PX003A

prime_field_polynomial_division_remainder_degree

Every actual constructed remainder is empty or has genuinely represented degree below the divisor degree, including constant divisors and empty inputs.

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.

Exact theorem in conservative defined notation

∀ p. ∀ ab. ∀ ac. ∀ L. ∀ bb. ∀ bc. ∀ d. ∀ qb. ∀ qc. ∀ q. ∀ rb. ∀ rc. ∀ R. Prime(p) → FpPolynomialDivisionExecution(p,ab,ac,L,bb,bc,d,qb,qc,q,rb,rc,R) → R = 0 ∨ (∃ x. FpRepresentedDegree(p,rb,rc,R,x) ∧ Lt(x,d))

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

FpRepresentedDegree(p,b,c,L,d) · 1FpPolynomialDivisionExecution(p,ab,ac,L,bb,bc,d,qb,qc,q,rb,rc,R) · 1Le(a,b) · 2Lt(a,b) · 1Prime(p) · 1

Actual proof prerequisites

polynomial_quotient_length_boundsprime_field_polynomial_trim_bounded_degreeprime_field_polynomial_trim_zero_prefix_remainder_boundprime_field_polynomial_quotient_prefix_remainder_zero
Original expanded first-order statement 
forall p ab ac L bb bc d qb qc q rb rc R. (~((p) = 1) /\ forall pfa_factor_left_division_degree_prime pfa_factor_right_division_degree_prime. (p) = pfa_factor_left_division_degree_prime * pfa_factor_right_division_degree_prime -> pfa_factor_left_division_degree_prime = 1 \/ pfa_factor_right_division_degree_prime = 1) -> (((forall fom_index_pfp_division_degree_executioninput. (exists fom_gap_pfp_division_degree_executioninput_index_bound. fom_gap_pfp_division_degree_executioninput_index_bound + S (fom_index_pfp_division_degree_executioninput) = L) -> exists fom_value_pfp_division_degree_executioninput. ((((exists fom_beta_height_pfp_division_degree_executioninput_entry. fom_beta_height_pfp_division_degree_executioninput_entry + S (fom_value_pfp_division_degree_executioninput) = S ((S (fom_index_pfp_division_degree_executioninput)) * ac)) /\ exists fom_beta_quotient_pfp_division_degree_executioninput_entry. ab = fom_beta_quotient_pfp_division_degree_executioninput_entry * S ((S (fom_index_pfp_division_degree_executioninput)) * ac) + (fom_value_pfp_division_degree_executioninput))) /\ (exists fom_gap_pfp_division_degree_executioninput_value_bound. fom_gap_pfp_division_degree_executioninput_value_bound + S (fom_value_pfp_division_degree_executioninput) = p))) /\ (((forall fom_index_pfp_division_degree_executiondivisor. (exists fom_gap_pfp_division_degree_executiondivisor_index_bound. fom_gap_pfp_division_degree_executiondivisor_index_bound + S (fom_index_pfp_division_degree_executiondivisor) = S (d)) -> exists fom_value_pfp_division_degree_executiondivisor. ((((exists fom_beta_height_pfp_division_degree_executiondivisor_entry. fom_beta_height_pfp_division_degree_executiondivisor_entry + S (fom_value_pfp_division_degree_executiondivisor) = S ((S (fom_index_pfp_division_degree_executiondivisor)) * bc)) /\ exists fom_beta_quotient_pfp_division_degree_executiondivisor_entry. bb = fom_beta_quotient_pfp_division_degree_executiondivisor_entry * S ((S (fom_index_pfp_division_degree_executiondivisor)) * bc) + (fom_value_pfp_division_degree_executiondivisor))) /\ (exists fom_gap_pfp_division_degree_executiondivisor_value_bound. fom_gap_pfp_division_degree_executiondivisor_value_bound + S (fom_value_pfp_division_degree_executiondivisor) = p))) /\ (((((((q)=0) /\ ((exists pfc_gap_division_degree_executionlengthshort. pfc_gap_division_degree_executionlengthshort+(L)=(d))))) \/ (((~((q)=0)) /\ (((q)+(d)=(L)))))) /\ ((exists pfd_head_division_degree_execution pfd_inverse_division_degree_execution pfd_product_code_division_degree_execution pfd_product_scale_division_degree_execution pfd_residual_code_division_degree_execution pfd_residual_scale_division_degree_execution pfd_cut_division_degree_execution. ((((exists ff_h_pfp_division_degree_executionhead. ff_h_pfp_division_degree_executionhead + S (pfd_head_division_degree_execution) = S ((S (0)) * bc)) /\ exists ff_q_pfp_division_degree_executionhead. bb = ff_q_pfp_division_degree_executionhead * S ((S (0)) * bc) + (pfd_head_division_degree_execution))) /\ (((((~((pfd_head_division_degree_execution) = 0)) /\ ((((exists pfa_gap_division_degree_executioninversemultiplicationleft. pfa_gap_division_degree_executioninversemultiplicationleft + S (pfd_head_division_degree_execution) = (p)) /\ (((exists pfa_gap_division_degree_executioninversemultiplicationright. pfa_gap_division_degree_executioninversemultiplicationright + S (pfd_inverse_division_degree_execution) = (p)) /\ ((((exists pfa_gap_division_degree_executioninversemultiplicationresultbound. pfa_gap_division_degree_executioninversemultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_division_degree_executioninversemultiplicationresultcongruence pfa_offset_right_division_degree_executioninversemultiplicationresultcongruence. ((pfd_head_division_degree_execution) * (pfd_inverse_division_degree_execution)) + (p) * pfa_offset_left_division_degree_executioninversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_division_degree_executioninversemultiplicationresultcongruence)))))))))))) /\ (((forall pfd_index_division_degree_executionquotient. (exists pfa_gap_division_degree_executionquotientbound. pfa_gap_division_degree_executionquotientbound + S (pfd_index_division_degree_executionquotient) = (q)) -> exists pfd_value_division_degree_executionquotient. ((((exists ff_h_pfp_division_degree_executionquotiententry. ff_h_pfp_division_degree_executionquotiententry + S (pfd_value_division_degree_executionquotient) = S ((S (pfd_index_division_degree_executionquotient)) * qc)) /\ exists ff_q_pfp_division_degree_executionquotiententry. qb = ff_q_pfp_division_degree_executionquotiententry * S ((S (pfd_index_division_degree_executionquotient)) * qc) + (pfd_value_division_degree_executionquotient))) /\ ((exists pfd_input_division_degree_executionquotientstep pfd_previous_division_degree_executionquotientstep pfd_difference_division_degree_executionquotientstep. ((((exists ff_h_pfp_division_degree_executionquotientstepinput. ff_h_pfp_division_degree_executionquotientstepinput + S (pfd_input_division_degree_executionquotientstep) = S ((S (pfd_index_division_degree_executionquotient)) * ac)) /\ exists ff_q_pfp_division_degree_executionquotientstepinput. ab = ff_q_pfp_division_degree_executionquotientstepinput * S ((S (pfd_index_division_degree_executionquotient)) * ac) + (pfd_input_division_degree_executionquotientstep))) /\ (((exists pfc_terms_code_division_degree_executionquotientstepprevious pfc_terms_scale_division_degree_executionquotientstepprevious pfc_natural_sum_division_degree_executionquotientstepprevious. ((forall pfc_index_division_degree_executionquotientsteppreviousdiagonal. (exists pfa_gap_division_degree_executionquotientsteppreviousdiagonalbound. pfa_gap_division_degree_executionquotientsteppreviousdiagonalbound + S (pfc_index_division_degree_executionquotientsteppreviousdiagonal) = (S (pfd_index_division_degree_executionquotient))) -> exists pfc_value_division_degree_executionquotientsteppreviousdiagonal. ((((exists ff_h_pfp_division_degree_executionquotientsteppreviousdiagonalentry. ff_h_pfp_division_degree_executionquotientsteppreviousdiagonalentry + S (pfc_value_division_degree_executionquotientsteppreviousdiagonal) = S ((S (pfc_index_division_degree_executionquotientsteppreviousdiagonal)) * pfc_terms_scale_division_degree_executionquotientstepprevious)) /\ exists ff_q_pfp_division_degree_executionquotientsteppreviousdiagonalentry. pfc_terms_code_division_degree_executionquotientstepprevious = ff_q_pfp_division_degree_executionquotientsteppreviousdiagonalentry * S ((S (pfc_index_division_degree_executionquotientsteppreviousdiagonal)) * pfc_terms_scale_division_degree_executionquotientstepprevious) + (pfc_value_division_degree_executionquotientsteppreviousdiagonal))) /\ ((exists pfc_complement_division_degree_executionquotientsteppreviousdiagonalterm pfc_left_division_degree_executionquotientsteppreviousdiagonalterm pfc_right_division_degree_executionquotientsteppreviousdiagonalterm. (((pfc_index_division_degree_executionquotientsteppreviousdiagonal)+pfc_complement_division_degree_executionquotientsteppreviousdiagonalterm=(pfd_index_division_degree_executionquotient)) /\ ((((((exists pfa_gap_division_degree_executionquotientsteppreviousdiagonaltermleftinside. pfa_gap_division_degree_executionquotientsteppreviousdiagonaltermleftinside + S (pfc_index_division_degree_executionquotientsteppreviousdiagonal) = (pfd_index_division_degree_executionquotient)) /\ ((((exists ff_h_pfp_division_degree_executionquotientsteppreviousdiagonaltermleftentry. ff_h_pfp_division_degree_executionquotientsteppreviousdiagonaltermleftentry + S (pfc_left_division_degree_executionquotientsteppreviousdiagonalterm) = S ((S (pfc_index_division_degree_executionquotientsteppreviousdiagonal)) * qc)) /\ exists ff_q_pfp_division_degree_executionquotientsteppreviousdiagonaltermleftentry. qb = ff_q_pfp_division_degree_executionquotientsteppreviousdiagonaltermleftentry * S ((S (pfc_index_division_degree_executionquotientsteppreviousdiagonal)) * qc) + (pfc_left_division_degree_executionquotientsteppreviousdiagonalterm)))))) \/ (((exists pfc_gap_division_degree_executionquotientsteppreviousdiagonaltermleftoutside. pfc_gap_division_degree_executionquotientsteppreviousdiagonaltermleftoutside+(pfd_index_division_degree_executionquotient)=(pfc_index_division_degree_executionquotientsteppreviousdiagonal)) /\ (((pfc_left_division_degree_executionquotientsteppreviousdiagonalterm)=0))))) /\ ((((((exists pfa_gap_division_degree_executionquotientsteppreviousdiagonaltermrightinside. pfa_gap_division_degree_executionquotientsteppreviousdiagonaltermrightinside + S (pfc_complement_division_degree_executionquotientsteppreviousdiagonalterm) = (S (d))) /\ ((((exists ff_h_pfp_division_degree_executionquotientsteppreviousdiagonaltermrightentry. ff_h_pfp_division_degree_executionquotientsteppreviousdiagonaltermrightentry + S (pfc_right_division_degree_executionquotientsteppreviousdiagonalterm) = S ((S (pfc_complement_division_degree_executionquotientsteppreviousdiagonalterm)) * bc)) /\ exists ff_q_pfp_division_degree_executionquotientsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_division_degree_executionquotientsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_division_degree_executionquotientsteppreviousdiagonalterm)) * bc) + (pfc_right_division_degree_executionquotientsteppreviousdiagonalterm)))))) \/ (((exists pfc_gap_division_degree_executionquotientsteppreviousdiagonaltermrightoutside. pfc_gap_division_degree_executionquotientsteppreviousdiagonaltermrightoutside+(S (d))=(pfc_complement_division_degree_executionquotientsteppreviousdiagonalterm)) /\ (((pfc_right_division_degree_executionquotientsteppreviousdiagonalterm)=0))))) /\ (((pfc_value_division_degree_executionquotientsteppreviousdiagonal)=pfc_left_division_degree_executionquotientsteppreviousdiagonalterm*pfc_right_division_degree_executionquotientsteppreviousdiagonalterm))))))))))) /\ (((exists fs_u_pfc_division_degree_executionquotientstepprevioussum fs_v_pfc_division_degree_executionquotientstepprevioussum. ((((exists fs_h_pfc_division_degree_executionquotientstepprevioussum_body_start. fs_h_pfc_division_degree_executionquotientstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_degree_executionquotientstepprevioussum)) /\ exists fs_q_pfc_division_degree_executionquotientstepprevioussum_body_start. fs_u_pfc_division_degree_executionquotientstepprevioussum = fs_q_pfc_division_degree_executionquotientstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_division_degree_executionquotientstepprevioussum) + (0))) /\ ((((exists fs_h_pfc_division_degree_executionquotientstepprevioussum_body_terminal. fs_h_pfc_division_degree_executionquotientstepprevioussum_body_terminal + S (pfc_natural_sum_division_degree_executionquotientstepprevious) = S ((S (S (pfd_index_division_degree_executionquotient))) * fs_v_pfc_division_degree_executionquotientstepprevioussum)) /\ exists fs_q_pfc_division_degree_executionquotientstepprevioussum_body_terminal. fs_u_pfc_division_degree_executionquotientstepprevioussum = fs_q_pfc_division_degree_executionquotientstepprevioussum_body_terminal * S ((S (S (pfd_index_division_degree_executionquotient))) * fs_v_pfc_division_degree_executionquotientstepprevioussum) + (pfc_natural_sum_division_degree_executionquotientstepprevious))) /\ forall fs_i_pfc_division_degree_executionquotientstepprevioussum_body_steps. (exists fs_lt_pfc_division_degree_executionquotientstepprevioussum_body_steps_bound. fs_lt_pfc_division_degree_executionquotientstepprevioussum_body_steps_bound + S fs_i_pfc_division_degree_executionquotientstepprevioussum_body_steps = S (pfd_index_division_degree_executionquotient)) -> exists fs_a_pfc_division_degree_executionquotientstepprevioussum_body_steps fs_r_pfc_division_degree_executionquotientstepprevioussum_body_steps fs_s_pfc_division_degree_executionquotientstepprevioussum_body_steps. ((((exists fs_h_pfc_division_degree_executionquotientstepprevioussum_body_steps_summand. fs_h_pfc_division_degree_executionquotientstepprevioussum_body_steps_summand + S (fs_a_pfc_division_degree_executionquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_degree_executionquotientstepprevioussum_body_steps)) * pfc_terms_scale_division_degree_executionquotientstepprevious)) /\ exists fs_q_pfc_division_degree_executionquotientstepprevioussum_body_steps_summand. pfc_terms_code_division_degree_executionquotientstepprevious = fs_q_pfc_division_degree_executionquotientstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_division_degree_executionquotientstepprevioussum_body_steps)) * pfc_terms_scale_division_degree_executionquotientstepprevious) + (fs_a_pfc_division_degree_executionquotientstepprevioussum_body_steps))) /\ ((((exists fs_h_pfc_division_degree_executionquotientstepprevioussum_body_steps_partial. fs_h_pfc_division_degree_executionquotientstepprevioussum_body_steps_partial + S (fs_r_pfc_division_degree_executionquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_division_degree_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_division_degree_executionquotientstepprevioussum)) /\ exists fs_q_pfc_division_degree_executionquotientstepprevioussum_body_steps_partial. fs_u_pfc_division_degree_executionquotientstepprevioussum = fs_q_pfc_division_degree_executionquotientstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_division_degree_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_division_degree_executionquotientstepprevioussum) + (fs_r_pfc_division_degree_executionquotientstepprevioussum_body_steps))) /\ ((((exists fs_h_pfc_division_degree_executionquotientstepprevioussum_body_steps_successor. fs_h_pfc_division_degree_executionquotientstepprevioussum_body_steps_successor + S (fs_s_pfc_division_degree_executionquotientstepprevioussum_body_steps) = S ((S (S fs_i_pfc_division_degree_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_division_degree_executionquotientstepprevioussum)) /\ exists fs_q_pfc_division_degree_executionquotientstepprevioussum_body_steps_successor. fs_u_pfc_division_degree_executionquotientstepprevioussum = fs_q_pfc_division_degree_executionquotientstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_division_degree_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_division_degree_executionquotientstepprevioussum) + (fs_s_pfc_division_degree_executionquotientstepprevioussum_body_steps))) /\ fs_s_pfc_division_degree_executionquotientstepprevioussum_body_steps = fs_r_pfc_division_degree_executionquotientstepprevioussum_body_steps + fs_a_pfc_division_degree_executionquotientstepprevioussum_body_steps)))))) /\ ((((exists pfa_gap_division_degree_executionquotientsteppreviousresiduebound. pfa_gap_division_degree_executionquotientsteppreviousresiduebound + S (pfd_previous_division_degree_executionquotientstep) = (p)) /\ ((exists pfa_offset_left_division_degree_executionquotientsteppreviousresiduecongruence pfa_offset_right_division_degree_executionquotientsteppreviousresiduecongruence. (pfc_natural_sum_division_degree_executionquotientstepprevious) + (p) * pfa_offset_left_division_degree_executionquotientsteppreviousresiduecongruence = (pfd_previous_division_degree_executionquotientstep) + (p) * pfa_offset_right_division_degree_executionquotientsteppreviousresiduecongruence))))))))) /\ (((((exists pfa_gap_division_degree_executionquotientstepsubtractleft. pfa_gap_division_degree_executionquotientstepsubtractleft + S (pfd_previous_division_degree_executionquotientstep) = (p)) /\ (((exists pfa_gap_division_degree_executionquotientstepsubtractright. pfa_gap_division_degree_executionquotientstepsubtractright + S (pfd_difference_division_degree_executionquotientstep) = (p)) /\ ((((exists pfa_gap_division_degree_executionquotientstepsubtractresultbound. pfa_gap_division_degree_executionquotientstepsubtractresultbound + S (pfd_input_division_degree_executionquotientstep) = (p)) /\ ((exists pfa_offset_left_division_degree_executionquotientstepsubtractresultcongruence pfa_offset_right_division_degree_executionquotientstepsubtractresultcongruence. ((pfd_previous_division_degree_executionquotientstep) + (pfd_difference_division_degree_executionquotientstep)) + (p) * pfa_offset_left_division_degree_executionquotientstepsubtractresultcongruence = (pfd_input_division_degree_executionquotientstep) + (p) * pfa_offset_right_division_degree_executionquotientstepsubtractresultcongruence))))))))) /\ ((((exists pfa_gap_division_degree_executionquotientstepmultiplyleft. pfa_gap_division_degree_executionquotientstepmultiplyleft + S (pfd_inverse_division_degree_execution) = (p)) /\ (((exists pfa_gap_division_degree_executionquotientstepmultiplyright. pfa_gap_division_degree_executionquotientstepmultiplyright + S (pfd_difference_division_degree_executionquotientstep) = (p)) /\ ((((exists pfa_gap_division_degree_executionquotientstepmultiplyresultbound. pfa_gap_division_degree_executionquotientstepmultiplyresultbound + S (pfd_value_division_degree_executionquotient) = (p)) /\ ((exists pfa_offset_left_division_degree_executionquotientstepmultiplyresultcongruence pfa_offset_right_division_degree_executionquotientstepmultiplyresultcongruence. ((pfd_inverse_division_degree_execution) * (pfd_difference_division_degree_executionquotientstep)) + (p) * pfa_offset_left_division_degree_executionquotientstepmultiplyresultcongruence = (pfd_value_division_degree_executionquotient) + (p) * pfa_offset_right_division_degree_executionquotientstepmultiplyresultcongruence))))))))))))))))))) /\ (((forall pfc_index_division_degree_executionproduct. (exists pfa_gap_division_degree_executionproductbound. pfa_gap_division_degree_executionproductbound + S (pfc_index_division_degree_executionproduct) = (L)) -> exists pfc_value_division_degree_executionproduct. ((((exists ff_h_pfp_division_degree_executionproductentry. ff_h_pfp_division_degree_executionproductentry + S (pfc_value_division_degree_executionproduct) = S ((S (pfc_index_division_degree_executionproduct)) * pfd_product_scale_division_degree_execution)) /\ exists ff_q_pfp_division_degree_executionproductentry. pfd_product_code_division_degree_execution = ff_q_pfp_division_degree_executionproductentry * S ((S (pfc_index_division_degree_executionproduct)) * pfd_product_scale_division_degree_execution) + (pfc_value_division_degree_executionproduct))) /\ ((exists pfc_terms_code_division_degree_executionproductcoefficient pfc_terms_scale_division_degree_executionproductcoefficient pfc_natural_sum_division_degree_executionproductcoefficient. ((forall pfc_index_division_degree_executionproductcoefficientdiagonal. (exists pfa_gap_division_degree_executionproductcoefficientdiagonalbound. pfa_gap_division_degree_executionproductcoefficientdiagonalbound + S (pfc_index_division_degree_executionproductcoefficientdiagonal) = (S (pfc_index_division_degree_executionproduct))) -> exists pfc_value_division_degree_executionproductcoefficientdiagonal. ((((exists ff_h_pfp_division_degree_executionproductcoefficientdiagonalentry. ff_h_pfp_division_degree_executionproductcoefficientdiagonalentry + S (pfc_value_division_degree_executionproductcoefficientdiagonal) = S ((S (pfc_index_division_degree_executionproductcoefficientdiagonal)) * pfc_terms_scale_division_degree_executionproductcoefficient)) /\ exists ff_q_pfp_division_degree_executionproductcoefficientdiagonalentry. pfc_terms_code_division_degree_executionproductcoefficient = ff_q_pfp_division_degree_executionproductcoefficientdiagonalentry * S ((S (pfc_index_division_degree_executionproductcoefficientdiagonal)) * pfc_terms_scale_division_degree_executionproductcoefficient) + (pfc_value_division_degree_executionproductcoefficientdiagonal))) /\ ((exists pfc_complement_division_degree_executionproductcoefficientdiagonalterm pfc_left_division_degree_executionproductcoefficientdiagonalterm pfc_right_division_degree_executionproductcoefficientdiagonalterm. (((pfc_index_division_degree_executionproductcoefficientdiagonal)+pfc_complement_division_degree_executionproductcoefficientdiagonalterm=(pfc_index_division_degree_executionproduct)) /\ ((((((exists pfa_gap_division_degree_executionproductcoefficientdiagonaltermleftinside. pfa_gap_division_degree_executionproductcoefficientdiagonaltermleftinside + S (pfc_index_division_degree_executionproductcoefficientdiagonal) = (q)) /\ ((((exists ff_h_pfp_division_degree_executionproductcoefficientdiagonaltermleftentry. ff_h_pfp_division_degree_executionproductcoefficientdiagonaltermleftentry + S (pfc_left_division_degree_executionproductcoefficientdiagonalterm) = S ((S (pfc_index_division_degree_executionproductcoefficientdiagonal)) * qc)) /\ exists ff_q_pfp_division_degree_executionproductcoefficientdiagonaltermleftentry. qb = ff_q_pfp_division_degree_executionproductcoefficientdiagonaltermleftentry * S ((S (pfc_index_division_degree_executionproductcoefficientdiagonal)) * qc) + (pfc_left_division_degree_executionproductcoefficientdiagonalterm)))))) \/ (((exists pfc_gap_division_degree_executionproductcoefficientdiagonaltermleftoutside. pfc_gap_division_degree_executionproductcoefficientdiagonaltermleftoutside+(q)=(pfc_index_division_degree_executionproductcoefficientdiagonal)) /\ (((pfc_left_division_degree_executionproductcoefficientdiagonalterm)=0))))) /\ ((((((exists pfa_gap_division_degree_executionproductcoefficientdiagonaltermrightinside. pfa_gap_division_degree_executionproductcoefficientdiagonaltermrightinside + S (pfc_complement_division_degree_executionproductcoefficientdiagonalterm) = (S (d))) /\ ((((exists ff_h_pfp_division_degree_executionproductcoefficientdiagonaltermrightentry. ff_h_pfp_division_degree_executionproductcoefficientdiagonaltermrightentry + S (pfc_right_division_degree_executionproductcoefficientdiagonalterm) = S ((S (pfc_complement_division_degree_executionproductcoefficientdiagonalterm)) * bc)) /\ exists ff_q_pfp_division_degree_executionproductcoefficientdiagonaltermrightentry. bb = ff_q_pfp_division_degree_executionproductcoefficientdiagonaltermrightentry * S ((S (pfc_complement_division_degree_executionproductcoefficientdiagonalterm)) * bc) + (pfc_right_division_degree_executionproductcoefficientdiagonalterm)))))) \/ (((exists pfc_gap_division_degree_executionproductcoefficientdiagonaltermrightoutside. pfc_gap_division_degree_executionproductcoefficientdiagonaltermrightoutside+(S (d))=(pfc_complement_division_degree_executionproductcoefficientdiagonalterm)) /\ (((pfc_right_division_degree_executionproductcoefficientdiagonalterm)=0))))) /\ (((pfc_value_division_degree_executionproductcoefficientdiagonal)=pfc_left_division_degree_executionproductcoefficientdiagonalterm*pfc_right_division_degree_executionproductcoefficientdiagonalterm))))))))))) /\ (((exists fs_u_pfc_division_degree_executionproductcoefficientsum fs_v_pfc_division_degree_executionproductcoefficientsum. ((((exists fs_h_pfc_division_degree_executionproductcoefficientsum_body_start. fs_h_pfc_division_degree_executionproductcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_degree_executionproductcoefficientsum)) /\ exists fs_q_pfc_division_degree_executionproductcoefficientsum_body_start. fs_u_pfc_division_degree_executionproductcoefficientsum = fs_q_pfc_division_degree_executionproductcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_division_degree_executionproductcoefficientsum) + (0))) /\ ((((exists fs_h_pfc_division_degree_executionproductcoefficientsum_body_terminal. fs_h_pfc_division_degree_executionproductcoefficientsum_body_terminal + S (pfc_natural_sum_division_degree_executionproductcoefficient) = S ((S (S (pfc_index_division_degree_executionproduct))) * fs_v_pfc_division_degree_executionproductcoefficientsum)) /\ exists fs_q_pfc_division_degree_executionproductcoefficientsum_body_terminal. fs_u_pfc_division_degree_executionproductcoefficientsum = fs_q_pfc_division_degree_executionproductcoefficientsum_body_terminal * S ((S (S (pfc_index_division_degree_executionproduct))) * fs_v_pfc_division_degree_executionproductcoefficientsum) + (pfc_natural_sum_division_degree_executionproductcoefficient))) /\ forall fs_i_pfc_division_degree_executionproductcoefficientsum_body_steps. (exists fs_lt_pfc_division_degree_executionproductcoefficientsum_body_steps_bound. fs_lt_pfc_division_degree_executionproductcoefficientsum_body_steps_bound + S