Two actual convolution outputs represent the same formal polynomial whenever their respective factors do, with all four representation lengths independent. A genuine mixed product is constructed from canonical inputs supplied by the actual products; no output identity or extra field hypothesis is assumed.
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 BB BC I CB CC K. (~(p=0)) -> (forall pfrep_power_congruent_both_left pfrep_left_congruent_both_left pfrep_right_congruent_both_left. ((exists pfrep_position_congruent_both_leftfirst. ((pfrep_position_congruent_both_leftfirst+S (pfrep_power_congruent_both_left)=(L)) /\ ((((exists ff_h_pfp_congruent_both_leftfirstentry. ff_h_pfp_congruent_both_leftfirstentry + S (pfrep_left_congruent_both_left) = S ((S (pfrep_position_congruent_both_leftfirst)) * ac)) /\ exists ff_q_pfp_congruent_both_leftfirstentry. ab = ff_q_pfp_congruent_both_leftfirstentry * S ((S (pfrep_position_congruent_both_leftfirst)) * ac) + (pfrep_left_congruent_both_left)))))) \/ (((exists pfrep_gap_congruent_both_leftfirstoutside. pfrep_gap_congruent_both_leftfirstoutside+(L)=(pfrep_power_congruent_both_left)) /\ (((pfrep_left_congruent_both_left)=0))))) -> ((exists pfrep_position_congruent_both_leftsecond. ((pfrep_position_congruent_both_leftsecond+S (pfrep_power_congruent_both_left)=(H)) /\ ((((exists ff_h_pfp_congruent_both_leftsecondentry. ff_h_pfp_congruent_both_leftsecondentry + S (pfrep_right_congruent_both_left) = S ((S (pfrep_position_congruent_both_leftsecond)) * AC)) /\ exists ff_q_pfp_congruent_both_leftsecondentry. AB = ff_q_pfp_congruent_both_leftsecondentry * S ((S (pfrep_position_congruent_both_leftsecond)) * AC) + (pfrep_right_congruent_both_left)))))) \/ (((exists pfrep_gap_congruent_both_leftsecondoutside. pfrep_gap_congruent_both_leftsecondoutside+(H)=(pfrep_power_congruent_both_left)) /\ (((pfrep_right_congruent_both_left)=0))))) -> pfrep_left_congruent_both_left=pfrep_right_congruent_both_left) -> (forall pfrep_power_congruent_both_right pfrep_left_congruent_both_right pfrep_right_congruent_both_right. ((exists pfrep_position_congruent_both_rightfirst. ((pfrep_position_congruent_both_rightfirst+S (pfrep_power_congruent_both_right)=(M)) /\ ((((exists ff_h_pfp_congruent_both_rightfirstentry. ff_h_pfp_congruent_both_rightfirstentry + S (pfrep_left_congruent_both_right) = S ((S (pfrep_position_congruent_both_rightfirst)) * bc)) /\ exists ff_q_pfp_congruent_both_rightfirstentry. bb = ff_q_pfp_congruent_both_rightfirstentry * S ((S (pfrep_position_congruent_both_rightfirst)) * bc) + (pfrep_left_congruent_both_right)))))) \/ (((exists pfrep_gap_congruent_both_rightfirstoutside. pfrep_gap_congruent_both_rightfirstoutside+(M)=(pfrep_power_congruent_both_right)) /\ (((pfrep_left_congruent_both_right)=0))))) -> ((exists pfrep_position_congruent_both_rightsecond. ((pfrep_position_congruent_both_rightsecond+S (pfrep_power_congruent_both_right)=(I)) /\ ((((exists ff_h_pfp_congruent_both_rightsecondentry. ff_h_pfp_congruent_both_rightsecondentry + S (pfrep_right_congruent_both_right) = S ((S (pfrep_position_congruent_both_rightsecond)) * BC)) /\ exists ff_q_pfp_congruent_both_rightsecondentry. BB = ff_q_pfp_congruent_both_rightsecondentry * S ((S (pfrep_position_congruent_both_rightsecond)) * BC) + (pfrep_right_congruent_both_right)))))) \/ (((exists pfrep_gap_congruent_both_rightsecondoutside. pfrep_gap_congruent_both_rightsecondoutside+(I)=(pfrep_power_congruent_both_right)) /\ (((pfrep_right_congruent_both_right)=0))))) -> pfrep_left_congruent_both_right=pfrep_right_congruent_both_right) -> (((forall fom_index_pfp_congruent_both_originalleft. (exists fom_gap_pfp_congruent_both_originalleft_index_bound. fom_gap_pfp_congruent_both_originalleft_index_bound + S (fom_index_pfp_congruent_both_originalleft) = L) -> exists fom_value_pfp_congruent_both_originalleft. ((((exists fom_beta_height_pfp_congruent_both_originalleft_entry. fom_beta_height_pfp_congruent_both_originalleft_entry + S (fom_value_pfp_congruent_both_originalleft) = S ((S (fom_index_pfp_congruent_both_originalleft)) * ac)) /\ exists fom_beta_quotient_pfp_congruent_both_originalleft_entry. ab = fom_beta_quotient_pfp_congruent_both_originalleft_entry * S ((S (fom_index_pfp_congruent_both_originalleft)) * ac) + (fom_value_pfp_congruent_both_originalleft))) /\ (exists fom_gap_pfp_congruent_both_originalleft_value_bound. fom_gap_pfp_congruent_both_originalleft_value_bound + S (fom_value_pfp_congruent_both_originalleft) = p))) /\ (((forall fom_index_pfp_congruent_both_originalright. (exists fom_gap_pfp_congruent_both_originalright_index_bound. fom_gap_pfp_congruent_both_originalright_index_bound + S (fom_index_pfp_congruent_both_originalright) = M) -> exists fom_value_pfp_congruent_both_originalright. ((((exists fom_beta_height_pfp_congruent_both_originalright_entry. fom_beta_height_pfp_congruent_both_originalright_entry + S (fom_value_pfp_congruent_both_originalright) = S ((S (fom_index_pfp_congruent_both_originalright)) * bc)) /\ exists fom_beta_quotient_pfp_congruent_both_originalright_entry. bb = fom_beta_quotient_pfp_congruent_both_originalright_entry * S ((S (fom_index_pfp_congruent_both_originalright)) * bc) + (fom_value_pfp_congruent_both_originalright))) /\ (exists fom_gap_pfp_congruent_both_originalright_value_bound. fom_gap_pfp_congruent_both_originalright_value_bound + S (fom_value_pfp_congruent_both_originalright) = p))) /\ (((((((L)=0 \/ (M)=0) /\ (((N)=0)))) \/ (((~((L)=0)) /\ (((~((M)=0)) /\ (((L)+(M)=S (N)))))))) /\ ((forall pfc_index_congruent_both_originalcoefficients. (exists pfa_gap_congruent_both_originalcoefficientsbound. pfa_gap_congruent_both_originalcoefficientsbound + S (pfc_index_congruent_both_originalcoefficients) = (N)) -> exists pfc_value_congruent_both_originalcoefficients. ((((exists ff_h_pfp_congruent_both_originalcoefficientsentry. ff_h_pfp_congruent_both_originalcoefficientsentry + S (pfc_value_congruent_both_originalcoefficients) = S ((S (pfc_index_congruent_both_originalcoefficients)) * cc)) /\ exists ff_q_pfp_congruent_both_originalcoefficientsentry. cb = ff_q_pfp_congruent_both_originalcoefficientsentry * S ((S (pfc_index_congruent_both_originalcoefficients)) * cc) + (pfc_value_congruent_both_originalcoefficients))) /\ ((exists pfc_terms_code_congruent_both_originalcoefficientscoefficient pfc_terms_scale_congruent_both_originalcoefficientscoefficient pfc_natural_sum_congruent_both_originalcoefficientscoefficient. ((forall pfc_index_congruent_both_originalcoefficientscoefficientdiagonal. (exists pfa_gap_congruent_both_originalcoefficientscoefficientdiagonalbound. pfa_gap_congruent_both_originalcoefficientscoefficientdiagonalbound + S (pfc_index_congruent_both_originalcoefficientscoefficientdiagonal) = (S (pfc_index_congruent_both_originalcoefficients))) -> exists pfc_value_congruent_both_originalcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_congruent_both_originalcoefficientscoefficientdiagonalentry. ff_h_pfp_congruent_both_originalcoefficientscoefficientdiagonalentry + S (pfc_value_congruent_both_originalcoefficientscoefficientdiagonal) = S ((S (pfc_index_congruent_both_originalcoefficientscoefficientdiagonal)) * pfc_terms_scale_congruent_both_originalcoefficientscoefficient)) /\ exists ff_q_pfp_congruent_both_originalcoefficientscoefficientdiagonalentry. pfc_terms_code_congruent_both_originalcoefficientscoefficient = ff_q_pfp_congruent_both_originalcoefficientscoefficientdiagonalentry * S ((S (pfc_index_congruent_both_originalcoefficientscoefficientdiagonal)) * pfc_terms_scale_congruent_both_originalcoefficientscoefficient) + (pfc_value_congruent_both_originalcoefficientscoefficientdiagonal))) /\ ((exists pfc_complement_congruent_both_originalcoefficientscoefficientdiagonalterm pfc_left_congruent_both_originalcoefficientscoefficientdiagonalterm