial_division_residual_data_functional (x2)", "specialize prime_field_polynomial_division_residual_data_functional (x3)", "specialize prime_field_polynomial_division_residual_data_functional (x4)", "specialize prime_field_polynomial_division_residual_data_functional (x5)", "specialize prime_field_polynomial_division_residual_data_functional (x6)", "specialize prime_field_polynomial_division_residual_data_functional (rb)", "specialize prime_field_polynomial_division_residual_data_functional (rc)", "specialize prime_field_polynomial_division_residual_data_functional (R)", "specialize prime_field_polynomial_division_residual_data_functional (x9)", "specialize prime_field_polynomial_division_residual_data_functional (x10)", "specialize prime_field_polynomial_division_residual_data_functional (x11)", "specialize prime_field_polynomial_division_residual_data_functional (x12)", "specialize prime_field_polynomial_division_residual_data_functional (x13)", "specialize prime_field_polynomial_division_residual_data_functional (RB)", "specialize prime_field_polynomial_division_residual_data_functional (RC)", "specialize prime_field_polynomial_division_residual_data_functional (K)", "apply prime_field_polynomial_division_residual_data_functional", "exact hquotients_right_right_right", "split", "exact hfirst_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right_right_left", "split", "exact hfirst_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_left", "exact hfirst_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right", "rewrite hquotients_right_right_left", "rewrite hquotients_right_right_left", "split", "exact hsecond_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right_right_left", "split", "exact hsecond_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_left", "exact hsecond_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right", "cases hremainder", "cases hremainder_right", "split", "exact hquotients_right_right_left", "split", "exact hquotients_right_right_right", "split", "exact hremainder_right_left", "exact hremainder_right_right" ], "script_sha256": "a882c0deab1fa8131c75b5f7d79871c916cec590480ef1e4de35ac476f00b12a", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_division_uniqueness_candidate.py", "sha256": "6a9d9ebe1f72202743e5df2c069b9aa367fdb3d61108f1d9354cdc9276ab2d15" }, "statement": "forall p ab ac L bb bc d qb qc q rb rc R QB QC Q RB RC K. (((forall fom_index_pfp_execution_unique_firstinput. (exists fom_gap_pfp_execution_unique_firstinput_index_bound. fom_gap_pfp_execution_unique_firstinput_index_bound + S (fom_index_pfp_execution_unique_firstinput) = L) -> exists fom_value_pfp_execution_unique_firstinput. ((((exists fom_beta_height_pfp_execution_unique_firstinput_entry. fom_beta_height_pfp_execution_unique_firstinput_entry + S (fom_value_pfp_execution_unique_firstinput) = S ((S (fom_index_pfp_execution_unique_firstinput)) * ac)) /\\ exists fom_beta_quotient_pfp_execution_unique_firstinput_entry. ab = fom_beta_quotient_pfp_execution_unique_firstinput_entry * S ((S (fom_index_pfp_execution_unique_firstinput)) * ac) + (fom_value_pfp_execution_unique_firstinput))) /\\ (exists fom_gap_pfp_execution_unique_firstinput_value_bound. fom_gap_pfp_execution_unique_firstinput_value_bound + S (fom_value_pfp_execution_unique_firstinput) = p))) /\\ (((forall fom_index_pfp_execution_unique_firstdivisor. (exists fom_gap_pfp_execution_unique_firstdivisor_index_bound. fom_gap_pfp_execution_unique_firstdivisor_index_bound + S (fom_index_pfp_execution_unique_firstdivisor) = S (d)) -> exists fom_value_pfp_execution_unique_firstdivisor. ((((exists fom_beta_height_pfp_execution_unique_firstdivisor_entry. fom_beta_height_pfp_execution_unique_firstdivisor_entry + S (fom_value_pfp_execution_unique_firstdivisor) = S ((S (fom_index_pfp_execution_unique_firstdivisor)) * bc)) /\\ exists fom_beta_quotient_pfp_execution_unique_firstdivisor_entry. bb = fom_beta_quotient_pfp_execution_unique_firstdivisor_entry * S ((S (fom_index_pfp_execution_unique_firstdivisor)) * bc) + (fom_value_pfp_execution_unique_firstdivisor))) /\\ (exists fom_gap_pfp_execution_unique_firstdivisor_value_bound. fom_gap_pfp_execution_unique_firstdivisor_value_bound + S (fom_value_pfp_execution_unique_firstdivisor) = p))) /\\ (((((((q)=0) /\\ ((exists pfc_gap_execution_unique_firstlengthshort. pfc_gap_execution_unique_firstlengthshort+(L)=(d))))) \\/ (((~((q)=0)) /\\ (((q)+(d)=(L)))))) /\\ ((exists pfd_head_execution_unique_first pfd_inverse_execution_unique_first pfd_product_code_execution_unique_first pfd_product_scale_execution_unique_first pfd_residual_code_execution_unique_first pfd_residual_scale_execution_unique_first pfd_cut_execution_unique_first. ((((exists ff_h_pfp_execution_unique_firsthead. ff_h_pfp_execution_unique_firsthead + S (pfd_head_execution_unique_first) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_execution_unique_firsthead. bb = ff_q_pfp_execution_unique_firsthead * S ((S (0)) * bc) + (pfd_head_execution_unique_first))) /\\ (((((~((pfd_head_execution_unique_first) = 0)) /\\ ((((exists pfa_gap_execution_unique_firstinversemultiplicationleft. pfa_gap_execution_unique_firstinversemultiplicationleft + S (pfd_head_execution_unique_first) = (p)) /\\ (((exists pfa_gap_execution_unique_firstinversemultiplicationright. pfa_gap_execution_unique_firstinversemultiplicationright + S (pfd_inverse_execution_unique_first) = (p)) /\\ ((((exists pfa_gap_execution_unique_firstinversemultiplicationresultbound. pfa_gap_execution_unique_firstinversemultiplicationresultbound + S (1) = (p)) /\\ ((exists pfa_offset_left_execution_unique_firstinversemultiplicationresultcongruence pfa_offset_right_execution_unique_firstinversemultiplicationresultcongruence. ((pfd_head_execution_unique_first) * (pfd_inverse_execution_unique_first)) + (p) * pfa_offset_left_execution_unique_firstinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_execution_unique_firstinversemultiplicationresultcongruence)))))))))))) /\\ (((forall pfd_index_execution_unique_firstquotient. (exists pfa_gap_execution_unique_firstquotientbound. pfa_gap_execution_unique_firstquotientbound + S (pfd_index_execution_unique_firstquotient) = (q)) -> exists pfd_value_execution_unique_firstquotient. ((((exists ff_h_pfp_execution_unique_firstquotiententry. ff_h_pfp_execution_unique_firstquotiententry + S (pfd_value_execution_unique_firstquotient) = S ((S (pfd_index_execution_unique_firstquotient)) * qc)) /\\ exists ff_q_pfp_execution_unique_firstquotiententry. qb = ff_q_pfp_execution_unique_firstquotiententry * S ((S (pfd_index_execution_unique_firstquotient)) * qc) + (pfd_value_execution_unique_firstquotient))) /\\ ((exists pfd_input_execution_unique_firstquotientstep pfd_previous_execution_unique_firstquotientstep pfd_difference_execution_unique_firstquotientstep. ((((exists ff_h_pfp_execution_unique_firstquotientstepinput. ff_h_pfp_execution_unique_firstquotientstepinput + S (pfd_input_execution_unique_firstquotientstep) = S ((S (pfd_index_execution_unique_firstquotient)) * ac)) /\\ exists ff_q_pfp_execution_unique_firstquotientstepinput. ab = ff_q_pfp_execution_unique_firstquotientstepinput * S ((S (pfd_index_execution_unique_firstquotient)) * ac) + (pfd_input_execution_unique_firstquotientstep))) /\\ (((exists pfc_terms_code_execution_unique_firstquotientstepprevious pfc_terms_scale_execution_unique_firstquotientstepprevious pfc_natural_sum_execution_unique_firstquotientstepprevious. ((forall pfc_index_execution_unique_firstquotientsteppreviousdiagonal. (exists pfa_gap_execution_unique_firstquotientsteppreviousdiagonalbound. pfa_gap_execution_unique_firstquotientsteppreviousdiagonalbound + S (pfc_index_execution_unique_firstquotientsteppreviousdiagonal) = (S (pfd_index_execution_unique_firstquotient))) -> exists pfc_value_execution_unique_firstquotientsteppreviousdiagonal. ((((exists ff_h_pfp_execution_unique_firstquotientsteppreviousdiagonalentry. ff_h_pfp_execution_unique_firstquotientsteppreviousdiagonalentry + S (pfc_value_execution_unique_firstquotientsteppreviousdiagonal) = S ((S (pfc_index_execution_unique_firstquotientsteppreviousdiagonal)) * pfc_terms_scale_execution_unique_firstquotientstepprevious)) /\\ exists ff_q_pfp_execution_unique_firstquotientsteppreviousdiagonalentry. pfc_terms_code_execution_unique_firstquotientstepprevious = ff_q_pfp_execution_unique_firstquotientsteppreviousdiagonalentry * S ((S (pfc_index_execution_unique_firstquotientsteppreviousdiagonal)) * pfc_terms_scale_execution_unique_firstquotientstepprevious) + (pfc_value_execution_unique_firstquotientsteppreviousdiagonal))) /\\ ((exists pfc_complement_execution_unique_firstquotientsteppreviousdiagonalterm pfc_left_execution_unique_firstquotientsteppreviousdiagonalterm pfc_right_execution_unique_firstquotientsteppreviousdiagonalterm. (((pfc_index_execution_unique_firstquotientsteppreviousdiagonal)+pfc_complement_execution_unique_firstquotientsteppreviousdiagonalterm=(pfd_index_execution_unique_firstquotient)) /\\ ((((((exists pfa_gap_execution_unique_firstquotientsteppreviousdiagonaltermleftinside. pfa_gap_execution_unique_firstquotientsteppreviousdiagonaltermleftinside + S (pfc_index_execution_unique_firstquotientsteppreviousdiagonal) = (pfd_index_execution_unique_firstquotient)) /\\ ((((exists ff_h_pfp_execution_unique_firstquotientsteppreviousdiagonaltermleftentry. ff_h_pfp_execution_unique_firstquotientsteppreviousdiagonaltermleftentry + S (pfc_left_execution_unique_firstquotientsteppreviousdiagonalterm) = S ((S (pfc_index_execution_unique_firstquotientsteppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_execution_unique_firstquotientsteppreviousdiagonaltermleftentry. qb = ff_q_pfp_execution_unique_firstquotientsteppreviousdiagonaltermleftentry * S ((S (pfc_index_execution_unique_firstquotientsteppreviousdiagonal)) * qc) + (pfc_left_execution_unique_firstquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_execution_unique_firstquotientsteppreviousdiagonaltermleftoutside. pfc_gap_execution_unique_firstquotientsteppreviousdiagonaltermleftoutside+(pfd_index_execution_unique_firstquotient)=(pfc_index_execution_unique_firstquotientsteppreviousdiagonal)) /\\ (((pfc_left_execution_unique_firstquotientsteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_execution_unique_firstquotientsteppreviousdiagonaltermrightinside. pfa_gap_execution_unique_firstquotientsteppreviousdiagonaltermrightinside + S (pfc_complement_execution_unique_firstquotientsteppreviousdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_execution_unique_firstquotientsteppreviousdiagonaltermrightentry. ff_h_pfp_execution_unique_firstquotientsteppreviousdiagonaltermrightentry + S (pfc_right_execution_unique_firstquotientsteppreviousdiagonalterm) = S ((S (pfc_complement_execution_unique_firstquotientsteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_execution_unique_firstquotientsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_execution_unique_firstquotientsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_execution_unique_firstquotientsteppreviousdiagonalterm)) * bc) + (pfc_right_execution_unique_firstquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_execution_unique_firstquotientsteppreviousdiagonaltermrightoutside. pfc_gap_execution_unique_firstquotientsteppreviousdiagonaltermrightoutside+(S (d))=(pfc_complement_execution_unique_firstquotientsteppreviousdiagonalterm)) /\\ (((pfc_right_execution_unique_firstquotientsteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_execution_unique_firstquotientsteppreviousdiagonal)=pfc_left_execution_unique_firstquotientsteppreviousdiagonalterm*pfc_right_execution_unique_firstquotientsteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_execution_unique_firstquotientstepprevioussum fs_v_pfc_execution_unique_firstquotientstepprevioussum. ((((exists fs_h_pfc_execution_unique_firstquotientstepprevioussum_body_start. fs_h_pfc_execution_unique_firstquotientstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_unique_firstquotientstepprevioussum)) /\\ exists fs_q_pfc_execution_unique_firstquotientstepprevioussum_body_start. fs_u_pfc_execution_unique_firstquotientstepprevioussum = fs_q_pfc_execution_unique_firstquotientstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_execution_unique_firstquotientstepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_execution_unique_firstquotientstepprevioussum_body_terminal. fs_h_pfc_execution_unique_firstquotientstepprevioussum_body_terminal + S (pfc_natural_sum_execution_unique_firstquotientstepprevious) = S ((S (S (pfd_index_execution_unique_firstquotient))) * fs_v_pfc_execution_unique_firstquotientstepprevioussum)) /\\ exists fs_q_pfc_execution_unique_firstquotientstepprevioussum_body_terminal. fs_u_pfc_execution_unique_firstquotientstepprevioussum = fs_q_pfc_execution_unique_firstquotientstepprevioussum_body_terminal * S ((S (S (pfd_index_execution_unique_firstquotient))) * fs_v_pfc_execution_unique_firstquotientstepprevioussum) + (pfc_natural_sum_execution_unique_firstquotientstepprevious))) /\\ forall fs_i_pfc_execution_unique_firstquotientstepprevioussum_body_steps. (exists fs_lt_pfc_execution_unique_firstquotientstepprevioussum_body_steps_bound. fs_lt_pfc_execution_unique_firstquotientstepprevioussum_body_steps_bound + S fs_i_pfc_execution_unique_firstquotientstepprevioussum_body_steps = S (pfd_index_execution_unique_firstquotient)) -> exists fs_a_pfc_execution_unique_firstquotientstepprevioussum_body_steps fs_r_pfc_execution_unique_firstquotientstepprevioussum_body_steps fs_s_pfc_execution_unique_firstquotientstepprevioussum_body_steps. ((((exists fs_h_pfc_execution_unique_firstquotientstepprevioussum_body_steps_summand. fs_h_pfc_execution_unique_firstquotientstepprevioussum_body_steps_summand + S (fs_a_pfc_execution_unique_firstquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_execution_unique_firstquotientstepprevioussum_body_steps)) * pfc_terms_scale_execution_unique_firstquotientstepprevious)) /\\ exists fs_q_pfc_execution_unique_firstquotientstepprevioussum_body_steps_summand. pfc_terms_code_execution_unique_firstquotientstepprevious = fs_q_pfc_execution_unique_firstquotientstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_execution_unique_firstquotientstepprevioussum_body_steps)) * pfc_terms_scale_execution_unique_firstquotientstepprevious) + (fs_a_pfc_execution_unique_firstquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_execution_unique_firstquotientstepprevioussum_body_steps_partial. fs_h_pfc_execution_unique_firstquotientstepprevioussum_body_steps_partial + S (fs_r_pfc_execution_unique_firstquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_execution_unique_firstquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_unique_firstquotientstepprevioussum)) /\\ exists fs_q_pfc_execution_unique_firstquotientstepprevioussum_body_steps_partial. fs_u_pfc_execution_unique_firstquotientstepprevioussum = fs_q_pfc_execution_unique_firstquotientstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_execution_unique_firstquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_unique_firstquotientstepprevioussum) + (fs_r_pfc_execution_unique_firstquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_execution_unique_firstquotientstepprevioussum_body_steps_successor. fs_h_pfc_execution_unique_firstquotientstepprevioussum_body_steps_successor + S (fs_s_pfc_execution_unique_firstquotientstepprevioussum_body_steps) = S ((S (S fs_i_pfc_execution_unique_firstquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_unique_firstquotientstepprevioussum)) /\\ exists fs_q_pfc_execution_unique_firstquotientstepprevioussum_body_steps_successor. fs_u_pfc_execution_unique_firstquotientstepprevioussum = fs_q_pfc_execution_unique_firstquotientstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_execution_unique_firstquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_unique_firstquotientstepprevioussum) + (fs_s_pfc_execution_unique_firstquotientstepprevioussum_body_steps))) /\\ fs_s_pfc_execution_unique_firstquotientstepprevioussum_body_steps = fs_r_pfc_execution_unique_firstquotientstepprevioussum_body_steps + fs_a_pfc_execution_unique_firstquotientstepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_execution_unique_firstquotientsteppreviousresiduebound. pfa_gap_execution_unique_firstquotientsteppreviousresiduebound + S (pfd_previous_execution_unique_firstquotientstep) = (p)) /\\ ((exists pfa_offset_left_execution_unique_firstquotientsteppreviousresiduecongruence pfa_offset_right_execution_unique_firstquotientsteppreviousresiduecongruence. (pfc_natural_sum_execution_unique_firstquotientstepprevious) + (p) * pfa_offset_left_execution_unique_firstquotientsteppreviousresiduecongruence = (pfd_previous_execution_unique_firstquotientstep) + (p) * pfa_offset_right_execution_unique_firstquotientsteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_execution_unique_firstquotientstepsubtractleft. pfa_gap_execution_unique_firstquotientstepsubtractleft + S (pfd_previous_execution_unique_firstquotientstep) = (p)) /\\ (((exists pfa_gap_execution_unique_firstquotientstepsubtractright. pfa_gap_execution_unique_firstquotientstepsubtractright + S (pfd_difference_execution_unique_firstquotientstep) = (p)) /\\ ((((exists pfa_gap_execution_unique_firstquotientstepsubtractresultbound. pfa_gap_execution_unique_firstquotientstepsubtractresultbound + S (pfd_input_execution_unique_firstquotientstep) = (p)) /\\ ((exists pfa_offset_left_execution_unique_firstquotientstepsubtractresultcongruence pfa_offset_right_execution_unique_firstquotientstepsubtractresultcongruence. ((pfd_previous_execution_unique_firstquotientstep) + (pfd_difference_execution_unique_firstquotientstep)) + (p) * pfa_offset_left_execution_unique_firstquotientstepsubtractresultcongruence = (pfd_input_execution_unique_firstquotientstep) + (p) * pfa_offset_right_execution_unique_firstquotientstepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_execution_unique_firstquotientstepmultiplyleft. pfa_gap_execution_unique_firstquotientstepmultiplyleft + S (pfd_inverse_execution_unique_first) = (p)) /\\ (((exists pfa_gap_execution_unique_firstquotientstepmultiplyright. pfa_gap_execution_unique_firstquotientstepmultiplyright + S (pfd_difference_execution_unique_firstquotientstep) = (p)) /\\ ((((exists pfa_gap_execution_unique_firstquotientstepmultiplyresultbound. pfa_gap_execution_unique_firstquotientstepmultiplyresultbound + S (pfd_value_execution_unique_firstquotient) = (p)) /\\ ((exists pfa_offset_left_execution_unique_firstquotientstepmultiplyresultcongruence pfa_offset_right_execution_unique_firstquotientstepmultiplyresultcongruence. ((pfd_inverse_execution_unique_first) * (pfd_difference_execution_unique_firstquotientstep)) + (p) * pfa_offset_left_execution_unique_firstquotientstepmultiplyresultcongruence = (pfd_value_execution_unique_firstquotient) + (p) * pfa_offset_right_execution_unique_firstquotientstepmultiplyresultcongruence))))))))))))))))))) /\\ (((forall pfc_index_execution_unique_firstproduct. (exists pfa_gap_execution_unique_firstproductbound. pfa_gap_execution_unique_firstproductbound + S (pfc_index_execution_unique_firstproduct) = (L)) -> exists pfc_value_execution_unique_firstproduct. ((((exists ff_h_pfp_execution_unique_firstproductentry. ff_h_pfp_execution_unique_firstproductentry + S (pfc_value_execution_unique_firstproduct) = S ((S (pfc_index_execution_unique_firstproduct)) * pfd_product_scale_execution_unique_first)) /\\ exists ff_q_pfp_execution_unique_firstproductentry. pfd_product_code_execution_unique_first = ff_q_pfp_execution_unique_firstproductentry * S ((S (pfc_index_execution_unique_firstproduct)) * pfd_product_scale_execution_unique_first) + (pfc_value_execution_unique_firstproduct))) /\\ ((exists pfc_terms_code_execution_unique_firstproductcoefficient pfc_terms_scale_execution_unique_firstproductcoefficient pfc_natural_sum_execution_unique_firstproductcoefficient. ((forall pfc_index_execution_unique_firstproductcoefficientdiagonal. (exists pfa_gap_execution_unique_firstproductcoefficientdiagonalbound. pfa_gap_execution_unique_firstproductcoefficientdiagonalbound + S (pfc_index_execution_unique_firstproductcoefficientdiagonal) = (S (pfc_index_execution_unique_firstproduct))) -> exists pfc_value_execution_unique_firstproductcoefficientdiagonal. ((((exists ff_h_pfp_execution_unique_firstproductcoefficientdiagonalentry. ff_h_pfp_execution_unique_firstproductcoefficientdiagonalentry + S (pfc_value_execution_unique_firstproductcoefficientdiagonal) = S ((S (pfc_index_execution_unique_firstproductcoefficientdiagonal)) * pfc_terms_scale_execution_unique_firstproductcoefficient)) /\\ exists ff_q_pfp_execution_unique_firstproductcoefficientdiagonalentry. pfc_terms_code_execution_unique_firstproductcoefficient = ff_q_pfp_execution_unique_firstproductcoefficientdiagonalentry * S ((S (pfc_index_execution_unique_firstproductcoefficientdiagonal)) * pfc_terms_scale_execution_unique_firstproductcoefficient) + (pfc_value_execution_unique_firstproductcoefficientdiagonal))) /\\ ((exists pfc_complement_execution_unique_firstproductcoefficientdiagonalterm pfc_left_execution_unique_firstproductcoefficientdiagonalterm pfc_right_execution_unique_firstproductcoefficientdiagonalterm. (((pfc_index_execution_unique_firstproductcoefficientdiagonal)+pfc_complement_execution_unique_firstproductcoefficientdiagonalterm=(pfc_index_execution_unique_firstproduct)) /\\ ((((((exists pfa_gap_execution_unique_firstproductcoefficientdiagonaltermleftinside. pfa_gap_execution_unique_firstproductcoefficientdiagonaltermleftinside + S (pfc_index_execution_unique_firstproductcoefficientdiagonal) = (q)) /\\ ((((exists ff_h_pfp_execution_unique_firstproductcoefficientdiagonaltermleftentry. ff_h_pfp_execution_unique_firstproductcoefficientdiagonaltermleftentry + S (pfc_left_execution_unique_firstproductcoefficientdiagonalterm) = S ((S (pfc_index_execution_unique_firstproductcoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_execution_unique_firstproductcoefficientdiagonaltermleftentry. qb = ff_q_pfp_execution_unique_firstproductcoefficientdiagonaltermleftentry * S ((S (pfc_index_execution_unique_firstproductcoefficientdiagonal)) * qc) + (pfc_left_execution_unique_firstproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_unique_firstproductcoefficientdiagonaltermleftoutside. pfc_gap_execution_unique_firstproductcoefficientdiagonaltermleftoutside+(q)=(pfc_index_execution_unique_firstproductcoefficientdiagonal)) /\\ (((pfc_left_execution_unique_firstproductcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_execution_unique_firstproductcoefficientdiagonaltermrightinside. pfa_gap_execution_unique_firstproductcoefficientdiagonaltermrightinside + S (pfc_complement_execution_unique_firstproductcoefficientdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_execution_unique_firstproductcoefficientdiagonaltermrightentry. ff_h_pfp_execution_unique_firstproductcoefficientdiagonaltermrightentry + S (pfc_right_execution_unique_firstproductcoefficientdiagonalterm) = S ((S (pfc_complement_execution_unique_firstproductcoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_execution_unique_firstproductcoefficientdiagonaltermrightentry. bb = ff_q_pfp_execution_unique_firstproductcoefficientdiagonaltermrightentry * S ((S (pfc_complement_execution_unique_firstproductcoefficientdiagonalterm)) * bc) + (pfc_right_execution_unique_firstproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_unique_firstproductcoefficientdiagonaltermrightoutside. pfc_gap_execution_unique_firstproductcoefficientdiagonaltermrightoutside+(S (d))=(pfc_complement_execution_unique_firstproductcoefficientdiagonalterm)) /\\ (((pfc_right_execution_unique_firstproductcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_execution_unique_firstproductcoefficientdiagonal)=pfc_left_execution_unique_firstproductcoefficientdiagonalterm*pfc_right_execution_unique_firstproductcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_execution_unique_firstproductcoefficientsum fs_v_pfc_execution_unique_firstproductcoefficientsum. ((((exists fs_h_pfc_execution_unique_firstproductcoefficientsum_body_start. fs_h_pfc_execution_unique_firstproductcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_unique_firstproductcoefficientsum)) /\\ exists fs_q_pfc_execution_unique_firstproductcoefficientsum_body_start. fs_u_pfc_execution_unique_firstproductcoefficientsum = fs_q_pfc_execution_unique_firstproductcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_execution_unique_firstproductcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_execution_unique_firstproductcoefficientsum_body_terminal. fs_h_pfc_execution_unique_firstproductcoefficientsum_body_terminal + S (pfc_natural_sum_execution_unique_firstproductcoefficient) = S ((S (S (pfc_index_execution_unique_firstproduct))) * fs_v_pfc_execution_unique_firstproductcoefficientsum)) /\\ exists fs_q_pfc_execution_unique_firstproductcoefficientsum_body_terminal. fs_u_pfc_execution_unique_firstproductcoefficientsum = fs_q_pfc_execution_unique_firstproductcoefficientsum_body_terminal * S ((S (S (pfc_index_execution_unique_firstproduct))) * fs_v_pfc_execution_unique_firstproductcoefficientsum) + (pfc_natural_sum_execution_unique_firstproductcoefficient))) /\\ forall fs_i_pfc_execution_unique_firstproductcoefficientsum_body_steps. (exists fs_lt_pfc_execution_unique_firstproductcoefficientsum_body_steps_bound. fs_lt_pfc_execution_unique_firstproductcoefficientsum_body_steps_bound + S fs_i_pfc_execution_unique_firstproductcoefficientsum_body_steps = S (pfc_index_execution_unique_firstproduct)) -> exists fs_a_pfc_execution_unique_firstproductcoefficientsum_body_steps fs_r_pfc_execution_unique_firstproductcoefficientsum_body_steps fs_s_pfc_execution_unique_firstproductcoefficientsum_body_steps. ((((exists fs_h_pfc_execution_unique_firstproductcoefficientsum_body_steps_summand. fs_h_pfc_execution_unique_firstproductcoefficientsum_body_steps_summand + S (fs_a_pfc_execution_unique_firstproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_unique_firstproductcoefficientsum_body_steps)) * pfc_terms_scale_execution_unique_firstproductcoefficient)) /\\ exists fs_q_pfc_execution_unique_firstproductcoefficientsum_body_steps_summand. pfc_terms_code_execution_unique_firstproductcoefficient = fs_q_pfc_execution_unique_firstproductcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_execution_unique_firstproductcoefficientsum_body_steps)) * pfc_terms_scale_execution_unique_firstproductcoefficient) + (fs_a_pfc_execution_unique_firstproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_unique_firstproductcoefficientsum_body_steps_partial. fs_h_pfc_execution_unique_firstproductcoefficientsum_body_steps_partial + S (fs_r_pfc_execution_unique_firstproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_unique_firstproductcoefficientsum_body_steps)) * fs_v_pfc_execution_unique_firstproductcoefficientsum)) /\\ exists fs_q_pfc_execution_unique_firstproductcoefficientsum_body_steps_partial. fs_u_pfc_execution_unique_firstproductcoefficientsum = fs_q_pfc_execution_unique_firstproductcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_execution_unique_firstproductcoefficientsum_body_steps)) * fs_v_pfc_execution_unique_firstproductcoefficientsum) + (fs_r_pfc_execution_unique_firstproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_unique_firstproductcoefficientsum_body_steps_successor. fs_h_pfc_execution_unique_firstproductcoefficientsum_body_steps_successor + S (fs_s_pfc_execution_unique_firstproductcoefficientsum_body_steps) = S ((S (S fs_i_pfc_execution_unique_firstproductcoefficientsum_body_steps)) * fs_v_pfc_execution_unique_firstproductcoefficientsum)) /\\ exists fs_q_pfc_execution_unique_firstproductcoefficientsum_body_steps_successor. fs_u_pfc_execution_unique_firstproductcoefficientsum = fs_q_pfc_execution_unique_firstproductcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_execution_unique_firstproductcoefficientsum_body_steps)) * fs_v_pfc_execution_unique_firstproductcoefficientsum) + (fs_s_pfc_execution_unique_firstproductcoefficientsum_body_steps))) /\\ fs_s_pfc_execution_unique_firstproductcoefficientsum_body_steps = fs_r_pfc_execution_unique_firstproductcoefficientsum_body_steps + fs_a_pfc_execution_unique_firstproductcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_execution_unique_firstproductcoefficientresiduebound. pfa_gap_execution_unique_firstproductcoefficientresiduebound + S (pfc_value_execution_unique_firstproduct) = (p)) /\\ ((exists pfa_offset_left_execution_unique_firstproductcoefficientresiduecongruence pfa_offset_right_execution_unique_firstproductcoefficientresiduecongruence. (pfc_natural_sum_execution_unique_firstproductcoefficient) + (p) * pfa_offset_left_execution_unique_firstproductcoefficientresiduecongruence = (pfc_value_execution_unique_firstproduct) + (p) * pfa_offset_right_execution_unique_firstproductcoefficientresiduecongruence)))))))))))) /\\ (((forall pfs_index_execution_unique_firstdifference. (exists pfa_gap_execution_unique_firstdifferenceindex. pfa_gap_execution_unique_firstdifferenceindex + S (pfs_index_execution_unique_firstdifference) = (L)) -> exists pfs_left_execution_unique_firstdifference pfs_right_execution_unique_firstdifference pfs_result_execution_unique_firstdifference. ((((exists ff_h_pfp_execution_unique_firstdifferenceleft. ff_h_pfp_execution_unique_firstdifferenceleft + S (pfs_left_execution_unique_firstdifference) = S ((S (pfs_index_execution_unique_firstdifference)) * ac)) /\\ exists ff_q_pfp_execution_unique_firstdifferenceleft. ab = ff_q_pfp_execution_unique_firstdifferenceleft * S ((S (pfs_index_execution_unique_firstdifference)) * ac) + (pfs_left_execution_unique_firstdifference))) /\\ (((((exists ff_h_pfp_execution_unique_firstdifferenceright. ff_h_pfp_execution_unique_firstdifferenceright + S (pfs_right_execution_unique_firstdifference) = S ((S (pfs_index_execution_unique_firstdifference)) * pfd_product_scale_execution_unique_first)) /\\ exists ff_q_pfp_execution_unique_firstdifferenceright. pfd_product_code_execution_unique_first = ff_q_pfp_execution_unique_firstdifferenceright * S ((S (pfs_index_execution_unique_firstdifference)) * pfd_product_scale_execution_unique_first) + (pfs_right_execution_unique_firstdifference))) /\\ (((((exists ff_h_pfp_execution_unique_firstdifferenceresult. ff_h_pfp_execution_unique_firstdifferenceresult + S (pfs_result_execution_unique_firstdifference) = S ((S (pfs_index_execution_unique_firstdifference)) * pfd_residual_scale_execution_unique_first)) /\\ exists ff_q_pfp_execution_unique_firstdifferenceresult. pfd_residual_code_execution_unique_first = ff_q_pfp_execution_unique_firstdifferenceresult * S ((S (pfs_index_execution_unique_firstdifference)) * pfd_residual_scale_execution_unique_first) + (pfs_result_execution_unique_firstdifference))) /\\ ((((exists pfa_gap_execution_unique_firstdifferenceoperationleft. pfa_gap_execution_unique_firstdifferenceoperationleft + S (pfs_right_execution_unique_firstdifference) = (p)) /\\ (((exists pfa_gap_execution_unique_firstdifferenceoperationright. pfa_gap_execution_unique_firstdifferenceoperationright + S (pfs_result_execution_unique_firstdifference) = (p)) /\\ ((((exists pfa_gap_execution_unique_firstdifferenceoperationresultbound. pfa_gap_execution_unique_firstdifferenceoperationresultbound + S (pfs_left_execution_unique_firstdifference) = (p)) /\\ ((exists pfa_offset_left_execution_unique_firstdifferenceoperationresultcongruence pfa_offset_right_execution_unique_firstdifferenceoperationresultcongruence. ((pfs_right_execution_unique_firstdifference) + (pfs_result_execution_unique_firstdifference)) + (p) * pfa_offset_left_execution_unique_firstdifferenceoperationresultcongruence = (pfs_left_execution_unique_firstdifference) + (p) * pfa_offset_right_execution_unique_firstdifferenceoperationresultcongruence)))))))))))))))) /\\ (((((L)=(pfd_cut_execution_unique_first)+(R)) /\\ (((forall fom_index_pfp_execution_unique_firsttriminput. (exists fom_gap_pfp_execution_unique_firsttriminput_index_bound. fom_gap_pfp_execution_unique_firsttriminput_index_bound + S (fom_index_pfp_execution_unique_firsttriminput) = L) -> exists fom_value_pfp_execution_unique_firsttriminput. ((((exists fom_beta_height_pfp_execution_unique_firsttriminput_entry. fom_beta_height_pfp_execution_unique_firsttriminput_entry + S (fom_value_pfp_execution_unique_firsttriminput) = S ((S (fom_index_pfp_execution_unique_firsttriminput)) * pfd_residual_scale_execution_unique_first)) /\\ exists fom_beta_quotient_pfp_execution_unique_firsttriminput_entry. pfd_residual_code_execution_unique_first = fom_beta_quotient_pfp_execution_unique_firsttriminput_entry * S ((S (fom_index_pfp_execution_unique_firsttriminput)) * pfd_residual_scale_execution_unique_first) + (fom_value_pfp_execution_unique_firsttriminput))) /\\ (exists fom_gap_pfp_execution_unique_firsttriminput_value_bound. fom_gap_pfp_execution_unique_firsttriminput_value_bound + S (fom_value_pfp_execution_unique_firsttriminput) = p))) /\\ (((forall pfp_repeat_index_execution_unique_firsttrimremoved. (exists pfa_gap_execution_unique_firsttrimremovedindex. pfa_gap_execution_unique_firsttrimremovedindex + S (pfp_repeat_index_execution_unique_firsttrimremoved) = (pfd_cut_execution_unique_first)) -> (((exists ff_h_pfp_execution_unique_firsttrimremovedentry. ff_h_pfp_execution_unique_firsttrimremovedentry + S (0) = S ((S (pfp_repeat_index_execution_unique_firsttrimremoved)) * pfd_residual_scale_execution_unique_first)) /\\ exists ff_q_pfp_execution_unique_firsttrimremovedentry. pfd_residual_code_execution_unique_first = ff_q_pfp_execution_unique_firsttrimremovedentry * S ((S (pfp_repeat_index_execution_unique_firsttrimremoved)) * pfd_residual_scale_execution_unique_first) + (0)))) /\\ (((forall pftrim_index_execution_unique_firsttrimsuffix pftrim_value_execution_unique_firsttrimsuffix. (exists pfa_gap_execution_unique_firsttrimsuffixbound. pfa_gap_execution_unique_firsttrimsuffixbound + S (pftrim_index_execution_unique_firsttrimsuffix) = (R)) -> (((exists ff_h_pfp_execution_unique_firsttrimsuffixsource. ff_h_pfp_execution_unique_firsttrimsuffixsource + S (pftrim_value_execution_unique_firsttrimsuffix) = S ((S ((pfd_cut_execution_unique_first)+pftrim_index_execution_unique_firsttrimsuffix)) * pfd_residual_scale_execution_unique_first)) /\\ exists ff_q_pfp_execution_unique_firsttrimsuffixsource. pfd_residual_code_execution_unique_first = ff_q_pfp_execution_unique_firsttrimsuffixsource * S ((S ((pfd_cut_execution_unique_first)+pftrim_index_execution_unique_firsttrimsuffix)) * pfd_residual_scale_execution_unique_first) + (pftrim_value_execution_unique_firsttrimsuffix))) -> (((exists ff_h_pfp_execution_unique_firsttrimsuffixoutput. ff_h_pfp_execution_unique_firsttrimsuffixoutput + S (pftrim_value_execution_unique_firsttrimsuffix) = S ((S (pftrim_index_execution_unique_firsttrimsuffix)) * rc)) /\\ exists ff_q_pfp_execution_unique_firsttrimsuffixoutput. rb = ff_q_pfp_execution_unique_firsttrimsuffixoutput * S ((S (pftrim_index_execution_unique_firsttrimsuffix)) * rc) + (pftrim_value_execution_unique_firsttrimsuffix)))) /\\ (((R)=0 \\/ (exists pftrim_leading_execution_unique_firsttrimnormal. ((((exists ff_h_pfp_execution_unique_firsttrimnormalentry. ff_h_pfp_execution_unique_firsttrimnormalentry + S (pftrim_leading_execution_unique_firsttrimnormal) = S ((S (0)) * rc)) /\\ exists ff_q_pfp_execution_unique_firsttrimnormalentry. rb = ff_q_pfp_execution_unique_firsttrimnormalentry * S ((S (0)) * rc) + (pftrim_leading_execution_unique_firsttrimnormal))) /\\ ((~(pftrim_leading_execution_unique_firsttrimnormal=0))))))))))))))))))))))))))))))))) -> (((forall fom_index_pfp_execution_unique_secondinput. (exists fom_gap_pfp_execution_unique_secondinput_index_bound. fom_gap_pfp_execution_unique_secondinput_index_bound + S (fom_index_pfp_execution_unique_secondinput) = L) -> exists fom_value_pfp_execution_unique_secondinput. ((((exists fom_beta_height_pfp_execution_unique_secondinput_entry. fom_beta_height_pfp_execution_unique_secondinput_entry + S (fom_value_pfp_execution_unique_secondinput) = S ((S (fom_index_pfp_execution_unique_secondinput)) * ac)) /\\ exists fom_beta_quotient_pfp_execution_unique_secondinput_entry. ab = fom_beta_quotient_pfp_execution_unique_secondinput_entry * S ((S (fom_index_pfp_execution_unique_secondinput)) * ac) + (fom_value_pfp_execution_unique_secondinput))) /\\ (exists fom_gap_pfp_execution_unique_secondinput_value_bound. fom_gap_pfp_execution_unique_secondinput_value_bound + S (fom_value_pfp_execution_unique_secondinput) = p))) /\\ (((forall fom_index_pfp_execution_unique_seconddivisor. (exists fom_gap_pfp_execution_unique_seconddivisor_index_bound. fom_gap_pfp_execution_unique_seconddivisor_index_bound + S (fom_index_pfp_execution_unique_seconddivisor) = S (d)) -> exists fom_value_pfp_execution_unique_seconddivisor. ((((exists fom_beta_height_pfp_execution_unique_seconddivisor_entry. fom_beta_height_pfp_execution_unique_seconddivisor_entry + S (fom_value_pfp_execution_unique_seconddivisor) = S ((S (fom_index_pfp_execution_unique_seconddivisor)) * bc)) /\\ exists fom_beta_quotient_pfp_execution_unique_seconddivisor_entry. bb = fom_beta_quotient_pfp_execution_unique_seconddivisor_entry * S ((S (fom_index_pfp_execution_unique_seconddivisor)) * bc) + (fom_value_pfp_execution_unique_seconddivisor))) /\\ (exists fom_gap_pfp_execution_unique_seconddivisor_value_bound. fom_gap_pfp_execution_unique_seconddivisor_value_bound + S (fom_value_pfp_execution_unique_seconddivisor) = p))) /\\ (((((((Q)=0) /\\ ((exists pfc_gap_execution_unique_secondlengthshort. pfc_gap_execution_unique_secondlengthshort+(L)=(d))))) \\/ (((~((Q)=0)) /\\ (((Q)+(d)=(L)))))) /\\ ((exists pfd_head_execution_unique_second pfd_inverse_execution_unique_second pfd_product_code_execution_unique_second pfd_product_scale_execution_unique_second pfd_residual_code_execution_unique_second pfd_residual_scale_execution_unique_second pfd_cut_execution_unique_second. ((((exists ff_h_pfp_execution_unique_secondhead. ff_h_pfp_execution_unique_secondhead + S (pfd_head_execution_unique_second) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_execution_unique_secondhead. bb = ff_q_pfp_execution_unique_secondhead * S ((S (0)) * bc) + (pfd_head_execution_unique_second))) /\\ (((((~((pfd_head_execution_unique_second) = 0)) /\\ ((((exists pfa_gap_execution_unique_secondinversemultiplicationleft. pfa_gap_execution_unique_secondinversemultiplicationleft + S (pfd_head_execution_unique_second) = (p)) /\\ (((exists pfa_gap_execution_unique_secondinversemultiplicationright. pfa_gap_execution_unique_secondinversemultiplicationright + S (pfd_inverse_execution_unique_second) = (p)) /\\ ((((exists pfa_gap_execution_unique_secondinversemultiplicationresultbound. pfa_gap_execution_unique_secondinversemultiplicationresultbound + S (1) = (p)) /\\ ((exists pfa_offset_left_execution_unique_secondinversemultiplicationresultcongruence pfa_offset_right_execution_unique_secondinversemultiplicationresultcongruence. ((pfd_head_execution_unique_second) * (pfd_inverse_execution_unique_second)) + (p) * pfa_offset_left_execution_unique_secondinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_execution_unique_secondinversemultiplicationresultcongruence)))))))))))) /\\ (((forall pfd_index_execution_unique_secondquotient. (exists pfa_gap_execution_unique_secondquotientbound. pfa_gap_execution_unique_secondquotientbound + S (pfd_index_execution_unique_secondquotient) = (Q)) -> exists pfd_value_execution_unique_secondquotient. ((((exists ff_h_pfp_execution_unique_secondquotiententry. ff_h_pfp_execution_unique_secondquotiententry + S (pfd_value_execution_unique_secondquotient) = S ((S (pfd_index_execution_unique_secondquotient)) * QC)) /\\ exists ff_q_pfp_execution_unique_secondquotiententry. QB = ff_q_pfp_execution_unique_secondquotiententry * S ((S (pfd_index_execution_unique_secondquotient)) * QC) + (pfd_value_execution_unique_secondquotient))) /\\ ((exists pfd_input_execution_unique_secondquotientstep pfd_previous_execution_unique_secondquotientstep pfd_difference_execution_unique_secondquotientstep. ((((exists ff_h_pfp_execution_unique_secondquotientstepinput. ff_h_pfp_execution_unique_secondquotientstepinput + S (pfd_input_execution_unique_secondquotientstep) = S ((S (pfd_index_execution_unique_secondquotient)) * ac)) /\\ exists ff_q_pfp_execution_unique_secondquotientstepinput. ab = ff_q_pfp_execution_unique_secondquotientstepinput * S ((S (pfd_index_execution_unique_secondquotient)) * ac) + (pfd_input_execution_unique_secondquotientstep))) /\\ (((exists pfc_terms_code_execution_unique_secondquotientstepprevious pfc_terms_scale_execution_unique_secondquotientstepprevious pfc_natural_sum_execution_unique_secondquotientstepprevious. ((forall pfc_index_execution_unique_secondquotientsteppreviousdiagonal. (exists pfa_gap_execution_unique_secondquotientsteppreviousdiagonalbound. pfa_gap_execution_unique_secondquotientsteppreviousdiagonalbound + S (pfc_index_execution_unique_secondquotientsteppreviousdiagonal) = (S (pfd_index_execution_unique_secondquotient))) -> exists pfc_value_execution_unique_secondquotientsteppreviousdiagonal. ((((exists ff_h_pfp_execution_unique_secondquotientsteppreviousdiagonalentry. ff_h_pfp_execution_unique_secondquotientsteppreviousdiagonalentry + S (pfc_value_execution_unique_secondquotientsteppreviousdiagonal) = S ((S (pfc_index_execution_unique_secondquotientsteppreviousdiagonal)) * pfc_terms_scale_execution_unique_secondquotientstepprevious)) /\\ exists ff_q_pfp_execution_unique_secondquotientsteppreviousdiagonalentry. pfc_terms_code_execution_unique_secondquotientstepprevious = ff_q_pfp_execution_unique_secondquotientsteppreviousdiagonalentry * S ((S (pfc_index_execution_unique_secondquotientsteppreviousdiagonal)) * pfc_terms_scale_execution_unique_secondquotientstepprevious) + (pfc_value_execution_unique_secondquotientsteppreviousdiagonal))) /\\ ((exists pfc_complement_execution_unique_secondquotientsteppreviousdiagonalterm pfc_left_execution_unique_secondquotientsteppreviousdiagonalterm pfc_right_execution_unique_secondquotientsteppreviousdiagonalterm. (((pfc_index_execution_unique_secondquotientsteppreviousdiagonal)+pfc_complement_execution_unique_secondquotientsteppreviousdiagonalterm=(pfd_index_execution_unique_secondquotient)) /\\ ((((((exists pfa_gap_execution_unique_secondquotientsteppreviousdiagonaltermleftinside. pfa_gap_execution_unique_secondquotientsteppreviousdiagonaltermleftinside + S (pfc_index_execution_unique_secondquotientsteppreviousdiagonal) = (pfd_index_execution_unique_secondquotient)) /\\ ((((exists ff_h_pfp_execution_unique_secondquotientsteppreviousdiagonaltermleftentry. ff_h_pfp_execution_unique_secondquotientsteppreviousdiagonaltermleftentry + S (pfc_left_execution_unique_secondquotientsteppreviousdiagonalterm) = S ((S (pfc_index_execution_unique_secondquotientsteppreviousdiagonal)) * QC)) /\\ exists ff_q_pfp_execution_unique_secondquotientsteppreviousdiagonaltermleftentry. QB = ff_q_pfp_execution_unique_secondquotientsteppreviousdiagonaltermleftentry * S ((S (pfc_index_execution_unique_secondquotientsteppreviousdiagonal)) * QC) + (pfc_left_execution_unique_secondquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_execution_unique_secondquotientsteppreviousdiagonaltermleftoutside. pfc_gap_execution_unique_secondquotientsteppreviousdiagonaltermleftoutside+(pfd_index_execution_unique_secondquotient)=(pfc_index_execution_unique_secondquotientsteppreviousdiagonal)) /\\ (((pfc_left_execution_unique_secondquotientsteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_execution_unique_secondquotientsteppreviousdiagonaltermrightinside. pfa_gap_execution_unique_secondquotientsteppreviousdiagonaltermrightinside + S (pfc_complement_execution_unique_secondquotientsteppreviousdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_execution_unique_secondquotientsteppreviousdiagonaltermrightentry. ff_h_pfp_execution_unique_secondquotientsteppreviousdiagonaltermrightentry + S (pfc_right_execution_unique_secondquotientsteppreviousdiagonalterm) = S ((S (pfc_complement_execution_unique_secondquotientsteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_execution_unique_secondquotientsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_execution_unique_secondquotientsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_execution_unique_secondquotientsteppreviousdiagonalterm)) * bc) + (pfc_right_execution_unique_secondquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_execution_unique_secondquotientsteppreviousdiagonaltermrightoutside. pfc_gap_execution_unique_secondquotientsteppreviousdiagonaltermrightoutside+(S (d))=(pfc_complement_execution_unique_secondquotientsteppreviousdiagonalterm)) /\\ (((pfc_right_execution_unique_secondquotientsteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_execution_unique_secondquotientsteppreviousdiagonal)=pfc_left_execution_unique_secondquotientsteppreviousdiagonalterm*pfc_right_execution_unique_secondquotientsteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_execution_unique_secondquotientstepprevioussum fs_v_pfc_execution_unique_secondquotientstepprevioussum. ((((exists fs_h_pfc_execution_unique_secondquotientstepprevioussum_body_start. fs_h_pfc_execution_unique_secondquotientstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_unique_secondquotientstepprevioussum)) /\\ exists fs_q_pfc_execution_unique_secondquotientstepprevioussum_body_start. fs_u_pfc_execution_unique_secondquotientstepprevioussum = fs_q_pfc_execution_unique_secondquotientstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_execution_unique_secondquotientstepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_execution_unique_secondquotientstepprevioussum_body_terminal. fs_h_pfc_execution_unique_secondquotientstepprevioussum_body_terminal + S (pfc_natural_sum_execution_unique_secondquotientstepprevious) = S ((S (S (pfd_index_execution_unique_secondquotient))) * fs_v_pfc_execution_unique_secondquotientstepprevioussum)) /\\ exists fs_q_pfc_execution_unique_secondquotientstepprevioussum_body_terminal. fs_u_pfc_execution_unique_secondquotientstepprevioussum = fs_q_pfc_execution_unique_secondquotientstepprevioussum_body_terminal * S ((S (S (pfd_index_execution_unique_secondquotient))) * fs_v_pfc_execution_unique_secondquotientstepprevioussum) + (pfc_natural_sum_execution_unique_secondquotientstepprevious))) /\\ forall fs_i_pfc_execution_unique_secondquotientstepprevioussum_body_steps. (exists fs_lt_pfc_execution_unique_secondquotientstepprevioussum_body_steps_bound. fs_lt_pfc_execution_unique_secondquotientstepprevioussum_body_steps_bound + S fs_i_pfc_execution_unique_secondquotientstepprevioussum_body_steps = S (pfd_index_execution_unique_secondquotient)) -> exists fs_a_pfc_execution_unique_secondquotientstepprevioussum_body_steps fs_r_pfc_execution_unique_secondquotientstepprevioussum_body_steps fs_s_pfc_execution_unique_secondquotientstepprevioussum_body_steps. ((((exists fs_h_pfc_execution_unique_secondquotientstepprevioussum_body_steps_summand. fs_h_pfc_execution_unique_secondquotientstepprevioussum_body_steps_summand + S (fs_a_pfc_execution_unique_secondquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_execution_unique_secondquotientstepprevioussum_body_steps)) * pfc_terms_scale_execution_unique_secondquotientstepprevious)) /\\ exists fs_q_pfc_execution_unique_secondquotientstepprevioussum_body_steps_summand. pfc_terms_code_execution_unique_secondquotientstepprevious = fs_q_pfc_execution_unique_secondquotientstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_execution_unique_secondquotientstepprevioussum_body_steps)) * pfc_terms_scale_execution_unique_secondquotientstepprevious) + (fs_a_pfc_execution_unique_secondquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_execution_unique_secondquotientstepprevioussum_body_steps_partial. fs_h_pfc_execution_unique_secondquotientstepprevioussum_body_steps_partial + S (fs_r_pfc_execution_unique_secondquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_execution_unique_secondquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_unique_secondquotientstepprevioussum)) /\\ exists fs_q_pfc_execution_unique_secondquotientstepprevioussum_body_steps_partial. fs_u_pfc_execution_unique_secondquotientstepprevioussum = fs_q_pfc_execution_unique_secondquotientstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_execution_unique_secondquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_unique_secondquotientstepprevioussum) + (fs_r_pfc_execution_unique_secondquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_execution_unique_secondquotientstepprevioussum_body_steps_successor. fs_h_pfc_execution_unique_secondquotientstepprevioussum_body_steps_successor + S (fs_s_pfc_execution_unique_secondquotientstepprevioussum_body_steps) = S ((S (S fs_i_pfc_execution_unique_secondquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_unique_secondquotientstepprevioussum)) /\\ exists fs_q_pfc_execution_unique_secondquotientstepprevioussum_body_steps_successor. fs_u_pfc_execution_unique_secondquotientstepprevioussum = fs_q_pfc_execution_unique_secondquotientstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_execution_unique_secondquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_unique_secondquotientstepprevioussum) + (fs_s_pfc_execution_unique_secondquotientstepprevioussum_body_steps))) /\\ fs_s_pfc_execution_unique_secondquotientstepprevioussum_body_steps = fs_r_pfc_execution_unique_secondquotientstepprevioussum_body_steps + fs_a_pfc_execution_unique_secondquotientstepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_execution_unique_secondquotientsteppreviousresiduebound. pfa_gap_execution_unique_secondquotientsteppreviousresiduebound + S (pfd_previous_execution_unique_secondquotientstep) = (p)) /\\ ((exists pfa_offset_left_execution_unique_secondquotientsteppreviousresiduecongruence pfa_offset_right_execution_unique_secondquotientsteppreviousresiduecongruence. (pfc_natural_sum_execution_unique_secondquotientstepprevious) + (p) * pfa_offset_left_execution_unique_secondquotientsteppreviousresiduecongruence = (pfd_previous_execution_unique_secondquotientstep) + (p) * pfa_offset_right_execution_unique_secondquotientsteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_execution_unique_secondquotientstepsubtractleft. pfa_gap_execution_unique_secondquotientstepsubtractleft + S (pfd_previous_execution_unique_secondquotientstep) = (p)) /\\ (((exists pfa_gap_execution_unique_secondquotientstepsubtractright. pfa_gap_execution_unique_secondquotientstepsubtractright + S (pfd_difference_execution_unique_secondquotientstep) = (p)) /\\ ((((exists pfa_gap_execution_unique_secondquotientstepsubtractresultbound. pfa_gap_execution_unique_secondquotientstepsubtractresultbound + S (pfd_input_execution_unique_secondquotientstep) = (p)) /\\ ((exists pfa_offset_left_execution_unique_secondquotientstepsubtractresultcongruence pfa_offset_right_execution_unique_secondquotientstepsubtractresultcongruence. ((pfd_previous_execution_unique_secondquotientstep) + (pfd_difference_execution_unique_secondquotientstep)) + (p) * pfa_offset_left_execution_unique_secondquotientstepsubtractresultcongruence = (pfd_input_execution_unique_secondquotientstep) + (p) * pfa_offset_right_execution_unique_secondquotientstepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_execution_unique_secondquotientstepmultiplyleft. pfa_gap_execution_unique_secondquotientstepmultiplyleft + S (pfd_inverse_execution_unique_second) = (p)) /\\ (((exists pfa_gap_execution_unique_secondquotientstepmultiplyright. pfa_gap_execution_unique_secondquotientstepmultiplyright + S (pfd_difference_execution_unique_secondquotientstep) = (p)) /\\ ((((exists pfa_gap_execution_unique_secondquotientstepmultiplyresultbound. pfa_gap_execution_unique_secondquotientstepmultiplyresultbound + S (pfd_value_execution_unique_secondquotient) = (p)) /\\ ((exists pfa_offset_left_execution_unique_secondquotientstepmultiplyresultcongruence pfa_offset_right_execution_unique_secondquotientstepmultiplyresultcongruence. ((pfd_inverse_execution_unique_second) * (pfd_difference_execution_unique_secondquotientstep)) + (p) * pfa_offset_left_execution_unique_secondquotientstepmultiplyresultcongruence = (pfd_value_execution_unique_secondquotient) + (p) * pfa_offset_right_execution_unique_secondquotientstepmultiplyresultcongruence))))))))))))))))))) /\\ (((forall pfc_index_execution_unique_secondproduct. (exists pfa_gap_execution_unique_secondproductbound. pfa_gap_execution_unique_secondproductbound + S (pfc_index_execution_unique_secondproduct) = (L)) -> exists pfc_value_execution_unique_secondproduct. ((((exists ff_h_pfp_execution_unique_secondproductentry. ff_h_pfp_execution_unique_secondproductentry + S (pfc_value_execution_unique_secondproduct) = S ((S (pfc_index_execution_unique_secondproduct)) * pfd_product_scale_execution_unique_second)) /\\ exists ff_q_pfp_execution_unique_secondproductentry. pfd_product_code_execution_unique_second = ff_q_pfp_execution_unique_secondproductentry * S ((S (pfc_index_execution_unique_secondproduct)) * pfd_product_scale_execution_unique_second) + (pfc_value_execution_unique_secondproduct))) /\\ ((exists pfc_terms_code_execution_unique_secondproductcoefficient pfc_terms_scale_execution_unique_secondproductcoefficient pfc_natural_sum_execution_unique_secondproductcoefficient. ((forall pfc_index_execution_unique_secondproductcoefficientdiagonal. (exists pfa_gap_execution_unique_secondproductcoefficientdiagonalbound. pfa_gap_execution_unique_secondproductcoefficientdiagonalbound + S (pfc_index_execution_unique_secondproductcoefficientdiagonal) = (S (pfc_index_execution_unique_secondproduct))) -> exists pfc_value_execution_unique_secondproductcoefficientdiagonal. ((((exists ff_h_pfp_execution_unique_secondproductcoefficientdiagonalentry. ff_h_pfp_execution_unique_secondproductcoefficientdiagonalentry + S (pfc_value_execution_unique_secondproductcoefficientdiagonal) = S ((S (pfc_index_execution_unique_secondproductcoefficientdiagonal)) * pfc_terms_scale_execution_unique_secondproductcoefficient)) /\\ exists ff_q_pfp_execution_unique_secondproductcoefficientdiagonalentry. pfc_terms_code_execution_unique_secondproductcoefficient = ff_q_pfp_execution_unique_secondproductcoefficientdiagonalentry * S ((S (pfc_index_execution_unique_secondproductcoefficientdiagonal)) * pfc_terms_scale_execution_unique_secondproductcoefficient) + (pfc_value_execution_unique_secondproductcoefficientdiagonal))) /\\ ((exists pfc_complement_execution_unique_secondproductcoefficientdiagonalterm pfc_left_execution_unique_secondproductcoefficientdiagonalterm pfc_right_execution_unique_secondproductcoefficientdiagonalterm. (((pfc_index_execution_unique_secondproductcoefficientdiagonal)+pfc_complement_execution_unique_secondproductcoefficientdiagonalterm=(pfc_index_execution_unique_secondproduct)) /\\ ((((((exists pfa_gap_execution_unique_secondproductcoefficientdiagonaltermleftinside. pfa_gap_execution_unique_secondproductcoefficientdiagonaltermleftinside + S (pfc_index_execution_unique_secondproductcoefficientdiagonal) = (Q)) /\\ ((((exists ff_h_pfp_execution_unique_secondproductcoefficientdiagonaltermleftentry. ff_h_pfp_execution_unique_secondproductcoefficientdiagonaltermleftentry + S (pfc_left_execution_unique_secondproductcoefficientdiagonalterm) = S ((S (pfc_index_execution_unique_secondproductcoefficientdiagonal)) * QC)) /\\ exists ff_q_pfp_execution_unique_secondproductcoefficientdiagonaltermleftentry. QB = ff_q_pfp_execution_unique_secondproductcoefficientdiagonaltermleftentry * S ((S (pfc_index_execution_unique_secondproductcoefficientdiagonal)) * QC) + (pfc_left_execution_unique_secondproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_unique_secondproductcoefficientdiagonaltermleftoutside. pfc_gap_execution_unique_secondproductcoefficientdiagonaltermleftoutside+(Q)=(pfc_index_execution_unique_secondproductcoefficientdiagonal)) /\\ (((pfc_left_execution_unique_secondproductcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_execution_unique_secondproductcoefficientdiagonaltermrightinside. pfa_gap_execution_unique_secondproductcoefficientdiagonaltermrightinside + S (pfc_complement_execution_unique_secondproductcoefficientdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_execution_unique_secondproductcoefficientdiagonaltermrightentry. ff_h_pfp_execution_unique_secondproductcoefficientdiagonaltermrightentry + S (pfc_right_execution_unique_secondproductcoefficientdiagonalterm) = S ((S (pfc_complement_execution_unique_secondproductcoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_execution_unique_secondproductcoefficientdiagonaltermrightentry. bb = ff_q_pfp_execution_unique_secondproductcoefficientdiagonaltermrightentry * S ((S (pfc_complement_execution_unique_secondproductcoefficientdiagonalterm)) * bc) + (pfc_right_execution_unique_secondproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_unique_secondproductcoefficientdiagonaltermrightoutside. pfc_gap_execution_unique_secondproductcoefficientdiagonaltermrightoutside+(S (d))=(pfc_complement_execution_unique_secondproductcoefficientdiagonalterm)) /\\ (((pfc_right_execution_unique_secondproductcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_execution_unique_secondproductcoefficientdiagonal)=pfc_left_execution_unique_secondproductcoefficientdiagonalterm*pfc_right_execution_unique_secondproductcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_execution_unique_secondproductcoefficientsum fs_v_pfc_execution_unique_secondproductcoefficientsum. ((((exists fs_h_pfc_execution_unique_secondproductcoefficientsum_body_start. fs_h_pfc_execution_unique_secondproductcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_unique_secondproductcoefficientsum)) /\\ exists fs_q_pfc_execution_unique_secondproductcoefficientsum_body_start. fs_u_pfc_execution_unique_secondproductcoefficientsum = fs_q_pfc_execution_unique_secondproductcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_execution_unique_secondproductcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_execution_unique_secondproductcoefficientsum_body_terminal. fs_h_pfc_execution_unique_secondproductcoefficientsum_body_terminal + S (pfc_natural_sum_execution_unique_secondproductcoefficient) = S ((S (S (pfc_index_execution_unique_secondproduct))) * fs_v_pfc_execution_unique_secondproductcoefficientsum)) /\\ exists fs_q_pfc_execution_unique_secondproductcoefficientsum_body_terminal. fs_u_pfc_execution_unique_secondproductcoefficientsum = fs_q_pfc_execution_unique_secondproductcoefficientsum_body_terminal * S ((S (S (pfc_index_execution_unique_secondproduct))) * fs_v_pfc_execution_unique_secondproductcoefficientsum) + (pfc_natural_sum_execution_unique_secondproductcoefficient))) /\\ forall fs_i_pfc_execution_unique_secondproductcoefficientsum_body_steps. (exists fs_lt_pfc_execution_unique_secondproductcoefficientsum_body_steps_bound. fs_lt_pfc_execution_unique_secondproductcoefficientsum_body_steps_bound + S fs_i_pfc_execution_unique_secondproductcoefficientsum_body_steps = S (pfc_index_execution_unique_secondproduct)) -> exists fs_a_pfc_execution_unique_secondproductcoefficientsum_body_steps fs_r_pfc_execution_unique_secondproductcoefficientsum_body_steps fs_s_pfc_execution_unique_secondproductcoefficientsum_body_steps. ((((exists fs_h_pfc_execution_unique_secondproductcoefficientsum_body_steps_summand. fs_h_pfc_execution_unique_secondproductcoefficientsum_body_steps_summand + S (fs_a_pfc_execution_unique_secondproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_unique_secondproductcoefficientsum_body_steps)) * pfc_terms_scale_execution_unique_secondproductcoefficient)) /\\ exists fs_q_pfc_execution_unique_secondproductcoefficientsum_body_steps_summand. pfc_terms_code_execution_unique_secondproductcoefficient = fs_q_pfc_execution_unique_secondproductcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_execution_unique_secondproductcoefficientsum_body_steps)) * pfc_terms_scale_execution_unique_secondproductcoefficient) + (fs_a_pfc_execution_unique_secondproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_unique_secondproductcoefficientsum_body_steps_partial. fs_h_pfc_execution_unique_secondproductcoefficientsum_body_steps_partial + S (fs_r_pfc_execution_unique_secondproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_unique_secondproductcoefficientsum_body_steps)) * fs_v_pfc_execution_unique_secondproductcoefficientsum)) /\\ exists fs_q_pfc_execution_unique_secondproductcoefficientsum_body_steps_partial. fs_u_pfc_execution_unique_secondproductcoefficientsum = fs_q_pfc_execution_unique_secondproductcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_execution_unique_secondproductcoefficientsum_body_steps)) * fs_v_pfc_execution_unique_secondproductcoefficientsum) + (fs_r_pfc_execution_unique_secondproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_unique_secondproductcoefficientsum_body_steps_successor. fs_h_pfc_execution_unique_secondproductcoefficientsum_body_steps_successor + S (fs_s_pfc_execution_unique_secondproductcoefficientsum_body_steps) = S ((S (S fs_i_pfc_execution_unique_secondproductcoefficientsum_body_steps)) * fs_v_pfc_execution_unique_secondproductcoefficientsum)) /\\ exists fs_q_pfc_execution_unique_secondproductcoefficientsum_body_steps_successor. fs_u_pfc_execution_unique_secondproductcoefficientsum = fs_q_pfc_execution_unique_secondproductcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_execution_unique_secondproductcoefficientsum_body_steps)) * fs_v_pfc_execution_unique_secondproductcoefficientsum) + (fs_s_pfc_execution_unique_secondproductcoefficientsum_body_steps))) /\\ fs_s_pfc_execution_unique_secondproductcoefficientsum_body_steps = fs_r_pfc_execution_unique_secondproductcoefficientsum_body_steps + fs_a_pfc_execution_unique_secondproductcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_execution_unique_secondproductcoefficientresiduebound. pfa_gap_execution_unique_secondproductcoefficientresiduebound + S (pfc_value_execution_unique_secondproduct) = (p)) /\\ ((exists pfa_offset_left_execution_unique_secondproductcoefficientresiduecongruence pfa_offset_right_execution_unique_secondproductcoefficientresiduecongruence. (pfc_natural_sum_execution_unique_secondproductcoefficient) + (p) * pfa_offset_left_execution_unique_secondproductcoefficientresiduecongruence = (pfc_value_execution_unique_secondproduct) + (p) * pfa_offset_right_execution_unique_secondproductcoefficientresiduecongruence)))))))))))) /\\ (((forall pfs_index_execution_unique_seconddifference. (exists pfa_gap_execution_unique_seconddifferenceindex. pfa_gap_execution_unique_seconddifferenceindex + S (pfs_index_execution_unique_seconddifference) = (L)) -> exists pfs_left_execution_unique_seconddifference pfs_right_execution_unique_seconddifference pfs_result_execution_unique_seconddifference. ((((exists ff_h_pfp_execution_unique_seconddifferenceleft. ff_h_pfp_execution_unique_seconddifferenceleft + S (pfs_left_execution_unique_seconddifference) = S ((S (pfs_index_execution_unique_seconddifference)) * ac)) /\\ exists ff_q_pfp_execution_unique_seconddifferenceleft. ab = ff_q_pfp_execution_unique_seconddifferenceleft * S ((S (pfs_index_execution_unique_seconddifference)) * ac) + (pfs_left_execution_unique_seconddifference))) /\\ (((((exists ff_h_pfp_execution_unique_seconddifferenceright. ff_h_pfp_execution_unique_seconddifferenceright + S (pfs_right_execution_unique_seconddifference) = S ((S (pfs_index_execution_unique_seconddifference)) * pfd_product_scale_execution_unique_second)) /\\ exists ff_q_pfp_execution_unique_seconddifferenceright. pfd_product_code_execution_unique_second = ff_q_pfp_execution_unique_seconddifferenceright * S ((S (pfs_index_execution_unique_seconddifference)) * pfd_product_scale_execution_unique_second) + (pfs_right_execution_unique_seconddifference))) /\\ (((((exists ff_h_pfp_execution_unique_seconddifferenceresult. ff_h_pfp_execution_unique_seconddifferenceresult + S (pfs_result_execution_unique_seconddifference) = S ((S (pfs_index_execution_unique_seconddifference)) * pfd_residual_scale_execution_unique_second)) /\\ exists ff_q_pfp_execution_unique_seconddifferenceresult. pfd_residual_code_execution_unique_second = ff_q_pfp_execution_unique_seconddifferenceresult * S ((S (pfs_index_execution_unique_seconddifference)) * pfd_residual_scale_execution_unique_second) + (pfs_result_execution_unique_seconddifference))) /\\ ((((exists pfa_gap_execution_unique_seconddifferenceoperationleft. pfa_gap_execution_unique_seconddifferenceoperationleft + S (pfs_right_execution_unique_seconddifference) = (p)) /\\ (((exists pfa_gap_execution_unique_seconddifferenceoperationright. pfa_gap_execution_unique_seconddifferenceoperationright + S (pfs_result_execution_unique_seconddifference) = (p)) /\\ ((((exists pfa_gap_execution_unique_seconddifferenceoperationresultbound. pfa_gap_execution_unique_seconddifferenceoperationresultbound + S (pfs_left_execution_unique_seconddifference) = (p)) /\\ ((exists pfa_offset_left_execution_unique_seconddifferenceoperationresultcongruence pfa_offset_right_execution_unique_seconddifferenceoperationresultcongruence. ((pfs_right_execution_unique_seconddifference) + (pfs_result_execution_unique_seconddifference)) + (p) * pfa_offset_left_execution_unique_seconddifferenceoperationresultcongruence = (pfs_left_execution_unique_seconddifference) + (p) * pfa_offset_right_execution_unique_seconddifferenceoperationresultcongruence)))))))))))))))) /\\ (((((L)=(pfd_cut_execution_unique_second)+(K)) /\\ (((forall fom_index_pfp_execution_unique_secondtriminput. (exists fom_gap_pfp_execution_unique_secondtriminput_index_bound. fom_gap_pfp_execution_unique_secondtriminput_index_bound + S (fom_index_pfp_execution_unique_secondtriminput) = L) -> exists fom_value_pfp_execution_unique_secondtriminput. ((((exists fom_beta_height_pfp_execution_unique_secondtriminput_entry. fom_beta_height_pfp_execution_unique_secondtriminput_entry + S (fom_value_pfp_execution_unique_secondtriminput) = S ((S (fom_index_pfp_execution_unique_secondtriminput)) * pfd_residual_scale_execution_unique_second)) /\\ exists fom_beta_quotient_pfp_execution_unique_secondtriminput_entry. pfd_residual_code_execution_unique_second = fom_beta_quotient_pfp_execution_unique_secondtriminput_entry * S ((S (fom_index_pfp_execution_unique_secondtriminput)) * pfd_residual_scale_execution_unique_second) + (fom_value_pfp_execution_unique_secondtriminput))) /\\ (exists fom_gap_pfp_execution_unique_secondtriminput_value_bound. fom_gap_pfp_execution_unique_secondtriminput_value_bound + S (fom_value_pfp_execution_unique_secondtriminput) = p))) /\\ (((forall pfp_repeat_index_execution_unique_secondtrimremoved. (exists pfa_gap_execution_unique_secondtrimremovedindex. pfa_gap_execution_unique_secondtrimremovedindex + S (pfp_repeat_index_execution_unique_secondtrimremoved) = (pfd_cut_execution_unique_second)) -> (((exists ff_h_pfp_execution_unique_secondtrimremovedentry. ff_h_pfp_execution_unique_secondtrimremovedentry + S (0) = S ((S (pfp_repeat_index_execution_unique_secondtrimremoved)) * pfd_residual_scale_execution_unique_second)) /\\ exists ff_q_pfp_execution_unique_secondtrimremovedentry. pfd_residual_code_execution_unique_second = ff_q_pfp_execution_unique_secondtrimremovedentry * S ((S (pfp_repeat_index_execution_unique_secondtrimremoved)) * pfd_residual_scale_execution_unique_second) + (0)))) /\\ (((forall pftrim_index_execution_unique_secondtrimsuffix pftrim_value_execution_unique_secondtrimsuffix. (exists pfa_gap_execution_unique_secondtrimsuffixbound. pfa_gap_execution_unique_secondtrimsuffixbound + S (pftrim_index_execution_unique_secondtrimsuffix) = (K)) -> (((exists ff_h_pfp_execution_unique_secondtrimsuffixsource. ff_h_pfp_execution_unique_secondtrimsuffixsource + S (pftrim_value_execution_unique_secondtrimsuffix) = S ((S ((pfd_cut_execution_unique_second)+pftrim_index_execution_unique_secondtrimsuffix)) * pfd_residual_scale_execution_unique_second)) /\\ exists ff_q_pfp_execution_unique_secondtrimsuffixsource. pfd_residual_code_execution_unique_second = ff_q_pfp_execution_unique_secondtrimsuffixsource * S ((S ((pfd_cut_execution_unique_second)+pftrim_index_execution_unique_secondtrimsuffix)) * pfd_residual_scale_execution_unique_second) + (pftrim_value_execution_unique_secondtrimsuffix))) -> (((exists ff_h_pfp_execution_unique_secondtrimsuffixoutput. ff_h_pfp_execution_unique_secondtrimsuffixoutput + S (pftrim_value_execution_unique_secondtrimsuffix) = S ((S (pftrim_index_execution_unique_secondtrimsuffix)) * RC)) /\\ exists ff_q_pfp_execution_unique_secondtrimsuffixoutput. RB = ff_q_pfp_execution_unique_secondtrimsuffixoutput * S ((S (pftrim_index_execution_unique_secondtrimsuffix)) * RC) + (pftrim_value_execution_unique_secondtrimsuffix)))) /\\ (((K)=0 \\/ (exists pftrim_leading_execution_unique_secondtrimnormal. ((((exists ff_h_pfp_execution_unique_secondtrimnormalentry. ff_h_pfp_execution_unique_secondtrimnormalentry + S (pftrim_leading_execution_unique_secondtrimnormal) = S ((S (0)) * RC)) /\\ exists ff_q_pfp_execution_unique_secondtrimnormalentry. RB = ff_q_pfp_execution_unique_secondtrimnormalentry * S ((S (0)) * RC) + (pftrim_leading_execution_unique_secondtrimnormal))) /\\ ((~(pftrim_leading_execution_unique_secondtrimnormal=0))))))))))))))))))))))))))))))))) -> ((((q)=(Q)) /\\ (((forall mdr_i_pfp_execution_unique_outputsquotient mdr_a_pfp_execution_unique_outputsquotient. (exists mdr_gap_pfp_execution_unique_outputsquotientb. mdr_gap_pfp_execution_unique_outputsquotientb + S (mdr_i_pfp_execution_unique_outputsquotient) = (q)) -> (((exists ff_h_mdr_pfp_execution_unique_outputsquotiento. ff_h_mdr_pfp_execution_unique_outputsquotiento + S (mdr_a_pfp_execution_unique_outputsquotient) = S ((S (mdr_i_pfp_execution_unique_outputsquotient)) * qc)) /\\ exists ff_q_mdr_pfp_execution_unique_outputsquotiento. qb = ff_q_mdr_pfp_execution_unique_outputsquotiento * S ((S (mdr_i_pfp_execution_unique_outputsquotient)) * qc) + (mdr_a_pfp_execution_unique_outputsquotient))) -> (((exists ff_h_mdr_pfp_execution_unique_outputsquotientn. ff_h_mdr_pfp_execution_unique_outputsquotientn + S (mdr_a_pfp_execution_unique_outputsquotient) = S ((S (mdr_i_pfp_execution_unique_outputsquotient)) * QC)) /\\ exists ff_q_mdr_pfp_execution_unique_outputsquotientn. QB = ff_q_mdr_pfp_execution_unique_outputsquotientn * S ((S (mdr_i_pfp_execution_unique_outputsquotient)) * QC) + (mdr_a_pfp_execution_unique_outputsquotient)))) /\\ ((((R)=(K)) /\\ ((forall mdr_i_pfp_execution_unique_outputsremainder mdr_a_pfp_execution_unique_outputsremainder. (exists mdr_gap_pfp_execution_unique_outputsremainderb. mdr_gap_pfp_execution_unique_outputsremainderb + S (mdr_i_pfp_execution_unique_outputsremainder) = (R)) -> (((exists ff_h_mdr_pfp_execution_unique_outputsremaindero. ff_h_mdr_pfp_execution_unique_outputsremaindero + S (mdr_a_pfp_execution_unique_outputsremainder) = S ((S (mdr_i_pfp_execution_unique_outputsremainder)) * rc)) /\\ exists ff_q_mdr_pfp_execution_unique_outputsremaindero. rb = ff_q_mdr_pfp_execution_unique_outputsremaindero * S ((S (mdr_i_pfp_execution_unique_outputsremainder)) * rc) + (mdr_a_pfp_execution_unique_outputsremainder))) -> (((exists ff_h_mdr_pfp_execution_unique_outputsremaindern. ff_h_mdr_pfp_execution_unique_outputsremaindern + S (mdr_a_pfp_execution_unique_outputsremainder) = S ((S (mdr_i_pfp_execution_unique_outputsremainder)) * RC)) /\\ exists ff_q_mdr_pfp_execution_unique_outputsremaindern. RB = ff_q_mdr_pfp_execution_unique_outputsremaindern * S ((S (mdr_i_pfp_execution_unique_outputsremainder)) * RC) + (mdr_a_pfp_execution_unique_outputsremainder)))))))))))", "statement_sha256": "b14ad2149cd34386887dcac50cb06b7df7014500b1ab918fac7967976b6042fe", "summary": "Two actual executions on the same annotated inputs agree in quotient and remainder lengths and decoded coefficients, not in beta codes.", "summary_sha256": "07bfebf137b231554f82da2fdeabcbf2da7151443c10b8ee6785f92ffda1213b" }, { "alpha_v33_frontier_enrollment": { "body_receipt_sha256": "aadc684c63faf6bf8df384d265e6ed1de45a71eeb7d83fa9bd19e8c6afbfb8e0", "bundle_campaign": "polynomial-euclidean-division", "bundle_node_id": 344, "bundle_sha256": "6ae667d8518e4dbe722bb08ad1b08715a0d282c2893e533c8133d770fe861dcf", "campaign": "polynomial-euclidean-division", "first_enrolled_version": "v33", "parent_catalog_sha256": "41b9f387d88a5a4f0fe5ee2bd5578f37a27a4657b0a80f1a1a2cb5109f69a623", "rfc_sha256": "38d609ffe5dbf7c36cd0e5112dcfb219934bac3c8af09da5d96f30844ae44895", "source_sha256": "6a9d9ebe1f72202743e5df2c069b9aa367fdb3d61108f1d9354cdc9276ab2d15", "test_sha256": "e3d9c9f2439383a50dd1d9b5f9d1569c8e7e3bbef2a5a2ed64f3bb5f2384ef46" }, "body_checked": true, "body_receipt": { "command_count": 65, "dependency_count": 2, "dne_command_count": 0, "name": "prime_field_polynomial_division_execution_exists_unique", "proof_depth": 54, "proof_edges": 75, "proof_nodes": 76, "proof_objects": 76, "reused_objects": 0, "status": "kernel_checked_dependency_curried_body" }, "checked_use": true, "dependencies": [ "prime_field_polynomial_division_execution_exists", "prime_field_polynomial_division_execution_functional" ], "dependencies_sha256": "333cd6e65c4a150a590ee7b1dfdbf3a7c4cd7d13a6139cb070e472c1d369613a", "empty_context_closure": { "body_proof_depth": 54, "body_proof_nodes": 76, "bundle_campaign": "polynomial-euclidean-division", "bundle_dependency_edge_count": 1071, "bundle_node_count": 377, "bundle_node_id": 344, "bundle_path": "research/arithmetic-library/artifacts/prime-field-polynomial-euclidean-division-proof-bundle-v1.json", "bundle_root_id": 376, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "6ae667d8518e4dbe722bb08ad1b08715a0d282c2893e533c8133d770fe861dcf", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": "0ac4c1f5ca519e7db039365ff2a703f8772e22e58376d4c55a3f7777e08565fc", "status": "checked" }, "enrollment_index": 4060, "enrollment_origin": "ha", "evidence_links": [ { "document_sha256": "6a9d9ebe1f72202743e5df2c069b9aa367fdb3d61108f1d9354cdc9276ab2d15", "kind": "alpha_v33_frontier_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_division_uniqueness_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "e3d9c9f2439383a50dd1d9b5f9d1569c8e7e3bbef2a5a2ed64f3bb5f2384ef46", "kind": "alpha_v33_frontier_executable_audit", "path": "peano-lab/py/tests/test_campaign_research_v33_closure.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "38d609ffe5dbf7c36cd0e5112dcfb219934bac3c8af09da5d96f30844ae44895", "kind": "alpha_v33_frontier_campaign_rfc", "path": "research/arithmetic-library/prime-field-polynomial-euclidean-division-rfc-v1.md", "role": "reviewed_constructive_campaign_contract", "selector": "document" }, { "document_sha256": "6ae667d8518e4dbe722bb08ad1b08715a0d282c2893e533c8133d770fe861dcf", "kind": "alpha_v33_complete_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/prime-field-polynomial-euclidean-division-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=344]" }, { "document_sha256": "cea85e5c595a021061fb50997df7ad489c8905ab563695209dadc8235f6762cb", "kind": "alpha_v33_live_original_kernel_and_lean_receipt", "path": "research/arithmetic-library/artifacts/alpha-v33-research-receipt-v1.json", "role": "fresh_release_proof_verification", "selector": "families[slug=polynomial-euclidean-division]" }, { "document_sha256": "41b9f387d88a5a4f0fe5ee2bd5578f37a27a4657b0a80f1a1a2cb5109f69a623", "kind": "sealed_alpha_v32_parent", "path": "artifacts/peano-library/alpha/catalog-v32.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "polynomial-euclidean-division", "logical_spec_sha256": "1eca45e2114a2deac97b7617b95d7254f1c2a4f1d8224ae269ac5b230bd99b3f", "membership": "alpha_only", "name": "prime_field_polynomial_division_execution_exists_unique", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro ab", "intro ac", "intro L", "intro bb", "intro bc", "intro d", "intro hprime", "intro hinput", "intro hdivisor", "have hexecution : exists qb qc q rb rc R. (((forall fom_index_pfp_execution_exists_unique_choseninput. (exists fom_gap_pfp_execution_exists_unique_choseninput_index_bound. fom_gap_pfp_execution_exists_unique_choseninput_index_bound + S (fom_index_pfp_execution_exists_unique_choseninput) = L) -> exists fom_value_pfp_execution_exists_unique_choseninput. ((((exists fom_beta_height_pfp_execution_exists_unique_choseninput_entry. fom_beta_height_pfp_execution_exists_unique_choseninput_entry + S (fom_value_pfp_execution_exists_unique_choseninput) = S ((S (fom_index_pfp_execution_exists_unique_choseninput)) * ac)) /\\ exists fom_beta_quotient_pfp_execution_exists_unique_choseninput_entry. ab = fom_beta_quotient_pfp_execution_exists_unique_choseninput_entry * S ((S (fom_index_pfp_execution_exists_unique_choseninput)) * ac) + (fom_value_pfp_execution_exists_unique_choseninput))) /\\ (exists fom_gap_pfp_execution_exists_unique_choseninput_value_bound. fom_gap_pfp_execution_exists_unique_choseninput_value_bound + S (fom_value_pfp_execution_exists_unique_choseninput) = p))) /\\ (((forall fom_index_pfp_execution_exists_unique_chosendivisor. (exists fom_gap_pfp_execution_exists_unique_chosendivisor_index_bound. fom_gap_pfp_execution_exists_unique_chosendivisor_index_bound + S (fom_index_pfp_execution_exists_unique_chosendivisor) = S (d)) -> exists fom_value_pfp_execution_exists_unique_chosendivisor. ((((exists fom_beta_height_pfp_execution_exists_unique_chosendivisor_entry. fom_beta_height_pfp_execution_exists_unique_chosendivisor_entry + S (fom_value_pfp_execution_exists_unique_chosendivisor) = S ((S (fom_index_pfp_execution_exists_unique_chosendivisor)) * bc)) /\\ exists fom_beta_quotient_pfp_execution_exists_unique_chosendivisor_entry. bb = fom_beta_quotient_pfp_execution_exists_unique_chosendivisor_entry * S ((S (fom_index_pfp_execution_exists_unique_chosendivisor)) * bc) + (fom_value_pfp_execution_exists_unique_chosendivisor))) /\\ (exists fom_gap_pfp_execution_exists_unique_chosendivisor_value_bound. fom_gap_pfp_execution_exists_unique_chosendivisor_value_bound + S (fom_value_pfp_execution_exists_unique_chosendivisor) = p))) /\\ (((((((q)=0) /\\ ((exists pfc_gap_execution_exists_unique_chosenlengthshort. pfc_gap_execution_exists_unique_chosenlengthshort+(L)=(d))))) \\/ (((~((q)=0)) /\\ (((q)+(d)=(L)))))) /\\ ((exists pfd_head_execution_exists_unique_chosen pfd_inverse_execution_exists_unique_chosen pfd_product_code_execution_exists_unique_chosen pfd_product_scale_execution_exists_unique_chosen pfd_residual_code_execution_exists_unique_chosen pfd_residual_scale_execution_exists_unique_chosen pfd_cut_execution_exists_unique_chosen. ((((exists ff_h_pfp_execution_exists_unique_chosenhead. ff_h_pfp_execution_exists_unique_chosenhead + S (pfd_head_execution_exists_unique_chosen) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_execution_exists_unique_chosenhead. bb = ff_q_pfp_execution_exists_unique_chosenhead * S ((S (0)) * bc) + (pfd_head_execution_exists_unique_chosen))) /\\ (((((~((pfd_head_execution_exists_unique_chosen) = 0)) /\\ ((((exists pfa_gap_execution_exists_unique_choseninversemultiplicationleft. pfa_gap_execution_exists_unique_choseninversemultiplicationleft + S (pfd_head_execution_exists_unique_chosen) = (p)) /\\ (((exists pfa_gap_execution_exists_unique_choseninversemultiplicationright. pfa_gap_execution_exists_unique_choseninversemultiplicationright + S (pfd_inverse_execution_exists_unique_chosen) = (p)) /\\ ((((exists pfa_gap_execution_exists_unique_choseninversemultiplicationresultbound. pfa_gap_execution_exists_unique_choseninversemultiplicationresultbound + S (1) = (p)) /\\ ((exists pfa_offset_left_execution_exists_unique_choseninversemultiplicationresultcongruence pfa_offset_right_execution_exists_unique_choseninversemultiplicationresultcongruence. ((pfd_head_execution_exists_unique_chosen) * (pfd_inverse_execution_exists_unique_chosen)) + (p) * pfa_offset_left_execution_exists_unique_choseninversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_execution_exists_unique_choseninversemultiplicationresultcongruence)))))))))))) /\\ (((forall pfd_index_execution_exists_unique_chosenquotient. (exists pfa_gap_execution_exists_unique_chosenquotientbound. pfa_gap_execution_exists_unique_chosenquotientbound + S (pfd_index_execution_exists_unique_chosenquotient) = (q)) -> exists pfd_value_execution_exists_unique_chosenquotient. ((((exists ff_h_pfp_execution_exists_unique_chosenquotiententry. ff_h_pfp_execution_exists_unique_chosenquotiententry + S (pfd_value_execution_exists_unique_chosenquotient) = S ((S (pfd_index_execution_exists_unique_chosenquotient)) * qc)) /\\ exists ff_q_pfp_execution_exists_unique_chosenquotiententry. qb = ff_q_pfp_execution_exists_unique_chosenquotiententry * S ((S (pfd_index_execution_exists_unique_chosenquotient)) * qc) + (pfd_value_execution_exists_unique_chosenquotient))) /\\ ((exists pfd_input_execution_exists_unique_chosenquotientstep pfd_previous_execution_exists_unique_chosenquotientstep pfd_difference_execution_exists_unique_chosenquotientstep. ((((exists ff_h_pfp_execution_exists_unique_chosenquotientstepinput. ff_h_pfp_execution_exists_unique_chosenquotientstepinput + S (pfd_input_execution_exists_unique_chosenquotientstep) = S ((S (pfd_index_execution_exists_unique_chosenquotient)) * ac)) /\\ exists ff_q_pfp_execution_exists_unique_chosenquotientstepinput. ab = ff_q_pfp_execution_exists_unique_chosenquotientstepinput * S ((S (pfd_index_execution_exists_unique_chosenquotient)) * ac) + (pfd_input_execution_exists_unique_chosenquotientstep))) /\\ (((exists pfc_terms_code_execution_exists_unique_chosenquotientstepprevious pfc_terms_scale_execution_exists_unique_chosenquotientstepprevious pfc_natural_sum_execution_exists_unique_chosenquotientstepprevious. ((forall pfc_index_execution_exists_unique_chosenquotientsteppreviousdiagonal. (exists pfa_gap_execution_exists_unique_chosenquotientsteppreviousdiagonalbound. pfa_gap_execution_exists_unique_chosenquotientsteppreviousdiagonalbound + S (pfc_index_execution_exists_unique_chosenquotientsteppreviousdiagonal) = (S (pfd_index_execution_exists_unique_chosenquotient))) -> exists pfc_value_execution_exists_unique_chosenquotientsteppreviousdiagonal. ((((exists ff_h_pfp_execution_exists_unique_chosenquotientsteppreviousdiagonalentry. ff_h_pfp_execution_exists_unique_chosenquotientsteppreviousdiagonalentry + S (pfc_value_execution_exists_unique_chosenquotientsteppreviousdiagonal) = S ((S (pfc_index_execution_exists_unique_chosenquotientsteppreviousdiagonal)) * pfc_terms_scale_execution_exists_unique_chosenquotientstepprevious)) /\\ exists ff_q_pfp_execution_exists_unique_chosenquotientsteppreviousdiagonalentry. pfc_terms_code_execution_exists_unique_chosenquotientstepprevious = ff_q_pfp_execution_exists_unique_chosenquotientsteppreviousdiagonalentry * S ((S (pfc_index_execution_exists_unique_chosenquotientsteppreviousdiagonal)) * pfc_terms_scale_execution_exists_unique_chosenquotientstepprevious) + (pfc_value_execution_exists_unique_chosenquotientsteppreviousdiagonal))) /\\ ((exists pfc_complement_execution_exists_unique_chosenquotientsteppreviousdiagonalterm pfc_left_execution_exists_unique_chosenquotientsteppreviousdiagonalterm pfc_right_execution_exists_unique_chosenquotientsteppreviousdiagonalterm. (((pfc_index_execution_exists_unique_chosenquotientsteppreviousdiagonal)+pfc_complement_execution_exists_unique_chosenquotientsteppreviousdiagonalterm=(pfd_index_execution_exists_unique_chosenquotient)) /\\ ((((((exists pfa_gap_execution_exists_unique_chosenquotientsteppreviousdiagonaltermleftinside. pfa_gap_execution_exists_unique_chosenquotientsteppreviousdiagonaltermleftinside + S (pfc_index_execution_exists_unique_chosenquotientsteppreviousdiagonal) = (pfd_index_execution_exists_unique_chosenquotient)) /\\ ((((exists ff_h_pfp_execution_exists_unique_chosenquotientsteppreviousdiagonaltermleftentry. ff_h_pfp_execution_exists_unique_chosenquotientsteppreviousdiagonaltermleftentry + S (pfc_left_execution_exists_unique_chosenquotientsteppreviousdiagonalterm) = S ((S (pfc_index_execution_exists_unique_chosenquotientsteppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_execution_exists_unique_chosenquotientsteppreviousdiagonaltermleftentry. qb = ff_q_pfp_execution_exists_unique_chosenquotientsteppreviousdiagonaltermleftentry * S ((S (pfc_index_execution_exists_unique_chosenquotientsteppreviousdiagonal)) * qc) + (pfc_left_execution_exists_unique_chosenquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_execution_exists_unique_chosenquotientsteppreviousdiagonaltermleftoutside. pfc_gap_execution_exists_unique_chosenquotientsteppreviousdiagonaltermleftoutside+(pfd_index_execution_exists_unique_chosenquotient)=(pfc_index_execution_exists_unique_chosenquotientsteppreviousdiagonal)) /\\ (((pfc_left_execution_exists_unique_chosenquotientsteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_execution_exists_unique_chosenquotientsteppreviousdiagonaltermrightinside. pfa_gap_execution_exists_unique_chosenquotientsteppreviousdiagonaltermrightinside + S (pfc_complement_execution_exists_unique_chosenquotientsteppreviousdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_execution_exists_unique_chosenquotientsteppreviousdiagonaltermrightentry. ff_h_pfp_execution_exists_unique_chosenquotientsteppreviousdiagonaltermrightentry + S (pfc_right_execution_exists_unique_chosenquotientsteppreviousdiagonalterm) = S ((S (pfc_complement_execution_exists_unique_chosenquotientsteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_execution_exists_unique_chosenquotientsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_execution_exists_unique_chosenquotientsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_execution_exists_unique_chosenquotientsteppreviousdiagonalterm)) * bc) + (pfc_right_execution_exists_unique_chosenquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_execution_exists_unique_chosenquotientsteppreviousdiagonaltermrightoutside. pfc_gap_execution_exists_unique_chosenquotientsteppreviousdiagonaltermrightoutside+(S (d))=(pfc_complement_execution_exists_unique_chosenquotientsteppreviousdiagonalterm)) /\\ (((pfc_right_execution_exists_unique_chosenquotientsteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_execution_exists_unique_chosenquotientsteppreviousdiagonal)=pfc_left_execution_exists_unique_chosenquotientsteppreviousdiagonalterm*pfc_right_execution_exists_unique_chosenquotientsteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_execution_exists_unique_chosenquotientstepprevioussum fs_v_pfc_execution_exists_unique_chosenquotientstepprevioussum. ((((exists fs_h_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_start. fs_h_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_exists_unique_chosenquotientstepprevioussum)) /\\ exists fs_q_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_start. fs_u_pfc_execution_exists_unique_chosenquotientstepprevioussum = fs_q_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_execution_exists_unique_chosenquotientstepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_terminal. fs_h_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_terminal + S (pfc_natural_sum_execution_exists_unique_chosenquotientstepprevious) = S ((S (S (pfd_index_execution_exists_unique_chosenquotient))) * fs_v_pfc_execution_exists_unique_chosenquotientstepprevioussum)) /\\ exists fs_q_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_terminal. fs_u_pfc_execution_exists_unique_chosenquotientstepprevioussum = fs_q_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_terminal * S ((S (S (pfd_index_execution_exists_unique_chosenquotient))) * fs_v_pfc_execution_exists_unique_chosenquotientstepprevioussum) + (pfc_natural_sum_execution_exists_unique_chosenquotientstepprevious))) /\\ forall fs_i_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_steps. (exists fs_lt_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_steps_bound. fs_lt_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_steps_bound + S fs_i_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_steps = S (pfd_index_execution_exists_unique_chosenquotient)) -> exists fs_a_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_steps fs_r_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_steps fs_s_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_steps. ((((exists fs_h_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_steps_summand. fs_h_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_steps_summand + S (fs_a_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_steps)) * pfc_terms_scale_execution_exists_unique_chosenquotientstepprevious)) /\\ exists fs_q_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_steps_summand. pfc_terms_code_execution_exists_unique_chosenquotientstepprevious = fs_q_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_steps)) * pfc_terms_scale_execution_exists_unique_chosenquotientstepprevious) + (fs_a_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_steps_partial. fs_h_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_steps_partial + S (fs_r_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_exists_unique_chosenquotientstepprevioussum)) /\\ exists fs_q_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_steps_partial. fs_u_pfc_execution_exists_unique_chosenquotientstepprevioussum = fs_q_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_exists_unique_chosenquotientstepprevioussum) + (fs_r_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_steps_successor. fs_h_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_steps_successor + S (fs_s_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_steps) = S ((S (S fs_i_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_exists_unique_chosenquotientstepprevioussum)) /\\ exists fs_q_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_steps_successor. fs_u_pfc_execution_exists_unique_chosenquotientstepprevioussum = fs_q_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_exists_unique_chosenquotientstepprevioussum) + (fs_s_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_steps))) /\\ fs_s_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_steps = fs_r_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_steps + fs_a_pfc_execution_exists_unique_chosenquotientstepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_execution_exists_unique_chosenquotientsteppreviousresiduebound. pfa_gap_execution_exists_unique_chosenquotientsteppreviousresiduebound + S (pfd_previous_execution_exists_unique_chosenquotientstep) = (p)) /\\ ((exists pfa_offset_left_execution_exists_unique_chosenquotientsteppreviousresiduecongruence pfa_offset_right_execution_exists_unique_chosenquotientsteppreviousresiduecongruence. (pfc_natural_sum_execution_exists_unique_chosenquotientstepprevious) + (p) * pfa_offset_left_execution_exists_unique_chosenquotientsteppreviousresiduecongruence = (pfd_previous_execution_exists_unique_chosenquotientstep) + (p) * pfa_offset_right_execution_exists_unique_chosenquotientsteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_execution_exists_unique_chosenquotientstepsubtractleft. pfa_gap_execution_exists_unique_chosenquotientstepsubtractleft + S (pfd_previous_execution_exists_unique_chosenquotientstep) = (p)) /\\ (((exists pfa_gap_execution_exists_unique_chosenquotientstepsubtractright. pfa_gap_execution_exists_unique_chosenquotientstepsubtractright + S (pfd_difference_execution_exists_unique_chosenquotientstep) = (p)) /\\ ((((exists pfa_gap_execution_exists_unique_chosenquotientstepsubtractresultbound. pfa_gap_execution_exists_unique_chosenquotientstepsubtractresultbound + S (pfd_input_execution_exists_unique_chosenquotientstep) = (p)) /\\ ((exists pfa_offset_left_execution_exists_unique_chosenquotientstepsubtractresultcongruence pfa_offset_right_execution_exists_unique_chosenquotientstepsubtractresultcongruence. ((pfd_previous_execution_exists_unique_chosenquotientstep) + (pfd_difference_execution_exists_unique_chosenquotientstep)) + (p) * pfa_offset_left_execution_exists_unique_chosenquotientstepsubtractresultcongruence = (pfd_input_execution_exists_unique_chosenquotientstep) + (p) * pfa_offset_right_execution_exists_unique_chosenquotientstepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_execution_exists_unique_chosenquotientstepmultiplyleft. pfa_gap_execution_exists_unique_chosenquotientstepmultiplyleft + S (pfd_inverse_execution_exists_unique_chosen) = (p)) /\\ (((exists pfa_gap_execution_exists_unique_chosenquotientstepmultiplyright. pfa_gap_execution_exists_unique_chosenquotientstepmultiplyright + S (pfd_difference_execution_exists_unique_chosenquotientstep) = (p)) /\\ ((((exists pfa_gap_execution_exists_unique_chosenquotientstepmultiplyresultbound. pfa_gap_execution_exists_unique_chosenquotientstepmultiplyresultbound + S (pfd_value_execution_exists_unique_chosenquotient) = (p)) /\\ ((exists pfa_offset_left_execution_exists_unique_chosenquotientstepmultiplyresultcongruence pfa_offset_right_execution_exists_unique_chosenquotientstepmultiplyresultcongruence. ((pfd_inverse_execution_exists_unique_chosen) * (pfd_difference_execution_exists_unique_chosenquotientstep)) + (p) * pfa_offset_left_execution_exists_unique_chosenquotientstepmultiplyresultcongruence = (pfd_value_execution_exists_unique_chosenquotient) + (p) * pfa_offset_right_execution_exists_unique_chosenquotientstepmultiplyresultcongruence))))))))))))))))))) /\\ (((forall pfc_index_execution_exists_unique_chosenproduct. (exists pfa_gap_execution_exists_unique_chosenproductbound. pfa_gap_execution_exists_unique_chosenproductbound + S (pfc_index_execution_exists_unique_chosenproduct) = (L)) -> exists pfc_value_execution_exists_unique_chosenproduct. ((((exists ff_h_pfp_execution_exists_unique_chosenproductentry. ff_h_pfp_execution_exists_unique_chosenproductentry + S (pfc_value_execution_exists_unique_chosenproduct) = S ((S (pfc_index_execution_exists_unique_chosenproduct)) * pfd_product_scale_execution_exists_unique_chosen)) /\\ exists ff_q_pfp_execution_exists_unique_chosenproductentry. pfd_product_code_execution_exists_unique_chosen = ff_q_pfp_execution_exists_unique_chosenproductentry * S ((S (pfc_index_execution_exists_unique_chosenproduct)) * pfd_product_scale_execution_exists_unique_chosen) + (pfc_value_execution_exists_unique_chosenproduct))) /\\ ((exists pfc_terms_code_execution_exists_unique_chosenproductcoefficient pfc_terms_scale_execution_exists_unique_chosenproductcoefficient pfc_natural_sum_execution_exists_unique_chosenproductcoefficient. ((forall pfc_index_execution_exists_unique_chosenproductcoefficientdiagonal. (exists pfa_gap_execution_exists_unique_chosenproductcoefficientdiagonalbound. pfa_gap_execution_exists_unique_chosenproductcoefficientdiagonalbound + S (pfc_index_execution_exists_unique_chosenproductcoefficientdiagonal) = (S (pfc_index_execution_exists_unique_chosenproduct))) -> exists pfc_value_execution_exists_unique_chosenproductcoefficientdiagonal. ((((exists ff_h_pfp_execution_exists_unique_chosenproductcoefficientdiagonalentry. ff_h_pfp_execution_exists_unique_chosenproductcoefficientdiagonalentry + S (pfc_value_execution_exists_unique_chosenproductcoefficientdiagonal) = S ((S (pfc_index_execution_exists_unique_chosenproductcoefficientdiagonal)) * pfc_terms_scale_execution_exists_unique_chosenproductcoefficient)) /\\ exists ff_q_pfp_execution_exists_unique_chosenproductcoefficientdiagonalentry. pfc_terms_code_execution_exists_unique_chosenproductcoefficient = ff_q_pfp_execution_exists_unique_chosenproductcoefficientdiagonalentry * S ((S (pfc_index_execution_exists_unique_chosenproductcoefficientdiagonal)) * pfc_terms_scale_execution_exists_unique_chosenproductcoefficient) + (pfc_value_execution_exists_unique_chosenproductcoefficientdiagonal))) /\\ ((exists pfc_complement_execution_exists_unique_chosenproductcoefficientdiagonalterm pfc_left_execution_exists_unique_chosenproductcoefficientdiagonalterm pfc_right_execution_exists_unique_chosenproductcoefficientdiagonalterm. (((pfc_index_execution_exists_unique_chosenproductcoefficientdiagonal)+pfc_complement_execution_exists_unique_chosenproductcoefficientdiagonalterm=(pfc_index_execution_exists_unique_chosenproduct)) /\\ ((((((exists pfa_gap_execution_exists_unique_chosenproductcoefficientdiagonaltermleftinside. pfa_gap_execution_exists_unique_chosenproductcoefficientdiagonaltermleftinside + S (pfc_index_execution_exists_unique_chosenproductcoefficientdiagonal) = (q)) /\\ ((((exists ff_h_pfp_execution_exists_unique_chosenproductcoefficientdiagonaltermleftentry. ff_h_pfp_execution_exists_unique_chosenproductcoefficientdiagonaltermleftentry + S (pfc_left_execution_exists_unique_chosenproductcoefficientdiagonalterm) = S ((S (pfc_index_execution_exists_unique_chosenproductcoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_execution_exists_unique_chosenproductcoefficientdiagonaltermleftentry. qb = ff_q_pfp_execution_exists_unique_chosenproductcoefficientdiagonaltermleftentry * S ((S (pfc_index_execution_exists_unique_chosenproductcoefficientdiagonal)) * qc) + (pfc_left_execution_exists_unique_chosenproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_exists_unique_chosenproductcoefficientdiagonaltermleftoutside. pfc_gap_execution_exists_unique_chosenproductcoefficientdiagonaltermleftoutside+(q)=(pfc_index_execution_exists_unique_chosenproductcoefficientdiagonal)) /\\ (((pfc_left_execution_exists_unique_chosenproductcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_execution_exists_unique_chosenproductcoefficientdiagonaltermrightinside. pfa_gap_execution_exists_unique_chosenproductcoefficientdiagonaltermrightinside + S (pfc_complement_execution_exists_unique_chosenproductcoefficientdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_execution_exists_unique_chosenproductcoefficientdiagonaltermrightentry. ff_h_pfp_execution_exists_unique_chosenproductcoefficientdiagonaltermrightentry + S (pfc_right_execution_exists_unique_chosenproductcoefficientdiagonalterm) = S ((S (pfc_complement_execution_exists_unique_chosenproductcoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_execution_exists_unique_chosenproductcoefficientdiagonaltermrightentry. bb = ff_q_pfp_execution_exists_unique_chosenproductcoefficientdiagonaltermrightentry * S ((S (pfc_complement_execution_exists_unique_chosenproductcoefficientdiagonalterm)) * bc) + (pfc_right_execution_exists_unique_chosenproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_exists_unique_chosenproductcoefficientdiagonaltermrightoutside. pfc_gap_execution_exists_unique_chosenproductcoefficientdiagonaltermrightoutside+(S (d))=(pfc_complement_execution_exists_unique_chosenproductcoefficientdiagonalterm)) /\\ (((pfc_right_execution_exists_unique_chosenproductcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_execution_exists_unique_chosenproductcoefficientdiagonal)=pfc_left_execution_exists_unique_chosenproductcoefficientdiagonalterm*pfc_right_execution_exists_unique_chosenproductcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_execution_exists_unique_chosenproductcoefficientsum fs_v_pfc_execution_exists_unique_chosenproductcoefficientsum. ((((exists fs_h_pfc_execution_exists_unique_chosenproductcoefficientsum_body_start. fs_h_pfc_execution_exists_unique_chosenproductcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_exists_unique_chosenproductcoefficientsum)) /\\ exists fs_q_pfc_execution_exists_unique_chosenproductcoefficientsum_body_start. fs_u_pfc_execution_exists_unique_chosenproductcoefficientsum = fs_q_pfc_execution_exists_unique_chosenproductcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_execution_exists_unique_chosenproductcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_execution_exists_unique_chosenproductcoefficientsum_body_terminal. fs_h_pfc_execution_exists_unique_chosenproductcoefficientsum_body_terminal + S (pfc_natural_sum_execution_exists_unique_chosenproductcoefficient) = S ((S (S (pfc_index_execution_exists_unique_chosenproduct))) * fs_v_pfc_execution_exists_unique_chosenproductcoefficientsum)) /\\ exists fs_q_pfc_execution_exists_unique_chosenproductcoefficientsum_body_terminal. fs_u_pfc_execution_exists_unique_chosenproductcoefficientsum = fs_q_pfc_execution_exists_unique_chosenproductcoefficientsum_body_terminal * S ((S (S (pfc_index_execution_exists_unique_chosenproduct))) * fs_v_pfc_execution_exists_unique_chosenproductcoefficientsum) + (pfc_natural_sum_execution_exists_unique_chosenproductcoefficient))) /\\ forall fs_i_pfc_execution_exists_unique_chosenproductcoefficientsum_body_steps. (exists fs_lt_pfc_execution_exists_unique_chosenproductcoefficientsum_body_steps_bound. fs_lt_pfc_execution_exists_unique_chosenproductcoefficientsum_body_steps_bound + S fs_i_pfc_execution_exists_unique_chosenproductcoefficientsum_body_steps = S (pfc_index_execution_exists_unique_chosenproduct)) -> exists fs_a_pfc_execution_exists_unique_chosenproductcoefficientsum_body_steps fs_r_pfc_execution_exists_unique_chosenproductcoefficientsum_body_steps fs_s_pfc_execution_exists_unique_chosenproductcoefficientsum_body_steps. ((((exists fs_h_pfc_execution_exists_unique_chosenproductcoefficientsum_body_steps_summand. fs_h_pfc_execution_exists_unique_chosenproductcoefficientsum_body_steps_summand + S (fs_a_pfc_execution_exists_unique_chosenproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_exists_unique_chosenproductcoefficientsum_body_steps)) * pfc_terms_scale_execution_exists_unique_chosenproductcoefficient)) /\\ exists fs_q_pfc_execution_exists_unique_chosenproductcoefficientsum_body_steps_summand. pfc_terms_code_execution_exists_unique_chosenproductcoefficient = fs_q_pfc_execution_exists_unique_chosenproductcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_execution_exists_unique_chosenproductcoefficientsum_body_steps)) * pfc_terms_scale_execution_exists_unique_chosenproductcoefficient) + (fs_a_pfc_execution_exists_unique_chosenproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_exists_unique_chosenproductcoefficientsum_body_steps_partial. fs_h_pfc_execution_exists_unique_chosenproductcoefficientsum_body_steps_partial + S (fs_r_pfc_execution_exists_unique_chosenproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_exists_unique_chosenproductcoefficientsum_body_steps)) * fs_v_pfc_execution_exists_unique_chosenproductcoefficientsum)) /\\ exists fs_q_pfc_execution_exists_unique_chosenproductcoefficientsum_body_steps_partial. fs_u_pfc_execution_exists_unique_chosenproductcoefficientsum = fs_q_pfc_execution_exists_unique_chosenproductcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_execution_exists_unique_chosenproductcoefficientsum_body_steps)) * fs_v_pfc_execution_exists_unique_chosenproductcoefficientsum) + (fs_r_pfc_execution_exists_unique_chosenproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_exists_unique_chosenproductcoefficientsum_body_steps_successor. fs_h_pfc_execution_exists_unique_chosenproductcoefficientsum_body_steps_successor + S (fs_s_pfc_execution_exists_unique_chosenproductcoefficientsum_body_steps) = S ((S (S fs_i_pfc_execution_exists_unique_chosenproductcoefficientsum_body_steps)) * fs_v_pfc_execution_exists_unique_chosenproductcoefficientsum)) /\\ exists fs_q_pfc_execution_exists_unique_chosenproductcoefficientsum_body_steps_successor. fs_u_pfc_execution_exists_unique_chosenproductcoefficientsum = fs_q_pfc_execution_exists_unique_chosenproductcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_execution_exists_unique_chosenproductcoefficientsum_body_steps)) * fs_v_pfc_execution_exists_unique_chosenproductcoefficientsum) + (fs_s_pfc_execution_exists_unique_chosenproductcoefficientsum_body_steps))) /\\ fs_s_pfc_execution_exists_unique_chosenproductcoefficientsum_body_steps = fs_r_pfc_execution_exists_unique_chosenproductcoefficientsum_body_steps + fs_a_pfc_execution_exists_unique_chosenproductcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_execution_exists_unique_chosenproductcoefficientresiduebound. pfa_gap_execution_exists_unique_chosenproductcoefficientresiduebound + S (pfc_value_execution_exists_unique_chosenproduct) = (p)) /\\ ((exists pfa_offset_left_execution_exists_unique_chosenproductcoefficientresiduecongruence pfa_offset_right_execution_exists_unique_chosenproductcoefficientresiduecongruence. (pfc_natural_sum_execution_exists_unique_chosenproductcoefficient) + (p) * pfa_offset_left_execution_exists_unique_chosenproductcoefficientresiduecongruence = (pfc_value_execution_exists_unique_chosenproduct) + (p) * pfa_offset_right_execution_exists_unique_chosenproductcoefficientresiduecongruence)))))))))))) /\\ (((forall pfs_index_execution_exists_unique_chosendifference. (exists pfa_gap_execution_exists_unique_chosendifferenceindex. pfa_gap_execution_exists_unique_chosendifferenceindex + S (pfs_index_execution_exists_unique_chosendifference) = (L)) -> exists pfs_left_execution_exists_unique_chosendifference pfs_right_execution_exists_unique_chosendifference pfs_result_execution_exists_unique_chosendifference. ((((exists ff_h_pfp_execution_exists_unique_chosendifferenceleft. ff_h_pfp_execution_exists_unique_chosendifferenceleft + S (pfs_left_execution_exists_unique_chosendifference) = S ((S (pfs_index_execution_exists_unique_chosendifference)) * ac)) /\\ exists ff_q_pfp_execution_exists_unique_chosendifferenceleft. ab = ff_q_pfp_execution_exists_unique_chosendifferenceleft * S ((S (pfs_index_execution_exists_unique_chosendifference)) * ac) + (pfs_left_execution_exists_unique_chosendifference))) /\\ (((((exists ff_h_pfp_execution_exists_unique_chosendifferenceright. ff_h_pfp_execution_exists_unique_chosendifferenceright + S (pfs_right_execution_exists_unique_chosendifference) = S ((S (pfs_index_execution_exists_unique_chosendifference)) * pfd_product_scale_execution_exists_unique_chosen)) /\\ exists ff_q_pfp_execution_exists_unique_chosendifferenceright. pfd_product_code_execution_exists_unique_chosen = ff_q_pfp_execution_exists_unique_chosendifferenceright * S ((S (pfs_index_execution_exists_unique_chosendifference)) * pfd_product_scale_execution_exists_unique_chosen) + (pfs_right_execution_exists_unique_chosendifference))) /\\ (((((exists ff_h_pfp_execution_exists_unique_chosendifferenceresult. ff_h_pfp_execution_exists_unique_chosendifferenceresult + S (pfs_result_execution_exists_unique_chosendifference) = S ((S (pfs_index_execution_exists_unique_chosendifference)) * pfd_residual_scale_execution_exists_unique_chosen)) /\\ exists ff_q_pfp_execution_exists_unique_chosendifferenceresult. pfd_residual_code_execution_exists_unique_chosen = ff_q_pfp_execution_exists_unique_chosendifferenceresult * S ((S (pfs_index_execution_exists_unique_chosendifference)) * pfd_residual_scale_execution_exists_unique_chosen) + (pfs_result_execution_exists_unique_chosendifference))) /\\ ((((exists pfa_gap_execution_exists_unique_chosendifferenceoperationleft. pfa_gap_execution_exists_unique_chosendifferenceoperationleft + S (pfs_right_execution_exists_unique_chosendifference) = (p)) /\\ (((exists pfa_gap_execution_exists_unique_chosendifferenceoperationright. pfa_gap_execution_exists_unique_chosendifferenceoperationright + S (pfs_result_execution_exists_unique_chosendifference) = (p)) /\\ ((((exists pfa_gap_execution_exists_unique_chosendifferenceoperationresultbound. pfa_gap_execution_exists_unique_chosendifferenceoperationresultbound + S (pfs_left_execution_exists_unique_chosendifference) = (p)) /\\ ((exists pfa_offset_left_execution_exists_unique_chosendifferenceoperationresultcongruence pfa_offset_right_execution_exists_unique_chosendifferenceoperationresultcongruence. ((pfs_right_execution_exists_unique_chosendifference) + (pfs_result_execution_exists_unique_chosendifference)) + (p) * pfa_offset_left_execution_exists_unique_chosendifferenceoperationresultcongruence = (pfs_left_execution_exists_unique_chosendifference) + (p) * pfa_offset_right_execution_exists_unique_chosendifferenceoperationresultcongruence)))))))))))))))) /\\ (((((L)=(pfd_cut_execution_exists_unique_chosen)+(R)) /\\ (((forall fom_index_pfp_execution_exists_unique_chosentriminput. (exists fom_gap_pfp_execution_exists_unique_chosentriminput_index_bound. fom_gap_pfp_execution_exists_unique_chosentriminput_index_bound + S (fom_index_pfp_execution_exists_unique_chosentriminput) = L) -> exists fom_value_pfp_execution_exists_unique_chosentriminput. ((((exists fom_beta_height_pfp_execution_exists_unique_chosentriminput_entry. fom_beta_height_pfp_execution_exists_unique_chosentriminput_entry + S (fom_value_pfp_execution_exists_unique_chosentriminput) = S ((S (fom_index_pfp_execution_exists_unique_chosentriminput)) * pfd_residual_scale_execution_exists_unique_chosen)) /\\ exists fom_beta_quotient_pfp_execution_exists_unique_chosentriminput_entry. pfd_residual_code_execution_exists_unique_chosen = fom_beta_quotient_pfp_execution_exists_unique_chosentriminput_entry * S ((S (fom_index_pfp_execution_exists_unique_chosentriminput)) * pfd_residual_scale_execution_exists_unique_chosen) + (fom_value_pfp_execution_exists_unique_chosentriminput))) /\\ (exists fom_gap_pfp_execution_exists_unique_chosentriminput_value_bound. fom_gap_pfp_execution_exists_unique_chosentriminput_value_bound + S (fom_value_pfp_execution_exists_unique_chosentriminput) = p))) /\\ (((forall pfp_repeat_index_execution_exists_unique_chosentrimremoved. (exists pfa_gap_execution_exists_unique_chosentrimremovedindex. pfa_gap_execution_exists_unique_chosentrimremovedindex + S (pfp_repeat_index_execution_exists_unique_chosentrimremoved) = (pfd_cut_execution_exists_unique_chosen)) -> (((exists ff_h_pfp_execution_exists_unique_chosentrimremovedentry. ff_h_pfp_execution_exists_unique_chosentrimremovedentry + S (0) = S ((S (pfp_repeat_index_execution_exists_unique_chosentrimremoved)) * pfd_residual_scale_execution_exists_unique_chosen)) /\\ exists ff_q_pfp_execution_exists_unique_chosentrimremovedentry. pfd_residual_code_execution_exists_unique_chosen = ff_q_pfp_execution_exists_unique_chosentrimremovedentry * S ((S (pfp_repeat_index_execution_exists_unique_chosentrimremoved)) * pfd_residual_scale_execution_exists_unique_chosen) + (0)))) /\\ (((forall pftrim_index_execution_exists_unique_chosentrimsuffix pftrim_value_execution_exists_unique_chosentrimsuffix. (exists pfa_gap_execution_exists_unique_chosentrimsuffixbound. pfa_gap_execution_exists_unique_chosentrimsuffixbound + S (pftrim_index_execution_exists_unique_chosentrimsuffix) = (R)) -> (((exists ff_h_pfp_execution_exists_unique_chosentrimsuffixsource. ff_h_pfp_execution_exists_unique_chosentrimsuffixsource + S (pftrim_value_execution_exists_unique_chosentrimsuffix) = S ((S ((pfd_cut_execution_exists_unique_chosen)+pftrim_index_execution_exists_unique_chosentrimsuffix)) * pfd_residual_scale_execution_exists_unique_chosen)) /\\ exists ff_q_pfp_execution_exists_unique_chosentrimsuffixsource. pfd_residual_code_execution_exists_unique_chosen = ff_q_pfp_execution_exists_unique_chosentrimsuffixsource * S ((S ((pfd_cut_execution_exists_unique_chosen)+pftrim_index_execution_exists_unique_chosentrimsuffix)) * pfd_residual_scale_execution_exists_unique_chosen) + (pftrim_value_execution_exists_unique_chosentrimsuffix))) -> (((exists ff_h_pfp_execution_exists_unique_chosentrimsuffixoutput. ff_h_pfp_execution_exists_unique_chosentrimsuffixoutput + S (pftrim_value_execution_exists_unique_chosentrimsuffix) = S ((S (pftrim_index_execution_exists_unique_chosentrimsuffix)) * rc)) /\\ exists ff_q_pfp_execution_exists_unique_chosentrimsuffixoutput. rb = ff_q_pfp_execution_exists_unique_chosentrimsuffixoutput * S ((S (pftrim_index_execution_exists_unique_chosentrimsuffix)) * rc) + (pftrim_value_execution_exists_unique_chosentrimsuffix)))) /\\ (((R)=0 \\/ (exists pftrim_leading_execution_exists_unique_chosentrimnormal. ((((exists ff_h_pfp_execution_exists_unique_chosentrimnormalentry. ff_h_pfp_execution_exists_unique_chosentrimnormalentry + S (pftrim_leading_execution_exists_unique_chosentrimnormal) = S ((S (0)) * rc)) /\\ exists ff_q_pfp_execution_exists_unique_chosentrimnormalentry. rb = ff_q_pfp_execution_exists_unique_chosentrimnormalentry * S ((S (0)) * rc) + (pftrim_leading_execution_exists_unique_chosentrimnormal))) /\\ ((~(pftrim_leading_execution_exists_unique_chosentrimnormal=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 (bb)", "specialize prime_field_polynomial_division_execution_exists (bc)", "specialize prime_field_polynomial_division_execution_exists (d)", "apply prime_field_polynomial_division_execution_exists", "exact hprime", "exact hinput", "exact hdivisor", "cases hexecution", "cases hexecution_witness", "cases hexecution_witness_witness", "cases hexecution_witness_witness_witness", "cases hexecution_witness_witness_witness_witness", "cases hexecution_witness_witness_witness_witness_witness", "exists x", "exists x1", "exists x2", "exists x3", "exists x4", "exists x5", "split", "exact hexecution_witness_witness_witness_witness_witness_witness", "intro QB", "intro QC", "intro Q", "intro RB", "intro RC", "intro K", "intro hother", "specialize prime_field_polynomial_division_execution_functional (p)", "specialize prime_field_polynomial_division_execution_functional (ab)", "specialize prime_field_polynomial_division_execution_functional (ac)", "specialize prime_field_polynomial_division_execution_functional (L)", "specialize prime_field_polynomial_division_execution_functional (bb)", "specialize prime_field_polynomial_division_execution_functional (bc)", "specialize prime_field_polynomial_division_execution_functional (d)", "specialize prime_field_polynomial_division_execution_functional (x)", "specialize