fs_i_pfc_division_degree_executionproductcoefficientsum_body_steps = S (pfc_index_division_degree_executionproduct)) -> exists fs_a_pfc_division_degree_executionproductcoefficientsum_body_steps fs_r_pfc_division_degree_executionproductcoefficientsum_body_steps fs_s_pfc_division_degree_executionproductcoefficientsum_body_steps. ((((exists fs_h_pfc_division_degree_executionproductcoefficientsum_body_steps_summand. fs_h_pfc_division_degree_executionproductcoefficientsum_body_steps_summand + S (fs_a_pfc_division_degree_executionproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_division_degree_executionproductcoefficientsum_body_steps)) * pfc_terms_scale_division_degree_executionproductcoefficient)) /\ exists fs_q_pfc_division_degree_executionproductcoefficientsum_body_steps_summand. pfc_terms_code_division_degree_executionproductcoefficient = fs_q_pfc_division_degree_executionproductcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_division_degree_executionproductcoefficientsum_body_steps)) * pfc_terms_scale_division_degree_executionproductcoefficient) + (fs_a_pfc_division_degree_executionproductcoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_division_degree_executionproductcoefficientsum_body_steps_partial. fs_h_pfc_division_degree_executionproductcoefficientsum_body_steps_partial + S (fs_r_pfc_division_degree_executionproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_division_degree_executionproductcoefficientsum_body_steps)) * fs_v_pfc_division_degree_executionproductcoefficientsum)) /\ exists fs_q_pfc_division_degree_executionproductcoefficientsum_body_steps_partial. fs_u_pfc_division_degree_executionproductcoefficientsum = fs_q_pfc_division_degree_executionproductcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_division_degree_executionproductcoefficientsum_body_steps)) * fs_v_pfc_division_degree_executionproductcoefficientsum) + (fs_r_pfc_division_degree_executionproductcoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_division_degree_executionproductcoefficientsum_body_steps_successor. fs_h_pfc_division_degree_executionproductcoefficientsum_body_steps_successor + S (fs_s_pfc_division_degree_executionproductcoefficientsum_body_steps) = S ((S (S fs_i_pfc_division_degree_executionproductcoefficientsum_body_steps)) * fs_v_pfc_division_degree_executionproductcoefficientsum)) /\ exists fs_q_pfc_division_degree_executionproductcoefficientsum_body_steps_successor. fs_u_pfc_division_degree_executionproductcoefficientsum = fs_q_pfc_division_degree_executionproductcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_division_degree_executionproductcoefficientsum_body_steps)) * fs_v_pfc_division_degree_executionproductcoefficientsum) + (fs_s_pfc_division_degree_executionproductcoefficientsum_body_steps))) /\ fs_s_pfc_division_degree_executionproductcoefficientsum_body_steps = fs_r_pfc_division_degree_executionproductcoefficientsum_body_steps + fs_a_pfc_division_degree_executionproductcoefficientsum_body_steps)))))) /\ ((((exists pfa_gap_division_degree_executionproductcoefficientresiduebound. pfa_gap_division_degree_executionproductcoefficientresiduebound + S (pfc_value_division_degree_executionproduct) = (p)) /\ ((exists pfa_offset_left_division_degree_executionproductcoefficientresiduecongruence pfa_offset_right_division_degree_executionproductcoefficientresiduecongruence. (pfc_natural_sum_division_degree_executionproductcoefficient) + (p) * pfa_offset_left_division_degree_executionproductcoefficientresiduecongruence = (pfc_value_division_degree_executionproduct) + (p) * pfa_offset_right_division_degree_executionproductcoefficientresiduecongruence)))))))))))) /\ (((forall pfs_index_division_degree_executiondifference. (exists pfa_gap_division_degree_executiondifferenceindex. pfa_gap_division_degree_executiondifferenceindex + S (pfs_index_division_degree_executiondifference) = (L)) -> exists pfs_left_division_degree_executiondifference pfs_right_division_degree_executiondifference pfs_result_division_degree_executiondifference. ((((exists ff_h_pfp_division_degree_executiondifferenceleft. ff_h_pfp_division_degree_executiondifferenceleft + S (pfs_left_division_degree_executiondifference) = S ((S (pfs_index_division_degree_executiondifference)) * ac)) /\ exists ff_q_pfp_division_degree_executiondifferenceleft. ab = ff_q_pfp_division_degree_executiondifferenceleft * S ((S (pfs_index_division_degree_executiondifference)) * ac) + (pfs_left_division_degree_executiondifference))) /\ (((((exists ff_h_pfp_division_degree_executiondifferenceright. ff_h_pfp_division_degree_executiondifferenceright + S (pfs_right_division_degree_executiondifference) = S ((S (pfs_index_division_degree_executiondifference)) * pfd_product_scale_division_degree_execution)) /\ exists ff_q_pfp_division_degree_executiondifferenceright. pfd_product_code_division_degree_execution = ff_q_pfp_division_degree_executiondifferenceright * S ((S (pfs_index_division_degree_executiondifference)) * pfd_product_scale_division_degree_execution) + (pfs_right_division_degree_executiondifference))) /\ (((((exists ff_h_pfp_division_degree_executiondifferenceresult. ff_h_pfp_division_degree_executiondifferenceresult + S (pfs_result_division_degree_executiondifference) = S ((S (pfs_index_division_degree_executiondifference)) * pfd_residual_scale_division_degree_execution)) /\ exists ff_q_pfp_division_degree_executiondifferenceresult. pfd_residual_code_division_degree_execution = ff_q_pfp_division_degree_executiondifferenceresult * S ((S (pfs_index_division_degree_executiondifference)) * pfd_residual_scale_division_degree_execution) + (pfs_result_division_degree_executiondifference))) /\ ((((exists pfa_gap_division_degree_executiondifferenceoperationleft. pfa_gap_division_degree_executiondifferenceoperationleft + S (pfs_right_division_degree_executiondifference) = (p)) /\ (((exists pfa_gap_division_degree_executiondifferenceoperationright. pfa_gap_division_degree_executiondifferenceoperationright + S (pfs_result_division_degree_executiondifference) = (p)) /\ ((((exists pfa_gap_division_degree_executiondifferenceoperationresultbound. pfa_gap_division_degree_executiondifferenceoperationresultbound + S (pfs_left_division_degree_executiondifference) = (p)) /\ ((exists pfa_offset_left_division_degree_executiondifferenceoperationresultcongruence