% TIMEFORMAT='%3R'; { time (exec 2>&1; /home/martin/bin/satallax -E /home/martin/.isabelle/contrib/e-2.5-1/x86_64-linux/eprover -p tstp -t 5 /home/martin/judgement-day/tptp-thf/tptp/Fundamental_Theorem_Algebra/prob_1053__5377816_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:35:33.856

% Could-be-implicit typings (4)
thf(ty_n_t__Polynomial__Opoly_It__Nat__Onat_J, type,
    poly_nat : $tType).
thf(ty_n_t__Polynomial__Opoly_Itf__a_J, type,
    poly_a : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).
thf(ty_n_tf__a, type,
    a : $tType).

% Explicit typings (42)
thf(sy_c_Cancellation_Oiterate__add_001t__Nat__Onat, type,
    iterate_add_nat : nat > nat > nat).
thf(sy_c_Cancellation_Oiterate__add_001tf__a, type,
    iterate_add_a : nat > a > a).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Oconstant_001tf__a_001tf__a, type,
    fundam236050252nt_a_a : (a > a) > $o).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Ooffset__poly_001t__Nat__Onat, type,
    fundam170929432ly_nat : poly_nat > nat > poly_nat).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Ooffset__poly_001tf__a, type,
    fundam1358810038poly_a : poly_a > a > poly_a).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Opsize_001t__Nat__Onat, type,
    fundam1567013434ze_nat : poly_nat > nat).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Opsize_001tf__a, type,
    fundam247907092size_a : poly_a > nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat, type,
    one_one_nat : nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    one_one_poly_nat : poly_nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_Itf__a_J, type,
    one_one_poly_a : poly_a).
thf(sy_c_Groups_Oone__class_Oone_001tf__a, type,
    one_one_a : a).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    zero_zero_poly_nat : poly_nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_Itf__a_J, type,
    zero_zero_poly_a : poly_a).
thf(sy_c_Groups_Ozero__class_Ozero_001tf__a, type,
    zero_zero_a : a).
thf(sy_c_If_001t__Nat__Onat, type,
    if_nat : $o > nat > nat > nat).
thf(sy_c_If_001tf__a, type,
    if_a : $o > a > a > a).
thf(sy_c_Nat_OSuc, type,
    suc : nat > nat).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat, type,
    ord_less_eq_nat : nat > nat > $o).
thf(sy_c_Polynomial_Odegree_001t__Nat__Onat, type,
    degree_nat : poly_nat > nat).
thf(sy_c_Polynomial_Odegree_001tf__a, type,
    degree_a : poly_a > nat).
thf(sy_c_Polynomial_Omonom_001t__Nat__Onat, type,
    monom_nat : nat > nat > poly_nat).
thf(sy_c_Polynomial_Omonom_001tf__a, type,
    monom_a : a > nat > poly_a).
thf(sy_c_Polynomial_Oorder_001tf__a, type,
    order_a : a > poly_a > nat).
thf(sy_c_Polynomial_OpCons_001t__Nat__Onat, type,
    pCons_nat : nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_OpCons_001tf__a, type,
    pCons_a : a > poly_a > poly_a).
thf(sy_c_Polynomial_Opoly_001t__Nat__Onat, type,
    poly_nat2 : poly_nat > nat > nat).
thf(sy_c_Polynomial_Opoly_001tf__a, type,
    poly_a2 : poly_a > a > a).
thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Nat__Onat, type,
    coeff_nat : poly_nat > nat > nat).
thf(sy_c_Polynomial_Opoly_Ocoeff_001tf__a, type,
    coeff_a : poly_a > nat > a).
thf(sy_c_Polynomial_Oreflect__poly_001t__Nat__Onat, type,
    reflect_poly_nat : poly_nat > poly_nat).
thf(sy_c_Polynomial_Oreflect__poly_001tf__a, type,
    reflect_poly_a : poly_a > poly_a).
thf(sy_c_Polynomial_Osmult_001t__Nat__Onat, type,
    smult_nat : nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_Osmult_001tf__a, type,
    smult_a : a > poly_a > poly_a).
thf(sy_c_Polynomial_Osynthetic__div_001t__Nat__Onat, type,
    synthetic_div_nat : poly_nat > nat > poly_nat).
thf(sy_c_Polynomial_Osynthetic__div_001tf__a, type,
    synthetic_div_a : poly_a > a > poly_a).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat, type,
    dvd_dvd_nat : nat > nat > $o).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    dvd_dvd_poly_nat : poly_nat > poly_nat > $o).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_Itf__a_J, type,
    dvd_dvd_poly_a : poly_a > poly_a > $o).
thf(sy_c_Rings_Odvd__class_Odvd_001tf__a, type,
    dvd_dvd_a : a > a > $o).
thf(sy_v_p, type,
    p : poly_a).

