Formal coefficient equivalence of the left factor preserves two actual products at arbitrary representation lengths, including empty factors; actual leading padding is recovered in the appropriate direction.
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 M cb cc N AB AC H CB CC K. (~(p=0)) -> (forall pfrep_power_congruent_factor_left pfrep_left_congruent_factor_left pfrep_right_congruent_factor_left. ((exists pfrep_position_congruent_factor_leftfirst. ((pfrep_position_congruent_factor_leftfirst+S (pfrep_power_congruent_factor_left)=(L)) /\ ((((exists ff_h_pfp_congruent_factor_leftfirstentry. ff_h_pfp_congruent_factor_leftfirstentry + S (pfrep_left_congruent_factor_left) = S ((S (pfrep_position_congruent_factor_leftfirst)) * ac)) /\ exists ff_q_pfp_congruent_factor_leftfirstentry. ab = ff_q_pfp_congruent_factor_leftfirstentry * S ((S (pfrep_position_congruent_factor_leftfirst)) * ac) + (pfrep_left_congruent_factor_left)))))) \/ (((exists pfrep_gap_congruent_factor_leftfirstoutside. pfrep_gap_congruent_factor_leftfirstoutside+(L)=(pfrep_power_congruent_factor_left)) /\ (((pfrep_left_congruent_factor_left)=0))))) -> ((exists pfrep_position_congruent_factor_leftsecond. ((pfrep_position_congruent_factor_leftsecond+S (pfrep_power_congruent_factor_left)=(H)) /\ ((((exists ff_h_pfp_congruent_factor_leftsecondentry. ff_h_pfp_congruent_factor_leftsecondentry + S (pfrep_right_congruent_factor_left) = S ((S (pfrep_position_congruent_factor_leftsecond)) * AC)) /\ exists ff_q_pfp_congruent_factor_leftsecondentry. AB = ff_q_pfp_congruent_factor_leftsecondentry * S ((S (pfrep_position_congruent_factor_leftsecond)) * AC) + (pfrep_right_congruent_factor_left)))))) \/ (((exists pfrep_gap_congruent_factor_leftsecondoutside. pfrep_gap_congruent_factor_leftsecondoutside+(H)=(pfrep_power_congruent_factor_left)) /\ (((pfrep_right_congruent_factor_left)=0))))) -> pfrep_left_congruent_factor_left=pfrep_right_congruent_factor_left) -> (((forall fom_index_pfp_congruent_original_leftleft. (exists fom_gap_pfp_congruent_original_leftleft_index_bound. fom_gap_pfp_congruent_original_leftleft_index_bound + S (fom_index_pfp_congruent_original_leftleft) = L) -> exists fom_value_pfp_congruent_original_leftleft. ((((exists fom_beta_height_pfp_congruent_original_leftleft_entry. fom_beta_height_pfp_congruent_original_leftleft_entry + S (fom_value_pfp_congruent_original_leftleft) = S ((S (fom_index_pfp_congruent_original_leftleft)) * ac)) /\ exists fom_beta_quotient_pfp_congruent_original_leftleft_entry. ab = fom_beta_quotient_pfp_congruent_original_leftleft_entry * S ((S (fom_index_pfp_congruent_original_leftleft)) * ac) + (fom_value_pfp_congruent_original_leftleft))) /\ (exists fom_gap_pfp_congruent_original_leftleft_value_bound. fom_gap_pfp_congruent_original_leftleft_value_bound + S (fom_value_pfp_congruent_original_leftleft) = p))) /\ (((forall fom_index_pfp_congruent_original_leftright. (exists fom_gap_pfp_congruent_original_leftright_index_bound. fom_gap_pfp_congruent_original_leftright_index_bound + S (fom_index_pfp_congruent_original_leftright) = M) -> exists fom_value_pfp_congruent_original_leftright. ((((exists fom_beta_height_pfp_congruent_original_leftright_entry. fom_beta_height_pfp_congruent_original_leftright_entry + S (fom_value_pfp_congruent_original_leftright) = S ((S (fom_index_pfp_congruent_original_leftright)) * bc)) /\ exists fom_beta_quotient_pfp_congruent_original_leftright_entry. bb = fom_beta_quotient_pfp_congruent_original_leftright_entry * S ((S (fom_index_pfp_congruent_original_leftright)) * bc) + (fom_value_pfp_congruent_original_leftright))) /\ (exists fom_gap_pfp_congruent_original_leftright_value_bound. fom_gap_pfp_congruent_original_leftright_value_bound + S (fom_value_pfp_congruent_original_leftright) = p))) /\ (((((((L)=0 \/ (M)=0) /\ (((N)=0)))) \/ (((~((L)=0)) /\ (((~((M)=0)) /\ (((L)+(M)=S (N)))))))) /\ ((forall pfc_index_congruent_original_leftcoefficients. (exists pfa_gap_congruent_original_leftcoefficientsbound. pfa_gap_congruent_original_leftcoefficientsbound + S (pfc_index_congruent_original_leftcoefficients) = (N)) -> exists