% 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_101__5367950_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:27:07.692

% Could-be-implicit typings (5)
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    poly_poly_a : $tType).
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 (32)
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_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_It__Polynomial__Opoly_Itf__a_J_J, type,
    zero_z2096148049poly_a : poly_poly_a).
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_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_Polynomial_Odegree_001t__Nat__Onat, type,
    degree_nat : poly_nat > nat).
thf(sy_c_Polynomial_Odegree_001t__Polynomial__Opoly_Itf__a_J, type,
    degree_poly_a : poly_poly_a > nat).
thf(sy_c_Polynomial_Odegree_001tf__a, type,
    degree_a : poly_a > nat).
thf(sy_c_Polynomial_Ois__zero_001tf__a, type,
    is_zero_a : poly_a > $o).
thf(sy_c_Polynomial_Omap__poly_001t__Nat__Onat_001t__Nat__Onat, type,
    map_poly_nat_nat : (nat > nat) > poly_nat > poly_nat).
thf(sy_c_Polynomial_Omap__poly_001t__Nat__Onat_001tf__a, type,
    map_poly_nat_a : (nat > a) > poly_nat > poly_a).
thf(sy_c_Polynomial_Omap__poly_001tf__a_001tf__a, type,
    map_poly_a_a : (a > a) > poly_a > poly_a).
thf(sy_c_Polynomial_OpCons_001t__Nat__Onat, type,
    pCons_nat : nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_OpCons_001t__Polynomial__Opoly_Itf__a_J, type,
    pCons_poly_a : poly_a > poly_poly_a > poly_poly_a).
thf(sy_c_Polynomial_OpCons_001tf__a, type,
    pCons_a : a > poly_a > poly_a).
thf(sy_c_Polynomial_Opcompose_001t__Nat__Onat, type,
    pcompose_nat : poly_nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_Opoly_001t__Nat__Onat, type,
    poly_nat2 : poly_nat > nat > nat).
thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Nat__Onat, type,
    coeff_nat : poly_nat > nat > nat).
thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Polynomial__Opoly_Itf__a_J, type,
    coeff_poly_a : poly_poly_a > nat > poly_a).
thf(sy_c_Polynomial_Opoly_Ocoeff_001tf__a, type,
    coeff_a : poly_a > nat > a).
thf(sy_c_Polynomial_Opoly__cutoff_001tf__a, type,
    poly_cutoff_a : nat > poly_a > poly_a).
thf(sy_c_Polynomial_Opoly__shift_001tf__a, type,
    poly_shift_a : nat > poly_a > poly_a).
thf(sy_c_Polynomial_Oreflect__poly_001t__Nat__Onat, type,
    reflect_poly_nat : poly_nat > poly_nat).
thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_Itf__a_J, type,
    reflect_poly_poly_a : poly_poly_a > poly_poly_a).
thf(sy_c_Polynomial_Oreflect__poly_001tf__a, type,
    reflect_poly_a : poly_a > poly_a).
thf(sy_v_p, type,
    p : poly_a).