% Relevant facts (152)
thf(fact_0_that, axiom,
    ((fundam236050252nt_a_a @ (poly_a2 @ p)))). % that
thf(fact_1_zero__natural_Orsp, axiom,
    ((zero_zero_nat = zero_zero_nat))). % zero_natural.rsp
thf(fact_2_zero__reorient, axiom,
    ((![X : a]: ((zero_zero_a = X) = (X = zero_zero_a))))). % zero_reorient
thf(fact_3_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_4_degree__offset__poly, axiom,
    ((![P : poly_nat, H : nat]: ((degree_nat @ (fundam170929432ly_nat @ P @ H)) = (degree_nat @ P))))). % degree_offset_poly
thf(fact_5_degree__offset__poly, axiom,
    ((![P : poly_a, H : a]: ((degree_a @ (fundam1358810038poly_a @ P @ H)) = (degree_a @ P))))). % degree_offset_poly
thf(fact_6_degree__smult__eq, axiom,
    ((![A : a, P : poly_a]: (((A = zero_zero_a) => ((degree_a @ (smult_a @ A @ P)) = zero_zero_nat)) & ((~ ((A = zero_zero_a))) => ((degree_a @ (smult_a @ A @ P)) = (degree_a @ P))))))). % degree_smult_eq
thf(fact_7_degree__smult__eq, axiom,
    ((![A : nat, P : poly_nat]: (((A = zero_zero_nat) => ((degree_nat @ (smult_nat @ A @ P)) = zero_zero_nat)) & ((~ ((A = zero_zero_nat))) => ((degree_nat @ (smult_nat @ A @ P)) = (degree_nat @ P))))))). % degree_smult_eq
thf(fact_8_iterate__add__simps_I1_J, axiom,
    ((![A : a]: ((iterate_add_a @ zero_zero_nat @ A) = zero_zero_a)))). % iterate_add_simps(1)
thf(fact_9_iterate__add__simps_I1_J, axiom,
    ((![A : nat]: ((iterate_add_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % iterate_add_simps(1)
thf(fact_10_degree__0, axiom,
    (((degree_a @ zero_zero_poly_a) = zero_zero_nat))). % degree_0
thf(fact_11_degree__0, axiom,
    (((degree_nat @ zero_zero_poly_nat) = zero_zero_nat))). % degree_0
thf(fact_12_degree__1, axiom,
    (((degree_a @ one_one_poly_a) = zero_zero_nat))). % degree_1
thf(fact_13_degree__1, axiom,
    (((degree_nat @ one_one_poly_nat) = zero_zero_nat))). % degree_1
thf(fact_14_smult__1__left, axiom,
    ((![P : poly_a]: ((smult_a @ one_one_a @ P) = P)))). % smult_1_left
thf(fact_15_smult__1__left, axiom,
    ((![P : poly_nat]: ((smult_nat @ one_one_nat @ P) = P)))). % smult_1_left
thf(fact_16_smult__0__right, axiom,
    ((![A : nat]: ((smult_nat @ A @ zero_zero_poly_nat) = zero_zero_poly_nat)))). % smult_0_right
thf(fact_17_smult__0__right, axiom,
    ((![A : a]: ((smult_a @ A @ zero_zero_poly_a) = zero_zero_poly_a)))). % smult_0_right
thf(fact_18_iterate__add__empty, axiom,
    ((![N : nat]: ((iterate_add_a @ N @ zero_zero_a) = zero_zero_a)))). % iterate_add_empty
thf(fact_19_iterate__add__empty, axiom,
    ((![N : nat]: ((iterate_add_nat @ N @ zero_zero_nat) = zero_zero_nat)))). % iterate_add_empty
thf(fact_20_smult__eq__0__iff, axiom,
    ((![A : nat, P : poly_nat]: (((smult_nat @ A @ P) = zero_zero_poly_nat) = (((A = zero_zero_nat)) | ((P = zero_zero_poly_nat))))))). % smult_eq_0_iff
thf(fact_21_smult__eq__0__iff, axiom,
    ((![A : a, P : poly_a]: (((smult_a @ A @ P) = zero_zero_poly_a) = (((A = zero_zero_a)) | ((P = zero_zero_poly_a))))))). % smult_eq_0_iff
thf(fact_22_smult__0__left, axiom,
    ((![P : poly_nat]: ((smult_nat @ zero_zero_nat @ P) = zero_zero_poly_nat)))). % smult_0_left
thf(fact_23_smult__0__left, axiom,
    ((![P : poly_a]: ((smult_a @ zero_zero_a @ P) = zero_zero_poly_a)))). % smult_0_left
thf(fact_24_poly__0, axiom,
    ((![X : nat]: ((poly_nat2 @ zero_zero_poly_nat @ X) = zero_zero_nat)))). % poly_0