pfc_value_congruent_original_leftcoefficients. ((((exists ff_h_pfp_congruent_original_leftcoefficientsentry. ff_h_pfp_congruent_original_leftcoefficientsentry + S (pfc_value_congruent_original_leftcoefficients) = S ((S (pfc_index_congruent_original_leftcoefficients)) * cc)) /\ exists ff_q_pfp_congruent_original_leftcoefficientsentry. cb = ff_q_pfp_congruent_original_leftcoefficientsentry * S ((S (pfc_index_congruent_original_leftcoefficients)) * cc) + (pfc_value_congruent_original_leftcoefficients))) /\ ((exists pfc_terms_code_congruent_original_leftcoefficientscoefficient pfc_terms_scale_congruent_original_leftcoefficientscoefficient pfc_natural_sum_congruent_original_leftcoefficientscoefficient. ((forall pfc_index_congruent_original_leftcoefficientscoefficientdiagonal. (exists pfa_gap_congruent_original_leftcoefficientscoefficientdiagonalbound. pfa_gap_congruent_original_leftcoefficientscoefficientdiagonalbound + S (pfc_index_congruent_original_leftcoefficientscoefficientdiagonal) = (S (pfc_index_congruent_original_leftcoefficients))) -> exists pfc_value_congruent_original_leftcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_congruent_original_leftcoefficientscoefficientdiagonalentry. ff_h_pfp_congruent_original_leftcoefficientscoefficientdiagonalentry + S (pfc_value_congruent_original_leftcoefficientscoefficientdiagonal) = S ((S (pfc_index_congruent_original_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_congruent_original_leftcoefficientscoefficient)) /\ exists ff_q_pfp_congruent_original_leftcoefficientscoefficientdiagonalentry. pfc_terms_code_congruent_original_leftcoefficientscoefficient = ff_q_pfp_congruent_original_leftcoefficientscoefficientdiagonalentry * S ((S (pfc_index_congruent_original_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_congruent_original_leftcoefficientscoefficient) + (pfc_value_congruent_original_leftcoefficientscoefficientdiagonal))) /\ ((exists pfc_complement_congruent_original_leftcoefficientscoefficientdiagonalterm pfc_left_congruent_original_leftcoefficientscoefficientdiagonalterm pfc_right_congruent_original_leftcoefficientscoefficientdiagonalterm. (((pfc_index_congruent_original_leftcoefficientscoefficientdiagonal)+pfc_complement_congruent_original_leftcoefficientscoefficientdiagonalterm=(pfc_index_congruent_original_leftcoefficients)) /\ ((((((exists pfa_gap_congruent_original_leftcoefficientscoefficientdiagonaltermleftinside. pfa_gap_congruent_original_leftcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_congruent_original_leftcoefficientscoefficientdiagonal) = (L)) /\ ((((exists ff_h_pfp_congruent_original_leftcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_congruent_original_leftcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_congruent_original_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_congruent_original_leftcoefficientscoefficientdiagonal)) * ac)) /\ exists ff_q_pfp_congruent_original_leftcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_congruent_original_leftcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_congruent_original_leftcoefficientscoefficientdiagonal)) * ac) + (pfc_left_congruent_original_leftcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_congruent_original_leftcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_congruent_original_leftcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_congruent_original_leftcoefficientscoefficientdiagonal)) /\ (((pfc_left_congruent_original_leftcoefficientscoefficientdiagonalterm)=0))))) /\ ((((((exists pfa_gap_congruent_original_leftcoefficientscoefficientdiagonaltermrightinside. pfa_gap_congruent_original_leftcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_congruent_original_leftcoefficientscoefficientdiagonalterm) = (M)) /\ ((((exists ff_h_pfp_congruent_original_leftcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_congruent_original_leftcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_congruent_original_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_congruent_original_leftcoefficientscoefficientdiagonalterm)) * bc)) /\ exists