Construct an actual intermediate product and compose the two proved factor-padding compatibilities; both empty and nonempty factors are covered.
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 t BB BC s CB CC K. (~(p=0)) -> (((forall pfp_repeat_index_both_left_paddingzeros. (exists pfa_gap_both_left_paddingzerosindex. pfa_gap_both_left_paddingzerosindex + S (pfp_repeat_index_both_left_paddingzeros) = (t)) -> (((exists ff_h_pfp_both_left_paddingzerosentry. ff_h_pfp_both_left_paddingzerosentry + S (0) = S ((S (pfp_repeat_index_both_left_paddingzeros)) * AC)) /\ exists ff_q_pfp_both_left_paddingzerosentry. AB = ff_q_pfp_both_left_paddingzerosentry * S ((S (pfp_repeat_index_both_left_paddingzeros)) * AC) + (0)))) /\ ((forall pfrep_index_both_left_padding pfrep_value_both_left_padding. (exists pfa_gap_both_left_paddingbound. pfa_gap_both_left_paddingbound + S (pfrep_index_both_left_padding) = (L)) -> (((exists ff_h_pfp_both_left_paddinginput. ff_h_pfp_both_left_paddinginput + S (pfrep_value_both_left_padding) = S ((S (pfrep_index_both_left_padding)) * ac)) /\ exists ff_q_pfp_both_left_paddinginput. ab = ff_q_pfp_both_left_paddinginput * S ((S (pfrep_index_both_left_padding)) * ac) + (pfrep_value_both_left_padding))) -> (((exists ff_h_pfp_both_left_paddingoutput. ff_h_pfp_both_left_paddingoutput + S (pfrep_value_both_left_padding) = S ((S ((t)+pfrep_index_both_left_padding)) * AC)) /\ exists ff_q_pfp_both_left_paddingoutput. AB = ff_q_pfp_both_left_paddingoutput * S ((S ((t)+pfrep_index_both_left_padding)) * AC) + (pfrep_value_both_left_padding))))))) -> (((forall pfp_repeat_index_both_right_paddingzeros. (exists pfa_gap_both_right_paddingzerosindex. pfa_gap_both_right_paddingzerosindex + S (pfp_repeat_index_both_right_paddingzeros) = (s)) -> (((exists ff_h_pfp_both_right_paddingzerosentry. ff_h_pfp_both_right_paddingzerosentry + S (0) = S ((S (pfp_repeat_index_both_right_paddingzeros)) * BC)) /\ exists ff_q_pfp_both_right_paddingzerosentry. BB = ff_q_pfp_both_right_paddingzerosentry * S ((S (pfp_repeat_index_both_right_paddingzeros)) * BC) + (0)))) /\ ((forall pfrep_index_both_right_padding pfrep_value_both_right_padding. (exists pfa_gap_both_right_paddingbound. pfa_gap_both_right_paddingbound + S (pfrep_index_both_right_padding) = (M)) -> (((exists ff_h_pfp_both_right_paddinginput. ff_h_pfp_both_right_paddinginput + S (pfrep_value_both_right_padding) = S ((S (pfrep_index_both_right_padding)) * bc)) /\ exists ff_q_pfp_both_right_paddinginput. bb = ff_q_pfp_both_right_paddinginput * S ((S (pfrep_index_both_right_padding)) * bc) + (pfrep_value_both_right_padding))) -> (((exists ff_h_pfp_both_right_paddingoutput. ff_h_pfp_both_right_paddingoutput + S (pfrep_value_both_right_padding) = S ((S ((s)+pfrep_index_both_right_padding)) * BC)) /\ exists ff_q_pfp_both_right_paddingoutput. BB = ff_q_pfp_both_right_paddingoutput * S ((S ((s)+pfrep_index_both_right_padding)) * BC) + (pfrep_value_both_right_padding))))))) -> (((forall fom_index_pfp_both_original_productleft. (exists fom_gap_pfp_both_original_productleft_index_bound. fom_gap_pfp_both_original_productleft_index_bound + S (fom_index_pfp_both_original_productleft) = L) -> exists fom_value_pfp_both_original_productleft. ((((exists fom_beta_height_pfp_both_original_productleft_entry. fom_beta_height_pfp_both_original_productleft_entry + S (fom_value_pfp_both_original_productleft) = S ((S (fom_index_pfp_both_original_productleft)) * ac)) /\ exists fom_beta_quotient_pfp_both_original_productleft_entry. ab = fom_beta_quotient_pfp_both_original_productleft_entry * S ((S (fom_index_pfp_both_original_productleft)) * ac) + (fom_value_pfp_both_original_productleft))) /\ (exists fom_gap_pfp_both_original_productleft_value_bound. fom_gap_pfp_both_original_productleft_value_bound + S (fom_value_pfp_both_original_productleft) = p))) /\ (((forall fom_index_pfp_both_original_productright. (exists fom_gap_pfp_both_original_productright_index_bound. fom_gap_pfp_both_original_productright_index_bound + S (fom_index_pfp_both_original_productright) = M) -> exists fom_value_pfp_both_original_productright. ((((exists fom_beta_height_pfp_both_original_productright_entry. fom_beta_height_pfp_both_original_productright_entry + S (fom_value_pfp_both_original_productright) = S ((S (fom_index_pfp_both_original_productright)) * bc)) /\ exists fom_beta_quotient_pfp_both_original_productright_entry. bb = fom_beta_quotient_pfp_both_original_productright_entry * S ((S (fom_index_pfp_both_original_productright)) * bc) + (fom_value_pfp_both_original_productright))) /\ (exists fom_gap_pfp_both_original_productright_value_bound. fom_gap_pfp_both_original_productright_value_bound + S (fom_value_pfp_both_original_productright) = p))) /\ (((((((L)=0 \/ (M)=0) /\ (((N)=0)))) \/ (((~((L)=0)) /\ (((~((M)=0)) /\ (((L)+(M)=S (N)))))))) /\ ((forall pfc_index_both_original_productcoefficients. (exists pfa_gap_both_original_productcoefficientsbound. pfa_gap_both_original_productcoefficientsbound + S (pfc_index_both_original_productcoefficients) = (N)) -> exists pfc_value_both_original_productcoefficients. ((((exists ff_h_pfp_both_original_productcoefficientsentry. ff_h_pfp_both_original_productcoefficientsentry + S (pfc_value_both_original_productcoefficients) = S ((S (pfc_index_both_original_productcoefficients)) * cc)) /\ exists ff_q_pfp_both_original_productcoefficientsentry. cb = ff_q_pfp_both_original_productcoefficientsentry * S ((S (pfc_index_both_original_productcoefficients)) * cc) + (pfc_value_both_original_productcoefficients))) /\ ((exists pfc_terms_code_both_original_productcoefficientscoefficient pfc_terms_scale_both_original_productcoefficientscoefficient pfc_natural_sum_both_original_productcoefficientscoefficient. ((forall pfc_index_both_original_productcoefficientscoefficientdiagonal. (exists pfa_gap_both_original_productcoefficientscoefficientdiagonalbound. pfa_gap_both_original_productcoefficientscoefficientdiagonalbound + S (pfc_index_both_original_productcoefficientscoefficientdiagonal) = (S (pfc_index_both_original_productcoefficients))) -> exists pfc_value_both_original_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_both_original_productcoefficientscoefficientdiagonalentry. ff_h_pfp_both_original_productcoefficientscoefficientdiagonalentry + S (pfc_value_both_original_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_both_original_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_both_original_productcoefficientscoefficient)) /\ exists ff_q_pfp_both_original_productcoefficientscoefficientdiagonalentry. pfc_terms_code_both_original_productcoefficientscoefficient = ff_q_pfp_both_original_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_both_original_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_both_original_productcoefficientscoefficient) + (pfc_value_both_original_productcoefficientscoefficientdiagonal))) /\ ((exists pfc_complement_both_original_productcoefficientscoefficientdiagonalterm pfc_left_both_original_productcoefficientscoefficientdiagonalterm pfc_right_both_original_productcoefficientscoefficientdiagonalterm. (((pfc_index_both_original_productcoefficientscoefficientdiagonal)+pfc_complement_both_original_productcoefficientscoefficientdiagonalterm=(pfc_index_both_original_productcoefficients)) /\ ((((((exists pfa_gap_both_original_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_both_original_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_both_original_productcoefficientscoefficientdiagonal) = (L)) /\ ((((exists ff_h_pfp_both_original_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_both_original_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_both_original_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_both_original_productcoefficientscoefficientdiagonal)) * ac)) /\ exists ff_q_pfp_both_original_productcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_both_original_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_both_original_productcoefficientscoefficientdiagonal)) * ac) + (pfc_left_both_original_productcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_both_original_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_both_original_productcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_both_original_productcoefficientscoefficientdiagonal)) /\ (((pfc_left_both_original_productcoefficientscoefficientdiagonalterm)=0))))) /\ ((((((exists pfa_gap_both_original_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_both_original_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_both_original_productcoefficientscoefficientdiagonalterm) = (M)) /\ ((((exists ff_h_pfp_both_original_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_both_original_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_both_original_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_both_original_productcoefficientscoefficientdiagonalterm)) * bc)) /\ exists ff_q_pfp_both_original_productcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_both_original_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_both_original_productcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_both_original_productcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_both_original_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_both_original_productcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_both_original_productcoefficientscoefficientdiagonalterm)) /\ (((pfc_right_both_original_productcoefficientscoefficientdiagonalterm)=0))))) /\ (((pfc_value_both_original_productcoefficientscoefficientdiagonal)=pfc_left_both_original_productcoefficientscoefficientdiagonalterm*pfc_right_both_original_productcoefficientscoefficientdiagonalterm))))))))))) /\ (((exists fs_u_pfc_both_original_productcoefficientscoefficientsum fs_v_pfc_both_original_productcoefficientscoefficientsum. ((((exists fs_h_pfc_both_original_productcoefficientscoefficientsum_body_start. fs_h_pfc_both_original_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_both_original_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_both_original_productcoefficientscoefficientsum_body_start. fs_u_pfc_both_original_productcoefficientscoefficientsum = fs_q_pfc_both_original_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_both_original_productcoefficientscoefficientsum) + (0))) /\ ((((exists fs_h_pfc_both_original_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_both_original_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_both_original_productcoefficientscoefficient) = S ((S (S (pfc_index_both_original_productcoefficients))) * fs_v_pfc_both_original_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_both_original_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_both_original_productcoefficientscoefficientsum = fs_q_pfc_both_original_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_both_original_productcoefficients))) * fs_v_pfc_both_original_productcoefficientscoefficientsum) + (pfc_natural_sum_both_original_productcoefficientscoefficient))) /\ forall fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_both_original_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_both_original_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps = S (pfc_index_both_original_productcoefficients)) -> exists fs_a_pfc_both_original_productcoefficientscoefficientsum_body_steps fs_r_pfc_both_original_productcoefficientscoefficientsum_body_steps fs_s_pfc_both_original_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_both_original_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_both_original_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_both_original_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_both_original_productcoefficientscoefficient)) /\ exists fs_q_pfc_both_original_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_both_original_productcoefficientscoefficient = fs_q_pfc_both_original_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_both_original_productcoefficientscoefficient) + (fs_a_pfc_both_original_productcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_both_original_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_both_original_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_both_original_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_original_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_both_original_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_both_original_productcoefficientscoefficientsum = fs_q_pfc_both_original_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_original_productcoefficientscoefficientsum) + (fs_r_pfc_both_original_productcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_both_original_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_both_original_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_both_original_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_original_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_both_original_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_both_original_productcoefficientscoefficientsum = fs_q_pfc_both_original_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_both_original_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_original_productcoefficientscoefficientsum) + (fs_s_pfc_both_original_productcoefficientscoefficientsum_body_steps))) /\ fs_s_pfc_both_original_productcoefficientscoefficientsum_body_steps = fs_r_pfc_both_original_productcoefficientscoefficientsum_body_steps + fs_a_pfc_both_original_productcoefficientscoefficientsum_body_steps)))))) /\ ((((exists pfa_gap_both_original_productcoefficientscoefficientresiduebound. pfa_gap_both_original_productcoefficientscoefficientresiduebound + S (pfc_value_both_original_productcoefficients) = (p)) /\ ((exists pfa_offset_left_both_original_productcoefficientscoefficientresiduecongruence pfa_offset_right_both_original_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_both_original_productcoefficientscoefficient) + (p) * pfa_offset_left_both_original_productcoefficientscoefficientresiduecongruence = (pfc_value_both_original_productcoefficients) + (p) * pfa_offset_right_both_original_productcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_both_padded_productleft. (exists fom_gap_pfp_both_padded_productleft_index_bound. fom_gap_pfp_both_padded_productleft_index_bound + S (fom_index_pfp_both_padded_productleft) = t+L) -> exists fom_value_pfp_both_padded_productleft. ((((exists fom_beta_height_pfp_both_padded_productleft_entry. fom_beta_height_pfp_both_padded_productleft_entry + S (fom_value_pfp_both_padded_productleft) = S ((S (fom_index_pfp_both_padded_productleft)) * AC)) /\ exists fom_beta_quotient_pfp_both_padded_productleft_entry. AB = fom_beta_quotient_pfp_both_padded_productleft_entry * S ((S (fom_index_pfp_both_padded_productleft)) * AC) + (fom_value_pfp_both_padded_productleft))) /\ (exists fom_gap_pfp_both_padded_productleft_value_bound. fom_gap_pfp_both_padded_productleft_value_bound + S (fom_value_pfp_both_padded_productleft) = p))) /\ (((forall fom_index_pfp_both_padded_productright. (exists fom_gap_pfp_both_padded_productright_index_bound. fom_gap_pfp_both_padded_productright_index_bound + S (fom_index_pfp_both_padded_productright) = s+M) -> exists fom_value_pfp_both_padded_productright. ((((exists fom_beta_height_pfp_both_padded_productright_entry. fom_beta_height_pfp_both_padded_productright_entry + S (fom_value_pfp_both_padded_productright) = S ((S (fom_index_pfp_both_padded_productright)) * BC)) /\ exists fom_beta_quotient_pfp_both_padded_productright_entry. BB = fom_beta_quotient_pfp_both_padded_productright_entry * S ((S (fom_index_pfp_both_padded_productright)) * BC) + (fom_value_pfp_both_padded_productright))) /\ (exists fom_gap_pfp_both_padded_productright_value_bound. fom_gap_pfp_both_padded_productright_value_bound + S (fom_value_pfp_both_padded_productright) = p))) /\ (((((((t+L)=0 \/ (s+M)=0) /\ (((K)=0)))) \/ (((~((t+L)=0)) /\ (((~((s+M)=0)) /\ (((t+L)+(s+M)=S (K)))))))) /\ ((forall pfc_index_both_padded_productcoefficients. (exists pfa_gap_both_padded_productcoefficientsbound. pfa_gap_both_padded_productcoefficientsbound + S (pfc_index_both_padded_productcoefficients) = (K)) -> exists pfc_value_both_padded_productcoefficients. ((((exists ff_h_pfp_both_padded_productcoefficientsentry. ff_h_pfp_both_padded_productcoefficientsentry + S (pfc_value_both_padded_productcoefficients) = S ((S (pfc_index_both_padded_productcoefficients)) * CC)) /\ exists ff_q_pfp_both_padded_productcoefficientsentry. CB = ff_q_pfp_both_padded_productcoefficientsentry * S ((S (pfc_index_both_padded_productcoefficients)) * CC) + (pfc_value_both_padded_productcoefficients))) /\ ((exists pfc_terms_code_both_padded_productcoefficientscoefficient pfc_terms_scale_both_padded_productcoefficientscoefficient pfc_natural_sum_both_padded_productcoefficientscoefficient. ((forall pfc_index_both_padded_productcoefficientscoefficientdiagonal. (exists pfa_gap_both_padded_productcoefficientscoefficientdiagonalbound. pfa_gap_both_padded_productcoefficientscoefficientdiagonalbound + S (pfc_index_both_padded_productcoefficientscoefficientdiagonal) = (S (pfc_index_both_padded_productcoefficients))) -> exists pfc_value_both_padded_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_both_padded_productcoefficientscoefficientdiagonalentry. ff_h_pfp_both_padded_productcoefficientscoefficientdiagonalentry + S (pfc_value_both_padded_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_both_padded_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_both_padded_productcoefficientscoefficient)) /\ exists