pfc_right_congruent_both_originalcoefficientscoefficientdiagonalterm. (((pfc_index_congruent_both_originalcoefficientscoefficientdiagonal)+pfc_complement_congruent_both_originalcoefficientscoefficientdiagonalterm=(pfc_index_congruent_both_originalcoefficients)) /\ ((((((exists pfa_gap_congruent_both_originalcoefficientscoefficientdiagonaltermleftinside. pfa_gap_congruent_both_originalcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_congruent_both_originalcoefficientscoefficientdiagonal) = (L)) /\ ((((exists ff_h_pfp_congruent_both_originalcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_congruent_both_originalcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_congruent_both_originalcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_congruent_both_originalcoefficientscoefficientdiagonal)) * ac)) /\ exists ff_q_pfp_congruent_both_originalcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_congruent_both_originalcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_congruent_both_originalcoefficientscoefficientdiagonal)) * ac) + (pfc_left_congruent_both_originalcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_congruent_both_originalcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_congruent_both_originalcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_congruent_both_originalcoefficientscoefficientdiagonal)) /\ (((pfc_left_congruent_both_originalcoefficientscoefficientdiagonalterm)=0))))) /\ ((((((exists pfa_gap_congruent_both_originalcoefficientscoefficientdiagonaltermrightinside. pfa_gap_congruent_both_originalcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_congruent_both_originalcoefficientscoefficientdiagonalterm) = (M)) /\ ((((exists ff_h_pfp_congruent_both_originalcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_congruent_both_originalcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_congruent_both_originalcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_congruent_both_originalcoefficientscoefficientdiagonalterm)) * bc)) /\ exists ff_q_pfp_congruent_both_originalcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_congruent_both_originalcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_congruent_both_originalcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_congruent_both_originalcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_congruent_both_originalcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_congruent_both_originalcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_congruent_both_originalcoefficientscoefficientdiagonalterm)) /\ (((pfc_right_congruent_both_originalcoefficientscoefficientdiagonalterm)=0))))) /\ (((pfc_value_congruent_both_originalcoefficientscoefficientdiagonal)=pfc_left_congruent_both_originalcoefficientscoefficientdiagonalterm*pfc_right_congruent_both_originalcoefficientscoefficientdiagonalterm))))))))))) /\ (((exists fs_u_pfc_congruent_both_originalcoefficientscoefficientsum fs_v_pfc_congruent_both_originalcoefficientscoefficientsum. ((((exists fs_h_pfc_congruent_both_originalcoefficientscoefficientsum_body_start. fs_h_pfc_congruent_both_originalcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_congruent_both_originalcoefficientscoefficientsum)) /\ exists fs_q_pfc_congruent_both_originalcoefficientscoefficientsum_body_start. fs_u_pfc_congruent_both_originalcoefficientscoefficientsum = fs_q_pfc_congruent_both_originalcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_congruent_both_originalcoefficientscoefficientsum) + (0))) /\ ((((exists fs_h_pfc_congruent_both_originalcoefficientscoefficientsum_body_terminal. fs_h_pfc_congruent_both_originalcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_congruent_both_originalcoefficientscoefficient) = S ((S (S (pfc_index_congruent_both_originalcoefficients))) * fs_v_pfc_congruent_both_originalcoefficientscoefficientsum)) /\ exists fs_q_pfc_congruent_both_originalcoefficientscoefficientsum_body_terminal. fs_u_pfc_congruent_both_originalcoefficientscoefficientsum = fs_q_pfc_congruent_both_originalcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_congruent_both_originalcoefficients))) * fs_v_pfc_congruent_both_originalcoefficientscoefficientsum) + (pfc_natural_sum_congruent_both_originalcoefficientscoefficient))) /\ forall fs_i_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps = S (pfc_index_congruent_both_originalcoefficients)) -> exists fs_a_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps fs_r_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps fs_s_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_congruent_both_originalcoefficientscoefficient)) /\ exists fs_q_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_congruent_both_originalcoefficientscoefficient = fs_q_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_congruent_both_originalcoefficientscoefficient) + (fs_a_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_both_originalcoefficientscoefficientsum)) /\ exists fs_q_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_congruent_both_originalcoefficientscoefficientsum = fs_q_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_both_originalcoefficientscoefficientsum) + (fs_r_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_both_originalcoefficientscoefficientsum)) /\ exists fs_q_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_congruent_both_originalcoefficientscoefficientsum = fs_q_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_both_originalcoefficientscoefficientsum) + (fs_s_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps))) /\ fs_s_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps = fs_r_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps + fs_a_pfc_congruent_both_originalcoefficientscoefficientsum_body_steps)))))) /\ ((((exists pfa_gap_congruent_both_originalcoefficientscoefficientresiduebound. pfa_gap_congruent_both_originalcoefficientscoefficientresiduebound + S (pfc_value_congruent_both_originalcoefficients) = (p)) /\ ((exists pfa_offset_left_congruent_both_originalcoefficientscoefficientresiduecongruence pfa_offset_right_congruent_both_originalcoefficientscoefficientresiduecongruence. (pfc_natural_sum_congruent_both_originalcoefficientscoefficient) + (p) * pfa_offset_left_congruent_both_originalcoefficientscoefficientresiduecongruence = (pfc_value_congruent_both_originalcoefficients) + (p) * pfa_offset_right_congruent_both_originalcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_congruent_both_otherleft. (exists fom_gap_pfp_congruent_both_otherleft_index_bound. fom_gap_pfp_congruent_both_otherleft_index_bound + S (fom_index_pfp_congruent_both_otherleft) = H) -> exists fom_value_pfp_congruent_both_otherleft. ((((exists fom_beta_height_pfp_congruent_both_otherleft_entry. fom_beta_height_pfp_congruent_both_otherleft_entry + S (fom_value_pfp_congruent_both_otherleft) = S ((S (fom_index_pfp_congruent_both_otherleft)) * AC)) /\ exists fom_beta_quotient_pfp_congruent_both_otherleft_entry. AB = fom_beta_quotient_pfp_congruent_both_otherleft_entry * S ((S (fom_index_pfp_congruent_both_otherleft)) * AC) + (fom_value_pfp_congruent_both_otherleft))) /\ (exists fom_gap_pfp_congruent_both_otherleft_value_bound. fom_gap_pfp_congruent_both_otherleft_value_bound + S (fom_value_pfp_congruent_both_otherleft) = p))) /\ (((forall fom_index_pfp_congruent_both_otherright. (exists fom_gap_pfp_congruent_both_otherright_index_bound. fom_gap_pfp_congruent_both_otherright_index_bound + S (fom_index_pfp_congruent_both_otherright) = I) -> exists fom_value_pfp_congruent_both_otherright. ((((exists fom_beta_height_pfp_congruent_both_otherright_entry. fom_beta_height_pfp_congruent_both_otherright_entry + S (fom_value_pfp_congruent_both_otherright) = S ((S (fom_index_pfp_congruent_both_otherright)) * BC)) /\ exists fom_beta_quotient_pfp_congruent_both_otherright_entry. BB = fom_beta_quotient_pfp_congruent_both_otherright_entry * S ((S (fom_index_pfp_congruent_both_otherright)) * BC) + (fom_value_pfp_congruent_both_otherright))) /\ (exists fom_gap_pfp_congruent_both_otherright_value_bound. fom_gap_pfp_congruent_both_otherright_value_bound + S (fom_value_pfp_congruent_both_otherright) = p))) /\ (((((((H)=0 \/ (I)=0) /\ (((K)=0)))) \/ (((~((H)=0)) /\ (((~((I)=0)) /\ (((H)+(I)=S (K)))))))) /\ ((forall pfc_index_congruent_both_othercoefficients. (exists pfa_gap_congruent_both_othercoefficientsbound. pfa_gap_congruent_both_othercoefficientsbound + S (pfc_index_congruent_both_othercoefficients) = (K)) -> exists pfc_value_congruent_both_othercoefficients. ((((exists ff_h_pfp_congruent_both_othercoefficientsentry. ff_h_pfp_congruent_both_othercoefficientsentry + S (pfc_value_congruent_both_othercoefficients) = S ((S (pfc_index_congruent_both_othercoefficients)) * CC)) /\ exists ff_q_pfp_congruent_both_othercoefficientsentry. CB = ff_q_pfp_congruent_both_othercoefficientsentry * S ((S (pfc_index_congruent_both_othercoefficients)) * CC) + (pfc_value_congruent_both_othercoefficients))) /\ ((exists pfc_terms_code_congruent_both_othercoefficientscoefficient pfc_terms_scale_congruent_both_othercoefficientscoefficient pfc_natural_sum_congruent_both_othercoefficientscoefficient. ((forall pfc_index_congruent_both_othercoefficientscoefficientdiagonal. (exists pfa_gap_congruent_both_othercoefficientscoefficientdiagonalbound. pfa_gap_congruent_both_othercoefficientscoefficientdiagonalbound + S (pfc_index_congruent_both_othercoefficientscoefficientdiagonal) = (S (pfc_index_congruent_both_othercoefficients))) -> exists pfc_value_congruent_both_othercoefficientscoefficientdiagonal. ((((exists ff_h_pfp_congruent_both_othercoefficientscoefficientdiagonalentry. ff_h_pfp_congruent_both_othercoefficientscoefficientdiagonalentry + S (pfc_value_congruent_both_othercoefficientscoefficientdiagonal) = S ((S (pfc_index_congruent_both_othercoefficientscoefficientdiagonal)) * pfc_terms_scale_congruent_both_othercoefficientscoefficient)) /\ exists ff_q_pfp_congruent_both_othercoefficientscoefficientdiagonalentry. pfc_terms_code_congruent_both_othercoefficientscoefficient = ff_q_pfp_congruent_both_othercoefficientscoefficientdiagonalentry * S ((S (pfc_index_congruent_both_othercoefficientscoefficientdiagonal)) * pfc_terms_scale_congruent_both_othercoefficientscoefficient) + (pfc_value_congruent_both_othercoefficientscoefficientdiagonal))) /\ ((exists pfc_complement_congruent_both_othercoefficientscoefficientdiagonalterm pfc_left_congruent_both_othercoefficientscoefficientdiagonalterm pfc_right_congruent_both_othercoefficientscoefficientdiagonalterm. (((pfc_index_congruent_both_othercoefficientscoefficientdiagonal)+pfc_complement_congruent_both_othercoefficientscoefficientdiagonalterm=(pfc_index_congruent_both_othercoefficients)) /\ ((((((exists pfa_gap_congruent_both_othercoefficientscoefficientdiagonaltermleftinside. pfa_gap_congruent_both_othercoefficientscoefficientdiagonaltermleftinside + S (pfc_index_congruent_both_othercoefficientscoefficientdiagonal) = (H)) /\ ((((exists ff_h_pfp_congruent_both_othercoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_congruent_both_othercoefficientscoefficientdiagonaltermleftentry + S (pfc_left_congruent_both_othercoefficientscoefficientdiagonalterm) = S ((S (pfc_index_congruent_both_othercoefficientscoefficientdiagonal)) * AC)) /\ exists ff_q_pfp_congruent_both_othercoefficientscoefficientdiagonaltermleftentry. AB = ff_q_pfp_congruent_both_othercoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_congruent_both_othercoefficientscoefficientdiagonal)) * AC) + (pfc_left_congruent_both_othercoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_congruent_both_othercoefficientscoefficientdiagonaltermleftoutside. pfc_gap_congruent_both_othercoefficientscoefficientdiagonaltermleftoutside+(H)=(pfc_index_congruent_both_othercoefficientscoefficientdiagonal)) /\ (((pfc_left_congruent_both_othercoefficientscoefficientdiagonalterm)=0))))) /\ ((((((exists pfa_gap_congruent_both_othercoefficientscoefficientdiagonaltermrightinside. pfa_gap_congruent_both_othercoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_congruent_both_othercoefficientscoefficientdiagonalterm) = (I)) /\ ((((exists ff_h_pfp_congruent_both_othercoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_congruent_both_othercoefficientscoefficientdiagonaltermrightentry + S (pfc_right_congruent_both_othercoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_congruent_both_othercoefficientscoefficientdiagonalterm)) * BC)) /\ exists ff_q_pfp_congruent_both_othercoefficientscoefficientdiagonaltermrightentry. BB = ff_q_pfp_congruent_both_othercoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_congruent_both_othercoefficientscoefficientdiagonalterm)) * BC) + (pfc_right_congruent_both_othercoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_congruent_both_othercoefficientscoefficientdiagonaltermrightoutside. pfc_gap_congruent_both_othercoefficientscoefficientdiagonaltermrightoutside+(I)=(pfc_complement_congruent_both_othercoefficientscoefficientdiagonalterm)) /\ (((pfc_right_congruent_both_othercoefficientscoefficientdiagonalterm)=0))))) /\ (((pfc_value_congruent_both_othercoefficientscoefficientdiagonal)=pfc_left_congruent_both_othercoefficientscoefficientdiagonalterm*pfc_right_congruent_both_othercoefficientscoefficientdiagonalterm))))))))))) /\ (((exists fs_u_pfc_congruent_both_othercoefficientscoefficientsum fs_v_pfc_congruent_both_othercoefficientscoefficientsum. ((((exists fs_h_pfc_congruent_both_othercoefficientscoefficientsum_body_start. fs_h_pfc_congruent_both_othercoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_congruent_both_othercoefficientscoefficientsum)) /\ exists fs_q_pfc_congruent_both_othercoefficientscoefficientsum_body_start. fs_u_pfc_congruent_both_othercoefficientscoefficientsum = fs_q_pfc_congruent_both_othercoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_congruent_both_othercoefficientscoefficientsum) + (0))) /\ ((((exists fs_h_pfc_congruent_both_othercoefficientscoefficientsum_body_terminal. fs_h_pfc_congruent_both_othercoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_congruent_both_othercoefficientscoefficient) = S ((S (S (pfc_index_congruent_both_othercoefficients))) * fs_v_pfc_congruent_both_othercoefficientscoefficientsum)) /\ exists fs_q_pfc_congruent_both_othercoefficientscoefficientsum_body_terminal. fs_u_pfc_congruent_both_othercoefficientscoefficientsum = fs_q_pfc_congruent_both_othercoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_congruent_both_othercoefficients))) * fs_v_pfc_congruent_both_othercoefficientscoefficientsum) + (pfc_natural_sum_congruent_both_othercoefficientscoefficient))) /\ forall fs_i_pfc_congruent_both_othercoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_congruent_both_othercoefficientscoefficientsum_body_steps = S (pfc_index_congruent_both_othercoefficients)) -> exists fs_a_pfc_congruent_both_othercoefficientscoefficientsum_body_steps fs_r_pfc_congruent_both_othercoefficientscoefficientsum_body_steps fs_s_pfc_congruent_both_othercoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_summand. fs_h_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_congruent_both_othercoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_congruent_both_othercoefficientscoefficientsum_body_steps)) * pfc_terms_scale_congruent_both_othercoefficientscoefficient)) /\ exists fs_q_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_summand. pfc_terms_code_congruent_both_othercoefficientscoefficient = fs_q_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_congruent_both_othercoefficientscoefficientsum_body_steps)) * pfc_terms_scale_congruent_both_othercoefficientscoefficient) + (fs_a_pfc_congruent_both_othercoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_partial. fs_h_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_congruent_both_othercoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_congruent_both_othercoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_both_othercoefficientscoefficientsum)) /\ exists fs_q_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_partial. fs_u_pfc_congruent_both_othercoefficientscoefficientsum = fs_q_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_congruent_both_othercoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_both_othercoefficientscoefficientsum) + (fs_r_pfc_congruent_both_othercoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_successor. fs_h_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_congruent_both_othercoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_congruent_both_othercoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_both_othercoefficientscoefficientsum)) /\ exists fs_q_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_successor. fs_u_pfc_congruent_both_othercoefficientscoefficientsum = fs_q_pfc_congruent_both_othercoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_congruent_both_othercoefficientscoefficientsum_body_steps)) * fs_v_pfc_congruent_both_othercoefficientscoefficientsum) + (fs_s_pfc_congruent_both_othercoefficientscoefficientsum_body_steps))) /\ fs_s_pfc_congruent_both_othercoefficientscoefficientsum_body_steps = fs_r_pfc_congruent_both_othercoefficientscoefficientsum_body_steps + fs_a_pfc_congruent_both_othercoefficientscoefficientsum_body_steps)))))) /\ ((((exists pfa_gap_congruent_both_othercoefficientscoefficientresiduebound. pfa_gap_congruent_both_othercoefficientscoefficientresiduebound + S (pfc_value_congruent_both_othercoefficients) = (p)) /\ ((exists pfa_offset_left_congruent_both_othercoefficientscoefficientresiduecongruence pfa_offset_right_congruent_both_othercoefficientscoefficientresiduecongruence. (pfc_natural_sum_congruent_both_othercoefficientscoefficient) + (p) * pfa_offset_left_congruent_both_othercoefficientscoefficientresiduecongruence = (pfc_value_congruent_both_othercoefficients) + (p) * pfa_offset_right_congruent_both_othercoefficientscoefficientresiduecongruence))))))))))))))))))) -> (forall pfrep_power_congruent_both_result pfrep_left_congruent_both_result pfrep_right_congruent_both_result. ((exists pfrep_position_congruent_both_resultfirst. ((pfrep_position_congruent_both_resultfirst+S (pfrep_power_congruent_both_result)=(N)) /\ ((((exists ff_h_pfp_congruent_both_resultfirstentry. ff_h_pfp_congruent_both_resultfirstentry + S (pfrep_left_congruent_both_result) = S ((S (pfrep_position_congruent_both_resultfirst)) * cc)) /\ exists ff_q_pfp_congruent_both_resultfirstentry. cb = ff_q_pfp_congruent_both_resultfirstentry * S ((S (pfrep_position_congruent_both_resultfirst)) * cc) + (pfrep_left_congruent_both_result)))))) \/ (((exists pfrep_gap_congruent_both_resultfirstoutside. pfrep_gap_congruent_both_resultfirstoutside+(N)=(pfrep_power_congruent_both_result)) /\ (((pfrep_left_congruent_both_result)=0))))) -> ((exists pfrep_position_congruent_both_resultsecond. ((pfrep_position_congruent_both_resultsecond+S (pfrep_power_congruent_both_result)=(K)) /\ ((((exists ff_h_pfp_congruent_both_resultsecondentry. ff_h_pfp_congruent_both_resultsecondentry + S (pfrep_right_congruent_both_result) = S ((S (pfrep_position_congruent_both_resultsecond)) * CC)) /\ exists ff_q_pfp_congruent_both_resultsecondentry. CB = ff_q_pfp_congruent_both_resultsecondentry * S ((S (pfrep_position_congruent_both_resultsecond)) * CC) + (pfrep_right_congruent_both_result)))))) \/ (((exists pfrep_gap_congruent_both_resultsecondoutside. pfrep_gap_congruent_both_resultsecondoutside+(K)=(pfrep_power_congruent_both_result)) /\ (((pfrep_right_congruent_both_result)=0))))) -> pfrep_left_congruent_both_result=pfrep_right_congruent_both_result)
Complete tactic proof in conservative notation
All 107 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 convolution at length exists.