Formal coefficient equivalence of the right factor preserves two actual products at arbitrary representation lengths, including empty factors; actual leading padding is recovered in the appropriate direction.
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 BB BC H CB CC K. (~(p=0)) -> (forall pfrep_power_congruent_factor_right pfrep_left_congruent_factor_right pfrep_right_congruent_factor_right. ((exists pfrep_position_congruent_factor_rightfirst. ((pfrep_position_congruent_factor_rightfirst+S (pfrep_power_congruent_factor_right)=(M)) /\ ((((exists ff_h_pfp_congruent_factor_rightfirstentry. ff_h_pfp_congruent_factor_rightfirstentry + S (pfrep_left_congruent_factor_right) = S ((S (pfrep_position_congruent_factor_rightfirst)) * bc)) /\ exists ff_q_pfp_congruent_factor_rightfirstentry. bb = ff_q_pfp_congruent_factor_rightfirstentry * S ((S (pfrep_position_congruent_factor_rightfirst)) * bc) + (pfrep_left_congruent_factor_right)))))) \/ (((exists pfrep_gap_congruent_factor_rightfirstoutside. pfrep_gap_congruent_factor_rightfirstoutside+(M)=(pfrep_power_congruent_factor_right)) /\ (((pfrep_left_congruent_factor_right)=0))))) -> ((exists pfrep_position_congruent_factor_rightsecond. ((pfrep_position_congruent_factor_rightsecond+S (pfrep_power_congruent_factor_right)=(H)) /\ ((((exists ff_h_pfp_congruent_factor_rightsecondentry. ff_h_pfp_congruent_factor_rightsecondentry + S (pfrep_right_congruent_factor_right) = S ((S (pfrep_position_congruent_factor_rightsecond)) * BC)) /\ exists ff_q_pfp_congruent_factor_rightsecondentry. BB = ff_q_pfp_congruent_factor_rightsecondentry * S ((S (pfrep_position_congruent_factor_rightsecond)) * BC) + (pfrep_right_congruent_factor_right)))))) \/ (((exists pfrep_gap_congruent_factor_rightsecondoutside. pfrep_gap_congruent_factor_rightsecondoutside+(H)=(pfrep_power_congruent_factor_right)) /\ (((pfrep_right_congruent_factor_right)=0))))) -> pfrep_left_congruent_factor_right=pfrep_right_congruent_factor_right) -> (((forall fom_index_pfp_congruent_original_rightleft. (exists fom_gap_pfp_congruent_original_rightleft_index_bound. fom_gap_pfp_congruent_original_rightleft_index_bound + S (fom_index_pfp_congruent_original_rightleft) = L) -> exists fom_value_pfp_congruent_original_rightleft. ((((exists fom_beta_height_pfp_congruent_original_rightleft_entry. fom_beta_height_pfp_congruent_original_rightleft_entry + S (fom_value_pfp_congruent_original_rightleft) = S ((S (fom_index_pfp_congruent_original_rightleft)) * ac)) /\ exists fom_beta_quotient_pfp_congruent_original_rightleft_entry. ab = fom_beta_quotient_pfp_congruent_original_rightleft_entry * S ((S (fom_index_pfp_congruent_original_rightleft)) * ac) + (fom_value_pfp_congruent_original_rightleft))) /\ (exists fom_gap_pfp_congruent_original_rightleft_value_bound. fom_gap_pfp_congruent_original_rightleft_value_bound + S (fom_value_pfp_congruent_original_rightleft) = p))) /\ (((forall fom_index_pfp_congruent_original_rightright. (exists fom_gap_pfp_congruent_original_rightright_index_bound. fom_gap_pfp_congruent_original_rightright_index_bound + S (fom_index_pfp_congruent_original_rightright) = M) -> exists fom_value_pfp_congruent_original_rightright. ((((exists fom_beta_height_pfp_congruent_original_rightright_entry. fom_beta_height_pfp_congruent_original_rightright_entry + S (fom_value_pfp_congruent_original_rightright) = S ((S (fom_index_pfp_congruent_original_rightright)) * bc)) /\ exists fom_beta_quotient_pfp_congruent_original_rightright_entry. bb = fom_beta_quotient_pfp_congruent_original_rightright_entry * S ((S (fom_index_pfp_congruent_original_rightright)) * bc) + (fom_value_pfp_congruent_original_rightright))) /\ (exists fom_gap_pfp_congruent_original_rightright_value_bound. fom_gap_pfp_congruent_original_rightright_value_bound + S (fom_value_pfp_congruent_original_rightright) = p))) /\ (((((((L)=0 \/ (M)=0) /\ (((N)=0)))) \/ (((~((L)=0)) /\ (((~((M)=0)) /\ (((L)+(M)=S (N)))))))) /\ ((forall pfc_index_congruent_original_rightcoefficients. (exists pfa_gap_congruent_original_rightcoefficientsbound. pfa_gap_congruent_original_rightcoefficientsbound + S (pfc_index_congruent_original_rightcoefficients) = (N)) -> exists