thf(fact_25_poly__0, axiom,
    ((![X : a]: ((poly_a2 @ zero_zero_poly_a @ X) = zero_zero_a)))). % poly_0
thf(fact_26_poly__1, axiom,
    ((![X : a]: ((poly_a2 @ one_one_poly_a @ X) = one_one_a)))). % poly_1
thf(fact_27_poly__1, axiom,
    ((![X : nat]: ((poly_nat2 @ one_one_poly_nat @ X) = one_one_nat)))). % poly_1
thf(fact_28_one__reorient, axiom,
    ((![X : nat]: ((one_one_nat = X) = (X = one_one_nat))))). % one_reorient
thf(fact_29_poly__eq__poly__eq__iff, axiom,
    ((![P : poly_a, Q : poly_a]: (((poly_a2 @ P) = (poly_a2 @ Q)) = (P = Q))))). % poly_eq_poly_eq_iff
thf(fact_30_offset__poly__0, axiom,
    ((![H : a]: ((fundam1358810038poly_a @ zero_zero_poly_a @ H) = zero_zero_poly_a)))). % offset_poly_0
thf(fact_31_offset__poly__eq__0__iff, axiom,
    ((![P : poly_a, H : a]: (((fundam1358810038poly_a @ P @ H) = zero_zero_poly_a) = (P = zero_zero_poly_a))))). % offset_poly_eq_0_iff
thf(fact_32_poly__all__0__iff__0, axiom,
    ((![P : poly_a]: ((![X2 : a]: ((poly_a2 @ P @ X2) = zero_zero_a)) = (P = zero_zero_poly_a))))). % poly_all_0_iff_0
thf(fact_33_constant__def, axiom,
    ((fundam236050252nt_a_a = (^[F : a > a]: (![X2 : a]: (![Y : a]: ((F @ X2) = (F @ Y)))))))). % constant_def
thf(fact_34_zero__neq__one, axiom,
    ((~ ((zero_zero_nat = one_one_nat))))). % zero_neq_one
thf(fact_35_zero__neq__one, axiom,
    ((~ ((zero_zero_a = one_one_a))))). % zero_neq_one
thf(fact_36_synthetic__div__eq__0__iff, axiom,
    ((![P : poly_a, C : a]: (((synthetic_div_a @ P @ C) = zero_zero_poly_a) = ((degree_a @ P) = zero_zero_nat))))). % synthetic_div_eq_0_iff
thf(fact_37_synthetic__div__eq__0__iff, axiom,
    ((![P : poly_nat, C : nat]: (((synthetic_div_nat @ P @ C) = zero_zero_poly_nat) = ((degree_nat @ P) = zero_zero_nat))))). % synthetic_div_eq_0_iff
thf(fact_38_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_nat]: (((poly_nat2 @ (reflect_poly_nat @ P) @ zero_zero_nat) = zero_zero_nat) = (P = zero_zero_poly_nat))))). % reflect_poly_at_0_eq_0_iff
thf(fact_39_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_a]: (((poly_a2 @ (reflect_poly_a @ P) @ zero_zero_a) = zero_zero_a) = (P = zero_zero_poly_a))))). % reflect_poly_at_0_eq_0_iff
thf(fact_40_reflect__poly__smult, axiom,
    ((![C : nat, P : poly_nat]: ((reflect_poly_nat @ (smult_nat @ C @ P)) = (smult_nat @ C @ (reflect_poly_nat @ P)))))). % reflect_poly_smult
thf(fact_41_reflect__poly__smult, axiom,
    ((![C : a, P : poly_a]: ((reflect_poly_a @ (smult_a @ C @ P)) = (smult_a @ C @ (reflect_poly_a @ P)))))). % reflect_poly_smult
thf(fact_42_poly__reflect__poly__0, axiom,
    ((![P : poly_nat]: ((poly_nat2 @ (reflect_poly_nat @ P) @ zero_zero_nat) = (coeff_nat @ P @ (degree_nat @ P)))))). % poly_reflect_poly_0
thf(fact_43_poly__reflect__poly__0, axiom,
    ((![P : poly_a]: ((poly_a2 @ (reflect_poly_a @ P) @ zero_zero_a) = (coeff_a @ P @ (degree_a @ P)))))). % poly_reflect_poly_0
thf(fact_44_degree__reflect__poly__eq, axiom,
    ((![P : poly_nat]: ((~ (((coeff_nat @ P @ zero_zero_nat) = zero_zero_nat))) => ((degree_nat @ (reflect_poly_nat @ P)) = (degree_nat @ P)))))). % degree_reflect_poly_eq
thf(fact_45_degree__reflect__poly__eq, axiom,
    ((![P : poly_a]: ((~ (((coeff_a @ P @ zero_zero_nat) = zero_zero_a))) => ((degree_a @ (reflect_poly_a @ P)) = (degree_a @ P)))))). % degree_reflect_poly_eq
