prime_field_polynomial_bezout_from_right_multiple

Exact expanded first-order arithmetic statement

forall p ab ac A bb bc B gb gc G. (~((p) = 1) /\ forall pfa_factor_left_terminal_prime pfa_factor_right_terminal_prime. (p) = pfa_factor_left_terminal_prime * pfa_factor_right_terminal_prime -> pfa_factor_left_terminal_prime = 1 \/ pfa_factor_right_terminal_prime = 1) -> (forall fom_index_pfp_terminal_other_bound. (exists fom_gap_pfp_terminal_other_bound_index_bound. fom_gap_pfp_terminal_other_bound_index_bound + S (fom_index_pfp_terminal_other_bound) = B) -> exists fom_value_pfp_terminal_other_bound. ((((exists fom_beta_height_pfp_terminal_other_bound_entry. fom_beta_height_pfp_terminal_other_bound_entry + S (fom_value_pfp_terminal_other_bound) = S ((S (fom_index_pfp_terminal_other_bound)) * bc)) /\ exists fom_beta_quotient_pfp_terminal_other_bound_entry. bb = fom_beta_quotient_pfp_terminal_other_bound_entry * S ((S (fom_index_pfp_terminal_other_bound)) * bc) + (fom_value_pfp_terminal_other_bound))) /\ (exists fom_gap_pfp_terminal_other_bound_value_bound. fom_gap_pfp_terminal_other_bound_value_bound + S (fom_value_pfp_terminal_other_bound) = p))) -> (((forall fom_index_pfp_terminal_input_canonical. (exists fom_gap_pfp_terminal_input_canonical_index_bound. fom_gap_pfp_terminal_input_canonical_index_bound + S (fom_index_pfp_terminal_input_canonical) = G) -> exists fom_value_pfp_terminal_input_canonical. ((((exists fom_beta_height_pfp_terminal_input_canonical_entry. fom_beta_height_pfp_terminal_input_canonical_entry + S (fom_value_pfp_terminal_input_canonical) = S ((S (fom_index_pfp_terminal_input_canonical)) * gc)) /\ exists fom_beta_quotient_pfp_terminal_input_canonical_entry. gb = fom_beta_quotient_pfp_terminal_input_canonical_entry * S ((S (fom_index_pfp_terminal_input_canonical)) * gc) + (fom_value_pfp_terminal_input_canonical))) /\ (exists fom_gap_pfp_terminal_input_canonical_value_bound. fom_gap_pfp_terminal_input_canonical_value_bound + S (fom_value_pfp_terminal_input_canonical) = p))) /\ ((exists pfrd_qb_terminal_input pfrd_qc_terminal_input pfrd_qlen_terminal_input pfrd_pb_terminal_input pfrd_pc_terminal_input pfrd_plen_terminal_input. ((((forall fom_index_pfp_terminal_input_productleft. (exists fom_gap_pfp_terminal_input_productleft_index_bound. fom_gap_pfp_terminal_input_productleft_index_bound + S (fom_index_pfp_terminal_input_productleft) = pfrd_qlen_terminal_input) -> exists fom_value_pfp_terminal_input_productleft. ((((exists fom_beta_height_pfp_terminal_input_productleft_entry. fom_beta_height_pfp_terminal_input_productleft_entry + S (fom_value_pfp_terminal_input_productleft) = S ((S (fom_index_pfp_terminal_input_productleft)) * pfrd_qc_terminal_input)) /\ exists fom_beta_quotient_pfp_terminal_input_productleft_entry. pfrd_qb_terminal_input = fom_beta_quotient_pfp_terminal_input_productleft_entry * S ((S (fom_index_pfp_terminal_input_productleft)) * pfrd_qc_terminal_input) + (fom_value_pfp_terminal_input_productleft))) /\ (exists fom_gap_pfp_terminal_input_productleft_value_bound. fom_gap_pfp_terminal_input_productleft_value_bound + S (fom_value_pfp_terminal_input_productleft) = p))) /\ (((forall fom_index_pfp_terminal_input_productright. (exists fom_gap_pfp_terminal_input_productright_index_bound. fom_gap_pfp_terminal_input_productright_index_bound + S (fom_index_pfp_terminal_input_productright) = A) -> exists fom_value_pfp_terminal_input_productright. ((((exists fom_beta_height_pfp_terminal_input_productright_entry. fom_beta_height_pfp_terminal_input_productright_entry + S (fom_value_pfp_terminal_input_productright) = S ((S (fom_index_pfp_terminal_input_productright)) * ac)) /\ exists fom_beta_quotient_pfp_terminal_input_productright_entry. ab = fom_beta_quotient_pfp_terminal_input_productright_entry * S ((S (fom_index_pfp_terminal_input_productright)) * ac) + (fom_value_pfp_terminal_input_productright))) /\ (exists fom_gap_pfp_terminal_input_productright_value_bound. fom_gap_pfp_terminal_input_productright_value_bound + S (fom_value_pfp_terminal_input_productright) = p))) /\ (((((((pfrd_qlen_terminal_input)=0 \/ (A)=0) /\ (((pfrd_plen_terminal_input)=0)))) \/ (((~((pfrd_qlen_terminal_input)=0)) /\ (((~((A)=0)) /\ (((pfrd_qlen_terminal_input)+(A)=S (pfrd_plen_terminal_input)))))))) /\ ((forall pfc_index_terminal_input_productcoefficients. (exists pfa_gap_terminal_input_productcoefficientsbound. pfa_gap_terminal_input_productcoefficientsbound + S (pfc_index_terminal_input_productcoefficients) = (pfrd_plen_terminal_input)) -> exists pfc_value_terminal_input_productcoefficients. ((((exists ff_h_pfp_terminal_input_productcoefficientsentry. ff_h_pfp_terminal_input_productcoefficientsentry + S (pfc_value_terminal_input_productcoefficients) = S ((S (pfc_index_terminal_input_productcoefficients)) * pfrd_pc_terminal_input)) /\ exists ff_q_pfp_terminal_input_productcoefficientsentry. pfrd_pb_terminal_input = ff_q_pfp_terminal_input_productcoefficientsentry * S ((S (pfc_index_terminal_input_productcoefficients)) * pfrd_pc_terminal_input) + (pfc_value_terminal_input_productcoefficients))) /\ ((exists pfc_terms_code_terminal_input_productcoefficientscoefficient pfc_terms_scale_terminal_input_productcoefficientscoefficient pfc_natural_sum_terminal_input_productcoefficientscoefficient. ((forall pfc_index_terminal_input_productcoefficientscoefficientdiagonal. (exists pfa_gap_terminal_input_productcoefficientscoefficientdiagonalbound. pfa_gap_terminal_input_productcoefficientscoefficientdiagonalbound + S (pfc_index_terminal_input_productcoefficientscoefficientdiagonal) = (S (pfc_index_terminal_input_productcoefficients))) -> exists pfc_value_terminal_input_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_terminal_input_productcoefficientscoefficientdiagonalentry. ff_h_pfp_terminal_input_productcoefficientscoefficientdiagonalentry + S (pfc_value_terminal_input_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_terminal_input_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_terminal_input_productcoefficientscoefficient)) /\ exists ff_q_pfp_terminal_input_productcoefficientscoefficientdiagonalentry. pfc_terms_code_terminal_input_productcoefficientscoefficient = ff_q_pfp_terminal_input_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_terminal_input_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_terminal_input_productcoefficientscoefficient) + (pfc_value_terminal_input_productcoefficientscoefficientdiagonal))) /\ ((exists pfc_complement_terminal_input_productcoefficientscoefficientdiagonalterm