Two actual executions on the same annotated inputs agree in quotient and remainder lengths and decoded coefficients, not in beta codes.
Alpha v34 checked-use · first admitted v33 · independently kernel and Lean verified; not Stable
Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.
Coefficients are highest-degree-first. The divisor has a nonzero decoded head; primality supplies its actual inverse. Empty quotients and remainders are included. Functionality compares the constructed execution lengths and decoded coefficients, never arbitrary beta codes. Formal polynomial equivalence compares every coefficient, not evaluations on a finite field. The formal identity and remainder-degree bound are proved separately, not assumed by the execution graph. Arbitrary quotient/remainder-pair uniqueness from a formal identity, multiplication associativity, gcd/Bezout, irreducible-polynomial existence, and the full G091 prime-power-field goal remain open. The seven displayed new names are conservative first-order notation, not new kernel primitives.
forall p ab ac L bb bc d qb qc q rb rc R QB QC Q RB RC K. (((forall fom_index_pfp_execution_unique_firstinput. (exists fom_gap_pfp_execution_unique_firstinput_index_bound. fom_gap_pfp_execution_unique_firstinput_index_bound + S (fom_index_pfp_execution_unique_firstinput) = L) -> exists fom_value_pfp_execution_unique_firstinput. ((((exists fom_beta_height_pfp_execution_unique_firstinput_entry. fom_beta_height_pfp_execution_unique_firstinput_entry + S (fom_value_pfp_execution_unique_firstinput) = S ((S (fom_index_pfp_execution_unique_firstinput)) * ac)) /\ exists fom_beta_quotient_pfp_execution_unique_firstinput_entry. ab = fom_beta_quotient_pfp_execution_unique_firstinput_entry * S ((S (fom_index_pfp_execution_unique_firstinput)) * ac) + (fom_value_pfp_execution_unique_firstinput))) /\ (exists fom_gap_pfp_execution_unique_firstinput_value_bound. fom_gap_pfp_execution_unique_firstinput_value_bound + S (fom_value_pfp_execution_unique_firstinput) = p))) /\ (((forall fom_index_pfp_execution_unique_firstdivisor. (exists fom_gap_pfp_execution_unique_firstdivisor_index_bound. fom_gap_pfp_execution_unique_firstdivisor_index_bound + S (fom_index_pfp_execution_unique_firstdivisor) = S (d)) -> exists fom_value_pfp_execution_unique_firstdivisor. ((((exists fom_beta_height_pfp_execution_unique_firstdivisor_entry. fom_beta_height_pfp_execution_unique_firstdivisor_entry + S (fom_value_pfp_execution_unique_firstdivisor) = S ((S (fom_index_pfp_execution_unique_firstdivisor)) * bc)) /\ exists fom_beta_quotient_pfp_execution_unique_firstdivisor_entry. bb = fom_beta_quotient_pfp_execution_unique_firstdivisor_entry * S ((S (fom_index_pfp_execution_unique_firstdivisor)) * bc) + (fom_value_pfp_execution_unique_firstdivisor))) /\ (exists fom_gap_pfp_execution_unique_firstdivisor_value_bound. fom_gap_pfp_execution_unique_firstdivisor_value_bound + S (fom_value_pfp_execution_unique_firstdivisor) = p))) /\ (((((((q)=0) /\ ((exists pfc_gap_execution_unique_firstlengthshort. pfc_gap_execution_unique_firstlengthshort+(L)=(d))))) \/ (((~((q)=0)) /\ (((q)+(d)=(L)))))) /\ ((exists pfd_head_execution_unique_first pfd_inverse_execution_unique_first pfd_product_code_execution_unique_first pfd_product_scale_execution_unique_first pfd_residual_code_execution_unique_first pfd_residual_scale_execution_unique_first pfd_cut_execution_unique_first. ((((exists ff_h_pfp_execution_unique_firsthead. ff_h_pfp_execution_unique_firsthead + S (pfd_head_execution_unique_first) = S ((S (0)) * bc)) /\ exists ff_q_pfp_execution_unique_firsthead. bb = ff_q_pfp_execution_unique_firsthead * S ((S (0)) * bc) + (pfd_head_execution_unique_first))) /\ (((((~((pfd_head_execution_unique_first) = 0)) /\ ((((exists pfa_gap_execution_unique_firstinversemultiplicationleft. pfa_gap_execution_unique_firstinversemultiplicationleft + S (pfd_head_execution_unique_first) = (p)) /\ (((exists pfa_gap_execution_unique_firstinversemultiplicationright. pfa_gap_execution_unique_firstinversemultiplicationright + S (pfd_inverse_execution_unique_first) = (p)) /\ ((((exists pfa_gap_execution_unique_firstinversemultiplicationresultbound. pfa_gap_execution_unique_firstinversemultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_execution_unique_firstinversemultiplicationresultcongruence pfa_offset_right_execution_unique_firstinversemultiplicationresultcongruence. ((pfd_head_execution_unique_first) * (pfd_inverse_execution_unique_first)) + (p) * pfa_offset_left_execution_unique_firstinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_execution_unique_firstinversemultiplicationresultcongruence)))))))))))) /\ (((forall pfd_index_execution_unique_firstquotient. (exists pfa_gap_execution_unique_firstquotientbound. pfa_gap_execution_unique_firstquotientbound + S (pfd_index_execution_unique_firstquotient) = (q)) -> exists pfd_value_execution_unique_firstquotient. ((((exists ff_h_pfp_execution_unique_firstquotiententry. ff_h_pfp_execution_unique_firstquotiententry + S (pfd_value_execution_unique_firstquotient) = S ((S (pfd_index_execution_unique_firstquotient)) * qc)) /\ exists ff_q_pfp_execution_unique_firstquotiententry. qb = ff_q_pfp_execution_unique_firstquotiententry * S ((S (pfd_index_execution_unique_firstquotient)) * qc) + (pfd_value_execution_unique_firstquotient))) /\ ((exists pfd_input_execution_unique_firstquotientstep pfd_previous_execution_unique_firstquotientstep pfd_difference_execution_unique_firstquotientstep. ((((exists ff_h_pfp_execution_unique_firstquotientstepinput. ff_h_pfp_execution_unique_firstquotientstepinput + S (pfd_input_execution_unique_firstquotientstep) = S ((S (pfd_index_execution_unique_firstquotient)) * ac)) /\ exists ff_q_pfp_execution_unique_firstquotientstepinput. ab = ff_q_pfp_execution_unique_firstquotientstepinput * S ((S (pfd_index_execution_unique_firstquotient)) * ac) + (pfd_input_execution_unique_firstquotientstep))) /\ (((exists pfc_terms_code_execution_unique_firstquotientstepprevious pfc_terms_scale_execution_unique_firstquotientstepprevious pfc_natural_sum_execution_unique_firstquotientstepprevious. ((forall pfc_index_execution_unique_firstquotientsteppreviousdiagonal. (exists pfa_gap_execution_unique_firstquotientsteppreviousdiagonalbound. pfa_gap_execution_unique_firstquotientsteppreviousdiagonalbound + S (pfc_index_execution_unique_firstquotientsteppreviousdiagonal) = (S (pfd_index_execution_unique_firstquotient))) -> exists pfc_value_execution_unique_firstquotientsteppreviousdiagonal. ((((exists ff_h_pfp_execution_unique_firstquotientsteppreviousdiagonalentry. ff_h_pfp_execution_unique_firstquotientsteppreviousdiagonalentry + S (pfc_value_execution_unique_firstquotientsteppreviousdiagonal) = S ((S (pfc_index_execution_unique_firstquotientsteppreviousdiagonal)) * pfc_terms_scale_execution_unique_firstquotientstepprevious)) /\ exists ff_q_pfp_execution_unique_firstquotientsteppreviousdiagonalentry. pfc_terms_code_execution_unique_firstquotientstepprevious = ff_q_pfp_execution_unique_firstquotientsteppreviousdiagonalentry * S ((S (pfc_index_execution_unique_firstquotientsteppreviousdiagonal)) * pfc_terms_scale_execution_unique_firstquotientstepprevious) + (pfc_value_execution_unique_firstquotientsteppreviousdiagonal))) /\ ((exists pfc_complement_execution_unique_firstquotientsteppreviousdiagonalterm pfc_left_execution_unique_firstquotientsteppreviousdiagonalterm pfc_right_execution_unique_firstquotientsteppreviousdiagonalterm. (((pfc_index_execution_unique_firstquotientsteppreviousdiagonal)+pfc_complement_execution_unique_firstquotientsteppreviousdiagonalterm=(pfd_index_execution_unique_firstquotient)) /\ ((((((exists pfa_gap_execution_unique_firstquotientsteppreviousdiagonaltermleftinside. pfa_gap_execution_unique_firstquotientsteppreviousdiagonaltermleftinside + S (pfc_index_execution_unique_firstquotientsteppreviousdiagonal) = (pfd_index_execution_unique_firstquotient)) /\ ((((exists ff_h_pfp_execution_unique_firstquotientsteppreviousdiagonaltermleftentry. ff_h_pfp_execution_unique_firstquotientsteppreviousdiagonaltermleftentry + S (pfc_left_execution_unique_firstquotientsteppreviousdiagonalterm) = S ((S (pfc_index_execution_unique_firstquotientsteppreviousdiagonal)) * qc)) /\ exists ff_q_pfp_execution_unique_firstquotientsteppreviousdiagonaltermleftentry. qb = ff_q_pfp_execution_unique_firstquotientsteppreviousdiagonaltermleftentry * S ((S (pfc_index_execution_unique_firstquotientsteppreviousdiagonal)) * qc) + (pfc_left_execution_unique_firstquotientsteppreviousdiagonalterm)))))) \/ (((exists pfc_gap_execution_unique_firstquotientsteppreviousdiagonaltermleftoutside. pfc_gap_execution_unique_firstquotientsteppreviousdiagonaltermleftoutside+(pfd_index_execution_unique_firstquotient)=(pfc_index_execution_unique_firstquotientsteppreviousdiagonal)) /\ (((pfc_left_execution_unique_firstquotientsteppreviousdiagonalterm)=0))))) /\ ((((((exists pfa_gap_execution_unique_firstquotientsteppreviousdiagonaltermrightinside. pfa_gap_execution_unique_firstquotientsteppreviousdiagonaltermrightinside + S (pfc_complement_execution_unique_firstquotientsteppreviousdiagonalterm) = (S (d))) /\ ((((exists ff_h_pfp_execution_unique_firstquotientsteppreviousdiagonaltermrightentry. ff_h_pfp_execution_unique_firstquotientsteppreviousdiagonaltermrightentry + S (pfc_right_execution_unique_firstquotientsteppreviousdiagonalterm) = S ((S (pfc_complement_execution_unique_firstquotientsteppreviousdiagonalterm)) * bc)) /\ exists ff_q_pfp_execution_unique_firstquotientsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_execution_unique_firstquotientsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_execution_unique_firstquotientsteppreviousdiagonalterm)) * bc) + (pfc_right_execution_unique_firstquotientsteppreviousdiagonalterm)))))) \/ (((exists pfc_gap_execution_unique_firstquotientsteppreviousdiagonaltermrightoutside. pfc_gap_execution_unique_firstquotientsteppreviousdiagonaltermrightoutside+(S (d))=(pfc_complement_execution_unique_firstquotientsteppreviousdiagonalterm)) /\ (((pfc_right_execution_unique_firstquotientsteppreviousdiagonalterm)=0))))) /\ (((pfc_value_execution_unique_firstquotientsteppreviousdiagonal)=pfc_left_execution_unique_firstquotientsteppreviousdiagonalterm*pfc_right_execution_unique_firstquotientsteppreviousdiagonalterm))))))))))) /\ (((exists fs_u_pfc_execution_unique_firstquotientstepprevioussum fs_v_pfc_execution_unique_firstquotientstepprevioussum. ((((exists fs_h_pfc_execution_unique_firstquotientstepprevioussum_body_start. fs_h_pfc_execution_unique_firstquotientstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_unique_firstquotientstepprevioussum)) /\ exists fs_q_pfc_execution_unique_firstquotientstepprevioussum_body_start. fs_u_pfc_execution_unique_firstquotientstepprevioussum = fs_q_pfc_execution_unique_firstquotientstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_execution_unique_firstquotientstepprevioussum) + (0))) /\ ((((exists fs_h_pfc_execution_unique_firstquotientstepprevioussum_body_terminal. fs_h_pfc_execution_unique_firstquotientstepprevioussum_body_terminal + S (pfc_natural_sum_execution_unique_firstquotientstepprevious) = S ((S (S (pfd_index_execution_unique_firstquotient))) * fs_v_pfc_execution_unique_firstquotientstepprevioussum)) /\ exists fs_q_pfc_execution_unique_firstquotientstepprevioussum_body_terminal. fs_u_pfc_execution_unique_firstquotientstepprevioussum = fs_q_pfc_execution_unique_firstquotientstepprevioussum_body_terminal * S ((S (S (pfd_index_execution_unique_firstquotient))) * fs_v_pfc_execution_unique_firstquotientstepprevioussum) + (pfc_natural_sum_execution_unique_firstquotientstepprevious))) /\ forall fs_i_pfc_execution_unique_firstquotientstepprevioussum_body_steps. (exists fs_lt_pfc_execution_unique_firstquotientstepprevioussum_body_steps_bound. fs_lt_pfc_execution_unique_firstquotientstepprevioussum_body_steps_bound + S fs_i_pfc_execution_unique_firstquotientstepprevioussum_body_steps = S (pfd_index_execution_unique_firstquotient)) -> exists fs_a_pfc_execution_unique_firstquotientstepprevioussum_body_steps fs_r_pfc_execution_unique_firstquotientstepprevioussum_body_steps fs_s_pfc_execution_unique_firstquotientstepprevioussum_body_steps. ((((exists fs_h_pfc_execution_unique_firstquotientstepprevioussum_body_steps_summand. fs_h_pfc_execution_unique_firstquotientstepprevioussum_body_steps_summand + S (fs_a_pfc_execution_unique_firstquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_execution_unique_firstquotientstepprevioussum_body_steps)) * pfc_terms_scale_execution_unique_firstquotientstepprevious)) /\ exists fs_q_pfc_execution_unique_firstquotientstepprevioussum_body_steps_summand. pfc_terms_code_execution_unique_firstquotientstepprevious = fs_q_pfc_execution_unique_firstquotientstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_execution_unique_firstquotientstepprevioussum_body_steps)) * pfc_terms_scale_execution_unique_firstquotientstepprevious) + (fs_a_pfc_execution_unique_firstquotientstepprevioussum_body_steps))) /\ ((((exists fs_h_pfc_execution_unique_firstquotientstepprevioussum_body_steps_partial. fs_h_pfc_execution_unique_firstquotientstepprevioussum_body_steps_partial + S (fs_r_pfc_execution_unique_firstquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_execution_unique_firstquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_unique_firstquotientstepprevioussum)) /\ exists fs_q_pfc_execution_unique_firstquotientstepprevioussum_body_steps_partial. fs_u_pfc_execution_unique_firstquotientstepprevioussum = fs_q_pfc_execution_unique_firstquotientstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_execution_unique_firstquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_unique_firstquotientstepprevioussum) + (fs_r_pfc_execution_unique_firstquotientstepprevioussum_body_steps))) /\ ((((exists fs_h_pfc_execution_unique_firstquotientstepprevioussum_body_steps_successor. fs_h_pfc_execution_unique_firstquotientstepprevioussum_body_steps_successor + S (fs_s_pfc_execution_unique_firstquotientstepprevioussum_body_steps) = S ((S (S fs_i_pfc_execution_unique_firstquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_unique_firstquotientstepprevioussum)) /\ exists fs_q_pfc_execution_unique_firstquotientstepprevioussum_body_steps_successor. fs_u_pfc_execution_unique_firstquotientstepprevioussum = fs_q_pfc_execution_unique_firstquotientstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_execution_unique_firstquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_unique_firstquotientstepprevioussum) + (fs_s_pfc_execution_unique_firstquotientstepprevioussum_body_steps))) /\ fs_s_pfc_execution_unique_firstquotientstepprevioussum_body_steps = fs_r_pfc_execution_unique_firstquotientstepprevioussum_body_steps + fs_a_pfc_execution_unique_firstquotientstepprevioussum_body_steps)))))) /\ ((((exists pfa_gap_execution_unique_firstquotientsteppreviousresiduebound. pfa_gap_execution_unique_firstquotientsteppreviousresiduebound + S (pfd_previous_execution_unique_firstquotientstep) = (p)) /\ ((exists pfa_offset_left_execution_unique_firstquotientsteppreviousresiduecongruence pfa_offset_right_execution_unique_firstquotientsteppreviousresiduecongruence. (pfc_natural_sum_execution_unique_firstquotientstepprevious) + (p) * pfa_offset_left_execution_unique_firstquotientsteppreviousresiduecongruence = (pfd_previous_execution_unique_firstquotientstep) + (p) * pfa_offset_right_execution_unique_firstquotientsteppreviousresiduecongruence))))))))) /\ (((((exists pfa_gap_execution_unique_firstquotientstepsubtractleft. pfa_gap_execution_unique_firstquotientstepsubtractleft + S (pfd_previous_execution_unique_firstquotientstep) = (p)) /\ (((exists pfa_gap_execution_unique_firstquotientstepsubtractright. pfa_gap_execution_unique_firstquotientstepsubtractright + S (pfd_difference_execution_unique_firstquotientstep) = (p)) /\ ((((exists pfa_gap_execution_unique_firstquotientstepsubtractresultbound. pfa_gap_execution_unique_firstquotientstepsubtractresultbound + S (pfd_input_execution_unique_firstquotientstep) = (p)) /\ ((exists pfa_offset_left_execution_unique_firstquotientstepsubtractresultcongruence pfa_offset_right_execution_unique_firstquotientstepsubtractresultcongruence. ((pfd_previous_execution_unique_firstquotientstep) + (pfd_difference_execution_unique_firstquotientstep)) + (p) * pfa_offset_left_execution_unique_firstquotientstepsubtractresultcongruence = (pfd_input_execution_unique_firstquotientstep) + (p) * pfa_offset_right_execution_unique_firstquotientstepsubtractresultcongruence))))))))) /\ ((((exists pfa_gap_execution_unique_firstquotientstepmultiplyleft. pfa_gap_execution_unique_firstquotientstepmultiplyleft + S (pfd_inverse_execution_unique_first) = (p)) /\ (((exists pfa_gap_execution_unique_firstquotientstepmultiplyright. pfa_gap_execution_unique_firstquotientstepmultiplyright + S (pfd_difference_execution_unique_firstquotientstep) = (p)) /\ ((((exists pfa_gap_execution_unique_firstquotientstepmultiplyresultbound. pfa_gap_execution_unique_firstquotientstepmultiplyresultbound + S (pfd_value_execution_unique_firstquotient) = (p)) /\ ((exists pfa_offset_left_execution_unique_firstquotientstepmultiplyresultcongruence pfa_offset_right_execution_unique_firstquotientstepmultiplyresultcongruence. ((pfd_inverse_execution_unique_first) * (pfd_difference_execution_unique_firstquotientstep)) + (p) * pfa_offset_left_execution_unique_firstquotientstepmultiplyresultcongruence = (pfd_value_execution_unique_firstquotient) + (p) * pfa_offset_right_execution_unique_firstquotientstepmultiplyresultcongruence))))))))))))))))))) /\ (((forall pfc_index_execution_unique_firstproduct. (exists