% Relevant facts (129)
thf(fact_0_degree__0, axiom,
    (((degree_a @ zero_zero_poly_a) = zero_zero_nat))). % degree_0
thf(fact_1_psize__def, axiom,
    ((fundam247907092size_a = (^[P : poly_a]: (if_nat @ (P = zero_zero_poly_a) @ zero_zero_nat @ (suc @ (degree_a @ P))))))). % psize_def
thf(fact_2_nat_Oinject, axiom,
    ((![X2 : nat, Y2 : nat]: (((suc @ X2) = (suc @ Y2)) = (X2 = Y2))))). % nat.inject
thf(fact_3_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_4_nat_Odistinct_I1_J, axiom,
    ((![X2 : nat]: (~ ((zero_zero_nat = (suc @ X2))))))). % nat.distinct(1)
thf(fact_5_old_Onat_Odistinct_I2_J, axiom,
    ((![Nat2 : nat]: (~ (((suc @ Nat2) = zero_zero_nat)))))). % old.nat.distinct(2)
thf(fact_6_old_Onat_Odistinct_I1_J, axiom,
    ((![Nat2 : nat]: (~ ((zero_zero_nat = (suc @ Nat2))))))). % old.nat.distinct(1)
thf(fact_7_nat_OdiscI, axiom,
    ((![Nat : nat, X2 : nat]: ((Nat = (suc @ X2)) => (~ ((Nat = zero_zero_nat))))))). % nat.discI
thf(fact_8_nat__induct, axiom,
    ((![P2 : nat > $o, N : nat]: ((P2 @ zero_zero_nat) => ((![N2 : nat]: ((P2 @ N2) => (P2 @ (suc @ N2)))) => (P2 @ N)))))). % nat_induct
thf(fact_9_diff__induct, axiom,
    ((![P2 : nat > nat > $o, M : nat, N : nat]: ((![X : nat]: (P2 @ X @ zero_zero_nat)) => ((![Y : nat]: (P2 @ zero_zero_nat @ (suc @ Y))) => ((![X : nat, Y : nat]: ((P2 @ X @ Y) => (P2 @ (suc @ X) @ (suc @ Y)))) => (P2 @ M @ N))))))). % diff_induct
thf(fact_10_zero__reorient, axiom,
    ((![X3 : poly_a]: ((zero_zero_poly_a = X3) = (X3 = zero_zero_poly_a))))). % zero_reorient
thf(fact_11_zero__reorient, axiom,
    ((![X3 : nat]: ((zero_zero_nat = X3) = (X3 = zero_zero_nat))))). % zero_reorient
thf(fact_12_n__not__Suc__n, axiom,
    ((![N : nat]: (~ ((N = (suc @ N))))))). % n_not_Suc_n
thf(fact_13_Suc__inject, axiom,
    ((![X3 : nat, Y3 : nat]: (((suc @ X3) = (suc @ Y3)) => (X3 = Y3))))). % Suc_inject
thf(fact_14_not0__implies__Suc, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (?[M2 : nat]: (N = (suc @ M2))))))). % not0_implies_Suc
thf(fact_15_old_Onat_Oinducts, axiom,
    ((![P2 : nat > $o, Nat : nat]: ((P2 @ zero_zero_nat) => ((![Nat3 : nat]: ((P2 @ Nat3) => (P2 @ (suc @ Nat3)))) => (P2 @ Nat)))))). % old.nat.inducts
thf(fact_16_old_Onat_Oexhaust, axiom,
    ((![Y3 : nat]: ((~ ((Y3 = zero_zero_nat))) => (~ ((![Nat3 : nat]: (~ ((Y3 = (suc @ Nat3))))))))))). % old.nat.exhaust
thf(fact_17_Zero__not__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_not_Suc
thf(fact_18_Zero__neq__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_neq_Suc
thf(fact_19_Suc__neq__Zero, axiom,
    ((![M : nat]: (~ (((suc @ M) = zero_zero_nat)))))). % Suc_neq_Zero
thf(fact_20_zero__induct, axiom,
    ((![P2 : nat > $o, K : nat]: ((P2 @ K) => ((![N2 : nat]: ((P2 @ (suc @ N2)) => (P2 @ N2))) => (P2 @ zero_zero_nat)))))). % zero_induct
