prime_field_polynomial_quotient_step_prefix_functional

Exact expanded first-order arithmetic statement

forall p k ab ac bb bc M qb qc QB QC i q r. (forall mdr_i_pfp_step_unique_prefix mdr_a_pfp_step_unique_prefix. (exists mdr_gap_pfp_step_unique_prefixb. mdr_gap_pfp_step_unique_prefixb + S (mdr_i_pfp_step_unique_prefix) = (i)) -> (((exists ff_h_mdr_pfp_step_unique_prefixo. ff_h_mdr_pfp_step_unique_prefixo + S (mdr_a_pfp_step_unique_prefix) = S ((S (mdr_i_pfp_step_unique_prefix)) * qc)) /\ exists ff_q_mdr_pfp_step_unique_prefixo. qb = ff_q_mdr_pfp_step_unique_prefixo * S ((S (mdr_i_pfp_step_unique_prefix)) * qc) + (mdr_a_pfp_step_unique_prefix))) -> (((exists ff_h_mdr_pfp_step_unique_prefixn. ff_h_mdr_pfp_step_unique_prefixn + S (mdr_a_pfp_step_unique_prefix) = S ((S (mdr_i_pfp_step_unique_prefix)) * QC)) /\ exists ff_q_mdr_pfp_step_unique_prefixn. QB = ff_q_mdr_pfp_step_unique_prefixn * S ((S (mdr_i_pfp_step_unique_prefix)) * QC) + (mdr_a_pfp_step_unique_prefix)))) -> (exists pfd_input_step_unique_old pfd_previous_step_unique_old pfd_difference_step_unique_old. ((((exists ff_h_pfp_step_unique_oldinput. ff_h_pfp_step_unique_oldinput + S (pfd_input_step_unique_old) = S ((S (i)) * ac)) /\ exists ff_q_pfp_step_unique_oldinput. ab = ff_q_pfp_step_unique_oldinput * S ((S (i)) * ac) + (pfd_input_step_unique_old))) /\ (((exists pfc_terms_code_step_unique_oldprevious pfc_terms_scale_step_unique_oldprevious pfc_natural_sum_step_unique_oldprevious. ((forall pfc_index_step_unique_oldpreviousdiagonal. (exists pfa_gap_step_unique_oldpreviousdiagonalbound. pfa_gap_step_unique_oldpreviousdiagonalbound + S (pfc_index_step_unique_oldpreviousdiagonal) = (S (i))) -> exists pfc_value_step_unique_oldpreviousdiagonal. ((((exists ff_h_pfp_step_unique_oldpreviousdiagonalentry. ff_h_pfp_step_unique_oldpreviousdiagonalentry + S (pfc_value_step_unique_oldpreviousdiagonal) = S ((S (pfc_index_step_unique_oldpreviousdiagonal)) * pfc_terms_scale_step_unique_oldprevious)) /\ exists ff_q_pfp_step_unique_oldpreviousdiagonalentry. pfc_terms_code_step_unique_oldprevious = ff_q_pfp_step_unique_oldpreviousdiagonalentry * S ((S (pfc_index_step_unique_oldpreviousdiagonal)) * pfc_terms_scale_step_unique_oldprevious) + (pfc_value_step_unique_oldpreviousdiagonal))) /\ ((exists pfc_complement_step_unique_oldpreviousdiagonalterm pfc_left_step_unique_oldpreviousdiagonalterm pfc_right_step_unique_oldpreviousdiagonalterm. (((pfc_index_step_unique_oldpreviousdiagonal)+pfc_complement_step_unique_oldpreviousdiagonalterm=(i)) /\ ((((((exists pfa_gap_step_unique_oldpreviousdiagonaltermleftinside. pfa_gap_step_unique_oldpreviousdiagonaltermleftinside + S (pfc_index_step_unique_oldpreviousdiagonal) = (i)) /\ ((((exists ff_h_pfp_step_unique_oldpreviousdiagonaltermleftentry. ff_h_pfp_step_unique_oldpreviousdiagonaltermleftentry + S (pfc_left_step_unique_oldpreviousdiagonalterm) = S ((S (pfc_index_step_unique_oldpreviousdiagonal)) * qc)) /\ exists ff_q_pfp_step_unique_oldpreviousdiagonaltermleftentry. qb = ff_q_pfp_step_unique_oldpreviousdiagonaltermleftentry * S ((S (pfc_index_step_unique_oldpreviousdiagonal)) * qc) + (pfc_left_step_unique_oldpreviousdiagonalterm)))))) \/ (((exists pfc_gap_step_unique_oldpreviousdiagonaltermleftoutside. pfc_gap_step_unique_oldpreviousdiagonaltermleftoutside+(i)=(pfc_index_step_unique_oldpreviousdiagonal)) /\ (((pfc_left_step_unique_oldpreviousdiagonalterm)=0))))) /\ ((((((exists pfa_gap_step_unique_oldpreviousdiagonaltermrightinside. pfa_gap_step_unique_oldpreviousdiagonaltermrightinside + S (pfc_complement_step_unique_oldpreviousdiagonalterm) = (M)) /\ ((((exists ff_h_pfp_step_unique_oldpreviousdiagonaltermrightentry. ff_h_pfp_step_unique_oldpreviousdiagonaltermrightentry + S (pfc_right_step_unique_oldpreviousdiagonalterm) = S ((S (pfc_complement_step_unique_oldpreviousdiagonalterm)) * bc)) /\ exists ff_q_pfp_step_unique_oldpreviousdiagonaltermrightentry. bb = ff_q_pfp_step_unique_oldpreviousdiagonaltermrightentry * S ((S (pfc_complement_step_unique_oldpreviousdiagonalterm)) * bc) + (pfc_right_step_unique_oldpreviousdiagonalterm)))))) \/ (((exists pfc_gap_step_unique_oldpreviousdiagonaltermrightoutside. pfc_gap_step_unique_oldpreviousdiagonaltermrightoutside+(M)=(pfc_complement_step_unique_oldpreviousdiagonalterm)) /\ (((pfc_right_step_unique_oldpreviousdiagonalterm)=0))))) /\ (((pfc_value_step_unique_oldpreviousdiagonal)=pfc_left_step_unique_oldpreviousdiagonalterm*pfc_right_step_unique_oldpreviousdiagonalterm))))))))))) /\ (((exists fs_u_pfc_step_unique_oldprevioussum fs_v_pfc_step_unique_oldprevioussum. ((((exists fs_h_pfc_step_unique_oldprevioussum_body_start. fs_h_pfc_step_unique_oldprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_step_unique_oldprevioussum)) /\ exists fs_q_pfc_step_unique_oldprevioussum_body_start. fs_u_pfc_step_unique_oldprevioussum = fs_q_pfc_step_unique_oldprevioussum_body_start * S ((S (0)) * fs_v_pfc_step_unique_oldprevioussum) + (0))) /\ ((((exists fs_h_pfc_step_unique_oldprevioussum_body_terminal. fs_h_pfc_step_unique_oldprevioussum_body_terminal + S (pfc_natural_sum_step_unique_oldprevious) = S ((S (S (i))) * fs_v_pfc_step_unique_oldprevioussum)) /\ exists fs_q_pfc_step_unique_oldprevioussum_body_terminal. fs_u_pfc_step_unique_oldprevioussum = fs_q_pfc_step_unique_oldprevioussum_body_terminal * S ((S (S (i))) * fs_v_pfc_step_unique_oldprevioussum) + (pfc_natural_sum_step_unique_oldprevious))) /\ forall fs_i_pfc_step_unique_oldprevioussum_body_steps. (exists fs_lt_pfc_step_unique_oldprevioussum_body_steps_bound. fs_lt_pfc_step_unique_oldprevioussum_body_steps_bound + S fs_i_pfc_step_unique_oldprevioussum_body_steps = S (i)) -> exists fs_a_pfc_step_unique_oldprevioussum_body_steps fs_r_pfc_step_unique_oldprevioussum_body_steps fs_s_pfc_step_unique_oldprevioussum_body_steps. ((((exists fs_h_pfc_step_unique_oldprevioussum_body_steps_summand. fs_h_pfc_step_unique_oldprevioussum_body_steps_summand + S (fs_a_pfc_step_unique_oldprevioussum_body_steps) = S ((S (fs_i_pfc_step_unique_oldprevioussum_body_steps)) * pfc_terms_scale_step_unique_oldprevious)) /\ exists