Construct all three genuine proper-length left products and then prove their coefficient-addition identity; the product witnesses and the distributive conclusion are outputs, never input assumptions.
Alpha v34 checked-use · first admitted v33 · independently kernel and Lean verified; not Stable
Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.
Coefficients are highest-degree-first. The divisor has a nonzero decoded head; primality supplies its actual inverse. Empty quotients and remainders are included. Functionality compares the constructed execution lengths and decoded coefficients, never arbitrary beta codes. Formal polynomial equivalence compares every coefficient, not evaluations on a finite field. The formal identity and remainder-degree bound are proved separately, not assumed by the execution graph. Arbitrary quotient/remainder-pair uniqueness from a formal identity, multiplication associativity, gcd/Bezout, irreducible-polynomial existence, and the full G091 prime-power-field goal remain open. The seven displayed new names are conservative first-order notation, not new kernel primitives.
forall p ab ac bb bc cb cc L db dc M. (~(p=0)) -> (forall fom_index_pfp_products_left_fixed_input. (exists fom_gap_pfp_products_left_fixed_input_index_bound. fom_gap_pfp_products_left_fixed_input_index_bound + S (fom_index_pfp_products_left_fixed_input) = M) -> exists fom_value_pfp_products_left_fixed_input. ((((exists fom_beta_height_pfp_products_left_fixed_input_entry. fom_beta_height_pfp_products_left_fixed_input_entry + S (fom_value_pfp_products_left_fixed_input) = S ((S (fom_index_pfp_products_left_fixed_input)) * dc)) /\ exists fom_beta_quotient_pfp_products_left_fixed_input_entry. db = fom_beta_quotient_pfp_products_left_fixed_input_entry * S ((S (fom_index_pfp_products_left_fixed_input)) * dc) + (fom_value_pfp_products_left_fixed_input))) /\ (exists fom_gap_pfp_products_left_fixed_input_value_bound. fom_gap_pfp_products_left_fixed_input_value_bound + S (fom_value_pfp_products_left_fixed_input) = p))) -> (forall pfp_index_products_left_input_addition. (exists pfa_gap_products_left_input_additionindex. pfa_gap_products_left_input_additionindex + S (pfp_index_products_left_input_addition) = (L)) -> exists pfp_left_products_left_input_addition pfp_right_products_left_input_addition pfp_value_products_left_input_addition. ((((exists ff_h_pfp_products_left_input_additionleft. ff_h_pfp_products_left_input_additionleft + S (pfp_left_products_left_input_addition) = S ((S (pfp_index_products_left_input_addition)) * ac)) /\ exists ff_q_pfp_products_left_input_additionleft. ab = ff_q_pfp_products_left_input_additionleft * S ((S (pfp_index_products_left_input_addition)) * ac) + (pfp_left_products_left_input_addition))) /\ (((((exists ff_h_pfp_products_left_input_additionright. ff_h_pfp_products_left_input_additionright + S (pfp_right_products_left_input_addition) = S ((S (pfp_index_products_left_input_addition)) * bc)) /\ exists ff_q_pfp_products_left_input_additionright. bb = ff_q_pfp_products_left_input_additionright * S ((S (pfp_index_products_left_input_addition)) * bc) + (pfp_right_products_left_input_addition))) /\ (((((exists ff_h_pfp_products_left_input_additiontarget. ff_h_pfp_products_left_input_additiontarget + S (pfp_value_products_left_input_addition) = S ((S (pfp_index_products_left_input_addition)) * cc)) /\ exists ff_q_pfp_products_left_input_additiontarget. cb = ff_q_pfp_products_left_input_additiontarget * S ((S (pfp_index_products_left_input_addition)) * cc) + (pfp_value_products_left_input_addition))) /\ ((((exists pfa_gap_products_left_input_additionoperationleft. pfa_gap_products_left_input_additionoperationleft + S (pfp_left_products_left_input_addition) = (p)) /\ (((exists pfa_gap_products_left_input_additionoperationright. pfa_gap_products_left_input_additionoperationright + S (pfp_right_products_left_input_addition) = (p)) /\ ((((exists pfa_gap_products_left_input_additionoperationresultbound. pfa_gap_products_left_input_additionoperationresultbound + S (pfp_value_products_left_input_addition) = (p)) /\ ((exists pfa_offset_left_products_left_input_additionoperationresultcongruence pfa_offset_right_products_left_input_additionoperationresultcongruence. ((pfp_left_products_left_input_addition) + (pfp_right_products_left_input_addition)) + (p) * pfa_offset_left_products_left_input_additionoperationresultcongruence = (pfp_value_products_left_input_addition) + (p) * pfa_offset_right_products_left_input_additionoperationresultcongruence)))))))))))))))) -> (exists N ub uc vb vc wb wc. ((((forall fom_index_pfp_products_left_result_ubleft. (exists fom_gap_pfp_products_left_result_ubleft_index_bound. fom_gap_pfp_products_left_result_ubleft_index_bound + S (fom_index_pfp_products_left_result_ubleft) = M) -> exists fom_value_pfp_products_left_result_ubleft. ((((exists