ff_q_pfp_both_padded_productcoefficientscoefficientdiagonalentry. pfc_terms_code_both_padded_productcoefficientscoefficient = ff_q_pfp_both_padded_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_both_padded_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_both_padded_productcoefficientscoefficient) + (pfc_value_both_padded_productcoefficientscoefficientdiagonal))) /\ ((exists pfc_complement_both_padded_productcoefficientscoefficientdiagonalterm pfc_left_both_padded_productcoefficientscoefficientdiagonalterm pfc_right_both_padded_productcoefficientscoefficientdiagonalterm. (((pfc_index_both_padded_productcoefficientscoefficientdiagonal)+pfc_complement_both_padded_productcoefficientscoefficientdiagonalterm=(pfc_index_both_padded_productcoefficients)) /\ ((((((exists pfa_gap_both_padded_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_both_padded_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_both_padded_productcoefficientscoefficientdiagonal) = (t+L)) /\ ((((exists ff_h_pfp_both_padded_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_both_padded_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_both_padded_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_both_padded_productcoefficientscoefficientdiagonal)) * AC)) /\ exists ff_q_pfp_both_padded_productcoefficientscoefficientdiagonaltermleftentry. AB = ff_q_pfp_both_padded_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_both_padded_productcoefficientscoefficientdiagonal)) * AC) + (pfc_left_both_padded_productcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_both_padded_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_both_padded_productcoefficientscoefficientdiagonaltermleftoutside+(t+L)=(pfc_index_both_padded_productcoefficientscoefficientdiagonal)) /\ (((pfc_left_both_padded_productcoefficientscoefficientdiagonalterm)=0))))) /\ ((((((exists pfa_gap_both_padded_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_both_padded_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_both_padded_productcoefficientscoefficientdiagonalterm) = (s+M)) /\ ((((exists ff_h_pfp_both_padded_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_both_padded_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_both_padded_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_both_padded_productcoefficientscoefficientdiagonalterm)) * BC)) /\ exists ff_q_pfp_both_padded_productcoefficientscoefficientdiagonaltermrightentry. BB = ff_q_pfp_both_padded_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_both_padded_productcoefficientscoefficientdiagonalterm)) * BC) + (pfc_right_both_padded_productcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_both_padded_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_both_padded_productcoefficientscoefficientdiagonaltermrightoutside+(s+M)=(pfc_complement_both_padded_productcoefficientscoefficientdiagonalterm)) /\ (((pfc_right_both_padded_productcoefficientscoefficientdiagonalterm)=0))))) /\ (((pfc_value_both_padded_productcoefficientscoefficientdiagonal)=pfc_left_both_padded_productcoefficientscoefficientdiagonalterm*pfc_right_both_padded_productcoefficientscoefficientdiagonalterm))))))))))) /\ (((exists fs_u_pfc_both_padded_productcoefficientscoefficientsum fs_v_pfc_both_padded_productcoefficientscoefficientsum. ((((exists fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_start. fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_both_padded_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_start. fs_u_pfc_both_padded_productcoefficientscoefficientsum = fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_both_padded_productcoefficientscoefficientsum) + (0))) /\ ((((exists fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_both_padded_productcoefficientscoefficient) = S ((S (S (pfc_index_both_padded_productcoefficients))) * fs_v_pfc_both_padded_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_both_padded_productcoefficientscoefficientsum = fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_both_padded_productcoefficients))) * fs_v_pfc_both_padded_productcoefficientscoefficientsum) + (pfc_natural_sum_both_padded_productcoefficientscoefficient))) /\ forall fs_i_pfc_both_padded_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_both_padded_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_both_padded_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_both_padded_productcoefficientscoefficientsum_body_steps = S (pfc_index_both_padded_productcoefficients)) -> exists fs_a_pfc_both_padded_productcoefficientscoefficientsum_body_steps fs_r_pfc_both_padded_productcoefficientscoefficientsum_body_steps fs_s_pfc_both_padded_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_both_padded_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_both_padded_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_both_padded_productcoefficientscoefficient)) /\ exists fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_both_padded_productcoefficientscoefficient = fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_both_padded_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_both_padded_productcoefficientscoefficient) + (fs_a_pfc_both_padded_productcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_both_padded_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_both_padded_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_padded_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_both_padded_productcoefficientscoefficientsum = fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_both_padded_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_padded_productcoefficientscoefficientsum) + (fs_r_pfc_both_padded_productcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_both_padded_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_both_padded_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_both_padded_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_padded_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_both_padded_productcoefficientscoefficientsum = fs_q_pfc_both_padded_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_both_padded_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_both_padded_productcoefficientscoefficientsum) + (fs_s_pfc_both_padded_productcoefficientscoefficientsum_body_steps))) /\ fs_s_pfc_both_padded_productcoefficientscoefficientsum_body_steps = fs_r_pfc_both_padded_productcoefficientscoefficientsum_body_steps + fs_a_pfc_both_padded_productcoefficientscoefficientsum_body_steps)))))) /\ ((((exists pfa_gap_both_padded_productcoefficientscoefficientresiduebound. pfa_gap_both_padded_productcoefficientscoefficientresiduebound + S (pfc_value_both_padded_productcoefficients) = (p)) /\ ((exists pfa_offset_left_both_padded_productcoefficientscoefficientresiduecongruence pfa_offset_right_both_padded_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_both_padded_productcoefficientscoefficient) + (p) * pfa_offset_left_both_padded_productcoefficientscoefficientresiduecongruence = (pfc_value_both_padded_productcoefficients) + (p) * pfa_offset_right_both_padded_productcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (forall pfrep_power_both_equivalent_result pfrep_left_both_equivalent_result pfrep_right_both_equivalent_result. ((exists pfrep_position_both_equivalent_resultfirst. ((pfrep_position_both_equivalent_resultfirst+S (pfrep_power_both_equivalent_result)=(N)) /\ ((((exists ff_h_pfp_both_equivalent_resultfirstentry. ff_h_pfp_both_equivalent_resultfirstentry + S (pfrep_left_both_equivalent_result) = S ((S (pfrep_position_both_equivalent_resultfirst)) * cc)) /\ exists ff_q_pfp_both_equivalent_resultfirstentry. cb = ff_q_pfp_both_equivalent_resultfirstentry * S ((S (pfrep_position_both_equivalent_resultfirst)) * cc) + (pfrep_left_both_equivalent_result)))))) \/ (((exists pfrep_gap_both_equivalent_resultfirstoutside. pfrep_gap_both_equivalent_resultfirstoutside+(N)=(pfrep_power_both_equivalent_result)) /\ (((pfrep_left_both_equivalent_result)=0))))) -> ((exists pfrep_position_both_equivalent_resultsecond. ((pfrep_position_both_equivalent_resultsecond+S (pfrep_power_both_equivalent_result)=(K)) /\ ((((exists ff_h_pfp_both_equivalent_resultsecondentry. ff_h_pfp_both_equivalent_resultsecondentry + S (pfrep_right_both_equivalent_result) = S ((S (pfrep_position_both_equivalent_resultsecond)) * CC)) /\ exists ff_q_pfp_both_equivalent_resultsecondentry. CB = ff_q_pfp_both_equivalent_resultsecondentry * S ((S (pfrep_position_both_equivalent_resultsecond)) * CC) + (pfrep_right_both_equivalent_result)))))) \/ (((exists pfrep_gap_both_equivalent_resultsecondoutside. pfrep_gap_both_equivalent_resultsecondoutside+(K)=(pfrep_power_both_equivalent_result)) /\ (((pfrep_right_both_equivalent_result)=0))))) -> pfrep_left_both_equivalent_result=pfrep_right_both_equivalent_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.