fs_q_pfc_step_unique_oldprevioussum_body_steps_summand. pfc_terms_code_step_unique_oldprevious = fs_q_pfc_step_unique_oldprevioussum_body_steps_summand * S ((S (fs_i_pfc_step_unique_oldprevioussum_body_steps)) * pfc_terms_scale_step_unique_oldprevious) + (fs_a_pfc_step_unique_oldprevioussum_body_steps))) /\ ((((exists fs_h_pfc_step_unique_oldprevioussum_body_steps_partial. fs_h_pfc_step_unique_oldprevioussum_body_steps_partial + S (fs_r_pfc_step_unique_oldprevioussum_body_steps) = S ((S (fs_i_pfc_step_unique_oldprevioussum_body_steps)) * fs_v_pfc_step_unique_oldprevioussum)) /\ exists fs_q_pfc_step_unique_oldprevioussum_body_steps_partial. fs_u_pfc_step_unique_oldprevioussum = fs_q_pfc_step_unique_oldprevioussum_body_steps_partial * S ((S (fs_i_pfc_step_unique_oldprevioussum_body_steps)) * fs_v_pfc_step_unique_oldprevioussum) + (fs_r_pfc_step_unique_oldprevioussum_body_steps))) /\ ((((exists fs_h_pfc_step_unique_oldprevioussum_body_steps_successor. fs_h_pfc_step_unique_oldprevioussum_body_steps_successor + S (fs_s_pfc_step_unique_oldprevioussum_body_steps) = S ((S (S fs_i_pfc_step_unique_oldprevioussum_body_steps)) * fs_v_pfc_step_unique_oldprevioussum)) /\ exists fs_q_pfc_step_unique_oldprevioussum_body_steps_successor. fs_u_pfc_step_unique_oldprevioussum = fs_q_pfc_step_unique_oldprevioussum_body_steps_successor * S ((S (S fs_i_pfc_step_unique_oldprevioussum_body_steps)) * fs_v_pfc_step_unique_oldprevioussum) + (fs_s_pfc_step_unique_oldprevioussum_body_steps))) /\ fs_s_pfc_step_unique_oldprevioussum_body_steps = fs_r_pfc_step_unique_oldprevioussum_body_steps + fs_a_pfc_step_unique_oldprevioussum_body_steps)))))) /\ ((((exists pfa_gap_step_unique_oldpreviousresiduebound. pfa_gap_step_unique_oldpreviousresiduebound + S (pfd_previous_step_unique_old) = (p)) /\ ((exists pfa_offset_left_step_unique_oldpreviousresiduecongruence pfa_offset_right_step_unique_oldpreviousresiduecongruence. (pfc_natural_sum_step_unique_oldprevious) + (p) * pfa_offset_left_step_unique_oldpreviousresiduecongruence = (pfd_previous_step_unique_old) + (p) * pfa_offset_right_step_unique_oldpreviousresiduecongruence))))))))) /\ (((((exists pfa_gap_step_unique_oldsubtractleft. pfa_gap_step_unique_oldsubtractleft + S (pfd_previous_step_unique_old) = (p)) /\ (((exists pfa_gap_step_unique_oldsubtractright. pfa_gap_step_unique_oldsubtractright + S (pfd_difference_step_unique_old) = (p)) /\ ((((exists pfa_gap_step_unique_oldsubtractresultbound. pfa_gap_step_unique_oldsubtractresultbound + S (pfd_input_step_unique_old) = (p)) /\ ((exists pfa_offset_left_step_unique_oldsubtractresultcongruence pfa_offset_right_step_unique_oldsubtractresultcongruence. ((pfd_previous_step_unique_old) + (pfd_difference_step_unique_old)) + (p) * pfa_offset_left_step_unique_oldsubtractresultcongruence = (pfd_input_step_unique_old) + (p) * pfa_offset_right_step_unique_oldsubtractresultcongruence))))))))) /\ ((((exists pfa_gap_step_unique_oldmultiplyleft. pfa_gap_step_unique_oldmultiplyleft + S (k) = (p)) /\ (((exists pfa_gap_step_unique_oldmultiplyright. pfa_gap_step_unique_oldmultiplyright + S (pfd_difference_step_unique_old) = (p)) /\ ((((exists pfa_gap_step_unique_oldmultiplyresultbound. pfa_gap_step_unique_oldmultiplyresultbound + S (q) = (p)) /\ ((exists pfa_offset_left_step_unique_oldmultiplyresultcongruence pfa_offset_right_step_unique_oldmultiplyresultcongruence. ((k) * (pfd_difference_step_unique_old)) + (p) * pfa_offset_left_step_unique_oldmultiplyresultcongruence = (q) + (p) * pfa_offset_right_step_unique_oldmultiplyresultcongruence)))))))))))))))) -> (exists pfd_input_step_unique_new pfd_previous_step_unique_new pfd_difference_step_unique_new. ((((exists ff_h_pfp_step_unique_newinput. ff_h_pfp_step_unique_newinput + S (pfd_input_step_unique_new) = S ((S (i)) * ac)) /\ exists ff_q_pfp_step_unique_newinput. ab = ff_q_pfp_step_unique_newinput * S ((S (i)) * ac) + (pfd_input_step_unique_new))) /\ (((exists pfc_terms_code_step_unique_newprevious pfc_terms_scale_step_unique_newprevious pfc_natural_sum_step_unique_newprevious. ((forall pfc_index_step_unique_newpreviousdiagonal. (exists pfa_gap_step_unique_newpreviousdiagonalbound. pfa_gap_step_unique_newpreviousdiagonalbound + S (pfc_index_step_unique_newpreviousdiagonal) = (S (i))) -> exists pfc_value_step_unique_newpreviousdiagonal. ((((exists ff_h_pfp_step_unique_newpreviousdiagonalentry. ff_h_pfp_step_unique_newpreviousdiagonalentry + S (pfc_value_step_unique_newpreviousdiagonal) = S ((S (pfc_index_step_unique_newpreviousdiagonal)) * pfc_terms_scale_step_unique_newprevious)) /\ exists ff_q_pfp_step_unique_newpreviousdiagonalentry. pfc_terms_code_step_unique_newprevious = ff_q_pfp_step_unique_newpreviousdiagonalentry * S ((S (pfc_index_step_unique_newpreviousdiagonal)) * pfc_terms_scale_step_unique_newprevious) + (pfc_value_step_unique_newpreviousdiagonal))) /\ ((exists pfc_complement_step_unique_newpreviousdiagonalterm pfc_left_step_unique_newpreviousdiagonalterm pfc_right_step_unique_newpreviousdiagonalterm. (((pfc_index_step_unique_newpreviousdiagonal)+pfc_complement_step_unique_newpreviousdiagonalterm=(i)) /\ ((((((exists pfa_gap_step_unique_newpreviousdiagonaltermleftinside. pfa_gap_step_unique_newpreviousdiagonaltermleftinside + S (pfc_index_step_unique_newpreviousdiagonal) = (i)) /\ ((((exists ff_h_pfp_step_unique_newpreviousdiagonaltermleftentry. ff_h_pfp_step_unique_newpreviousdiagonaltermleftentry + S (pfc_left_step_unique_newpreviousdiagonalterm) = S ((S (pfc_index_step_unique_newpreviousdiagonal)) * QC)) /\ exists ff_q_pfp_step_unique_newpreviousdiagonaltermleftentry. QB = ff_q_pfp_step_unique_newpreviousdiagonaltermleftentry * S ((S (pfc_index_step_unique_newpreviousdiagonal)) * QC) + (pfc_left_step_unique_newpreviousdiagonalterm)))))) \/ (((exists pfc_gap_step_unique_newpreviousdiagonaltermleftoutside. pfc_gap_step_unique_newpreviousdiagonaltermleftoutside+(i)=(pfc_index_step_unique_newpreviousdiagonal)) /\ (((pfc_left_step_unique_newpreviousdiagonalterm)=0))))) /\ ((((((exists pfa_gap_step_unique_newpreviousdiagonaltermrightinside. pfa_gap_step_unique_newpreviousdiagonaltermrightinside + S (pfc_complement_step_unique_newpreviousdiagonalterm) = (M)) /\ ((((exists ff_h_pfp_step_unique_newpreviousdiagonaltermrightentry. ff_h_pfp_step_unique_newpreviousdiagonaltermrightentry + S (pfc_right_step_unique_newpreviousdiagonalterm) = S ((S (pfc_complement_step_unique_newpreviousdiagonalterm)) * bc)) /\ exists ff_q_pfp_step_unique_newpreviousdiagonaltermrightentry. bb = ff_q_pfp_step_unique_newpreviousdiagonaltermrightentry * S ((S (pfc_complement_step_unique_newpreviousdiagonalterm)) * bc) + (pfc_right_step_unique_newpreviousdiagonalterm)))))) \/ (((exists pfc_gap_step_unique_newpreviousdiagonaltermrightoutside. pfc_gap_step_unique_newpreviousdiagonaltermrightoutside+(M)=(pfc_complement_step_unique_newpreviousdiagonalterm)) /\ (((pfc_right_step_unique_newpreviousdiagonalterm)=0))))) /\ (((pfc_value_step_unique_newpreviousdiagonal)=pfc_left_step_unique_newpreviousdiagonalterm*pfc_right_step_unique_newpreviousdiagonalterm))))))))))) /\ (((exists fs_u_pfc_step_unique_newprevioussum fs_v_pfc_step_unique_newprevioussum. ((((exists fs_h_pfc_step_unique_newprevioussum_body_start. fs_h_pfc_step_unique_newprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_step_unique_newprevioussum)) /\ exists fs_q_pfc_step_unique_newprevioussum_body_start. fs_u_pfc_step_unique_newprevioussum = fs_q_pfc_step_unique_newprevioussum_body_start * S ((S (0)) * fs_v_pfc_step_unique_newprevioussum) + (0))) /\ ((((exists fs_h_pfc_step_unique_newprevioussum_body_terminal. fs_h_pfc_step_unique_newprevioussum_body_terminal + S (pfc_natural_sum_step_unique_newprevious) = S ((S (S (i))) * fs_v_pfc_step_unique_newprevioussum)) /\ exists fs_q_pfc_step_unique_newprevioussum_body_terminal. fs_u_pfc_step_unique_newprevioussum = fs_q_pfc_step_unique_newprevioussum_body_terminal * S ((S (S (i))) * fs_v_pfc_step_unique_newprevioussum) + (pfc_natural_sum_step_unique_newprevious))) /\ forall fs_i_pfc_step_unique_newprevioussum_body_steps. (exists fs_lt_pfc_step_unique_newprevioussum_body_steps_bound. fs_lt_pfc_step_unique_newprevioussum_body_steps_bound + S fs_i_pfc_step_unique_newprevioussum_body_steps = S (i)) -> exists fs_a_pfc_step_unique_newprevioussum_body_steps fs_r_pfc_step_unique_newprevioussum_body_steps fs_s_pfc_step_unique_newprevioussum_body_steps. ((((exists fs_h_pfc_step_unique_newprevioussum_body_steps_summand. fs_h_pfc_step_unique_newprevioussum_body_steps_summand + S (fs_a_pfc_step_unique_newprevioussum_body_steps) = S ((S (fs_i_pfc_step_unique_newprevioussum_body_steps)) * pfc_terms_scale_step_unique_newprevious)) /\ exists fs_q_pfc_step_unique_newprevioussum_body_steps_summand. pfc_terms_code_step_unique_newprevious = fs_q_pfc_step_unique_newprevioussum_body_steps_summand * S ((S (fs_i_pfc_step_unique_newprevioussum_body_steps)) * pfc_terms_scale_step_unique_newprevious) + (fs_a_pfc_step_unique_newprevioussum_body_steps))) /\ ((((exists fs_h_pfc_step_unique_newprevioussum_body_steps_partial. fs_h_pfc_step_unique_newprevioussum_body_steps_partial + S (fs_r_pfc_step_unique_newprevioussum_body_steps) = S ((S (fs_i_pfc_step_unique_newprevioussum_body_steps)) * fs_v_pfc_step_unique_newprevioussum)) /\ exists fs_q_pfc_step_unique_newprevioussum_body_steps_partial. fs_u_pfc_step_unique_newprevioussum = fs_q_pfc_step_unique_newprevioussum_body_steps_partial * S ((S (fs_i_pfc_step_unique_newprevioussum_body_steps)) * fs_v_pfc_step_unique_newprevioussum) + (fs_r_pfc_step_unique_newprevioussum_body_steps))) /\ ((((exists fs_h_pfc_step_unique_newprevioussum_body_steps_successor. fs_h_pfc_step_unique_newprevioussum_body_steps_successor + S (fs_s_pfc_step_unique_newprevioussum_body_steps) = S ((S (S fs_i_pfc_step_unique_newprevioussum_body_steps)) * fs_v_pfc_step_unique_newprevioussum)) /\ exists fs_q_pfc_step_unique_newprevioussum_body_steps_successor. fs_u_pfc_step_unique_newprevioussum = fs_q_pfc_step_unique_newprevioussum_body_steps_successor * S ((S (S fs_i_pfc_step_unique_newprevioussum_body_steps)) * fs_v_pfc_step_unique_newprevioussum) + (fs_s_pfc_step_unique_newprevioussum_body_steps))) /\ fs_s_pfc_step_unique_newprevioussum_body_steps = fs_r_pfc_step_unique_newprevioussum_body_steps + fs_a_pfc_step_unique_newprevioussum_body_steps)))))) /\ ((((exists pfa_gap_step_unique_newpreviousresiduebound. pfa_gap_step_unique_newpreviousresiduebound + S (pfd_previous_step_unique_new) = (p)) /\ ((exists pfa_offset_left_step_unique_newpreviousresiduecongruence pfa_offset_right_step_unique_newpreviousresiduecongruence. (pfc_natural_sum_step_unique_newprevious) + (p) * pfa_offset_left_step_unique_newpreviousresiduecongruence = (pfd_previous_step_unique_new) + (p) * pfa_offset_right_step_unique_newpreviousresiduecongruence))))))))) /\ (((((exists pfa_gap_step_unique_newsubtractleft. pfa_gap_step_unique_newsubtractleft + S (pfd_previous_step_unique_new) = (p)) /\ (((exists pfa_gap_step_unique_newsubtractright. pfa_gap_step_unique_newsubtractright + S (pfd_difference_step_unique_new) = (p)) /\ ((((exists pfa_gap_step_unique_newsubtractresultbound. pfa_gap_step_unique_newsubtractresultbound + S (pfd_input_step_unique_new) = (p)) /\ ((exists pfa_offset_left_step_unique_newsubtractresultcongruence pfa_offset_right_step_unique_newsubtractresultcongruence. ((pfd_previous_step_unique_new) + (pfd_difference_step_unique_new)) + (p) * pfa_offset_left_step_unique_newsubtractresultcongruence = (pfd_input_step_unique_new) + (p) * pfa_offset_right_step_unique_newsubtractresultcongruence))))))))) /\ ((((exists pfa_gap_step_unique_newmultiplyleft. pfa_gap_step_unique_newmultiplyleft + S (k) = (p)) /\ (((exists pfa_gap_step_unique_newmultiplyright. pfa_gap_step_unique_newmultiplyright + S (pfd_difference_step_unique_new) = (p)) /\ ((((exists pfa_gap_step_unique_newmultiplyresultbound. pfa_gap_step_unique_newmultiplyresultbound + S (r) = (p)) /\ ((exists pfa_offset_left_step_unique_newmultiplyresultcongruence pfa_offset_right_step_unique_newmultiplyresultcongruence. ((k) * (pfd_difference_step_unique_new)) + (p) * pfa_offset_left_step_unique_newmultiplyresultcongruence = (r) + (p) * pfa_offset_right_step_unique_newmultiplyresultcongruence)))))))))))))))) -> q=r