pfc_left_terminal_input_productcoefficientscoefficientdiagonalterm pfc_right_terminal_input_productcoefficientscoefficientdiagonalterm. (((pfc_index_terminal_input_productcoefficientscoefficientdiagonal)+pfc_complement_terminal_input_productcoefficientscoefficientdiagonalterm=(pfc_index_terminal_input_productcoefficients)) /\ ((((((exists pfa_gap_terminal_input_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_terminal_input_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_terminal_input_productcoefficientscoefficientdiagonal) = (pfrd_qlen_terminal_input)) /\ ((((exists ff_h_pfp_terminal_input_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_terminal_input_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_terminal_input_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_terminal_input_productcoefficientscoefficientdiagonal)) * pfrd_qc_terminal_input)) /\ exists ff_q_pfp_terminal_input_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_terminal_input = ff_q_pfp_terminal_input_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_terminal_input_productcoefficientscoefficientdiagonal)) * pfrd_qc_terminal_input) + (pfc_left_terminal_input_productcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_terminal_input_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_terminal_input_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_terminal_input)=(pfc_index_terminal_input_productcoefficientscoefficientdiagonal)) /\ (((pfc_left_terminal_input_productcoefficientscoefficientdiagonalterm)=0))))) /\ ((((((exists pfa_gap_terminal_input_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_terminal_input_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_terminal_input_productcoefficientscoefficientdiagonalterm) = (A)) /\ ((((exists ff_h_pfp_terminal_input_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_terminal_input_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_terminal_input_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_terminal_input_productcoefficientscoefficientdiagonalterm)) * ac)) /\ exists ff_q_pfp_terminal_input_productcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_terminal_input_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_terminal_input_productcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_terminal_input_productcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_terminal_input_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_terminal_input_productcoefficientscoefficientdiagonaltermrightoutside+(A)=(pfc_complement_terminal_input_productcoefficientscoefficientdiagonalterm)) /\ (((pfc_right_terminal_input_productcoefficientscoefficientdiagonalterm)=0))))) /\ (((pfc_value_terminal_input_productcoefficientscoefficientdiagonal)=pfc_left_terminal_input_productcoefficientscoefficientdiagonalterm*pfc_right_terminal_input_productcoefficientscoefficientdiagonalterm))))))))))) /\ (((exists fs_u_pfc_terminal_input_productcoefficientscoefficientsum fs_v_pfc_terminal_input_productcoefficientscoefficientsum. ((((exists fs_h_pfc_terminal_input_productcoefficientscoefficientsum_body_start. fs_h_pfc_terminal_input_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_terminal_input_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_terminal_input_productcoefficientscoefficientsum_body_start. fs_u_pfc_terminal_input_productcoefficientscoefficientsum = fs_q_pfc_terminal_input_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_terminal_input_productcoefficientscoefficientsum) + (0))) /\ ((((exists fs_h_pfc_terminal_input_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_terminal_input_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_terminal_input_productcoefficientscoefficient) = S ((S (S (pfc_index_terminal_input_productcoefficients))) * fs_v_pfc_terminal_input_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_terminal_input_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_terminal_input_productcoefficientscoefficientsum = fs_q_pfc_terminal_input_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_terminal_input_productcoefficients))) * fs_v_pfc_terminal_input_productcoefficientscoefficientsum) + (pfc_natural_sum_terminal_input_productcoefficientscoefficient))) /\ forall fs_i_pfc_terminal_input_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_terminal_input_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_terminal_input_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_terminal_input_productcoefficientscoefficientsum_body_steps = S (pfc_index_terminal_input_productcoefficients)) -> exists fs_a_pfc_terminal_input_productcoefficientscoefficientsum_body_steps fs_r_pfc_terminal_input_productcoefficientscoefficientsum_body_steps fs_s_pfc_terminal_input_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_terminal_input_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_terminal_input_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_terminal_input_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_terminal_input_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_terminal_input_productcoefficientscoefficient)) /\ exists fs_q_pfc_terminal_input_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_terminal_input_productcoefficientscoefficient = fs_q_pfc_terminal_input_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_terminal_input_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_terminal_input_productcoefficientscoefficient) + (fs_a_pfc_terminal_input_productcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_terminal_input_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_terminal_input_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_terminal_input_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_terminal_input_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_terminal_input_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_terminal_input_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_terminal_input_productcoefficientscoefficientsum = fs_q_pfc_terminal_input_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_terminal_input_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_terminal_input_productcoefficientscoefficientsum) + (fs_r_pfc_terminal_input_productcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_terminal_input_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_terminal_input_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_terminal_input_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_terminal_input_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_terminal_input_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_terminal_input_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_terminal_input_productcoefficientscoefficientsum = fs_q_pfc_terminal_input_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_terminal_input_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_terminal_input_productcoefficientscoefficientsum) + (fs_s_pfc_terminal_input_productcoefficientscoefficientsum_body_steps))) /\ fs_s_pfc_terminal_input_productcoefficientscoefficientsum_body_steps = fs_r_pfc_terminal_input_productcoefficientscoefficientsum_body_steps + fs_a_pfc_terminal_input_productcoefficientscoefficientsum_body_steps)))))) /\ ((((exists pfa_gap_terminal_input_productcoefficientscoefficientresiduebound. pfa_gap_terminal_input_productcoefficientscoefficientresiduebound + S (pfc_value_terminal_input_productcoefficients) = (p)) /\ ((exists pfa_offset_left_terminal_input_productcoefficientscoefficientresiduecongruence pfa_offset_right_terminal_input_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_terminal_input_productcoefficientscoefficient) + (p) * pfa_offset_left_terminal_input_productcoefficientscoefficientresiduecongruence = (pfc_value_terminal_input_productcoefficients) + (p) * pfa_offset_right_terminal_input_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\ ((forall pfrep_power_terminal_input_target pfrep_left_terminal_input_target pfrep_right_terminal_input_target. ((exists pfrep_position_terminal_input_targetfirst. ((pfrep_position_terminal_input_targetfirst+S (pfrep_power_terminal_input_target)=(pfrd_plen_terminal_input)) /\ ((((exists ff_h_pfp_terminal_input_targetfirstentry. ff_h_pfp_terminal_input_targetfirstentry + S (pfrep_left_terminal_input_target) = S ((S (pfrep_position_terminal_input_targetfirst)) * pfrd_pc_terminal_input)) /\ exists ff_q_pfp_terminal_input_targetfirstentry. pfrd_pb_terminal_input = ff_q_pfp_terminal_input_targetfirstentry * S ((S (pfrep_position_terminal_input_targetfirst)) * pfrd_pc_terminal_input) + (pfrep_left_terminal_input_target)))))) \/ (((exists pfrep_gap_terminal_input_targetfirstoutside. pfrep_gap_terminal_input_targetfirstoutside+(pfrd_plen_terminal_input)=(pfrep_power_terminal_input_target)) /\ (((pfrep_left_terminal_input_target)=0))))) -> ((exists pfrep_position_terminal_input_targetsecond. ((pfrep_position_terminal_input_targetsecond+S (pfrep_power_terminal_input_target)=(G)) /\ ((((exists ff_h_pfp_terminal_input_targetsecondentry. ff_h_pfp_terminal_input_targetsecondentry + S (pfrep_right_terminal_input_target) = S ((S (pfrep_position_terminal_input_targetsecond)) * gc)) /\ exists ff_q_pfp_terminal_input_targetsecondentry. gb = ff_q_pfp_terminal_input_targetsecondentry * S ((S (pfrep_position_terminal_input_targetsecond)) * gc) + (pfrep_right_terminal_input_target)))))) \/ (((exists pfrep_gap_terminal_input_targetsecondoutside. pfrep_gap_terminal_input_targetsecondoutside+(G)=(pfrep_power_terminal_input_target)) /\ (((pfrep_right_terminal_input_target)=0))))) -> pfrep_left_terminal_input_target=pfrep_right_terminal_input_target))))))) -> (exists ub uc U. exists pfbz_left_code_terminal_result pfbz_left_scale_terminal_result pfbz_left_length_terminal_result pfbz_right_code_terminal_result pfbz_right_scale_terminal_result pfbz_right_length_terminal_result. ((((forall fom_index_pfp_terminal_result_left_productleft. (exists fom_gap_pfp_terminal_result_left_productleft_index_bound. fom_gap_pfp_terminal_result_left_productleft_index_bound + S (fom_index_pfp_terminal_result_left_productleft) = U) -> exists fom_value_pfp_terminal_result_left_productleft. ((((exists fom_beta_height_pfp_terminal_result_left_productleft_entry. fom_beta_height_pfp_terminal_result_left_productleft_entry + S (fom_value_pfp_terminal_result_left_productleft) = S ((S (fom_index_pfp_terminal_result_left_productleft)) * uc)) /\ exists fom_beta_quotient_pfp_terminal_result_left_productleft_entry. ub = fom_beta_quotient_pfp_terminal_result_left_productleft_entry * S ((S (fom_index_pfp_terminal_result_left_productleft)) * uc) + (fom_value_pfp_terminal_result_left_productleft))) /\ (exists fom_gap_pfp_terminal_result_left_productleft_value_bound. fom_gap_pfp_terminal_result_left_productleft_value_bound + S (fom_value_pfp_terminal_result_left_productleft) = p))) /\ (((forall fom_index_pfp_terminal_result_left_productright. (exists fom_gap_pfp_terminal_result_left_productright_index_bound. fom_gap_pfp_terminal_result_left_productright_index_bound + S (fom_index_pfp_terminal_result_left_productright) = A) -> exists fom_value_pfp_terminal_result_left_productright. ((((exists fom_beta_height_pfp_terminal_result_left_productright_entry. fom_beta_height_pfp_terminal_result_left_productright_entry + S (fom_value_pfp_terminal_result_left_productright) = S ((S (fom_index_pfp_terminal_result_left_productright)) * ac)) /\ exists fom_beta_quotient_pfp_terminal_result_left_productright_entry. ab = fom_beta_quotient_pfp_terminal_result_left_productright_entry * S ((S (fom_index_pfp_terminal_result_left_productright)) * ac) + (fom_value_pfp_terminal_result_left_productright))) /\ (exists fom_gap_pfp_terminal_result_left_productright_value_bound. fom_gap_pfp_terminal_result_left_productright_value_bound + S (fom_value_pfp_terminal_result_left_productright) = p))) /\ (((((((U)=0 \/ (A)=0) /\ (((pfbz_left_length_terminal_result)=0)))) \/ (((~((U)=0)) /\ (((~((A)=0)) /\ (((U)+(A)=S (pfbz_left_length_terminal_result)))))))) /\ ((forall pfc_index_terminal_result_left_productcoefficients. (exists pfa_gap_terminal_result_left_productcoefficientsbound. pfa_gap_terminal_result_left_productcoefficientsbound + S (pfc_index_terminal_result_left_productcoefficients) = (pfbz_left_length_terminal_result)) -> exists pfc_value_terminal_result_left_productcoefficients. ((((exists ff_h_pfp_terminal_result_left_productcoefficientsentry. ff_h_pfp_terminal_result_left_productcoefficientsentry + S (pfc_value_terminal_result_left_productcoefficients) = S ((S (pfc_index_terminal_result_left_productcoefficients)) * pfbz_left_scale_terminal_result)) /\ exists ff_q_pfp_terminal_result_left_productcoefficientsentry. pfbz_left_code_terminal_result = ff_q_pfp_terminal_result_left_productcoefficientsentry * S ((S (pfc_index_terminal_result_left_productcoefficients)) * pfbz_left_scale_terminal_result) + (pfc_value_terminal_result_left_productcoefficients))) /\ ((exists pfc_terms_code_terminal_result_left_productcoefficientscoefficient pfc_terms_scale_terminal_result_left_productcoefficientscoefficient