prime_field_polynomial_division_execution_functional (x1)", "specialize prime_field_polynomial_division_execution_functional (x2)", "specialize prime_field_polynomial_division_execution_functional (x3)", "specialize prime_field_polynomial_division_execution_functional (x4)", "specialize prime_field_polynomial_division_execution_functional (x5)", "specialize prime_field_polynomial_division_execution_functional (QB)", "specialize prime_field_polynomial_division_execution_functional (QC)", "specialize prime_field_polynomial_division_execution_functional (Q)", "specialize prime_field_polynomial_division_execution_functional (RB)", "specialize prime_field_polynomial_division_execution_functional (RC)", "specialize prime_field_polynomial_division_execution_functional (K)", "apply prime_field_polynomial_division_execution_functional", "exact hexecution_witness_witness_witness_witness_witness_witness", "exact hother" ], "script_sha256": "e3aa241f366df1802117815fa23d89ba22effb74a5e8af1d8a722b5c86b8b09c", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_division_uniqueness_candidate.py", "sha256": "6a9d9ebe1f72202743e5df2c069b9aa367fdb3d61108f1d9354cdc9276ab2d15" }, "statement": "forall p ab ac L bb bc d. (~((p) = 1) /\\ forall pfa_factor_left_execution_exists_unique_prime pfa_factor_right_execution_exists_unique_prime. (p) = pfa_factor_left_execution_exists_unique_prime * pfa_factor_right_execution_exists_unique_prime -> pfa_factor_left_execution_exists_unique_prime = 1 \\/ pfa_factor_right_execution_exists_unique_prime = 1) -> (forall fom_index_pfp_execution_exists_unique_input. (exists fom_gap_pfp_execution_exists_unique_input_index_bound. fom_gap_pfp_execution_exists_unique_input_index_bound + S (fom_index_pfp_execution_exists_unique_input) = L) -> exists fom_value_pfp_execution_exists_unique_input. ((((exists fom_beta_height_pfp_execution_exists_unique_input_entry. fom_beta_height_pfp_execution_exists_unique_input_entry + S (fom_value_pfp_execution_exists_unique_input) = S ((S (fom_index_pfp_execution_exists_unique_input)) * ac)) /\\ exists fom_beta_quotient_pfp_execution_exists_unique_input_entry. ab = fom_beta_quotient_pfp_execution_exists_unique_input_entry * S ((S (fom_index_pfp_execution_exists_unique_input)) * ac) + (fom_value_pfp_execution_exists_unique_input))) /\\ (exists fom_gap_pfp_execution_exists_unique_input_value_bound. fom_gap_pfp_execution_exists_unique_input_value_bound + S (fom_value_pfp_execution_exists_unique_input) = p))) -> ((((S d)=S (d)) /\\ (((forall fom_index_pfp_execution_exists_unique_divisorcoefficients. (exists fom_gap_pfp_execution_exists_unique_divisorcoefficients_index_bound. fom_gap_pfp_execution_exists_unique_divisorcoefficients_index_bound + S (fom_index_pfp_execution_exists_unique_divisorcoefficients) = S d) -> exists fom_value_pfp_execution_exists_unique_divisorcoefficients. ((((exists fom_beta_height_pfp_execution_exists_unique_divisorcoefficients_entry. fom_beta_height_pfp_execution_exists_unique_divisorcoefficients_entry + S (fom_value_pfp_execution_exists_unique_divisorcoefficients) = S ((S (fom_index_pfp_execution_exists_unique_divisorcoefficients)) * bc)) /\\ exists fom_beta_quotient_pfp_execution_exists_unique_divisorcoefficients_entry. bb = fom_beta_quotient_pfp_execution_exists_unique_divisorcoefficients_entry * S ((S (fom_index_pfp_execution_exists_unique_divisorcoefficients)) * bc) + (fom_value_pfp_execution_exists_unique_divisorcoefficients))) /\\ (exists fom_gap_pfp_execution_exists_unique_divisorcoefficients_value_bound. fom_gap_pfp_execution_exists_unique_divisorcoefficients_value_bound + S (fom_value_pfp_execution_exists_unique_divisorcoefficients) = p))) /\\ ((exists pfd_leading_execution_exists_unique_divisor. ((((exists ff_h_pfp_execution_exists_unique_divisorentry. ff_h_pfp_execution_exists_unique_divisorentry + S (pfd_leading_execution_exists_unique_divisor) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_execution_exists_unique_divisorentry. bb = ff_q_pfp_execution_exists_unique_divisorentry * S ((S (0)) * bc) + (pfd_leading_execution_exists_unique_divisor))) /\\ ((~(pfd_leading_execution_exists_unique_divisor=0)))))))))) -> exists qb qc q rb rc R. (((((forall fom_index_pfp_execution_exists_unique_actualinput. (exists fom_gap_pfp_execution_exists_unique_actualinput_index_bound. fom_gap_pfp_execution_exists_unique_actualinput_index_bound + S (fom_index_pfp_execution_exists_unique_actualinput) = L) -> exists fom_value_pfp_execution_exists_unique_actualinput. ((((exists fom_beta_height_pfp_execution_exists_unique_actualinput_entry. fom_beta_height_pfp_execution_exists_unique_actualinput_entry + S (fom_value_pfp_execution_exists_unique_actualinput) = S ((S (fom_index_pfp_execution_exists_unique_actualinput)) * ac)) /\\ exists fom_beta_quotient_pfp_execution_exists_unique_actualinput_entry. ab = fom_beta_quotient_pfp_execution_exists_unique_actualinput_entry * S ((S (fom_index_pfp_execution_exists_unique_actualinput)) * ac) + (fom_value_pfp_execution_exists_unique_actualinput))) /\\ (exists fom_gap_pfp_execution_exists_unique_actualinput_value_bound. fom_gap_pfp_execution_exists_unique_actualinput_value_bound + S (fom_value_pfp_execution_exists_unique_actualinput) = p))) /\\ (((forall fom_index_pfp_execution_exists_unique_actualdivisor. (exists fom_gap_pfp_execution_exists_unique_actualdivisor_index_bound. fom_gap_pfp_execution_exists_unique_actualdivisor_index_bound + S (fom_index_pfp_execution_exists_unique_actualdivisor) = S (d)) -> exists fom_value_pfp_execution_exists_unique_actualdivisor. ((((exists fom_beta_height_pfp_execution_exists_unique_actualdivisor_entry. fom_beta_height_pfp_execution_exists_unique_actualdivisor_entry + S (fom_value_pfp_execution_exists_unique_actualdivisor) = S ((S (fom_index_pfp_execution_exists_unique_actualdivisor)) * bc)) /\\ exists fom_beta_quotient_pfp_execution_exists_unique_actualdivisor_entry. bb = fom_beta_quotient_pfp_execution_exists_unique_actualdivisor_entry * S ((S (fom_index_pfp_execution_exists_unique_actualdivisor)) * bc) + (fom_value_pfp_execution_exists_unique_actualdivisor))) /\\ (exists fom_gap_pfp_execution_exists_unique_actualdivisor_value_bound. fom_gap_pfp_execution_exists_unique_actualdivisor_value_bound + S (fom_value_pfp_execution_exists_unique_actualdivisor) = p))) /\\ (((((((q)=0) /\\ ((exists pfc_gap_execution_exists_unique_actuallengthshort. pfc_gap_execution_exists_unique_actuallengthshort+(L)=(d))))) \\/ (((~((q)=0)) /\\ (((q)+(d)=(L)))))) /\\ ((exists pfd_head_execution_exists_unique_actual pfd_inverse_execution_exists_unique_actual pfd_product_code_execution_exists_unique_actual pfd_product_scale_execution_exists_unique_actual pfd_residual_code_execution_exists_unique_actual pfd_residual_scale_execution_exists_unique_actual pfd_cut_execution_exists_unique_actual. ((((exists ff_h_pfp_execution_exists_unique_actualhead. ff_h_pfp_execution_exists_unique_actualhead + S (pfd_head_execution_exists_unique_actual) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_execution_exists_unique_actualhead. bb = ff_q_pfp_execution_exists_unique_actualhead * S ((S (0)) * bc) + (pfd_head_execution_exists_unique_actual))) /\\ (((((~((pfd_head_execution_exists_unique_actual) = 0)) /\\ ((((exists pfa_gap_execution_exists_unique_actualinversemultiplicationleft. pfa_gap_execution_exists_unique_actualinversemultiplicationleft + S (pfd_head_execution_exists_unique_actual) = (p)) /\\ (((exists pfa_gap_execution_exists_unique_actualinversemultiplicationright. pfa_gap_execution_exists_unique_actualinversemultiplicationright + S (pfd_inverse_execution_exists_unique_actual) = (p)) /\\ ((((exists pfa_gap_execution_exists_unique_actualinversemultiplicationresultbound. pfa_gap_execution_exists_unique_actualinversemultiplicationresultbound + S (1) = (p)) /\\ ((exists pfa_offset_left_execution_exists_unique_actualinversemultiplicationresultcongruence pfa_offset_right_execution_exists_unique_actualinversemultiplicationresultcongruence. ((pfd_head_execution_exists_unique_actual) * (pfd_inverse_execution_exists_unique_actual)) + (p) * pfa_offset_left_execution_exists_unique_actualinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_execution_exists_unique_actualinversemultiplicationresultcongruence)))))))))))) /\\ (((forall pfd_index_execution_exists_unique_actualquotient. (exists pfa_gap_execution_exists_unique_actualquotientbound. pfa_gap_execution_exists_unique_actualquotientbound + S (pfd_index_execution_exists_unique_actualquotient) = (q)) -> exists pfd_value_execution_exists_unique_actualquotient. ((((exists ff_h_pfp_execution_exists_unique_actualquotiententry. ff_h_pfp_execution_exists_unique_actualquotiententry + S (pfd_value_execution_exists_unique_actualquotient) = S ((S (pfd_index_execution_exists_unique_actualquotient)) * qc)) /\\ exists ff_q_pfp_execution_exists_unique_actualquotiententry. qb = ff_q_pfp_execution_exists_unique_actualquotiententry * S ((S (pfd_index_execution_exists_unique_actualquotient)) * qc) + (pfd_value_execution_exists_unique_actualquotient))) /\\ ((exists pfd_input_execution_exists_unique_actualquotientstep pfd_previous_execution_exists_unique_actualquotientstep pfd_difference_execution_exists_unique_actualquotientstep. ((((exists ff_h_pfp_execution_exists_unique_actualquotientstepinput. ff_h_pfp_execution_exists_unique_actualquotientstepinput + S (pfd_input_execution_exists_unique_actualquotientstep) = S ((S (pfd_index_execution_exists_unique_actualquotient)) * ac)) /\\ exists ff_q_pfp_execution_exists_unique_actualquotientstepinput. ab = ff_q_pfp_execution_exists_unique_actualquotientstepinput * S ((S (pfd_index_execution_exists_unique_actualquotient)) * ac) + (pfd_input_execution_exists_unique_actualquotientstep))) /\\ (((exists pfc_terms_code_execution_exists_unique_actualquotientstepprevious pfc_terms_scale_execution_exists_unique_actualquotientstepprevious pfc_natural_sum_execution_exists_unique_actualquotientstepprevious. ((forall pfc_index_execution_exists_unique_actualquotientsteppreviousdiagonal. (exists pfa_gap_execution_exists_unique_actualquotientsteppreviousdiagonalbound. pfa_gap_execution_exists_unique_actualquotientsteppreviousdiagonalbound + S (pfc_index_execution_exists_unique_actualquotientsteppreviousdiagonal) = (S (pfd_index_execution_exists_unique_actualquotient))) -> exists pfc_value_execution_exists_unique_actualquotientsteppreviousdiagonal. ((((exists ff_h_pfp_execution_exists_unique_actualquotientsteppreviousdiagonalentry. ff_h_pfp_execution_exists_unique_actualquotientsteppreviousdiagonalentry + S (pfc_value_execution_exists_unique_actualquotientsteppreviousdiagonal) = S ((S (pfc_index_execution_exists_unique_actualquotientsteppreviousdiagonal)) * pfc_terms_scale_execution_exists_unique_actualquotientstepprevious)) /\\ exists ff_q_pfp_execution_exists_unique_actualquotientsteppreviousdiagonalentry. pfc_terms_code_execution_exists_unique_actualquotientstepprevious = ff_q_pfp_execution_exists_unique_actualquotientsteppreviousdiagonalentry * S ((S (pfc_index_execution_exists_unique_actualquotientsteppreviousdiagonal)) * pfc_terms_scale_execution_exists_unique_actualquotientstepprevious) + (pfc_value_execution_exists_unique_actualquotientsteppreviousdiagonal))) /\\ ((exists pfc_complement_execution_exists_unique_actualquotientsteppreviousdiagonalterm pfc_left_execution_exists_unique_actualquotientsteppreviousdiagonalterm pfc_right_execution_exists_unique_actualquotientsteppreviousdiagonalterm. (((pfc_index_execution_exists_unique_actualquotientsteppreviousdiagonal)+pfc_complement_execution_exists_unique_actualquotientsteppreviousdiagonalterm=(pfd_index_execution_exists_unique_actualquotient)) /\\ ((((((exists pfa_gap_execution_exists_unique_actualquotientsteppreviousdiagonaltermleftinside. pfa_gap_execution_exists_unique_actualquotientsteppreviousdiagonaltermleftinside + S (pfc_index_execution_exists_unique_actualquotientsteppreviousdiagonal) = (pfd_index_execution_exists_unique_actualquotient)) /\\ ((((exists ff_h_pfp_execution_exists_unique_actualquotientsteppreviousdiagonaltermleftentry. ff_h_pfp_execution_exists_unique_actualquotientsteppreviousdiagonaltermleftentry + S (pfc_left_execution_exists_unique_actualquotientsteppreviousdiagonalterm) = S ((S (pfc_index_execution_exists_unique_actualquotientsteppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_execution_exists_unique_actualquotientsteppreviousdiagonaltermleftentry. qb = ff_q_pfp_execution_exists_unique_actualquotientsteppreviousdiagonaltermleftentry * S ((S (pfc_index_execution_exists_unique_actualquotientsteppreviousdiagonal)) * qc) + (pfc_left_execution_exists_unique_actualquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_execution_exists_unique_actualquotientsteppreviousdiagonaltermleftoutside. pfc_gap_execution_exists_unique_actualquotientsteppreviousdiagonaltermleftoutside+(pfd_index_execution_exists_unique_actualquotient)=(pfc_index_execution_exists_unique_actualquotientsteppreviousdiagonal)) /\\ (((pfc_left_execution_exists_unique_actualquotientsteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_execution_exists_unique_actualquotientsteppreviousdiagonaltermrightinside. pfa_gap_execution_exists_unique_actualquotientsteppreviousdiagonaltermrightinside + S (pfc_complement_execution_exists_unique_actualquotientsteppreviousdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_execution_exists_unique_actualquotientsteppreviousdiagonaltermrightentry. ff_h_pfp_execution_exists_unique_actualquotientsteppreviousdiagonaltermrightentry + S (pfc_right_execution_exists_unique_actualquotientsteppreviousdiagonalterm) = S ((S (pfc_complement_execution_exists_unique_actualquotientsteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_execution_exists_unique_actualquotientsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_execution_exists_unique_actualquotientsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_execution_exists_unique_actualquotientsteppreviousdiagonalterm)) * bc) + (pfc_right_execution_exists_unique_actualquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_execution_exists_unique_actualquotientsteppreviousdiagonaltermrightoutside. pfc_gap_execution_exists_unique_actualquotientsteppreviousdiagonaltermrightoutside+(S (d))=(pfc_complement_execution_exists_unique_actualquotientsteppreviousdiagonalterm)) /\\ (((pfc_right_execution_exists_unique_actualquotientsteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_execution_exists_unique_actualquotientsteppreviousdiagonal)=pfc_left_execution_exists_unique_actualquotientsteppreviousdiagonalterm*pfc_right_execution_exists_unique_actualquotientsteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_execution_exists_unique_actualquotientstepprevioussum fs_v_pfc_execution_exists_unique_actualquotientstepprevioussum. ((((exists fs_h_pfc_execution_exists_unique_actualquotientstepprevioussum_body_start. fs_h_pfc_execution_exists_unique_actualquotientstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_exists_unique_actualquotientstepprevioussum)) /\\ exists fs_q_pfc_execution_exists_unique_actualquotientstepprevioussum_body_start. fs_u_pfc_execution_exists_unique_actualquotientstepprevioussum = fs_q_pfc_execution_exists_unique_actualquotientstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_execution_exists_unique_actualquotientstepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_execution_exists_unique_actualquotientstepprevioussum_body_terminal. fs_h_pfc_execution_exists_unique_actualquotientstepprevioussum_body_terminal + S (pfc_natural_sum_execution_exists_unique_actualquotientstepprevious) = S ((S (S (pfd_index_execution_exists_unique_actualquotient))) * fs_v_pfc_execution_exists_unique_actualquotientstepprevioussum)) /\\ exists fs_q_pfc_execution_exists_unique_actualquotientstepprevioussum_body_terminal. fs_u_pfc_execution_exists_unique_actualquotientstepprevioussum = fs_q_pfc_execution_exists_unique_actualquotientstepprevioussum_body_terminal * S ((S (S (pfd_index_execution_exists_unique_actualquotient))) * fs_v_pfc_execution_exists_unique_actualquotientstepprevioussum) + (pfc_natural_sum_execution_exists_unique_actualquotientstepprevious))) /\\ forall fs_i_pfc_execution_exists_unique_actualquotientstepprevioussum_body_steps. (exists fs_lt_pfc_execution_exists_unique_actualquotientstepprevioussum_body_steps_bound. fs_lt_pfc_execution_exists_unique_actualquotientstepprevioussum_body_steps_bound + S fs_i_pfc_execution_exists_unique_actualquotientstepprevioussum_body_steps = S (pfd_index_execution_exists_unique_actualquotient)) -> exists fs_a_pfc_execution_exists_unique_actualquotientstepprevioussum_body_steps fs_r_pfc_execution_exists_unique_actualquotientstepprevioussum_body_steps fs_s_pfc_execution_exists_unique_actualquotientstepprevioussum_body_steps. ((((exists fs_h_pfc_execution_exists_unique_actualquotientstepprevioussum_body_steps_summand. fs_h_pfc_execution_exists_unique_actualquotientstepprevioussum_body_steps_summand + S (fs_a_pfc_execution_exists_unique_actualquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_execution_exists_unique_actualquotientstepprevioussum_body_steps)) * pfc_terms_scale_execution_exists_unique_actualquotientstepprevious)) /\\ exists fs_q_pfc_execution_exists_unique_actualquotientstepprevioussum_body_steps_summand. pfc_terms_code_execution_exists_unique_actualquotientstepprevious = fs_q_pfc_execution_exists_unique_actualquotientstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_execution_exists_unique_actualquotientstepprevioussum_body_steps)) * pfc_terms_scale_execution_exists_unique_actualquotientstepprevious) + (fs_a_pfc_execution_exists_unique_actualquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_execution_exists_unique_actualquotientstepprevioussum_body_steps_partial. fs_h_pfc_execution_exists_unique_actualquotientstepprevioussum_body_steps_partial + S (fs_r_pfc_execution_exists_unique_actualquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_execution_exists_unique_actualquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_exists_unique_actualquotientstepprevioussum)) /\\ exists fs_q_pfc_execution_exists_unique_actualquotientstepprevioussum_body_steps_partial. fs_u_pfc_execution_exists_unique_actualquotientstepprevioussum = fs_q_pfc_execution_exists_unique_actualquotientstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_execution_exists_unique_actualquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_exists_unique_actualquotientstepprevioussum) + (fs_r_pfc_execution_exists_unique_actualquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_execution_exists_unique_actualquotientstepprevioussum_body_steps_successor. fs_h_pfc_execution_exists_unique_actualquotientstepprevioussum_body_steps_successor + S (fs_s_pfc_execution_exists_unique_actualquotientstepprevioussum_body_steps) = S ((S (S fs_i_pfc_execution_exists_unique_actualquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_exists_unique_actualquotientstepprevioussum)) /\\ exists fs_q_pfc_execution_exists_unique_actualquotientstepprevioussum_body_steps_successor. fs_u_pfc_execution_exists_unique_actualquotientstepprevioussum = fs_q_pfc_execution_exists_unique_actualquotientstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_execution_exists_unique_actualquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_exists_unique_actualquotientstepprevioussum) + (fs_s_pfc_execution_exists_unique_actualquotientstepprevioussum_body_steps))) /\\ fs_s_pfc_execution_exists_unique_actualquotientstepprevioussum_body_steps = fs_r_pfc_execution_exists_unique_actualquotientstepprevioussum_body_steps + fs_a_pfc_execution_exists_unique_actualquotientstepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_execution_exists_unique_actualquotientsteppreviousresiduebound. pfa_gap_execution_exists_unique_actualquotientsteppreviousresiduebound + S (pfd_previous_execution_exists_unique_actualquotientstep) = (p)) /\\ ((exists pfa_offset_left_execution_exists_unique_actualquotientsteppreviousresiduecongruence pfa_offset_right_execution_exists_unique_actualquotientsteppreviousresiduecongruence. (pfc_natural_sum_execution_exists_unique_actualquotientstepprevious) + (p) * pfa_offset_left_execution_exists_unique_actualquotientsteppreviousresiduecongruence = (pfd_previous_execution_exists_unique_actualquotientstep) + (p) * pfa_offset_right_execution_exists_unique_actualquotientsteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_execution_exists_unique_actualquotientstepsubtractleft. pfa_gap_execution_exists_unique_actualquotientstepsubtractleft + S (pfd_previous_execution_exists_unique_actualquotientstep) = (p)) /\\ (((exists pfa_gap_execution_exists_unique_actualquotientstepsubtractright. pfa_gap_execution_exists_unique_actualquotientstepsubtractright + S (pfd_difference_execution_exists_unique_actualquotientstep) = (p)) /\\ ((((exists pfa_gap_execution_exists_unique_actualquotientstepsubtractresultbound. pfa_gap_execution_exists_unique_actualquotientstepsubtractresultbound + S (pfd_input_execution_exists_unique_actualquotientstep) = (p)) /\\ ((exists pfa_offset_left_execution_exists_unique_actualquotientstepsubtractresultcongruence pfa_offset_right_execution_exists_unique_actualquotientstepsubtractresultcongruence. ((pfd_previous_execution_exists_unique_actualquotientstep) + (pfd_difference_execution_exists_unique_actualquotientstep)) + (p) * pfa_offset_left_execution_exists_unique_actualquotientstepsubtractresultcongruence = (pfd_input_execution_exists_unique_actualquotientstep) + (p) * pfa_offset_right_execution_exists_unique_actualquotientstepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_execution_exists_unique_actualquotientstepmultiplyleft. pfa_gap_execution_exists_unique_actualquotientstepmultiplyleft + S (pfd_inverse_execution_exists_unique_actual) = (p)) /\\ (((exists pfa_gap_execution_exists_unique_actualquotientstepmultiplyright. pfa_gap_execution_exists_unique_actualquotientstepmultiplyright + S (pfd_difference_execution_exists_unique_actualquotientstep) = (p)) /\\ ((((exists pfa_gap_execution_exists_unique_actualquotientstepmultiplyresultbound. pfa_gap_execution_exists_unique_actualquotientstepmultiplyresultbound + S (pfd_value_execution_exists_unique_actualquotient) = (p)) /\\ ((exists pfa_offset_left_execution_exists_unique_actualquotientstepmultiplyresultcongruence pfa_offset_right_execution_exists_unique_actualquotientstepmultiplyresultcongruence. ((pfd_inverse_execution_exists_unique_actual) * (pfd_difference_execution_exists_unique_actualquotientstep)) + (p) * pfa_offset_left_execution_exists_unique_actualquotientstepmultiplyresultcongruence = (pfd_value_execution_exists_unique_actualquotient) + (p) * pfa_offset_right_execution_exists_unique_actualquotientstepmultiplyresultcongruence))))))))))))))))))) /\\ (((forall pfc_index_execution_exists_unique_actualproduct. (exists pfa_gap_execution_exists_unique_actualproductbound. pfa_gap_execution_exists_unique_actualproductbound + S (pfc_index_execution_exists_unique_actualproduct) = (L)) -> exists pfc_value_execution_exists_unique_actualproduct. ((((exists ff_h_pfp_execution_exists_unique_actualproductentry. ff_h_pfp_execution_exists_unique_actualproductentry + S (pfc_value_execution_exists_unique_actualproduct) = S ((S (pfc_index_execution_exists_unique_actualproduct)) * pfd_product_scale_execution_exists_unique_actual)) /\\ exists ff_q_pfp_execution_exists_unique_actualproductentry. pfd_product_code_execution_exists_unique_actual = ff_q_pfp_execution_exists_unique_actualproductentry * S ((S (pfc_index_execution_exists_unique_actualproduct)) * pfd_product_scale_execution_exists_unique_actual) + (pfc_value_execution_exists_unique_actualproduct))) /\\ ((exists pfc_terms_code_execution_exists_unique_actualproductcoefficient pfc_terms_scale_execution_exists_unique_actualproductcoefficient pfc_natural_sum_execution_exists_unique_actualproductcoefficient. ((forall pfc_index_execution_exists_unique_actualproductcoefficientdiagonal. (exists pfa_gap_execution_exists_unique_actualproductcoefficientdiagonalbound. pfa_gap_execution_exists_unique_actualproductcoefficientdiagonalbound + S (pfc_index_execution_exists_unique_actualproductcoefficientdiagonal) = (S (pfc_index_execution_exists_unique_actualproduct))) -> exists pfc_value_execution_exists_unique_actualproductcoefficientdiagonal. ((((exists ff_h_pfp_execution_exists_unique_actualproductcoefficientdiagonalentry. ff_h_pfp_execution_exists_unique_actualproductcoefficientdiagonalentry + S (pfc_value_execution_exists_unique_actualproductcoefficientdiagonal) = S ((S (pfc_index_execution_exists_unique_actualproductcoefficientdiagonal)) * pfc_terms_scale_execution_exists_unique_actualproductcoefficient)) /\\ exists ff_q_pfp_execution_exists_unique_actualproductcoefficientdiagonalentry. pfc_terms_code_execution_exists_unique_actualproductcoefficient = ff_q_pfp_execution_exists_unique_actualproductcoefficientdiagonalentry * S ((S (pfc_index_execution_exists_unique_actualproductcoefficientdiagonal)) * pfc_terms_scale_execution_exists_unique_actualproductcoefficient) + (pfc_value_execution_exists_unique_actualproductcoefficientdiagonal))) /\\ ((exists pfc_complement_execution_exists_unique_actualproductcoefficientdiagonalterm pfc_left_execution_exists_unique_actualproductcoefficientdiagonalterm pfc_right_execution_exists_unique_actualproductcoefficientdiagonalterm. (((pfc_index_execution_exists_unique_actualproductcoefficientdiagonal)+pfc_complement_execution_exists_unique_actualproductcoefficientdiagonalterm=(pfc_index_execution_exists_unique_actualproduct)) /\\ ((((((exists pfa_gap_execution_exists_unique_actualproductcoefficientdiagonaltermleftinside. pfa_gap_execution_exists_unique_actualproductcoefficientdiagonaltermleftinside + S (pfc_index_execution_exists_unique_actualproductcoefficientdiagonal) = (q)) /\\ ((((exists ff_h_pfp_execution_exists_unique_actualproductcoefficientdiagonaltermleftentry. ff_h_pfp_execution_exists_unique_actualproductcoefficientdiagonaltermleftentry + S (pfc_left_execution_exists_unique_actualproductcoefficientdiagonalterm) = S ((S (pfc_index_execution_exists_unique_actualproductcoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_execution_exists_unique_actualproductcoefficientdiagonaltermleftentry. qb = ff_q_pfp_execution_exists_unique_actualproductcoefficientdiagonaltermleftentry * S ((S (pfc_index_execution_exists_unique_actualproductcoefficientdiagonal)) * qc) + (pfc_left_execution_exists_unique_actualproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_exists_unique_actualproductcoefficientdiagonaltermleftoutside. pfc_gap_execution_exists_unique_actualproductcoefficientdiagonaltermleftoutside+(q)=(pfc_index_execution_exists_unique_actualproductcoefficientdiagonal)) /\\ (((pfc_left_execution_exists_unique_actualproductcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_execution_exists_unique_actualproductcoefficientdiagonaltermrightinside. pfa_gap_execution_exists_unique_actualproductcoefficientdiagonaltermrightinside + S (pfc_complement_execution_exists_unique_actualproductcoefficientdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_execution_exists_unique_actualproductcoefficientdiagonaltermrightentry. ff_h_pfp_execution_exists_unique_actualproductcoefficientdiagonaltermrightentry + S (pfc_right_execution_exists_unique_actualproductcoefficientdiagonalterm) = S ((S (pfc_complement_execution_exists_unique_actualproductcoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_execution_exists_unique_actualproductcoefficientdiagonaltermrightentry. bb = ff_q_pfp_execution_exists_unique_actualproductcoefficientdiagonaltermrightentry * S ((S (pfc_complement_execution_exists_unique_actualproductcoefficientdiagonalterm)) * bc) + (pfc_right_execution_exists_unique_actualproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_exists_unique_actualproductcoefficientdiagonaltermrightoutside. pfc_gap_execution_exists_unique_actualproductcoefficientdiagonaltermrightoutside+(S (d))=(pfc_complement_execution_exists_unique_actualproductcoefficientdiagonalterm)) /\\ (((pfc_right_execution_exists_unique_actualproductcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_execution_exists_unique_actualproductcoefficientdiagonal)=pfc_left_execution_exists_unique_actualproductcoefficientdiagonalterm*pfc_right_execution_exists_unique_actualproductcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_execution_exists_unique_actualproductcoefficientsum fs_v_pfc_execution_exists_unique_actualproductcoefficientsum. ((((exists fs_h_pfc_execution_exists_unique_actualproductcoefficientsum_body_start. fs_h_pfc_execution_exists_unique_actualproductcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_exists_unique_actualproductcoefficientsum)) /\\ exists fs_q_pfc_execution_exists_unique_actualproductcoefficientsum_body_start. fs_u_pfc_execution_exists_unique_actualproductcoefficientsum = fs_q_pfc_execution_exists_unique_actualproductcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_execution_exists_unique_actualproductcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_execution_exists_unique_actualproductcoefficientsum_body_terminal. fs_h_pfc_execution_exists_unique_actualproductcoefficientsum_body_terminal + S (pfc_natural_sum_execution_exists_unique_actualproductcoefficient) = S ((S (S (pfc_index_execution_exists_unique_actualproduct))) * fs_v_pfc_execution_exists_unique_actualproductcoefficientsum)) /\\ exists fs_q_pfc_execution_exists_unique_actualproductcoefficientsum_body_terminal. fs_u_pfc_execution_exists_unique_actualproductcoefficientsum = fs_q_pfc_execution_exists_unique_actualproductcoefficientsum_body_terminal * S ((S (S (pfc_index_execution_exists_unique_actualproduct))) * fs_v_pfc_execution_exists_unique_actualproductcoefficientsum) + (pfc_natural_sum_execution_exists_unique_actualproductcoefficient))) /\\ forall fs_i_pfc_execution_exists_unique_actualproductcoefficientsum_body_steps. (exists fs_lt_pfc_execution_exists_unique_actualproductcoefficientsum_body_steps_bound. fs_lt_pfc_execution_exists_unique_actualproductcoefficientsum_body_steps_bound + S fs_i_pfc_execution_exists_unique_actualproductcoefficientsum_body_steps = S (pfc_index_execution_exists_unique_actualproduct)) -> exists fs_a_pfc_execution_exists_unique_actualproductcoefficientsum_body_steps fs_r_pfc_execution_exists_unique_actualproductcoefficientsum_body_steps fs_s_pfc_execution_exists_unique_actualproductcoefficientsum_body_steps. ((((exists fs_h_pfc_execution_exists_unique_actualproductcoefficientsum_body_steps_summand. fs_h_pfc_execution_exists_unique_actualproductcoefficientsum_body_steps_summand + S (fs_a_pfc_execution_exists_unique_actualproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_exists_unique_actualproductcoefficientsum_body_steps)) * pfc_terms_scale_execution_exists_unique_actualproductcoefficient)) /\\ exists fs_q_pfc_execution_exists_unique_actualproductcoefficientsum_body_steps_summand. pfc_terms_code_execution_exists_unique_actualproductcoefficient = fs_q_pfc_execution_exists_unique_actualproductcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_execution_exists_unique_actualproductcoefficientsum_body_steps)) * pfc_terms_scale_execution_exists_unique_actualproductcoefficient) + (fs_a_pfc_execution_exists_unique_actualproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_exists_unique_actualproductcoefficientsum_body_steps_partial. fs_h_pfc_execution_exists_unique_actualproductcoefficientsum_body_steps_partial + S (fs_r_pfc_execution_exists_unique_actualproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_exists_unique_actualproductcoefficientsum_body_steps)) * fs_v_pfc_execution_exists_unique_actualproductcoefficientsum)) /\\ exists fs_q_pfc_execution_exists_unique_actualproductcoefficientsum_body_steps_partial. fs_u_pfc_execution_exists_unique_actualproductcoefficientsum = fs_q_pfc_execution_exists_unique_actualproductcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_execution_exists_unique_actualproductcoefficientsum_body_steps)) * fs_v_pfc_execution_exists_unique_actualproductcoefficientsum) + (fs_r_pfc_execution_exists_unique_actualproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_exists_unique_actualproductcoefficientsum_body_steps_successor. fs_h_pfc_execution_exists_unique_actualproductcoefficientsum_body_steps_successor + S (fs_s_pfc_execution_exists_unique_actualproductcoefficientsum_body_steps) = S ((S (S fs_i_pfc_execution_exists_unique_actualproductcoefficientsum_body_steps)) * fs_v_pfc_execution_exists_unique_actualproductcoefficientsum)) /\\ exists fs_q_pfc_execution_exists_unique_actualproductcoefficientsum_body_steps_successor. fs_u_pfc_execution_exists_unique_actualproductcoefficientsum = fs_q_pfc_execution_exists_unique_actualproductcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_execution_exists_unique_actualproductcoefficientsum_body_steps)) * fs_v_pfc_execution_exists_unique_actualproductcoefficientsum) + (fs_s_pfc_execution_exists_unique_actualproductcoefficientsum_body_steps))) /\\ fs_s_pfc_execution_exists_unique_actualproductcoefficientsum_body_steps = fs_r_pfc_execution_exists_unique_actualproductcoefficientsum_body_steps + fs_a_pfc_execution_exists_unique_actualproductcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_execution_exists_unique_actualproductcoefficientresiduebound. pfa_gap_execution_exists_unique_actualproductcoefficientresiduebound + S (pfc_value_execution_exists_unique_actualproduct) = (p)) /\\ ((exists pfa_offset_left_execution_exists_unique_actualproductcoefficientresiduecongruence pfa_offset_right_execution_exists_unique_actualproductcoefficientresiduecongruence. (pfc_natural_sum_execution_exists_unique_actualproductcoefficient) + (p) * pfa_offset_left_execution_exists_unique_actualproductcoefficientresiduecongruence = (pfc_value_execution_exists_unique_actualproduct) + (p) * pfa_offset_right_execution_exists_unique_actualproductcoefficientresiduecongruence)))))))))))) /\\ (((forall pfs_index_execution_exists_unique_actualdifference. (exists pfa_gap_execution_exists_unique_actualdifferenceindex. pfa_gap_execution_exists_unique_actualdifferenceindex + S (pfs_index_execution_exists_unique_actualdifference) = (L)) -> exists pfs_left_execution_exists_unique_actualdifference pfs_right_execution_exists_unique_actualdifference pfs_result_execution_exists_unique_actualdifference. ((((exists ff_h_pfp_execution_exists_unique_actualdifferenceleft. ff_h_pfp_execution_exists_unique_actualdifferenceleft + S (pfs_left_execution_exists_unique_actualdifference) = S ((S (pfs_index_execution_exists_unique_actualdifference)) * ac)) /\\ exists ff_q_pfp_execution_exists_unique_actualdifferenceleft. ab = ff_q_pfp_execution_exists_unique_actualdifferenceleft * S ((S (pfs_index_execution_exists_unique_actualdifference)) * ac) + (pfs_left_execution_exists_unique_actualdifference))) /\\ (((((exists ff_h_pfp_execution_exists_unique_actualdifferenceright. ff_h_pfp_execution_exists_unique_actualdifferenceright + S (pfs_right_execution_exists_unique_actualdifference) = S ((S (pfs_index_execution_exists_unique_actualdifference)) * pfd_product_scale_execution_exists_unique_actual)) /\\ exists ff_q_pfp_execution_exists_unique_actualdifferenceright. pfd_product_code_execution_exists_unique_actual = ff_q_pfp_execution_exists_unique_actualdifferenceright * S ((S (pfs_index_execution_exists_unique_actualdifference)) * pfd_product_scale_execution_exists_unique_actual) + (pfs_right_execution_exists_unique_actualdifference))) /\\ (((((exists ff_h_pfp_execution_exists_unique_actualdifferenceresult. ff_h_pfp_execution_exists_unique_actualdifferenceresult + S (pfs_result_execution_exists_unique_actualdifference) = S ((S (pfs_index_execution_exists_unique_actualdifference)) * pfd_residual_scale_execution_exists_unique_actual)) /\\ exists ff_q_pfp_execution_exists_unique_actualdifferenceresult. pfd_residual_code_execution_exists_unique_actual = ff_q_pfp_execution_exists_unique_actualdifferenceresult * S ((S (pfs_index_execution_exists_unique_actualdifference)) * pfd_residual_scale_execution_exists_unique_actual) + (pfs_result_execution_exists_unique_actualdifference))) /\\ ((((exists pfa_gap_execution_exists_unique_actualdifferenceoperationleft. pfa_gap_execution_exists_unique_actualdifferenceoperationleft + S (pfs_right_execution_exists_unique_actualdifference) = (p)) /\\ (((exists pfa_gap_execution_exists_unique_actualdifferenceoperationright. pfa_gap_execution_exists_unique_actualdifferenceoperationright + S (pfs_result_execution_exists_unique_actualdifference) = (p)) /\\ ((((exists pfa_gap_execution_exists_unique_actualdifferenceoperationresultbound. pfa_gap_execution_exists_unique_actualdifferenceoperationresultbound + S (pfs_left_execution_exists_unique_actualdifference) = (p)) /\\ ((exists pfa_offset_left_execution_exists_unique_actualdifferenceoperationresultcongruence pfa_offset_right_execution_exists_unique_actualdifferenceoperationresultcongruence. ((pfs_right_execution_exists_unique_actualdifference) + (pfs_result_execution_exists_unique_actualdifference)) + (p) * pfa_offset_left_execution_exists_unique_actualdifferenceoperationresultcongruence = (pfs_left_execution_exists_unique_actualdifference) + (p) * pfa_offset_right_execution_exists_unique_actualdifferenceoperationresultcongruence)))))))))))))))) /\\ (((((L)=(pfd_cut_execution_exists_unique_actual)+(R)) /\\ (((forall fom_index_pfp_execution_exists_unique_actualtriminput. (exists fom_gap_pfp_execution_exists_unique_actualtriminput_index_bound. fom_gap_pfp_execution_exists_unique_actualtriminput_index_bound + S (fom_index_pfp_execution_exists_unique_actualtriminput) = L) -> exists fom_value_pfp_execution_exists_unique_actualtriminput. ((((exists fom_beta_height_pfp_execution_exists_unique_actualtriminput_entry. fom_beta_height_pfp_execution_exists_unique_actualtriminput_entry + S (fom_value_pfp_execution_exists_unique_actualtriminput) = S ((S (fom_index_pfp_execution_exists_unique_actualtriminput)) * pfd_residual_scale_execution_exists_unique_actual)) /\\ exists fom_beta_quotient_pfp_execution_exists_unique_actualtriminput_entry. pfd_residual_code_execution_exists_unique_actual = fom_beta_quotient_pfp_execution_exists_unique_actualtriminput_entry * S ((S (fom_index_pfp_execution_exists_unique_actualtriminput)) * pfd_residual_scale_execution_exists_unique_actual) + (fom_value_pfp_execution_exists_unique_actualtriminput))) /\\ (exists fom_gap_pfp_execution_exists_unique_actualtriminput_value_bound. fom_gap_pfp_execution_exists_unique_actualtriminput_value_bound + S (fom_value_pfp_execution_exists_unique_actualtriminput) = p))) /\\ (((forall pfp_repeat_index_execution_exists_unique_actualtrimremoved. (exists pfa_gap_execution_exists_unique_actualtrimremovedindex. pfa_gap_execution_exists_unique_actualtrimremovedindex + S (pfp_repeat_index_execution_exists_unique_actualtrimremoved) = (pfd_cut_execution_exists_unique_actual)) -> (((exists ff_h_pfp_execution_exists_unique_actualtrimremovedentry. ff_h_pfp_execution_exists_unique_actualtrimremovedentry + S (0) = S ((S (pfp_repeat_index_execution_exists_unique_actualtrimremoved)) * pfd_residual_scale_execution_exists_unique_actual)) /\\ exists ff_q_pfp_execution_exists_unique_actualtrimremovedentry. pfd_residual_code_execution_exists_unique_actual = ff_q_pfp_execution_exists_unique_actualtrimremovedentry * S ((S (pfp_repeat_index_execution_exists_unique_actualtrimremoved)) * pfd_residual_scale_execution_exists_unique_actual) + (0)))) /\\ (((forall pftrim_index_execution_exists_unique_actualtrimsuffix pftrim_value_execution_exists_unique_actualtrimsuffix. (exists pfa_gap_execution_exists_unique_actualtrimsuffixbound. pfa_gap_execution_exists_unique_actualtrimsuffixbound + S (pftrim_index_execution_exists_unique_actualtrimsuffix) = (R)) -> (((exists ff_h_pfp_execution_exists_unique_actualtrimsuffixsource. ff_h_pfp_execution_exists_unique_actualtrimsuffixsource + S (pftrim_value_execution_exists_unique_actualtrimsuffix) = S ((S ((pfd_cut_execution_exists_unique_actual)+pftrim_index_execution_exists_unique_actualtrimsuffix)) * pfd_residual_scale_execution_exists_unique_actual)) /\\ exists ff_q_pfp_execution_exists_unique_actualtrimsuffixsource. pfd_residual_code_execution_exists_unique_actual = ff_q_pfp_execution_exists_unique_actualtrimsuffixsource * S ((S ((pfd_cut_execution_exists_unique_actual)+pftrim_index_execution_exists_unique_actualtrimsuffix)) * pfd_residual_scale_execution_exists_unique_actual) + (pftrim_value_execution_exists_unique_actualtrimsuffix))) -> (((exists ff_h_pfp_execution_exists_unique_actualtrimsuffixoutput. ff_h_pfp_execution_exists_unique_actualtrimsuffixoutput + S (pftrim_value_execution_exists_unique_actualtrimsuffix) = S ((S (pftrim_index_execution_exists_unique_actualtrimsuffix)) * rc)) /\\ exists ff_q_pfp_execution_exists_unique_actualtrimsuffixoutput. rb = ff_q_pfp_execution_exists_unique_actualtrimsuffixoutput * S ((S (pftrim_index_execution_exists_unique_actualtrimsuffix)) * rc) + (pftrim_value_execution_exists_unique_actualtrimsuffix)))) /\\ (((R)=0 \\/ (exists pftrim_leading_execution_exists_unique_actualtrimnormal. ((((exists ff_h_pfp_execution_exists_unique_actualtrimnormalentry. ff_h_pfp_execution_exists_unique_actualtrimnormalentry + S (pftrim_leading_execution_exists_unique_actualtrimnormal) = S ((S (0)) * rc)) /\\ exists ff_q_pfp_execution_exists_unique_actualtrimnormalentry. rb = ff_q_pfp_execution_exists_unique_actualtrimnormalentry * S ((S (0)) * rc) + (pftrim_leading_execution_exists_unique_actualtrimnormal))) /\\ ((~(pftrim_leading_execution_exists_unique_actualtrimnormal=0))))))))))))))))))))))))))))))))) /\\ ((forall QB QC Q RB RC K. (((forall fom_index_pfp_execution_exists_unique_otherinput. (exists fom_gap_pfp_execution_exists_unique_otherinput_index_bound. fom_gap_pfp_execution_exists_unique_otherinput_index_bound + S (fom_index_pfp_execution_exists_unique_otherinput) = L) -> exists fom_value_pfp_execution_exists_unique_otherinput. ((((exists fom_beta_height_pfp_execution_exists_unique_otherinput_entry. fom_beta_height_pfp_execution_exists_unique_otherinput_entry + S (fom_value_pfp_execution_exists_unique_otherinput) = S ((S (fom_index_pfp_execution_exists_unique_otherinput)) * ac)) /\\ exists fom_beta_quotient_pfp_execution_exists_unique_otherinput_entry. ab = fom_beta_quotient_pfp_execution_exists_unique_otherinput_entry * S ((S (fom_index_pfp_execution_exists_unique_otherinput)) * ac) + (fom_value_pfp_execution_exists_unique_otherinput))) /\\ (exists fom_gap_pfp_execution_exists_unique_otherinput_value_bound. fom_gap_pfp_execution_exists_unique_otherinput_value_bound + S (fom_value_pfp_execution_exists_unique_otherinput) = p))) /\\ (((forall fom_index_pfp_execution_exists_unique_otherdivisor. (exists fom_gap_pfp_execution_exists_unique_otherdivisor_index_bound. fom_gap_pfp_execution_exists_unique_otherdivisor_index_bound + S (fom_index_pfp_execution_exists_unique_otherdivisor) = S (d)) -> exists fom_value_pfp_execution_exists_unique_otherdivisor. ((((exists fom_beta_height_pfp_execution_exists_unique_otherdivisor_entry. fom_beta_height_pfp_execution_exists_unique_otherdivisor_entry + S (fom_value_pfp_execution_exists_unique_otherdivisor) = S ((S (fom_index_pfp_execution_exists_unique_otherdivisor)) * bc)) /\\ exists fom_beta_quotient_pfp_execution_exists_unique_otherdivisor_entry. bb = fom_beta_quotient_pfp_execution_exists_unique_otherdivisor_entry * S ((S (fom_index_pfp_execution_exists_unique_otherdivisor)) * bc) + (fom_value_pfp_execution_exists_unique_otherdivisor))) /\\ (exists fom_gap_pfp_execution_exists_unique_otherdivisor_value_bound. fom_gap_pfp_execution_exists_unique_otherdivisor_value_bound + S (fom_value_pfp_execution_exists_unique_otherdivisor) = p))) /\\ (((((((Q)=0) /\\ ((exists pfc_gap_execution_exists_unique_otherlengthshort. pfc_gap_execution_exists_unique_otherlengthshort+(L)=(d))))) \\/ (((~((Q)=0)) /\\ (((Q)+(d)=(L)))))) /\\ ((exists pfd_head_execution_exists_unique_other pfd_inverse_execution_exists_unique_other pfd_product_code_execution_exists_unique_other pfd_product_scale_execution_exists_unique_other pfd_residual_code_execution_exists_unique_other pfd_residual_scale_execution_exists_unique_other pfd_cut_execution_exists_unique_other. ((((exists ff_h_pfp_execution_exists_unique_otherhead. ff_h_pfp_execution_exists_unique_otherhead + S (pfd_head_execution_exists_unique_other) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_execution_exists_unique_otherhead. bb = ff_q_pfp_execution_exists_unique_otherhead * S ((S (0)) * bc) + (pfd_head_execution_exists_unique_other))) /\\ (((((~((pfd_head_execution_exists_unique_other) = 0)) /\\ ((((exists pfa_gap_execution_exists_unique_otherinversemultiplicationleft. pfa_gap_execution_exists_unique_otherinversemultiplicationleft + S (pfd_head_execution_exists_unique_other) = (p)) /\\ (((exists pfa_gap_execution_exists_unique_otherinversemultiplicationright. pfa_gap_execution_exists_unique_otherinversemultiplicationright + S (pfd_inverse_execution_exists_unique_other) = (p)) /\\ ((((exists pfa_gap_execution_exists_unique_otherinversemultiplicationresultbound. pfa_gap_execution_exists_unique_otherinversemultiplicationresultbound + S (1) = (p)) /\\ ((exists pfa_offset_left_execution_exists_unique_otherinversemultiplicationresultcongruence pfa_offset_right_execution_exists_unique_otherinversemultiplicationresultcongruence. ((pfd_head_execution_exists_unique_other) * (pfd_inverse_execution_exists_unique_other)) + (p) * pfa_offset_left_execution_exists_unique_otherinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_execution_exists_unique_otherinversemultiplicationresultcongruence)))))))))))) /\\ (((forall pfd_index_execution_exists_unique_otherquotient. (exists pfa_gap_execution_exists_unique_otherquotientbound. pfa_gap_execution_exists_unique_otherquotientbound + S (pfd_index_execution_exists_unique_otherquotient) = (Q)) -> exists pfd_value_execution_exists_unique_otherquotient. ((((exists ff_h_pfp_execution_exists_unique_otherquotiententry. ff_h_pfp_execution_exists_unique_otherquotiententry + S (pfd_value_execution_exists_unique_otherquotient) = S ((S (pfd_index_execution_exists_unique_otherquotient)) * QC)) /\\ exists ff_q_pfp_execution_exists_unique_otherquotiententry. QB = ff_q_pfp_execution_exists_unique_otherquotiententry * S ((S (pfd_index_execution_exists_unique_otherquotient)) * QC) + (pfd_value_execution_exists_unique_otherquotient))) /\\ ((exists pfd_input_execution_exists_unique_otherquotientstep pfd_previous_execution_exists_unique_otherquotientstep pfd_difference_execution_exists_unique_otherquotientstep. ((((exists ff_h_pfp_execution_exists_unique_otherquotientstepinput. ff_h_pfp_execution_exists_unique_otherquotientstepinput + S (pfd_input_execution_exists_unique_otherquotientstep) = S ((S (pfd_index_execution_exists_unique_otherquotient)) * ac)) /\\ exists ff_q_pfp_execution_exists_unique_otherquotientstepinput. ab = ff_q_pfp_execution_exists_unique_otherquotientstepinput * S ((S (pfd_index_execution_exists_unique_otherquotient)) * ac) + (pfd_input_execution_exists_unique_otherquotientstep))) /\\ (((exists pfc_terms_code_execution_exists_unique_otherquotientstepprevious pfc_terms_scale_execution_exists_unique_otherquotientstepprevious pfc_natural_sum_execution_exists_unique_otherquotientstepprevious. ((forall pfc_index_execution_exists_unique_otherquotientsteppreviousdiagonal. (exists pfa_gap_execution_exists_unique_otherquotientsteppreviousdiagonalbound. pfa_gap_execution_exists_unique_otherquotientsteppreviousdiagonalbound + S (pfc_index_execution_exists_unique_otherquotientsteppreviousdiagonal) = (S (pfd_index_execution_exists_unique_otherquotient))) -> exists pfc_value_execution_exists_unique_otherquotientsteppreviousdiagonal. ((((exists ff_h_pfp_execution_exists_unique_otherquotientsteppreviousdiagonalentry. ff_h_pfp_execution_exists_unique_otherquotientsteppreviousdiagonalentry + S (pfc_value_execution_exists_unique_otherquotientsteppreviousdiagonal) = S ((S (pfc_index_execution_exists_unique_otherquotientsteppreviousdiagonal)) * pfc_terms_scale_execution_exists_unique_otherquotientstepprevious)) /\\ exists ff_q_pfp_execution_exists_unique_otherquotientsteppreviousdiagonalentry. pfc_terms_code_execution_exists_unique_otherquotientstepprevious = ff_q_pfp_execution_exists_unique_otherquotientsteppreviousdiagonalentry * S ((S (pfc_index_execution_exists_unique_otherquotientsteppreviousdiagonal)) * pfc_terms_scale_execution_exists_unique_otherquotientstepprevious) + (pfc_value_execution_exists_unique_otherquotientsteppreviousdiagonal))) /\\ ((exists pfc_complement_execution_exists_unique_otherquotientsteppreviousdiagonalterm pfc_left_execution_exists_unique_otherquotientsteppreviousdiagonalterm pfc_right_execution_exists_unique_otherquotientsteppreviousdiagonalterm. (((pfc_index_execution_exists_unique_otherquotientsteppreviousdiagonal)+pfc_complement_execution_exists_unique_otherquotientsteppreviousdiagonalterm=(pfd_index_execution_exists_unique_otherquotient)) /\\ ((((((exists pfa_gap_execution_exists_unique_otherquotientsteppreviousdiagonaltermleftinside. pfa_gap_execution_exists_unique_otherquotientsteppreviousdiagonaltermleftinside + S (pfc_index_execution_exists_unique_otherquotientsteppreviousdiagonal) = (pfd_index_execution_exists_unique_otherquotient)) /\\ ((((exists ff_h_pfp_execution_exists_unique_otherquotientsteppreviousdiagonaltermleftentry. ff_h_pfp_execution_exists_unique_otherquotientsteppreviousdiagonaltermleftentry + S (pfc_left_execution_exists_unique_otherquotientsteppreviousdiagonalterm) = S ((S (pfc_index_execution_exists_unique_otherquotientsteppreviousdiagonal)) * QC)) /\\ exists ff_q_pfp_execution_exists_unique_otherquotientsteppreviousdiagonaltermleftentry. QB = ff_q_pfp_execution_exists_unique_otherquotientsteppreviousdiagonaltermleftentry * S ((S (pfc_index_execution_exists_unique_otherquotientsteppreviousdiagonal)) * QC) + (pfc_left_execution_exists_unique_otherquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_execution_exists_unique_otherquotientsteppreviousdiagonaltermleftoutside. pfc_gap_execution_exists_unique_otherquotientsteppreviousdiagonaltermleftoutside+(pfd_index_execution_exists_unique_otherquotient)=(pfc_index_execution_exists_unique_otherquotientsteppreviousdiagonal)) /\\ (((pfc_left_execution_exists_unique_otherquotientsteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_execution_exists_unique_otherquotientsteppreviousdiagonaltermrightinside. pfa_gap_execution_exists_unique_otherquotientsteppreviousdiagonaltermrightinside + S (pfc_complement_execution_exists_unique_otherquotientsteppreviousdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_execution_exists_unique_otherquotientsteppreviousdiagonaltermrightentry. ff_h_pfp_execution_exists_unique_otherquotientsteppreviousdiagonaltermrightentry + S (pfc_right_execution_exists_unique_otherquotientsteppreviousdiagonalterm) = S ((S (pfc_complement_execution_exists_unique_otherquotientsteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_execution_exists_unique_otherquotientsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_execution_exists_unique_otherquotientsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_execution_exists_unique_otherquotientsteppreviousdiagonalterm)) * bc) + (pfc_right_execution_exists_unique_otherquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_execution_exists_unique_otherquotientsteppreviousdiagonaltermrightoutside. pfc_gap_execution_exists_unique_otherquotientsteppreviousdiagonaltermrightoutside+(S (d))=(pfc_complement_execution_exists_unique_otherquotientsteppreviousdiagonalterm)) /\\ (((pfc_right_execution_exists_unique_otherquotientsteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_execution_exists_unique_otherquotientsteppreviousdiagonal)=pfc_left_execution_exists_unique_otherquotientsteppreviousdiagonalterm*pfc_right_execution_exists_unique_otherquotientsteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_execution_exists_unique_otherquotientstepprevioussum fs_v_pfc_execution_exists_unique_otherquotientstepprevioussum. ((((exists fs_h_pfc_execution_exists_unique_otherquotientstepprevioussum_body_start. fs_h_pfc_execution_exists_unique_otherquotientstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_exists_unique_otherquotientstepprevioussum)) /\\ exists fs_q_pfc_execution_exists_unique_otherquotientstepprevioussum_body_start. fs_u_pfc_execution_exists_unique_otherquotientstepprevioussum = fs_q_pfc_execution_exists_unique_otherquotientstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_execution_exists_unique_otherquotientstepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_execution_exists_unique_otherquotientstepprevioussum_body_terminal. fs_h_pfc_execution_exists_unique_otherquotientstepprevioussum_body_terminal + S (pfc_natural_sum_execution_exists_unique_otherquotientstepprevious) = S ((S (S (pfd_index_execution_exists_unique_otherquotient))) * fs_v_pfc_execution_exists_unique_otherquotientstepprevioussum)) /\\ exists fs_q_pfc_execution_exists_unique_otherquotientstepprevioussum_body_terminal. fs_u_pfc_execution_exists_unique_otherquotientstepprevioussum = fs_q_pfc_execution_exists_unique_otherquotientstepprevioussum_body_terminal * S ((S (S (pfd_index_execution_exists_unique_otherquotient))) * fs_v_pfc_execution_exists_unique_otherquotientstepprevioussum) + (pfc_natural_sum_execution_exists_unique_otherquotientstepprevious))) /\\ forall fs_i_pfc_execution_exists_unique_otherquotientstepprevioussum_body_steps. (exists fs_lt_pfc_execution_exists_unique_otherquotientstepprevioussum_body_steps_bound. fs_lt_pfc_execution_exists_unique_otherquotientstepprevioussum_body_steps_bound + S fs_i_pfc_execution_exists_unique_otherquotientstepprevioussum_body_steps = S (pfd_index_execution_exists_unique_otherquotient)) -> exists fs_a_pfc_execution_exists_unique_otherquotientstepprevioussum_body_steps fs_r_pfc_execution_exists_unique_otherquotientstepprevioussum_body_steps fs_s_pfc_execution_exists_unique_otherquotientstepprevioussum_body_steps. ((((exists fs_h_pfc_execution_exists_unique_otherquotientstepprevioussum_body_steps_summand. fs_h_pfc_execution_exists_unique_otherquotientstepprevioussum_body_steps_summand + S (fs_a_pfc_execution_exists_unique_otherquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_execution_exists_unique_otherquotientstepprevioussum_body_steps)) * pfc_terms_scale_execution_exists_unique_otherquotientstepprevious)) /\\ exists fs_q_pfc_execution_exists_unique_otherquotientstepprevioussum_body_steps_summand. pfc_terms_code_execution_exists_unique_otherquotientstepprevious = fs_q_pfc_execution_exists_unique_otherquotientstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_execution_exists_unique_otherquotientstepprevioussum_body_steps)) * pfc_terms_scale_execution_exists_unique_otherquotientstepprevious) + (fs_a_pfc_execution_exists_unique_otherquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_execution_exists_unique_otherquotientstepprevioussum_body_steps_partial. fs_h_pfc_execution_exists_unique_otherquotientstepprevioussum_body_steps_partial + S (fs_r_pfc_execution_exists_unique_otherquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_execution_exists_unique_otherquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_exists_unique_otherquotientstepprevioussum)) /\\ exists fs_q_pfc_execution_exists_unique_otherquotientstepprevioussum_body_steps_partial. fs_u_pfc_execution_exists_unique_otherquotientstepprevioussum = fs_q_pfc_execution_exists_unique_otherquotientstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_execution_exists_unique_otherquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_exists_unique_otherquotientstepprevioussum) + (fs_r_pfc_execution_exists_unique_otherquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_execution_exists_unique_otherquotientstepprevioussum_body_steps_successor. fs_h_pfc_execution_exists_unique_otherquotientstepprevioussum_body_steps_successor + S (fs_s_pfc_execution_exists_unique_otherquotientstepprevioussum_body_steps) = S ((S (S fs_i_pfc_execution_exists_unique_otherquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_exists_unique_otherquotientstepprevioussum)) /\\ exists fs_q_pfc_execution_exists_unique_otherquotientstepprevioussum_body_steps_successor. fs_u_pfc_execution_exists_unique_otherquotientstepprevioussum = fs_q_pfc_execution_exists_unique_otherquotientstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_execution_exists_unique_otherquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_exists_unique_otherquotientstepprevioussum) + (fs_s_pfc_execution_exists_unique_otherquotientstepprevioussum_body_steps))) /\\ fs_s_pfc_execution_exists_unique_otherquotientstepprevioussum_body_steps = fs_r_pfc_execution_exists_unique_otherquotientstepprevioussum_body_steps + fs_a_pfc_execution_exists_unique_otherquotientstepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_execution_exists_unique_otherquotientsteppreviousresiduebound. pfa_gap_execution_exists_unique_otherquotientsteppreviousresiduebound + S (pfd_previous_execution_exists_unique_otherquotientstep) = (p)) /\\ ((exists pfa_offset_left_execution_exists_unique_otherquotientsteppreviousresiduecongruence pfa_offset_right_execution_exists_unique_otherquotientsteppreviousresiduecongruence. (pfc_natural_sum_execution_exists_unique_otherquotientstepprevious) + (p) * pfa_offset_left_execution_exists_unique_otherquotientsteppreviousresiduecongruence = (pfd_previous_execution_exists_unique_otherquotientstep) + (p) * pfa_offset_right_execution_exists_unique_otherquotientsteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_execution_exists_unique_otherquotientstepsubtractleft. pfa_gap_execution_exists_unique_otherquotientstepsubtractleft + S (pfd_previous_execution_exists_unique_otherquotientstep) = (p)) /\\ (((exists pfa_gap_execution_exists_unique_otherquotientstepsubtractright. pfa_gap_execution_exists_unique_otherquotientstepsubtractright + S (pfd_difference_execution_exists_unique_otherquotientstep) = (p)) /\\ ((((exists pfa_gap_execution_exists_unique_otherquotientstepsubtractresultbound. pfa_gap_execution_exists_unique_otherquotientstepsubtractresultbound + S (pfd_input_execution_exists_unique_otherquotientstep) = (p)) /\\ ((exists pfa_offset_left_execution_exists_unique_otherquotientstepsubtractresultcongruence pfa_offset_right_execution_exists_unique_otherquotientstepsubtractresultcongruence. ((pfd_previous_execution_exists_unique_otherquotientstep) + (pfd_difference_execution_exists_unique_otherquotientstep)) + (p) * pfa_offset_left_execution_exists_unique_otherquotientstepsubtractresultcongruence = (pfd_input_execution_exists_unique_otherquotientstep) + (p) * pfa_offset_right_execution_exists_unique_otherquotientstepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_execution_exists_unique_otherquotientstepmultiplyleft. pfa_gap_execution_exists_unique_otherquotientstepmultiplyleft + S (pfd_inverse_execution_exists_unique_other) = (p)) /\\ (((exists pfa_gap_execution_exists_unique_otherquotientstepmultiplyright. pfa_gap_execution_exists_unique_otherquotientstepmultiplyright + S (pfd_difference_execution_exists_unique_otherquotientstep) = (p)) /\\ ((((exists pfa_gap_execution_exists_unique_otherquotientstepmultiplyresultbound. pfa_gap_execution_exists_unique_otherquotientstepmultiplyresultbound + S (pfd_value_execution_exists_unique_otherquotient) = (p)) /\\ ((exists pfa_offset_left_execution_exists_unique_otherquotientstepmultiplyresultcongruence pfa_offset_right_execution_exists_unique_otherquotientstepmultiplyresultcongruence. ((pfd_inverse_execution_exists_unique_other) * (pfd_difference_execution_exists_unique_otherquotientstep)) + (p) * pfa_offset_left_execution_exists_unique_otherquotientstepmultiplyresultcongruence = (pfd_value_execution_exists_unique_otherquotient) + (p) * pfa_offset_right_execution_exists_unique_otherquotientstepmultiplyresultcongruence))))))))))))))))))) /\\ (((forall pfc_index_execution_exists_unique_otherproduct. (exists pfa_gap_execution_exists_unique_otherproductbound. pfa_gap_execution_exists_unique_otherproductbound + S (pfc_index_execution_exists_unique_otherproduct) = (L)) -> exists pfc_value_execution_exists_unique_otherproduct. ((((exists ff_h_pfp_execution_exists_unique_otherproductentry. ff_h_pfp_execution_exists_unique_otherproductentry + S (pfc_value_execution_exists_unique_otherproduct) = S ((S (pfc_index_execution_exists_unique_otherproduct)) * pfd_product_scale_execution_exists_unique_other)) /\\ exists ff_q_pfp_execution_exists_unique_otherproductentry. pfd_product_code_execution_exists_unique_other = ff_q_pfp_execution_exists_unique_otherproductentry * S ((S (pfc_index_execution_exists_unique_otherproduct)) * pfd_product_scale_execution_exists_unique_other) + (pfc_value_execution_exists_unique_otherproduct))) /\\ ((exists pfc_terms_code_execution_exists_unique_otherproductcoefficient pfc_terms_scale_execution_exists_unique_otherproductcoefficient pfc_natural_sum_execution_exists_unique_otherproductcoefficient. ((forall pfc_index_execution_exists_unique_otherproductcoefficientdiagonal. (exists pfa_gap_execution_exists_unique_otherproductcoefficientdiagonalbound. pfa_gap_execution_exists_unique_otherproductcoefficientdiagonalbound + S (pfc_index_execution_exists_unique_otherproductcoefficientdiagonal) = (S (pfc_index_execution_exists_unique_otherproduct))) -> exists pfc_value_execution_exists_unique_otherproductcoefficientdiagonal. ((((exists ff_h_pfp_execution_exists_unique_otherproductcoefficientdiagonalentry. ff_h_pfp_execution_exists_unique_otherproductcoefficientdiagonalentry + S (pfc_value_execution_exists_unique_otherproductcoefficientdiagonal) = S ((S (pfc_index_execution_exists_unique_otherproductcoefficientdiagonal)) * pfc_terms_scale_execution_exists_unique_otherproductcoefficient)) /\\ exists ff_q_pfp_execution_exists_unique_otherproductcoefficientdiagonalentry. pfc_terms_code_execution_exists_unique_otherproductcoefficient = ff_q_pfp_execution_exists_unique_otherproductcoefficientdiagonalentry * S ((S (pfc_index_execution_exists_unique_otherproductcoefficientdiagonal)) * pfc_terms_scale_execution_exists_unique_otherproductcoefficient) + (pfc_value_execution_exists_unique_otherproductcoefficientdiagonal))) /\\ ((exists pfc_complement_execution_exists_unique_otherproductcoefficientdiagonalterm pfc_left_execution_exists_unique_otherproductcoefficientdiagonalterm pfc_right_execution_exists_unique_otherproductcoefficientdiagonalterm. (((pfc_index_execution_exists_unique_otherproductcoefficientdiagonal)+pfc_complement_execution_exists_unique_otherproductcoefficientdiagonalterm=(pfc_index_execution_exists_unique_otherproduct)) /\\ ((((((exists pfa_gap_execution_exists_unique_otherproductcoefficientdiagonaltermleftinside. pfa_gap_execution_exists_unique_otherproductcoefficientdiagonaltermleftinside + S (pfc_index_execution_exists_unique_otherproductcoefficientdiagonal) = (Q)) /\\ ((((exists ff_h_pfp_execution_exists_unique_otherproductcoefficientdiagonaltermleftentry. ff_h_pfp_execution_exists_unique_otherproductcoefficientdiagonaltermleftentry + S (pfc_left_execution_exists_unique_otherproductcoefficientdiagonalterm) = S ((S (pfc_index_execution_exists_unique_otherproductcoefficientdiagonal)) * QC)) /\\ exists ff_q_pfp_execution_exists_unique_otherproductcoefficientdiagonaltermleftentry. QB = ff_q_pfp_execution_exists_unique_otherproductcoefficientdiagonaltermleftentry * S ((S (pfc_index_execution_exists_unique_otherproductcoefficientdiagonal)) * QC) + (pfc_left_execution_exists_unique_otherproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_exists_unique_otherproductcoefficientdiagonaltermleftoutside. pfc_gap_execution_exists_unique_otherproductcoefficientdiagonaltermleftoutside+(Q)=(pfc_index_execution_exists_unique_otherproductcoefficientdiagonal)) /\\ (((pfc_left_execution_exists_unique_otherproductcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_execution_exists_unique_otherproductcoefficientdiagonaltermrightinside. pfa_gap_execution_exists_unique_otherproductcoefficientdiagonaltermrightinside + S (pfc_complement_execution_exists_unique_otherproductcoefficientdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_execution_exists_unique_otherproductcoefficientdiagonaltermrightentry. ff_h_pfp_execution_exists_unique_otherproductcoefficientdiagonaltermrightentry + S (pfc_right_execution_exists_unique_otherproductcoefficientdiagonalterm) = S ((S (pfc_complement_execution_exists_unique_otherproductcoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_execution_exists_unique_otherproductcoefficientdiagonaltermrightentry. bb = ff_q_pfp_execution_exists_unique_otherproductcoefficientdiagonaltermrightentry * S ((S (pfc_complement_execution_exists_unique_otherproductcoefficientdiagonalterm)) * bc) + (pfc_right_execution_exists_unique_otherproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_exists_unique_otherproductcoefficientdiagonaltermrightoutside. pfc_gap_execution_exists_unique_otherproductcoefficientdiagonaltermrightoutside+(S (d))=(pfc_complement_execution_exists_unique_otherproductcoefficientdiagonalterm)) /\\ (((pfc_right_execution_exists_unique_otherproductcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_execution_exists_unique_otherproductcoefficientdiagonal)=pfc_left_execution_exists_unique_otherproductcoefficientdiagonalterm*pfc_right_execution_exists_unique_otherproductcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_execution_exists_unique_otherproductcoefficientsum fs_v_pfc_execution_exists_unique_otherproductcoefficientsum. ((((exists fs_h_pfc_execution_exists_unique_otherproductcoefficientsum_body_start. fs_h_pfc_execution_exists_unique_otherproductcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_exists_unique_otherproductcoefficientsum)) /\\ exists fs_q_pfc_execution_exists_unique_otherproductcoefficientsum_body_start. fs_u_pfc_execution_exists_unique_otherproductcoefficientsum = fs_q_pfc_execution_exists_unique_otherproductcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_execution_exists_unique_otherproductcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_execution_exists_unique_otherproductcoefficientsum_body_terminal. fs_h_pfc_execution_exists_unique_otherproductcoefficientsum_body_terminal + S (pfc_natural_sum_execution_exists_unique_otherproductcoefficient) = S ((S (S (pfc_index_execution_exists_unique_otherproduct))) * fs_v_pfc_execution_exists_unique_otherproductcoefficientsum)) /\\ exists fs_q_pfc_execution_exists_unique_otherproductcoefficientsum_body_terminal. fs_u_pfc_execution_exists_unique_otherproductcoefficientsum = fs_q_pfc_execution_exists_unique_otherproductcoefficientsum_body_terminal * S ((S (S (pfc_index_execution_exists_unique_otherproduct))) * fs_v_pfc_execution_exists_unique_otherproductcoefficientsum) + (pfc_natural_sum_execution_exists_unique_otherproductcoefficient))) /\\ forall fs_i_pfc_execution_exists_unique_otherproductcoefficientsum_body_steps. (exists fs_lt_pfc_execution_exists_unique_otherproductcoefficientsum_body_steps_bound. fs_lt_pfc_execution_exists_unique_otherproductcoefficientsum_body_steps_bound + S fs_i_pfc_execution_exists_unique_otherproductcoefficientsum_body_steps = S (pfc_index_execution_exists_unique_otherproduct)) -> exists fs_a_pfc_execution_exists_unique_otherproductcoefficientsum_body_steps fs_r_pfc_execution_exists_unique_otherproductcoefficientsum_body_steps fs_s_pfc_execution_exists_unique_otherproductcoefficientsum_body_steps. ((((exists fs_h_pfc_execution_exists_unique_otherproductcoefficientsum_body_steps_summand. fs_h_pfc_execution_exists_unique_otherproductcoefficientsum_body_steps_summand + S (fs_a_pfc_execution_exists_unique_otherproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_exists_unique_otherproductcoefficientsum_body_steps)) * pfc_terms_scale_execution_exists_unique_otherproductcoefficient)) /\\ exists fs_q_pfc_execution_exists_unique_otherproductcoefficientsum_body_steps_summand. pfc_terms_code_execution_exists_unique_otherproductcoefficient = fs_q_pfc_execution_exists_unique_otherproductcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_execution_exists_unique_otherproductcoefficientsum_body_steps)) * pfc_terms_scale_execution_exists_unique_otherproductcoefficient) + (fs_a_pfc_execution_exists_unique_otherproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_exists_unique_otherproductcoefficientsum_body_steps_partial. fs_h_pfc_execution_exists_unique_otherproductcoefficientsum_body_steps_partial + S (fs_r_pfc_execution_exists_unique_otherproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_exists_unique_otherproductcoefficientsum_body_steps)) * fs_v_pfc_execution_exists_unique_otherproductcoefficientsum)) /\\ exists fs_q_pfc_execution_exists_unique_otherproductcoefficientsum_body_steps_partial. fs_u_pfc_execution_exists_unique_otherproductcoefficientsum = fs_q_pfc_execution_exists_unique_otherproductcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_execution_exists_unique_otherproductcoefficientsum_body_steps)) * fs_v_pfc_execution_exists_unique_otherproductcoefficientsum) + (fs_r_pfc_execution_exists_unique_otherproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_exists_unique_otherproductcoefficientsum_body_steps_successor. fs_h_pfc_execution_exists_unique_otherproductcoefficientsum_body_steps_successor + S (fs_s_pfc_execution_exists_unique_otherproductcoefficientsum_body_steps) = S ((S (S fs_i_pfc_execution_exists_unique_otherproductcoefficientsum_body_steps)) * fs_v_pfc_execution_exists_unique_otherproductcoefficientsum)) /\\ exists fs_q_pfc_execution_exists_unique_otherproductcoefficientsum_body_steps_successor. fs_u_pfc_execution_exists_unique_otherproductcoefficientsum = fs_q_pfc_execution_exists_unique_otherproductcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_execution_exists_unique_otherproductcoefficientsum_body_steps)) * fs_v_pfc_execution_exists_unique_otherproductcoefficientsum) + (fs_s_pfc_execution_exists_unique_otherproductcoefficientsum_body_steps))) /\\ fs_s_pfc_execution_exists_unique_otherproductcoefficientsum_body_steps = fs_r_pfc_execution_exists_unique_otherproductcoefficientsum_body_steps + fs_a_pfc_execution_exists_unique_otherproductcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_execution_exists_unique_otherproductcoefficientresiduebound. pfa_gap_execution_exists_unique_otherproductcoefficientresiduebound + S (pfc_value_execution_exists_unique_otherproduct) = (p)) /\\ ((exists pfa_offset_left_execution_exists_unique_otherproductcoefficientresiduecongruence pfa_offset_right_execution_exists_unique_otherproductcoefficientresiduecongruence. (pfc_natural_sum_execution_exists_unique_otherproductcoefficient) + (p) * pfa_offset_left_execution_exists_unique_otherproductcoefficientresiduecongruence = (pfc_value_execution_exists_unique_otherproduct) + (p) * pfa_offset_right_execution_exists_unique_otherproductcoefficientresiduecongruence)))))))))))) /\\ (((forall pfs_index_execution_exists_unique_otherdifference. (exists pfa_gap_execution_exists_unique_otherdifferenceindex. pfa_gap_execution_exists_unique_otherdifferenceindex + S (pfs_index_execution_exists_unique_otherdifference) = (L)) -> exists pfs_left_execution_exists_unique_otherdifference pfs_right_execution_exists_unique_otherdifference pfs_result_execution_exists_unique_otherdifference. ((((exists ff_h_pfp_execution_exists_unique_otherdifferenceleft. ff_h_pfp_execution_exists_unique_otherdifferenceleft + S (pfs_left_execution_exists_unique_otherdifference) = S ((S (pfs_index_execution_exists_unique_otherdifference)) * ac)) /\\ exists ff_q_pfp_execution_exists_unique_otherdifferenceleft. ab = ff_q_pfp_execution_exists_unique_otherdifferenceleft * S ((S (pfs_index_execution_exists_unique_otherdifference)) * ac) + (pfs_left_execution_exists_unique_otherdifference))) /\\ (((((exists ff_h_pfp_execution_exists_unique_otherdifferenceright. ff_h_pfp_execution_exists_unique_otherdifferenceright + S (pfs_right_execution_exists_unique_otherdifference) = S ((S (pfs_index_execution_exists_unique_otherdifference)) * pfd_product_scale_execution_exists_unique_other)) /\\ exists ff_q_pfp_execution_exists_unique_otherdifferenceright. pfd_product_code_execution_exists_unique_other = ff_q_pfp_execution_exists_unique_otherdifferenceright * S ((S (pfs_index_execution_exists_unique_otherdifference)) * pfd_product_scale_execution_exists_unique_other) + (pfs_right_execution_exists_unique_otherdifference))) /\\ (((((exists ff_h_pfp_execution_exists_unique_otherdifferenceresult. ff_h_pfp_execution_exists_unique_otherdifferenceresult + S (pfs_result_execution_exists_unique_otherdifference) = S ((S (pfs_index_execution_exists_unique_otherdifference)) * pfd_residual_scale_execution_exists_unique_other)) /\\ exists ff_q_pfp_execution_exists_unique_otherdifferenceresult. pfd_residual_code_execution_exists_unique_other = ff_q_pfp_execution_exists_unique_otherdifferenceresult * S ((S (pfs_index_execution_exists_unique_otherdifference)) * pfd_residual_scale_execution_exists_unique_other) + (pfs_result_execution_exists_unique_otherdifference))) /\\ ((((exists pfa_gap_execution_exists_unique_otherdifferenceoperationleft. pfa_gap_execution_exists_unique_otherdifferenceoperationleft + S (pfs_right_execution_exists_unique_otherdifference) = (p)) /\\ (((exists pfa_gap_execution_exists_unique_otherdifferenceoperationright. pfa_gap_execution_exists_unique_otherdifferenceoperationright + S (pfs_result_execution_exists_unique_otherdifference) = (p)) /\\ ((((exists pfa_gap_execution_exists_unique_otherdifferenceoperationresultbound. pfa_gap_execution_exists_unique_otherdifferenceoperationresultbound + S (pfs_left_execution_exists_unique_otherdifference) = (p)) /\\ ((exists pfa_offset_left_execution_exists_unique_otherdifferenceoperationresultcongruence pfa_offset_right_execution_exists_unique_otherdifferenceoperationresultcongruence. ((pfs_right_execution_exists_unique_otherdifference) + (pfs_result_execution_exists_unique_otherdifference)) + (p) * pfa_offset_left_execution_exists_unique_otherdifferenceoperationresultcongruence = (pfs_left_execution_exists_unique_otherdifference) + (p) * pfa_offset_right_execution_exists_unique_otherdifferenceoperationresultcongruence)))))))))))))))) /\\ (((((L)=(pfd_cut_execution_exists_unique_other)+(K)) /\\ (((forall fom_index_pfp_execution_exists_unique_othertriminput. (exists fom_gap_pfp_execution_exists_unique_othertriminput_index_bound. fom_gap_pfp_execution_exists_unique_othertriminput_index_bound + S (fom_index_pfp_execution_exists_unique_othertriminput) = L) -> exists fom_value_pfp_execution_exists_unique_othertriminput. ((((exists fom_beta_height_pfp_execution_exists_unique_othertriminput_entry. fom_beta_height_pfp_execution_exists_unique_othertriminput_entry + S (fom_value_pfp_execution_exists_unique_othertriminput) = S ((S (fom_index_pfp_execution_exists_unique_othertriminput)) * pfd_residual_scale_execution_exists_unique_other)) /\\ exists fom_beta_quotient_pfp_execution_exists_unique_othertriminput_entry. pfd_residual_code_execution_exists_unique_other = fom_beta_quotient_pfp_execution_exists_unique_othertriminput_entry * S ((S (fom_index_pfp_execution_exists_unique_othertriminput)) * pfd_residual_scale_execution_exists_unique_other) + (fom_value_pfp_execution_exists_unique_othertriminput))) /\\ (exists fom_gap_pfp_execution_exists_unique_othertriminput_value_bound. fom_gap_pfp_execution_exists_unique_othertriminput_value_bound + S (fom_value_pfp_execution_exists_unique_othertriminput) = p))) /\\ (((forall pfp_repeat_index_execution_exists_unique_othertrimremoved. (exists pfa_gap_execution_exists_unique_othertrimremovedindex. pfa_gap_execution_exists_unique_othertrimremovedindex + S (pfp_repeat_index_execution_exists_unique_othertrimremoved) = (pfd_cut_execution_exists_unique_other)) -> (((exists ff_h_pfp_execution_exists_unique_othertrimremovedentry. ff_h_pfp_execution_exists_unique_othertrimremovedentry + S (0) = S ((S (pfp_repeat_index_execution_exists_unique_othertrimremoved)) * pfd_residual_scale_execution_exists_unique_other)) /\\ exists ff_q_pfp_execution_exists_unique_othertrimremovedentry. pfd_residual_code_execution_exists_unique_other = ff_q_pfp_execution_exists_unique_othertrimremovedentry * S ((S (pfp_repeat_index_execution_exists_unique_othertrimremoved)) * pfd_residual_scale_execution_exists_unique_other) + (0)))) /\\ (((forall pftrim_index_execution_exists_unique_othertrimsuffix pftrim_value_execution_exists_unique_othertrimsuffix. (exists pfa_gap_execution_exists_unique_othertrimsuffixbound. pfa_gap_execution_exists_unique_othertrimsuffixbound + S (pftrim_index_execution_exists_unique_othertrimsuffix) = (K)) -> (((exists ff_h_pfp_execution_exists_unique_othertrimsuffixsource. ff_h_pfp_execution_exists_unique_othertrimsuffixsource + S (pftrim_value_execution_exists_unique_othertrimsuffix) = S ((S ((pfd_cut_execution_exists_unique_other)+pftrim_index_execution_exists_unique_othertrimsuffix)) * pfd_residual_scale_execution_exists_unique_other)) /\\ exists ff_q_pfp_execution_exists_unique_othertrimsuffixsource. pfd_residual_code_execution_exists_unique_other = ff_q_pfp_execution_exists_unique_othertrimsuffixsource * S ((S ((pfd_cut_execution_exists_unique_other)+pftrim_index_execution_exists_unique_othertrimsuffix)) * pfd_residual_scale_execution_exists_unique_other) + (pftrim_value_execution_exists_unique_othertrimsuffix))) -> (((exists ff_h_pfp_execution_exists_unique_othertrimsuffixoutput. ff_h_pfp_execution_exists_unique_othertrimsuffixoutput + S (pftrim_value_execution_exists_unique_othertrimsuffix) = S ((S (pftrim_index_execution_exists_unique_othertrimsuffix)) * RC)) /\\ exists ff_q_pfp_execution_exists_unique_othertrimsuffixoutput. RB = ff_q_pfp_execution_exists_unique_othertrimsuffixoutput * S ((S (pftrim_index_execution_exists_unique_othertrimsuffix)) * RC) + (pftrim_value_execution_exists_unique_othertrimsuffix)))) /\\ (((K)=0 \\/ (exists pftrim_leading_execution_exists_unique_othertrimnormal. ((((exists ff_h_pfp_execution_exists_unique_othertrimnormalentry. ff_h_pfp_execution_exists_unique_othertrimnormalentry + S (pftrim_leading_execution_exists_unique_othertrimnormal) = S ((S (0)) * RC)) /\\ exists ff_q_pfp_execution_exists_unique_othertrimnormalentry. RB = ff_q_pfp_execution_exists_unique_othertrimnormalentry * S ((S (0)) * RC) + (pftrim_leading_execution_exists_unique_othertrimnormal))) /\\ ((~(pftrim_leading_execution_exists_unique_othertrimnormal=0))))))))))))))))))))))))))))))))) -> ((((q)=(Q)) /\\ (((forall mdr_i_pfp_execution_exists_unique_equalquotient mdr_a_pfp_execution_exists_unique_equalquotient. 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(exists pfa_gap_shift_paddingbound. pfa_gap_shift_paddingbound + S (pfrep_index_shift_padding) = (L)) -> (((exists ff_h_pfp_shift_paddinginput. ff_h_pfp_shift_paddinginput + S (pfrep_value_shift_padding) = S ((S (pfrep_index_shift_padding)) * c)) /\\ exists ff_q_pfp_shift_paddinginput. b = ff_q_pfp_shift_paddinginput * S ((S (pfrep_index_shift_padding)) * c) + (pfrep_value_shift_padding))) -> (((exists ff_h_pfp_shift_paddingoutput. ff_h_pfp_shift_paddingoutput + S (pfrep_value_shift_padding) = S ((S ((t)+pfrep_index_shift_padding)) * C)) /\\ exists ff_q_pfp_shift_paddingoutput. B = ff_q_pfp_shift_paddingoutput * S ((S ((t)+pfrep_index_shift_padding)) * C) + (pfrep_value_shift_padding))))))) -> ((((exists pfa_gap_shift_originalinside. pfa_gap_shift_originalinside + S (i) = (L)) /\\ ((((exists ff_h_pfp_shift_originalentry. ff_h_pfp_shift_originalentry + S (a) = S ((S (i)) * c)) /\\ exists ff_q_pfp_shift_originalentry. b = ff_q_pfp_shift_originalentry * S ((S (i)) * c) + (a)))))) \\/ (((exists pfc_gap_shift_originaloutside. pfc_gap_shift_originaloutside+(L)=(i)) /\\ (((a)=0))))) -> ((((exists pfa_gap_shift_resultinside. pfa_gap_shift_resultinside + S (t+i) = (t+L)) /\\ ((((exists ff_h_pfp_shift_resultentry. ff_h_pfp_shift_resultentry + S (a) = S ((S (t+i)) * C)) /\\ exists ff_q_pfp_shift_resultentry. 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(((forall pfp_repeat_index_before_paddingzeros. (exists pfa_gap_before_paddingzerosindex. pfa_gap_before_paddingzerosindex + S (pfp_repeat_index_before_paddingzeros) = (t)) -> (((exists ff_h_pfp_before_paddingzerosentry. ff_h_pfp_before_paddingzerosentry + S (0) = S ((S (pfp_repeat_index_before_paddingzeros)) * C)) /\\ exists ff_q_pfp_before_paddingzerosentry. B = ff_q_pfp_before_paddingzerosentry * S ((S (pfp_repeat_index_before_paddingzeros)) * C) + (0)))) /\\ ((forall pfrep_index_before_padding pfrep_value_before_padding. (exists pfa_gap_before_paddingbound. pfa_gap_before_paddingbound + S (pfrep_index_before_padding) = (L)) -> (((exists ff_h_pfp_before_paddinginput. ff_h_pfp_before_paddinginput + S (pfrep_value_before_padding) = S ((S (pfrep_index_before_padding)) * c)) /\\ exists ff_q_pfp_before_paddinginput. b = ff_q_pfp_before_paddinginput * S ((S (pfrep_index_before_padding)) * c) + (pfrep_value_before_padding))) -> (((exists ff_h_pfp_before_paddingoutput. ff_h_pfp_before_paddingoutput + S (pfrep_value_before_padding) = S ((S ((t)+pfrep_index_before_padding)) * C)) /\\ exists ff_q_pfp_before_paddingoutput. B = ff_q_pfp_before_paddingoutput * S ((S ((t)+pfrep_index_before_padding)) * C) + (pfrep_value_before_padding))))))) -> (exists pfa_gap_before_bound. pfa_gap_before_bound + S (i) = (t)) -> ((((exists pfa_gap_before_zeroinside. pfa_gap_before_zeroinside + S (i) = (t+L)) /\\ ((((exists ff_h_pfp_before_zeroentry. ff_h_pfp_before_zeroentry + S (0) = S ((S (i)) * C)) /\\ exists ff_q_pfp_before_zeroentry. 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B = ff_q_pfp_padded_zero_datazerosentry * S ((S (pfp_repeat_index_padded_zero_datazeros)) * C) + (0)))) /\\ ((forall pfrep_index_padded_zero_data pfrep_value_padded_zero_data. (exists pfa_gap_padded_zero_databound. pfa_gap_padded_zero_databound + S (pfrep_index_padded_zero_data) = (L)) -> (((exists ff_h_pfp_padded_zero_datainput. ff_h_pfp_padded_zero_datainput + S (pfrep_value_padded_zero_data) = S ((S (pfrep_index_padded_zero_data)) * c)) /\\ exists ff_q_pfp_padded_zero_datainput. b = ff_q_pfp_padded_zero_datainput * S ((S (pfrep_index_padded_zero_data)) * c) + (pfrep_value_padded_zero_data))) -> (((exists ff_h_pfp_padded_zero_dataoutput. ff_h_pfp_padded_zero_dataoutput + S (pfrep_value_padded_zero_data) = S ((S ((t)+pfrep_index_padded_zero_data)) * C)) /\\ exists ff_q_pfp_padded_zero_dataoutput. B = ff_q_pfp_padded_zero_dataoutput * S ((S ((t)+pfrep_index_padded_zero_data)) * C) + (pfrep_value_padded_zero_data))))))) -> (forall pfp_repeat_index_padded_zero_result. 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(exists pfa_gap_sum_pad_prefixzerosindex. pfa_gap_sum_pad_prefixzerosindex + S (pfp_repeat_index_sum_pad_prefixzeros) = (t)) -> (((exists ff_h_pfp_sum_pad_prefixzerosentry. ff_h_pfp_sum_pad_prefixzerosentry + S (0) = S ((S (pfp_repeat_index_sum_pad_prefixzeros)) * C)) /\\ exists ff_q_pfp_sum_pad_prefixzerosentry. B = ff_q_pfp_sum_pad_prefixzerosentry * S ((S (pfp_repeat_index_sum_pad_prefixzeros)) * C) + (0)))) /\\ ((forall pfrep_index_sum_pad_prefix pfrep_value_sum_pad_prefix. (exists pfa_gap_sum_pad_prefixbound. pfa_gap_sum_pad_prefixbound + S (pfrep_index_sum_pad_prefix) = (L)) -> (((exists ff_h_pfp_sum_pad_prefixinput. ff_h_pfp_sum_pad_prefixinput + S (pfrep_value_sum_pad_prefix) = S ((S (pfrep_index_sum_pad_prefix)) * c)) /\\ exists ff_q_pfp_sum_pad_prefixinput. b = ff_q_pfp_sum_pad_prefixinput * S ((S (pfrep_index_sum_pad_prefix)) * c) + (pfrep_value_sum_pad_prefix))) -> (((exists ff_h_pfp_sum_pad_prefixoutput. ff_h_pfp_sum_pad_prefixoutput + S (pfrep_value_sum_pad_prefix) = S ((S ((t)+pfrep_index_sum_pad_prefix)) * C)) /\\ exists ff_q_pfp_sum_pad_prefixoutput. B = ff_q_pfp_sum_pad_prefixoutput * S ((S ((t)+pfrep_index_sum_pad_prefix)) * C) + (pfrep_value_sum_pad_prefix))))))", "split", "exact hpad_left", "intro i", "intro a", "intro hi", "intro ha", "specialize hpad_right (i)", "specialize hpad_right (a)", "apply hpad_right", "specialize le_succ (S i)", "specialize le_succ (L)", "apply le_succ", "exact hi", "exact ha", "have hfirst : exists a u. ((((exists ff_h_pfp_sum_pad_firstentry. ff_h_pfp_sum_pad_firstentry + S (a) = S ((S (L)) * c)) /\\ exists ff_q_pfp_sum_pad_firstentry. b = ff_q_pfp_sum_pad_firstentry * S ((S (L)) * c) + (a))) /\\ (((exists fs_u_pfc_sum_pad_firstprefix fs_v_pfc_sum_pad_firstprefix. ((((exists fs_h_pfc_sum_pad_firstprefix_body_start. fs_h_pfc_sum_pad_firstprefix_body_start + S (0) = S ((S (0)) * fs_v_pfc_sum_pad_firstprefix)) /\\ exists fs_q_pfc_sum_pad_firstprefix_body_start. fs_u_pfc_sum_pad_firstprefix = fs_q_pfc_sum_pad_firstprefix_body_start * S ((S (0)) * fs_v_pfc_sum_pad_firstprefix) + (0))) /\\ ((((exists fs_h_pfc_sum_pad_firstprefix_body_terminal. fs_h_pfc_sum_pad_firstprefix_body_terminal + S (u) = S ((S (L)) * fs_v_pfc_sum_pad_firstprefix)) /\\ exists fs_q_pfc_sum_pad_firstprefix_body_terminal. fs_u_pfc_sum_pad_firstprefix = fs_q_pfc_sum_pad_firstprefix_body_terminal * S ((S (L)) * fs_v_pfc_sum_pad_firstprefix) + (u))) /\\ forall fs_i_pfc_sum_pad_firstprefix_body_steps. (exists fs_lt_pfc_sum_pad_firstprefix_body_steps_bound. fs_lt_pfc_sum_pad_firstprefix_body_steps_bound + S fs_i_pfc_sum_pad_firstprefix_body_steps = L) -> exists fs_a_pfc_sum_pad_firstprefix_body_steps fs_r_pfc_sum_pad_firstprefix_body_steps fs_s_pfc_sum_pad_firstprefix_body_steps. ((((exists fs_h_pfc_sum_pad_firstprefix_body_steps_summand. fs_h_pfc_sum_pad_firstprefix_body_steps_summand + S (fs_a_pfc_sum_pad_firstprefix_body_steps) = S ((S (fs_i_pfc_sum_pad_firstprefix_body_steps)) * c)) /\\ exists fs_q_pfc_sum_pad_firstprefix_body_steps_summand. b = fs_q_pfc_sum_pad_firstprefix_body_steps_summand * S ((S (fs_i_pfc_sum_pad_firstprefix_body_steps)) * c) + (fs_a_pfc_sum_pad_firstprefix_body_steps))) /\\ ((((exists fs_h_pfc_sum_pad_firstprefix_body_steps_partial. fs_h_pfc_sum_pad_firstprefix_body_steps_partial + S (fs_r_pfc_sum_pad_firstprefix_body_steps) = S ((S (fs_i_pfc_sum_pad_firstprefix_body_steps)) * fs_v_pfc_sum_pad_firstprefix)) /\\ exists fs_q_pfc_sum_pad_firstprefix_body_steps_partial. fs_u_pfc_sum_pad_firstprefix = fs_q_pfc_sum_pad_firstprefix_body_steps_partial * S ((S (fs_i_pfc_sum_pad_firstprefix_body_steps)) * fs_v_pfc_sum_pad_firstprefix) + (fs_r_pfc_sum_pad_firstprefix_body_steps))) /\\ ((((exists fs_h_pfc_sum_pad_firstprefix_body_steps_successor. fs_h_pfc_sum_pad_firstprefix_body_steps_successor + S (fs_s_pfc_sum_pad_firstprefix_body_steps) = S ((S (S fs_i_pfc_sum_pad_firstprefix_body_steps)) * fs_v_pfc_sum_pad_firstprefix)) /\\ exists fs_q_pfc_sum_pad_firstprefix_body_steps_successor. fs_u_pfc_sum_pad_firstprefix = fs_q_pfc_sum_pad_firstprefix_body_steps_successor * S ((S (S fs_i_pfc_sum_pad_firstprefix_body_steps)) * fs_v_pfc_sum_pad_firstprefix) + (fs_s_pfc_sum_pad_firstprefix_body_steps))) /\\ fs_s_pfc_sum_pad_firstprefix_body_steps = fs_r_pfc_sum_pad_firstprefix_body_steps + fs_a_pfc_sum_pad_firstprefix_body_steps)))))) /\\ (((n)=u+a)))))", "specialize beta_sum_succ_decompose (b)", "specialize beta_sum_succ_decompose (c)", "specialize beta_sum_succ_decompose (L)", "specialize beta_sum_succ_decompose (n)", "apply beta_sum_succ_decompose", "exact hs", "cases hfirst", "cases hfirst_witness", "cases hfirst_witness_witness", "cases hfirst_witness_witness_right", "have hlength : t+S L=S (t+L)", "simp", "rewrite hlength at ht", "rewrite hlength at ht", "rewrite hlength at ht", "have hsecond : exists a u. ((((exists ff_h_pfp_sum_pad_secondentry. ff_h_pfp_sum_pad_secondentry + S (a) = S ((S (t+L)) * C)) /\\ exists ff_q_pfp_sum_pad_secondentry. B = ff_q_pfp_sum_pad_secondentry * S ((S (t+L)) * C) + (a))) /\\ (((exists fs_u_pfc_sum_pad_secondprefix fs_v_pfc_sum_pad_secondprefix. ((((exists fs_h_pfc_sum_pad_secondprefix_body_start. fs_h_pfc_sum_pad_secondprefix_body_start + S (0) = S ((S (0)) * fs_v_pfc_sum_pad_secondprefix)) /\\ exists fs_q_pfc_sum_pad_secondprefix_body_start. fs_u_pfc_sum_pad_secondprefix = fs_q_pfc_sum_pad_secondprefix_body_start * S ((S (0)) * fs_v_pfc_sum_pad_secondprefix) + (0))) /\\ ((((exists fs_h_pfc_sum_pad_secondprefix_body_terminal. fs_h_pfc_sum_pad_secondprefix_body_terminal + S (u) = S ((S (t+L)) * fs_v_pfc_sum_pad_secondprefix)) /\\ exists fs_q_pfc_sum_pad_secondprefix_body_terminal. fs_u_pfc_sum_pad_secondprefix = fs_q_pfc_sum_pad_secondprefix_body_terminal * S ((S (t+L)) * fs_v_pfc_sum_pad_secondprefix) + (u))) /\\ forall fs_i_pfc_sum_pad_secondprefix_body_steps. (exists fs_lt_pfc_sum_pad_secondprefix_body_steps_bound. fs_lt_pfc_sum_pad_secondprefix_body_steps_bound + S fs_i_pfc_sum_pad_secondprefix_body_steps = t+L) -> exists fs_a_pfc_sum_pad_secondprefix_body_steps fs_r_pfc_sum_pad_secondprefix_body_steps fs_s_pfc_sum_pad_secondprefix_body_steps. ((((exists fs_h_pfc_sum_pad_secondprefix_body_steps_summand. fs_h_pfc_sum_pad_secondprefix_body_steps_summand + S (fs_a_pfc_sum_pad_secondprefix_body_steps) = S ((S (fs_i_pfc_sum_pad_secondprefix_body_steps)) * C)) /\\ exists fs_q_pfc_sum_pad_secondprefix_body_steps_summand. B = fs_q_pfc_sum_pad_secondprefix_body_steps_summand * S ((S (fs_i_pfc_sum_pad_secondprefix_body_steps)) * C) + (fs_a_pfc_sum_pad_secondprefix_body_steps))) /\\ ((((exists fs_h_pfc_sum_pad_secondprefix_body_steps_partial. fs_h_pfc_sum_pad_secondprefix_body_steps_partial + S (fs_r_pfc_sum_pad_secondprefix_body_steps) = S ((S (fs_i_pfc_sum_pad_secondprefix_body_steps)) * fs_v_pfc_sum_pad_secondprefix)) /\\ exists fs_q_pfc_sum_pad_secondprefix_body_steps_partial. fs_u_pfc_sum_pad_secondprefix = fs_q_pfc_sum_pad_secondprefix_body_steps_partial * S ((S (fs_i_pfc_sum_pad_secondprefix_body_steps)) * fs_v_pfc_sum_pad_secondprefix) + (fs_r_pfc_sum_pad_secondprefix_body_steps))) /\\ ((((exists fs_h_pfc_sum_pad_secondprefix_body_steps_successor. fs_h_pfc_sum_pad_secondprefix_body_steps_successor + S (fs_s_pfc_sum_pad_secondprefix_body_steps) = S ((S (S fs_i_pfc_sum_pad_secondprefix_body_steps)) * fs_v_pfc_sum_pad_secondprefix)) /\\ exists fs_q_pfc_sum_pad_secondprefix_body_steps_successor. fs_u_pfc_sum_pad_secondprefix = fs_q_pfc_sum_pad_secondprefix_body_steps_successor * S ((S (S fs_i_pfc_sum_pad_secondprefix_body_steps)) * fs_v_pfc_sum_pad_secondprefix) + (fs_s_pfc_sum_pad_secondprefix_body_steps))) /\\ fs_s_pfc_sum_pad_secondprefix_body_steps = fs_r_pfc_sum_pad_secondprefix_body_steps + fs_a_pfc_sum_pad_secondprefix_body_steps)))))) /\\ (((m)=u+a)))))", "specialize beta_sum_succ_decompose (B)", "specialize beta_sum_succ_decompose (C)", "specialize beta_sum_succ_decompose (t+L)", "specialize beta_sum_succ_decompose (m)", "apply beta_sum_succ_decompose", "exact ht", "cases hsecond", "cases hsecond_witness", "cases hsecond_witness_witness", "cases hsecond_witness_witness_right", "have hprefix : x3=x1", "specialize IH (x1)", "specialize IH (x3)", "apply IH", "exact hp", "exact hfirst_witness_witness_right_left", "exact hsecond_witness_witness_right_left", "have hentry : x2=x", "specialize beta_at_unique (B)", "specialize beta_at_unique (C)", "specialize beta_at_unique (t+L)", "specialize beta_at_unique (x2)", "specialize beta_at_unique (x)", "apply beta_at_unique", "exact hsecond_witness_witness_left", "specialize hpad_right (L)", "specialize hpad_right (x)", "apply hpad_right", "specialize le_refl (S L)", "apply le_refl", "exact hfirst_witness_witness_left", "trans x3+x2", "exact hsecond_witness_witness_right_right", "trans x1+x", "congr", "exact hprefix", "exact hentry", "symm", "exact hfirst_witness_witness_right_right" ], "script_sha256": "7c16b876b112017c50606a127fa6ec1d2f914c043f0fefa14c4984e2d4de12d9", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_convolution_padding_candidate.py", "sha256": "2d874ecfb35a5db0aecdeb07b549464efebad9072c363113aa5a0a977845d007" }, "statement": "forall b c B C t L n m. (((forall pfp_repeat_index_sum_pad_datazeros. (exists pfa_gap_sum_pad_datazerosindex. pfa_gap_sum_pad_datazerosindex + S (pfp_repeat_index_sum_pad_datazeros) = (t)) -> (((exists ff_h_pfp_sum_pad_datazerosentry. ff_h_pfp_sum_pad_datazerosentry + S (0) = S ((S (pfp_repeat_index_sum_pad_datazeros)) * C)) /\\ exists ff_q_pfp_sum_pad_datazerosentry. B = ff_q_pfp_sum_pad_datazerosentry * S ((S (pfp_repeat_index_sum_pad_datazeros)) * C) + (0)))) /\\ ((forall pfrep_index_sum_pad_data pfrep_value_sum_pad_data. (exists pfa_gap_sum_pad_databound. pfa_gap_sum_pad_databound + S (pfrep_index_sum_pad_data) = (L)) -> (((exists ff_h_pfp_sum_pad_datainput. ff_h_pfp_sum_pad_datainput + S (pfrep_value_sum_pad_data) = S ((S (pfrep_index_sum_pad_data)) * c)) /\\ exists ff_q_pfp_sum_pad_datainput. b = ff_q_pfp_sum_pad_datainput * S ((S (pfrep_index_sum_pad_data)) * c) + (pfrep_value_sum_pad_data))) -> (((exists ff_h_pfp_sum_pad_dataoutput. ff_h_pfp_sum_pad_dataoutput + S (pfrep_value_sum_pad_data) = S ((S ((t)+pfrep_index_sum_pad_data)) * C)) /\\ exists ff_q_pfp_sum_pad_dataoutput. B = ff_q_pfp_sum_pad_dataoutput * S ((S ((t)+pfrep_index_sum_pad_data)) * C) + (pfrep_value_sum_pad_data))))))) -> (exists fs_u_pfc_sum_pad_original fs_v_pfc_sum_pad_original. ((((exists fs_h_pfc_sum_pad_original_body_start. fs_h_pfc_sum_pad_original_body_start + S (0) = S ((S (0)) * fs_v_pfc_sum_pad_original)) /\\ exists fs_q_pfc_sum_pad_original_body_start. fs_u_pfc_sum_pad_original = fs_q_pfc_sum_pad_original_body_start * S ((S (0)) * fs_v_pfc_sum_pad_original) + (0))) /\\ ((((exists fs_h_pfc_sum_pad_original_body_terminal. fs_h_pfc_sum_pad_original_body_terminal + S (n) = S ((S (L)) * fs_v_pfc_sum_pad_original)) /\\ exists fs_q_pfc_sum_pad_original_body_terminal. fs_u_pfc_sum_pad_original = fs_q_pfc_sum_pad_original_body_terminal * S ((S (L)) * fs_v_pfc_sum_pad_original) + (n))) /\\ forall fs_i_pfc_sum_pad_original_body_steps. (exists fs_lt_pfc_sum_pad_original_body_steps_bound. fs_lt_pfc_sum_pad_original_body_steps_bound + S fs_i_pfc_sum_pad_original_body_steps = L) -> exists fs_a_pfc_sum_pad_original_body_steps fs_r_pfc_sum_pad_original_body_steps fs_s_pfc_sum_pad_original_body_steps. ((((exists fs_h_pfc_sum_pad_original_body_steps_summand. fs_h_pfc_sum_pad_original_body_steps_summand + S (fs_a_pfc_sum_pad_original_body_steps) = S ((S (fs_i_pfc_sum_pad_original_body_steps)) * c)) /\\ exists fs_q_pfc_sum_pad_original_body_steps_summand. b = fs_q_pfc_sum_pad_original_body_steps_summand * S ((S (fs_i_pfc_sum_pad_original_body_steps)) * c) + (fs_a_pfc_sum_pad_original_body_steps))) /\\ ((((exists fs_h_pfc_sum_pad_original_body_steps_partial. fs_h_pfc_sum_pad_original_body_steps_partial + S (fs_r_pfc_sum_pad_original_body_steps) = S ((S (fs_i_pfc_sum_pad_original_body_steps)) * fs_v_pfc_sum_pad_original)) /\\ exists fs_q_pfc_sum_pad_original_body_steps_partial. fs_u_pfc_sum_pad_original = fs_q_pfc_sum_pad_original_body_steps_partial * S ((S (fs_i_pfc_sum_pad_original_body_steps)) * fs_v_pfc_sum_pad_original) + (fs_r_pfc_sum_pad_original_body_steps))) /\\ ((((exists fs_h_pfc_sum_pad_original_body_steps_successor. fs_h_pfc_sum_pad_original_body_steps_successor + S (fs_s_pfc_sum_pad_original_body_steps) = S ((S (S fs_i_pfc_sum_pad_original_body_steps)) * fs_v_pfc_sum_pad_original)) /\\ exists fs_q_pfc_sum_pad_original_body_steps_successor. fs_u_pfc_sum_pad_original = fs_q_pfc_sum_pad_original_body_steps_successor * S ((S (S fs_i_pfc_sum_pad_original_body_steps)) * fs_v_pfc_sum_pad_original) + (fs_s_pfc_sum_pad_original_body_steps))) /\\ fs_s_pfc_sum_pad_original_body_steps = fs_r_pfc_sum_pad_original_body_steps + fs_a_pfc_sum_pad_original_body_steps)))))) -> (exists fs_u_pfc_sum_pad_actual fs_v_pfc_sum_pad_actual. ((((exists fs_h_pfc_sum_pad_actual_body_start. fs_h_pfc_sum_pad_actual_body_start + S (0) = S ((S (0)) * fs_v_pfc_sum_pad_actual)) /\\ exists fs_q_pfc_sum_pad_actual_body_start. fs_u_pfc_sum_pad_actual = fs_q_pfc_sum_pad_actual_body_start * S ((S (0)) * fs_v_pfc_sum_pad_actual) + (0))) /\\ ((((exists fs_h_pfc_sum_pad_actual_body_terminal. fs_h_pfc_sum_pad_actual_body_terminal + S (m) = S ((S (t+L)) * fs_v_pfc_sum_pad_actual)) /\\ exists fs_q_pfc_sum_pad_actual_body_terminal. fs_u_pfc_sum_pad_actual = fs_q_pfc_sum_pad_actual_body_terminal * S ((S (t+L)) * fs_v_pfc_sum_pad_actual) + (m))) /\\ forall fs_i_pfc_sum_pad_actual_body_steps. (exists fs_lt_pfc_sum_pad_actual_body_steps_bound. fs_lt_pfc_sum_pad_actual_body_steps_bound + S fs_i_pfc_sum_pad_actual_body_steps = t+L) -> exists fs_a_pfc_sum_pad_actual_body_steps fs_r_pfc_sum_pad_actual_body_steps fs_s_pfc_sum_pad_actual_body_steps. ((((exists fs_h_pfc_sum_pad_actual_body_steps_summand. fs_h_pfc_sum_pad_actual_body_steps_summand + S (fs_a_pfc_sum_pad_actual_body_steps) = S ((S (fs_i_pfc_sum_pad_actual_body_steps)) * C)) /\\ exists fs_q_pfc_sum_pad_actual_body_steps_summand. B = fs_q_pfc_sum_pad_actual_body_steps_summand * S ((S (fs_i_pfc_sum_pad_actual_body_steps)) * C) + (fs_a_pfc_sum_pad_actual_body_steps))) /\\ ((((exists fs_h_pfc_sum_pad_actual_body_steps_partial. fs_h_pfc_sum_pad_actual_body_steps_partial + S (fs_r_pfc_sum_pad_actual_body_steps) = S ((S (fs_i_pfc_sum_pad_actual_body_steps)) * fs_v_pfc_sum_pad_actual)) /\\ exists fs_q_pfc_sum_pad_actual_body_steps_partial. fs_u_pfc_sum_pad_actual = fs_q_pfc_sum_pad_actual_body_steps_partial * S ((S (fs_i_pfc_sum_pad_actual_body_steps)) * fs_v_pfc_sum_pad_actual) + (fs_r_pfc_sum_pad_actual_body_steps))) /\\ ((((exists fs_h_pfc_sum_pad_actual_body_steps_successor. fs_h_pfc_sum_pad_actual_body_steps_successor + S (fs_s_pfc_sum_pad_actual_body_steps) = S ((S (S fs_i_pfc_sum_pad_actual_body_steps)) * fs_v_pfc_sum_pad_actual)) /\\ exists fs_q_pfc_sum_pad_actual_body_steps_successor. fs_u_pfc_sum_pad_actual = fs_q_pfc_sum_pad_actual_body_steps_successor * S ((S (S fs_i_pfc_sum_pad_actual_body_steps)) * fs_v_pfc_sum_pad_actual) + (fs_s_pfc_sum_pad_actual_body_steps))) /\\ fs_s_pfc_sum_pad_actual_body_steps = fs_r_pfc_sum_pad_actual_body_steps + fs_a_pfc_sum_pad_actual_body_steps)))))) -> (m=n)", "statement_sha256": "b9aafc6855dabfd96d024c6f57547baecc7f36d8cd95b883daa740d4e63a33ee", "summary": "Two actual natural sum traces have equal totals when one term prefix is the genuine leading-zero padding of the other.", "summary_sha256": "d04104178c1030bdf34969818b4ab80adcce45cbab5c7165a12906d0d216361a" }, { "alpha_v33_frontier_enrollment": { "body_receipt_sha256": "7cb394920c0f6d04931535825caaefd9588aa36f272a8d89a0a437767a1fee91", "bundle_campaign": "polynomial-euclidean-division", "bundle_node_id": 349, "bundle_sha256": "6ae667d8518e4dbe722bb08ad1b08715a0d282c2893e533c8133d770fe861dcf", "campaign": "polynomial-euclidean-division", "first_enrolled_version": "v33", "parent_catalog_sha256": "41b9f387d88a5a4f0fe5ee2bd5578f37a27a4657b0a80f1a1a2cb5109f69a623", "rfc_sha256": 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"intro hz", "intro hs", "intro ht", "have hlength : L+0=L", "simp", "rewrite hlength at ht", "rewrite hlength at ht", "rewrite hlength at ht", "specialize beta_sum_functional (B)", "specialize beta_sum_functional (C)", "specialize beta_sum_functional (L)", "specialize beta_sum_functional (m)", "specialize beta_sum_functional (n)", "apply beta_sum_functional", "exact ht", "specialize beta_sum_transport_prefix (b)", "specialize beta_sum_transport_prefix (c)", "specialize beta_sum_transport_prefix (B)", "specialize beta_sum_transport_prefix (C)", "specialize beta_sum_transport_prefix (L)", "specialize beta_sum_transport_prefix (n)", "apply beta_sum_transport_prefix", "exact hs", "exact he", "intro n", "intro m", "intro he", "intro hz", "intro hs", "intro ht", "have hlength : L+S t=S (L+t)", "simp", "rewrite hlength at ht", "rewrite hlength at ht", "rewrite hlength at ht", "have hd : exists a u. ((((exists ff_h_pfp_sum_tail_decompositionentry. ff_h_pfp_sum_tail_decompositionentry + S (a) = S ((S (L+t)) * C)) /\\ exists ff_q_pfp_sum_tail_decompositionentry. B = ff_q_pfp_sum_tail_decompositionentry * S ((S (L+t)) * C) + (a))) /\\ (((exists fs_u_pfc_sum_tail_decompositionprefix fs_v_pfc_sum_tail_decompositionprefix. ((((exists fs_h_pfc_sum_tail_decompositionprefix_body_start. fs_h_pfc_sum_tail_decompositionprefix_body_start + S (0) = S ((S (0)) * fs_v_pfc_sum_tail_decompositionprefix)) /\\ exists fs_q_pfc_sum_tail_decompositionprefix_body_start. fs_u_pfc_sum_tail_decompositionprefix = fs_q_pfc_sum_tail_decompositionprefix_body_start * S ((S (0)) * fs_v_pfc_sum_tail_decompositionprefix) + (0))) /\\ ((((exists fs_h_pfc_sum_tail_decompositionprefix_body_terminal. fs_h_pfc_sum_tail_decompositionprefix_body_terminal + S (u) = S ((S (L+t)) * fs_v_pfc_sum_tail_decompositionprefix)) /\\ exists fs_q_pfc_sum_tail_decompositionprefix_body_terminal. fs_u_pfc_sum_tail_decompositionprefix = fs_q_pfc_sum_tail_decompositionprefix_body_terminal * S ((S (L+t)) * fs_v_pfc_sum_tail_decompositionprefix) + (u))) /\\ forall fs_i_pfc_sum_tail_decompositionprefix_body_steps. (exists fs_lt_pfc_sum_tail_decompositionprefix_body_steps_bound. fs_lt_pfc_sum_tail_decompositionprefix_body_steps_bound + S fs_i_pfc_sum_tail_decompositionprefix_body_steps = L+t) -> exists fs_a_pfc_sum_tail_decompositionprefix_body_steps fs_r_pfc_sum_tail_decompositionprefix_body_steps fs_s_pfc_sum_tail_decompositionprefix_body_steps. ((((exists fs_h_pfc_sum_tail_decompositionprefix_body_steps_summand. fs_h_pfc_sum_tail_decompositionprefix_body_steps_summand + S (fs_a_pfc_sum_tail_decompositionprefix_body_steps) = S ((S (fs_i_pfc_sum_tail_decompositionprefix_body_steps)) * C)) /\\ exists fs_q_pfc_sum_tail_decompositionprefix_body_steps_summand. B = fs_q_pfc_sum_tail_decompositionprefix_body_steps_summand * S ((S (fs_i_pfc_sum_tail_decompositionprefix_body_steps)) * C) + (fs_a_pfc_sum_tail_decompositionprefix_body_steps))) /\\ ((((exists fs_h_pfc_sum_tail_decompositionprefix_body_steps_partial. fs_h_pfc_sum_tail_decompositionprefix_body_steps_partial + S (fs_r_pfc_sum_tail_decompositionprefix_body_steps) = S ((S (fs_i_pfc_sum_tail_decompositionprefix_body_steps)) * fs_v_pfc_sum_tail_decompositionprefix)) /\\ exists fs_q_pfc_sum_tail_decompositionprefix_body_steps_partial. fs_u_pfc_sum_tail_decompositionprefix = fs_q_pfc_sum_tail_decompositionprefix_body_steps_partial * S ((S (fs_i_pfc_sum_tail_decompositionprefix_body_steps)) * fs_v_pfc_sum_tail_decompositionprefix) + (fs_r_pfc_sum_tail_decompositionprefix_body_steps))) /\\ ((((exists fs_h_pfc_sum_tail_decompositionprefix_body_steps_successor. fs_h_pfc_sum_tail_decompositionprefix_body_steps_successor + S (fs_s_pfc_sum_tail_decompositionprefix_body_steps) = S ((S (S fs_i_pfc_sum_tail_decompositionprefix_body_steps)) * fs_v_pfc_sum_tail_decompositionprefix)) /\\ exists fs_q_pfc_sum_tail_decompositionprefix_body_steps_successor. fs_u_pfc_sum_tail_decompositionprefix = fs_q_pfc_sum_tail_decompositionprefix_body_steps_successor * S ((S (S fs_i_pfc_sum_tail_decompositionprefix_body_steps)) * fs_v_pfc_sum_tail_decompositionprefix) + (fs_s_pfc_sum_tail_decompositionprefix_body_steps))) /\\ fs_s_pfc_sum_tail_decompositionprefix_body_steps = fs_r_pfc_sum_tail_decompositionprefix_body_steps + fs_a_pfc_sum_tail_decompositionprefix_body_steps)))))) /\\ (((m)=u+a)))))", "specialize beta_sum_succ_decompose (B)", "specialize beta_sum_succ_decompose (C)", "specialize beta_sum_succ_decompose (L+t)", "specialize beta_sum_succ_decompose (m)", "apply beta_sum_succ_decompose", "exact ht", "cases hd", "cases hd_witness", "cases hd_witness_witness", "cases hd_witness_witness_right", "have hprefix : x1=n", "specialize IH (n)", "specialize IH (x1)", "apply IH", "exact he", "intro i", "intro hi", "specialize hz (i)", "apply hz", "specialize le_succ (S i)", "specialize le_succ (t)", "apply le_succ", "exact hi", "exact hs", "exact hd_witness_witness_right_left", "have hzero : x=0", "specialize beta_at_unique (B)", "specialize beta_at_unique (C)", "specialize beta_at_unique (L+t)", "specialize beta_at_unique (x)", "specialize beta_at_unique (0)", "apply beta_at_unique", "exact hd_witness_witness_left", "specialize hz (t)", "apply hz", "specialize le_refl (S t)", "apply le_refl", "trans x1+x", "exact hd_witness_witness_right_right", "trans x1", "rewrite hzero", "simp", "exact hprefix" ], "script_sha256": "49c5df7a593fa17225dbf1521ea4f9f5eb5ba0fe89cec7c02d7bccb3698dcb76", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_convolution_padding_candidate.py", "sha256": "2d874ecfb35a5db0aecdeb07b549464efebad9072c363113aa5a0a977845d007" }, "statement": "forall b c B C L t n m. (forall mdr_i_pfp_sum_tail_equal mdr_a_pfp_sum_tail_equal. (exists mdr_gap_pfp_sum_tail_equalb. mdr_gap_pfp_sum_tail_equalb + S (mdr_i_pfp_sum_tail_equal) = (L)) -> (((exists ff_h_mdr_pfp_sum_tail_equalo. ff_h_mdr_pfp_sum_tail_equalo + S (mdr_a_pfp_sum_tail_equal) = S ((S (mdr_i_pfp_sum_tail_equal)) * c)) /\\ exists ff_q_mdr_pfp_sum_tail_equalo. b = ff_q_mdr_pfp_sum_tail_equalo * S ((S (mdr_i_pfp_sum_tail_equal)) * c) + (mdr_a_pfp_sum_tail_equal))) -> (((exists ff_h_mdr_pfp_sum_tail_equaln. ff_h_mdr_pfp_sum_tail_equaln + S (mdr_a_pfp_sum_tail_equal) = S ((S (mdr_i_pfp_sum_tail_equal)) * C)) /\\ exists ff_q_mdr_pfp_sum_tail_equaln. B = ff_q_mdr_pfp_sum_tail_equaln * S ((S (mdr_i_pfp_sum_tail_equal)) * C) + (mdr_a_pfp_sum_tail_equal)))) -> (forall pfpad_tail_index_sum_tail_zero. (exists pfa_gap_sum_tail_zerobound. pfa_gap_sum_tail_zerobound + S (pfpad_tail_index_sum_tail_zero) = (t)) -> (((exists ff_h_pfp_sum_tail_zerozero. ff_h_pfp_sum_tail_zerozero + S (0) = S ((S ((L)+pfpad_tail_index_sum_tail_zero)) * C)) /\\ exists ff_q_pfp_sum_tail_zerozero. B = ff_q_pfp_sum_tail_zerozero * S ((S ((L)+pfpad_tail_index_sum_tail_zero)) * C) + (0)))) -> (exists fs_u_pfc_sum_tail_original fs_v_pfc_sum_tail_original. ((((exists fs_h_pfc_sum_tail_original_body_start. fs_h_pfc_sum_tail_original_body_start + S (0) = S ((S (0)) * fs_v_pfc_sum_tail_original)) /\\ exists fs_q_pfc_sum_tail_original_body_start. fs_u_pfc_sum_tail_original = fs_q_pfc_sum_tail_original_body_start * S ((S (0)) * fs_v_pfc_sum_tail_original) + (0))) /\\ ((((exists fs_h_pfc_sum_tail_original_body_terminal. fs_h_pfc_sum_tail_original_body_terminal + S (n) = S ((S (L)) * fs_v_pfc_sum_tail_original)) /\\ exists fs_q_pfc_sum_tail_original_body_terminal. fs_u_pfc_sum_tail_original = fs_q_pfc_sum_tail_original_body_terminal * S ((S (L)) * fs_v_pfc_sum_tail_original) + (n))) /\\ forall fs_i_pfc_sum_tail_original_body_steps. (exists fs_lt_pfc_sum_tail_original_body_steps_bound. fs_lt_pfc_sum_tail_original_body_steps_bound + S fs_i_pfc_sum_tail_original_body_steps = L) -> exists fs_a_pfc_sum_tail_original_body_steps fs_r_pfc_sum_tail_original_body_steps fs_s_pfc_sum_tail_original_body_steps. ((((exists fs_h_pfc_sum_tail_original_body_steps_summand. fs_h_pfc_sum_tail_original_body_steps_summand + S (fs_a_pfc_sum_tail_original_body_steps) = S ((S (fs_i_pfc_sum_tail_original_body_steps)) * c)) /\\ exists fs_q_pfc_sum_tail_original_body_steps_summand. b = fs_q_pfc_sum_tail_original_body_steps_summand * S ((S (fs_i_pfc_sum_tail_original_body_steps)) * c) + (fs_a_pfc_sum_tail_original_body_steps))) /\\ ((((exists fs_h_pfc_sum_tail_original_body_steps_partial. fs_h_pfc_sum_tail_original_body_steps_partial + S (fs_r_pfc_sum_tail_original_body_steps) = S ((S (fs_i_pfc_sum_tail_original_body_steps)) * fs_v_pfc_sum_tail_original)) /\\ exists fs_q_pfc_sum_tail_original_body_steps_partial. fs_u_pfc_sum_tail_original = fs_q_pfc_sum_tail_original_body_steps_partial * S ((S (fs_i_pfc_sum_tail_original_body_steps)) * fs_v_pfc_sum_tail_original) + (fs_r_pfc_sum_tail_original_body_steps))) /\\ ((((exists fs_h_pfc_sum_tail_original_body_steps_successor. fs_h_pfc_sum_tail_original_body_steps_successor + S (fs_s_pfc_sum_tail_original_body_steps) = S ((S (S fs_i_pfc_sum_tail_original_body_steps)) * fs_v_pfc_sum_tail_original)) /\\ exists fs_q_pfc_sum_tail_original_body_steps_successor. fs_u_pfc_sum_tail_original = fs_q_pfc_sum_tail_original_body_steps_successor * S ((S (S fs_i_pfc_sum_tail_original_body_steps)) * fs_v_pfc_sum_tail_original) + (fs_s_pfc_sum_tail_original_body_steps))) /\\ fs_s_pfc_sum_tail_original_body_steps = fs_r_pfc_sum_tail_original_body_steps + fs_a_pfc_sum_tail_original_body_steps)))))) -> (exists fs_u_pfc_sum_tail_actual fs_v_pfc_sum_tail_actual. ((((exists fs_h_pfc_sum_tail_actual_body_start. fs_h_pfc_sum_tail_actual_body_start + S (0) = S ((S (0)) * fs_v_pfc_sum_tail_actual)) /\\ exists fs_q_pfc_sum_tail_actual_body_start. fs_u_pfc_sum_tail_actual = fs_q_pfc_sum_tail_actual_body_start * S ((S (0)) * fs_v_pfc_sum_tail_actual) + (0))) /\\ ((((exists fs_h_pfc_sum_tail_actual_body_terminal. fs_h_pfc_sum_tail_actual_body_terminal + S (m) = S ((S (L+t)) * fs_v_pfc_sum_tail_actual)) /\\ exists fs_q_pfc_sum_tail_actual_body_terminal. fs_u_pfc_sum_tail_actual = fs_q_pfc_sum_tail_actual_body_terminal * S ((S (L+t)) * fs_v_pfc_sum_tail_actual) + (m))) /\\ forall fs_i_pfc_sum_tail_actual_body_steps. (exists fs_lt_pfc_sum_tail_actual_body_steps_bound. fs_lt_pfc_sum_tail_actual_body_steps_bound + S fs_i_pfc_sum_tail_actual_body_steps = L+t) -> exists fs_a_pfc_sum_tail_actual_body_steps fs_r_pfc_sum_tail_actual_body_steps fs_s_pfc_sum_tail_actual_body_steps. ((((exists fs_h_pfc_sum_tail_actual_body_steps_summand. fs_h_pfc_sum_tail_actual_body_steps_summand + S (fs_a_pfc_sum_tail_actual_body_steps) = S ((S (fs_i_pfc_sum_tail_actual_body_steps)) * C)) /\\ exists fs_q_pfc_sum_tail_actual_body_steps_summand. 