pfc_value_congruent_original_rightcoefficients. ((((exists ff_h_pfp_congruent_original_rightcoefficientsentry. ff_h_pfp_congruent_original_rightcoefficientsentry + S (pfc_value_congruent_original_rightcoefficients) = S ((S (pfc_index_congruent_original_rightcoefficients)) * cc)) /\ exists ff_q_pfp_congruent_original_rightcoefficientsentry. cb = ff_q_pfp_congruent_original_rightcoefficientsentry * S ((S (pfc_index_congruent_original_rightcoefficients)) * cc) + (pfc_value_congruent_original_rightcoefficients))) /\ ((exists pfc_terms_code_congruent_original_rightcoefficientscoefficient pfc_terms_scale_congruent_original_rightcoefficientscoefficient pfc_natural_sum_congruent_original_rightcoefficientscoefficient. ((forall pfc_index_congruent_original_rightcoefficientscoefficientdiagonal. (exists pfa_gap_congruent_original_rightcoefficientscoefficientdiagonalbound. pfa_gap_congruent_original_rightcoefficientscoefficientdiagonalbound + S (pfc_index_congruent_original_rightcoefficientscoefficientdiagonal) = (S (pfc_index_congruent_original_rightcoefficients))) -> exists pfc_value_congruent_original_rightcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_congruent_original_rightcoefficientscoefficientdiagonalentry. ff_h_pfp_congruent_original_rightcoefficientscoefficientdiagonalentry + S (pfc_value_congruent_original_rightcoefficientscoefficientdiagonal) = S ((S (pfc_index_congruent_original_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_congruent_original_rightcoefficientscoefficient)) /\ exists ff_q_pfp_congruent_original_rightcoefficientscoefficientdiagonalentry. pfc_terms_code_congruent_original_rightcoefficientscoefficient = ff_q_pfp_congruent_original_rightcoefficientscoefficientdiagonalentry * S ((S (pfc_index_congruent_original_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_congruent_original_rightcoefficientscoefficient) + (pfc_value_congruent_original_rightcoefficientscoefficientdiagonal))) /\ ((exists pfc_complement_congruent_original_rightcoefficientscoefficientdiagonalterm pfc_left_congruent_original_rightcoefficientscoefficientdiagonalterm pfc_right_congruent_original_rightcoefficientscoefficientdiagonalterm. (((pfc_index_congruent_original_rightcoefficientscoefficientdiagonal)+pfc_complement_congruent_original_rightcoefficientscoefficientdiagonalterm=(pfc_index_congruent_original_rightcoefficients)) /\ ((((((exists pfa_gap_congruent_original_rightcoefficientscoefficientdiagonaltermleftinside. pfa_gap_congruent_original_rightcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_congruent_original_rightcoefficientscoefficientdiagonal) = (L)) /\ ((((exists ff_h_pfp_congruent_original_rightcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_congruent_original_rightcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_congruent_original_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_congruent_original_rightcoefficientscoefficientdiagonal)) * ac)) /\ exists ff_q_pfp_congruent_original_rightcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_congruent_original_rightcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_congruent_original_rightcoefficientscoefficientdiagonal)) * ac) + (pfc_left_congruent_original_rightcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_congruent_original_rightcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_congruent_original_rightcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_congruent_original_rightcoefficientscoefficientdiagonal)) /\ (((pfc_left_congruent_original_rightcoefficientscoefficientdiagonalterm)=0))))) /\ ((((((exists pfa_gap_congruent_original_rightcoefficientscoefficientdiagonaltermrightinside. pfa_gap_congruent_original_rightcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_congruent_original_rightcoefficientscoefficientdiagonalterm) = (M)) /\ ((((exists ff_h_pfp_congruent_original_rightcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_congruent_original_rightcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_congruent_original_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_congruent_original_rightcoefficientscoefficientdiagonalterm)) * bc)) /\ exists ff_q_pfp_congruent_original_rightcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_congruent_original_rightcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_congruent_original_rightcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_congruent_original_rightcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_congruent_original_rightcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_congruent_original_rightcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_congruent_original_rightcoefficientscoefficientdiagonalterm)) /\ (((pfc_right_congruent_original_rightcoefficientscoefficientdiagonalterm)=0))))) /\ (((pfc_value_congruent_original_rightcoefficientscoefficientdiagonal)=pfc_left_congruent_original_rightcoefficientscoefficientdiagonalterm*pfc_right_congruent_original_rightcoefficientscoefficientdiagonalterm))))))))))) /\ (((exists fs_u_pfc_congruent_original_rightcoefficientscoefficientsum fs_v_pfc_congruent_original_rightcoefficientscoefficientsum. ((((exists fs_h_pfc_congruent_original_rightcoefficientscoefficientsum_body_start. fs_h_pfc_congruent_original_rightcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_congruent_original_rightcoefficientscoefficientsum)) /\ exists fs_q_pfc_congruent_original_rightcoefficientscoefficientsum_body_start. fs_u_pfc_congruent_original_rightcoefficientscoefficientsum = fs_q_pfc_congruent_original_rightcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_congruent_original_rightcoefficientscoefficientsum) + (0))) /\ ((((exists fs_h_pfc_congruent_original_rightcoefficientscoefficientsum_body_terminal. fs_h_pfc_congruent_original_rightcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_congruent_original_rightcoefficientscoefficient) = S ((S (S (pfc_index_congruent_original_rightcoefficients))) * fs_v_pfc_congruent_original_rightcoefficientscoefficientsum)) /\ exists fs_q_pfc_congruent_original_rightcoefficientscoefficientsum_body_terminal. fs_u_pfc_congruent_original_rightcoefficientscoefficientsum = fs_q_pfc_congruent_original_rightcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_congruent_original_rightcoefficients))) * fs_v_pfc_congruent_original_rightcoefficientscoefficientsum) + (pfc_natural_sum_congruent_original_rightcoefficientscoefficient))) /\ forall fs_i_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps = S (pfc_index_congruent_original_rightcoefficients)) -> exists fs_a_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps fs_r_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps fs_s_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_congruent_original_rightcoefficientscoefficient)) /\ exists fs_q_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_congruent_original_rightcoefficientscoefficient = fs_q_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_congruent_original_rightcoefficientscoefficient) + (fs_a_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_original_rightcoefficientscoefficientsum)) /\ exists fs_q_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_congruent_original_rightcoefficientscoefficientsum = fs_q_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_original_rightcoefficientscoefficientsum) + (fs_r_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_original_rightcoefficientscoefficientsum)) /\ exists fs_q_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_congruent_original_rightcoefficientscoefficientsum = fs_q_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_original_rightcoefficientscoefficientsum) + (fs_s_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps))) /\ fs_s_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps = fs_r_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps + fs_a_pfc_congruent_original_rightcoefficientscoefficientsum_body_steps)))))) /\ ((((exists pfa_gap_congruent_original_rightcoefficientscoefficientresiduebound. pfa_gap_congruent_original_rightcoefficientscoefficientresiduebound + S (pfc_value_congruent_original_rightcoefficients) = (p)) /\ ((exists pfa_offset_left_congruent_original_rightcoefficientscoefficientresiduecongruence