fom_beta_height_pfp_products_left_result_ubleft_entry. fom_beta_height_pfp_products_left_result_ubleft_entry + S (fom_value_pfp_products_left_result_ubleft) = S ((S (fom_index_pfp_products_left_result_ubleft)) * dc)) /\ exists fom_beta_quotient_pfp_products_left_result_ubleft_entry. db = fom_beta_quotient_pfp_products_left_result_ubleft_entry * S ((S (fom_index_pfp_products_left_result_ubleft)) * dc) + (fom_value_pfp_products_left_result_ubleft))) /\ (exists fom_gap_pfp_products_left_result_ubleft_value_bound. fom_gap_pfp_products_left_result_ubleft_value_bound + S (fom_value_pfp_products_left_result_ubleft) = p))) /\ (((forall fom_index_pfp_products_left_result_ubright. (exists fom_gap_pfp_products_left_result_ubright_index_bound. fom_gap_pfp_products_left_result_ubright_index_bound + S (fom_index_pfp_products_left_result_ubright) = L) -> exists fom_value_pfp_products_left_result_ubright. ((((exists fom_beta_height_pfp_products_left_result_ubright_entry. fom_beta_height_pfp_products_left_result_ubright_entry + S (fom_value_pfp_products_left_result_ubright) = S ((S (fom_index_pfp_products_left_result_ubright)) * ac)) /\ exists fom_beta_quotient_pfp_products_left_result_ubright_entry. ab = fom_beta_quotient_pfp_products_left_result_ubright_entry * S ((S (fom_index_pfp_products_left_result_ubright)) * ac) + (fom_value_pfp_products_left_result_ubright))) /\ (exists fom_gap_pfp_products_left_result_ubright_value_bound. fom_gap_pfp_products_left_result_ubright_value_bound + S (fom_value_pfp_products_left_result_ubright) = p))) /\ (((((((M)=0 \/ (L)=0) /\ (((N)=0)))) \/ (((~((M)=0)) /\ (((~((L)=0)) /\ (((M)+(L)=S (N)))))))) /\ ((forall pfc_index_products_left_result_ubcoefficients. (exists pfa_gap_products_left_result_ubcoefficientsbound. pfa_gap_products_left_result_ubcoefficientsbound + S (pfc_index_products_left_result_ubcoefficients) = (N)) -> exists pfc_value_products_left_result_ubcoefficients. ((((exists ff_h_pfp_products_left_result_ubcoefficientsentry. ff_h_pfp_products_left_result_ubcoefficientsentry + S (pfc_value_products_left_result_ubcoefficients) = S ((S (pfc_index_products_left_result_ubcoefficients)) * uc)) /\ exists ff_q_pfp_products_left_result_ubcoefficientsentry. ub = ff_q_pfp_products_left_result_ubcoefficientsentry * S ((S (pfc_index_products_left_result_ubcoefficients)) * uc) + (pfc_value_products_left_result_ubcoefficients))) /\ ((exists pfc_terms_code_products_left_result_ubcoefficientscoefficient pfc_terms_scale_products_left_result_ubcoefficientscoefficient pfc_natural_sum_products_left_result_ubcoefficientscoefficient. ((forall pfc_index_products_left_result_ubcoefficientscoefficientdiagonal. (exists pfa_gap_products_left_result_ubcoefficientscoefficientdiagonalbound. pfa_gap_products_left_result_ubcoefficientscoefficientdiagonalbound + S (pfc_index_products_left_result_ubcoefficientscoefficientdiagonal) = (S (pfc_index_products_left_result_ubcoefficients))) -> exists pfc_value_products_left_result_ubcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_products_left_result_ubcoefficientscoefficientdiagonalentry. ff_h_pfp_products_left_result_ubcoefficientscoefficientdiagonalentry + S (pfc_value_products_left_result_ubcoefficientscoefficientdiagonal) = S ((S (pfc_index_products_left_result_ubcoefficientscoefficientdiagonal)) * pfc_terms_scale_products_left_result_ubcoefficientscoefficient)) /\ exists ff_q_pfp_products_left_result_ubcoefficientscoefficientdiagonalentry. pfc_terms_code_products_left_result_ubcoefficientscoefficient = ff_q_pfp_products_left_result_ubcoefficientscoefficientdiagonalentry * S ((S (pfc_index_products_left_result_ubcoefficientscoefficientdiagonal)) * pfc_terms_scale_products_left_result_ubcoefficientscoefficient) + (pfc_value_products_left_result_ubcoefficientscoefficientdiagonal))) /\ ((exists pfc_complement_products_left_result_ubcoefficientscoefficientdiagonalterm pfc_left_products_left_result_ubcoefficientscoefficientdiagonalterm pfc_right_products_left_result_ubcoefficientscoefficientdiagonalterm. (((pfc_index_products_left_result_ubcoefficientscoefficientdiagonal)+pfc_complement_products_left_result_ubcoefficientscoefficientdiagonalterm=(pfc_index_products_left_result_ubcoefficients)) /\ ((((((exists pfa_gap_products_left_result_ubcoefficientscoefficientdiagonaltermleftinside. pfa_gap_products_left_result_ubcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_products_left_result_ubcoefficientscoefficientdiagonal) = (M)) /\ ((((exists ff_h_pfp_products_left_result_ubcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_products_left_result_ubcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_products_left_result_ubcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_products_left_result_ubcoefficientscoefficientdiagonal)) * dc)) /\ exists ff_q_pfp_products_left_result_ubcoefficientscoefficientdiagonaltermleftentry. db = ff_q_pfp_products_left_result_ubcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_products_left_result_ubcoefficientscoefficientdiagonal)) * dc) + (pfc_left_products_left_result_ubcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_products_left_result_ubcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_products_left_result_ubcoefficientscoefficientdiagonaltermleftoutside+(M)=(pfc_index_products_left_result_ubcoefficientscoefficientdiagonal)) /\ (((pfc_left_products_left_result_ubcoefficientscoefficientdiagonalterm)=0))))) /\ ((((((exists pfa_gap_products_left_result_ubcoefficientscoefficientdiagonaltermrightinside. pfa_gap_products_left_result_ubcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_products_left_result_ubcoefficientscoefficientdiagonalterm) = (L)) /\ ((((exists ff_h_pfp_products_left_result_ubcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_products_left_result_ubcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_products_left_result_ubcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_products_left_result_ubcoefficientscoefficientdiagonalterm)) * ac)) /\ exists ff_q_pfp_products_left_result_ubcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_products_left_result_ubcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_products_left_result_ubcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_products_left_result_ubcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_products_left_result_ubcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_products_left_result_ubcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_products_left_result_ubcoefficientscoefficientdiagonalterm)) /\ (((pfc_right_products_left_result_ubcoefficientscoefficientdiagonalterm)=0))))) /\ (((pfc_value_products_left_result_ubcoefficientscoefficientdiagonal)=pfc_left_products_left_result_ubcoefficientscoefficientdiagonalterm*pfc_right_products_left_result_ubcoefficientscoefficientdiagonalterm))))))))))) /\ (((exists fs_u_pfc_products_left_result_ubcoefficientscoefficientsum fs_v_pfc_products_left_result_ubcoefficientscoefficientsum. ((((exists fs_h_pfc_products_left_result_ubcoefficientscoefficientsum_body_start. fs_h_pfc_products_left_result_ubcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_products_left_result_ubcoefficientscoefficientsum)) /\ exists fs_q_pfc_products_left_result_ubcoefficientscoefficientsum_body_start. fs_u_pfc_products_left_result_ubcoefficientscoefficientsum = fs_q_pfc_products_left_result_ubcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_products_left_result_ubcoefficientscoefficientsum) + (0))) /\ ((((exists fs_h_pfc_products_left_result_ubcoefficientscoefficientsum_body_terminal. fs_h_pfc_products_left_result_ubcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_products_left_result_ubcoefficientscoefficient) = S ((S (S (pfc_index_products_left_result_ubcoefficients))) * fs_v_pfc_products_left_result_ubcoefficientscoefficientsum)) /\ exists fs_q_pfc_products_left_result_ubcoefficientscoefficientsum_body_terminal. fs_u_pfc_products_left_result_ubcoefficientscoefficientsum = fs_q_pfc_products_left_result_ubcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_products_left_result_ubcoefficients))) * fs_v_pfc_products_left_result_ubcoefficientscoefficientsum) + (pfc_natural_sum_products_left_result_ubcoefficientscoefficient))) /\ forall fs_i_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps = S (pfc_index_products_left_result_ubcoefficients)) -> exists fs_a_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps fs_r_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps fs_s_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_products_left_result_ubcoefficientscoefficient)) /\ exists fs_q_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_products_left_result_ubcoefficientscoefficient = fs_q_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_products_left_result_ubcoefficientscoefficient) + (fs_a_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_result_ubcoefficientscoefficientsum)) /\ exists fs_q_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_products_left_result_ubcoefficientscoefficientsum = fs_q_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_result_ubcoefficientscoefficientsum) + (fs_r_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_result_ubcoefficientscoefficientsum)) /\ exists fs_q_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_products_left_result_ubcoefficientscoefficientsum = fs_q_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_result_ubcoefficientscoefficientsum) + (fs_s_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps))) /\ fs_s_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps = fs_r_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps + fs_a_pfc_products_left_result_ubcoefficientscoefficientsum_body_steps)))))) /\ ((((exists pfa_gap_products_left_result_ubcoefficientscoefficientresiduebound. pfa_gap_products_left_result_ubcoefficientscoefficientresiduebound + S (pfc_value_products_left_result_ubcoefficients) = (p)) /\ ((exists pfa_offset_left_products_left_result_ubcoefficientscoefficientresiduecongruence pfa_offset_right_products_left_result_ubcoefficientscoefficientresiduecongruence. (pfc_natural_sum_products_left_result_ubcoefficientscoefficient) + (p) * pfa_offset_left_products_left_result_ubcoefficientscoefficientresiduecongruence = (pfc_value_products_left_result_ubcoefficients) + (p) * pfa_offset_right_products_left_result_ubcoefficientscoefficientresiduecongruence))))))))))))))))))) /\ (((((forall fom_index_pfp_products_left_result_vbleft. (exists fom_gap_pfp_products_left_result_vbleft_index_bound. fom_gap_pfp_products_left_result_vbleft_index_bound + S (fom_index_pfp_products_left_result_vbleft) = M) -> exists fom_value_pfp_products_left_result_vbleft. ((((exists fom_beta_height_pfp_products_left_result_vbleft_entry. fom_beta_height_pfp_products_left_result_vbleft_entry + S (fom_value_pfp_products_left_result_vbleft) = S ((S (fom_index_pfp_products_left_result_vbleft)) * dc)) /\ exists fom_beta_quotient_pfp_products_left_result_vbleft_entry. db = fom_beta_quotient_pfp_products_left_result_vbleft_entry * S ((S (fom_index_pfp_products_left_result_vbleft)) * dc) + (fom_value_pfp_products_left_result_vbleft))) /\ (exists fom_gap_pfp_products_left_result_vbleft_value_bound. fom_gap_pfp_products_left_result_vbleft_value_bound + S (fom_value_pfp_products_left_result_vbleft) = p))) /\ (((forall fom_index_pfp_products_left_result_vbright. (exists fom_gap_pfp_products_left_result_vbright_index_bound. fom_gap_pfp_products_left_result_vbright_index_bound + S (fom_index_pfp_products_left_result_vbright) = L) -> exists fom_value_pfp_products_left_result_vbright. ((((exists fom_beta_height_pfp_products_left_result_vbright_entry. fom_beta_height_pfp_products_left_result_vbright_entry + S (fom_value_pfp_products_left_result_vbright) = S ((S (fom_index_pfp_products_left_result_vbright)) * bc)) /\ exists fom_beta_quotient_pfp_products_left_result_vbright_entry. bb = fom_beta_quotient_pfp_products_left_result_vbright_entry * S ((S (fom_index_pfp_products_left_result_vbright)) * bc) + (fom_value_pfp_products_left_result_vbright))) /\ (exists fom_gap_pfp_products_left_result_vbright_value_bound. fom_gap_pfp_products_left_result_vbright_value_bound + S (fom_value_pfp_products_left_result_vbright) = p))) /\ (((((((M)=0 \/ (L)=0) /\ (((N)=0)))) \/ (((~((M)=0)) /\ (((~((L)=0)) /\ (((M)+(L)=S (N)))))))) /\ ((forall pfc_index_products_left_result_vbcoefficients. (exists pfa_gap_products_left_result_vbcoefficientsbound. pfa_gap_products_left_result_vbcoefficientsbound + S (pfc_index_products_left_result_vbcoefficients) = (N)) -> exists pfc_value_products_left_result_vbcoefficients. ((((exists ff_h_pfp_products_left_result_vbcoefficientsentry. ff_h_pfp_products_left_result_vbcoefficientsentry + S (pfc_value_products_left_result_vbcoefficients) = S ((S (pfc_index_products_left_result_vbcoefficients)) * vc)) /\ exists ff_q_pfp_products_left_result_vbcoefficientsentry. vb = ff_q_pfp_products_left_result_vbcoefficientsentry * S ((S (pfc_index_products_left_result_vbcoefficients)) * vc) + (pfc_value_products_left_result_vbcoefficients))) /\ ((exists pfc_terms_code_products_left_result_vbcoefficientscoefficient pfc_terms_scale_products_left_result_vbcoefficientscoefficient pfc_natural_sum_products_left_result_vbcoefficientscoefficient. ((forall pfc_index_products_left_result_vbcoefficientscoefficientdiagonal. (exists pfa_gap_products_left_result_vbcoefficientscoefficientdiagonalbound. pfa_gap_products_left_result_vbcoefficientscoefficientdiagonalbound + S (pfc_index_products_left_result_vbcoefficientscoefficientdiagonal) = (S (pfc_index_products_left_result_vbcoefficients))) -> exists pfc_value_products_left_result_vbcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_products_left_result_vbcoefficientscoefficientdiagonalentry. ff_h_pfp_products_left_result_vbcoefficientscoefficientdiagonalentry + S (pfc_value_products_left_result_vbcoefficientscoefficientdiagonal) = S ((S (pfc_index_products_left_result_vbcoefficientscoefficientdiagonal)) * pfc_terms_scale_products_left_result_vbcoefficientscoefficient)) /\ exists ff_q_pfp_products_left_result_vbcoefficientscoefficientdiagonalentry. pfc_terms_code_products_left_result_vbcoefficientscoefficient = ff_q_pfp_products_left_result_vbcoefficientscoefficientdiagonalentry * S ((S (pfc_index_products_left_result_vbcoefficientscoefficientdiagonal)) * pfc_terms_scale_products_left_result_vbcoefficientscoefficient) + (pfc_value_products_left_result_vbcoefficientscoefficientdiagonal))) /\ ((exists pfc_complement_products_left_result_vbcoefficientscoefficientdiagonalterm pfc_left_products_left_result_vbcoefficientscoefficientdiagonalterm