pfc_natural_sum_terminal_result_left_productcoefficientscoefficient. ((forall pfc_index_terminal_result_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_terminal_result_left_productcoefficientscoefficientdiagonalbound. pfa_gap_terminal_result_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_terminal_result_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_terminal_result_left_productcoefficients))) -> exists pfc_value_terminal_result_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_terminal_result_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_terminal_result_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_terminal_result_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_terminal_result_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_terminal_result_left_productcoefficientscoefficient)) /\ exists ff_q_pfp_terminal_result_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_terminal_result_left_productcoefficientscoefficient = ff_q_pfp_terminal_result_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_terminal_result_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_terminal_result_left_productcoefficientscoefficient) + (pfc_value_terminal_result_left_productcoefficientscoefficientdiagonal))) /\ ((exists pfc_complement_terminal_result_left_productcoefficientscoefficientdiagonalterm pfc_left_terminal_result_left_productcoefficientscoefficientdiagonalterm pfc_right_terminal_result_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_terminal_result_left_productcoefficientscoefficientdiagonal)+pfc_complement_terminal_result_left_productcoefficientscoefficientdiagonalterm=(pfc_index_terminal_result_left_productcoefficients)) /\ ((((((exists pfa_gap_terminal_result_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_terminal_result_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_terminal_result_left_productcoefficientscoefficientdiagonal) = (U)) /\ ((((exists ff_h_pfp_terminal_result_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_terminal_result_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_terminal_result_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_terminal_result_left_productcoefficientscoefficientdiagonal)) * uc)) /\ exists ff_q_pfp_terminal_result_left_productcoefficientscoefficientdiagonaltermleftentry. ub = ff_q_pfp_terminal_result_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_terminal_result_left_productcoefficientscoefficientdiagonal)) * uc) + (pfc_left_terminal_result_left_productcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_terminal_result_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_terminal_result_left_productcoefficientscoefficientdiagonaltermleftoutside+(U)=(pfc_index_terminal_result_left_productcoefficientscoefficientdiagonal)) /\ (((pfc_left_terminal_result_left_productcoefficientscoefficientdiagonalterm)=0))))) /\ ((((((exists pfa_gap_terminal_result_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_terminal_result_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_terminal_result_left_productcoefficientscoefficientdiagonalterm) = (A)) /\ ((((exists ff_h_pfp_terminal_result_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_terminal_result_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_terminal_result_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_terminal_result_left_productcoefficientscoefficientdiagonalterm)) * ac)) /\ exists ff_q_pfp_terminal_result_left_productcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_terminal_result_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_terminal_result_left_productcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_terminal_result_left_productcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_terminal_result_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_terminal_result_left_productcoefficientscoefficientdiagonaltermrightoutside+(A)=(pfc_complement_terminal_result_left_productcoefficientscoefficientdiagonalterm)) /\ (((pfc_right_terminal_result_left_productcoefficientscoefficientdiagonalterm)=0))))) /\ (((pfc_value_terminal_result_left_productcoefficientscoefficientdiagonal)=pfc_left_terminal_result_left_productcoefficientscoefficientdiagonalterm*pfc_right_terminal_result_left_productcoefficientscoefficientdiagonalterm))))))))))) /\ (((exists fs_u_pfc_terminal_result_left_productcoefficientscoefficientsum fs_v_pfc_terminal_result_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_terminal_result_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_terminal_result_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_terminal_result_left_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_terminal_result_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_terminal_result_left_productcoefficientscoefficientsum = fs_q_pfc_terminal_result_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_terminal_result_left_productcoefficientscoefficientsum) + (0))) /\ ((((exists fs_h_pfc_terminal_result_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_terminal_result_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_terminal_result_left_productcoefficientscoefficient) = S ((S (S (pfc_index_terminal_result_left_productcoefficients))) * fs_v_pfc_terminal_result_left_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_terminal_result_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_terminal_result_left_productcoefficientscoefficientsum = fs_q_pfc_terminal_result_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_terminal_result_left_productcoefficients))) * fs_v_pfc_terminal_result_left_productcoefficientscoefficientsum) + (pfc_natural_sum_terminal_result_left_productcoefficientscoefficient))) /\ forall fs_i_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_terminal_result_left_productcoefficients)) -> exists fs_a_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_terminal_result_left_productcoefficientscoefficient)) /\ exists fs_q_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_terminal_result_left_productcoefficientscoefficient = fs_q_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_terminal_result_left_productcoefficientscoefficient) + (fs_a_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_terminal_result_left_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_terminal_result_left_productcoefficientscoefficientsum = fs_q_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_terminal_result_left_productcoefficientscoefficientsum) + (fs_r_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_terminal_result_left_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_terminal_result_left_productcoefficientscoefficientsum = fs_q_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_terminal_result_left_productcoefficientscoefficientsum) + (fs_s_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps))) /\ fs_s_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps)))))) /\ ((((exists pfa_gap_terminal_result_left_productcoefficientscoefficientresiduebound. pfa_gap_terminal_result_left_productcoefficientscoefficientresiduebound + S (pfc_value_terminal_result_left_productcoefficients) = (p)) /\ ((exists pfa_offset_left_terminal_result_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_terminal_result_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_terminal_result_left_productcoefficientscoefficient) + (p) * pfa_offset_left_terminal_result_left_productcoefficientscoefficientresiduecongruence = (pfc_value_terminal_result_left_productcoefficients) + (p) * pfa_offset_right_terminal_result_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\ (((((forall fom_index_pfp_terminal_result_right_productleft. (exists fom_gap_pfp_terminal_result_right_productleft_index_bound. fom_gap_pfp_terminal_result_right_productleft_index_bound + S (fom_index_pfp_terminal_result_right_productleft) = 0) -> exists fom_value_pfp_terminal_result_right_productleft. ((((exists fom_beta_height_pfp_terminal_result_right_productleft_entry. fom_beta_height_pfp_terminal_result_right_productleft_entry + S (fom_value_pfp_terminal_result_right_productleft) = S ((S (fom_index_pfp_terminal_result_right_productleft)) * 0)) /\ exists fom_beta_quotient_pfp_terminal_result_right_productleft_entry. 0 = fom_beta_quotient_pfp_terminal_result_right_productleft_entry * S ((S (fom_index_pfp_terminal_result_right_productleft)) * 0) + (fom_value_pfp_terminal_result_right_productleft))) /\ (exists fom_gap_pfp_terminal_result_right_productleft_value_bound. fom_gap_pfp_terminal_result_right_productleft_value_bound + S (fom_value_pfp_terminal_result_right_productleft) = p))) /\ (((forall fom_index_pfp_terminal_result_right_productright. (exists fom_gap_pfp_terminal_result_right_productright_index_bound. fom_gap_pfp_terminal_result_right_productright_index_bound + S (fom_index_pfp_terminal_result_right_productright) = B) -> exists fom_value_pfp_terminal_result_right_productright. ((((exists fom_beta_height_pfp_terminal_result_right_productright_entry. fom_beta_height_pfp_terminal_result_right_productright_entry + S (fom_value_pfp_terminal_result_right_productright) = S ((S (fom_index_pfp_terminal_result_right_productright)) * bc)) /\ exists fom_beta_quotient_pfp_terminal_result_right_productright_entry. bb = fom_beta_quotient_pfp_terminal_result_right_productright_entry * S ((S (fom_index_pfp_terminal_result_right_productright)) * bc) + (fom_value_pfp_terminal_result_right_productright))) /\ (exists fom_gap_pfp_terminal_result_right_productright_value_bound. fom_gap_pfp_terminal_result_right_productright_value_bound + S (fom_value_pfp_terminal_result_right_productright) = p))) /\ (((((((0)=0 \/ (B)=0) /\ (((pfbz_right_length_terminal_result)=0)))) \/ (((~((0)=0)) /\ (((~((B)=0)) /\ (((0)+(B)=S (pfbz_right_length_terminal_result)))))))) /\ ((forall pfc_index_terminal_result_right_productcoefficients. (exists pfa_gap_terminal_result_right_productcoefficientsbound. pfa_gap_terminal_result_right_productcoefficientsbound + S (pfc_index_terminal_result_right_productcoefficients) = (pfbz_right_length_terminal_result)) -> exists pfc_value_terminal_result_right_productcoefficients. ((((exists ff_h_pfp_terminal_result_right_productcoefficientsentry. ff_h_pfp_terminal_result_right_productcoefficientsentry + S (pfc_value_terminal_result_right_productcoefficients) = S ((S (pfc_index_terminal_result_right_productcoefficients)) * pfbz_right_scale_terminal_result)) /\ exists ff_q_pfp_terminal_result_right_productcoefficientsentry. pfbz_right_code_terminal_result = ff_q_pfp_terminal_result_right_productcoefficientsentry * S ((S (pfc_index_terminal_result_right_productcoefficients)) * pfbz_right_scale_terminal_result) + (pfc_value_terminal_result_right_productcoefficients))) /\ ((exists pfc_terms_code_terminal_result_right_productcoefficientscoefficient pfc_terms_scale_terminal_result_right_productcoefficientscoefficient pfc_natural_sum_terminal_result_right_productcoefficientscoefficient. ((forall pfc_index_terminal_result_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_terminal_result_right_productcoefficientscoefficientdiagonalbound. pfa_gap_terminal_result_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_terminal_result_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_terminal_result_right_productcoefficients))) -> exists pfc_value_terminal_result_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_terminal_result_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_terminal_result_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_terminal_result_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_terminal_result_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_terminal_result_right_productcoefficientscoefficient)) /\ exists ff_q_pfp_terminal_result_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_terminal_result_right_productcoefficientscoefficient = ff_q_pfp_terminal_result_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_terminal_result_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_terminal_result_right_productcoefficientscoefficient) + (pfc_value_terminal_result_right_productcoefficientscoefficientdiagonal))) /\ ((exists pfc_complement_terminal_result_right_productcoefficientscoefficientdiagonalterm pfc_left_terminal_result_right_productcoefficientscoefficientdiagonalterm pfc_right_terminal_result_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_terminal_result_right_productcoefficientscoefficientdiagonal)+pfc_complement_terminal_result_right_productcoefficientscoefficientdiagonalterm=(pfc_index_terminal_result_right_productcoefficients)) /\ ((((((exists pfa_gap_terminal_result_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_terminal_result_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_terminal_result_right_productcoefficientscoefficientdiagonal) = (0)) /\ ((((exists ff_h_pfp_terminal_result_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_terminal_result_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_terminal_result_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_terminal_result_right_productcoefficientscoefficientdiagonal)) * 0)) /\ exists ff_q_pfp_terminal_result_right_productcoefficientscoefficientdiagonaltermleftentry. 