ff_q_pfp_congruent_original_leftcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_congruent_original_leftcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_congruent_original_leftcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_congruent_original_leftcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_congruent_original_leftcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_congruent_original_leftcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_congruent_original_leftcoefficientscoefficientdiagonalterm)) /\ (((pfc_right_congruent_original_leftcoefficientscoefficientdiagonalterm)=0))))) /\ (((pfc_value_congruent_original_leftcoefficientscoefficientdiagonal)=pfc_left_congruent_original_leftcoefficientscoefficientdiagonalterm*pfc_right_congruent_original_leftcoefficientscoefficientdiagonalterm))))))))))) /\ (((exists fs_u_pfc_congruent_original_leftcoefficientscoefficientsum fs_v_pfc_congruent_original_leftcoefficientscoefficientsum. ((((exists fs_h_pfc_congruent_original_leftcoefficientscoefficientsum_body_start. fs_h_pfc_congruent_original_leftcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_congruent_original_leftcoefficientscoefficientsum)) /\ exists fs_q_pfc_congruent_original_leftcoefficientscoefficientsum_body_start. fs_u_pfc_congruent_original_leftcoefficientscoefficientsum = fs_q_pfc_congruent_original_leftcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_congruent_original_leftcoefficientscoefficientsum) + (0))) /\ ((((exists fs_h_pfc_congruent_original_leftcoefficientscoefficientsum_body_terminal. fs_h_pfc_congruent_original_leftcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_congruent_original_leftcoefficientscoefficient) = S ((S (S (pfc_index_congruent_original_leftcoefficients))) * fs_v_pfc_congruent_original_leftcoefficientscoefficientsum)) /\ exists fs_q_pfc_congruent_original_leftcoefficientscoefficientsum_body_terminal. fs_u_pfc_congruent_original_leftcoefficientscoefficientsum = fs_q_pfc_congruent_original_leftcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_congruent_original_leftcoefficients))) * fs_v_pfc_congruent_original_leftcoefficientscoefficientsum) + (pfc_natural_sum_congruent_original_leftcoefficientscoefficient))) /\ forall fs_i_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps = S (pfc_index_congruent_original_leftcoefficients)) -> exists fs_a_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps fs_r_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps fs_s_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_congruent_original_leftcoefficientscoefficient)) /\ exists fs_q_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_congruent_original_leftcoefficientscoefficient = fs_q_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_congruent_original_leftcoefficientscoefficient) + (fs_a_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_original_leftcoefficientscoefficientsum)) /\ exists fs_q_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_congruent_original_leftcoefficientscoefficientsum = fs_q_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_original_leftcoefficientscoefficientsum) + (fs_r_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_original_leftcoefficientscoefficientsum)) /\ exists fs_q_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_congruent_original_leftcoefficientscoefficientsum = fs_q_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_original_leftcoefficientscoefficientsum) + (fs_s_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps))) /\ fs_s_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps = fs_r_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps + fs_a_pfc_congruent_original_leftcoefficientscoefficientsum_body_steps)))))) /\ ((((exists pfa_gap_congruent_original_leftcoefficientscoefficientresiduebound. pfa_gap_congruent_original_leftcoefficientscoefficientresiduebound + S (pfc_value_congruent_original_leftcoefficients) = (p)) /\ ((exists pfa_offset_left_congruent_original_leftcoefficientscoefficientresiduecongruence pfa_offset_right_congruent_original_leftcoefficientscoefficientresiduecongruence. (pfc_natural_sum_congruent_original_leftcoefficientscoefficient) + (p) * pfa_offset_left_congruent_original_leftcoefficientscoefficientresiduecongruence = (pfc_value_congruent_original_leftcoefficients) + (p) * pfa_offset_right_congruent_original_leftcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_congruent_other_leftleft. (exists fom_gap_pfp_congruent_other_leftleft_index_bound. fom_gap_pfp_congruent_other_leftleft_index_bound + S (fom_index_pfp_congruent_other_leftleft) = H) -> exists fom_value_pfp_congruent_other_leftleft. ((((exists fom_beta_height_pfp_congruent_other_leftleft_entry. fom_beta_height_pfp_congruent_other_leftleft_entry + S (fom_value_pfp_congruent_other_leftleft) = S ((S (fom_index_pfp_congruent_other_leftleft)) * AC)) /\ exists fom_beta_quotient_pfp_congruent_other_leftleft_entry. AB = fom_beta_quotient_pfp_congruent_other_leftleft_entry * S ((S (fom_index_pfp_congruent_other_leftleft)) * AC) + (fom_value_pfp_congruent_other_leftleft))) /\ (exists fom_gap_pfp_congruent_other_leftleft_value_bound. fom_gap_pfp_congruent_other_leftleft_value_bound + S (fom_value_pfp_congruent_other_leftleft) = p))) /\ (((forall fom_index_pfp_congruent_other_leftright. (exists fom_gap_pfp_congruent_other_leftright_index_bound. fom_gap_pfp_congruent_other_leftright_index_bound + S (fom_index_pfp_congruent_other_leftright) = M) -> exists fom_value_pfp_congruent_other_leftright. ((((exists fom_beta_height_pfp_congruent_other_leftright_entry. fom_beta_height_pfp_congruent_other_leftright_entry + S (fom_value_pfp_congruent_other_leftright) = S ((S (fom_index_pfp_congruent_other_leftright)) * bc)) /\ exists fom_beta_quotient_pfp_congruent_other_leftright_entry. bb = fom_beta_quotient_pfp_congruent_other_leftright_entry * S ((S (fom_index_pfp_congruent_other_leftright)) * bc) + (fom_value_pfp_congruent_other_leftright))) /\ (exists fom_gap_pfp_congruent_other_leftright_value_bound. fom_gap_pfp_congruent_other_leftright_value_bound + S (fom_value_pfp_congruent_other_leftright) = p))) /\ (((((((H)=0 \/ (M)=0) /\ (((K)=0)))) \/ (((~((H)=0)) /\ (((~((M)=0)) /\ (((H)+(M)=S (K)))))))) /\ ((forall pfc_index_congruent_other_leftcoefficients. (exists pfa_gap_congruent_other_leftcoefficientsbound. pfa_gap_congruent_other_leftcoefficientsbound + S (pfc_index_congruent_other_leftcoefficients) = (K)) -> exists pfc_value_congruent_other_leftcoefficients. ((((exists ff_h_pfp_congruent_other_leftcoefficientsentry. ff_h_pfp_congruent_other_leftcoefficientsentry + S (pfc_value_congruent_other_leftcoefficients) = S ((S (pfc_index_congruent_other_leftcoefficients)) * CC)) /\ exists ff_q_pfp_congruent_other_leftcoefficientsentry. CB = ff_q_pfp_congruent_other_leftcoefficientsentry * S ((S (pfc_index_congruent_other_leftcoefficients)) * CC) + (pfc_value_congruent_other_leftcoefficients))) /\ ((exists pfc_terms_code_congruent_other_leftcoefficientscoefficient pfc_terms_scale_congruent_other_leftcoefficientscoefficient pfc_natural_sum_congruent_other_leftcoefficientscoefficient. ((forall pfc_index_congruent_other_leftcoefficientscoefficientdiagonal. (exists pfa_gap_congruent_other_leftcoefficientscoefficientdiagonalbound. pfa_gap_congruent_other_leftcoefficientscoefficientdiagonalbound + S (pfc_index_congruent_other_leftcoefficientscoefficientdiagonal) = (S (pfc_index_congruent_other_leftcoefficients))) -> exists pfc_value_congruent_other_leftcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_congruent_other_leftcoefficientscoefficientdiagonalentry. ff_h_pfp_congruent_other_leftcoefficientscoefficientdiagonalentry + S (pfc_value_congruent_other_leftcoefficientscoefficientdiagonal) = S ((S (pfc_index_congruent_other_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_congruent_other_leftcoefficientscoefficient)) /\ exists ff_q_pfp_congruent_other_leftcoefficientscoefficientdiagonalentry. pfc_terms_code_congruent_other_leftcoefficientscoefficient = ff_q_pfp_congruent_other_leftcoefficientscoefficientdiagonalentry * S ((S (pfc_index_congruent_other_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_congruent_other_leftcoefficientscoefficient) + (pfc_value_congruent_other_leftcoefficientscoefficientdiagonal))) /\ ((exists pfc_complement_congruent_other_leftcoefficientscoefficientdiagonalterm