thf(fact_46_coeff__0__reflect__poly__0__iff, axiom,
    ((![P : poly_nat]: (((coeff_nat @ (reflect_poly_nat @ P) @ zero_zero_nat) = zero_zero_nat) = (P = zero_zero_poly_nat))))). % coeff_0_reflect_poly_0_iff
thf(fact_47_coeff__0__reflect__poly__0__iff, axiom,
    ((![P : poly_a]: (((coeff_a @ (reflect_poly_a @ P) @ zero_zero_nat) = zero_zero_a) = (P = zero_zero_poly_a))))). % coeff_0_reflect_poly_0_iff
thf(fact_48_order__root, axiom,
    ((![P : poly_a, A : a]: (((poly_a2 @ P @ A) = zero_zero_a) = (((P = zero_zero_poly_a)) | ((~ (((order_a @ A @ P) = zero_zero_nat))))))))). % order_root
thf(fact_49_psize__def, axiom,
    ((fundam247907092size_a = (^[P2 : poly_a]: (if_nat @ (P2 = zero_zero_poly_a) @ zero_zero_nat @ (suc @ (degree_a @ P2))))))). % psize_def
thf(fact_50_psize__def, axiom,
    ((fundam1567013434ze_nat = (^[P2 : poly_nat]: (if_nat @ (P2 = zero_zero_poly_nat) @ zero_zero_nat @ (suc @ (degree_nat @ P2))))))). % psize_def
thf(fact_51_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_52_nat_Oinject, axiom,
    ((![X22 : nat, Y2 : nat]: (((suc @ X22) = (suc @ Y2)) = (X22 = Y2))))). % nat.inject
thf(fact_53_dvd__0__left__iff, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) = (A = zero_zero_nat))))). % dvd_0_left_iff
thf(fact_54_dvd__0__left__iff, axiom,
    ((![A : a]: ((dvd_dvd_a @ zero_zero_a @ A) = (A = zero_zero_a))))). % dvd_0_left_iff
thf(fact_55_dvd__0__right, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ zero_zero_nat)))). % dvd_0_right
thf(fact_56_dvd__0__right, axiom,
    ((![A : a]: (dvd_dvd_a @ A @ zero_zero_a)))). % dvd_0_right
thf(fact_57_coeff__0, axiom,
    ((![N : nat]: ((coeff_nat @ zero_zero_poly_nat @ N) = zero_zero_nat)))). % coeff_0
thf(fact_58_coeff__0, axiom,
    ((![N : nat]: ((coeff_a @ zero_zero_poly_a @ N) = zero_zero_a)))). % coeff_0
thf(fact_59_leading__coeff__0__iff, axiom,
    ((![P : poly_nat]: (((coeff_nat @ P @ (degree_nat @ P)) = zero_zero_nat) = (P = zero_zero_poly_nat))))). % leading_coeff_0_iff
thf(fact_60_leading__coeff__0__iff, axiom,
    ((![P : poly_a]: (((coeff_a @ P @ (degree_a @ P)) = zero_zero_a) = (P = zero_zero_poly_a))))). % leading_coeff_0_iff
thf(fact_61_reflect__poly__reflect__poly, axiom,
    ((![P : poly_nat]: ((~ (((coeff_nat @ P @ zero_zero_nat) = zero_zero_nat))) => ((reflect_poly_nat @ (reflect_poly_nat @ P)) = P))))). % reflect_poly_reflect_poly
thf(fact_62_reflect__poly__reflect__poly, axiom,
    ((![P : poly_a]: ((~ (((coeff_a @ P @ zero_zero_nat) = zero_zero_a))) => ((reflect_poly_a @ (reflect_poly_a @ P)) = P))))). % reflect_poly_reflect_poly
thf(fact_63_lead__coeff__1, axiom,
    (((coeff_a @ one_one_poly_a @ (degree_a @ one_one_poly_a)) = one_one_a))). % lead_coeff_1
thf(fact_64_lead__coeff__1, axiom,
    (((coeff_nat @ one_one_poly_nat @ (degree_nat @ one_one_poly_nat)) = one_one_nat))). % lead_coeff_1
thf(fact_65_coeff__0__reflect__poly, axiom,
    ((![P : poly_a]: ((coeff_a @ (reflect_poly_a @ P) @ zero_zero_nat) = (coeff_a @ P @ (degree_a @ P)))))). % coeff_0_reflect_poly
thf(fact_66_coeff__0__reflect__poly, axiom,
    ((![P : poly_nat]: ((coeff_nat @ (reflect_poly_nat @ P) @ zero_zero_nat) = (coeff_nat @ P @ (degree_nat @ P)))))). % coeff_0_reflect_poly
thf(fact_67_dvd__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ B @ C) => (dvd_dvd_nat @ A @ C)))))). % dvd_trans