0 = ff_q_pfp_terminal_result_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_terminal_result_right_productcoefficientscoefficientdiagonal)) * 0) + (pfc_left_terminal_result_right_productcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_terminal_result_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_terminal_result_right_productcoefficientscoefficientdiagonaltermleftoutside+(0)=(pfc_index_terminal_result_right_productcoefficientscoefficientdiagonal)) /\ (((pfc_left_terminal_result_right_productcoefficientscoefficientdiagonalterm)=0))))) /\ ((((((exists pfa_gap_terminal_result_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_terminal_result_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_terminal_result_right_productcoefficientscoefficientdiagonalterm) = (B)) /\ ((((exists ff_h_pfp_terminal_result_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_terminal_result_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_terminal_result_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_terminal_result_right_productcoefficientscoefficientdiagonalterm)) * bc)) /\ exists ff_q_pfp_terminal_result_right_productcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_terminal_result_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_terminal_result_right_productcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_terminal_result_right_productcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_terminal_result_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_terminal_result_right_productcoefficientscoefficientdiagonaltermrightoutside+(B)=(pfc_complement_terminal_result_right_productcoefficientscoefficientdiagonalterm)) /\ (((pfc_right_terminal_result_right_productcoefficientscoefficientdiagonalterm)=0))))) /\ (((pfc_value_terminal_result_right_productcoefficientscoefficientdiagonal)=pfc_left_terminal_result_right_productcoefficientscoefficientdiagonalterm*pfc_right_terminal_result_right_productcoefficientscoefficientdiagonalterm))))))))))) /\ (((exists fs_u_pfc_terminal_result_right_productcoefficientscoefficientsum fs_v_pfc_terminal_result_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_terminal_result_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_terminal_result_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_terminal_result_right_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_terminal_result_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_terminal_result_right_productcoefficientscoefficientsum = fs_q_pfc_terminal_result_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_terminal_result_right_productcoefficientscoefficientsum) + (0))) /\ ((((exists fs_h_pfc_terminal_result_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_terminal_result_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_terminal_result_right_productcoefficientscoefficient) = S ((S (S (pfc_index_terminal_result_right_productcoefficients))) * fs_v_pfc_terminal_result_right_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_terminal_result_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_terminal_result_right_productcoefficientscoefficientsum = fs_q_pfc_terminal_result_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_terminal_result_right_productcoefficients))) * fs_v_pfc_terminal_result_right_productcoefficientscoefficientsum) + (pfc_natural_sum_terminal_result_right_productcoefficientscoefficient))) /\ forall fs_i_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_terminal_result_right_productcoefficients)) -> exists fs_a_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_terminal_result_right_productcoefficientscoefficient)) /\ exists fs_q_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_terminal_result_right_productcoefficientscoefficient = fs_q_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_terminal_result_right_productcoefficientscoefficient) + (fs_a_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_terminal_result_right_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_terminal_result_right_productcoefficientscoefficientsum = fs_q_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_terminal_result_right_productcoefficientscoefficientsum) + (fs_r_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_terminal_result_right_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_terminal_result_right_productcoefficientscoefficientsum = fs_q_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_terminal_result_right_productcoefficientscoefficientsum) + (fs_s_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps))) /\ fs_s_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps)))))) /\ ((((exists pfa_gap_terminal_result_right_productcoefficientscoefficientresiduebound. pfa_gap_terminal_result_right_productcoefficientscoefficientresiduebound + S (pfc_value_terminal_result_right_productcoefficients) = (p)) /\ ((exists pfa_offset_left_terminal_result_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_terminal_result_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_terminal_result_right_productcoefficientscoefficient) + (p) * pfa_offset_left_terminal_result_right_productcoefficientscoefficientresiduecongruence = (pfc_value_terminal_result_right_productcoefficients) + (p) * pfa_offset_right_terminal_result_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\ ((((forall fom_index_pfp_terminal_result_sum_left_bounded. (exists fom_gap_pfp_terminal_result_sum_left_bounded_index_bound. fom_gap_pfp_terminal_result_sum_left_bounded_index_bound + S (fom_index_pfp_terminal_result_sum_left_bounded) = pfbz_left_length_terminal_result) -> exists fom_value_pfp_terminal_result_sum_left_bounded. ((((exists fom_beta_height_pfp_terminal_result_sum_left_bounded_entry. fom_beta_height_pfp_terminal_result_sum_left_bounded_entry + S (fom_value_pfp_terminal_result_sum_left_bounded) = S ((S (fom_index_pfp_terminal_result_sum_left_bounded)) * pfbz_left_scale_terminal_result)) /\ exists fom_beta_quotient_pfp_terminal_result_sum_left_bounded_entry. pfbz_left_code_terminal_result = fom_beta_quotient_pfp_terminal_result_sum_left_bounded_entry * S ((S (fom_index_pfp_terminal_result_sum_left_bounded)) * pfbz_left_scale_terminal_result) + (fom_value_pfp_terminal_result_sum_left_bounded))) /\ (exists fom_gap_pfp_terminal_result_sum_left_bounded_value_bound. fom_gap_pfp_terminal_result_sum_left_bounded_value_bound + S (fom_value_pfp_terminal_result_sum_left_bounded) = p))) /\ (((forall fom_index_pfp_terminal_result_sum_right_bounded. (exists fom_gap_pfp_terminal_result_sum_right_bounded_index_bound. fom_gap_pfp_terminal_result_sum_right_bounded_index_bound + S (fom_index_pfp_terminal_result_sum_right_bounded) = pfbz_right_length_terminal_result) -> exists fom_value_pfp_terminal_result_sum_right_bounded. ((((exists fom_beta_height_pfp_terminal_result_sum_right_bounded_entry. fom_beta_height_pfp_terminal_result_sum_right_bounded_entry + S (fom_value_pfp_terminal_result_sum_right_bounded) = S ((S (fom_index_pfp_terminal_result_sum_right_bounded)) * pfbz_right_scale_terminal_result)) /\ exists fom_beta_quotient_pfp_terminal_result_sum_right_bounded_entry. pfbz_right_code_terminal_result = fom_beta_quotient_pfp_terminal_result_sum_right_bounded_entry * S ((S (fom_index_pfp_terminal_result_sum_right_bounded)) * pfbz_right_scale_terminal_result) + (fom_value_pfp_terminal_result_sum_right_bounded))) /\ (exists fom_gap_pfp_terminal_result_sum_right_bounded_value_bound. fom_gap_pfp_terminal_result_sum_right_bounded_value_bound + S (fom_value_pfp_terminal_result_sum_right_bounded) = p))) /\ (((forall fom_index_pfp_terminal_result_sum_result_bounded. (exists fom_gap_pfp_terminal_result_sum_result_bounded_index_bound. fom_gap_pfp_terminal_result_sum_result_bounded_index_bound + S (fom_index_pfp_terminal_result_sum_result_bounded) = G) -> exists fom_value_pfp_terminal_result_sum_result_bounded. ((((exists fom_beta_height_pfp_terminal_result_sum_result_bounded_entry. fom_beta_height_pfp_terminal_result_sum_result_bounded_entry + S (fom_value_pfp_terminal_result_sum_result_bounded) = S ((S (fom_index_pfp_terminal_result_sum_result_bounded)) * gc)) /\ exists fom_beta_quotient_pfp_terminal_result_sum_result_bounded_entry. gb = fom_beta_quotient_pfp_terminal_result_sum_result_bounded_entry * S ((S (fom_index_pfp_terminal_result_sum_result_bounded)) * gc) + (fom_value_pfp_terminal_result_sum_result_bounded))) /\ (exists fom_gap_pfp_terminal_result_sum_result_bounded_value_bound. fom_gap_pfp_terminal_result_sum_result_bounded_value_bound + S (fom_value_pfp_terminal_result_sum_result_bounded) = p))) /\ ((exists pfaa_left_b_terminal_result_sum pfaa_left_c_terminal_result_sum pfaa_right_b_terminal_result_sum pfaa_right_c_terminal_result_sum pfaa_sum_b_terminal_result_sum pfaa_sum_c_terminal_result_sum pfaa_length_terminal_result_sum. ((((forall pfrep_power_terminal_result_sum_witness_common_left pfrep_left_terminal_result_sum_witness_common_left pfrep_right_terminal_result_sum_witness_common_left. ((exists pfrep_position_terminal_result_sum_witness_common_leftfirst. ((pfrep_position_terminal_result_sum_witness_common_leftfirst+S (pfrep_power_terminal_result_sum_witness_common_left)=(pfbz_left_length_terminal_result)) /\ ((((exists ff_h_pfp_terminal_result_sum_witness_common_leftfirstentry. ff_h_pfp_terminal_result_sum_witness_common_leftfirstentry + S (pfrep_left_terminal_result_sum_witness_common_left) = S ((S (pfrep_position_terminal_result_sum_witness_common_leftfirst)) * pfbz_left_scale_terminal_result)) /\ exists ff_q_pfp_terminal_result_sum_witness_common_leftfirstentry. pfbz_left_code_terminal_result = ff_q_pfp_terminal_result_sum_witness_common_leftfirstentry * S ((S (pfrep_position_terminal_result_sum_witness_common_leftfirst)) * pfbz_left_scale_terminal_result) + (pfrep_left_terminal_result_sum_witness_common_left)))))) \/ (((exists pfrep_gap_terminal_result_sum_witness_common_leftfirstoutside. pfrep_gap_terminal_result_sum_witness_common_leftfirstoutside+(pfbz_left_length_terminal_result)=(pfrep_power_terminal_result_sum_witness_common_left)) /\ (((pfrep_left_terminal_result_sum_witness_common_left)=0))))) -> ((exists pfrep_position_terminal_result_sum_witness_common_leftsecond. ((pfrep_position_terminal_result_sum_witness_common_leftsecond+S (pfrep_power_terminal_result_sum_witness_common_left)=(pfaa_length_terminal_result_sum)) /\ ((((exists ff_h_pfp_terminal_result_sum_witness_common_leftsecondentry. ff_h_pfp_terminal_result_sum_witness_common_leftsecondentry + S (pfrep_right_terminal_result_sum_witness_common_left) = S ((S (pfrep_position_terminal_result_sum_witness_common_leftsecond)) * pfaa_left_c_terminal_result_sum)) /\ exists ff_q_pfp_terminal_result_sum_witness_common_leftsecondentry. pfaa_left_b_terminal_result_sum = ff_q_pfp_terminal_result_sum_witness_common_leftsecondentry * S ((S (pfrep_position_terminal_result_sum_witness_common_leftsecond)) * pfaa_left_c_terminal_result_sum) + (pfrep_right_terminal_result_sum_witness_common_left)))))) \/ (((exists pfrep_gap_terminal_result_sum_witness_common_leftsecondoutside. pfrep_gap_terminal_result_sum_witness_common_leftsecondoutside+(pfaa_length_terminal_result_sum)=(pfrep_power_terminal_result_sum_witness_common_left)) /\ (((pfrep_right_terminal_result_sum_witness_common_left)=0))))) -> pfrep_left_terminal_result_sum_witness_common_left=pfrep_right_terminal_result_sum_witness_common_left) /\ ((forall pfrep_power_terminal_result_sum_witness_common_right pfrep_left_terminal_result_sum_witness_common_right pfrep_right_terminal_result_sum_witness_common_right. ((exists pfrep_position_terminal_result_sum_witness_common_rightfirst. ((pfrep_position_terminal_result_sum_witness_common_rightfirst+S (pfrep_power_terminal_result_sum_witness_common_right)=(pfbz_right_length_terminal_result)) /\ ((((exists ff_h_pfp_terminal_result_sum_witness_common_rightfirstentry. ff_h_pfp_terminal_result_sum_witness_common_rightfirstentry + S (pfrep_left_terminal_result_sum_witness_common_right) = S ((S (pfrep_position_terminal_result_sum_witness_common_rightfirst)) * pfbz_right_scale_terminal_result)) /\ exists ff_q_pfp_terminal_result_sum_witness_common_rightfirstentry. pfbz_right_code_terminal_result = ff_q_pfp_terminal_result_sum_witness_common_rightfirstentry * S ((S (pfrep_position_terminal_result_sum_witness_common_rightfirst)) * pfbz_right_scale_terminal_result) + (pfrep_left_terminal_result_sum_witness_common_right)))))) \/ (((exists pfrep_gap_terminal_result_sum_witness_common_rightfirstoutside. pfrep_gap_terminal_result_sum_witness_common_rightfirstoutside+(pfbz_right_length_terminal_result)=(pfrep_power_terminal_result_sum_witness_common_right)) /\ (((pfrep_left_terminal_result_sum_witness_common_right)=0))))) -> ((exists pfrep_position_terminal_result_sum_witness_common_rightsecond. ((pfrep_position_terminal_result_sum_witness_common_rightsecond+S (pfrep_power_terminal_result_sum_witness_common_right)=(pfaa_length_terminal_result_sum)) /\ ((((exists ff_h_pfp_terminal_result_sum_witness_common_rightsecondentry. ff_h_pfp_terminal_result_sum_witness_common_rightsecondentry + S (pfrep_right_terminal_result_sum_witness_common_right) = S ((S (pfrep_position_terminal_result_sum_witness_common_rightsecond)) * pfaa_right_c_terminal_result_sum)) /\ exists ff_q_pfp_terminal_result_sum_witness_common_rightsecondentry. pfaa_right_b_terminal_result_sum = ff_q_pfp_terminal_result_sum_witness_common_rightsecondentry * S ((S (pfrep_position_terminal_result_sum_witness_common_rightsecond)) * pfaa_right_c_terminal_result_sum) + (pfrep_right_terminal_result_sum_witness_common_right)))))) \/ (((exists pfrep_gap_terminal_result_sum_witness_common_rightsecondoutside. pfrep_gap_terminal_result_sum_witness_common_rightsecondoutside+(pfaa_length_terminal_result_sum)=(pfrep_power_terminal_result_sum_witness_common_right)) /\ (((pfrep_right_terminal_result_sum_witness_common_right)=0))))) -> pfrep_left_terminal_result_sum_witness_common_right=pfrep_right_terminal_result_sum_witness_common_right)))) /\ (((forall pfp_index_terminal_result_sum_witness_operation. (exists pfa_gap_terminal_result_sum_witness_operationindex. pfa_gap_terminal_result_sum_witness_operationindex + S (pfp_index_terminal_result_sum_witness_operation) = (pfaa_length_terminal_result_sum)) -> exists pfp_left_terminal_result_sum_witness_operation pfp_right_terminal_result_sum_witness_operation pfp_value_terminal_result_sum_witness_operation. ((((exists ff_h_pfp_terminal_result_sum_witness_operationleft. ff_h_pfp_terminal_result_sum_witness_operationleft + S (pfp_left_terminal_result_sum_witness_operation) = S ((S (pfp_index_terminal_result_sum_witness_operation)) * pfaa_left_c_terminal_result_sum)) /\ exists ff_q_pfp_terminal_result_sum_witness_operationleft. pfaa_left_b_terminal_result_sum = ff_q_pfp_terminal_result_sum_witness_operationleft * S ((S (pfp_index_terminal_result_sum_witness_operation)) * pfaa_left_c_terminal_result_sum) + (pfp_left_terminal_result_sum_witness_operation))) /\ (((((exists ff_h_pfp_terminal_result_sum_witness_operationright. ff_h_pfp_terminal_result_sum_witness_operationright + S (pfp_right_terminal_result_sum_witness_operation) = S ((S (pfp_index_terminal_result_sum_witness_operation)) * pfaa_right_c_terminal_result_sum)) /\ exists ff_q_pfp_terminal_result_sum_witness_operationright. pfaa_right_b_terminal_result_sum = ff_q_pfp_terminal_result_sum_witness_operationright * S ((S (pfp_index_terminal_result_sum_witness_operation)) * pfaa_right_c_terminal_result_sum) + (pfp_right_terminal_result_sum_witness_operation))) /\ (((((exists ff_h_pfp_terminal_result_sum_witness_operationtarget. ff_h_pfp_terminal_result_sum_witness_operationtarget + S (pfp_value_terminal_result_sum_witness_operation) = S ((S (pfp_index_terminal_result_sum_witness_operation)) * pfaa_sum_c_terminal_result_sum)) /\ exists ff_q_pfp_terminal_result_sum_witness_operationtarget. pfaa_sum_b_terminal_result_sum = ff_q_pfp_terminal_result_sum_witness_operationtarget * S ((S (pfp_index_terminal_result_sum_witness_operation)) * pfaa_sum_c_terminal_result_sum) + (pfp_value_terminal_result_sum_witness_operation))) /\ ((((exists pfa_gap_terminal_result_sum_witness_operationoperationleft. pfa_gap_terminal_result_sum_witness_operationoperationleft + S (pfp_left_terminal_result_sum_witness_operation) = (p)) /\ (((exists pfa_gap_terminal_result_sum_witness_operationoperationright. pfa_gap_terminal_result_sum_witness_operationoperationright + S (pfp_right_terminal_result_sum_witness_operation) = (p)) /\ ((((exists pfa_gap_terminal_result_sum_witness_operationoperationresultbound. pfa_gap_terminal_result_sum_witness_operationoperationresultbound + S (pfp_value_terminal_result_sum_witness_operation) = (p)) /\ ((exists pfa_offset_left_terminal_result_sum_witness_operationoperationresultcongruence pfa_offset_right_terminal_result_sum_witness_operationoperationresultcongruence. ((pfp_left_terminal_result_sum_witness_operation) + (pfp_right_terminal_result_sum_witness_operation)) + (p) * pfa_offset_left_terminal_result_sum_witness_operationoperationresultcongruence = (pfp_value_terminal_result_sum_witness_operation) + (p) * pfa_offset_right_terminal_result_sum_witness_operationoperationresultcongruence)))))))))))))))) /\ ((forall pfrep_power_terminal_result_sum_witness_output pfrep_left_terminal_result_sum_witness_output pfrep_right_terminal_result_sum_witness_output. ((exists pfrep_position_terminal_result_sum_witness_outputfirst. ((pfrep_position_terminal_result_sum_witness_outputfirst+S (pfrep_power_terminal_result_sum_witness_output)=(pfaa_length_terminal_result_sum)) /\ ((((exists ff_h_pfp_terminal_result_sum_witness_outputfirstentry. ff_h_pfp_terminal_result_sum_witness_outputfirstentry + S (pfrep_left_terminal_result_sum_witness_output) = S ((S (pfrep_position_terminal_result_sum_witness_outputfirst)) * pfaa_sum_c_terminal_result_sum)) /\ exists ff_q_pfp_terminal_result_sum_witness_outputfirstentry. pfaa_sum_b_terminal_result_sum = ff_q_pfp_terminal_result_sum_witness_outputfirstentry * S ((S (pfrep_position_terminal_result_sum_witness_outputfirst)) * pfaa_sum_c_terminal_result_sum) + (pfrep_left_terminal_result_sum_witness_output)))))) \/ (((exists pfrep_gap_terminal_result_sum_witness_outputfirstoutside. pfrep_gap_terminal_result_sum_witness_outputfirstoutside+(pfaa_length_terminal_result_sum)=(pfrep_power_terminal_result_sum_witness_output)) /\ (((pfrep_left_terminal_result_sum_witness_output)=0))))) -> ((exists pfrep_position_terminal_result_sum_witness_outputsecond. ((pfrep_position_terminal_result_sum_witness_outputsecond+S (pfrep_power_terminal_result_sum_witness_output)=(G)) /\ ((((exists ff_h_pfp_terminal_result_sum_witness_outputsecondentry. ff_h_pfp_terminal_result_sum_witness_outputsecondentry + S (pfrep_right_terminal_result_sum_witness_output) = S ((S (pfrep_position_terminal_result_sum_witness_outputsecond)) * gc)) /\ exists ff_q_pfp_terminal_result_sum_witness_outputsecondentry. gb = ff_q_pfp_terminal_result_sum_witness_outputsecondentry * S ((S (pfrep_position_terminal_result_sum_witness_outputsecond)) * gc) + (pfrep_right_terminal_result_sum_witness_output)))))) \/ (((exists pfrep_gap_terminal_result_sum_witness_outputsecondoutside. pfrep_gap_terminal_result_sum_witness_outputsecondoutside+(G)=(pfrep_power_terminal_result_sum_witness_output)) /\ (((pfrep_right_terminal_result_sum_witness_output)=0))))) -> pfrep_left_terminal_result_sum_witness_output=pfrep_right_terminal_result_sum_witness_output))))))))))))))))))