pfa_offset_right_congruent_original_rightcoefficientscoefficientresiduecongruence. (pfc_natural_sum_congruent_original_rightcoefficientscoefficient) + (p) * pfa_offset_left_congruent_original_rightcoefficientscoefficientresiduecongruence = (pfc_value_congruent_original_rightcoefficients) + (p) * pfa_offset_right_congruent_original_rightcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_congruent_other_rightleft. (exists fom_gap_pfp_congruent_other_rightleft_index_bound. fom_gap_pfp_congruent_other_rightleft_index_bound + S (fom_index_pfp_congruent_other_rightleft) = L) -> exists fom_value_pfp_congruent_other_rightleft. ((((exists fom_beta_height_pfp_congruent_other_rightleft_entry. fom_beta_height_pfp_congruent_other_rightleft_entry + S (fom_value_pfp_congruent_other_rightleft) = S ((S (fom_index_pfp_congruent_other_rightleft)) * ac)) /\ exists fom_beta_quotient_pfp_congruent_other_rightleft_entry. ab = fom_beta_quotient_pfp_congruent_other_rightleft_entry * S ((S (fom_index_pfp_congruent_other_rightleft)) * ac) + (fom_value_pfp_congruent_other_rightleft))) /\ (exists fom_gap_pfp_congruent_other_rightleft_value_bound. fom_gap_pfp_congruent_other_rightleft_value_bound + S (fom_value_pfp_congruent_other_rightleft) = p))) /\ (((forall fom_index_pfp_congruent_other_rightright. (exists fom_gap_pfp_congruent_other_rightright_index_bound. fom_gap_pfp_congruent_other_rightright_index_bound + S (fom_index_pfp_congruent_other_rightright) = H) -> exists fom_value_pfp_congruent_other_rightright. ((((exists fom_beta_height_pfp_congruent_other_rightright_entry. fom_beta_height_pfp_congruent_other_rightright_entry + S (fom_value_pfp_congruent_other_rightright) = S ((S (fom_index_pfp_congruent_other_rightright)) * BC)) /\ exists fom_beta_quotient_pfp_congruent_other_rightright_entry. BB = fom_beta_quotient_pfp_congruent_other_rightright_entry * S ((S (fom_index_pfp_congruent_other_rightright)) * BC) + (fom_value_pfp_congruent_other_rightright))) /\ (exists fom_gap_pfp_congruent_other_rightright_value_bound. fom_gap_pfp_congruent_other_rightright_value_bound + S (fom_value_pfp_congruent_other_rightright) = p))) /\ (((((((L)=0 \/ (H)=0) /\ (((K)=0)))) \/ (((~((L)=0)) /\ (((~((H)=0)) /\ (((L)+(H)=S (K)))))))) /\ ((forall pfc_index_congruent_other_rightcoefficients. (exists pfa_gap_congruent_other_rightcoefficientsbound. pfa_gap_congruent_other_rightcoefficientsbound + S (pfc_index_congruent_other_rightcoefficients) = (K)) -> exists pfc_value_congruent_other_rightcoefficients. ((((exists ff_h_pfp_congruent_other_rightcoefficientsentry. ff_h_pfp_congruent_other_rightcoefficientsentry + S (pfc_value_congruent_other_rightcoefficients) = S ((S (pfc_index_congruent_other_rightcoefficients)) * CC)) /\ exists ff_q_pfp_congruent_other_rightcoefficientsentry. CB = ff_q_pfp_congruent_other_rightcoefficientsentry * S ((S (pfc_index_congruent_other_rightcoefficients)) * CC) + (pfc_value_congruent_other_rightcoefficients))) /\ ((exists pfc_terms_code_congruent_other_rightcoefficientscoefficient pfc_terms_scale_congruent_other_rightcoefficientscoefficient pfc_natural_sum_congruent_other_rightcoefficientscoefficient. ((forall pfc_index_congruent_other_rightcoefficientscoefficientdiagonal. (exists pfa_gap_congruent_other_rightcoefficientscoefficientdiagonalbound. pfa_gap_congruent_other_rightcoefficientscoefficientdiagonalbound + S (pfc_index_congruent_other_rightcoefficientscoefficientdiagonal) = (S (pfc_index_congruent_other_rightcoefficients))) -> exists pfc_value_congruent_other_rightcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_congruent_other_rightcoefficientscoefficientdiagonalentry. ff_h_pfp_congruent_other_rightcoefficientscoefficientdiagonalentry + S (pfc_value_congruent_other_rightcoefficientscoefficientdiagonal) = S ((S (pfc_index_congruent_other_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_congruent_other_rightcoefficientscoefficient)) /\ exists ff_q_pfp_congruent_other_rightcoefficientscoefficientdiagonalentry. pfc_terms_code_congruent_other_rightcoefficientscoefficient = ff_q_pfp_congruent_other_rightcoefficientscoefficientdiagonalentry * S ((S (pfc_index_congruent_other_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_congruent_other_rightcoefficientscoefficient) + (pfc_value_congruent_other_rightcoefficientscoefficientdiagonal))) /\ ((exists pfc_complement_congruent_other_rightcoefficientscoefficientdiagonalterm pfc_left_congruent_other_rightcoefficientscoefficientdiagonalterm pfc_right_congruent_other_rightcoefficientscoefficientdiagonalterm. (((pfc_index_congruent_other_rightcoefficientscoefficientdiagonal)+pfc_complement_congruent_other_rightcoefficientscoefficientdiagonalterm=(pfc_index_congruent_other_rightcoefficients)) /\ ((((((exists pfa_gap_congruent_other_rightcoefficientscoefficientdiagonaltermleftinside. pfa_gap_congruent_other_rightcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_congruent_other_rightcoefficientscoefficientdiagonal) = (L)) /\ ((((exists ff_h_pfp_congruent_other_rightcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_congruent_other_rightcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_congruent_other_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_congruent_other_rightcoefficientscoefficientdiagonal)) * ac)) /\ exists ff_q_pfp_congruent_other_rightcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_congruent_other_rightcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_congruent_other_rightcoefficientscoefficientdiagonal)) * ac) + (pfc_left_congruent_other_rightcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_congruent_other_rightcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_congruent_other_rightcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_congruent_other_rightcoefficientscoefficientdiagonal)) /\ (((pfc_left_congruent_other_rightcoefficientscoefficientdiagonalterm)=0))))) /\ ((((((exists pfa_gap_congruent_other_rightcoefficientscoefficientdiagonaltermrightinside. pfa_gap_congruent_other_rightcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_congruent_other_rightcoefficientscoefficientdiagonalterm) = (H)) /\ ((((exists ff_h_pfp_congruent_other_rightcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_congruent_other_rightcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_congruent_other_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_congruent_other_rightcoefficientscoefficientdiagonalterm)) * BC)) /\ exists ff_q_pfp_congruent_other_rightcoefficientscoefficientdiagonaltermrightentry. BB = ff_q_pfp_congruent_other_rightcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_congruent_other_rightcoefficientscoefficientdiagonalterm)) * BC) + (pfc_right_congruent_other_rightcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_congruent_other_rightcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_congruent_other_rightcoefficientscoefficientdiagonaltermrightoutside+(H)=(pfc_complement_congruent_other_rightcoefficientscoefficientdiagonalterm)) /\ (((pfc_right_congruent_other_rightcoefficientscoefficientdiagonalterm)=0))))) /\ (((pfc_value_congruent_other_rightcoefficientscoefficientdiagonal)=pfc_left_congruent_other_rightcoefficientscoefficientdiagonalterm*pfc_right_congruent_other_rightcoefficientscoefficientdiagonalterm))))))))))) /\ (((exists fs_u_pfc_congruent_other_rightcoefficientscoefficientsum fs_v_pfc_congruent_other_rightcoefficientscoefficientsum. ((((exists fs_h_pfc_congruent_other_rightcoefficientscoefficientsum_body_start. fs_h_pfc_congruent_other_rightcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_congruent_other_rightcoefficientscoefficientsum)) /\ exists fs_q_pfc_congruent_other_rightcoefficientscoefficientsum_body_start. fs_u_pfc_congruent_other_rightcoefficientscoefficientsum = fs_q_pfc_congruent_other_rightcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_congruent_other_rightcoefficientscoefficientsum) + (0))) /\ ((((exists fs_h_pfc_congruent_other_rightcoefficientscoefficientsum_body_terminal. fs_h_pfc_congruent_other_rightcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_congruent_other_rightcoefficientscoefficient) = S ((S (S (pfc_index_congruent_other_rightcoefficients))) * fs_v_pfc_congruent_other_rightcoefficientscoefficientsum)) /\ exists fs_q_pfc_congruent_other_rightcoefficientscoefficientsum_body_terminal. fs_u_pfc_congruent_other_rightcoefficientscoefficientsum = fs_q_pfc_congruent_other_rightcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_congruent_other_rightcoefficients))) * fs_v_pfc_congruent_other_rightcoefficientscoefficientsum) + (pfc_natural_sum_congruent_other_rightcoefficientscoefficient))) /\ forall fs_i_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps = S (pfc_index_congruent_other_rightcoefficients)) -> exists fs_a_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps fs_r_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps fs_s_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_congruent_other_rightcoefficientscoefficient)) /\ exists fs_q_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_congruent_other_rightcoefficientscoefficient = fs_q_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_congruent_other_rightcoefficientscoefficient) + (fs_a_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_other_rightcoefficientscoefficientsum)) /\ exists fs_q_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_congruent_other_rightcoefficientscoefficientsum = fs_q_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_other_rightcoefficientscoefficientsum) + (fs_r_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_other_rightcoefficientscoefficientsum)) /\ exists fs_q_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_congruent_other_rightcoefficientscoefficientsum = fs_q_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_other_rightcoefficientscoefficientsum) + (fs_s_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps))) /\ fs_s_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps = fs_r_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps + fs_a_pfc_congruent_other_rightcoefficientscoefficientsum_body_steps)))))) /\ ((((exists pfa_gap_congruent_other_rightcoefficientscoefficientresiduebound. pfa_gap_congruent_other_rightcoefficientscoefficientresiduebound + S (pfc_value_congruent_other_rightcoefficients) = (p)) /\ ((exists pfa_offset_left_congruent_other_rightcoefficientscoefficientresiduecongruence pfa_offset_right_congruent_other_rightcoefficientscoefficientresiduecongruence. (pfc_natural_sum_congruent_other_rightcoefficientscoefficient) + (p) * pfa_offset_left_congruent_other_rightcoefficientscoefficientresiduecongruence = (pfc_value_congruent_other_rightcoefficients) + (p) * pfa_offset_right_congruent_other_rightcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (forall pfrep_power_congruent_output_right pfrep_left_congruent_output_right pfrep_right_congruent_output_right. ((exists pfrep_position_congruent_output_rightfirst. ((pfrep_position_congruent_output_rightfirst+S (pfrep_power_congruent_output_right)=(N)) /\ ((((exists ff_h_pfp_congruent_output_rightfirstentry. ff_h_pfp_congruent_output_rightfirstentry + S (pfrep_left_congruent_output_right) = S ((S (pfrep_position_congruent_output_rightfirst)) * cc)) /\ exists ff_q_pfp_congruent_output_rightfirstentry. cb = ff_q_pfp_congruent_output_rightfirstentry * S ((S (pfrep_position_congruent_output_rightfirst)) * cc) + (pfrep_left_congruent_output_right)))))) \/ (((exists pfrep_gap_congruent_output_rightfirstoutside. pfrep_gap_congruent_output_rightfirstoutside+(N)=(pfrep_power_congruent_output_right)) /\ (((pfrep_left_congruent_output_right)=0))))) -> ((exists pfrep_position_congruent_output_rightsecond. ((pfrep_position_congruent_output_rightsecond+S (pfrep_power_congruent_output_right)=(K)) /\ ((((exists ff_h_pfp_congruent_output_rightsecondentry. ff_h_pfp_congruent_output_rightsecondentry + S (pfrep_right_congruent_output_right) = S ((S (pfrep_position_congruent_output_rightsecond)) * CC)) /\ exists ff_q_pfp_congruent_output_rightsecondentry. CB = ff_q_pfp_congruent_output_rightsecondentry * S ((S (pfrep_position_congruent_output_rightsecond)) * CC) + (pfrep_right_congruent_output_right)))))) \/ (((exists pfrep_gap_congruent_output_rightsecondoutside. pfrep_gap_congruent_output_rightsecondoutside+(K)=(pfrep_power_congruent_output_right)) /\ (((pfrep_right_congruent_output_right)=0))))) -> pfrep_left_congruent_output_right=pfrep_right_congruent_output_right)
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.