pfa_gap_execution_unique_firstproductbound. pfa_gap_execution_unique_firstproductbound + S (pfc_index_execution_unique_firstproduct) = (L)) -> exists pfc_value_execution_unique_firstproduct. ((((exists ff_h_pfp_execution_unique_firstproductentry. ff_h_pfp_execution_unique_firstproductentry + S (pfc_value_execution_unique_firstproduct) = S ((S (pfc_index_execution_unique_firstproduct)) * pfd_product_scale_execution_unique_first)) /\ exists ff_q_pfp_execution_unique_firstproductentry. pfd_product_code_execution_unique_first = ff_q_pfp_execution_unique_firstproductentry * S ((S (pfc_index_execution_unique_firstproduct)) * pfd_product_scale_execution_unique_first) + (pfc_value_execution_unique_firstproduct))) /\ ((exists pfc_terms_code_execution_unique_firstproductcoefficient pfc_terms_scale_execution_unique_firstproductcoefficient pfc_natural_sum_execution_unique_firstproductcoefficient. ((forall pfc_index_execution_unique_firstproductcoefficientdiagonal. (exists pfa_gap_execution_unique_firstproductcoefficientdiagonalbound. pfa_gap_execution_unique_firstproductcoefficientdiagonalbound + S (pfc_index_execution_unique_firstproductcoefficientdiagonal) = (S (pfc_index_execution_unique_firstproduct))) -> exists pfc_value_execution_unique_firstproductcoefficientdiagonal. ((((exists ff_h_pfp_execution_unique_firstproductcoefficientdiagonalentry. ff_h_pfp_execution_unique_firstproductcoefficientdiagonalentry + S (pfc_value_execution_unique_firstproductcoefficientdiagonal) = S ((S (pfc_index_execution_unique_firstproductcoefficientdiagonal)) * pfc_terms_scale_execution_unique_firstproductcoefficient)) /\ exists ff_q_pfp_execution_unique_firstproductcoefficientdiagonalentry. pfc_terms_code_execution_unique_firstproductcoefficient = ff_q_pfp_execution_unique_firstproductcoefficientdiagonalentry * S ((S (pfc_index_execution_unique_firstproductcoefficientdiagonal)) * pfc_terms_scale_execution_unique_firstproductcoefficient) + (pfc_value_execution_unique_firstproductcoefficientdiagonal))) /\ ((exists pfc_complement_execution_unique_firstproductcoefficientdiagonalterm pfc_left_execution_unique_firstproductcoefficientdiagonalterm pfc_right_execution_unique_firstproductcoefficientdiagonalterm. (((pfc_index_execution_unique_firstproductcoefficientdiagonal)+pfc_complement_execution_unique_firstproductcoefficientdiagonalterm=(pfc_index_execution_unique_firstproduct)) /\ ((((((exists pfa_gap_execution_unique_firstproductcoefficientdiagonaltermleftinside. pfa_gap_execution_unique_firstproductcoefficientdiagonaltermleftinside + S (pfc_index_execution_unique_firstproductcoefficientdiagonal) = (q)) /\ ((((exists ff_h_pfp_execution_unique_firstproductcoefficientdiagonaltermleftentry. ff_h_pfp_execution_unique_firstproductcoefficientdiagonaltermleftentry + S (pfc_left_execution_unique_firstproductcoefficientdiagonalterm) = S ((S (pfc_index_execution_unique_firstproductcoefficientdiagonal)) * qc)) /\ exists ff_q_pfp_execution_unique_firstproductcoefficientdiagonaltermleftentry. qb = ff_q_pfp_execution_unique_firstproductcoefficientdiagonaltermleftentry * S ((S (pfc_index_execution_unique_firstproductcoefficientdiagonal)) * qc) + (pfc_left_execution_unique_firstproductcoefficientdiagonalterm)))))) \/ (((exists pfc_gap_execution_unique_firstproductcoefficientdiagonaltermleftoutside. pfc_gap_execution_unique_firstproductcoefficientdiagonaltermleftoutside+(q)=(pfc_index_execution_unique_firstproductcoefficientdiagonal)) /\ (((pfc_left_execution_unique_firstproductcoefficientdiagonalterm)=0))))) /\ ((((((exists pfa_gap_execution_unique_firstproductcoefficientdiagonaltermrightinside. pfa_gap_execution_unique_firstproductcoefficientdiagonaltermrightinside + S (pfc_complement_execution_unique_firstproductcoefficientdiagonalterm) = (S (d))) /\ ((((exists ff_h_pfp_execution_unique_firstproductcoefficientdiagonaltermrightentry. ff_h_pfp_execution_unique_firstproductcoefficientdiagonaltermrightentry + S (pfc_right_execution_unique_firstproductcoefficientdiagonalterm) = S ((S (pfc_complement_execution_unique_firstproductcoefficientdiagonalterm)) * bc)) /\ exists ff_q_pfp_execution_unique_firstproductcoefficientdiagonaltermrightentry. bb = ff_q_pfp_execution_unique_firstproductcoefficientdiagonaltermrightentry * S ((S (pfc_complement_execution_unique_firstproductcoefficientdiagonalterm)) * bc) + (pfc_right_execution_unique_firstproductcoefficientdiagonalterm)))))) \/ (((exists pfc_gap_execution_unique_firstproductcoefficientdiagonaltermrightoutside. pfc_gap_execution_unique_firstproductcoefficientdiagonaltermrightoutside+(S (d))=(pfc_complement_execution_unique_firstproductcoefficientdiagonalterm)) /\ (((pfc_right_execution_unique_firstproductcoefficientdiagonalterm)=0))))) /\ (((pfc_value_execution_unique_firstproductcoefficientdiagonal)=pfc_left_execution_unique_firstproductcoefficientdiagonalterm*pfc_right_execution_unique_firstproductcoefficientdiagonalterm))))))))))) /\ (((exists fs_u_pfc_execution_unique_firstproductcoefficientsum fs_v_pfc_execution_unique_firstproductcoefficientsum. ((((exists fs_h_pfc_execution_unique_firstproductcoefficientsum_body_start. fs_h_pfc_execution_unique_firstproductcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_unique_firstproductcoefficientsum)) /\ exists fs_q_pfc_execution_unique_firstproductcoefficientsum_body_start. fs_u_pfc_execution_unique_firstproductcoefficientsum = fs_q_pfc_execution_unique_firstproductcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_execution_unique_firstproductcoefficientsum) + (0))) /\ ((((exists fs_h_pfc_execution_unique_firstproductcoefficientsum_body_terminal. fs_h_pfc_execution_unique_firstproductcoefficientsum_body_terminal + S (pfc_natural_sum_execution_unique_firstproductcoefficient) = S ((S (S (pfc_index_execution_unique_firstproduct))) * fs_v_pfc_execution_unique_firstproductcoefficientsum)) /\ exists fs_q_pfc_execution_unique_firstproductcoefficientsum_body_terminal. fs_u_pfc_execution_unique_firstproductcoefficientsum = fs_q_pfc_execution_unique_firstproductcoefficientsum_body_terminal * S ((S (S (pfc_index_execution_unique_firstproduct))) * fs_v_pfc_execution_unique_firstproductcoefficientsum) + (pfc_natural_sum_execution_unique_firstproductcoefficient))) /\ forall fs_i_pfc_execution_unique_firstproductcoefficientsum_body_steps. (exists fs_lt_pfc_execution_unique_firstproductcoefficientsum_body_steps_bound. fs_lt_pfc_execution_unique_firstproductcoefficientsum_body_steps_bound + S fs_i_pfc_execution_unique_firstproductcoefficientsum_body_steps = S (pfc_index_execution_unique_firstproduct)) -> exists fs_a_pfc_execution_unique_firstproductcoefficientsum_body_steps fs_r_pfc_execution_unique_firstproductcoefficientsum_body_steps fs_s_pfc_execution_unique_firstproductcoefficientsum_body_steps. ((((exists fs_h_pfc_execution_unique_firstproductcoefficientsum_body_steps_summand. fs_h_pfc_execution_unique_firstproductcoefficientsum_body_steps_summand + S (fs_a_pfc_execution_unique_firstproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_unique_firstproductcoefficientsum_body_steps)) * pfc_terms_scale_execution_unique_firstproductcoefficient)) /\ exists fs_q_pfc_execution_unique_firstproductcoefficientsum_body_steps_summand. pfc_terms_code_execution_unique_firstproductcoefficient = fs_q_pfc_execution_unique_firstproductcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_execution_unique_firstproductcoefficientsum_body_steps)) * pfc_terms_scale_execution_unique_firstproductcoefficient) + (fs_a_pfc_execution_unique_firstproductcoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_execution_unique_firstproductcoefficientsum_body_steps_partial. fs_h_pfc_execution_unique_firstproductcoefficientsum_body_steps_partial + S (fs_r_pfc_execution_unique_firstproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_unique_firstproductcoefficientsum_body_steps)) * fs_v_pfc_execution_unique_firstproductcoefficientsum)) /\ exists fs_q_pfc_execution_unique_firstproductcoefficientsum_body_steps_partial. fs_u_pfc_execution_unique_firstproductcoefficientsum = fs_q_pfc_execution_unique_firstproductcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_execution_unique_firstproductcoefficientsum_body_steps)) * fs_v_pfc_execution_unique_firstproductcoefficientsum) + (fs_r_pfc_execution_unique_firstproductcoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_execution_unique_firstproductcoefficientsum_body_steps_successor. fs_h_pfc_execution_unique_firstproductcoefficientsum_body_steps_successor + S (fs_s_pfc_execution_unique_firstproductcoefficientsum_body_steps) = S ((S (S fs_i_pfc_execution_unique_firstproductcoefficientsum_body_steps)) * fs_v_pfc_execution_unique_firstproductcoefficientsum)) /\ exists fs_q_pfc_execution_unique_firstproductcoefficientsum_body_steps_successor. fs_u_pfc_execution_unique_firstproductcoefficientsum = fs_q_pfc_execution_unique_firstproductcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_execution_unique_firstproductcoefficientsum_body_steps)) * fs_v_pfc_execution_unique_firstproductcoefficientsum) + (fs_s_pfc_execution_unique_firstproductcoefficientsum_body_steps))) /\ fs_s_pfc_execution_unique_firstproductcoefficientsum_body_steps = fs_r_pfc_execution_unique_firstproductcoefficientsum_body_steps + fs_a_pfc_execution_unique_firstproductcoefficientsum_body_steps)))))) /\ ((((exists pfa_gap_execution_unique_firstproductcoefficientresiduebound. pfa_gap_execution_unique_firstproductcoefficientresiduebound + S (pfc_value_execution_unique_firstproduct) = (p)) /\ ((exists pfa_offset_left_execution_unique_firstproductcoefficientresiduecongruence pfa_offset_right_execution_unique_firstproductcoefficientresiduecongruence. (pfc_natural_sum_execution_unique_firstproductcoefficient) + (p) * pfa_offset_left_execution_unique_firstproductcoefficientresiduecongruence = (pfc_value_execution_unique_firstproduct) + (p) * pfa_offset_right_execution_unique_firstproductcoefficientresiduecongruence)))))))))))) /\ (((forall pfs_index_execution_unique_firstdifference. (exists pfa_gap_execution_unique_firstdifferenceindex. pfa_gap_execution_unique_firstdifferenceindex + S (pfs_index_execution_unique_firstdifference) = (L)) -> exists pfs_left_execution_unique_firstdifference pfs_right_execution_unique_firstdifference pfs_result_execution_unique_firstdifference. ((((exists ff_h_pfp_execution_unique_firstdifferenceleft. ff_h_pfp_execution_unique_firstdifferenceleft + S (pfs_left_execution_unique_firstdifference) = S ((S (pfs_index_execution_unique_firstdifference)) * ac)) /\ exists ff_q_pfp_execution_unique_firstdifferenceleft. ab = ff_q_pfp_execution_unique_firstdifferenceleft * S ((S (pfs_index_execution_unique_firstdifference)) * ac) + (pfs_left_execution_unique_firstdifference))) /\ (((((exists ff_h_pfp_execution_unique_firstdifferenceright. ff_h_pfp_execution_unique_firstdifferenceright + S (pfs_right_execution_unique_firstdifference) = S ((S (pfs_index_execution_unique_firstdifference)) * pfd_product_scale_execution_unique_first)) /\ exists ff_q_pfp_execution_unique_firstdifferenceright. pfd_product_code_execution_unique_first = ff_q_pfp_execution_unique_firstdifferenceright * S ((S (pfs_index_execution_unique_firstdifference)) * pfd_product_scale_execution_unique_first) + (pfs_right_execution_unique_firstdifference))) /\ (((((exists ff_h_pfp_execution_unique_firstdifferenceresult. ff_h_pfp_execution_unique_firstdifferenceresult + S (pfs_result_execution_unique_firstdifference) = S ((S (pfs_index_execution_unique_firstdifference)) * pfd_residual_scale_execution_unique_first)) /\ exists ff_q_pfp_execution_unique_firstdifferenceresult. pfd_residual_code_execution_unique_first = ff_q_pfp_execution_unique_firstdifferenceresult * S ((S (pfs_index_execution_unique_firstdifference)) * pfd_residual_scale_execution_unique_first) + (pfs_result_execution_unique_firstdifference))) /\ ((((exists pfa_gap_execution_unique_firstdifferenceoperationleft. pfa_gap_execution_unique_firstdifferenceoperationleft + S (pfs_right_execution_unique_firstdifference) = (p)) /\ (((exists pfa_gap_execution_unique_firstdifferenceoperationright. pfa_gap_execution_unique_firstdifferenceoperationright + S (pfs_result_execution_unique_firstdifference) = (p)) /\ ((((exists pfa_gap_execution_unique_firstdifferenceoperationresultbound. pfa_gap_execution_unique_firstdifferenceoperationresultbound + S (pfs_left_execution_unique_firstdifference) = (p)) /\ ((exists pfa_offset_left_execution_unique_firstdifferenceoperationresultcongruence pfa_offset_right_execution_unique_firstdifferenceoperationresultcongruence. ((pfs_right_execution_unique_firstdifference) + (pfs_result_execution_unique_firstdifference)) + (p) * pfa_offset_left_execution_unique_firstdifferenceoperationresultcongruence = (pfs_left_execution_unique_firstdifference) + (p) * pfa_offset_right_execution_unique_firstdifferenceoperationresultcongruence)))))))))))))))) /\ (((((L)=(pfd_cut_execution_unique_first)+(R)) /\ (((forall fom_index_pfp_execution_unique_firsttriminput. (exists fom_gap_pfp_execution_unique_firsttriminput_index_bound. fom_gap_pfp_execution_unique_firsttriminput_index_bound + S (fom_index_pfp_execution_unique_firsttriminput) = L) -> exists fom_value_pfp_execution_unique_firsttriminput. ((((exists fom_beta_height_pfp_execution_unique_firsttriminput_entry. fom_beta_height_pfp_execution_unique_firsttriminput_entry + S (fom_value_pfp_execution_unique_firsttriminput) = S ((S (fom_index_pfp_execution_unique_firsttriminput)) * pfd_residual_scale_execution_unique_first)) /\ exists fom_beta_quotient_pfp_execution_unique_firsttriminput_entry. pfd_residual_code_execution_unique_first = fom_beta_quotient_pfp_execution_unique_firsttriminput_entry * S ((S (fom_index_pfp_execution_unique_firsttriminput)) * pfd_residual_scale_execution_unique_first) + (fom_value_pfp_execution_unique_firsttriminput))) /\ (exists fom_gap_pfp_execution_unique_firsttriminput_value_bound. fom_gap_pfp_execution_unique_firsttriminput_value_bound + S (fom_value_pfp_execution_unique_firsttriminput) = p))) /\ (((forall pfp_repeat_index_execution_unique_firsttrimremoved. (exists pfa_gap_execution_unique_firsttrimremovedindex. pfa_gap_execution_unique_firsttrimremovedindex + S (pfp_repeat_index_execution_unique_firsttrimremoved) = (pfd_cut_execution_unique_first)) -> (((exists ff_h_pfp_execution_unique_firsttrimremovedentry. ff_h_pfp_execution_unique_firsttrimremovedentry + S (0) = S ((S (pfp_repeat_index_execution_unique_firsttrimremoved)) * pfd_residual_scale_execution_unique_first)) /\ exists ff_q_pfp_execution_unique_firsttrimremovedentry. pfd_residual_code_execution_unique_first = ff_q_pfp_execution_unique_firsttrimremovedentry * S ((S (pfp_repeat_index_execution_unique_firsttrimremoved)) * pfd_residual_scale_execution_unique_first) + (0)))) /\ (((forall pftrim_index_execution_unique_firsttrimsuffix pftrim_value_execution_unique_firsttrimsuffix. (exists pfa_gap_execution_unique_firsttrimsuffixbound. pfa_gap_execution_unique_firsttrimsuffixbound + S (pftrim_index_execution_unique_firsttrimsuffix) = (R)) -> (((exists ff_h_pfp_execution_unique_firsttrimsuffixsource. ff_h_pfp_execution_unique_firsttrimsuffixsource + S (pftrim_value_execution_unique_firsttrimsuffix) = S ((S ((pfd_cut_execution_unique_first)+pftrim_index_execution_unique_firsttrimsuffix)) * pfd_residual_scale_execution_unique_first)) /\ exists ff_q_pfp_execution_unique_firsttrimsuffixsource. pfd_residual_code_execution_unique_first = ff_q_pfp_execution_unique_firsttrimsuffixsource * S ((S ((pfd_cut_execution_unique_first)+pftrim_index_execution_unique_firsttrimsuffix)) * pfd_residual_scale_execution_unique_first) + (pftrim_value_execution_unique_firsttrimsuffix))) -> (((exists ff_h_pfp_execution_unique_firsttrimsuffixoutput. ff_h_pfp_execution_unique_firsttrimsuffixoutput + S (pftrim_value_execution_unique_firsttrimsuffix) = S ((S (pftrim_index_execution_unique_firsttrimsuffix)) * rc)) /\ exists ff_q_pfp_execution_unique_firsttrimsuffixoutput. rb = ff_q_pfp_execution_unique_firsttrimsuffixoutput * S ((S (pftrim_index_execution_unique_firsttrimsuffix)) * rc) + (pftrim_value_execution_unique_firsttrimsuffix)))) /\ (((R)=0 \/ (exists pftrim_leading_execution_unique_firsttrimnormal. ((((exists ff_h_pfp_execution_unique_firsttrimnormalentry. ff_h_pfp_execution_unique_firsttrimnormalentry + S (pftrim_leading_execution_unique_firsttrimnormal) = S ((S (0)) * rc)) /\ exists ff_q_pfp_execution_unique_firsttrimnormalentry. rb = ff_q_pfp_execution_unique_firsttrimnormalentry * S ((S (0)) * rc) + (pftrim_leading_execution_unique_firsttrimnormal))) /\ ((~(pftrim_leading_execution_unique_firsttrimnormal=0))))))))))))))))))))))))))))))))) -> (((forall fom_index_pfp_execution_unique_secondinput. (exists fom_gap_pfp_execution_unique_secondinput_index_bound. fom_gap_pfp_execution_unique_secondinput_index_bound + S (fom_index_pfp_execution_unique_secondinput) = L) -> exists fom_value_pfp_execution_unique_secondinput. ((((exists fom_beta_height_pfp_execution_unique_secondinput_entry. fom_beta_height_pfp_execution_unique_secondinput_entry + S (fom_value_pfp_execution_unique_secondinput) = S ((S (fom_index_pfp_execution_unique_secondinput)) * ac)) /\ exists fom_beta_quotient_pfp_execution_unique_secondinput_entry. ab = fom_beta_quotient_pfp_execution_unique_secondinput_entry * S ((S (fom_index_pfp_execution_unique_secondinput)) * ac) + (fom_value_pfp_execution_unique_secondinput))) /\ (exists fom_gap_pfp_execution_unique_secondinput_value_bound. fom_gap_pfp_execution_unique_secondinput_value_bound + S (fom_value_pfp_execution_unique_secondinput) = p))) /\ (((forall fom_index_pfp_execution_unique_seconddivisor. (exists