Every requested ambient output prefix of the three actual right products satisfies actual coefficientwise addition, including N=0 and prefixes extending past product support.
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 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. ((pfp_left_prefix_add_right_result) + (pfp_right_prefix_add_right_result)) + (p) * pfa_offset_left_prefix_add_right_resultoperationresultcongruence = (pfp_value_prefix_add_right_result) + (p) * pfa_offset_right_prefix_add_right_resultoperationresultcongruence))))))))))))))))
Complete tactic proof in conservative notation
All 71 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.