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.
Alpha v34 checked-use · first admitted v33 · independently kernel and Lean verified; not Stable
Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.
Coefficients are highest-degree-first. The divisor has a nonzero decoded head; primality supplies its actual inverse. Empty quotients and remainders are included. Functionality compares the constructed execution lengths and decoded coefficients, never arbitrary beta codes. Formal polynomial equivalence compares every coefficient, not evaluations on a finite field. The formal identity and remainder-degree bound are proved separately, not assumed by the execution graph. Arbitrary quotient/remainder-pair uniqueness from a formal identity, multiplication associativity, gcd/Bezout, irreducible-polynomial existence, and the full G091 prime-power-field goal remain open. The seven displayed new names are conservative first-order notation, not new kernel primitives.
forall p ab ac L bb bc d qb qc q rb rc R. (~((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))))))))))))))))))))
Complete tactic proof in conservative notation
All 96 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.
This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.
Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.