pfa_offset_right_division_degree_executiondifferenceoperationresultcongruence. ((pfs_right_division_degree_executiondifference) + (pfs_result_division_degree_executiondifference)) + (p) * pfa_offset_left_division_degree_executiondifferenceoperationresultcongruence = (pfs_left_division_degree_executiondifference) + (p) * pfa_offset_right_division_degree_executiondifferenceoperationresultcongruence)))))))))))))))) /\ (((((L)=(pfd_cut_division_degree_execution)+(R)) /\ (((forall fom_index_pfp_division_degree_executiontriminput. (exists fom_gap_pfp_division_degree_executiontriminput_index_bound. fom_gap_pfp_division_degree_executiontriminput_index_bound + S (fom_index_pfp_division_degree_executiontriminput) = L) -> exists fom_value_pfp_division_degree_executiontriminput. ((((exists fom_beta_height_pfp_division_degree_executiontriminput_entry. fom_beta_height_pfp_division_degree_executiontriminput_entry + S (fom_value_pfp_division_degree_executiontriminput) = S ((S (fom_index_pfp_division_degree_executiontriminput)) * pfd_residual_scale_division_degree_execution)) /\ exists fom_beta_quotient_pfp_division_degree_executiontriminput_entry. pfd_residual_code_division_degree_execution = fom_beta_quotient_pfp_division_degree_executiontriminput_entry * S ((S (fom_index_pfp_division_degree_executiontriminput)) * pfd_residual_scale_division_degree_execution) + (fom_value_pfp_division_degree_executiontriminput))) /\ (exists fom_gap_pfp_division_degree_executiontriminput_value_bound. fom_gap_pfp_division_degree_executiontriminput_value_bound + S (fom_value_pfp_division_degree_executiontriminput) = p))) /\ (((forall pfp_repeat_index_division_degree_executiontrimremoved. (exists pfa_gap_division_degree_executiontrimremovedindex. pfa_gap_division_degree_executiontrimremovedindex + S (pfp_repeat_index_division_degree_executiontrimremoved) = (pfd_cut_division_degree_execution)) -> (((exists ff_h_pfp_division_degree_executiontrimremovedentry. ff_h_pfp_division_degree_executiontrimremovedentry + S (0) = S ((S (pfp_repeat_index_division_degree_executiontrimremoved)) * pfd_residual_scale_division_degree_execution)) /\ exists ff_q_pfp_division_degree_executiontrimremovedentry. pfd_residual_code_division_degree_execution = ff_q_pfp_division_degree_executiontrimremovedentry * S ((S (pfp_repeat_index_division_degree_executiontrimremoved)) * pfd_residual_scale_division_degree_execution) + (0)))) /\ (((forall pftrim_index_division_degree_executiontrimsuffix pftrim_value_division_degree_executiontrimsuffix. (exists pfa_gap_division_degree_executiontrimsuffixbound. pfa_gap_division_degree_executiontrimsuffixbound + S (pftrim_index_division_degree_executiontrimsuffix) = (R)) -> (((exists ff_h_pfp_division_degree_executiontrimsuffixsource. ff_h_pfp_division_degree_executiontrimsuffixsource + S (pftrim_value_division_degree_executiontrimsuffix) = S ((S ((pfd_cut_division_degree_execution)+pftrim_index_division_degree_executiontrimsuffix)) * pfd_residual_scale_division_degree_execution)) /\ exists ff_q_pfp_division_degree_executiontrimsuffixsource. pfd_residual_code_division_degree_execution = ff_q_pfp_division_degree_executiontrimsuffixsource * S ((S ((pfd_cut_division_degree_execution)+pftrim_index_division_degree_executiontrimsuffix)) * pfd_residual_scale_division_degree_execution) + (pftrim_value_division_degree_executiontrimsuffix))) -> (((exists ff_h_pfp_division_degree_executiontrimsuffixoutput. ff_h_pfp_division_degree_executiontrimsuffixoutput + S (pftrim_value_division_degree_executiontrimsuffix) = S ((S (pftrim_index_division_degree_executiontrimsuffix)) * rc)) /\ exists ff_q_pfp_division_degree_executiontrimsuffixoutput. rb = ff_q_pfp_division_degree_executiontrimsuffixoutput * S ((S (pftrim_index_division_degree_executiontrimsuffix)) * rc) + (pftrim_value_division_degree_executiontrimsuffix)))) /\ (((R)=0 \/ (exists pftrim_leading_division_degree_executiontrimnormal. ((((exists ff_h_pfp_division_degree_executiontrimnormalentry. ff_h_pfp_division_degree_executiontrimnormalentry + S (pftrim_leading_division_degree_executiontrimnormal) = S ((S (0)) * rc)) /\ exists ff_q_pfp_division_degree_executiontrimnormalentry. rb = ff_q_pfp_division_degree_executiontrimnormalentry * S ((S (0)) * rc) + (pftrim_leading_division_degree_executiontrimnormal))) /\ ((~(pftrim_leading_division_degree_executiontrimnormal=0))))))))))))))))))))))))))))))))) -> ((R)=0 \/ (exists pfd_remainder_degree_division_degree_result. (((((R)=S (pfd_remainder_degree_division_degree_result)) /\ (((forall fom_index_pfp_division_degree_resultrepresentedcoefficients. (exists fom_gap_pfp_division_degree_resultrepresentedcoefficients_index_bound. fom_gap_pfp_division_degree_resultrepresentedcoefficients_index_bound + S (fom_index_pfp_division_degree_resultrepresentedcoefficients) = R) -> exists fom_value_pfp_division_degree_resultrepresentedcoefficients. ((((exists fom_beta_height_pfp_division_degree_resultrepresentedcoefficients_entry. fom_beta_height_pfp_division_degree_resultrepresentedcoefficients_entry + S (fom_value_pfp_division_degree_resultrepresentedcoefficients) = S ((S (fom_index_pfp_division_degree_resultrepresentedcoefficients)) * rc)) /\ exists fom_beta_quotient_pfp_division_degree_resultrepresentedcoefficients_entry. rb = fom_beta_quotient_pfp_division_degree_resultrepresentedcoefficients_entry * S ((S (fom_index_pfp_division_degree_resultrepresentedcoefficients)) * rc) + (fom_value_pfp_division_degree_resultrepresentedcoefficients))) /\ (exists fom_gap_pfp_division_degree_resultrepresentedcoefficients_value_bound. fom_gap_pfp_division_degree_resultrepresentedcoefficients_value_bound + S (fom_value_pfp_division_degree_resultrepresentedcoefficients) = p))) /\ ((exists pfd_leading_division_degree_resultrepresented. ((((exists ff_h_pfp_division_degree_resultrepresentedentry. ff_h_pfp_division_degree_resultrepresentedentry + S (pfd_leading_division_degree_resultrepresented) = S ((S (0)) * rc)) /\ exists ff_q_pfp_division_degree_resultrepresentedentry. rb = ff_q_pfp_division_degree_resultrepresentedentry * S ((S (0)) * rc) + (pfd_leading_division_degree_resultrepresented))) /\ ((~(pfd_leading_division_degree_resultrepresented=0)))))))))) /\ ((exists pfa_gap_division_degree_resultstrict. pfa_gap_division_degree_resultstrict + S (pfd_remainder_degree_division_degree_result) = (d))))))