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fs_i_pfc_sum_tail_actual_body_steps)) * fs_v_pfc_sum_tail_actual) + (fs_s_pfc_sum_tail_actual_body_steps))) /\\ fs_s_pfc_sum_tail_actual_body_steps = fs_r_pfc_sum_tail_actual_body_steps + fs_a_pfc_sum_tail_actual_body_steps)))))) -> (m=n)", "statement_sha256": "7c18a21ce0406dbd76d857906b2844a0287b96e3933e685a44484ed5d096a074", "summary": "Appending a genuinely all-zero tail to an independently recoded natural summand prefix preserves its actual sum.", "summary_sha256": "b97df0a906dee418341e4fbd80661c01bef26ebae79919209a6041056b62d023" }, { "alpha_v33_frontier_enrollment": { "body_receipt_sha256": "3085ee8cb9db2ab1410ff93b157d1d819f915d1ed50aa91b028a77d5bf8e0f8f", "bundle_campaign": "polynomial-euclidean-division", "bundle_node_id": 350, "bundle_sha256": "6ae667d8518e4dbe722bb08ad1b08715a0d282c2893e533c8133d770fe861dcf", "campaign": "polynomial-euclidean-division", "first_enrolled_version": "v33", "parent_catalog_sha256": 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(((forall pfp_repeat_index_term_padding_leftzeros. (exists pfa_gap_term_padding_leftzerosindex. pfa_gap_term_padding_leftzerosindex + S (pfp_repeat_index_term_padding_leftzeros) = (t)) -> (((exists ff_h_pfp_term_padding_leftzerosentry. ff_h_pfp_term_padding_leftzerosentry + S (0) = S ((S (pfp_repeat_index_term_padding_leftzeros)) * AC)) /\\ exists ff_q_pfp_term_padding_leftzerosentry. AB = ff_q_pfp_term_padding_leftzerosentry * S ((S (pfp_repeat_index_term_padding_leftzeros)) * AC) + (0)))) /\\ ((forall pfrep_index_term_padding_left pfrep_value_term_padding_left. (exists pfa_gap_term_padding_leftbound. pfa_gap_term_padding_leftbound + S (pfrep_index_term_padding_left) = (L)) -> (((exists ff_h_pfp_term_padding_leftinput. ff_h_pfp_term_padding_leftinput + S (pfrep_value_term_padding_left) = S ((S (pfrep_index_term_padding_left)) * ac)) /\\ exists ff_q_pfp_term_padding_leftinput. ab = ff_q_pfp_term_padding_leftinput * S ((S (pfrep_index_term_padding_left)) * ac) + (pfrep_value_term_padding_left))) -> (((exists ff_h_pfp_term_padding_leftoutput. ff_h_pfp_term_padding_leftoutput + S (pfrep_value_term_padding_left) = S ((S ((t)+pfrep_index_term_padding_left)) * AC)) /\\ exists ff_q_pfp_term_padding_leftoutput. AB = ff_q_pfp_term_padding_leftoutput * S ((S ((t)+pfrep_index_term_padding_left)) * AC) + (pfrep_value_term_padding_left))))))) -> (exists pfc_complement_term_original_left pfc_left_term_original_left pfc_right_term_original_left. (((j)+pfc_complement_term_original_left=(i)) /\\ ((((((exists pfa_gap_term_original_leftleftinside. pfa_gap_term_original_leftleftinside + S (j) = (L)) /\\ ((((exists ff_h_pfp_term_original_leftleftentry. ff_h_pfp_term_original_leftleftentry + S (pfc_left_term_original_left) = S ((S (j)) * ac)) /\\ exists ff_q_pfp_term_original_leftleftentry. ab = ff_q_pfp_term_original_leftleftentry * S ((S (j)) * ac) + (pfc_left_term_original_left)))))) \\/ (((exists pfc_gap_term_original_leftleftoutside. pfc_gap_term_original_leftleftoutside+(L)=(j)) /\\ (((pfc_left_term_original_left)=0))))) /\\ ((((((exists pfa_gap_term_original_leftrightinside. pfa_gap_term_original_leftrightinside + S (pfc_complement_term_original_left) = (M)) /\\ ((((exists ff_h_pfp_term_original_leftrightentry. ff_h_pfp_term_original_leftrightentry + S (pfc_right_term_original_left) = S ((S (pfc_complement_term_original_left)) * bc)) /\\ exists ff_q_pfp_term_original_leftrightentry. bb = ff_q_pfp_term_original_leftrightentry * S ((S (pfc_complement_term_original_left)) * bc) + (pfc_right_term_original_left)))))) \\/ (((exists pfc_gap_term_original_leftrightoutside. pfc_gap_term_original_leftrightoutside+(M)=(pfc_complement_term_original_left)) /\\ (((pfc_right_term_original_left)=0))))) /\\ (((z)=pfc_left_term_original_left*pfc_right_term_original_left)))))))) -> (exists pfc_complement_term_shifted_left pfc_left_term_shifted_left pfc_right_term_shifted_left. (((t+j)+pfc_complement_term_shifted_left=(t+i)) /\\ ((((((exists pfa_gap_term_shifted_leftleftinside. pfa_gap_term_shifted_leftleftinside + S (t+j) = (t+L)) /\\ ((((exists ff_h_pfp_term_shifted_leftleftentry. ff_h_pfp_term_shifted_leftleftentry + S (pfc_left_term_shifted_left) = S ((S (t+j)) * AC)) /\\ exists ff_q_pfp_term_shifted_leftleftentry. AB = ff_q_pfp_term_shifted_leftleftentry * S ((S (t+j)) * AC) + (pfc_left_term_shifted_left)))))) \\/ (((exists pfc_gap_term_shifted_leftleftoutside. pfc_gap_term_shifted_leftleftoutside+(t+L)=(t+j)) /\\ (((pfc_left_term_shifted_left)=0))))) /\\ ((((((exists pfa_gap_term_shifted_leftrightinside. pfa_gap_term_shifted_leftrightinside + S (pfc_complement_term_shifted_left) = (M)) /\\ ((((exists ff_h_pfp_term_shifted_leftrightentry. ff_h_pfp_term_shifted_leftrightentry + S (pfc_right_term_shifted_left) = S ((S (pfc_complement_term_shifted_left)) * bc)) /\\ exists ff_q_pfp_term_shifted_leftrightentry. bb = ff_q_pfp_term_shifted_leftrightentry * S ((S (pfc_complement_term_shifted_left)) * bc) + (pfc_right_term_shifted_left)))))) \\/ (((exists pfc_gap_term_shifted_leftrightoutside. pfc_gap_term_shifted_leftrightoutside+(M)=(pfc_complement_term_shifted_left)) /\\ (((pfc_right_term_shifted_left)=0))))) /\\ (((z)=pfc_left_term_shifted_left*pfc_right_term_shifted_left))))))))", 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(((forall pfp_repeat_index_term_padding_rightzeros. (exists pfa_gap_term_padding_rightzerosindex. pfa_gap_term_padding_rightzerosindex + S (pfp_repeat_index_term_padding_rightzeros) = (t)) -> (((exists ff_h_pfp_term_padding_rightzerosentry. ff_h_pfp_term_padding_rightzerosentry + S (0) = S ((S (pfp_repeat_index_term_padding_rightzeros)) * BC)) /\\ exists ff_q_pfp_term_padding_rightzerosentry. BB = ff_q_pfp_term_padding_rightzerosentry * S ((S (pfp_repeat_index_term_padding_rightzeros)) * BC) + (0)))) /\\ ((forall pfrep_index_term_padding_right pfrep_value_term_padding_right. (exists pfa_gap_term_padding_rightbound. pfa_gap_term_padding_rightbound + S (pfrep_index_term_padding_right) = (M)) -> (((exists ff_h_pfp_term_padding_rightinput. ff_h_pfp_term_padding_rightinput + S (pfrep_value_term_padding_right) = S ((S (pfrep_index_term_padding_right)) * bc)) /\\ exists ff_q_pfp_term_padding_rightinput. bb = ff_q_pfp_term_padding_rightinput * S ((S (pfrep_index_term_padding_right)) * bc) + (pfrep_value_term_padding_right))) -> (((exists ff_h_pfp_term_padding_rightoutput. ff_h_pfp_term_padding_rightoutput + S (pfrep_value_term_padding_right) = S ((S ((t)+pfrep_index_term_padding_right)) * BC)) /\\ exists ff_q_pfp_term_padding_rightoutput. BB = ff_q_pfp_term_padding_rightoutput * S ((S ((t)+pfrep_index_term_padding_right)) * BC) + (pfrep_value_term_padding_right))))))) -> (exists pfc_complement_term_original_right pfc_left_term_original_right pfc_right_term_original_right. (((j)+pfc_complement_term_original_right=(i)) /\\ ((((((exists pfa_gap_term_original_rightleftinside. pfa_gap_term_original_rightleftinside + S (j) = (L)) /\\ ((((exists ff_h_pfp_term_original_rightleftentry. ff_h_pfp_term_original_rightleftentry + S (pfc_left_term_original_right) = S ((S (j)) * ac)) /\\ exists ff_q_pfp_term_original_rightleftentry. ab = ff_q_pfp_term_original_rightleftentry * S ((S (j)) * ac) + (pfc_left_term_original_right)))))) \\/ (((exists pfc_gap_term_original_rightleftoutside. pfc_gap_term_original_rightleftoutside+(L)=(j)) /\\ (((pfc_left_term_original_right)=0))))) /\\ ((((((exists pfa_gap_term_original_rightrightinside. pfa_gap_term_original_rightrightinside + S (pfc_complement_term_original_right) = (M)) /\\ ((((exists ff_h_pfp_term_original_rightrightentry. ff_h_pfp_term_original_rightrightentry + S (pfc_right_term_original_right) = S ((S (pfc_complement_term_original_right)) * bc)) /\\ exists ff_q_pfp_term_original_rightrightentry. bb = ff_q_pfp_term_original_rightrightentry * S ((S (pfc_complement_term_original_right)) * bc) + (pfc_right_term_original_right)))))) \\/ (((exists pfc_gap_term_original_rightrightoutside. pfc_gap_term_original_rightrightoutside+(M)=(pfc_complement_term_original_right)) /\\ (((pfc_right_term_original_right)=0))))) /\\ (((z)=pfc_left_term_original_right*pfc_right_term_original_right)))))))) -> (exists pfc_complement_term_shifted_right pfc_left_term_shifted_right pfc_right_term_shifted_right. (((j)+pfc_complement_term_shifted_right=(t+i)) /\\ ((((((exists pfa_gap_term_shifted_rightleftinside. pfa_gap_term_shifted_rightleftinside + S (j) = (L)) /\\ ((((exists ff_h_pfp_term_shifted_rightleftentry. ff_h_pfp_term_shifted_rightleftentry + S (pfc_left_term_shifted_right) = S ((S (j)) * ac)) /\\ exists ff_q_pfp_term_shifted_rightleftentry. ab = ff_q_pfp_term_shifted_rightleftentry * S ((S (j)) * ac) + (pfc_left_term_shifted_right)))))) \\/ (((exists pfc_gap_term_shifted_rightleftoutside. pfc_gap_term_shifted_rightleftoutside+(L)=(j)) /\\ (((pfc_left_term_shifted_right)=0))))) /\\ ((((((exists pfa_gap_term_shifted_rightrightinside. pfa_gap_term_shifted_rightrightinside + S (pfc_complement_term_shifted_right) = (t+M)) /\\ ((((exists ff_h_pfp_term_shifted_rightrightentry. ff_h_pfp_term_shifted_rightrightentry + S (pfc_right_term_shifted_right) = S ((S (pfc_complement_term_shifted_right)) * BC)) /\\ exists ff_q_pfp_term_shifted_rightrightentry. BB = ff_q_pfp_term_shifted_rightrightentry * S ((S (pfc_complement_term_shifted_right)) * BC) + (pfc_right_term_shifted_right)))))) \\/ (((exists pfc_gap_term_shifted_rightrightoutside. pfc_gap_term_shifted_rightrightoutside+(t+M)=(pfc_complement_term_shifted_right)) /\\ (((pfc_right_term_shifted_right)=0))))) /\\ (((z)=pfc_left_term_shifted_right*pfc_right_term_shifted_right))))))))", "statement_sha256": "aa1b0f2cd1b6e32d462465b6ae3282f1d422806c299678d6abdb34338b656465", "summary": "A genuine right factor leading-zero padding shifts the antidiagonal position while preserving each actual natural product term.", "summary_sha256": "eeb70daffa9aa156eda85cbc4fb910b1581c80d52b8fbd9208c10fa41b975bea" }, { "alpha_v33_frontier_enrollment": { "body_receipt_sha256": "f0c5c706aa2a96be05ff72f69896546a95ad22345fd3e58a42db83e3b4074034", "bundle_campaign": "polynomial-euclidean-division", "bundle_node_id": 352, "bundle_sha256": "6ae667d8518e4dbe722bb08ad1b08715a0d282c2893e533c8133d770fe861dcf", 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(((forall pfp_repeat_index_zero_term_pad_leftzeros. (exists pfa_gap_zero_term_pad_leftzerosindex. pfa_gap_zero_term_pad_leftzerosindex + S (pfp_repeat_index_zero_term_pad_leftzeros) = (t)) -> (((exists ff_h_pfp_zero_term_pad_leftzerosentry. ff_h_pfp_zero_term_pad_leftzerosentry + S (0) = S ((S (pfp_repeat_index_zero_term_pad_leftzeros)) * AC)) /\\ exists ff_q_pfp_zero_term_pad_leftzerosentry. AB = ff_q_pfp_zero_term_pad_leftzerosentry * S ((S (pfp_repeat_index_zero_term_pad_leftzeros)) * AC) + (0)))) /\\ ((forall pfrep_index_zero_term_pad_left pfrep_value_zero_term_pad_left. (exists pfa_gap_zero_term_pad_leftbound. pfa_gap_zero_term_pad_leftbound + S (pfrep_index_zero_term_pad_left) = (L)) -> (((exists ff_h_pfp_zero_term_pad_leftinput. ff_h_pfp_zero_term_pad_leftinput + S (pfrep_value_zero_term_pad_left) = S ((S (pfrep_index_zero_term_pad_left)) * ac)) /\\ exists ff_q_pfp_zero_term_pad_leftinput. ab = ff_q_pfp_zero_term_pad_leftinput * S ((S (pfrep_index_zero_term_pad_left)) * ac) + (pfrep_value_zero_term_pad_left))) -> (((exists ff_h_pfp_zero_term_pad_leftoutput. ff_h_pfp_zero_term_pad_leftoutput + S (pfrep_value_zero_term_pad_left) = S ((S ((t)+pfrep_index_zero_term_pad_left)) * AC)) /\\ exists ff_q_pfp_zero_term_pad_leftoutput. AB = ff_q_pfp_zero_term_pad_leftoutput * S ((S ((t)+pfrep_index_zero_term_pad_left)) * AC) + (pfrep_value_zero_term_pad_left))))))) -> (exists pfa_gap_term_zero_left_bound. pfa_gap_term_zero_left_bound + S (j) = (t)) -> (exists pfc_complement_zero_term_actual_left pfc_left_zero_term_actual_left pfc_right_zero_term_actual_left. (((j)+pfc_complement_zero_term_actual_left=(i)) /\\ ((((((exists pfa_gap_zero_term_actual_leftleftinside. pfa_gap_zero_term_actual_leftleftinside + S (j) = (t+L)) /\\ ((((exists ff_h_pfp_zero_term_actual_leftleftentry. ff_h_pfp_zero_term_actual_leftleftentry + S (pfc_left_zero_term_actual_left) = S ((S (j)) * AC)) /\\ exists ff_q_pfp_zero_term_actual_leftleftentry. AB = ff_q_pfp_zero_term_actual_leftleftentry * S ((S (j)) * AC) + (pfc_left_zero_term_actual_left)))))) \\/ (((exists pfc_gap_zero_term_actual_leftleftoutside. pfc_gap_zero_term_actual_leftleftoutside+(t+L)=(j)) /\\ (((pfc_left_zero_term_actual_left)=0))))) /\\ ((((((exists pfa_gap_zero_term_actual_leftrightinside. pfa_gap_zero_term_actual_leftrightinside + S (pfc_complement_zero_term_actual_left) = (M)) /\\ ((((exists ff_h_pfp_zero_term_actual_leftrightentry. ff_h_pfp_zero_term_actual_leftrightentry + S (pfc_right_zero_term_actual_left) = S ((S (pfc_complement_zero_term_actual_left)) * bc)) /\\ exists ff_q_pfp_zero_term_actual_leftrightentry. bb = ff_q_pfp_zero_term_actual_leftrightentry * S ((S (pfc_complement_zero_term_actual_left)) * bc) + (pfc_right_zero_term_actual_left)))))) \\/ (((exists pfc_gap_zero_term_actual_leftrightoutside. pfc_gap_zero_term_actual_leftrightoutside+(M)=(pfc_complement_zero_term_actual_left)) /\\ (((pfc_right_zero_term_actual_left)=0))))) /\\ 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"statement": "forall ab ac L bb bc M BB BC t i j z. (((forall pfp_repeat_index_zero_term_pad_rightzeros. (exists pfa_gap_zero_term_pad_rightzerosindex. pfa_gap_zero_term_pad_rightzerosindex + S (pfp_repeat_index_zero_term_pad_rightzeros) = (t)) -> (((exists ff_h_pfp_zero_term_pad_rightzerosentry. ff_h_pfp_zero_term_pad_rightzerosentry + S (0) = S ((S (pfp_repeat_index_zero_term_pad_rightzeros)) * BC)) /\\ exists ff_q_pfp_zero_term_pad_rightzerosentry. BB = ff_q_pfp_zero_term_pad_rightzerosentry * S ((S (pfp_repeat_index_zero_term_pad_rightzeros)) * BC) + (0)))) /\\ ((forall pfrep_index_zero_term_pad_right pfrep_value_zero_term_pad_right. (exists pfa_gap_zero_term_pad_rightbound. pfa_gap_zero_term_pad_rightbound + S (pfrep_index_zero_term_pad_right) = (M)) -> (((exists ff_h_pfp_zero_term_pad_rightinput. ff_h_pfp_zero_term_pad_rightinput + S (pfrep_value_zero_term_pad_right) = S ((S (pfrep_index_zero_term_pad_right)) * bc)) /\\ exists ff_q_pfp_zero_term_pad_rightinput. bb = ff_q_pfp_zero_term_pad_rightinput * S ((S (pfrep_index_zero_term_pad_right)) * bc) + (pfrep_value_zero_term_pad_right))) -> (((exists ff_h_pfp_zero_term_pad_rightoutput. ff_h_pfp_zero_term_pad_rightoutput + S (pfrep_value_zero_term_pad_right) = S ((S ((t)+pfrep_index_zero_term_pad_right)) * BC)) /\\ exists ff_q_pfp_zero_term_pad_rightoutput. BB = ff_q_pfp_zero_term_pad_rightoutput * S ((S ((t)+pfrep_index_zero_term_pad_right)) * BC) + (pfrep_value_zero_term_pad_right))))))) -> (exists pfa_gap_term_zero_right_bound. pfa_gap_term_zero_right_bound + S (i) = (t+j)) -> (exists pfc_complement_zero_term_actual_right pfc_left_zero_term_actual_right pfc_right_zero_term_actual_right. (((j)+pfc_complement_zero_term_actual_right=(i)) /\\ ((((((exists pfa_gap_zero_term_actual_rightleftinside. pfa_gap_zero_term_actual_rightleftinside + S (j) = (L)) /\\ ((((exists ff_h_pfp_zero_term_actual_rightleftentry. ff_h_pfp_zero_term_actual_rightleftentry + S (pfc_left_zero_term_actual_right) = S ((S (j)) * ac)) /\\ exists ff_q_pfp_zero_term_actual_rightleftentry. ab = ff_q_pfp_zero_term_actual_rightleftentry * S ((S (j)) * ac) + (pfc_left_zero_term_actual_right)))))) \\/ (((exists pfc_gap_zero_term_actual_rightleftoutside. pfc_gap_zero_term_actual_rightleftoutside+(L)=(j)) /\\ (((pfc_left_zero_term_actual_right)=0))))) /\\ ((((((exists pfa_gap_zero_term_actual_rightrightinside. pfa_gap_zero_term_actual_rightrightinside + S (pfc_complement_zero_term_actual_right) = (t+M)) /\\ ((((exists ff_h_pfp_zero_term_actual_rightrightentry. ff_h_pfp_zero_term_actual_rightrightentry + S (pfc_right_zero_term_actual_right) = S ((S (pfc_complement_zero_term_actual_right)) * BC)) /\\ exists ff_q_pfp_zero_term_actual_rightrightentry. BB = ff_q_pfp_zero_term_actual_rightrightentry * S ((S (pfc_complement_zero_term_actual_right)) * BC) + (pfc_right_zero_term_actual_right)))))) \\/ (((exists pfc_gap_zero_term_actual_rightrightoutside. pfc_gap_zero_term_actual_rightrightoutside+(t+M)=(pfc_complement_zero_term_actual_right)) /\\ (((pfc_right_zero_term_actual_right)=0))))) /\\ (((z)=pfc_left_zero_term_actual_right*pfc_right_zero_term_actual_right)))))))) -> (z=0)", "statement_sha256": "9c213827d3a42f4b0a1d2536e2ef2339bd9382d3a2a16d648742116bd10c925c", "summary": "A genuine antidiagonal term is zero when its right factor index lies in the actual leading padding block.", "summary_sha256": "de5e13124ef653ac9a99ceadfa315cc5e4e56de6a7f9ef3f737cc6ad86306069" }, { "alpha_v33_frontier_enrollment": { "body_receipt_sha256": "2f7497efc9bee768d1628abac4938676947ffe0cb472a4a49a00488bf73a94a6", "bundle_campaign": "polynomial-euclidean-division", "bundle_node_id": 354, "bundle_sha256": 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(((j)+pfc_complement_diagonal_copy_old=(i)) /\\ ((((((exists pfa_gap_diagonal_copy_oldleftinside. pfa_gap_diagonal_copy_oldleftinside + S (j) = (L)) /\\ ((((exists ff_h_pfp_diagonal_copy_oldleftentry. ff_h_pfp_diagonal_copy_oldleftentry + S (pfc_left_diagonal_copy_old) = S ((S (j)) * ac)) /\\ exists ff_q_pfp_diagonal_copy_oldleftentry. ab = ff_q_pfp_diagonal_copy_oldleftentry * S ((S (j)) * ac) + (pfc_left_diagonal_copy_old)))))) \\/ (((exists pfc_gap_diagonal_copy_oldleftoutside. pfc_gap_diagonal_copy_oldleftoutside+(L)=(j)) /\\ (((pfc_left_diagonal_copy_old)=0))))) /\\ ((((((exists pfa_gap_diagonal_copy_oldrightinside. pfa_gap_diagonal_copy_oldrightinside + S (pfc_complement_diagonal_copy_old) = (M)) /\\ ((((exists ff_h_pfp_diagonal_copy_oldrightentry. ff_h_pfp_diagonal_copy_oldrightentry + S (pfc_right_diagonal_copy_old) = S ((S (pfc_complement_diagonal_copy_old)) * bc)) /\\ exists ff_q_pfp_diagonal_copy_oldrightentry. bb = ff_q_pfp_diagonal_copy_oldrightentry * S ((S (pfc_complement_diagonal_copy_old)) * bc) + (pfc_right_diagonal_copy_old)))))) \\/ (((exists pfc_gap_diagonal_copy_oldrightoutside. pfc_gap_diagonal_copy_oldrightoutside+(M)=(pfc_complement_diagonal_copy_old)) /\\ (((pfc_right_diagonal_copy_old)=0))))) /\\ (((a)=pfc_left_diagonal_copy_old*pfc_right_diagonal_copy_old)))))))", "specialize polynomial_diagonal_prefix_entry (ab)", "specialize polynomial_diagonal_prefix_entry (ac)", "specialize polynomial_diagonal_prefix_entry (L)", "specialize polynomial_diagonal_prefix_entry (bb)", "specialize polynomial_diagonal_prefix_entry (bc)", "specialize polynomial_diagonal_prefix_entry (M)", "specialize polynomial_diagonal_prefix_entry (i)", "specialize polynomial_diagonal_prefix_entry (db)", "specialize polynomial_diagonal_prefix_entry (dc)", "specialize polynomial_diagonal_prefix_entry (S i)", "specialize polynomial_diagonal_prefix_entry (j)", "specialize polynomial_diagonal_prefix_entry (a)", "apply polynomial_diagonal_prefix_entry", "exact hold", "exact hj", "exact ha", "have hv : exists z. ((((exists ff_h_pfp_diagonal_copy_entry. ff_h_pfp_diagonal_copy_entry + S (z) = S ((S (t+j)) * ec)) /\\ exists ff_q_pfp_diagonal_copy_entry. eb = ff_q_pfp_diagonal_copy_entry * S ((S (t+j)) * ec) + (z))) /\\ ((exists pfc_complement_diagonal_copy_actual pfc_left_diagonal_copy_actual pfc_right_diagonal_copy_actual. (((t+j)+pfc_complement_diagonal_copy_actual=(t+i)) /\\ ((((((exists pfa_gap_diagonal_copy_actualleftinside. pfa_gap_diagonal_copy_actualleftinside + S (t+j) = (t+L)) /\\ ((((exists ff_h_pfp_diagonal_copy_actualleftentry. ff_h_pfp_diagonal_copy_actualleftentry + S (pfc_left_diagonal_copy_actual) = S ((S (t+j)) * AC)) /\\ exists ff_q_pfp_diagonal_copy_actualleftentry. AB = ff_q_pfp_diagonal_copy_actualleftentry * S ((S (t+j)) * AC) + (pfc_left_diagonal_copy_actual)))))) \\/ (((exists pfc_gap_diagonal_copy_actualleftoutside. pfc_gap_diagonal_copy_actualleftoutside+(t+L)=(t+j)) /\\ (((pfc_left_diagonal_copy_actual)=0))))) /\\ ((((((exists pfa_gap_diagonal_copy_actualrightinside. pfa_gap_diagonal_copy_actualrightinside + S (pfc_complement_diagonal_copy_actual) = (M)) /\\ ((((exists ff_h_pfp_diagonal_copy_actualrightentry. ff_h_pfp_diagonal_copy_actualrightentry + S (pfc_right_diagonal_copy_actual) = S ((S (pfc_complement_diagonal_copy_actual)) * bc)) /\\ exists ff_q_pfp_diagonal_copy_actualrightentry. bb = ff_q_pfp_diagonal_copy_actualrightentry * S ((S (pfc_complement_diagonal_copy_actual)) * bc) + (pfc_right_diagonal_copy_actual)))))) \\/ (((exists pfc_gap_diagonal_copy_actualrightoutside. pfc_gap_diagonal_copy_actualrightoutside+(M)=(pfc_complement_diagonal_copy_actual)) /\\ (((pfc_right_diagonal_copy_actual)=0))))) /\\ (((z)=pfc_left_diagonal_copy_actual*pfc_right_diagonal_copy_actual))))))))))", "specialize hnew (t+j)", "apply hnew", "specialize succ_le_succ (t+j)", "specialize succ_le_succ (t+i)", "apply succ_le_succ", "specialize add_le_add_left (j)", "specialize add_le_add_left (i)", "specialize add_le_add_left (t)", "apply add_le_add_left", "specialize le_of_succ_le_succ (j)", "specialize le_of_succ_le_succ (i)", "apply le_of_succ_le_succ", "exact hj", "cases hv", "cases hv_witness", "have heq : x=a", "specialize polynomial_diagonal_term_functional (AB)", "specialize polynomial_diagonal_term_functional (AC)", "specialize polynomial_diagonal_term_functional (t+L)", "specialize polynomial_diagonal_term_functional (bb)", "specialize polynomial_diagonal_term_functional (bc)", "specialize polynomial_diagonal_term_functional (M)", "specialize polynomial_diagonal_term_functional (t+i)", "specialize polynomial_diagonal_term_functional (t+j)", "specialize polynomial_diagonal_term_functional (x)", "specialize polynomial_diagonal_term_functional (a)", "apply polynomial_diagonal_term_functional", "exact hv_witness_right", "specialize polynomial_diagonal_term_left_padding_left (ab)", "specialize polynomial_diagonal_term_left_padding_left (ac)", "specialize polynomial_diagonal_term_left_padding_left (L)", "specialize polynomial_diagonal_term_left_padding_left (bb)", "specialize polynomial_diagonal_term_left_padding_left (bc)", "specialize polynomial_diagonal_term_left_padding_left (M)", "specialize polynomial_diagonal_term_left_padding_left (AB)", "specialize polynomial_diagonal_term_left_padding_left (AC)", "specialize polynomial_diagonal_term_left_padding_left (t)", "specialize polynomial_diagonal_term_left_padding_left (i)", "specialize polynomial_diagonal_term_left_padding_left (j)", "specialize polynomial_diagonal_term_left_padding_left (a)", "apply polynomial_diagonal_term_left_padding_left", "exact hpad", "exact hterm", "rewrite heq at hv_witness_left", "rewrite heq at hv_witness_left", "exact hv_witness_left" ], "script_sha256": "45ed08a7467abb1365683086f9b2beb6313be5e84bb1aa72e8ebcf928efacea2", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_convolution_padding_candidate.py", "sha256": "2d874ecfb35a5db0aecdeb07b549464efebad9072c363113aa5a0a977845d007" }, "statement": "forall ab ac L bb bc M AB AC t i db dc eb ec. (((forall pfp_repeat_index_diagonal_actual_padding_leftzeros. (exists pfa_gap_diagonal_actual_padding_leftzerosindex. pfa_gap_diagonal_actual_padding_leftzerosindex + S (pfp_repeat_index_diagonal_actual_padding_leftzeros) = (t)) -> (((exists ff_h_pfp_diagonal_actual_padding_leftzerosentry. ff_h_pfp_diagonal_actual_padding_leftzerosentry + S (0) = S ((S (pfp_repeat_index_diagonal_actual_padding_leftzeros)) * AC)) /\\ exists ff_q_pfp_diagonal_actual_padding_leftzerosentry. AB = ff_q_pfp_diagonal_actual_padding_leftzerosentry * S ((S (pfp_repeat_index_diagonal_actual_padding_leftzeros)) * AC) + (0)))) /\\ ((forall pfrep_index_diagonal_actual_padding_left pfrep_value_diagonal_actual_padding_left. (exists pfa_gap_diagonal_actual_padding_leftbound. pfa_gap_diagonal_actual_padding_leftbound + S (pfrep_index_diagonal_actual_padding_left) = (L)) -> (((exists ff_h_pfp_diagonal_actual_padding_leftinput. ff_h_pfp_diagonal_actual_padding_leftinput + S (pfrep_value_diagonal_actual_padding_left) = S ((S (pfrep_index_diagonal_actual_padding_left)) * ac)) /\\ exists ff_q_pfp_diagonal_actual_padding_leftinput. ab = ff_q_pfp_diagonal_actual_padding_leftinput * S ((S (pfrep_index_diagonal_actual_padding_left)) * ac) + (pfrep_value_diagonal_actual_padding_left))) -> (((exists ff_h_pfp_diagonal_actual_padding_leftoutput. ff_h_pfp_diagonal_actual_padding_leftoutput + S (pfrep_value_diagonal_actual_padding_left) = S ((S ((t)+pfrep_index_diagonal_actual_padding_left)) * AC)) /\\ exists ff_q_pfp_diagonal_actual_padding_leftoutput. AB = ff_q_pfp_diagonal_actual_padding_leftoutput * S ((S ((t)+pfrep_index_diagonal_actual_padding_left)) * AC) + (pfrep_value_diagonal_actual_padding_left))))))) -> (forall pfc_index_diagonal_pad_old_left. (exists pfa_gap_diagonal_pad_old_leftbound. pfa_gap_diagonal_pad_old_leftbound + S (pfc_index_diagonal_pad_old_left) = (S i)) -> exists pfc_value_diagonal_pad_old_left. ((((exists ff_h_pfp_diagonal_pad_old_leftentry. ff_h_pfp_diagonal_pad_old_leftentry + S (pfc_value_diagonal_pad_old_left) = S ((S (pfc_index_diagonal_pad_old_left)) * dc)) /\\ exists ff_q_pfp_diagonal_pad_old_leftentry. db = ff_q_pfp_diagonal_pad_old_leftentry * S ((S (pfc_index_diagonal_pad_old_left)) * dc) + (pfc_value_diagonal_pad_old_left))) /\\ ((exists pfc_complement_diagonal_pad_old_leftterm pfc_left_diagonal_pad_old_leftterm pfc_right_diagonal_pad_old_leftterm. (((pfc_index_diagonal_pad_old_left)+pfc_complement_diagonal_pad_old_leftterm=(i)) /\\ ((((((exists pfa_gap_diagonal_pad_old_lefttermleftinside. pfa_gap_diagonal_pad_old_lefttermleftinside + S (pfc_index_diagonal_pad_old_left) = (L)) /\\ ((((exists ff_h_pfp_diagonal_pad_old_lefttermleftentry. ff_h_pfp_diagonal_pad_old_lefttermleftentry + S (pfc_left_diagonal_pad_old_leftterm) = S ((S (pfc_index_diagonal_pad_old_left)) * ac)) /\\ exists ff_q_pfp_diagonal_pad_old_lefttermleftentry. ab = ff_q_pfp_diagonal_pad_old_lefttermleftentry * S ((S (pfc_index_diagonal_pad_old_left)) * ac) + (pfc_left_diagonal_pad_old_leftterm)))))) \\/ (((exists pfc_gap_diagonal_pad_old_lefttermleftoutside. pfc_gap_diagonal_pad_old_lefttermleftoutside+(L)=(pfc_index_diagonal_pad_old_left)) /\\ (((pfc_left_diagonal_pad_old_leftterm)=0))))) /\\ ((((((exists pfa_gap_diagonal_pad_old_lefttermrightinside. pfa_gap_diagonal_pad_old_lefttermrightinside + S (pfc_complement_diagonal_pad_old_leftterm) = (M)) /\\ ((((exists ff_h_pfp_diagonal_pad_old_lefttermrightentry. ff_h_pfp_diagonal_pad_old_lefttermrightentry + S (pfc_right_diagonal_pad_old_leftterm) = S ((S (pfc_complement_diagonal_pad_old_leftterm)) * bc)) /\\ exists ff_q_pfp_diagonal_pad_old_lefttermrightentry. bb = ff_q_pfp_diagonal_pad_old_lefttermrightentry * S ((S (pfc_complement_diagonal_pad_old_leftterm)) * bc) + (pfc_right_diagonal_pad_old_leftterm)))))) \\/ (((exists pfc_gap_diagonal_pad_old_lefttermrightoutside. pfc_gap_diagonal_pad_old_lefttermrightoutside+(M)=(pfc_complement_diagonal_pad_old_leftterm)) /\\ (((pfc_right_diagonal_pad_old_leftterm)=0))))) /\\ (((pfc_value_diagonal_pad_old_left)=pfc_left_diagonal_pad_old_leftterm*pfc_right_diagonal_pad_old_leftterm))))))))))) -> (forall pfc_index_diagonal_pad_new_left. (exists pfa_gap_diagonal_pad_new_leftbound. pfa_gap_diagonal_pad_new_leftbound + S (pfc_index_diagonal_pad_new_left) = (S (t+i))) -> exists pfc_value_diagonal_pad_new_left. ((((exists ff_h_pfp_diagonal_pad_new_leftentry. ff_h_pfp_diagonal_pad_new_leftentry + S (pfc_value_diagonal_pad_new_left) = S ((S (pfc_index_diagonal_pad_new_left)) * ec)) /\\ exists ff_q_pfp_diagonal_pad_new_leftentry. eb = ff_q_pfp_diagonal_pad_new_leftentry * S ((S (pfc_index_diagonal_pad_new_left)) * ec) + (pfc_value_diagonal_pad_new_left))) /\\ ((exists pfc_complement_diagonal_pad_new_leftterm pfc_left_diagonal_pad_new_leftterm pfc_right_diagonal_pad_new_leftterm. (((pfc_index_diagonal_pad_new_left)+pfc_complement_diagonal_pad_new_leftterm=(t+i)) /\\ ((((((exists pfa_gap_diagonal_pad_new_lefttermleftinside. pfa_gap_diagonal_pad_new_lefttermleftinside + S (pfc_index_diagonal_pad_new_left) = (t+L)) /\\ ((((exists ff_h_pfp_diagonal_pad_new_lefttermleftentry. ff_h_pfp_diagonal_pad_new_lefttermleftentry + S (pfc_left_diagonal_pad_new_leftterm) = S ((S (pfc_index_diagonal_pad_new_left)) * AC)) /\\ exists ff_q_pfp_diagonal_pad_new_lefttermleftentry. AB = ff_q_pfp_diagonal_pad_new_lefttermleftentry * S ((S (pfc_index_diagonal_pad_new_left)) * AC) + (pfc_left_diagonal_pad_new_leftterm)))))) \\/ (((exists pfc_gap_diagonal_pad_new_lefttermleftoutside. pfc_gap_diagonal_pad_new_lefttermleftoutside+(t+L)=(pfc_index_diagonal_pad_new_left)) /\\ (((pfc_left_diagonal_pad_new_leftterm)=0))))) /\\ ((((((exists pfa_gap_diagonal_pad_new_lefttermrightinside. pfa_gap_diagonal_pad_new_lefttermrightinside + S (pfc_complement_diagonal_pad_new_leftterm) = (M)) /\\ ((((exists ff_h_pfp_diagonal_pad_new_lefttermrightentry. ff_h_pfp_diagonal_pad_new_lefttermrightentry + S (pfc_right_diagonal_pad_new_leftterm) = S ((S (pfc_complement_diagonal_pad_new_leftterm)) * bc)) /\\ exists ff_q_pfp_diagonal_pad_new_lefttermrightentry. bb = ff_q_pfp_diagonal_pad_new_lefttermrightentry * S ((S (pfc_complement_diagonal_pad_new_leftterm)) * bc) + (pfc_right_diagonal_pad_new_leftterm)))))) \\/ (((exists pfc_gap_diagonal_pad_new_lefttermrightoutside. pfc_gap_diagonal_pad_new_lefttermrightoutside+(M)=(pfc_complement_diagonal_pad_new_leftterm)) /\\ (((pfc_right_diagonal_pad_new_leftterm)=0))))) /\\ (((pfc_value_diagonal_pad_new_left)=pfc_left_diagonal_pad_new_leftterm*pfc_right_diagonal_pad_new_leftterm))))))))))) -> (((forall pfp_repeat_index_diagonal_pad_result_leftzeros. (exists pfa_gap_diagonal_pad_result_leftzerosindex. pfa_gap_diagonal_pad_result_leftzerosindex + S (pfp_repeat_index_diagonal_pad_result_leftzeros) = (t)) -> (((exists ff_h_pfp_diagonal_pad_result_leftzerosentry. ff_h_pfp_diagonal_pad_result_leftzerosentry + S (0) = S ((S (pfp_repeat_index_diagonal_pad_result_leftzeros)) * ec)) /\\ exists ff_q_pfp_diagonal_pad_result_leftzerosentry. eb = ff_q_pfp_diagonal_pad_result_leftzerosentry * S ((S (pfp_repeat_index_diagonal_pad_result_leftzeros)) * ec) + (0)))) /\\ ((forall pfrep_index_diagonal_pad_result_left pfrep_value_diagonal_pad_result_left. (exists pfa_gap_diagonal_pad_result_leftbound. pfa_gap_diagonal_pad_result_leftbound + S (pfrep_index_diagonal_pad_result_left) = (S i)) -> (((exists ff_h_pfp_diagonal_pad_result_leftinput. ff_h_pfp_diagonal_pad_result_leftinput + S (pfrep_value_diagonal_pad_result_left) = S ((S (pfrep_index_diagonal_pad_result_left)) * dc)) /\\ exists ff_q_pfp_diagonal_pad_result_leftinput. db = ff_q_pfp_diagonal_pad_result_leftinput * S ((S (pfrep_index_diagonal_pad_result_left)) * dc) + (pfrep_value_diagonal_pad_result_left))) -> (((exists ff_h_pfp_diagonal_pad_result_leftoutput. ff_h_pfp_diagonal_pad_result_leftoutput + S (pfrep_value_diagonal_pad_result_left) = S ((S ((t)+pfrep_index_diagonal_pad_result_left)) * ec)) /\\ exists ff_q_pfp_diagonal_pad_result_leftoutput. eb = ff_q_pfp_diagonal_pad_result_leftoutput * S ((S ((t)+pfrep_index_diagonal_pad_result_left)) * ec) + (pfrep_value_diagonal_pad_result_left)))))))", "statement_sha256": 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"intro ha", "have hterm : exists pfc_complement_diagonal_copy_old pfc_left_diagonal_copy_old pfc_right_diagonal_copy_old. (((j)+pfc_complement_diagonal_copy_old=(i)) /\\ ((((((exists pfa_gap_diagonal_copy_oldleftinside. pfa_gap_diagonal_copy_oldleftinside + S (j) = (L)) /\\ ((((exists ff_h_pfp_diagonal_copy_oldleftentry. ff_h_pfp_diagonal_copy_oldleftentry + S (pfc_left_diagonal_copy_old) = S ((S (j)) * ac)) /\\ exists ff_q_pfp_diagonal_copy_oldleftentry. ab = ff_q_pfp_diagonal_copy_oldleftentry * S ((S (j)) * ac) + (pfc_left_diagonal_copy_old)))))) \\/ (((exists pfc_gap_diagonal_copy_oldleftoutside. pfc_gap_diagonal_copy_oldleftoutside+(L)=(j)) /\\ (((pfc_left_diagonal_copy_old)=0))))) /\\ ((((((exists pfa_gap_diagonal_copy_oldrightinside. pfa_gap_diagonal_copy_oldrightinside + S (pfc_complement_diagonal_copy_old) = (M)) /\\ ((((exists ff_h_pfp_diagonal_copy_oldrightentry. ff_h_pfp_diagonal_copy_oldrightentry + S (pfc_right_diagonal_copy_old) = S ((S (pfc_complement_diagonal_copy_old)) * bc)) /\\ exists ff_q_pfp_diagonal_copy_oldrightentry. bb = ff_q_pfp_diagonal_copy_oldrightentry * S ((S (pfc_complement_diagonal_copy_old)) * bc) + (pfc_right_diagonal_copy_old)))))) \\/ (((exists pfc_gap_diagonal_copy_oldrightoutside. pfc_gap_diagonal_copy_oldrightoutside+(M)=(pfc_complement_diagonal_copy_old)) /\\ (((pfc_right_diagonal_copy_old)=0))))) /\\ (((a)=pfc_left_diagonal_copy_old*pfc_right_diagonal_copy_old)))))))", "specialize polynomial_diagonal_prefix_entry (ab)", "specialize polynomial_diagonal_prefix_entry (ac)", "specialize polynomial_diagonal_prefix_entry (L)", "specialize polynomial_diagonal_prefix_entry (bb)", "specialize polynomial_diagonal_prefix_entry (bc)", "specialize polynomial_diagonal_prefix_entry (M)", "specialize polynomial_diagonal_prefix_entry (i)", "specialize polynomial_diagonal_prefix_entry (db)", "specialize polynomial_diagonal_prefix_entry (dc)", "specialize polynomial_diagonal_prefix_entry (S i)", "specialize polynomial_diagonal_prefix_entry (j)", "specialize polynomial_diagonal_prefix_entry (a)", "apply polynomial_diagonal_prefix_entry", "exact hold", "exact hj", "exact ha", "have hv : exists z. ((((exists ff_h_pfp_diagonal_copy_entry. ff_h_pfp_diagonal_copy_entry + S (z) = S ((S (j)) * ec)) /\\ exists ff_q_pfp_diagonal_copy_entry. eb = ff_q_pfp_diagonal_copy_entry * S ((S (j)) * ec) + (z))) /\\ ((exists pfc_complement_diagonal_copy_actual pfc_left_diagonal_copy_actual pfc_right_diagonal_copy_actual. (((j)+pfc_complement_diagonal_copy_actual=(t+i)) /\\ ((((((exists pfa_gap_diagonal_copy_actualleftinside. pfa_gap_diagonal_copy_actualleftinside + S (j) = (L)) /\\ ((((exists ff_h_pfp_diagonal_copy_actualleftentry. ff_h_pfp_diagonal_copy_actualleftentry + S (pfc_left_diagonal_copy_actual) = S ((S (j)) * ac)) /\\ exists ff_q_pfp_diagonal_copy_actualleftentry. ab = ff_q_pfp_diagonal_copy_actualleftentry * S ((S (j)) * ac) + (pfc_left_diagonal_copy_actual)))))) \\/ (((exists pfc_gap_diagonal_copy_actualleftoutside. pfc_gap_diagonal_copy_actualleftoutside+(L)=(j)) /\\ (((pfc_left_diagonal_copy_actual)=0))))) /\\ ((((((exists pfa_gap_diagonal_copy_actualrightinside. pfa_gap_diagonal_copy_actualrightinside + S (pfc_complement_diagonal_copy_actual) = (t+M)) /\\ ((((exists ff_h_pfp_diagonal_copy_actualrightentry. ff_h_pfp_diagonal_copy_actualrightentry + S (pfc_right_diagonal_copy_actual) = S ((S (pfc_complement_diagonal_copy_actual)) * BC)) /\\ exists ff_q_pfp_diagonal_copy_actualrightentry. BB = ff_q_pfp_diagonal_copy_actualrightentry * S ((S (pfc_complement_diagonal_copy_actual)) * BC) + (pfc_right_diagonal_copy_actual)))))) \\/ (((exists pfc_gap_diagonal_copy_actualrightoutside. pfc_gap_diagonal_copy_actualrightoutside+(t+M)=(pfc_complement_diagonal_copy_actual)) /\\ (((pfc_right_diagonal_copy_actual)=0))))) /\\ (((z)=pfc_left_diagonal_copy_actual*pfc_right_diagonal_copy_actual))))))))))", "specialize hnew (j)", "apply hnew", "specialize le_trans (S j)", "specialize le_trans (S i)", "specialize le_trans (S (t+i))", "apply le_trans", "exact hj", "specialize succ_le_succ (i)", "specialize succ_le_succ (t+i)", "apply succ_le_succ", "specialize le_add_left (i)", "specialize le_add_left (t)", "apply le_add_left", "cases hv", "cases hv_witness", "have heq : x=a", "specialize polynomial_diagonal_term_functional (ab)", "specialize polynomial_diagonal_term_functional (ac)", "specialize polynomial_diagonal_term_functional (L)", "specialize polynomial_diagonal_term_functional (BB)", "specialize polynomial_diagonal_term_functional (BC)", "specialize polynomial_diagonal_term_functional (t+M)", "specialize polynomial_diagonal_term_functional (t+i)", "specialize polynomial_diagonal_term_functional (j)", "specialize polynomial_diagonal_term_functional (x)", "specialize polynomial_diagonal_term_functional (a)", "apply polynomial_diagonal_term_functional", "exact hv_witness_right", "specialize polynomial_diagonal_term_left_padding_right (ab)", "specialize polynomial_diagonal_term_left_padding_right (ac)", "specialize polynomial_diagonal_term_left_padding_right (L)", "specialize polynomial_diagonal_term_left_padding_right (bb)", "specialize polynomial_diagonal_term_left_padding_right (bc)", "specialize polynomial_diagonal_term_left_padding_right (M)", "specialize polynomial_diagonal_term_left_padding_right (BB)", "specialize polynomial_diagonal_term_left_padding_right (BC)", "specialize polynomial_diagonal_term_left_padding_right (t)", "specialize polynomial_diagonal_term_left_padding_right (i)", "specialize polynomial_diagonal_term_left_padding_right (j)", "specialize polynomial_diagonal_term_left_padding_right (a)", "apply polynomial_diagonal_term_left_padding_right", "exact hpad", "exact hterm", "rewrite heq at hv_witness_left", "rewrite heq at hv_witness_left", "exact hv_witness_left", "intro j", "intro hj", "have hv : exists z. ((((exists ff_h_pfp_diagonal_tail_entry. ff_h_pfp_diagonal_tail_entry + S (z) = S ((S (S i+j)) * ec)) /\\ exists ff_q_pfp_diagonal_tail_entry. eb = ff_q_pfp_diagonal_tail_entry * S ((S (S i+j)) * ec) + (z))) /\\ ((exists pfc_complement_diagonal_tail_actual pfc_left_diagonal_tail_actual pfc_right_diagonal_tail_actual. (((S i+j)+pfc_complement_diagonal_tail_actual=(t+i)) /\\ ((((((exists pfa_gap_diagonal_tail_actualleftinside. pfa_gap_diagonal_tail_actualleftinside + S (S i+j) = (L)) /\\ ((((exists ff_h_pfp_diagonal_tail_actualleftentry. ff_h_pfp_diagonal_tail_actualleftentry + S (pfc_left_diagonal_tail_actual) = S ((S (S i+j)) * ac)) /\\ exists ff_q_pfp_diagonal_tail_actualleftentry. ab = ff_q_pfp_diagonal_tail_actualleftentry * S ((S (S i+j)) * ac) + (pfc_left_diagonal_tail_actual)))))) \\/ (((exists pfc_gap_diagonal_tail_actualleftoutside. pfc_gap_diagonal_tail_actualleftoutside+(L)=(S i+j)) /\\ (((pfc_left_diagonal_tail_actual)=0))))) /\\ ((((((exists pfa_gap_diagonal_tail_actualrightinside. pfa_gap_diagonal_tail_actualrightinside + S (pfc_complement_diagonal_tail_actual) = (t+M)) /\\ ((((exists ff_h_pfp_diagonal_tail_actualrightentry. ff_h_pfp_diagonal_tail_actualrightentry + S (pfc_right_diagonal_tail_actual) = S ((S (pfc_complement_diagonal_tail_actual)) * BC)) /\\ exists ff_q_pfp_diagonal_tail_actualrightentry. BB = ff_q_pfp_diagonal_tail_actualrightentry * S ((S (pfc_complement_diagonal_tail_actual)) * BC) + (pfc_right_diagonal_tail_actual)))))) \\/ (((exists pfc_gap_diagonal_tail_actualrightoutside. pfc_gap_diagonal_tail_actualrightoutside+(t+M)=(pfc_complement_diagonal_tail_actual)) /\\ (((pfc_right_diagonal_tail_actual)=0))))) /\\ (((z)=pfc_left_diagonal_tail_actual*pfc_right_diagonal_tail_actual))))))))))", "specialize hnew (S i+j)", "apply hnew", "have hlength : S i+t=S (t+i)", "simp [add_succ_left,add_comm]", "have hbound : exists pfa_gap_diagonal_tail_bound. pfa_gap_diagonal_tail_bound + S (S i+j) = (S i+t)", "specialize matrix_recursive_lt_add_left (j)", "specialize matrix_recursive_lt_add_left (t)", "specialize matrix_recursive_lt_add_left (S i)", "apply matrix_recursive_lt_add_left", "exact hj", "rewrite hlength at hbound", "exact hbound", "cases hv", "cases hv_witness", "have hz : x=0", "specialize polynomial_diagonal_term_left_padding_zero_right (ab)", "specialize polynomial_diagonal_term_left_padding_zero_right (ac)", "specialize polynomial_diagonal_term_left_padding_zero_right (L)", "specialize polynomial_diagonal_term_left_padding_zero_right (bb)", "specialize polynomial_diagonal_term_left_padding_zero_right (bc)", "specialize polynomial_diagonal_term_left_padding_zero_right (M)", "specialize polynomial_diagonal_term_left_padding_zero_right (BB)", "specialize polynomial_diagonal_term_left_padding_zero_right (BC)", "specialize polynomial_diagonal_term_left_padding_zero_right (t)", "specialize polynomial_diagonal_term_left_padding_zero_right (t+i)", "specialize polynomial_diagonal_term_left_padding_zero_right (S i+j)", "specialize polynomial_diagonal_term_left_padding_zero_right (x)", "apply polynomial_diagonal_term_left_padding_zero_right", "exact hpad", "specialize matrix_recursive_lt_add_left (i)", "specialize matrix_recursive_lt_add_left (S i+j)", "specialize matrix_recursive_lt_add_left (t)", "apply matrix_recursive_lt_add_left", "exists j", "specialize add_comm (j)", "specialize add_comm (S i)", "apply add_comm", "exact hv_witness_right", "rewrite hz at hv_witness_left", "rewrite hz at hv_witness_left", "exact hv_witness_left" ], "script_sha256": "c55a2048734e443bde13a76f080fac63294e9c62feaddeb4360985caef91f624", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_convolution_padding_candidate.py", "sha256": "2d874ecfb35a5db0aecdeb07b549464efebad9072c363113aa5a0a977845d007" }, "statement": "forall ab ac L bb bc M BB BC t i db dc eb ec. (((forall pfp_repeat_index_diagonal_actual_padding_rightzeros. (exists pfa_gap_diagonal_actual_padding_rightzerosindex. pfa_gap_diagonal_actual_padding_rightzerosindex + S (pfp_repeat_index_diagonal_actual_padding_rightzeros) = (t)) -> (((exists ff_h_pfp_diagonal_actual_padding_rightzerosentry. ff_h_pfp_diagonal_actual_padding_rightzerosentry + S (0) = S ((S (pfp_repeat_index_diagonal_actual_padding_rightzeros)) * BC)) /\\ exists ff_q_pfp_diagonal_actual_padding_rightzerosentry. BB = ff_q_pfp_diagonal_actual_padding_rightzerosentry * S ((S (pfp_repeat_index_diagonal_actual_padding_rightzeros)) * BC) + (0)))) /\\ ((forall pfrep_index_diagonal_actual_padding_right pfrep_value_diagonal_actual_padding_right. (exists pfa_gap_diagonal_actual_padding_rightbound. pfa_gap_diagonal_actual_padding_rightbound + S (pfrep_index_diagonal_actual_padding_right) = (M)) -> (((exists ff_h_pfp_diagonal_actual_padding_rightinput. ff_h_pfp_diagonal_actual_padding_rightinput + S (pfrep_value_diagonal_actual_padding_right) = S ((S (pfrep_index_diagonal_actual_padding_right)) * bc)) /\\ exists ff_q_pfp_diagonal_actual_padding_rightinput. bb = ff_q_pfp_diagonal_actual_padding_rightinput * S ((S (pfrep_index_diagonal_actual_padding_right)) * bc) + (pfrep_value_diagonal_actual_padding_right))) -> (((exists ff_h_pfp_diagonal_actual_padding_rightoutput. ff_h_pfp_diagonal_actual_padding_rightoutput + S (pfrep_value_diagonal_actual_padding_right) = S ((S ((t)+pfrep_index_diagonal_actual_padding_right)) * BC)) /\\ exists ff_q_pfp_diagonal_actual_padding_rightoutput. BB = ff_q_pfp_diagonal_actual_padding_rightoutput * S ((S ((t)+pfrep_index_diagonal_actual_padding_right)) * BC) + (pfrep_value_diagonal_actual_padding_right))))))) -> (forall pfc_index_diagonal_pad_old_right. (exists pfa_gap_diagonal_pad_old_rightbound. pfa_gap_diagonal_pad_old_rightbound + S (pfc_index_diagonal_pad_old_right) = (S i)) -> exists pfc_value_diagonal_pad_old_right. ((((exists ff_h_pfp_diagonal_pad_old_rightentry. ff_h_pfp_diagonal_pad_old_rightentry + S (pfc_value_diagonal_pad_old_right) = S ((S (pfc_index_diagonal_pad_old_right)) * dc)) /\\ exists ff_q_pfp_diagonal_pad_old_rightentry. db = ff_q_pfp_diagonal_pad_old_rightentry * S ((S (pfc_index_diagonal_pad_old_right)) * dc) + (pfc_value_diagonal_pad_old_right))) /\\ ((exists pfc_complement_diagonal_pad_old_rightterm pfc_left_diagonal_pad_old_rightterm pfc_right_diagonal_pad_old_rightterm. (((pfc_index_diagonal_pad_old_right)+pfc_complement_diagonal_pad_old_rightterm=(i)) /\\ ((((((exists pfa_gap_diagonal_pad_old_righttermleftinside. pfa_gap_diagonal_pad_old_righttermleftinside + S (pfc_index_diagonal_pad_old_right) = (L)) /\\ ((((exists ff_h_pfp_diagonal_pad_old_righttermleftentry. ff_h_pfp_diagonal_pad_old_righttermleftentry + S (pfc_left_diagonal_pad_old_rightterm) = S ((S (pfc_index_diagonal_pad_old_right)) * ac)) /\\ exists ff_q_pfp_diagonal_pad_old_righttermleftentry. ab = ff_q_pfp_diagonal_pad_old_righttermleftentry * S ((S (pfc_index_diagonal_pad_old_right)) * ac) + (pfc_left_diagonal_pad_old_rightterm)))))) \\/ (((exists pfc_gap_diagonal_pad_old_righttermleftoutside. pfc_gap_diagonal_pad_old_righttermleftoutside+(L)=(pfc_index_diagonal_pad_old_right)) /\\ (((pfc_left_diagonal_pad_old_rightterm)=0))))) /\\ ((((((exists pfa_gap_diagonal_pad_old_righttermrightinside. pfa_gap_diagonal_pad_old_righttermrightinside + S (pfc_complement_diagonal_pad_old_rightterm) = (M)) /\\ ((((exists ff_h_pfp_diagonal_pad_old_righttermrightentry. ff_h_pfp_diagonal_pad_old_righttermrightentry + S (pfc_right_diagonal_pad_old_rightterm) = S ((S (pfc_complement_diagonal_pad_old_rightterm)) * bc)) /\\ exists ff_q_pfp_diagonal_pad_old_righttermrightentry. bb = ff_q_pfp_diagonal_pad_old_righttermrightentry * S ((S (pfc_complement_diagonal_pad_old_rightterm)) * bc) + (pfc_right_diagonal_pad_old_rightterm)))))) \\/ (((exists pfc_gap_diagonal_pad_old_righttermrightoutside. pfc_gap_diagonal_pad_old_righttermrightoutside+(M)=(pfc_complement_diagonal_pad_old_rightterm)) /\\ (((pfc_right_diagonal_pad_old_rightterm)=0))))) /\\ (((pfc_value_diagonal_pad_old_right)=pfc_left_diagonal_pad_old_rightterm*pfc_right_diagonal_pad_old_rightterm))))))))))) -> (forall pfc_index_diagonal_pad_new_right. (exists pfa_gap_diagonal_pad_new_rightbound. pfa_gap_diagonal_pad_new_rightbound + S (pfc_index_diagonal_pad_new_right) = (S (t+i))) -> exists pfc_value_diagonal_pad_new_right. ((((exists ff_h_pfp_diagonal_pad_new_rightentry. ff_h_pfp_diagonal_pad_new_rightentry + S (pfc_value_diagonal_pad_new_right) = S ((S (pfc_index_diagonal_pad_new_right)) * ec)) /\\ exists ff_q_pfp_diagonal_pad_new_rightentry. eb = ff_q_pfp_diagonal_pad_new_rightentry * S ((S (pfc_index_diagonal_pad_new_right)) * ec) + (pfc_value_diagonal_pad_new_right))) /\\ ((exists pfc_complement_diagonal_pad_new_rightterm pfc_left_diagonal_pad_new_rightterm pfc_right_diagonal_pad_new_rightterm. (((pfc_index_diagonal_pad_new_right)+pfc_complement_diagonal_pad_new_rightterm=(t+i)) /\\ ((((((exists pfa_gap_diagonal_pad_new_righttermleftinside. pfa_gap_diagonal_pad_new_righttermleftinside + S (pfc_index_diagonal_pad_new_right) = (L)) /\\ ((((exists ff_h_pfp_diagonal_pad_new_righttermleftentry. ff_h_pfp_diagonal_pad_new_righttermleftentry + S (pfc_left_diagonal_pad_new_rightterm) = S ((S (pfc_index_diagonal_pad_new_right)) * ac)) /\\ exists ff_q_pfp_diagonal_pad_new_righttermleftentry. ab = ff_q_pfp_diagonal_pad_new_righttermleftentry * S ((S (pfc_index_diagonal_pad_new_right)) * ac) + (pfc_left_diagonal_pad_new_rightterm)))))) \\/ (((exists pfc_gap_diagonal_pad_new_righttermleftoutside. pfc_gap_diagonal_pad_new_righttermleftoutside+(L)=(pfc_index_diagonal_pad_new_right)) /\\ (((pfc_left_diagonal_pad_new_rightterm)=0))))) /\\ ((((((exists pfa_gap_diagonal_pad_new_righttermrightinside. pfa_gap_diagonal_pad_new_righttermrightinside + S (pfc_complement_diagonal_pad_new_rightterm) = (t+M)) /\\ ((((exists ff_h_pfp_diagonal_pad_new_righttermrightentry. ff_h_pfp_diagonal_pad_new_righttermrightentry + S (pfc_right_diagonal_pad_new_rightterm) = S ((S (pfc_complement_diagonal_pad_new_rightterm)) * BC)) /\\ exists ff_q_pfp_diagonal_pad_new_righttermrightentry. BB = ff_q_pfp_diagonal_pad_new_righttermrightentry * S ((S (pfc_complement_diagonal_pad_new_rightterm)) * BC) + (pfc_right_diagonal_pad_new_rightterm)))))) \\/ (((exists pfc_gap_diagonal_pad_new_righttermrightoutside. pfc_gap_diagonal_pad_new_righttermrightoutside+(t+M)=(pfc_complement_diagonal_pad_new_rightterm)) /\\ (((pfc_right_diagonal_pad_new_rightterm)=0))))) /\\ (((pfc_value_diagonal_pad_new_right)=pfc_left_diagonal_pad_new_rightterm*pfc_right_diagonal_pad_new_rightterm))))))))))) -> (((forall mdr_i_pfp_diagonal_pad_result_equal mdr_a_pfp_diagonal_pad_result_equal. (exists mdr_gap_pfp_diagonal_pad_result_equalb. mdr_gap_pfp_diagonal_pad_result_equalb + S (mdr_i_pfp_diagonal_pad_result_equal) = (S i)) -> (((exists ff_h_mdr_pfp_diagonal_pad_result_equalo. ff_h_mdr_pfp_diagonal_pad_result_equalo + S (mdr_a_pfp_diagonal_pad_result_equal) = S ((S (mdr_i_pfp_diagonal_pad_result_equal)) * dc)) /\\ exists ff_q_mdr_pfp_diagonal_pad_result_equalo. db = ff_q_mdr_pfp_diagonal_pad_result_equalo * S ((S (mdr_i_pfp_diagonal_pad_result_equal)) * dc) + (mdr_a_pfp_diagonal_pad_result_equal))) -> (((exists ff_h_mdr_pfp_diagonal_pad_result_equaln. ff_h_mdr_pfp_diagonal_pad_result_equaln + S (mdr_a_pfp_diagonal_pad_result_equal) = S ((S (mdr_i_pfp_diagonal_pad_result_equal)) * ec)) /\\ exists ff_q_mdr_pfp_diagonal_pad_result_equaln. eb = ff_q_mdr_pfp_diagonal_pad_result_equaln * S ((S (mdr_i_pfp_diagonal_pad_result_equal)) * ec) + (mdr_a_pfp_diagonal_pad_result_equal)))) /\\ ((forall pfpad_tail_index_diagonal_pad_result_tail. (exists pfa_gap_diagonal_pad_result_tailbound. pfa_gap_diagonal_pad_result_tailbound + S (pfpad_tail_index_diagonal_pad_result_tail) = (t)) -> (((exists ff_h_pfp_diagonal_pad_result_tailzero. ff_h_pfp_diagonal_pad_result_tailzero + S (0) = S ((S ((S i)+pfpad_tail_index_diagonal_pad_result_tail)) * ec)) /\\ exists ff_q_pfp_diagonal_pad_result_tailzero. eb = ff_q_pfp_diagonal_pad_result_tailzero * S ((S ((S i)+pfpad_tail_index_diagonal_pad_result_tail)) * ec) + (0)))))))", "statement_sha256": "0006a71b829d1fa8cfd50ed7e0c9bc81d50ee7b72bfe9255631465f6a846786a", "summary": "The two actual antidiagonal tables differ by a proved trailing zero block and exact copied natural summands.", "summary_sha256": "d3bad996108c5c3a9d1f47b9f0339618e109e6b40d5c9bdfd4a80dd043925277" }, { "alpha_v33_frontier_enrollment": { "body_receipt_sha256": "717ff93dec9bfe4a9820901d618efb14f8220a6d226250c7980bfd340914c94b", "bundle_campaign": "polynomial-euclidean-division", "bundle_node_id": 356, "bundle_sha256": 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"e2bf28e20e5395060bb023143b05e8c4ac1e8bfbfdc670c9037525e60cd57a5e", "membership": "alpha_only", "name": "prime_field_convolution_coefficient_left_padding_left", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro ab", "intro ac", "intro L", "intro bb", "intro bc", "intro M", "intro AB", "intro AC", "intro t", "intro i", "intro r", "intro hpad", "intro hc", "cases hc", "cases hc_witness", "cases hc_witness_witness", "cases hc_witness_witness_witness", "cases hc_witness_witness_witness_right", "have hd : exists eb ec. forall pfc_index_coefficient_padded_diagonal_left. (exists pfa_gap_coefficient_padded_diagonal_leftbound. pfa_gap_coefficient_padded_diagonal_leftbound + S (pfc_index_coefficient_padded_diagonal_left) = (S (t+i))) -> exists pfc_value_coefficient_padded_diagonal_left. ((((exists ff_h_pfp_coefficient_padded_diagonal_leftentry. ff_h_pfp_coefficient_padded_diagonal_leftentry + S (pfc_value_coefficient_padded_diagonal_left) = S ((S (pfc_index_coefficient_padded_diagonal_left)) * ec)) /\\ exists ff_q_pfp_coefficient_padded_diagonal_leftentry. eb = ff_q_pfp_coefficient_padded_diagonal_leftentry * S ((S (pfc_index_coefficient_padded_diagonal_left)) * ec) + (pfc_value_coefficient_padded_diagonal_left))) /\\ ((exists pfc_complement_coefficient_padded_diagonal_leftterm pfc_left_coefficient_padded_diagonal_leftterm pfc_right_coefficient_padded_diagonal_leftterm. (((pfc_index_coefficient_padded_diagonal_left)+pfc_complement_coefficient_padded_diagonal_leftterm=(t+i)) /\\ ((((((exists pfa_gap_coefficient_padded_diagonal_lefttermleftinside. pfa_gap_coefficient_padded_diagonal_lefttermleftinside + S (pfc_index_coefficient_padded_diagonal_left) = (t+L)) /\\ ((((exists ff_h_pfp_coefficient_padded_diagonal_lefttermleftentry. ff_h_pfp_coefficient_padded_diagonal_lefttermleftentry + S (pfc_left_coefficient_padded_diagonal_leftterm) = S ((S (pfc_index_coefficient_padded_diagonal_left)) * AC)) /\\ exists ff_q_pfp_coefficient_padded_diagonal_lefttermleftentry. AB = ff_q_pfp_coefficient_padded_diagonal_lefttermleftentry * S ((S (pfc_index_coefficient_padded_diagonal_left)) * AC) + (pfc_left_coefficient_padded_diagonal_leftterm)))))) \\/ (((exists pfc_gap_coefficient_padded_diagonal_lefttermleftoutside. pfc_gap_coefficient_padded_diagonal_lefttermleftoutside+(t+L)=(pfc_index_coefficient_padded_diagonal_left)) /\\ (((pfc_left_coefficient_padded_diagonal_leftterm)=0))))) /\\ ((((((exists pfa_gap_coefficient_padded_diagonal_lefttermrightinside. pfa_gap_coefficient_padded_diagonal_lefttermrightinside + S (pfc_complement_coefficient_padded_diagonal_leftterm) = (M)) /\\ ((((exists ff_h_pfp_coefficient_padded_diagonal_lefttermrightentry. ff_h_pfp_coefficient_padded_diagonal_lefttermrightentry + S (pfc_right_coefficient_padded_diagonal_leftterm) = S ((S (pfc_complement_coefficient_padded_diagonal_leftterm)) * bc)) /\\ exists ff_q_pfp_coefficient_padded_diagonal_lefttermrightentry. bb = ff_q_pfp_coefficient_padded_diagonal_lefttermrightentry * S ((S (pfc_complement_coefficient_padded_diagonal_leftterm)) * bc) + (pfc_right_coefficient_padded_diagonal_leftterm)))))) \\/ (((exists pfc_gap_coefficient_padded_diagonal_lefttermrightoutside. pfc_gap_coefficient_padded_diagonal_lefttermrightoutside+(M)=(pfc_complement_coefficient_padded_diagonal_leftterm)) /\\ (((pfc_right_coefficient_padded_diagonal_leftterm)=0))))) /\\ (((pfc_value_coefficient_padded_diagonal_left)=pfc_left_coefficient_padded_diagonal_leftterm*pfc_right_coefficient_padded_diagonal_leftterm))))))))))", "specialize polynomial_diagonal_prefix_exists (AB)", "specialize polynomial_diagonal_prefix_exists (AC)", "specialize polynomial_diagonal_prefix_exists (t+L)", "specialize polynomial_diagonal_prefix_exists (bb)", "specialize polynomial_diagonal_prefix_exists (bc)", "specialize polynomial_diagonal_prefix_exists (M)", "specialize polynomial_diagonal_prefix_exists (t+i)", "apply polynomial_diagonal_prefix_exists", "cases hd", "cases hd_witness", "have hs : exists n. exists fs_u_pfc_coefficient_padded_sum_left fs_v_pfc_coefficient_padded_sum_left. ((((exists fs_h_pfc_coefficient_padded_sum_left_body_start. fs_h_pfc_coefficient_padded_sum_left_body_start + S (0) = S ((S (0)) * fs_v_pfc_coefficient_padded_sum_left)) /\\ exists fs_q_pfc_coefficient_padded_sum_left_body_start. fs_u_pfc_coefficient_padded_sum_left = fs_q_pfc_coefficient_padded_sum_left_body_start * S ((S (0)) * fs_v_pfc_coefficient_padded_sum_left) + (0))) /\\ ((((exists fs_h_pfc_coefficient_padded_sum_left_body_terminal. fs_h_pfc_coefficient_padded_sum_left_body_terminal + S (n) = S ((S (S (t+i))) * fs_v_pfc_coefficient_padded_sum_left)) /\\ exists fs_q_pfc_coefficient_padded_sum_left_body_terminal. fs_u_pfc_coefficient_padded_sum_left = fs_q_pfc_coefficient_padded_sum_left_body_terminal * S ((S (S (t+i))) * fs_v_pfc_coefficient_padded_sum_left) + (n))) /\\ forall fs_i_pfc_coefficient_padded_sum_left_body_steps. (exists fs_lt_pfc_coefficient_padded_sum_left_body_steps_bound. fs_lt_pfc_coefficient_padded_sum_left_body_steps_bound + S fs_i_pfc_coefficient_padded_sum_left_body_steps = S (t+i)) -> exists fs_a_pfc_coefficient_padded_sum_left_body_steps fs_r_pfc_coefficient_padded_sum_left_body_steps fs_s_pfc_coefficient_padded_sum_left_body_steps. ((((exists fs_h_pfc_coefficient_padded_sum_left_body_steps_summand. fs_h_pfc_coefficient_padded_sum_left_body_steps_summand + S (fs_a_pfc_coefficient_padded_sum_left_body_steps) = S ((S (fs_i_pfc_coefficient_padded_sum_left_body_steps)) * x4)) /\\ exists fs_q_pfc_coefficient_padded_sum_left_body_steps_summand. x3 = fs_q_pfc_coefficient_padded_sum_left_body_steps_summand * S ((S (fs_i_pfc_coefficient_padded_sum_left_body_steps)) * x4) + (fs_a_pfc_coefficient_padded_sum_left_body_steps))) /\\ ((((exists fs_h_pfc_coefficient_padded_sum_left_body_steps_partial. fs_h_pfc_coefficient_padded_sum_left_body_steps_partial + S (fs_r_pfc_coefficient_padded_sum_left_body_steps) = S ((S (fs_i_pfc_coefficient_padded_sum_left_body_steps)) * fs_v_pfc_coefficient_padded_sum_left)) /\\ exists fs_q_pfc_coefficient_padded_sum_left_body_steps_partial. fs_u_pfc_coefficient_padded_sum_left = fs_q_pfc_coefficient_padded_sum_left_body_steps_partial * S ((S (fs_i_pfc_coefficient_padded_sum_left_body_steps)) * fs_v_pfc_coefficient_padded_sum_left) + (fs_r_pfc_coefficient_padded_sum_left_body_steps))) /\\ ((((exists fs_h_pfc_coefficient_padded_sum_left_body_steps_successor. fs_h_pfc_coefficient_padded_sum_left_body_steps_successor + S (fs_s_pfc_coefficient_padded_sum_left_body_steps) = S ((S (S fs_i_pfc_coefficient_padded_sum_left_body_steps)) * fs_v_pfc_coefficient_padded_sum_left)) /\\ exists fs_q_pfc_coefficient_padded_sum_left_body_steps_successor. fs_u_pfc_coefficient_padded_sum_left = fs_q_pfc_coefficient_padded_sum_left_body_steps_successor * S ((S (S fs_i_pfc_coefficient_padded_sum_left_body_steps)) * fs_v_pfc_coefficient_padded_sum_left) + (fs_s_pfc_coefficient_padded_sum_left_body_steps))) /\\ fs_s_pfc_coefficient_padded_sum_left_body_steps = fs_r_pfc_coefficient_padded_sum_left_body_steps + fs_a_pfc_coefficient_padded_sum_left_body_steps)))))", "specialize beta_sum_exists (x3)", "specialize beta_sum_exists (x4)", "specialize beta_sum_exists (S (t+i))", "apply beta_sum_exists", "cases hs", "have hdata : ((forall pfp_repeat_index_coefficient_diagonal_paddingzeros. (exists pfa_gap_coefficient_diagonal_paddingzerosindex. pfa_gap_coefficient_diagonal_paddingzerosindex + S (pfp_repeat_index_coefficient_diagonal_paddingzeros) = (t)) -> (((exists ff_h_pfp_coefficient_diagonal_paddingzerosentry. ff_h_pfp_coefficient_diagonal_paddingzerosentry + S (0) = S ((S (pfp_repeat_index_coefficient_diagonal_paddingzeros)) * x4)) /\\ exists ff_q_pfp_coefficient_diagonal_paddingzerosentry. x3 = ff_q_pfp_coefficient_diagonal_paddingzerosentry * S ((S (pfp_repeat_index_coefficient_diagonal_paddingzeros)) * x4) + (0)))) /\\ ((forall pfrep_index_coefficient_diagonal_padding pfrep_value_coefficient_diagonal_padding. (exists pfa_gap_coefficient_diagonal_paddingbound. pfa_gap_coefficient_diagonal_paddingbound + S (pfrep_index_coefficient_diagonal_padding) = (S i)) -> (((exists ff_h_pfp_coefficient_diagonal_paddinginput. ff_h_pfp_coefficient_diagonal_paddinginput + S (pfrep_value_coefficient_diagonal_padding) = S ((S (pfrep_index_coefficient_diagonal_padding)) * x1)) /\\ exists ff_q_pfp_coefficient_diagonal_paddinginput. x = ff_q_pfp_coefficient_diagonal_paddinginput * S ((S (pfrep_index_coefficient_diagonal_padding)) * x1) + (pfrep_value_coefficient_diagonal_padding))) -> (((exists ff_h_pfp_coefficient_diagonal_paddingoutput. ff_h_pfp_coefficient_diagonal_paddingoutput + S (pfrep_value_coefficient_diagonal_padding) = S ((S ((t)+pfrep_index_coefficient_diagonal_padding)) * x4)) /\\ exists ff_q_pfp_coefficient_diagonal_paddingoutput. x3 = ff_q_pfp_coefficient_diagonal_paddingoutput * S ((S ((t)+pfrep_index_coefficient_diagonal_padding)) * x4) + (pfrep_value_coefficient_diagonal_padding))))))", "specialize polynomial_diagonal_left_padding_left (ab)", "specialize polynomial_diagonal_left_padding_left (ac)", "specialize polynomial_diagonal_left_padding_left (L)", "specialize polynomial_diagonal_left_padding_left (bb)", "specialize polynomial_diagonal_left_padding_left (bc)", "specialize polynomial_diagonal_left_padding_left (M)", "specialize polynomial_diagonal_left_padding_left (AB)", "specialize polynomial_diagonal_left_padding_left (AC)", "specialize polynomial_diagonal_left_padding_left (t)", "specialize polynomial_diagonal_left_padding_left (i)", "specialize polynomial_diagonal_left_padding_left (x)", "specialize polynomial_diagonal_left_padding_left (x1)", "specialize polynomial_diagonal_left_padding_left (x3)", "specialize polynomial_diagonal_left_padding_left (x4)", "apply polynomial_diagonal_left_padding_left", "exact hpad", "exact hc_witness_witness_witness_left", "exact hd_witness_witness", "have heq : x5=x2", "specialize polynomial_left_pad_natural_sum_invariant (x)", "specialize polynomial_left_pad_natural_sum_invariant (x1)", "specialize polynomial_left_pad_natural_sum_invariant (x3)", "specialize polynomial_left_pad_natural_sum_invariant (x4)", "specialize polynomial_left_pad_natural_sum_invariant (t)", "specialize polynomial_left_pad_natural_sum_invariant (S i)", "specialize polynomial_left_pad_natural_sum_invariant (x2)", "specialize polynomial_left_pad_natural_sum_invariant (x5)", "apply polynomial_left_pad_natural_sum_invariant", "exact hdata", "exact hc_witness_witness_witness_right_left", "have hlength : t+S i=S (t+i)", "simp", "rewrite hlength", "rewrite hlength", "rewrite hlength", "exact hs_witness", "exists x3", "exists x4", "exists x2", "split", "exact hd_witness_witness", "split", "rewrite heq at hs_witness", "rewrite heq at hs_witness", "exact hs_witness", "exact hc_witness_witness_witness_right_right" ], "script_sha256": "1e4fdeccf97595b9e336385abc6995ec716f2636f514139b3b6577c102e33660", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_convolution_padding_candidate.py", "sha256": "2d874ecfb35a5db0aecdeb07b549464efebad9072c363113aa5a0a977845d007" }, "statement": "forall p ab ac L bb bc M AB AC t i r. (((forall pfp_repeat_index_coefficient_padding_leftzeros. (exists pfa_gap_coefficient_padding_leftzerosindex. pfa_gap_coefficient_padding_leftzerosindex + S (pfp_repeat_index_coefficient_padding_leftzeros) = (t)) -> (((exists ff_h_pfp_coefficient_padding_leftzerosentry. ff_h_pfp_coefficient_padding_leftzerosentry + S (0) = S ((S (pfp_repeat_index_coefficient_padding_leftzeros)) * AC)) /\\ exists ff_q_pfp_coefficient_padding_leftzerosentry. AB = ff_q_pfp_coefficient_padding_leftzerosentry * S ((S (pfp_repeat_index_coefficient_padding_leftzeros)) * AC) + (0)))) /\\ ((forall pfrep_index_coefficient_padding_left pfrep_value_coefficient_padding_left. (exists pfa_gap_coefficient_padding_leftbound. pfa_gap_coefficient_padding_leftbound + S (pfrep_index_coefficient_padding_left) = (L)) -> (((exists ff_h_pfp_coefficient_padding_leftinput. ff_h_pfp_coefficient_padding_leftinput + S (pfrep_value_coefficient_padding_left) = S ((S (pfrep_index_coefficient_padding_left)) * ac)) /\\ exists ff_q_pfp_coefficient_padding_leftinput. ab = ff_q_pfp_coefficient_padding_leftinput * S ((S (pfrep_index_coefficient_padding_left)) * ac) + (pfrep_value_coefficient_padding_left))) -> (((exists ff_h_pfp_coefficient_padding_leftoutput. ff_h_pfp_coefficient_padding_leftoutput + S (pfrep_value_coefficient_padding_left) = S ((S ((t)+pfrep_index_coefficient_padding_left)) * AC)) /\\ exists ff_q_pfp_coefficient_padding_leftoutput. AB = ff_q_pfp_coefficient_padding_leftoutput * S ((S ((t)+pfrep_index_coefficient_padding_left)) * AC) + (pfrep_value_coefficient_padding_left))))))) -> (exists pfc_terms_code_coefficient_original_left pfc_terms_scale_coefficient_original_left pfc_natural_sum_coefficient_original_left. ((forall pfc_index_coefficient_original_leftdiagonal. (exists pfa_gap_coefficient_original_leftdiagonalbound. pfa_gap_coefficient_original_leftdiagonalbound + S (pfc_index_coefficient_original_leftdiagonal) = (S (i))) -> exists pfc_value_coefficient_original_leftdiagonal. ((((exists ff_h_pfp_coefficient_original_leftdiagonalentry. ff_h_pfp_coefficient_original_leftdiagonalentry + S (pfc_value_coefficient_original_leftdiagonal) = S ((S (pfc_index_coefficient_original_leftdiagonal)) * pfc_terms_scale_coefficient_original_left)) /\\ exists ff_q_pfp_coefficient_original_leftdiagonalentry. pfc_terms_code_coefficient_original_left = ff_q_pfp_coefficient_original_leftdiagonalentry * S ((S (pfc_index_coefficient_original_leftdiagonal)) * pfc_terms_scale_coefficient_original_left) + (pfc_value_coefficient_original_leftdiagonal))) /\\ ((exists pfc_complement_coefficient_original_leftdiagonalterm pfc_left_coefficient_original_leftdiagonalterm pfc_right_coefficient_original_leftdiagonalterm. (((pfc_index_coefficient_original_leftdiagonal)+pfc_complement_coefficient_original_leftdiagonalterm=(i)) /\\ ((((((exists pfa_gap_coefficient_original_leftdiagonaltermleftinside. pfa_gap_coefficient_original_leftdiagonaltermleftinside + S (pfc_index_coefficient_original_leftdiagonal) = (L)) /\\ ((((exists ff_h_pfp_coefficient_original_leftdiagonaltermleftentry. ff_h_pfp_coefficient_original_leftdiagonaltermleftentry + S (pfc_left_coefficient_original_leftdiagonalterm) = S ((S (pfc_index_coefficient_original_leftdiagonal)) * ac)) /\\ exists ff_q_pfp_coefficient_original_leftdiagonaltermleftentry. ab = ff_q_pfp_coefficient_original_leftdiagonaltermleftentry * S ((S (pfc_index_coefficient_original_leftdiagonal)) * ac) + (pfc_left_coefficient_original_leftdiagonalterm)))))) \\/ (((exists pfc_gap_coefficient_original_leftdiagonaltermleftoutside. pfc_gap_coefficient_original_leftdiagonaltermleftoutside+(L)=(pfc_index_coefficient_original_leftdiagonal)) /\\ (((pfc_left_coefficient_original_leftdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_coefficient_original_leftdiagonaltermrightinside. pfa_gap_coefficient_original_leftdiagonaltermrightinside + S (pfc_complement_coefficient_original_leftdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_coefficient_original_leftdiagonaltermrightentry. ff_h_pfp_coefficient_original_leftdiagonaltermrightentry + S (pfc_right_coefficient_original_leftdiagonalterm) = S ((S (pfc_complement_coefficient_original_leftdiagonalterm)) * bc)) /\\ exists ff_q_pfp_coefficient_original_leftdiagonaltermrightentry. bb = ff_q_pfp_coefficient_original_leftdiagonaltermrightentry * S ((S (pfc_complement_coefficient_original_leftdiagonalterm)) * bc) + (pfc_right_coefficient_original_leftdiagonalterm)))))) \\/ (((exists pfc_gap_coefficient_original_leftdiagonaltermrightoutside. pfc_gap_coefficient_original_leftdiagonaltermrightoutside+(M)=(pfc_complement_coefficient_original_leftdiagonalterm)) /\\ (((pfc_right_coefficient_original_leftdiagonalterm)=0))))) /\\ (((pfc_value_coefficient_original_leftdiagonal)=pfc_left_coefficient_original_leftdiagonalterm*pfc_right_coefficient_original_leftdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_coefficient_original_leftsum fs_v_pfc_coefficient_original_leftsum. ((((exists fs_h_pfc_coefficient_original_leftsum_body_start. fs_h_pfc_coefficient_original_leftsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_coefficient_original_leftsum)) /\\ exists fs_q_pfc_coefficient_original_leftsum_body_start. fs_u_pfc_coefficient_original_leftsum = fs_q_pfc_coefficient_original_leftsum_body_start * S ((S (0)) * fs_v_pfc_coefficient_original_leftsum) + (0))) /\\ ((((exists fs_h_pfc_coefficient_original_leftsum_body_terminal. fs_h_pfc_coefficient_original_leftsum_body_terminal + S (pfc_natural_sum_coefficient_original_left) = S ((S (S (i))) * fs_v_pfc_coefficient_original_leftsum)) /\\ exists fs_q_pfc_coefficient_original_leftsum_body_terminal. fs_u_pfc_coefficient_original_leftsum = fs_q_pfc_coefficient_original_leftsum_body_terminal * S ((S (S (i))) * fs_v_pfc_coefficient_original_leftsum) + (pfc_natural_sum_coefficient_original_left))) /\\ forall fs_i_pfc_coefficient_original_leftsum_body_steps. (exists fs_lt_pfc_coefficient_original_leftsum_body_steps_bound. fs_lt_pfc_coefficient_original_leftsum_body_steps_bound + S fs_i_pfc_coefficient_original_leftsum_body_steps = S (i)) -> exists fs_a_pfc_coefficient_original_leftsum_body_steps fs_r_pfc_coefficient_original_leftsum_body_steps fs_s_pfc_coefficient_original_leftsum_body_steps. ((((exists fs_h_pfc_coefficient_original_leftsum_body_steps_summand. fs_h_pfc_coefficient_original_leftsum_body_steps_summand + S (fs_a_pfc_coefficient_original_leftsum_body_steps) = S ((S (fs_i_pfc_coefficient_original_leftsum_body_steps)) * pfc_terms_scale_coefficient_original_left)) /\\ exists fs_q_pfc_coefficient_original_leftsum_body_steps_summand. pfc_terms_code_coefficient_original_left = fs_q_pfc_coefficient_original_leftsum_body_steps_summand * S ((S (fs_i_pfc_coefficient_original_leftsum_body_steps)) * pfc_terms_scale_coefficient_original_left) + (fs_a_pfc_coefficient_original_leftsum_body_steps))) /\\ ((((exists fs_h_pfc_coefficient_original_leftsum_body_steps_partial. fs_h_pfc_coefficient_original_leftsum_body_steps_partial + S (fs_r_pfc_coefficient_original_leftsum_body_steps) = S ((S (fs_i_pfc_coefficient_original_leftsum_body_steps)) * fs_v_pfc_coefficient_original_leftsum)) /\\ exists fs_q_pfc_coefficient_original_leftsum_body_steps_partial. fs_u_pfc_coefficient_original_leftsum = fs_q_pfc_coefficient_original_leftsum_body_steps_partial * S ((S (fs_i_pfc_coefficient_original_leftsum_body_steps)) * fs_v_pfc_coefficient_original_leftsum) + (fs_r_pfc_coefficient_original_leftsum_body_steps))) /\\ ((((exists fs_h_pfc_coefficient_original_leftsum_body_steps_successor. fs_h_pfc_coefficient_original_leftsum_body_steps_successor + S (fs_s_pfc_coefficient_original_leftsum_body_steps) = S ((S (S fs_i_pfc_coefficient_original_leftsum_body_steps)) * fs_v_pfc_coefficient_original_leftsum)) /\\ exists fs_q_pfc_coefficient_original_leftsum_body_steps_successor. fs_u_pfc_coefficient_original_leftsum = fs_q_pfc_coefficient_original_leftsum_body_steps_successor * S ((S (S fs_i_pfc_coefficient_original_leftsum_body_steps)) * fs_v_pfc_coefficient_original_leftsum) + (fs_s_pfc_coefficient_original_leftsum_body_steps))) /\\ fs_s_pfc_coefficient_original_leftsum_body_steps = fs_r_pfc_coefficient_original_leftsum_body_steps + fs_a_pfc_coefficient_original_leftsum_body_steps)))))) /\\ ((((exists pfa_gap_coefficient_original_leftresiduebound. pfa_gap_coefficient_original_leftresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_coefficient_original_leftresiduecongruence pfa_offset_right_coefficient_original_leftresiduecongruence. (pfc_natural_sum_coefficient_original_left) + (p) * pfa_offset_left_coefficient_original_leftresiduecongruence = (r) + (p) * pfa_offset_right_coefficient_original_leftresiduecongruence))))))))) -> (exists pfc_terms_code_coefficient_shifted_left pfc_terms_scale_coefficient_shifted_left pfc_natural_sum_coefficient_shifted_left. ((forall pfc_index_coefficient_shifted_leftdiagonal. (exists pfa_gap_coefficient_shifted_leftdiagonalbound. pfa_gap_coefficient_shifted_leftdiagonalbound + S (pfc_index_coefficient_shifted_leftdiagonal) = (S (t+i))) -> exists pfc_value_coefficient_shifted_leftdiagonal. ((((exists ff_h_pfp_coefficient_shifted_leftdiagonalentry. ff_h_pfp_coefficient_shifted_leftdiagonalentry + S (pfc_value_coefficient_shifted_leftdiagonal) = S ((S (pfc_index_coefficient_shifted_leftdiagonal)) * pfc_terms_scale_coefficient_shifted_left)) /\\ exists ff_q_pfp_coefficient_shifted_leftdiagonalentry. pfc_terms_code_coefficient_shifted_left = ff_q_pfp_coefficient_shifted_leftdiagonalentry * S ((S (pfc_index_coefficient_shifted_leftdiagonal)) * pfc_terms_scale_coefficient_shifted_left) + (pfc_value_coefficient_shifted_leftdiagonal))) /\\ ((exists pfc_complement_coefficient_shifted_leftdiagonalterm pfc_left_coefficient_shifted_leftdiagonalterm pfc_right_coefficient_shifted_leftdiagonalterm. (((pfc_index_coefficient_shifted_leftdiagonal)+pfc_complement_coefficient_shifted_leftdiagonalterm=(t+i)) /\\ ((((((exists pfa_gap_coefficient_shifted_leftdiagonaltermleftinside. pfa_gap_coefficient_shifted_leftdiagonaltermleftinside + S (pfc_index_coefficient_shifted_leftdiagonal) = (t+L)) /\\ ((((exists ff_h_pfp_coefficient_shifted_leftdiagonaltermleftentry. ff_h_pfp_coefficient_shifted_leftdiagonaltermleftentry + S (pfc_left_coefficient_shifted_leftdiagonalterm) = S ((S (pfc_index_coefficient_shifted_leftdiagonal)) * AC)) /\\ exists ff_q_pfp_coefficient_shifted_leftdiagonaltermleftentry. AB = ff_q_pfp_coefficient_shifted_leftdiagonaltermleftentry * S ((S (pfc_index_coefficient_shifted_leftdiagonal)) * AC) + (pfc_left_coefficient_shifted_leftdiagonalterm)))))) \\/ (((exists pfc_gap_coefficient_shifted_leftdiagonaltermleftoutside. pfc_gap_coefficient_shifted_leftdiagonaltermleftoutside+(t+L)=(pfc_index_coefficient_shifted_leftdiagonal)) /\\ (((pfc_left_coefficient_shifted_leftdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_coefficient_shifted_leftdiagonaltermrightinside. pfa_gap_coefficient_shifted_leftdiagonaltermrightinside + S (pfc_complement_coefficient_shifted_leftdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_coefficient_shifted_leftdiagonaltermrightentry. ff_h_pfp_coefficient_shifted_leftdiagonaltermrightentry + S (pfc_right_coefficient_shifted_leftdiagonalterm) = S ((S (pfc_complement_coefficient_shifted_leftdiagonalterm)) * bc)) /\\ exists ff_q_pfp_coefficient_shifted_leftdiagonaltermrightentry. bb = ff_q_pfp_coefficient_shifted_leftdiagonaltermrightentry * S ((S (pfc_complement_coefficient_shifted_leftdiagonalterm)) * bc) + (pfc_right_coefficient_shifted_leftdiagonalterm)))))) \\/ (((exists pfc_gap_coefficient_shifted_leftdiagonaltermrightoutside. pfc_gap_coefficient_shifted_leftdiagonaltermrightoutside+(M)=(pfc_complement_coefficient_shifted_leftdiagonalterm)) /\\ (((pfc_right_coefficient_shifted_leftdiagonalterm)=0))))) /\\ (((pfc_value_coefficient_shifted_leftdiagonal)=pfc_left_coefficient_shifted_leftdiagonalterm*pfc_right_coefficient_shifted_leftdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_coefficient_shifted_leftsum fs_v_pfc_coefficient_shifted_leftsum. ((((exists fs_h_pfc_coefficient_shifted_leftsum_body_start. fs_h_pfc_coefficient_shifted_leftsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_coefficient_shifted_leftsum)) /\\ exists fs_q_pfc_coefficient_shifted_leftsum_body_start. fs_u_pfc_coefficient_shifted_leftsum = fs_q_pfc_coefficient_shifted_leftsum_body_start * S ((S (0)) * fs_v_pfc_coefficient_shifted_leftsum) + (0))) /\\ ((((exists fs_h_pfc_coefficient_shifted_leftsum_body_terminal. fs_h_pfc_coefficient_shifted_leftsum_body_terminal + S (pfc_natural_sum_coefficient_shifted_left) = S ((S (S (t+i))) * fs_v_pfc_coefficient_shifted_leftsum)) /\\ exists fs_q_pfc_coefficient_shifted_leftsum_body_terminal. fs_u_pfc_coefficient_shifted_leftsum = fs_q_pfc_coefficient_shifted_leftsum_body_terminal * S ((S (S (t+i))) * fs_v_pfc_coefficient_shifted_leftsum) + (pfc_natural_sum_coefficient_shifted_left))) /\\ forall fs_i_pfc_coefficient_shifted_leftsum_body_steps. (exists fs_lt_pfc_coefficient_shifted_leftsum_body_steps_bound. fs_lt_pfc_coefficient_shifted_leftsum_body_steps_bound + S fs_i_pfc_coefficient_shifted_leftsum_body_steps = S (t+i)) -> exists fs_a_pfc_coefficient_shifted_leftsum_body_steps fs_r_pfc_coefficient_shifted_leftsum_body_steps fs_s_pfc_coefficient_shifted_leftsum_body_steps. ((((exists fs_h_pfc_coefficient_shifted_leftsum_body_steps_summand. fs_h_pfc_coefficient_shifted_leftsum_body_steps_summand + S (fs_a_pfc_coefficient_shifted_leftsum_body_steps) = S ((S (fs_i_pfc_coefficient_shifted_leftsum_body_steps)) * pfc_terms_scale_coefficient_shifted_left)) /\\ exists fs_q_pfc_coefficient_shifted_leftsum_body_steps_summand. pfc_terms_code_coefficient_shifted_left = fs_q_pfc_coefficient_shifted_leftsum_body_steps_summand * S ((S (fs_i_pfc_coefficient_shifted_leftsum_body_steps)) * pfc_terms_scale_coefficient_shifted_left) + (fs_a_pfc_coefficient_shifted_leftsum_body_steps))) /\\ ((((exists fs_h_pfc_coefficient_shifted_leftsum_body_steps_partial. fs_h_pfc_coefficient_shifted_leftsum_body_steps_partial + S (fs_r_pfc_coefficient_shifted_leftsum_body_steps) = S ((S (fs_i_pfc_coefficient_shifted_leftsum_body_steps)) * fs_v_pfc_coefficient_shifted_leftsum)) /\\ exists fs_q_pfc_coefficient_shifted_leftsum_body_steps_partial. fs_u_pfc_coefficient_shifted_leftsum = fs_q_pfc_coefficient_shifted_leftsum_body_steps_partial * S ((S (fs_i_pfc_coefficient_shifted_leftsum_body_steps)) * fs_v_pfc_coefficient_shifted_leftsum) + (fs_r_pfc_coefficient_shifted_leftsum_body_steps))) /\\ ((((exists fs_h_pfc_coefficient_shifted_leftsum_body_steps_successor. fs_h_pfc_coefficient_shifted_leftsum_body_steps_successor + S (fs_s_pfc_coefficient_shifted_leftsum_body_steps) = S ((S (S fs_i_pfc_coefficient_shifted_leftsum_body_steps)) * fs_v_pfc_coefficient_shifted_leftsum)) /\\ exists fs_q_pfc_coefficient_shifted_leftsum_body_steps_successor. fs_u_pfc_coefficient_shifted_leftsum = fs_q_pfc_coefficient_shifted_leftsum_body_steps_successor * S ((S (S fs_i_pfc_coefficient_shifted_leftsum_body_steps)) * fs_v_pfc_coefficient_shifted_leftsum) + (fs_s_pfc_coefficient_shifted_leftsum_body_steps))) /\\ fs_s_pfc_coefficient_shifted_leftsum_body_steps = fs_r_pfc_coefficient_shifted_leftsum_body_steps + fs_a_pfc_coefficient_shifted_leftsum_body_steps)))))) /\\ ((((exists pfa_gap_coefficient_shifted_leftresiduebound. pfa_gap_coefficient_shifted_leftresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_coefficient_shifted_leftresiduecongruence pfa_offset_right_coefficient_shifted_leftresiduecongruence. 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"script": [ "intro p", "intro ab", "intro ac", "intro L", "intro bb", "intro bc", "intro M", "intro BB", "intro BC", "intro t", "intro i", "intro r", "intro hpad", "intro hc", "cases hc", "cases hc_witness", "cases hc_witness_witness", "cases hc_witness_witness_witness", "cases hc_witness_witness_witness_right", "have hd : exists eb ec. forall pfc_index_coefficient_padded_diagonal_right. (exists pfa_gap_coefficient_padded_diagonal_rightbound. pfa_gap_coefficient_padded_diagonal_rightbound + S (pfc_index_coefficient_padded_diagonal_right) = (S (t+i))) -> exists pfc_value_coefficient_padded_diagonal_right. ((((exists ff_h_pfp_coefficient_padded_diagonal_rightentry. ff_h_pfp_coefficient_padded_diagonal_rightentry + S (pfc_value_coefficient_padded_diagonal_right) = S ((S (pfc_index_coefficient_padded_diagonal_right)) * ec)) /\\ exists ff_q_pfp_coefficient_padded_diagonal_rightentry. eb = ff_q_pfp_coefficient_padded_diagonal_rightentry * S ((S (pfc_index_coefficient_padded_diagonal_right)) * ec) + (pfc_value_coefficient_padded_diagonal_right))) /\\ ((exists pfc_complement_coefficient_padded_diagonal_rightterm pfc_left_coefficient_padded_diagonal_rightterm pfc_right_coefficient_padded_diagonal_rightterm. (((pfc_index_coefficient_padded_diagonal_right)+pfc_complement_coefficient_padded_diagonal_rightterm=(t+i)) /\\ ((((((exists pfa_gap_coefficient_padded_diagonal_righttermleftinside. pfa_gap_coefficient_padded_diagonal_righttermleftinside + S (pfc_index_coefficient_padded_diagonal_right) = (L)) /\\ ((((exists ff_h_pfp_coefficient_padded_diagonal_righttermleftentry. ff_h_pfp_coefficient_padded_diagonal_righttermleftentry + S (pfc_left_coefficient_padded_diagonal_rightterm) = S ((S (pfc_index_coefficient_padded_diagonal_right)) * ac)) /\\ exists ff_q_pfp_coefficient_padded_diagonal_righttermleftentry. ab = ff_q_pfp_coefficient_padded_diagonal_righttermleftentry * S ((S (pfc_index_coefficient_padded_diagonal_right)) * ac) + (pfc_left_coefficient_padded_diagonal_rightterm)))))) \\/ (((exists pfc_gap_coefficient_padded_diagonal_righttermleftoutside. pfc_gap_coefficient_padded_diagonal_righttermleftoutside+(L)=(pfc_index_coefficient_padded_diagonal_right)) /\\ (((pfc_left_coefficient_padded_diagonal_rightterm)=0))))) /\\ ((((((exists pfa_gap_coefficient_padded_diagonal_righttermrightinside. pfa_gap_coefficient_padded_diagonal_righttermrightinside + S (pfc_complement_coefficient_padded_diagonal_rightterm) = (t+M)) /\\ ((((exists ff_h_pfp_coefficient_padded_diagonal_righttermrightentry. ff_h_pfp_coefficient_padded_diagonal_righttermrightentry + S (pfc_right_coefficient_padded_diagonal_rightterm) = S ((S (pfc_complement_coefficient_padded_diagonal_rightterm)) * BC)) /\\ exists ff_q_pfp_coefficient_padded_diagonal_righttermrightentry. BB = ff_q_pfp_coefficient_padded_diagonal_righttermrightentry * S ((S (pfc_complement_coefficient_padded_diagonal_rightterm)) * BC) + (pfc_right_coefficient_padded_diagonal_rightterm)))))) \\/ (((exists pfc_gap_coefficient_padded_diagonal_righttermrightoutside. pfc_gap_coefficient_padded_diagonal_righttermrightoutside+(t+M)=(pfc_complement_coefficient_padded_diagonal_rightterm)) /\\ (((pfc_right_coefficient_padded_diagonal_rightterm)=0))))) /\\ (((pfc_value_coefficient_padded_diagonal_right)=pfc_left_coefficient_padded_diagonal_rightterm*pfc_right_coefficient_padded_diagonal_rightterm))))))))))", "specialize polynomial_diagonal_prefix_exists (ab)", "specialize polynomial_diagonal_prefix_exists (ac)", "specialize polynomial_diagonal_prefix_exists (L)", "specialize polynomial_diagonal_prefix_exists (BB)", "specialize polynomial_diagonal_prefix_exists (BC)", "specialize polynomial_diagonal_prefix_exists (t+M)", "specialize polynomial_diagonal_prefix_exists (t+i)", "apply polynomial_diagonal_prefix_exists", "cases hd", "cases hd_witness", "have hs : exists n. exists fs_u_pfc_coefficient_padded_sum_right fs_v_pfc_coefficient_padded_sum_right. ((((exists fs_h_pfc_coefficient_padded_sum_right_body_start. fs_h_pfc_coefficient_padded_sum_right_body_start + S (0) = S ((S (0)) * fs_v_pfc_coefficient_padded_sum_right)) /\\ exists fs_q_pfc_coefficient_padded_sum_right_body_start. fs_u_pfc_coefficient_padded_sum_right = fs_q_pfc_coefficient_padded_sum_right_body_start * S ((S (0)) * fs_v_pfc_coefficient_padded_sum_right) + (0))) /\\ ((((exists fs_h_pfc_coefficient_padded_sum_right_body_terminal. fs_h_pfc_coefficient_padded_sum_right_body_terminal + S (n) = S ((S (S (t+i))) * fs_v_pfc_coefficient_padded_sum_right)) /\\ exists fs_q_pfc_coefficient_padded_sum_right_body_terminal. fs_u_pfc_coefficient_padded_sum_right = fs_q_pfc_coefficient_padded_sum_right_body_terminal * S ((S (S (t+i))) * fs_v_pfc_coefficient_padded_sum_right) + (n))) /\\ forall fs_i_pfc_coefficient_padded_sum_right_body_steps. (exists fs_lt_pfc_coefficient_padded_sum_right_body_steps_bound. fs_lt_pfc_coefficient_padded_sum_right_body_steps_bound + S fs_i_pfc_coefficient_padded_sum_right_body_steps = S (t+i)) -> exists fs_a_pfc_coefficient_padded_sum_right_body_steps fs_r_pfc_coefficient_padded_sum_right_body_steps fs_s_pfc_coefficient_padded_sum_right_body_steps. ((((exists fs_h_pfc_coefficient_padded_sum_right_body_steps_summand. fs_h_pfc_coefficient_padded_sum_right_body_steps_summand + S (fs_a_pfc_coefficient_padded_sum_right_body_steps) = S ((S (fs_i_pfc_coefficient_padded_sum_right_body_steps)) * x4)) /\\ exists fs_q_pfc_coefficient_padded_sum_right_body_steps_summand. x3 = fs_q_pfc_coefficient_padded_sum_right_body_steps_summand * S ((S (fs_i_pfc_coefficient_padded_sum_right_body_steps)) * x4) + (fs_a_pfc_coefficient_padded_sum_right_body_steps))) /\\ ((((exists fs_h_pfc_coefficient_padded_sum_right_body_steps_partial. fs_h_pfc_coefficient_padded_sum_right_body_steps_partial + S (fs_r_pfc_coefficient_padded_sum_right_body_steps) = S ((S (fs_i_pfc_coefficient_padded_sum_right_body_steps)) * fs_v_pfc_coefficient_padded_sum_right)) /\\ exists fs_q_pfc_coefficient_padded_sum_right_body_steps_partial. fs_u_pfc_coefficient_padded_sum_right = fs_q_pfc_coefficient_padded_sum_right_body_steps_partial * S ((S (fs_i_pfc_coefficient_padded_sum_right_body_steps)) * fs_v_pfc_coefficient_padded_sum_right) + (fs_r_pfc_coefficient_padded_sum_right_body_steps))) /\\ ((((exists fs_h_pfc_coefficient_padded_sum_right_body_steps_successor. fs_h_pfc_coefficient_padded_sum_right_body_steps_successor + S (fs_s_pfc_coefficient_padded_sum_right_body_steps) = S ((S (S fs_i_pfc_coefficient_padded_sum_right_body_steps)) * fs_v_pfc_coefficient_padded_sum_right)) /\\ exists fs_q_pfc_coefficient_padded_sum_right_body_steps_successor. fs_u_pfc_coefficient_padded_sum_right = fs_q_pfc_coefficient_padded_sum_right_body_steps_successor * S ((S (S fs_i_pfc_coefficient_padded_sum_right_body_steps)) * fs_v_pfc_coefficient_padded_sum_right) + (fs_s_pfc_coefficient_padded_sum_right_body_steps))) /\\ fs_s_pfc_coefficient_padded_sum_right_body_steps = fs_r_pfc_coefficient_padded_sum_right_body_steps + fs_a_pfc_coefficient_padded_sum_right_body_steps)))))", "specialize beta_sum_exists (x3)", "specialize beta_sum_exists (x4)", "specialize beta_sum_exists (S (t+i))", "apply beta_sum_exists", "cases hs", "have hdata : ((forall mdr_i_pfp_coefficient_diagonal_equal mdr_a_pfp_coefficient_diagonal_equal. (exists mdr_gap_pfp_coefficient_diagonal_equalb. mdr_gap_pfp_coefficient_diagonal_equalb + S (mdr_i_pfp_coefficient_diagonal_equal) = (S i)) -> (((exists ff_h_mdr_pfp_coefficient_diagonal_equalo. ff_h_mdr_pfp_coefficient_diagonal_equalo + S (mdr_a_pfp_coefficient_diagonal_equal) = S ((S (mdr_i_pfp_coefficient_diagonal_equal)) * x1)) /\\ exists ff_q_mdr_pfp_coefficient_diagonal_equalo. x = ff_q_mdr_pfp_coefficient_diagonal_equalo * S ((S (mdr_i_pfp_coefficient_diagonal_equal)) * x1) + (mdr_a_pfp_coefficient_diagonal_equal))) -> (((exists ff_h_mdr_pfp_coefficient_diagonal_equaln. ff_h_mdr_pfp_coefficient_diagonal_equaln + S (mdr_a_pfp_coefficient_diagonal_equal) = S ((S (mdr_i_pfp_coefficient_diagonal_equal)) * x4)) /\\ exists ff_q_mdr_pfp_coefficient_diagonal_equaln. x3 = ff_q_mdr_pfp_coefficient_diagonal_equaln * S ((S (mdr_i_pfp_coefficient_diagonal_equal)) * x4) + (mdr_a_pfp_coefficient_diagonal_equal)))) /\\ ((forall pfpad_tail_index_coefficient_diagonal_tail. (exists pfa_gap_coefficient_diagonal_tailbound. pfa_gap_coefficient_diagonal_tailbound + S (pfpad_tail_index_coefficient_diagonal_tail) = (t)) -> (((exists ff_h_pfp_coefficient_diagonal_tailzero. ff_h_pfp_coefficient_diagonal_tailzero + S (0) = S ((S ((S i)+pfpad_tail_index_coefficient_diagonal_tail)) * x4)) /\\ exists ff_q_pfp_coefficient_diagonal_tailzero. x3 = ff_q_pfp_coefficient_diagonal_tailzero * S ((S ((S i)+pfpad_tail_index_coefficient_diagonal_tail)) * x4) + (0))))))", "specialize polynomial_diagonal_left_padding_right (ab)", "specialize polynomial_diagonal_left_padding_right (ac)", "specialize polynomial_diagonal_left_padding_right (L)", "specialize polynomial_diagonal_left_padding_right (bb)", "specialize polynomial_diagonal_left_padding_right (bc)", "specialize polynomial_diagonal_left_padding_right (M)", "specialize polynomial_diagonal_left_padding_right (BB)", "specialize polynomial_diagonal_left_padding_right (BC)", "specialize polynomial_diagonal_left_padding_right (t)", "specialize polynomial_diagonal_left_padding_right (i)", "specialize polynomial_diagonal_left_padding_right (x)", "specialize polynomial_diagonal_left_padding_right (x1)", "specialize polynomial_diagonal_left_padding_right (x3)", "specialize polynomial_diagonal_left_padding_right (x4)", "apply polynomial_diagonal_left_padding_right", "exact hpad", "exact hc_witness_witness_witness_left", "exact hd_witness_witness", "have heq : x5=x2", "cases hdata", "specialize polynomial_zero_tail_natural_sum_invariant (x)", "specialize polynomial_zero_tail_natural_sum_invariant (x1)", "specialize polynomial_zero_tail_natural_sum_invariant (x3)", "specialize polynomial_zero_tail_natural_sum_invariant (x4)", "specialize polynomial_zero_tail_natural_sum_invariant (S i)", "specialize polynomial_zero_tail_natural_sum_invariant (t)", "specialize polynomial_zero_tail_natural_sum_invariant (x2)", "specialize polynomial_zero_tail_natural_sum_invariant (x5)", "apply polynomial_zero_tail_natural_sum_invariant", "exact hdata_left", "exact hdata_right", "exact hc_witness_witness_witness_right_left", "have hlength : S i+t=S (t+i)", "simp [add_succ_left,add_comm]", "rewrite hlength", "rewrite hlength", "rewrite hlength", "exact hs_witness", "exists x3", "exists x4", "exists x2", "split", "exact hd_witness_witness", "split", "rewrite heq at hs_witness", "rewrite heq at hs_witness", "exact hs_witness", "exact hc_witness_witness_witness_right_right" ], "script_sha256": "9c3fa8433c96216f0a645581312c9024695b5290b3d8612abea93bc643f39cd2", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_convolution_padding_candidate.py", "sha256": "2d874ecfb35a5db0aecdeb07b549464efebad9072c363113aa5a0a977845d007" }, "statement": "forall p ab ac L bb bc M BB BC t i r. (((forall pfp_repeat_index_coefficient_padding_rightzeros. (exists pfa_gap_coefficient_padding_rightzerosindex. pfa_gap_coefficient_padding_rightzerosindex + S (pfp_repeat_index_coefficient_padding_rightzeros) = (t)) -> (((exists ff_h_pfp_coefficient_padding_rightzerosentry. ff_h_pfp_coefficient_padding_rightzerosentry + S (0) = S ((S (pfp_repeat_index_coefficient_padding_rightzeros)) * BC)) /\\ exists ff_q_pfp_coefficient_padding_rightzerosentry. BB = ff_q_pfp_coefficient_padding_rightzerosentry * S ((S (pfp_repeat_index_coefficient_padding_rightzeros)) * BC) + (0)))) /\\ ((forall pfrep_index_coefficient_padding_right pfrep_value_coefficient_padding_right. (exists pfa_gap_coefficient_padding_rightbound. pfa_gap_coefficient_padding_rightbound + S (pfrep_index_coefficient_padding_right) = (M)) -> (((exists ff_h_pfp_coefficient_padding_rightinput. ff_h_pfp_coefficient_padding_rightinput + S (pfrep_value_coefficient_padding_right) = S ((S (pfrep_index_coefficient_padding_right)) * bc)) /\\ exists ff_q_pfp_coefficient_padding_rightinput. bb = ff_q_pfp_coefficient_padding_rightinput * S ((S (pfrep_index_coefficient_padding_right)) * bc) + (pfrep_value_coefficient_padding_right))) -> (((exists ff_h_pfp_coefficient_padding_rightoutput. ff_h_pfp_coefficient_padding_rightoutput + S (pfrep_value_coefficient_padding_right) = S ((S ((t)+pfrep_index_coefficient_padding_right)) * BC)) /\\ exists ff_q_pfp_coefficient_padding_rightoutput. BB = ff_q_pfp_coefficient_padding_rightoutput * S ((S ((t)+pfrep_index_coefficient_padding_right)) * BC) + (pfrep_value_coefficient_padding_right))))))) -> (exists pfc_terms_code_coefficient_original_right pfc_terms_scale_coefficient_original_right pfc_natural_sum_coefficient_original_right. ((forall pfc_index_coefficient_original_rightdiagonal. (exists pfa_gap_coefficient_original_rightdiagonalbound. pfa_gap_coefficient_original_rightdiagonalbound + S (pfc_index_coefficient_original_rightdiagonal) = (S (i))) -> exists pfc_value_coefficient_original_rightdiagonal. ((((exists ff_h_pfp_coefficient_original_rightdiagonalentry. ff_h_pfp_coefficient_original_rightdiagonalentry + S (pfc_value_coefficient_original_rightdiagonal) = S ((S (pfc_index_coefficient_original_rightdiagonal)) * pfc_terms_scale_coefficient_original_right)) /\\ exists ff_q_pfp_coefficient_original_rightdiagonalentry. pfc_terms_code_coefficient_original_right = ff_q_pfp_coefficient_original_rightdiagonalentry * S ((S (pfc_index_coefficient_original_rightdiagonal)) * pfc_terms_scale_coefficient_original_right) + (pfc_value_coefficient_original_rightdiagonal))) /\\ ((exists pfc_complement_coefficient_original_rightdiagonalterm pfc_left_coefficient_original_rightdiagonalterm pfc_right_coefficient_original_rightdiagonalterm. (((pfc_index_coefficient_original_rightdiagonal)+pfc_complement_coefficient_original_rightdiagonalterm=(i)) /\\ ((((((exists pfa_gap_coefficient_original_rightdiagonaltermleftinside. pfa_gap_coefficient_original_rightdiagonaltermleftinside + S (pfc_index_coefficient_original_rightdiagonal) = (L)) /\\ ((((exists ff_h_pfp_coefficient_original_rightdiagonaltermleftentry. ff_h_pfp_coefficient_original_rightdiagonaltermleftentry + S (pfc_left_coefficient_original_rightdiagonalterm) = S ((S (pfc_index_coefficient_original_rightdiagonal)) * ac)) /\\ exists ff_q_pfp_coefficient_original_rightdiagonaltermleftentry. ab = ff_q_pfp_coefficient_original_rightdiagonaltermleftentry * S ((S (pfc_index_coefficient_original_rightdiagonal)) * ac) + (pfc_left_coefficient_original_rightdiagonalterm)))))) \\/ (((exists pfc_gap_coefficient_original_rightdiagonaltermleftoutside. pfc_gap_coefficient_original_rightdiagonaltermleftoutside+(L)=(pfc_index_coefficient_original_rightdiagonal)) /\\ (((pfc_left_coefficient_original_rightdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_coefficient_original_rightdiagonaltermrightinside. pfa_gap_coefficient_original_rightdiagonaltermrightinside + S (pfc_complement_coefficient_original_rightdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_coefficient_original_rightdiagonaltermrightentry. ff_h_pfp_coefficient_original_rightdiagonaltermrightentry + S (pfc_right_coefficient_original_rightdiagonalterm) = S ((S (pfc_complement_coefficient_original_rightdiagonalterm)) * bc)) /\\ exists ff_q_pfp_coefficient_original_rightdiagonaltermrightentry. bb = ff_q_pfp_coefficient_original_rightdiagonaltermrightentry * S ((S (pfc_complement_coefficient_original_rightdiagonalterm)) * bc) + (pfc_right_coefficient_original_rightdiagonalterm)))))) \\/ (((exists pfc_gap_coefficient_original_rightdiagonaltermrightoutside. pfc_gap_coefficient_original_rightdiagonaltermrightoutside+(M)=(pfc_complement_coefficient_original_rightdiagonalterm)) /\\ (((pfc_right_coefficient_original_rightdiagonalterm)=0))))) /\\ (((pfc_value_coefficient_original_rightdiagonal)=pfc_left_coefficient_original_rightdiagonalterm*pfc_right_coefficient_original_rightdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_coefficient_original_rightsum fs_v_pfc_coefficient_original_rightsum. ((((exists fs_h_pfc_coefficient_original_rightsum_body_start. fs_h_pfc_coefficient_original_rightsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_coefficient_original_rightsum)) /\\ exists fs_q_pfc_coefficient_original_rightsum_body_start. fs_u_pfc_coefficient_original_rightsum = fs_q_pfc_coefficient_original_rightsum_body_start * S ((S (0)) * fs_v_pfc_coefficient_original_rightsum) + (0))) /\\ ((((exists fs_h_pfc_coefficient_original_rightsum_body_terminal. fs_h_pfc_coefficient_original_rightsum_body_terminal + S (pfc_natural_sum_coefficient_original_right) = S ((S (S (i))) * fs_v_pfc_coefficient_original_rightsum)) /\\ exists fs_q_pfc_coefficient_original_rightsum_body_terminal. fs_u_pfc_coefficient_original_rightsum = fs_q_pfc_coefficient_original_rightsum_body_terminal * S ((S (S (i))) * fs_v_pfc_coefficient_original_rightsum) + (pfc_natural_sum_coefficient_original_right))) /\\ forall fs_i_pfc_coefficient_original_rightsum_body_steps. (exists fs_lt_pfc_coefficient_original_rightsum_body_steps_bound. fs_lt_pfc_coefficient_original_rightsum_body_steps_bound + S fs_i_pfc_coefficient_original_rightsum_body_steps = S (i)) -> exists fs_a_pfc_coefficient_original_rightsum_body_steps fs_r_pfc_coefficient_original_rightsum_body_steps fs_s_pfc_coefficient_original_rightsum_body_steps. ((((exists fs_h_pfc_coefficient_original_rightsum_body_steps_summand. fs_h_pfc_coefficient_original_rightsum_body_steps_summand + S (fs_a_pfc_coefficient_original_rightsum_body_steps) = S ((S (fs_i_pfc_coefficient_original_rightsum_body_steps)) * pfc_terms_scale_coefficient_original_right)) /\\ exists fs_q_pfc_coefficient_original_rightsum_body_steps_summand. pfc_terms_code_coefficient_original_right = fs_q_pfc_coefficient_original_rightsum_body_steps_summand * S ((S (fs_i_pfc_coefficient_original_rightsum_body_steps)) * pfc_terms_scale_coefficient_original_right) + (fs_a_pfc_coefficient_original_rightsum_body_steps))) /\\ ((((exists fs_h_pfc_coefficient_original_rightsum_body_steps_partial. fs_h_pfc_coefficient_original_rightsum_body_steps_partial + S (fs_r_pfc_coefficient_original_rightsum_body_steps) = S ((S (fs_i_pfc_coefficient_original_rightsum_body_steps)) * fs_v_pfc_coefficient_original_rightsum)) /\\ exists fs_q_pfc_coefficient_original_rightsum_body_steps_partial. fs_u_pfc_coefficient_original_rightsum = fs_q_pfc_coefficient_original_rightsum_body_steps_partial * S ((S (fs_i_pfc_coefficient_original_rightsum_body_steps)) * fs_v_pfc_coefficient_original_rightsum) + (fs_r_pfc_coefficient_original_rightsum_body_steps))) /\\ ((((exists fs_h_pfc_coefficient_original_rightsum_body_steps_successor. fs_h_pfc_coefficient_original_rightsum_body_steps_successor + S (fs_s_pfc_coefficient_original_rightsum_body_steps) = S ((S (S fs_i_pfc_coefficient_original_rightsum_body_steps)) * fs_v_pfc_coefficient_original_rightsum)) /\\ exists fs_q_pfc_coefficient_original_rightsum_body_steps_successor. fs_u_pfc_coefficient_original_rightsum = fs_q_pfc_coefficient_original_rightsum_body_steps_successor * S ((S (S fs_i_pfc_coefficient_original_rightsum_body_steps)) * fs_v_pfc_coefficient_original_rightsum) + (fs_s_pfc_coefficient_original_rightsum_body_steps))) /\\ fs_s_pfc_coefficient_original_rightsum_body_steps = fs_r_pfc_coefficient_original_rightsum_body_steps + fs_a_pfc_coefficient_original_rightsum_body_steps)))))) /\\ ((((exists pfa_gap_coefficient_original_rightresiduebound. pfa_gap_coefficient_original_rightresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_coefficient_original_rightresiduecongruence pfa_offset_right_coefficient_original_rightresiduecongruence. (pfc_natural_sum_coefficient_original_right) + (p) * pfa_offset_left_coefficient_original_rightresiduecongruence = (r) + (p) * pfa_offset_right_coefficient_original_rightresiduecongruence))))))))) -> (exists pfc_terms_code_coefficient_shifted_right pfc_terms_scale_coefficient_shifted_right pfc_natural_sum_coefficient_shifted_right. ((forall pfc_index_coefficient_shifted_rightdiagonal. (exists pfa_gap_coefficient_shifted_rightdiagonalbound. pfa_gap_coefficient_shifted_rightdiagonalbound + S (pfc_index_coefficient_shifted_rightdiagonal) = (S (t+i))) -> exists pfc_value_coefficient_shifted_rightdiagonal. ((((exists ff_h_pfp_coefficient_shifted_rightdiagonalentry. ff_h_pfp_coefficient_shifted_rightdiagonalentry + S (pfc_value_coefficient_shifted_rightdiagonal) = S ((S (pfc_index_coefficient_shifted_rightdiagonal)) * pfc_terms_scale_coefficient_shifted_right)) /\\ exists ff_q_pfp_coefficient_shifted_rightdiagonalentry. pfc_terms_code_coefficient_shifted_right = ff_q_pfp_coefficient_shifted_rightdiagonalentry * S ((S (pfc_index_coefficient_shifted_rightdiagonal)) * pfc_terms_scale_coefficient_shifted_right) + (pfc_value_coefficient_shifted_rightdiagonal))) /\\ ((exists pfc_complement_coefficient_shifted_rightdiagonalterm pfc_left_coefficient_shifted_rightdiagonalterm pfc_right_coefficient_shifted_rightdiagonalterm. (((pfc_index_coefficient_shifted_rightdiagonal)+pfc_complement_coefficient_shifted_rightdiagonalterm=(t+i)) /\\ ((((((exists pfa_gap_coefficient_shifted_rightdiagonaltermleftinside. pfa_gap_coefficient_shifted_rightdiagonaltermleftinside + S (pfc_index_coefficient_shifted_rightdiagonal) = (L)) /\\ ((((exists ff_h_pfp_coefficient_shifted_rightdiagonaltermleftentry. ff_h_pfp_coefficient_shifted_rightdiagonaltermleftentry + S (pfc_left_coefficient_shifted_rightdiagonalterm) = S ((S (pfc_index_coefficient_shifted_rightdiagonal)) * ac)) /\\ exists ff_q_pfp_coefficient_shifted_rightdiagonaltermleftentry. ab = ff_q_pfp_coefficient_shifted_rightdiagonaltermleftentry * S ((S (pfc_index_coefficient_shifted_rightdiagonal)) * ac) + (pfc_left_coefficient_shifted_rightdiagonalterm)))))) \\/ (((exists pfc_gap_coefficient_shifted_rightdiagonaltermleftoutside. pfc_gap_coefficient_shifted_rightdiagonaltermleftoutside+(L)=(pfc_index_coefficient_shifted_rightdiagonal)) /\\ (((pfc_left_coefficient_shifted_rightdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_coefficient_shifted_rightdiagonaltermrightinside. pfa_gap_coefficient_shifted_rightdiagonaltermrightinside + S (pfc_complement_coefficient_shifted_rightdiagonalterm) = (t+M)) /\\ ((((exists ff_h_pfp_coefficient_shifted_rightdiagonaltermrightentry. ff_h_pfp_coefficient_shifted_rightdiagonaltermrightentry + S (pfc_right_coefficient_shifted_rightdiagonalterm) = S ((S (pfc_complement_coefficient_shifted_rightdiagonalterm)) * BC)) /\\ exists ff_q_pfp_coefficient_shifted_rightdiagonaltermrightentry. BB = ff_q_pfp_coefficient_shifted_rightdiagonaltermrightentry * S ((S (pfc_complement_coefficient_shifted_rightdiagonalterm)) * BC) + (pfc_right_coefficient_shifted_rightdiagonalterm)))))) \\/ (((exists pfc_gap_coefficient_shifted_rightdiagonaltermrightoutside. pfc_gap_coefficient_shifted_rightdiagonaltermrightoutside+(t+M)=(pfc_complement_coefficient_shifted_rightdiagonalterm)) /\\ (((pfc_right_coefficient_shifted_rightdiagonalterm)=0))))) /\\ (((pfc_value_coefficient_shifted_rightdiagonal)=pfc_left_coefficient_shifted_rightdiagonalterm*pfc_right_coefficient_shifted_rightdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_coefficient_shifted_rightsum fs_v_pfc_coefficient_shifted_rightsum. ((((exists fs_h_pfc_coefficient_shifted_rightsum_body_start. fs_h_pfc_coefficient_shifted_rightsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_coefficient_shifted_rightsum)) /\\ exists fs_q_pfc_coefficient_shifted_rightsum_body_start. fs_u_pfc_coefficient_shifted_rightsum = fs_q_pfc_coefficient_shifted_rightsum_body_start * S ((S (0)) * fs_v_pfc_coefficient_shifted_rightsum) + (0))) /\\ ((((exists fs_h_pfc_coefficient_shifted_rightsum_body_terminal. fs_h_pfc_coefficient_shifted_rightsum_body_terminal + S (pfc_natural_sum_coefficient_shifted_right) = S ((S (S (t+i))) * fs_v_pfc_coefficient_shifted_rightsum)) /\\ exists fs_q_pfc_coefficient_shifted_rightsum_body_terminal. fs_u_pfc_coefficient_shifted_rightsum = fs_q_pfc_coefficient_shifted_rightsum_body_terminal * S ((S (S (t+i))) * fs_v_pfc_coefficient_shifted_rightsum) + (pfc_natural_sum_coefficient_shifted_right))) /\\ forall fs_i_pfc_coefficient_shifted_rightsum_body_steps. (exists fs_lt_pfc_coefficient_shifted_rightsum_body_steps_bound. fs_lt_pfc_coefficient_shifted_rightsum_body_steps_bound + S fs_i_pfc_coefficient_shifted_rightsum_body_steps = S (t+i)) -> exists fs_a_pfc_coefficient_shifted_rightsum_body_steps fs_r_pfc_coefficient_shifted_rightsum_body_steps fs_s_pfc_coefficient_shifted_rightsum_body_steps. ((((exists fs_h_pfc_coefficient_shifted_rightsum_body_steps_summand. fs_h_pfc_coefficient_shifted_rightsum_body_steps_summand + S (fs_a_pfc_coefficient_shifted_rightsum_body_steps) = S ((S (fs_i_pfc_coefficient_shifted_rightsum_body_steps)) * pfc_terms_scale_coefficient_shifted_right)) /\\ exists fs_q_pfc_coefficient_shifted_rightsum_body_steps_summand. pfc_terms_code_coefficient_shifted_right = fs_q_pfc_coefficient_shifted_rightsum_body_steps_summand * S ((S (fs_i_pfc_coefficient_shifted_rightsum_body_steps)) * pfc_terms_scale_coefficient_shifted_right) + (fs_a_pfc_coefficient_shifted_rightsum_body_steps))) /\\ ((((exists fs_h_pfc_coefficient_shifted_rightsum_body_steps_partial. fs_h_pfc_coefficient_shifted_rightsum_body_steps_partial + S (fs_r_pfc_coefficient_shifted_rightsum_body_steps) = S ((S (fs_i_pfc_coefficient_shifted_rightsum_body_steps)) * fs_v_pfc_coefficient_shifted_rightsum)) /\\ exists fs_q_pfc_coefficient_shifted_rightsum_body_steps_partial. fs_u_pfc_coefficient_shifted_rightsum = fs_q_pfc_coefficient_shifted_rightsum_body_steps_partial * S ((S (fs_i_pfc_coefficient_shifted_rightsum_body_steps)) * fs_v_pfc_coefficient_shifted_rightsum) + (fs_r_pfc_coefficient_shifted_rightsum_body_steps))) /\\ ((((exists fs_h_pfc_coefficient_shifted_rightsum_body_steps_successor. fs_h_pfc_coefficient_shifted_rightsum_body_steps_successor + S (fs_s_pfc_coefficient_shifted_rightsum_body_steps) = S ((S (S fs_i_pfc_coefficient_shifted_rightsum_body_steps)) * fs_v_pfc_coefficient_shifted_rightsum)) /\\ exists fs_q_pfc_coefficient_shifted_rightsum_body_steps_successor. fs_u_pfc_coefficient_shifted_rightsum = fs_q_pfc_coefficient_shifted_rightsum_body_steps_successor * S ((S (S fs_i_pfc_coefficient_shifted_rightsum_body_steps)) * fs_v_pfc_coefficient_shifted_rightsum) + (fs_s_pfc_coefficient_shifted_rightsum_body_steps))) /\\ fs_s_pfc_coefficient_shifted_rightsum_body_steps = fs_r_pfc_coefficient_shifted_rightsum_body_steps + fs_a_pfc_coefficient_shifted_rightsum_body_steps)))))) /\\ ((((exists pfa_gap_coefficient_shifted_rightresiduebound. pfa_gap_coefficient_shifted_rightresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_coefficient_shifted_rightresiduecongruence pfa_offset_right_coefficient_shifted_rightresiduecongruence. (pfc_natural_sum_coefficient_shifted_right) + (p) * pfa_offset_left_coefficient_shifted_rightresiduecongruence = (r) + (p) * pfa_offset_right_coefficient_shifted_rightresiduecongruence)))))))))", "statement_sha256": "d990a071c03b8ea62cd9f5a733f316441903a2c6659959fa349b2b16b4acf297", "summary": "Construct an actual padded antidiagonal table and actual sum trace, proving the shifted coefficient has the same canonical residue without assuming an equality of sums.", "summary_sha256": "9bc366c01bbdfe7d1d1ef6b9d1a956b89f3ae6712985593f5e40c7a00dda5ac8" }, { "alpha_v33_frontier_enrollment": { "body_receipt_sha256": "15e9478ea9a23bcd8fe78c2b09625e3bc2b5e12135efb202ba883386836977e1", "bundle_campaign": "polynomial-euclidean-division", "bundle_node_id": 358, "bundle_sha256": "6ae667d8518e4dbe722bb08ad1b08715a0d282c2893e533c8133d770fe861dcf", "campaign": "polynomial-euclidean-division", "first_enrolled_version": "v33", "parent_catalog_sha256": 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"ha" ], "script": [ "intro p", "intro ab", "intro ac", "intro L", "intro bb", "intro bc", "intro M", "intro AB", "intro AC", "intro t", "intro i", "intro r", "intro hp", "intro hpad", "intro hi", "intro hc", "cases hc", "cases hc_witness", "cases hc_witness_witness", "cases hc_witness_witness_witness", "cases hc_witness_witness_witness_right", "have hz : forall pfp_repeat_index_before_coefficient_zero_diagonal_left. (exists pfa_gap_before_coefficient_zero_diagonal_leftindex. pfa_gap_before_coefficient_zero_diagonal_leftindex + S (pfp_repeat_index_before_coefficient_zero_diagonal_left) = (S i)) -> (((exists ff_h_pfp_before_coefficient_zero_diagonal_leftentry. ff_h_pfp_before_coefficient_zero_diagonal_leftentry + S (0) = S ((S (pfp_repeat_index_before_coefficient_zero_diagonal_left)) * x1)) /\\ exists ff_q_pfp_before_coefficient_zero_diagonal_leftentry. x = ff_q_pfp_before_coefficient_zero_diagonal_leftentry * S ((S (pfp_repeat_index_before_coefficient_zero_diagonal_left)) * x1) + (0)))", "intro j", "intro hj", "have ht : exists z. ((((exists ff_h_pfp_before_coefficient_entry_left. ff_h_pfp_before_coefficient_entry_left + S (z) = S ((S (j)) * x1)) /\\ exists ff_q_pfp_before_coefficient_entry_left. x = ff_q_pfp_before_coefficient_entry_left * S ((S (j)) * x1) + (z))) /\\ ((exists pfc_complement_before_coefficient_term_left pfc_left_before_coefficient_term_left pfc_right_before_coefficient_term_left. (((j)+pfc_complement_before_coefficient_term_left=(i)) /\\ ((((((exists pfa_gap_before_coefficient_term_leftleftinside. pfa_gap_before_coefficient_term_leftleftinside + S (j) = (t+L)) /\\ ((((exists ff_h_pfp_before_coefficient_term_leftleftentry. ff_h_pfp_before_coefficient_term_leftleftentry + S (pfc_left_before_coefficient_term_left) = S ((S (j)) * AC)) /\\ exists ff_q_pfp_before_coefficient_term_leftleftentry. AB = ff_q_pfp_before_coefficient_term_leftleftentry * S ((S (j)) * AC) + (pfc_left_before_coefficient_term_left)))))) \\/ (((exists pfc_gap_before_coefficient_term_leftleftoutside. pfc_gap_before_coefficient_term_leftleftoutside+(t+L)=(j)) /\\ (((pfc_left_before_coefficient_term_left)=0))))) /\\ ((((((exists pfa_gap_before_coefficient_term_leftrightinside. pfa_gap_before_coefficient_term_leftrightinside + S (pfc_complement_before_coefficient_term_left) = (M)) /\\ ((((exists ff_h_pfp_before_coefficient_term_leftrightentry. ff_h_pfp_before_coefficient_term_leftrightentry + S (pfc_right_before_coefficient_term_left) = S ((S (pfc_complement_before_coefficient_term_left)) * bc)) /\\ exists ff_q_pfp_before_coefficient_term_leftrightentry. bb = ff_q_pfp_before_coefficient_term_leftrightentry * S ((S (pfc_complement_before_coefficient_term_left)) * bc) + (pfc_right_before_coefficient_term_left)))))) \\/ (((exists pfc_gap_before_coefficient_term_leftrightoutside. pfc_gap_before_coefficient_term_leftrightoutside+(M)=(pfc_complement_before_coefficient_term_left)) /\\ (((pfc_right_before_coefficient_term_left)=0))))) /\\ (((z)=pfc_left_before_coefficient_term_left*pfc_right_before_coefficient_term_left))))))))))", "specialize hc_witness_witness_witness_left (j)", "apply hc_witness_witness_witness_left", "exact hj", "cases ht", "cases ht_witness", "have heq : x3=0", "specialize polynomial_diagonal_term_left_padding_zero_left (ab)", "specialize polynomial_diagonal_term_left_padding_zero_left (ac)", "specialize polynomial_diagonal_term_left_padding_zero_left (L)", "specialize polynomial_diagonal_term_left_padding_zero_left (bb)", "specialize polynomial_diagonal_term_left_padding_zero_left (bc)", "specialize polynomial_diagonal_term_left_padding_zero_left (M)", "specialize polynomial_diagonal_term_left_padding_zero_left (AB)", "specialize polynomial_diagonal_term_left_padding_zero_left (AC)", "specialize polynomial_diagonal_term_left_padding_zero_left (t)", "specialize polynomial_diagonal_term_left_padding_zero_left (i)", "specialize polynomial_diagonal_term_left_padding_zero_left (j)", "specialize polynomial_diagonal_term_left_padding_zero_left (x3)", "apply polynomial_diagonal_term_left_padding_zero_left", "exact hpad", "specialize le_trans (S j)", "specialize le_trans (S i)", "specialize le_trans (t)", "apply le_trans", "exact hj", "exact hi", "exact ht_witness_right", "rewrite heq at ht_witness_left", "rewrite heq at ht_witness_left", "exact ht_witness_left", "have hsum : x2=0", "trans (S i)*0", "specialize beta_repeat_sum_exact (x)", "specialize beta_repeat_sum_exact (x1)", "specialize beta_repeat_sum_exact (0)", "specialize beta_repeat_sum_exact (S i)", "specialize beta_repeat_sum_exact (x2)", "apply beta_repeat_sum_exact", "exact hz", "exact hc_witness_witness_witness_right_left", "simp", "rewrite hsum at hc_witness_witness_witness_right_right", "specialize prime_field_residue_bounded_value (p)", "specialize prime_field_residue_bounded_value (0)", "specialize prime_field_residue_bounded_value (r)", "apply prime_field_residue_bounded_value", "specialize one_le_of_ne_zero (p)", "apply one_le_of_ne_zero", "exact hp", "exact hc_witness_witness_witness_right_right" ], "script_sha256": "4f64526cb9dcff7e09bb5f69615d6b77d8c6873556d4f1590580b73ff8f94f0d", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_convolution_padding_candidate.py", "sha256": "2d874ecfb35a5db0aecdeb07b549464efebad9072c363113aa5a0a977845d007" }, "statement": "forall p ab ac L bb bc M AB AC t i r. (~(p=0)) -> (((forall pfp_repeat_index_before_coefficient_pad_leftzeros. (exists pfa_gap_before_coefficient_pad_leftzerosindex. pfa_gap_before_coefficient_pad_leftzerosindex + S (pfp_repeat_index_before_coefficient_pad_leftzeros) = (t)) -> (((exists ff_h_pfp_before_coefficient_pad_leftzerosentry. ff_h_pfp_before_coefficient_pad_leftzerosentry + S (0) = S ((S (pfp_repeat_index_before_coefficient_pad_leftzeros)) * AC)) /\\ exists ff_q_pfp_before_coefficient_pad_leftzerosentry. AB = ff_q_pfp_before_coefficient_pad_leftzerosentry * S ((S (pfp_repeat_index_before_coefficient_pad_leftzeros)) * AC) + (0)))) /\\ ((forall pfrep_index_before_coefficient_pad_left pfrep_value_before_coefficient_pad_left. (exists pfa_gap_before_coefficient_pad_leftbound. pfa_gap_before_coefficient_pad_leftbound + S (pfrep_index_before_coefficient_pad_left) = (L)) -> (((exists ff_h_pfp_before_coefficient_pad_leftinput. ff_h_pfp_before_coefficient_pad_leftinput + S (pfrep_value_before_coefficient_pad_left) = S ((S (pfrep_index_before_coefficient_pad_left)) * ac)) /\\ exists ff_q_pfp_before_coefficient_pad_leftinput. ab = ff_q_pfp_before_coefficient_pad_leftinput * S ((S (pfrep_index_before_coefficient_pad_left)) * ac) + (pfrep_value_before_coefficient_pad_left))) -> (((exists ff_h_pfp_before_coefficient_pad_leftoutput. ff_h_pfp_before_coefficient_pad_leftoutput + S (pfrep_value_before_coefficient_pad_left) = S ((S ((t)+pfrep_index_before_coefficient_pad_left)) * AC)) /\\ exists ff_q_pfp_before_coefficient_pad_leftoutput. AB = ff_q_pfp_before_coefficient_pad_leftoutput * S ((S ((t)+pfrep_index_before_coefficient_pad_left)) * AC) + (pfrep_value_before_coefficient_pad_left))))))) -> (exists pfa_gap_before_coefficient_index_left. pfa_gap_before_coefficient_index_left + S (i) = (t)) -> (exists pfc_terms_code_before_coefficient_actual_left pfc_terms_scale_before_coefficient_actual_left pfc_natural_sum_before_coefficient_actual_left. ((forall pfc_index_before_coefficient_actual_leftdiagonal. (exists pfa_gap_before_coefficient_actual_leftdiagonalbound. pfa_gap_before_coefficient_actual_leftdiagonalbound + S (pfc_index_before_coefficient_actual_leftdiagonal) = (S (i))) -> exists pfc_value_before_coefficient_actual_leftdiagonal. ((((exists ff_h_pfp_before_coefficient_actual_leftdiagonalentry. ff_h_pfp_before_coefficient_actual_leftdiagonalentry + S (pfc_value_before_coefficient_actual_leftdiagonal) = S ((S (pfc_index_before_coefficient_actual_leftdiagonal)) * pfc_terms_scale_before_coefficient_actual_left)) /\\ exists ff_q_pfp_before_coefficient_actual_leftdiagonalentry. pfc_terms_code_before_coefficient_actual_left = ff_q_pfp_before_coefficient_actual_leftdiagonalentry * S ((S (pfc_index_before_coefficient_actual_leftdiagonal)) * pfc_terms_scale_before_coefficient_actual_left) + (pfc_value_before_coefficient_actual_leftdiagonal))) /\\ ((exists pfc_complement_before_coefficient_actual_leftdiagonalterm pfc_left_before_coefficient_actual_leftdiagonalterm pfc_right_before_coefficient_actual_leftdiagonalterm. (((pfc_index_before_coefficient_actual_leftdiagonal)+pfc_complement_before_coefficient_actual_leftdiagonalterm=(i)) /\\ ((((((exists pfa_gap_before_coefficient_actual_leftdiagonaltermleftinside. pfa_gap_before_coefficient_actual_leftdiagonaltermleftinside + S (pfc_index_before_coefficient_actual_leftdiagonal) = (t+L)) /\\ ((((exists ff_h_pfp_before_coefficient_actual_leftdiagonaltermleftentry. ff_h_pfp_before_coefficient_actual_leftdiagonaltermleftentry + S (pfc_left_before_coefficient_actual_leftdiagonalterm) = S ((S (pfc_index_before_coefficient_actual_leftdiagonal)) * AC)) /\\ exists ff_q_pfp_before_coefficient_actual_leftdiagonaltermleftentry. AB = ff_q_pfp_before_coefficient_actual_leftdiagonaltermleftentry * S ((S (pfc_index_before_coefficient_actual_leftdiagonal)) * AC) + (pfc_left_before_coefficient_actual_leftdiagonalterm)))))) \\/ (((exists pfc_gap_before_coefficient_actual_leftdiagonaltermleftoutside. pfc_gap_before_coefficient_actual_leftdiagonaltermleftoutside+(t+L)=(pfc_index_before_coefficient_actual_leftdiagonal)) /\\ (((pfc_left_before_coefficient_actual_leftdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_before_coefficient_actual_leftdiagonaltermrightinside. pfa_gap_before_coefficient_actual_leftdiagonaltermrightinside + S (pfc_complement_before_coefficient_actual_leftdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_before_coefficient_actual_leftdiagonaltermrightentry. ff_h_pfp_before_coefficient_actual_leftdiagonaltermrightentry + S (pfc_right_before_coefficient_actual_leftdiagonalterm) = S ((S (pfc_complement_before_coefficient_actual_leftdiagonalterm)) * bc)) /\\ exists ff_q_pfp_before_coefficient_actual_leftdiagonaltermrightentry. bb = ff_q_pfp_before_coefficient_actual_leftdiagonaltermrightentry * S ((S (pfc_complement_before_coefficient_actual_leftdiagonalterm)) * bc) + (pfc_right_before_coefficient_actual_leftdiagonalterm)))))) \\/ (((exists pfc_gap_before_coefficient_actual_leftdiagonaltermrightoutside. pfc_gap_before_coefficient_actual_leftdiagonaltermrightoutside+(M)=(pfc_complement_before_coefficient_actual_leftdiagonalterm)) /\\ (((pfc_right_before_coefficient_actual_leftdiagonalterm)=0))))) /\\ (((pfc_value_before_coefficient_actual_leftdiagonal)=pfc_left_before_coefficient_actual_leftdiagonalterm*pfc_right_before_coefficient_actual_leftdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_before_coefficient_actual_leftsum fs_v_pfc_before_coefficient_actual_leftsum. ((((exists fs_h_pfc_before_coefficient_actual_leftsum_body_start. fs_h_pfc_before_coefficient_actual_leftsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_before_coefficient_actual_leftsum)) /\\ exists fs_q_pfc_before_coefficient_actual_leftsum_body_start. fs_u_pfc_before_coefficient_actual_leftsum = fs_q_pfc_before_coefficient_actual_leftsum_body_start * S ((S (0)) * fs_v_pfc_before_coefficient_actual_leftsum) + (0))) /\\ ((((exists fs_h_pfc_before_coefficient_actual_leftsum_body_terminal. fs_h_pfc_before_coefficient_actual_leftsum_body_terminal + S (pfc_natural_sum_before_coefficient_actual_left) = S ((S (S (i))) * fs_v_pfc_before_coefficient_actual_leftsum)) /\\ exists fs_q_pfc_before_coefficient_actual_leftsum_body_terminal. fs_u_pfc_before_coefficient_actual_leftsum = fs_q_pfc_before_coefficient_actual_leftsum_body_terminal * S ((S (S (i))) * fs_v_pfc_before_coefficient_actual_leftsum) + (pfc_natural_sum_before_coefficient_actual_left))) /\\ forall fs_i_pfc_before_coefficient_actual_leftsum_body_steps. (exists fs_lt_pfc_before_coefficient_actual_leftsum_body_steps_bound. fs_lt_pfc_before_coefficient_actual_leftsum_body_steps_bound + S fs_i_pfc_before_coefficient_actual_leftsum_body_steps = S (i)) -> exists fs_a_pfc_before_coefficient_actual_leftsum_body_steps fs_r_pfc_before_coefficient_actual_leftsum_body_steps fs_s_pfc_before_coefficient_actual_leftsum_body_steps. ((((exists fs_h_pfc_before_coefficient_actual_leftsum_body_steps_summand. fs_h_pfc_before_coefficient_actual_leftsum_body_steps_summand + S (fs_a_pfc_before_coefficient_actual_leftsum_body_steps) = S ((S (fs_i_pfc_before_coefficient_actual_leftsum_body_steps)) * pfc_terms_scale_before_coefficient_actual_left)) /\\ exists fs_q_pfc_before_coefficient_actual_leftsum_body_steps_summand. pfc_terms_code_before_coefficient_actual_left = fs_q_pfc_before_coefficient_actual_leftsum_body_steps_summand * S ((S (fs_i_pfc_before_coefficient_actual_leftsum_body_steps)) * pfc_terms_scale_before_coefficient_actual_left) + (fs_a_pfc_before_coefficient_actual_leftsum_body_steps))) /\\ ((((exists fs_h_pfc_before_coefficient_actual_leftsum_body_steps_partial. fs_h_pfc_before_coefficient_actual_leftsum_body_steps_partial + S (fs_r_pfc_before_coefficient_actual_leftsum_body_steps) = S ((S (fs_i_pfc_before_coefficient_actual_leftsum_body_steps)) * fs_v_pfc_before_coefficient_actual_leftsum)) /\\ exists fs_q_pfc_before_coefficient_actual_leftsum_body_steps_partial. fs_u_pfc_before_coefficient_actual_leftsum = fs_q_pfc_before_coefficient_actual_leftsum_body_steps_partial * S ((S (fs_i_pfc_before_coefficient_actual_leftsum_body_steps)) * fs_v_pfc_before_coefficient_actual_leftsum) + (fs_r_pfc_before_coefficient_actual_leftsum_body_steps))) /\\ ((((exists fs_h_pfc_before_coefficient_actual_leftsum_body_steps_successor. fs_h_pfc_before_coefficient_actual_leftsum_body_steps_successor + S (fs_s_pfc_before_coefficient_actual_leftsum_body_steps) = S ((S (S fs_i_pfc_before_coefficient_actual_leftsum_body_steps)) * fs_v_pfc_before_coefficient_actual_leftsum)) /\\ exists fs_q_pfc_before_coefficient_actual_leftsum_body_steps_successor. fs_u_pfc_before_coefficient_actual_leftsum = fs_q_pfc_before_coefficient_actual_leftsum_body_steps_successor * S ((S (S fs_i_pfc_before_coefficient_actual_leftsum_body_steps)) * fs_v_pfc_before_coefficient_actual_leftsum) + (fs_s_pfc_before_coefficient_actual_leftsum_body_steps))) /\\ fs_s_pfc_before_coefficient_actual_leftsum_body_steps = fs_r_pfc_before_coefficient_actual_leftsum_body_steps + fs_a_pfc_before_coefficient_actual_leftsum_body_steps)))))) /\\ ((((exists pfa_gap_before_coefficient_actual_leftresiduebound. pfa_gap_before_coefficient_actual_leftresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_before_coefficient_actual_leftresiduecongruence pfa_offset_right_before_coefficient_actual_leftresiduecongruence. (pfc_natural_sum_before_coefficient_actual_left) + (p) * pfa_offset_left_before_coefficient_actual_leftresiduecongruence = (r) + (p) * pfa_offset_right_before_coefficient_actual_leftresiduecongruence))))))))) -> (r=0)", "statement_sha256": "19b82b626bcfea24b5666c0739db4d22bb1f88a79324c177dbc74af78fe9c958", "summary": "Every actual convolution coefficient before the added leading block is zero at a nonzero modulus.", "summary_sha256": "e065d5bd54f1b5430755ea13261fc3aed4ff88747bcecb2452e430b293120b5c" }, { "alpha_v33_frontier_enrollment": { "body_receipt_sha256": "511417e41bb8abbda22a1165e268bde75ba7e16b781e2a37095aa8fef058cd6e", "bundle_campaign": "polynomial-euclidean-division", "bundle_node_id": 359, "bundle_sha256": "6ae667d8518e4dbe722bb08ad1b08715a0d282c2893e533c8133d770fe861dcf", "campaign": "polynomial-euclidean-division", "first_enrolled_version": "v33", "parent_catalog_sha256": "41b9f387d88a5a4f0fe5ee2bd5578f37a27a4657b0a80f1a1a2cb5109f69a623", "rfc_sha256": 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BB", "intro BC", "intro t", "intro i", "intro r", "intro hp", "intro hpad", "intro hi", "intro hc", "cases hc", "cases hc_witness", "cases hc_witness_witness", "cases hc_witness_witness_witness", "cases hc_witness_witness_witness_right", "have hz : forall pfp_repeat_index_before_coefficient_zero_diagonal_right. (exists pfa_gap_before_coefficient_zero_diagonal_rightindex. pfa_gap_before_coefficient_zero_diagonal_rightindex + S (pfp_repeat_index_before_coefficient_zero_diagonal_right) = (S i)) -> (((exists ff_h_pfp_before_coefficient_zero_diagonal_rightentry. ff_h_pfp_before_coefficient_zero_diagonal_rightentry + S (0) = S ((S (pfp_repeat_index_before_coefficient_zero_diagonal_right)) * x1)) /\\ exists ff_q_pfp_before_coefficient_zero_diagonal_rightentry. x = ff_q_pfp_before_coefficient_zero_diagonal_rightentry * S ((S (pfp_repeat_index_before_coefficient_zero_diagonal_right)) * x1) + (0)))", "intro j", "intro hj", "have ht : exists z. ((((exists ff_h_pfp_before_coefficient_entry_right. ff_h_pfp_before_coefficient_entry_right + S (z) = S ((S (j)) * x1)) /\\ exists ff_q_pfp_before_coefficient_entry_right. x = ff_q_pfp_before_coefficient_entry_right * S ((S (j)) * x1) + (z))) /\\ ((exists pfc_complement_before_coefficient_term_right pfc_left_before_coefficient_term_right pfc_right_before_coefficient_term_right. (((j)+pfc_complement_before_coefficient_term_right=(i)) /\\ ((((((exists pfa_gap_before_coefficient_term_rightleftinside. pfa_gap_before_coefficient_term_rightleftinside + S (j) = (L)) /\\ ((((exists ff_h_pfp_before_coefficient_term_rightleftentry. ff_h_pfp_before_coefficient_term_rightleftentry + S (pfc_left_before_coefficient_term_right) = S ((S (j)) * ac)) /\\ exists ff_q_pfp_before_coefficient_term_rightleftentry. ab = ff_q_pfp_before_coefficient_term_rightleftentry * S ((S (j)) * ac) + (pfc_left_before_coefficient_term_right)))))) \\/ (((exists pfc_gap_before_coefficient_term_rightleftoutside. pfc_gap_before_coefficient_term_rightleftoutside+(L)=(j)) /\\ (((pfc_left_before_coefficient_term_right)=0))))) /\\ ((((((exists pfa_gap_before_coefficient_term_rightrightinside. pfa_gap_before_coefficient_term_rightrightinside + S (pfc_complement_before_coefficient_term_right) = (t+M)) /\\ ((((exists ff_h_pfp_before_coefficient_term_rightrightentry. ff_h_pfp_before_coefficient_term_rightrightentry + S (pfc_right_before_coefficient_term_right) = S ((S (pfc_complement_before_coefficient_term_right)) * BC)) /\\ exists ff_q_pfp_before_coefficient_term_rightrightentry. BB = ff_q_pfp_before_coefficient_term_rightrightentry * S ((S (pfc_complement_before_coefficient_term_right)) * BC) + (pfc_right_before_coefficient_term_right)))))) \\/ (((exists pfc_gap_before_coefficient_term_rightrightoutside. pfc_gap_before_coefficient_term_rightrightoutside+(t+M)=(pfc_complement_before_coefficient_term_right)) /\\ (((pfc_right_before_coefficient_term_right)=0))))) /\\ (((z)=pfc_left_before_coefficient_term_right*pfc_right_before_coefficient_term_right))))))))))", "specialize hc_witness_witness_witness_left (j)", "apply hc_witness_witness_witness_left", "exact hj", "cases ht", "cases ht_witness", "have heq : x3=0", "specialize polynomial_diagonal_term_left_padding_zero_right (ab)", "specialize polynomial_diagonal_term_left_padding_zero_right (ac)", "specialize polynomial_diagonal_term_left_padding_zero_right (L)", "specialize polynomial_diagonal_term_left_padding_zero_right (bb)", "specialize polynomial_diagonal_term_left_padding_zero_right (bc)", "specialize polynomial_diagonal_term_left_padding_zero_right (M)", "specialize polynomial_diagonal_term_left_padding_zero_right (BB)", "specialize polynomial_diagonal_term_left_padding_zero_right (BC)", "specialize polynomial_diagonal_term_left_padding_zero_right (t)", "specialize polynomial_diagonal_term_left_padding_zero_right (i)", "specialize polynomial_diagonal_term_left_padding_zero_right (j)", "specialize polynomial_diagonal_term_left_padding_zero_right (x3)", "apply polynomial_diagonal_term_left_padding_zero_right", "exact hpad", "specialize le_trans (S i)", "specialize le_trans (t)", "specialize le_trans (t+j)", "apply le_trans", "exact hi", "specialize le_add_right (t)", "specialize le_add_right (j)", "apply le_add_right", "exact ht_witness_right", "rewrite heq at ht_witness_left", "rewrite heq at ht_witness_left", "exact ht_witness_left", "have hsum : x2=0", "trans (S i)*0", "specialize beta_repeat_sum_exact (x)", "specialize beta_repeat_sum_exact (x1)", "specialize beta_repeat_sum_exact (0)", "specialize beta_repeat_sum_exact (S i)", "specialize beta_repeat_sum_exact (x2)", "apply beta_repeat_sum_exact", "exact hz", "exact hc_witness_witness_witness_right_left", "simp", "rewrite hsum at hc_witness_witness_witness_right_right", "specialize prime_field_residue_bounded_value (p)", "specialize prime_field_residue_bounded_value (0)", "specialize prime_field_residue_bounded_value (r)", "apply prime_field_residue_bounded_value", "specialize one_le_of_ne_zero (p)", "apply one_le_of_ne_zero", "exact hp", "exact hc_witness_witness_witness_right_right" ], "script_sha256": "c7a29027320dbfb4ca0214fd354eeaa5ddd0c77839f8a8888e82b59c1a492bd1", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_convolution_padding_candidate.py", "sha256": "2d874ecfb35a5db0aecdeb07b549464efebad9072c363113aa5a0a977845d007" }, "statement": "forall p ab ac L bb bc M BB BC t i r. (~(p=0)) -> (((forall pfp_repeat_index_before_coefficient_pad_rightzeros. (exists pfa_gap_before_coefficient_pad_rightzerosindex. pfa_gap_before_coefficient_pad_rightzerosindex + S (pfp_repeat_index_before_coefficient_pad_rightzeros) = (t)) -> (((exists ff_h_pfp_before_coefficient_pad_rightzerosentry. ff_h_pfp_before_coefficient_pad_rightzerosentry + S (0) = S ((S (pfp_repeat_index_before_coefficient_pad_rightzeros)) * BC)) /\\ exists ff_q_pfp_before_coefficient_pad_rightzerosentry. BB = ff_q_pfp_before_coefficient_pad_rightzerosentry * S ((S (pfp_repeat_index_before_coefficient_pad_rightzeros)) * BC) + (0)))) /\\ ((forall pfrep_index_before_coefficient_pad_right pfrep_value_before_coefficient_pad_right. (exists pfa_gap_before_coefficient_pad_rightbound. pfa_gap_before_coefficient_pad_rightbound + S (pfrep_index_before_coefficient_pad_right) = (M)) -> (((exists ff_h_pfp_before_coefficient_pad_rightinput. ff_h_pfp_before_coefficient_pad_rightinput + S (pfrep_value_before_coefficient_pad_right) = S ((S (pfrep_index_before_coefficient_pad_right)) * bc)) /\\ exists ff_q_pfp_before_coefficient_pad_rightinput. bb = ff_q_pfp_before_coefficient_pad_rightinput * S ((S (pfrep_index_before_coefficient_pad_right)) * bc) + (pfrep_value_before_coefficient_pad_right))) -> (((exists ff_h_pfp_before_coefficient_pad_rightoutput. ff_h_pfp_before_coefficient_pad_rightoutput + S (pfrep_value_before_coefficient_pad_right) = S ((S ((t)+pfrep_index_before_coefficient_pad_right)) * BC)) /\\ exists ff_q_pfp_before_coefficient_pad_rightoutput. BB = ff_q_pfp_before_coefficient_pad_rightoutput * S ((S ((t)+pfrep_index_before_coefficient_pad_right)) * BC) + (pfrep_value_before_coefficient_pad_right))))))) -> (exists pfa_gap_before_coefficient_index_right. pfa_gap_before_coefficient_index_right + S (i) = (t)) -> (exists pfc_terms_code_before_coefficient_actual_right pfc_terms_scale_before_coefficient_actual_right pfc_natural_sum_before_coefficient_actual_right. ((forall pfc_index_before_coefficient_actual_rightdiagonal. (exists pfa_gap_before_coefficient_actual_rightdiagonalbound. pfa_gap_before_coefficient_actual_rightdiagonalbound + S (pfc_index_before_coefficient_actual_rightdiagonal) = (S (i))) -> exists pfc_value_before_coefficient_actual_rightdiagonal. ((((exists ff_h_pfp_before_coefficient_actual_rightdiagonalentry. ff_h_pfp_before_coefficient_actual_rightdiagonalentry + S (pfc_value_before_coefficient_actual_rightdiagonal) = S ((S (pfc_index_before_coefficient_actual_rightdiagonal)) * pfc_terms_scale_before_coefficient_actual_right)) /\\ exists ff_q_pfp_before_coefficient_actual_rightdiagonalentry. pfc_terms_code_before_coefficient_actual_right = ff_q_pfp_before_coefficient_actual_rightdiagonalentry * S ((S (pfc_index_before_coefficient_actual_rightdiagonal)) * pfc_terms_scale_before_coefficient_actual_right) + (pfc_value_before_coefficient_actual_rightdiagonal))) /\\ ((exists pfc_complement_before_coefficient_actual_rightdiagonalterm pfc_left_before_coefficient_actual_rightdiagonalterm pfc_right_before_coefficient_actual_rightdiagonalterm. (((pfc_index_before_coefficient_actual_rightdiagonal)+pfc_complement_before_coefficient_actual_rightdiagonalterm=(i)) /\\ ((((((exists pfa_gap_before_coefficient_actual_rightdiagonaltermleftinside. pfa_gap_before_coefficient_actual_rightdiagonaltermleftinside + S (pfc_index_before_coefficient_actual_rightdiagonal) = (L)) /\\ ((((exists ff_h_pfp_before_coefficient_actual_rightdiagonaltermleftentry. ff_h_pfp_before_coefficient_actual_rightdiagonaltermleftentry + S (pfc_left_before_coefficient_actual_rightdiagonalterm) = S ((S (pfc_index_before_coefficient_actual_rightdiagonal)) * ac)) /\\ exists ff_q_pfp_before_coefficient_actual_rightdiagonaltermleftentry. ab = ff_q_pfp_before_coefficient_actual_rightdiagonaltermleftentry * S ((S (pfc_index_before_coefficient_actual_rightdiagonal)) * ac) + (pfc_left_before_coefficient_actual_rightdiagonalterm)))))) \\/ (((exists pfc_gap_before_coefficient_actual_rightdiagonaltermleftoutside. pfc_gap_before_coefficient_actual_rightdiagonaltermleftoutside+(L)=(pfc_index_before_coefficient_actual_rightdiagonal)) /\\ (((pfc_left_before_coefficient_actual_rightdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_before_coefficient_actual_rightdiagonaltermrightinside. pfa_gap_before_coefficient_actual_rightdiagonaltermrightinside + S (pfc_complement_before_coefficient_actual_rightdiagonalterm) = (t+M)) /\\ ((((exists ff_h_pfp_before_coefficient_actual_rightdiagonaltermrightentry. ff_h_pfp_before_coefficient_actual_rightdiagonaltermrightentry + S (pfc_right_before_coefficient_actual_rightdiagonalterm) = S ((S (pfc_complement_before_coefficient_actual_rightdiagonalterm)) * BC)) /\\ exists ff_q_pfp_before_coefficient_actual_rightdiagonaltermrightentry. BB = ff_q_pfp_before_coefficient_actual_rightdiagonaltermrightentry * S ((S (pfc_complement_before_coefficient_actual_rightdiagonalterm)) * BC) + (pfc_right_before_coefficient_actual_rightdiagonalterm)))))) \\/ (((exists pfc_gap_before_coefficient_actual_rightdiagonaltermrightoutside. pfc_gap_before_coefficient_actual_rightdiagonaltermrightoutside+(t+M)=(pfc_complement_before_coefficient_actual_rightdiagonalterm)) /\\ (((pfc_right_before_coefficient_actual_rightdiagonalterm)=0))))) /\\ (((pfc_value_before_coefficient_actual_rightdiagonal)=pfc_left_before_coefficient_actual_rightdiagonalterm*pfc_right_before_coefficient_actual_rightdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_before_coefficient_actual_rightsum fs_v_pfc_before_coefficient_actual_rightsum. ((((exists fs_h_pfc_before_coefficient_actual_rightsum_body_start. fs_h_pfc_before_coefficient_actual_rightsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_before_coefficient_actual_rightsum)) /\\ exists fs_q_pfc_before_coefficient_actual_rightsum_body_start. fs_u_pfc_before_coefficient_actual_rightsum = fs_q_pfc_before_coefficient_actual_rightsum_body_start * S ((S (0)) * fs_v_pfc_before_coefficient_actual_rightsum) + (0))) /\\ ((((exists fs_h_pfc_before_coefficient_actual_rightsum_body_terminal. fs_h_pfc_before_coefficient_actual_rightsum_body_terminal + S (pfc_natural_sum_before_coefficient_actual_right) = S ((S (S (i))) * fs_v_pfc_before_coefficient_actual_rightsum)) /\\ exists fs_q_pfc_before_coefficient_actual_rightsum_body_terminal. fs_u_pfc_before_coefficient_actual_rightsum = fs_q_pfc_before_coefficient_actual_rightsum_body_terminal * S ((S (S (i))) * fs_v_pfc_before_coefficient_actual_rightsum) + (pfc_natural_sum_before_coefficient_actual_right))) /\\ forall fs_i_pfc_before_coefficient_actual_rightsum_body_steps. (exists fs_lt_pfc_before_coefficient_actual_rightsum_body_steps_bound. fs_lt_pfc_before_coefficient_actual_rightsum_body_steps_bound + S fs_i_pfc_before_coefficient_actual_rightsum_body_steps = S (i)) -> exists fs_a_pfc_before_coefficient_actual_rightsum_body_steps fs_r_pfc_before_coefficient_actual_rightsum_body_steps fs_s_pfc_before_coefficient_actual_rightsum_body_steps. ((((exists fs_h_pfc_before_coefficient_actual_rightsum_body_steps_summand. fs_h_pfc_before_coefficient_actual_rightsum_body_steps_summand + S (fs_a_pfc_before_coefficient_actual_rightsum_body_steps) = S ((S (fs_i_pfc_before_coefficient_actual_rightsum_body_steps)) * pfc_terms_scale_before_coefficient_actual_right)) /\\ exists fs_q_pfc_before_coefficient_actual_rightsum_body_steps_summand. pfc_terms_code_before_coefficient_actual_right = fs_q_pfc_before_coefficient_actual_rightsum_body_steps_summand * S ((S (fs_i_pfc_before_coefficient_actual_rightsum_body_steps)) * pfc_terms_scale_before_coefficient_actual_right) + (fs_a_pfc_before_coefficient_actual_rightsum_body_steps))) /\\ ((((exists fs_h_pfc_before_coefficient_actual_rightsum_body_steps_partial. fs_h_pfc_before_coefficient_actual_rightsum_body_steps_partial + S (fs_r_pfc_before_coefficient_actual_rightsum_body_steps) = S ((S (fs_i_pfc_before_coefficient_actual_rightsum_body_steps)) * fs_v_pfc_before_coefficient_actual_rightsum)) /\\ exists fs_q_pfc_before_coefficient_actual_rightsum_body_steps_partial. fs_u_pfc_before_coefficient_actual_rightsum = fs_q_pfc_before_coefficient_actual_rightsum_body_steps_partial * S ((S (fs_i_pfc_before_coefficient_actual_rightsum_body_steps)) * fs_v_pfc_before_coefficient_actual_rightsum) + (fs_r_pfc_before_coefficient_actual_rightsum_body_steps))) /\\ ((((exists fs_h_pfc_before_coefficient_actual_rightsum_body_steps_successor. fs_h_pfc_before_coefficient_actual_rightsum_body_steps_successor + S (fs_s_pfc_before_coefficient_actual_rightsum_body_steps) = S ((S (S fs_i_pfc_before_coefficient_actual_rightsum_body_steps)) * fs_v_pfc_before_coefficient_actual_rightsum)) /\\ exists fs_q_pfc_before_coefficient_actual_rightsum_body_steps_successor. fs_u_pfc_before_coefficient_actual_rightsum = fs_q_pfc_before_coefficient_actual_rightsum_body_steps_successor * S ((S (S fs_i_pfc_before_coefficient_actual_rightsum_body_steps)) * fs_v_pfc_before_coefficient_actual_rightsum) + (fs_s_pfc_before_coefficient_actual_rightsum_body_steps))) /\\ fs_s_pfc_before_coefficient_actual_rightsum_body_steps = fs_r_pfc_before_coefficient_actual_rightsum_body_steps + fs_a_pfc_before_coefficient_actual_rightsum_body_steps)))))) /\\ ((((exists pfa_gap_before_coefficient_actual_rightresiduebound. pfa_gap_before_coefficient_actual_rightresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_before_coefficient_actual_rightresiduecongruence pfa_offset_right_before_coefficient_actual_rightresiduecongruence. 