fom_gap_pfp_execution_unique_seconddivisor_index_bound. fom_gap_pfp_execution_unique_seconddivisor_index_bound + S (fom_index_pfp_execution_unique_seconddivisor) = S (d)) -> exists fom_value_pfp_execution_unique_seconddivisor. ((((exists fom_beta_height_pfp_execution_unique_seconddivisor_entry. fom_beta_height_pfp_execution_unique_seconddivisor_entry + S (fom_value_pfp_execution_unique_seconddivisor) = S ((S (fom_index_pfp_execution_unique_seconddivisor)) * bc)) /\ exists fom_beta_quotient_pfp_execution_unique_seconddivisor_entry. bb = fom_beta_quotient_pfp_execution_unique_seconddivisor_entry * S ((S (fom_index_pfp_execution_unique_seconddivisor)) * bc) + (fom_value_pfp_execution_unique_seconddivisor))) /\ (exists fom_gap_pfp_execution_unique_seconddivisor_value_bound. fom_gap_pfp_execution_unique_seconddivisor_value_bound + S (fom_value_pfp_execution_unique_seconddivisor) = p))) /\ (((((((Q)=0) /\ ((exists pfc_gap_execution_unique_secondlengthshort. pfc_gap_execution_unique_secondlengthshort+(L)=(d))))) \/ (((~((Q)=0)) /\ (((Q)+(d)=(L)))))) /\ ((exists pfd_head_execution_unique_second pfd_inverse_execution_unique_second pfd_product_code_execution_unique_second pfd_product_scale_execution_unique_second pfd_residual_code_execution_unique_second pfd_residual_scale_execution_unique_second pfd_cut_execution_unique_second. ((((exists ff_h_pfp_execution_unique_secondhead. ff_h_pfp_execution_unique_secondhead + S (pfd_head_execution_unique_second) = S ((S (0)) * bc)) /\ exists ff_q_pfp_execution_unique_secondhead. bb = ff_q_pfp_execution_unique_secondhead * S ((S (0)) * bc) + (pfd_head_execution_unique_second))) /\ (((((~((pfd_head_execution_unique_second) = 0)) /\ ((((exists pfa_gap_execution_unique_secondinversemultiplicationleft. pfa_gap_execution_unique_secondinversemultiplicationleft + S (pfd_head_execution_unique_second) = (p)) /\ (((exists pfa_gap_execution_unique_secondinversemultiplicationright. pfa_gap_execution_unique_secondinversemultiplicationright + S (pfd_inverse_execution_unique_second) = (p)) /\ ((((exists pfa_gap_execution_unique_secondinversemultiplicationresultbound. pfa_gap_execution_unique_secondinversemultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_execution_unique_secondinversemultiplicationresultcongruence pfa_offset_right_execution_unique_secondinversemultiplicationresultcongruence. ((pfd_head_execution_unique_second) * (pfd_inverse_execution_unique_second)) + (p) * pfa_offset_left_execution_unique_secondinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_execution_unique_secondinversemultiplicationresultcongruence)))))))))))) /\ (((forall pfd_index_execution_unique_secondquotient. (exists pfa_gap_execution_unique_secondquotientbound. pfa_gap_execution_unique_secondquotientbound + S (pfd_index_execution_unique_secondquotient) = (Q)) -> exists pfd_value_execution_unique_secondquotient. ((((exists ff_h_pfp_execution_unique_secondquotiententry. ff_h_pfp_execution_unique_secondquotiententry + S (pfd_value_execution_unique_secondquotient) = S ((S (pfd_index_execution_unique_secondquotient)) * QC)) /\ exists ff_q_pfp_execution_unique_secondquotiententry. QB = ff_q_pfp_execution_unique_secondquotiententry * S ((S (pfd_index_execution_unique_secondquotient)) * QC) + (pfd_value_execution_unique_secondquotient))) /\ ((exists pfd_input_execution_unique_secondquotientstep pfd_previous_execution_unique_secondquotientstep pfd_difference_execution_unique_secondquotientstep. ((((exists ff_h_pfp_execution_unique_secondquotientstepinput. ff_h_pfp_execution_unique_secondquotientstepinput + S (pfd_input_execution_unique_secondquotientstep) = S ((S (pfd_index_execution_unique_secondquotient)) * ac)) /\ exists ff_q_pfp_execution_unique_secondquotientstepinput. ab = ff_q_pfp_execution_unique_secondquotientstepinput * S ((S (pfd_index_execution_unique_secondquotient)) * ac) + (pfd_input_execution_unique_secondquotientstep))) /\ (((exists pfc_terms_code_execution_unique_secondquotientstepprevious pfc_terms_scale_execution_unique_secondquotientstepprevious pfc_natural_sum_execution_unique_secondquotientstepprevious. ((forall pfc_index_execution_unique_secondquotientsteppreviousdiagonal. (exists pfa_gap_execution_unique_secondquotientsteppreviousdiagonalbound. pfa_gap_execution_unique_secondquotientsteppreviousdiagonalbound + S (pfc_index_execution_unique_secondquotientsteppreviousdiagonal) = (S (pfd_index_execution_unique_secondquotient))) -> exists pfc_value_execution_unique_secondquotientsteppreviousdiagonal. ((((exists ff_h_pfp_execution_unique_secondquotientsteppreviousdiagonalentry. ff_h_pfp_execution_unique_secondquotientsteppreviousdiagonalentry + S (pfc_value_execution_unique_secondquotientsteppreviousdiagonal) = S ((S (pfc_index_execution_unique_secondquotientsteppreviousdiagonal)) * pfc_terms_scale_execution_unique_secondquotientstepprevious)) /\ exists ff_q_pfp_execution_unique_secondquotientsteppreviousdiagonalentry. pfc_terms_code_execution_unique_secondquotientstepprevious = ff_q_pfp_execution_unique_secondquotientsteppreviousdiagonalentry * S ((S (pfc_index_execution_unique_secondquotientsteppreviousdiagonal)) * pfc_terms_scale_execution_unique_secondquotientstepprevious) + (pfc_value_execution_unique_secondquotientsteppreviousdiagonal))) /\ ((exists pfc_complement_execution_unique_secondquotientsteppreviousdiagonalterm pfc_left_execution_unique_secondquotientsteppreviousdiagonalterm pfc_right_execution_unique_secondquotientsteppreviousdiagonalterm. (((pfc_index_execution_unique_secondquotientsteppreviousdiagonal)+pfc_complement_execution_unique_secondquotientsteppreviousdiagonalterm=(pfd_index_execution_unique_secondquotient)) /\ ((((((exists pfa_gap_execution_unique_secondquotientsteppreviousdiagonaltermleftinside. pfa_gap_execution_unique_secondquotientsteppreviousdiagonaltermleftinside + S (pfc_index_execution_unique_secondquotientsteppreviousdiagonal) = (pfd_index_execution_unique_secondquotient)) /\ ((((exists ff_h_pfp_execution_unique_secondquotientsteppreviousdiagonaltermleftentry. ff_h_pfp_execution_unique_secondquotientsteppreviousdiagonaltermleftentry + S (pfc_left_execution_unique_secondquotientsteppreviousdiagonalterm) = S ((S (pfc_index_execution_unique_secondquotientsteppreviousdiagonal)) * QC)) /\ exists ff_q_pfp_execution_unique_secondquotientsteppreviousdiagonaltermleftentry. QB = ff_q_pfp_execution_unique_secondquotientsteppreviousdiagonaltermleftentry * S ((S (pfc_index_execution_unique_secondquotientsteppreviousdiagonal)) * QC) + (pfc_left_execution_unique_secondquotientsteppreviousdiagonalterm)))))) \/ (((exists pfc_gap_execution_unique_secondquotientsteppreviousdiagonaltermleftoutside. pfc_gap_execution_unique_secondquotientsteppreviousdiagonaltermleftoutside+(pfd_index_execution_unique_secondquotient)=(pfc_index_execution_unique_secondquotientsteppreviousdiagonal)) /\ (((pfc_left_execution_unique_secondquotientsteppreviousdiagonalterm)=0))))) /\ ((((((exists pfa_gap_execution_unique_secondquotientsteppreviousdiagonaltermrightinside. pfa_gap_execution_unique_secondquotientsteppreviousdiagonaltermrightinside + S (pfc_complement_execution_unique_secondquotientsteppreviousdiagonalterm) = (S (d))) /\ ((((exists ff_h_pfp_execution_unique_secondquotientsteppreviousdiagonaltermrightentry. ff_h_pfp_execution_unique_secondquotientsteppreviousdiagonaltermrightentry + S (pfc_right_execution_unique_secondquotientsteppreviousdiagonalterm) = S ((S (pfc_complement_execution_unique_secondquotientsteppreviousdiagonalterm)) * bc)) /\ exists ff_q_pfp_execution_unique_secondquotientsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_execution_unique_secondquotientsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_execution_unique_secondquotientsteppreviousdiagonalterm)) * bc) + (pfc_right_execution_unique_secondquotientsteppreviousdiagonalterm)))))) \/ (((exists pfc_gap_execution_unique_secondquotientsteppreviousdiagonaltermrightoutside. pfc_gap_execution_unique_secondquotientsteppreviousdiagonaltermrightoutside+(S (d))=(pfc_complement_execution_unique_secondquotientsteppreviousdiagonalterm)) /\ (((pfc_right_execution_unique_secondquotientsteppreviousdiagonalterm)=0))))) /\ (((pfc_value_execution_unique_secondquotientsteppreviousdiagonal)=pfc_left_execution_unique_secondquotientsteppreviousdiagonalterm*pfc_right_execution_unique_secondquotientsteppreviousdiagonalterm))))))))))) /\ (((exists fs_u_pfc_execution_unique_secondquotientstepprevioussum fs_v_pfc_execution_unique_secondquotientstepprevioussum. ((((exists fs_h_pfc_execution_unique_secondquotientstepprevioussum_body_start. fs_h_pfc_execution_unique_secondquotientstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_unique_secondquotientstepprevioussum)) /\ exists fs_q_pfc_execution_unique_secondquotientstepprevioussum_body_start. fs_u_pfc_execution_unique_secondquotientstepprevioussum = fs_q_pfc_execution_unique_secondquotientstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_execution_unique_secondquotientstepprevioussum) + (0))) /\ ((((exists fs_h_pfc_execution_unique_secondquotientstepprevioussum_body_terminal. fs_h_pfc_execution_unique_secondquotientstepprevioussum_body_terminal + S (pfc_natural_sum_execution_unique_secondquotientstepprevious) = S ((S (S (pfd_index_execution_unique_secondquotient))) * fs_v_pfc_execution_unique_secondquotientstepprevioussum)) /\ exists fs_q_pfc_execution_unique_secondquotientstepprevioussum_body_terminal. fs_u_pfc_execution_unique_secondquotientstepprevioussum = fs_q_pfc_execution_unique_secondquotientstepprevioussum_body_terminal * S ((S (S (pfd_index_execution_unique_secondquotient))) * fs_v_pfc_execution_unique_secondquotientstepprevioussum) + (pfc_natural_sum_execution_unique_secondquotientstepprevious))) /\ forall fs_i_pfc_execution_unique_secondquotientstepprevioussum_body_steps. (exists fs_lt_pfc_execution_unique_secondquotientstepprevioussum_body_steps_bound. fs_lt_pfc_execution_unique_secondquotientstepprevioussum_body_steps_bound + S fs_i_pfc_execution_unique_secondquotientstepprevioussum_body_steps = S (pfd_index_execution_unique_secondquotient)) -> exists fs_a_pfc_execution_unique_secondquotientstepprevioussum_body_steps fs_r_pfc_execution_unique_secondquotientstepprevioussum_body_steps fs_s_pfc_execution_unique_secondquotientstepprevioussum_body_steps. ((((exists fs_h_pfc_execution_unique_secondquotientstepprevioussum_body_steps_summand. fs_h_pfc_execution_unique_secondquotientstepprevioussum_body_steps_summand + S (fs_a_pfc_execution_unique_secondquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_execution_unique_secondquotientstepprevioussum_body_steps)) * pfc_terms_scale_execution_unique_secondquotientstepprevious)) /\ exists fs_q_pfc_execution_unique_secondquotientstepprevioussum_body_steps_summand. pfc_terms_code_execution_unique_secondquotientstepprevious = fs_q_pfc_execution_unique_secondquotientstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_execution_unique_secondquotientstepprevioussum_body_steps)) * pfc_terms_scale_execution_unique_secondquotientstepprevious) + (fs_a_pfc_execution_unique_secondquotientstepprevioussum_body_steps))) /\ ((((exists fs_h_pfc_execution_unique_secondquotientstepprevioussum_body_steps_partial. fs_h_pfc_execution_unique_secondquotientstepprevioussum_body_steps_partial + S (fs_r_pfc_execution_unique_secondquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_execution_unique_secondquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_unique_secondquotientstepprevioussum)) /\ exists fs_q_pfc_execution_unique_secondquotientstepprevioussum_body_steps_partial. fs_u_pfc_execution_unique_secondquotientstepprevioussum = fs_q_pfc_execution_unique_secondquotientstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_execution_unique_secondquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_unique_secondquotientstepprevioussum) + (fs_r_pfc_execution_unique_secondquotientstepprevioussum_body_steps))) /\ ((((exists fs_h_pfc_execution_unique_secondquotientstepprevioussum_body_steps_successor. fs_h_pfc_execution_unique_secondquotientstepprevioussum_body_steps_successor + S (fs_s_pfc_execution_unique_secondquotientstepprevioussum_body_steps) = S ((S (S fs_i_pfc_execution_unique_secondquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_unique_secondquotientstepprevioussum)) /\ exists fs_q_pfc_execution_unique_secondquotientstepprevioussum_body_steps_successor. fs_u_pfc_execution_unique_secondquotientstepprevioussum = fs_q_pfc_execution_unique_secondquotientstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_execution_unique_secondquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_unique_secondquotientstepprevioussum) + (fs_s_pfc_execution_unique_secondquotientstepprevioussum_body_steps))) /\ fs_s_pfc_execution_unique_secondquotientstepprevioussum_body_steps = fs_r_pfc_execution_unique_secondquotientstepprevioussum_body_steps + fs_a_pfc_execution_unique_secondquotientstepprevioussum_body_steps)))))) /\ ((((exists pfa_gap_execution_unique_secondquotientsteppreviousresiduebound. pfa_gap_execution_unique_secondquotientsteppreviousresiduebound + S (pfd_previous_execution_unique_secondquotientstep) = (p)) /\ ((exists pfa_offset_left_execution_unique_secondquotientsteppreviousresiduecongruence pfa_offset_right_execution_unique_secondquotientsteppreviousresiduecongruence. (pfc_natural_sum_execution_unique_secondquotientstepprevious) + (p) * pfa_offset_left_execution_unique_secondquotientsteppreviousresiduecongruence = (pfd_previous_execution_unique_secondquotientstep) + (p) * pfa_offset_right_execution_unique_secondquotientsteppreviousresiduecongruence))))))))) /\ (((((exists pfa_gap_execution_unique_secondquotientstepsubtractleft. pfa_gap_execution_unique_secondquotientstepsubtractleft + S (pfd_previous_execution_unique_secondquotientstep) = (p)) /\ (((exists pfa_gap_execution_unique_secondquotientstepsubtractright. pfa_gap_execution_unique_secondquotientstepsubtractright + S (pfd_difference_execution_unique_secondquotientstep) = (p)) /\ ((((exists pfa_gap_execution_unique_secondquotientstepsubtractresultbound. pfa_gap_execution_unique_secondquotientstepsubtractresultbound + S (pfd_input_execution_unique_secondquotientstep) = (p)) /\ ((exists pfa_offset_left_execution_unique_secondquotientstepsubtractresultcongruence pfa_offset_right_execution_unique_secondquotientstepsubtractresultcongruence. ((pfd_previous_execution_unique_secondquotientstep) + (pfd_difference_execution_unique_secondquotientstep)) + (p) * pfa_offset_left_execution_unique_secondquotientstepsubtractresultcongruence = (pfd_input_execution_unique_secondquotientstep) + (p) * pfa_offset_right_execution_unique_secondquotientstepsubtractresultcongruence))))))))) /\ ((((exists pfa_gap_execution_unique_secondquotientstepmultiplyleft. pfa_gap_execution_unique_secondquotientstepmultiplyleft + S (pfd_inverse_execution_unique_second) = (p)) /\ (((exists pfa_gap_execution_unique_secondquotientstepmultiplyright. pfa_gap_execution_unique_secondquotientstepmultiplyright + S (pfd_difference_execution_unique_secondquotientstep) = (p)) /\ ((((exists pfa_gap_execution_unique_secondquotientstepmultiplyresultbound. pfa_gap_execution_unique_secondquotientstepmultiplyresultbound + S (pfd_value_execution_unique_secondquotient) = (p)) /\ ((exists pfa_offset_left_execution_unique_secondquotientstepmultiplyresultcongruence pfa_offset_right_execution_unique_secondquotientstepmultiplyresultcongruence. ((pfd_inverse_execution_unique_second) * (pfd_difference_execution_unique_secondquotientstep)) + (p) * pfa_offset_left_execution_unique_secondquotientstepmultiplyresultcongruence = (pfd_value_execution_unique_secondquotient) + (p) * pfa_offset_right_execution_unique_secondquotientstepmultiplyresultcongruence))))))))))))))))))) /\ (((forall pfc_index_execution_unique_secondproduct. (exists pfa_gap_execution_unique_secondproductbound. pfa_gap_execution_unique_secondproductbound + S (pfc_index_execution_unique_secondproduct) = (L)) -> exists pfc_value_execution_unique_secondproduct. ((((exists ff_h_pfp_execution_unique_secondproductentry. ff_h_pfp_execution_unique_secondproductentry + S (pfc_value_execution_unique_secondproduct) = S ((S (pfc_index_execution_unique_secondproduct)) * pfd_product_scale_execution_unique_second)) /\ exists ff_q_pfp_execution_unique_secondproductentry. pfd_product_code_execution_unique_second = ff_q_pfp_execution_unique_secondproductentry * S ((S (pfc_index_execution_unique_secondproduct)) * pfd_product_scale_execution_unique_second) + (pfc_value_execution_unique_secondproduct))) /\ ((exists pfc_terms_code_execution_unique_secondproductcoefficient pfc_terms_scale_execution_unique_secondproductcoefficient pfc_natural_sum_execution_unique_secondproductcoefficient. ((forall pfc_index_execution_unique_secondproductcoefficientdiagonal. (exists pfa_gap_execution_unique_secondproductcoefficientdiagonalbound. pfa_gap_execution_unique_secondproductcoefficientdiagonalbound + S (pfc_index_execution_unique_secondproductcoefficientdiagonal) = (S (pfc_index_execution_unique_secondproduct))) -> exists pfc_value_execution_unique_secondproductcoefficientdiagonal. ((((exists ff_h_pfp_execution_unique_secondproductcoefficientdiagonalentry. ff_h_pfp_execution_unique_secondproductcoefficientdiagonalentry + S (pfc_value_execution_unique_secondproductcoefficientdiagonal) = S ((S (pfc_index_execution_unique_secondproductcoefficientdiagonal)) * pfc_terms_scale_execution_unique_secondproductcoefficient)) /\ exists ff_q_pfp_execution_unique_secondproductcoefficientdiagonalentry. pfc_terms_code_execution_unique_secondproductcoefficient = ff_q_pfp_execution_unique_secondproductcoefficientdiagonalentry * S ((S (pfc_index_execution_unique_secondproductcoefficientdiagonal)) * pfc_terms_scale_execution_unique_secondproductcoefficient) + (pfc_value_execution_unique_secondproductcoefficientdiagonal))) /\ ((exists pfc_complement_execution_unique_secondproductcoefficientdiagonalterm pfc_left_execution_unique_secondproductcoefficientdiagonalterm pfc_right_execution_unique_secondproductcoefficientdiagonalterm. (((pfc_index_execution_unique_secondproductcoefficientdiagonal)+pfc_complement_execution_unique_secondproductcoefficientdiagonalterm=(pfc_index_execution_unique_secondproduct)) /\ ((((((exists pfa_gap_execution_unique_secondproductcoefficientdiagonaltermleftinside. pfa_gap_execution_unique_secondproductcoefficientdiagonaltermleftinside + S (pfc_index_execution_unique_secondproductcoefficientdiagonal) = (Q)) /\ ((((exists ff_h_pfp_execution_unique_secondproductcoefficientdiagonaltermleftentry. ff_h_pfp_execution_unique_secondproductcoefficientdiagonaltermleftentry + S (pfc_left_execution_unique_secondproductcoefficientdiagonalterm) = S ((S (pfc_index_execution_unique_secondproductcoefficientdiagonal)) * QC)) /\ exists ff_q_pfp_execution_unique_secondproductcoefficientdiagonaltermleftentry. QB = ff_q_pfp_execution_unique_secondproductcoefficientdiagonaltermleftentry * S ((S (pfc_index_execution_unique_secondproductcoefficientdiagonal)) * QC) + (pfc_left_execution_unique_secondproductcoefficientdiagonalterm)))))) \/ (((exists pfc_gap_execution_unique_secondproductcoefficientdiagonaltermleftoutside. pfc_gap_execution_unique_secondproductcoefficientdiagonaltermleftoutside+(Q)=(pfc_index_execution_unique_secondproductcoefficientdiagonal)) /\ (((pfc_left_execution_unique_secondproductcoefficientdiagonalterm)=0))))) /\ ((((((exists pfa_gap_execution_unique_secondproductcoefficientdiagonaltermrightinside. pfa_gap_execution_unique_secondproductcoefficientdiagonaltermrightinside + S (pfc_complement_execution_unique_secondproductcoefficientdiagonalterm) = (S (d))) /\ ((((exists ff_h_pfp_execution_unique_secondproductcoefficientdiagonaltermrightentry. ff_h_pfp_execution_unique_secondproductcoefficientdiagonaltermrightentry + S (pfc_right_execution_unique_secondproductcoefficientdiagonalterm) = S ((S (pfc_complement_execution_unique_secondproductcoefficientdiagonalterm)) * bc)) /\ exists ff_q_pfp_execution_unique_secondproductcoefficientdiagonaltermrightentry. bb = ff_q_pfp_execution_unique_secondproductcoefficientdiagonaltermrightentry * S ((S (pfc_complement_execution_unique_secondproductcoefficientdiagonalterm)) * bc) + (pfc_right_execution_unique_secondproductcoefficientdiagonalterm)))))) \/ (((exists pfc_gap_execution_unique_secondproductcoefficientdiagonaltermrightoutside. pfc_gap_execution_unique_secondproductcoefficientdiagonaltermrightoutside+(S (d))=(pfc_complement_execution_unique_secondproductcoefficientdiagonalterm)) /\ (((pfc_right_execution_unique_secondproductcoefficientdiagonalterm)=0))))) /\ (((pfc_value_execution_unique_secondproductcoefficientdiagonal)=pfc_left_execution_unique_secondproductcoefficientdiagonalterm*pfc_right_execution_unique_secondproductcoefficientdiagonalterm))))))))))) /\ (((exists fs_u_pfc_execution_unique_secondproductcoefficientsum fs_v_pfc_execution_unique_secondproductcoefficientsum. ((((exists fs_h_pfc_execution_unique_secondproductcoefficientsum_body_start. fs_h_pfc_execution_unique_secondproductcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_unique_secondproductcoefficientsum)) /\ exists fs_q_pfc_execution_unique_secondproductcoefficientsum_body_start. fs_u_pfc_execution_unique_secondproductcoefficientsum = fs_q_pfc_execution_unique_secondproductcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_execution_unique_secondproductcoefficientsum) + (0))) /\ ((((exists fs_h_pfc_execution_unique_secondproductcoefficientsum_body_terminal. fs_h_pfc_execution_unique_secondproductcoefficientsum_body_terminal + S (pfc_natural_sum_execution_unique_secondproductcoefficient) = S ((S (S (pfc_index_execution_unique_secondproduct))) * fs_v_pfc_execution_unique_secondproductcoefficientsum)) /\ exists fs_q_pfc_execution_unique_secondproductcoefficientsum_body_terminal. fs_u_pfc_execution_unique_secondproductcoefficientsum = fs_q_pfc_execution_unique_secondproductcoefficientsum_body_terminal * S ((S (S (pfc_index_execution_unique_secondproduct))) * fs_v_pfc_execution_unique_secondproductcoefficientsum) + (pfc_natural_sum_execution_unique_secondproductcoefficient))) /\ forall fs_i_pfc_execution_unique_secondproductcoefficientsum_body_steps. (exists fs_lt_pfc_execution_unique_secondproductcoefficientsum_body_steps_bound. fs_lt_pfc_execution_unique_secondproductcoefficientsum_body_steps_bound + S fs_i_pfc_execution_unique_secondproductcoefficientsum_body_steps = S (pfc_index_execution_unique_secondproduct)) -> exists fs_a_pfc_execution_unique_secondproductcoefficientsum_body_steps fs_r_pfc_execution_unique_secondproductcoefficientsum_body_steps fs_s_pfc_execution_unique_secondproductcoefficientsum_body_steps. ((((exists fs_h_pfc_execution_unique_secondproductcoefficientsum_body_steps_summand. fs_h_pfc_execution_unique_secondproductcoefficientsum_body_steps_summand + S (fs_a_pfc_execution_unique_secondproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_unique_secondproductcoefficientsum_body_steps)) * pfc_terms_scale_execution_unique_secondproductcoefficient)) /\ exists fs_q_pfc_execution_unique_secondproductcoefficientsum_body_steps_summand. pfc_terms_code_execution_unique_secondproductcoefficient = fs_q_pfc_execution_unique_secondproductcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_execution_unique_secondproductcoefficientsum_body_steps)) * pfc_terms_scale_execution_unique_secondproductcoefficient) + (fs_a_pfc_execution_unique_secondproductcoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_execution_unique_secondproductcoefficientsum_body_steps_partial. fs_h_pfc_execution_unique_secondproductcoefficientsum_body_steps_partial + S (fs_r_pfc_execution_unique_secondproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_unique_secondproductcoefficientsum_body_steps)) * fs_v_pfc_execution_unique_secondproductcoefficientsum)) /\ exists fs_q_pfc_execution_unique_secondproductcoefficientsum_body_steps_partial. fs_u_pfc_execution_unique_secondproductcoefficientsum = fs_q_pfc_execution_unique_secondproductcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_execution_unique_secondproductcoefficientsum_body_steps)) * fs_v_pfc_execution_unique_secondproductcoefficientsum) + (fs_r_pfc_execution_unique_secondproductcoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_execution_unique_secondproductcoefficientsum_body_steps_successor. fs_h_pfc_execution_unique_secondproductcoefficientsum_body_steps_successor + S (fs_s_pfc_execution_unique_secondproductcoefficientsum_body_steps) = S ((S (S fs_i_pfc_execution_unique_secondproductcoefficientsum_body_steps)) * fs_v_pfc_execution_unique_secondproductcoefficientsum)) /\ exists fs_q_pfc_execution_unique_secondproductcoefficientsum_body_steps_successor. fs_u_pfc_execution_unique_secondproductcoefficientsum = fs_q_pfc_execution_unique_secondproductcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_execution_unique_secondproductcoefficientsum_body_steps)) * fs_v_pfc_execution_unique_secondproductcoefficientsum) + (fs_s_pfc_execution_unique_secondproductcoefficientsum_body_steps))) /\ fs_s_pfc_execution_unique_secondproductcoefficientsum_body_steps = fs_r_pfc_execution_unique_secondproductcoefficientsum_body_steps + fs_a_pfc_execution_unique_secondproductcoefficientsum_body_steps)))))) /\ ((((exists pfa_gap_execution_unique_secondproductcoefficientresiduebound. pfa_gap_execution_unique_secondproductcoefficientresiduebound + S (pfc_value_execution_unique_secondproduct) = (p)) /\ ((exists pfa_offset_left_execution_unique_secondproductcoefficientresiduecongruence pfa_offset_right_execution_unique_secondproductcoefficientresiduecongruence. (pfc_natural_sum_execution_unique_secondproductcoefficient) + (p) * pfa_offset_left_execution_unique_secondproductcoefficientresiduecongruence = (pfc_value_execution_unique_secondproduct) + (p) * pfa_offset_right_execution_unique_secondproductcoefficientresiduecongruence)))))))))))) /\ (((forall pfs_index_execution_unique_seconddifference. (exists pfa_gap_execution_unique_seconddifferenceindex. pfa_gap_execution_unique_seconddifferenceindex + S (pfs_index_execution_unique_seconddifference) = (L)) -> exists pfs_left_execution_unique_seconddifference pfs_right_execution_unique_seconddifference pfs_result_execution_unique_seconddifference. ((((exists ff_h_pfp_execution_unique_seconddifferenceleft. ff_h_pfp_execution_unique_seconddifferenceleft + S (pfs_left_execution_unique_seconddifference) = S ((S (pfs_index_execution_unique_seconddifference)) * ac)) /\ exists ff_q_pfp_execution_unique_seconddifferenceleft. ab = ff_q_pfp_execution_unique_seconddifferenceleft * S ((S (pfs_index_execution_unique_seconddifference)) * ac) + (pfs_left_execution_unique_seconddifference))) /\ (((((exists ff_h_pfp_execution_unique_seconddifferenceright. ff_h_pfp_execution_unique_seconddifferenceright + S (pfs_right_execution_unique_seconddifference) = S ((S (pfs_index_execution_unique_seconddifference)) * pfd_product_scale_execution_unique_second)) /\ exists ff_q_pfp_execution_unique_seconddifferenceright. pfd_product_code_execution_unique_second = ff_q_pfp_execution_unique_seconddifferenceright * S ((S (pfs_index_execution_unique_seconddifference)) * pfd_product_scale_execution_unique_second) + (pfs_right_execution_unique_seconddifference))) /\ (((((exists ff_h_pfp_execution_unique_seconddifferenceresult. ff_h_pfp_execution_unique_seconddifferenceresult + S (pfs_result_execution_unique_seconddifference) = S ((S (pfs_index_execution_unique_seconddifference)) * pfd_residual_scale_execution_unique_second)) /\ exists ff_q_pfp_execution_unique_seconddifferenceresult. pfd_residual_code_execution_unique_second = ff_q_pfp_execution_unique_seconddifferenceresult * S ((S (pfs_index_execution_unique_seconddifference)) * pfd_residual_scale_execution_unique_second) + (pfs_result_execution_unique_seconddifference))) /\ ((((exists pfa_gap_execution_unique_seconddifferenceoperationleft. pfa_gap_execution_unique_seconddifferenceoperationleft + S (pfs_right_execution_unique_seconddifference) = (p)) /\ (((exists pfa_gap_execution_unique_seconddifferenceoperationright. pfa_gap_execution_unique_seconddifferenceoperationright + S (pfs_result_execution_unique_seconddifference) = (p)) /\ ((((exists pfa_gap_execution_unique_seconddifferenceoperationresultbound. pfa_gap_execution_unique_seconddifferenceoperationresultbound + S (pfs_left_execution_unique_seconddifference) = (p)) /\ ((exists pfa_offset_left_execution_unique_seconddifferenceoperationresultcongruence pfa_offset_right_execution_unique_seconddifferenceoperationresultcongruence. ((pfs_right_execution_unique_seconddifference) + (pfs_result_execution_unique_seconddifference)) + (p) * pfa_offset_left_execution_unique_seconddifferenceoperationresultcongruence = (pfs_left_execution_unique_seconddifference) + (p) * pfa_offset_right_execution_unique_seconddifferenceoperationresultcongruence)))))))))))))))) /\ (((((L)=(pfd_cut_execution_unique_second)+(K)) /\ (((forall fom_index_pfp_execution_unique_secondtriminput. (exists fom_gap_pfp_execution_unique_secondtriminput_index_bound. fom_gap_pfp_execution_unique_secondtriminput_index_bound + S (fom_index_pfp_execution_unique_secondtriminput) = L) -> exists fom_value_pfp_execution_unique_secondtriminput. ((((exists fom_beta_height_pfp_execution_unique_secondtriminput_entry. fom_beta_height_pfp_execution_unique_secondtriminput_entry + S (fom_value_pfp_execution_unique_secondtriminput) = S ((S (fom_index_pfp_execution_unique_secondtriminput)) * pfd_residual_scale_execution_unique_second)) /\ exists fom_beta_quotient_pfp_execution_unique_secondtriminput_entry. pfd_residual_code_execution_unique_second = fom_beta_quotient_pfp_execution_unique_secondtriminput_entry * S ((S (fom_index_pfp_execution_unique_secondtriminput)) * pfd_residual_scale_execution_unique_second) + (fom_value_pfp_execution_unique_secondtriminput))) /\ (exists fom_gap_pfp_execution_unique_secondtriminput_value_bound. fom_gap_pfp_execution_unique_secondtriminput_value_bound + S (fom_value_pfp_execution_unique_secondtriminput) = p))) /\ (((forall pfp_repeat_index_execution_unique_secondtrimremoved. (exists pfa_gap_execution_unique_secondtrimremovedindex. pfa_gap_execution_unique_secondtrimremovedindex + S (pfp_repeat_index_execution_unique_secondtrimremoved) = (pfd_cut_execution_unique_second)) -> (((exists ff_h_pfp_execution_unique_secondtrimremovedentry. ff_h_pfp_execution_unique_secondtrimremovedentry + S (0) = S ((S (pfp_repeat_index_execution_unique_secondtrimremoved)) * pfd_residual_scale_execution_unique_second)) /\ exists ff_q_pfp_execution_unique_secondtrimremovedentry. pfd_residual_code_execution_unique_second = ff_q_pfp_execution_unique_secondtrimremovedentry * S ((S (pfp_repeat_index_execution_unique_secondtrimremoved)) * pfd_residual_scale_execution_unique_second) + (0)))) /\ (((forall pftrim_index_execution_unique_secondtrimsuffix pftrim_value_execution_unique_secondtrimsuffix. (exists pfa_gap_execution_unique_secondtrimsuffixbound. pfa_gap_execution_unique_secondtrimsuffixbound + S (pftrim_index_execution_unique_secondtrimsuffix) = (K)) -> (((exists ff_h_pfp_execution_unique_secondtrimsuffixsource. ff_h_pfp_execution_unique_secondtrimsuffixsource + S (pftrim_value_execution_unique_secondtrimsuffix) = S ((S ((pfd_cut_execution_unique_second)+pftrim_index_execution_unique_secondtrimsuffix)) * pfd_residual_scale_execution_unique_second)) /\ exists ff_q_pfp_execution_unique_secondtrimsuffixsource. pfd_residual_code_execution_unique_second = ff_q_pfp_execution_unique_secondtrimsuffixsource * S ((S ((pfd_cut_execution_unique_second)+pftrim_index_execution_unique_secondtrimsuffix)) * pfd_residual_scale_execution_unique_second) + (pftrim_value_execution_unique_secondtrimsuffix))) -> (((exists ff_h_pfp_execution_unique_secondtrimsuffixoutput. ff_h_pfp_execution_unique_secondtrimsuffixoutput + S (pftrim_value_execution_unique_secondtrimsuffix) = S ((S (pftrim_index_execution_unique_secondtrimsuffix)) * RC)) /\ exists ff_q_pfp_execution_unique_secondtrimsuffixoutput. RB = ff_q_pfp_execution_unique_secondtrimsuffixoutput * S ((S (pftrim_index_execution_unique_secondtrimsuffix)) * RC) + (pftrim_value_execution_unique_secondtrimsuffix)))) /\ (((K)=0 \/ (exists pftrim_leading_execution_unique_secondtrimnormal. ((((exists ff_h_pfp_execution_unique_secondtrimnormalentry. ff_h_pfp_execution_unique_secondtrimnormalentry + S (pftrim_leading_execution_unique_secondtrimnormal) = S ((S (0)) * RC)) /\ exists ff_q_pfp_execution_unique_secondtrimnormalentry. RB = ff_q_pfp_execution_unique_secondtrimnormalentry * S ((S (0)) * RC) + (pftrim_leading_execution_unique_secondtrimnormal))) /\ ((~(pftrim_leading_execution_unique_secondtrimnormal=0))))))))))))))))))))))))))))))))) -> ((((q)=(Q)) /\ (((forall mdr_i_pfp_execution_unique_outputsquotient mdr_a_pfp_execution_unique_outputsquotient. (exists mdr_gap_pfp_execution_unique_outputsquotientb. mdr_gap_pfp_execution_unique_outputsquotientb + S (mdr_i_pfp_execution_unique_outputsquotient) = (q)) -> (((exists ff_h_mdr_pfp_execution_unique_outputsquotiento. ff_h_mdr_pfp_execution_unique_outputsquotiento + S (mdr_a_pfp_execution_unique_outputsquotient) = S ((S (mdr_i_pfp_execution_unique_outputsquotient)) * qc)) /\ exists ff_q_mdr_pfp_execution_unique_outputsquotiento. qb = ff_q_mdr_pfp_execution_unique_outputsquotiento * S ((S (mdr_i_pfp_execution_unique_outputsquotient)) * qc) + (mdr_a_pfp_execution_unique_outputsquotient))) -> (((exists ff_h_mdr_pfp_execution_unique_outputsquotientn. ff_h_mdr_pfp_execution_unique_outputsquotientn + S (mdr_a_pfp_execution_unique_outputsquotient) = S ((S (mdr_i_pfp_execution_unique_outputsquotient)) * QC)) /\ exists ff_q_mdr_pfp_execution_unique_outputsquotientn. QB = ff_q_mdr_pfp_execution_unique_outputsquotientn * S ((S (mdr_i_pfp_execution_unique_outputsquotient)) * QC) + (mdr_a_pfp_execution_unique_outputsquotient)))) /\ ((((R)=(K)) /\ ((forall mdr_i_pfp_execution_unique_outputsremainder mdr_a_pfp_execution_unique_outputsremainder. (exists mdr_gap_pfp_execution_unique_outputsremainderb. mdr_gap_pfp_execution_unique_outputsremainderb + S (mdr_i_pfp_execution_unique_outputsremainder) = (R)) -> (((exists ff_h_mdr_pfp_execution_unique_outputsremaindero. ff_h_mdr_pfp_execution_unique_outputsremaindero + S (mdr_a_pfp_execution_unique_outputsremainder) = S ((S (mdr_i_pfp_execution_unique_outputsremainder)) * rc)) /\ exists ff_q_mdr_pfp_execution_unique_outputsremaindero. rb = ff_q_mdr_pfp_execution_unique_outputsremaindero * S ((S (mdr_i_pfp_execution_unique_outputsremainder)) * rc) + (mdr_a_pfp_execution_unique_outputsremainder))) -> (((exists ff_h_mdr_pfp_execution_unique_outputsremaindern. ff_h_mdr_pfp_execution_unique_outputsremaindern + S (mdr_a_pfp_execution_unique_outputsremainder) = S ((S (mdr_i_pfp_execution_unique_outputsremainder)) * RC)) /\ exists ff_q_mdr_pfp_execution_unique_outputsremaindern. RB = ff_q_mdr_pfp_execution_unique_outputsremaindern * S ((S (mdr_i_pfp_execution_unique_outputsremainder)) * RC) + (mdr_a_pfp_execution_unique_outputsremainder)))))))))))
Complete tactic proof in conservative notation
All 139 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.
This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.
Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.