thf(fact_21_exists__least__lemma, axiom,
    ((![P2 : nat > $o]: ((~ ((P2 @ zero_zero_nat))) => ((?[X_1 : nat]: (P2 @ X_1)) => (?[N2 : nat]: ((~ ((P2 @ N2))) & (P2 @ (suc @ N2))))))))). % exists_least_lemma
thf(fact_22_is__zero__null, axiom,
    ((is_zero_a = (^[P : poly_a]: (P = zero_zero_poly_a))))). % is_zero_null
thf(fact_23_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_cutoff_a @ N @ zero_zero_poly_a) = zero_zero_poly_a)))). % poly_cutoff_0
thf(fact_24_degree__pCons__eq__if, axiom,
    ((![P3 : poly_a, A : a]: (((P3 = zero_zero_poly_a) => ((degree_a @ (pCons_a @ A @ P3)) = zero_zero_nat)) & ((~ ((P3 = zero_zero_poly_a))) => ((degree_a @ (pCons_a @ A @ P3)) = (suc @ (degree_a @ P3)))))))). % degree_pCons_eq_if
thf(fact_25_poly__shift__0, axiom,
    ((![N : nat]: ((poly_shift_a @ N @ zero_zero_poly_a) = zero_zero_poly_a)))). % poly_shift_0
thf(fact_26_leading__coeff__0__iff, axiom,
    ((![P3 : poly_a]: (((coeff_a @ P3 @ (degree_a @ P3)) = zero_zero_a) = (P3 = zero_zero_poly_a))))). % leading_coeff_0_iff
thf(fact_27_leading__coeff__0__iff, axiom,
    ((![P3 : poly_poly_a]: (((coeff_poly_a @ P3 @ (degree_poly_a @ P3)) = zero_zero_poly_a) = (P3 = zero_z2096148049poly_a))))). % leading_coeff_0_iff
thf(fact_28_leading__coeff__0__iff, axiom,
    ((![P3 : poly_nat]: (((coeff_nat @ P3 @ (degree_nat @ P3)) = zero_zero_nat) = (P3 = zero_zero_poly_nat))))). % leading_coeff_0_iff
thf(fact_29_coeff__0, axiom,
    ((![N : nat]: ((coeff_poly_a @ zero_z2096148049poly_a @ N) = zero_zero_poly_a)))). % coeff_0
thf(fact_30_coeff__0, axiom,
    ((![N : nat]: ((coeff_nat @ zero_zero_poly_nat @ N) = zero_zero_nat)))). % coeff_0
thf(fact_31_coeff__0, axiom,
    ((![N : nat]: ((coeff_a @ zero_zero_poly_a @ N) = zero_zero_a)))). % coeff_0
thf(fact_32_pCons__0__0, axiom,
    (((pCons_a @ zero_zero_a @ zero_zero_poly_a) = zero_zero_poly_a))). % pCons_0_0
thf(fact_33_pCons__0__0, axiom,
    (((pCons_poly_a @ zero_zero_poly_a @ zero_z2096148049poly_a) = zero_z2096148049poly_a))). % pCons_0_0
thf(fact_34_pCons__0__0, axiom,
    (((pCons_nat @ zero_zero_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % pCons_0_0
thf(fact_35_pCons__eq__0__iff, axiom,
    ((![A : poly_a, P3 : poly_poly_a]: (((pCons_poly_a @ A @ P3) = zero_z2096148049poly_a) = (((A = zero_zero_poly_a)) & ((P3 = zero_z2096148049poly_a))))))). % pCons_eq_0_iff
thf(fact_36_pCons__eq__0__iff, axiom,
    ((![A : nat, P3 : poly_nat]: (((pCons_nat @ A @ P3) = zero_zero_poly_nat) = (((A = zero_zero_nat)) & ((P3 = zero_zero_poly_nat))))))). % pCons_eq_0_iff
thf(fact_37_pCons__eq__0__iff, axiom,
    ((![A : a, P3 : poly_a]: (((pCons_a @ A @ P3) = zero_zero_poly_a) = (((A = zero_zero_a)) & ((P3 = zero_zero_poly_a))))))). % pCons_eq_0_iff