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"012942552bc109df42630e4a350deb2ca2b207d34b01ea53762c756f6ddb852e", "membership": "alpha_only", "name": "prime_field_polynomial_convolution_left_padding_nonempty_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 N", "intro AB", "intro AC", "intro t", "intro CB", "intro CC", "intro K", "intro hp", "intro hL", "intro hM", "intro hpad", "intro hc", "intro hn", "cases hc", "cases hc_right", "cases hc_right_right", "cases hn", "cases hn_right", "cases hn_right_right", "have hlength : K=t+N", "specialize polynomial_product_length_left_padding_left (L)", "specialize polynomial_product_length_left_padding_left (M)", "specialize polynomial_product_length_left_padding_left (N)", "specialize polynomial_product_length_left_padding_left (t)", "specialize polynomial_product_length_left_padding_left (K)", "apply polynomial_product_length_left_padding_left", "exact hc_right_right_left", "exact hL", "exact hM", "exact hn_right_right_left", "split", "exact hlength", "split", "intro i", "intro hi", "have hv : exists r. ((((exists ff_h_pfp_nonempty_zero_entry_left. ff_h_pfp_nonempty_zero_entry_left + S (r) = S ((S (i)) * CC)) /\\ exists ff_q_pfp_nonempty_zero_entry_left. CB = ff_q_pfp_nonempty_zero_entry_left * S ((S (i)) * CC) + (r))) /\\ ((exists pfc_terms_code_nonempty_zero_coefficient_left pfc_terms_scale_nonempty_zero_coefficient_left pfc_natural_sum_nonempty_zero_coefficient_left. ((forall pfc_index_nonempty_zero_coefficient_leftdiagonal. (exists pfa_gap_nonempty_zero_coefficient_leftdiagonalbound. pfa_gap_nonempty_zero_coefficient_leftdiagonalbound + S (pfc_index_nonempty_zero_coefficient_leftdiagonal) = (S (i))) -> exists pfc_value_nonempty_zero_coefficient_leftdiagonal. ((((exists ff_h_pfp_nonempty_zero_coefficient_leftdiagonalentry. ff_h_pfp_nonempty_zero_coefficient_leftdiagonalentry + S (pfc_value_nonempty_zero_coefficient_leftdiagonal) = S ((S (pfc_index_nonempty_zero_coefficient_leftdiagonal)) * pfc_terms_scale_nonempty_zero_coefficient_left)) /\\ exists ff_q_pfp_nonempty_zero_coefficient_leftdiagonalentry. pfc_terms_code_nonempty_zero_coefficient_left = ff_q_pfp_nonempty_zero_coefficient_leftdiagonalentry * S ((S (pfc_index_nonempty_zero_coefficient_leftdiagonal)) * pfc_terms_scale_nonempty_zero_coefficient_left) + (pfc_value_nonempty_zero_coefficient_leftdiagonal))) /\\ ((exists pfc_complement_nonempty_zero_coefficient_leftdiagonalterm pfc_left_nonempty_zero_coefficient_leftdiagonalterm pfc_right_nonempty_zero_coefficient_leftdiagonalterm. (((pfc_index_nonempty_zero_coefficient_leftdiagonal)+pfc_complement_nonempty_zero_coefficient_leftdiagonalterm=(i)) /\\ ((((((exists pfa_gap_nonempty_zero_coefficient_leftdiagonaltermleftinside. pfa_gap_nonempty_zero_coefficient_leftdiagonaltermleftinside + S (pfc_index_nonempty_zero_coefficient_leftdiagonal) = (t+L)) /\\ ((((exists ff_h_pfp_nonempty_zero_coefficient_leftdiagonaltermleftentry. ff_h_pfp_nonempty_zero_coefficient_leftdiagonaltermleftentry + S (pfc_left_nonempty_zero_coefficient_leftdiagonalterm) = S ((S (pfc_index_nonempty_zero_coefficient_leftdiagonal)) * AC)) /\\ exists ff_q_pfp_nonempty_zero_coefficient_leftdiagonaltermleftentry. AB = ff_q_pfp_nonempty_zero_coefficient_leftdiagonaltermleftentry * S ((S (pfc_index_nonempty_zero_coefficient_leftdiagonal)) * AC) + (pfc_left_nonempty_zero_coefficient_leftdiagonalterm)))))) \\/ (((exists pfc_gap_nonempty_zero_coefficient_leftdiagonaltermleftoutside. pfc_gap_nonempty_zero_coefficient_leftdiagonaltermleftoutside+(t+L)=(pfc_index_nonempty_zero_coefficient_leftdiagonal)) /\\ (((pfc_left_nonempty_zero_coefficient_leftdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_nonempty_zero_coefficient_leftdiagonaltermrightinside. pfa_gap_nonempty_zero_coefficient_leftdiagonaltermrightinside + S (pfc_complement_nonempty_zero_coefficient_leftdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_nonempty_zero_coefficient_leftdiagonaltermrightentry. ff_h_pfp_nonempty_zero_coefficient_leftdiagonaltermrightentry + S (pfc_right_nonempty_zero_coefficient_leftdiagonalterm) = S ((S (pfc_complement_nonempty_zero_coefficient_leftdiagonalterm)) * bc)) /\\ exists ff_q_pfp_nonempty_zero_coefficient_leftdiagonaltermrightentry. bb = ff_q_pfp_nonempty_zero_coefficient_leftdiagonaltermrightentry * S ((S (pfc_complement_nonempty_zero_coefficient_leftdiagonalterm)) * bc) + (pfc_right_nonempty_zero_coefficient_leftdiagonalterm)))))) \\/ (((exists pfc_gap_nonempty_zero_coefficient_leftdiagonaltermrightoutside. pfc_gap_nonempty_zero_coefficient_leftdiagonaltermrightoutside+(M)=(pfc_complement_nonempty_zero_coefficient_leftdiagonalterm)) /\\ (((pfc_right_nonempty_zero_coefficient_leftdiagonalterm)=0))))) /\\ (((pfc_value_nonempty_zero_coefficient_leftdiagonal)=pfc_left_nonempty_zero_coefficient_leftdiagonalterm*pfc_right_nonempty_zero_coefficient_leftdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_nonempty_zero_coefficient_leftsum fs_v_pfc_nonempty_zero_coefficient_leftsum. ((((exists fs_h_pfc_nonempty_zero_coefficient_leftsum_body_start. fs_h_pfc_nonempty_zero_coefficient_leftsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_nonempty_zero_coefficient_leftsum)) /\\ exists fs_q_pfc_nonempty_zero_coefficient_leftsum_body_start. fs_u_pfc_nonempty_zero_coefficient_leftsum = fs_q_pfc_nonempty_zero_coefficient_leftsum_body_start * S ((S (0)) * fs_v_pfc_nonempty_zero_coefficient_leftsum) + (0))) /\\ ((((exists fs_h_pfc_nonempty_zero_coefficient_leftsum_body_terminal. fs_h_pfc_nonempty_zero_coefficient_leftsum_body_terminal + S (pfc_natural_sum_nonempty_zero_coefficient_left) = S ((S (S (i))) * fs_v_pfc_nonempty_zero_coefficient_leftsum)) /\\ exists fs_q_pfc_nonempty_zero_coefficient_leftsum_body_terminal. fs_u_pfc_nonempty_zero_coefficient_leftsum = fs_q_pfc_nonempty_zero_coefficient_leftsum_body_terminal * S ((S (S (i))) * fs_v_pfc_nonempty_zero_coefficient_leftsum) + (pfc_natural_sum_nonempty_zero_coefficient_left))) /\\ forall fs_i_pfc_nonempty_zero_coefficient_leftsum_body_steps. (exists fs_lt_pfc_nonempty_zero_coefficient_leftsum_body_steps_bound. fs_lt_pfc_nonempty_zero_coefficient_leftsum_body_steps_bound + S fs_i_pfc_nonempty_zero_coefficient_leftsum_body_steps = S (i)) -> exists fs_a_pfc_nonempty_zero_coefficient_leftsum_body_steps fs_r_pfc_nonempty_zero_coefficient_leftsum_body_steps fs_s_pfc_nonempty_zero_coefficient_leftsum_body_steps. ((((exists fs_h_pfc_nonempty_zero_coefficient_leftsum_body_steps_summand. fs_h_pfc_nonempty_zero_coefficient_leftsum_body_steps_summand + S (fs_a_pfc_nonempty_zero_coefficient_leftsum_body_steps) = S ((S (fs_i_pfc_nonempty_zero_coefficient_leftsum_body_steps)) * pfc_terms_scale_nonempty_zero_coefficient_left)) /\\ exists fs_q_pfc_nonempty_zero_coefficient_leftsum_body_steps_summand. pfc_terms_code_nonempty_zero_coefficient_left = fs_q_pfc_nonempty_zero_coefficient_leftsum_body_steps_summand * S ((S (fs_i_pfc_nonempty_zero_coefficient_leftsum_body_steps)) * pfc_terms_scale_nonempty_zero_coefficient_left) + (fs_a_pfc_nonempty_zero_coefficient_leftsum_body_steps))) /\\ ((((exists fs_h_pfc_nonempty_zero_coefficient_leftsum_body_steps_partial. fs_h_pfc_nonempty_zero_coefficient_leftsum_body_steps_partial + S (fs_r_pfc_nonempty_zero_coefficient_leftsum_body_steps) = S ((S (fs_i_pfc_nonempty_zero_coefficient_leftsum_body_steps)) * fs_v_pfc_nonempty_zero_coefficient_leftsum)) /\\ exists fs_q_pfc_nonempty_zero_coefficient_leftsum_body_steps_partial. fs_u_pfc_nonempty_zero_coefficient_leftsum = fs_q_pfc_nonempty_zero_coefficient_leftsum_body_steps_partial * S ((S (fs_i_pfc_nonempty_zero_coefficient_leftsum_body_steps)) * fs_v_pfc_nonempty_zero_coefficient_leftsum) + (fs_r_pfc_nonempty_zero_coefficient_leftsum_body_steps))) /\\ ((((exists fs_h_pfc_nonempty_zero_coefficient_leftsum_body_steps_successor. fs_h_pfc_nonempty_zero_coefficient_leftsum_body_steps_successor + S (fs_s_pfc_nonempty_zero_coefficient_leftsum_body_steps) = S ((S (S fs_i_pfc_nonempty_zero_coefficient_leftsum_body_steps)) * fs_v_pfc_nonempty_zero_coefficient_leftsum)) /\\ exists fs_q_pfc_nonempty_zero_coefficient_leftsum_body_steps_successor. fs_u_pfc_nonempty_zero_coefficient_leftsum = fs_q_pfc_nonempty_zero_coefficient_leftsum_body_steps_successor * S ((S (S fs_i_pfc_nonempty_zero_coefficient_leftsum_body_steps)) * fs_v_pfc_nonempty_zero_coefficient_leftsum) + (fs_s_pfc_nonempty_zero_coefficient_leftsum_body_steps))) /\\ fs_s_pfc_nonempty_zero_coefficient_leftsum_body_steps = fs_r_pfc_nonempty_zero_coefficient_leftsum_body_steps + fs_a_pfc_nonempty_zero_coefficient_leftsum_body_steps)))))) /\\ ((((exists pfa_gap_nonempty_zero_coefficient_leftresiduebound. pfa_gap_nonempty_zero_coefficient_leftresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_nonempty_zero_coefficient_leftresiduecongruence pfa_offset_right_nonempty_zero_coefficient_leftresiduecongruence. (pfc_natural_sum_nonempty_zero_coefficient_left) + (p) * pfa_offset_left_nonempty_zero_coefficient_leftresiduecongruence = (r) + (p) * pfa_offset_right_nonempty_zero_coefficient_leftresiduecongruence)))))))))))", "specialize hn_right_right_right (i)", "apply hn_right_right_right", "specialize le_trans (S i)", "specialize le_trans (t)", "specialize le_trans (K)", "apply le_trans", "exact hi", "rewrite hlength", "specialize le_add_right (t)", "specialize le_add_right (N)", "apply le_add_right", "cases hv", "cases hv_witness", "have hzero : x=0", "specialize prime_field_convolution_coefficient_before_left_padding_left (p)", "specialize prime_field_convolution_coefficient_before_left_padding_left (ab)", "specialize prime_field_convolution_coefficient_before_left_padding_left (ac)", "specialize prime_field_convolution_coefficient_before_left_padding_left (L)", "specialize prime_field_convolution_coefficient_before_left_padding_left (bb)", "specialize prime_field_convolution_coefficient_before_left_padding_left (bc)", "specialize prime_field_convolution_coefficient_before_left_padding_left (M)", "specialize prime_field_convolution_coefficient_before_left_padding_left (AB)", "specialize prime_field_convolution_coefficient_before_left_padding_left (AC)", "specialize prime_field_convolution_coefficient_before_left_padding_left (t)", "specialize prime_field_convolution_coefficient_before_left_padding_left (i)", "specialize prime_field_convolution_coefficient_before_left_padding_left (x)", "apply prime_field_convolution_coefficient_before_left_padding_left", "exact hp", "exact hpad", "exact hi", "exact hv_witness_right", "rewrite hzero at hv_witness_left", "rewrite hzero at hv_witness_left", "exact hv_witness_left", "intro i", "intro a", "intro hi", "intro ha", "have hcoefficient : exists pfc_terms_code_nonempty_original_coefficient_left pfc_terms_scale_nonempty_original_coefficient_left pfc_natural_sum_nonempty_original_coefficient_left. ((forall pfc_index_nonempty_original_coefficient_leftdiagonal. (exists pfa_gap_nonempty_original_coefficient_leftdiagonalbound. pfa_gap_nonempty_original_coefficient_leftdiagonalbound + S (pfc_index_nonempty_original_coefficient_leftdiagonal) = (S (i))) -> exists pfc_value_nonempty_original_coefficient_leftdiagonal. ((((exists ff_h_pfp_nonempty_original_coefficient_leftdiagonalentry. ff_h_pfp_nonempty_original_coefficient_leftdiagonalentry + S (pfc_value_nonempty_original_coefficient_leftdiagonal) = S ((S (pfc_index_nonempty_original_coefficient_leftdiagonal)) * pfc_terms_scale_nonempty_original_coefficient_left)) /\\ exists ff_q_pfp_nonempty_original_coefficient_leftdiagonalentry. pfc_terms_code_nonempty_original_coefficient_left = ff_q_pfp_nonempty_original_coefficient_leftdiagonalentry * S ((S (pfc_index_nonempty_original_coefficient_leftdiagonal)) * pfc_terms_scale_nonempty_original_coefficient_left) + (pfc_value_nonempty_original_coefficient_leftdiagonal))) /\\ ((exists pfc_complement_nonempty_original_coefficient_leftdiagonalterm pfc_left_nonempty_original_coefficient_leftdiagonalterm pfc_right_nonempty_original_coefficient_leftdiagonalterm. (((pfc_index_nonempty_original_coefficient_leftdiagonal)+pfc_complement_nonempty_original_coefficient_leftdiagonalterm=(i)) /\\ ((((((exists pfa_gap_nonempty_original_coefficient_leftdiagonaltermleftinside. pfa_gap_nonempty_original_coefficient_leftdiagonaltermleftinside + S (pfc_index_nonempty_original_coefficient_leftdiagonal) = (L)) /\\ ((((exists ff_h_pfp_nonempty_original_coefficient_leftdiagonaltermleftentry. ff_h_pfp_nonempty_original_coefficient_leftdiagonaltermleftentry + S (pfc_left_nonempty_original_coefficient_leftdiagonalterm) = S ((S (pfc_index_nonempty_original_coefficient_leftdiagonal)) * ac)) /\\ exists ff_q_pfp_nonempty_original_coefficient_leftdiagonaltermleftentry. ab = ff_q_pfp_nonempty_original_coefficient_leftdiagonaltermleftentry * S ((S (pfc_index_nonempty_original_coefficient_leftdiagonal)) * ac) + (pfc_left_nonempty_original_coefficient_leftdiagonalterm)))))) \\/ (((exists pfc_gap_nonempty_original_coefficient_leftdiagonaltermleftoutside. pfc_gap_nonempty_original_coefficient_leftdiagonaltermleftoutside+(L)=(pfc_index_nonempty_original_coefficient_leftdiagonal)) /\\ (((pfc_left_nonempty_original_coefficient_leftdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_nonempty_original_coefficient_leftdiagonaltermrightinside. pfa_gap_nonempty_original_coefficient_leftdiagonaltermrightinside + S (pfc_complement_nonempty_original_coefficient_leftdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_nonempty_original_coefficient_leftdiagonaltermrightentry. ff_h_pfp_nonempty_original_coefficient_leftdiagonaltermrightentry + S (pfc_right_nonempty_original_coefficient_leftdiagonalterm) = S ((S (pfc_complement_nonempty_original_coefficient_leftdiagonalterm)) * bc)) /\\ exists ff_q_pfp_nonempty_original_coefficient_leftdiagonaltermrightentry. bb = ff_q_pfp_nonempty_original_coefficient_leftdiagonaltermrightentry * S ((S (pfc_complement_nonempty_original_coefficient_leftdiagonalterm)) * bc) + (pfc_right_nonempty_original_coefficient_leftdiagonalterm)))))) \\/ (((exists pfc_gap_nonempty_original_coefficient_leftdiagonaltermrightoutside. pfc_gap_nonempty_original_coefficient_leftdiagonaltermrightoutside+(M)=(pfc_complement_nonempty_original_coefficient_leftdiagonalterm)) /\\ (((pfc_right_nonempty_original_coefficient_leftdiagonalterm)=0))))) /\\ (((pfc_value_nonempty_original_coefficient_leftdiagonal)=pfc_left_nonempty_original_coefficient_leftdiagonalterm*pfc_right_nonempty_original_coefficient_leftdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_nonempty_original_coefficient_leftsum fs_v_pfc_nonempty_original_coefficient_leftsum. ((((exists fs_h_pfc_nonempty_original_coefficient_leftsum_body_start. fs_h_pfc_nonempty_original_coefficient_leftsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_nonempty_original_coefficient_leftsum)) /\\ exists fs_q_pfc_nonempty_original_coefficient_leftsum_body_start. fs_u_pfc_nonempty_original_coefficient_leftsum = fs_q_pfc_nonempty_original_coefficient_leftsum_body_start * S ((S (0)) * fs_v_pfc_nonempty_original_coefficient_leftsum) + (0))) /\\ ((((exists fs_h_pfc_nonempty_original_coefficient_leftsum_body_terminal. fs_h_pfc_nonempty_original_coefficient_leftsum_body_terminal + S (pfc_natural_sum_nonempty_original_coefficient_left) = S ((S (S (i))) * fs_v_pfc_nonempty_original_coefficient_leftsum)) /\\ exists fs_q_pfc_nonempty_original_coefficient_leftsum_body_terminal. fs_u_pfc_nonempty_original_coefficient_leftsum = fs_q_pfc_nonempty_original_coefficient_leftsum_body_terminal * S ((S (S (i))) * fs_v_pfc_nonempty_original_coefficient_leftsum) + (pfc_natural_sum_nonempty_original_coefficient_left))) /\\ forall fs_i_pfc_nonempty_original_coefficient_leftsum_body_steps. (exists fs_lt_pfc_nonempty_original_coefficient_leftsum_body_steps_bound. fs_lt_pfc_nonempty_original_coefficient_leftsum_body_steps_bound + S fs_i_pfc_nonempty_original_coefficient_leftsum_body_steps = S (i)) -> exists fs_a_pfc_nonempty_original_coefficient_leftsum_body_steps fs_r_pfc_nonempty_original_coefficient_leftsum_body_steps fs_s_pfc_nonempty_original_coefficient_leftsum_body_steps. ((((exists fs_h_pfc_nonempty_original_coefficient_leftsum_body_steps_summand. fs_h_pfc_nonempty_original_coefficient_leftsum_body_steps_summand + S (fs_a_pfc_nonempty_original_coefficient_leftsum_body_steps) = S ((S (fs_i_pfc_nonempty_original_coefficient_leftsum_body_steps)) * pfc_terms_scale_nonempty_original_coefficient_left)) /\\ exists fs_q_pfc_nonempty_original_coefficient_leftsum_body_steps_summand. pfc_terms_code_nonempty_original_coefficient_left = fs_q_pfc_nonempty_original_coefficient_leftsum_body_steps_summand * S ((S (fs_i_pfc_nonempty_original_coefficient_leftsum_body_steps)) * pfc_terms_scale_nonempty_original_coefficient_left) + (fs_a_pfc_nonempty_original_coefficient_leftsum_body_steps))) /\\ ((((exists fs_h_pfc_nonempty_original_coefficient_leftsum_body_steps_partial. fs_h_pfc_nonempty_original_coefficient_leftsum_body_steps_partial + S (fs_r_pfc_nonempty_original_coefficient_leftsum_body_steps) = S ((S (fs_i_pfc_nonempty_original_coefficient_leftsum_body_steps)) * fs_v_pfc_nonempty_original_coefficient_leftsum)) /\\ exists fs_q_pfc_nonempty_original_coefficient_leftsum_body_steps_partial. fs_u_pfc_nonempty_original_coefficient_leftsum = fs_q_pfc_nonempty_original_coefficient_leftsum_body_steps_partial * S ((S (fs_i_pfc_nonempty_original_coefficient_leftsum_body_steps)) * fs_v_pfc_nonempty_original_coefficient_leftsum) + (fs_r_pfc_nonempty_original_coefficient_leftsum_body_steps))) /\\ ((((exists fs_h_pfc_nonempty_original_coefficient_leftsum_body_steps_successor. fs_h_pfc_nonempty_original_coefficient_leftsum_body_steps_successor + S (fs_s_pfc_nonempty_original_coefficient_leftsum_body_steps) = S ((S (S fs_i_pfc_nonempty_original_coefficient_leftsum_body_steps)) * fs_v_pfc_nonempty_original_coefficient_leftsum)) /\\ exists fs_q_pfc_nonempty_original_coefficient_leftsum_body_steps_successor. fs_u_pfc_nonempty_original_coefficient_leftsum = fs_q_pfc_nonempty_original_coefficient_leftsum_body_steps_successor * S ((S (S fs_i_pfc_nonempty_original_coefficient_leftsum_body_steps)) * fs_v_pfc_nonempty_original_coefficient_leftsum) + (fs_s_pfc_nonempty_original_coefficient_leftsum_body_steps))) /\\ fs_s_pfc_nonempty_original_coefficient_leftsum_body_steps = fs_r_pfc_nonempty_original_coefficient_leftsum_body_steps + fs_a_pfc_nonempty_original_coefficient_leftsum_body_steps)))))) /\\ ((((exists pfa_gap_nonempty_original_coefficient_leftresiduebound. pfa_gap_nonempty_original_coefficient_leftresiduebound + S (a) = (p)) /\\ ((exists pfa_offset_left_nonempty_original_coefficient_leftresiduecongruence pfa_offset_right_nonempty_original_coefficient_leftresiduecongruence. (pfc_natural_sum_nonempty_original_coefficient_left) + (p) * pfa_offset_left_nonempty_original_coefficient_leftresiduecongruence = (a) + (p) * pfa_offset_right_nonempty_original_coefficient_leftresiduecongruence))))))))", "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 hv : exists r. ((((exists ff_h_pfp_nonempty_shifted_entry_left. ff_h_pfp_nonempty_shifted_entry_left + S (r) = S ((S (t+i)) * CC)) /\\ exists ff_q_pfp_nonempty_shifted_entry_left. CB = ff_q_pfp_nonempty_shifted_entry_left * S ((S (t+i)) * CC) + (r))) /\\ ((exists pfc_terms_code_nonempty_shifted_coefficient_left pfc_terms_scale_nonempty_shifted_coefficient_left pfc_natural_sum_nonempty_shifted_coefficient_left. ((forall pfc_index_nonempty_shifted_coefficient_leftdiagonal. (exists pfa_gap_nonempty_shifted_coefficient_leftdiagonalbound. pfa_gap_nonempty_shifted_coefficient_leftdiagonalbound + S (pfc_index_nonempty_shifted_coefficient_leftdiagonal) = (S (t+i))) -> exists pfc_value_nonempty_shifted_coefficient_leftdiagonal. ((((exists ff_h_pfp_nonempty_shifted_coefficient_leftdiagonalentry. ff_h_pfp_nonempty_shifted_coefficient_leftdiagonalentry + S (pfc_value_nonempty_shifted_coefficient_leftdiagonal) = S ((S (pfc_index_nonempty_shifted_coefficient_leftdiagonal)) * pfc_terms_scale_nonempty_shifted_coefficient_left)) /\\ exists ff_q_pfp_nonempty_shifted_coefficient_leftdiagonalentry. pfc_terms_code_nonempty_shifted_coefficient_left = ff_q_pfp_nonempty_shifted_coefficient_leftdiagonalentry * S ((S (pfc_index_nonempty_shifted_coefficient_leftdiagonal)) * pfc_terms_scale_nonempty_shifted_coefficient_left) + (pfc_value_nonempty_shifted_coefficient_leftdiagonal))) /\\ ((exists pfc_complement_nonempty_shifted_coefficient_leftdiagonalterm pfc_left_nonempty_shifted_coefficient_leftdiagonalterm pfc_right_nonempty_shifted_coefficient_leftdiagonalterm. (((pfc_index_nonempty_shifted_coefficient_leftdiagonal)+pfc_complement_nonempty_shifted_coefficient_leftdiagonalterm=(t+i)) /\\ ((((((exists pfa_gap_nonempty_shifted_coefficient_leftdiagonaltermleftinside. pfa_gap_nonempty_shifted_coefficient_leftdiagonaltermleftinside + S (pfc_index_nonempty_shifted_coefficient_leftdiagonal) = (t+L)) /\\ ((((exists ff_h_pfp_nonempty_shifted_coefficient_leftdiagonaltermleftentry. ff_h_pfp_nonempty_shifted_coefficient_leftdiagonaltermleftentry + S (pfc_left_nonempty_shifted_coefficient_leftdiagonalterm) = S ((S (pfc_index_nonempty_shifted_coefficient_leftdiagonal)) * AC)) /\\ exists ff_q_pfp_nonempty_shifted_coefficient_leftdiagonaltermleftentry. AB = ff_q_pfp_nonempty_shifted_coefficient_leftdiagonaltermleftentry * S ((S (pfc_index_nonempty_shifted_coefficient_leftdiagonal)) * AC) + (pfc_left_nonempty_shifted_coefficient_leftdiagonalterm)))))) \\/ (((exists pfc_gap_nonempty_shifted_coefficient_leftdiagonaltermleftoutside. pfc_gap_nonempty_shifted_coefficient_leftdiagonaltermleftoutside+(t+L)=(pfc_index_nonempty_shifted_coefficient_leftdiagonal)) /\\ (((pfc_left_nonempty_shifted_coefficient_leftdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_nonempty_shifted_coefficient_leftdiagonaltermrightinside. pfa_gap_nonempty_shifted_coefficient_leftdiagonaltermrightinside + S (pfc_complement_nonempty_shifted_coefficient_leftdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_nonempty_shifted_coefficient_leftdiagonaltermrightentry. ff_h_pfp_nonempty_shifted_coefficient_leftdiagonaltermrightentry + S (pfc_right_nonempty_shifted_coefficient_leftdiagonalterm) = S ((S (pfc_complement_nonempty_shifted_coefficient_leftdiagonalterm)) * bc)) /\\ exists ff_q_pfp_nonempty_shifted_coefficient_leftdiagonaltermrightentry. bb = ff_q_pfp_nonempty_shifted_coefficient_leftdiagonaltermrightentry * S ((S (pfc_complement_nonempty_shifted_coefficient_leftdiagonalterm)) * bc) + (pfc_right_nonempty_shifted_coefficient_leftdiagonalterm)))))) \\/ (((exists pfc_gap_nonempty_shifted_coefficient_leftdiagonaltermrightoutside. pfc_gap_nonempty_shifted_coefficient_leftdiagonaltermrightoutside+(M)=(pfc_complement_nonempty_shifted_coefficient_leftdiagonalterm)) /\\ (((pfc_right_nonempty_shifted_coefficient_leftdiagonalterm)=0))))) /\\ (((pfc_value_nonempty_shifted_coefficient_leftdiagonal)=pfc_left_nonempty_shifted_coefficient_leftdiagonalterm*pfc_right_nonempty_shifted_coefficient_leftdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_nonempty_shifted_coefficient_leftsum fs_v_pfc_nonempty_shifted_coefficient_leftsum. ((((exists fs_h_pfc_nonempty_shifted_coefficient_leftsum_body_start. fs_h_pfc_nonempty_shifted_coefficient_leftsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_nonempty_shifted_coefficient_leftsum)) /\\ exists fs_q_pfc_nonempty_shifted_coefficient_leftsum_body_start. fs_u_pfc_nonempty_shifted_coefficient_leftsum = fs_q_pfc_nonempty_shifted_coefficient_leftsum_body_start * S ((S (0)) * fs_v_pfc_nonempty_shifted_coefficient_leftsum) + (0))) /\\ ((((exists fs_h_pfc_nonempty_shifted_coefficient_leftsum_body_terminal. fs_h_pfc_nonempty_shifted_coefficient_leftsum_body_terminal + S (pfc_natural_sum_nonempty_shifted_coefficient_left) = S ((S (S (t+i))) * fs_v_pfc_nonempty_shifted_coefficient_leftsum)) /\\ exists fs_q_pfc_nonempty_shifted_coefficient_leftsum_body_terminal. fs_u_pfc_nonempty_shifted_coefficient_leftsum = fs_q_pfc_nonempty_shifted_coefficient_leftsum_body_terminal * S ((S (S (t+i))) * fs_v_pfc_nonempty_shifted_coefficient_leftsum) + (pfc_natural_sum_nonempty_shifted_coefficient_left))) /\\ forall fs_i_pfc_nonempty_shifted_coefficient_leftsum_body_steps. (exists fs_lt_pfc_nonempty_shifted_coefficient_leftsum_body_steps_bound. fs_lt_pfc_nonempty_shifted_coefficient_leftsum_body_steps_bound + S fs_i_pfc_nonempty_shifted_coefficient_leftsum_body_steps = S (t+i)) -> exists fs_a_pfc_nonempty_shifted_coefficient_leftsum_body_steps fs_r_pfc_nonempty_shifted_coefficient_leftsum_body_steps fs_s_pfc_nonempty_shifted_coefficient_leftsum_body_steps. ((((exists fs_h_pfc_nonempty_shifted_coefficient_leftsum_body_steps_summand. fs_h_pfc_nonempty_shifted_coefficient_leftsum_body_steps_summand + S (fs_a_pfc_nonempty_shifted_coefficient_leftsum_body_steps) = S ((S (fs_i_pfc_nonempty_shifted_coefficient_leftsum_body_steps)) * pfc_terms_scale_nonempty_shifted_coefficient_left)) /\\ exists fs_q_pfc_nonempty_shifted_coefficient_leftsum_body_steps_summand. pfc_terms_code_nonempty_shifted_coefficient_left = fs_q_pfc_nonempty_shifted_coefficient_leftsum_body_steps_summand * S ((S (fs_i_pfc_nonempty_shifted_coefficient_leftsum_body_steps)) * pfc_terms_scale_nonempty_shifted_coefficient_left) + (fs_a_pfc_nonempty_shifted_coefficient_leftsum_body_steps))) /\\ ((((exists fs_h_pfc_nonempty_shifted_coefficient_leftsum_body_steps_partial. fs_h_pfc_nonempty_shifted_coefficient_leftsum_body_steps_partial + S (fs_r_pfc_nonempty_shifted_coefficient_leftsum_body_steps) = S ((S (fs_i_pfc_nonempty_shifted_coefficient_leftsum_body_steps)) * fs_v_pfc_nonempty_shifted_coefficient_leftsum)) /\\ exists fs_q_pfc_nonempty_shifted_coefficient_leftsum_body_steps_partial. fs_u_pfc_nonempty_shifted_coefficient_leftsum = fs_q_pfc_nonempty_shifted_coefficient_leftsum_body_steps_partial * S ((S (fs_i_pfc_nonempty_shifted_coefficient_leftsum_body_steps)) * fs_v_pfc_nonempty_shifted_coefficient_leftsum) + (fs_r_pfc_nonempty_shifted_coefficient_leftsum_body_steps))) /\\ ((((exists fs_h_pfc_nonempty_shifted_coefficient_leftsum_body_steps_successor. fs_h_pfc_nonempty_shifted_coefficient_leftsum_body_steps_successor + S (fs_s_pfc_nonempty_shifted_coefficient_leftsum_body_steps) = S ((S (S fs_i_pfc_nonempty_shifted_coefficient_leftsum_body_steps)) * fs_v_pfc_nonempty_shifted_coefficient_leftsum)) /\\ exists fs_q_pfc_nonempty_shifted_coefficient_leftsum_body_steps_successor. fs_u_pfc_nonempty_shifted_coefficient_leftsum = fs_q_pfc_nonempty_shifted_coefficient_leftsum_body_steps_successor * S ((S (S fs_i_pfc_nonempty_shifted_coefficient_leftsum_body_steps)) * fs_v_pfc_nonempty_shifted_coefficient_leftsum) + (fs_s_pfc_nonempty_shifted_coefficient_leftsum_body_steps))) /\\ fs_s_pfc_nonempty_shifted_coefficient_leftsum_body_steps = fs_r_pfc_nonempty_shifted_coefficient_leftsum_body_steps + fs_a_pfc_nonempty_shifted_coefficient_leftsum_body_steps)))))) /\\ ((((exists pfa_gap_nonempty_shifted_coefficient_leftresiduebound. pfa_gap_nonempty_shifted_coefficient_leftresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_nonempty_shifted_coefficient_leftresiduecongruence pfa_offset_right_nonempty_shifted_coefficient_leftresiduecongruence. (pfc_natural_sum_nonempty_shifted_coefficient_left) + (p) * pfa_offset_left_nonempty_shifted_coefficient_leftresiduecongruence = (r) + (p) * pfa_offset_right_nonempty_shifted_coefficient_leftresiduecongruence)))))))))))", "specialize hn_right_right_right (t+i)", "apply hn_right_right_right", "rewrite hlength", "specialize matrix_recursive_lt_add_left (i)", "specialize matrix_recursive_lt_add_left (N)", "specialize matrix_recursive_lt_add_left (t)", "apply matrix_recursive_lt_add_left", "exact hi", "cases hv", "cases hv_witness", "have heq : x=a", "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 (t+L)", "specialize prime_field_convolution_coefficient_functional (bb)", "specialize prime_field_convolution_coefficient_functional (bc)", "specialize prime_field_convolution_coefficient_functional (M)", "specialize prime_field_convolution_coefficient_functional (t+i)", "specialize prime_field_convolution_coefficient_functional (x)", "specialize prime_field_convolution_coefficient_functional (a)", "apply prime_field_convolution_coefficient_functional", "exact hv_witness_right", "specialize prime_field_convolution_coefficient_left_padding_left (p)", "specialize prime_field_convolution_coefficient_left_padding_left (ab)", "specialize prime_field_convolution_coefficient_left_padding_left (ac)", "specialize prime_field_convolution_coefficient_left_padding_left (L)", "specialize prime_field_convolution_coefficient_left_padding_left (bb)", "specialize prime_field_convolution_coefficient_left_padding_left (bc)", "specialize prime_field_convolution_coefficient_left_padding_left (M)", "specialize prime_field_convolution_coefficient_left_padding_left (AB)", "specialize prime_field_convolution_coefficient_left_padding_left (AC)", "specialize prime_field_convolution_coefficient_left_padding_left (t)", "specialize prime_field_convolution_coefficient_left_padding_left (i)", "specialize prime_field_convolution_coefficient_left_padding_left (a)", "apply prime_field_convolution_coefficient_left_padding_left", "exact hpad", "exact hcoefficient", "rewrite heq at hv_witness_left", "rewrite heq at hv_witness_left", "exact hv_witness_left" ], "script_sha256": "9bf8e5668da8f0e9cf6b5844b6372be6c432df2104211b7d3e841aab84c6e880", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_convolution_padding_candidate.py", "sha256": "2d874ecfb35a5db0aecdeb07b549464efebad9072c363113aa5a0a977845d007" }, "statement": "forall p ab ac L bb bc M cb cc N AB AC t CB CC K. (~(p=0)) -> (~(L=0)) -> (~(M=0)) -> (((forall pfp_repeat_index_nonempty_factor_padding_leftzeros. (exists pfa_gap_nonempty_factor_padding_leftzerosindex. pfa_gap_nonempty_factor_padding_leftzerosindex + S (pfp_repeat_index_nonempty_factor_padding_leftzeros) = (t)) -> (((exists ff_h_pfp_nonempty_factor_padding_leftzerosentry. ff_h_pfp_nonempty_factor_padding_leftzerosentry + S (0) = S ((S (pfp_repeat_index_nonempty_factor_padding_leftzeros)) * AC)) /\\ exists ff_q_pfp_nonempty_factor_padding_leftzerosentry. AB = ff_q_pfp_nonempty_factor_padding_leftzerosentry * S ((S (pfp_repeat_index_nonempty_factor_padding_leftzeros)) * AC) + (0)))) /\\ ((forall pfrep_index_nonempty_factor_padding_left pfrep_value_nonempty_factor_padding_left. (exists pfa_gap_nonempty_factor_padding_leftbound. pfa_gap_nonempty_factor_padding_leftbound + S (pfrep_index_nonempty_factor_padding_left) = (L)) -> (((exists ff_h_pfp_nonempty_factor_padding_leftinput. ff_h_pfp_nonempty_factor_padding_leftinput + S (pfrep_value_nonempty_factor_padding_left) = S ((S (pfrep_index_nonempty_factor_padding_left)) * ac)) /\\ exists ff_q_pfp_nonempty_factor_padding_leftinput. ab = ff_q_pfp_nonempty_factor_padding_leftinput * S ((S (pfrep_index_nonempty_factor_padding_left)) * ac) + (pfrep_value_nonempty_factor_padding_left))) -> (((exists ff_h_pfp_nonempty_factor_padding_leftoutput. ff_h_pfp_nonempty_factor_padding_leftoutput + S (pfrep_value_nonempty_factor_padding_left) = S ((S ((t)+pfrep_index_nonempty_factor_padding_left)) * AC)) /\\ exists ff_q_pfp_nonempty_factor_padding_leftoutput. AB = ff_q_pfp_nonempty_factor_padding_leftoutput * S ((S ((t)+pfrep_index_nonempty_factor_padding_left)) * AC) + (pfrep_value_nonempty_factor_padding_left))))))) -> (((forall fom_index_pfp_nonempty_old_leftleft. (exists fom_gap_pfp_nonempty_old_leftleft_index_bound. fom_gap_pfp_nonempty_old_leftleft_index_bound + S (fom_index_pfp_nonempty_old_leftleft) = L) -> exists fom_value_pfp_nonempty_old_leftleft. ((((exists fom_beta_height_pfp_nonempty_old_leftleft_entry. fom_beta_height_pfp_nonempty_old_leftleft_entry + S (fom_value_pfp_nonempty_old_leftleft) = S ((S (fom_index_pfp_nonempty_old_leftleft)) * ac)) /\\ exists fom_beta_quotient_pfp_nonempty_old_leftleft_entry. ab = fom_beta_quotient_pfp_nonempty_old_leftleft_entry * S ((S (fom_index_pfp_nonempty_old_leftleft)) * ac) + (fom_value_pfp_nonempty_old_leftleft))) /\\ (exists fom_gap_pfp_nonempty_old_leftleft_value_bound. fom_gap_pfp_nonempty_old_leftleft_value_bound + S (fom_value_pfp_nonempty_old_leftleft) = p))) /\\ (((forall fom_index_pfp_nonempty_old_leftright. (exists fom_gap_pfp_nonempty_old_leftright_index_bound. fom_gap_pfp_nonempty_old_leftright_index_bound + S (fom_index_pfp_nonempty_old_leftright) = M) -> exists fom_value_pfp_nonempty_old_leftright. ((((exists fom_beta_height_pfp_nonempty_old_leftright_entry. fom_beta_height_pfp_nonempty_old_leftright_entry + S (fom_value_pfp_nonempty_old_leftright) = S ((S (fom_index_pfp_nonempty_old_leftright)) * bc)) /\\ exists fom_beta_quotient_pfp_nonempty_old_leftright_entry. bb = fom_beta_quotient_pfp_nonempty_old_leftright_entry * S ((S (fom_index_pfp_nonempty_old_leftright)) * bc) + (fom_value_pfp_nonempty_old_leftright))) /\\ (exists fom_gap_pfp_nonempty_old_leftright_value_bound. fom_gap_pfp_nonempty_old_leftright_value_bound + S (fom_value_pfp_nonempty_old_leftright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_nonempty_old_leftcoefficients. (exists pfa_gap_nonempty_old_leftcoefficientsbound. pfa_gap_nonempty_old_leftcoefficientsbound + S (pfc_index_nonempty_old_leftcoefficients) = (N)) -> exists pfc_value_nonempty_old_leftcoefficients. ((((exists ff_h_pfp_nonempty_old_leftcoefficientsentry. ff_h_pfp_nonempty_old_leftcoefficientsentry + S (pfc_value_nonempty_old_leftcoefficients) = S ((S (pfc_index_nonempty_old_leftcoefficients)) * cc)) /\\ exists ff_q_pfp_nonempty_old_leftcoefficientsentry. cb = ff_q_pfp_nonempty_old_leftcoefficientsentry * S ((S (pfc_index_nonempty_old_leftcoefficients)) * cc) + (pfc_value_nonempty_old_leftcoefficients))) /\\ ((exists pfc_terms_code_nonempty_old_leftcoefficientscoefficient pfc_terms_scale_nonempty_old_leftcoefficientscoefficient pfc_natural_sum_nonempty_old_leftcoefficientscoefficient. ((forall pfc_index_nonempty_old_leftcoefficientscoefficientdiagonal. (exists pfa_gap_nonempty_old_leftcoefficientscoefficientdiagonalbound. pfa_gap_nonempty_old_leftcoefficientscoefficientdiagonalbound + S (pfc_index_nonempty_old_leftcoefficientscoefficientdiagonal) = (S (pfc_index_nonempty_old_leftcoefficients))) -> exists pfc_value_nonempty_old_leftcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_nonempty_old_leftcoefficientscoefficientdiagonalentry. ff_h_pfp_nonempty_old_leftcoefficientscoefficientdiagonalentry + S (pfc_value_nonempty_old_leftcoefficientscoefficientdiagonal) = S ((S (pfc_index_nonempty_old_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_nonempty_old_leftcoefficientscoefficient)) /\\ exists ff_q_pfp_nonempty_old_leftcoefficientscoefficientdiagonalentry. pfc_terms_code_nonempty_old_leftcoefficientscoefficient = ff_q_pfp_nonempty_old_leftcoefficientscoefficientdiagonalentry * S ((S (pfc_index_nonempty_old_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_nonempty_old_leftcoefficientscoefficient) + (pfc_value_nonempty_old_leftcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_nonempty_old_leftcoefficientscoefficientdiagonalterm pfc_left_nonempty_old_leftcoefficientscoefficientdiagonalterm pfc_right_nonempty_old_leftcoefficientscoefficientdiagonalterm. (((pfc_index_nonempty_old_leftcoefficientscoefficientdiagonal)+pfc_complement_nonempty_old_leftcoefficientscoefficientdiagonalterm=(pfc_index_nonempty_old_leftcoefficients)) /\\ ((((((exists pfa_gap_nonempty_old_leftcoefficientscoefficientdiagonaltermleftinside. pfa_gap_nonempty_old_leftcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_nonempty_old_leftcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_nonempty_old_leftcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_nonempty_old_leftcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_nonempty_old_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_nonempty_old_leftcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_nonempty_old_leftcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_nonempty_old_leftcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_nonempty_old_leftcoefficientscoefficientdiagonal)) * ac) + (pfc_left_nonempty_old_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_nonempty_old_leftcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_nonempty_old_leftcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_nonempty_old_leftcoefficientscoefficientdiagonal)) /\\ (((pfc_left_nonempty_old_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_nonempty_old_leftcoefficientscoefficientdiagonaltermrightinside. pfa_gap_nonempty_old_leftcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_nonempty_old_leftcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_nonempty_old_leftcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_nonempty_old_leftcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_nonempty_old_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_nonempty_old_leftcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_nonempty_old_leftcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_nonempty_old_leftcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_nonempty_old_leftcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_nonempty_old_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_nonempty_old_leftcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_nonempty_old_leftcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_nonempty_old_leftcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_nonempty_old_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_nonempty_old_leftcoefficientscoefficientdiagonal)=pfc_left_nonempty_old_leftcoefficientscoefficientdiagonalterm*pfc_right_nonempty_old_leftcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_nonempty_old_leftcoefficientscoefficientsum fs_v_pfc_nonempty_old_leftcoefficientscoefficientsum. ((((exists fs_h_pfc_nonempty_old_leftcoefficientscoefficientsum_body_start. fs_h_pfc_nonempty_old_leftcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_nonempty_old_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_nonempty_old_leftcoefficientscoefficientsum_body_start. fs_u_pfc_nonempty_old_leftcoefficientscoefficientsum = fs_q_pfc_nonempty_old_leftcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_nonempty_old_leftcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_nonempty_old_leftcoefficientscoefficientsum_body_terminal. fs_h_pfc_nonempty_old_leftcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_nonempty_old_leftcoefficientscoefficient) = S ((S (S (pfc_index_nonempty_old_leftcoefficients))) * fs_v_pfc_nonempty_old_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_nonempty_old_leftcoefficientscoefficientsum_body_terminal. fs_u_pfc_nonempty_old_leftcoefficientscoefficientsum = fs_q_pfc_nonempty_old_leftcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_nonempty_old_leftcoefficients))) * fs_v_pfc_nonempty_old_leftcoefficientscoefficientsum) + (pfc_natural_sum_nonempty_old_leftcoefficientscoefficient))) /\\ forall fs_i_pfc_nonempty_old_leftcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_nonempty_old_leftcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_nonempty_old_leftcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_nonempty_old_leftcoefficientscoefficientsum_body_steps = S (pfc_index_nonempty_old_leftcoefficients)) -> exists fs_a_pfc_nonempty_old_leftcoefficientscoefficientsum_body_steps fs_r_pfc_nonempty_old_leftcoefficientscoefficientsum_body_steps fs_s_pfc_nonempty_old_leftcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_nonempty_old_leftcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_nonempty_old_leftcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_nonempty_old_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_nonempty_old_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_nonempty_old_leftcoefficientscoefficient)) /\\ exists fs_q_pfc_nonempty_old_leftcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_nonempty_old_leftcoefficientscoefficient = fs_q_pfc_nonempty_old_leftcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_nonempty_old_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_nonempty_old_leftcoefficientscoefficient) + (fs_a_pfc_nonempty_old_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_nonempty_old_leftcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_nonempty_old_leftcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_nonempty_old_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_nonempty_old_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_nonempty_old_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_nonempty_old_leftcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_nonempty_old_leftcoefficientscoefficientsum = fs_q_pfc_nonempty_old_leftcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_nonempty_old_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_nonempty_old_leftcoefficientscoefficientsum) + (fs_r_pfc_nonempty_old_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_nonempty_old_leftcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_nonempty_old_leftcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_nonempty_old_leftcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_nonempty_old_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_nonempty_old_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_nonempty_old_leftcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_nonempty_old_leftcoefficientscoefficientsum = fs_q_pfc_nonempty_old_leftcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_nonempty_old_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_nonempty_old_leftcoefficientscoefficientsum) + (fs_s_pfc_nonempty_old_leftcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_nonempty_old_leftcoefficientscoefficientsum_body_steps = fs_r_pfc_nonempty_old_leftcoefficientscoefficientsum_body_steps + fs_a_pfc_nonempty_old_leftcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_nonempty_old_leftcoefficientscoefficientresiduebound. pfa_gap_nonempty_old_leftcoefficientscoefficientresiduebound + S (pfc_value_nonempty_old_leftcoefficients) = (p)) /\\ ((exists pfa_offset_left_nonempty_old_leftcoefficientscoefficientresiduecongruence pfa_offset_right_nonempty_old_leftcoefficientscoefficientresiduecongruence. (pfc_natural_sum_nonempty_old_leftcoefficientscoefficient) + (p) * pfa_offset_left_nonempty_old_leftcoefficientscoefficientresiduecongruence = (pfc_value_nonempty_old_leftcoefficients) + (p) * pfa_offset_right_nonempty_old_leftcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_nonempty_new_leftleft. (exists fom_gap_pfp_nonempty_new_leftleft_index_bound. fom_gap_pfp_nonempty_new_leftleft_index_bound + S (fom_index_pfp_nonempty_new_leftleft) = t+L) -> exists fom_value_pfp_nonempty_new_leftleft. ((((exists fom_beta_height_pfp_nonempty_new_leftleft_entry. fom_beta_height_pfp_nonempty_new_leftleft_entry + S (fom_value_pfp_nonempty_new_leftleft) = S ((S (fom_index_pfp_nonempty_new_leftleft)) * AC)) /\\ exists fom_beta_quotient_pfp_nonempty_new_leftleft_entry. AB = fom_beta_quotient_pfp_nonempty_new_leftleft_entry * S ((S (fom_index_pfp_nonempty_new_leftleft)) * AC) + (fom_value_pfp_nonempty_new_leftleft))) /\\ (exists fom_gap_pfp_nonempty_new_leftleft_value_bound. fom_gap_pfp_nonempty_new_leftleft_value_bound + S (fom_value_pfp_nonempty_new_leftleft) = p))) /\\ (((forall fom_index_pfp_nonempty_new_leftright. (exists fom_gap_pfp_nonempty_new_leftright_index_bound. fom_gap_pfp_nonempty_new_leftright_index_bound + S (fom_index_pfp_nonempty_new_leftright) = M) -> exists fom_value_pfp_nonempty_new_leftright. ((((exists fom_beta_height_pfp_nonempty_new_leftright_entry. fom_beta_height_pfp_nonempty_new_leftright_entry + S (fom_value_pfp_nonempty_new_leftright) = S ((S (fom_index_pfp_nonempty_new_leftright)) * bc)) /\\ exists fom_beta_quotient_pfp_nonempty_new_leftright_entry. bb = fom_beta_quotient_pfp_nonempty_new_leftright_entry * S ((S (fom_index_pfp_nonempty_new_leftright)) * bc) + (fom_value_pfp_nonempty_new_leftright))) /\\ (exists fom_gap_pfp_nonempty_new_leftright_value_bound. fom_gap_pfp_nonempty_new_leftright_value_bound + S (fom_value_pfp_nonempty_new_leftright) = p))) /\\ (((((((t+L)=0 \\/ (M)=0) /\\ (((K)=0)))) \\/ (((~((t+L)=0)) /\\ (((~((M)=0)) /\\ (((t+L)+(M)=S (K)))))))) /\\ ((forall pfc_index_nonempty_new_leftcoefficients. (exists pfa_gap_nonempty_new_leftcoefficientsbound. pfa_gap_nonempty_new_leftcoefficientsbound + S (pfc_index_nonempty_new_leftcoefficients) = (K)) -> exists pfc_value_nonempty_new_leftcoefficients. ((((exists ff_h_pfp_nonempty_new_leftcoefficientsentry. ff_h_pfp_nonempty_new_leftcoefficientsentry + S (pfc_value_nonempty_new_leftcoefficients) = S ((S (pfc_index_nonempty_new_leftcoefficients)) * CC)) /\\ exists ff_q_pfp_nonempty_new_leftcoefficientsentry. CB = ff_q_pfp_nonempty_new_leftcoefficientsentry * S ((S (pfc_index_nonempty_new_leftcoefficients)) * CC) + (pfc_value_nonempty_new_leftcoefficients))) /\\ ((exists pfc_terms_code_nonempty_new_leftcoefficientscoefficient pfc_terms_scale_nonempty_new_leftcoefficientscoefficient pfc_natural_sum_nonempty_new_leftcoefficientscoefficient. ((forall pfc_index_nonempty_new_leftcoefficientscoefficientdiagonal. (exists pfa_gap_nonempty_new_leftcoefficientscoefficientdiagonalbound. pfa_gap_nonempty_new_leftcoefficientscoefficientdiagonalbound + S (pfc_index_nonempty_new_leftcoefficientscoefficientdiagonal) = (S (pfc_index_nonempty_new_leftcoefficients))) -> exists pfc_value_nonempty_new_leftcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_nonempty_new_leftcoefficientscoefficientdiagonalentry. ff_h_pfp_nonempty_new_leftcoefficientscoefficientdiagonalentry + S (pfc_value_nonempty_new_leftcoefficientscoefficientdiagonal) = S ((S (pfc_index_nonempty_new_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_nonempty_new_leftcoefficientscoefficient)) /\\ exists ff_q_pfp_nonempty_new_leftcoefficientscoefficientdiagonalentry. pfc_terms_code_nonempty_new_leftcoefficientscoefficient = ff_q_pfp_nonempty_new_leftcoefficientscoefficientdiagonalentry * S ((S (pfc_index_nonempty_new_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_nonempty_new_leftcoefficientscoefficient) + (pfc_value_nonempty_new_leftcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_nonempty_new_leftcoefficientscoefficientdiagonalterm pfc_left_nonempty_new_leftcoefficientscoefficientdiagonalterm pfc_right_nonempty_new_leftcoefficientscoefficientdiagonalterm. (((pfc_index_nonempty_new_leftcoefficientscoefficientdiagonal)+pfc_complement_nonempty_new_leftcoefficientscoefficientdiagonalterm=(pfc_index_nonempty_new_leftcoefficients)) /\\ ((((((exists pfa_gap_nonempty_new_leftcoefficientscoefficientdiagonaltermleftinside. pfa_gap_nonempty_new_leftcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_nonempty_new_leftcoefficientscoefficientdiagonal) = (t+L)) /\\ ((((exists ff_h_pfp_nonempty_new_leftcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_nonempty_new_leftcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_nonempty_new_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_nonempty_new_leftcoefficientscoefficientdiagonal)) * AC)) /\\ exists ff_q_pfp_nonempty_new_leftcoefficientscoefficientdiagonaltermleftentry. AB = ff_q_pfp_nonempty_new_leftcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_nonempty_new_leftcoefficientscoefficientdiagonal)) * AC) + (pfc_left_nonempty_new_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_nonempty_new_leftcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_nonempty_new_leftcoefficientscoefficientdiagonaltermleftoutside+(t+L)=(pfc_index_nonempty_new_leftcoefficientscoefficientdiagonal)) /\\ (((pfc_left_nonempty_new_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_nonempty_new_leftcoefficientscoefficientdiagonaltermrightinside. pfa_gap_nonempty_new_leftcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_nonempty_new_leftcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_nonempty_new_leftcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_nonempty_new_leftcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_nonempty_new_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_nonempty_new_leftcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_nonempty_new_leftcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_nonempty_new_leftcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_nonempty_new_leftcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_nonempty_new_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_nonempty_new_leftcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_nonempty_new_leftcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_nonempty_new_leftcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_nonempty_new_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_nonempty_new_leftcoefficientscoefficientdiagonal)=pfc_left_nonempty_new_leftcoefficientscoefficientdiagonalterm*pfc_right_nonempty_new_leftcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_nonempty_new_leftcoefficientscoefficientsum fs_v_pfc_nonempty_new_leftcoefficientscoefficientsum. ((((exists fs_h_pfc_nonempty_new_leftcoefficientscoefficientsum_body_start. fs_h_pfc_nonempty_new_leftcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_nonempty_new_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_nonempty_new_leftcoefficientscoefficientsum_body_start. fs_u_pfc_nonempty_new_leftcoefficientscoefficientsum = fs_q_pfc_nonempty_new_leftcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_nonempty_new_leftcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_nonempty_new_leftcoefficientscoefficientsum_body_terminal. fs_h_pfc_nonempty_new_leftcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_nonempty_new_leftcoefficientscoefficient) = S ((S (S (pfc_index_nonempty_new_leftcoefficients))) * fs_v_pfc_nonempty_new_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_nonempty_new_leftcoefficientscoefficientsum_body_terminal. fs_u_pfc_nonempty_new_leftcoefficientscoefficientsum = fs_q_pfc_nonempty_new_leftcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_nonempty_new_leftcoefficients))) * fs_v_pfc_nonempty_new_leftcoefficientscoefficientsum) + (pfc_natural_sum_nonempty_new_leftcoefficientscoefficient))) /\\ forall fs_i_pfc_nonempty_new_leftcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_nonempty_new_leftcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_nonempty_new_leftcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_nonempty_new_leftcoefficientscoefficientsum_body_steps = S (pfc_index_nonempty_new_leftcoefficients)) -> exists fs_a_pfc_nonempty_new_leftcoefficientscoefficientsum_body_steps fs_r_pfc_nonempty_new_leftcoefficientscoefficientsum_body_steps fs_s_pfc_nonempty_new_leftcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_nonempty_new_leftcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_nonempty_new_leftcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_nonempty_new_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_nonempty_new_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_nonempty_new_leftcoefficientscoefficient)) /\\ exists fs_q_pfc_nonempty_new_leftcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_nonempty_new_leftcoefficientscoefficient = fs_q_pfc_nonempty_new_leftcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_nonempty_new_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_nonempty_new_leftcoefficientscoefficient) + (fs_a_pfc_nonempty_new_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_nonempty_new_leftcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_nonempty_new_leftcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_nonempty_new_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_nonempty_new_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_nonempty_new_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_nonempty_new_leftcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_nonempty_new_leftcoefficientscoefficientsum = fs_q_pfc_nonempty_new_leftcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_nonempty_new_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_nonempty_new_leftcoefficientscoefficientsum) + (fs_r_pfc_nonempty_new_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_nonempty_new_leftcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_nonempty_new_leftcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_nonempty_new_leftcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_nonempty_new_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_nonempty_new_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_nonempty_new_leftcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_nonempty_new_leftcoefficientscoefficientsum = fs_q_pfc_nonempty_new_leftcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_nonempty_new_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_nonempty_new_leftcoefficientscoefficientsum) + (fs_s_pfc_nonempty_new_leftcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_nonempty_new_leftcoefficientscoefficientsum_body_steps = fs_r_pfc_nonempty_new_leftcoefficientscoefficientsum_body_steps + fs_a_pfc_nonempty_new_leftcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_nonempty_new_leftcoefficientscoefficientresiduebound. pfa_gap_nonempty_new_leftcoefficientscoefficientresiduebound + S (pfc_value_nonempty_new_leftcoefficients) = (p)) /\\ ((exists pfa_offset_left_nonempty_new_leftcoefficientscoefficientresiduecongruence pfa_offset_right_nonempty_new_leftcoefficientscoefficientresiduecongruence. (pfc_natural_sum_nonempty_new_leftcoefficientscoefficient) + (p) * pfa_offset_left_nonempty_new_leftcoefficientscoefficientresiduecongruence = (pfc_value_nonempty_new_leftcoefficients) + (p) * pfa_offset_right_nonempty_new_leftcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((K=t+N) /\\ ((((forall pfp_repeat_index_nonempty_product_padding_leftzeros. (exists pfa_gap_nonempty_product_padding_leftzerosindex. pfa_gap_nonempty_product_padding_leftzerosindex + S (pfp_repeat_index_nonempty_product_padding_leftzeros) = (t)) -> (((exists ff_h_pfp_nonempty_product_padding_leftzerosentry. ff_h_pfp_nonempty_product_padding_leftzerosentry + S (0) = S ((S (pfp_repeat_index_nonempty_product_padding_leftzeros)) * CC)) /\\ exists ff_q_pfp_nonempty_product_padding_leftzerosentry. CB = ff_q_pfp_nonempty_product_padding_leftzerosentry * S ((S (pfp_repeat_index_nonempty_product_padding_leftzeros)) * CC) + (0)))) /\\ ((forall pfrep_index_nonempty_product_padding_left pfrep_value_nonempty_product_padding_left. 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"alpha_only", "name": "prime_field_polynomial_convolution_left_padding_nonempty_right", "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 t", "intro CB", "intro CC", "intro K", "intro hp", "intro hL", "intro hM", "intro hpad", "intro hc", "intro hn", "cases hc", "cases hc_right", "cases hc_right_right", "cases hn", "cases hn_right", "cases hn_right_right", "have hlength : K=t+N", "specialize polynomial_product_length_left_padding_right (L)", "specialize polynomial_product_length_left_padding_right (M)", "specialize polynomial_product_length_left_padding_right (N)", "specialize polynomial_product_length_left_padding_right (t)", "specialize polynomial_product_length_left_padding_right (K)", "apply polynomial_product_length_left_padding_right", "exact hc_right_right_left", "exact hL", "exact hM", "exact hn_right_right_left", "split", "exact hlength", "split", "intro i", "intro hi", "have hv : exists r. ((((exists ff_h_pfp_nonempty_zero_entry_right. ff_h_pfp_nonempty_zero_entry_right + S (r) = S ((S (i)) * CC)) /\\ exists ff_q_pfp_nonempty_zero_entry_right. CB = ff_q_pfp_nonempty_zero_entry_right * S ((S (i)) * CC) + (r))) /\\ ((exists pfc_terms_code_nonempty_zero_coefficient_right pfc_terms_scale_nonempty_zero_coefficient_right pfc_natural_sum_nonempty_zero_coefficient_right. ((forall pfc_index_nonempty_zero_coefficient_rightdiagonal. (exists pfa_gap_nonempty_zero_coefficient_rightdiagonalbound. pfa_gap_nonempty_zero_coefficient_rightdiagonalbound + S (pfc_index_nonempty_zero_coefficient_rightdiagonal) = (S (i))) -> exists pfc_value_nonempty_zero_coefficient_rightdiagonal. ((((exists ff_h_pfp_nonempty_zero_coefficient_rightdiagonalentry. ff_h_pfp_nonempty_zero_coefficient_rightdiagonalentry + S (pfc_value_nonempty_zero_coefficient_rightdiagonal) = S ((S (pfc_index_nonempty_zero_coefficient_rightdiagonal)) * pfc_terms_scale_nonempty_zero_coefficient_right)) /\\ exists ff_q_pfp_nonempty_zero_coefficient_rightdiagonalentry. pfc_terms_code_nonempty_zero_coefficient_right = ff_q_pfp_nonempty_zero_coefficient_rightdiagonalentry * S ((S (pfc_index_nonempty_zero_coefficient_rightdiagonal)) * pfc_terms_scale_nonempty_zero_coefficient_right) + (pfc_value_nonempty_zero_coefficient_rightdiagonal))) /\\ ((exists pfc_complement_nonempty_zero_coefficient_rightdiagonalterm pfc_left_nonempty_zero_coefficient_rightdiagonalterm pfc_right_nonempty_zero_coefficient_rightdiagonalterm. (((pfc_index_nonempty_zero_coefficient_rightdiagonal)+pfc_complement_nonempty_zero_coefficient_rightdiagonalterm=(i)) /\\ ((((((exists pfa_gap_nonempty_zero_coefficient_rightdiagonaltermleftinside. pfa_gap_nonempty_zero_coefficient_rightdiagonaltermleftinside + S (pfc_index_nonempty_zero_coefficient_rightdiagonal) = (L)) /\\ ((((exists ff_h_pfp_nonempty_zero_coefficient_rightdiagonaltermleftentry. ff_h_pfp_nonempty_zero_coefficient_rightdiagonaltermleftentry + S (pfc_left_nonempty_zero_coefficient_rightdiagonalterm) = S ((S (pfc_index_nonempty_zero_coefficient_rightdiagonal)) * ac)) /\\ exists ff_q_pfp_nonempty_zero_coefficient_rightdiagonaltermleftentry. ab = ff_q_pfp_nonempty_zero_coefficient_rightdiagonaltermleftentry * S ((S (pfc_index_nonempty_zero_coefficient_rightdiagonal)) * ac) + (pfc_left_nonempty_zero_coefficient_rightdiagonalterm)))))) \\/ (((exists pfc_gap_nonempty_zero_coefficient_rightdiagonaltermleftoutside. pfc_gap_nonempty_zero_coefficient_rightdiagonaltermleftoutside+(L)=(pfc_index_nonempty_zero_coefficient_rightdiagonal)) /\\ (((pfc_left_nonempty_zero_coefficient_rightdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_nonempty_zero_coefficient_rightdiagonaltermrightinside. pfa_gap_nonempty_zero_coefficient_rightdiagonaltermrightinside + S (pfc_complement_nonempty_zero_coefficient_rightdiagonalterm) = (t+M)) /\\ ((((exists ff_h_pfp_nonempty_zero_coefficient_rightdiagonaltermrightentry. ff_h_pfp_nonempty_zero_coefficient_rightdiagonaltermrightentry + S (pfc_right_nonempty_zero_coefficient_rightdiagonalterm) = S ((S (pfc_complement_nonempty_zero_coefficient_rightdiagonalterm)) * BC)) /\\ exists ff_q_pfp_nonempty_zero_coefficient_rightdiagonaltermrightentry. BB = ff_q_pfp_nonempty_zero_coefficient_rightdiagonaltermrightentry * S ((S (pfc_complement_nonempty_zero_coefficient_rightdiagonalterm)) * BC) + (pfc_right_nonempty_zero_coefficient_rightdiagonalterm)))))) \\/ (((exists pfc_gap_nonempty_zero_coefficient_rightdiagonaltermrightoutside. pfc_gap_nonempty_zero_coefficient_rightdiagonaltermrightoutside+(t+M)=(pfc_complement_nonempty_zero_coefficient_rightdiagonalterm)) /\\ (((pfc_right_nonempty_zero_coefficient_rightdiagonalterm)=0))))) /\\ (((pfc_value_nonempty_zero_coefficient_rightdiagonal)=pfc_left_nonempty_zero_coefficient_rightdiagonalterm*pfc_right_nonempty_zero_coefficient_rightdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_nonempty_zero_coefficient_rightsum fs_v_pfc_nonempty_zero_coefficient_rightsum. ((((exists fs_h_pfc_nonempty_zero_coefficient_rightsum_body_start. fs_h_pfc_nonempty_zero_coefficient_rightsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_nonempty_zero_coefficient_rightsum)) /\\ exists fs_q_pfc_nonempty_zero_coefficient_rightsum_body_start. fs_u_pfc_nonempty_zero_coefficient_rightsum = fs_q_pfc_nonempty_zero_coefficient_rightsum_body_start * S ((S (0)) * fs_v_pfc_nonempty_zero_coefficient_rightsum) + (0))) /\\ ((((exists fs_h_pfc_nonempty_zero_coefficient_rightsum_body_terminal. fs_h_pfc_nonempty_zero_coefficient_rightsum_body_terminal + S (pfc_natural_sum_nonempty_zero_coefficient_right) = S ((S (S (i))) * fs_v_pfc_nonempty_zero_coefficient_rightsum)) /\\ exists fs_q_pfc_nonempty_zero_coefficient_rightsum_body_terminal. fs_u_pfc_nonempty_zero_coefficient_rightsum = fs_q_pfc_nonempty_zero_coefficient_rightsum_body_terminal * S ((S (S (i))) * fs_v_pfc_nonempty_zero_coefficient_rightsum) + (pfc_natural_sum_nonempty_zero_coefficient_right))) /\\ forall fs_i_pfc_nonempty_zero_coefficient_rightsum_body_steps. (exists fs_lt_pfc_nonempty_zero_coefficient_rightsum_body_steps_bound. fs_lt_pfc_nonempty_zero_coefficient_rightsum_body_steps_bound + S fs_i_pfc_nonempty_zero_coefficient_rightsum_body_steps = S (i)) -> exists fs_a_pfc_nonempty_zero_coefficient_rightsum_body_steps fs_r_pfc_nonempty_zero_coefficient_rightsum_body_steps fs_s_pfc_nonempty_zero_coefficient_rightsum_body_steps. ((((exists fs_h_pfc_nonempty_zero_coefficient_rightsum_body_steps_summand. fs_h_pfc_nonempty_zero_coefficient_rightsum_body_steps_summand + S (fs_a_pfc_nonempty_zero_coefficient_rightsum_body_steps) = S ((S (fs_i_pfc_nonempty_zero_coefficient_rightsum_body_steps)) * pfc_terms_scale_nonempty_zero_coefficient_right)) /\\ exists fs_q_pfc_nonempty_zero_coefficient_rightsum_body_steps_summand. pfc_terms_code_nonempty_zero_coefficient_right = fs_q_pfc_nonempty_zero_coefficient_rightsum_body_steps_summand * S ((S (fs_i_pfc_nonempty_zero_coefficient_rightsum_body_steps)) * pfc_terms_scale_nonempty_zero_coefficient_right) + (fs_a_pfc_nonempty_zero_coefficient_rightsum_body_steps))) /\\ ((((exists fs_h_pfc_nonempty_zero_coefficient_rightsum_body_steps_partial. fs_h_pfc_nonempty_zero_coefficient_rightsum_body_steps_partial + S (fs_r_pfc_nonempty_zero_coefficient_rightsum_body_steps) = S ((S (fs_i_pfc_nonempty_zero_coefficient_rightsum_body_steps)) * fs_v_pfc_nonempty_zero_coefficient_rightsum)) /\\ exists fs_q_pfc_nonempty_zero_coefficient_rightsum_body_steps_partial. fs_u_pfc_nonempty_zero_coefficient_rightsum = fs_q_pfc_nonempty_zero_coefficient_rightsum_body_steps_partial * S ((S (fs_i_pfc_nonempty_zero_coefficient_rightsum_body_steps)) * fs_v_pfc_nonempty_zero_coefficient_rightsum) + (fs_r_pfc_nonempty_zero_coefficient_rightsum_body_steps))) /\\ ((((exists fs_h_pfc_nonempty_zero_coefficient_rightsum_body_steps_successor. fs_h_pfc_nonempty_zero_coefficient_rightsum_body_steps_successor + S (fs_s_pfc_nonempty_zero_coefficient_rightsum_body_steps) = S ((S (S fs_i_pfc_nonempty_zero_coefficient_rightsum_body_steps)) * fs_v_pfc_nonempty_zero_coefficient_rightsum)) /\\ exists fs_q_pfc_nonempty_zero_coefficient_rightsum_body_steps_successor. fs_u_pfc_nonempty_zero_coefficient_rightsum = fs_q_pfc_nonempty_zero_coefficient_rightsum_body_steps_successor * S ((S (S fs_i_pfc_nonempty_zero_coefficient_rightsum_body_steps)) * fs_v_pfc_nonempty_zero_coefficient_rightsum) + (fs_s_pfc_nonempty_zero_coefficient_rightsum_body_steps))) /\\ fs_s_pfc_nonempty_zero_coefficient_rightsum_body_steps = fs_r_pfc_nonempty_zero_coefficient_rightsum_body_steps + fs_a_pfc_nonempty_zero_coefficient_rightsum_body_steps)))))) /\\ ((((exists pfa_gap_nonempty_zero_coefficient_rightresiduebound. pfa_gap_nonempty_zero_coefficient_rightresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_nonempty_zero_coefficient_rightresiduecongruence pfa_offset_right_nonempty_zero_coefficient_rightresiduecongruence. (pfc_natural_sum_nonempty_zero_coefficient_right) + (p) * pfa_offset_left_nonempty_zero_coefficient_rightresiduecongruence = (r) + (p) * pfa_offset_right_nonempty_zero_coefficient_rightresiduecongruence)))))))))))", "specialize hn_right_right_right (i)", "apply hn_right_right_right", "specialize le_trans (S i)", "specialize le_trans (t)", "specialize le_trans (K)", "apply le_trans", "exact hi", "rewrite hlength", "specialize le_add_right (t)", "specialize le_add_right (N)", "apply le_add_right", "cases hv", "cases hv_witness", "have hzero : x=0", "specialize prime_field_convolution_coefficient_before_left_padding_right (p)", "specialize prime_field_convolution_coefficient_before_left_padding_right (ab)", "specialize prime_field_convolution_coefficient_before_left_padding_right (ac)", "specialize prime_field_convolution_coefficient_before_left_padding_right (L)", "specialize prime_field_convolution_coefficient_before_left_padding_right (bb)", "specialize prime_field_convolution_coefficient_before_left_padding_right (bc)", "specialize prime_field_convolution_coefficient_before_left_padding_right (M)", "specialize prime_field_convolution_coefficient_before_left_padding_right (BB)", "specialize prime_field_convolution_coefficient_before_left_padding_right (BC)", "specialize prime_field_convolution_coefficient_before_left_padding_right (t)", "specialize prime_field_convolution_coefficient_before_left_padding_right (i)", "specialize prime_field_convolution_coefficient_before_left_padding_right (x)", "apply prime_field_convolution_coefficient_before_left_padding_right", "exact hp", "exact hpad", "exact hi", "exact hv_witness_right", "rewrite hzero at hv_witness_left", "rewrite hzero at hv_witness_left", "exact hv_witness_left", "intro i", "intro a", "intro hi", "intro ha", "have hcoefficient : exists pfc_terms_code_nonempty_original_coefficient_right pfc_terms_scale_nonempty_original_coefficient_right pfc_natural_sum_nonempty_original_coefficient_right. ((forall pfc_index_nonempty_original_coefficient_rightdiagonal. (exists pfa_gap_nonempty_original_coefficient_rightdiagonalbound. pfa_gap_nonempty_original_coefficient_rightdiagonalbound + S (pfc_index_nonempty_original_coefficient_rightdiagonal) = (S (i))) -> exists pfc_value_nonempty_original_coefficient_rightdiagonal. ((((exists ff_h_pfp_nonempty_original_coefficient_rightdiagonalentry. ff_h_pfp_nonempty_original_coefficient_rightdiagonalentry + S (pfc_value_nonempty_original_coefficient_rightdiagonal) = S ((S (pfc_index_nonempty_original_coefficient_rightdiagonal)) * pfc_terms_scale_nonempty_original_coefficient_right)) /\\ exists ff_q_pfp_nonempty_original_coefficient_rightdiagonalentry. pfc_terms_code_nonempty_original_coefficient_right = ff_q_pfp_nonempty_original_coefficient_rightdiagonalentry * S ((S (pfc_index_nonempty_original_coefficient_rightdiagonal)) * pfc_terms_scale_nonempty_original_coefficient_right) + (pfc_value_nonempty_original_coefficient_rightdiagonal))) /\\ ((exists pfc_complement_nonempty_original_coefficient_rightdiagonalterm pfc_left_nonempty_original_coefficient_rightdiagonalterm pfc_right_nonempty_original_coefficient_rightdiagonalterm. (((pfc_index_nonempty_original_coefficient_rightdiagonal)+pfc_complement_nonempty_original_coefficient_rightdiagonalterm=(i)) /\\ ((((((exists pfa_gap_nonempty_original_coefficient_rightdiagonaltermleftinside. pfa_gap_nonempty_original_coefficient_rightdiagonaltermleftinside + S (pfc_index_nonempty_original_coefficient_rightdiagonal) = (L)) /\\ ((((exists ff_h_pfp_nonempty_original_coefficient_rightdiagonaltermleftentry. ff_h_pfp_nonempty_original_coefficient_rightdiagonaltermleftentry + S (pfc_left_nonempty_original_coefficient_rightdiagonalterm) = S ((S (pfc_index_nonempty_original_coefficient_rightdiagonal)) * ac)) /\\ exists ff_q_pfp_nonempty_original_coefficient_rightdiagonaltermleftentry. ab = ff_q_pfp_nonempty_original_coefficient_rightdiagonaltermleftentry * S ((S (pfc_index_nonempty_original_coefficient_rightdiagonal)) * ac) + (pfc_left_nonempty_original_coefficient_rightdiagonalterm)))))) \\/ (((exists pfc_gap_nonempty_original_coefficient_rightdiagonaltermleftoutside. pfc_gap_nonempty_original_coefficient_rightdiagonaltermleftoutside+(L)=(pfc_index_nonempty_original_coefficient_rightdiagonal)) /\\ (((pfc_left_nonempty_original_coefficient_rightdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_nonempty_original_coefficient_rightdiagonaltermrightinside. pfa_gap_nonempty_original_coefficient_rightdiagonaltermrightinside + S (pfc_complement_nonempty_original_coefficient_rightdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_nonempty_original_coefficient_rightdiagonaltermrightentry. ff_h_pfp_nonempty_original_coefficient_rightdiagonaltermrightentry + S (pfc_right_nonempty_original_coefficient_rightdiagonalterm) = S ((S (pfc_complement_nonempty_original_coefficient_rightdiagonalterm)) * bc)) /\\ exists ff_q_pfp_nonempty_original_coefficient_rightdiagonaltermrightentry. bb = ff_q_pfp_nonempty_original_coefficient_rightdiagonaltermrightentry * S ((S (pfc_complement_nonempty_original_coefficient_rightdiagonalterm)) * bc) + (pfc_right_nonempty_original_coefficient_rightdiagonalterm)))))) \\/ (((exists pfc_gap_nonempty_original_coefficient_rightdiagonaltermrightoutside. pfc_gap_nonempty_original_coefficient_rightdiagonaltermrightoutside+(M)=(pfc_complement_nonempty_original_coefficient_rightdiagonalterm)) /\\ (((pfc_right_nonempty_original_coefficient_rightdiagonalterm)=0))))) /\\ (((pfc_value_nonempty_original_coefficient_rightdiagonal)=pfc_left_nonempty_original_coefficient_rightdiagonalterm*pfc_right_nonempty_original_coefficient_rightdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_nonempty_original_coefficient_rightsum fs_v_pfc_nonempty_original_coefficient_rightsum. ((((exists fs_h_pfc_nonempty_original_coefficient_rightsum_body_start. fs_h_pfc_nonempty_original_coefficient_rightsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_nonempty_original_coefficient_rightsum)) /\\ exists fs_q_pfc_nonempty_original_coefficient_rightsum_body_start. fs_u_pfc_nonempty_original_coefficient_rightsum = fs_q_pfc_nonempty_original_coefficient_rightsum_body_start * S ((S (0)) * fs_v_pfc_nonempty_original_coefficient_rightsum) + (0))) /\\ ((((exists fs_h_pfc_nonempty_original_coefficient_rightsum_body_terminal. fs_h_pfc_nonempty_original_coefficient_rightsum_body_terminal + S (pfc_natural_sum_nonempty_original_coefficient_right) = S ((S (S (i))) * fs_v_pfc_nonempty_original_coefficient_rightsum)) /\\ exists fs_q_pfc_nonempty_original_coefficient_rightsum_body_terminal. fs_u_pfc_nonempty_original_coefficient_rightsum = fs_q_pfc_nonempty_original_coefficient_rightsum_body_terminal * S ((S (S (i))) * fs_v_pfc_nonempty_original_coefficient_rightsum) + (pfc_natural_sum_nonempty_original_coefficient_right))) /\\ forall fs_i_pfc_nonempty_original_coefficient_rightsum_body_steps. (exists fs_lt_pfc_nonempty_original_coefficient_rightsum_body_steps_bound. fs_lt_pfc_nonempty_original_coefficient_rightsum_body_steps_bound + S fs_i_pfc_nonempty_original_coefficient_rightsum_body_steps = S (i)) -> exists fs_a_pfc_nonempty_original_coefficient_rightsum_body_steps fs_r_pfc_nonempty_original_coefficient_rightsum_body_steps fs_s_pfc_nonempty_original_coefficient_rightsum_body_steps. ((((exists fs_h_pfc_nonempty_original_coefficient_rightsum_body_steps_summand. fs_h_pfc_nonempty_original_coefficient_rightsum_body_steps_summand + S (fs_a_pfc_nonempty_original_coefficient_rightsum_body_steps) = S ((S (fs_i_pfc_nonempty_original_coefficient_rightsum_body_steps)) * pfc_terms_scale_nonempty_original_coefficient_right)) /\\ exists fs_q_pfc_nonempty_original_coefficient_rightsum_body_steps_summand. pfc_terms_code_nonempty_original_coefficient_right = fs_q_pfc_nonempty_original_coefficient_rightsum_body_steps_summand * S ((S (fs_i_pfc_nonempty_original_coefficient_rightsum_body_steps)) * pfc_terms_scale_nonempty_original_coefficient_right) + (fs_a_pfc_nonempty_original_coefficient_rightsum_body_steps))) /\\ ((((exists fs_h_pfc_nonempty_original_coefficient_rightsum_body_steps_partial. fs_h_pfc_nonempty_original_coefficient_rightsum_body_steps_partial + S (fs_r_pfc_nonempty_original_coefficient_rightsum_body_steps) = S ((S (fs_i_pfc_nonempty_original_coefficient_rightsum_body_steps)) * fs_v_pfc_nonempty_original_coefficient_rightsum)) /\\ exists fs_q_pfc_nonempty_original_coefficient_rightsum_body_steps_partial. fs_u_pfc_nonempty_original_coefficient_rightsum = fs_q_pfc_nonempty_original_coefficient_rightsum_body_steps_partial * S ((S (fs_i_pfc_nonempty_original_coefficient_rightsum_body_steps)) * fs_v_pfc_nonempty_original_coefficient_rightsum) + (fs_r_pfc_nonempty_original_coefficient_rightsum_body_steps))) /\\ ((((exists fs_h_pfc_nonempty_original_coefficient_rightsum_body_steps_successor. fs_h_pfc_nonempty_original_coefficient_rightsum_body_steps_successor + S (fs_s_pfc_nonempty_original_coefficient_rightsum_body_steps) = S ((S (S fs_i_pfc_nonempty_original_coefficient_rightsum_body_steps)) * fs_v_pfc_nonempty_original_coefficient_rightsum)) /\\ exists fs_q_pfc_nonempty_original_coefficient_rightsum_body_steps_successor. fs_u_pfc_nonempty_original_coefficient_rightsum = fs_q_pfc_nonempty_original_coefficient_rightsum_body_steps_successor * S ((S (S fs_i_pfc_nonempty_original_coefficient_rightsum_body_steps)) * fs_v_pfc_nonempty_original_coefficient_rightsum) + (fs_s_pfc_nonempty_original_coefficient_rightsum_body_steps))) /\\ fs_s_pfc_nonempty_original_coefficient_rightsum_body_steps = fs_r_pfc_nonempty_original_coefficient_rightsum_body_steps + fs_a_pfc_nonempty_original_coefficient_rightsum_body_steps)))))) /\\ ((((exists pfa_gap_nonempty_original_coefficient_rightresiduebound. pfa_gap_nonempty_original_coefficient_rightresiduebound + S (a) = (p)) /\\ ((exists pfa_offset_left_nonempty_original_coefficient_rightresiduecongruence pfa_offset_right_nonempty_original_coefficient_rightresiduecongruence. (pfc_natural_sum_nonempty_original_coefficient_right) + (p) * pfa_offset_left_nonempty_original_coefficient_rightresiduecongruence = (a) + (p) * pfa_offset_right_nonempty_original_coefficient_rightresiduecongruence))))))))", "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 hv : exists r. ((((exists ff_h_pfp_nonempty_shifted_entry_right. ff_h_pfp_nonempty_shifted_entry_right + S (r) = S ((S (t+i)) * CC)) /\\ exists ff_q_pfp_nonempty_shifted_entry_right. CB = ff_q_pfp_nonempty_shifted_entry_right * S ((S (t+i)) * CC) + (r))) /\\ ((exists pfc_terms_code_nonempty_shifted_coefficient_right pfc_terms_scale_nonempty_shifted_coefficient_right pfc_natural_sum_nonempty_shifted_coefficient_right. ((forall pfc_index_nonempty_shifted_coefficient_rightdiagonal. (exists pfa_gap_nonempty_shifted_coefficient_rightdiagonalbound. pfa_gap_nonempty_shifted_coefficient_rightdiagonalbound + S (pfc_index_nonempty_shifted_coefficient_rightdiagonal) = (S (t+i))) -> exists pfc_value_nonempty_shifted_coefficient_rightdiagonal. ((((exists ff_h_pfp_nonempty_shifted_coefficient_rightdiagonalentry. ff_h_pfp_nonempty_shifted_coefficient_rightdiagonalentry + S (pfc_value_nonempty_shifted_coefficient_rightdiagonal) = S ((S (pfc_index_nonempty_shifted_coefficient_rightdiagonal)) * pfc_terms_scale_nonempty_shifted_coefficient_right)) /\\ exists ff_q_pfp_nonempty_shifted_coefficient_rightdiagonalentry. pfc_terms_code_nonempty_shifted_coefficient_right = ff_q_pfp_nonempty_shifted_coefficient_rightdiagonalentry * S ((S (pfc_index_nonempty_shifted_coefficient_rightdiagonal)) * pfc_terms_scale_nonempty_shifted_coefficient_right) + (pfc_value_nonempty_shifted_coefficient_rightdiagonal))) /\\ ((exists pfc_complement_nonempty_shifted_coefficient_rightdiagonalterm pfc_left_nonempty_shifted_coefficient_rightdiagonalterm pfc_right_nonempty_shifted_coefficient_rightdiagonalterm. (((pfc_index_nonempty_shifted_coefficient_rightdiagonal)+pfc_complement_nonempty_shifted_coefficient_rightdiagonalterm=(t+i)) /\\ ((((((exists pfa_gap_nonempty_shifted_coefficient_rightdiagonaltermleftinside. pfa_gap_nonempty_shifted_coefficient_rightdiagonaltermleftinside + S (pfc_index_nonempty_shifted_coefficient_rightdiagonal) = (L)) /\\ ((((exists ff_h_pfp_nonempty_shifted_coefficient_rightdiagonaltermleftentry. ff_h_pfp_nonempty_shifted_coefficient_rightdiagonaltermleftentry + S (pfc_left_nonempty_shifted_coefficient_rightdiagonalterm) = S ((S (pfc_index_nonempty_shifted_coefficient_rightdiagonal)) * ac)) /\\ exists ff_q_pfp_nonempty_shifted_coefficient_rightdiagonaltermleftentry. ab = ff_q_pfp_nonempty_shifted_coefficient_rightdiagonaltermleftentry * S ((S (pfc_index_nonempty_shifted_coefficient_rightdiagonal)) * ac) + (pfc_left_nonempty_shifted_coefficient_rightdiagonalterm)))))) \\/ (((exists pfc_gap_nonempty_shifted_coefficient_rightdiagonaltermleftoutside. pfc_gap_nonempty_shifted_coefficient_rightdiagonaltermleftoutside+(L)=(pfc_index_nonempty_shifted_coefficient_rightdiagonal)) /\\ (((pfc_left_nonempty_shifted_coefficient_rightdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_nonempty_shifted_coefficient_rightdiagonaltermrightinside. pfa_gap_nonempty_shifted_coefficient_rightdiagonaltermrightinside + S (pfc_complement_nonempty_shifted_coefficient_rightdiagonalterm) = (t+M)) /\\ ((((exists ff_h_pfp_nonempty_shifted_coefficient_rightdiagonaltermrightentry. ff_h_pfp_nonempty_shifted_coefficient_rightdiagonaltermrightentry + S (pfc_right_nonempty_shifted_coefficient_rightdiagonalterm) = S ((S (pfc_complement_nonempty_shifted_coefficient_rightdiagonalterm)) * BC)) /\\ exists ff_q_pfp_nonempty_shifted_coefficient_rightdiagonaltermrightentry. BB = ff_q_pfp_nonempty_shifted_coefficient_rightdiagonaltermrightentry * S ((S (pfc_complement_nonempty_shifted_coefficient_rightdiagonalterm)) * BC) + (pfc_right_nonempty_shifted_coefficient_rightdiagonalterm)))))) \\/ (((exists pfc_gap_nonempty_shifted_coefficient_rightdiagonaltermrightoutside. pfc_gap_nonempty_shifted_coefficient_rightdiagonaltermrightoutside+(t+M)=(pfc_complement_nonempty_shifted_coefficient_rightdiagonalterm)) /\\ (((pfc_right_nonempty_shifted_coefficient_rightdiagonalterm)=0))))) /\\ (((pfc_value_nonempty_shifted_coefficient_rightdiagonal)=pfc_left_nonempty_shifted_coefficient_rightdiagonalterm*pfc_right_nonempty_shifted_coefficient_rightdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_nonempty_shifted_coefficient_rightsum fs_v_pfc_nonempty_shifted_coefficient_rightsum. ((((exists fs_h_pfc_nonempty_shifted_coefficient_rightsum_body_start. fs_h_pfc_nonempty_shifted_coefficient_rightsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_nonempty_shifted_coefficient_rightsum)) /\\ exists fs_q_pfc_nonempty_shifted_coefficient_rightsum_body_start. fs_u_pfc_nonempty_shifted_coefficient_rightsum = fs_q_pfc_nonempty_shifted_coefficient_rightsum_body_start * S ((S (0)) * fs_v_pfc_nonempty_shifted_coefficient_rightsum) + (0))) /\\ ((((exists fs_h_pfc_nonempty_shifted_coefficient_rightsum_body_terminal. fs_h_pfc_nonempty_shifted_coefficient_rightsum_body_terminal + S (pfc_natural_sum_nonempty_shifted_coefficient_right) = S ((S (S (t+i))) * fs_v_pfc_nonempty_shifted_coefficient_rightsum)) /\\ exists fs_q_pfc_nonempty_shifted_coefficient_rightsum_body_terminal. fs_u_pfc_nonempty_shifted_coefficient_rightsum = fs_q_pfc_nonempty_shifted_coefficient_rightsum_body_terminal * S ((S (S (t+i))) * fs_v_pfc_nonempty_shifted_coefficient_rightsum) + (pfc_natural_sum_nonempty_shifted_coefficient_right))) /\\ forall fs_i_pfc_nonempty_shifted_coefficient_rightsum_body_steps. (exists fs_lt_pfc_nonempty_shifted_coefficient_rightsum_body_steps_bound. fs_lt_pfc_nonempty_shifted_coefficient_rightsum_body_steps_bound + S fs_i_pfc_nonempty_shifted_coefficient_rightsum_body_steps = S (t+i)) -> exists fs_a_pfc_nonempty_shifted_coefficient_rightsum_body_steps fs_r_pfc_nonempty_shifted_coefficient_rightsum_body_steps fs_s_pfc_nonempty_shifted_coefficient_rightsum_body_steps. ((((exists fs_h_pfc_nonempty_shifted_coefficient_rightsum_body_steps_summand. fs_h_pfc_nonempty_shifted_coefficient_rightsum_body_steps_summand + S (fs_a_pfc_nonempty_shifted_coefficient_rightsum_body_steps) = S ((S (fs_i_pfc_nonempty_shifted_coefficient_rightsum_body_steps)) * pfc_terms_scale_nonempty_shifted_coefficient_right)) /\\ exists fs_q_pfc_nonempty_shifted_coefficient_rightsum_body_steps_summand. pfc_terms_code_nonempty_shifted_coefficient_right = fs_q_pfc_nonempty_shifted_coefficient_rightsum_body_steps_summand * S ((S (fs_i_pfc_nonempty_shifted_coefficient_rightsum_body_steps)) * pfc_terms_scale_nonempty_shifted_coefficient_right) + (fs_a_pfc_nonempty_shifted_coefficient_rightsum_body_steps))) /\\ ((((exists fs_h_pfc_nonempty_shifted_coefficient_rightsum_body_steps_partial. fs_h_pfc_nonempty_shifted_coefficient_rightsum_body_steps_partial + S (fs_r_pfc_nonempty_shifted_coefficient_rightsum_body_steps) = S ((S (fs_i_pfc_nonempty_shifted_coefficient_rightsum_body_steps)) * fs_v_pfc_nonempty_shifted_coefficient_rightsum)) /\\ exists fs_q_pfc_nonempty_shifted_coefficient_rightsum_body_steps_partial. fs_u_pfc_nonempty_shifted_coefficient_rightsum = fs_q_pfc_nonempty_shifted_coefficient_rightsum_body_steps_partial * S ((S (fs_i_pfc_nonempty_shifted_coefficient_rightsum_body_steps)) * fs_v_pfc_nonempty_shifted_coefficient_rightsum) + (fs_r_pfc_nonempty_shifted_coefficient_rightsum_body_steps))) /\\ ((((exists fs_h_pfc_nonempty_shifted_coefficient_rightsum_body_steps_successor. fs_h_pfc_nonempty_shifted_coefficient_rightsum_body_steps_successor + S (fs_s_pfc_nonempty_shifted_coefficient_rightsum_body_steps) = S ((S (S fs_i_pfc_nonempty_shifted_coefficient_rightsum_body_steps)) * fs_v_pfc_nonempty_shifted_coefficient_rightsum)) /\\ exists fs_q_pfc_nonempty_shifted_coefficient_rightsum_body_steps_successor. fs_u_pfc_nonempty_shifted_coefficient_rightsum = fs_q_pfc_nonempty_shifted_coefficient_rightsum_body_steps_successor * S ((S (S fs_i_pfc_nonempty_shifted_coefficient_rightsum_body_steps)) * fs_v_pfc_nonempty_shifted_coefficient_rightsum) + (fs_s_pfc_nonempty_shifted_coefficient_rightsum_body_steps))) /\\ fs_s_pfc_nonempty_shifted_coefficient_rightsum_body_steps = fs_r_pfc_nonempty_shifted_coefficient_rightsum_body_steps + fs_a_pfc_nonempty_shifted_coefficient_rightsum_body_steps)))))) /\\ ((((exists pfa_gap_nonempty_shifted_coefficient_rightresiduebound. pfa_gap_nonempty_shifted_coefficient_rightresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_nonempty_shifted_coefficient_rightresiduecongruence pfa_offset_right_nonempty_shifted_coefficient_rightresiduecongruence. (pfc_natural_sum_nonempty_shifted_coefficient_right) + (p) * pfa_offset_left_nonempty_shifted_coefficient_rightresiduecongruence = (r) + (p) * pfa_offset_right_nonempty_shifted_coefficient_rightresiduecongruence)))))))))))", "specialize hn_right_right_right (t+i)", "apply hn_right_right_right", "rewrite hlength", "specialize matrix_recursive_lt_add_left (i)", "specialize matrix_recursive_lt_add_left (N)", "specialize matrix_recursive_lt_add_left (t)", "apply matrix_recursive_lt_add_left", "exact hi", "cases hv", "cases hv_witness", "have heq : x=a", "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 (t+M)", "specialize prime_field_convolution_coefficient_functional (t+i)", "specialize prime_field_convolution_coefficient_functional (x)", "specialize prime_field_convolution_coefficient_functional (a)", "apply prime_field_convolution_coefficient_functional", "exact hv_witness_right", "specialize prime_field_convolution_coefficient_left_padding_right (p)", "specialize prime_field_convolution_coefficient_left_padding_right (ab)", "specialize prime_field_convolution_coefficient_left_padding_right (ac)", "specialize prime_field_convolution_coefficient_left_padding_right (L)", "specialize prime_field_convolution_coefficient_left_padding_right (bb)", "specialize prime_field_convolution_coefficient_left_padding_right (bc)", "specialize prime_field_convolution_coefficient_left_padding_right (M)", "specialize prime_field_convolution_coefficient_left_padding_right (BB)", "specialize prime_field_convolution_coefficient_left_padding_right (BC)", "specialize prime_field_convolution_coefficient_left_padding_right (t)", "specialize prime_field_convolution_coefficient_left_padding_right (i)", "specialize prime_field_convolution_coefficient_left_padding_right (a)", "apply prime_field_convolution_coefficient_left_padding_right", "exact hpad", "exact hcoefficient", "rewrite heq at hv_witness_left", "rewrite heq at hv_witness_left", "exact hv_witness_left" ], "script_sha256": "704803179e20454d4c560c2fe47ddc8e25b789f4d7ac697baf03fe0d0a115481", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_convolution_padding_candidate.py", "sha256": "2d874ecfb35a5db0aecdeb07b549464efebad9072c363113aa5a0a977845d007" }, "statement": "forall p ab ac L bb bc M cb cc N BB BC t CB CC K. (~(p=0)) -> (~(L=0)) -> (~(M=0)) -> (((forall pfp_repeat_index_nonempty_factor_padding_rightzeros. (exists pfa_gap_nonempty_factor_padding_rightzerosindex. pfa_gap_nonempty_factor_padding_rightzerosindex + S (pfp_repeat_index_nonempty_factor_padding_rightzeros) = (t)) -> (((exists ff_h_pfp_nonempty_factor_padding_rightzerosentry. ff_h_pfp_nonempty_factor_padding_rightzerosentry + S (0) = S ((S (pfp_repeat_index_nonempty_factor_padding_rightzeros)) * BC)) /\\ exists ff_q_pfp_nonempty_factor_padding_rightzerosentry. BB = ff_q_pfp_nonempty_factor_padding_rightzerosentry * S ((S (pfp_repeat_index_nonempty_factor_padding_rightzeros)) * BC) + (0)))) /\\ ((forall pfrep_index_nonempty_factor_padding_right pfrep_value_nonempty_factor_padding_right. (exists pfa_gap_nonempty_factor_padding_rightbound. pfa_gap_nonempty_factor_padding_rightbound + S (pfrep_index_nonempty_factor_padding_right) = (M)) -> (((exists ff_h_pfp_nonempty_factor_padding_rightinput. ff_h_pfp_nonempty_factor_padding_rightinput + S (pfrep_value_nonempty_factor_padding_right) = S ((S (pfrep_index_nonempty_factor_padding_right)) * bc)) /\\ exists ff_q_pfp_nonempty_factor_padding_rightinput. bb = ff_q_pfp_nonempty_factor_padding_rightinput * S ((S (pfrep_index_nonempty_factor_padding_right)) * bc) + (pfrep_value_nonempty_factor_padding_right))) -> (((exists ff_h_pfp_nonempty_factor_padding_rightoutput. ff_h_pfp_nonempty_factor_padding_rightoutput + S (pfrep_value_nonempty_factor_padding_right) = S ((S ((t)+pfrep_index_nonempty_factor_padding_right)) * BC)) /\\ exists ff_q_pfp_nonempty_factor_padding_rightoutput. BB = ff_q_pfp_nonempty_factor_padding_rightoutput * S ((S ((t)+pfrep_index_nonempty_factor_padding_right)) * BC) + (pfrep_value_nonempty_factor_padding_right))))))) -> (((forall fom_index_pfp_nonempty_old_rightleft. (exists fom_gap_pfp_nonempty_old_rightleft_index_bound. fom_gap_pfp_nonempty_old_rightleft_index_bound + S (fom_index_pfp_nonempty_old_rightleft) = L) -> exists fom_value_pfp_nonempty_old_rightleft. ((((exists fom_beta_height_pfp_nonempty_old_rightleft_entry. fom_beta_height_pfp_nonempty_old_rightleft_entry + S (fom_value_pfp_nonempty_old_rightleft) = S ((S (fom_index_pfp_nonempty_old_rightleft)) * ac)) /\\ exists fom_beta_quotient_pfp_nonempty_old_rightleft_entry. ab = fom_beta_quotient_pfp_nonempty_old_rightleft_entry * S ((S (fom_index_pfp_nonempty_old_rightleft)) * ac) + (fom_value_pfp_nonempty_old_rightleft))) /\\ (exists fom_gap_pfp_nonempty_old_rightleft_value_bound. fom_gap_pfp_nonempty_old_rightleft_value_bound + S (fom_value_pfp_nonempty_old_rightleft) = p))) /\\ (((forall fom_index_pfp_nonempty_old_rightright. (exists fom_gap_pfp_nonempty_old_rightright_index_bound. fom_gap_pfp_nonempty_old_rightright_index_bound + S (fom_index_pfp_nonempty_old_rightright) = M) -> exists fom_value_pfp_nonempty_old_rightright. ((((exists fom_beta_height_pfp_nonempty_old_rightright_entry. fom_beta_height_pfp_nonempty_old_rightright_entry + S (fom_value_pfp_nonempty_old_rightright) = S ((S (fom_index_pfp_nonempty_old_rightright)) * bc)) /\\ exists fom_beta_quotient_pfp_nonempty_old_rightright_entry. bb = fom_beta_quotient_pfp_nonempty_old_rightright_entry * S ((S (fom_index_pfp_nonempty_old_rightright)) * bc) + (fom_value_pfp_nonempty_old_rightright))) /\\ (exists fom_gap_pfp_nonempty_old_rightright_value_bound. fom_gap_pfp_nonempty_old_rightright_value_bound + S (fom_value_pfp_nonempty_old_rightright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_nonempty_old_rightcoefficients. (exists pfa_gap_nonempty_old_rightcoefficientsbound. pfa_gap_nonempty_old_rightcoefficientsbound + S (pfc_index_nonempty_old_rightcoefficients) = (N)) -> exists pfc_value_nonempty_old_rightcoefficients. ((((exists ff_h_pfp_nonempty_old_rightcoefficientsentry. ff_h_pfp_nonempty_old_rightcoefficientsentry + S (pfc_value_nonempty_old_rightcoefficients) = S ((S (pfc_index_nonempty_old_rightcoefficients)) * cc)) /\\ exists ff_q_pfp_nonempty_old_rightcoefficientsentry. cb = ff_q_pfp_nonempty_old_rightcoefficientsentry * S ((S (pfc_index_nonempty_old_rightcoefficients)) * cc) + (pfc_value_nonempty_old_rightcoefficients))) /\\ ((exists pfc_terms_code_nonempty_old_rightcoefficientscoefficient pfc_terms_scale_nonempty_old_rightcoefficientscoefficient pfc_natural_sum_nonempty_old_rightcoefficientscoefficient. ((forall pfc_index_nonempty_old_rightcoefficientscoefficientdiagonal. (exists pfa_gap_nonempty_old_rightcoefficientscoefficientdiagonalbound. pfa_gap_nonempty_old_rightcoefficientscoefficientdiagonalbound + S (pfc_index_nonempty_old_rightcoefficientscoefficientdiagonal) = (S (pfc_index_nonempty_old_rightcoefficients))) -> exists pfc_value_nonempty_old_rightcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_nonempty_old_rightcoefficientscoefficientdiagonalentry. ff_h_pfp_nonempty_old_rightcoefficientscoefficientdiagonalentry + S (pfc_value_nonempty_old_rightcoefficientscoefficientdiagonal) = S ((S (pfc_index_nonempty_old_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_nonempty_old_rightcoefficientscoefficient)) /\\ exists ff_q_pfp_nonempty_old_rightcoefficientscoefficientdiagonalentry. pfc_terms_code_nonempty_old_rightcoefficientscoefficient = ff_q_pfp_nonempty_old_rightcoefficientscoefficientdiagonalentry * S ((S (pfc_index_nonempty_old_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_nonempty_old_rightcoefficientscoefficient) + (pfc_value_nonempty_old_rightcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_nonempty_old_rightcoefficientscoefficientdiagonalterm pfc_left_nonempty_old_rightcoefficientscoefficientdiagonalterm pfc_right_nonempty_old_rightcoefficientscoefficientdiagonalterm. (((pfc_index_nonempty_old_rightcoefficientscoefficientdiagonal)+pfc_complement_nonempty_old_rightcoefficientscoefficientdiagonalterm=(pfc_index_nonempty_old_rightcoefficients)) /\\ ((((((exists pfa_gap_nonempty_old_rightcoefficientscoefficientdiagonaltermleftinside. pfa_gap_nonempty_old_rightcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_nonempty_old_rightcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_nonempty_old_rightcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_nonempty_old_rightcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_nonempty_old_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_nonempty_old_rightcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_nonempty_old_rightcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_nonempty_old_rightcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_nonempty_old_rightcoefficientscoefficientdiagonal)) * ac) + (pfc_left_nonempty_old_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_nonempty_old_rightcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_nonempty_old_rightcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_nonempty_old_rightcoefficientscoefficientdiagonal)) /\\ (((pfc_left_nonempty_old_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_nonempty_old_rightcoefficientscoefficientdiagonaltermrightinside. pfa_gap_nonempty_old_rightcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_nonempty_old_rightcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_nonempty_old_rightcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_nonempty_old_rightcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_nonempty_old_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_nonempty_old_rightcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_nonempty_old_rightcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_nonempty_old_rightcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_nonempty_old_rightcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_nonempty_old_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_nonempty_old_rightcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_nonempty_old_rightcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_nonempty_old_rightcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_nonempty_old_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_nonempty_old_rightcoefficientscoefficientdiagonal)=pfc_left_nonempty_old_rightcoefficientscoefficientdiagonalterm*pfc_right_nonempty_old_rightcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_nonempty_old_rightcoefficientscoefficientsum fs_v_pfc_nonempty_old_rightcoefficientscoefficientsum. ((((exists fs_h_pfc_nonempty_old_rightcoefficientscoefficientsum_body_start. fs_h_pfc_nonempty_old_rightcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_nonempty_old_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_nonempty_old_rightcoefficientscoefficientsum_body_start. fs_u_pfc_nonempty_old_rightcoefficientscoefficientsum = fs_q_pfc_nonempty_old_rightcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_nonempty_old_rightcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_nonempty_old_rightcoefficientscoefficientsum_body_terminal. fs_h_pfc_nonempty_old_rightcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_nonempty_old_rightcoefficientscoefficient) = S ((S (S (pfc_index_nonempty_old_rightcoefficients))) * fs_v_pfc_nonempty_old_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_nonempty_old_rightcoefficientscoefficientsum_body_terminal. fs_u_pfc_nonempty_old_rightcoefficientscoefficientsum = fs_q_pfc_nonempty_old_rightcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_nonempty_old_rightcoefficients))) * fs_v_pfc_nonempty_old_rightcoefficientscoefficientsum) + (pfc_natural_sum_nonempty_old_rightcoefficientscoefficient))) /\\ forall fs_i_pfc_nonempty_old_rightcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_nonempty_old_rightcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_nonempty_old_rightcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_nonempty_old_rightcoefficientscoefficientsum_body_steps = S (pfc_index_nonempty_old_rightcoefficients)) -> exists fs_a_pfc_nonempty_old_rightcoefficientscoefficientsum_body_steps fs_r_pfc_nonempty_old_rightcoefficientscoefficientsum_body_steps fs_s_pfc_nonempty_old_rightcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_nonempty_old_rightcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_nonempty_old_rightcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_nonempty_old_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_nonempty_old_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_nonempty_old_rightcoefficientscoefficient)) /\\ exists fs_q_pfc_nonempty_old_rightcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_nonempty_old_rightcoefficientscoefficient = fs_q_pfc_nonempty_old_rightcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_nonempty_old_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_nonempty_old_rightcoefficientscoefficient) + (fs_a_pfc_nonempty_old_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_nonempty_old_rightcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_nonempty_old_rightcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_nonempty_old_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_nonempty_old_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_nonempty_old_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_nonempty_old_rightcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_nonempty_old_rightcoefficientscoefficientsum = fs_q_pfc_nonempty_old_rightcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_nonempty_old_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_nonempty_old_rightcoefficientscoefficientsum) + (fs_r_pfc_nonempty_old_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_nonempty_old_rightcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_nonempty_old_rightcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_nonempty_old_rightcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_nonempty_old_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_nonempty_old_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_nonempty_old_rightcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_nonempty_old_rightcoefficientscoefficientsum = fs_q_pfc_nonempty_old_rightcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_nonempty_old_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_nonempty_old_rightcoefficientscoefficientsum) + (fs_s_pfc_nonempty_old_rightcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_nonempty_old_rightcoefficientscoefficientsum_body_steps = fs_r_pfc_nonempty_old_rightcoefficientscoefficientsum_body_steps + fs_a_pfc_nonempty_old_rightcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_nonempty_old_rightcoefficientscoefficientresiduebound. pfa_gap_nonempty_old_rightcoefficientscoefficientresiduebound + S (pfc_value_nonempty_old_rightcoefficients) = (p)) /\\ ((exists pfa_offset_left_nonempty_old_rightcoefficientscoefficientresiduecongruence pfa_offset_right_nonempty_old_rightcoefficientscoefficientresiduecongruence. (pfc_natural_sum_nonempty_old_rightcoefficientscoefficient) + (p) * pfa_offset_left_nonempty_old_rightcoefficientscoefficientresiduecongruence = (pfc_value_nonempty_old_rightcoefficients) + (p) * pfa_offset_right_nonempty_old_rightcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_nonempty_new_rightleft. (exists fom_gap_pfp_nonempty_new_rightleft_index_bound. fom_gap_pfp_nonempty_new_rightleft_index_bound + S (fom_index_pfp_nonempty_new_rightleft) = L) -> exists fom_value_pfp_nonempty_new_rightleft. ((((exists fom_beta_height_pfp_nonempty_new_rightleft_entry. fom_beta_height_pfp_nonempty_new_rightleft_entry + S (fom_value_pfp_nonempty_new_rightleft) = S ((S (fom_index_pfp_nonempty_new_rightleft)) * ac)) /\\ exists fom_beta_quotient_pfp_nonempty_new_rightleft_entry. ab = fom_beta_quotient_pfp_nonempty_new_rightleft_entry * S ((S (fom_index_pfp_nonempty_new_rightleft)) * ac) + (fom_value_pfp_nonempty_new_rightleft))) /\\ (exists fom_gap_pfp_nonempty_new_rightleft_value_bound. fom_gap_pfp_nonempty_new_rightleft_value_bound + S (fom_value_pfp_nonempty_new_rightleft) = p))) /\\ (((forall fom_index_pfp_nonempty_new_rightright. (exists fom_gap_pfp_nonempty_new_rightright_index_bound. fom_gap_pfp_nonempty_new_rightright_index_bound + S (fom_index_pfp_nonempty_new_rightright) = t+M) -> exists fom_value_pfp_nonempty_new_rightright. ((((exists fom_beta_height_pfp_nonempty_new_rightright_entry. fom_beta_height_pfp_nonempty_new_rightright_entry + S (fom_value_pfp_nonempty_new_rightright) = S ((S (fom_index_pfp_nonempty_new_rightright)) * BC)) /\\ exists fom_beta_quotient_pfp_nonempty_new_rightright_entry. BB = fom_beta_quotient_pfp_nonempty_new_rightright_entry * S ((S (fom_index_pfp_nonempty_new_rightright)) * BC) + (fom_value_pfp_nonempty_new_rightright))) /\\ (exists fom_gap_pfp_nonempty_new_rightright_value_bound. fom_gap_pfp_nonempty_new_rightright_value_bound + S (fom_value_pfp_nonempty_new_rightright) = p))) /\\ (((((((L)=0 \\/ (t+M)=0) /\\ (((K)=0)))) \\/ (((~((L)=0)) /\\ (((~((t+M)=0)) /\\ (((L)+(t+M)=S (K)))))))) /\\ ((forall pfc_index_nonempty_new_rightcoefficients. (exists pfa_gap_nonempty_new_rightcoefficientsbound. pfa_gap_nonempty_new_rightcoefficientsbound + S (pfc_index_nonempty_new_rightcoefficients) = (K)) -> exists pfc_value_nonempty_new_rightcoefficients. ((((exists ff_h_pfp_nonempty_new_rightcoefficientsentry. ff_h_pfp_nonempty_new_rightcoefficientsentry + S (pfc_value_nonempty_new_rightcoefficients) = S ((S (pfc_index_nonempty_new_rightcoefficients)) * CC)) /\\ exists ff_q_pfp_nonempty_new_rightcoefficientsentry. CB = ff_q_pfp_nonempty_new_rightcoefficientsentry * S ((S (pfc_index_nonempty_new_rightcoefficients)) * CC) + (pfc_value_nonempty_new_rightcoefficients))) /\\ ((exists pfc_terms_code_nonempty_new_rightcoefficientscoefficient pfc_terms_scale_nonempty_new_rightcoefficientscoefficient pfc_natural_sum_nonempty_new_rightcoefficientscoefficient. ((forall pfc_index_nonempty_new_rightcoefficientscoefficientdiagonal. (exists pfa_gap_nonempty_new_rightcoefficientscoefficientdiagonalbound. pfa_gap_nonempty_new_rightcoefficientscoefficientdiagonalbound + S (pfc_index_nonempty_new_rightcoefficientscoefficientdiagonal) = (S (pfc_index_nonempty_new_rightcoefficients))) -> exists pfc_value_nonempty_new_rightcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_nonempty_new_rightcoefficientscoefficientdiagonalentry. ff_h_pfp_nonempty_new_rightcoefficientscoefficientdiagonalentry + S (pfc_value_nonempty_new_rightcoefficientscoefficientdiagonal) = S ((S (pfc_index_nonempty_new_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_nonempty_new_rightcoefficientscoefficient)) /\\ exists ff_q_pfp_nonempty_new_rightcoefficientscoefficientdiagonalentry. pfc_terms_code_nonempty_new_rightcoefficientscoefficient = ff_q_pfp_nonempty_new_rightcoefficientscoefficientdiagonalentry * S ((S (pfc_index_nonempty_new_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_nonempty_new_rightcoefficientscoefficient) + (pfc_value_nonempty_new_rightcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_nonempty_new_rightcoefficientscoefficientdiagonalterm pfc_left_nonempty_new_rightcoefficientscoefficientdiagonalterm pfc_right_nonempty_new_rightcoefficientscoefficientdiagonalterm. (((pfc_index_nonempty_new_rightcoefficientscoefficientdiagonal)+pfc_complement_nonempty_new_rightcoefficientscoefficientdiagonalterm=(pfc_index_nonempty_new_rightcoefficients)) /\\ ((((((exists pfa_gap_nonempty_new_rightcoefficientscoefficientdiagonaltermleftinside. pfa_gap_nonempty_new_rightcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_nonempty_new_rightcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_nonempty_new_rightcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_nonempty_new_rightcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_nonempty_new_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_nonempty_new_rightcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_nonempty_new_rightcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_nonempty_new_rightcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_nonempty_new_rightcoefficientscoefficientdiagonal)) * ac) + (pfc_left_nonempty_new_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_nonempty_new_rightcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_nonempty_new_rightcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_nonempty_new_rightcoefficientscoefficientdiagonal)) /\\ (((pfc_left_nonempty_new_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_nonempty_new_rightcoefficientscoefficientdiagonaltermrightinside. pfa_gap_nonempty_new_rightcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_nonempty_new_rightcoefficientscoefficientdiagonalterm) = (t+M)) /\\ ((((exists ff_h_pfp_nonempty_new_rightcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_nonempty_new_rightcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_nonempty_new_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_nonempty_new_rightcoefficientscoefficientdiagonalterm)) * BC)) /\\ exists ff_q_pfp_nonempty_new_rightcoefficientscoefficientdiagonaltermrightentry. BB = ff_q_pfp_nonempty_new_rightcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_nonempty_new_rightcoefficientscoefficientdiagonalterm)) * BC) + (pfc_right_nonempty_new_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_nonempty_new_rightcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_nonempty_new_rightcoefficientscoefficientdiagonaltermrightoutside+(t+M)=(pfc_complement_nonempty_new_rightcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_nonempty_new_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_nonempty_new_rightcoefficientscoefficientdiagonal)=pfc_left_nonempty_new_rightcoefficientscoefficientdiagonalterm*pfc_right_nonempty_new_rightcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_nonempty_new_rightcoefficientscoefficientsum fs_v_pfc_nonempty_new_rightcoefficientscoefficientsum. ((((exists fs_h_pfc_nonempty_new_rightcoefficientscoefficientsum_body_start. fs_h_pfc_nonempty_new_rightcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_nonempty_new_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_nonempty_new_rightcoefficientscoefficientsum_body_start. fs_u_pfc_nonempty_new_rightcoefficientscoefficientsum = fs_q_pfc_nonempty_new_rightcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_nonempty_new_rightcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_nonempty_new_rightcoefficientscoefficientsum_body_terminal. fs_h_pfc_nonempty_new_rightcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_nonempty_new_rightcoefficientscoefficient) = S ((S (S (pfc_index_nonempty_new_rightcoefficients))) * fs_v_pfc_nonempty_new_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_nonempty_new_rightcoefficientscoefficientsum_body_terminal. fs_u_pfc_nonempty_new_rightcoefficientscoefficientsum = fs_q_pfc_nonempty_new_rightcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_nonempty_new_rightcoefficients))) * fs_v_pfc_nonempty_new_rightcoefficientscoefficientsum) + (pfc_natural_sum_nonempty_new_rightcoefficientscoefficient))) /\\ forall fs_i_pfc_nonempty_new_rightcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_nonempty_new_rightcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_nonempty_new_rightcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_nonempty_new_rightcoefficientscoefficientsum_body_steps = S (pfc_index_nonempty_new_rightcoefficients)) -> exists fs_a_pfc_nonempty_new_rightcoefficientscoefficientsum_body_steps fs_r_pfc_nonempty_new_rightcoefficientscoefficientsum_body_steps fs_s_pfc_nonempty_new_rightcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_nonempty_new_rightcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_nonempty_new_rightcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_nonempty_new_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_nonempty_new_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_nonempty_new_rightcoefficientscoefficient)) /\\ exists fs_q_pfc_nonempty_new_rightcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_nonempty_new_rightcoefficientscoefficient = fs_q_pfc_nonempty_new_rightcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_nonempty_new_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_nonempty_new_rightcoefficientscoefficient) + (fs_a_pfc_nonempty_new_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_nonempty_new_rightcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_nonempty_new_rightcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_nonempty_new_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_nonempty_new_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_nonempty_new_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_nonempty_new_rightcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_nonempty_new_rightcoefficientscoefficientsum = fs_q_pfc_nonempty_new_rightcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_nonempty_new_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_nonempty_new_rightcoefficientscoefficientsum) + (fs_r_pfc_nonempty_new_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_nonempty_new_rightcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_nonempty_new_rightcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_nonempty_new_rightcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_nonempty_new_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_nonempty_new_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_nonempty_new_rightcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_nonempty_new_rightcoefficientscoefficientsum = fs_q_pfc_nonempty_new_rightcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_nonempty_new_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_nonempty_new_rightcoefficientscoefficientsum) + (fs_s_pfc_nonempty_new_rightcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_nonempty_new_rightcoefficientscoefficientsum_body_steps = fs_r_pfc_nonempty_new_rightcoefficientscoefficientsum_body_steps + fs_a_pfc_nonempty_new_rightcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_nonempty_new_rightcoefficientscoefficientresiduebound. pfa_gap_nonempty_new_rightcoefficientscoefficientresiduebound + S (pfc_value_nonempty_new_rightcoefficients) = (p)) /\\ ((exists pfa_offset_left_nonempty_new_rightcoefficientscoefficientresiduecongruence pfa_offset_right_nonempty_new_rightcoefficientscoefficientresiduecongruence. (pfc_natural_sum_nonempty_new_rightcoefficientscoefficient) + (p) * pfa_offset_left_nonempty_new_rightcoefficientscoefficientresiduecongruence = (pfc_value_nonempty_new_rightcoefficients) + (p) * pfa_offset_right_nonempty_new_rightcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((K=t+N) /\\ ((((forall pfp_repeat_index_nonempty_product_padding_rightzeros. (exists pfa_gap_nonempty_product_padding_rightzerosindex. pfa_gap_nonempty_product_padding_rightzerosindex + S (pfp_repeat_index_nonempty_product_padding_rightzeros) = (t)) -> (((exists ff_h_pfp_nonempty_product_padding_rightzerosentry. ff_h_pfp_nonempty_product_padding_rightzerosentry + S (0) = S ((S (pfp_repeat_index_nonempty_product_padding_rightzeros)) * CC)) /\\ exists ff_q_pfp_nonempty_product_padding_rightzerosentry. CB = ff_q_pfp_nonempty_product_padding_rightzerosentry * S ((S (pfp_repeat_index_nonempty_product_padding_rightzeros)) * CC) + (0)))) /\\ ((forall pfrep_index_nonempty_product_padding_right pfrep_value_nonempty_product_padding_right. (exists pfa_gap_nonempty_product_padding_rightbound. pfa_gap_nonempty_product_padding_rightbound + S (pfrep_index_nonempty_product_padding_right) = (N)) -> (((exists ff_h_pfp_nonempty_product_padding_rightinput. ff_h_pfp_nonempty_product_padding_rightinput + S (pfrep_value_nonempty_product_padding_right) = S ((S (pfrep_index_nonempty_product_padding_right)) * cc)) /\\ exists ff_q_pfp_nonempty_product_padding_rightinput. cb = ff_q_pfp_nonempty_product_padding_rightinput * S ((S (pfrep_index_nonempty_product_padding_right)) * cc) + (pfrep_value_nonempty_product_padding_right))) -> (((exists ff_h_pfp_nonempty_product_padding_rightoutput. ff_h_pfp_nonempty_product_padding_rightoutput + S (pfrep_value_nonempty_product_padding_right) = S ((S ((t)+pfrep_index_nonempty_product_padding_right)) * CC)) /\\ exists ff_q_pfp_nonempty_product_padding_rightoutput. 