pfc_right_products_left_result_vbcoefficientscoefficientdiagonalterm. (((pfc_index_products_left_result_vbcoefficientscoefficientdiagonal)+pfc_complement_products_left_result_vbcoefficientscoefficientdiagonalterm=(pfc_index_products_left_result_vbcoefficients)) /\ ((((((exists pfa_gap_products_left_result_vbcoefficientscoefficientdiagonaltermleftinside. pfa_gap_products_left_result_vbcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_products_left_result_vbcoefficientscoefficientdiagonal) = (M)) /\ ((((exists ff_h_pfp_products_left_result_vbcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_products_left_result_vbcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_products_left_result_vbcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_products_left_result_vbcoefficientscoefficientdiagonal)) * dc)) /\ exists ff_q_pfp_products_left_result_vbcoefficientscoefficientdiagonaltermleftentry. db = ff_q_pfp_products_left_result_vbcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_products_left_result_vbcoefficientscoefficientdiagonal)) * dc) + (pfc_left_products_left_result_vbcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_products_left_result_vbcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_products_left_result_vbcoefficientscoefficientdiagonaltermleftoutside+(M)=(pfc_index_products_left_result_vbcoefficientscoefficientdiagonal)) /\ (((pfc_left_products_left_result_vbcoefficientscoefficientdiagonalterm)=0))))) /\ ((((((exists pfa_gap_products_left_result_vbcoefficientscoefficientdiagonaltermrightinside. pfa_gap_products_left_result_vbcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_products_left_result_vbcoefficientscoefficientdiagonalterm) = (L)) /\ ((((exists ff_h_pfp_products_left_result_vbcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_products_left_result_vbcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_products_left_result_vbcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_products_left_result_vbcoefficientscoefficientdiagonalterm)) * bc)) /\ exists ff_q_pfp_products_left_result_vbcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_products_left_result_vbcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_products_left_result_vbcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_products_left_result_vbcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_products_left_result_vbcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_products_left_result_vbcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_products_left_result_vbcoefficientscoefficientdiagonalterm)) /\ (((pfc_right_products_left_result_vbcoefficientscoefficientdiagonalterm)=0))))) /\ (((pfc_value_products_left_result_vbcoefficientscoefficientdiagonal)=pfc_left_products_left_result_vbcoefficientscoefficientdiagonalterm*pfc_right_products_left_result_vbcoefficientscoefficientdiagonalterm))))))))))) /\ (((exists fs_u_pfc_products_left_result_vbcoefficientscoefficientsum fs_v_pfc_products_left_result_vbcoefficientscoefficientsum. ((((exists fs_h_pfc_products_left_result_vbcoefficientscoefficientsum_body_start. fs_h_pfc_products_left_result_vbcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_products_left_result_vbcoefficientscoefficientsum)) /\ exists fs_q_pfc_products_left_result_vbcoefficientscoefficientsum_body_start. fs_u_pfc_products_left_result_vbcoefficientscoefficientsum = fs_q_pfc_products_left_result_vbcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_products_left_result_vbcoefficientscoefficientsum) + (0))) /\ ((((exists fs_h_pfc_products_left_result_vbcoefficientscoefficientsum_body_terminal. fs_h_pfc_products_left_result_vbcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_products_left_result_vbcoefficientscoefficient) = S ((S (S (pfc_index_products_left_result_vbcoefficients))) * fs_v_pfc_products_left_result_vbcoefficientscoefficientsum)) /\ exists fs_q_pfc_products_left_result_vbcoefficientscoefficientsum_body_terminal. fs_u_pfc_products_left_result_vbcoefficientscoefficientsum = fs_q_pfc_products_left_result_vbcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_products_left_result_vbcoefficients))) * fs_v_pfc_products_left_result_vbcoefficientscoefficientsum) + (pfc_natural_sum_products_left_result_vbcoefficientscoefficient))) /\ forall fs_i_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps = S (pfc_index_products_left_result_vbcoefficients)) -> exists fs_a_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps fs_r_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps fs_s_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_products_left_result_vbcoefficientscoefficient)) /\ exists fs_q_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_products_left_result_vbcoefficientscoefficient = fs_q_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_products_left_result_vbcoefficientscoefficient) + (fs_a_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_result_vbcoefficientscoefficientsum)) /\ exists fs_q_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_products_left_result_vbcoefficientscoefficientsum = fs_q_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_result_vbcoefficientscoefficientsum) + (fs_r_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_result_vbcoefficientscoefficientsum)) /\ exists fs_q_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_products_left_result_vbcoefficientscoefficientsum = fs_q_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_result_vbcoefficientscoefficientsum) + (fs_s_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps))) /\ fs_s_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps = fs_r_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps + fs_a_pfc_products_left_result_vbcoefficientscoefficientsum_body_steps)))))) /\ ((((exists pfa_gap_products_left_result_vbcoefficientscoefficientresiduebound. pfa_gap_products_left_result_vbcoefficientscoefficientresiduebound + S (pfc_value_products_left_result_vbcoefficients) = (p)) /\ ((exists pfa_offset_left_products_left_result_vbcoefficientscoefficientresiduecongruence pfa_offset_right_products_left_result_vbcoefficientscoefficientresiduecongruence. (pfc_natural_sum_products_left_result_vbcoefficientscoefficient) + (p) * pfa_offset_left_products_left_result_vbcoefficientscoefficientresiduecongruence = (pfc_value_products_left_result_vbcoefficients) + (p) * pfa_offset_right_products_left_result_vbcoefficientscoefficientresiduecongruence))))))))))))))))))) /\ (((((forall fom_index_pfp_products_left_result_wbleft. (exists fom_gap_pfp_products_left_result_wbleft_index_bound. fom_gap_pfp_products_left_result_wbleft_index_bound + S (fom_index_pfp_products_left_result_wbleft) = M) -> exists fom_value_pfp_products_left_result_wbleft. ((((exists fom_beta_height_pfp_products_left_result_wbleft_entry. fom_beta_height_pfp_products_left_result_wbleft_entry + S (fom_value_pfp_products_left_result_wbleft) = S ((S (fom_index_pfp_products_left_result_wbleft)) * dc)) /\ exists fom_beta_quotient_pfp_products_left_result_wbleft_entry. db = fom_beta_quotient_pfp_products_left_result_wbleft_entry * S ((S (fom_index_pfp_products_left_result_wbleft)) * dc) + (fom_value_pfp_products_left_result_wbleft))) /\ (exists fom_gap_pfp_products_left_result_wbleft_value_bound. fom_gap_pfp_products_left_result_wbleft_value_bound + S (fom_value_pfp_products_left_result_wbleft) = p))) /\ (((forall fom_index_pfp_products_left_result_wbright. (exists fom_gap_pfp_products_left_result_wbright_index_bound. fom_gap_pfp_products_left_result_wbright_index_bound + S (fom_index_pfp_products_left_result_wbright) = L) -> exists fom_value_pfp_products_left_result_wbright. ((((exists fom_beta_height_pfp_products_left_result_wbright_entry. fom_beta_height_pfp_products_left_result_wbright_entry + S (fom_value_pfp_products_left_result_wbright) = S ((S (fom_index_pfp_products_left_result_wbright)) * cc)) /\ exists fom_beta_quotient_pfp_products_left_result_wbright_entry. cb = fom_beta_quotient_pfp_products_left_result_wbright_entry * S ((S (fom_index_pfp_products_left_result_wbright)) * cc) + (fom_value_pfp_products_left_result_wbright))) /\ (exists fom_gap_pfp_products_left_result_wbright_value_bound. fom_gap_pfp_products_left_result_wbright_value_bound + S (fom_value_pfp_products_left_result_wbright) = p))) /\ (((((((M)=0 \/ (L)=0) /\ (((N)=0)))) \/ (((~((M)=0)) /\ (((~((L)=0)) /\ (((M)+(L)=S (N)))))))) /\ ((forall pfc_index_products_left_result_wbcoefficients. (exists pfa_gap_products_left_result_wbcoefficientsbound. pfa_gap_products_left_result_wbcoefficientsbound + S (pfc_index_products_left_result_wbcoefficients) = (N)) -> exists pfc_value_products_left_result_wbcoefficients. ((((exists ff_h_pfp_products_left_result_wbcoefficientsentry. ff_h_pfp_products_left_result_wbcoefficientsentry + S (pfc_value_products_left_result_wbcoefficients) = S ((S (pfc_index_products_left_result_wbcoefficients)) * wc)) /\ exists ff_q_pfp_products_left_result_wbcoefficientsentry. wb = ff_q_pfp_products_left_result_wbcoefficientsentry * S ((S (pfc_index_products_left_result_wbcoefficients)) * wc) + (pfc_value_products_left_result_wbcoefficients))) /\ ((exists pfc_terms_code_products_left_result_wbcoefficientscoefficient pfc_terms_scale_products_left_result_wbcoefficientscoefficient pfc_natural_sum_products_left_result_wbcoefficientscoefficient. ((forall pfc_index_products_left_result_wbcoefficientscoefficientdiagonal. (exists pfa_gap_products_left_result_wbcoefficientscoefficientdiagonalbound. pfa_gap_products_left_result_wbcoefficientscoefficientdiagonalbound + S (pfc_index_products_left_result_wbcoefficientscoefficientdiagonal) = (S (pfc_index_products_left_result_wbcoefficients))) -> exists pfc_value_products_left_result_wbcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_products_left_result_wbcoefficientscoefficientdiagonalentry. ff_h_pfp_products_left_result_wbcoefficientscoefficientdiagonalentry + S (pfc_value_products_left_result_wbcoefficientscoefficientdiagonal) = S ((S (pfc_index_products_left_result_wbcoefficientscoefficientdiagonal)) * pfc_terms_scale_products_left_result_wbcoefficientscoefficient)) /\ exists ff_q_pfp_products_left_result_wbcoefficientscoefficientdiagonalentry. pfc_terms_code_products_left_result_wbcoefficientscoefficient = ff_q_pfp_products_left_result_wbcoefficientscoefficientdiagonalentry * S ((S (pfc_index_products_left_result_wbcoefficientscoefficientdiagonal)) * pfc_terms_scale_products_left_result_wbcoefficientscoefficient) + (pfc_value_products_left_result_wbcoefficientscoefficientdiagonal))) /\ ((exists pfc_complement_products_left_result_wbcoefficientscoefficientdiagonalterm pfc_left_products_left_result_wbcoefficientscoefficientdiagonalterm pfc_right_products_left_result_wbcoefficientscoefficientdiagonalterm. (((pfc_index_products_left_result_wbcoefficientscoefficientdiagonal)+pfc_complement_products_left_result_wbcoefficientscoefficientdiagonalterm=(pfc_index_products_left_result_wbcoefficients)) /\ ((((((exists pfa_gap_products_left_result_wbcoefficientscoefficientdiagonaltermleftinside. pfa_gap_products_left_result_wbcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_products_left_result_wbcoefficientscoefficientdiagonal) = (M)) /\ ((((exists ff_h_pfp_products_left_result_wbcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_products_left_result_wbcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_products_left_result_wbcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_products_left_result_wbcoefficientscoefficientdiagonal)) * dc)) /\ exists ff_q_pfp_products_left_result_wbcoefficientscoefficientdiagonaltermleftentry. db = ff_q_pfp_products_left_result_wbcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_products_left_result_wbcoefficientscoefficientdiagonal)) * dc) + (pfc_left_products_left_result_wbcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_products_left_result_wbcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_products_left_result_wbcoefficientscoefficientdiagonaltermleftoutside+(M)=(pfc_index_products_left_result_wbcoefficientscoefficientdiagonal)) /\ (((pfc_left_products_left_result_wbcoefficientscoefficientdiagonalterm)=0))))) /\ ((((((exists pfa_gap_products_left_result_wbcoefficientscoefficientdiagonaltermrightinside. pfa_gap_products_left_result_wbcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_products_left_result_wbcoefficientscoefficientdiagonalterm) = (L)) /\ ((((exists ff_h_pfp_products_left_result_wbcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_products_left_result_wbcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_products_left_result_wbcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_products_left_result_wbcoefficientscoefficientdiagonalterm)) * cc)) /\ exists ff_q_pfp_products_left_result_wbcoefficientscoefficientdiagonaltermrightentry. cb = ff_q_pfp_products_left_result_wbcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_products_left_result_wbcoefficientscoefficientdiagonalterm)) * cc) + (pfc_right_products_left_result_wbcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_products_left_result_wbcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_products_left_result_wbcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_products_left_result_wbcoefficientscoefficientdiagonalterm)) /\ (((pfc_right_products_left_result_wbcoefficientscoefficientdiagonalterm)=0))))) /\ (((pfc_value_products_left_result_wbcoefficientscoefficientdiagonal)=pfc_left_products_left_result_wbcoefficientscoefficientdiagonalterm*pfc_right_products_left_result_wbcoefficientscoefficientdiagonalterm))))))))))) /\ (((exists fs_u_pfc_products_left_result_wbcoefficientscoefficientsum fs_v_pfc_products_left_result_wbcoefficientscoefficientsum. ((((exists fs_h_pfc_products_left_result_wbcoefficientscoefficientsum_body_start. fs_h_pfc_products_left_result_wbcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_products_left_result_wbcoefficientscoefficientsum)) /\ exists fs_q_pfc_products_left_result_wbcoefficientscoefficientsum_body_start. fs_u_pfc_products_left_result_wbcoefficientscoefficientsum = fs_q_pfc_products_left_result_wbcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_products_left_result_wbcoefficientscoefficientsum) + (0))) /\ ((((exists fs_h_pfc_products_left_result_wbcoefficientscoefficientsum_body_terminal. fs_h_pfc_products_left_result_wbcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_products_left_result_wbcoefficientscoefficient) = S ((S (S (pfc_index_products_left_result_wbcoefficients))) * fs_v_pfc_products_left_result_wbcoefficientscoefficientsum)) /\ exists fs_q_pfc_products_left_result_wbcoefficientscoefficientsum_body_terminal. fs_u_pfc_products_left_result_wbcoefficientscoefficientsum = fs_q_pfc_products_left_result_wbcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_products_left_result_wbcoefficients))) * fs_v_pfc_products_left_result_wbcoefficientscoefficientsum) + (pfc_natural_sum_products_left_result_wbcoefficientscoefficient))) /\ forall fs_i_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps = S (pfc_index_products_left_result_wbcoefficients)) -> exists fs_a_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps fs_r_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps fs_s_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_products_left_result_wbcoefficientscoefficient)) /\ exists fs_q_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_products_left_result_wbcoefficientscoefficient = fs_q_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_products_left_result_wbcoefficientscoefficient) + (fs_a_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_result_wbcoefficientscoefficientsum)) /\ exists fs_q_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_products_left_result_wbcoefficientscoefficientsum = fs_q_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_result_wbcoefficientscoefficientsum) + (fs_r_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_result_wbcoefficientscoefficientsum)) /\ exists fs_q_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_products_left_result_wbcoefficientscoefficientsum = fs_q_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_products_left_result_wbcoefficientscoefficientsum) + (fs_s_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps))) /\ fs_s_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps = fs_r_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps + fs_a_pfc_products_left_result_wbcoefficientscoefficientsum_body_steps)))))) /\ ((((exists pfa_gap_products_left_result_wbcoefficientscoefficientresiduebound. pfa_gap_products_left_result_wbcoefficientscoefficientresiduebound + S (pfc_value_products_left_result_wbcoefficients) = (p)) /\ ((exists pfa_offset_left_products_left_result_wbcoefficientscoefficientresiduecongruence pfa_offset_right_products_left_result_wbcoefficientscoefficientresiduecongruence. (pfc_natural_sum_products_left_result_wbcoefficientscoefficient) + (p) * pfa_offset_left_products_left_result_wbcoefficientscoefficientresiduecongruence = (pfc_value_products_left_result_wbcoefficients) + (p) * pfa_offset_right_products_left_result_wbcoefficientscoefficientresiduecongruence))))))))))))))))))) /\ ((forall pfp_index_products_left_addition. (exists pfa_gap_products_left_additionindex. pfa_gap_products_left_additionindex + S (pfp_index_products_left_addition) = (N)) -> exists pfp_left_products_left_addition pfp_right_products_left_addition pfp_value_products_left_addition. ((((exists ff_h_pfp_products_left_additionleft. ff_h_pfp_products_left_additionleft + S (pfp_left_products_left_addition) = S ((S (pfp_index_products_left_addition)) * uc)) /\ exists ff_q_pfp_products_left_additionleft. ub = ff_q_pfp_products_left_additionleft * S ((S (pfp_index_products_left_addition)) * uc) + (pfp_left_products_left_addition))) /\ (((((exists ff_h_pfp_products_left_additionright. ff_h_pfp_products_left_additionright + S (pfp_right_products_left_addition) = S ((S (pfp_index_products_left_addition)) * vc)) /\ exists ff_q_pfp_products_left_additionright. vb = ff_q_pfp_products_left_additionright * S ((S (pfp_index_products_left_addition)) * vc) + (pfp_right_products_left_addition))) /\ (((((exists ff_h_pfp_products_left_additiontarget. ff_h_pfp_products_left_additiontarget + S (pfp_value_products_left_addition) = S ((S (pfp_index_products_left_addition)) * wc)) /\ exists ff_q_pfp_products_left_additiontarget. wb = ff_q_pfp_products_left_additiontarget * S ((S (pfp_index_products_left_addition)) * wc) + (pfp_value_products_left_addition))) /\ ((((exists pfa_gap_products_left_additionoperationleft. pfa_gap_products_left_additionoperationleft + S (pfp_left_products_left_addition) = (p)) /\ (((exists pfa_gap_products_left_additionoperationright. pfa_gap_products_left_additionoperationright + S (pfp_right_products_left_addition) = (p)) /\ ((((exists pfa_gap_products_left_additionoperationresultbound. pfa_gap_products_left_additionoperationresultbound + S (pfp_value_products_left_addition) = (p)) /\ ((exists pfa_offset_left_products_left_additionoperationresultcongruence pfa_offset_right_products_left_additionoperationresultcongruence. ((pfp_left_products_left_addition) + (pfp_right_products_left_addition)) + (p) * pfa_offset_left_products_left_additionoperationresultcongruence = (pfp_value_products_left_addition) + (p) * pfa_offset_right_products_left_additionoperationresultcongruence)))))))))))))))))))))))
Complete tactic proof in conservative notation
All 116 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.
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.
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.