thf(fact_38_lead__coeff__pCons_I2_J, axiom,
    ((![P3 : poly_a, A : a]: ((P3 = zero_zero_poly_a) => ((coeff_a @ (pCons_a @ A @ P3) @ (degree_a @ (pCons_a @ A @ P3))) = A))))). % lead_coeff_pCons(2)
thf(fact_39_lead__coeff__pCons_I1_J, axiom,
    ((![P3 : poly_a, A : a]: ((~ ((P3 = zero_zero_poly_a))) => ((coeff_a @ (pCons_a @ A @ P3) @ (degree_a @ (pCons_a @ A @ P3))) = (coeff_a @ P3 @ (degree_a @ P3))))))). % lead_coeff_pCons(1)
thf(fact_40_poly__induct2, axiom,
    ((![P2 : poly_a > poly_a > $o, P3 : poly_a, Q : poly_a]: ((P2 @ zero_zero_poly_a @ zero_zero_poly_a) => ((![A2 : a, P4 : poly_a, B : a, Q2 : poly_a]: ((P2 @ P4 @ Q2) => (P2 @ (pCons_a @ A2 @ P4) @ (pCons_a @ B @ Q2)))) => (P2 @ P3 @ Q)))))). % poly_induct2
thf(fact_41_pCons__induct, axiom,
    ((![P2 : poly_poly_a > $o, P3 : poly_poly_a]: ((P2 @ zero_z2096148049poly_a) => ((![A2 : poly_a, P4 : poly_poly_a]: (((~ ((A2 = zero_zero_poly_a))) | (~ ((P4 = zero_z2096148049poly_a)))) => ((P2 @ P4) => (P2 @ (pCons_poly_a @ A2 @ P4))))) => (P2 @ P3)))))). % pCons_induct
thf(fact_42_pCons__induct, axiom,
    ((![P2 : poly_nat > $o, P3 : poly_nat]: ((P2 @ zero_zero_poly_nat) => ((![A2 : nat, P4 : poly_nat]: (((~ ((A2 = zero_zero_nat))) | (~ ((P4 = zero_zero_poly_nat)))) => ((P2 @ P4) => (P2 @ (pCons_nat @ A2 @ P4))))) => (P2 @ P3)))))). % pCons_induct
thf(fact_43_pCons__induct, axiom,
    ((![P2 : poly_a > $o, P3 : poly_a]: ((P2 @ zero_zero_poly_a) => ((![A2 : a, P4 : poly_a]: (((~ ((A2 = zero_zero_a))) | (~ ((P4 = zero_zero_poly_a)))) => ((P2 @ P4) => (P2 @ (pCons_a @ A2 @ P4))))) => (P2 @ P3)))))). % pCons_induct
thf(fact_44_zero__poly_Orep__eq, axiom,
    (((coeff_poly_a @ zero_z2096148049poly_a) = (^[Uu : nat]: zero_zero_poly_a)))). % zero_poly.rep_eq
thf(fact_45_zero__poly_Orep__eq, axiom,
    (((coeff_nat @ zero_zero_poly_nat) = (^[Uu : nat]: zero_zero_nat)))). % zero_poly.rep_eq
thf(fact_46_zero__poly_Orep__eq, axiom,
    (((coeff_a @ zero_zero_poly_a) = (^[Uu : nat]: zero_zero_a)))). % zero_poly.rep_eq
thf(fact_47_leading__coeff__neq__0, axiom,
    ((![P3 : poly_poly_a]: ((~ ((P3 = zero_z2096148049poly_a))) => (~ (((coeff_poly_a @ P3 @ (degree_poly_a @ P3)) = zero_zero_poly_a))))))). % leading_coeff_neq_0
thf(fact_48_leading__coeff__neq__0, axiom,
    ((![P3 : poly_nat]: ((~ ((P3 = zero_zero_poly_nat))) => (~ (((coeff_nat @ P3 @ (degree_nat @ P3)) = zero_zero_nat))))))). % leading_coeff_neq_0
thf(fact_49_leading__coeff__neq__0, axiom,
    ((![P3 : poly_a]: ((~ ((P3 = zero_zero_poly_a))) => (~ (((coeff_a @ P3 @ (degree_a @ P3)) = zero_zero_a))))))). % leading_coeff_neq_0
thf(fact_50_degree__pCons__0, axiom,
    ((![A : a]: ((degree_a @ (pCons_a @ A @ zero_zero_poly_a)) = zero_zero_nat)))). % degree_pCons_0