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"polynomial-euclidean-division", "logical_spec_sha256": "1848be2eebad8420873856a70594d865f0869e3e2d4ec2d1533ea6912354a563", "membership": "alpha_only", "name": "prime_field_polynomial_convolution_left_padding_equivalent_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 N", "intro AB", "intro AC", "intro t", "intro CB", "intro CC", "intro K", "intro hp", "intro hpad", "intro hc", "intro hn", "have hL : L=0 \\/ ~(L=0)", "specialize eq_decidable (L)", "specialize eq_decidable (0)", "apply eq_decidable", "cases hL", "have hz : forall pfp_repeat_index_empty_factor_zero_leftleft. (exists pfa_gap_empty_factor_zero_leftleftindex. pfa_gap_empty_factor_zero_leftleftindex + S (pfp_repeat_index_empty_factor_zero_leftleft) = (L)) -> (((exists ff_h_pfp_empty_factor_zero_leftleftentry. ff_h_pfp_empty_factor_zero_leftleftentry + S (0) = S ((S (pfp_repeat_index_empty_factor_zero_leftleft)) * ac)) /\\ exists ff_q_pfp_empty_factor_zero_leftleftentry. ab = ff_q_pfp_empty_factor_zero_leftleftentry * S ((S (pfp_repeat_index_empty_factor_zero_leftleft)) * ac) + (0)))", "intro j", "intro hj", "rewrite hL_left at hj", "exfalso", "specialize matrix_rank_no_index_below_zero (j)", "apply matrix_rank_no_index_below_zero", "exact hj", "have hc0 : forall pfp_repeat_index_empty_original_product_leftleft. (exists pfa_gap_empty_original_product_leftleftindex. pfa_gap_empty_original_product_leftleftindex + S (pfp_repeat_index_empty_original_product_leftleft) = (N)) -> (((exists ff_h_pfp_empty_original_product_leftleftentry. ff_h_pfp_empty_original_product_leftleftentry + S (0) = S ((S (pfp_repeat_index_empty_original_product_leftleft)) * cc)) /\\ exists ff_q_pfp_empty_original_product_leftleftentry. cb = ff_q_pfp_empty_original_product_leftleftentry * S ((S (pfp_repeat_index_empty_original_product_leftleft)) * cc) + (0)))", "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", "have hn0 : forall pfp_repeat_index_empty_padded_product_leftleft. (exists pfa_gap_empty_padded_product_leftleftindex. pfa_gap_empty_padded_product_leftleftindex + S (pfp_repeat_index_empty_padded_product_leftleft) = (K)) -> (((exists ff_h_pfp_empty_padded_product_leftleftentry. ff_h_pfp_empty_padded_product_leftleftentry + S (0) = S ((S (pfp_repeat_index_empty_padded_product_leftleft)) * CC)) /\\ exists ff_q_pfp_empty_padded_product_leftleftentry. CB = ff_q_pfp_empty_padded_product_leftleftentry * S ((S (pfp_repeat_index_empty_padded_product_leftleft)) * CC) + (0)))", "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 (t+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 (K)", "apply prime_field_polynomial_convolution_zero_left", "exact hp", "specialize polynomial_left_pad_zero_prefix (ab)", "specialize polynomial_left_pad_zero_prefix (ac)", "specialize polynomial_left_pad_zero_prefix (L)", "specialize polynomial_left_pad_zero_prefix (t)", "specialize polynomial_left_pad_zero_prefix (AB)", "specialize polynomial_left_pad_zero_prefix (AC)", "apply polynomial_left_pad_zero_prefix", "exact hz", "exact hpad", "exact hn", "have hec : forall pfrep_power_empty_first_equivalence_leftleft pfrep_left_empty_first_equivalence_leftleft pfrep_right_empty_first_equivalence_leftleft. ((exists pfrep_position_empty_first_equivalence_leftleftfirst. ((pfrep_position_empty_first_equivalence_leftleftfirst+S (pfrep_power_empty_first_equivalence_leftleft)=(N)) /\\ ((((exists ff_h_pfp_empty_first_equivalence_leftleftfirstentry. ff_h_pfp_empty_first_equivalence_leftleftfirstentry + S (pfrep_left_empty_first_equivalence_leftleft) = S ((S (pfrep_position_empty_first_equivalence_leftleftfirst)) * cc)) /\\ exists ff_q_pfp_empty_first_equivalence_leftleftfirstentry. cb = ff_q_pfp_empty_first_equivalence_leftleftfirstentry * S ((S (pfrep_position_empty_first_equivalence_leftleftfirst)) * cc) + (pfrep_left_empty_first_equivalence_leftleft)))))) \\/ (((exists pfrep_gap_empty_first_equivalence_leftleftfirstoutside. pfrep_gap_empty_first_equivalence_leftleftfirstoutside+(N)=(pfrep_power_empty_first_equivalence_leftleft)) /\\ (((pfrep_left_empty_first_equivalence_leftleft)=0))))) -> ((exists pfrep_position_empty_first_equivalence_leftleftsecond. ((pfrep_position_empty_first_equivalence_leftleftsecond+S (pfrep_power_empty_first_equivalence_leftleft)=(0)) /\\ ((((exists ff_h_pfp_empty_first_equivalence_leftleftsecondentry. ff_h_pfp_empty_first_equivalence_leftleftsecondentry + S (pfrep_right_empty_first_equivalence_leftleft) = S ((S (pfrep_position_empty_first_equivalence_leftleftsecond)) * 0)) /\\ exists ff_q_pfp_empty_first_equivalence_leftleftsecondentry. 0 = ff_q_pfp_empty_first_equivalence_leftleftsecondentry * S ((S (pfrep_position_empty_first_equivalence_leftleftsecond)) * 0) + (pfrep_right_empty_first_equivalence_leftleft)))))) \\/ (((exists pfrep_gap_empty_first_equivalence_leftleftsecondoutside. pfrep_gap_empty_first_equivalence_leftleftsecondoutside+(0)=(pfrep_power_empty_first_equivalence_leftleft)) /\\ (((pfrep_right_empty_first_equivalence_leftleft)=0))))) -> pfrep_left_empty_first_equivalence_leftleft=pfrep_right_empty_first_equivalence_leftleft", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (cb)", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (cc)", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (N)", "apply prime_field_polynomial_zero_prefix_equivalent_empty", "exact hc0", "have hen : forall pfrep_power_empty_second_equivalence_leftleft pfrep_left_empty_second_equivalence_leftleft pfrep_right_empty_second_equivalence_leftleft. ((exists pfrep_position_empty_second_equivalence_leftleftfirst. ((pfrep_position_empty_second_equivalence_leftleftfirst+S (pfrep_power_empty_second_equivalence_leftleft)=(K)) /\\ ((((exists ff_h_pfp_empty_second_equivalence_leftleftfirstentry. ff_h_pfp_empty_second_equivalence_leftleftfirstentry + S (pfrep_left_empty_second_equivalence_leftleft) = S ((S (pfrep_position_empty_second_equivalence_leftleftfirst)) * CC)) /\\ exists ff_q_pfp_empty_second_equivalence_leftleftfirstentry. CB = ff_q_pfp_empty_second_equivalence_leftleftfirstentry * S ((S (pfrep_position_empty_second_equivalence_leftleftfirst)) * CC) + (pfrep_left_empty_second_equivalence_leftleft)))))) \\/ (((exists pfrep_gap_empty_second_equivalence_leftleftfirstoutside. pfrep_gap_empty_second_equivalence_leftleftfirstoutside+(K)=(pfrep_power_empty_second_equivalence_leftleft)) /\\ (((pfrep_left_empty_second_equivalence_leftleft)=0))))) -> ((exists pfrep_position_empty_second_equivalence_leftleftsecond. ((pfrep_position_empty_second_equivalence_leftleftsecond+S (pfrep_power_empty_second_equivalence_leftleft)=(0)) /\\ ((((exists ff_h_pfp_empty_second_equivalence_leftleftsecondentry. ff_h_pfp_empty_second_equivalence_leftleftsecondentry + S (pfrep_right_empty_second_equivalence_leftleft) = S ((S (pfrep_position_empty_second_equivalence_leftleftsecond)) * 0)) /\\ exists ff_q_pfp_empty_second_equivalence_leftleftsecondentry. 0 = ff_q_pfp_empty_second_equivalence_leftleftsecondentry * S ((S (pfrep_position_empty_second_equivalence_leftleftsecond)) * 0) + (pfrep_right_empty_second_equivalence_leftleft)))))) \\/ (((exists pfrep_gap_empty_second_equivalence_leftleftsecondoutside. pfrep_gap_empty_second_equivalence_leftleftsecondoutside+(0)=(pfrep_power_empty_second_equivalence_leftleft)) /\\ (((pfrep_right_empty_second_equivalence_leftleft)=0))))) -> pfrep_left_empty_second_equivalence_leftleft=pfrep_right_empty_second_equivalence_leftleft", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (CB)", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (CC)", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (K)", "apply prime_field_polynomial_zero_prefix_equivalent_empty", "exact hn0", "specialize prime_field_polynomial_equivalent_transitive (cb)", "specialize prime_field_polynomial_equivalent_transitive (cc)", "specialize prime_field_polynomial_equivalent_transitive (N)", "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 (CB)", "specialize prime_field_polynomial_equivalent_transitive (CC)", "specialize prime_field_polynomial_equivalent_transitive (K)", "apply prime_field_polynomial_equivalent_transitive", "exact hec", "specialize prime_field_polynomial_equivalent_symmetric (CB)", "specialize prime_field_polynomial_equivalent_symmetric (CC)", "specialize prime_field_polynomial_equivalent_symmetric (K)", "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", "exact hen", "have hM : M=0 \\/ ~(M=0)", "specialize eq_decidable (M)", "specialize eq_decidable (0)", "apply eq_decidable", "cases hM", "have hz : forall pfp_repeat_index_empty_factor_zero_leftright. (exists pfa_gap_empty_factor_zero_leftrightindex. pfa_gap_empty_factor_zero_leftrightindex + S (pfp_repeat_index_empty_factor_zero_leftright) = (M)) -> (((exists ff_h_pfp_empty_factor_zero_leftrightentry. ff_h_pfp_empty_factor_zero_leftrightentry + S (0) = S ((S (pfp_repeat_index_empty_factor_zero_leftright)) * bc)) /\\ exists ff_q_pfp_empty_factor_zero_leftrightentry. bb = ff_q_pfp_empty_factor_zero_leftrightentry * S ((S (pfp_repeat_index_empty_factor_zero_leftright)) * bc) + (0)))", "intro j", "intro hj", "rewrite hM_left at hj", "exfalso", "specialize matrix_rank_no_index_below_zero (j)", "apply matrix_rank_no_index_below_zero", "exact hj", "have hc0 : forall pfp_repeat_index_empty_original_product_leftright. (exists pfa_gap_empty_original_product_leftrightindex. pfa_gap_empty_original_product_leftrightindex + S (pfp_repeat_index_empty_original_product_leftright) = (N)) -> (((exists ff_h_pfp_empty_original_product_leftrightentry. ff_h_pfp_empty_original_product_leftrightentry + S (0) = S ((S (pfp_repeat_index_empty_original_product_leftright)) * cc)) /\\ exists ff_q_pfp_empty_original_product_leftrightentry. cb = ff_q_pfp_empty_original_product_leftrightentry * S ((S (pfp_repeat_index_empty_original_product_leftright)) * cc) + (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 (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", "have hn0 : forall pfp_repeat_index_empty_padded_product_leftright. (exists pfa_gap_empty_padded_product_leftrightindex. pfa_gap_empty_padded_product_leftrightindex + S (pfp_repeat_index_empty_padded_product_leftright) = (K)) -> (((exists ff_h_pfp_empty_padded_product_leftrightentry. ff_h_pfp_empty_padded_product_leftrightentry + S (0) = S ((S (pfp_repeat_index_empty_padded_product_leftright)) * CC)) /\\ exists ff_q_pfp_empty_padded_product_leftrightentry. CB = ff_q_pfp_empty_padded_product_leftrightentry * S ((S (pfp_repeat_index_empty_padded_product_leftright)) * CC) + (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 (t+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 (K)", "apply prime_field_polynomial_convolution_zero_right", "exact hp", "exact hz", "exact hn", "have hec : forall pfrep_power_empty_first_equivalence_leftright pfrep_left_empty_first_equivalence_leftright pfrep_right_empty_first_equivalence_leftright. ((exists pfrep_position_empty_first_equivalence_leftrightfirst. ((pfrep_position_empty_first_equivalence_leftrightfirst+S (pfrep_power_empty_first_equivalence_leftright)=(N)) /\\ ((((exists ff_h_pfp_empty_first_equivalence_leftrightfirstentry. ff_h_pfp_empty_first_equivalence_leftrightfirstentry + S (pfrep_left_empty_first_equivalence_leftright) = S ((S (pfrep_position_empty_first_equivalence_leftrightfirst)) * cc)) /\\ exists ff_q_pfp_empty_first_equivalence_leftrightfirstentry. cb = ff_q_pfp_empty_first_equivalence_leftrightfirstentry * S ((S (pfrep_position_empty_first_equivalence_leftrightfirst)) * cc) + (pfrep_left_empty_first_equivalence_leftright)))))) \\/ (((exists pfrep_gap_empty_first_equivalence_leftrightfirstoutside. pfrep_gap_empty_first_equivalence_leftrightfirstoutside+(N)=(pfrep_power_empty_first_equivalence_leftright)) /\\ (((pfrep_left_empty_first_equivalence_leftright)=0))))) -> ((exists pfrep_position_empty_first_equivalence_leftrightsecond. ((pfrep_position_empty_first_equivalence_leftrightsecond+S (pfrep_power_empty_first_equivalence_leftright)=(0)) /\\ ((((exists ff_h_pfp_empty_first_equivalence_leftrightsecondentry. ff_h_pfp_empty_first_equivalence_leftrightsecondentry + S (pfrep_right_empty_first_equivalence_leftright) = S ((S (pfrep_position_empty_first_equivalence_leftrightsecond)) * 0)) /\\ exists ff_q_pfp_empty_first_equivalence_leftrightsecondentry. 0 = ff_q_pfp_empty_first_equivalence_leftrightsecondentry * S ((S (pfrep_position_empty_first_equivalence_leftrightsecond)) * 0) + (pfrep_right_empty_first_equivalence_leftright)))))) \\/ (((exists pfrep_gap_empty_first_equivalence_leftrightsecondoutside. pfrep_gap_empty_first_equivalence_leftrightsecondoutside+(0)=(pfrep_power_empty_first_equivalence_leftright)) /\\ (((pfrep_right_empty_first_equivalence_leftright)=0))))) -> pfrep_left_empty_first_equivalence_leftright=pfrep_right_empty_first_equivalence_leftright", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (cb)", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (cc)", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (N)", "apply prime_field_polynomial_zero_prefix_equivalent_empty", "exact hc0", "have hen : forall pfrep_power_empty_second_equivalence_leftright pfrep_left_empty_second_equivalence_leftright pfrep_right_empty_second_equivalence_leftright. ((exists pfrep_position_empty_second_equivalence_leftrightfirst. ((pfrep_position_empty_second_equivalence_leftrightfirst+S (pfrep_power_empty_second_equivalence_leftright)=(K)) /\\ ((((exists ff_h_pfp_empty_second_equivalence_leftrightfirstentry. ff_h_pfp_empty_second_equivalence_leftrightfirstentry + S (pfrep_left_empty_second_equivalence_leftright) = S ((S (pfrep_position_empty_second_equivalence_leftrightfirst)) * CC)) /\\ exists ff_q_pfp_empty_second_equivalence_leftrightfirstentry. CB = ff_q_pfp_empty_second_equivalence_leftrightfirstentry * S ((S (pfrep_position_empty_second_equivalence_leftrightfirst)) * CC) + (pfrep_left_empty_second_equivalence_leftright)))))) \\/ (((exists pfrep_gap_empty_second_equivalence_leftrightfirstoutside. pfrep_gap_empty_second_equivalence_leftrightfirstoutside+(K)=(pfrep_power_empty_second_equivalence_leftright)) /\\ (((pfrep_left_empty_second_equivalence_leftright)=0))))) -> ((exists pfrep_position_empty_second_equivalence_leftrightsecond. ((pfrep_position_empty_second_equivalence_leftrightsecond+S (pfrep_power_empty_second_equivalence_leftright)=(0)) /\\ ((((exists ff_h_pfp_empty_second_equivalence_leftrightsecondentry. ff_h_pfp_empty_second_equivalence_leftrightsecondentry + S (pfrep_right_empty_second_equivalence_leftright) = S ((S (pfrep_position_empty_second_equivalence_leftrightsecond)) * 0)) /\\ exists ff_q_pfp_empty_second_equivalence_leftrightsecondentry. 0 = ff_q_pfp_empty_second_equivalence_leftrightsecondentry * S ((S (pfrep_position_empty_second_equivalence_leftrightsecond)) * 0) + (pfrep_right_empty_second_equivalence_leftright)))))) \\/ (((exists pfrep_gap_empty_second_equivalence_leftrightsecondoutside. pfrep_gap_empty_second_equivalence_leftrightsecondoutside+(0)=(pfrep_power_empty_second_equivalence_leftright)) /\\ (((pfrep_right_empty_second_equivalence_leftright)=0))))) -> pfrep_left_empty_second_equivalence_leftright=pfrep_right_empty_second_equivalence_leftright", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (CB)", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (CC)", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (K)", "apply prime_field_polynomial_zero_prefix_equivalent_empty", "exact hn0", "specialize prime_field_polynomial_equivalent_transitive (cb)", "specialize prime_field_polynomial_equivalent_transitive (cc)", "specialize prime_field_polynomial_equivalent_transitive (N)", "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 (CB)", "specialize prime_field_polynomial_equivalent_transitive (CC)", "specialize prime_field_polynomial_equivalent_transitive (K)", "apply prime_field_polynomial_equivalent_transitive", "exact hec", "specialize prime_field_polynomial_equivalent_symmetric (CB)", "specialize prime_field_polynomial_equivalent_symmetric (CC)", "specialize prime_field_polynomial_equivalent_symmetric (K)", "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", "exact hen", "have hd : ((K=t+N) /\\ ((((forall pfp_repeat_index_product_equivalent_padding_leftzeros. (exists pfa_gap_product_equivalent_padding_leftzerosindex. pfa_gap_product_equivalent_padding_leftzerosindex + S (pfp_repeat_index_product_equivalent_padding_leftzeros) = (t)) -> (((exists ff_h_pfp_product_equivalent_padding_leftzerosentry. ff_h_pfp_product_equivalent_padding_leftzerosentry + S (0) = S ((S (pfp_repeat_index_product_equivalent_padding_leftzeros)) * CC)) /\\ exists ff_q_pfp_product_equivalent_padding_leftzerosentry. CB = ff_q_pfp_product_equivalent_padding_leftzerosentry * S ((S (pfp_repeat_index_product_equivalent_padding_leftzeros)) * CC) + (0)))) /\\ ((forall pfrep_index_product_equivalent_padding_left pfrep_value_product_equivalent_padding_left. (exists pfa_gap_product_equivalent_padding_leftbound. pfa_gap_product_equivalent_padding_leftbound + S (pfrep_index_product_equivalent_padding_left) = (N)) -> (((exists ff_h_pfp_product_equivalent_padding_leftinput. ff_h_pfp_product_equivalent_padding_leftinput + S (pfrep_value_product_equivalent_padding_left) = S ((S (pfrep_index_product_equivalent_padding_left)) * cc)) /\\ exists ff_q_pfp_product_equivalent_padding_leftinput. cb = ff_q_pfp_product_equivalent_padding_leftinput * S ((S (pfrep_index_product_equivalent_padding_left)) * cc) + (pfrep_value_product_equivalent_padding_left))) -> (((exists ff_h_pfp_product_equivalent_padding_leftoutput. ff_h_pfp_product_equivalent_padding_leftoutput + S (pfrep_value_product_equivalent_padding_left) = S ((S ((t)+pfrep_index_product_equivalent_padding_left)) * CC)) /\\ exists ff_q_pfp_product_equivalent_padding_leftoutput. CB = ff_q_pfp_product_equivalent_padding_leftoutput * S ((S ((t)+pfrep_index_product_equivalent_padding_left)) * CC) + (pfrep_value_product_equivalent_padding_left)))))))))", "specialize prime_field_polynomial_convolution_left_padding_nonempty_left (p)", "specialize prime_field_polynomial_convolution_left_padding_nonempty_left (ab)", "specialize prime_field_polynomial_convolution_left_padding_nonempty_left (ac)", "specialize prime_field_polynomial_convolution_left_padding_nonempty_left (L)", "specialize prime_field_polynomial_convolution_left_padding_nonempty_left (bb)", "specialize prime_field_polynomial_convolution_left_padding_nonempty_left (bc)", "specialize prime_field_polynomial_convolution_left_padding_nonempty_left (M)", "specialize prime_field_polynomial_convolution_left_padding_nonempty_left (cb)", "specialize prime_field_polynomial_convolution_left_padding_nonempty_left (cc)", "specialize prime_field_polynomial_convolution_left_padding_nonempty_left (N)", "specialize prime_field_polynomial_convolution_left_padding_nonempty_left (AB)", "specialize prime_field_polynomial_convolution_left_padding_nonempty_left (AC)", "specialize prime_field_polynomial_convolution_left_padding_nonempty_left (t)", "specialize prime_field_polynomial_convolution_left_padding_nonempty_left (CB)", "specialize prime_field_polynomial_convolution_left_padding_nonempty_left (CC)", "specialize prime_field_polynomial_convolution_left_padding_nonempty_left (K)", "apply prime_field_polynomial_convolution_left_padding_nonempty_left", "exact hp", "exact hL_right", "exact hM_right", "exact hpad", "exact hc", "exact hn", "cases hd", "rewrite hd_left", "rewrite hd_left", "specialize prime_field_polynomial_left_pad_equivalent (cb)", "specialize prime_field_polynomial_left_pad_equivalent (cc)", "specialize prime_field_polynomial_left_pad_equivalent (N)", "specialize prime_field_polynomial_left_pad_equivalent (t)", "specialize prime_field_polynomial_left_pad_equivalent (CB)", "specialize prime_field_polynomial_left_pad_equivalent (CC)", "apply prime_field_polynomial_left_pad_equivalent", "exact hd_right" ], "script_sha256": "aca634e846acc065f0544196cf8947988b9f5932a00cbc083d58f4efb2d8cf37", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_convolution_padding_candidate.py", "sha256": "2d874ecfb35a5db0aecdeb07b549464efebad9072c363113aa5a0a977845d007" }, "statement": "forall p ab ac L bb bc M cb cc N AB AC t CB CC K. (~(p=0)) -> (((forall pfp_repeat_index_product_factor_padding_leftzeros. (exists pfa_gap_product_factor_padding_leftzerosindex. pfa_gap_product_factor_padding_leftzerosindex + S (pfp_repeat_index_product_factor_padding_leftzeros) = (t)) -> (((exists ff_h_pfp_product_factor_padding_leftzerosentry. ff_h_pfp_product_factor_padding_leftzerosentry + S (0) = S ((S (pfp_repeat_index_product_factor_padding_leftzeros)) * AC)) /\\ exists ff_q_pfp_product_factor_padding_leftzerosentry. AB = ff_q_pfp_product_factor_padding_leftzerosentry * S ((S (pfp_repeat_index_product_factor_padding_leftzeros)) * AC) + (0)))) /\\ ((forall pfrep_index_product_factor_padding_left pfrep_value_product_factor_padding_left. (exists pfa_gap_product_factor_padding_leftbound. pfa_gap_product_factor_padding_leftbound + S (pfrep_index_product_factor_padding_left) = (L)) -> (((exists ff_h_pfp_product_factor_padding_leftinput. ff_h_pfp_product_factor_padding_leftinput + S (pfrep_value_product_factor_padding_left) = S ((S (pfrep_index_product_factor_padding_left)) * ac)) /\\ exists ff_q_pfp_product_factor_padding_leftinput. ab = ff_q_pfp_product_factor_padding_leftinput * S ((S (pfrep_index_product_factor_padding_left)) * ac) + (pfrep_value_product_factor_padding_left))) -> (((exists ff_h_pfp_product_factor_padding_leftoutput. ff_h_pfp_product_factor_padding_leftoutput + S (pfrep_value_product_factor_padding_left) = S ((S ((t)+pfrep_index_product_factor_padding_left)) * AC)) /\\ exists ff_q_pfp_product_factor_padding_leftoutput. AB = ff_q_pfp_product_factor_padding_leftoutput * S ((S ((t)+pfrep_index_product_factor_padding_left)) * AC) + (pfrep_value_product_factor_padding_left))))))) -> (((forall fom_index_pfp_product_original_leftleft. (exists fom_gap_pfp_product_original_leftleft_index_bound. fom_gap_pfp_product_original_leftleft_index_bound + S (fom_index_pfp_product_original_leftleft) = L) -> exists fom_value_pfp_product_original_leftleft. ((((exists fom_beta_height_pfp_product_original_leftleft_entry. fom_beta_height_pfp_product_original_leftleft_entry + S (fom_value_pfp_product_original_leftleft) = S ((S (fom_index_pfp_product_original_leftleft)) * ac)) /\\ exists fom_beta_quotient_pfp_product_original_leftleft_entry. ab = fom_beta_quotient_pfp_product_original_leftleft_entry * S ((S (fom_index_pfp_product_original_leftleft)) * ac) + (fom_value_pfp_product_original_leftleft))) /\\ (exists fom_gap_pfp_product_original_leftleft_value_bound. fom_gap_pfp_product_original_leftleft_value_bound + S (fom_value_pfp_product_original_leftleft) = p))) /\\ (((forall fom_index_pfp_product_original_leftright. (exists fom_gap_pfp_product_original_leftright_index_bound. fom_gap_pfp_product_original_leftright_index_bound + S (fom_index_pfp_product_original_leftright) = M) -> exists fom_value_pfp_product_original_leftright. ((((exists fom_beta_height_pfp_product_original_leftright_entry. fom_beta_height_pfp_product_original_leftright_entry + S (fom_value_pfp_product_original_leftright) = S ((S (fom_index_pfp_product_original_leftright)) * bc)) /\\ exists fom_beta_quotient_pfp_product_original_leftright_entry. bb = fom_beta_quotient_pfp_product_original_leftright_entry * S ((S (fom_index_pfp_product_original_leftright)) * bc) + (fom_value_pfp_product_original_leftright))) /\\ (exists fom_gap_pfp_product_original_leftright_value_bound. fom_gap_pfp_product_original_leftright_value_bound + S (fom_value_pfp_product_original_leftright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_product_original_leftcoefficients. (exists pfa_gap_product_original_leftcoefficientsbound. pfa_gap_product_original_leftcoefficientsbound + S (pfc_index_product_original_leftcoefficients) = (N)) -> exists pfc_value_product_original_leftcoefficients. ((((exists ff_h_pfp_product_original_leftcoefficientsentry. ff_h_pfp_product_original_leftcoefficientsentry + S (pfc_value_product_original_leftcoefficients) = S ((S (pfc_index_product_original_leftcoefficients)) * cc)) /\\ exists ff_q_pfp_product_original_leftcoefficientsentry. cb = ff_q_pfp_product_original_leftcoefficientsentry * S ((S (pfc_index_product_original_leftcoefficients)) * cc) + (pfc_value_product_original_leftcoefficients))) /\\ ((exists pfc_terms_code_product_original_leftcoefficientscoefficient pfc_terms_scale_product_original_leftcoefficientscoefficient pfc_natural_sum_product_original_leftcoefficientscoefficient. ((forall pfc_index_product_original_leftcoefficientscoefficientdiagonal. (exists pfa_gap_product_original_leftcoefficientscoefficientdiagonalbound. pfa_gap_product_original_leftcoefficientscoefficientdiagonalbound + S (pfc_index_product_original_leftcoefficientscoefficientdiagonal) = (S (pfc_index_product_original_leftcoefficients))) -> exists pfc_value_product_original_leftcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_product_original_leftcoefficientscoefficientdiagonalentry. ff_h_pfp_product_original_leftcoefficientscoefficientdiagonalentry + S (pfc_value_product_original_leftcoefficientscoefficientdiagonal) = S ((S (pfc_index_product_original_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_product_original_leftcoefficientscoefficient)) /\\ exists ff_q_pfp_product_original_leftcoefficientscoefficientdiagonalentry. pfc_terms_code_product_original_leftcoefficientscoefficient = ff_q_pfp_product_original_leftcoefficientscoefficientdiagonalentry * S ((S (pfc_index_product_original_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_product_original_leftcoefficientscoefficient) + (pfc_value_product_original_leftcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_product_original_leftcoefficientscoefficientdiagonalterm pfc_left_product_original_leftcoefficientscoefficientdiagonalterm pfc_right_product_original_leftcoefficientscoefficientdiagonalterm. (((pfc_index_product_original_leftcoefficientscoefficientdiagonal)+pfc_complement_product_original_leftcoefficientscoefficientdiagonalterm=(pfc_index_product_original_leftcoefficients)) /\\ ((((((exists pfa_gap_product_original_leftcoefficientscoefficientdiagonaltermleftinside. pfa_gap_product_original_leftcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_product_original_leftcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_product_original_leftcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_product_original_leftcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_product_original_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_product_original_leftcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_product_original_leftcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_product_original_leftcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_product_original_leftcoefficientscoefficientdiagonal)) * ac) + (pfc_left_product_original_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_product_original_leftcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_product_original_leftcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_product_original_leftcoefficientscoefficientdiagonal)) /\\ (((pfc_left_product_original_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_product_original_leftcoefficientscoefficientdiagonaltermrightinside. pfa_gap_product_original_leftcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_product_original_leftcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_product_original_leftcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_product_original_leftcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_product_original_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_product_original_leftcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_product_original_leftcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_product_original_leftcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_product_original_leftcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_product_original_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_product_original_leftcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_product_original_leftcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_product_original_leftcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_product_original_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_product_original_leftcoefficientscoefficientdiagonal)=pfc_left_product_original_leftcoefficientscoefficientdiagonalterm*pfc_right_product_original_leftcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_product_original_leftcoefficientscoefficientsum fs_v_pfc_product_original_leftcoefficientscoefficientsum. ((((exists fs_h_pfc_product_original_leftcoefficientscoefficientsum_body_start. fs_h_pfc_product_original_leftcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_product_original_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_product_original_leftcoefficientscoefficientsum_body_start. fs_u_pfc_product_original_leftcoefficientscoefficientsum = fs_q_pfc_product_original_leftcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_product_original_leftcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_product_original_leftcoefficientscoefficientsum_body_terminal. fs_h_pfc_product_original_leftcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_product_original_leftcoefficientscoefficient) = S ((S (S (pfc_index_product_original_leftcoefficients))) * fs_v_pfc_product_original_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_product_original_leftcoefficientscoefficientsum_body_terminal. fs_u_pfc_product_original_leftcoefficientscoefficientsum = fs_q_pfc_product_original_leftcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_product_original_leftcoefficients))) * fs_v_pfc_product_original_leftcoefficientscoefficientsum) + (pfc_natural_sum_product_original_leftcoefficientscoefficient))) /\\ forall fs_i_pfc_product_original_leftcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_product_original_leftcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_product_original_leftcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_product_original_leftcoefficientscoefficientsum_body_steps = S (pfc_index_product_original_leftcoefficients)) -> exists fs_a_pfc_product_original_leftcoefficientscoefficientsum_body_steps fs_r_pfc_product_original_leftcoefficientscoefficientsum_body_steps fs_s_pfc_product_original_leftcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_product_original_leftcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_product_original_leftcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_product_original_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_product_original_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_product_original_leftcoefficientscoefficient)) /\\ exists fs_q_pfc_product_original_leftcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_product_original_leftcoefficientscoefficient = fs_q_pfc_product_original_leftcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_product_original_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_product_original_leftcoefficientscoefficient) + (fs_a_pfc_product_original_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_product_original_leftcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_product_original_leftcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_product_original_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_product_original_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_product_original_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_product_original_leftcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_product_original_leftcoefficientscoefficientsum = fs_q_pfc_product_original_leftcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_product_original_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_product_original_leftcoefficientscoefficientsum) + (fs_r_pfc_product_original_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_product_original_leftcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_product_original_leftcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_product_original_leftcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_product_original_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_product_original_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_product_original_leftcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_product_original_leftcoefficientscoefficientsum = fs_q_pfc_product_original_leftcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_product_original_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_product_original_leftcoefficientscoefficientsum) + (fs_s_pfc_product_original_leftcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_product_original_leftcoefficientscoefficientsum_body_steps = fs_r_pfc_product_original_leftcoefficientscoefficientsum_body_steps + fs_a_pfc_product_original_leftcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_product_original_leftcoefficientscoefficientresiduebound. pfa_gap_product_original_leftcoefficientscoefficientresiduebound + S (pfc_value_product_original_leftcoefficients) = (p)) /\\ ((exists pfa_offset_left_product_original_leftcoefficientscoefficientresiduecongruence pfa_offset_right_product_original_leftcoefficientscoefficientresiduecongruence. (pfc_natural_sum_product_original_leftcoefficientscoefficient) + (p) * pfa_offset_left_product_original_leftcoefficientscoefficientresiduecongruence = (pfc_value_product_original_leftcoefficients) + (p) * pfa_offset_right_product_original_leftcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_product_padded_leftleft. (exists fom_gap_pfp_product_padded_leftleft_index_bound. fom_gap_pfp_product_padded_leftleft_index_bound + S (fom_index_pfp_product_padded_leftleft) = t+L) -> exists fom_value_pfp_product_padded_leftleft. ((((exists fom_beta_height_pfp_product_padded_leftleft_entry. fom_beta_height_pfp_product_padded_leftleft_entry + S (fom_value_pfp_product_padded_leftleft) = S ((S (fom_index_pfp_product_padded_leftleft)) * AC)) /\\ exists fom_beta_quotient_pfp_product_padded_leftleft_entry. AB = fom_beta_quotient_pfp_product_padded_leftleft_entry * S ((S (fom_index_pfp_product_padded_leftleft)) * AC) + (fom_value_pfp_product_padded_leftleft))) /\\ (exists fom_gap_pfp_product_padded_leftleft_value_bound. fom_gap_pfp_product_padded_leftleft_value_bound + S (fom_value_pfp_product_padded_leftleft) = p))) /\\ (((forall fom_index_pfp_product_padded_leftright. (exists fom_gap_pfp_product_padded_leftright_index_bound. fom_gap_pfp_product_padded_leftright_index_bound + S (fom_index_pfp_product_padded_leftright) = M) -> exists fom_value_pfp_product_padded_leftright. ((((exists fom_beta_height_pfp_product_padded_leftright_entry. fom_beta_height_pfp_product_padded_leftright_entry + S (fom_value_pfp_product_padded_leftright) = S ((S (fom_index_pfp_product_padded_leftright)) * bc)) /\\ exists fom_beta_quotient_pfp_product_padded_leftright_entry. bb = fom_beta_quotient_pfp_product_padded_leftright_entry * S ((S (fom_index_pfp_product_padded_leftright)) * bc) + (fom_value_pfp_product_padded_leftright))) /\\ (exists fom_gap_pfp_product_padded_leftright_value_bound. fom_gap_pfp_product_padded_leftright_value_bound + S (fom_value_pfp_product_padded_leftright) = p))) /\\ (((((((t+L)=0 \\/ (M)=0) /\\ (((K)=0)))) \\/ (((~((t+L)=0)) /\\ (((~((M)=0)) /\\ (((t+L)+(M)=S (K)))))))) /\\ ((forall pfc_index_product_padded_leftcoefficients. (exists pfa_gap_product_padded_leftcoefficientsbound. pfa_gap_product_padded_leftcoefficientsbound + S (pfc_index_product_padded_leftcoefficients) = (K)) -> exists pfc_value_product_padded_leftcoefficients. ((((exists ff_h_pfp_product_padded_leftcoefficientsentry. ff_h_pfp_product_padded_leftcoefficientsentry + S (pfc_value_product_padded_leftcoefficients) = S ((S (pfc_index_product_padded_leftcoefficients)) * CC)) /\\ exists ff_q_pfp_product_padded_leftcoefficientsentry. CB = ff_q_pfp_product_padded_leftcoefficientsentry * S ((S (pfc_index_product_padded_leftcoefficients)) * CC) + (pfc_value_product_padded_leftcoefficients))) /\\ ((exists pfc_terms_code_product_padded_leftcoefficientscoefficient pfc_terms_scale_product_padded_leftcoefficientscoefficient pfc_natural_sum_product_padded_leftcoefficientscoefficient. ((forall pfc_index_product_padded_leftcoefficientscoefficientdiagonal. (exists pfa_gap_product_padded_leftcoefficientscoefficientdiagonalbound. pfa_gap_product_padded_leftcoefficientscoefficientdiagonalbound + S (pfc_index_product_padded_leftcoefficientscoefficientdiagonal) = (S (pfc_index_product_padded_leftcoefficients))) -> exists pfc_value_product_padded_leftcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_product_padded_leftcoefficientscoefficientdiagonalentry. ff_h_pfp_product_padded_leftcoefficientscoefficientdiagonalentry + S (pfc_value_product_padded_leftcoefficientscoefficientdiagonal) = S ((S (pfc_index_product_padded_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_product_padded_leftcoefficientscoefficient)) /\\ exists ff_q_pfp_product_padded_leftcoefficientscoefficientdiagonalentry. pfc_terms_code_product_padded_leftcoefficientscoefficient = ff_q_pfp_product_padded_leftcoefficientscoefficientdiagonalentry * S ((S (pfc_index_product_padded_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_product_padded_leftcoefficientscoefficient) + (pfc_value_product_padded_leftcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_product_padded_leftcoefficientscoefficientdiagonalterm pfc_left_product_padded_leftcoefficientscoefficientdiagonalterm pfc_right_product_padded_leftcoefficientscoefficientdiagonalterm. (((pfc_index_product_padded_leftcoefficientscoefficientdiagonal)+pfc_complement_product_padded_leftcoefficientscoefficientdiagonalterm=(pfc_index_product_padded_leftcoefficients)) /\\ ((((((exists pfa_gap_product_padded_leftcoefficientscoefficientdiagonaltermleftinside. pfa_gap_product_padded_leftcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_product_padded_leftcoefficientscoefficientdiagonal) = (t+L)) /\\ ((((exists ff_h_pfp_product_padded_leftcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_product_padded_leftcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_product_padded_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_product_padded_leftcoefficientscoefficientdiagonal)) * AC)) /\\ exists ff_q_pfp_product_padded_leftcoefficientscoefficientdiagonaltermleftentry. AB = ff_q_pfp_product_padded_leftcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_product_padded_leftcoefficientscoefficientdiagonal)) * AC) + (pfc_left_product_padded_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_product_padded_leftcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_product_padded_leftcoefficientscoefficientdiagonaltermleftoutside+(t+L)=(pfc_index_product_padded_leftcoefficientscoefficientdiagonal)) /\\ (((pfc_left_product_padded_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_product_padded_leftcoefficientscoefficientdiagonaltermrightinside. pfa_gap_product_padded_leftcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_product_padded_leftcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_product_padded_leftcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_product_padded_leftcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_product_padded_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_product_padded_leftcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_product_padded_leftcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_product_padded_leftcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_product_padded_leftcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_product_padded_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_product_padded_leftcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_product_padded_leftcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_product_padded_leftcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_product_padded_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_product_padded_leftcoefficientscoefficientdiagonal)=pfc_left_product_padded_leftcoefficientscoefficientdiagonalterm*pfc_right_product_padded_leftcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_product_padded_leftcoefficientscoefficientsum fs_v_pfc_product_padded_leftcoefficientscoefficientsum. ((((exists fs_h_pfc_product_padded_leftcoefficientscoefficientsum_body_start. fs_h_pfc_product_padded_leftcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_product_padded_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_product_padded_leftcoefficientscoefficientsum_body_start. fs_u_pfc_product_padded_leftcoefficientscoefficientsum = fs_q_pfc_product_padded_leftcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_product_padded_leftcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_product_padded_leftcoefficientscoefficientsum_body_terminal. fs_h_pfc_product_padded_leftcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_product_padded_leftcoefficientscoefficient) = S ((S (S (pfc_index_product_padded_leftcoefficients))) * fs_v_pfc_product_padded_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_product_padded_leftcoefficientscoefficientsum_body_terminal. fs_u_pfc_product_padded_leftcoefficientscoefficientsum = fs_q_pfc_product_padded_leftcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_product_padded_leftcoefficients))) * fs_v_pfc_product_padded_leftcoefficientscoefficientsum) + (pfc_natural_sum_product_padded_leftcoefficientscoefficient))) /\\ forall fs_i_pfc_product_padded_leftcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_product_padded_leftcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_product_padded_leftcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_product_padded_leftcoefficientscoefficientsum_body_steps = S (pfc_index_product_padded_leftcoefficients)) -> exists fs_a_pfc_product_padded_leftcoefficientscoefficientsum_body_steps fs_r_pfc_product_padded_leftcoefficientscoefficientsum_body_steps fs_s_pfc_product_padded_leftcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_product_padded_leftcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_product_padded_leftcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_product_padded_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_product_padded_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_product_padded_leftcoefficientscoefficient)) /\\ exists fs_q_pfc_product_padded_leftcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_product_padded_leftcoefficientscoefficient = fs_q_pfc_product_padded_leftcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_product_padded_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_product_padded_leftcoefficientscoefficient) + (fs_a_pfc_product_padded_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_product_padded_leftcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_product_padded_leftcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_product_padded_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_product_padded_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_product_padded_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_product_padded_leftcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_product_padded_leftcoefficientscoefficientsum = fs_q_pfc_product_padded_leftcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_product_padded_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_product_padded_leftcoefficientscoefficientsum) + (fs_r_pfc_product_padded_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_product_padded_leftcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_product_padded_leftcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_product_padded_leftcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_product_padded_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_product_padded_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_product_padded_leftcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_product_padded_leftcoefficientscoefficientsum = fs_q_pfc_product_padded_leftcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_product_padded_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_product_padded_leftcoefficientscoefficientsum) + (fs_s_pfc_product_padded_leftcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_product_padded_leftcoefficientscoefficientsum_body_steps = fs_r_pfc_product_padded_leftcoefficientscoefficientsum_body_steps + fs_a_pfc_product_padded_leftcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_product_padded_leftcoefficientscoefficientresiduebound. pfa_gap_product_padded_leftcoefficientscoefficientresiduebound + S (pfc_value_product_padded_leftcoefficients) = (p)) /\\ ((exists pfa_offset_left_product_padded_leftcoefficientscoefficientresiduecongruence pfa_offset_right_product_padded_leftcoefficientscoefficientresiduecongruence. (pfc_natural_sum_product_padded_leftcoefficientscoefficient) + (p) * pfa_offset_left_product_padded_leftcoefficientscoefficientresiduecongruence = (pfc_value_product_padded_leftcoefficients) + (p) * pfa_offset_right_product_padded_leftcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (forall pfrep_power_product_equivalent_left pfrep_left_product_equivalent_left pfrep_right_product_equivalent_left. ((exists pfrep_position_product_equivalent_leftfirst. ((pfrep_position_product_equivalent_leftfirst+S (pfrep_power_product_equivalent_left)=(N)) /\\ ((((exists ff_h_pfp_product_equivalent_leftfirstentry. ff_h_pfp_product_equivalent_leftfirstentry + S (pfrep_left_product_equivalent_left) = S ((S (pfrep_position_product_equivalent_leftfirst)) * cc)) /\\ exists ff_q_pfp_product_equivalent_leftfirstentry. cb = ff_q_pfp_product_equivalent_leftfirstentry * S ((S (pfrep_position_product_equivalent_leftfirst)) * cc) + (pfrep_left_product_equivalent_left)))))) \\/ (((exists pfrep_gap_product_equivalent_leftfirstoutside. pfrep_gap_product_equivalent_leftfirstoutside+(N)=(pfrep_power_product_equivalent_left)) /\\ (((pfrep_left_product_equivalent_left)=0))))) -> ((exists pfrep_position_product_equivalent_leftsecond. ((pfrep_position_product_equivalent_leftsecond+S (pfrep_power_product_equivalent_left)=(K)) /\\ ((((exists ff_h_pfp_product_equivalent_leftsecondentry. ff_h_pfp_product_equivalent_leftsecondentry + S (pfrep_right_product_equivalent_left) = S ((S (pfrep_position_product_equivalent_leftsecond)) * CC)) /\\ exists ff_q_pfp_product_equivalent_leftsecondentry. CB = ff_q_pfp_product_equivalent_leftsecondentry * S ((S (pfrep_position_product_equivalent_leftsecond)) * CC) + (pfrep_right_product_equivalent_left)))))) \\/ (((exists pfrep_gap_product_equivalent_leftsecondoutside. pfrep_gap_product_equivalent_leftsecondoutside+(K)=(pfrep_power_product_equivalent_left)) /\\ (((pfrep_right_product_equivalent_left)=0))))) -> pfrep_left_product_equivalent_left=pfrep_right_product_equivalent_left)", "statement_sha256": "65232322cfd2e6204ecc0b416d0ae2c2654ca3b5eae55ee0798a496173285c55", "summary": "Genuine leading-zero padding of the left factor preserves the formal polynomial product, including empty factors whose proper product lengths need not differ by the padding count.", "summary_sha256": "f86c41dd656e34bb91f4da8b3fc68a0a81d14b423d49cc2177daf4eda0dc07e0" }, { "alpha_v33_frontier_enrollment": { "body_receipt_sha256": "2a562cadbb804f5d0428dda43334f45f599055bfd6bc12511f4532e3760b58b3", "bundle_campaign": "polynomial-euclidean-division", 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"prime_field_polynomial_convolution_zero_left", "prime_field_polynomial_convolution_zero_right", "polynomial_left_pad_zero_prefix", "prime_field_polynomial_zero_prefix_equivalent_empty", "prime_field_polynomial_equivalent_transitive", "prime_field_polynomial_equivalent_symmetric", "prime_field_polynomial_convolution_left_padding_nonempty_right", "prime_field_polynomial_left_pad_equivalent" ], "dependencies_sha256": "62d6274d9b91130d894d244473100d4d18ad126aeb027b89b200b6894ccc1135", "empty_context_closure": { "body_proof_depth": 59, "body_proof_nodes": 310, "bundle_campaign": "polynomial-euclidean-division", "bundle_dependency_edge_count": 1071, "bundle_node_count": 377, "bundle_node_id": 365, "bundle_path": "research/arithmetic-library/artifacts/prime-field-polynomial-euclidean-division-proof-bundle-v1.json", "bundle_root_id": 376, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "6ae667d8518e4dbe722bb08ad1b08715a0d282c2893e533c8133d770fe861dcf", 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"sealed_alpha_v32_parent", "path": "artifacts/peano-library/alpha/catalog-v32.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "polynomial-euclidean-division", "logical_spec_sha256": "b89c5fd3ab747f68873f1f67ecaa9aea2b4ab47d55a6bac10895117870032c44", "membership": "alpha_only", "name": "prime_field_polynomial_convolution_left_padding_equivalent_right", "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 t", "intro CB", "intro CC", "intro K", "intro hp", "intro hpad", "intro hc", "intro hn", "have hL : L=0 \\/ ~(L=0)", "specialize eq_decidable (L)", "specialize eq_decidable (0)", "apply eq_decidable", "cases hL", "have hz : forall pfp_repeat_index_empty_factor_zero_rightleft. (exists pfa_gap_empty_factor_zero_rightleftindex. pfa_gap_empty_factor_zero_rightleftindex + S (pfp_repeat_index_empty_factor_zero_rightleft) = (L)) -> (((exists ff_h_pfp_empty_factor_zero_rightleftentry. ff_h_pfp_empty_factor_zero_rightleftentry + S (0) = S ((S (pfp_repeat_index_empty_factor_zero_rightleft)) * ac)) /\\ exists ff_q_pfp_empty_factor_zero_rightleftentry. ab = ff_q_pfp_empty_factor_zero_rightleftentry * S ((S (pfp_repeat_index_empty_factor_zero_rightleft)) * ac) + (0)))", "intro j", "intro hj", "rewrite hL_left at hj", "exfalso", "specialize matrix_rank_no_index_below_zero (j)", "apply matrix_rank_no_index_below_zero", "exact hj", "have hc0 : forall pfp_repeat_index_empty_original_product_rightleft. (exists pfa_gap_empty_original_product_rightleftindex. pfa_gap_empty_original_product_rightleftindex + S (pfp_repeat_index_empty_original_product_rightleft) = (N)) -> (((exists ff_h_pfp_empty_original_product_rightleftentry. ff_h_pfp_empty_original_product_rightleftentry + S (0) = S ((S (pfp_repeat_index_empty_original_product_rightleft)) * cc)) /\\ exists ff_q_pfp_empty_original_product_rightleftentry. cb = ff_q_pfp_empty_original_product_rightleftentry * S ((S (pfp_repeat_index_empty_original_product_rightleft)) * cc) + (0)))", "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", "have hn0 : forall pfp_repeat_index_empty_padded_product_rightleft. (exists pfa_gap_empty_padded_product_rightleftindex. pfa_gap_empty_padded_product_rightleftindex + S (pfp_repeat_index_empty_padded_product_rightleft) = (K)) -> (((exists ff_h_pfp_empty_padded_product_rightleftentry. ff_h_pfp_empty_padded_product_rightleftentry + S (0) = S ((S (pfp_repeat_index_empty_padded_product_rightleft)) * CC)) /\\ exists ff_q_pfp_empty_padded_product_rightleftentry. CB = ff_q_pfp_empty_padded_product_rightleftentry * S ((S (pfp_repeat_index_empty_padded_product_rightleft)) * CC) + (0)))", "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 (t+M)", "specialize prime_field_polynomial_convolution_zero_left (CB)", "specialize prime_field_polynomial_convolution_zero_left (CC)", "specialize prime_field_polynomial_convolution_zero_left (K)", "apply prime_field_polynomial_convolution_zero_left", "exact hp", "exact hz", "exact hn", "have hec : forall pfrep_power_empty_first_equivalence_rightleft pfrep_left_empty_first_equivalence_rightleft pfrep_right_empty_first_equivalence_rightleft. ((exists pfrep_position_empty_first_equivalence_rightleftfirst. ((pfrep_position_empty_first_equivalence_rightleftfirst+S (pfrep_power_empty_first_equivalence_rightleft)=(N)) /\\ ((((exists ff_h_pfp_empty_first_equivalence_rightleftfirstentry. ff_h_pfp_empty_first_equivalence_rightleftfirstentry + S (pfrep_left_empty_first_equivalence_rightleft) = S ((S (pfrep_position_empty_first_equivalence_rightleftfirst)) * cc)) /\\ exists ff_q_pfp_empty_first_equivalence_rightleftfirstentry. cb = ff_q_pfp_empty_first_equivalence_rightleftfirstentry * S ((S (pfrep_position_empty_first_equivalence_rightleftfirst)) * cc) + (pfrep_left_empty_first_equivalence_rightleft)))))) \\/ (((exists pfrep_gap_empty_first_equivalence_rightleftfirstoutside. pfrep_gap_empty_first_equivalence_rightleftfirstoutside+(N)=(pfrep_power_empty_first_equivalence_rightleft)) /\\ (((pfrep_left_empty_first_equivalence_rightleft)=0))))) -> ((exists pfrep_position_empty_first_equivalence_rightleftsecond. ((pfrep_position_empty_first_equivalence_rightleftsecond+S (pfrep_power_empty_first_equivalence_rightleft)=(0)) /\\ ((((exists ff_h_pfp_empty_first_equivalence_rightleftsecondentry. ff_h_pfp_empty_first_equivalence_rightleftsecondentry + S (pfrep_right_empty_first_equivalence_rightleft) = S ((S (pfrep_position_empty_first_equivalence_rightleftsecond)) * 0)) /\\ exists ff_q_pfp_empty_first_equivalence_rightleftsecondentry. 0 = ff_q_pfp_empty_first_equivalence_rightleftsecondentry * S ((S (pfrep_position_empty_first_equivalence_rightleftsecond)) * 0) + (pfrep_right_empty_first_equivalence_rightleft)))))) \\/ (((exists pfrep_gap_empty_first_equivalence_rightleftsecondoutside. pfrep_gap_empty_first_equivalence_rightleftsecondoutside+(0)=(pfrep_power_empty_first_equivalence_rightleft)) /\\ (((pfrep_right_empty_first_equivalence_rightleft)=0))))) -> pfrep_left_empty_first_equivalence_rightleft=pfrep_right_empty_first_equivalence_rightleft", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (cb)", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (cc)", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (N)", "apply prime_field_polynomial_zero_prefix_equivalent_empty", "exact hc0", "have hen : forall pfrep_power_empty_second_equivalence_rightleft pfrep_left_empty_second_equivalence_rightleft pfrep_right_empty_second_equivalence_rightleft. ((exists pfrep_position_empty_second_equivalence_rightleftfirst. ((pfrep_position_empty_second_equivalence_rightleftfirst+S (pfrep_power_empty_second_equivalence_rightleft)=(K)) /\\ ((((exists ff_h_pfp_empty_second_equivalence_rightleftfirstentry. ff_h_pfp_empty_second_equivalence_rightleftfirstentry + S (pfrep_left_empty_second_equivalence_rightleft) = S ((S (pfrep_position_empty_second_equivalence_rightleftfirst)) * CC)) /\\ exists ff_q_pfp_empty_second_equivalence_rightleftfirstentry. CB = ff_q_pfp_empty_second_equivalence_rightleftfirstentry * S ((S (pfrep_position_empty_second_equivalence_rightleftfirst)) * CC) + (pfrep_left_empty_second_equivalence_rightleft)))))) \\/ (((exists pfrep_gap_empty_second_equivalence_rightleftfirstoutside. pfrep_gap_empty_second_equivalence_rightleftfirstoutside+(K)=(pfrep_power_empty_second_equivalence_rightleft)) /\\ (((pfrep_left_empty_second_equivalence_rightleft)=0))))) -> ((exists pfrep_position_empty_second_equivalence_rightleftsecond. ((pfrep_position_empty_second_equivalence_rightleftsecond+S (pfrep_power_empty_second_equivalence_rightleft)=(0)) /\\ ((((exists ff_h_pfp_empty_second_equivalence_rightleftsecondentry. ff_h_pfp_empty_second_equivalence_rightleftsecondentry + S (pfrep_right_empty_second_equivalence_rightleft) = S ((S (pfrep_position_empty_second_equivalence_rightleftsecond)) * 0)) /\\ exists ff_q_pfp_empty_second_equivalence_rightleftsecondentry. 0 = ff_q_pfp_empty_second_equivalence_rightleftsecondentry * S ((S (pfrep_position_empty_second_equivalence_rightleftsecond)) * 0) + (pfrep_right_empty_second_equivalence_rightleft)))))) \\/ (((exists pfrep_gap_empty_second_equivalence_rightleftsecondoutside. pfrep_gap_empty_second_equivalence_rightleftsecondoutside+(0)=(pfrep_power_empty_second_equivalence_rightleft)) /\\ (((pfrep_right_empty_second_equivalence_rightleft)=0))))) -> pfrep_left_empty_second_equivalence_rightleft=pfrep_right_empty_second_equivalence_rightleft", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (CB)", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (CC)", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (K)", "apply prime_field_polynomial_zero_prefix_equivalent_empty", "exact hn0", "specialize prime_field_polynomial_equivalent_transitive (cb)", "specialize prime_field_polynomial_equivalent_transitive (cc)", "specialize prime_field_polynomial_equivalent_transitive (N)", "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 (CB)", "specialize prime_field_polynomial_equivalent_transitive (CC)", "specialize prime_field_polynomial_equivalent_transitive (K)", "apply prime_field_polynomial_equivalent_transitive", "exact hec", "specialize prime_field_polynomial_equivalent_symmetric (CB)", "specialize prime_field_polynomial_equivalent_symmetric (CC)", "specialize prime_field_polynomial_equivalent_symmetric (K)", "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", "exact hen", "have hM : M=0 \\/ ~(M=0)", "specialize eq_decidable (M)", "specialize eq_decidable (0)", "apply eq_decidable", "cases hM", "have hz : forall pfp_repeat_index_empty_factor_zero_rightright. (exists pfa_gap_empty_factor_zero_rightrightindex. pfa_gap_empty_factor_zero_rightrightindex + S (pfp_repeat_index_empty_factor_zero_rightright) = (M)) -> (((exists ff_h_pfp_empty_factor_zero_rightrightentry. ff_h_pfp_empty_factor_zero_rightrightentry + S (0) = S ((S (pfp_repeat_index_empty_factor_zero_rightright)) * bc)) /\\ exists ff_q_pfp_empty_factor_zero_rightrightentry. bb = ff_q_pfp_empty_factor_zero_rightrightentry * S ((S (pfp_repeat_index_empty_factor_zero_rightright)) * bc) + (0)))", "intro j", "intro hj", "rewrite hM_left at hj", "exfalso", "specialize matrix_rank_no_index_below_zero (j)", "apply matrix_rank_no_index_below_zero", "exact hj", "have hc0 : forall pfp_repeat_index_empty_original_product_rightright. (exists pfa_gap_empty_original_product_rightrightindex. pfa_gap_empty_original_product_rightrightindex + S (pfp_repeat_index_empty_original_product_rightright) = (N)) -> (((exists ff_h_pfp_empty_original_product_rightrightentry. ff_h_pfp_empty_original_product_rightrightentry + S (0) = S ((S (pfp_repeat_index_empty_original_product_rightright)) * cc)) /\\ exists ff_q_pfp_empty_original_product_rightrightentry. cb = ff_q_pfp_empty_original_product_rightrightentry * S ((S (pfp_repeat_index_empty_original_product_rightright)) * cc) + (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 (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", "have hn0 : forall pfp_repeat_index_empty_padded_product_rightright. (exists pfa_gap_empty_padded_product_rightrightindex. pfa_gap_empty_padded_product_rightrightindex + S (pfp_repeat_index_empty_padded_product_rightright) = (K)) -> (((exists ff_h_pfp_empty_padded_product_rightrightentry. ff_h_pfp_empty_padded_product_rightrightentry + S (0) = S ((S (pfp_repeat_index_empty_padded_product_rightright)) * CC)) /\\ exists ff_q_pfp_empty_padded_product_rightrightentry. CB = ff_q_pfp_empty_padded_product_rightrightentry * S ((S (pfp_repeat_index_empty_padded_product_rightright)) * CC) + (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 (BB)", "specialize prime_field_polynomial_convolution_zero_right (BC)", "specialize prime_field_polynomial_convolution_zero_right (t+M)", "specialize prime_field_polynomial_convolution_zero_right (CB)", "specialize prime_field_polynomial_convolution_zero_right (CC)", "specialize prime_field_polynomial_convolution_zero_right (K)", "apply prime_field_polynomial_convolution_zero_right", "exact hp", "specialize polynomial_left_pad_zero_prefix (bb)", "specialize polynomial_left_pad_zero_prefix (bc)", "specialize polynomial_left_pad_zero_prefix (M)", "specialize polynomial_left_pad_zero_prefix (t)", "specialize polynomial_left_pad_zero_prefix (BB)", "specialize polynomial_left_pad_zero_prefix (BC)", "apply polynomial_left_pad_zero_prefix", "exact hz", "exact hpad", "exact hn", "have hec : forall pfrep_power_empty_first_equivalence_rightright pfrep_left_empty_first_equivalence_rightright pfrep_right_empty_first_equivalence_rightright. ((exists pfrep_position_empty_first_equivalence_rightrightfirst. ((pfrep_position_empty_first_equivalence_rightrightfirst+S (pfrep_power_empty_first_equivalence_rightright)=(N)) /\\ ((((exists ff_h_pfp_empty_first_equivalence_rightrightfirstentry. ff_h_pfp_empty_first_equivalence_rightrightfirstentry + S (pfrep_left_empty_first_equivalence_rightright) = S ((S (pfrep_position_empty_first_equivalence_rightrightfirst)) * cc)) /\\ exists ff_q_pfp_empty_first_equivalence_rightrightfirstentry. cb = ff_q_pfp_empty_first_equivalence_rightrightfirstentry * S ((S (pfrep_position_empty_first_equivalence_rightrightfirst)) * cc) + (pfrep_left_empty_first_equivalence_rightright)))))) \\/ (((exists pfrep_gap_empty_first_equivalence_rightrightfirstoutside. pfrep_gap_empty_first_equivalence_rightrightfirstoutside+(N)=(pfrep_power_empty_first_equivalence_rightright)) /\\ (((pfrep_left_empty_first_equivalence_rightright)=0))))) -> ((exists pfrep_position_empty_first_equivalence_rightrightsecond. ((pfrep_position_empty_first_equivalence_rightrightsecond+S (pfrep_power_empty_first_equivalence_rightright)=(0)) /\\ ((((exists ff_h_pfp_empty_first_equivalence_rightrightsecondentry. ff_h_pfp_empty_first_equivalence_rightrightsecondentry + S (pfrep_right_empty_first_equivalence_rightright) = S ((S (pfrep_position_empty_first_equivalence_rightrightsecond)) * 0)) /\\ exists ff_q_pfp_empty_first_equivalence_rightrightsecondentry. 0 = ff_q_pfp_empty_first_equivalence_rightrightsecondentry * S ((S (pfrep_position_empty_first_equivalence_rightrightsecond)) * 0) + (pfrep_right_empty_first_equivalence_rightright)))))) \\/ (((exists pfrep_gap_empty_first_equivalence_rightrightsecondoutside. pfrep_gap_empty_first_equivalence_rightrightsecondoutside+(0)=(pfrep_power_empty_first_equivalence_rightright)) /\\ (((pfrep_right_empty_first_equivalence_rightright)=0))))) -> pfrep_left_empty_first_equivalence_rightright=pfrep_right_empty_first_equivalence_rightright", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (cb)", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (cc)", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (N)", "apply prime_field_polynomial_zero_prefix_equivalent_empty", "exact hc0", "have hen : forall pfrep_power_empty_second_equivalence_rightright pfrep_left_empty_second_equivalence_rightright pfrep_right_empty_second_equivalence_rightright. ((exists pfrep_position_empty_second_equivalence_rightrightfirst. ((pfrep_position_empty_second_equivalence_rightrightfirst+S (pfrep_power_empty_second_equivalence_rightright)=(K)) /\\ ((((exists ff_h_pfp_empty_second_equivalence_rightrightfirstentry. ff_h_pfp_empty_second_equivalence_rightrightfirstentry + S (pfrep_left_empty_second_equivalence_rightright) = S ((S (pfrep_position_empty_second_equivalence_rightrightfirst)) * CC)) /\\ exists ff_q_pfp_empty_second_equivalence_rightrightfirstentry. CB = ff_q_pfp_empty_second_equivalence_rightrightfirstentry * S ((S (pfrep_position_empty_second_equivalence_rightrightfirst)) * CC) + (pfrep_left_empty_second_equivalence_rightright)))))) \\/ (((exists pfrep_gap_empty_second_equivalence_rightrightfirstoutside. pfrep_gap_empty_second_equivalence_rightrightfirstoutside+(K)=(pfrep_power_empty_second_equivalence_rightright)) /\\ (((pfrep_left_empty_second_equivalence_rightright)=0))))) -> ((exists pfrep_position_empty_second_equivalence_rightrightsecond. ((pfrep_position_empty_second_equivalence_rightrightsecond+S (pfrep_power_empty_second_equivalence_rightright)=(0)) /\\ ((((exists ff_h_pfp_empty_second_equivalence_rightrightsecondentry. ff_h_pfp_empty_second_equivalence_rightrightsecondentry + S (pfrep_right_empty_second_equivalence_rightright) = S ((S (pfrep_position_empty_second_equivalence_rightrightsecond)) * 0)) /\\ exists ff_q_pfp_empty_second_equivalence_rightrightsecondentry. 0 = ff_q_pfp_empty_second_equivalence_rightrightsecondentry * S ((S (pfrep_position_empty_second_equivalence_rightrightsecond)) * 0) + (pfrep_right_empty_second_equivalence_rightright)))))) \\/ (((exists pfrep_gap_empty_second_equivalence_rightrightsecondoutside. pfrep_gap_empty_second_equivalence_rightrightsecondoutside+(0)=(pfrep_power_empty_second_equivalence_rightright)) /\\ (((pfrep_right_empty_second_equivalence_rightright)=0))))) -> pfrep_left_empty_second_equivalence_rightright=pfrep_right_empty_second_equivalence_rightright", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (CB)", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (CC)", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (K)", "apply prime_field_polynomial_zero_prefix_equivalent_empty", "exact hn0", "specialize prime_field_polynomial_equivalent_transitive (cb)", "specialize prime_field_polynomial_equivalent_transitive (cc)", "specialize prime_field_polynomial_equivalent_transitive (N)", "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 (CB)", "specialize prime_field_polynomial_equivalent_transitive (CC)", "specialize prime_field_polynomial_equivalent_transitive (K)", "apply prime_field_polynomial_equivalent_transitive", "exact hec", "specialize prime_field_polynomial_equivalent_symmetric (CB)", "specialize prime_field_polynomial_equivalent_symmetric (CC)", "specialize prime_field_polynomial_equivalent_symmetric (K)", "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", "exact hen", "have hd : ((K=t+N) /\\ ((((forall pfp_repeat_index_product_equivalent_padding_rightzeros. (exists pfa_gap_product_equivalent_padding_rightzerosindex. pfa_gap_product_equivalent_padding_rightzerosindex + S (pfp_repeat_index_product_equivalent_padding_rightzeros) = (t)) -> (((exists ff_h_pfp_product_equivalent_padding_rightzerosentry. ff_h_pfp_product_equivalent_padding_rightzerosentry + S (0) = S ((S (pfp_repeat_index_product_equivalent_padding_rightzeros)) * CC)) /\\ exists ff_q_pfp_product_equivalent_padding_rightzerosentry. CB = ff_q_pfp_product_equivalent_padding_rightzerosentry * S ((S (pfp_repeat_index_product_equivalent_padding_rightzeros)) * CC) + (0)))) /\\ ((forall pfrep_index_product_equivalent_padding_right pfrep_value_product_equivalent_padding_right. (exists pfa_gap_product_equivalent_padding_rightbound. pfa_gap_product_equivalent_padding_rightbound + S (pfrep_index_product_equivalent_padding_right) = (N)) -> (((exists ff_h_pfp_product_equivalent_padding_rightinput. ff_h_pfp_product_equivalent_padding_rightinput + S (pfrep_value_product_equivalent_padding_right) = S ((S (pfrep_index_product_equivalent_padding_right)) * cc)) /\\ exists ff_q_pfp_product_equivalent_padding_rightinput. cb = ff_q_pfp_product_equivalent_padding_rightinput * S ((S (pfrep_index_product_equivalent_padding_right)) * cc) + (pfrep_value_product_equivalent_padding_right))) -> (((exists ff_h_pfp_product_equivalent_padding_rightoutput. ff_h_pfp_product_equivalent_padding_rightoutput + S (pfrep_value_product_equivalent_padding_right) = S ((S ((t)+pfrep_index_product_equivalent_padding_right)) * CC)) /\\ exists ff_q_pfp_product_equivalent_padding_rightoutput. CB = ff_q_pfp_product_equivalent_padding_rightoutput * S ((S ((t)+pfrep_index_product_equivalent_padding_right)) * CC) + (pfrep_value_product_equivalent_padding_right)))))))))", "specialize prime_field_polynomial_convolution_left_padding_nonempty_right (p)", "specialize prime_field_polynomial_convolution_left_padding_nonempty_right (ab)", "specialize prime_field_polynomial_convolution_left_padding_nonempty_right (ac)", "specialize prime_field_polynomial_convolution_left_padding_nonempty_right (L)", "specialize prime_field_polynomial_convolution_left_padding_nonempty_right (bb)", "specialize prime_field_polynomial_convolution_left_padding_nonempty_right (bc)", "specialize prime_field_polynomial_convolution_left_padding_nonempty_right (M)", "specialize prime_field_polynomial_convolution_left_padding_nonempty_right (cb)", "specialize prime_field_polynomial_convolution_left_padding_nonempty_right (cc)", "specialize prime_field_polynomial_convolution_left_padding_nonempty_right (N)", "specialize prime_field_polynomial_convolution_left_padding_nonempty_right (BB)", "specialize prime_field_polynomial_convolution_left_padding_nonempty_right (BC)", "specialize prime_field_polynomial_convolution_left_padding_nonempty_right (t)", "specialize prime_field_polynomial_convolution_left_padding_nonempty_right (CB)", "specialize prime_field_polynomial_convolution_left_padding_nonempty_right (CC)", "specialize prime_field_polynomial_convolution_left_padding_nonempty_right (K)", "apply prime_field_polynomial_convolution_left_padding_nonempty_right", "exact hp", "exact hL_right", "exact hM_right", "exact hpad", "exact hc", "exact hn", "cases hd", "rewrite hd_left", "rewrite hd_left", "specialize prime_field_polynomial_left_pad_equivalent (cb)", "specialize prime_field_polynomial_left_pad_equivalent (cc)", "specialize prime_field_polynomial_left_pad_equivalent (N)", "specialize prime_field_polynomial_left_pad_equivalent (t)", "specialize prime_field_polynomial_left_pad_equivalent (CB)", "specialize prime_field_polynomial_left_pad_equivalent (CC)", "apply prime_field_polynomial_left_pad_equivalent", "exact hd_right" ], "script_sha256": "f8f25338eab3502cb4f589f02199fedc707f9d907ca0f4763bb5499c56a58381", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_convolution_padding_candidate.py", "sha256": "2d874ecfb35a5db0aecdeb07b549464efebad9072c363113aa5a0a977845d007" }, "statement": "forall p ab ac L bb bc M cb cc N BB BC t CB CC K. (~(p=0)) -> (((forall pfp_repeat_index_product_factor_padding_rightzeros. (exists pfa_gap_product_factor_padding_rightzerosindex. pfa_gap_product_factor_padding_rightzerosindex + S (pfp_repeat_index_product_factor_padding_rightzeros) = (t)) -> (((exists ff_h_pfp_product_factor_padding_rightzerosentry. ff_h_pfp_product_factor_padding_rightzerosentry + S (0) = S ((S (pfp_repeat_index_product_factor_padding_rightzeros)) * BC)) /\\ exists ff_q_pfp_product_factor_padding_rightzerosentry. BB = ff_q_pfp_product_factor_padding_rightzerosentry * S ((S (pfp_repeat_index_product_factor_padding_rightzeros)) * BC) + (0)))) /\\ ((forall pfrep_index_product_factor_padding_right pfrep_value_product_factor_padding_right. (exists pfa_gap_product_factor_padding_rightbound. pfa_gap_product_factor_padding_rightbound + S (pfrep_index_product_factor_padding_right) = (M)) -> (((exists ff_h_pfp_product_factor_padding_rightinput. ff_h_pfp_product_factor_padding_rightinput + S (pfrep_value_product_factor_padding_right) = S ((S (pfrep_index_product_factor_padding_right)) * bc)) /\\ exists ff_q_pfp_product_factor_padding_rightinput. bb = ff_q_pfp_product_factor_padding_rightinput * S ((S (pfrep_index_product_factor_padding_right)) * bc) + (pfrep_value_product_factor_padding_right))) -> (((exists ff_h_pfp_product_factor_padding_rightoutput. ff_h_pfp_product_factor_padding_rightoutput + S (pfrep_value_product_factor_padding_right) = S ((S ((t)+pfrep_index_product_factor_padding_right)) * BC)) /\\ exists ff_q_pfp_product_factor_padding_rightoutput. BB = ff_q_pfp_product_factor_padding_rightoutput * S ((S ((t)+pfrep_index_product_factor_padding_right)) * BC) + (pfrep_value_product_factor_padding_right))))))) -> (((forall fom_index_pfp_product_original_rightleft. (exists fom_gap_pfp_product_original_rightleft_index_bound. fom_gap_pfp_product_original_rightleft_index_bound + S (fom_index_pfp_product_original_rightleft) = L) -> exists fom_value_pfp_product_original_rightleft. ((((exists fom_beta_height_pfp_product_original_rightleft_entry. fom_beta_height_pfp_product_original_rightleft_entry + S (fom_value_pfp_product_original_rightleft) = S ((S (fom_index_pfp_product_original_rightleft)) * ac)) /\\ exists fom_beta_quotient_pfp_product_original_rightleft_entry. ab = fom_beta_quotient_pfp_product_original_rightleft_entry * S ((S (fom_index_pfp_product_original_rightleft)) * ac) + (fom_value_pfp_product_original_rightleft))) /\\ (exists fom_gap_pfp_product_original_rightleft_value_bound. fom_gap_pfp_product_original_rightleft_value_bound + S (fom_value_pfp_product_original_rightleft) = p))) /\\ (((forall fom_index_pfp_product_original_rightright. (exists fom_gap_pfp_product_original_rightright_index_bound. fom_gap_pfp_product_original_rightright_index_bound + S (fom_index_pfp_product_original_rightright) = M) -> exists fom_value_pfp_product_original_rightright. ((((exists fom_beta_height_pfp_product_original_rightright_entry. fom_beta_height_pfp_product_original_rightright_entry + S (fom_value_pfp_product_original_rightright) = S ((S (fom_index_pfp_product_original_rightright)) * bc)) /\\ exists fom_beta_quotient_pfp_product_original_rightright_entry. bb = fom_beta_quotient_pfp_product_original_rightright_entry * S ((S (fom_index_pfp_product_original_rightright)) * bc) + (fom_value_pfp_product_original_rightright))) /\\ (exists fom_gap_pfp_product_original_rightright_value_bound. fom_gap_pfp_product_original_rightright_value_bound + S (fom_value_pfp_product_original_rightright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_product_original_rightcoefficients. (exists pfa_gap_product_original_rightcoefficientsbound. pfa_gap_product_original_rightcoefficientsbound + S (pfc_index_product_original_rightcoefficients) = (N)) -> exists pfc_value_product_original_rightcoefficients. ((((exists ff_h_pfp_product_original_rightcoefficientsentry. ff_h_pfp_product_original_rightcoefficientsentry + S (pfc_value_product_original_rightcoefficients) = S ((S (pfc_index_product_original_rightcoefficients)) * cc)) /\\ exists ff_q_pfp_product_original_rightcoefficientsentry. cb = ff_q_pfp_product_original_rightcoefficientsentry * S ((S (pfc_index_product_original_rightcoefficients)) * cc) + (pfc_value_product_original_rightcoefficients))) /\\ ((exists pfc_terms_code_product_original_rightcoefficientscoefficient pfc_terms_scale_product_original_rightcoefficientscoefficient pfc_natural_sum_product_original_rightcoefficientscoefficient. ((forall pfc_index_product_original_rightcoefficientscoefficientdiagonal. (exists pfa_gap_product_original_rightcoefficientscoefficientdiagonalbound. pfa_gap_product_original_rightcoefficientscoefficientdiagonalbound + S (pfc_index_product_original_rightcoefficientscoefficientdiagonal) = (S (pfc_index_product_original_rightcoefficients))) -> exists pfc_value_product_original_rightcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_product_original_rightcoefficientscoefficientdiagonalentry. ff_h_pfp_product_original_rightcoefficientscoefficientdiagonalentry + S (pfc_value_product_original_rightcoefficientscoefficientdiagonal) = S ((S (pfc_index_product_original_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_product_original_rightcoefficientscoefficient)) /\\ exists ff_q_pfp_product_original_rightcoefficientscoefficientdiagonalentry. pfc_terms_code_product_original_rightcoefficientscoefficient = ff_q_pfp_product_original_rightcoefficientscoefficientdiagonalentry * S ((S (pfc_index_product_original_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_product_original_rightcoefficientscoefficient) + (pfc_value_product_original_rightcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_product_original_rightcoefficientscoefficientdiagonalterm pfc_left_product_original_rightcoefficientscoefficientdiagonalterm pfc_right_product_original_rightcoefficientscoefficientdiagonalterm. (((pfc_index_product_original_rightcoefficientscoefficientdiagonal)+pfc_complement_product_original_rightcoefficientscoefficientdiagonalterm=(pfc_index_product_original_rightcoefficients)) /\\ ((((((exists pfa_gap_product_original_rightcoefficientscoefficientdiagonaltermleftinside. pfa_gap_product_original_rightcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_product_original_rightcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_product_original_rightcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_product_original_rightcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_product_original_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_product_original_rightcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_product_original_rightcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_product_original_rightcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_product_original_rightcoefficientscoefficientdiagonal)) * ac) + (pfc_left_product_original_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_product_original_rightcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_product_original_rightcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_product_original_rightcoefficientscoefficientdiagonal)) /\\ (((pfc_left_product_original_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_product_original_rightcoefficientscoefficientdiagonaltermrightinside. pfa_gap_product_original_rightcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_product_original_rightcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_product_original_rightcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_product_original_rightcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_product_original_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_product_original_rightcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_product_original_rightcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_product_original_rightcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_product_original_rightcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_product_original_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_product_original_rightcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_product_original_rightcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_product_original_rightcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_product_original_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_product_original_rightcoefficientscoefficientdiagonal)=pfc_left_product_original_rightcoefficientscoefficientdiagonalterm*pfc_right_product_original_rightcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_product_original_rightcoefficientscoefficientsum fs_v_pfc_product_original_rightcoefficientscoefficientsum. ((((exists fs_h_pfc_product_original_rightcoefficientscoefficientsum_body_start. fs_h_pfc_product_original_rightcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_product_original_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_product_original_rightcoefficientscoefficientsum_body_start. fs_u_pfc_product_original_rightcoefficientscoefficientsum = fs_q_pfc_product_original_rightcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_product_original_rightcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_product_original_rightcoefficientscoefficientsum_body_terminal. fs_h_pfc_product_original_rightcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_product_original_rightcoefficientscoefficient) = S ((S (S (pfc_index_product_original_rightcoefficients))) * fs_v_pfc_product_original_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_product_original_rightcoefficientscoefficientsum_body_terminal. fs_u_pfc_product_original_rightcoefficientscoefficientsum = fs_q_pfc_product_original_rightcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_product_original_rightcoefficients))) * fs_v_pfc_product_original_rightcoefficientscoefficientsum) + (pfc_natural_sum_product_original_rightcoefficientscoefficient))) /\\ forall fs_i_pfc_product_original_rightcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_product_original_rightcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_product_original_rightcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_product_original_rightcoefficientscoefficientsum_body_steps = S (pfc_index_product_original_rightcoefficients)) -> exists fs_a_pfc_product_original_rightcoefficientscoefficientsum_body_steps fs_r_pfc_product_original_rightcoefficientscoefficientsum_body_steps fs_s_pfc_product_original_rightcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_product_original_rightcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_product_original_rightcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_product_original_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_product_original_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_product_original_rightcoefficientscoefficient)) /\\ exists fs_q_pfc_product_original_rightcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_product_original_rightcoefficientscoefficient = fs_q_pfc_product_original_rightcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_product_original_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_product_original_rightcoefficientscoefficient) + (fs_a_pfc_product_original_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_product_original_rightcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_product_original_rightcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_product_original_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_product_original_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_product_original_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_product_original_rightcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_product_original_rightcoefficientscoefficientsum = fs_q_pfc_product_original_rightcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_product_original_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_product_original_rightcoefficientscoefficientsum) + (fs_r_pfc_product_original_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_product_original_rightcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_product_original_rightcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_product_original_rightcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_product_original_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_product_original_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_product_original_rightcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_product_original_rightcoefficientscoefficientsum = fs_q_pfc_product_original_rightcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_product_original_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_product_original_rightcoefficientscoefficientsum) + (fs_s_pfc_product_original_rightcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_product_original_rightcoefficientscoefficientsum_body_steps = fs_r_pfc_product_original_rightcoefficientscoefficientsum_body_steps + fs_a_pfc_product_original_rightcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_product_original_rightcoefficientscoefficientresiduebound. pfa_gap_product_original_rightcoefficientscoefficientresiduebound + S (pfc_value_product_original_rightcoefficients) = (p)) /\\ ((exists pfa_offset_left_product_original_rightcoefficientscoefficientresiduecongruence pfa_offset_right_product_original_rightcoefficientscoefficientresiduecongruence. (pfc_natural_sum_product_original_rightcoefficientscoefficient) + (p) * pfa_offset_left_product_original_rightcoefficientscoefficientresiduecongruence = (pfc_value_product_original_rightcoefficients) + (p) * pfa_offset_right_product_original_rightcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_product_padded_rightleft. (exists fom_gap_pfp_product_padded_rightleft_index_bound. fom_gap_pfp_product_padded_rightleft_index_bound + S (fom_index_pfp_product_padded_rightleft) = L) -> exists fom_value_pfp_product_padded_rightleft. ((((exists fom_beta_height_pfp_product_padded_rightleft_entry. fom_beta_height_pfp_product_padded_rightleft_entry + S (fom_value_pfp_product_padded_rightleft) = S ((S (fom_index_pfp_product_padded_rightleft)) * ac)) /\\ exists fom_beta_quotient_pfp_product_padded_rightleft_entry. ab = fom_beta_quotient_pfp_product_padded_rightleft_entry * S ((S (fom_index_pfp_product_padded_rightleft)) * ac) + (fom_value_pfp_product_padded_rightleft))) /\\ (exists fom_gap_pfp_product_padded_rightleft_value_bound. fom_gap_pfp_product_padded_rightleft_value_bound + S (fom_value_pfp_product_padded_rightleft) = p))) /\\ (((forall fom_index_pfp_product_padded_rightright. (exists fom_gap_pfp_product_padded_rightright_index_bound. fom_gap_pfp_product_padded_rightright_index_bound + S (fom_index_pfp_product_padded_rightright) = t+M) -> exists fom_value_pfp_product_padded_rightright. ((((exists fom_beta_height_pfp_product_padded_rightright_entry. fom_beta_height_pfp_product_padded_rightright_entry + S (fom_value_pfp_product_padded_rightright) = S ((S (fom_index_pfp_product_padded_rightright)) * BC)) /\\ exists fom_beta_quotient_pfp_product_padded_rightright_entry. BB = fom_beta_quotient_pfp_product_padded_rightright_entry * S ((S (fom_index_pfp_product_padded_rightright)) * BC) + (fom_value_pfp_product_padded_rightright))) /\\ (exists fom_gap_pfp_product_padded_rightright_value_bound. fom_gap_pfp_product_padded_rightright_value_bound + S (fom_value_pfp_product_padded_rightright) = p))) /\\ (((((((L)=0 \\/ (t+M)=0) /\\ (((K)=0)))) \\/ (((~((L)=0)) /\\ (((~((t+M)=0)) /\\ (((L)+(t+M)=S (K)))))))) /\\ ((forall pfc_index_product_padded_rightcoefficients. (exists pfa_gap_product_padded_rightcoefficientsbound. pfa_gap_product_padded_rightcoefficientsbound + S (pfc_index_product_padded_rightcoefficients) = (K)) -> exists pfc_value_product_padded_rightcoefficients. ((((exists ff_h_pfp_product_padded_rightcoefficientsentry. ff_h_pfp_product_padded_rightcoefficientsentry + S (pfc_value_product_padded_rightcoefficients) = S ((S (pfc_index_product_padded_rightcoefficients)) * CC)) /\\ exists ff_q_pfp_product_padded_rightcoefficientsentry. CB = ff_q_pfp_product_padded_rightcoefficientsentry * S ((S (pfc_index_product_padded_rightcoefficients)) * CC) + (pfc_value_product_padded_rightcoefficients))) /\\ ((exists pfc_terms_code_product_padded_rightcoefficientscoefficient pfc_terms_scale_product_padded_rightcoefficientscoefficient pfc_natural_sum_product_padded_rightcoefficientscoefficient. ((forall pfc_index_product_padded_rightcoefficientscoefficientdiagonal. (exists pfa_gap_product_padded_rightcoefficientscoefficientdiagonalbound. pfa_gap_product_padded_rightcoefficientscoefficientdiagonalbound + S (pfc_index_product_padded_rightcoefficientscoefficientdiagonal) = (S (pfc_index_product_padded_rightcoefficients))) -> exists pfc_value_product_padded_rightcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_product_padded_rightcoefficientscoefficientdiagonalentry. ff_h_pfp_product_padded_rightcoefficientscoefficientdiagonalentry + S (pfc_value_product_padded_rightcoefficientscoefficientdiagonal) = S ((S (pfc_index_product_padded_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_product_padded_rightcoefficientscoefficient)) /\\ exists ff_q_pfp_product_padded_rightcoefficientscoefficientdiagonalentry. pfc_terms_code_product_padded_rightcoefficientscoefficient = ff_q_pfp_product_padded_rightcoefficientscoefficientdiagonalentry * S ((S (pfc_index_product_padded_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_product_padded_rightcoefficientscoefficient) + (pfc_value_product_padded_rightcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_product_padded_rightcoefficientscoefficientdiagonalterm pfc_left_product_padded_rightcoefficientscoefficientdiagonalterm pfc_right_product_padded_rightcoefficientscoefficientdiagonalterm. (((pfc_index_product_padded_rightcoefficientscoefficientdiagonal)+pfc_complement_product_padded_rightcoefficientscoefficientdiagonalterm=(pfc_index_product_padded_rightcoefficients)) /\\ ((((((exists pfa_gap_product_padded_rightcoefficientscoefficientdiagonaltermleftinside. pfa_gap_product_padded_rightcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_product_padded_rightcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_product_padded_rightcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_product_padded_rightcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_product_padded_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_product_padded_rightcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_product_padded_rightcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_product_padded_rightcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_product_padded_rightcoefficientscoefficientdiagonal)) * ac) + (pfc_left_product_padded_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_product_padded_rightcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_product_padded_rightcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_product_padded_rightcoefficientscoefficientdiagonal)) /\\ (((pfc_left_product_padded_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_product_padded_rightcoefficientscoefficientdiagonaltermrightinside. pfa_gap_product_padded_rightcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_product_padded_rightcoefficientscoefficientdiagonalterm) = (t+M)) /\\ ((((exists ff_h_pfp_product_padded_rightcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_product_padded_rightcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_product_padded_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_product_padded_rightcoefficientscoefficientdiagonalterm)) * BC)) /\\ exists ff_q_pfp_product_padded_rightcoefficientscoefficientdiagonaltermrightentry. BB = ff_q_pfp_product_padded_rightcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_product_padded_rightcoefficientscoefficientdiagonalterm)) * BC) + (pfc_right_product_padded_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_product_padded_rightcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_product_padded_rightcoefficientscoefficientdiagonaltermrightoutside+(t+M)=(pfc_complement_product_padded_rightcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_product_padded_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_product_padded_rightcoefficientscoefficientdiagonal)=pfc_left_product_padded_rightcoefficientscoefficientdiagonalterm*pfc_right_product_padded_rightcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_product_padded_rightcoefficientscoefficientsum fs_v_pfc_product_padded_rightcoefficientscoefficientsum. ((((exists fs_h_pfc_product_padded_rightcoefficientscoefficientsum_body_start. fs_h_pfc_product_padded_rightcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_product_padded_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_product_padded_rightcoefficientscoefficientsum_body_start. fs_u_pfc_product_padded_rightcoefficientscoefficientsum = fs_q_pfc_product_padded_rightcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_product_padded_rightcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_product_padded_rightcoefficientscoefficientsum_body_terminal. fs_h_pfc_product_padded_rightcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_product_padded_rightcoefficientscoefficient) = S ((S (S (pfc_index_product_padded_rightcoefficients))) * fs_v_pfc_product_padded_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_product_padded_rightcoefficientscoefficientsum_body_terminal. fs_u_pfc_product_padded_rightcoefficientscoefficientsum = fs_q_pfc_product_padded_rightcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_product_padded_rightcoefficients))) * fs_v_pfc_product_padded_rightcoefficientscoefficientsum) + (pfc_natural_sum_product_padded_rightcoefficientscoefficient))) /\\ forall fs_i_pfc_product_padded_rightcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_product_padded_rightcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_product_padded_rightcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_product_padded_rightcoefficientscoefficientsum_body_steps = S (pfc_index_product_padded_rightcoefficients)) -> exists fs_a_pfc_product_padded_rightcoefficientscoefficientsum_body_steps fs_r_pfc_product_padded_rightcoefficientscoefficientsum_body_steps fs_s_pfc_product_padded_rightcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_product_padded_rightcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_product_padded_rightcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_product_padded_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_product_padded_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_product_padded_rightcoefficientscoefficient)) /\\ exists fs_q_pfc_product_padded_rightcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_product_padded_rightcoefficientscoefficient = fs_q_pfc_product_padded_rightcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_product_padded_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_product_padded_rightcoefficientscoefficient) + (fs_a_pfc_product_padded_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_product_padded_rightcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_product_padded_rightcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_product_padded_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_product_padded_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_product_padded_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_product_padded_rightcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_product_padded_rightcoefficientscoefficientsum = fs_q_pfc_product_padded_rightcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_product_padded_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_product_padded_rightcoefficientscoefficientsum) + (fs_r_pfc_product_padded_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_product_padded_rightcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_product_padded_rightcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_product_padded_rightcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_product_padded_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_product_padded_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_product_padded_rightcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_product_padded_rightcoefficientscoefficientsum = fs_q_pfc_product_padded_rightcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_product_padded_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_product_padded_rightcoefficientscoefficientsum) + (fs_s_pfc_product_padded_rightcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_product_padded_rightcoefficientscoefficientsum_body_steps = fs_r_pfc_product_padded_rightcoefficientscoefficientsum_body_steps + fs_a_pfc_product_padded_rightcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_product_padded_rightcoefficientscoefficientresiduebound. pfa_gap_product_padded_rightcoefficientscoefficientresiduebound + S (pfc_value_product_padded_rightcoefficients) = (p)) /\\ ((exists pfa_offset_left_product_padded_rightcoefficientscoefficientresiduecongruence pfa_offset_right_product_padded_rightcoefficientscoefficientresiduecongruence. (pfc_natural_sum_product_padded_rightcoefficientscoefficient) + (p) * pfa_offset_left_product_padded_rightcoefficientscoefficientresiduecongruence = (pfc_value_product_padded_rightcoefficients) + (p) * pfa_offset_right_product_padded_rightcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (forall pfrep_power_product_equivalent_right pfrep_left_product_equivalent_right pfrep_right_product_equivalent_right. ((exists pfrep_position_product_equivalent_rightfirst. ((pfrep_position_product_equivalent_rightfirst+S (pfrep_power_product_equivalent_right)=(N)) /\\ ((((exists ff_h_pfp_product_equivalent_rightfirstentry. ff_h_pfp_product_equivalent_rightfirstentry + S (pfrep_left_product_equivalent_right) = S ((S (pfrep_position_product_equivalent_rightfirst)) * cc)) /\\ exists ff_q_pfp_product_equivalent_rightfirstentry. cb = ff_q_pfp_product_equivalent_rightfirstentry * S ((S (pfrep_position_product_equivalent_rightfirst)) * cc) + (pfrep_left_product_equivalent_right)))))) \\/ (((exists pfrep_gap_product_equivalent_rightfirstoutside. pfrep_gap_product_equivalent_rightfirstoutside+(N)=(pfrep_power_product_equivalent_right)) /\\ (((pfrep_left_product_equivalent_right)=0))))) -> ((exists pfrep_position_product_equivalent_rightsecond. ((pfrep_position_product_equivalent_rightsecond+S (pfrep_power_product_equivalent_right)=(K)) /\\ ((((exists ff_h_pfp_product_equivalent_rightsecondentry. ff_h_pfp_product_equivalent_rightsecondentry + S (pfrep_right_product_equivalent_right) = S ((S (pfrep_position_product_equivalent_rightsecond)) * CC)) /\\ exists ff_q_pfp_product_equivalent_rightsecondentry. CB = ff_q_pfp_product_equivalent_rightsecondentry * S ((S (pfrep_position_product_equivalent_rightsecond)) * CC) + (pfrep_right_product_equivalent_right)))))) \\/ (((exists pfrep_gap_product_equivalent_rightsecondoutside. pfrep_gap_product_equivalent_rightsecondoutside+(K)=(pfrep_power_product_equivalent_right)) /\\ (((pfrep_right_product_equivalent_right)=0))))) -> pfrep_left_product_equivalent_right=pfrep_right_product_equivalent_right)", "statement_sha256": "d91d1f5b7bb88f00089654637c02ffa4381c38b95aaba8988c8cd74d6c8e1701", "summary": "Genuine leading-zero padding of the right factor preserves the formal polynomial product, including empty factors whose proper product lengths need not differ by the padding count.", "summary_sha256": "803fdb5cbf35e36e1e371d67be14dc0cab7a69f4b7b6134f4cd384d01658303e" }, { "alpha_v33_frontier_enrollment": { "body_receipt_sha256": "a6dc7caed11ad1fcedfb3be1931de3de935032b6f16ded7f3fe275ec3b14993e", "bundle_campaign": "polynomial-euclidean-division", 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"reviewed_constructive_campaign_contract", "selector": "document" }, { "document_sha256": "6ae667d8518e4dbe722bb08ad1b08715a0d282c2893e533c8133d770fe861dcf", "kind": "alpha_v33_complete_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/prime-field-polynomial-euclidean-division-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=366]" }, { "document_sha256": "cea85e5c595a021061fb50997df7ad489c8905ab563695209dadc8235f6762cb", "kind": "alpha_v33_live_original_kernel_and_lean_receipt", "path": "research/arithmetic-library/artifacts/alpha-v33-research-receipt-v1.json", "role": "fresh_release_proof_verification", "selector": "families[slug=polynomial-euclidean-division]" }, { "document_sha256": "41b9f387d88a5a4f0fe5ee2bd5578f37a27a4657b0a80f1a1a2cb5109f69a623", "kind": "sealed_alpha_v32_parent", "path": "artifacts/peano-library/alpha/catalog-v32.