pfc_left_congruent_other_leftcoefficientscoefficientdiagonalterm pfc_right_congruent_other_leftcoefficientscoefficientdiagonalterm. (((pfc_index_congruent_other_leftcoefficientscoefficientdiagonal)+pfc_complement_congruent_other_leftcoefficientscoefficientdiagonalterm=(pfc_index_congruent_other_leftcoefficients)) /\ ((((((exists pfa_gap_congruent_other_leftcoefficientscoefficientdiagonaltermleftinside. pfa_gap_congruent_other_leftcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_congruent_other_leftcoefficientscoefficientdiagonal) = (H)) /\ ((((exists ff_h_pfp_congruent_other_leftcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_congruent_other_leftcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_congruent_other_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_congruent_other_leftcoefficientscoefficientdiagonal)) * AC)) /\ exists ff_q_pfp_congruent_other_leftcoefficientscoefficientdiagonaltermleftentry. AB = ff_q_pfp_congruent_other_leftcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_congruent_other_leftcoefficientscoefficientdiagonal)) * AC) + (pfc_left_congruent_other_leftcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_congruent_other_leftcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_congruent_other_leftcoefficientscoefficientdiagonaltermleftoutside+(H)=(pfc_index_congruent_other_leftcoefficientscoefficientdiagonal)) /\ (((pfc_left_congruent_other_leftcoefficientscoefficientdiagonalterm)=0))))) /\ ((((((exists pfa_gap_congruent_other_leftcoefficientscoefficientdiagonaltermrightinside. pfa_gap_congruent_other_leftcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_congruent_other_leftcoefficientscoefficientdiagonalterm) = (M)) /\ ((((exists ff_h_pfp_congruent_other_leftcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_congruent_other_leftcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_congruent_other_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_congruent_other_leftcoefficientscoefficientdiagonalterm)) * bc)) /\ exists ff_q_pfp_congruent_other_leftcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_congruent_other_leftcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_congruent_other_leftcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_congruent_other_leftcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_congruent_other_leftcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_congruent_other_leftcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_congruent_other_leftcoefficientscoefficientdiagonalterm)) /\ (((pfc_right_congruent_other_leftcoefficientscoefficientdiagonalterm)=0))))) /\ (((pfc_value_congruent_other_leftcoefficientscoefficientdiagonal)=pfc_left_congruent_other_leftcoefficientscoefficientdiagonalterm*pfc_right_congruent_other_leftcoefficientscoefficientdiagonalterm))))))))))) /\ (((exists fs_u_pfc_congruent_other_leftcoefficientscoefficientsum fs_v_pfc_congruent_other_leftcoefficientscoefficientsum. ((((exists fs_h_pfc_congruent_other_leftcoefficientscoefficientsum_body_start. fs_h_pfc_congruent_other_leftcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_congruent_other_leftcoefficientscoefficientsum)) /\ exists fs_q_pfc_congruent_other_leftcoefficientscoefficientsum_body_start. fs_u_pfc_congruent_other_leftcoefficientscoefficientsum = fs_q_pfc_congruent_other_leftcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_congruent_other_leftcoefficientscoefficientsum) + (0))) /\ ((((exists fs_h_pfc_congruent_other_leftcoefficientscoefficientsum_body_terminal. fs_h_pfc_congruent_other_leftcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_congruent_other_leftcoefficientscoefficient) = S ((S (S (pfc_index_congruent_other_leftcoefficients))) * fs_v_pfc_congruent_other_leftcoefficientscoefficientsum)) /\ exists fs_q_pfc_congruent_other_leftcoefficientscoefficientsum_body_terminal. fs_u_pfc_congruent_other_leftcoefficientscoefficientsum = fs_q_pfc_congruent_other_leftcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_congruent_other_leftcoefficients))) * fs_v_pfc_congruent_other_leftcoefficientscoefficientsum) + (pfc_natural_sum_congruent_other_leftcoefficientscoefficient))) /\ forall fs_i_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps = S (pfc_index_congruent_other_leftcoefficients)) -> exists fs_a_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps fs_r_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps fs_s_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_congruent_other_leftcoefficientscoefficient)) /\ exists fs_q_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_congruent_other_leftcoefficientscoefficient = fs_q_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_congruent_other_leftcoefficientscoefficient) + (fs_a_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_other_leftcoefficientscoefficientsum)) /\ exists fs_q_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_congruent_other_leftcoefficientscoefficientsum = fs_q_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_other_leftcoefficientscoefficientsum) + (fs_r_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_other_leftcoefficientscoefficientsum)) /\ exists fs_q_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_congruent_other_leftcoefficientscoefficientsum = fs_q_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_other_leftcoefficientscoefficientsum) + (fs_s_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps))) /\ fs_s_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps = fs_r_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps + fs_a_pfc_congruent_other_leftcoefficientscoefficientsum_body_steps)))))) /\ ((((exists pfa_gap_congruent_other_leftcoefficientscoefficientresiduebound. pfa_gap_congruent_other_leftcoefficientscoefficientresiduebound + S (pfc_value_congruent_other_leftcoefficients) = (p)) /\ ((exists pfa_offset_left_congruent_other_leftcoefficientscoefficientresiduecongruence pfa_offset_right_congruent_other_leftcoefficientscoefficientresiduecongruence. (pfc_natural_sum_congruent_other_leftcoefficientscoefficient) + (p) * pfa_offset_left_congruent_other_leftcoefficientscoefficientresiduecongruence = (pfc_value_congruent_other_leftcoefficients) + (p) * pfa_offset_right_congruent_other_leftcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (forall pfrep_power_congruent_output_left pfrep_left_congruent_output_left pfrep_right_congruent_output_left. ((exists pfrep_position_congruent_output_leftfirst. ((pfrep_position_congruent_output_leftfirst+S (pfrep_power_congruent_output_left)=(N)) /\ ((((exists ff_h_pfp_congruent_output_leftfirstentry. ff_h_pfp_congruent_output_leftfirstentry + S (pfrep_left_congruent_output_left) = S ((S (pfrep_position_congruent_output_leftfirst)) * cc)) /\ exists ff_q_pfp_congruent_output_leftfirstentry. cb = ff_q_pfp_congruent_output_leftfirstentry * S ((S (pfrep_position_congruent_output_leftfirst)) * cc) + (pfrep_left_congruent_output_left)))))) \/ (((exists pfrep_gap_congruent_output_leftfirstoutside. pfrep_gap_congruent_output_leftfirstoutside+(N)=(pfrep_power_congruent_output_left)) /\ (((pfrep_left_congruent_output_left)=0))))) -> ((exists pfrep_position_congruent_output_leftsecond. ((pfrep_position_congruent_output_leftsecond+S (pfrep_power_congruent_output_left)=(K)) /\ ((((exists ff_h_pfp_congruent_output_leftsecondentry. ff_h_pfp_congruent_output_leftsecondentry + S (pfrep_right_congruent_output_left) = S ((S (pfrep_position_congruent_output_leftsecond)) * CC)) /\ exists ff_q_pfp_congruent_output_leftsecondentry. CB = ff_q_pfp_congruent_output_leftsecondentry * S ((S (pfrep_position_congruent_output_leftsecond)) * CC) + (pfrep_right_congruent_output_left)))))) \/ (((exists pfrep_gap_congruent_output_leftsecondoutside. pfrep_gap_congruent_output_leftsecondoutside+(K)=(pfrep_power_congruent_output_left)) /\ (((pfrep_right_congruent_output_left)=0))))) -> pfrep_left_congruent_output_left=pfrep_right_congruent_output_left)
Complete tactic proof in conservative notation
All 117 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.
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime field polynomial equivalent implies left pad.
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime field polynomial equivalent implies left pad.