Complete tactic proof in conservative notation

All 86 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.

Read the argument

Proof checkpoints

86 script commands · 11 reading checkpoints · 1 local claims

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.

Named ingredients (4)
polynomial quotient length boundsprime field polynomial trim bounded degreeprime field polynomial trim zero prefix remainder boundprime field polynomial quotient prefix remainder zero
01Fix variables and assumptionsL1–10

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro p
  2. L2
    intro ab
  3. L3
    intro ac
  4. L4
    intro L
  5. L5
    intro bb
  6. L6
    intro bc
  7. L7
    intro d
  8. L8
    intro qb
  9. L9
    intro qc
  10. L10
    intro q
02Fix variables and assumptionsL11–15

Work with arbitrary variables or the premises of the current implication.

  1. L11
    intro rb
  2. L12
    intro rc
  3. L13
    intro R
  4. L14
    intro hp
  5. L15
    intro h
03Separate the logical casesL16–25

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L16
    cases h
  2. L17
    cases h_right
  3. L18
    cases h_right_right
  4. L19
    cases h_right_right_right
  5. L20
    cases h_right_right_right_witness
  6. L21
    cases h_right_right_right_witness_witness
  7. L22
    cases h_right_right_right_witness_witness_witness
  8. L23
    cases h_right_right_right_witness_witness_witness_witness
  9. L24
    cases h_right_right_right_witness_witness_witness_witness_witness
  10. L25
    cases h_right_right_right_witness_witness_witness_witness_witness_witness
04Separate the logical casesL26–30

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L26
    cases h_right_right_right_witness_witness_witness_witness_witness_witness_witness
  2. L27
    cases h_right_right_right_witness_witness_witness_witness_witness_witness_witness_right
  3. L28
    cases h_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right
  4. L29
    cases h_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right_right
  5. L30
    cases h_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right_right_right
05Establish hboundsL31–36

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply polynomial quotient length bounds.

  1. L31
    have hbounds : Le(q,L) ∧ Le(L,q + d)Definitions: Le(q,L)Le(L,q + d)Original native command in the exact edition
  2. L32
    specialize polynomial_quotient_length_bounds (L)
  3. L33
    specialize polynomial_quotient_length_bounds (d)
  4. L34
    specialize polynomial_quotient_length_bounds (q)
  5. L35
    apply polynomial_quotient_length_bounds
  6. L36
    exact h_right_right_left
06Separate the logical casesL37–37

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L37
    cases hbounds
07Use earlier factsL38–47

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L38
    specialize prime_field_polynomial_trim_bounded_degree (p)
  2. L39
    specialize prime_field_polynomial_trim_bounded_degree (x4)
  3. L40
    specialize prime_field_polynomial_trim_bounded_degree (x5)
  4. L41
    specialize prime_field_polynomial_trim_bounded_degree (L)
  5. L42
    specialize prime_field_polynomial_trim_bounded_degree (x6)
  6. L43
    specialize prime_field_polynomial_trim_bounded_degree (rb)
  7. L44
    specialize prime_field_polynomial_trim_bounded_degree (rc)
  8. L45
    specialize prime_field_polynomial_trim_bounded_degree (R)
  9. L46
    specialize prime_field_polynomial_trim_bounded_degree (d)
  10. L47
    apply prime_field_polynomial_trim_bounded_degree
08Use earlier factsL48–57

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L48
    exact h_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right
  2. L49
    specialize prime_field_polynomial_trim_zero_prefix_remainder_bound (p)
  3. L50
    specialize prime_field_polynomial_trim_zero_prefix_remainder_bound (x4)
  4. L51
    specialize prime_field_polynomial_trim_zero_prefix_remainder_bound (x5)
  5. L52
    specialize prime_field_polynomial_trim_zero_prefix_remainder_bound (L)
  6. L53
    specialize prime_field_polynomial_trim_zero_prefix_remainder_bound (x6)
  7. L54
    specialize prime_field_polynomial_trim_zero_prefix_remainder_bound (rb)
  8. L55
    specialize prime_field_polynomial_trim_zero_prefix_remainder_bound (rc)
  9. L56
    specialize prime_field_polynomial_trim_zero_prefix_remainder_bound (R)
  10. L57
    specialize prime_field_polynomial_trim_zero_prefix_remainder_bound (q)
09Use earlier factsL58–67

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L58
    specialize prime_field_polynomial_trim_zero_prefix_remainder_bound (d)
  2. L59
    apply prime_field_polynomial_trim_zero_prefix_remainder_bound
  3. L60
    exact hbounds_left
  4. L61
    exact hbounds_right
  5. L62
    specialize prime_field_polynomial_quotient_prefix_remainder_zero (p)
  6. L63
    specialize prime_field_polynomial_quotient_prefix_remainder_zero (x1)
  7. L64
    specialize prime_field_polynomial_quotient_prefix_remainder_zero (ab)
  8. L65
    specialize prime_field_polynomial_quotient_prefix_remainder_zero (ac)
  9. L66
    specialize prime_field_polynomial_quotient_prefix_remainder_zero (bb)
  10. L67
    specialize prime_field_polynomial_quotient_prefix_remainder_zero (bc)
10Use earlier factsL68–77

