prime_field_polynomial_right_divides_aligned_add

Exact expanded first-order arithmetic statement

forall p db dc J ab ac L bb bc M rb rc N. (~((p) = 1) /\ forall pfa_factor_left_add_divides_prime pfa_factor_right_add_divides_prime. (p) = pfa_factor_left_add_divides_prime * pfa_factor_right_add_divides_prime -> pfa_factor_left_add_divides_prime = 1 \/ pfa_factor_right_add_divides_prime = 1) -> (((forall fom_index_pfp_add_divides_left_canonical. (exists fom_gap_pfp_add_divides_left_canonical_index_bound. fom_gap_pfp_add_divides_left_canonical_index_bound + S (fom_index_pfp_add_divides_left_canonical) = L) -> exists fom_value_pfp_add_divides_left_canonical. ((((exists fom_beta_height_pfp_add_divides_left_canonical_entry. fom_beta_height_pfp_add_divides_left_canonical_entry + S (fom_value_pfp_add_divides_left_canonical) = S ((S (fom_index_pfp_add_divides_left_canonical)) * ac)) /\ exists fom_beta_quotient_pfp_add_divides_left_canonical_entry. ab = fom_beta_quotient_pfp_add_divides_left_canonical_entry * S ((S (fom_index_pfp_add_divides_left_canonical)) * ac) + (fom_value_pfp_add_divides_left_canonical))) /\ (exists fom_gap_pfp_add_divides_left_canonical_value_bound. fom_gap_pfp_add_divides_left_canonical_value_bound + S (fom_value_pfp_add_divides_left_canonical) = p))) /\ ((exists pfrd_qb_add_divides_left pfrd_qc_add_divides_left pfrd_qlen_add_divides_left pfrd_pb_add_divides_left pfrd_pc_add_divides_left pfrd_plen_add_divides_left. ((((forall fom_index_pfp_add_divides_left_productleft. (exists fom_gap_pfp_add_divides_left_productleft_index_bound. fom_gap_pfp_add_divides_left_productleft_index_bound + S (fom_index_pfp_add_divides_left_productleft) = pfrd_qlen_add_divides_left) -> exists fom_value_pfp_add_divides_left_productleft. ((((exists fom_beta_height_pfp_add_divides_left_productleft_entry. fom_beta_height_pfp_add_divides_left_productleft_entry + S (fom_value_pfp_add_divides_left_productleft) = S ((S (fom_index_pfp_add_divides_left_productleft)) * pfrd_qc_add_divides_left)) /\ exists fom_beta_quotient_pfp_add_divides_left_productleft_entry. pfrd_qb_add_divides_left = fom_beta_quotient_pfp_add_divides_left_productleft_entry * S ((S (fom_index_pfp_add_divides_left_productleft)) * pfrd_qc_add_divides_left) + (fom_value_pfp_add_divides_left_productleft))) /\ (exists fom_gap_pfp_add_divides_left_productleft_value_bound. fom_gap_pfp_add_divides_left_productleft_value_bound + S (fom_value_pfp_add_divides_left_productleft) = p))) /\ (((forall fom_index_pfp_add_divides_left_productright. (exists fom_gap_pfp_add_divides_left_productright_index_bound. fom_gap_pfp_add_divides_left_productright_index_bound + S (fom_index_pfp_add_divides_left_productright) = J) -> exists fom_value_pfp_add_divides_left_productright. ((((exists fom_beta_height_pfp_add_divides_left_productright_entry. fom_beta_height_pfp_add_divides_left_productright_entry + S (fom_value_pfp_add_divides_left_productright) = S ((S (fom_index_pfp_add_divides_left_productright)) * dc)) /\ exists fom_beta_quotient_pfp_add_divides_left_productright_entry. db = fom_beta_quotient_pfp_add_divides_left_productright_entry * S ((S (fom_index_pfp_add_divides_left_productright)) * dc) + (fom_value_pfp_add_divides_left_productright))) /\ (exists fom_gap_pfp_add_divides_left_productright_value_bound. fom_gap_pfp_add_divides_left_productright_value_bound + S (fom_value_pfp_add_divides_left_productright) = p))) /\ (((((((pfrd_qlen_add_divides_left)=0 \/ (J)=0) /\ (((pfrd_plen_add_divides_left)=0)))) \/ (((~((pfrd_qlen_add_divides_left)=0)) /\ (((~((J)=0)) /\ (((pfrd_qlen_add_divides_left)+(J)=S (pfrd_plen_add_divides_left)))))))) /\ ((forall pfc_index_add_divides_left_productcoefficients. (exists pfa_gap_add_divides_left_productcoefficientsbound. pfa_gap_add_divides_left_productcoefficientsbound + S (pfc_index_add_divides_left_productcoefficients) = (pfrd_plen_add_divides_left)) -> exists pfc_value_add_divides_left_productcoefficients. ((((exists ff_h_pfp_add_divides_left_productcoefficientsentry. ff_h_pfp_add_divides_left_productcoefficientsentry + S (pfc_value_add_divides_left_productcoefficients) = S ((S (pfc_index_add_divides_left_productcoefficients)) * pfrd_pc_add_divides_left)) /\ exists ff_q_pfp_add_divides_left_productcoefficientsentry. pfrd_pb_add_divides_left = ff_q_pfp_add_divides_left_productcoefficientsentry * S ((S (pfc_index_add_divides_left_productcoefficients)) * pfrd_pc_add_divides_left) + (pfc_value_add_divides_left_productcoefficients))) /\ ((exists pfc_terms_code_add_divides_left_productcoefficientscoefficient pfc_terms_scale_add_divides_left_productcoefficientscoefficient pfc_natural_sum_add_divides_left_productcoefficientscoefficient. ((forall pfc_index_add_divides_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_add_divides_left_productcoefficientscoefficientdiagonalbound. pfa_gap_add_divides_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_add_divides_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_add_divides_left_productcoefficients))) -> exists pfc_value_add_divides_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_add_divides_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_add_divides_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_add_divides_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_add_divides_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_add_divides_left_productcoefficientscoefficient)) /\ exists ff_q_pfp_add_divides_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_add_divides_left_productcoefficientscoefficient = ff_q_pfp_add_divides_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_add_divides_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_add_divides_left_productcoefficientscoefficient) + (pfc_value_add_divides_left_productcoefficientscoefficientdiagonal))) /\ ((exists pfc_complement_add_divides_left_productcoefficientscoefficientdiagonalterm pfc_left_add_divides_left_productcoefficientscoefficientdiagonalterm pfc_right_add_divides_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_add_divides_left_productcoefficientscoefficientdiagonal)+pfc_complement_add_divides_left_productcoefficientscoefficientdiagonalterm=(pfc_index_add_divides_left_productcoefficients)) /\ ((((((exists pfa_gap_add_divides_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_add_divides_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_add_divides_left_productcoefficientscoefficientdiagonal) = (pfrd_qlen_add_divides_left)) /\ ((((exists ff_h_pfp_add_divides_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_add_divides_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_add_divides_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_add_divides_left_productcoefficientscoefficientdiagonal)) * pfrd_qc_add_divides_left)) /\ exists ff_q_pfp_add_divides_left_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_add_divides_left = ff_q_pfp_add_divides_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_add_divides_left_productcoefficientscoefficientdiagonal)) * pfrd_qc_add_divides_left) + (pfc_left_add_divides_left_productcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_add_divides_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_add_divides_left_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_add_divides_left)=(pfc_index_add_divides_left_productcoefficientscoefficientdiagonal)) /\ (((pfc_left_add_divides_left_productcoefficientscoefficientdiagonalterm)=0))))) /\ ((((((exists pfa_gap_add_divides_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_add_divides_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_add_divides_left_productcoefficientscoefficientdiagonalterm) = (J)) /\ ((((exists ff_h_pfp_add_divides_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_add_divides_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_add_divides_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_add_divides_left_productcoefficientscoefficientdiagonalterm)) * dc)) /\ exists ff_q_pfp_add_divides_left_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_add_divides_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_add_divides_left_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_add_divides_left_productcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_add_divides_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_add_divides_left_productcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_add_divides_left_productcoefficientscoefficientdiagonalterm)) /\ (((pfc_right_add_divides_left_productcoefficientscoefficientdiagonalterm)=0))))) /\ (((pfc_value_add_divides_left_productcoefficientscoefficientdiagonal)=pfc_left_add_divides_left_productcoefficientscoefficientdiagonalterm*pfc_right_add_divides_left_productcoefficientscoefficientdiagonalterm))))))))))) /\ (((exists fs_u_pfc_add_divides_left_productcoefficientscoefficientsum fs_v_pfc_add_divides_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_add_divides_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_add_divides_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_add_divides_left_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_add_divides_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_add_divides_left_productcoefficientscoefficientsum = fs_q_pfc_add_divides_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_add_divides_left_productcoefficientscoefficientsum) + (0))) /\ ((((exists fs_h_pfc_add_divides_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_add_divides_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_add_divides_left_productcoefficientscoefficient) = S ((S (S (pfc_index_add_divides_left_productcoefficients))) * fs_v_pfc_add_divides_left_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_add_divides_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_add_divides_left_productcoefficientscoefficientsum = fs_q_pfc_add_divides_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_add_divides_left_productcoefficients))) * fs_v_pfc_add_divides_left_productcoefficientscoefficientsum) + (pfc_natural_sum_add_divides_left_productcoefficientscoefficient))) /\ forall fs_i_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_add_divides_left_productcoefficients)) -> exists fs_a_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_add_divides_left_productcoefficientscoefficient)) /\ exists fs_q_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_add_divides_left_productcoefficientscoefficient = fs_q_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_add_divides_left_productcoefficientscoefficient) + (fs_a_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_add_divides_left_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_add_divides_left_productcoefficientscoefficientsum = fs_q_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_add_divides_left_productcoefficientscoefficientsum) + (fs_r_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_add_divides_left_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_add_divides_left_productcoefficientscoefficientsum = fs_q_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_add_divides_left_productcoefficientscoefficientsum) + (fs_s_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps))) /\ fs_s_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps)))))) /\ ((((exists pfa_gap_add_divides_left_productcoefficientscoefficientresiduebound. pfa_gap_add_divides_left_productcoefficientscoefficientresiduebound + S (pfc_value_add_divides_left_productcoefficients) = (p)) /\ ((exists pfa_offset_left_add_divides_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_add_divides_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_add_divides_left_productcoefficientscoefficient) + (p) * pfa_offset_left_add_divides_left_productcoefficientscoefficientresiduecongruence = (pfc_value_add_divides_left_productcoefficients) + (p) * pfa_offset_right_add_divides_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\ ((forall pfrep_power_add_divides_left_target pfrep_left_add_divides_left_target pfrep_right_add_divides_left_target. ((exists pfrep_position_add_divides_left_targetfirst. ((pfrep_position_add_divides_left_targetfirst+S (pfrep_power_add_divides_left_target)=(pfrd_plen_add_divides_left)) /\ ((((exists ff_h_pfp_add_divides_left_targetfirstentry. ff_h_pfp_add_divides_left_targetfirstentry + S (pfrep_left_add_divides_left_target) = S ((S (pfrep_position_add_divides_left_targetfirst)) * pfrd_pc_add_divides_left)) /\ exists ff_q_pfp_add_divides_left_targetfirstentry. pfrd_pb_add_divides_left = ff_q_pfp_add_divides_left_targetfirstentry * S ((S (pfrep_position_add_divides_left_targetfirst)) * pfrd_pc_add_divides_left) + (pfrep_left_add_divides_left_target)))))) \/ (((exists pfrep_gap_add_divides_left_targetfirstoutside. pfrep_gap_add_divides_left_targetfirstoutside+(pfrd_plen_add_divides_left)=(pfrep_power_add_divides_left_target)) /\ (((pfrep_left_add_divides_left_target)=0))))) -> ((exists pfrep_position_add_divides_left_targetsecond. ((pfrep_position_add_divides_left_targetsecond+S (pfrep_power_add_divides_left_target)=(L)) /\ ((((exists ff_h_pfp_add_divides_left_targetsecondentry. ff_h_pfp_add_divides_left_targetsecondentry + S (pfrep_right_add_divides_left_target) = S ((S (pfrep_position_add_divides_left_targetsecond)) * ac)) /\ exists ff_q_pfp_add_divides_left_targetsecondentry. ab = ff_q_pfp_add_divides_left_targetsecondentry * S ((S (pfrep_position_add_divides_left_targetsecond)) * ac) + (pfrep_right_add_divides_left_target)))))) \/ (((exists pfrep_gap_add_divides_left_targetsecondoutside. pfrep_gap_add_divides_left_targetsecondoutside+(L)=(pfrep_power_add_divides_left_target)) /\ (((pfrep_right_add_divides_left_target)=0))))) -> pfrep_left_add_divides_left_target=pfrep_right_add_divides_left_target))))))) -> (((forall fom_index_pfp_add_divides_right_canonical. (exists fom_gap_pfp_add_divides_right_canonical_index_bound. fom_gap_pfp_add_divides_right_canonical_index_bound + S (fom_index_pfp_add_divides_right_canonical) = M) -> exists fom_value_pfp_add_divides_right_canonical. ((((exists fom_beta_height_pfp_add_divides_right_canonical_entry. fom_beta_height_pfp_add_divides_right_canonical_entry + S (fom_value_pfp_add_divides_right_canonical) = S ((S (fom_index_pfp_add_divides_right_canonical)) * bc)) /\ exists fom_beta_quotient_pfp_add_divides_right_canonical_entry. bb = fom_beta_quotient_pfp_add_divides_right_canonical_entry * S ((S (fom_index_pfp_add_divides_right_canonical)) * bc) + (fom_value_pfp_add_divides_right_canonical))) /\ (exists fom_gap_pfp_add_divides_right_canonical_value_bound. fom_gap_pfp_add_divides_right_canonical_value_bound + S (fom_value_pfp_add_divides_right_canonical) = p))) /\ ((exists pfrd_qb_add_divides_right pfrd_qc_add_divides_right pfrd_qlen_add_divides_right pfrd_pb_add_divides_right pfrd_pc_add_divides_right pfrd_plen_add_divides_right. ((((forall fom_index_pfp_add_divides_right_productleft. (exists fom_gap_pfp_add_divides_right_productleft_index_bound. fom_gap_pfp_add_divides_right_productleft_index_bound + S (fom_index_pfp_add_divides_right_productleft) = pfrd_qlen_add_divides_right) -> exists fom_value_pfp_add_divides_right_productleft. ((((exists fom_beta_height_pfp_add_divides_right_productleft_entry. fom_beta_height_pfp_add_divides_right_productleft_entry + S (fom_value_pfp_add_divides_right_productleft) = S ((S (fom_index_pfp_add_divides_right_productleft)) * pfrd_qc_add_divides_right)) /\ exists fom_beta_quotient_pfp_add_divides_right_productleft_entry. pfrd_qb_add_divides_right = fom_beta_quotient_pfp_add_divides_right_productleft_entry * S ((S (fom_index_pfp_add_divides_right_productleft)) * pfrd_qc_add_divides_right) + (fom_value_pfp_add_divides_right_productleft))) /\ (exists fom_gap_pfp_add_divides_right_productleft_value_bound. fom_gap_pfp_add_divides_right_productleft_value_bound + S (fom_value_pfp_add_divides_right_productleft) = p))) /\ (((forall fom_index_pfp_add_divides_right_productright. (exists fom_gap_pfp_add_divides_right_productright_index_bound. fom_gap_pfp_add_divides_right_productright_index_bound + S (fom_index_pfp_add_divides_right_productright) = J) -> exists fom_value_pfp_add_divides_right_productright. ((((exists fom_beta_height_pfp_add_divides_right_productright_entry. fom_beta_height_pfp_add_divides_right_productright_entry + S (fom_value_pfp_add_divides_right_productright) = S ((S (fom_index_pfp_add_divides_right_productright)) * dc)) /\ exists fom_beta_quotient_pfp_add_divides_right_productright_entry. db = fom_beta_quotient_pfp_add_divides_right_productright_entry * S ((S (fom_index_pfp_add_divides_right_productright)) * dc) + (fom_value_pfp_add_divides_right_productright))) /\ (exists fom_gap_pfp_add_divides_right_productright_value_bound. fom_gap_pfp_add_divides_right_productright_value_bound + S (fom_value_pfp_add_divides_right_productright) = p))) /\ (((((((pfrd_qlen_add_divides_right)=0 \/ (J)=0) /\ (((pfrd_plen_add_divides_right)=0)))) \/ (((~((pfrd_qlen_add_divides_right)=0)) /\ (((~((J)=0)) /\ (((pfrd_qlen_add_divides_right)+(J)=S (pfrd_plen_add_divides_right)))))))) /\ ((forall pfc_index_add_divides_right_productcoefficients. (exists pfa_gap_add_divides_right_productcoefficientsbound. pfa_gap_add_divides_right_productcoefficientsbound + S (pfc_index_add_divides_right_productcoefficients) = (pfrd_plen_add_divides_right)) -> exists pfc_value_add_divides_right_productcoefficients. ((((exists ff_h_pfp_add_divides_right_productcoefficientsentry. ff_h_pfp_add_divides_right_productcoefficientsentry + S (pfc_value_add_divides_right_productcoefficients) = S ((S (pfc_index_add_divides_right_productcoefficients)) * pfrd_pc_add_divides_right)) /\ exists ff_q_pfp_add_divides_right_productcoefficientsentry. pfrd_pb_add_divides_right = ff_q_pfp_add_divides_right_productcoefficientsentry * S ((S (pfc_index_add_divides_right_productcoefficients)) * pfrd_pc_add_divides_right) + (pfc_value_add_divides_right_productcoefficients))) /\ ((exists pfc_terms_code_add_divides_right_productcoefficientscoefficient pfc_terms_scale_add_divides_right_productcoefficientscoefficient pfc_natural_sum_add_divides_right_productcoefficientscoefficient. ((forall pfc_index_add_divides_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_add_divides_right_productcoefficientscoefficientdiagonalbound. pfa_gap_add_divides_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_add_divides_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_add_divides_right_productcoefficients))) -> exists pfc_value_add_divides_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_add_divides_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_add_divides_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_add_divides_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_add_divides_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_add_divides_right_productcoefficientscoefficient)) /\ exists ff_q_pfp_add_divides_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_add_divides_right_productcoefficientscoefficient = ff_q_pfp_add_divides_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_add_divides_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_add_divides_right_productcoefficientscoefficient) + (pfc_value_add_divides_right_productcoefficientscoefficientdiagonal))) /\ ((exists pfc_complement_add_divides_right_productcoefficientscoefficientdiagonalterm pfc_left_add_divides_right_productcoefficientscoefficientdiagonalterm pfc_right_add_divides_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_add_divides_right_productcoefficientscoefficientdiagonal)+pfc_complement_add_divides_right_productcoefficientscoefficientdiagonalterm=(pfc_index_add_divides_right_productcoefficients)) /\ ((((((exists pfa_gap_add_divides_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_add_divides_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_add_divides_right_productcoefficientscoefficientdiagonal) = (pfrd_qlen_add_divides_right)) /\ ((((exists ff_h_pfp_add_divides_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_add_divides_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_add_divides_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_add_divides_right_productcoefficientscoefficientdiagonal)) * pfrd_qc_add_divides_right)) /\ exists ff_q_pfp_add_divides_right_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_add_divides_right = ff_q_pfp_add_divides_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_add_divides_right_productcoefficientscoefficientdiagonal)) * pfrd_qc_add_divides_right) + (pfc_left_add_divides_right_productcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_add_divides_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_add_divides_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_add_divides_right)=(pfc_index_add_divides_right_productcoefficientscoefficientdiagonal)) /\ (((pfc_left_add_divides_right_productcoefficientscoefficientdiagonalterm)=0))))) /\ ((((((exists pfa_gap_add_divides_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_add_divides_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_add_divides_right_productcoefficientscoefficientdiagonalterm) = (J)) /\ ((((exists ff_h_pfp_add_divides_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_add_divides_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_add_divides_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_add_divides_right_productcoefficientscoefficientdiagonalterm)) * dc)) /\ exists ff_q_pfp_add_divides_right_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_add_divides_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_add_divides_right_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_add_divides_right_productcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_add_divides_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_add_divides_right_productcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_add_divides_right_productcoefficientscoefficientdiagonalterm)) /\ (((pfc_right_add_divides_right_productcoefficientscoefficientdiagonalterm)=0))))) /\ (((pfc_value_add_divides_right_productcoefficientscoefficientdiagonal)=pfc_left_add_divides_right_productcoefficientscoefficientdiagonalterm*pfc_right_add_divides_right_productcoefficientscoefficientdiagonalterm))))))))))) /\ (((exists fs_u_pfc_add_divides_right_productcoefficientscoefficientsum fs_v_pfc_add_divides_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_add_divides_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_add_divides_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_add_divides_right_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_add_divides_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_add_divides_right_productcoefficientscoefficientsum = fs_q_pfc_add_divides_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_add_divides_right_productcoefficientscoefficientsum) + (0))) /\ ((((exists fs_h_pfc_add_divides_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_add_divides_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_add_divides_right_productcoefficientscoefficient) = S ((S (S (pfc_index_add_divides_right_productcoefficients))) * fs_v_pfc_add_divides_right_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_add_divides_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_add_divides_right_productcoefficientscoefficientsum = fs_q_pfc_add_divides_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_add_divides_right_productcoefficients))) * fs_v_pfc_add_divides_right_productcoefficientscoefficientsum) + (pfc_natural_sum_add_divides_right_productcoefficientscoefficient))) /\ forall fs_i_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_add_divides_right_productcoefficients)) -> exists fs_a_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_add_divides_right_productcoefficientscoefficient)) /\ exists fs_q_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_add_divides_right_productcoefficientscoefficient = fs_q_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_add_divides_right_productcoefficientscoefficient) + (fs_a_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_add_divides_right_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_add_divides_right_productcoefficientscoefficientsum = fs_q_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_add_divides_right_productcoefficientscoefficientsum) + (fs_r_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_add_divides_right_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_add_divides_right_productcoefficientscoefficientsum = fs_q_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_add_divides_right_productcoefficientscoefficientsum) + (fs_s_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps))) /\ fs_s_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps)))))) /\ ((((exists pfa_gap_add_divides_right_productcoefficientscoefficientresiduebound. pfa_gap_add_divides_right_productcoefficientscoefficientresiduebound + S (pfc_value_add_divides_right_productcoefficients) = (p)) /\ ((exists pfa_offset_left_add_divides_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_add_divides_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_add_divides_right_productcoefficientscoefficient) + (p) * pfa_offset_left_add_divides_right_productcoefficientscoefficientresiduecongruence = (pfc_value_add_divides_right_productcoefficients) + (p) * pfa_offset_right_add_divides_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\ ((forall pfrep_power_add_divides_right_target pfrep_left_add_divides_right_target pfrep_right_add_divides_right_target. ((exists pfrep_position_add_divides_right_targetfirst. ((pfrep_position_add_divides_right_targetfirst+S (pfrep_power_add_divides_right_target)=(pfrd_plen_add_divides_right)) /\ ((((exists ff_h_pfp_add_divides_right_targetfirstentry. ff_h_pfp_add_divides_right_targetfirstentry + S (pfrep_left_add_divides_right_target) = S ((S (pfrep_position_add_divides_right_targetfirst)) * pfrd_pc_add_divides_right)) /\ exists ff_q_pfp_add_divides_right_targetfirstentry. pfrd_pb_add_divides_right = ff_q_pfp_add_divides_right_targetfirstentry * S ((S (pfrep_position_add_divides_right_targetfirst)) * pfrd_pc_add_divides_right) + (pfrep_left_add_divides_right_target)))))) \/ (((exists pfrep_gap_add_divides_right_targetfirstoutside. pfrep_gap_add_divides_right_targetfirstoutside+(pfrd_plen_add_divides_right)=(pfrep_power_add_divides_right_target)) /\ (((pfrep_left_add_divides_right_target)=0))))) -> ((exists pfrep_position_add_divides_right_targetsecond. ((pfrep_position_add_divides_right_targetsecond+S (pfrep_power_add_divides_right_target)=(M)) /\ ((((exists ff_h_pfp_add_divides_right_targetsecondentry. ff_h_pfp_add_divides_right_targetsecondentry + S (pfrep_right_add_divides_right_target) = S ((S (pfrep_position_add_divides_right_targetsecond)) * bc)) /\ exists ff_q_pfp_add_divides_right_targetsecondentry. bb = ff_q_pfp_add_divides_right_targetsecondentry * S ((S (pfrep_position_add_divides_right_targetsecond)) * bc) + (pfrep_right_add_divides_right_target)))))) \/ (((exists pfrep_gap_add_divides_right_targetsecondoutside. pfrep_gap_add_divides_right_targetsecondoutside+(M)=(pfrep_power_add_divides_right_target)) /\ (((pfrep_right_add_divides_right_target)=0))))) -> pfrep_left_add_divides_right_target=pfrep_right_add_divides_right_target))))))) -> (((forall fom_index_pfp_add_divides_operation_left_bounded. (exists fom_gap_pfp_add_divides_operation_left_bounded_index_bound. fom_gap_pfp_add_divides_operation_left_bounded_index_bound + S (fom_index_pfp_add_divides_operation_left_bounded) = L) -> exists fom_value_pfp_add_divides_operation_left_bounded. ((((exists fom_beta_height_pfp_add_divides_operation_left_bounded_entry. fom_beta_height_pfp_add_divides_operation_left_bounded_entry + S (fom_value_pfp_add_divides_operation_left_bounded) = S ((S (fom_index_pfp_add_divides_operation_left_bounded)) * ac)) /\ exists fom_beta_quotient_pfp_add_divides_operation_left_bounded_entry. ab = fom_beta_quotient_pfp_add_divides_operation_left_bounded_entry * S ((S (fom_index_pfp_add_divides_operation_left_bounded)) * ac) + (fom_value_pfp_add_divides_operation_left_bounded))) /\ (exists fom_gap_pfp_add_divides_operation_left_bounded_value_bound. fom_gap_pfp_add_divides_operation_left_bounded_value_bound + S (fom_value_pfp_add_divides_operation_left_bounded) = p))) /\ (((forall fom_index_pfp_add_divides_operation_right_bounded. (exists fom_gap_pfp_add_divides_operation_right_bounded_index_bound. fom_gap_pfp_add_divides_operation_right_bounded_index_bound + S (fom_index_pfp_add_divides_operation_right_bounded) = M) -> exists fom_value_pfp_add_divides_operation_right_bounded. ((((exists fom_beta_height_pfp_add_divides_operation_right_bounded_entry. fom_beta_height_pfp_add_divides_operation_right_bounded_entry + S (fom_value_pfp_add_divides_operation_right_bounded) = S ((S (fom_index_pfp_add_divides_operation_right_bounded)) * bc)) /\ exists fom_beta_quotient_pfp_add_divides_operation_right_bounded_entry. bb = fom_beta_quotient_pfp_add_divides_operation_right_bounded_entry * S ((S (fom_index_pfp_add_divides_operation_right_bounded)) * bc) + (fom_value_pfp_add_divides_operation_right_bounded))) /\ (exists fom_gap_pfp_add_divides_operation_right_bounded_value_bound. fom_gap_pfp_add_divides_operation_right_bounded_value_bound + S (fom_value_pfp_add_divides_operation_right_bounded) = p))) /\ (((forall fom_index_pfp_add_divides_operation_result_bounded. (exists fom_gap_pfp_add_divides_operation_result_bounded_index_bound. fom_gap_pfp_add_divides_operation_result_bounded_index_bound + S (fom_index_pfp_add_divides_operation_result_bounded) = N) -> exists fom_value_pfp_add_divides_operation_result_bounded. ((((exists fom_beta_height_pfp_add_divides_operation_result_bounded_entry. fom_beta_height_pfp_add_divides_operation_result_bounded_entry + S (fom_value_pfp_add_divides_operation_result_bounded) = S ((S (fom_index_pfp_add_divides_operation_result_bounded)) * rc)) /\ exists fom_beta_quotient_pfp_add_divides_operation_result_bounded_entry. rb = fom_beta_quotient_pfp_add_divides_operation_result_bounded_entry * S ((S (fom_index_pfp_add_divides_operation_result_bounded)) * rc) + (fom_value_pfp_add_divides_operation_result_bounded))) /\ (exists fom_gap_pfp_add_divides_operation_result_bounded_value_bound. fom_gap_pfp_add_divides_operation_result_bounded_value_bound + S (fom_value_pfp_add_divides_operation_result_bounded) = p))) /\ ((exists pfaa_left_b_add_divides_operation pfaa_left_c_add_divides_operation pfaa_right_b_add_divides_operation pfaa_right_c_add_divides_operation pfaa_sum_b_add_divides_operation pfaa_sum_c_add_divides_operation pfaa_length_add_divides_operation. ((((forall pfrep_power_add_divides_operation_witness_common_left pfrep_left_add_divides_operation_witness_common_left pfrep_right_add_divides_operation_witness_common_left. ((exists pfrep_position_add_divides_operation_witness_common_leftfirst. ((pfrep_position_add_divides_operation_witness_common_leftfirst+S (pfrep_power_add_divides_operation_witness_common_left)=(L)) /\ ((((exists ff_h_pfp_add_divides_operation_witness_common_leftfirstentry. ff_h_pfp_add_divides_operation_witness_common_leftfirstentry + S (pfrep_left_add_divides_operation_witness_common_left) = S ((S (pfrep_position_add_divides_operation_witness_common_leftfirst)) * ac)) /\ exists ff_q_pfp_add_divides_operation_witness_common_leftfirstentry. ab = ff_q_pfp_add_divides_operation_witness_common_leftfirstentry * S ((S (pfrep_position_add_divides_operation_witness_common_leftfirst)) * ac) + (pfrep_left_add_divides_operation_witness_common_left)))))) \/ (((exists pfrep_gap_add_divides_operation_witness_common_leftfirstoutside. pfrep_gap_add_divides_operation_witness_common_leftfirstoutside+(L)=(pfrep_power_add_divides_operation_witness_common_left)) /\ (((pfrep_left_add_divides_operation_witness_common_left)=0))))) -> ((exists pfrep_position_add_divides_operation_witness_common_leftsecond. ((pfrep_position_add_divides_operation_witness_common_leftsecond+S (pfrep_power_add_divides_operation_witness_common_left)=(pfaa_length_add_divides_operation)) /\ ((((exists ff_h_pfp_add_divides_operation_witness_common_leftsecondentry. ff_h_pfp_add_divides_operation_witness_common_leftsecondentry + S (pfrep_right_add_divides_operation_witness_common_left) = S ((S (pfrep_position_add_divides_operation_witness_common_leftsecond)) * pfaa_left_c_add_divides_operation)) /\ exists ff_q_pfp_add_divides_operation_witness_common_leftsecondentry. pfaa_left_b_add_divides_operation = ff_q_pfp_add_divides_operation_witness_common_leftsecondentry * S ((S (pfrep_position_add_divides_operation_witness_common_leftsecond)) * pfaa_left_c_add_divides_operation) + (pfrep_right_add_divides_operation_witness_common_left)))))) \/ (((exists pfrep_gap_add_divides_operation_witness_common_leftsecondoutside. pfrep_gap_add_divides_operation_witness_common_leftsecondoutside+(pfaa_length_add_divides_operation)=(pfrep_power_add_divides_operation_witness_common_left)) /\ (((pfrep_right_add_divides_operation_witness_common_left)=0))))) -> pfrep_left_add_divides_operation_witness_common_left=pfrep_right_add_divides_operation_witness_common_left) /\ ((forall pfrep_power_add_divides_operation_witness_common_right pfrep_left_add_divides_operation_witness_common_right pfrep_right_add_divides_operation_witness_common_right. ((exists pfrep_position_add_divides_operation_witness_common_rightfirst. ((pfrep_position_add_divides_operation_witness_common_rightfirst+S (pfrep_power_add_divides_operation_witness_common_right)=(M)) /\ ((((exists ff_h_pfp_add_divides_operation_witness_common_rightfirstentry. ff_h_pfp_add_divides_operation_witness_common_rightfirstentry + S (pfrep_left_add_divides_operation_witness_common_right) = S ((S (pfrep_position_add_divides_operation_witness_common_rightfirst)) * bc)) /\ exists ff_q_pfp_add_divides_operation_witness_common_rightfirstentry. bb = ff_q_pfp_add_divides_operation_witness_common_rightfirstentry * S ((S (pfrep_position_add_divides_operation_witness_common_rightfirst)) * bc) + (pfrep_left_add_divides_operation_witness_common_right)))))) \/ (((exists pfrep_gap_add_divides_operation_witness_common_rightfirstoutside. pfrep_gap_add_divides_operation_witness_common_rightfirstoutside+(M)=(pfrep_power_add_divides_operation_witness_common_right)) /\ (((pfrep_left_add_divides_operation_witness_common_right)=0))))) -> ((exists pfrep_position_add_divides_operation_witness_common_rightsecond. ((pfrep_position_add_divides_operation_witness_common_rightsecond+S (pfrep_power_add_divides_operation_witness_common_right)=(pfaa_length_add_divides_operation)) /\ ((((exists ff_h_pfp_add_divides_operation_witness_common_rightsecondentry. ff_h_pfp_add_divides_operation_witness_common_rightsecondentry + S (pfrep_right_add_divides_operation_witness_common_right) = S ((S (pfrep_position_add_divides_operation_witness_common_rightsecond)) * pfaa_right_c_add_divides_operation)) /\ exists ff_q_pfp_add_divides_operation_witness_common_rightsecondentry. pfaa_right_b_add_divides_operation = ff_q_pfp_add_divides_operation_witness_common_rightsecondentry * S ((S (pfrep_position_add_divides_operation_witness_common_rightsecond)) * pfaa_right_c_add_divides_operation) + (pfrep_right_add_divides_operation_witness_common_right)))))) \/ (((exists pfrep_gap_add_divides_operation_witness_common_rightsecondoutside. pfrep_gap_add_divides_operation_witness_common_rightsecondoutside+(pfaa_length_add_divides_operation)=(pfrep_power_add_divides_operation_witness_common_right)) /\ (((pfrep_right_add_divides_operation_witness_common_right)=0))))) -> pfrep_left_add_divides_operation_witness_common_right=pfrep_right_add_divides_operation_witness_common_right)))) /\ (((forall pfp_index_add_divides_operation_witness_operation. (exists pfa_gap_add_divides_operation_witness_operationindex. pfa_gap_add_divides_operation_witness_operationindex + S (pfp_index_add_divides_operation_witness_operation) = (pfaa_length_add_divides_operation)) -> exists pfp_left_add_divides_operation_witness_operation pfp_right_add_divides_operation_witness_operation pfp_value_add_divides_operation_witness_operation. ((((exists ff_h_pfp_add_divides_operation_witness_operationleft. ff_h_pfp_add_divides_operation_witness_operationleft + S (pfp_left_add_divides_operation_witness_operation) = S ((S (pfp_index_add_divides_operation_witness_operation)) * pfaa_left_c_add_divides_operation)) /\ exists ff_q_pfp_add_divides_operation_witness_operationleft. pfaa_left_b_add_divides_operation = ff_q_pfp_add_divides_operation_witness_operationleft * S ((S (pfp_index_add_divides_operation_witness_operation)) * pfaa_left_c_add_divides_operation) + (pfp_left_add_divides_operation_witness_operation))) /\ (((((exists ff_h_pfp_add_divides_operation_witness_operationright. ff_h_pfp_add_divides_operation_witness_operationright + S (pfp_right_add_divides_operation_witness_operation) = S ((S (pfp_index_add_divides_operation_witness_operation)) * pfaa_right_c_add_divides_operation)) /\ exists ff_q_pfp_add_divides_operation_witness_operationright. pfaa_right_b_add_divides_operation = ff_q_pfp_add_divides_operation_witness_operationright * S ((S (pfp_index_add_divides_operation_witness_operation)) * pfaa_right_c_add_divides_operation) + (pfp_right_add_divides_operation_witness_operation))) /\ (((((exists ff_h_pfp_add_divides_operation_witness_operationtarget. ff_h_pfp_add_divides_operation_witness_operationtarget + S (pfp_value_add_divides_operation_witness_operation) = S ((S (pfp_index_add_divides_operation_witness_operation)) * pfaa_sum_c_add_divides_operation)) /\ exists ff_q_pfp_add_divides_operation_witness_operationtarget. pfaa_sum_b_add_divides_operation = ff_q_pfp_add_divides_operation_witness_operationtarget * S ((S (pfp_index_add_divides_operation_witness_operation)) * pfaa_sum_c_add_divides_operation) + (pfp_value_add_divides_operation_witness_operation))) /\ ((((exists pfa_gap_add_divides_operation_witness_operationoperationleft. pfa_gap_add_divides_operation_witness_operationoperationleft + S (pfp_left_add_divides_operation_witness_operation) = (p)) /\ (((exists pfa_gap_add_divides_operation_witness_operationoperationright. pfa_gap_add_divides_operation_witness_operationoperationright + S (pfp_right_add_divides_operation_witness_operation) = (p)) /\ ((((exists pfa_gap_add_divides_operation_witness_operationoperationresultbound. pfa_gap_add_divides_operation_witness_operationoperationresultbound + S (pfp_value_add_divides_operation_witness_operation) = (p)) /\ ((exists pfa_offset_left_add_divides_operation_witness_operationoperationresultcongruence pfa_offset_right_add_divides_operation_witness_operationoperationresultcongruence. ((pfp_left_add_divides_operation_witness_operation) + (pfp_right_add_divides_operation_witness_operation)) + (p) * pfa_offset_left_add_divides_operation_witness_operationoperationresultcongruence = (pfp_value_add_divides_operation_witness_operation) + (p) * pfa_offset_right_add_divides_operation_witness_operationoperationresultcongruence)))))))))))))))) /\ ((forall pfrep_power_add_divides_operation_witness_output pfrep_left_add_divides_operation_witness_output pfrep_right_add_divides_operation_witness_output. ((exists pfrep_position_add_divides_operation_witness_outputfirst. ((pfrep_position_add_divides_operation_witness_outputfirst+S (pfrep_power_add_divides_operation_witness_output)=(pfaa_length_add_divides_operation)) /\ ((((exists ff_h_pfp_add_divides_operation_witness_outputfirstentry. ff_h_pfp_add_divides_operation_witness_outputfirstentry + S (pfrep_left_add_divides_operation_witness_output) = S ((S (pfrep_position_add_divides_operation_witness_outputfirst)) * pfaa_sum_c_add_divides_operation)) /\ exists ff_q_pfp_add_divides_operation_witness_outputfirstentry. pfaa_sum_b_add_divides_operation = ff_q_pfp_add_divides_operation_witness_outputfirstentry * S ((S (pfrep_position_add_divides_operation_witness_outputfirst)) * pfaa_sum_c_add_divides_operation) + (pfrep_left_add_divides_operation_witness_output)))))) \/ (((exists pfrep_gap_add_divides_operation_witness_outputfirstoutside. pfrep_gap_add_divides_operation_witness_outputfirstoutside+(pfaa_length_add_divides_operation)=(pfrep_power_add_divides_operation_witness_output)) /\ (((pfrep_left_add_divides_operation_witness_output)=0))))) -> ((exists pfrep_position_add_divides_operation_witness_outputsecond. ((pfrep_position_add_divides_operation_witness_outputsecond+S (pfrep_power_add_divides_operation_witness_output)=(N)) /\ ((((exists ff_h_pfp_add_divides_operation_witness_outputsecondentry. ff_h_pfp_add_divides_operation_witness_outputsecondentry + S (pfrep_right_add_divides_operation_witness_output) = S ((S (pfrep_position_add_divides_operation_witness_outputsecond)) * rc)) /\ exists ff_q_pfp_add_divides_operation_witness_outputsecondentry. rb = ff_q_pfp_add_divides_operation_witness_outputsecondentry * S ((S (pfrep_position_add_divides_operation_witness_outputsecond)) * rc) + (pfrep_right_add_divides_operation_witness_output)))))) \/ (((exists pfrep_gap_add_divides_operation_witness_outputsecondoutside. pfrep_gap_add_divides_operation_witness_outputsecondoutside+(N)=(pfrep_power_add_divides_operation_witness_output)) /\ (((pfrep_right_add_divides_operation_witness_output)=0))))) -> pfrep_left_add_divides_operation_witness_output=pfrep_right_add_divides_operation_witness_output))))))))))))) -> (((forall fom_index_pfp_add_divides_result_canonical. (exists fom_gap_pfp_add_divides_result_canonical_index_bound. fom_gap_pfp_add_divides_result_canonical_index_bound + S (fom_index_pfp_add_divides_result_canonical) = N) -> exists fom_value_pfp_add_divides_result_canonical. ((((exists fom_beta_height_pfp_add_divides_result_canonical_entry. fom_beta_height_pfp_add_divides_result_canonical_entry + S (fom_value_pfp_add_divides_result_canonical) = S ((S (fom_index_pfp_add_divides_result_canonical)) * rc)) /\ exists fom_beta_quotient_pfp_add_divides_result_canonical_entry. rb = fom_beta_quotient_pfp_add_divides_result_canonical_entry * S ((S (fom_index_pfp_add_divides_result_canonical)) * rc) + (fom_value_pfp_add_divides_result_canonical))) /\ (exists fom_gap_pfp_add_divides_result_canonical_value_bound. fom_gap_pfp_add_divides_result_canonical_value_bound + S (fom_value_pfp_add_divides_result_canonical) = p))) /\ ((exists pfrd_qb_add_divides_result pfrd_qc_add_divides_result pfrd_qlen_add_divides_result pfrd_pb_add_divides_result pfrd_pc_add_divides_result pfrd_plen_add_divides_result. ((((forall fom_index_pfp_add_divides_result_productleft. (exists fom_gap_pfp_add_divides_result_productleft_index_bound. fom_gap_pfp_add_divides_result_productleft_index_bound + S (fom_index_pfp_add_divides_result_productleft) = pfrd_qlen_add_divides_result) -> exists fom_value_pfp_add_divides_result_productleft. ((((exists fom_beta_height_pfp_add_divides_result_productleft_entry. fom_beta_height_pfp_add_divides_result_productleft_entry + S (fom_value_pfp_add_divides_result_productleft) = S ((S (fom_index_pfp_add_divides_result_productleft)) * pfrd_qc_add_divides_result)) /\ exists fom_beta_quotient_pfp_add_divides_result_productleft_entry. pfrd_qb_add_divides_result = fom_beta_quotient_pfp_add_divides_result_productleft_entry * S ((S (fom_index_pfp_add_divides_result_productleft)) * pfrd_qc_add_divides_result) + (fom_value_pfp_add_divides_result_productleft))) /\ (exists fom_gap_pfp_add_divides_result_productleft_value_bound. fom_gap_pfp_add_divides_result_productleft_value_bound + S (fom_value_pfp_add_divides_result_productleft) = p))) /\ (((forall fom_index_pfp_add_divides_result_productright. (exists fom_gap_pfp_add_divides_result_productright_index_bound. fom_gap_pfp_add_divides_result_productright_index_bound + S (fom_index_pfp_add_divides_result_productright) = J) -> exists fom_value_pfp_add_divides_result_productright. ((((exists fom_beta_height_pfp_add_divides_result_productright_entry. fom_beta_height_pfp_add_divides_result_productright_entry + S (fom_value_pfp_add_divides_result_productright) = S ((S (fom_index_pfp_add_divides_result_productright)) * dc)) /\ exists fom_beta_quotient_pfp_add_divides_result_productright_entry. db = fom_beta_quotient_pfp_add_divides_result_productright_entry * S ((S (fom_index_pfp_add_divides_result_productright)) * dc) + (fom_value_pfp_add_divides_result_productright))) /\ (exists fom_gap_pfp_add_divides_result_productright_value_bound. fom_gap_pfp_add_divides_result_productright_value_bound + S (fom_value_pfp_add_divides_result_productright) = p))) /\ (((((((pfrd_qlen_add_divides_result)=0 \/ (J)=0) /\ (((pfrd_plen_add_divides_result)=0)))) \/ (((~((pfrd_qlen_add_divides_result)=0)) /\ (((~((J)=0)) /\ (((pfrd_qlen_add_divides_result)+(J)=S (pfrd_plen_add_divides_result)))))))) /\ ((forall pfc_index_add_divides_result_productcoefficients. (exists pfa_gap_add_divides_result_productcoefficientsbound. pfa_gap_add_divides_result_productcoefficientsbound + S (pfc_index_add_divides_result_productcoefficients) = (pfrd_plen_add_divides_result)) -> exists pfc_value_add_divides_result_productcoefficients. ((((exists ff_h_pfp_add_divides_result_productcoefficientsentry. ff_h_pfp_add_divides_result_productcoefficientsentry + S (pfc_value_add_divides_result_productcoefficients) = S ((S (pfc_index_add_divides_result_productcoefficients)) * pfrd_pc_add_divides_result)) /\ exists ff_q_pfp_add_divides_result_productcoefficientsentry. pfrd_pb_add_divides_result = ff_q_pfp_add_divides_result_productcoefficientsentry * S ((S (pfc_index_add_divides_result_productcoefficients)) * pfrd_pc_add_divides_result) + (pfc_value_add_divides_result_productcoefficients))) /\ ((exists pfc_terms_code_add_divides_result_productcoefficientscoefficient pfc_terms_scale_add_divides_result_productcoefficientscoefficient pfc_natural_sum_add_divides_result_productcoefficientscoefficient. ((forall pfc_index_add_divides_result_productcoefficientscoefficientdiagonal. (exists pfa_gap_add_divides_result_productcoefficientscoefficientdiagonalbound. pfa_gap_add_divides_result_productcoefficientscoefficientdiagonalbound + S (pfc_index_add_divides_result_productcoefficientscoefficientdiagonal) = (S (pfc_index_add_divides_result_productcoefficients))) -> exists pfc_value_add_divides_result_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_add_divides_result_productcoefficientscoefficientdiagonalentry. ff_h_pfp_add_divides_result_productcoefficientscoefficientdiagonalentry + S (pfc_value_add_divides_result_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_add_divides_result_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_add_divides_result_productcoefficientscoefficient)) /\ exists ff_q_pfp_add_divides_result_productcoefficientscoefficientdiagonalentry. pfc_terms_code_add_divides_result_productcoefficientscoefficient = ff_q_pfp_add_divides_result_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_add_divides_result_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_add_divides_result_productcoefficientscoefficient) + (pfc_value_add_divides_result_productcoefficientscoefficientdiagonal))) /\ ((exists pfc_complement_add_divides_result_productcoefficientscoefficientdiagonalterm pfc_left_add_divides_result_productcoefficientscoefficientdiagonalterm pfc_right_add_divides_result_productcoefficientscoefficientdiagonalterm. (((pfc_index_add_divides_result_productcoefficientscoefficientdiagonal)+pfc_complement_add_divides_result_productcoefficientscoefficientdiagonalterm=(pfc_index_add_divides_result_productcoefficients)) /\ ((((((exists pfa_gap_add_divides_result_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_add_divides_result_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_add_divides_result_productcoefficientscoefficientdiagonal) = (pfrd_qlen_add_divides_result)) /\ ((((exists ff_h_pfp_add_divides_result_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_add_divides_result_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_add_divides_result_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_add_divides_result_productcoefficientscoefficientdiagonal)) * pfrd_qc_add_divides_result)) /\ exists ff_q_pfp_add_divides_result_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_add_divides_result = ff_q_pfp_add_divides_result_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_add_divides_result_productcoefficientscoefficientdiagonal)) * pfrd_qc_add_divides_result) + (pfc_left_add_divides_result_productcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_add_divides_result_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_add_divides_result_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_add_divides_result)=(pfc_index_add_divides_result_productcoefficientscoefficientdiagonal)) /\ (((pfc_left_add_divides_result_productcoefficientscoefficientdiagonalterm)=0))))) /\ ((((((exists pfa_gap_add_divides_result_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_add_divides_result_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_add_divides_result_productcoefficientscoefficientdiagonalterm) = (J)) /\ ((((exists ff_h_pfp_add_divides_result_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_add_divides_result_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_add_divides_result_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_add_divides_result_productcoefficientscoefficientdiagonalterm)) * dc)) /\ exists ff_q_pfp_add_divides_result_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_add_divides_result_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_add_divides_result_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_add_divides_result_productcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_add_divides_result_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_add_divides_result_productcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_add_divides_result_productcoefficientscoefficientdiagonalterm)) /\ (((pfc_right_add_divides_result_productcoefficientscoefficientdiagonalterm)=0))))) /\ (((pfc_value_add_divides_result_productcoefficientscoefficientdiagonal)=pfc_left_add_divides_result_productcoefficientscoefficientdiagonalterm*pfc_right_add_divides_result_productcoefficientscoefficientdiagonalterm))))))))))) /\ (((exists fs_u_pfc_add_divides_result_productcoefficientscoefficientsum fs_v_pfc_add_divides_result_productcoefficientscoefficientsum. ((((exists fs_h_pfc_add_divides_result_productcoefficientscoefficientsum_body_start. fs_h_pfc_add_divides_result_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_add_divides_result_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_add_divides_result_productcoefficientscoefficientsum_body_start. fs_u_pfc_add_divides_result_productcoefficientscoefficientsum = fs_q_pfc_add_divides_result_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_add_divides_result_productcoefficientscoefficientsum) + (0))) /\ ((((exists fs_h_pfc_add_divides_result_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_add_divides_result_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_add_divides_result_productcoefficientscoefficient) = S ((S (S (pfc_index_add_divides_result_productcoefficients))) * fs_v_pfc_add_divides_result_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_add_divides_result_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_add_divides_result_productcoefficientscoefficientsum = fs_q_pfc_add_divides_result_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_add_divides_result_productcoefficients))) * fs_v_pfc_add_divides_result_productcoefficientscoefficientsum) + (pfc_natural_sum_add_divides_result_productcoefficientscoefficient))) /\ forall fs_i_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps = S (pfc_index_add_divides_result_productcoefficients)) -> exists fs_a_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps fs_r_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps fs_s_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_add_divides_result_productcoefficientscoefficient)) /\ exists fs_q_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_add_divides_result_productcoefficientscoefficient = fs_q_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_add_divides_result_productcoefficientscoefficient) + (fs_a_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_add_divides_result_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_add_divides_result_productcoefficientscoefficientsum = fs_q_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_add_divides_result_productcoefficientscoefficientsum) + (fs_r_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_add_divides_result_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_add_divides_result_productcoefficientscoefficientsum = fs_q_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_add_divides_result_productcoefficientscoefficientsum) + (fs_s_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps))) /\ fs_s_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps = fs_r_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps + fs_a_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps)))))) /\ ((((exists pfa_gap_add_divides_result_productcoefficientscoefficientresiduebound. pfa_gap_add_divides_result_productcoefficientscoefficientresiduebound + S (pfc_value_add_divides_result_productcoefficients) = (p)) /\ ((exists pfa_offset_left_add_divides_result_productcoefficientscoefficientresiduecongruence pfa_offset_right_add_divides_result_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_add_divides_result_productcoefficientscoefficient) + (p) * pfa_offset_left_add_divides_result_productcoefficientscoefficientresiduecongruence = (pfc_value_add_divides_result_productcoefficients) + (p) * pfa_offset_right_add_divides_result_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\ ((forall pfrep_power_add_divides_result_target pfrep_left_add_divides_result_target pfrep_right_add_divides_result_target. ((exists pfrep_position_add_divides_result_targetfirst. ((pfrep_position_add_divides_result_targetfirst+S (pfrep_power_add_divides_result_target)=(pfrd_plen_add_divides_result)) /\ ((((exists ff_h_pfp_add_divides_result_targetfirstentry. ff_h_pfp_add_divides_result_targetfirstentry + S (pfrep_left_add_divides_result_target) = S ((S (pfrep_position_add_divides_result_targetfirst)) * pfrd_pc_add_divides_result)) /\ exists ff_q_pfp_add_divides_result_targetfirstentry. pfrd_pb_add_divides_result = ff_q_pfp_add_divides_result_targetfirstentry * S ((S (pfrep_position_add_divides_result_targetfirst)) * pfrd_pc_add_divides_result) + (pfrep_left_add_divides_result_target)))))) \/ (((exists pfrep_gap_add_divides_result_targetfirstoutside. pfrep_gap_add_divides_result_targetfirstoutside+(pfrd_plen_add_divides_result)=(pfrep_power_add_divides_result_target)) /\ (((pfrep_left_add_divides_result_target)=0))))) -> ((exists pfrep_position_add_divides_result_targetsecond. ((pfrep_position_add_divides_result_targetsecond+S (pfrep_power_add_divides_result_target)=(N)) /\ ((((exists ff_h_pfp_add_divides_result_targetsecondentry. ff_h_pfp_add_divides_result_targetsecondentry + S (pfrep_right_add_divides_result_target) = S ((S (pfrep_position_add_divides_result_targetsecond)) * rc)) /\ exists ff_q_pfp_add_divides_result_targetsecondentry. rb = ff_q_pfp_add_divides_result_targetsecondentry * S ((S (pfrep_position_add_divides_result_targetsecond)) * rc) + (pfrep_right_add_divides_result_target)))))) \/ (((exists pfrep_gap_add_divides_result_targetsecondoutside. pfrep_gap_add_divides_result_targetsecondoutside+(N)=(pfrep_power_add_divides_result_target)) /\ (((pfrep_right_add_divides_result_target)=0))))) -> pfrep_left_add_divides_result_target=pfrep_right_add_divides_result_target)))))))