The actual ambient product table agrees with the input throughout the computed quotient prefix, including the vacuous zero-length case.
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 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. (exists mdr_gap_pfp_division_match_table_resultb. mdr_gap_pfp_division_match_table_resultb + S (mdr_i_pfp_division_match_table_result) = (N)) -> (((exists ff_h_mdr_pfp_division_match_table_resulto. ff_h_mdr_pfp_division_match_table_resulto + S (mdr_a_pfp_division_match_table_result) = S ((S (mdr_i_pfp_division_match_table_result)) * ac)) /\ exists ff_q_mdr_pfp_division_match_table_resulto. ab = ff_q_mdr_pfp_division_match_table_resulto * S ((S (mdr_i_pfp_division_match_table_result)) * ac) + (mdr_a_pfp_division_match_table_result))) -> (((exists ff_h_mdr_pfp_division_match_table_resultn. ff_h_mdr_pfp_division_match_table_resultn + S (mdr_a_pfp_division_match_table_result) = S ((S (mdr_i_pfp_division_match_table_result)) * pc)) /\ exists ff_q_mdr_pfp_division_match_table_resultn. pb = ff_q_mdr_pfp_division_match_table_resultn * S ((S (mdr_i_pfp_division_match_table_result)) * pc) + (mdr_a_pfp_division_match_table_result))))
Complete tactic proof in conservative notation
All 68 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.