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "polynomial-euclidean-division", "logical_spec_sha256": "bcf2391490859d0c88a1e7f9da4b69771a8be576977b2f13645dec6ae2cf2317", "membership": "alpha_only", "name": "prime_field_polynomial_convolution_both_left_paddings_equivalent", "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 AB", "intro AC", "intro t", "intro BB", "intro BC", "intro s", "intro CB", "intro CC", "intro K", "intro hp", "intro hA", "intro hB", "intro hc", "intro hn", "have hcopy : ((forall fom_index_pfp_both_original_productleft. (exists fom_gap_pfp_both_original_productleft_index_bound. fom_gap_pfp_both_original_productleft_index_bound + S (fom_index_pfp_both_original_productleft) = L) -> exists fom_value_pfp_both_original_productleft. ((((exists fom_beta_height_pfp_both_original_productleft_entry. fom_beta_height_pfp_both_original_productleft_entry + S (fom_value_pfp_both_original_productleft) = S ((S (fom_index_pfp_both_original_productleft)) * ac)) /\\ exists fom_beta_quotient_pfp_both_original_productleft_entry. ab = fom_beta_quotient_pfp_both_original_productleft_entry * S ((S (fom_index_pfp_both_original_productleft)) * ac) + (fom_value_pfp_both_original_productleft))) /\\ (exists fom_gap_pfp_both_original_productleft_value_bound. fom_gap_pfp_both_original_productleft_value_bound + S (fom_value_pfp_both_original_productleft) = p))) /\\ (((forall fom_index_pfp_both_original_productright. (exists fom_gap_pfp_both_original_productright_index_bound. fom_gap_pfp_both_original_productright_index_bound + S (fom_index_pfp_both_original_productright) = M) -> exists fom_value_pfp_both_original_productright. ((((exists fom_beta_height_pfp_both_original_productright_entry. fom_beta_height_pfp_both_original_productright_entry + S (fom_value_pfp_both_original_productright) = S ((S (fom_index_pfp_both_original_productright)) * bc)) /\\ exists fom_beta_quotient_pfp_both_original_productright_entry. bb = fom_beta_quotient_pfp_both_original_productright_entry * S ((S (fom_index_pfp_both_original_productright)) * bc) + (fom_value_pfp_both_original_productright))) /\\ (exists fom_gap_pfp_both_original_productright_value_bound. fom_gap_pfp_both_original_productright_value_bound + S (fom_value_pfp_both_original_productright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_both_original_productcoefficients. (exists pfa_gap_both_original_productcoefficientsbound. pfa_gap_both_original_productcoefficientsbound + S (pfc_index_both_original_productcoefficients) = (N)) -> exists pfc_value_both_original_productcoefficients. ((((exists ff_h_pfp_both_original_productcoefficientsentry. ff_h_pfp_both_original_productcoefficientsentry + S (pfc_value_both_original_productcoefficients) = S ((S (pfc_index_both_original_productcoefficients)) * cc)) /\\ exists ff_q_pfp_both_original_productcoefficientsentry. cb = ff_q_pfp_both_original_productcoefficientsentry * S ((S (pfc_index_both_original_productcoefficients)) * cc) + (pfc_value_both_original_productcoefficients))) /\\ ((exists pfc_terms_code_both_original_productcoefficientscoefficient pfc_terms_scale_both_original_productcoefficientscoefficient pfc_natural_sum_both_original_productcoefficientscoefficient. ((forall pfc_index_both_original_productcoefficientscoefficientdiagonal. (exists pfa_gap_both_original_productcoefficientscoefficientdiagonalbound. pfa_gap_both_original_productcoefficientscoefficientdiagonalbound + S (pfc_index_both_original_productcoefficientscoefficientdiagonal) = (S (pfc_index_both_original_productcoefficients))) -> exists pfc_value_both_original_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_both_original_productcoefficientscoefficientdiagonalentry. ff_h_pfp_both_original_productcoefficientscoefficientdiagonalentry + S (pfc_value_both_original_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_both_original_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_both_original_productcoefficientscoefficient)) /\\ exists ff_q_pfp_both_original_productcoefficientscoefficientdiagonalentry. pfc_terms_code_both_original_productcoefficientscoefficient = ff_q_pfp_both_original_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_both_original_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_both_original_productcoefficientscoefficient) + (pfc_value_both_original_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_both_original_productcoefficientscoefficientdiagonalterm pfc_left_both_original_productcoefficientscoefficientdiagonalterm pfc_right_both_original_productcoefficientscoefficientdiagonalterm. (((pfc_index_both_original_productcoefficientscoefficientdiagonal)+pfc_complement_both_original_productcoefficientscoefficientdiagonalterm=(pfc_index_both_original_productcoefficients)) /\\ ((((((exists pfa_gap_both_original_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_both_original_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_both_original_productcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_both_original_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_both_original_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_both_original_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_both_original_productcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_both_original_productcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_both_original_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_both_original_productcoefficientscoefficientdiagonal)) * ac) + (pfc_left_both_original_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_both_original_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_both_original_productcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_both_original_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_both_original_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_both_original_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_both_original_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_both_original_productcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_both_original_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_both_original_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_both_original_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_both_original_productcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_both_original_productcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_both_original_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_both_original_productcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_both_original_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_both_original_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_both_original_productcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_both_original_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_both_original_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_both_original_productcoefficientscoefficientdiagonal)=pfc_left_both_original_productcoefficientscoefficientdiagonalterm*pfc_right_both_original_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_both_original_productcoefficientscoefficientsum fs_v_pfc_both_original_productcoefficientscoefficientsum. ((((exists fs_h_pfc_both_original_productcoefficientscoefficientsum_body_start. fs_h_pfc_both_original_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_both_original_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_original_productcoefficientscoefficientsum_body_start. fs_u_pfc_both_original_productcoefficientscoefficientsum = fs_q_pfc_both_original_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_both_original_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_both_original_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_both_original_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_both_original_productcoefficientscoefficient) = S ((S (S (pfc_index_both_original_productcoefficients))) * fs_v_pfc_both_original_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_original_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_both_original_productcoefficientscoefficientsum = fs_q_pfc_both_original_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_both_original_productcoefficients))) * fs_v_pfc_both_original_productcoefficientscoefficientsum) + (pfc_natural_sum_both_original_productcoefficientscoefficient))) /\\ forall fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_both_original_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_both_original_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps = S (pfc_index_both_original_productcoefficients)) -> exists fs_a_pfc_both_original_productcoefficientscoefficientsum_body_steps fs_r_pfc_both_original_productcoefficientscoefficientsum_body_steps fs_s_pfc_both_original_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_both_original_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_both_original_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_both_original_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_both_original_productcoefficientscoefficient)) /\\ exists fs_q_pfc_both_original_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_both_original_productcoefficientscoefficient = fs_q_pfc_both_original_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_both_original_productcoefficientscoefficient) + (fs_a_pfc_both_original_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_both_original_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_both_original_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_both_original_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_original_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_original_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_both_original_productcoefficientscoefficientsum = fs_q_pfc_both_original_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_original_productcoefficientscoefficientsum) + (fs_r_pfc_both_original_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_both_original_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_both_original_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_both_original_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_original_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_original_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_both_original_productcoefficientscoefficientsum = fs_q_pfc_both_original_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_original_productcoefficientscoefficientsum) + (fs_s_pfc_both_original_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_both_original_productcoefficientscoefficientsum_body_steps = fs_r_pfc_both_original_productcoefficientscoefficientsum_body_steps + fs_a_pfc_both_original_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_both_original_productcoefficientscoefficientresiduebound. pfa_gap_both_original_productcoefficientscoefficientresiduebound + S (pfc_value_both_original_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_both_original_productcoefficientscoefficientresiduecongruence pfa_offset_right_both_original_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_both_original_productcoefficientscoefficient) + (p) * pfa_offset_left_both_original_productcoefficientscoefficientresiduecongruence = (pfc_value_both_original_productcoefficients) + (p) * pfa_offset_right_both_original_productcoefficientscoefficientresiduecongruence))))))))))))))))))", "exact hc", "have hnewcopy : ((forall fom_index_pfp_both_padded_productleft. (exists fom_gap_pfp_both_padded_productleft_index_bound. fom_gap_pfp_both_padded_productleft_index_bound + S (fom_index_pfp_both_padded_productleft) = t+L) -> exists fom_value_pfp_both_padded_productleft. ((((exists fom_beta_height_pfp_both_padded_productleft_entry. fom_beta_height_pfp_both_padded_productleft_entry + S (fom_value_pfp_both_padded_productleft) = S ((S (fom_index_pfp_both_padded_productleft)) * AC)) /\\ exists fom_beta_quotient_pfp_both_padded_productleft_entry. AB = fom_beta_quotient_pfp_both_padded_productleft_entry * S ((S (fom_index_pfp_both_padded_productleft)) * AC) + (fom_value_pfp_both_padded_productleft))) /\\ (exists fom_gap_pfp_both_padded_productleft_value_bound. fom_gap_pfp_both_padded_productleft_value_bound + S (fom_value_pfp_both_padded_productleft) = p))) /\\ (((forall fom_index_pfp_both_padded_productright. (exists fom_gap_pfp_both_padded_productright_index_bound. fom_gap_pfp_both_padded_productright_index_bound + S (fom_index_pfp_both_padded_productright) = s+M) -> exists fom_value_pfp_both_padded_productright. ((((exists fom_beta_height_pfp_both_padded_productright_entry. fom_beta_height_pfp_both_padded_productright_entry + S (fom_value_pfp_both_padded_productright) = S ((S (fom_index_pfp_both_padded_productright)) * BC)) /\\ exists fom_beta_quotient_pfp_both_padded_productright_entry. BB = fom_beta_quotient_pfp_both_padded_productright_entry * S ((S (fom_index_pfp_both_padded_productright)) * BC) + (fom_value_pfp_both_padded_productright))) /\\ (exists fom_gap_pfp_both_padded_productright_value_bound. fom_gap_pfp_both_padded_productright_value_bound + S (fom_value_pfp_both_padded_productright) = p))) /\\ (((((((t+L)=0 \\/ (s+M)=0) /\\ (((K)=0)))) \\/ (((~((t+L)=0)) /\\ (((~((s+M)=0)) /\\ (((t+L)+(s+M)=S (K)))))))) /\\ ((forall pfc_index_both_padded_productcoefficients. (exists pfa_gap_both_padded_productcoefficientsbound. pfa_gap_both_padded_productcoefficientsbound + S (pfc_index_both_padded_productcoefficients) = (K)) -> exists pfc_value_both_padded_productcoefficients. ((((exists ff_h_pfp_both_padded_productcoefficientsentry. ff_h_pfp_both_padded_productcoefficientsentry + S (pfc_value_both_padded_productcoefficients) = S ((S (pfc_index_both_padded_productcoefficients)) * CC)) /\\ exists ff_q_pfp_both_padded_productcoefficientsentry. CB = ff_q_pfp_both_padded_productcoefficientsentry * S ((S (pfc_index_both_padded_productcoefficients)) * CC) + (pfc_value_both_padded_productcoefficients))) /\\ ((exists pfc_terms_code_both_padded_productcoefficientscoefficient pfc_terms_scale_both_padded_productcoefficientscoefficient pfc_natural_sum_both_padded_productcoefficientscoefficient. ((forall pfc_index_both_padded_productcoefficientscoefficientdiagonal. (exists pfa_gap_both_padded_productcoefficientscoefficientdiagonalbound. pfa_gap_both_padded_productcoefficientscoefficientdiagonalbound + S (pfc_index_both_padded_productcoefficientscoefficientdiagonal) = (S (pfc_index_both_padded_productcoefficients))) -> exists pfc_value_both_padded_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_both_padded_productcoefficientscoefficientdiagonalentry. ff_h_pfp_both_padded_productcoefficientscoefficientdiagonalentry + S (pfc_value_both_padded_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_both_padded_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_both_padded_productcoefficientscoefficient)) /\\ exists ff_q_pfp_both_padded_productcoefficientscoefficientdiagonalentry. pfc_terms_code_both_padded_productcoefficientscoefficient = ff_q_pfp_both_padded_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_both_padded_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_both_padded_productcoefficientscoefficient) + (pfc_value_both_padded_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_both_padded_productcoefficientscoefficientdiagonalterm pfc_left_both_padded_productcoefficientscoefficientdiagonalterm pfc_right_both_padded_productcoefficientscoefficientdiagonalterm. (((pfc_index_both_padded_productcoefficientscoefficientdiagonal)+pfc_complement_both_padded_productcoefficientscoefficientdiagonalterm=(pfc_index_both_padded_productcoefficients)) /\\ ((((((exists pfa_gap_both_padded_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_both_padded_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_both_padded_productcoefficientscoefficientdiagonal) = (t+L)) /\\ ((((exists ff_h_pfp_both_padded_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_both_padded_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_both_padded_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_both_padded_productcoefficientscoefficientdiagonal)) * AC)) /\\ exists ff_q_pfp_both_padded_productcoefficientscoefficientdiagonaltermleftentry. AB = ff_q_pfp_both_padded_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_both_padded_productcoefficientscoefficientdiagonal)) * AC) + (pfc_left_both_padded_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_both_padded_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_both_padded_productcoefficientscoefficientdiagonaltermleftoutside+(t+L)=(pfc_index_both_padded_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_both_padded_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_both_padded_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_both_padded_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_both_padded_productcoefficientscoefficientdiagonalterm) = (s+M)) /\\ ((((exists ff_h_pfp_both_padded_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_both_padded_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_both_padded_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_both_padded_productcoefficientscoefficientdiagonalterm)) * BC)) /\\ exists ff_q_pfp_both_padded_productcoefficientscoefficientdiagonaltermrightentry. BB = ff_q_pfp_both_padded_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_both_padded_productcoefficientscoefficientdiagonalterm)) * BC) + (pfc_right_both_padded_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_both_padded_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_both_padded_productcoefficientscoefficientdiagonaltermrightoutside+(s+M)=(pfc_complement_both_padded_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_both_padded_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_both_padded_productcoefficientscoefficientdiagonal)=pfc_left_both_padded_productcoefficientscoefficientdiagonalterm*pfc_right_both_padded_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_both_padded_productcoefficientscoefficientsum fs_v_pfc_both_padded_productcoefficientscoefficientsum. ((((exists fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_start. fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_both_padded_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_start. fs_u_pfc_both_padded_productcoefficientscoefficientsum = fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_both_padded_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_both_padded_productcoefficientscoefficient) = S ((S (S (pfc_index_both_padded_productcoefficients))) * fs_v_pfc_both_padded_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_both_padded_productcoefficientscoefficientsum = fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_both_padded_productcoefficients))) * fs_v_pfc_both_padded_productcoefficientscoefficientsum) + (pfc_natural_sum_both_padded_productcoefficientscoefficient))) /\\ forall fs_i_pfc_both_padded_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_both_padded_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_both_padded_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_both_padded_productcoefficientscoefficientsum_body_steps = S (pfc_index_both_padded_productcoefficients)) -> exists fs_a_pfc_both_padded_productcoefficientscoefficientsum_body_steps fs_r_pfc_both_padded_productcoefficientscoefficientsum_body_steps fs_s_pfc_both_padded_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_both_padded_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_both_padded_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_both_padded_productcoefficientscoefficient)) /\\ exists fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_both_padded_productcoefficientscoefficient = fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_both_padded_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_both_padded_productcoefficientscoefficient) + (fs_a_pfc_both_padded_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_both_padded_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_both_padded_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_padded_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_both_padded_productcoefficientscoefficientsum = fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_both_padded_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_padded_productcoefficientscoefficientsum) + (fs_r_pfc_both_padded_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_both_padded_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_both_padded_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_padded_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_both_padded_productcoefficientscoefficientsum = fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_both_padded_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_padded_productcoefficientscoefficientsum) + (fs_s_pfc_both_padded_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_both_padded_productcoefficientscoefficientsum_body_steps = fs_r_pfc_both_padded_productcoefficientscoefficientsum_body_steps + fs_a_pfc_both_padded_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_both_padded_productcoefficientscoefficientresiduebound. pfa_gap_both_padded_productcoefficientscoefficientresiduebound + S (pfc_value_both_padded_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_both_padded_productcoefficientscoefficientresiduecongruence pfa_offset_right_both_padded_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_both_padded_productcoefficientscoefficient) + (p) * pfa_offset_left_both_padded_productcoefficientscoefficientresiduecongruence = (pfc_value_both_padded_productcoefficients) + (p) * pfa_offset_right_both_padded_productcoefficientscoefficientresiduecongruence))))))))))))))))))", "exact hn", "cases hcopy", "cases hcopy_right", "cases hcopy_right_right", "cases hnewcopy", "cases hnewcopy_right", "cases hnewcopy_right_right", "have hlength : exists J. ((((t+L)=0 \\/ (M)=0) /\\ (((J)=0)))) \\/ (((~((t+L)=0)) /\\ (((~((M)=0)) /\\ (((t+L)+(M)=S (J)))))))", "specialize polynomial_product_length_exists (t+L)", "specialize polynomial_product_length_exists (M)", "apply polynomial_product_length_exists", "cases hlength", "have hmiddle : exists db dc. ((forall fom_index_pfp_both_middle_productleft. (exists fom_gap_pfp_both_middle_productleft_index_bound. fom_gap_pfp_both_middle_productleft_index_bound + S (fom_index_pfp_both_middle_productleft) = t+L) -> exists fom_value_pfp_both_middle_productleft. ((((exists fom_beta_height_pfp_both_middle_productleft_entry. fom_beta_height_pfp_both_middle_productleft_entry + S (fom_value_pfp_both_middle_productleft) = S ((S (fom_index_pfp_both_middle_productleft)) * AC)) /\\ exists fom_beta_quotient_pfp_both_middle_productleft_entry. AB = fom_beta_quotient_pfp_both_middle_productleft_entry * S ((S (fom_index_pfp_both_middle_productleft)) * AC) + (fom_value_pfp_both_middle_productleft))) /\\ (exists fom_gap_pfp_both_middle_productleft_value_bound. fom_gap_pfp_both_middle_productleft_value_bound + S (fom_value_pfp_both_middle_productleft) = p))) /\\ (((forall fom_index_pfp_both_middle_productright. (exists fom_gap_pfp_both_middle_productright_index_bound. fom_gap_pfp_both_middle_productright_index_bound + S (fom_index_pfp_both_middle_productright) = M) -> exists fom_value_pfp_both_middle_productright. ((((exists fom_beta_height_pfp_both_middle_productright_entry. fom_beta_height_pfp_both_middle_productright_entry + S (fom_value_pfp_both_middle_productright) = S ((S (fom_index_pfp_both_middle_productright)) * bc)) /\\ exists fom_beta_quotient_pfp_both_middle_productright_entry. bb = fom_beta_quotient_pfp_both_middle_productright_entry * S ((S (fom_index_pfp_both_middle_productright)) * bc) + (fom_value_pfp_both_middle_productright))) /\\ (exists fom_gap_pfp_both_middle_productright_value_bound. fom_gap_pfp_both_middle_productright_value_bound + S (fom_value_pfp_both_middle_productright) = p))) /\\ (((((((t+L)=0 \\/ (M)=0) /\\ (((x)=0)))) \\/ (((~((t+L)=0)) /\\ (((~((M)=0)) /\\ (((t+L)+(M)=S (x)))))))) /\\ ((forall pfc_index_both_middle_productcoefficients. (exists pfa_gap_both_middle_productcoefficientsbound. pfa_gap_both_middle_productcoefficientsbound + S (pfc_index_both_middle_productcoefficients) = (x)) -> exists pfc_value_both_middle_productcoefficients. ((((exists ff_h_pfp_both_middle_productcoefficientsentry. ff_h_pfp_both_middle_productcoefficientsentry + S (pfc_value_both_middle_productcoefficients) = S ((S (pfc_index_both_middle_productcoefficients)) * dc)) /\\ exists ff_q_pfp_both_middle_productcoefficientsentry. db = ff_q_pfp_both_middle_productcoefficientsentry * S ((S (pfc_index_both_middle_productcoefficients)) * dc) + (pfc_value_both_middle_productcoefficients))) /\\ ((exists pfc_terms_code_both_middle_productcoefficientscoefficient pfc_terms_scale_both_middle_productcoefficientscoefficient pfc_natural_sum_both_middle_productcoefficientscoefficient. ((forall pfc_index_both_middle_productcoefficientscoefficientdiagonal. (exists pfa_gap_both_middle_productcoefficientscoefficientdiagonalbound. pfa_gap_both_middle_productcoefficientscoefficientdiagonalbound + S (pfc_index_both_middle_productcoefficientscoefficientdiagonal) = (S (pfc_index_both_middle_productcoefficients))) -> exists pfc_value_both_middle_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_both_middle_productcoefficientscoefficientdiagonalentry. ff_h_pfp_both_middle_productcoefficientscoefficientdiagonalentry + S (pfc_value_both_middle_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_both_middle_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_both_middle_productcoefficientscoefficient)) /\\ exists ff_q_pfp_both_middle_productcoefficientscoefficientdiagonalentry. pfc_terms_code_both_middle_productcoefficientscoefficient = ff_q_pfp_both_middle_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_both_middle_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_both_middle_productcoefficientscoefficient) + (pfc_value_both_middle_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_both_middle_productcoefficientscoefficientdiagonalterm pfc_left_both_middle_productcoefficientscoefficientdiagonalterm pfc_right_both_middle_productcoefficientscoefficientdiagonalterm. (((pfc_index_both_middle_productcoefficientscoefficientdiagonal)+pfc_complement_both_middle_productcoefficientscoefficientdiagonalterm=(pfc_index_both_middle_productcoefficients)) /\\ ((((((exists pfa_gap_both_middle_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_both_middle_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_both_middle_productcoefficientscoefficientdiagonal) = (t+L)) /\\ ((((exists ff_h_pfp_both_middle_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_both_middle_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_both_middle_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_both_middle_productcoefficientscoefficientdiagonal)) * AC)) /\\ exists ff_q_pfp_both_middle_productcoefficientscoefficientdiagonaltermleftentry. AB = ff_q_pfp_both_middle_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_both_middle_productcoefficientscoefficientdiagonal)) * AC) + (pfc_left_both_middle_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_both_middle_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_both_middle_productcoefficientscoefficientdiagonaltermleftoutside+(t+L)=(pfc_index_both_middle_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_both_middle_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_both_middle_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_both_middle_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_both_middle_productcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_both_middle_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_both_middle_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_both_middle_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_both_middle_productcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_both_middle_productcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_both_middle_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_both_middle_productcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_both_middle_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_both_middle_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_both_middle_productcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_both_middle_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_both_middle_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_both_middle_productcoefficientscoefficientdiagonal)=pfc_left_both_middle_productcoefficientscoefficientdiagonalterm*pfc_right_both_middle_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_both_middle_productcoefficientscoefficientsum fs_v_pfc_both_middle_productcoefficientscoefficientsum. ((((exists fs_h_pfc_both_middle_productcoefficientscoefficientsum_body_start. fs_h_pfc_both_middle_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_both_middle_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_middle_productcoefficientscoefficientsum_body_start. fs_u_pfc_both_middle_productcoefficientscoefficientsum = fs_q_pfc_both_middle_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_both_middle_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_both_middle_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_both_middle_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_both_middle_productcoefficientscoefficient) = S ((S (S (pfc_index_both_middle_productcoefficients))) * fs_v_pfc_both_middle_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_middle_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_both_middle_productcoefficientscoefficientsum = fs_q_pfc_both_middle_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_both_middle_productcoefficients))) * fs_v_pfc_both_middle_productcoefficientscoefficientsum) + (pfc_natural_sum_both_middle_productcoefficientscoefficient))) /\\ forall fs_i_pfc_both_middle_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_both_middle_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_both_middle_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_both_middle_productcoefficientscoefficientsum_body_steps = S (pfc_index_both_middle_productcoefficients)) -> exists fs_a_pfc_both_middle_productcoefficientscoefficientsum_body_steps fs_r_pfc_both_middle_productcoefficientscoefficientsum_body_steps fs_s_pfc_both_middle_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_both_middle_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_both_middle_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_both_middle_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_both_middle_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_both_middle_productcoefficientscoefficient)) /\\ exists fs_q_pfc_both_middle_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_both_middle_productcoefficientscoefficient = fs_q_pfc_both_middle_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_both_middle_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_both_middle_productcoefficientscoefficient) + (fs_a_pfc_both_middle_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_both_middle_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_both_middle_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_both_middle_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_both_middle_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_middle_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_middle_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_both_middle_productcoefficientscoefficientsum = fs_q_pfc_both_middle_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_both_middle_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_middle_productcoefficientscoefficientsum) + (fs_r_pfc_both_middle_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_both_middle_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_both_middle_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_both_middle_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_both_middle_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_middle_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_middle_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_both_middle_productcoefficientscoefficientsum = fs_q_pfc_both_middle_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_both_middle_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_middle_productcoefficientscoefficientsum) + (fs_s_pfc_both_middle_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_both_middle_productcoefficientscoefficientsum_body_steps = fs_r_pfc_both_middle_productcoefficientscoefficientsum_body_steps + fs_a_pfc_both_middle_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_both_middle_productcoefficientscoefficientresiduebound. pfa_gap_both_middle_productcoefficientscoefficientresiduebound + S (pfc_value_both_middle_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_both_middle_productcoefficientscoefficientresiduecongruence pfa_offset_right_both_middle_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_both_middle_productcoefficientscoefficient) + (p) * pfa_offset_left_both_middle_productcoefficientscoefficientresiduecongruence = (pfc_value_both_middle_productcoefficients) + (p) * pfa_offset_right_both_middle_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 (t+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 (M)", "specialize prime_field_polynomial_convolution_at_length_exists (x)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hp", "exact hnewcopy_left", "exact hcopy_right_left", "exact hlength_witness", "cases hmiddle", "cases hmiddle_witness", "specialize prime_field_polynomial_equivalent_transitive (cb)", "specialize prime_field_polynomial_equivalent_transitive (cc)", "specialize prime_field_polynomial_equivalent_transitive (N)", "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 (CB)", "specialize prime_field_polynomial_equivalent_transitive (CC)", "specialize prime_field_polynomial_equivalent_transitive (K)", "apply prime_field_polynomial_equivalent_transitive", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (p)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (ab)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (ac)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (L)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (bb)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (bc)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (M)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (cb)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (cc)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (N)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (AB)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (AC)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (t)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (x1)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (x2)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (x)", "apply prime_field_polynomial_convolution_left_padding_equivalent_left", "exact hp", "exact hA", "exact hc", "exact hmiddle_witness_witness", "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 (t+L)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (bb)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (bc)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (M)", "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 (BB)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (BC)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (s)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (CB)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (CC)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (K)", "apply prime_field_polynomial_convolution_left_padding_equivalent_right", "exact hp", "exact hB", "exact hmiddle_witness_witness", "exact hn" ], "script_sha256": "8ed2614d011f76a8e210592d62f9acab1f74da45d0f7f8c4eaf331549b84297d", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_convolution_padding_candidate.py", "sha256": "2d874ecfb35a5db0aecdeb07b549464efebad9072c363113aa5a0a977845d007" }, "statement": "forall p ab ac L bb bc M cb cc N AB AC t BB BC s CB CC K. (~(p=0)) -> (((forall pfp_repeat_index_both_left_paddingzeros. (exists pfa_gap_both_left_paddingzerosindex. pfa_gap_both_left_paddingzerosindex + S (pfp_repeat_index_both_left_paddingzeros) = (t)) -> (((exists ff_h_pfp_both_left_paddingzerosentry. ff_h_pfp_both_left_paddingzerosentry + S (0) = S ((S (pfp_repeat_index_both_left_paddingzeros)) * AC)) /\\ exists ff_q_pfp_both_left_paddingzerosentry. AB = ff_q_pfp_both_left_paddingzerosentry * S ((S (pfp_repeat_index_both_left_paddingzeros)) * AC) + (0)))) /\\ ((forall pfrep_index_both_left_padding pfrep_value_both_left_padding. (exists pfa_gap_both_left_paddingbound. pfa_gap_both_left_paddingbound + S (pfrep_index_both_left_padding) = (L)) -> (((exists ff_h_pfp_both_left_paddinginput. ff_h_pfp_both_left_paddinginput + S (pfrep_value_both_left_padding) = S ((S (pfrep_index_both_left_padding)) * ac)) /\\ exists ff_q_pfp_both_left_paddinginput. ab = ff_q_pfp_both_left_paddinginput * S ((S (pfrep_index_both_left_padding)) * ac) + (pfrep_value_both_left_padding))) -> (((exists ff_h_pfp_both_left_paddingoutput. ff_h_pfp_both_left_paddingoutput + S (pfrep_value_both_left_padding) = S ((S ((t)+pfrep_index_both_left_padding)) * AC)) /\\ exists ff_q_pfp_both_left_paddingoutput. AB = ff_q_pfp_both_left_paddingoutput * S ((S ((t)+pfrep_index_both_left_padding)) * AC) + (pfrep_value_both_left_padding))))))) -> (((forall pfp_repeat_index_both_right_paddingzeros. (exists pfa_gap_both_right_paddingzerosindex. pfa_gap_both_right_paddingzerosindex + S (pfp_repeat_index_both_right_paddingzeros) = (s)) -> (((exists ff_h_pfp_both_right_paddingzerosentry. ff_h_pfp_both_right_paddingzerosentry + S (0) = S ((S (pfp_repeat_index_both_right_paddingzeros)) * BC)) /\\ exists ff_q_pfp_both_right_paddingzerosentry. BB = ff_q_pfp_both_right_paddingzerosentry * S ((S (pfp_repeat_index_both_right_paddingzeros)) * BC) + (0)))) /\\ ((forall pfrep_index_both_right_padding pfrep_value_both_right_padding. (exists pfa_gap_both_right_paddingbound. pfa_gap_both_right_paddingbound + S (pfrep_index_both_right_padding) = (M)) -> (((exists ff_h_pfp_both_right_paddinginput. ff_h_pfp_both_right_paddinginput + S (pfrep_value_both_right_padding) = S ((S (pfrep_index_both_right_padding)) * bc)) /\\ exists ff_q_pfp_both_right_paddinginput. bb = ff_q_pfp_both_right_paddinginput * S ((S (pfrep_index_both_right_padding)) * bc) + (pfrep_value_both_right_padding))) -> (((exists ff_h_pfp_both_right_paddingoutput. ff_h_pfp_both_right_paddingoutput + S (pfrep_value_both_right_padding) = S ((S ((s)+pfrep_index_both_right_padding)) * BC)) /\\ exists ff_q_pfp_both_right_paddingoutput. BB = ff_q_pfp_both_right_paddingoutput * S ((S ((s)+pfrep_index_both_right_padding)) * BC) + (pfrep_value_both_right_padding))))))) -> (((forall fom_index_pfp_both_original_productleft. (exists fom_gap_pfp_both_original_productleft_index_bound. fom_gap_pfp_both_original_productleft_index_bound + S (fom_index_pfp_both_original_productleft) = L) -> exists fom_value_pfp_both_original_productleft. ((((exists fom_beta_height_pfp_both_original_productleft_entry. fom_beta_height_pfp_both_original_productleft_entry + S (fom_value_pfp_both_original_productleft) = S ((S (fom_index_pfp_both_original_productleft)) * ac)) /\\ exists fom_beta_quotient_pfp_both_original_productleft_entry. ab = fom_beta_quotient_pfp_both_original_productleft_entry * S ((S (fom_index_pfp_both_original_productleft)) * ac) + (fom_value_pfp_both_original_productleft))) /\\ (exists fom_gap_pfp_both_original_productleft_value_bound. fom_gap_pfp_both_original_productleft_value_bound + S (fom_value_pfp_both_original_productleft) = p))) /\\ (((forall fom_index_pfp_both_original_productright. (exists fom_gap_pfp_both_original_productright_index_bound. fom_gap_pfp_both_original_productright_index_bound + S (fom_index_pfp_both_original_productright) = M) -> exists fom_value_pfp_both_original_productright. ((((exists fom_beta_height_pfp_both_original_productright_entry. fom_beta_height_pfp_both_original_productright_entry + S (fom_value_pfp_both_original_productright) = S ((S (fom_index_pfp_both_original_productright)) * bc)) /\\ exists fom_beta_quotient_pfp_both_original_productright_entry. bb = fom_beta_quotient_pfp_both_original_productright_entry * S ((S (fom_index_pfp_both_original_productright)) * bc) + (fom_value_pfp_both_original_productright))) /\\ (exists fom_gap_pfp_both_original_productright_value_bound. fom_gap_pfp_both_original_productright_value_bound + S (fom_value_pfp_both_original_productright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_both_original_productcoefficients. (exists pfa_gap_both_original_productcoefficientsbound. pfa_gap_both_original_productcoefficientsbound + S (pfc_index_both_original_productcoefficients) = (N)) -> exists pfc_value_both_original_productcoefficients. ((((exists ff_h_pfp_both_original_productcoefficientsentry. ff_h_pfp_both_original_productcoefficientsentry + S (pfc_value_both_original_productcoefficients) = S ((S (pfc_index_both_original_productcoefficients)) * cc)) /\\ exists ff_q_pfp_both_original_productcoefficientsentry. cb = ff_q_pfp_both_original_productcoefficientsentry * S ((S (pfc_index_both_original_productcoefficients)) * cc) + (pfc_value_both_original_productcoefficients))) /\\ ((exists pfc_terms_code_both_original_productcoefficientscoefficient pfc_terms_scale_both_original_productcoefficientscoefficient pfc_natural_sum_both_original_productcoefficientscoefficient. ((forall pfc_index_both_original_productcoefficientscoefficientdiagonal. (exists pfa_gap_both_original_productcoefficientscoefficientdiagonalbound. pfa_gap_both_original_productcoefficientscoefficientdiagonalbound + S (pfc_index_both_original_productcoefficientscoefficientdiagonal) = (S (pfc_index_both_original_productcoefficients))) -> exists pfc_value_both_original_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_both_original_productcoefficientscoefficientdiagonalentry. ff_h_pfp_both_original_productcoefficientscoefficientdiagonalentry + S (pfc_value_both_original_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_both_original_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_both_original_productcoefficientscoefficient)) /\\ exists ff_q_pfp_both_original_productcoefficientscoefficientdiagonalentry. pfc_terms_code_both_original_productcoefficientscoefficient = ff_q_pfp_both_original_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_both_original_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_both_original_productcoefficientscoefficient) + (pfc_value_both_original_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_both_original_productcoefficientscoefficientdiagonalterm pfc_left_both_original_productcoefficientscoefficientdiagonalterm pfc_right_both_original_productcoefficientscoefficientdiagonalterm. (((pfc_index_both_original_productcoefficientscoefficientdiagonal)+pfc_complement_both_original_productcoefficientscoefficientdiagonalterm=(pfc_index_both_original_productcoefficients)) /\\ ((((((exists pfa_gap_both_original_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_both_original_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_both_original_productcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_both_original_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_both_original_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_both_original_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_both_original_productcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_both_original_productcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_both_original_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_both_original_productcoefficientscoefficientdiagonal)) * ac) + (pfc_left_both_original_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_both_original_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_both_original_productcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_both_original_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_both_original_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_both_original_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_both_original_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_both_original_productcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_both_original_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_both_original_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_both_original_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_both_original_productcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_both_original_productcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_both_original_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_both_original_productcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_both_original_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_both_original_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_both_original_productcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_both_original_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_both_original_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_both_original_productcoefficientscoefficientdiagonal)=pfc_left_both_original_productcoefficientscoefficientdiagonalterm*pfc_right_both_original_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_both_original_productcoefficientscoefficientsum fs_v_pfc_both_original_productcoefficientscoefficientsum. ((((exists fs_h_pfc_both_original_productcoefficientscoefficientsum_body_start. fs_h_pfc_both_original_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_both_original_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_original_productcoefficientscoefficientsum_body_start. fs_u_pfc_both_original_productcoefficientscoefficientsum = fs_q_pfc_both_original_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_both_original_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_both_original_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_both_original_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_both_original_productcoefficientscoefficient) = S ((S (S (pfc_index_both_original_productcoefficients))) * fs_v_pfc_both_original_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_original_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_both_original_productcoefficientscoefficientsum = fs_q_pfc_both_original_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_both_original_productcoefficients))) * fs_v_pfc_both_original_productcoefficientscoefficientsum) + (pfc_natural_sum_both_original_productcoefficientscoefficient))) /\\ forall fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_both_original_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_both_original_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps = S (pfc_index_both_original_productcoefficients)) -> exists fs_a_pfc_both_original_productcoefficientscoefficientsum_body_steps fs_r_pfc_both_original_productcoefficientscoefficientsum_body_steps fs_s_pfc_both_original_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_both_original_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_both_original_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_both_original_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_both_original_productcoefficientscoefficient)) /\\ exists fs_q_pfc_both_original_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_both_original_productcoefficientscoefficient = fs_q_pfc_both_original_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_both_original_productcoefficientscoefficient) + (fs_a_pfc_both_original_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_both_original_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_both_original_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_both_original_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_original_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_original_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_both_original_productcoefficientscoefficientsum = fs_q_pfc_both_original_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_original_productcoefficientscoefficientsum) + (fs_r_pfc_both_original_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_both_original_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_both_original_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_both_original_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_original_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_original_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_both_original_productcoefficientscoefficientsum = fs_q_pfc_both_original_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_original_productcoefficientscoefficientsum) + (fs_s_pfc_both_original_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_both_original_productcoefficientscoefficientsum_body_steps = fs_r_pfc_both_original_productcoefficientscoefficientsum_body_steps + fs_a_pfc_both_original_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_both_original_productcoefficientscoefficientresiduebound. pfa_gap_both_original_productcoefficientscoefficientresiduebound + S (pfc_value_both_original_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_both_original_productcoefficientscoefficientresiduecongruence pfa_offset_right_both_original_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_both_original_productcoefficientscoefficient) + (p) * pfa_offset_left_both_original_productcoefficientscoefficientresiduecongruence = (pfc_value_both_original_productcoefficients) + (p) * pfa_offset_right_both_original_productcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_both_padded_productleft. (exists fom_gap_pfp_both_padded_productleft_index_bound. fom_gap_pfp_both_padded_productleft_index_bound + S (fom_index_pfp_both_padded_productleft) = t+L) -> exists fom_value_pfp_both_padded_productleft. ((((exists fom_beta_height_pfp_both_padded_productleft_entry. fom_beta_height_pfp_both_padded_productleft_entry + S (fom_value_pfp_both_padded_productleft) = S ((S (fom_index_pfp_both_padded_productleft)) * AC)) /\\ exists fom_beta_quotient_pfp_both_padded_productleft_entry. AB = fom_beta_quotient_pfp_both_padded_productleft_entry * S ((S (fom_index_pfp_both_padded_productleft)) * AC) + (fom_value_pfp_both_padded_productleft))) /\\ (exists fom_gap_pfp_both_padded_productleft_value_bound. fom_gap_pfp_both_padded_productleft_value_bound + S (fom_value_pfp_both_padded_productleft) = p))) /\\ (((forall fom_index_pfp_both_padded_productright. (exists fom_gap_pfp_both_padded_productright_index_bound. fom_gap_pfp_both_padded_productright_index_bound + S (fom_index_pfp_both_padded_productright) = s+M) -> exists fom_value_pfp_both_padded_productright. ((((exists fom_beta_height_pfp_both_padded_productright_entry. fom_beta_height_pfp_both_padded_productright_entry + S (fom_value_pfp_both_padded_productright) = S ((S (fom_index_pfp_both_padded_productright)) * BC)) /\\ exists fom_beta_quotient_pfp_both_padded_productright_entry. BB = fom_beta_quotient_pfp_both_padded_productright_entry * S ((S (fom_index_pfp_both_padded_productright)) * BC) + (fom_value_pfp_both_padded_productright))) /\\ (exists fom_gap_pfp_both_padded_productright_value_bound. fom_gap_pfp_both_padded_productright_value_bound + S (fom_value_pfp_both_padded_productright) = p))) /\\ (((((((t+L)=0 \\/ (s+M)=0) /\\ (((K)=0)))) \\/ (((~((t+L)=0)) /\\ (((~((s+M)=0)) /\\ (((t+L)+(s+M)=S (K)))))))) /\\ ((forall pfc_index_both_padded_productcoefficients. (exists pfa_gap_both_padded_productcoefficientsbound. pfa_gap_both_padded_productcoefficientsbound + S (pfc_index_both_padded_productcoefficients) = (K)) -> exists pfc_value_both_padded_productcoefficients. ((((exists ff_h_pfp_both_padded_productcoefficientsentry. ff_h_pfp_both_padded_productcoefficientsentry + S (pfc_value_both_padded_productcoefficients) = S ((S (pfc_index_both_padded_productcoefficients)) * CC)) /\\ exists ff_q_pfp_both_padded_productcoefficientsentry. CB = ff_q_pfp_both_padded_productcoefficientsentry * S ((S (pfc_index_both_padded_productcoefficients)) * CC) + (pfc_value_both_padded_productcoefficients))) /\\ ((exists pfc_terms_code_both_padded_productcoefficientscoefficient pfc_terms_scale_both_padded_productcoefficientscoefficient pfc_natural_sum_both_padded_productcoefficientscoefficient. ((forall pfc_index_both_padded_productcoefficientscoefficientdiagonal. (exists pfa_gap_both_padded_productcoefficientscoefficientdiagonalbound. pfa_gap_both_padded_productcoefficientscoefficientdiagonalbound + S (pfc_index_both_padded_productcoefficientscoefficientdiagonal) = (S (pfc_index_both_padded_productcoefficients))) -> exists pfc_value_both_padded_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_both_padded_productcoefficientscoefficientdiagonalentry. ff_h_pfp_both_padded_productcoefficientscoefficientdiagonalentry + S (pfc_value_both_padded_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_both_padded_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_both_padded_productcoefficientscoefficient)) /\\ exists ff_q_pfp_both_padded_productcoefficientscoefficientdiagonalentry. pfc_terms_code_both_padded_productcoefficientscoefficient = ff_q_pfp_both_padded_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_both_padded_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_both_padded_productcoefficientscoefficient) + (pfc_value_both_padded_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_both_padded_productcoefficientscoefficientdiagonalterm pfc_left_both_padded_productcoefficientscoefficientdiagonalterm pfc_right_both_padded_productcoefficientscoefficientdiagonalterm. (((pfc_index_both_padded_productcoefficientscoefficientdiagonal)+pfc_complement_both_padded_productcoefficientscoefficientdiagonalterm=(pfc_index_both_padded_productcoefficients)) /\\ ((((((exists pfa_gap_both_padded_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_both_padded_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_both_padded_productcoefficientscoefficientdiagonal) = (t+L)) /\\ ((((exists ff_h_pfp_both_padded_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_both_padded_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_both_padded_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_both_padded_productcoefficientscoefficientdiagonal)) * AC)) /\\ exists ff_q_pfp_both_padded_productcoefficientscoefficientdiagonaltermleftentry. AB = ff_q_pfp_both_padded_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_both_padded_productcoefficientscoefficientdiagonal)) * AC) + (pfc_left_both_padded_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_both_padded_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_both_padded_productcoefficientscoefficientdiagonaltermleftoutside+(t+L)=(pfc_index_both_padded_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_both_padded_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_both_padded_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_both_padded_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_both_padded_productcoefficientscoefficientdiagonalterm) = (s+M)) /\\ ((((exists ff_h_pfp_both_padded_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_both_padded_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_both_padded_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_both_padded_productcoefficientscoefficientdiagonalterm)) * BC)) /\\ exists ff_q_pfp_both_padded_productcoefficientscoefficientdiagonaltermrightentry. BB = ff_q_pfp_both_padded_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_both_padded_productcoefficientscoefficientdiagonalterm)) * BC) + (pfc_right_both_padded_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_both_padded_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_both_padded_productcoefficientscoefficientdiagonaltermrightoutside+(s+M)=(pfc_complement_both_padded_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_both_padded_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_both_padded_productcoefficientscoefficientdiagonal)=pfc_left_both_padded_productcoefficientscoefficientdiagonalterm*pfc_right_both_padded_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_both_padded_productcoefficientscoefficientsum fs_v_pfc_both_padded_productcoefficientscoefficientsum. ((((exists fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_start. fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_both_padded_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_start. fs_u_pfc_both_padded_productcoefficientscoefficientsum = fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_both_padded_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_both_padded_productcoefficientscoefficient) = S ((S (S (pfc_index_both_padded_productcoefficients))) * fs_v_pfc_both_padded_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_both_padded_productcoefficientscoefficientsum = fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_both_padded_productcoefficients))) * fs_v_pfc_both_padded_productcoefficientscoefficientsum) + (pfc_natural_sum_both_padded_productcoefficientscoefficient))) /\\ forall fs_i_pfc_both_padded_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_both_padded_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_both_padded_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_both_padded_productcoefficientscoefficientsum_body_steps = S (pfc_index_both_padded_productcoefficients)) -> exists fs_a_pfc_both_padded_productcoefficientscoefficientsum_body_steps fs_r_pfc_both_padded_productcoefficientscoefficientsum_body_steps fs_s_pfc_both_padded_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_both_padded_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_both_padded_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_both_padded_productcoefficientscoefficient)) /\\ exists fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_both_padded_productcoefficientscoefficient = fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_both_padded_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_both_padded_productcoefficientscoefficient) + (fs_a_pfc_both_padded_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_both_padded_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_both_padded_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_padded_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_both_padded_productcoefficientscoefficientsum = fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_both_padded_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_padded_productcoefficientscoefficientsum) + (fs_r_pfc_both_padded_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_both_padded_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_both_padded_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_padded_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_both_padded_productcoefficientscoefficientsum = fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_both_padded_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_padded_productcoefficientscoefficientsum) + (fs_s_pfc_both_padded_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_both_padded_productcoefficientscoefficientsum_body_steps = fs_r_pfc_both_padded_productcoefficientscoefficientsum_body_steps + fs_a_pfc_both_padded_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_both_padded_productcoefficientscoefficientresiduebound. pfa_gap_both_padded_productcoefficientscoefficientresiduebound + S (pfc_value_both_padded_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_both_padded_productcoefficientscoefficientresiduecongruence pfa_offset_right_both_padded_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_both_padded_productcoefficientscoefficient) + (p) * pfa_offset_left_both_padded_productcoefficientscoefficientresiduecongruence = (pfc_value_both_padded_productcoefficients) + (p) * pfa_offset_right_both_padded_productcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (forall pfrep_power_both_equivalent_result pfrep_left_both_equivalent_result pfrep_right_both_equivalent_result. ((exists pfrep_position_both_equivalent_resultfirst. ((pfrep_position_both_equivalent_resultfirst+S (pfrep_power_both_equivalent_result)=(N)) /\\ ((((exists ff_h_pfp_both_equivalent_resultfirstentry. ff_h_pfp_both_equivalent_resultfirstentry + S (pfrep_left_both_equivalent_result) = S ((S (pfrep_position_both_equivalent_resultfirst)) * cc)) /\\ exists ff_q_pfp_both_equivalent_resultfirstentry. cb = ff_q_pfp_both_equivalent_resultfirstentry * S ((S (pfrep_position_both_equivalent_resultfirst)) * cc) + (pfrep_left_both_equivalent_result)))))) \\/ (((exists pfrep_gap_both_equivalent_resultfirstoutside. pfrep_gap_both_equivalent_resultfirstoutside+(N)=(pfrep_power_both_equivalent_result)) /\\ (((pfrep_left_both_equivalent_result)=0))))) -> ((exists pfrep_position_both_equivalent_resultsecond. ((pfrep_position_both_equivalent_resultsecond+S (pfrep_power_both_equivalent_result)=(K)) /\\ ((((exists ff_h_pfp_both_equivalent_resultsecondentry. ff_h_pfp_both_equivalent_resultsecondentry + S (pfrep_right_both_equivalent_result) = S ((S (pfrep_position_both_equivalent_resultsecond)) * CC)) /\\ exists ff_q_pfp_both_equivalent_resultsecondentry. CB = ff_q_pfp_both_equivalent_resultsecondentry * S ((S (pfrep_position_both_equivalent_resultsecond)) * CC) + (pfrep_right_both_equivalent_result)))))) \\/ (((exists pfrep_gap_both_equivalent_resultsecondoutside. pfrep_gap_both_equivalent_resultsecondoutside+(K)=(pfrep_power_both_equivalent_result)) /\\ (((pfrep_right_both_equivalent_result)=0))))) -> pfrep_left_both_equivalent_result=pfrep_right_both_equivalent_result)", "statement_sha256": "fbefa6c478ac7028d2c60d742799660f05d010578ce4ed30b0f72f6f0af237d6", "summary": "Construct an actual intermediate product and compose the two proved factor-padding compatibilities; both empty and nonempty factors are covered.", "summary_sha256": "ecb341bb396637e73799f8fe813380510b5e33e1330edd775900c3bf8f972359" }, { "alpha_v33_frontier_enrollment": { "body_receipt_sha256": "87d8852b7d18eb019aa98ca406e7e725b0aa705f6d2a37b808c5f1fd668c82b2", "bundle_campaign": "polynomial-euclidean-division", "bundle_node_id": 367, "bundle_sha256": 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"polynomial-euclidean-division", "logical_spec_sha256": "7a18e42d4b9037c8ed51158d01342fa7312f430d18723d70fe6e24be0deecd89", "membership": "alpha_only", "name": "prime_field_polynomial_convolution_both_left_paddings_exists", "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 AB", "intro AC", "intro t", "intro BB", "intro BC", "intro s", "intro hprime", "intro hA", "intro hB", "intro hc", "have hcopy : ((forall fom_index_pfp_both_original_productleft. (exists fom_gap_pfp_both_original_productleft_index_bound. fom_gap_pfp_both_original_productleft_index_bound + S (fom_index_pfp_both_original_productleft) = L) -> exists fom_value_pfp_both_original_productleft. ((((exists fom_beta_height_pfp_both_original_productleft_entry. fom_beta_height_pfp_both_original_productleft_entry + S (fom_value_pfp_both_original_productleft) = S ((S (fom_index_pfp_both_original_productleft)) * ac)) /\\ exists fom_beta_quotient_pfp_both_original_productleft_entry. ab = fom_beta_quotient_pfp_both_original_productleft_entry * S ((S (fom_index_pfp_both_original_productleft)) * ac) + (fom_value_pfp_both_original_productleft))) /\\ (exists fom_gap_pfp_both_original_productleft_value_bound. fom_gap_pfp_both_original_productleft_value_bound + S (fom_value_pfp_both_original_productleft) = p))) /\\ (((forall fom_index_pfp_both_original_productright. (exists fom_gap_pfp_both_original_productright_index_bound. fom_gap_pfp_both_original_productright_index_bound + S (fom_index_pfp_both_original_productright) = M) -> exists fom_value_pfp_both_original_productright. ((((exists fom_beta_height_pfp_both_original_productright_entry. fom_beta_height_pfp_both_original_productright_entry + S (fom_value_pfp_both_original_productright) = S ((S (fom_index_pfp_both_original_productright)) * bc)) /\\ exists fom_beta_quotient_pfp_both_original_productright_entry. bb = fom_beta_quotient_pfp_both_original_productright_entry * S ((S (fom_index_pfp_both_original_productright)) * bc) + (fom_value_pfp_both_original_productright))) /\\ (exists fom_gap_pfp_both_original_productright_value_bound. fom_gap_pfp_both_original_productright_value_bound + S (fom_value_pfp_both_original_productright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_both_original_productcoefficients. (exists pfa_gap_both_original_productcoefficientsbound. pfa_gap_both_original_productcoefficientsbound + S (pfc_index_both_original_productcoefficients) = (N)) -> exists pfc_value_both_original_productcoefficients. ((((exists ff_h_pfp_both_original_productcoefficientsentry. ff_h_pfp_both_original_productcoefficientsentry + S (pfc_value_both_original_productcoefficients) = S ((S (pfc_index_both_original_productcoefficients)) * cc)) /\\ exists ff_q_pfp_both_original_productcoefficientsentry. cb = ff_q_pfp_both_original_productcoefficientsentry * S ((S (pfc_index_both_original_productcoefficients)) * cc) + (pfc_value_both_original_productcoefficients))) /\\ ((exists pfc_terms_code_both_original_productcoefficientscoefficient pfc_terms_scale_both_original_productcoefficientscoefficient pfc_natural_sum_both_original_productcoefficientscoefficient. ((forall pfc_index_both_original_productcoefficientscoefficientdiagonal. (exists pfa_gap_both_original_productcoefficientscoefficientdiagonalbound. pfa_gap_both_original_productcoefficientscoefficientdiagonalbound + S (pfc_index_both_original_productcoefficientscoefficientdiagonal) = (S (pfc_index_both_original_productcoefficients))) -> exists pfc_value_both_original_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_both_original_productcoefficientscoefficientdiagonalentry. ff_h_pfp_both_original_productcoefficientscoefficientdiagonalentry + S (pfc_value_both_original_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_both_original_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_both_original_productcoefficientscoefficient)) /\\ exists ff_q_pfp_both_original_productcoefficientscoefficientdiagonalentry. pfc_terms_code_both_original_productcoefficientscoefficient = ff_q_pfp_both_original_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_both_original_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_both_original_productcoefficientscoefficient) + (pfc_value_both_original_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_both_original_productcoefficientscoefficientdiagonalterm pfc_left_both_original_productcoefficientscoefficientdiagonalterm pfc_right_both_original_productcoefficientscoefficientdiagonalterm. (((pfc_index_both_original_productcoefficientscoefficientdiagonal)+pfc_complement_both_original_productcoefficientscoefficientdiagonalterm=(pfc_index_both_original_productcoefficients)) /\\ ((((((exists pfa_gap_both_original_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_both_original_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_both_original_productcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_both_original_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_both_original_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_both_original_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_both_original_productcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_both_original_productcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_both_original_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_both_original_productcoefficientscoefficientdiagonal)) * ac) + (pfc_left_both_original_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_both_original_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_both_original_productcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_both_original_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_both_original_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_both_original_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_both_original_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_both_original_productcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_both_original_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_both_original_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_both_original_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_both_original_productcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_both_original_productcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_both_original_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_both_original_productcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_both_original_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_both_original_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_both_original_productcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_both_original_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_both_original_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_both_original_productcoefficientscoefficientdiagonal)=pfc_left_both_original_productcoefficientscoefficientdiagonalterm*pfc_right_both_original_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_both_original_productcoefficientscoefficientsum fs_v_pfc_both_original_productcoefficientscoefficientsum. ((((exists fs_h_pfc_both_original_productcoefficientscoefficientsum_body_start. fs_h_pfc_both_original_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_both_original_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_original_productcoefficientscoefficientsum_body_start. fs_u_pfc_both_original_productcoefficientscoefficientsum = fs_q_pfc_both_original_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_both_original_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_both_original_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_both_original_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_both_original_productcoefficientscoefficient) = S ((S (S (pfc_index_both_original_productcoefficients))) * fs_v_pfc_both_original_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_original_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_both_original_productcoefficientscoefficientsum = fs_q_pfc_both_original_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_both_original_productcoefficients))) * fs_v_pfc_both_original_productcoefficientscoefficientsum) + (pfc_natural_sum_both_original_productcoefficientscoefficient))) /\\ forall fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_both_original_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_both_original_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps = S (pfc_index_both_original_productcoefficients)) -> exists fs_a_pfc_both_original_productcoefficientscoefficientsum_body_steps fs_r_pfc_both_original_productcoefficientscoefficientsum_body_steps fs_s_pfc_both_original_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_both_original_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_both_original_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_both_original_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_both_original_productcoefficientscoefficient)) /\\ exists fs_q_pfc_both_original_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_both_original_productcoefficientscoefficient = fs_q_pfc_both_original_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_both_original_productcoefficientscoefficient) + (fs_a_pfc_both_original_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_both_original_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_both_original_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_both_original_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_original_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_original_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_both_original_productcoefficientscoefficientsum = fs_q_pfc_both_original_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_original_productcoefficientscoefficientsum) + (fs_r_pfc_both_original_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_both_original_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_both_original_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_both_original_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_original_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_original_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_both_original_productcoefficientscoefficientsum = fs_q_pfc_both_original_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_original_productcoefficientscoefficientsum) + (fs_s_pfc_both_original_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_both_original_productcoefficientscoefficientsum_body_steps = fs_r_pfc_both_original_productcoefficientscoefficientsum_body_steps + fs_a_pfc_both_original_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_both_original_productcoefficientscoefficientresiduebound. pfa_gap_both_original_productcoefficientscoefficientresiduebound + S (pfc_value_both_original_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_both_original_productcoefficientscoefficientresiduecongruence pfa_offset_right_both_original_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_both_original_productcoefficientscoefficient) + (p) * pfa_offset_left_both_original_productcoefficientscoefficientresiduecongruence = (pfc_value_both_original_productcoefficients) + (p) * pfa_offset_right_both_original_productcoefficientscoefficientresiduecongruence))))))))))))))))))", "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. ((((t+L)=0 \\/ (s+M)=0) /\\ (((K)=0)))) \\/ (((~((t+L)=0)) /\\ (((~((s+M)=0)) /\\ (((t+L)+(s+M)=S (K)))))))", "specialize polynomial_product_length_exists (t+L)", "specialize polynomial_product_length_exists (s+M)", "apply polynomial_product_length_exists", "cases hlength", "have hnew : exists CB CC. ((forall fom_index_pfp_both_exists_productleft. (exists fom_gap_pfp_both_exists_productleft_index_bound. fom_gap_pfp_both_exists_productleft_index_bound + S (fom_index_pfp_both_exists_productleft) = t+L) -> exists fom_value_pfp_both_exists_productleft. ((((exists fom_beta_height_pfp_both_exists_productleft_entry. fom_beta_height_pfp_both_exists_productleft_entry + S (fom_value_pfp_both_exists_productleft) = S ((S (fom_index_pfp_both_exists_productleft)) * AC)) /\\ exists fom_beta_quotient_pfp_both_exists_productleft_entry. AB = fom_beta_quotient_pfp_both_exists_productleft_entry * S ((S (fom_index_pfp_both_exists_productleft)) * AC) + (fom_value_pfp_both_exists_productleft))) /\\ (exists fom_gap_pfp_both_exists_productleft_value_bound. fom_gap_pfp_both_exists_productleft_value_bound + S (fom_value_pfp_both_exists_productleft) = p))) /\\ (((forall fom_index_pfp_both_exists_productright. (exists fom_gap_pfp_both_exists_productright_index_bound. fom_gap_pfp_both_exists_productright_index_bound + S (fom_index_pfp_both_exists_productright) = s+M) -> exists fom_value_pfp_both_exists_productright. ((((exists fom_beta_height_pfp_both_exists_productright_entry. fom_beta_height_pfp_both_exists_productright_entry + S (fom_value_pfp_both_exists_productright) = S ((S (fom_index_pfp_both_exists_productright)) * BC)) /\\ exists fom_beta_quotient_pfp_both_exists_productright_entry. BB = fom_beta_quotient_pfp_both_exists_productright_entry * S ((S (fom_index_pfp_both_exists_productright)) * BC) + (fom_value_pfp_both_exists_productright))) /\\ (exists fom_gap_pfp_both_exists_productright_value_bound. fom_gap_pfp_both_exists_productright_value_bound + S (fom_value_pfp_both_exists_productright) = p))) /\\ (((((((t+L)=0 \\/ (s+M)=0) /\\ (((x)=0)))) \\/ (((~((t+L)=0)) /\\ (((~((s+M)=0)) /\\ (((t+L)+(s+M)=S (x)))))))) /\\ ((forall pfc_index_both_exists_productcoefficients. (exists pfa_gap_both_exists_productcoefficientsbound. pfa_gap_both_exists_productcoefficientsbound + S (pfc_index_both_exists_productcoefficients) = (x)) -> exists pfc_value_both_exists_productcoefficients. ((((exists ff_h_pfp_both_exists_productcoefficientsentry. ff_h_pfp_both_exists_productcoefficientsentry + S (pfc_value_both_exists_productcoefficients) = S ((S (pfc_index_both_exists_productcoefficients)) * CC)) /\\ exists ff_q_pfp_both_exists_productcoefficientsentry. CB = ff_q_pfp_both_exists_productcoefficientsentry * S ((S (pfc_index_both_exists_productcoefficients)) * CC) + (pfc_value_both_exists_productcoefficients))) /\\ ((exists pfc_terms_code_both_exists_productcoefficientscoefficient pfc_terms_scale_both_exists_productcoefficientscoefficient pfc_natural_sum_both_exists_productcoefficientscoefficient. ((forall pfc_index_both_exists_productcoefficientscoefficientdiagonal. (exists pfa_gap_both_exists_productcoefficientscoefficientdiagonalbound. pfa_gap_both_exists_productcoefficientscoefficientdiagonalbound + S (pfc_index_both_exists_productcoefficientscoefficientdiagonal) = (S (pfc_index_both_exists_productcoefficients))) -> exists pfc_value_both_exists_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_both_exists_productcoefficientscoefficientdiagonalentry. ff_h_pfp_both_exists_productcoefficientscoefficientdiagonalentry + S (pfc_value_both_exists_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_both_exists_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_both_exists_productcoefficientscoefficient)) /\\ exists ff_q_pfp_both_exists_productcoefficientscoefficientdiagonalentry. pfc_terms_code_both_exists_productcoefficientscoefficient = ff_q_pfp_both_exists_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_both_exists_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_both_exists_productcoefficientscoefficient) + (pfc_value_both_exists_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_both_exists_productcoefficientscoefficientdiagonalterm pfc_left_both_exists_productcoefficientscoefficientdiagonalterm pfc_right_both_exists_productcoefficientscoefficientdiagonalterm. (((pfc_index_both_exists_productcoefficientscoefficientdiagonal)+pfc_complement_both_exists_productcoefficientscoefficientdiagonalterm=(pfc_index_both_exists_productcoefficients)) /\\ ((((((exists pfa_gap_both_exists_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_both_exists_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_both_exists_productcoefficientscoefficientdiagonal) = (t+L)) /\\ ((((exists ff_h_pfp_both_exists_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_both_exists_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_both_exists_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_both_exists_productcoefficientscoefficientdiagonal)) * AC)) /\\ exists ff_q_pfp_both_exists_productcoefficientscoefficientdiagonaltermleftentry. AB = ff_q_pfp_both_exists_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_both_exists_productcoefficientscoefficientdiagonal)) * AC) + (pfc_left_both_exists_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_both_exists_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_both_exists_productcoefficientscoefficientdiagonaltermleftoutside+(t+L)=(pfc_index_both_exists_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_both_exists_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_both_exists_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_both_exists_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_both_exists_productcoefficientscoefficientdiagonalterm) = (s+M)) /\\ ((((exists ff_h_pfp_both_exists_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_both_exists_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_both_exists_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_both_exists_productcoefficientscoefficientdiagonalterm)) * BC)) /\\ exists ff_q_pfp_both_exists_productcoefficientscoefficientdiagonaltermrightentry. BB = ff_q_pfp_both_exists_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_both_exists_productcoefficientscoefficientdiagonalterm)) * BC) + (pfc_right_both_exists_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_both_exists_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_both_exists_productcoefficientscoefficientdiagonaltermrightoutside+(s+M)=(pfc_complement_both_exists_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_both_exists_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_both_exists_productcoefficientscoefficientdiagonal)=pfc_left_both_exists_productcoefficientscoefficientdiagonalterm*pfc_right_both_exists_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_both_exists_productcoefficientscoefficientsum fs_v_pfc_both_exists_productcoefficientscoefficientsum. ((((exists fs_h_pfc_both_exists_productcoefficientscoefficientsum_body_start. fs_h_pfc_both_exists_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_both_exists_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_exists_productcoefficientscoefficientsum_body_start. fs_u_pfc_both_exists_productcoefficientscoefficientsum = fs_q_pfc_both_exists_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_both_exists_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_both_exists_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_both_exists_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_both_exists_productcoefficientscoefficient) = S ((S (S (pfc_index_both_exists_productcoefficients))) * fs_v_pfc_both_exists_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_exists_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_both_exists_productcoefficientscoefficientsum = fs_q_pfc_both_exists_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_both_exists_productcoefficients))) * fs_v_pfc_both_exists_productcoefficientscoefficientsum) + (pfc_natural_sum_both_exists_productcoefficientscoefficient))) /\\ forall fs_i_pfc_both_exists_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_both_exists_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_both_exists_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_both_exists_productcoefficientscoefficientsum_body_steps = S (pfc_index_both_exists_productcoefficients)) -> exists fs_a_pfc_both_exists_productcoefficientscoefficientsum_body_steps fs_r_pfc_both_exists_productcoefficientscoefficientsum_body_steps fs_s_pfc_both_exists_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_both_exists_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_both_exists_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_both_exists_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_both_exists_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_both_exists_productcoefficientscoefficient)) /\\ exists fs_q_pfc_both_exists_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_both_exists_productcoefficientscoefficient = fs_q_pfc_both_exists_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_both_exists_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_both_exists_productcoefficientscoefficient) + (fs_a_pfc_both_exists_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_both_exists_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_both_exists_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_both_exists_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_both_exists_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_exists_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_exists_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_both_exists_productcoefficientscoefficientsum = fs_q_pfc_both_exists_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_both_exists_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_exists_productcoefficientscoefficientsum) + (fs_r_pfc_both_exists_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_both_exists_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_both_exists_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_both_exists_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_both_exists_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_exists_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_exists_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_both_exists_productcoefficientscoefficientsum = fs_q_pfc_both_exists_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_both_exists_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_exists_productcoefficientscoefficientsum) + (fs_s_pfc_both_exists_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_both_exists_productcoefficientscoefficientsum_body_steps = fs_r_pfc_both_exists_productcoefficientscoefficientsum_body_steps + fs_a_pfc_both_exists_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_both_exists_productcoefficientscoefficientresiduebound. pfa_gap_both_exists_productcoefficientscoefficientresiduebound + S (pfc_value_both_exists_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_both_exists_productcoefficientscoefficientresiduecongruence pfa_offset_right_both_exists_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_both_exists_productcoefficientscoefficient) + (p) * pfa_offset_left_both_exists_productcoefficientscoefficientresiduecongruence = (pfc_value_both_exists_productcoefficients) + (p) * pfa_offset_right_both_exists_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 (t+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", "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 (t)", "specialize prime_field_polynomial_left_pad_bounded (AB)", "specialize prime_field_polynomial_left_pad_bounded (AC)", "apply prime_field_polynomial_left_pad_bounded", "exact hprime", "exact hcopy_left", "exact hA", "specialize prime_field_polynomial_left_pad_bounded (p)", "specialize prime_field_polynomial_left_pad_bounded (bb)", "specialize prime_field_polynomial_left_pad_bounded (bc)", "specialize prime_field_polynomial_left_pad_bounded (M)", "specialize prime_field_polynomial_left_pad_bounded (s)", "specialize prime_field_polynomial_left_pad_bounded (BB)", "specialize prime_field_polynomial_left_pad_bounded (BC)", "apply prime_field_polynomial_left_pad_bounded", "exact hprime", "exact hcopy_right_left", "exact hB", "exact hlength_witness", "cases hnew", "cases hnew_witness", "exists x", "exists x1", "exists x2", "split", "exact hnew_witness_witness", "specialize prime_field_polynomial_convolution_both_left_paddings_equivalent (p)", "specialize prime_field_polynomial_convolution_both_left_paddings_equivalent (ab)", "specialize prime_field_polynomial_convolution_both_left_paddings_equivalent (ac)", "specialize prime_field_polynomial_convolution_both_left_paddings_equivalent (L)", "specialize prime_field_polynomial_convolution_both_left_paddings_equivalent (bb)", "specialize prime_field_polynomial_convolution_both_left_paddings_equivalent (bc)", "specialize prime_field_polynomial_convolution_both_left_paddings_equivalent (M)", "specialize prime_field_polynomial_convolution_both_left_paddings_equivalent (cb)", "specialize prime_field_polynomial_convolution_both_left_paddings_equivalent (cc)", "specialize prime_field_polynomial_convolution_both_left_paddings_equivalent (N)", "specialize prime_field_polynomial_convolution_both_left_paddings_equivalent (AB)", "specialize prime_field_polynomial_convolution_both_left_paddings_equivalent (AC)", "specialize prime_field_polynomial_convolution_both_left_paddings_equivalent (t)", "specialize prime_field_polynomial_convolution_both_left_paddings_equivalent (BB)", "specialize prime_field_polynomial_convolution_both_left_paddings_equivalent (BC)", "specialize prime_field_polynomial_convolution_both_left_paddings_equivalent (s)", "specialize prime_field_polynomial_convolution_both_left_paddings_equivalent (x1)", "specialize prime_field_polynomial_convolution_both_left_paddings_equivalent (x2)", "specialize prime_field_polynomial_convolution_both_left_paddings_equivalent (x)", "apply prime_field_polynomial_convolution_both_left_paddings_equivalent", "exact hp", "exact hA", "exact hB", "exact hc", "exact hnew_witness_witness" ], "script_sha256": "3ba2c1390b30431607c2902eca44018b21b6c3610a2dac53eccaed923cf59354", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_convolution_padding_candidate.py", "sha256": "2d874ecfb35a5db0aecdeb07b549464efebad9072c363113aa5a0a977845d007" }, "statement": "forall p ab ac L bb bc M cb cc N AB AC t BB BC s. (~((p) = 1) /\\ forall pfa_factor_left_both_exists_prime pfa_factor_right_both_exists_prime. (p) = pfa_factor_left_both_exists_prime * pfa_factor_right_both_exists_prime -> pfa_factor_left_both_exists_prime = 1 \\/ pfa_factor_right_both_exists_prime = 1) -> (((forall pfp_repeat_index_both_left_paddingzeros. (exists pfa_gap_both_left_paddingzerosindex. pfa_gap_both_left_paddingzerosindex + S (pfp_repeat_index_both_left_paddingzeros) = (t)) -> (((exists ff_h_pfp_both_left_paddingzerosentry. ff_h_pfp_both_left_paddingzerosentry + S (0) = S ((S (pfp_repeat_index_both_left_paddingzeros)) * AC)) /\\ exists ff_q_pfp_both_left_paddingzerosentry. AB = ff_q_pfp_both_left_paddingzerosentry * S ((S (pfp_repeat_index_both_left_paddingzeros)) * AC) + (0)))) /\\ ((forall pfrep_index_both_left_padding pfrep_value_both_left_padding. (exists pfa_gap_both_left_paddingbound. pfa_gap_both_left_paddingbound + S (pfrep_index_both_left_padding) = (L)) -> (((exists ff_h_pfp_both_left_paddinginput. ff_h_pfp_both_left_paddinginput + S (pfrep_value_both_left_padding) = S ((S (pfrep_index_both_left_padding)) * ac)) /\\ exists ff_q_pfp_both_left_paddinginput. ab = ff_q_pfp_both_left_paddinginput * S ((S (pfrep_index_both_left_padding)) * ac) + (pfrep_value_both_left_padding))) -> (((exists ff_h_pfp_both_left_paddingoutput. ff_h_pfp_both_left_paddingoutput + S (pfrep_value_both_left_padding) = S ((S ((t)+pfrep_index_both_left_padding)) * AC)) /\\ exists ff_q_pfp_both_left_paddingoutput. AB = ff_q_pfp_both_left_paddingoutput * S ((S ((t)+pfrep_index_both_left_padding)) * AC) + (pfrep_value_both_left_padding))))))) -> (((forall pfp_repeat_index_both_right_paddingzeros. (exists pfa_gap_both_right_paddingzerosindex. pfa_gap_both_right_paddingzerosindex + S (pfp_repeat_index_both_right_paddingzeros) = (s)) -> (((exists ff_h_pfp_both_right_paddingzerosentry. ff_h_pfp_both_right_paddingzerosentry + S (0) = S ((S (pfp_repeat_index_both_right_paddingzeros)) * BC)) /\\ exists ff_q_pfp_both_right_paddingzerosentry. BB = ff_q_pfp_both_right_paddingzerosentry * S ((S (pfp_repeat_index_both_right_paddingzeros)) * BC) + (0)))) /\\ ((forall pfrep_index_both_right_padding pfrep_value_both_right_padding. (exists pfa_gap_both_right_paddingbound. pfa_gap_both_right_paddingbound + S (pfrep_index_both_right_padding) = (M)) -> (((exists ff_h_pfp_both_right_paddinginput. ff_h_pfp_both_right_paddinginput + S (pfrep_value_both_right_padding) = S ((S (pfrep_index_both_right_padding)) * bc)) /\\ exists ff_q_pfp_both_right_paddinginput. bb = ff_q_pfp_both_right_paddinginput * S ((S (pfrep_index_both_right_padding)) * bc) + (pfrep_value_both_right_padding))) -> (((exists ff_h_pfp_both_right_paddingoutput. ff_h_pfp_both_right_paddingoutput + S (pfrep_value_both_right_padding) = S ((S ((s)+pfrep_index_both_right_padding)) * BC)) /\\ exists ff_q_pfp_both_right_paddingoutput. BB = ff_q_pfp_both_right_paddingoutput * S ((S ((s)+pfrep_index_both_right_padding)) * BC) + (pfrep_value_both_right_padding))))))) -> (((forall fom_index_pfp_both_original_productleft. (exists fom_gap_pfp_both_original_productleft_index_bound. fom_gap_pfp_both_original_productleft_index_bound + S (fom_index_pfp_both_original_productleft) = L) -> exists fom_value_pfp_both_original_productleft. ((((exists fom_beta_height_pfp_both_original_productleft_entry. fom_beta_height_pfp_both_original_productleft_entry + S (fom_value_pfp_both_original_productleft) = S ((S (fom_index_pfp_both_original_productleft)) * ac)) /\\ exists fom_beta_quotient_pfp_both_original_productleft_entry. ab = fom_beta_quotient_pfp_both_original_productleft_entry * S ((S (fom_index_pfp_both_original_productleft)) * ac) + (fom_value_pfp_both_original_productleft))) /\\ (exists fom_gap_pfp_both_original_productleft_value_bound. fom_gap_pfp_both_original_productleft_value_bound + S (fom_value_pfp_both_original_productleft) = p))) /\\ (((forall fom_index_pfp_both_original_productright. (exists fom_gap_pfp_both_original_productright_index_bound. fom_gap_pfp_both_original_productright_index_bound + S (fom_index_pfp_both_original_productright) = M) -> exists fom_value_pfp_both_original_productright. ((((exists fom_beta_height_pfp_both_original_productright_entry. fom_beta_height_pfp_both_original_productright_entry + S (fom_value_pfp_both_original_productright) = S ((S (fom_index_pfp_both_original_productright)) * bc)) /\\ exists fom_beta_quotient_pfp_both_original_productright_entry. bb = fom_beta_quotient_pfp_both_original_productright_entry * S ((S (fom_index_pfp_both_original_productright)) * bc) + (fom_value_pfp_both_original_productright))) /\\ (exists fom_gap_pfp_both_original_productright_value_bound. fom_gap_pfp_both_original_productright_value_bound + S (fom_value_pfp_both_original_productright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_both_original_productcoefficients. (exists pfa_gap_both_original_productcoefficientsbound. pfa_gap_both_original_productcoefficientsbound + S (pfc_index_both_original_productcoefficients) = (N)) -> exists pfc_value_both_original_productcoefficients. ((((exists ff_h_pfp_both_original_productcoefficientsentry. ff_h_pfp_both_original_productcoefficientsentry + S (pfc_value_both_original_productcoefficients) = S ((S (pfc_index_both_original_productcoefficients)) * cc)) /\\ exists ff_q_pfp_both_original_productcoefficientsentry. cb = ff_q_pfp_both_original_productcoefficientsentry * S ((S (pfc_index_both_original_productcoefficients)) * cc) + (pfc_value_both_original_productcoefficients))) /\\ ((exists pfc_terms_code_both_original_productcoefficientscoefficient pfc_terms_scale_both_original_productcoefficientscoefficient pfc_natural_sum_both_original_productcoefficientscoefficient. ((forall pfc_index_both_original_productcoefficientscoefficientdiagonal. (exists pfa_gap_both_original_productcoefficientscoefficientdiagonalbound. pfa_gap_both_original_productcoefficientscoefficientdiagonalbound + S (pfc_index_both_original_productcoefficientscoefficientdiagonal) = (S (pfc_index_both_original_productcoefficients))) -> exists pfc_value_both_original_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_both_original_productcoefficientscoefficientdiagonalentry. ff_h_pfp_both_original_productcoefficientscoefficientdiagonalentry + S (pfc_value_both_original_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_both_original_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_both_original_productcoefficientscoefficient)) /\\ exists ff_q_pfp_both_original_productcoefficientscoefficientdiagonalentry. pfc_terms_code_both_original_productcoefficientscoefficient = ff_q_pfp_both_original_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_both_original_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_both_original_productcoefficientscoefficient) + (pfc_value_both_original_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_both_original_productcoefficientscoefficientdiagonalterm pfc_left_both_original_productcoefficientscoefficientdiagonalterm pfc_right_both_original_productcoefficientscoefficientdiagonalterm. (((pfc_index_both_original_productcoefficientscoefficientdiagonal)+pfc_complement_both_original_productcoefficientscoefficientdiagonalterm=(pfc_index_both_original_productcoefficients)) /\\ ((((((exists pfa_gap_both_original_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_both_original_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_both_original_productcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_both_original_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_both_original_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_both_original_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_both_original_productcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_both_original_productcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_both_original_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_both_original_productcoefficientscoefficientdiagonal)) * ac) + (pfc_left_both_original_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_both_original_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_both_original_productcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_both_original_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_both_original_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_both_original_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_both_original_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_both_original_productcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_both_original_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_both_original_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_both_original_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_both_original_productcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_both_original_productcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_both_original_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_both_original_productcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_both_original_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_both_original_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_both_original_productcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_both_original_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_both_original_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_both_original_productcoefficientscoefficientdiagonal)=pfc_left_both_original_productcoefficientscoefficientdiagonalterm*pfc_right_both_original_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_both_original_productcoefficientscoefficientsum fs_v_pfc_both_original_productcoefficientscoefficientsum. ((((exists fs_h_pfc_both_original_productcoefficientscoefficientsum_body_start. fs_h_pfc_both_original_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_both_original_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_original_productcoefficientscoefficientsum_body_start. fs_u_pfc_both_original_productcoefficientscoefficientsum = fs_q_pfc_both_original_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_both_original_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_both_original_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_both_original_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_both_original_productcoefficientscoefficient) = S ((S (S (pfc_index_both_original_productcoefficients))) * fs_v_pfc_both_original_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_original_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_both_original_productcoefficientscoefficientsum = fs_q_pfc_both_original_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_both_original_productcoefficients))) * fs_v_pfc_both_original_productcoefficientscoefficientsum) + (pfc_natural_sum_both_original_productcoefficientscoefficient))) /\\ forall fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_both_original_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_both_original_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps = S (pfc_index_both_original_productcoefficients)) -> exists fs_a_pfc_both_original_productcoefficientscoefficientsum_body_steps fs_r_pfc_both_original_productcoefficientscoefficientsum_body_steps fs_s_pfc_both_original_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_both_original_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_both_original_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_both_original_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_both_original_productcoefficientscoefficient)) /\\ exists fs_q_pfc_both_original_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_both_original_productcoefficientscoefficient = fs_q_pfc_both_original_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_both_original_productcoefficientscoefficient) + (fs_a_pfc_both_original_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_both_original_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_both_original_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_both_original_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_original_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_original_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_both_original_productcoefficientscoefficientsum = fs_q_pfc_both_original_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_original_productcoefficientscoefficientsum) + (fs_r_pfc_both_original_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_both_original_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_both_original_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_both_original_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_original_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_original_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_both_original_productcoefficientscoefficientsum = fs_q_pfc_both_original_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_original_productcoefficientscoefficientsum) + (fs_s_pfc_both_original_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_both_original_productcoefficientscoefficientsum_body_steps = fs_r_pfc_both_original_productcoefficientscoefficientsum_body_steps + fs_a_pfc_both_original_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_both_original_productcoefficientscoefficientresiduebound. pfa_gap_both_original_productcoefficientscoefficientresiduebound + S (pfc_value_both_original_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_both_original_productcoefficientscoefficientresiduecongruence pfa_offset_right_both_original_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_both_original_productcoefficientscoefficient) + (p) * pfa_offset_left_both_original_productcoefficientscoefficientresiduecongruence = (pfc_value_both_original_productcoefficients) + (p) * pfa_offset_right_both_original_productcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (exists K CB CC. ((((forall fom_index_pfp_both_padded_productleft. (exists fom_gap_pfp_both_padded_productleft_index_bound. fom_gap_pfp_both_padded_productleft_index_bound + S (fom_index_pfp_both_padded_productleft) = t+L) -> exists fom_value_pfp_both_padded_productleft. ((((exists fom_beta_height_pfp_both_padded_productleft_entry. fom_beta_height_pfp_both_padded_productleft_entry + S (fom_value_pfp_both_padded_productleft) = S ((S (fom_index_pfp_both_padded_productleft)) * AC)) /\\ exists fom_beta_quotient_pfp_both_padded_productleft_entry. AB = fom_beta_quotient_pfp_both_padded_productleft_entry * S ((S (fom_index_pfp_both_padded_productleft)) * AC) + (fom_value_pfp_both_padded_productleft))) /\\ (exists fom_gap_pfp_both_padded_productleft_value_bound. fom_gap_pfp_both_padded_productleft_value_bound + S (fom_value_pfp_both_padded_productleft) = p))) /\\ (((forall fom_index_pfp_both_padded_productright. (exists fom_gap_pfp_both_padded_productright_index_bound. fom_gap_pfp_both_padded_productright_index_bound + S (fom_index_pfp_both_padded_productright) = s+M) -> exists fom_value_pfp_both_padded_productright. ((((exists fom_beta_height_pfp_both_padded_productright_entry. fom_beta_height_pfp_both_padded_productright_entry + S (fom_value_pfp_both_padded_productright) = S ((S (fom_index_pfp_both_padded_productright)) * BC)) /\\ exists fom_beta_quotient_pfp_both_padded_productright_entry. BB = fom_beta_quotient_pfp_both_padded_productright_entry * S ((S (fom_index_pfp_both_padded_productright)) * BC) + (fom_value_pfp_both_padded_productright))) /\\ (exists fom_gap_pfp_both_padded_productright_value_bound. fom_gap_pfp_both_padded_productright_value_bound + S (fom_value_pfp_both_padded_productright) = p))) /\\ (((((((t+L)=0 \\/ (s+M)=0) /\\ (((K)=0)))) \\/ (((~((t+L)=0)) /\\ (((~((s+M)=0)) /\\ (((t+L)+(s+M)=S (K)))))))) /\\ ((forall pfc_index_both_padded_productcoefficients. (exists pfa_gap_both_padded_productcoefficientsbound. pfa_gap_both_padded_productcoefficientsbound + S (pfc_index_both_padded_productcoefficients) = (K)) -> exists pfc_value_both_padded_productcoefficients. ((((exists ff_h_pfp_both_padded_productcoefficientsentry. ff_h_pfp_both_padded_productcoefficientsentry + S (pfc_value_both_padded_productcoefficients) = S ((S (pfc_index_both_padded_productcoefficients)) * CC)) /\\ exists ff_q_pfp_both_padded_productcoefficientsentry. CB = ff_q_pfp_both_padded_productcoefficientsentry * S ((S (pfc_index_both_padded_productcoefficients)) * CC) + (pfc_value_both_padded_productcoefficients))) /\\ ((exists pfc_terms_code_both_padded_productcoefficientscoefficient pfc_terms_scale_both_padded_productcoefficientscoefficient pfc_natural_sum_both_padded_productcoefficientscoefficient. ((forall pfc_index_both_padded_productcoefficientscoefficientdiagonal. (exists pfa_gap_both_padded_productcoefficientscoefficientdiagonalbound. pfa_gap_both_padded_productcoefficientscoefficientdiagonalbound + S (pfc_index_both_padded_productcoefficientscoefficientdiagonal) = (S (pfc_index_both_padded_productcoefficients))) -> exists pfc_value_both_padded_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_both_padded_productcoefficientscoefficientdiagonalentry. ff_h_pfp_both_padded_productcoefficientscoefficientdiagonalentry + S (pfc_value_both_padded_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_both_padded_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_both_padded_productcoefficientscoefficient)) /\\ exists ff_q_pfp_both_padded_productcoefficientscoefficientdiagonalentry. pfc_terms_code_both_padded_productcoefficientscoefficient = ff_q_pfp_both_padded_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_both_padded_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_both_padded_productcoefficientscoefficient) + (pfc_value_both_padded_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_both_padded_productcoefficientscoefficientdiagonalterm pfc_left_both_padded_productcoefficientscoefficientdiagonalterm pfc_right_both_padded_productcoefficientscoefficientdiagonalterm. (((pfc_index_both_padded_productcoefficientscoefficientdiagonal)+pfc_complement_both_padded_productcoefficientscoefficientdiagonalterm=(pfc_index_both_padded_productcoefficients)) /\\ ((((((exists pfa_gap_both_padded_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_both_padded_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_both_padded_productcoefficientscoefficientdiagonal) = (t+L)) /\\ ((((exists ff_h_pfp_both_padded_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_both_padded_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_both_padded_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_both_padded_productcoefficientscoefficientdiagonal)) * AC)) /\\ exists ff_q_pfp_both_padded_productcoefficientscoefficientdiagonaltermleftentry. AB = ff_q_pfp_both_padded_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_both_padded_productcoefficientscoefficientdiagonal)) * AC) + (pfc_left_both_padded_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_both_padded_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_both_padded_productcoefficientscoefficientdiagonaltermleftoutside+(t+L)=(pfc_index_both_padded_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_both_padded_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_both_padded_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_both_padded_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_both_padded_productcoefficientscoefficientdiagonalterm) = (s+M)) /\\ ((((exists ff_h_pfp_both_padded_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_both_padded_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_both_padded_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_both_padded_productcoefficientscoefficientdiagonalterm)) * BC)) /\\ exists ff_q_pfp_both_padded_productcoefficientscoefficientdiagonaltermrightentry. BB = ff_q_pfp_both_padded_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_both_padded_productcoefficientscoefficientdiagonalterm)) * BC) + (pfc_right_both_padded_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_both_padded_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_both_padded_productcoefficientscoefficientdiagonaltermrightoutside+(s+M)=(pfc_complement_both_padded_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_both_padded_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_both_padded_productcoefficientscoefficientdiagonal)=pfc_left_both_padded_productcoefficientscoefficientdiagonalterm*pfc_right_both_padded_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_both_padded_productcoefficientscoefficientsum fs_v_pfc_both_padded_productcoefficientscoefficientsum. ((((exists fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_start. fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_both_padded_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_start. fs_u_pfc_both_padded_productcoefficientscoefficientsum = fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_both_padded_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_both_padded_productcoefficientscoefficient) = S ((S (S (pfc_index_both_padded_productcoefficients))) * fs_v_pfc_both_padded_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_both_padded_productcoefficientscoefficientsum = fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_both_padded_productcoefficients))) * fs_v_pfc_both_padded_productcoefficientscoefficientsum) + (pfc_natural_sum_both_padded_productcoefficientscoefficient))) /\\ forall fs_i_pfc_both_padded_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_both_padded_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_both_padded_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_both_padded_productcoefficientscoefficientsum_body_steps = S (pfc_index_both_padded_productcoefficients)) -> exists fs_a_pfc_both_padded_productcoefficientscoefficientsum_body_steps fs_r_pfc_both_padded_productcoefficientscoefficientsum_body_steps fs_s_pfc_both_padded_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_both_padded_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_both_padded_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_both_padded_productcoefficientscoefficient)) /\\ exists fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_both_padded_productcoefficientscoefficient = fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_both_padded_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_both_padded_productcoefficientscoefficient) + (fs_a_pfc_both_padded_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_both_padded_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_both_padded_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_padded_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_both_padded_productcoefficientscoefficientsum = fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_both_padded_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_padded_productcoefficientscoefficientsum) + (fs_r_pfc_both_padded_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_both_padded_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_both_padded_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_padded_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_both_padded_productcoefficientscoefficientsum = fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_both_padded_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_padded_productcoefficientscoefficientsum) + (fs_s_pfc_both_padded_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_both_padded_productcoefficientscoefficientsum_body_steps = fs_r_pfc_both_padded_productcoefficientscoefficientsum_body_steps + fs_a_pfc_both_padded_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_both_padded_productcoefficientscoefficientresiduebound. pfa_gap_both_padded_productcoefficientscoefficientresiduebound + S (pfc_value_both_padded_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_both_padded_productcoefficientscoefficientresiduecongruence pfa_offset_right_both_padded_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_both_padded_productcoefficientscoefficient) + (p) * pfa_offset_left_both_padded_productcoefficientscoefficientresiduecongruence = (pfc_value_both_padded_productcoefficients) + (p) * pfa_offset_right_both_padded_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_both_equivalent_result pfrep_left_both_equivalent_result pfrep_right_both_equivalent_result. ((exists pfrep_position_both_equivalent_resultfirst. ((pfrep_position_both_equivalent_resultfirst+S (pfrep_power_both_equivalent_result)=(N)) /\\ ((((exists ff_h_pfp_both_equivalent_resultfirstentry. ff_h_pfp_both_equivalent_resultfirstentry + S (pfrep_left_both_equivalent_result) = S ((S (pfrep_position_both_equivalent_resultfirst)) * cc)) /\\ exists ff_q_pfp_both_equivalent_resultfirstentry. cb = ff_q_pfp_both_equivalent_resultfirstentry * S ((S (pfrep_position_both_equivalent_resultfirst)) * cc) + (pfrep_left_both_equivalent_result)))))) \\/ (((exists pfrep_gap_both_equivalent_resultfirstoutside. pfrep_gap_both_equivalent_resultfirstoutside+(N)=(pfrep_power_both_equivalent_result)) /\\ (((pfrep_left_both_equivalent_result)=0))))) -> ((exists pfrep_position_both_equivalent_resultsecond. ((pfrep_position_both_equivalent_resultsecond+S (pfrep_power_both_equivalent_result)=(K)) /\\ ((((exists ff_h_pfp_both_equivalent_resultsecondentry. ff_h_pfp_both_equivalent_resultsecondentry + S (pfrep_right_both_equivalent_result) = S ((S (pfrep_position_both_equivalent_resultsecond)) * CC)) /\\ exists ff_q_pfp_both_equivalent_resultsecondentry. CB = ff_q_pfp_both_equivalent_resultsecondentry * S ((S (pfrep_position_both_equivalent_resultsecond)) * CC) + (pfrep_right_both_equivalent_result)))))) \\/ (((exists pfrep_gap_both_equivalent_resultsecondoutside. pfrep_gap_both_equivalent_resultsecondoutside+(K)=(pfrep_power_both_equivalent_result)) /\\ (((pfrep_right_both_equivalent_result)=0))))) -> pfrep_left_both_equivalent_result=pfrep_right_both_equivalent_result))))", "statement_sha256": "b79ee5e0362c752f6b0189437e25cacc49e7060037adf8837f3105db832f8ffd", "summary": "For actual leading-zero paddings over a prime field, construct a genuine proper-length product and prove its formal equivalence to the original product; no output certificate or identity is supplied.", "summary_sha256": "cc5c81d22115f6b022c52bab40993131de6b2de57606a1b3e034b096f88cedea" }, { "alpha_v33_frontier_enrollment": { "body_receipt_sha256": "030a306e8c889f334460f6904e24e5dd675e217139e8fef13e506ee358b86c7b", "bundle_campaign": "polynomial-euclidean-division", 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"prime_field_polynomial_equivalent_symmetric", "prime_field_polynomial_equivalent_transitive", "prime_field_polynomial_equivalent_implies_equal_same_length", "prime_field_polynomial_left_pad_transport" ], "dependencies_sha256": "c40b3b66ebcd645d588d45f5ff5f97a5de7b6dde9ec94a3e7879e9b11440bf5b", "empty_context_closure": { "body_proof_depth": 30, "body_proof_nodes": 78, "bundle_campaign": "polynomial-euclidean-division", "bundle_dependency_edge_count": 1071, "bundle_node_count": 377, "bundle_node_id": 368, "bundle_path": "research/arithmetic-library/artifacts/prime-field-polynomial-euclidean-division-proof-bundle-v1.json", "bundle_root_id": 376, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "6ae667d8518e4dbe722bb08ad1b08715a0d282c2893e533c8133d770fe861dcf", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": 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"catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "polynomial-euclidean-division", "logical_spec_sha256": "7c8ae473ce2efce4e0966bd63dfc8ad92cf20a8804b97edbdf11dbb078e05498", "membership": "alpha_only", "name": "prime_field_polynomial_equivalent_implies_left_pad", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro b", "intro c", "intro L", "intro t", "intro d", "intro e", "intro he", "have hp : exists B C. (((forall pfp_repeat_index_converse_constructedzeros. (exists pfa_gap_converse_constructedzerosindex. pfa_gap_converse_constructedzerosindex + S (pfp_repeat_index_converse_constructedzeros) = (t)) -> (((exists ff_h_pfp_converse_constructedzerosentry. ff_h_pfp_converse_constructedzerosentry + S (0) = S ((S (pfp_repeat_index_converse_constructedzeros)) * C)) /\\ exists ff_q_pfp_converse_constructedzerosentry. B = ff_q_pfp_converse_constructedzerosentry * S ((S (pfp_repeat_index_converse_constructedzeros)) * C) + (0)))) /\\ ((forall pfrep_index_converse_constructed pfrep_value_converse_constructed. (exists pfa_gap_converse_constructedbound. pfa_gap_converse_constructedbound + S (pfrep_index_converse_constructed) = (L)) -> (((exists ff_h_pfp_converse_constructedinput. ff_h_pfp_converse_constructedinput + S (pfrep_value_converse_constructed) = S ((S (pfrep_index_converse_constructed)) * c)) /\\ exists ff_q_pfp_converse_constructedinput. b = ff_q_pfp_converse_constructedinput * S ((S (pfrep_index_converse_constructed)) * c) + (pfrep_value_converse_constructed))) -> (((exists ff_h_pfp_converse_constructedoutput. ff_h_pfp_converse_constructedoutput + S (pfrep_value_converse_constructed) = S ((S ((t)+pfrep_index_converse_constructed)) * C)) /\\ exists ff_q_pfp_converse_constructedoutput. B = ff_q_pfp_converse_constructedoutput * S ((S ((t)+pfrep_index_converse_constructed)) * C) + (pfrep_value_converse_constructed)))))))", "specialize prime_field_polynomial_left_pad_exists (b)", "specialize prime_field_polynomial_left_pad_exists (c)", "specialize prime_field_polynomial_left_pad_exists (t)", "specialize prime_field_polynomial_left_pad_exists (L)", "apply prime_field_polynomial_left_pad_exists", "cases hp", "cases hp_witness", "have hs : forall pfrep_power_converse_source pfrep_left_converse_source pfrep_right_converse_source. ((exists pfrep_position_converse_sourcefirst. ((pfrep_position_converse_sourcefirst+S (pfrep_power_converse_source)=(L)) /\\ ((((exists ff_h_pfp_converse_sourcefirstentry. ff_h_pfp_converse_sourcefirstentry + S (pfrep_left_converse_source) = S ((S (pfrep_position_converse_sourcefirst)) * c)) /\\ exists ff_q_pfp_converse_sourcefirstentry. b = ff_q_pfp_converse_sourcefirstentry * S ((S (pfrep_position_converse_sourcefirst)) * c) + (pfrep_left_converse_source)))))) \\/ (((exists pfrep_gap_converse_sourcefirstoutside. pfrep_gap_converse_sourcefirstoutside+(L)=(pfrep_power_converse_source)) /\\ (((pfrep_left_converse_source)=0))))) -> ((exists pfrep_position_converse_sourcesecond. ((pfrep_position_converse_sourcesecond+S (pfrep_power_converse_source)=(t+L)) /\\ ((((exists ff_h_pfp_converse_sourcesecondentry. ff_h_pfp_converse_sourcesecondentry + S (pfrep_right_converse_source) = S ((S (pfrep_position_converse_sourcesecond)) * x1)) /\\ exists ff_q_pfp_converse_sourcesecondentry. x = ff_q_pfp_converse_sourcesecondentry * S ((S (pfrep_position_converse_sourcesecond)) * x1) + (pfrep_right_converse_source)))))) \\/ (((exists pfrep_gap_converse_sourcesecondoutside. pfrep_gap_converse_sourcesecondoutside+(t+L)=(pfrep_power_converse_source)) /\\ (((pfrep_right_converse_source)=0))))) -> pfrep_left_converse_source=pfrep_right_converse_source", "specialize prime_field_polynomial_left_pad_equivalent (b)", "specialize prime_field_polynomial_left_pad_equivalent (c)", "specialize prime_field_polynomial_left_pad_equivalent (L)", "specialize prime_field_polynomial_left_pad_equivalent (t)", "specialize prime_field_polynomial_left_pad_equivalent (x)", "specialize prime_field_polynomial_left_pad_equivalent (x1)", "apply prime_field_polynomial_left_pad_equivalent", "exact hp_witness_witness", "have hr : forall pfrep_power_converse_reverse pfrep_left_converse_reverse pfrep_right_converse_reverse. ((exists pfrep_position_converse_reversefirst. ((pfrep_position_converse_reversefirst+S (pfrep_power_converse_reverse)=(t+L)) /\\ ((((exists ff_h_pfp_converse_reversefirstentry. ff_h_pfp_converse_reversefirstentry + S (pfrep_left_converse_reverse) = S ((S (pfrep_position_converse_reversefirst)) * x1)) /\\ exists ff_q_pfp_converse_reversefirstentry. x = ff_q_pfp_converse_reversefirstentry * S ((S (pfrep_position_converse_reversefirst)) * x1) + (pfrep_left_converse_reverse)))))) \\/ (((exists pfrep_gap_converse_reversefirstoutside. pfrep_gap_converse_reversefirstoutside+(t+L)=(pfrep_power_converse_reverse)) /\\ (((pfrep_left_converse_reverse)=0))))) -> ((exists pfrep_position_converse_reversesecond. ((pfrep_position_converse_reversesecond+S (pfrep_power_converse_reverse)=(L)) /\\ ((((exists ff_h_pfp_converse_reversesecondentry. ff_h_pfp_converse_reversesecondentry + S (pfrep_right_converse_reverse) = S ((S (pfrep_position_converse_reversesecond)) * c)) /\\ exists ff_q_pfp_converse_reversesecondentry. b = ff_q_pfp_converse_reversesecondentry * S ((S (pfrep_position_converse_reversesecond)) * c) + (pfrep_right_converse_reverse)))))) \\/ (((exists pfrep_gap_converse_reversesecondoutside. pfrep_gap_converse_reversesecondoutside+(L)=(pfrep_power_converse_reverse)) /\\ (((pfrep_right_converse_reverse)=0))))) -> pfrep_left_converse_reverse=pfrep_right_converse_reverse", "specialize prime_field_polynomial_equivalent_symmetric (b)", "specialize prime_field_polynomial_equivalent_symmetric (c)", "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 (t+L)", "apply prime_field_polynomial_equivalent_symmetric", "exact hs", "have ht : forall pfrep_power_converse_target pfrep_left_converse_target pfrep_right_converse_target. ((exists pfrep_position_converse_targetfirst. ((pfrep_position_converse_targetfirst+S (pfrep_power_converse_target)=(t+L)) /\\ ((((exists ff_h_pfp_converse_targetfirstentry. ff_h_pfp_converse_targetfirstentry + S (pfrep_left_converse_target) = S ((S (pfrep_position_converse_targetfirst)) * x1)) /\\ exists ff_q_pfp_converse_targetfirstentry. x = ff_q_pfp_converse_targetfirstentry * S ((S (pfrep_position_converse_targetfirst)) * x1) + (pfrep_left_converse_target)))))) \\/ (((exists pfrep_gap_converse_targetfirstoutside. pfrep_gap_converse_targetfirstoutside+(t+L)=(pfrep_power_converse_target)) /\\ (((pfrep_left_converse_target)=0))))) -> ((exists pfrep_position_converse_targetsecond. ((pfrep_position_converse_targetsecond+S (pfrep_power_converse_target)=(t+L)) /\\ ((((exists ff_h_pfp_converse_targetsecondentry. ff_h_pfp_converse_targetsecondentry + S (pfrep_right_converse_target) = S ((S (pfrep_position_converse_targetsecond)) * e)) /\\ exists ff_q_pfp_converse_targetsecondentry. d = ff_q_pfp_converse_targetsecondentry * S ((S (pfrep_position_converse_targetsecond)) * e) + (pfrep_right_converse_target)))))) \\/ (((exists pfrep_gap_converse_targetsecondoutside. pfrep_gap_converse_targetsecondoutside+(t+L)=(pfrep_power_converse_target)) /\\ (((pfrep_right_converse_target)=0))))) -> pfrep_left_converse_target=pfrep_right_converse_target", "specialize prime_field_polynomial_equivalent_transitive (x)", "specialize prime_field_polynomial_equivalent_transitive (x1)", "specialize prime_field_polynomial_equivalent_transitive (t+L)", "specialize prime_field_polynomial_equivalent_transitive (b)", "specialize prime_field_polynomial_equivalent_transitive (c)", "specialize prime_field_polynomial_equivalent_transitive (L)", "specialize prime_field_polynomial_equivalent_transitive (d)", "specialize prime_field_polynomial_equivalent_transitive (e)", "specialize prime_field_polynomial_equivalent_transitive (t+L)", "apply prime_field_polynomial_equivalent_transitive", "exact hr", "exact he", "have hv : forall mdr_i_pfp_converse_values mdr_a_pfp_converse_values. (exists mdr_gap_pfp_converse_valuesb. mdr_gap_pfp_converse_valuesb + S (mdr_i_pfp_converse_values) = (t+L)) -> (((exists ff_h_mdr_pfp_converse_valueso. ff_h_mdr_pfp_converse_valueso + S (mdr_a_pfp_converse_values) = S ((S (mdr_i_pfp_converse_values)) * x1)) /\\ exists ff_q_mdr_pfp_converse_valueso. x = ff_q_mdr_pfp_converse_valueso * S ((S (mdr_i_pfp_converse_values)) * x1) + (mdr_a_pfp_converse_values))) -> (((exists ff_h_mdr_pfp_converse_valuesn. ff_h_mdr_pfp_converse_valuesn + S (mdr_a_pfp_converse_values) = S ((S (mdr_i_pfp_converse_values)) * e)) /\\ exists ff_q_mdr_pfp_converse_valuesn. d = ff_q_mdr_pfp_converse_valuesn * S ((S (mdr_i_pfp_converse_values)) * e) + (mdr_a_pfp_converse_values)))", "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 (d)", "specialize prime_field_polynomial_equivalent_implies_equal_same_length (e)", "specialize prime_field_polynomial_equivalent_implies_equal_same_length (t+L)", "apply prime_field_polynomial_equivalent_implies_equal_same_length", "exact ht", "specialize prime_field_polynomial_left_pad_transport (b)", "specialize prime_field_polynomial_left_pad_transport (c)", "specialize prime_field_polynomial_left_pad_transport (b)", "specialize prime_field_polynomial_left_pad_transport (c)", "specialize prime_field_polynomial_left_pad_transport (L)", "specialize prime_field_polynomial_left_pad_transport (t)", "specialize prime_field_polynomial_left_pad_transport (x)", "specialize prime_field_polynomial_left_pad_transport (x1)", "specialize prime_field_polynomial_left_pad_transport (d)", "specialize prime_field_polynomial_left_pad_transport (e)", "apply prime_field_polynomial_left_pad_transport", "intro i", "intro a", "intro hi", "intro ha", "exact ha", "exact hv", "exact hp_witness_witness" ], "script_sha256": "c63ee4085fec94e4d03e45a36adddd7824a5c35c075d6d4caa5c05cfa600c171", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_equivalence_candidate.py", "sha256": "929eb67318c8a09577fb9ebac277b82656abf04c82b97a417fff83f39e7bb373" }, "statement": "forall b c L t d e. (forall pfrep_power_converse_input pfrep_left_converse_input pfrep_right_converse_input. ((exists pfrep_position_converse_inputfirst. ((pfrep_position_converse_inputfirst+S (pfrep_power_converse_input)=(L)) /\\ ((((exists ff_h_pfp_converse_inputfirstentry. ff_h_pfp_converse_inputfirstentry + S (pfrep_left_converse_input) = S ((S (pfrep_position_converse_inputfirst)) * c)) /\\ exists ff_q_pfp_converse_inputfirstentry. b = ff_q_pfp_converse_inputfirstentry * S ((S (pfrep_position_converse_inputfirst)) * c) + (pfrep_left_converse_input)))))) \\/ (((exists pfrep_gap_converse_inputfirstoutside. pfrep_gap_converse_inputfirstoutside+(L)=(pfrep_power_converse_input)) /\\ (((pfrep_left_converse_input)=0))))) -> ((exists pfrep_position_converse_inputsecond. ((pfrep_position_converse_inputsecond+S (pfrep_power_converse_input)=(t+L)) /\\ ((((exists ff_h_pfp_converse_inputsecondentry. ff_h_pfp_converse_inputsecondentry + S (pfrep_right_converse_input) = S ((S (pfrep_position_converse_inputsecond)) * e)) /\\ exists ff_q_pfp_converse_inputsecondentry. d = ff_q_pfp_converse_inputsecondentry * S ((S (pfrep_position_converse_inputsecond)) * e) + (pfrep_right_converse_input)))))) \\/ (((exists pfrep_gap_converse_inputsecondoutside. pfrep_gap_converse_inputsecondoutside+(t+L)=(pfrep_power_converse_input)) /\\ (((pfrep_right_converse_input)=0))))) -> pfrep_left_converse_input=pfrep_right_converse_input) -> (((forall pfp_repeat_index_converse_resultzeros. (exists pfa_gap_converse_resultzerosindex. pfa_gap_converse_resultzerosindex + S (pfp_repeat_index_converse_resultzeros) = (t)) -> (((exists ff_h_pfp_converse_resultzerosentry. ff_h_pfp_converse_resultzerosentry + S (0) = S ((S (pfp_repeat_index_converse_resultzeros)) * e)) /\\ exists ff_q_pfp_converse_resultzerosentry. d = ff_q_pfp_converse_resultzerosentry * S ((S (pfp_repeat_index_converse_resultzeros)) * e) + (0)))) /\\ ((forall pfrep_index_converse_result pfrep_value_converse_result. (exists pfa_gap_converse_resultbound. pfa_gap_converse_resultbound + S (pfrep_index_converse_result) = (L)) -> (((exists ff_h_pfp_converse_resultinput. ff_h_pfp_converse_resultinput + S (pfrep_value_converse_result) = S ((S (pfrep_index_converse_result)) * c)) /\\ exists ff_q_pfp_converse_resultinput. b = ff_q_pfp_converse_resultinput * S ((S (pfrep_index_converse_result)) * c) + (pfrep_value_converse_result))) -> (((exists ff_h_pfp_converse_resultoutput. ff_h_pfp_converse_resultoutput + S (pfrep_value_converse_result) = S ((S ((t)+pfrep_index_converse_result)) * e)) /\\ exists ff_q_pfp_converse_resultoutput. d = ff_q_pfp_converse_resultoutput * S ((S ((t)+pfrep_index_converse_result)) * e) + (pfrep_value_converse_result)))))))", "statement_sha256": "e9b137b8b2e2d502cb4f5405a4cb90a0abcbb50de9a0df45ff51d5127761a25c", "summary": "Formal equivalence to a prefix of length t+L forces its actual leading-zero block and every copied source coefficient; construct a real padding and transport it 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(((forall pfp_repeat_index_add_output_constructedzeros. (exists pfa_gap_add_output_constructedzerosindex. pfa_gap_add_output_constructedzerosindex + S (pfp_repeat_index_add_output_constructedzeros) = (t)) -> (((exists ff_h_pfp_add_output_constructedzerosentry. ff_h_pfp_add_output_constructedzerosentry + S (0) = S ((S (pfp_repeat_index_add_output_constructedzeros)) * dc)) /\\ exists ff_q_pfp_add_output_constructedzerosentry. db = ff_q_pfp_add_output_constructedzerosentry * S ((S (pfp_repeat_index_add_output_constructedzeros)) * dc) + (0)))) /\\ ((forall pfrep_index_add_output_constructed pfrep_value_add_output_constructed. (exists pfa_gap_add_output_constructedbound. pfa_gap_add_output_constructedbound + S (pfrep_index_add_output_constructed) = (L)) -> (((exists ff_h_pfp_add_output_constructedinput. ff_h_pfp_add_output_constructedinput + S (pfrep_value_add_output_constructed) = S ((S (pfrep_index_add_output_constructed)) * cc)) /\\ exists ff_q_pfp_add_output_constructedinput. cb = ff_q_pfp_add_output_constructedinput * S ((S (pfrep_index_add_output_constructed)) * cc) + (pfrep_value_add_output_constructed))) -> (((exists ff_h_pfp_add_output_constructedoutput. ff_h_pfp_add_output_constructedoutput + S (pfrep_value_add_output_constructed) = S ((S ((t)+pfrep_index_add_output_constructed)) * dc)) /\\ exists ff_q_pfp_add_output_constructedoutput. db = ff_q_pfp_add_output_constructedoutput * S ((S ((t)+pfrep_index_add_output_constructed)) * dc) + (pfrep_value_add_output_constructed)))))))", "specialize prime_field_polynomial_left_pad_exists (cb)", "specialize prime_field_polynomial_left_pad_exists (cc)", "specialize prime_field_polynomial_left_pad_exists (t)", "specialize prime_field_polynomial_left_pad_exists (L)", "apply prime_field_polynomial_left_pad_exists", "cases hd", "cases hd_witness", "have hm : forall pfp_index_add_output_middle. (exists pfa_gap_add_output_middleindex. pfa_gap_add_output_middleindex + S (pfp_index_add_output_middle) = (t+L)) -> exists pfp_left_add_output_middle pfp_right_add_output_middle pfp_value_add_output_middle. ((((exists ff_h_pfp_add_output_middleleft. ff_h_pfp_add_output_middleleft + S (pfp_left_add_output_middle) = S ((S (pfp_index_add_output_middle)) * AC)) /\\ exists ff_q_pfp_add_output_middleleft. AB = ff_q_pfp_add_output_middleleft * S ((S (pfp_index_add_output_middle)) * AC) + (pfp_left_add_output_middle))) /\\ (((((exists ff_h_pfp_add_output_middleright. ff_h_pfp_add_output_middleright + S (pfp_right_add_output_middle) = S ((S (pfp_index_add_output_middle)) * BC)) /\\ exists ff_q_pfp_add_output_middleright. BB = ff_q_pfp_add_output_middleright * S ((S (pfp_index_add_output_middle)) * BC) + (pfp_right_add_output_middle))) /\\ (((((exists ff_h_pfp_add_output_middletarget. ff_h_pfp_add_output_middletarget + S (pfp_value_add_output_middle) = S ((S (pfp_index_add_output_middle)) * x1)) /\\ exists ff_q_pfp_add_output_middletarget. x = ff_q_pfp_add_output_middletarget * S ((S (pfp_index_add_output_middle)) * x1) + (pfp_value_add_output_middle))) /\\ ((((exists pfa_gap_add_output_middleoperationleft. pfa_gap_add_output_middleoperationleft + S (pfp_left_add_output_middle) = (p)) /\\ (((exists pfa_gap_add_output_middleoperationright. pfa_gap_add_output_middleoperationright + S (pfp_right_add_output_middle) = (p)) /\\ ((((exists pfa_gap_add_output_middleoperationresultbound. pfa_gap_add_output_middleoperationresultbound + S (pfp_value_add_output_middle) = (p)) /\\ ((exists pfa_offset_left_add_output_middleoperationresultcongruence pfa_offset_right_add_output_middleoperationresultcongruence. ((pfp_left_add_output_middle) + (pfp_right_add_output_middle)) + (p) * pfa_offset_left_add_output_middleoperationresultcongruence = (pfp_value_add_output_middle) + (p) * pfa_offset_right_add_output_middleoperationresultcongruence)))))))))))))))", "specialize prime_field_polynomial_add_left_pad_transport (p)", "specialize prime_field_polynomial_add_left_pad_transport (ab)", "specialize prime_field_polynomial_add_left_pad_transport (ac)", "specialize prime_field_polynomial_add_left_pad_transport (bb)", "specialize prime_field_polynomial_add_left_pad_transport (bc)", "specialize prime_field_polynomial_add_left_pad_transport (cb)", "specialize prime_field_polynomial_add_left_pad_transport (cc)", "specialize prime_field_polynomial_add_left_pad_transport (L)", "specialize prime_field_polynomial_add_left_pad_transport (t)", "specialize prime_field_polynomial_add_left_pad_transport (AB)", "specialize prime_field_polynomial_add_left_pad_transport (AC)", "specialize prime_field_polynomial_add_left_pad_transport (BB)", "specialize prime_field_polynomial_add_left_pad_transport (BC)", "specialize prime_field_polynomial_add_left_pad_transport (x)", "specialize prime_field_polynomial_add_left_pad_transport (x1)", "apply prime_field_polynomial_add_left_pad_transport", "exact hp", "exact ho", "exact hA", "exact hB", "exact hd_witness_witness", "have he : forall mdr_i_pfp_add_output_values mdr_a_pfp_add_output_values. (exists mdr_gap_pfp_add_output_valuesb. mdr_gap_pfp_add_output_valuesb + S (mdr_i_pfp_add_output_values) = (t+L)) -> (((exists ff_h_mdr_pfp_add_output_valueso. ff_h_mdr_pfp_add_output_valueso + S (mdr_a_pfp_add_output_values) = S ((S (mdr_i_pfp_add_output_values)) * x1)) /\\ exists ff_q_mdr_pfp_add_output_valueso. x = ff_q_mdr_pfp_add_output_valueso * S ((S (mdr_i_pfp_add_output_values)) * x1) + (mdr_a_pfp_add_output_values))) -> (((exists ff_h_mdr_pfp_add_output_valuesn. ff_h_mdr_pfp_add_output_valuesn + S (mdr_a_pfp_add_output_values) = S ((S (mdr_i_pfp_add_output_values)) * CC)) /\\ exists ff_q_mdr_pfp_add_output_valuesn. CB = ff_q_mdr_pfp_add_output_valuesn * S ((S (mdr_i_pfp_add_output_values)) * CC) + (mdr_a_pfp_add_output_values)))", "specialize prime_field_polynomial_add_functional (p)", "specialize prime_field_polynomial_add_functional (AB)", "specialize prime_field_polynomial_add_functional (AC)", "specialize prime_field_polynomial_add_functional (BB)", "specialize prime_field_polynomial_add_functional (BC)", "specialize prime_field_polynomial_add_functional (x)", "specialize prime_field_polynomial_add_functional (x1)", "specialize prime_field_polynomial_add_functional (CB)", "specialize prime_field_polynomial_add_functional (CC)", "specialize prime_field_polynomial_add_functional (t+L)", "apply prime_field_polynomial_add_functional", "exact hm", "exact hn", "specialize prime_field_polynomial_left_pad_transport (cb)", "specialize prime_field_polynomial_left_pad_transport (cc)", "specialize prime_field_polynomial_left_pad_transport (cb)", "specialize prime_field_polynomial_left_pad_transport (cc)", "specialize prime_field_polynomial_left_pad_transport (L)", "specialize prime_field_polynomial_left_pad_transport (t)", "specialize prime_field_polynomial_left_pad_transport (x)", "specialize prime_field_polynomial_left_pad_transport (x1)", "specialize prime_field_polynomial_left_pad_transport (CB)", "specialize prime_field_polynomial_left_pad_transport (CC)", "apply prime_field_polynomial_left_pad_transport", "intro i", "intro a", "intro hi", "intro ha", "exact ha", "exact he", "exact hd_witness_witness" ], "script_sha256": "94ce2913e9ca49a3516b00fd1cdcf6d21eeac351bb5e7b69f74a8c775b2635fa", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_equivalence_candidate.py", "sha256": "929eb67318c8a09577fb9ebac277b82656abf04c82b97a417fff83f39e7bb373" }, "statement": "forall p ab ac bb bc cb cc L t AB AC BB BC CB CC. (~((p) = 1) /\\ forall pfa_factor_left_add_output_prime pfa_factor_right_add_output_prime. (p) = pfa_factor_left_add_output_prime * pfa_factor_right_add_output_prime -> pfa_factor_left_add_output_prime = 1 \\/ pfa_factor_right_add_output_prime = 1) -> (forall pfp_index_add_output_original. (exists pfa_gap_add_output_originalindex. pfa_gap_add_output_originalindex + S (pfp_index_add_output_original) = (L)) -> exists pfp_left_add_output_original pfp_right_add_output_original pfp_value_add_output_original. ((((exists ff_h_pfp_add_output_originalleft. ff_h_pfp_add_output_originalleft + S (pfp_left_add_output_original) = S ((S (pfp_index_add_output_original)) * ac)) /\\ exists ff_q_pfp_add_output_originalleft. ab = ff_q_pfp_add_output_originalleft * S ((S (pfp_index_add_output_original)) * ac) + (pfp_left_add_output_original))) /\\ (((((exists ff_h_pfp_add_output_originalright. ff_h_pfp_add_output_originalright + S (pfp_right_add_output_original) = S ((S (pfp_index_add_output_original)) * bc)) /\\ exists ff_q_pfp_add_output_originalright. bb = ff_q_pfp_add_output_originalright * S ((S (pfp_index_add_output_original)) * bc) + (pfp_right_add_output_original))) /\\ (((((exists ff_h_pfp_add_output_originaltarget. ff_h_pfp_add_output_originaltarget + S (pfp_value_add_output_original) = S ((S (pfp_index_add_output_original)) * cc)) /\\ exists ff_q_pfp_add_output_originaltarget. cb = ff_q_pfp_add_output_originaltarget * S ((S (pfp_index_add_output_original)) * cc) + (pfp_value_add_output_original))) /\\ ((((exists pfa_gap_add_output_originaloperationleft. pfa_gap_add_output_originaloperationleft + S (pfp_left_add_output_original) = (p)) /\\ (((exists pfa_gap_add_output_originaloperationright. pfa_gap_add_output_originaloperationright + S (pfp_right_add_output_original) = (p)) /\\ ((((exists pfa_gap_add_output_originaloperationresultbound. pfa_gap_add_output_originaloperationresultbound + S (pfp_value_add_output_original) = (p)) /\\ ((exists pfa_offset_left_add_output_originaloperationresultcongruence pfa_offset_right_add_output_originaloperationresultcongruence. ((pfp_left_add_output_original) + (pfp_right_add_output_original)) + (p) * pfa_offset_left_add_output_originaloperationresultcongruence = (pfp_value_add_output_original) + (p) * pfa_offset_right_add_output_originaloperationresultcongruence)))))))))))))))) -> (((forall pfp_repeat_index_add_output_leftzeros. (exists pfa_gap_add_output_leftzerosindex. pfa_gap_add_output_leftzerosindex + S (pfp_repeat_index_add_output_leftzeros) = (t)) -> (((exists ff_h_pfp_add_output_leftzerosentry. ff_h_pfp_add_output_leftzerosentry + S (0) = S ((S (pfp_repeat_index_add_output_leftzeros)) * AC)) /\\ exists ff_q_pfp_add_output_leftzerosentry. AB = ff_q_pfp_add_output_leftzerosentry * S ((S (pfp_repeat_index_add_output_leftzeros)) * AC) + (0)))) /\\ ((forall pfrep_index_add_output_left pfrep_value_add_output_left. (exists pfa_gap_add_output_leftbound. pfa_gap_add_output_leftbound + S (pfrep_index_add_output_left) = (L)) -> (((exists ff_h_pfp_add_output_leftinput. ff_h_pfp_add_output_leftinput + S (pfrep_value_add_output_left) = S ((S (pfrep_index_add_output_left)) * ac)) /\\ exists ff_q_pfp_add_output_leftinput. ab = ff_q_pfp_add_output_leftinput * S ((S (pfrep_index_add_output_left)) * ac) + (pfrep_value_add_output_left))) -> (((exists ff_h_pfp_add_output_leftoutput. ff_h_pfp_add_output_leftoutput + S (pfrep_value_add_output_left) = S ((S ((t)+pfrep_index_add_output_left)) * AC)) /\\ exists ff_q_pfp_add_output_leftoutput. AB = ff_q_pfp_add_output_leftoutput * S ((S ((t)+pfrep_index_add_output_left)) * AC) + (pfrep_value_add_output_left))))))) -> (((forall pfp_repeat_index_add_output_rightzeros. (exists pfa_gap_add_output_rightzerosindex. pfa_gap_add_output_rightzerosindex + S (pfp_repeat_index_add_output_rightzeros) = (t)) -> (((exists ff_h_pfp_add_output_rightzerosentry. ff_h_pfp_add_output_rightzerosentry + S (0) = S ((S (pfp_repeat_index_add_output_rightzeros)) * BC)) /\\ exists ff_q_pfp_add_output_rightzerosentry. BB = ff_q_pfp_add_output_rightzerosentry * S ((S (pfp_repeat_index_add_output_rightzeros)) * BC) + (0)))) /\\ ((forall pfrep_index_add_output_right pfrep_value_add_output_right. (exists pfa_gap_add_output_rightbound. pfa_gap_add_output_rightbound + S (pfrep_index_add_output_right) = (L)) -> (((exists ff_h_pfp_add_output_rightinput. ff_h_pfp_add_output_rightinput + S (pfrep_value_add_output_right) = S ((S (pfrep_index_add_output_right)) * bc)) /\\ exists ff_q_pfp_add_output_rightinput. bb = ff_q_pfp_add_output_rightinput * S ((S (pfrep_index_add_output_right)) * bc) + (pfrep_value_add_output_right))) -> (((exists ff_h_pfp_add_output_rightoutput. ff_h_pfp_add_output_rightoutput + S (pfrep_value_add_output_right) = S ((S ((t)+pfrep_index_add_output_right)) * BC)) /\\ exists ff_q_pfp_add_output_rightoutput. BB = ff_q_pfp_add_output_rightoutput * S ((S ((t)+pfrep_index_add_output_right)) * BC) + (pfrep_value_add_output_right))))))) -> (forall pfp_index_add_output_padded. (exists pfa_gap_add_output_paddedindex. pfa_gap_add_output_paddedindex + S (pfp_index_add_output_padded) = (t+L)) -> exists pfp_left_add_output_padded pfp_right_add_output_padded pfp_value_add_output_padded. ((((exists ff_h_pfp_add_output_paddedleft. ff_h_pfp_add_output_paddedleft + S (pfp_left_add_output_padded) = S ((S (pfp_index_add_output_padded)) * AC)) /\\ exists ff_q_pfp_add_output_paddedleft. AB = ff_q_pfp_add_output_paddedleft * S ((S (pfp_index_add_output_padded)) * AC) + (pfp_left_add_output_padded))) /\\ (((((exists ff_h_pfp_add_output_paddedright. ff_h_pfp_add_output_paddedright + S (pfp_right_add_output_padded) = S ((S (pfp_index_add_output_padded)) * BC)) /\\ exists ff_q_pfp_add_output_paddedright. BB = ff_q_pfp_add_output_paddedright * S ((S (pfp_index_add_output_padded)) * BC) + (pfp_right_add_output_padded))) /\\ (((((exists ff_h_pfp_add_output_paddedtarget. ff_h_pfp_add_output_paddedtarget + S (pfp_value_add_output_padded) = S ((S (pfp_index_add_output_padded)) * CC)) /\\ exists ff_q_pfp_add_output_paddedtarget. CB = ff_q_pfp_add_output_paddedtarget * S ((S (pfp_index_add_output_padded)) * CC) + (pfp_value_add_output_padded))) /\\ ((((exists pfa_gap_add_output_paddedoperationleft. pfa_gap_add_output_paddedoperationleft + S (pfp_left_add_output_padded) = (p)) /\\ (((exists pfa_gap_add_output_paddedoperationright. pfa_gap_add_output_paddedoperationright + S (pfp_right_add_output_padded) = (p)) /\\ ((((exists pfa_gap_add_output_paddedoperationresultbound. pfa_gap_add_output_paddedoperationresultbound + S (pfp_value_add_output_padded) = (p)) /\\ ((exists pfa_offset_left_add_output_paddedoperationresultcongruence pfa_offset_right_add_output_paddedoperationresultcongruence. ((pfp_left_add_output_padded) + (pfp_right_add_output_padded)) + (p) * pfa_offset_left_add_output_paddedoperationresultcongruence = (pfp_value_add_output_padded) + (p) * pfa_offset_right_add_output_paddedoperationresultcongruence)))))))))))))))) -> (((forall pfp_repeat_index_add_output_resultzeros. (exists pfa_gap_add_output_resultzerosindex. pfa_gap_add_output_resultzerosindex + S (pfp_repeat_index_add_output_resultzeros) = (t)) -> (((exists ff_h_pfp_add_output_resultzerosentry. ff_h_pfp_add_output_resultzerosentry + S (0) = S ((S (pfp_repeat_index_add_output_resultzeros)) * CC)) /\\ exists ff_q_pfp_add_output_resultzerosentry. CB = ff_q_pfp_add_output_resultzerosentry * S ((S (pfp_repeat_index_add_output_resultzeros)) * CC) + (0)))) /\\ ((forall pfrep_index_add_output_result pfrep_value_add_output_result. (exists pfa_gap_add_output_resultbound. pfa_gap_add_output_resultbound + S (pfrep_index_add_output_result) = (L)) -> (((exists ff_h_pfp_add_output_resultinput. ff_h_pfp_add_output_resultinput + S (pfrep_value_add_output_result) = S ((S (pfrep_index_add_output_result)) * cc)) /\\ exists ff_q_pfp_add_output_resultinput. cb = ff_q_pfp_add_output_resultinput * S ((S (pfrep_index_add_output_result)) * cc) + (pfrep_value_add_output_result))) -> (((exists ff_h_pfp_add_output_resultoutput. ff_h_pfp_add_output_resultoutput + S (pfrep_value_add_output_result) = S ((S ((t)+pfrep_index_add_output_result)) * CC)) /\\ exists ff_q_pfp_add_output_resultoutput. CB = ff_q_pfp_add_output_resultoutput * S ((S ((t)+pfrep_index_add_output_result)) * CC) + (pfrep_value_add_output_result)))))))", "statement_sha256": "85020d34a4e4fc2798457c2c48b7de5d49d001b96529738a0f447eabd6027fca", "summary": "Actual add outputs inherit the genuine common leading-zero padding of their inputs: construct a padded original output, prove its operation, then identify the supplied output by functionality.", "summary_sha256": "5233edb5c06d095b4f597fde586bebe3b7b6c526e948f9349acb1afa16fd08f4" }, { "alpha_v33_frontier_enrollment": { "body_receipt_sha256": "9bec1fa221a8c20ac6e31d8c6db0bb4761401454e31f52258468054a04e53ebf", "bundle_campaign": "polynomial-euclidean-division", "bundle_node_id": 370, "bundle_sha256": "6ae667d8518e4dbe722bb08ad1b08715a0d282c2893e533c8133d770fe861dcf", "campaign": "polynomial-euclidean-division", "first_enrolled_version": "v33", "parent_catalog_sha256": "41b9f387d88a5a4f0fe5ee2bd5578f37a27a4657b0a80f1a1a2cb5109f69a623", "rfc_sha256": 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"intro AB", "intro AC", "intro BB", "intro BC", "intro CB", "intro CC", "intro hp", "intro ho", "intro hA", "intro hB", "intro hn", "have hd : exists db dc. (((forall pfp_repeat_index_subtract_output_constructedzeros. (exists pfa_gap_subtract_output_constructedzerosindex. pfa_gap_subtract_output_constructedzerosindex + S (pfp_repeat_index_subtract_output_constructedzeros) = (t)) -> (((exists ff_h_pfp_subtract_output_constructedzerosentry. ff_h_pfp_subtract_output_constructedzerosentry + S (0) = S ((S (pfp_repeat_index_subtract_output_constructedzeros)) * dc)) /\\ exists ff_q_pfp_subtract_output_constructedzerosentry. db = ff_q_pfp_subtract_output_constructedzerosentry * S ((S (pfp_repeat_index_subtract_output_constructedzeros)) * dc) + (0)))) /\\ ((forall pfrep_index_subtract_output_constructed pfrep_value_subtract_output_constructed. (exists pfa_gap_subtract_output_constructedbound. pfa_gap_subtract_output_constructedbound + S (pfrep_index_subtract_output_constructed) = (L)) -> (((exists ff_h_pfp_subtract_output_constructedinput. ff_h_pfp_subtract_output_constructedinput + S (pfrep_value_subtract_output_constructed) = S ((S (pfrep_index_subtract_output_constructed)) * cc)) /\\ exists ff_q_pfp_subtract_output_constructedinput. cb = ff_q_pfp_subtract_output_constructedinput * S ((S (pfrep_index_subtract_output_constructed)) * cc) + (pfrep_value_subtract_output_constructed))) -> (((exists ff_h_pfp_subtract_output_constructedoutput. ff_h_pfp_subtract_output_constructedoutput + S (pfrep_value_subtract_output_constructed) = S ((S ((t)+pfrep_index_subtract_output_constructed)) * dc)) /\\ exists ff_q_pfp_subtract_output_constructedoutput. db = ff_q_pfp_subtract_output_constructedoutput * S ((S ((t)+pfrep_index_subtract_output_constructed)) * dc) + (pfrep_value_subtract_output_constructed)))))))", "specialize prime_field_polynomial_left_pad_exists (cb)", "specialize prime_field_polynomial_left_pad_exists (cc)", "specialize prime_field_polynomial_left_pad_exists (t)", "specialize prime_field_polynomial_left_pad_exists (L)", "apply prime_field_polynomial_left_pad_exists", "cases hd", "cases hd_witness", "have hm : forall pfs_index_subtract_output_middle. (exists pfa_gap_subtract_output_middleindex. pfa_gap_subtract_output_middleindex + S (pfs_index_subtract_output_middle) = (t+L)) -> exists pfs_left_subtract_output_middle pfs_right_subtract_output_middle pfs_result_subtract_output_middle. ((((exists ff_h_pfp_subtract_output_middleleft. ff_h_pfp_subtract_output_middleleft + S (pfs_left_subtract_output_middle) = S ((S (pfs_index_subtract_output_middle)) * AC)) /\\ exists ff_q_pfp_subtract_output_middleleft. AB = ff_q_pfp_subtract_output_middleleft * S ((S (pfs_index_subtract_output_middle)) * AC) + (pfs_left_subtract_output_middle))) /\\ (((((exists ff_h_pfp_subtract_output_middleright. ff_h_pfp_subtract_output_middleright + S (pfs_right_subtract_output_middle) = S ((S (pfs_index_subtract_output_middle)) * BC)) /\\ exists ff_q_pfp_subtract_output_middleright. BB = ff_q_pfp_subtract_output_middleright * S ((S (pfs_index_subtract_output_middle)) * BC) + (pfs_right_subtract_output_middle))) /\\ (((((exists ff_h_pfp_subtract_output_middleresult. ff_h_pfp_subtract_output_middleresult + S (pfs_result_subtract_output_middle) = S ((S (pfs_index_subtract_output_middle)) * x1)) /\\ exists ff_q_pfp_subtract_output_middleresult. x = ff_q_pfp_subtract_output_middleresult * S ((S (pfs_index_subtract_output_middle)) * x1) + (pfs_result_subtract_output_middle))) /\\ ((((exists pfa_gap_subtract_output_middleoperationleft. pfa_gap_subtract_output_middleoperationleft + S (pfs_right_subtract_output_middle) = (p)) /\\ (((exists pfa_gap_subtract_output_middleoperationright. pfa_gap_subtract_output_middleoperationright + S (pfs_result_subtract_output_middle) = (p)) /\\ ((((exists pfa_gap_subtract_output_middleoperationresultbound. pfa_gap_subtract_output_middleoperationresultbound + S (pfs_left_subtract_output_middle) = (p)) /\\ ((exists pfa_offset_left_subtract_output_middleoperationresultcongruence pfa_offset_right_subtract_output_middleoperationresultcongruence. ((pfs_right_subtract_output_middle) + (pfs_result_subtract_output_middle)) + (p) * pfa_offset_left_subtract_output_middleoperationresultcongruence = (pfs_left_subtract_output_middle) + (p) * pfa_offset_right_subtract_output_middleoperationresultcongruence)))))))))))))))", "specialize prime_field_polynomial_subtract_left_pad_transport (p)", "specialize prime_field_polynomial_subtract_left_pad_transport (ab)", "specialize prime_field_polynomial_subtract_left_pad_transport (ac)", "specialize prime_field_polynomial_subtract_left_pad_transport (bb)", "specialize prime_field_polynomial_subtract_left_pad_transport (bc)", "specialize prime_field_polynomial_subtract_left_pad_transport (cb)", "specialize prime_field_polynomial_subtract_left_pad_transport (cc)", "specialize prime_field_polynomial_subtract_left_pad_transport (L)", "specialize prime_field_polynomial_subtract_left_pad_transport (t)", "specialize prime_field_polynomial_subtract_left_pad_transport (AB)", "specialize prime_field_polynomial_subtract_left_pad_transport (AC)", "specialize prime_field_polynomial_subtract_left_pad_transport (BB)", "specialize prime_field_polynomial_subtract_left_pad_transport (BC)", "specialize prime_field_polynomial_subtract_left_pad_transport (x)", "specialize prime_field_polynomial_subtract_left_pad_transport (x1)", "apply prime_field_polynomial_subtract_left_pad_transport", "exact hp", "exact ho", "exact hA", "exact hB", "exact hd_witness_witness", "have he : forall mdr_i_pfp_subtract_output_values mdr_a_pfp_subtract_output_values. (exists mdr_gap_pfp_subtract_output_valuesb. mdr_gap_pfp_subtract_output_valuesb + S (mdr_i_pfp_subtract_output_values) = (t+L)) -> (((exists ff_h_mdr_pfp_subtract_output_valueso. ff_h_mdr_pfp_subtract_output_valueso + S (mdr_a_pfp_subtract_output_values) = S ((S (mdr_i_pfp_subtract_output_values)) * x1)) /\\ exists ff_q_mdr_pfp_subtract_output_valueso. x = ff_q_mdr_pfp_subtract_output_valueso * S ((S (mdr_i_pfp_subtract_output_values)) * x1) + (mdr_a_pfp_subtract_output_values))) -> (((exists ff_h_mdr_pfp_subtract_output_valuesn. ff_h_mdr_pfp_subtract_output_valuesn + S (mdr_a_pfp_subtract_output_values) = S ((S (mdr_i_pfp_subtract_output_values)) * CC)) /\\ exists ff_q_mdr_pfp_subtract_output_valuesn. CB = ff_q_mdr_pfp_subtract_output_valuesn * S ((S (mdr_i_pfp_subtract_output_values)) * CC) + (mdr_a_pfp_subtract_output_values)))", "specialize prime_field_polynomial_subtract_functional (p)", "specialize prime_field_polynomial_subtract_functional (AB)", "specialize prime_field_polynomial_subtract_functional (AC)", "specialize prime_field_polynomial_subtract_functional (BB)", "specialize prime_field_polynomial_subtract_functional (BC)", "specialize prime_field_polynomial_subtract_functional (x)", "specialize prime_field_polynomial_subtract_functional (x1)", "specialize prime_field_polynomial_subtract_functional (CB)", "specialize prime_field_polynomial_subtract_functional (CC)", "specialize prime_field_polynomial_subtract_functional (t+L)", "apply prime_field_polynomial_subtract_functional", "exact hm", "exact hn", "specialize prime_field_polynomial_left_pad_transport (cb)", "specialize prime_field_polynomial_left_pad_transport (cc)", "specialize prime_field_polynomial_left_pad_transport (cb)", "specialize prime_field_polynomial_left_pad_transport (cc)", "specialize prime_field_polynomial_left_pad_transport (L)", "specialize prime_field_polynomial_left_pad_transport (t)", "specialize prime_field_polynomial_left_pad_transport (x)", "specialize prime_field_polynomial_left_pad_transport (x1)", "specialize prime_field_polynomial_left_pad_transport (CB)", "specialize prime_field_polynomial_left_pad_transport (CC)", "apply prime_field_polynomial_left_pad_transport", "intro i", "intro a", "intro hi", "intro ha", "exact ha", "exact he", "exact hd_witness_witness" ], "script_sha256": "0e0b4328d75e5678362f3b32c4be1a3187f3508eea0778057889540ba0055cf5", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_equivalence_candidate.py", "sha256": "929eb67318c8a09577fb9ebac277b82656abf04c82b97a417fff83f39e7bb373" }, "statement": "forall p ab ac bb bc cb cc L t AB AC BB BC CB CC. (~((p) = 1) /\\ forall pfa_factor_left_subtract_output_prime pfa_factor_right_subtract_output_prime. (p) = pfa_factor_left_subtract_output_prime * pfa_factor_right_subtract_output_prime -> pfa_factor_left_subtract_output_prime = 1 \\/ pfa_factor_right_subtract_output_prime = 1) -> (forall pfs_index_subtract_output_original. (exists pfa_gap_subtract_output_originalindex. pfa_gap_subtract_output_originalindex + S (pfs_index_subtract_output_original) = (L)) -> exists pfs_left_subtract_output_original pfs_right_subtract_output_original pfs_result_subtract_output_original. ((((exists ff_h_pfp_subtract_output_originalleft. ff_h_pfp_subtract_output_originalleft + S (pfs_left_subtract_output_original) = S ((S (pfs_index_subtract_output_original)) * ac)) /\\ exists ff_q_pfp_subtract_output_originalleft. ab = ff_q_pfp_subtract_output_originalleft * S ((S (pfs_index_subtract_output_original)) * ac) + (pfs_left_subtract_output_original))) /\\ (((((exists ff_h_pfp_subtract_output_originalright. ff_h_pfp_subtract_output_originalright + S (pfs_right_subtract_output_original) = S ((S (pfs_index_subtract_output_original)) * bc)) /\\ exists ff_q_pfp_subtract_output_originalright. bb = ff_q_pfp_subtract_output_originalright * S ((S (pfs_index_subtract_output_original)) * bc) + (pfs_right_subtract_output_original))) /\\ (((((exists ff_h_pfp_subtract_output_originalresult. ff_h_pfp_subtract_output_originalresult + S (pfs_result_subtract_output_original) = S ((S (pfs_index_subtract_output_original)) * cc)) /\\ exists ff_q_pfp_subtract_output_originalresult. cb = ff_q_pfp_subtract_output_originalresult * S ((S (pfs_index_subtract_output_original)) * cc) + (pfs_result_subtract_output_original))) /\\ ((((exists pfa_gap_subtract_output_originaloperationleft. pfa_gap_subtract_output_originaloperationleft + S (pfs_right_subtract_output_original) = (p)) /\\ (((exists pfa_gap_subtract_output_originaloperationright. pfa_gap_subtract_output_originaloperationright + S (pfs_result_subtract_output_original) = (p)) /\\ ((((exists pfa_gap_subtract_output_originaloperationresultbound. pfa_gap_subtract_output_originaloperationresultbound + S (pfs_left_subtract_output_original) = (p)) /\\ ((exists pfa_offset_left_subtract_output_originaloperationresultcongruence pfa_offset_right_subtract_output_originaloperationresultcongruence. ((pfs_right_subtract_output_original) + (pfs_result_subtract_output_original)) + (p) * pfa_offset_left_subtract_output_originaloperationresultcongruence = (pfs_left_subtract_output_original) + (p) * pfa_offset_right_subtract_output_originaloperationresultcongruence)))))))))))))))) -> (((forall pfp_repeat_index_subtract_output_leftzeros. (exists pfa_gap_subtract_output_leftzerosindex. pfa_gap_subtract_output_leftzerosindex + S (pfp_repeat_index_subtract_output_leftzeros) = (t)) -> (((exists ff_h_pfp_subtract_output_leftzerosentry. ff_h_pfp_subtract_output_leftzerosentry + S (0) = S ((S (pfp_repeat_index_subtract_output_leftzeros)) * AC)) /\\ exists ff_q_pfp_subtract_output_leftzerosentry. AB = ff_q_pfp_subtract_output_leftzerosentry * S ((S (pfp_repeat_index_subtract_output_leftzeros)) * AC) + (0)))) /\\ ((forall pfrep_index_subtract_output_left pfrep_value_subtract_output_left. (exists pfa_gap_subtract_output_leftbound. pfa_gap_subtract_output_leftbound + S (pfrep_index_subtract_output_left) = (L)) -> (((exists ff_h_pfp_subtract_output_leftinput. ff_h_pfp_subtract_output_leftinput + S (pfrep_value_subtract_output_left) = S ((S (pfrep_index_subtract_output_left)) * ac)) /\\ exists ff_q_pfp_subtract_output_leftinput. ab = ff_q_pfp_subtract_output_leftinput * S ((S (pfrep_index_subtract_output_left)) * ac) + (pfrep_value_subtract_output_left))) -> (((exists ff_h_pfp_subtract_output_leftoutput. ff_h_pfp_subtract_output_leftoutput + S (pfrep_value_subtract_output_left) = S ((S ((t)+pfrep_index_subtract_output_left)) * AC)) /\\ exists ff_q_pfp_subtract_output_leftoutput. AB = ff_q_pfp_subtract_output_leftoutput * S ((S ((t)+pfrep_index_subtract_output_left)) * AC) + (pfrep_value_subtract_output_left))))))) -> (((forall pfp_repeat_index_subtract_output_rightzeros. (exists pfa_gap_subtract_output_rightzerosindex. pfa_gap_subtract_output_rightzerosindex + S (pfp_repeat_index_subtract_output_rightzeros) = (t)) -> (((exists ff_h_pfp_subtract_output_rightzerosentry. ff_h_pfp_subtract_output_rightzerosentry + S (0) = S ((S (pfp_repeat_index_subtract_output_rightzeros)) * BC)) /\\ exists ff_q_pfp_subtract_output_rightzerosentry. BB = ff_q_pfp_subtract_output_rightzerosentry * S ((S (pfp_repeat_index_subtract_output_rightzeros)) * BC) + (0)))) /\\ ((forall pfrep_index_subtract_output_right pfrep_value_subtract_output_right. (exists pfa_gap_subtract_output_rightbound. pfa_gap_subtract_output_rightbound + S (pfrep_index_subtract_output_right) = (L)) -> (((exists ff_h_pfp_subtract_output_rightinput. ff_h_pfp_subtract_output_rightinput + S (pfrep_value_subtract_output_right) = S ((S (pfrep_index_subtract_output_right)) * bc)) /\\ exists ff_q_pfp_subtract_output_rightinput. bb = ff_q_pfp_subtract_output_rightinput * S ((S (pfrep_index_subtract_output_right)) * bc) + (pfrep_value_subtract_output_right))) -> (((exists ff_h_pfp_subtract_output_rightoutput. ff_h_pfp_subtract_output_rightoutput + S (pfrep_value_subtract_output_right) = S ((S ((t)+pfrep_index_subtract_output_right)) * BC)) /\\ exists ff_q_pfp_subtract_output_rightoutput. BB = ff_q_pfp_subtract_output_rightoutput * S ((S ((t)+pfrep_index_subtract_output_right)) * BC) + (pfrep_value_subtract_output_right))))))) -> (forall pfs_index_subtract_output_padded. (exists pfa_gap_subtract_output_paddedindex. pfa_gap_subtract_output_paddedindex + S (pfs_index_subtract_output_padded) = (t+L)) -> exists pfs_left_subtract_output_padded pfs_right_subtract_output_padded pfs_result_subtract_output_padded. ((((exists ff_h_pfp_subtract_output_paddedleft. ff_h_pfp_subtract_output_paddedleft + S (pfs_left_subtract_output_padded) = S ((S (pfs_index_subtract_output_padded)) * AC)) /\\ exists ff_q_pfp_subtract_output_paddedleft. AB = ff_q_pfp_subtract_output_paddedleft * S ((S (pfs_index_subtract_output_padded)) * AC) + (pfs_left_subtract_output_padded))) /\\ (((((exists ff_h_pfp_subtract_output_paddedright. ff_h_pfp_subtract_output_paddedright + S (pfs_right_subtract_output_padded) = S ((S (pfs_index_subtract_output_padded)) * BC)) /\\ exists ff_q_pfp_subtract_output_paddedright. BB = ff_q_pfp_subtract_output_paddedright * S ((S (pfs_index_subtract_output_padded)) * BC) + (pfs_right_subtract_output_padded))) /\\ (((((exists ff_h_pfp_subtract_output_paddedresult. ff_h_pfp_subtract_output_paddedresult + S (pfs_result_subtract_output_padded) = S ((S (pfs_index_subtract_output_padded)) * CC)) /\\ exists ff_q_pfp_subtract_output_paddedresult. CB = ff_q_pfp_subtract_output_paddedresult * S ((S (pfs_index_subtract_output_padded)) * CC) + (pfs_result_subtract_output_padded))) /\\ ((((exists pfa_gap_subtract_output_paddedoperationleft. pfa_gap_subtract_output_paddedoperationleft + S (pfs_right_subtract_output_padded) = (p)) /\\ (((exists pfa_gap_subtract_output_paddedoperationright. pfa_gap_subtract_output_paddedoperationright + S (pfs_result_subtract_output_padded) = (p)) /\\ ((((exists pfa_gap_subtract_output_paddedoperationresultbound. pfa_gap_subtract_output_paddedoperationresultbound + S (pfs_left_subtract_output_padded) = (p)) /\\ ((exists pfa_offset_left_subtract_output_paddedoperationresultcongruence pfa_offset_right_subtract_output_paddedoperationresultcongruence. ((pfs_right_subtract_output_padded) + (pfs_result_subtract_output_padded)) + (p) * pfa_offset_left_subtract_output_paddedoperationresultcongruence = (pfs_left_subtract_output_padded) + (p) * pfa_offset_right_subtract_output_paddedoperationresultcongruence)))))))))))))))) -> (((forall pfp_repeat_index_subtract_output_resultzeros. (exists pfa_gap_subtract_output_resultzerosindex. pfa_gap_subtract_output_resultzerosindex + S (pfp_repeat_index_subtract_output_resultzeros) = (t)) -> (((exists ff_h_pfp_subtract_output_resultzerosentry. ff_h_pfp_subtract_output_resultzerosentry + S (0) = S ((S (pfp_repeat_index_subtract_output_resultzeros)) * CC)) /\\ exists ff_q_pfp_subtract_output_resultzerosentry. CB = ff_q_pfp_subtract_output_resultzerosentry * S ((S (pfp_repeat_index_subtract_output_resultzeros)) * CC) + (0)))) /\\ ((forall pfrep_index_subtract_output_result pfrep_value_subtract_output_result. (exists pfa_gap_subtract_output_resultbound. pfa_gap_subtract_output_resultbound + S (pfrep_index_subtract_output_result) = (L)) -> (((exists ff_h_pfp_subtract_output_resultinput. ff_h_pfp_subtract_output_resultinput + S (pfrep_value_subtract_output_result) = S ((S (pfrep_index_subtract_output_result)) * cc)) /\\ exists ff_q_pfp_subtract_output_resultinput. cb = ff_q_pfp_subtract_output_resultinput * S ((S (pfrep_index_subtract_output_result)) * cc) + (pfrep_value_subtract_output_result))) -> (((exists ff_h_pfp_subtract_output_resultoutput. ff_h_pfp_subtract_output_resultoutput + S (pfrep_value_subtract_output_result) = S ((S ((t)+pfrep_index_subtract_output_result)) * CC)) /\\ exists ff_q_pfp_subtract_output_resultoutput. CB = ff_q_pfp_subtract_output_resultoutput * S ((S ((t)+pfrep_index_subtract_output_result)) * CC) + (pfrep_value_subtract_output_result)))))))", "statement_sha256": "ee00e0ee9f0ba58b96f8b35f521617f69ddc4ae4effae8388f85770f376f38cf", "summary": "Actual subtract outputs inherit the genuine common leading-zero padding of their inputs: construct a padded original output, prove its operation, then identify the supplied output by functionality.", "summary_sha256": "af9258ad3211b4880abfe3cda909c1f1b4dde9f2ae456ee88006e5766ce2baba" }, { "alpha_v33_frontier_enrollment": { "body_receipt_sha256": "9656a26b310d8c93e09d36b3d93ce22471c51ab9030287da2fdadda88079b6b8", "bundle_campaign": "polynomial-euclidean-division", "bundle_node_id": 371, "bundle_sha256": "6ae667d8518e4dbe722bb08ad1b08715a0d282c2893e533c8133d770fe861dcf", "campaign": "polynomial-euclidean-division", "first_enrolled_version": "v33", "parent_catalog_sha256": "41b9f387d88a5a4f0fe5ee2bd5578f37a27a4657b0a80f1a1a2cb5109f69a623", 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"intro L", "intro AB", "intro AC", "intro BB", "intro BC", "intro CB", "intro CC", "intro K", "intro hp", "intro hA", "intro hB", "intro ho", "intro hn", "have horder : L<=K \\/ K<=L", "specialize le_total (L)", "specialize le_total (K)", "apply le_total", "cases horder", "cases horder_left", "rewrite <- horder_left_witness at hA", "rewrite <- horder_left_witness at hA", "rewrite <- horder_left_witness at hB", "rewrite <- horder_left_witness at hB", "rewrite <- horder_left_witness at hn", "rewrite <- horder_left_witness", "rewrite <- horder_left_witness", "have hpadA : ((forall pfp_repeat_index_add_congruent_forward_hpadAzeros. (exists pfa_gap_add_congruent_forward_hpadAzerosindex. pfa_gap_add_congruent_forward_hpadAzerosindex + S (pfp_repeat_index_add_congruent_forward_hpadAzeros) = (x)) -> (((exists ff_h_pfp_add_congruent_forward_hpadAzerosentry. ff_h_pfp_add_congruent_forward_hpadAzerosentry + S (0) = S ((S (pfp_repeat_index_add_congruent_forward_hpadAzeros)) * AC)) /\\ exists ff_q_pfp_add_congruent_forward_hpadAzerosentry. AB = ff_q_pfp_add_congruent_forward_hpadAzerosentry * S ((S (pfp_repeat_index_add_congruent_forward_hpadAzeros)) * AC) + (0)))) /\\ ((forall pfrep_index_add_congruent_forward_hpadA pfrep_value_add_congruent_forward_hpadA. (exists pfa_gap_add_congruent_forward_hpadAbound. pfa_gap_add_congruent_forward_hpadAbound + S (pfrep_index_add_congruent_forward_hpadA) = (L)) -> (((exists ff_h_pfp_add_congruent_forward_hpadAinput. ff_h_pfp_add_congruent_forward_hpadAinput + S (pfrep_value_add_congruent_forward_hpadA) = S ((S (pfrep_index_add_congruent_forward_hpadA)) * ac)) /\\ exists ff_q_pfp_add_congruent_forward_hpadAinput. ab = ff_q_pfp_add_congruent_forward_hpadAinput * S ((S (pfrep_index_add_congruent_forward_hpadA)) * ac) + (pfrep_value_add_congruent_forward_hpadA))) -> (((exists ff_h_pfp_add_congruent_forward_hpadAoutput. ff_h_pfp_add_congruent_forward_hpadAoutput + S (pfrep_value_add_congruent_forward_hpadA) = S ((S ((x)+pfrep_index_add_congruent_forward_hpadA)) * AC)) /\\ exists ff_q_pfp_add_congruent_forward_hpadAoutput. AB = ff_q_pfp_add_congruent_forward_hpadAoutput * S ((S ((x)+pfrep_index_add_congruent_forward_hpadA)) * AC) + (pfrep_value_add_congruent_forward_hpadA))))))", "specialize prime_field_polynomial_equivalent_implies_left_pad (ab)", "specialize prime_field_polynomial_equivalent_implies_left_pad (ac)", "specialize prime_field_polynomial_equivalent_implies_left_pad (L)", "specialize prime_field_polynomial_equivalent_implies_left_pad (x)", "specialize prime_field_polynomial_equivalent_implies_left_pad (AB)", "specialize prime_field_polynomial_equivalent_implies_left_pad (AC)", "apply prime_field_polynomial_equivalent_implies_left_pad", "exact hA", "have hpadB : ((forall pfp_repeat_index_add_congruent_forward_hpadBzeros. (exists pfa_gap_add_congruent_forward_hpadBzerosindex. pfa_gap_add_congruent_forward_hpadBzerosindex + S (pfp_repeat_index_add_congruent_forward_hpadBzeros) = (x)) -> (((exists ff_h_pfp_add_congruent_forward_hpadBzerosentry. ff_h_pfp_add_congruent_forward_hpadBzerosentry + S (0) = S ((S (pfp_repeat_index_add_congruent_forward_hpadBzeros)) * BC)) /\\ exists ff_q_pfp_add_congruent_forward_hpadBzerosentry. BB = ff_q_pfp_add_congruent_forward_hpadBzerosentry * S ((S (pfp_repeat_index_add_congruent_forward_hpadBzeros)) * BC) + (0)))) /\\ ((forall pfrep_index_add_congruent_forward_hpadB pfrep_value_add_congruent_forward_hpadB. (exists pfa_gap_add_congruent_forward_hpadBbound. pfa_gap_add_congruent_forward_hpadBbound + S (pfrep_index_add_congruent_forward_hpadB) = (L)) -> (((exists ff_h_pfp_add_congruent_forward_hpadBinput. ff_h_pfp_add_congruent_forward_hpadBinput + S (pfrep_value_add_congruent_forward_hpadB) = S ((S (pfrep_index_add_congruent_forward_hpadB)) * bc)) /\\ exists ff_q_pfp_add_congruent_forward_hpadBinput. bb = ff_q_pfp_add_congruent_forward_hpadBinput * S ((S (pfrep_index_add_congruent_forward_hpadB)) * bc) + (pfrep_value_add_congruent_forward_hpadB))) -> (((exists ff_h_pfp_add_congruent_forward_hpadBoutput. ff_h_pfp_add_congruent_forward_hpadBoutput + S (pfrep_value_add_congruent_forward_hpadB) = S ((S ((x)+pfrep_index_add_congruent_forward_hpadB)) * BC)) /\\ exists ff_q_pfp_add_congruent_forward_hpadBoutput. BB = ff_q_pfp_add_congruent_forward_hpadBoutput * S ((S ((x)+pfrep_index_add_congruent_forward_hpadB)) * BC) + (pfrep_value_add_congruent_forward_hpadB))))))", "specialize prime_field_polynomial_equivalent_implies_left_pad (bb)", "specialize prime_field_polynomial_equivalent_implies_left_pad (bc)", "specialize prime_field_polynomial_equivalent_implies_left_pad (L)", "specialize prime_field_polynomial_equivalent_implies_left_pad (x)", "specialize prime_field_polynomial_equivalent_implies_left_pad (BB)", "specialize prime_field_polynomial_equivalent_implies_left_pad (BC)", "apply prime_field_polynomial_equivalent_implies_left_pad", "exact hB", "specialize prime_field_polynomial_left_pad_equivalent (cb)", "specialize prime_field_polynomial_left_pad_equivalent (cc)", "specialize prime_field_polynomial_left_pad_equivalent (L)", "specialize prime_field_polynomial_left_pad_equivalent (x)", "specialize prime_field_polynomial_left_pad_equivalent (CB)", "specialize prime_field_polynomial_left_pad_equivalent (CC)", "apply prime_field_polynomial_left_pad_equivalent", "specialize prime_field_polynomial_add_left_pad_output (p)", "specialize prime_field_polynomial_add_left_pad_output (ab)", "specialize prime_field_polynomial_add_left_pad_output (ac)", "specialize prime_field_polynomial_add_left_pad_output (bb)", "specialize prime_field_polynomial_add_left_pad_output (bc)", "specialize prime_field_polynomial_add_left_pad_output (cb)", "specialize prime_field_polynomial_add_left_pad_output (cc)", "specialize prime_field_polynomial_add_left_pad_output (L)", "specialize prime_field_polynomial_add_left_pad_output (x)", "specialize prime_field_polynomial_add_left_pad_output (AB)", "specialize prime_field_polynomial_add_left_pad_output (AC)", "specialize prime_field_polynomial_add_left_pad_output (BB)", "specialize prime_field_polynomial_add_left_pad_output (BC)", "specialize prime_field_polynomial_add_left_pad_output (CB)", "specialize prime_field_polynomial_add_left_pad_output (CC)", "apply prime_field_polynomial_add_left_pad_output", "exact hp", "exact ho", "exact hpadA", "exact hpadB", "exact hn", "cases horder_right", "rewrite <- horder_right_witness at hA", "rewrite <- horder_right_witness at hA", "rewrite <- horder_right_witness at hB", "rewrite <- horder_right_witness at hB", "rewrite <- horder_right_witness at ho", "rewrite <- horder_right_witness", "rewrite <- horder_right_witness", "have hpadA : ((forall pfp_repeat_index_add_congruent_reverse_hpadAzeros. (exists pfa_gap_add_congruent_reverse_hpadAzerosindex. pfa_gap_add_congruent_reverse_hpadAzerosindex + S (pfp_repeat_index_add_congruent_reverse_hpadAzeros) = (x)) -> (((exists ff_h_pfp_add_congruent_reverse_hpadAzerosentry. ff_h_pfp_add_congruent_reverse_hpadAzerosentry + S (0) = S ((S (pfp_repeat_index_add_congruent_reverse_hpadAzeros)) * ac)) /\\ exists ff_q_pfp_add_congruent_reverse_hpadAzerosentry. ab = ff_q_pfp_add_congruent_reverse_hpadAzerosentry * S ((S (pfp_repeat_index_add_congruent_reverse_hpadAzeros)) * ac) + (0)))) /\\ ((forall pfrep_index_add_congruent_reverse_hpadA pfrep_value_add_congruent_reverse_hpadA. (exists pfa_gap_add_congruent_reverse_hpadAbound. pfa_gap_add_congruent_reverse_hpadAbound + S (pfrep_index_add_congruent_reverse_hpadA) = (K)) -> (((exists ff_h_pfp_add_congruent_reverse_hpadAinput. ff_h_pfp_add_congruent_reverse_hpadAinput + S (pfrep_value_add_congruent_reverse_hpadA) = S ((S (pfrep_index_add_congruent_reverse_hpadA)) * AC)) /\\ exists ff_q_pfp_add_congruent_reverse_hpadAinput. AB = ff_q_pfp_add_congruent_reverse_hpadAinput * S ((S (pfrep_index_add_congruent_reverse_hpadA)) * AC) + (pfrep_value_add_congruent_reverse_hpadA))) -> (((exists ff_h_pfp_add_congruent_reverse_hpadAoutput. ff_h_pfp_add_congruent_reverse_hpadAoutput + S (pfrep_value_add_congruent_reverse_hpadA) = S ((S ((x)+pfrep_index_add_congruent_reverse_hpadA)) * ac)) /\\ exists ff_q_pfp_add_congruent_reverse_hpadAoutput. ab = ff_q_pfp_add_congruent_reverse_hpadAoutput * S ((S ((x)+pfrep_index_add_congruent_reverse_hpadA)) * ac) + (pfrep_value_add_congruent_reverse_hpadA))))))", "specialize prime_field_polynomial_equivalent_implies_left_pad (AB)", "specialize prime_field_polynomial_equivalent_implies_left_pad (AC)", "specialize prime_field_polynomial_equivalent_implies_left_pad (K)", "specialize prime_field_polynomial_equivalent_implies_left_pad (x)", "specialize prime_field_polynomial_equivalent_implies_left_pad (ab)", "specialize prime_field_polynomial_equivalent_implies_left_pad (ac)", "apply prime_field_polynomial_equivalent_implies_left_pad", "specialize prime_field_polynomial_equivalent_symmetric (ab)", "specialize prime_field_polynomial_equivalent_symmetric (ac)", "specialize prime_field_polynomial_equivalent_symmetric (x+K)", "specialize prime_field_polynomial_equivalent_symmetric (AB)", "specialize prime_field_polynomial_equivalent_symmetric (AC)", "specialize prime_field_polynomial_equivalent_symmetric (K)", "apply prime_field_polynomial_equivalent_symmetric", "exact hA", "have hpadB : ((forall pfp_repeat_index_add_congruent_reverse_hpadBzeros. (exists pfa_gap_add_congruent_reverse_hpadBzerosindex. pfa_gap_add_congruent_reverse_hpadBzerosindex + S (pfp_repeat_index_add_congruent_reverse_hpadBzeros) = (x)) -> (((exists ff_h_pfp_add_congruent_reverse_hpadBzerosentry. ff_h_pfp_add_congruent_reverse_hpadBzerosentry + S (0) = S ((S (pfp_repeat_index_add_congruent_reverse_hpadBzeros)) * bc)) /\\ exists ff_q_pfp_add_congruent_reverse_hpadBzerosentry. bb = ff_q_pfp_add_congruent_reverse_hpadBzerosentry * S ((S (pfp_repeat_index_add_congruent_reverse_hpadBzeros)) * bc) + (0)))) /\\ ((forall pfrep_index_add_congruent_reverse_hpadB pfrep_value_add_congruent_reverse_hpadB. (exists pfa_gap_add_congruent_reverse_hpadBbound. pfa_gap_add_congruent_reverse_hpadBbound + S (pfrep_index_add_congruent_reverse_hpadB) = (K)) -> (((exists ff_h_pfp_add_congruent_reverse_hpadBinput. ff_h_pfp_add_congruent_reverse_hpadBinput + S (pfrep_value_add_congruent_reverse_hpadB) = S ((S (pfrep_index_add_congruent_reverse_hpadB)) * BC)) /\\ exists ff_q_pfp_add_congruent_reverse_hpadBinput. BB = ff_q_pfp_add_congruent_reverse_hpadBinput * S ((S (pfrep_index_add_congruent_reverse_hpadB)) * BC) + (pfrep_value_add_congruent_reverse_hpadB))) -> (((exists ff_h_pfp_add_congruent_reverse_hpadBoutput. ff_h_pfp_add_congruent_reverse_hpadBoutput + S (pfrep_value_add_congruent_reverse_hpadB) = S ((S ((x)+pfrep_index_add_congruent_reverse_hpadB)) * bc)) /\\ exists ff_q_pfp_add_congruent_reverse_hpadBoutput. bb = ff_q_pfp_add_congruent_reverse_hpadBoutput * S ((S ((x)+pfrep_index_add_congruent_reverse_hpadB)) * bc) + (pfrep_value_add_congruent_reverse_hpadB))))))", "specialize prime_field_polynomial_equivalent_implies_left_pad (BB)", "specialize prime_field_polynomial_equivalent_implies_left_pad (BC)", "specialize prime_field_polynomial_equivalent_implies_left_pad (K)", "specialize prime_field_polynomial_equivalent_implies_left_pad (x)", "specialize prime_field_polynomial_equivalent_implies_left_pad (bb)", "specialize prime_field_polynomial_equivalent_implies_left_pad (bc)", "apply prime_field_polynomial_equivalent_implies_left_pad", "specialize prime_field_polynomial_equivalent_symmetric (bb)", "specialize prime_field_polynomial_equivalent_symmetric (bc)", "specialize prime_field_polynomial_equivalent_symmetric (x+K)", "specialize prime_field_polynomial_equivalent_symmetric (BB)", "specialize prime_field_polynomial_equivalent_symmetric (BC)", "specialize prime_field_polynomial_equivalent_symmetric (K)", "apply prime_field_polynomial_equivalent_symmetric", "exact hB", "specialize prime_field_polynomial_equivalent_symmetric (CB)", "specialize prime_field_polynomial_equivalent_symmetric (CC)", "specialize prime_field_polynomial_equivalent_symmetric (K)", "specialize prime_field_polynomial_equivalent_symmetric (cb)", "specialize prime_field_polynomial_equivalent_symmetric (cc)", "specialize prime_field_polynomial_equivalent_symmetric (x+K)", "apply prime_field_polynomial_equivalent_symmetric", "specialize prime_field_polynomial_left_pad_equivalent (CB)", "specialize prime_field_polynomial_left_pad_equivalent (CC)", "specialize prime_field_polynomial_left_pad_equivalent (K)", "specialize prime_field_polynomial_left_pad_equivalent (x)", "specialize prime_field_polynomial_left_pad_equivalent (cb)", "specialize prime_field_polynomial_left_pad_equivalent (cc)", "apply prime_field_polynomial_left_pad_equivalent", "specialize prime_field_polynomial_add_left_pad_output (p)", "specialize prime_field_polynomial_add_left_pad_output (AB)", "specialize prime_field_polynomial_add_left_pad_output (AC)", "specialize prime_field_polynomial_add_left_pad_output (BB)", "specialize prime_field_polynomial_add_left_pad_output (BC)", "specialize prime_field_polynomial_add_left_pad_output (CB)", "specialize prime_field_polynomial_add_left_pad_output (CC)", "specialize prime_field_polynomial_add_left_pad_output (K)", "specialize prime_field_polynomial_add_left_pad_output (x)", "specialize prime_field_polynomial_add_left_pad_output (ab)", "specialize prime_field_polynomial_add_left_pad_output (ac)", "specialize prime_field_polynomial_add_left_pad_output (bb)", "specialize prime_field_polynomial_add_left_pad_output (bc)", "specialize prime_field_polynomial_add_left_pad_output (cb)", "specialize prime_field_polynomial_add_left_pad_output (cc)", "apply prime_field_polynomial_add_left_pad_output", "exact hp", "exact hn", "exact hpadA", "exact hpadB", "exact ho" ], "script_sha256": "4724b3a0c501013eaf4a1fb6a6a40fdbadfa483d31ff75e8271298c2e4911665", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_equivalence_candidate.py", "sha256": "929eb67318c8a09577fb9ebac277b82656abf04c82b97a417fff83f39e7bb373" }, "statement": "forall p ab ac bb bc cb cc L AB AC BB BC CB CC K. (~((p) = 1) /\\ forall pfa_factor_left_add_congruent_prime pfa_factor_right_add_congruent_prime. (p) = pfa_factor_left_add_congruent_prime * pfa_factor_right_add_congruent_prime -> pfa_factor_left_add_congruent_prime = 1 \\/ pfa_factor_right_add_congruent_prime = 1) -> (forall pfrep_power_add_congruent_first pfrep_left_add_congruent_first pfrep_right_add_congruent_first. ((exists pfrep_position_add_congruent_firstfirst. ((pfrep_position_add_congruent_firstfirst+S (pfrep_power_add_congruent_first)=(L)) /\\ ((((exists ff_h_pfp_add_congruent_firstfirstentry. ff_h_pfp_add_congruent_firstfirstentry + S (pfrep_left_add_congruent_first) = S ((S (pfrep_position_add_congruent_firstfirst)) * ac)) /\\ exists ff_q_pfp_add_congruent_firstfirstentry. ab = ff_q_pfp_add_congruent_firstfirstentry * S ((S (pfrep_position_add_congruent_firstfirst)) * ac) + (pfrep_left_add_congruent_first)))))) \\/ (((exists pfrep_gap_add_congruent_firstfirstoutside. pfrep_gap_add_congruent_firstfirstoutside+(L)=(pfrep_power_add_congruent_first)) /\\ (((pfrep_left_add_congruent_first)=0))))) -> ((exists pfrep_position_add_congruent_firstsecond. ((pfrep_position_add_congruent_firstsecond+S (pfrep_power_add_congruent_first)=(K)) /\\ ((((exists ff_h_pfp_add_congruent_firstsecondentry. ff_h_pfp_add_congruent_firstsecondentry + S (pfrep_right_add_congruent_first) = S ((S (pfrep_position_add_congruent_firstsecond)) * AC)) /\\ exists ff_q_pfp_add_congruent_firstsecondentry. AB = ff_q_pfp_add_congruent_firstsecondentry * S ((S (pfrep_position_add_congruent_firstsecond)) * AC) + (pfrep_right_add_congruent_first)))))) \\/ (((exists pfrep_gap_add_congruent_firstsecondoutside. pfrep_gap_add_congruent_firstsecondoutside+(K)=(pfrep_power_add_congruent_first)) /\\ (((pfrep_right_add_congruent_first)=0))))) -> pfrep_left_add_congruent_first=pfrep_right_add_congruent_first) -> (forall pfrep_power_add_congruent_second pfrep_left_add_congruent_second pfrep_right_add_congruent_second. ((exists pfrep_position_add_congruent_secondfirst. ((pfrep_position_add_congruent_secondfirst+S (pfrep_power_add_congruent_second)=(L)) /\\ ((((exists ff_h_pfp_add_congruent_secondfirstentry. ff_h_pfp_add_congruent_secondfirstentry + S (pfrep_left_add_congruent_second) = S ((S (pfrep_position_add_congruent_secondfirst)) * bc)) /\\ exists ff_q_pfp_add_congruent_secondfirstentry. bb = ff_q_pfp_add_congruent_secondfirstentry * S ((S (pfrep_position_add_congruent_secondfirst)) * bc) + (pfrep_left_add_congruent_second)))))) \\/ (((exists pfrep_gap_add_congruent_secondfirstoutside. pfrep_gap_add_congruent_secondfirstoutside+(L)=(pfrep_power_add_congruent_second)) /\\ (((pfrep_left_add_congruent_second)=0))))) -> ((exists pfrep_position_add_congruent_secondsecond. ((pfrep_position_add_congruent_secondsecond+S (pfrep_power_add_congruent_second)=(K)) /\\ ((((exists ff_h_pfp_add_congruent_secondsecondentry. ff_h_pfp_add_congruent_secondsecondentry + S (pfrep_right_add_congruent_second) = S ((S (pfrep_position_add_congruent_secondsecond)) * BC)) /\\ exists ff_q_pfp_add_congruent_secondsecondentry. BB = ff_q_pfp_add_congruent_secondsecondentry * S ((S (pfrep_position_add_congruent_secondsecond)) * BC) + (pfrep_right_add_congruent_second)))))) \\/ (((exists pfrep_gap_add_congruent_secondsecondoutside. pfrep_gap_add_congruent_secondsecondoutside+(K)=(pfrep_power_add_congruent_second)) /\\ (((pfrep_right_add_congruent_second)=0))))) -> pfrep_left_add_congruent_second=pfrep_right_add_congruent_second) -> (forall pfp_index_add_congruent_original. (exists pfa_gap_add_congruent_originalindex. pfa_gap_add_congruent_originalindex + S (pfp_index_add_congruent_original) = (L)) -> exists pfp_left_add_congruent_original pfp_right_add_congruent_original pfp_value_add_congruent_original. ((((exists ff_h_pfp_add_congruent_originalleft. ff_h_pfp_add_congruent_originalleft + S (pfp_left_add_congruent_original) = S ((S (pfp_index_add_congruent_original)) * ac)) /\\ exists ff_q_pfp_add_congruent_originalleft. ab = ff_q_pfp_add_congruent_originalleft * S ((S (pfp_index_add_congruent_original)) * ac) + (pfp_left_add_congruent_original))) /\\ (((((exists ff_h_pfp_add_congruent_originalright. ff_h_pfp_add_congruent_originalright + S (pfp_right_add_congruent_original) = S ((S (pfp_index_add_congruent_original)) * bc)) /\\ exists ff_q_pfp_add_congruent_originalright. bb = ff_q_pfp_add_congruent_originalright * S ((S (pfp_index_add_congruent_original)) * bc) + (pfp_right_add_congruent_original))) /\\ (((((exists ff_h_pfp_add_congruent_originaltarget. ff_h_pfp_add_congruent_originaltarget + S (pfp_value_add_congruent_original) = S ((S (pfp_index_add_congruent_original)) * cc)) /\\ exists ff_q_pfp_add_congruent_originaltarget. cb = ff_q_pfp_add_congruent_originaltarget * S ((S (pfp_index_add_congruent_original)) * cc) + (pfp_value_add_congruent_original))) /\\ ((((exists pfa_gap_add_congruent_originaloperationleft. pfa_gap_add_congruent_originaloperationleft + S (pfp_left_add_congruent_original) = (p)) /\\ (((exists pfa_gap_add_congruent_originaloperationright. pfa_gap_add_congruent_originaloperationright + S (pfp_right_add_congruent_original) = (p)) /\\ ((((exists pfa_gap_add_congruent_originaloperationresultbound. pfa_gap_add_congruent_originaloperationresultbound + S (pfp_value_add_congruent_original) = (p)) /\\ ((exists pfa_offset_left_add_congruent_originaloperationresultcongruence pfa_offset_right_add_congruent_originaloperationresultcongruence. ((pfp_left_add_congruent_original) + (pfp_right_add_congruent_original)) + (p) * pfa_offset_left_add_congruent_originaloperationresultcongruence = (pfp_value_add_congruent_original) + (p) * pfa_offset_right_add_congruent_originaloperationresultcongruence)))))))))))))))) -> (forall pfp_index_add_congruent_other. (exists pfa_gap_add_congruent_otherindex. pfa_gap_add_congruent_otherindex + S (pfp_index_add_congruent_other) = (K)) -> exists pfp_left_add_congruent_other pfp_right_add_congruent_other pfp_value_add_congruent_other. ((((exists ff_h_pfp_add_congruent_otherleft. ff_h_pfp_add_congruent_otherleft + S (pfp_left_add_congruent_other) = S ((S (pfp_index_add_congruent_other)) * AC)) /\\ exists ff_q_pfp_add_congruent_otherleft. AB = ff_q_pfp_add_congruent_otherleft * S ((S (pfp_index_add_congruent_other)) * AC) + (pfp_left_add_congruent_other))) /\\ (((((exists ff_h_pfp_add_congruent_otherright. ff_h_pfp_add_congruent_otherright + S (pfp_right_add_congruent_other) = S ((S (pfp_index_add_congruent_other)) * BC)) /\\ exists ff_q_pfp_add_congruent_otherright. BB = ff_q_pfp_add_congruent_otherright * S ((S (pfp_index_add_congruent_other)) * BC) + (pfp_right_add_congruent_other))) /\\ (((((exists ff_h_pfp_add_congruent_othertarget. ff_h_pfp_add_congruent_othertarget + S (pfp_value_add_congruent_other) = S ((S (pfp_index_add_congruent_other)) * CC)) /\\ exists ff_q_pfp_add_congruent_othertarget. CB = ff_q_pfp_add_congruent_othertarget * S ((S (pfp_index_add_congruent_other)) * CC) + (pfp_value_add_congruent_other))) /\\ ((((exists pfa_gap_add_congruent_otheroperationleft. pfa_gap_add_congruent_otheroperationleft + S (pfp_left_add_congruent_other) = (p)) /\\ (((exists pfa_gap_add_congruent_otheroperationright. pfa_gap_add_congruent_otheroperationright + S (pfp_right_add_congruent_other) = (p)) /\\ ((((exists pfa_gap_add_congruent_otheroperationresultbound. pfa_gap_add_congruent_otheroperationresultbound + S (pfp_value_add_congruent_other) = (p)) /\\ ((exists pfa_offset_left_add_congruent_otheroperationresultcongruence pfa_offset_right_add_congruent_otheroperationresultcongruence. ((pfp_left_add_congruent_other) + (pfp_right_add_congruent_other)) + (p) * pfa_offset_left_add_congruent_otheroperationresultcongruence = (pfp_value_add_congruent_other) + (p) * pfa_offset_right_add_congruent_otheroperationresultcongruence)))))))))))))))) -> (forall pfrep_power_add_congruent_result pfrep_left_add_congruent_result pfrep_right_add_congruent_result. ((exists pfrep_position_add_congruent_resultfirst. ((pfrep_position_add_congruent_resultfirst+S (pfrep_power_add_congruent_result)=(L)) /\\ ((((exists ff_h_pfp_add_congruent_resultfirstentry. ff_h_pfp_add_congruent_resultfirstentry + S (pfrep_left_add_congruent_result) = S ((S (pfrep_position_add_congruent_resultfirst)) * cc)) /\\ exists ff_q_pfp_add_congruent_resultfirstentry. cb = ff_q_pfp_add_congruent_resultfirstentry * S ((S (pfrep_position_add_congruent_resultfirst)) * cc) + (pfrep_left_add_congruent_result)))))) \\/ (((exists pfrep_gap_add_congruent_resultfirstoutside. pfrep_gap_add_congruent_resultfirstoutside+(L)=(pfrep_power_add_congruent_result)) /\\ (((pfrep_left_add_congruent_result)=0))))) -> ((exists pfrep_position_add_congruent_resultsecond. ((pfrep_position_add_congruent_resultsecond+S (pfrep_power_add_congruent_result)=(K)) /\\ ((((exists ff_h_pfp_add_congruent_resultsecondentry. ff_h_pfp_add_congruent_resultsecondentry + S (pfrep_right_add_congruent_result) = S ((S (pfrep_position_add_congruent_resultsecond)) * CC)) /\\ exists ff_q_pfp_add_congruent_resultsecondentry. CB = ff_q_pfp_add_congruent_resultsecondentry * S ((S (pfrep_position_add_congruent_resultsecond)) * CC) + (pfrep_right_add_congruent_result)))))) \\/ (((exists pfrep_gap_add_congruent_resultsecondoutside. pfrep_gap_add_congruent_resultsecondoutside+(K)=(pfrep_power_add_congruent_result)) /\\ (((pfrep_right_add_congruent_result)=0))))) -> pfrep_left_add_congruent_result=pfrep_right_add_congruent_result)", "statement_sha256": "847a60b511d446febdc15c56231f1368a7993172939945b7b99ab297cb65c4fb", "summary": "Pairwise formal-equivalent inputs give formal-equivalent actual add outputs at either ordering of the two aligned lengths, including empty prefixes; no output equivalence or raw-code equality is assumed.", "summary_sha256": "70db060691ded5f76d61ea24bc87c7ed6d05f68a433d04066bcf338d09d9f281" }, { "alpha_v33_frontier_enrollment": { "body_receipt_sha256": "ba2596659c6557d3f8b9e31ad057cfe5e2cc88c7dd7ce383459b533bce44a969", "bundle_campaign": "polynomial-euclidean-division", "bundle_node_id": 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"98190b2324e31d00b702aa1da5ee5e9ff43e1f3142ed821016429eefa68cdd49", "membership": "alpha_only", "name": "prime_field_polynomial_subtract_equivalent_congruent", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro ab", "intro ac", "intro bb", "intro bc", "intro cb", "intro cc", "intro L", "intro AB", "intro AC", "intro BB", "intro BC", "intro CB", "intro CC", "intro K", "intro hp", "intro hA", "intro hB", "intro ho", "intro hn", "have horder : L<=K \\/ K<=L", "specialize le_total (L)", "specialize le_total (K)", "apply le_total", "cases horder", "cases horder_left", "rewrite <- horder_left_witness at hA", "rewrite <- horder_left_witness at hA", "rewrite <- horder_left_witness at hB", "rewrite <- horder_left_witness at hB", "rewrite <- horder_left_witness at hn", "rewrite <- horder_left_witness", "rewrite <- horder_left_witness", "have hpadA : ((forall pfp_repeat_index_subtract_congruent_forward_hpadAzeros. (exists pfa_gap_subtract_congruent_forward_hpadAzerosindex. pfa_gap_subtract_congruent_forward_hpadAzerosindex + S (pfp_repeat_index_subtract_congruent_forward_hpadAzeros) = (x)) -> (((exists ff_h_pfp_subtract_congruent_forward_hpadAzerosentry. ff_h_pfp_subtract_congruent_forward_hpadAzerosentry + S (0) = S ((S (pfp_repeat_index_subtract_congruent_forward_hpadAzeros)) * AC)) /\\ exists ff_q_pfp_subtract_congruent_forward_hpadAzerosentry. AB = ff_q_pfp_subtract_congruent_forward_hpadAzerosentry * S ((S (pfp_repeat_index_subtract_congruent_forward_hpadAzeros)) * AC) + (0)))) /\\ ((forall pfrep_index_subtract_congruent_forward_hpadA pfrep_value_subtract_congruent_forward_hpadA. (exists pfa_gap_subtract_congruent_forward_hpadAbound. pfa_gap_subtract_congruent_forward_hpadAbound + S (pfrep_index_subtract_congruent_forward_hpadA) = (L)) -> (((exists ff_h_pfp_subtract_congruent_forward_hpadAinput. ff_h_pfp_subtract_congruent_forward_hpadAinput + S (pfrep_value_subtract_congruent_forward_hpadA) = S ((S (pfrep_index_subtract_congruent_forward_hpadA)) * ac)) /\\ exists ff_q_pfp_subtract_congruent_forward_hpadAinput. ab = ff_q_pfp_subtract_congruent_forward_hpadAinput * S ((S (pfrep_index_subtract_congruent_forward_hpadA)) * ac) + (pfrep_value_subtract_congruent_forward_hpadA))) -> (((exists ff_h_pfp_subtract_congruent_forward_hpadAoutput. ff_h_pfp_subtract_congruent_forward_hpadAoutput + S (pfrep_value_subtract_congruent_forward_hpadA) = S ((S ((x)+pfrep_index_subtract_congruent_forward_hpadA)) * AC)) /\\ exists ff_q_pfp_subtract_congruent_forward_hpadAoutput. AB = ff_q_pfp_subtract_congruent_forward_hpadAoutput * S ((S ((x)+pfrep_index_subtract_congruent_forward_hpadA)) * AC) + (pfrep_value_subtract_congruent_forward_hpadA))))))", "specialize prime_field_polynomial_equivalent_implies_left_pad (ab)", "specialize prime_field_polynomial_equivalent_implies_left_pad (ac)", "specialize prime_field_polynomial_equivalent_implies_left_pad (L)", "specialize prime_field_polynomial_equivalent_implies_left_pad (x)", "specialize prime_field_polynomial_equivalent_implies_left_pad (AB)", "specialize prime_field_polynomial_equivalent_implies_left_pad (AC)", "apply prime_field_polynomial_equivalent_implies_left_pad", "exact hA", "have hpadB : ((forall pfp_repeat_index_subtract_congruent_forward_hpadBzeros. (exists pfa_gap_subtract_congruent_forward_hpadBzerosindex. pfa_gap_subtract_congruent_forward_hpadBzerosindex + S (pfp_repeat_index_subtract_congruent_forward_hpadBzeros) = (x)) -> (((exists ff_h_pfp_subtract_congruent_forward_hpadBzerosentry. ff_h_pfp_subtract_congruent_forward_hpadBzerosentry + S (0) = S ((S (pfp_repeat_index_subtract_congruent_forward_hpadBzeros)) * BC)) /\\ exists ff_q_pfp_subtract_congruent_forward_hpadBzerosentry. BB = ff_q_pfp_subtract_congruent_forward_hpadBzerosentry * S ((S (pfp_repeat_index_subtract_congruent_forward_hpadBzeros)) * BC) + (0)))) /\\ ((forall pfrep_index_subtract_congruent_forward_hpadB pfrep_value_subtract_congruent_forward_hpadB. (exists pfa_gap_subtract_congruent_forward_hpadBbound. pfa_gap_subtract_congruent_forward_hpadBbound + S (pfrep_index_subtract_congruent_forward_hpadB) = (L)) -> (((exists ff_h_pfp_subtract_congruent_forward_hpadBinput. ff_h_pfp_subtract_congruent_forward_hpadBinput + S (pfrep_value_subtract_congruent_forward_hpadB) = S ((S (pfrep_index_subtract_congruent_forward_hpadB)) * bc)) /\\ exists ff_q_pfp_subtract_congruent_forward_hpadBinput. bb = ff_q_pfp_subtract_congruent_forward_hpadBinput * S ((S (pfrep_index_subtract_congruent_forward_hpadB)) * bc) + (pfrep_value_subtract_congruent_forward_hpadB))) -> (((exists ff_h_pfp_subtract_congruent_forward_hpadBoutput. ff_h_pfp_subtract_congruent_forward_hpadBoutput + S (pfrep_value_subtract_congruent_forward_hpadB) = S ((S ((x)+pfrep_index_subtract_congruent_forward_hpadB)) * BC)) /\\ exists ff_q_pfp_subtract_congruent_forward_hpadBoutput. BB = ff_q_pfp_subtract_congruent_forward_hpadBoutput * S ((S ((x)+pfrep_index_subtract_congruent_forward_hpadB)) * BC) + (pfrep_value_subtract_congruent_forward_hpadB))))))", "specialize prime_field_polynomial_equivalent_implies_left_pad (bb)", "specialize prime_field_polynomial_equivalent_implies_left_pad (bc)", "specialize prime_field_polynomial_equivalent_implies_left_pad (L)", "specialize prime_field_polynomial_equivalent_implies_left_pad (x)", "specialize prime_field_polynomial_equivalent_implies_left_pad (BB)", "specialize prime_field_polynomial_equivalent_implies_left_pad (BC)", "apply prime_field_polynomial_equivalent_implies_left_pad", "exact hB", "specialize prime_field_polynomial_left_pad_equivalent (cb)", "specialize prime_field_polynomial_left_pad_equivalent (cc)", "specialize prime_field_polynomial_left_pad_equivalent (L)", "specialize prime_field_polynomial_left_pad_equivalent (x)", "specialize prime_field_polynomial_left_pad_equivalent (CB)", "specialize prime_field_polynomial_left_pad_equivalent (CC)", "apply prime_field_polynomial_left_pad_equivalent", "specialize prime_field_polynomial_subtract_left_pad_output (p)", "specialize prime_field_polynomial_subtract_left_pad_output (ab)", "specialize prime_field_polynomial_subtract_left_pad_output (ac)", "specialize prime_field_polynomial_subtract_left_pad_output (bb)", "specialize prime_field_polynomial_subtract_left_pad_output (bc)", "specialize prime_field_polynomial_subtract_left_pad_output (cb)", "specialize prime_field_polynomial_subtract_left_pad_output (cc)", "specialize prime_field_polynomial_subtract_left_pad_output (L)", "specialize prime_field_polynomial_subtract_left_pad_output (x)", "specialize prime_field_polynomial_subtract_left_pad_output (AB)", "specialize prime_field_polynomial_subtract_left_pad_output (AC)", "specialize prime_field_polynomial_subtract_left_pad_output (BB)", "specialize prime_field_polynomial_subtract_left_pad_output (BC)", "specialize prime_field_polynomial_subtract_left_pad_output (CB)", "specialize prime_field_polynomial_subtract_left_pad_output (CC)", "apply prime_field_polynomial_subtract_left_pad_output", "exact hp", "exact ho", "exact hpadA", "exact hpadB", "exact hn", "cases horder_right", "rewrite <- horder_right_witness at hA", "rewrite <- horder_right_witness at hA", "rewrite <- horder_right_witness at hB", "rewrite <- horder_right_witness at hB", "rewrite <- horder_right_witness at ho", "rewrite <- horder_right_witness", "rewrite <- horder_right_witness", "have hpadA : ((forall pfp_repeat_index_subtract_congruent_reverse_hpadAzeros. (exists pfa_gap_subtract_congruent_reverse_hpadAzerosindex. pfa_gap_subtract_congruent_reverse_hpadAzerosindex + S (pfp_repeat_index_subtract_congruent_reverse_hpadAzeros) = (x)) -> (((exists ff_h_pfp_subtract_congruent_reverse_hpadAzerosentry. ff_h_pfp_subtract_congruent_reverse_hpadAzerosentry + S (0) = S ((S (pfp_repeat_index_subtract_congruent_reverse_hpadAzeros)) * ac)) /\\ exists ff_q_pfp_subtract_congruent_reverse_hpadAzerosentry. ab = ff_q_pfp_subtract_congruent_reverse_hpadAzerosentry * S ((S (pfp_repeat_index_subtract_congruent_reverse_hpadAzeros)) * ac) + (0)))) /\\ ((forall pfrep_index_subtract_congruent_reverse_hpadA pfrep_value_subtract_congruent_reverse_hpadA. (exists pfa_gap_subtract_congruent_reverse_hpadAbound. pfa_gap_subtract_congruent_reverse_hpadAbound + S (pfrep_index_subtract_congruent_reverse_hpadA) = (K)) -> (((exists ff_h_pfp_subtract_congruent_reverse_hpadAinput. ff_h_pfp_subtract_congruent_reverse_hpadAinput + S (pfrep_value_subtract_congruent_reverse_hpadA) = S ((S (pfrep_index_subtract_congruent_reverse_hpadA)) * AC)) /\\ exists ff_q_pfp_subtract_congruent_reverse_hpadAinput. AB = ff_q_pfp_subtract_congruent_reverse_hpadAinput * S ((S (pfrep_index_subtract_congruent_reverse_hpadA)) * AC) + (pfrep_value_subtract_congruent_reverse_hpadA))) -> (((exists ff_h_pfp_subtract_congruent_reverse_hpadAoutput. ff_h_pfp_subtract_congruent_reverse_hpadAoutput + S (pfrep_value_subtract_congruent_reverse_hpadA) = S ((S ((x)+pfrep_index_subtract_congruent_reverse_hpadA)) * ac)) /\\ exists ff_q_pfp_subtract_congruent_reverse_hpadAoutput. ab = ff_q_pfp_subtract_congruent_reverse_hpadAoutput * S ((S ((x)+pfrep_index_subtract_congruent_reverse_hpadA)) * ac) + (pfrep_value_subtract_congruent_reverse_hpadA))))))", "specialize prime_field_polynomial_equivalent_implies_left_pad (AB)", "specialize prime_field_polynomial_equivalent_implies_left_pad (AC)", "specialize prime_field_polynomial_equivalent_implies_left_pad (K)", "specialize prime_field_polynomial_equivalent_implies_left_pad (x)", "specialize prime_field_polynomial_equivalent_implies_left_pad (ab)", "specialize prime_field_polynomial_equivalent_implies_left_pad (ac)", "apply prime_field_polynomial_equivalent_implies_left_pad", "specialize prime_field_polynomial_equivalent_symmetric (ab)", "specialize prime_field_polynomial_equivalent_symmetric (ac)", "specialize prime_field_polynomial_equivalent_symmetric (x+K)", "specialize prime_field_polynomial_equivalent_symmetric (AB)", "specialize prime_field_polynomial_equivalent_symmetric (AC)", "specialize prime_field_polynomial_equivalent_symmetric (K)", "apply prime_field_polynomial_equivalent_symmetric", "exact hA", "have hpadB : ((forall pfp_repeat_index_subtract_congruent_reverse_hpadBzeros. (exists pfa_gap_subtract_congruent_reverse_hpadBzerosindex. pfa_gap_subtract_congruent_reverse_hpadBzerosindex + S (pfp_repeat_index_subtract_congruent_reverse_hpadBzeros) = (x)) -> (((exists ff_h_pfp_subtract_congruent_reverse_hpadBzerosentry. ff_h_pfp_subtract_congruent_reverse_hpadBzerosentry + S (0) = S ((S (pfp_repeat_index_subtract_congruent_reverse_hpadBzeros)) * bc)) /\\ exists ff_q_pfp_subtract_congruent_reverse_hpadBzerosentry. bb = ff_q_pfp_subtract_congruent_reverse_hpadBzerosentry * S ((S (pfp_repeat_index_subtract_congruent_reverse_hpadBzeros)) * bc) + (0)))) /\\ ((forall pfrep_index_subtract_congruent_reverse_hpadB pfrep_value_subtract_congruent_reverse_hpadB. (exists pfa_gap_subtract_congruent_reverse_hpadBbound. pfa_gap_subtract_congruent_reverse_hpadBbound + S (pfrep_index_subtract_congruent_reverse_hpadB) = (K)) -> (((exists ff_h_pfp_subtract_congruent_reverse_hpadBinput. ff_h_pfp_subtract_congruent_reverse_hpadBinput + S (pfrep_value_subtract_congruent_reverse_hpadB) = S ((S (pfrep_index_subtract_congruent_reverse_hpadB)) * BC)) /\\ exists ff_q_pfp_subtract_congruent_reverse_hpadBinput. BB = ff_q_pfp_subtract_congruent_reverse_hpadBinput * S ((S (pfrep_index_subtract_congruent_reverse_hpadB)) * BC) + (pfrep_value_subtract_congruent_reverse_hpadB))) -> (((exists ff_h_pfp_subtract_congruent_reverse_hpadBoutput. ff_h_pfp_subtract_congruent_reverse_hpadBoutput + S (pfrep_value_subtract_congruent_reverse_hpadB) = S ((S ((x)+pfrep_index_subtract_congruent_reverse_hpadB)) * bc)) /\\ exists ff_q_pfp_subtract_congruent_reverse_hpadBoutput. bb = ff_q_pfp_subtract_congruent_reverse_hpadBoutput * S ((S ((x)+pfrep_index_subtract_congruent_reverse_hpadB)) * bc) + (pfrep_value_subtract_congruent_reverse_hpadB))))))", "specialize prime_field_polynomial_equivalent_implies_left_pad (BB)", "specialize prime_field_polynomial_equivalent_implies_left_pad (BC)", "specialize prime_field_polynomial_equivalent_implies_left_pad (K)", "specialize prime_field_polynomial_equivalent_implies_left_pad (x)", "specialize prime_field_polynomial_equivalent_implies_left_pad (bb)", "specialize prime_field_polynomial_equivalent_implies_left_pad (bc)", "apply prime_field_polynomial_equivalent_implies_left_pad", "specialize prime_field_polynomial_equivalent_symmetric (bb)", "specialize prime_field_polynomial_equivalent_symmetric (bc)", "specialize prime_field_polynomial_equivalent_symmetric (x+K)", "specialize prime_field_polynomial_equivalent_symmetric (BB)", "specialize prime_field_polynomial_equivalent_symmetric (BC)", "specialize prime_field_polynomial_equivalent_symmetric (K)", "apply prime_field_polynomial_equivalent_symmetric", "exact hB", "specialize prime_field_polynomial_equivalent_symmetric (CB)", "specialize prime_field_polynomial_equivalent_symmetric (CC)", "specialize prime_field_polynomial_equivalent_symmetric (K)", "specialize prime_field_polynomial_equivalent_symmetric (cb)", "specialize prime_field_polynomial_equivalent_symmetric (cc)", "specialize prime_field_polynomial_equivalent_symmetric (x+K)", "apply prime_field_polynomial_equivalent_symmetric", "specialize prime_field_polynomial_left_pad_equivalent (CB)", "specialize prime_field_polynomial_left_pad_equivalent (CC)", "specialize prime_field_polynomial_left_pad_equivalent (K)", "specialize prime_field_polynomial_left_pad_equivalent (x)", "specialize prime_field_polynomial_left_pad_equivalent (cb)", "specialize prime_field_polynomial_left_pad_equivalent (cc)", "apply prime_field_polynomial_left_pad_equivalent", "specialize prime_field_polynomial_subtract_left_pad_output (p)", "specialize prime_field_polynomial_subtract_left_pad_output (AB)", "specialize prime_field_polynomial_subtract_left_pad_output (AC)", "specialize prime_field_polynomial_subtract_left_pad_output (BB)", "specialize prime_field_polynomial_subtract_left_pad_output (BC)", "specialize prime_field_polynomial_subtract_left_pad_output (CB)", "specialize prime_field_polynomial_subtract_left_pad_output (CC)", "specialize prime_field_polynomial_subtract_left_pad_output (K)", "specialize prime_field_polynomial_subtract_left_pad_output (x)", "specialize prime_field_polynomial_subtract_left_pad_output (ab)", "specialize prime_field_polynomial_subtract_left_pad_output (ac)", "specialize prime_field_polynomial_subtract_left_pad_output (bb)", "specialize prime_field_polynomial_subtract_left_pad_output (bc)", "specialize prime_field_polynomial_subtract_left_pad_output (cb)", "specialize prime_field_polynomial_subtract_left_pad_output (cc)", "apply prime_field_polynomial_subtract_left_pad_output", "exact hp", "exact hn", "exact hpadA", "exact hpadB", "exact ho" ], "script_sha256": "a93898b6371ce615c8f74ce6d41049857bcde19559aa3de866830fbc0dc4b9b2", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_equivalence_candidate.py", "sha256": "929eb67318c8a09577fb9ebac277b82656abf04c82b97a417fff83f39e7bb373" }, "statement": "forall p ab ac bb bc cb cc L AB AC BB BC CB CC K. (~((p) = 1) /\\ forall pfa_factor_left_subtract_congruent_prime pfa_factor_right_subtract_congruent_prime. (p) = pfa_factor_left_subtract_congruent_prime * pfa_factor_right_subtract_congruent_prime -> pfa_factor_left_subtract_congruent_prime = 1 \\/ pfa_factor_right_subtract_congruent_prime = 1) -> (forall pfrep_power_subtract_congruent_first pfrep_left_subtract_congruent_first pfrep_right_subtract_congruent_first. ((exists pfrep_position_subtract_congruent_firstfirst. ((pfrep_position_subtract_congruent_firstfirst+S (pfrep_power_subtract_congruent_first)=(L)) /\\ ((((exists ff_h_pfp_subtract_congruent_firstfirstentry. ff_h_pfp_subtract_congruent_firstfirstentry + S (pfrep_left_subtract_congruent_first) = S ((S (pfrep_position_subtract_congruent_firstfirst)) * ac)) /\\ exists ff_q_pfp_subtract_congruent_firstfirstentry. ab = ff_q_pfp_subtract_congruent_firstfirstentry * S ((S (pfrep_position_subtract_congruent_firstfirst)) * ac) + (pfrep_left_subtract_congruent_first)))))) \\/ (((exists pfrep_gap_subtract_congruent_firstfirstoutside. pfrep_gap_subtract_congruent_firstfirstoutside+(L)=(pfrep_power_subtract_congruent_first)) /\\ (((pfrep_left_subtract_congruent_first)=0))))) -> ((exists pfrep_position_subtract_congruent_firstsecond. ((pfrep_position_subtract_congruent_firstsecond+S (pfrep_power_subtract_congruent_first)=(K)) /\\ ((((exists ff_h_pfp_subtract_congruent_firstsecondentry. ff_h_pfp_subtract_congruent_firstsecondentry + S (pfrep_right_subtract_congruent_first) = S ((S (pfrep_position_subtract_congruent_firstsecond)) * AC)) /\\ exists ff_q_pfp_subtract_congruent_firstsecondentry. AB = ff_q_pfp_subtract_congruent_firstsecondentry * S ((S (pfrep_position_subtract_congruent_firstsecond)) * AC) + (pfrep_right_subtract_congruent_first)))))) \\/ (((exists pfrep_gap_subtract_congruent_firstsecondoutside. pfrep_gap_subtract_congruent_firstsecondoutside+(K)=(pfrep_power_subtract_congruent_first)) /\\ (((pfrep_right_subtract_congruent_first)=0))))) -> pfrep_left_subtract_congruent_first=pfrep_right_subtract_congruent_first) -> (forall pfrep_power_subtract_congruent_second pfrep_left_subtract_congruent_second pfrep_right_subtract_congruent_second. ((exists pfrep_position_subtract_congruent_secondfirst. ((pfrep_position_subtract_congruent_secondfirst+S (pfrep_power_subtract_congruent_second)=(L)) /\\ ((((exists ff_h_pfp_subtract_congruent_secondfirstentry. ff_h_pfp_subtract_congruent_secondfirstentry + S (pfrep_left_subtract_congruent_second) = S ((S (pfrep_position_subtract_congruent_secondfirst)) * bc)) /\\ exists ff_q_pfp_subtract_congruent_secondfirstentry. bb = ff_q_pfp_subtract_congruent_secondfirstentry * S ((S (pfrep_position_subtract_congruent_secondfirst)) * bc) + (pfrep_left_subtract_congruent_second)))))) \\/ (((exists pfrep_gap_subtract_congruent_secondfirstoutside. pfrep_gap_subtract_congruent_secondfirstoutside+(L)=(pfrep_power_subtract_congruent_second)) /\\ (((pfrep_left_subtract_congruent_second)=0))))) -> ((exists pfrep_position_subtract_congruent_secondsecond. ((pfrep_position_subtract_congruent_secondsecond+S (pfrep_power_subtract_congruent_second)=(K)) /\\ ((((exists ff_h_pfp_subtract_congruent_secondsecondentry. ff_h_pfp_subtract_congruent_secondsecondentry + S (pfrep_right_subtract_congruent_second) = S ((S (pfrep_position_subtract_congruent_secondsecond)) * BC)) /\\ exists ff_q_pfp_subtract_congruent_secondsecondentry. BB = ff_q_pfp_subtract_congruent_secondsecondentry * S ((S (pfrep_position_subtract_congruent_secondsecond)) * BC) + (pfrep_right_subtract_congruent_second)))))) \\/ (((exists pfrep_gap_subtract_congruent_secondsecondoutside. pfrep_gap_subtract_congruent_secondsecondoutside+(K)=(pfrep_power_subtract_congruent_second)) /\\ (((pfrep_right_subtract_congruent_second)=0))))) -> pfrep_left_subtract_congruent_second=pfrep_right_subtract_congruent_second) -> (forall pfs_index_subtract_congruent_original. (exists pfa_gap_subtract_congruent_originalindex. pfa_gap_subtract_congruent_originalindex + S (pfs_index_subtract_congruent_original) = (L)) -> exists pfs_left_subtract_congruent_original pfs_right_subtract_congruent_original pfs_result_subtract_congruent_original. ((((exists ff_h_pfp_subtract_congruent_originalleft. ff_h_pfp_subtract_congruent_originalleft + S (pfs_left_subtract_congruent_original) = S ((S (pfs_index_subtract_congruent_original)) * ac)) /\\ exists ff_q_pfp_subtract_congruent_originalleft. ab = ff_q_pfp_subtract_congruent_originalleft * S ((S (pfs_index_subtract_congruent_original)) * ac) + (pfs_left_subtract_congruent_original))) /\\ (((((exists ff_h_pfp_subtract_congruent_originalright. ff_h_pfp_subtract_congruent_originalright + S (pfs_right_subtract_congruent_original) = S ((S (pfs_index_subtract_congruent_original)) * bc)) /\\ exists ff_q_pfp_subtract_congruent_originalright. bb = ff_q_pfp_subtract_congruent_originalright * S ((S (pfs_index_subtract_congruent_original)) * bc) + (pfs_right_subtract_congruent_original))) /\\ (((((exists ff_h_pfp_subtract_congruent_originalresult. ff_h_pfp_subtract_congruent_originalresult + S (pfs_result_subtract_congruent_original) = S ((S (pfs_index_subtract_congruent_original)) * cc)) /\\ exists ff_q_pfp_subtract_congruent_originalresult. cb = ff_q_pfp_subtract_congruent_originalresult * S ((S (pfs_index_subtract_congruent_original)) * cc) + (pfs_result_subtract_congruent_original))) /\\ ((((exists pfa_gap_subtract_congruent_originaloperationleft. pfa_gap_subtract_congruent_originaloperationleft + S (pfs_right_subtract_congruent_original) = (p)) /\\ (((exists pfa_gap_subtract_congruent_originaloperationright. pfa_gap_subtract_congruent_originaloperationright + S (pfs_result_subtract_congruent_original) = (p)) /\\ ((((exists pfa_gap_subtract_congruent_originaloperationresultbound. pfa_gap_subtract_congruent_originaloperationresultbound + S (pfs_left_subtract_congruent_original) = (p)) /\\ ((exists pfa_offset_left_subtract_congruent_originaloperationresultcongruence pfa_offset_right_subtract_congruent_originaloperationresultcongruence. ((pfs_right_subtract_congruent_original) + (pfs_result_subtract_congruent_original)) + (p) * pfa_offset_left_subtract_congruent_originaloperationresultcongruence = (pfs_left_subtract_congruent_original) + (p) * pfa_offset_right_subtract_congruent_originaloperationresultcongruence)))))))))))))))) -> (forall pfs_index_subtract_congruent_other. (exists pfa_gap_subtract_congruent_otherindex. pfa_gap_subtract_congruent_otherindex + S (pfs_index_subtract_congruent_other) = (K)) -> exists pfs_left_subtract_congruent_other pfs_right_subtract_congruent_other pfs_result_subtract_congruent_other. ((((exists ff_h_pfp_subtract_congruent_otherleft. ff_h_pfp_subtract_congruent_otherleft + S (pfs_left_subtract_congruent_other) = S ((S (pfs_index_subtract_congruent_other)) * AC)) /\\ exists ff_q_pfp_subtract_congruent_otherleft. AB = ff_q_pfp_subtract_congruent_otherleft * S ((S (pfs_index_subtract_congruent_other)) * AC) + (pfs_left_subtract_congruent_other))) /\\ (((((exists ff_h_pfp_subtract_congruent_otherright. ff_h_pfp_subtract_congruent_otherright + S (pfs_right_subtract_congruent_other) = S ((S (pfs_index_subtract_congruent_other)) * BC)) /\\ exists ff_q_pfp_subtract_congruent_otherright. BB = ff_q_pfp_subtract_congruent_otherright * S ((S (pfs_index_subtract_congruent_other)) * BC) + (pfs_right_subtract_congruent_other))) /\\ (((((exists ff_h_pfp_subtract_congruent_otherresult. ff_h_pfp_subtract_congruent_otherresult + S (pfs_result_subtract_congruent_other) = S ((S (pfs_index_subtract_congruent_other)) * CC)) /\\ exists ff_q_pfp_subtract_congruent_otherresult. CB = ff_q_pfp_subtract_congruent_otherresult * S ((S (pfs_index_subtract_congruent_other)) * CC) + (pfs_result_subtract_congruent_other))) /\\ ((((exists pfa_gap_subtract_congruent_otheroperationleft. pfa_gap_subtract_congruent_otheroperationleft + S (pfs_right_subtract_congruent_other) = (p)) /\\ (((exists pfa_gap_subtract_congruent_otheroperationright. pfa_gap_subtract_congruent_otheroperationright + S (pfs_result_subtract_congruent_other) = (p)) /\\ ((((exists pfa_gap_subtract_congruent_otheroperationresultbound. pfa_gap_subtract_congruent_otheroperationresultbound + S (pfs_left_subtract_congruent_other) = (p)) /\\ ((exists pfa_offset_left_subtract_congruent_otheroperationresultcongruence pfa_offset_right_subtract_congruent_otheroperationresultcongruence. ((pfs_right_subtract_congruent_other) + (pfs_result_subtract_congruent_other)) + (p) * pfa_offset_left_subtract_congruent_otheroperationresultcongruence = (pfs_left_subtract_congruent_other) + (p) * pfa_offset_right_subtract_congruent_otheroperationresultcongruence)))))))))))))))) -> (forall pfrep_power_subtract_congruent_result pfrep_left_subtract_congruent_result pfrep_right_subtract_congruent_result. ((exists pfrep_position_subtract_congruent_resultfirst. ((pfrep_position_subtract_congruent_resultfirst+S (pfrep_power_subtract_congruent_result)=(L)) /\\ ((((exists ff_h_pfp_subtract_congruent_resultfirstentry. ff_h_pfp_subtract_congruent_resultfirstentry + S (pfrep_left_subtract_congruent_result) = S ((S (pfrep_position_subtract_congruent_resultfirst)) * cc)) /\\ exists ff_q_pfp_subtract_congruent_resultfirstentry. cb = ff_q_pfp_subtract_congruent_resultfirstentry * S ((S (pfrep_position_subtract_congruent_resultfirst)) * cc) + (pfrep_left_subtract_congruent_result)))))) \\/ (((exists pfrep_gap_subtract_congruent_resultfirstoutside. pfrep_gap_subtract_congruent_resultfirstoutside+(L)=(pfrep_power_subtract_congruent_result)) /\\ (((pfrep_left_subtract_congruent_result)=0))))) -> ((exists pfrep_position_subtract_congruent_resultsecond. ((pfrep_position_subtract_congruent_resultsecond+S (pfrep_power_subtract_congruent_result)=(K)) /\\ ((((exists ff_h_pfp_subtract_congruent_resultsecondentry. ff_h_pfp_subtract_congruent_resultsecondentry + S (pfrep_right_subtract_congruent_result) = S ((S (pfrep_position_subtract_congruent_resultsecond)) * CC)) /\\ exists ff_q_pfp_subtract_congruent_resultsecondentry. CB = ff_q_pfp_subtract_congruent_resultsecondentry * S ((S (pfrep_position_subtract_congruent_resultsecond)) * CC) + (pfrep_right_subtract_congruent_result)))))) \\/ (((exists pfrep_gap_subtract_congruent_resultsecondoutside. pfrep_gap_subtract_congruent_resultsecondoutside+(K)=(pfrep_power_subtract_congruent_result)) /\\ (((pfrep_right_subtract_congruent_result)=0))))) -> pfrep_left_subtract_congruent_result=pfrep_right_subtract_congruent_result)", "statement_sha256": "b073daede7886ec70b68c11665fc2f70154db2696cd613e542d1e22900e5f2a3", "summary": "Pairwise formal-equivalent inputs give formal-equivalent actual subtract outputs at either ordering of the two aligned lengths, including empty prefixes; no output equivalence or raw-code equality is assumed.", "summary_sha256": "3c5a3ec5f7c616efd56de6a74f39ec609986961b0b45dcf59a2050151baa05f0" }, { "alpha_v33_frontier_enrollment": { "body_receipt_sha256": "3b99d2e3065cdb39c7bf5ab2dfe3febed2103a5ae021b9d3ac324f9922efdadd", "bundle_campaign": 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"polynomial-euclidean-division", "logical_spec_sha256": "2dd5cad69bddd83417f3299cb83d746c4facf5fd3a1a4f60f9ddc2ce627ef2a2", "membership": "alpha_only", "name": "prime_field_polynomial_convolution_equivalent_congruent_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 N", "intro AB", "intro AC", "intro H", "intro CB", "intro CC", "intro K", "intro hp", "intro he", "intro hc", "intro hd", "have horder : (exists pfc_gap_congruent_forward_left. pfc_gap_congruent_forward_left+(L)=(H)) \\/ (exists pfc_gap_congruent_backward_left. pfc_gap_congruent_backward_left+(H)=(L))", "specialize le_total (L)", "specialize le_total (H)", "apply le_total", "cases horder", "cases horder_left", "have hpadding : ((forall pfp_repeat_index_congruent_padding_leftzeros. (exists pfa_gap_congruent_padding_leftzerosindex. pfa_gap_congruent_padding_leftzerosindex + S (pfp_repeat_index_congruent_padding_leftzeros) = (x)) -> (((exists ff_h_pfp_congruent_padding_leftzerosentry. ff_h_pfp_congruent_padding_leftzerosentry + S (0) = S ((S (pfp_repeat_index_congruent_padding_leftzeros)) * AC)) /\\ exists ff_q_pfp_congruent_padding_leftzerosentry. AB = ff_q_pfp_congruent_padding_leftzerosentry * S ((S (pfp_repeat_index_congruent_padding_leftzeros)) * AC) + (0)))) /\\ ((forall pfrep_index_congruent_padding_left pfrep_value_congruent_padding_left. (exists pfa_gap_congruent_padding_leftbound. pfa_gap_congruent_padding_leftbound + S (pfrep_index_congruent_padding_left) = (L)) -> (((exists ff_h_pfp_congruent_padding_leftinput. ff_h_pfp_congruent_padding_leftinput + S (pfrep_value_congruent_padding_left) = S ((S (pfrep_index_congruent_padding_left)) * ac)) /\\ exists ff_q_pfp_congruent_padding_leftinput. ab = ff_q_pfp_congruent_padding_leftinput * S ((S (pfrep_index_congruent_padding_left)) * ac) + (pfrep_value_congruent_padding_left))) -> (((exists ff_h_pfp_congruent_padding_leftoutput. ff_h_pfp_congruent_padding_leftoutput + S (pfrep_value_congruent_padding_left) = S ((S ((x)+pfrep_index_congruent_padding_left)) * AC)) /\\ exists ff_q_pfp_congruent_padding_leftoutput. AB = ff_q_pfp_congruent_padding_leftoutput * S ((S ((x)+pfrep_index_congruent_padding_left)) * AC) + (pfrep_value_congruent_padding_left))))))", "specialize prime_field_polynomial_equivalent_implies_left_pad (ab)", "specialize prime_field_polynomial_equivalent_implies_left_pad (ac)", "specialize prime_field_polynomial_equivalent_implies_left_pad (L)", "specialize prime_field_polynomial_equivalent_implies_left_pad (x)", "specialize prime_field_polynomial_equivalent_implies_left_pad (AB)", "specialize prime_field_polynomial_equivalent_implies_left_pad (AC)", "apply prime_field_polynomial_equivalent_implies_left_pad", "rewrite horder_left_witness", "rewrite horder_left_witness", "exact he", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (p)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (ab)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (ac)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (L)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (bb)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (bc)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (M)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (cb)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (cc)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (N)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (AB)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (AC)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (x)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (CB)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (CC)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (K)", "apply prime_field_polynomial_convolution_left_padding_equivalent_left", "exact hp", "exact hpadding", "exact hc", "rewrite horder_left_witness", "rewrite horder_left_witness", "rewrite horder_left_witness", "rewrite horder_left_witness", "rewrite horder_left_witness", "rewrite horder_left_witness", "exact hd", "cases horder_right", "have hpadding : ((forall pfp_repeat_index_congruent_padding_leftzeros. (exists pfa_gap_congruent_padding_leftzerosindex. pfa_gap_congruent_padding_leftzerosindex + S (pfp_repeat_index_congruent_padding_leftzeros) = (x)) -> (((exists ff_h_pfp_congruent_padding_leftzerosentry. ff_h_pfp_congruent_padding_leftzerosentry + S (0) = S ((S (pfp_repeat_index_congruent_padding_leftzeros)) * ac)) /\\ exists ff_q_pfp_congruent_padding_leftzerosentry. ab = ff_q_pfp_congruent_padding_leftzerosentry * S ((S (pfp_repeat_index_congruent_padding_leftzeros)) * ac) + (0)))) /\\ ((forall pfrep_index_congruent_padding_left pfrep_value_congruent_padding_left. (exists pfa_gap_congruent_padding_leftbound. pfa_gap_congruent_padding_leftbound + S (pfrep_index_congruent_padding_left) = (H)) -> (((exists ff_h_pfp_congruent_padding_leftinput. ff_h_pfp_congruent_padding_leftinput + S (pfrep_value_congruent_padding_left) = S ((S (pfrep_index_congruent_padding_left)) * AC)) /\\ exists ff_q_pfp_congruent_padding_leftinput. AB = ff_q_pfp_congruent_padding_leftinput * S ((S (pfrep_index_congruent_padding_left)) * AC) + (pfrep_value_congruent_padding_left))) -> (((exists ff_h_pfp_congruent_padding_leftoutput. ff_h_pfp_congruent_padding_leftoutput + S (pfrep_value_congruent_padding_left) = S ((S ((x)+pfrep_index_congruent_padding_left)) * ac)) /\\ exists ff_q_pfp_congruent_padding_leftoutput. ab = ff_q_pfp_congruent_padding_leftoutput * S ((S ((x)+pfrep_index_congruent_padding_left)) * ac) + (pfrep_value_congruent_padding_left))))))", "specialize prime_field_polynomial_equivalent_implies_left_pad (AB)", "specialize prime_field_polynomial_equivalent_implies_left_pad (AC)", "specialize prime_field_polynomial_equivalent_implies_left_pad (H)", "specialize prime_field_polynomial_equivalent_implies_left_pad (x)", "specialize prime_field_polynomial_equivalent_implies_left_pad (ab)", "specialize prime_field_polynomial_equivalent_implies_left_pad (ac)", "apply prime_field_polynomial_equivalent_implies_left_pad", "rewrite horder_right_witness", "rewrite horder_right_witness", "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 (AB)", "specialize prime_field_polynomial_equivalent_symmetric (AC)", "specialize prime_field_polynomial_equivalent_symmetric (H)", "apply prime_field_polynomial_equivalent_symmetric", "exact he", "specialize prime_field_polynomial_equivalent_symmetric (CB)", "specialize prime_field_polynomial_equivalent_symmetric (CC)", "specialize prime_field_polynomial_equivalent_symmetric (K)", "specialize prime_field_polynomial_equivalent_symmetric (cb)", "specialize prime_field_polynomial_equivalent_symmetric (cc)", "specialize prime_field_polynomial_equivalent_symmetric (N)", "apply prime_field_polynomial_equivalent_symmetric", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (p)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (AB)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (AC)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (H)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (bb)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (bc)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (M)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (CB)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (CC)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (K)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (ab)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (ac)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (x)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (cb)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (cc)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_left (N)", "apply prime_field_polynomial_convolution_left_padding_equivalent_left", "exact hp", "exact hpadding", "exact hd", "rewrite horder_right_witness", "rewrite horder_right_witness", "rewrite horder_right_witness", "rewrite horder_right_witness", "rewrite horder_right_witness", "rewrite horder_right_witness", "exact hc" ], "script_sha256": "bd0303799984370ee6bb835c44cc87d3dde0af59428aacd6e705ebc624b898a4", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_convolution_congruence_candidate.py", "sha256": "effc4b2df9418d9d964fd34216c4c1c2a09d12dd885877165c6fed2e761a8b70" }, "statement": "forall p ab ac L bb bc M cb cc N AB AC H CB CC K. (~(p=0)) -> (forall pfrep_power_congruent_factor_left pfrep_left_congruent_factor_left pfrep_right_congruent_factor_left. ((exists pfrep_position_congruent_factor_leftfirst. ((pfrep_position_congruent_factor_leftfirst+S (pfrep_power_congruent_factor_left)=(L)) /\\ ((((exists ff_h_pfp_congruent_factor_leftfirstentry. ff_h_pfp_congruent_factor_leftfirstentry + S (pfrep_left_congruent_factor_left) = S ((S (pfrep_position_congruent_factor_leftfirst)) * ac)) /\\ exists ff_q_pfp_congruent_factor_leftfirstentry. ab = ff_q_pfp_congruent_factor_leftfirstentry * S ((S (pfrep_position_congruent_factor_leftfirst)) * ac) + (pfrep_left_congruent_factor_left)))))) \\/ (((exists pfrep_gap_congruent_factor_leftfirstoutside. pfrep_gap_congruent_factor_leftfirstoutside+(L)=(pfrep_power_congruent_factor_left)) /\\ (((pfrep_left_congruent_factor_left)=0))))) -> ((exists pfrep_position_congruent_factor_leftsecond. ((pfrep_position_congruent_factor_leftsecond+S (pfrep_power_congruent_factor_left)=(H)) /\\ ((((exists ff_h_pfp_congruent_factor_leftsecondentry. ff_h_pfp_congruent_factor_leftsecondentry + S (pfrep_right_congruent_factor_left) = S ((S (pfrep_position_congruent_factor_leftsecond)) * AC)) /\\ exists ff_q_pfp_congruent_factor_leftsecondentry. AB = ff_q_pfp_congruent_factor_leftsecondentry * S ((S (pfrep_position_congruent_factor_leftsecond)) * AC) + (pfrep_right_congruent_factor_left)))))) \\/ (((exists pfrep_gap_congruent_factor_leftsecondoutside. pfrep_gap_congruent_factor_leftsecondoutside+(H)=(pfrep_power_congruent_factor_left)) /\\ (((pfrep_right_congruent_factor_left)=0))))) -> pfrep_left_congruent_factor_left=pfrep_right_congruent_factor_left) -> (((forall fom_index_pfp_congruent_original_leftleft. (exists fom_gap_pfp_congruent_original_leftleft_index_bound. fom_gap_pfp_congruent_original_leftleft_index_bound + S (fom_index_pfp_congruent_original_leftleft) = L) -> exists fom_value_pfp_congruent_original_leftleft. ((((exists fom_beta_height_pfp_congruent_original_leftleft_entry. fom_beta_height_pfp_congruent_original_leftleft_entry + S (fom_value_pfp_congruent_original_leftleft) = S ((S (fom_index_pfp_congruent_original_leftleft)) * ac)) /\\ exists fom_beta_quotient_pfp_congruent_original_leftleft_entry. ab = fom_beta_quotient_pfp_congruent_original_leftleft_entry * S ((S (fom_index_pfp_congruent_original_leftleft)) * ac) + (fom_value_pfp_congruent_original_leftleft))) /\\ (exists fom_gap_pfp_congruent_original_leftleft_value_bound. fom_gap_pfp_congruent_original_leftleft_value_bound + S (fom_value_pfp_congruent_original_leftleft) = p))) /\\ (((forall fom_index_pfp_congruent_original_leftright. (exists fom_gap_pfp_congruent_original_leftright_index_bound. fom_gap_pfp_congruent_original_leftright_index_bound + S (fom_index_pfp_congruent_original_leftright) = M) -> exists fom_value_pfp_congruent_original_leftright. ((((exists fom_beta_height_pfp_congruent_original_leftright_entry. fom_beta_height_pfp_congruent_original_leftright_entry + S (fom_value_pfp_congruent_original_leftright) = S ((S (fom_index_pfp_congruent_original_leftright)) * bc)) /\\ exists fom_beta_quotient_pfp_congruent_original_leftright_entry. bb = fom_beta_quotient_pfp_congruent_original_leftright_entry * S ((S (fom_index_pfp_congruent_original_leftright)) * bc) + (fom_value_pfp_congruent_original_leftright))) /\\ (exists fom_gap_pfp_congruent_original_leftright_value_bound. fom_gap_pfp_congruent_original_leftright_value_bound + S (fom_value_pfp_congruent_original_leftright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_congruent_original_leftcoefficients. (exists pfa_gap_congruent_original_leftcoefficientsbound. pfa_gap_congruent_original_leftcoefficientsbound + S (pfc_index_congruent_original_leftcoefficients) = (N)) -> exists pfc_value_congruent_original_leftcoefficients. ((((exists ff_h_pfp_congruent_original_leftcoefficientsentry. ff_h_pfp_congruent_original_leftcoefficientsentry + S (pfc_value_congruent_original_leftcoefficients) = S ((S (pfc_index_congruent_original_leftcoefficients)) * cc)) /\\ exists ff_q_pfp_congruent_original_leftcoefficientsentry. cb = ff_q_pfp_congruent_original_leftcoefficientsentry * S ((S (pfc_index_congruent_original_leftcoefficients)) * cc) + (pfc_value_congruent_original_leftcoefficients))) /\\ ((exists pfc_terms_code_congruent_original_leftcoefficientscoefficient pfc_terms_scale_congruent_original_leftcoefficientscoefficient pfc_natural_sum_congruent_original_leftcoefficientscoefficient. ((forall pfc_index_congruent_original_leftcoefficientscoefficientdiagonal. (exists pfa_gap_congruent_original_leftcoefficientscoefficientdiagonalbound. pfa_gap_congruent_original_leftcoefficientscoefficientdiagonalbound + S (pfc_index_congruent_original_leftcoefficientscoefficientdiagonal) = (S (pfc_index_congruent_original_leftcoefficients))) -> exists pfc_value_congruent_original_leftcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_congruent_original_leftcoefficientscoefficientdiagonalentry. ff_h_pfp_congruent_original_leftcoefficientscoefficientdiagonalentry + S (pfc_value_congruent_original_leftcoefficientscoefficientdiagonal) = S ((S (pfc_index_congruent_original_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_congruent_original_leftcoefficientscoefficient)) /\\ exists ff_q_pfp_congruent_original_leftcoefficientscoefficientdiagonalentry. pfc_terms_code_congruent_original_leftcoefficientscoefficient = ff_q_pfp_congruent_original_leftcoefficientscoefficientdiagonalentry * S ((S (pfc_index_congruent_original_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_congruent_original_leftcoefficientscoefficient) + (pfc_value_congruent_original_leftcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_congruent_original_leftcoefficientscoefficientdiagonalterm pfc_left_congruent_original_leftcoefficientscoefficientdiagonalterm pfc_right_congruent_original_leftcoefficientscoefficientdiagonalterm. (((pfc_index_congruent_original_leftcoefficientscoefficientdiagonal)+pfc_complement_congruent_original_leftcoefficientscoefficientdiagonalterm=(pfc_index_congruent_original_leftcoefficients)) /\\ ((((((exists pfa_gap_congruent_original_leftcoefficientscoefficientdiagonaltermleftinside. pfa_gap_congruent_original_leftcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_congruent_original_leftcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_congruent_original_leftcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_congruent_original_leftcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_congruent_original_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_congruent_original_leftcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_congruent_original_leftcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_congruent_original_leftcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_congruent_original_leftcoefficientscoefficientdiagonal)) * ac) + (pfc_left_congruent_original_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_congruent_original_leftcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_congruent_original_leftcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_congruent_original_leftcoefficientscoefficientdiagonal)) /\\ (((pfc_left_congruent_original_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_congruent_original_leftcoefficientscoefficientdiagonaltermrightinside. pfa_gap_congruent_original_leftcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_congruent_original_leftcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_congruent_original_leftcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_congruent_original_leftcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_congruent_original_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_congruent_original_leftcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_congruent_original_leftcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_congruent_original_leftcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_congruent_original_leftcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_congruent_original_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_congruent_original_leftcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_congruent_original_leftcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_congruent_original_leftcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_congruent_original_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_congruent_original_leftcoefficientscoefficientdiagonal)=pfc_left_congruent_original_leftcoefficientscoefficientdiagonalterm*pfc_right_congruent_original_leftcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_congruent_original_leftcoefficientscoefficientsum fs_v_pfc_congruent_original_leftcoefficientscoefficientsum. ((((exists fs_h_pfc_congruent_original_leftcoefficientscoefficientsum_body_start. fs_h_pfc_congruent_original_leftcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_congruent_original_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_original_leftcoefficientscoefficientsum_body_start. fs_u_pfc_congruent_original_leftcoefficientscoefficientsum = fs_q_pfc_congruent_original_leftcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_congruent_original_leftcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_congruent_original_leftcoefficientscoefficientsum_body_terminal. fs_h_pfc_congruent_original_leftcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_congruent_original_leftcoefficientscoefficient) = S ((S (S (pfc_index_congruent_original_leftcoefficients))) * fs_v_pfc_congruent_original_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_original_leftcoefficientscoefficientsum_body_terminal. fs_u_pfc_congruent_original_leftcoefficientscoefficientsum = fs_q_pfc_congruent_original_leftcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_congruent_original_leftcoefficients))) * fs_v_pfc_congruent_original_leftcoefficientscoefficientsum) + (pfc_natural_sum_congruent_original_leftcoefficientscoefficient))) /\\ forall fs_i_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps = S (pfc_index_congruent_original_leftcoefficients)) -> exists fs_a_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps fs_r_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps fs_s_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_congruent_original_leftcoefficientscoefficient)) /\\ exists fs_q_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_congruent_original_leftcoefficientscoefficient = fs_q_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_congruent_original_leftcoefficientscoefficient) + (fs_a_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_original_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_congruent_original_leftcoefficientscoefficientsum = fs_q_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_original_leftcoefficientscoefficientsum) + (fs_r_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_original_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_congruent_original_leftcoefficientscoefficientsum = fs_q_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_original_leftcoefficientscoefficientsum) + (fs_s_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps = fs_r_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps + fs_a_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_congruent_original_leftcoefficientscoefficientresiduebound. pfa_gap_congruent_original_leftcoefficientscoefficientresiduebound + S (pfc_value_congruent_original_leftcoefficients) = (p)) /\\ ((exists pfa_offset_left_congruent_original_leftcoefficientscoefficientresiduecongruence pfa_offset_right_congruent_original_leftcoefficientscoefficientresiduecongruence. (pfc_natural_sum_congruent_original_leftcoefficientscoefficient) + (p) * pfa_offset_left_congruent_original_leftcoefficientscoefficientresiduecongruence = (pfc_value_congruent_original_leftcoefficients) + (p) * pfa_offset_right_congruent_original_leftcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_congruent_other_leftleft. (exists fom_gap_pfp_congruent_other_leftleft_index_bound. fom_gap_pfp_congruent_other_leftleft_index_bound + S (fom_index_pfp_congruent_other_leftleft) = H) -> exists fom_value_pfp_congruent_other_leftleft. ((((exists fom_beta_height_pfp_congruent_other_leftleft_entry. fom_beta_height_pfp_congruent_other_leftleft_entry + S (fom_value_pfp_congruent_other_leftleft) = S ((S (fom_index_pfp_congruent_other_leftleft)) * AC)) /\\ exists fom_beta_quotient_pfp_congruent_other_leftleft_entry. AB = fom_beta_quotient_pfp_congruent_other_leftleft_entry * S ((S (fom_index_pfp_congruent_other_leftleft)) * AC) + (fom_value_pfp_congruent_other_leftleft))) /\\ (exists fom_gap_pfp_congruent_other_leftleft_value_bound. fom_gap_pfp_congruent_other_leftleft_value_bound + S (fom_value_pfp_congruent_other_leftleft) = p))) /\\ (((forall fom_index_pfp_congruent_other_leftright. (exists fom_gap_pfp_congruent_other_leftright_index_bound. fom_gap_pfp_congruent_other_leftright_index_bound + S (fom_index_pfp_congruent_other_leftright) = M) -> exists fom_value_pfp_congruent_other_leftright. ((((exists fom_beta_height_pfp_congruent_other_leftright_entry. fom_beta_height_pfp_congruent_other_leftright_entry + S (fom_value_pfp_congruent_other_leftright) = S ((S (fom_index_pfp_congruent_other_leftright)) * bc)) /\\ exists fom_beta_quotient_pfp_congruent_other_leftright_entry. bb = fom_beta_quotient_pfp_congruent_other_leftright_entry * S ((S (fom_index_pfp_congruent_other_leftright)) * bc) + (fom_value_pfp_congruent_other_leftright))) /\\ (exists fom_gap_pfp_congruent_other_leftright_value_bound. fom_gap_pfp_congruent_other_leftright_value_bound + S (fom_value_pfp_congruent_other_leftright) = p))) /\\ (((((((H)=0 \\/ (M)=0) /\\ (((K)=0)))) \\/ (((~((H)=0)) /\\ (((~((M)=0)) /\\ (((H)+(M)=S (K)))))))) /\\ ((forall pfc_index_congruent_other_leftcoefficients. (exists pfa_gap_congruent_other_leftcoefficientsbound. pfa_gap_congruent_other_leftcoefficientsbound + S (pfc_index_congruent_other_leftcoefficients) = (K)) -> exists pfc_value_congruent_other_leftcoefficients. ((((exists ff_h_pfp_congruent_other_leftcoefficientsentry. ff_h_pfp_congruent_other_leftcoefficientsentry + S (pfc_value_congruent_other_leftcoefficients) = S ((S (pfc_index_congruent_other_leftcoefficients)) * CC)) /\\ exists ff_q_pfp_congruent_other_leftcoefficientsentry. CB = ff_q_pfp_congruent_other_leftcoefficientsentry * S ((S (pfc_index_congruent_other_leftcoefficients)) * CC) + (pfc_value_congruent_other_leftcoefficients))) /\\ ((exists pfc_terms_code_congruent_other_leftcoefficientscoefficient pfc_terms_scale_congruent_other_leftcoefficientscoefficient pfc_natural_sum_congruent_other_leftcoefficientscoefficient. ((forall pfc_index_congruent_other_leftcoefficientscoefficientdiagonal. (exists pfa_gap_congruent_other_leftcoefficientscoefficientdiagonalbound. pfa_gap_congruent_other_leftcoefficientscoefficientdiagonalbound + S (pfc_index_congruent_other_leftcoefficientscoefficientdiagonal) = (S (pfc_index_congruent_other_leftcoefficients))) -> exists pfc_value_congruent_other_leftcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_congruent_other_leftcoefficientscoefficientdiagonalentry. ff_h_pfp_congruent_other_leftcoefficientscoefficientdiagonalentry + S (pfc_value_congruent_other_leftcoefficientscoefficientdiagonal) = S ((S (pfc_index_congruent_other_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_congruent_other_leftcoefficientscoefficient)) /\\ exists ff_q_pfp_congruent_other_leftcoefficientscoefficientdiagonalentry. pfc_terms_code_congruent_other_leftcoefficientscoefficient = ff_q_pfp_congruent_other_leftcoefficientscoefficientdiagonalentry * S ((S (pfc_index_congruent_other_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_congruent_other_leftcoefficientscoefficient) + (pfc_value_congruent_other_leftcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_congruent_other_leftcoefficientscoefficientdiagonalterm pfc_left_congruent_other_leftcoefficientscoefficientdiagonalterm pfc_right_congruent_other_leftcoefficientscoefficientdiagonalterm. (((pfc_index_congruent_other_leftcoefficientscoefficientdiagonal)+pfc_complement_congruent_other_leftcoefficientscoefficientdiagonalterm=(pfc_index_congruent_other_leftcoefficients)) /\\ ((((((exists pfa_gap_congruent_other_leftcoefficientscoefficientdiagonaltermleftinside. pfa_gap_congruent_other_leftcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_congruent_other_leftcoefficientscoefficientdiagonal) = (H)) /\\ ((((exists ff_h_pfp_congruent_other_leftcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_congruent_other_leftcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_congruent_other_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_congruent_other_leftcoefficientscoefficientdiagonal)) * AC)) /\\ exists ff_q_pfp_congruent_other_leftcoefficientscoefficientdiagonaltermleftentry. AB = ff_q_pfp_congruent_other_leftcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_congruent_other_leftcoefficientscoefficientdiagonal)) * AC) + (pfc_left_congruent_other_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_congruent_other_leftcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_congruent_other_leftcoefficientscoefficientdiagonaltermleftoutside+(H)=(pfc_index_congruent_other_leftcoefficientscoefficientdiagonal)) /\\ (((pfc_left_congruent_other_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_congruent_other_leftcoefficientscoefficientdiagonaltermrightinside. pfa_gap_congruent_other_leftcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_congruent_other_leftcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_congruent_other_leftcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_congruent_other_leftcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_congruent_other_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_congruent_other_leftcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_congruent_other_leftcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_congruent_other_leftcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_congruent_other_leftcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_congruent_other_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_congruent_other_leftcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_congruent_other_leftcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_congruent_other_leftcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_congruent_other_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_congruent_other_leftcoefficientscoefficientdiagonal)=pfc_left_congruent_other_leftcoefficientscoefficientdiagonalterm*pfc_right_congruent_other_leftcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_congruent_other_leftcoefficientscoefficientsum fs_v_pfc_congruent_other_leftcoefficientscoefficientsum. ((((exists fs_h_pfc_congruent_other_leftcoefficientscoefficientsum_body_start. fs_h_pfc_congruent_other_leftcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_congruent_other_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_other_leftcoefficientscoefficientsum_body_start. fs_u_pfc_congruent_other_leftcoefficientscoefficientsum = fs_q_pfc_congruent_other_leftcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_congruent_other_leftcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_congruent_other_leftcoefficientscoefficientsum_body_terminal. fs_h_pfc_congruent_other_leftcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_congruent_other_leftcoefficientscoefficient) = S ((S (S (pfc_index_congruent_other_leftcoefficients))) * fs_v_pfc_congruent_other_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_other_leftcoefficientscoefficientsum_body_terminal. fs_u_pfc_congruent_other_leftcoefficientscoefficientsum = fs_q_pfc_congruent_other_leftcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_congruent_other_leftcoefficients))) * fs_v_pfc_congruent_other_leftcoefficientscoefficientsum) + (pfc_natural_sum_congruent_other_leftcoefficientscoefficient))) /\\ forall fs_i_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps = S (pfc_index_congruent_other_leftcoefficients)) -> exists fs_a_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps fs_r_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps fs_s_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_congruent_other_leftcoefficientscoefficient)) /\\ exists fs_q_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_congruent_other_leftcoefficientscoefficient = fs_q_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_congruent_other_leftcoefficientscoefficient) + (fs_a_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_other_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_congruent_other_leftcoefficientscoefficientsum = fs_q_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_other_leftcoefficientscoefficientsum) + (fs_r_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_other_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_congruent_other_leftcoefficientscoefficientsum = fs_q_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_other_leftcoefficientscoefficientsum) + (fs_s_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps = fs_r_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps + fs_a_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_congruent_other_leftcoefficientscoefficientresiduebound. pfa_gap_congruent_other_leftcoefficientscoefficientresiduebound + S (pfc_value_congruent_other_leftcoefficients) = (p)) /\\ ((exists pfa_offset_left_congruent_other_leftcoefficientscoefficientresiduecongruence pfa_offset_right_congruent_other_leftcoefficientscoefficientresiduecongruence. (pfc_natural_sum_congruent_other_leftcoefficientscoefficient) + (p) * pfa_offset_left_congruent_other_leftcoefficientscoefficientresiduecongruence = (pfc_value_congruent_other_leftcoefficients) + (p) * pfa_offset_right_congruent_other_leftcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (forall pfrep_power_congruent_output_left pfrep_left_congruent_output_left pfrep_right_congruent_output_left. ((exists pfrep_position_congruent_output_leftfirst. ((pfrep_position_congruent_output_leftfirst+S (pfrep_power_congruent_output_left)=(N)) /\\ ((((exists ff_h_pfp_congruent_output_leftfirstentry. ff_h_pfp_congruent_output_leftfirstentry + S (pfrep_left_congruent_output_left) = S ((S (pfrep_position_congruent_output_leftfirst)) * cc)) /\\ exists ff_q_pfp_congruent_output_leftfirstentry. cb = ff_q_pfp_congruent_output_leftfirstentry * S ((S (pfrep_position_congruent_output_leftfirst)) * cc) + (pfrep_left_congruent_output_left)))))) \\/ (((exists pfrep_gap_congruent_output_leftfirstoutside. pfrep_gap_congruent_output_leftfirstoutside+(N)=(pfrep_power_congruent_output_left)) /\\ (((pfrep_left_congruent_output_left)=0))))) -> ((exists pfrep_position_congruent_output_leftsecond. ((pfrep_position_congruent_output_leftsecond+S (pfrep_power_congruent_output_left)=(K)) /\\ ((((exists ff_h_pfp_congruent_output_leftsecondentry. ff_h_pfp_congruent_output_leftsecondentry + S (pfrep_right_congruent_output_left) = S ((S (pfrep_position_congruent_output_leftsecond)) * CC)) /\\ exists ff_q_pfp_congruent_output_leftsecondentry. CB = ff_q_pfp_congruent_output_leftsecondentry * S ((S (pfrep_position_congruent_output_leftsecond)) * CC) + (pfrep_right_congruent_output_left)))))) \\/ (((exists pfrep_gap_congruent_output_leftsecondoutside. pfrep_gap_congruent_output_leftsecondoutside+(K)=(pfrep_power_congruent_output_left)) /\\ (((pfrep_right_congruent_output_left)=0))))) -> pfrep_left_congruent_output_left=pfrep_right_congruent_output_left)", "statement_sha256": "b0dcf38a0cd17a373bd25e6c8af3273f9d195eac29d71073c318740a8e4c860b", "summary": "Formal coefficient equivalence of the left factor preserves two actual products at arbitrary representation lengths, including empty factors; actual leading padding is recovered in the appropriate direction.", "summary_sha256": "7f2276edd38253c83112664a1492831262fd25b5dfbfb26dd7f04dad268a23f6" }, { "alpha_v33_frontier_enrollment": { "body_receipt_sha256": "f7547157422b3f78b3034ce3653fdfad0e2ae05b558f47d65d3105c4e29d11b4", "bundle_campaign": "polynomial-euclidean-division", 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"polynomial-euclidean-division", "logical_spec_sha256": "f221a06c75facbe7f7e49e0f0d73a4ca5db4887d029f6fa179807838884d232e", "membership": "alpha_only", "name": "prime_field_polynomial_convolution_equivalent_congruent_right", "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 H", "intro CB", "intro CC", "intro K", "intro hp", "intro he", "intro hc", "intro hd", "have horder : (exists pfc_gap_congruent_forward_right. pfc_gap_congruent_forward_right+(M)=(H)) \\/ (exists pfc_gap_congruent_backward_right. pfc_gap_congruent_backward_right+(H)=(M))", "specialize le_total (M)", "specialize le_total (H)", "apply le_total", "cases horder", "cases horder_left", "have hpadding : ((forall pfp_repeat_index_congruent_padding_rightzeros. (exists pfa_gap_congruent_padding_rightzerosindex. pfa_gap_congruent_padding_rightzerosindex + S (pfp_repeat_index_congruent_padding_rightzeros) = (x)) -> (((exists ff_h_pfp_congruent_padding_rightzerosentry. ff_h_pfp_congruent_padding_rightzerosentry + S (0) = S ((S (pfp_repeat_index_congruent_padding_rightzeros)) * BC)) /\\ exists ff_q_pfp_congruent_padding_rightzerosentry. BB = ff_q_pfp_congruent_padding_rightzerosentry * S ((S (pfp_repeat_index_congruent_padding_rightzeros)) * BC) + (0)))) /\\ ((forall pfrep_index_congruent_padding_right pfrep_value_congruent_padding_right. (exists pfa_gap_congruent_padding_rightbound. pfa_gap_congruent_padding_rightbound + S (pfrep_index_congruent_padding_right) = (M)) -> (((exists ff_h_pfp_congruent_padding_rightinput. ff_h_pfp_congruent_padding_rightinput + S (pfrep_value_congruent_padding_right) = S ((S (pfrep_index_congruent_padding_right)) * bc)) /\\ exists ff_q_pfp_congruent_padding_rightinput. bb = ff_q_pfp_congruent_padding_rightinput * S ((S (pfrep_index_congruent_padding_right)) * bc) + (pfrep_value_congruent_padding_right))) -> (((exists ff_h_pfp_congruent_padding_rightoutput. ff_h_pfp_congruent_padding_rightoutput + S (pfrep_value_congruent_padding_right) = S ((S ((x)+pfrep_index_congruent_padding_right)) * BC)) /\\ exists ff_q_pfp_congruent_padding_rightoutput. BB = ff_q_pfp_congruent_padding_rightoutput * S ((S ((x)+pfrep_index_congruent_padding_right)) * BC) + (pfrep_value_congruent_padding_right))))))", "specialize prime_field_polynomial_equivalent_implies_left_pad (bb)", "specialize prime_field_polynomial_equivalent_implies_left_pad (bc)", "specialize prime_field_polynomial_equivalent_implies_left_pad (M)", "specialize prime_field_polynomial_equivalent_implies_left_pad (x)", "specialize prime_field_polynomial_equivalent_implies_left_pad (BB)", "specialize prime_field_polynomial_equivalent_implies_left_pad (BC)", "apply prime_field_polynomial_equivalent_implies_left_pad", "rewrite horder_left_witness", "rewrite horder_left_witness", "exact he", "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 (bb)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (bc)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (M)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (cb)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (cc)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (N)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (BB)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (BC)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (x)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (CB)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (CC)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (K)", "apply prime_field_polynomial_convolution_left_padding_equivalent_right", "exact hp", "exact hpadding", "exact hc", "rewrite horder_left_witness", "rewrite horder_left_witness", "rewrite horder_left_witness", "rewrite horder_left_witness", "rewrite horder_left_witness", "rewrite horder_left_witness", "exact hd", "cases horder_right", "have hpadding : ((forall pfp_repeat_index_congruent_padding_rightzeros. (exists pfa_gap_congruent_padding_rightzerosindex. pfa_gap_congruent_padding_rightzerosindex + S (pfp_repeat_index_congruent_padding_rightzeros) = (x)) -> (((exists ff_h_pfp_congruent_padding_rightzerosentry. ff_h_pfp_congruent_padding_rightzerosentry + S (0) = S ((S (pfp_repeat_index_congruent_padding_rightzeros)) * bc)) /\\ exists ff_q_pfp_congruent_padding_rightzerosentry. bb = ff_q_pfp_congruent_padding_rightzerosentry * S ((S (pfp_repeat_index_congruent_padding_rightzeros)) * bc) + (0)))) /\\ ((forall pfrep_index_congruent_padding_right pfrep_value_congruent_padding_right. (exists pfa_gap_congruent_padding_rightbound. pfa_gap_congruent_padding_rightbound + S (pfrep_index_congruent_padding_right) = (H)) -> (((exists ff_h_pfp_congruent_padding_rightinput. ff_h_pfp_congruent_padding_rightinput + S (pfrep_value_congruent_padding_right) = S ((S (pfrep_index_congruent_padding_right)) * BC)) /\\ exists ff_q_pfp_congruent_padding_rightinput. BB = ff_q_pfp_congruent_padding_rightinput * S ((S (pfrep_index_congruent_padding_right)) * BC) + (pfrep_value_congruent_padding_right))) -> (((exists ff_h_pfp_congruent_padding_rightoutput. ff_h_pfp_congruent_padding_rightoutput + S (pfrep_value_congruent_padding_right) = S ((S ((x)+pfrep_index_congruent_padding_right)) * bc)) /\\ exists ff_q_pfp_congruent_padding_rightoutput. bb = ff_q_pfp_congruent_padding_rightoutput * S ((S ((x)+pfrep_index_congruent_padding_right)) * bc) + (pfrep_value_congruent_padding_right))))))", "specialize prime_field_polynomial_equivalent_implies_left_pad (BB)", "specialize prime_field_polynomial_equivalent_implies_left_pad (BC)", "specialize prime_field_polynomial_equivalent_implies_left_pad (H)", "specialize prime_field_polynomial_equivalent_implies_left_pad (x)", "specialize prime_field_polynomial_equivalent_implies_left_pad (bb)", "specialize prime_field_polynomial_equivalent_implies_left_pad (bc)", "apply prime_field_polynomial_equivalent_implies_left_pad", "rewrite horder_right_witness", "rewrite horder_right_witness", "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 (BB)", "specialize prime_field_polynomial_equivalent_symmetric (BC)", "specialize prime_field_polynomial_equivalent_symmetric (H)", "apply prime_field_polynomial_equivalent_symmetric", "exact he", "specialize prime_field_polynomial_equivalent_symmetric (CB)", "specialize prime_field_polynomial_equivalent_symmetric (CC)", "specialize prime_field_polynomial_equivalent_symmetric (K)", "specialize prime_field_polynomial_equivalent_symmetric (cb)", "specialize prime_field_polynomial_equivalent_symmetric (cc)", "specialize prime_field_polynomial_equivalent_symmetric (N)", "apply prime_field_polynomial_equivalent_symmetric", "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 (BB)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (BC)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (H)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (CB)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (CC)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (K)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (bb)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (bc)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (x)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (cb)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (cc)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (N)", "apply prime_field_polynomial_convolution_left_padding_equivalent_right", "exact hp", "exact hpadding", "exact hd", "rewrite horder_right_witness", "rewrite horder_right_witness", "rewrite horder_right_witness", "rewrite horder_right_witness", "rewrite horder_right_witness", "rewrite horder_right_witness", "exact hc" ], "script_sha256": "a473bb35be94b9c8dc37fbea2bc48008c71dddf800d04e11036e58053355ed6e", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_convolution_congruence_candidate.py", "sha256": "effc4b2df9418d9d964fd34216c4c1c2a09d12dd885877165c6fed2e761a8b70" }, "statement": "forall p ab ac L bb bc M cb cc N BB BC H CB CC K. (~(p=0)) -> (forall pfrep_power_congruent_factor_right pfrep_left_congruent_factor_right pfrep_right_congruent_factor_right. ((exists pfrep_position_congruent_factor_rightfirst. ((pfrep_position_congruent_factor_rightfirst+S (pfrep_power_congruent_factor_right)=(M)) /\\ ((((exists ff_h_pfp_congruent_factor_rightfirstentry. ff_h_pfp_congruent_factor_rightfirstentry + S (pfrep_left_congruent_factor_right) = S ((S (pfrep_position_congruent_factor_rightfirst)) * bc)) /\\ exists ff_q_pfp_congruent_factor_rightfirstentry. bb = ff_q_pfp_congruent_factor_rightfirstentry * S ((S (pfrep_position_congruent_factor_rightfirst)) * bc) + (pfrep_left_congruent_factor_right)))))) \\/ (((exists pfrep_gap_congruent_factor_rightfirstoutside. pfrep_gap_congruent_factor_rightfirstoutside+(M)=(pfrep_power_congruent_factor_right)) /\\ (((pfrep_left_congruent_factor_right)=0))))) -> ((exists pfrep_position_congruent_factor_rightsecond. ((pfrep_position_congruent_factor_rightsecond+S (pfrep_power_congruent_factor_right)=(H)) /\\ ((((exists ff_h_pfp_congruent_factor_rightsecondentry. ff_h_pfp_congruent_factor_rightsecondentry + S (pfrep_right_congruent_factor_right) = S ((S (pfrep_position_congruent_factor_rightsecond)) * BC)) /\\ exists ff_q_pfp_congruent_factor_rightsecondentry. BB = ff_q_pfp_congruent_factor_rightsecondentry * S ((S (pfrep_position_congruent_factor_rightsecond)) * BC) + (pfrep_right_congruent_factor_right)))))) \\/ (((exists pfrep_gap_congruent_factor_rightsecondoutside. pfrep_gap_congruent_factor_rightsecondoutside+(H)=(pfrep_power_congruent_factor_right)) /\\ (((pfrep_right_congruent_factor_right)=0))))) -> pfrep_left_congruent_factor_right=pfrep_right_congruent_factor_right) -> (((forall fom_index_pfp_congruent_original_rightleft. (exists fom_gap_pfp_congruent_original_rightleft_index_bound. fom_gap_pfp_congruent_original_rightleft_index_bound + S (fom_index_pfp_congruent_original_rightleft) = L) -> exists fom_value_pfp_congruent_original_rightleft. ((((exists fom_beta_height_pfp_congruent_original_rightleft_entry. fom_beta_height_pfp_congruent_original_rightleft_entry + S (fom_value_pfp_congruent_original_rightleft) = S ((S (fom_index_pfp_congruent_original_rightleft)) * ac)) /\\ exists fom_beta_quotient_pfp_congruent_original_rightleft_entry. ab = fom_beta_quotient_pfp_congruent_original_rightleft_entry * S ((S (fom_index_pfp_congruent_original_rightleft)) * ac) + (fom_value_pfp_congruent_original_rightleft))) /\\ (exists fom_gap_pfp_congruent_original_rightleft_value_bound. fom_gap_pfp_congruent_original_rightleft_value_bound + S (fom_value_pfp_congruent_original_rightleft) = p))) /\\ (((forall fom_index_pfp_congruent_original_rightright. (exists fom_gap_pfp_congruent_original_rightright_index_bound. fom_gap_pfp_congruent_original_rightright_index_bound + S (fom_index_pfp_congruent_original_rightright) = M) -> exists fom_value_pfp_congruent_original_rightright. ((((exists fom_beta_height_pfp_congruent_original_rightright_entry. fom_beta_height_pfp_congruent_original_rightright_entry + S (fom_value_pfp_congruent_original_rightright) = S ((S (fom_index_pfp_congruent_original_rightright)) * bc)) /\\ exists fom_beta_quotient_pfp_congruent_original_rightright_entry. bb = fom_beta_quotient_pfp_congruent_original_rightright_entry * S ((S (fom_index_pfp_congruent_original_rightright)) * bc) + (fom_value_pfp_congruent_original_rightright))) /\\ (exists fom_gap_pfp_congruent_original_rightright_value_bound. fom_gap_pfp_congruent_original_rightright_value_bound + S (fom_value_pfp_congruent_original_rightright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_congruent_original_rightcoefficients. (exists pfa_gap_congruent_original_rightcoefficientsbound. pfa_gap_congruent_original_rightcoefficientsbound + S (pfc_index_congruent_original_rightcoefficients) = (N)) -> exists pfc_value_congruent_original_rightcoefficients. ((((exists ff_h_pfp_congruent_original_rightcoefficientsentry. ff_h_pfp_congruent_original_rightcoefficientsentry + S (pfc_value_congruent_original_rightcoefficients) = S ((S (pfc_index_congruent_original_rightcoefficients)) * cc)) /\\ exists ff_q_pfp_congruent_original_rightcoefficientsentry. cb = ff_q_pfp_congruent_original_rightcoefficientsentry * S ((S (pfc_index_congruent_original_rightcoefficients)) * cc) + (pfc_value_congruent_original_rightcoefficients))) /\\ ((exists pfc_terms_code_congruent_original_rightcoefficientscoefficient pfc_terms_scale_congruent_original_rightcoefficientscoefficient pfc_natural_sum_congruent_original_rightcoefficientscoefficient. ((forall pfc_index_congruent_original_rightcoefficientscoefficientdiagonal. (exists pfa_gap_congruent_original_rightcoefficientscoefficientdiagonalbound. pfa_gap_congruent_original_rightcoefficientscoefficientdiagonalbound + S (pfc_index_congruent_original_rightcoefficientscoefficientdiagonal) = (S (pfc_index_congruent_original_rightcoefficients))) -> exists pfc_value_congruent_original_rightcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_congruent_original_rightcoefficientscoefficientdiagonalentry. ff_h_pfp_congruent_original_rightcoefficientscoefficientdiagonalentry + S (pfc_value_congruent_original_rightcoefficientscoefficientdiagonal) = S ((S (pfc_index_congruent_original_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_congruent_original_rightcoefficientscoefficient)) /\\ exists ff_q_pfp_congruent_original_rightcoefficientscoefficientdiagonalentry. pfc_terms_code_congruent_original_rightcoefficientscoefficient = ff_q_pfp_congruent_original_rightcoefficientscoefficientdiagonalentry * S ((S (pfc_index_congruent_original_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_congruent_original_rightcoefficientscoefficient) + (pfc_value_congruent_original_rightcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_congruent_original_rightcoefficientscoefficientdiagonalterm pfc_left_congruent_original_rightcoefficientscoefficientdiagonalterm pfc_right_congruent_original_rightcoefficientscoefficientdiagonalterm. (((pfc_index_congruent_original_rightcoefficientscoefficientdiagonal)+pfc_complement_congruent_original_rightcoefficientscoefficientdiagonalterm=(pfc_index_congruent_original_rightcoefficients)) /\\ ((((((exists pfa_gap_congruent_original_rightcoefficientscoefficientdiagonaltermleftinside. pfa_gap_congruent_original_rightcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_congruent_original_rightcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_congruent_original_rightcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_congruent_original_rightcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_congruent_original_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_congruent_original_rightcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_congruent_original_rightcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_congruent_original_rightcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_congruent_original_rightcoefficientscoefficientdiagonal)) * ac) + (pfc_left_congruent_original_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_congruent_original_rightcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_congruent_original_rightcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_congruent_original_rightcoefficientscoefficientdiagonal)) /\\ (((pfc_left_congruent_original_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_congruent_original_rightcoefficientscoefficientdiagonaltermrightinside. pfa_gap_congruent_original_rightcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_congruent_original_rightcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_congruent_original_rightcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_congruent_original_rightcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_congruent_original_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_congruent_original_rightcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_congruent_original_rightcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_congruent_original_rightcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_congruent_original_rightcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_congruent_original_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_congruent_original_rightcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_congruent_original_rightcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_congruent_original_rightcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_congruent_original_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_congruent_original_rightcoefficientscoefficientdiagonal)=pfc_left_congruent_original_rightcoefficientscoefficientdiagonalterm*pfc_right_congruent_original_rightcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_congruent_original_rightcoefficientscoefficientsum fs_v_pfc_congruent_original_rightcoefficientscoefficientsum. ((((exists fs_h_pfc_congruent_original_rightcoefficientscoefficientsum_body_start. fs_h_pfc_congruent_original_rightcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_congruent_original_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_original_rightcoefficientscoefficientsum_body_start. fs_u_pfc_congruent_original_rightcoefficientscoefficientsum = fs_q_pfc_congruent_original_rightcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_congruent_original_rightcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_congruent_original_rightcoefficientscoefficientsum_body_terminal. fs_h_pfc_congruent_original_rightcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_congruent_original_rightcoefficientscoefficient) = S ((S (S (pfc_index_congruent_original_rightcoefficients))) * fs_v_pfc_congruent_original_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_original_rightcoefficientscoefficientsum_body_terminal. fs_u_pfc_congruent_original_rightcoefficientscoefficientsum = fs_q_pfc_congruent_original_rightcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_congruent_original_rightcoefficients))) * fs_v_pfc_congruent_original_rightcoefficientscoefficientsum) + (pfc_natural_sum_congruent_original_rightcoefficientscoefficient))) /\\ forall fs_i_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps = S (pfc_index_congruent_original_rightcoefficients)) -> exists fs_a_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps fs_r_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps fs_s_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_congruent_original_rightcoefficientscoefficient)) /\\ exists fs_q_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_congruent_original_rightcoefficientscoefficient = fs_q_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_congruent_original_rightcoefficientscoefficient) + (fs_a_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_original_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_congruent_original_rightcoefficientscoefficientsum = fs_q_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_original_rightcoefficientscoefficientsum) + (fs_r_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_original_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_congruent_original_rightcoefficientscoefficientsum = fs_q_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_original_rightcoefficientscoefficientsum) + (fs_s_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps = fs_r_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps + fs_a_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_congruent_original_rightcoefficientscoefficientresiduebound. pfa_gap_congruent_original_rightcoefficientscoefficientresiduebound + S (pfc_value_congruent_original_rightcoefficients) = (p)) /\\ ((exists pfa_offset_left_congruent_original_rightcoefficientscoefficientresiduecongruence pfa_offset_right_congruent_original_rightcoefficientscoefficientresiduecongruence. (pfc_natural_sum_congruent_original_rightcoefficientscoefficient) + (p) * pfa_offset_left_congruent_original_rightcoefficientscoefficientresiduecongruence = (pfc_value_congruent_original_rightcoefficients) + (p) * pfa_offset_right_congruent_original_rightcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_congruent_other_rightleft. (exists fom_gap_pfp_congruent_other_rightleft_index_bound. fom_gap_pfp_congruent_other_rightleft_index_bound + S (fom_index_pfp_congruent_other_rightleft) = L) -> exists fom_value_pfp_congruent_other_rightleft. ((((exists fom_beta_height_pfp_congruent_other_rightleft_entry. fom_beta_height_pfp_congruent_other_rightleft_entry + S (fom_value_pfp_congruent_other_rightleft) = S ((S (fom_index_pfp_congruent_other_rightleft)) * ac)) /\\ exists fom_beta_quotient_pfp_congruent_other_rightleft_entry. ab = fom_beta_quotient_pfp_congruent_other_rightleft_entry * S ((S (fom_index_pfp_congruent_other_rightleft)) * ac) + (fom_value_pfp_congruent_other_rightleft))) /\\ (exists fom_gap_pfp_congruent_other_rightleft_value_bound. fom_gap_pfp_congruent_other_rightleft_value_bound + S (fom_value_pfp_congruent_other_rightleft) = p))) /\\ (((forall fom_index_pfp_congruent_other_rightright. (exists fom_gap_pfp_congruent_other_rightright_index_bound. fom_gap_pfp_congruent_other_rightright_index_bound + S (fom_index_pfp_congruent_other_rightright) = H) -> exists fom_value_pfp_congruent_other_rightright. ((((exists fom_beta_height_pfp_congruent_other_rightright_entry. fom_beta_height_pfp_congruent_other_rightright_entry + S (fom_value_pfp_congruent_other_rightright) = S ((S (fom_index_pfp_congruent_other_rightright)) * BC)) /\\ exists fom_beta_quotient_pfp_congruent_other_rightright_entry. BB = fom_beta_quotient_pfp_congruent_other_rightright_entry * S ((S (fom_index_pfp_congruent_other_rightright)) * BC) + (fom_value_pfp_congruent_other_rightright))) /\\ (exists fom_gap_pfp_congruent_other_rightright_value_bound. fom_gap_pfp_congruent_other_rightright_value_bound + S (fom_value_pfp_congruent_other_rightright) = p))) /\\ (((((((L)=0 \\/ (H)=0) /\\ (((K)=0)))) \\/ (((~((L)=0)) /\\ (((~((H)=0)) /\\ (((L)+(H)=S (K)))))))) /\\ ((forall pfc_index_congruent_other_rightcoefficients. (exists pfa_gap_congruent_other_rightcoefficientsbound. pfa_gap_congruent_other_rightcoefficientsbound + S (pfc_index_congruent_other_rightcoefficients) = (K)) -> exists pfc_value_congruent_other_rightcoefficients. ((((exists ff_h_pfp_congruent_other_rightcoefficientsentry. ff_h_pfp_congruent_other_rightcoefficientsentry + S (pfc_value_congruent_other_rightcoefficients) = S ((S (pfc_index_congruent_other_rightcoefficients)) * CC)) /\\ exists ff_q_pfp_congruent_other_rightcoefficientsentry. CB = ff_q_pfp_congruent_other_rightcoefficientsentry * S ((S (pfc_index_congruent_other_rightcoefficients)) * CC) + (pfc_value_congruent_other_rightcoefficients))) /\\ ((exists pfc_terms_code_congruent_other_rightcoefficientscoefficient pfc_terms_scale_congruent_other_rightcoefficientscoefficient pfc_natural_sum_congruent_other_rightcoefficientscoefficient. ((forall pfc_index_congruent_other_rightcoefficientscoefficientdiagonal. (exists pfa_gap_congruent_other_rightcoefficientscoefficientdiagonalbound. pfa_gap_congruent_other_rightcoefficientscoefficientdiagonalbound + S (pfc_index_congruent_other_rightcoefficientscoefficientdiagonal) = (S (pfc_index_congruent_other_rightcoefficients))) -> exists pfc_value_congruent_other_rightcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_congruent_other_rightcoefficientscoefficientdiagonalentry. ff_h_pfp_congruent_other_rightcoefficientscoefficientdiagonalentry + S (pfc_value_congruent_other_rightcoefficientscoefficientdiagonal) = S ((S (pfc_index_congruent_other_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_congruent_other_rightcoefficientscoefficient)) /\\ exists ff_q_pfp_congruent_other_rightcoefficientscoefficientdiagonalentry. pfc_terms_code_congruent_other_rightcoefficientscoefficient = ff_q_pfp_congruent_other_rightcoefficientscoefficientdiagonalentry * S ((S (pfc_index_congruent_other_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_congruent_other_rightcoefficientscoefficient) + (pfc_value_congruent_other_rightcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_congruent_other_rightcoefficientscoefficientdiagonalterm pfc_left_congruent_other_rightcoefficientscoefficientdiagonalterm pfc_right_congruent_other_rightcoefficientscoefficientdiagonalterm. (((pfc_index_congruent_other_rightcoefficientscoefficientdiagonal)+pfc_complement_congruent_other_rightcoefficientscoefficientdiagonalterm=(pfc_index_congruent_other_rightcoefficients)) /\\ ((((((exists pfa_gap_congruent_other_rightcoefficientscoefficientdiagonaltermleftinside. pfa_gap_congruent_other_rightcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_congruent_other_rightcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_congruent_other_rightcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_congruent_other_rightcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_congruent_other_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_congruent_other_rightcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_congruent_other_rightcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_congruent_other_rightcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_congruent_other_rightcoefficientscoefficientdiagonal)) * ac) + (pfc_left_congruent_other_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_congruent_other_rightcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_congruent_other_rightcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_congruent_other_rightcoefficientscoefficientdiagonal)) /\\ (((pfc_left_congruent_other_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_congruent_other_rightcoefficientscoefficientdiagonaltermrightinside. pfa_gap_congruent_other_rightcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_congruent_other_rightcoefficientscoefficientdiagonalterm) = (H)) /\\ ((((exists ff_h_pfp_congruent_other_rightcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_congruent_other_rightcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_congruent_other_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_congruent_other_rightcoefficientscoefficientdiagonalterm)) * BC)) /\\ exists ff_q_pfp_congruent_other_rightcoefficientscoefficientdiagonaltermrightentry. BB = ff_q_pfp_congruent_other_rightcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_congruent_other_rightcoefficientscoefficientdiagonalterm)) * BC) + (pfc_right_congruent_other_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_congruent_other_rightcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_congruent_other_rightcoefficientscoefficientdiagonaltermrightoutside+(H)=(pfc_complement_congruent_other_rightcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_congruent_other_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_congruent_other_rightcoefficientscoefficientdiagonal)=pfc_left_congruent_other_rightcoefficientscoefficientdiagonalterm*pfc_right_congruent_other_rightcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_congruent_other_rightcoefficientscoefficientsum fs_v_pfc_congruent_other_rightcoefficientscoefficientsum. ((((exists fs_h_pfc_congruent_other_rightcoefficientscoefficientsum_body_start. fs_h_pfc_congruent_other_rightcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_congruent_other_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_other_rightcoefficientscoefficientsum_body_start. fs_u_pfc_congruent_other_rightcoefficientscoefficientsum = fs_q_pfc_congruent_other_rightcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_congruent_other_rightcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_congruent_other_rightcoefficientscoefficientsum_body_terminal. fs_h_pfc_congruent_other_rightcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_congruent_other_rightcoefficientscoefficient) = S ((S (S (pfc_index_congruent_other_rightcoefficients))) * fs_v_pfc_congruent_other_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_other_rightcoefficientscoefficientsum_body_terminal. fs_u_pfc_congruent_other_rightcoefficientscoefficientsum = fs_q_pfc_congruent_other_rightcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_congruent_other_rightcoefficients))) * fs_v_pfc_congruent_other_rightcoefficientscoefficientsum) + (pfc_natural_sum_congruent_other_rightcoefficientscoefficient))) /\\ forall fs_i_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps = S (pfc_index_congruent_other_rightcoefficients)) -> exists fs_a_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps fs_r_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps fs_s_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_congruent_other_rightcoefficientscoefficient)) /\\ exists fs_q_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_congruent_other_rightcoefficientscoefficient = fs_q_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_congruent_other_rightcoefficientscoefficient) + (fs_a_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_other_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_congruent_other_rightcoefficientscoefficientsum = fs_q_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_other_rightcoefficientscoefficientsum) + (fs_r_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_other_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_congruent_other_rightcoefficientscoefficientsum = fs_q_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_other_rightcoefficientscoefficientsum) + (fs_s_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps = fs_r_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps + fs_a_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_congruent_other_rightcoefficientscoefficientresiduebound. pfa_gap_congruent_other_rightcoefficientscoefficientresiduebound + S (pfc_value_congruent_other_rightcoefficients) = (p)) /\\ ((exists pfa_offset_left_congruent_other_rightcoefficientscoefficientresiduecongruence pfa_offset_right_congruent_other_rightcoefficientscoefficientresiduecongruence. (pfc_natural_sum_congruent_other_rightcoefficientscoefficient) + (p) * pfa_offset_left_congruent_other_rightcoefficientscoefficientresiduecongruence = (pfc_value_congruent_other_rightcoefficients) + (p) * pfa_offset_right_congruent_other_rightcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (forall pfrep_power_congruent_output_right pfrep_left_congruent_output_right pfrep_right_congruent_output_right. ((exists pfrep_position_congruent_output_rightfirst. ((pfrep_position_congruent_output_rightfirst+S (pfrep_power_congruent_output_right)=(N)) /\\ ((((exists ff_h_pfp_congruent_output_rightfirstentry. ff_h_pfp_congruent_output_rightfirstentry + S (pfrep_left_congruent_output_right) = S ((S (pfrep_position_congruent_output_rightfirst)) * cc)) /\\ exists ff_q_pfp_congruent_output_rightfirstentry. cb = ff_q_pfp_congruent_output_rightfirstentry * S ((S (pfrep_position_congruent_output_rightfirst)) * cc) + (pfrep_left_congruent_output_right)))))) \\/ (((exists pfrep_gap_congruent_output_rightfirstoutside. pfrep_gap_congruent_output_rightfirstoutside+(N)=(pfrep_power_congruent_output_right)) /\\ (((pfrep_left_congruent_output_right)=0))))) -> ((exists pfrep_position_congruent_output_rightsecond. ((pfrep_position_congruent_output_rightsecond+S (pfrep_power_congruent_output_right)=(K)) /\\ ((((exists ff_h_pfp_congruent_output_rightsecondentry. ff_h_pfp_congruent_output_rightsecondentry + S (pfrep_right_congruent_output_right) = S ((S (pfrep_position_congruent_output_rightsecond)) * CC)) /\\ exists ff_q_pfp_congruent_output_rightsecondentry. CB = ff_q_pfp_congruent_output_rightsecondentry * S ((S (pfrep_position_congruent_output_rightsecond)) * CC) + (pfrep_right_congruent_output_right)))))) \\/ (((exists pfrep_gap_congruent_output_rightsecondoutside. pfrep_gap_congruent_output_rightsecondoutside+(K)=(pfrep_power_congruent_output_right)) /\\ (((pfrep_right_congruent_output_right)=0))))) -> pfrep_left_congruent_output_right=pfrep_right_congruent_output_right)", "statement_sha256": "3ae339f6d8812f2be7f6bb5d4b3fad1a763835e69a939b88a775091c43266dd2", "summary": "Formal coefficient equivalence of the right factor preserves two actual products at arbitrary representation lengths, including empty factors; actual leading padding is recovered in the appropriate direction.", "summary_sha256": "d9264d3ae853edaa6e2645149845ca948941be3acce8b5b80afcc76b7a9444d7" }, { "alpha_v33_frontier_enrollment": { "body_receipt_sha256": "26a729421aecfd84f2a9f2043eaf20097a64d88f4497de2c3d2f1b6e19b8350a", "bundle_campaign": 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"catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "polynomial-euclidean-division", "logical_spec_sha256": "4239b266374b08ce3f975a9edcdf4f7147086dba0d567e618d208fb7b4d2c9b6", "membership": "alpha_only", "name": "prime_field_polynomial_convolution_equivalent_congruent", "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 AB", "intro AC", "intro H", "intro BB", "intro BC", "intro I", "intro CB", "intro CC", "intro K", "intro hp", "intro hA", "intro hB", "intro hc", "intro hd", "have hsource : ((forall fom_index_pfp_congruent_both_originalleft. (exists fom_gap_pfp_congruent_both_originalleft_index_bound. fom_gap_pfp_congruent_both_originalleft_index_bound + S (fom_index_pfp_congruent_both_originalleft) = L) -> exists fom_value_pfp_congruent_both_originalleft. ((((exists fom_beta_height_pfp_congruent_both_originalleft_entry. fom_beta_height_pfp_congruent_both_originalleft_entry + S (fom_value_pfp_congruent_both_originalleft) = S ((S (fom_index_pfp_congruent_both_originalleft)) * ac)) /\\ exists fom_beta_quotient_pfp_congruent_both_originalleft_entry. ab = fom_beta_quotient_pfp_congruent_both_originalleft_entry * S ((S (fom_index_pfp_congruent_both_originalleft)) * ac) + (fom_value_pfp_congruent_both_originalleft))) /\\ (exists fom_gap_pfp_congruent_both_originalleft_value_bound. fom_gap_pfp_congruent_both_originalleft_value_bound + S (fom_value_pfp_congruent_both_originalleft) = p))) /\\ (((forall fom_index_pfp_congruent_both_originalright. (exists fom_gap_pfp_congruent_both_originalright_index_bound. fom_gap_pfp_congruent_both_originalright_index_bound + S (fom_index_pfp_congruent_both_originalright) = M) -> exists fom_value_pfp_congruent_both_originalright. ((((exists fom_beta_height_pfp_congruent_both_originalright_entry. fom_beta_height_pfp_congruent_both_originalright_entry + S (fom_value_pfp_congruent_both_originalright) = S ((S (fom_index_pfp_congruent_both_originalright)) * bc)) /\\ exists fom_beta_quotient_pfp_congruent_both_originalright_entry. bb = fom_beta_quotient_pfp_congruent_both_originalright_entry * S ((S (fom_index_pfp_congruent_both_originalright)) * bc) + (fom_value_pfp_congruent_both_originalright))) /\\ (exists fom_gap_pfp_congruent_both_originalright_value_bound. fom_gap_pfp_congruent_both_originalright_value_bound + S (fom_value_pfp_congruent_both_originalright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_congruent_both_originalcoefficients. (exists pfa_gap_congruent_both_originalcoefficientsbound. pfa_gap_congruent_both_originalcoefficientsbound + S (pfc_index_congruent_both_originalcoefficients) = (N)) -> exists pfc_value_congruent_both_originalcoefficients. ((((exists ff_h_pfp_congruent_both_originalcoefficientsentry. ff_h_pfp_congruent_both_originalcoefficientsentry + S (pfc_value_congruent_both_originalcoefficients) = S ((S (pfc_index_congruent_both_originalcoefficients)) * cc)) /\\ exists ff_q_pfp_congruent_both_originalcoefficientsentry. cb = ff_q_pfp_congruent_both_originalcoefficientsentry * S ((S (pfc_index_congruent_both_originalcoefficients)) * cc) + (pfc_value_congruent_both_originalcoefficients))) /\\ ((exists pfc_terms_code_congruent_both_originalcoefficientscoefficient pfc_terms_scale_congruent_both_originalcoefficientscoefficient pfc_natural_sum_congruent_both_originalcoefficientscoefficient. ((forall pfc_index_congruent_both_originalcoefficientscoefficientdiagonal. (exists pfa_gap_congruent_both_originalcoefficientscoefficientdiagonalbound. pfa_gap_congruent_both_originalcoefficientscoefficientdiagonalbound + S (pfc_index_congruent_both_originalcoefficientscoefficientdiagonal) = (S (pfc_index_congruent_both_originalcoefficients))) -> exists pfc_value_congruent_both_originalcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_congruent_both_originalcoefficientscoefficientdiagonalentry. ff_h_pfp_congruent_both_originalcoefficientscoefficientdiagonalentry + S (pfc_value_congruent_both_originalcoefficientscoefficientdiagonal) = S ((S (pfc_index_congruent_both_originalcoefficientscoefficientdiagonal)) * pfc_terms_scale_congruent_both_originalcoefficientscoefficient)) /\\ exists ff_q_pfp_congruent_both_originalcoefficientscoefficientdiagonalentry. pfc_terms_code_congruent_both_originalcoefficientscoefficient = ff_q_pfp_congruent_both_originalcoefficientscoefficientdiagonalentry * S ((S (pfc_index_congruent_both_originalcoefficientscoefficientdiagonal)) * pfc_terms_scale_congruent_both_originalcoefficientscoefficient) + (pfc_value_congruent_both_originalcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_congruent_both_originalcoefficientscoefficientdiagonalterm pfc_left_congruent_both_originalcoefficientscoefficientdiagonalterm pfc_right_congruent_both_originalcoefficientscoefficientdiagonalterm. (((pfc_index_congruent_both_originalcoefficientscoefficientdiagonal)+pfc_complement_congruent_both_originalcoefficientscoefficientdiagonalterm=(pfc_index_congruent_both_originalcoefficients)) /\\ ((((((exists pfa_gap_congruent_both_originalcoefficientscoefficientdiagonaltermleftinside. pfa_gap_congruent_both_originalcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_congruent_both_originalcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_congruent_both_originalcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_congruent_both_originalcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_congruent_both_originalcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_congruent_both_originalcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_congruent_both_originalcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_congruent_both_originalcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_congruent_both_originalcoefficientscoefficientdiagonal)) * ac) + (pfc_left_congruent_both_originalcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_congruent_both_originalcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_congruent_both_originalcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_congruent_both_originalcoefficientscoefficientdiagonal)) /\\ (((pfc_left_congruent_both_originalcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_congruent_both_originalcoefficientscoefficientdiagonaltermrightinside. pfa_gap_congruent_both_originalcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_congruent_both_originalcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_congruent_both_originalcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_congruent_both_originalcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_congruent_both_originalcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_congruent_both_originalcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_congruent_both_originalcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_congruent_both_originalcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_congruent_both_originalcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_congruent_both_originalcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_congruent_both_originalcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_congruent_both_originalcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_congruent_both_originalcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_congruent_both_originalcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_congruent_both_originalcoefficientscoefficientdiagonal)=pfc_left_congruent_both_originalcoefficientscoefficientdiagonalterm*pfc_right_congruent_both_originalcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_congruent_both_originalcoefficientscoefficientsum fs_v_pfc_congruent_both_originalcoefficientscoefficientsum. ((((exists fs_h_pfc_congruent_both_originalcoefficientscoefficientsum_body_start. fs_h_pfc_congruent_both_originalcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_congruent_both_originalcoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_both_originalcoefficientscoefficientsum_body_start. fs_u_pfc_congruent_both_originalcoefficientscoefficientsum = fs_q_pfc_congruent_both_originalcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_congruent_both_originalcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_congruent_both_originalcoefficientscoefficientsum_body_terminal. fs_h_pfc_congruent_both_originalcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_congruent_both_originalcoefficientscoefficient) = S ((S (S (pfc_index_congruent_both_originalcoefficients))) * fs_v_pfc_congruent_both_originalcoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_both_originalcoefficientscoefficientsum_body_terminal. fs_u_pfc_congruent_both_originalcoefficientscoefficientsum = fs_q_pfc_congruent_both_originalcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_congruent_both_originalcoefficients))) * fs_v_pfc_congruent_both_originalcoefficientscoefficientsum) + (pfc_natural_sum_congruent_both_originalcoefficientscoefficient))) /\\ forall fs_i_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps = S (pfc_index_congruent_both_originalcoefficients)) -> exists fs_a_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps fs_r_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps fs_s_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_congruent_both_originalcoefficientscoefficient)) /\\ exists fs_q_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_congruent_both_originalcoefficientscoefficient = fs_q_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_congruent_both_originalcoefficientscoefficient) + (fs_a_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_both_originalcoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_congruent_both_originalcoefficientscoefficientsum = fs_q_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_both_originalcoefficientscoefficientsum) + (fs_r_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_both_originalcoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_congruent_both_originalcoefficientscoefficientsum = fs_q_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_both_originalcoefficientscoefficientsum) + (fs_s_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps = fs_r_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps + fs_a_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_congruent_both_originalcoefficientscoefficientresiduebound. pfa_gap_congruent_both_originalcoefficientscoefficientresiduebound + S (pfc_value_congruent_both_originalcoefficients) = (p)) /\\ ((exists pfa_offset_left_congruent_both_originalcoefficientscoefficientresiduecongruence pfa_offset_right_congruent_both_originalcoefficientscoefficientresiduecongruence. (pfc_natural_sum_congruent_both_originalcoefficientscoefficient) + (p) * pfa_offset_left_congruent_both_originalcoefficientscoefficientresiduecongruence = (pfc_value_congruent_both_originalcoefficients) + (p) * pfa_offset_right_congruent_both_originalcoefficientscoefficientresiduecongruence))))))))))))))))))", "exact hc", "cases hsource", "cases hsource_right", "cases hsource_right_right", "have htarget : ((forall fom_index_pfp_congruent_both_otherleft. (exists fom_gap_pfp_congruent_both_otherleft_index_bound. fom_gap_pfp_congruent_both_otherleft_index_bound + S (fom_index_pfp_congruent_both_otherleft) = H) -> exists fom_value_pfp_congruent_both_otherleft. ((((exists fom_beta_height_pfp_congruent_both_otherleft_entry. fom_beta_height_pfp_congruent_both_otherleft_entry + S (fom_value_pfp_congruent_both_otherleft) = S ((S (fom_index_pfp_congruent_both_otherleft)) * AC)) /\\ exists fom_beta_quotient_pfp_congruent_both_otherleft_entry. AB = fom_beta_quotient_pfp_congruent_both_otherleft_entry * S ((S (fom_index_pfp_congruent_both_otherleft)) * AC) + (fom_value_pfp_congruent_both_otherleft))) /\\ (exists fom_gap_pfp_congruent_both_otherleft_value_bound. fom_gap_pfp_congruent_both_otherleft_value_bound + S (fom_value_pfp_congruent_both_otherleft) = p))) /\\ (((forall fom_index_pfp_congruent_both_otherright. (exists fom_gap_pfp_congruent_both_otherright_index_bound. fom_gap_pfp_congruent_both_otherright_index_bound + S (fom_index_pfp_congruent_both_otherright) = I) -> exists fom_value_pfp_congruent_both_otherright. ((((exists fom_beta_height_pfp_congruent_both_otherright_entry. fom_beta_height_pfp_congruent_both_otherright_entry + S (fom_value_pfp_congruent_both_otherright) = S ((S (fom_index_pfp_congruent_both_otherright)) * BC)) /\\ exists fom_beta_quotient_pfp_congruent_both_otherright_entry. BB = fom_beta_quotient_pfp_congruent_both_otherright_entry * S ((S (fom_index_pfp_congruent_both_otherright)) * BC) + (fom_value_pfp_congruent_both_otherright))) /\\ (exists fom_gap_pfp_congruent_both_otherright_value_bound. fom_gap_pfp_congruent_both_otherright_value_bound + S (fom_value_pfp_congruent_both_otherright) = p))) /\\ (((((((H)=0 \\/ (I)=0) /\\ (((K)=0)))) \\/ (((~((H)=0)) /\\ (((~((I)=0)) /\\ (((H)+(I)=S (K)))))))) /\\ ((forall pfc_index_congruent_both_othercoefficients. (exists pfa_gap_congruent_both_othercoefficientsbound. pfa_gap_congruent_both_othercoefficientsbound + S (pfc_index_congruent_both_othercoefficients) = (K)) -> exists pfc_value_congruent_both_othercoefficients. ((((exists ff_h_pfp_congruent_both_othercoefficientsentry. ff_h_pfp_congruent_both_othercoefficientsentry + S (pfc_value_congruent_both_othercoefficients) = S ((S (pfc_index_congruent_both_othercoefficients)) * CC)) /\\ exists ff_q_pfp_congruent_both_othercoefficientsentry. CB = ff_q_pfp_congruent_both_othercoefficientsentry * S ((S (pfc_index_congruent_both_othercoefficients)) * CC) + (pfc_value_congruent_both_othercoefficients))) /\\ ((exists pfc_terms_code_congruent_both_othercoefficientscoefficient pfc_terms_scale_congruent_both_othercoefficientscoefficient pfc_natural_sum_congruent_both_othercoefficientscoefficient. ((forall pfc_index_congruent_both_othercoefficientscoefficientdiagonal. (exists pfa_gap_congruent_both_othercoefficientscoefficientdiagonalbound. pfa_gap_congruent_both_othercoefficientscoefficientdiagonalbound + S (pfc_index_congruent_both_othercoefficientscoefficientdiagonal) = (S (pfc_index_congruent_both_othercoefficients))) -> exists pfc_value_congruent_both_othercoefficientscoefficientdiagonal. ((((exists ff_h_pfp_congruent_both_othercoefficientscoefficientdiagonalentry. ff_h_pfp_congruent_both_othercoefficientscoefficientdiagonalentry + S (pfc_value_congruent_both_othercoefficientscoefficientdiagonal) = S ((S (pfc_index_congruent_both_othercoefficientscoefficientdiagonal)) * pfc_terms_scale_congruent_both_othercoefficientscoefficient)) /\\ exists ff_q_pfp_congruent_both_othercoefficientscoefficientdiagonalentry. pfc_terms_code_congruent_both_othercoefficientscoefficient = ff_q_pfp_congruent_both_othercoefficientscoefficientdiagonalentry * S ((S (pfc_index_congruent_both_othercoefficientscoefficientdiagonal)) * pfc_terms_scale_congruent_both_othercoefficientscoefficient) + (pfc_value_congruent_both_othercoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_congruent_both_othercoefficientscoefficientdiagonalterm pfc_left_congruent_both_othercoefficientscoefficientdiagonalterm pfc_right_congruent_both_othercoefficientscoefficientdiagonalterm. (((pfc_index_congruent_both_othercoefficientscoefficientdiagonal)+pfc_complement_congruent_both_othercoefficientscoefficientdiagonalterm=(pfc_index_congruent_both_othercoefficients)) /\\ ((((((exists pfa_gap_congruent_both_othercoefficientscoefficientdiagonaltermleftinside. pfa_gap_congruent_both_othercoefficientscoefficientdiagonaltermleftinside + S (pfc_index_congruent_both_othercoefficientscoefficientdiagonal) = (H)) /\\ ((((exists ff_h_pfp_congruent_both_othercoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_congruent_both_othercoefficientscoefficientdiagonaltermleftentry + S (pfc_left_congruent_both_othercoefficientscoefficientdiagonalterm) = S ((S (pfc_index_congruent_both_othercoefficientscoefficientdiagonal)) * AC)) /\\ exists ff_q_pfp_congruent_both_othercoefficientscoefficientdiagonaltermleftentry. AB = ff_q_pfp_congruent_both_othercoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_congruent_both_othercoefficientscoefficientdiagonal)) * AC) + (pfc_left_congruent_both_othercoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_congruent_both_othercoefficientscoefficientdiagonaltermleftoutside. pfc_gap_congruent_both_othercoefficientscoefficientdiagonaltermleftoutside+(H)=(pfc_index_congruent_both_othercoefficientscoefficientdiagonal)) /\\ (((pfc_left_congruent_both_othercoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_congruent_both_othercoefficientscoefficientdiagonaltermrightinside. pfa_gap_congruent_both_othercoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_congruent_both_othercoefficientscoefficientdiagonalterm) = (I)) /\\ ((((exists ff_h_pfp_congruent_both_othercoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_congruent_both_othercoefficientscoefficientdiagonaltermrightentry + S (pfc_right_congruent_both_othercoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_congruent_both_othercoefficientscoefficientdiagonalterm)) * BC)) /\\ exists ff_q_pfp_congruent_both_othercoefficientscoefficientdiagonaltermrightentry. BB = ff_q_pfp_congruent_both_othercoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_congruent_both_othercoefficientscoefficientdiagonalterm)) * BC) + (pfc_right_congruent_both_othercoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_congruent_both_othercoefficientscoefficientdiagonaltermrightoutside. pfc_gap_congruent_both_othercoefficientscoefficientdiagonaltermrightoutside+(I)=(pfc_complement_congruent_both_othercoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_congruent_both_othercoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_congruent_both_othercoefficientscoefficientdiagonal)=pfc_left_congruent_both_othercoefficientscoefficientdiagonalterm*pfc_right_congruent_both_othercoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_congruent_both_othercoefficientscoefficientsum fs_v_pfc_congruent_both_othercoefficientscoefficientsum. ((((exists fs_h_pfc_congruent_both_othercoefficientscoefficientsum_body_start. fs_h_pfc_congruent_both_othercoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_congruent_both_othercoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_both_othercoefficientscoefficientsum_body_start. fs_u_pfc_congruent_both_othercoefficientscoefficientsum = fs_q_pfc_congruent_both_othercoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_congruent_both_othercoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_congruent_both_othercoefficientscoefficientsum_body_terminal. fs_h_pfc_congruent_both_othercoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_congruent_both_othercoefficientscoefficient) = S ((S (S (pfc_index_congruent_both_othercoefficients))) * fs_v_pfc_congruent_both_othercoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_both_othercoefficientscoefficientsum_body_terminal. fs_u_pfc_congruent_both_othercoefficientscoefficientsum = fs_q_pfc_congruent_both_othercoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_congruent_both_othercoefficients))) * fs_v_pfc_congruent_both_othercoefficientscoefficientsum) + (pfc_natural_sum_congruent_both_othercoefficientscoefficient))) /\\ forall fs_i_pfc_congruent_both_othercoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_congruent_both_othercoefficientscoefficientsum_body_steps = S (pfc_index_congruent_both_othercoefficients)) -> exists fs_a_pfc_congruent_both_othercoefficientscoefficientsum_body_steps fs_r_pfc_congruent_both_othercoefficientscoefficientsum_body_steps fs_s_pfc_congruent_both_othercoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_summand. fs_h_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_congruent_both_othercoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_congruent_both_othercoefficientscoefficientsum_body_steps)) * pfc_terms_scale_congruent_both_othercoefficientscoefficient)) /\\ exists fs_q_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_summand. pfc_terms_code_congruent_both_othercoefficientscoefficient = fs_q_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_congruent_both_othercoefficientscoefficientsum_body_steps)) * pfc_terms_scale_congruent_both_othercoefficientscoefficient) + (fs_a_pfc_congruent_both_othercoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_partial. fs_h_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_congruent_both_othercoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_congruent_both_othercoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_both_othercoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_partial. fs_u_pfc_congruent_both_othercoefficientscoefficientsum = fs_q_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_congruent_both_othercoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_both_othercoefficientscoefficientsum) + (fs_r_pfc_congruent_both_othercoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_successor. fs_h_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_congruent_both_othercoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_congruent_both_othercoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_both_othercoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_successor. fs_u_pfc_congruent_both_othercoefficientscoefficientsum = fs_q_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_congruent_both_othercoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_both_othercoefficientscoefficientsum) + (fs_s_pfc_congruent_both_othercoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_congruent_both_othercoefficientscoefficientsum_body_steps = fs_r_pfc_congruent_both_othercoefficientscoefficientsum_body_steps + fs_a_pfc_congruent_both_othercoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_congruent_both_othercoefficientscoefficientresiduebound. pfa_gap_congruent_both_othercoefficientscoefficientresiduebound + S (pfc_value_congruent_both_othercoefficients) = (p)) /\\ ((exists pfa_offset_left_congruent_both_othercoefficientscoefficientresiduecongruence pfa_offset_right_congruent_both_othercoefficientscoefficientresiduecongruence. (pfc_natural_sum_congruent_both_othercoefficientscoefficient) + (p) * pfa_offset_left_congruent_both_othercoefficientscoefficientresiduecongruence = (pfc_value_congruent_both_othercoefficients) + (p) * pfa_offset_right_congruent_both_othercoefficientscoefficientresiduecongruence))))))))))))))))))", "exact hd", "cases htarget", "cases htarget_right", "cases htarget_right_right", "have hlength : exists J. ((((H)=0 \\/ (M)=0) /\\ (((J)=0)))) \\/ (((~((H)=0)) /\\ (((~((M)=0)) /\\ (((H)+(M)=S (J)))))))", "specialize polynomial_product_length_exists (H)", "specialize polynomial_product_length_exists (M)", "apply polynomial_product_length_exists", "cases hlength", "have hmiddle : exists db dc. ((forall fom_index_pfp_congruent_middle_productleft. (exists fom_gap_pfp_congruent_middle_productleft_index_bound. fom_gap_pfp_congruent_middle_productleft_index_bound + S (fom_index_pfp_congruent_middle_productleft) = H) -> exists fom_value_pfp_congruent_middle_productleft. ((((exists fom_beta_height_pfp_congruent_middle_productleft_entry. fom_beta_height_pfp_congruent_middle_productleft_entry + S (fom_value_pfp_congruent_middle_productleft) = S ((S (fom_index_pfp_congruent_middle_productleft)) * AC)) /\\ exists fom_beta_quotient_pfp_congruent_middle_productleft_entry. AB = fom_beta_quotient_pfp_congruent_middle_productleft_entry * S ((S (fom_index_pfp_congruent_middle_productleft)) * AC) + (fom_value_pfp_congruent_middle_productleft))) /\\ (exists fom_gap_pfp_congruent_middle_productleft_value_bound. fom_gap_pfp_congruent_middle_productleft_value_bound + S (fom_value_pfp_congruent_middle_productleft) = p))) /\\ (((forall fom_index_pfp_congruent_middle_productright. (exists fom_gap_pfp_congruent_middle_productright_index_bound. fom_gap_pfp_congruent_middle_productright_index_bound + S (fom_index_pfp_congruent_middle_productright) = M) -> exists fom_value_pfp_congruent_middle_productright. ((((exists fom_beta_height_pfp_congruent_middle_productright_entry. fom_beta_height_pfp_congruent_middle_productright_entry + S (fom_value_pfp_congruent_middle_productright) = S ((S (fom_index_pfp_congruent_middle_productright)) * bc)) /\\ exists fom_beta_quotient_pfp_congruent_middle_productright_entry. bb = fom_beta_quotient_pfp_congruent_middle_productright_entry * S ((S (fom_index_pfp_congruent_middle_productright)) * bc) + (fom_value_pfp_congruent_middle_productright))) /\\ (exists fom_gap_pfp_congruent_middle_productright_value_bound. fom_gap_pfp_congruent_middle_productright_value_bound + S (fom_value_pfp_congruent_middle_productright) = p))) /\\ (((((((H)=0 \\/ (M)=0) /\\ (((x)=0)))) \\/ (((~((H)=0)) /\\ (((~((M)=0)) /\\ (((H)+(M)=S (x)))))))) /\\ ((forall pfc_index_congruent_middle_productcoefficients. (exists pfa_gap_congruent_middle_productcoefficientsbound. pfa_gap_congruent_middle_productcoefficientsbound + S (pfc_index_congruent_middle_productcoefficients) = (x)) -> exists pfc_value_congruent_middle_productcoefficients. ((((exists ff_h_pfp_congruent_middle_productcoefficientsentry. ff_h_pfp_congruent_middle_productcoefficientsentry + S (pfc_value_congruent_middle_productcoefficients) = S ((S (pfc_index_congruent_middle_productcoefficients)) * dc)) /\\ exists ff_q_pfp_congruent_middle_productcoefficientsentry. db = ff_q_pfp_congruent_middle_productcoefficientsentry * S ((S (pfc_index_congruent_middle_productcoefficients)) * dc) + (pfc_value_congruent_middle_productcoefficients))) /\\ ((exists pfc_terms_code_congruent_middle_productcoefficientscoefficient pfc_terms_scale_congruent_middle_productcoefficientscoefficient pfc_natural_sum_congruent_middle_productcoefficientscoefficient. ((forall pfc_index_congruent_middle_productcoefficientscoefficientdiagonal. (exists pfa_gap_congruent_middle_productcoefficientscoefficientdiagonalbound. pfa_gap_congruent_middle_productcoefficientscoefficientdiagonalbound + S (pfc_index_congruent_middle_productcoefficientscoefficientdiagonal) = (S (pfc_index_congruent_middle_productcoefficients))) -> exists pfc_value_congruent_middle_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_congruent_middle_productcoefficientscoefficientdiagonalentry. ff_h_pfp_congruent_middle_productcoefficientscoefficientdiagonalentry + S (pfc_value_congruent_middle_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_congruent_middle_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_congruent_middle_productcoefficientscoefficient)) /\\ exists ff_q_pfp_congruent_middle_productcoefficientscoefficientdiagonalentry. pfc_terms_code_congruent_middle_productcoefficientscoefficient = ff_q_pfp_congruent_middle_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_congruent_middle_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_congruent_middle_productcoefficientscoefficient) + (pfc_value_congruent_middle_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_congruent_middle_productcoefficientscoefficientdiagonalterm pfc_left_congruent_middle_productcoefficientscoefficientdiagonalterm pfc_right_congruent_middle_productcoefficientscoefficientdiagonalterm. (((pfc_index_congruent_middle_productcoefficientscoefficientdiagonal)+pfc_complement_congruent_middle_productcoefficientscoefficientdiagonalterm=(pfc_index_congruent_middle_productcoefficients)) /\\ ((((((exists pfa_gap_congruent_middle_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_congruent_middle_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_congruent_middle_productcoefficientscoefficientdiagonal) = (H)) /\\ ((((exists ff_h_pfp_congruent_middle_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_congruent_middle_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_congruent_middle_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_congruent_middle_productcoefficientscoefficientdiagonal)) * AC)) /\\ exists ff_q_pfp_congruent_middle_productcoefficientscoefficientdiagonaltermleftentry. AB = ff_q_pfp_congruent_middle_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_congruent_middle_productcoefficientscoefficientdiagonal)) * AC) + (pfc_left_congruent_middle_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_congruent_middle_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_congruent_middle_productcoefficientscoefficientdiagonaltermleftoutside+(H)=(pfc_index_congruent_middle_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_congruent_middle_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_congruent_middle_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_congruent_middle_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_congruent_middle_productcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_congruent_middle_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_congruent_middle_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_congruent_middle_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_congruent_middle_productcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_congruent_middle_productcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_congruent_middle_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_congruent_middle_productcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_congruent_middle_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_congruent_middle_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_congruent_middle_productcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_congruent_middle_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_congruent_middle_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_congruent_middle_productcoefficientscoefficientdiagonal)=pfc_left_congruent_middle_productcoefficientscoefficientdiagonalterm*pfc_right_congruent_middle_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_congruent_middle_productcoefficientscoefficientsum fs_v_pfc_congruent_middle_productcoefficientscoefficientsum. ((((exists fs_h_pfc_congruent_middle_productcoefficientscoefficientsum_body_start. fs_h_pfc_congruent_middle_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_congruent_middle_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_middle_productcoefficientscoefficientsum_body_start. fs_u_pfc_congruent_middle_productcoefficientscoefficientsum = fs_q_pfc_congruent_middle_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_congruent_middle_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_congruent_middle_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_congruent_middle_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_congruent_middle_productcoefficientscoefficient) = S ((S (S (pfc_index_congruent_middle_productcoefficients))) * fs_v_pfc_congruent_middle_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_middle_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_congruent_middle_productcoefficientscoefficientsum = fs_q_pfc_congruent_middle_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_congruent_middle_productcoefficients))) * fs_v_pfc_congruent_middle_productcoefficientscoefficientsum) + (pfc_natural_sum_congruent_middle_productcoefficientscoefficient))) /\\ forall fs_i_pfc_congruent_middle_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_congruent_middle_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_congruent_middle_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_congruent_middle_productcoefficientscoefficientsum_body_steps = S (pfc_index_congruent_middle_productcoefficients)) -> exists fs_a_pfc_congruent_middle_productcoefficientscoefficientsum_body_steps fs_r_pfc_congruent_middle_productcoefficientscoefficientsum_body_steps fs_s_pfc_congruent_middle_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_congruent_middle_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_congruent_middle_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_congruent_middle_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_congruent_middle_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_congruent_middle_productcoefficientscoefficient)) /\\ exists fs_q_pfc_congruent_middle_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_congruent_middle_productcoefficientscoefficient = fs_q_pfc_congruent_middle_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_congruent_middle_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_congruent_middle_productcoefficientscoefficient) + (fs_a_pfc_congruent_middle_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_congruent_middle_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_congruent_middle_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_congruent_middle_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_congruent_middle_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_middle_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_middle_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_congruent_middle_productcoefficientscoefficientsum = fs_q_pfc_congruent_middle_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_congruent_middle_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_middle_productcoefficientscoefficientsum) + (fs_r_pfc_congruent_middle_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_congruent_middle_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_congruent_middle_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_congruent_middle_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_congruent_middle_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_middle_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_middle_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_congruent_middle_productcoefficientscoefficientsum = fs_q_pfc_congruent_middle_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_congruent_middle_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_middle_productcoefficientscoefficientsum) + (fs_s_pfc_congruent_middle_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_congruent_middle_productcoefficientscoefficientsum_body_steps = fs_r_pfc_congruent_middle_productcoefficientscoefficientsum_body_steps + fs_a_pfc_congruent_middle_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_congruent_middle_productcoefficientscoefficientresiduebound. pfa_gap_congruent_middle_productcoefficientscoefficientresiduebound + S (pfc_value_congruent_middle_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_congruent_middle_productcoefficientscoefficientresiduecongruence pfa_offset_right_congruent_middle_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_congruent_middle_productcoefficientscoefficient) + (p) * pfa_offset_left_congruent_middle_productcoefficientscoefficientresiduecongruence = (pfc_value_congruent_middle_productcoefficients) + (p) * pfa_offset_right_congruent_middle_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 (H)", "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 (x)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hp", "exact htarget_left", "exact hsource_right_left", "exact hlength_witness", "cases hmiddle", "cases hmiddle_witness", "specialize prime_field_polynomial_equivalent_transitive (cb)", "specialize prime_field_polynomial_equivalent_transitive (cc)", "specialize prime_field_polynomial_equivalent_transitive (N)", "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 (CB)", "specialize prime_field_polynomial_equivalent_transitive (CC)", "specialize prime_field_polynomial_equivalent_transitive (K)", "apply prime_field_polynomial_equivalent_transitive", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (p)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (ab)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (ac)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (L)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (bb)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (bc)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (M)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (cb)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (cc)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (N)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (AB)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (AC)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (H)", "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 (x)", "apply prime_field_polynomial_convolution_equivalent_congruent_left", "exact hp", "exact hA", "exact hc", "exact hmiddle_witness_witness", "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 (H)", "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 (M)", "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 (x)", "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 (I)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (CB)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (CC)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (K)", "apply prime_field_polynomial_convolution_equivalent_congruent_right", "exact hp", "exact hB", "exact hmiddle_witness_witness", "exact hd" ], "script_sha256": "fe558b9fc1877e7265863a62e009be470ba7b99c208f6d20f114c85a9f1545cb", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_convolution_congruence_candidate.py", "sha256": "effc4b2df9418d9d964fd34216c4c1c2a09d12dd885877165c6fed2e761a8b70" }, "statement": "forall p ab ac L bb bc M cb cc N AB AC H BB BC I CB CC K. (~(p=0)) -> (forall pfrep_power_congruent_both_left pfrep_left_congruent_both_left pfrep_right_congruent_both_left. ((exists pfrep_position_congruent_both_leftfirst. ((pfrep_position_congruent_both_leftfirst+S (pfrep_power_congruent_both_left)=(L)) /\\ ((((exists ff_h_pfp_congruent_both_leftfirstentry. ff_h_pfp_congruent_both_leftfirstentry + S (pfrep_left_congruent_both_left) = S ((S (pfrep_position_congruent_both_leftfirst)) * ac)) /\\ exists ff_q_pfp_congruent_both_leftfirstentry. ab = ff_q_pfp_congruent_both_leftfirstentry * S ((S (pfrep_position_congruent_both_leftfirst)) * ac) + (pfrep_left_congruent_both_left)))))) \\/ (((exists pfrep_gap_congruent_both_leftfirstoutside. pfrep_gap_congruent_both_leftfirstoutside+(L)=(pfrep_power_congruent_both_left)) /\\ (((pfrep_left_congruent_both_left)=0))))) -> ((exists pfrep_position_congruent_both_leftsecond. ((pfrep_position_congruent_both_leftsecond+S (pfrep_power_congruent_both_left)=(H)) /\\ ((((exists ff_h_pfp_congruent_both_leftsecondentry. ff_h_pfp_congruent_both_leftsecondentry + S (pfrep_right_congruent_both_left) = S ((S (pfrep_position_congruent_both_leftsecond)) * AC)) /\\ exists ff_q_pfp_congruent_both_leftsecondentry. AB = ff_q_pfp_congruent_both_leftsecondentry * S ((S (pfrep_position_congruent_both_leftsecond)) * AC) + (pfrep_right_congruent_both_left)))))) \\/ (((exists pfrep_gap_congruent_both_leftsecondoutside. pfrep_gap_congruent_both_leftsecondoutside+(H)=(pfrep_power_congruent_both_left)) /\\ (((pfrep_right_congruent_both_left)=0))))) -> pfrep_left_congruent_both_left=pfrep_right_congruent_both_left) -> (forall pfrep_power_congruent_both_right pfrep_left_congruent_both_right pfrep_right_congruent_both_right. ((exists pfrep_position_congruent_both_rightfirst. ((pfrep_position_congruent_both_rightfirst+S (pfrep_power_congruent_both_right)=(M)) /\\ ((((exists ff_h_pfp_congruent_both_rightfirstentry. ff_h_pfp_congruent_both_rightfirstentry + S (pfrep_left_congruent_both_right) = S ((S (pfrep_position_congruent_both_rightfirst)) * bc)) /\\ exists ff_q_pfp_congruent_both_rightfirstentry. bb = ff_q_pfp_congruent_both_rightfirstentry * S ((S (pfrep_position_congruent_both_rightfirst)) * bc) + (pfrep_left_congruent_both_right)))))) \\/ (((exists pfrep_gap_congruent_both_rightfirstoutside. pfrep_gap_congruent_both_rightfirstoutside+(M)=(pfrep_power_congruent_both_right)) /\\ (((pfrep_left_congruent_both_right)=0))))) -> ((exists pfrep_position_congruent_both_rightsecond. ((pfrep_position_congruent_both_rightsecond+S (pfrep_power_congruent_both_right)=(I)) /\\ ((((exists ff_h_pfp_congruent_both_rightsecondentry. ff_h_pfp_congruent_both_rightsecondentry + S (pfrep_right_congruent_both_right) = S ((S (pfrep_position_congruent_both_rightsecond)) * BC)) /\\ exists ff_q_pfp_congruent_both_rightsecondentry. BB = ff_q_pfp_congruent_both_rightsecondentry * S ((S (pfrep_position_congruent_both_rightsecond)) * BC) + (pfrep_right_congruent_both_right)))))) \\/ (((exists pfrep_gap_congruent_both_rightsecondoutside. pfrep_gap_congruent_both_rightsecondoutside+(I)=(pfrep_power_congruent_both_right)) /\\ (((pfrep_right_congruent_both_right)=0))))) -> pfrep_left_congruent_both_right=pfrep_right_congruent_both_right) -> (((forall fom_index_pfp_congruent_both_originalleft. (exists fom_gap_pfp_congruent_both_originalleft_index_bound. fom_gap_pfp_congruent_both_originalleft_index_bound + S (fom_index_pfp_congruent_both_originalleft) = L) -> exists fom_value_pfp_congruent_both_originalleft. ((((exists fom_beta_height_pfp_congruent_both_originalleft_entry. fom_beta_height_pfp_congruent_both_originalleft_entry + S (fom_value_pfp_congruent_both_originalleft) = S ((S (fom_index_pfp_congruent_both_originalleft)) * ac)) /\\ exists fom_beta_quotient_pfp_congruent_both_originalleft_entry. ab = fom_beta_quotient_pfp_congruent_both_originalleft_entry * S ((S (fom_index_pfp_congruent_both_originalleft)) * ac) + (fom_value_pfp_congruent_both_originalleft))) /\\ (exists fom_gap_pfp_congruent_both_originalleft_value_bound. fom_gap_pfp_congruent_both_originalleft_value_bound + S (fom_value_pfp_congruent_both_originalleft) = p))) /\\ (((forall fom_index_pfp_congruent_both_originalright. (exists fom_gap_pfp_congruent_both_originalright_index_bound. fom_gap_pfp_congruent_both_originalright_index_bound + S (fom_index_pfp_congruent_both_originalright) = M) -> exists fom_value_pfp_congruent_both_originalright. ((((exists fom_beta_height_pfp_congruent_both_originalright_entry. fom_beta_height_pfp_congruent_both_originalright_entry + S (fom_value_pfp_congruent_both_originalright) = S ((S (fom_index_pfp_congruent_both_originalright)) * bc)) /\\ exists fom_beta_quotient_pfp_congruent_both_originalright_entry. bb = fom_beta_quotient_pfp_congruent_both_originalright_entry * S ((S (fom_index_pfp_congruent_both_originalright)) * bc) + (fom_value_pfp_congruent_both_originalright))) /\\ (exists fom_gap_pfp_congruent_both_originalright_value_bound. fom_gap_pfp_congruent_both_originalright_value_bound + S (fom_value_pfp_congruent_both_originalright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_congruent_both_originalcoefficients. (exists pfa_gap_congruent_both_originalcoefficientsbound. pfa_gap_congruent_both_originalcoefficientsbound + S (pfc_index_congruent_both_originalcoefficients) = (N)) -> exists pfc_value_congruent_both_originalcoefficients. ((((exists ff_h_pfp_congruent_both_originalcoefficientsentry. ff_h_pfp_congruent_both_originalcoefficientsentry + S (pfc_value_congruent_both_originalcoefficients) = S ((S (pfc_index_congruent_both_originalcoefficients)) * cc)) /\\ exists ff_q_pfp_congruent_both_originalcoefficientsentry. cb = ff_q_pfp_congruent_both_originalcoefficientsentry * S ((S (pfc_index_congruent_both_originalcoefficients)) * cc) + (pfc_value_congruent_both_originalcoefficients))) /\\ ((exists pfc_terms_code_congruent_both_originalcoefficientscoefficient pfc_terms_scale_congruent_both_originalcoefficientscoefficient pfc_natural_sum_congruent_both_originalcoefficientscoefficient. ((forall pfc_index_congruent_both_originalcoefficientscoefficientdiagonal. (exists pfa_gap_congruent_both_originalcoefficientscoefficientdiagonalbound. pfa_gap_congruent_both_originalcoefficientscoefficientdiagonalbound + S (pfc_index_congruent_both_originalcoefficientscoefficientdiagonal) = (S (pfc_index_congruent_both_originalcoefficients))) -> exists pfc_value_congruent_both_originalcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_congruent_both_originalcoefficientscoefficientdiagonalentry. ff_h_pfp_congruent_both_originalcoefficientscoefficientdiagonalentry + S (pfc_value_congruent_both_originalcoefficientscoefficientdiagonal) = S ((S (pfc_index_congruent_both_originalcoefficientscoefficientdiagonal)) * pfc_terms_scale_congruent_both_originalcoefficientscoefficient)) /\\ exists ff_q_pfp_congruent_both_originalcoefficientscoefficientdiagonalentry. pfc_terms_code_congruent_both_originalcoefficientscoefficient = ff_q_pfp_congruent_both_originalcoefficientscoefficientdiagonalentry * S ((S (pfc_index_congruent_both_originalcoefficientscoefficientdiagonal)) * pfc_terms_scale_congruent_both_originalcoefficientscoefficient) + (pfc_value_congruent_both_originalcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_congruent_both_originalcoefficientscoefficientdiagonalterm pfc_left_congruent_both_originalcoefficientscoefficientdiagonalterm pfc_right_congruent_both_originalcoefficientscoefficientdiagonalterm. (((pfc_index_congruent_both_originalcoefficientscoefficientdiagonal)+pfc_complement_congruent_both_originalcoefficientscoefficientdiagonalterm=(pfc_index_congruent_both_originalcoefficients)) /\\ ((((((exists pfa_gap_congruent_both_originalcoefficientscoefficientdiagonaltermleftinside. pfa_gap_congruent_both_originalcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_congruent_both_originalcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_congruent_both_originalcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_congruent_both_originalcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_congruent_both_originalcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_congruent_both_originalcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_congruent_both_originalcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_congruent_both_originalcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_congruent_both_originalcoefficientscoefficientdiagonal)) * ac) + (pfc_left_congruent_both_originalcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_congruent_both_originalcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_congruent_both_originalcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_congruent_both_originalcoefficientscoefficientdiagonal)) /\\ (((pfc_left_congruent_both_originalcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_congruent_both_originalcoefficientscoefficientdiagonaltermrightinside. pfa_gap_congruent_both_originalcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_congruent_both_originalcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_congruent_both_originalcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_congruent_both_originalcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_congruent_both_originalcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_congruent_both_originalcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_congruent_both_originalcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_congruent_both_originalcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_congruent_both_originalcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_congruent_both_originalcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_congruent_both_originalcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_congruent_both_originalcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_congruent_both_originalcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_congruent_both_originalcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_congruent_both_originalcoefficientscoefficientdiagonal)=pfc_left_congruent_both_originalcoefficientscoefficientdiagonalterm*pfc_right_congruent_both_originalcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_congruent_both_originalcoefficientscoefficientsum fs_v_pfc_congruent_both_originalcoefficientscoefficientsum. ((((exists fs_h_pfc_congruent_both_originalcoefficientscoefficientsum_body_start. fs_h_pfc_congruent_both_originalcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_congruent_both_originalcoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_both_originalcoefficientscoefficientsum_body_start. fs_u_pfc_congruent_both_originalcoefficientscoefficientsum = fs_q_pfc_congruent_both_originalcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_congruent_both_originalcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_congruent_both_originalcoefficientscoefficientsum_body_terminal. fs_h_pfc_congruent_both_originalcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_congruent_both_originalcoefficientscoefficient) = S ((S (S (pfc_index_congruent_both_originalcoefficients))) * fs_v_pfc_congruent_both_originalcoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_both_originalcoefficientscoefficientsum_body_terminal. fs_u_pfc_congruent_both_originalcoefficientscoefficientsum = fs_q_pfc_congruent_both_originalcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_congruent_both_originalcoefficients))) * fs_v_pfc_congruent_both_originalcoefficientscoefficientsum) + (pfc_natural_sum_congruent_both_originalcoefficientscoefficient))) /\\ forall fs_i_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps = S (pfc_index_congruent_both_originalcoefficients)) -> exists fs_a_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps fs_r_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps fs_s_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_congruent_both_originalcoefficientscoefficient)) /\\ exists fs_q_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_congruent_both_originalcoefficientscoefficient = fs_q_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_congruent_both_originalcoefficientscoefficient) + (fs_a_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_both_originalcoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_congruent_both_originalcoefficientscoefficientsum = fs_q_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_both_originalcoefficientscoefficientsum) + (fs_r_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_both_originalcoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_congruent_both_originalcoefficientscoefficientsum = fs_q_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_both_originalcoefficientscoefficientsum) + (fs_s_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps = fs_r_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps + fs_a_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_congruent_both_originalcoefficientscoefficientresiduebound. pfa_gap_congruent_both_originalcoefficientscoefficientresiduebound + S (pfc_value_congruent_both_originalcoefficients) = (p)) /\\ ((exists pfa_offset_left_congruent_both_originalcoefficientscoefficientresiduecongruence pfa_offset_right_congruent_both_originalcoefficientscoefficientresiduecongruence. (pfc_natural_sum_congruent_both_originalcoefficientscoefficient) + (p) * pfa_offset_left_congruent_both_originalcoefficientscoefficientresiduecongruence = (pfc_value_congruent_both_originalcoefficients) + (p) * pfa_offset_right_congruent_both_originalcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_congruent_both_otherleft. (exists fom_gap_pfp_congruent_both_otherleft_index_bound. fom_gap_pfp_congruent_both_otherleft_index_bound + S (fom_index_pfp_congruent_both_otherleft) = H) -> exists fom_value_pfp_congruent_both_otherleft. ((((exists fom_beta_height_pfp_congruent_both_otherleft_entry. fom_beta_height_pfp_congruent_both_otherleft_entry + S (fom_value_pfp_congruent_both_otherleft) = S ((S (fom_index_pfp_congruent_both_otherleft)) * AC)) /\\ exists fom_beta_quotient_pfp_congruent_both_otherleft_entry. AB = fom_beta_quotient_pfp_congruent_both_otherleft_entry * S ((S (fom_index_pfp_congruent_both_otherleft)) * AC) + (fom_value_pfp_congruent_both_otherleft))) /\\ (exists fom_gap_pfp_congruent_both_otherleft_value_bound. fom_gap_pfp_congruent_both_otherleft_value_bound + S (fom_value_pfp_congruent_both_otherleft) = p))) /\\ (((forall fom_index_pfp_congruent_both_otherright. (exists fom_gap_pfp_congruent_both_otherright_index_bound. fom_gap_pfp_congruent_both_otherright_index_bound + S (fom_index_pfp_congruent_both_otherright) = I) -> exists fom_value_pfp_congruent_both_otherright. ((((exists fom_beta_height_pfp_congruent_both_otherright_entry. fom_beta_height_pfp_congruent_both_otherright_entry + S (fom_value_pfp_congruent_both_otherright) = S ((S (fom_index_pfp_congruent_both_otherright)) * BC)) /\\ exists fom_beta_quotient_pfp_congruent_both_otherright_entry. BB = fom_beta_quotient_pfp_congruent_both_otherright_entry * S ((S (fom_index_pfp_congruent_both_otherright)) * BC) + (fom_value_pfp_congruent_both_otherright))) /\\ (exists fom_gap_pfp_congruent_both_otherright_value_bound. fom_gap_pfp_congruent_both_otherright_value_bound + S (fom_value_pfp_congruent_both_otherright) = p))) /\\ (((((((H)=0 \\/ (I)=0) /\\ (((K)=0)))) \\/ (((~((H)=0)) /\\ (((~((I)=0)) /\\ (((H)+(I)=S (K)))))))) /\\ ((forall pfc_index_congruent_both_othercoefficients. (exists pfa_gap_congruent_both_othercoefficientsbound. pfa_gap_congruent_both_othercoefficientsbound + S (pfc_index_congruent_both_othercoefficients) = (K)) -> exists pfc_value_congruent_both_othercoefficients. ((((exists ff_h_pfp_congruent_both_othercoefficientsentry. ff_h_pfp_congruent_both_othercoefficientsentry + S (pfc_value_congruent_both_othercoefficients) = S ((S (pfc_index_congruent_both_othercoefficients)) * CC)) /\\ exists ff_q_pfp_congruent_both_othercoefficientsentry. CB = ff_q_pfp_congruent_both_othercoefficientsentry * S ((S (pfc_index_congruent_both_othercoefficients)) * CC) + (pfc_value_congruent_both_othercoefficients))) /\\ ((exists pfc_terms_code_congruent_both_othercoefficientscoefficient pfc_terms_scale_congruent_both_othercoefficientscoefficient pfc_natural_sum_congruent_both_othercoefficientscoefficient. ((forall pfc_index_congruent_both_othercoefficientscoefficientdiagonal. (exists pfa_gap_congruent_both_othercoefficientscoefficientdiagonalbound. pfa_gap_congruent_both_othercoefficientscoefficientdiagonalbound + S (pfc_index_congruent_both_othercoefficientscoefficientdiagonal) = (S (pfc_index_congruent_both_othercoefficients))) -> exists pfc_value_congruent_both_othercoefficientscoefficientdiagonal. ((((exists ff_h_pfp_congruent_both_othercoefficientscoefficientdiagonalentry. ff_h_pfp_congruent_both_othercoefficientscoefficientdiagonalentry + S (pfc_value_congruent_both_othercoefficientscoefficientdiagonal) = S ((S (pfc_index_congruent_both_othercoefficientscoefficientdiagonal)) * pfc_terms_scale_congruent_both_othercoefficientscoefficient)) /\\ exists ff_q_pfp_congruent_both_othercoefficientscoefficientdiagonalentry. pfc_terms_code_congruent_both_othercoefficientscoefficient = ff_q_pfp_congruent_both_othercoefficientscoefficientdiagonalentry * S ((S (pfc_index_congruent_both_othercoefficientscoefficientdiagonal)) * pfc_terms_scale_congruent_both_othercoefficientscoefficient) + (pfc_value_congruent_both_othercoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_congruent_both_othercoefficientscoefficientdiagonalterm pfc_left_congruent_both_othercoefficientscoefficientdiagonalterm pfc_right_congruent_both_othercoefficientscoefficientdiagonalterm. (((pfc_index_congruent_both_othercoefficientscoefficientdiagonal)+pfc_complement_congruent_both_othercoefficientscoefficientdiagonalterm=(pfc_index_congruent_both_othercoefficients)) /\\ ((((((exists pfa_gap_congruent_both_othercoefficientscoefficientdiagonaltermleftinside. pfa_gap_congruent_both_othercoefficientscoefficientdiagonaltermleftinside + S (pfc_index_congruent_both_othercoefficientscoefficientdiagonal) = (H)) /\\ ((((exists ff_h_pfp_congruent_both_othercoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_congruent_both_othercoefficientscoefficientdiagonaltermleftentry + S (pfc_left_congruent_both_othercoefficientscoefficientdiagonalterm) = S ((S (pfc_index_congruent_both_othercoefficientscoefficientdiagonal)) * AC)) /\\ exists ff_q_pfp_congruent_both_othercoefficientscoefficientdiagonaltermleftentry. AB = ff_q_pfp_congruent_both_othercoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_congruent_both_othercoefficientscoefficientdiagonal)) * AC) + (pfc_left_congruent_both_othercoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_congruent_both_othercoefficientscoefficientdiagonaltermleftoutside. pfc_gap_congruent_both_othercoefficientscoefficientdiagonaltermleftoutside+(H)=(pfc_index_congruent_both_othercoefficientscoefficientdiagonal)) /\\ (((pfc_left_congruent_both_othercoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_congruent_both_othercoefficientscoefficientdiagonaltermrightinside. pfa_gap_congruent_both_othercoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_congruent_both_othercoefficientscoefficientdiagonalterm) = (I)) /\\ ((((exists ff_h_pfp_congruent_both_othercoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_congruent_both_othercoefficientscoefficientdiagonaltermrightentry + S (pfc_right_congruent_both_othercoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_congruent_both_othercoefficientscoefficientdiagonalterm)) * BC)) /\\ exists ff_q_pfp_congruent_both_othercoefficientscoefficientdiagonaltermrightentry. BB = ff_q_pfp_congruent_both_othercoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_congruent_both_othercoefficientscoefficientdiagonalterm)) * BC) + (pfc_right_congruent_both_othercoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_congruent_both_othercoefficientscoefficientdiagonaltermrightoutside. pfc_gap_congruent_both_othercoefficientscoefficientdiagonaltermrightoutside+(I)=(pfc_complement_congruent_both_othercoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_congruent_both_othercoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_congruent_both_othercoefficientscoefficientdiagonal)=pfc_left_congruent_both_othercoefficientscoefficientdiagonalterm*pfc_right_congruent_both_othercoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_congruent_both_othercoefficientscoefficientsum fs_v_pfc_congruent_both_othercoefficientscoefficientsum. ((((exists fs_h_pfc_congruent_both_othercoefficientscoefficientsum_body_start. fs_h_pfc_congruent_both_othercoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_congruent_both_othercoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_both_othercoefficientscoefficientsum_body_start. fs_u_pfc_congruent_both_othercoefficientscoefficientsum = fs_q_pfc_congruent_both_othercoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_congruent_both_othercoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_congruent_both_othercoefficientscoefficientsum_body_terminal. fs_h_pfc_congruent_both_othercoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_congruent_both_othercoefficientscoefficient) = S ((S (S (pfc_index_congruent_both_othercoefficients))) * fs_v_pfc_congruent_both_othercoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_both_othercoefficientscoefficientsum_body_terminal. fs_u_pfc_congruent_both_othercoefficientscoefficientsum = fs_q_pfc_congruent_both_othercoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_congruent_both_othercoefficients))) * fs_v_pfc_congruent_both_othercoefficientscoefficientsum) + (pfc_natural_sum_congruent_both_othercoefficientscoefficient))) /\\ forall fs_i_pfc_congruent_both_othercoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_congruent_both_othercoefficientscoefficientsum_body_steps = S (pfc_index_congruent_both_othercoefficients)) -> exists fs_a_pfc_congruent_both_othercoefficientscoefficientsum_body_steps fs_r_pfc_congruent_both_othercoefficientscoefficientsum_body_steps fs_s_pfc_congruent_both_othercoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_summand. fs_h_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_congruent_both_othercoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_congruent_both_othercoefficientscoefficientsum_body_steps)) * pfc_terms_scale_congruent_both_othercoefficientscoefficient)) /\\ exists fs_q_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_summand. pfc_terms_code_congruent_both_othercoefficientscoefficient = fs_q_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_congruent_both_othercoefficientscoefficientsum_body_steps)) * pfc_terms_scale_congruent_both_othercoefficientscoefficient) + (fs_a_pfc_congruent_both_othercoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_partial. fs_h_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_congruent_both_othercoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_congruent_both_othercoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_both_othercoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_partial. fs_u_pfc_congruent_both_othercoefficientscoefficientsum = fs_q_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_congruent_both_othercoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_both_othercoefficientscoefficientsum) + (fs_r_pfc_congruent_both_othercoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_successor. fs_h_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_congruent_both_othercoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_congruent_both_othercoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_both_othercoefficientscoefficientsum)) /\\ exists fs_q_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_successor. fs_u_pfc_congruent_both_othercoefficientscoefficientsum = fs_q_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_congruent_both_othercoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_both_othercoefficientscoefficientsum) + (fs_s_pfc_congruent_both_othercoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_congruent_both_othercoefficientscoefficientsum_body_steps = fs_r_pfc_congruent_both_othercoefficientscoefficientsum_body_steps + fs_a_pfc_congruent_both_othercoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_congruent_both_othercoefficientscoefficientresiduebound. pfa_gap_congruent_both_othercoefficientscoefficientresiduebound + S (pfc_value_congruent_both_othercoefficients) = (p)) /\\ ((exists pfa_offset_left_congruent_both_othercoefficientscoefficientresiduecongruence pfa_offset_right_congruent_both_othercoefficientscoefficientresiduecongruence. (pfc_natural_sum_congruent_both_othercoefficientscoefficient) + (p) * pfa_offset_left_congruent_both_othercoefficientscoefficientresiduecongruence = (pfc_value_congruent_both_othercoefficients) + (p) * pfa_offset_right_congruent_both_othercoefficientscoefficientresiduecongruence))))))))))))))))))) -> (forall pfrep_power_congruent_both_result pfrep_left_congruent_both_result pfrep_right_congruent_both_result. ((exists pfrep_position_congruent_both_resultfirst. ((pfrep_position_congruent_both_resultfirst+S (pfrep_power_congruent_both_result)=(N)) /\\ ((((exists ff_h_pfp_congruent_both_resultfirstentry. ff_h_pfp_congruent_both_resultfirstentry + S (pfrep_left_congruent_both_result) = S ((S (pfrep_position_congruent_both_resultfirst)) * cc)) /\\ exists ff_q_pfp_congruent_both_resultfirstentry. cb = ff_q_pfp_congruent_both_resultfirstentry * S ((S (pfrep_position_congruent_both_resultfirst)) * cc) + (pfrep_left_congruent_both_result)))))) \\/ (((exists pfrep_gap_congruent_both_resultfirstoutside. pfrep_gap_congruent_both_resultfirstoutside+(N)=(pfrep_power_congruent_both_result)) /\\ (((pfrep_left_congruent_both_result)=0))))) -> ((exists pfrep_position_congruent_both_resultsecond. ((pfrep_position_congruent_both_resultsecond+S (pfrep_power_congruent_both_result)=(K)) /\\ ((((exists ff_h_pfp_congruent_both_resultsecondentry. ff_h_pfp_congruent_both_resultsecondentry + S (pfrep_right_congruent_both_result) = S ((S (pfrep_position_congruent_both_resultsecond)) * CC)) /\\ exists ff_q_pfp_congruent_both_resultsecondentry. CB = ff_q_pfp_congruent_both_resultsecondentry * S ((S (pfrep_position_congruent_both_resultsecond)) * CC) + (pfrep_right_congruent_both_result)))))) \\/ (((exists pfrep_gap_congruent_both_resultsecondoutside. pfrep_gap_congruent_both_resultsecondoutside+(K)=(pfrep_power_congruent_both_result)) /\\ (((pfrep_right_congruent_both_result)=0))))) -> pfrep_left_congruent_both_result=pfrep_right_congruent_both_result)", "statement_sha256": "d984fe3c378d4d4b02941d6f3a126324a2c7c26bf47f4d8ee7c37b2e55404446", "summary": "Two actual convolution outputs represent the same formal polynomial whenever their respective factors do, with all four representation lengths independent. A genuine mixed product is constructed from canonical inputs supplied by the actual products; no output identity or extra field hypothesis is assumed.", "summary_sha256": "d1a603ab119d72607b8fce5f462dbc92f0b300e07b97bb4595d34845bfe7b94f" } ]