thf(fact_68_dvd__refl, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ A)))). % dvd_refl
thf(fact_69_n__not__Suc__n, axiom,
    ((![N : nat]: (~ ((N = (suc @ N))))))). % n_not_Suc_n
thf(fact_70_Suc__inject, axiom,
    ((![X : nat, Y3 : nat]: (((suc @ X) = (suc @ Y3)) => (X = Y3))))). % Suc_inject
thf(fact_71_dvd__0__left, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) => (A = zero_zero_nat))))). % dvd_0_left
thf(fact_72_dvd__0__left, axiom,
    ((![A : a]: ((dvd_dvd_a @ zero_zero_a @ A) => (A = zero_zero_a))))). % dvd_0_left
thf(fact_73_one__dvd, axiom,
    ((![A : nat]: (dvd_dvd_nat @ one_one_nat @ A)))). % one_dvd
thf(fact_74_unit__imp__dvd, axiom,
    ((![B : nat, A : nat]: ((dvd_dvd_nat @ B @ one_one_nat) => (dvd_dvd_nat @ B @ A))))). % unit_imp_dvd
thf(fact_75_dvd__unit__imp__unit, axiom,
    ((![A : nat, B : nat]: ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ B @ one_one_nat) => (dvd_dvd_nat @ A @ one_one_nat)))))). % dvd_unit_imp_unit
thf(fact_76_not0__implies__Suc, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (?[M : nat]: (N = (suc @ M))))))). % not0_implies_Suc
thf(fact_77_old_Onat_Oinducts, axiom,
    ((![P3 : nat > $o, Nat : nat]: ((P3 @ zero_zero_nat) => ((![Nat3 : nat]: ((P3 @ Nat3) => (P3 @ (suc @ Nat3)))) => (P3 @ Nat)))))). % old.nat.inducts
thf(fact_78_old_Onat_Oexhaust, axiom,
    ((![Y3 : nat]: ((~ ((Y3 = zero_zero_nat))) => (~ ((![Nat3 : nat]: (~ ((Y3 = (suc @ Nat3))))))))))). % old.nat.exhaust
thf(fact_79_Zero__not__Suc, axiom,
    ((![M2 : nat]: (~ ((zero_zero_nat = (suc @ M2))))))). % Zero_not_Suc
thf(fact_80_Zero__neq__Suc, axiom,
    ((![M2 : nat]: (~ ((zero_zero_nat = (suc @ M2))))))). % Zero_neq_Suc
thf(fact_81_Suc__neq__Zero, axiom,
    ((![M2 : nat]: (~ (((suc @ M2) = zero_zero_nat)))))). % Suc_neq_Zero
thf(fact_82_zero__induct, axiom,
    ((![P3 : nat > $o, K : nat]: ((P3 @ K) => ((![N2 : nat]: ((P3 @ (suc @ N2)) => (P3 @ N2))) => (P3 @ zero_zero_nat)))))). % zero_induct
thf(fact_83_diff__induct, axiom,
    ((![P3 : nat > nat > $o, M2 : nat, N : nat]: ((![X3 : nat]: (P3 @ X3 @ zero_zero_nat)) => ((![Y4 : nat]: (P3 @ zero_zero_nat @ (suc @ Y4))) => ((![X3 : nat, Y4 : nat]: ((P3 @ X3 @ Y4) => (P3 @ (suc @ X3) @ (suc @ Y4)))) => (P3 @ M2 @ N))))))). % diff_induct
thf(fact_84_nat__induct, axiom,
    ((![P3 : nat > $o, N : nat]: ((P3 @ zero_zero_nat) => ((![N2 : nat]: ((P3 @ N2) => (P3 @ (suc @ N2)))) => (P3 @ N)))))). % nat_induct
thf(fact_85_nat_OdiscI, axiom,
    ((![Nat : nat, X22 : nat]: ((Nat = (suc @ X22)) => (~ ((Nat = zero_zero_nat))))))). % nat.discI
thf(fact_86_old_Onat_Odistinct_I1_J, axiom,
    ((![Nat2 : nat]: (~ ((zero_zero_nat = (suc @ Nat2))))))). % old.nat.distinct(1)
thf(fact_87_old_Onat_Odistinct_I2_J, axiom,
    ((![Nat2 : nat]: (~ (((suc @ Nat2) = zero_zero_nat)))))). % old.nat.distinct(2)
thf(fact_88_nat_Odistinct_I1_J, axiom,
    ((![X22 : nat]: (~ ((zero_zero_nat = (suc @ X22))))))). % nat.distinct(1)
thf(fact_89_One__nat__def, axiom,
    ((one_one_nat = (suc @ zero_zero_nat)))). % One_nat_def
thf(fact_90_dvd__smult, axiom,
    ((![P : poly_nat, Q : poly_nat, A : nat]: ((dvd_dvd_poly_nat @ P @ Q) => (dvd_dvd_poly_nat @ P @ (smult_nat @ A @ Q)))))). % dvd_smult
thf(fact_91_dvd__smult, axiom,
    ((![P : poly_a, Q : poly_a, A : a]: ((dvd_dvd_poly_a @ P @ Q) => (dvd_dvd_poly_a @ P @ (smult_a @ A @ Q)))))). % dvd_smult