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L68
    specialize prime_field_polynomial_quotient_prefix_remainder_zero (d)
  2. L69
    specialize prime_field_polynomial_quotient_prefix_remainder_zero (qb)
  3. L70
    specialize prime_field_polynomial_quotient_prefix_remainder_zero (qc)
  4. L71
    specialize prime_field_polynomial_quotient_prefix_remainder_zero (q)
  5. L72
    specialize prime_field_polynomial_quotient_prefix_remainder_zero (x)
  6. L73
    specialize prime_field_polynomial_quotient_prefix_remainder_zero (x2)
  7. L74
    specialize prime_field_polynomial_quotient_prefix_remainder_zero (x3)
  8. L75
    specialize prime_field_polynomial_quotient_prefix_remainder_zero (L)
  9. L76
    specialize prime_field_polynomial_quotient_prefix_remainder_zero (x4)
  10. L77
    specialize prime_field_polynomial_quotient_prefix_remainder_zero (x5)
11Use earlier factsL78–86

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L78
    apply prime_field_polynomial_quotient_prefix_remainder_zero
  2. L79
    exact hp
  3. L80
    exact h_right_right_right_witness_witness_witness_witness_witness_witness_witness_left
  4. L81
    exact h_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_left
  5. L82
    exact h_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right_left
  6. L83
    exact hbounds_left
  7. L84
    exact h_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right_right_left
  8. L85
    exact h_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_left
  9. L86
    exact h_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right

Library-wide reading audit

Original defined command ledger · 86 lines
  1. 0001intro p
  2. 0002intro ab
  3. 0003intro ac
  4. 0004intro L
  5. 0005intro bb
  6. 0006intro bc
  7. 0007intro d
  8. 0008intro qb
  9. 0009intro qc
  10. 0010intro q
  11. 0011intro rb
  12. 0012intro rc
  13. 0013intro R
  14. 0014intro hp
  15. 0015intro h
  16. 0016cases h
  17. 0017cases h_right
  18. 0018cases h_right_right
  19. 0019cases h_right_right_right
  20. 0020cases h_right_right_right_witness
  21. 0021cases h_right_right_right_witness_witness
  22. 0022cases h_right_right_right_witness_witness_witness
  23. 0023cases h_right_right_right_witness_witness_witness_witness
  24. 0024cases h_right_right_right_witness_witness_witness_witness_witness
  25. 0025cases h_right_right_right_witness_witness_witness_witness_witness_witness
  26. 0026cases h_right_right_right_witness_witness_witness_witness_witness_witness_witness
  27. 0027cases h_right_right_right_witness_witness_witness_witness_witness_witness_witness_right
  28. 0028cases h_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right
  29. 0029cases h_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right_right
  30. 0030cases h_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right_right_right
  31. 0031have hbounds : Le(q,L) ∧ Le(L,q + d)
  32. 0032specialize polynomial_quotient_length_bounds (L)
  33. 0033specialize polynomial_quotient_length_bounds (d)
  34. 0034specialize polynomial_quotient_length_bounds (q)
  35. 0035apply polynomial_quotient_length_bounds
  36. 0036exact h_right_right_left
  37. 0037cases hbounds
  38. 0038specialize prime_field_polynomial_trim_bounded_degree (p)
  39. 0039specialize prime_field_polynomial_trim_bounded_degree (x4)
  40. 0040specialize prime_field_polynomial_trim_bounded_degree (x5)
  41. 0041specialize prime_field_polynomial_trim_bounded_degree (L)
  42. 0042specialize prime_field_polynomial_trim_bounded_degree (x6)
  43. 0043specialize prime_field_polynomial_trim_bounded_degree (rb)
  44. 0044specialize prime_field_polynomial_trim_bounded_degree (rc)
  45. 0045specialize prime_field_polynomial_trim_bounded_degree (R)
  46. 0046specialize prime_field_polynomial_trim_bounded_degree (d)
  47. 0047apply prime_field_polynomial_trim_bounded_degree
  48. 0048exact h_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right
  49. 0049specialize prime_field_polynomial_trim_zero_prefix_remainder_bound (p)
  50. 0050specialize prime_field_polynomial_trim_zero_prefix_remainder_bound (x4)
  51. 0051specialize prime_field_polynomial_trim_zero_prefix_remainder_bound (x5)
  52. 0052specialize prime_field_polynomial_trim_zero_prefix_remainder_bound (L)
  53. 0053specialize prime_field_polynomial_trim_zero_prefix_remainder_bound (x6)
  54. 0054specialize prime_field_polynomial_trim_zero_prefix_remainder_bound (rb)
  55. 0055specialize prime_field_polynomial_trim_zero_prefix_remainder_bound (rc)
  56. 0056specialize prime_field_polynomial_trim_zero_prefix_remainder_bound (R)
  57. 0057specialize prime_field_polynomial_trim_zero_prefix_remainder_bound (q)
  58. 0058specialize prime_field_polynomial_trim_zero_prefix_remainder_bound (d)
  59. 0059apply prime_field_polynomial_trim_zero_prefix_remainder_bound
  60. 0060exact hbounds_left
  61. 0061exact hbounds_right
  62. 0062specialize prime_field_polynomial_quotient_prefix_remainder_zero (p)
  63. 0063specialize prime_field_polynomial_quotient_prefix_remainder_zero (x1)
  64. 0064specialize prime_field_polynomial_quotient_prefix_remainder_zero (ab)
  65. 0065specialize prime_field_polynomial_quotient_prefix_remainder_zero (ac)
  66. 0066specialize prime_field_polynomial_quotient_prefix_remainder_zero (bb)
  67. 0067specialize prime_field_polynomial_quotient_prefix_remainder_zero (bc)
  68. 0068specialize prime_field_polynomial_quotient_prefix_remainder_zero (d)
  69. 0069specialize prime_field_polynomial_quotient_prefix_remainder_zero (qb)
  70. 0070specialize prime_field_polynomial_quotient_prefix_remainder_zero (qc)
  71. 0071specialize prime_field_polynomial_quotient_prefix_remainder_zero (q)
  72. 0072specialize prime_field_polynomial_quotient_prefix_remainder_zero (x)
  73. 0073specialize prime_field_polynomial_quotient_prefix_remainder_zero (x2)
  74. 0074specialize prime_field_polynomial_quotient_prefix_remainder_zero (x3)
  75. 0075specialize prime_field_polynomial_quotient_prefix_remainder_zero (L)
  76. 0076specialize prime_field_polynomial_quotient_prefix_remainder_zero (x4)
  77. 0077specialize prime_field_polynomial_quotient_prefix_remainder_zero (x5)
  78. 0078apply prime_field_polynomial_quotient_prefix_remainder_zero
  79. 0079exact hp
  80. 0080exact h_right_right_right_witness_witness_witness_witness_witness_witness_witness_left
  81. 0081exact h_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_left
  82. 0082exact h_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right_left
  83. 0083exact hbounds_left
  84. 0084exact h_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right_right_left
  85. 0085exact h_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_left
  86. 0086exact h_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right