thf(fact_92_smult__dvd__cancel, axiom,
    ((![A : nat, P : poly_nat, Q : poly_nat]: ((dvd_dvd_poly_nat @ (smult_nat @ A @ P) @ Q) => (dvd_dvd_poly_nat @ P @ Q))))). % smult_dvd_cancel
thf(fact_93_smult__dvd__cancel, axiom,
    ((![A : a, P : poly_a, Q : poly_a]: ((dvd_dvd_poly_a @ (smult_a @ A @ P) @ Q) => (dvd_dvd_poly_a @ P @ Q))))). % smult_dvd_cancel
thf(fact_94_is__unit__smult__iff, axiom,
    ((![C : a, P : poly_a]: ((dvd_dvd_poly_a @ (smult_a @ C @ P) @ one_one_poly_a) = (((dvd_dvd_a @ C @ one_one_a)) & ((dvd_dvd_poly_a @ P @ one_one_poly_a))))))). % is_unit_smult_iff
thf(fact_95_is__unit__smult__iff, axiom,
    ((![C : nat, P : poly_nat]: ((dvd_dvd_poly_nat @ (smult_nat @ C @ P) @ one_one_poly_nat) = (((dvd_dvd_nat @ C @ one_one_nat)) & ((dvd_dvd_poly_nat @ P @ one_one_poly_nat))))))). % is_unit_smult_iff
thf(fact_96_zero__poly_Orep__eq, axiom,
    (((coeff_nat @ zero_zero_poly_nat) = (^[Uu : nat]: zero_zero_nat)))). % zero_poly.rep_eq
thf(fact_97_zero__poly_Orep__eq, axiom,
    (((coeff_a @ zero_zero_poly_a) = (^[Uu : nat]: zero_zero_a)))). % zero_poly.rep_eq
thf(fact_98_not__is__unit__0, axiom,
    ((~ ((dvd_dvd_nat @ zero_zero_nat @ one_one_nat))))). % not_is_unit_0
thf(fact_99_order__smult, axiom,
    ((![C : a, X : a, P : poly_a]: ((~ ((C = zero_zero_a))) => ((order_a @ X @ (smult_a @ C @ P)) = (order_a @ X @ P)))))). % order_smult
thf(fact_100_poly__0__coeff__0, axiom,
    ((![P : poly_nat]: ((poly_nat2 @ P @ zero_zero_nat) = (coeff_nat @ P @ zero_zero_nat))))). % poly_0_coeff_0
thf(fact_101_poly__0__coeff__0, axiom,
    ((![P : poly_a]: ((poly_a2 @ P @ zero_zero_a) = (coeff_a @ P @ zero_zero_nat))))). % poly_0_coeff_0
thf(fact_102_leading__coeff__neq__0, axiom,
    ((![P : poly_nat]: ((~ ((P = zero_zero_poly_nat))) => (~ (((coeff_nat @ P @ (degree_nat @ P)) = zero_zero_nat))))))). % leading_coeff_neq_0
thf(fact_103_leading__coeff__neq__0, axiom,
    ((![P : poly_a]: ((~ ((P = zero_zero_poly_a))) => (~ (((coeff_a @ P @ (degree_a @ P)) = zero_zero_a))))))). % leading_coeff_neq_0
thf(fact_104_order__0I, axiom,
    ((![P : poly_a, A : a]: ((~ (((poly_a2 @ P @ A) = zero_zero_a))) => ((order_a @ A @ P) = zero_zero_nat))))). % order_0I
thf(fact_105_exists__least__lemma, axiom,
    ((![P3 : nat > $o]: ((~ ((P3 @ zero_zero_nat))) => ((?[X_1 : nat]: (P3 @ X_1)) => (?[N2 : nat]: ((~ ((P3 @ N2))) & (P3 @ (suc @ N2))))))))). % exists_least_lemma
thf(fact_106_nat__dvd__1__iff__1, axiom,
    ((![M2 : nat]: ((dvd_dvd_nat @ M2 @ one_one_nat) = (M2 = one_one_nat))))). % nat_dvd_1_iff_1
thf(fact_107_dvd__1__left, axiom,
    ((![K : nat]: (dvd_dvd_nat @ (suc @ zero_zero_nat) @ K)))). % dvd_1_left
thf(fact_108_dvd__1__iff__1, axiom,
    ((![M2 : nat]: ((dvd_dvd_nat @ M2 @ (suc @ zero_zero_nat)) = (M2 = (suc @ zero_zero_nat)))))). % dvd_1_iff_1
thf(fact_109_monom__eq__0, axiom,
    ((![N : nat]: ((monom_nat @ zero_zero_nat @ N) = zero_zero_poly_nat)))). % monom_eq_0
thf(fact_110_monom__eq__0, axiom,
    ((![N : nat]: ((monom_a @ zero_zero_a @ N) = zero_zero_poly_a)))). % monom_eq_0
thf(fact_111_monom__eq__0__iff, axiom,
    ((![A : nat, N : nat]: (((monom_nat @ A @ N) = zero_zero_poly_nat) = (A = zero_zero_nat))))). % monom_eq_0_iff
thf(fact_112_monom__eq__0__iff, axiom,
    ((![A : a, N : nat]: (((monom_a @ A @ N) = zero_zero_poly_a) = (A = zero_zero_a))))). % monom_eq_0_iff
thf(fact_113_coeff__monom, axiom,
    ((![M2 : nat, N : nat, A : nat]: (((M2 = N) => ((coeff_nat @ (monom_nat @ A @ M2) @ N) = A)) & ((~ ((M2 = N))) => ((coeff_nat @ (monom_nat @ A @ M2) @ N) = zero_zero_nat)))))). % coeff_monom
thf(fact_114_coeff__monom, axiom,
    ((![M2 : nat, N : nat, A : a]: (((M2 = N) => ((coeff_a @ (monom_a @ A @ M2) @ N) = A)) & ((~ ((M2 = N))) => ((coeff_a @ (monom_a @ A @ M2) @ N) = zero_zero_a)))))). % coeff_monom
thf(fact_115_lead__coeff__monom, axiom,
    ((![C : a, N : nat]: ((coeff_a @ (monom_a @ C @ N) @ (degree_a @ (monom_a @ C @ N))) = C)))). % lead_coeff_monom
thf(fact_116_lead__coeff__monom, axiom,
    ((![C : nat, N : nat]: ((coeff_nat @ (monom_nat @ C @ N) @ (degree_nat @ (monom_nat @ C @ N))) = C)))). % lead_coeff_monom
thf(fact_117_order__0__monom, axiom,
    ((![C : a, N : nat]: ((~ ((C = zero_zero_a))) => ((order_a @ zero_zero_a @ (monom_a @ C @ N)) = N))))). % order_0_monom
thf(fact_118_monom__eq__1, axiom,
    (((monom_nat @ one_one_nat @ zero_zero_nat) = one_one_poly_nat))). % monom_eq_1
thf(fact_119_one__natural_Orsp, axiom,
    ((one_one_nat = one_one_nat))). % one_natural.rsp
thf(fact_120_monom__eq__iff_H, axiom,
    ((![C : nat, N : nat, D : nat, M2 : nat]: (((monom_nat @ C @ N) = (monom_nat @ D @ M2)) = (((C = D)) & ((((C = zero_zero_nat)) | ((N = M2))))))))). % monom_eq_iff'
thf(fact_121_monom__eq__iff_H, axiom,
    ((![C : a, N : nat, D : a, M2 : nat]: (((monom_a @ C @ N) = (monom_a @ D @ M2)) = (((C = D)) & ((((C = zero_zero_a)) | ((N = M2))))))))). % monom_eq_iff'
thf(fact_122_degree__monom__eq, axiom,
    ((![A : nat, N : nat]: ((~ ((A = zero_zero_nat))) => ((degree_nat @ (monom_nat @ A @ N)) = N))))). % degree_monom_eq
thf(fact_123_degree__monom__eq, axiom,
    ((![A : a, N : nat]: ((~ ((A = zero_zero_a))) => ((degree_a @ (monom_a @ A @ N)) = N))))). % degree_monom_eq
thf(fact_124_monom_Orep__eq, axiom,
    ((![X : nat, Xa : nat]: ((coeff_nat @ (monom_nat @ X @ Xa)) = (^[N3 : nat]: (if_nat @ (Xa = N3) @ X @ zero_zero_nat)))))). % monom.rep_eq
thf(fact_125_monom_Orep__eq, axiom,
    ((![X : a, Xa : nat]: ((coeff_a @ (monom_a @ X @ Xa)) = (^[N3 : nat]: (if_a @ (Xa = N3) @ X @ zero_zero_a)))))). % monom.rep_eq
thf(fact_126_monom__eq__1__iff, axiom,
    ((![C : nat, N : nat]: (((monom_nat @ C @ N) = one_one_poly_nat) = (((C = one_one_nat)) & ((N = zero_zero_nat))))))). % monom_eq_1_iff
thf(fact_127_monom__1__dvd__iff, axiom,
    ((![P : poly_a, N : nat]: ((~ ((P = zero_zero_poly_a))) => ((dvd_dvd_poly_a @ (monom_a @ one_one_a @ N) @ P) = (ord_less_eq_nat @ N @ (order_a @ zero_zero_a @ P))))))). % monom_1_dvd_iff
thf(fact_128_monom__1__dvd__iff_H, axiom,
    ((![N : nat, P : poly_nat]: ((dvd_dvd_poly_nat @ (monom_nat @ one_one_nat @ N) @ P) = (![K2 : nat]: (((ord_less_nat @ K2 @ N)) => (((coeff_nat @ P @ K2) = zero_zero_nat)))))))). % monom_1_dvd_iff'
thf(fact_129_monom__1__dvd__iff_H, axiom,
    ((![N : nat, P : poly_a]: ((dvd_dvd_poly_a @ (monom_a @ one_one_a @ N) @ P) = (![K2 : nat]: (((ord_less_nat @ K2 @ N)) => (((coeff_a @ P @ K2) = zero_zero_a)))))))). % monom_1_dvd_iff'
thf(fact_130_is__unit__const__poly__iff, axiom,
    ((![C : nat]: ((dvd_dvd_poly_nat @ (pCons_nat @ C @ zero_zero_poly_nat) @ one_one_poly_nat) = (dvd_dvd_nat @ C @ one_one_nat))))). % is_unit_const_poly_iff
thf(fact_131_le__zero__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_zero_eq
thf(fact_132_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_133_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_134_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_135_bot__nat__0_Onot__eq__extremum, axiom,
    ((![A : nat]: ((~ ((A = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ A))))). % bot_nat_0.not_eq_extremum
thf(fact_136_lessI, axiom,
    ((![N : nat]: (ord_less_nat @ N @ (suc @ N))))). % lessI
thf(fact_137_Suc__mono, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ M2 @ N) => (ord_less_nat @ (suc @ M2) @ (suc @ N)))))). % Suc_mono
thf(fact_138_Suc__less__eq, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ (suc @ M2) @ (suc @ N)) = (ord_less_nat @ M2 @ N))))). % Suc_less_eq
thf(fact_139_le0, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % le0
thf(fact_140_bot__nat__0_Oextremum, axiom,
    ((![A : nat]: (ord_less_eq_nat @ zero_zero_nat @ A)))). % bot_nat_0.extremum
thf(fact_141_Suc__le__mono, axiom,
    ((![N : nat, M2 : nat]: ((ord_less_eq_nat @ (suc @ N) @ (suc @ M2)) = (ord_less_eq_nat @ N @ M2))))). % Suc_le_mono
thf(fact_142_zero__less__Suc, axiom,
    ((![N : nat]: (ord_less_nat @ zero_zero_nat @ (suc @ N))))). % zero_less_Suc
thf(fact_143_less__Suc0, axiom,
    ((![N : nat]: ((ord_less_nat @ N @ (suc @ zero_zero_nat)) = (N = zero_zero_nat))))). % less_Suc0
thf(fact_144_less__one, axiom,
    ((![N : nat]: ((ord_less_nat @ N @ one_one_nat) = (N = zero_zero_nat))))). % less_one
thf(fact_145_pCons__eq__0__iff, axiom,
    ((![A : nat, P : poly_nat]: (((pCons_nat @ A @ P) = zero_zero_poly_nat) = (((A = zero_zero_nat)) & ((P = zero_zero_poly_nat))))))). % pCons_eq_0_iff
thf(fact_146_pCons__eq__0__iff, axiom,
    ((![A : a, P : poly_a]: (((pCons_a @ A @ P) = zero_zero_poly_a) = (((A = zero_zero_a)) & ((P = zero_zero_poly_a))))))). % pCons_eq_0_iff
thf(fact_147_pCons__0__0, axiom,
    (((pCons_nat @ zero_zero_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % pCons_0_0
thf(fact_148_pCons__0__0, axiom,
    (((pCons_a @ zero_zero_a @ zero_zero_poly_a) = zero_zero_poly_a))). % pCons_0_0
thf(fact_149_synthetic__div__pCons, axiom,
    ((![A : a, P : poly_a, C : a]: ((synthetic_div_a @ (pCons_a @ A @ P) @ C) = (pCons_a @ (poly_a2 @ P @ C) @ (synthetic_div_a @ P @ C)))))). % synthetic_div_pCons
thf(fact_150_lead__coeff__pCons_I2_J, axiom,
    ((![P : poly_a, A : a]: ((P = zero_zero_poly_a) => ((coeff_a @ (pCons_a @ A @ P) @ (degree_a @ (pCons_a @ A @ P))) = A))))). % lead_coeff_pCons(2)
thf(fact_151_lead__coeff__pCons_I2_J, axiom,
    ((![P : poly_nat, A : nat]: ((P = zero_zero_poly_nat) => ((coeff_nat @ (pCons_nat @ A @ P) @ (degree_nat @ (pCons_nat @ A @ P))) = A))))). % lead_coeff_pCons(2)

% Helper facts (5)
thf(help_If_2_1_If_001tf__a_T, axiom,
    ((![X : a, Y3 : a]: ((if_a @ $false @ X @ Y3) = Y3)))).
thf(help_If_1_1_If_001tf__a_T, axiom,
    ((![X : a, Y3 : a]: ((if_a @ $true @ X @ Y3) = X)))).
thf(help_If_3_1_If_001t__Nat__Onat_T, axiom,
    ((![P3 : $o]: ((P3 = $true) | (P3 = $false))))).
thf(help_If_2_1_If_001t__Nat__Onat_T, axiom,
    ((![X : nat, Y3 : nat]: ((if_nat @ $false @ X @ Y3) = Y3)))).
thf(help_If_1_1_If_001t__Nat__Onat_T, axiom,
    ((![X : nat, Y3 : nat]: ((if_nat @ $true @ X @ Y3) = X)))).

% Conjectures (1)
thf(conj_0, conjecture,
    (((degree_a @ p) = zero_zero_nat))).