thf(fact_51_degree__eq__zeroE, axiom,
    ((![P3 : poly_a]: (((degree_a @ P3) = zero_zero_nat) => (~ ((![A2 : a]: (~ ((P3 = (pCons_a @ A2 @ zero_zero_poly_a))))))))))). % degree_eq_zeroE
thf(fact_52_degree__pCons__eq, axiom,
    ((![P3 : poly_a, A : a]: ((~ ((P3 = zero_zero_poly_a))) => ((degree_a @ (pCons_a @ A @ P3)) = (suc @ (degree_a @ P3))))))). % degree_pCons_eq
thf(fact_53_degree__reflect__poly__eq, axiom,
    ((![P3 : poly_a]: ((~ (((coeff_a @ P3 @ zero_zero_nat) = zero_zero_a))) => ((degree_a @ (reflect_poly_a @ P3)) = (degree_a @ P3)))))). % degree_reflect_poly_eq
thf(fact_54_degree__reflect__poly__eq, axiom,
    ((![P3 : poly_poly_a]: ((~ (((coeff_poly_a @ P3 @ zero_zero_nat) = zero_zero_poly_a))) => ((degree_poly_a @ (reflect_poly_poly_a @ P3)) = (degree_poly_a @ P3)))))). % degree_reflect_poly_eq
thf(fact_55_degree__reflect__poly__eq, axiom,
    ((![P3 : poly_nat]: ((~ (((coeff_nat @ P3 @ zero_zero_nat) = zero_zero_nat))) => ((degree_nat @ (reflect_poly_nat @ P3)) = (degree_nat @ P3)))))). % degree_reflect_poly_eq
thf(fact_56_coeff__0__reflect__poly__0__iff, axiom,
    ((![P3 : poly_a]: (((coeff_a @ (reflect_poly_a @ P3) @ zero_zero_nat) = zero_zero_a) = (P3 = zero_zero_poly_a))))). % coeff_0_reflect_poly_0_iff
thf(fact_57_coeff__0__reflect__poly__0__iff, axiom,
    ((![P3 : poly_poly_a]: (((coeff_poly_a @ (reflect_poly_poly_a @ P3) @ zero_zero_nat) = zero_zero_poly_a) = (P3 = zero_z2096148049poly_a))))). % coeff_0_reflect_poly_0_iff
thf(fact_58_coeff__0__reflect__poly__0__iff, axiom,
    ((![P3 : poly_nat]: (((coeff_nat @ (reflect_poly_nat @ P3) @ zero_zero_nat) = zero_zero_nat) = (P3 = zero_zero_poly_nat))))). % coeff_0_reflect_poly_0_iff
thf(fact_59_coeff__0__reflect__poly, axiom,
    ((![P3 : poly_a]: ((coeff_a @ (reflect_poly_a @ P3) @ zero_zero_nat) = (coeff_a @ P3 @ (degree_a @ P3)))))). % coeff_0_reflect_poly
thf(fact_60_reflect__poly__reflect__poly, axiom,
    ((![P3 : poly_poly_a]: ((~ (((coeff_poly_a @ P3 @ zero_zero_nat) = zero_zero_poly_a))) => ((reflect_poly_poly_a @ (reflect_poly_poly_a @ P3)) = P3))))). % reflect_poly_reflect_poly
thf(fact_61_reflect__poly__reflect__poly, axiom,
    ((![P3 : poly_nat]: ((~ (((coeff_nat @ P3 @ zero_zero_nat) = zero_zero_nat))) => ((reflect_poly_nat @ (reflect_poly_nat @ P3)) = P3))))). % reflect_poly_reflect_poly
thf(fact_62_reflect__poly__0, axiom,
    (((reflect_poly_a @ zero_zero_poly_a) = zero_zero_poly_a))). % reflect_poly_0
thf(fact_63_reflect__poly__const, axiom,
    ((![A : a]: ((reflect_poly_a @ (pCons_a @ A @ zero_zero_poly_a)) = (pCons_a @ A @ zero_zero_poly_a))))). % reflect_poly_const
thf(fact_64_one__poly__eq__simps_I1_J, axiom,
    ((one_one_poly_nat = (pCons_nat @ one_one_nat @ zero_zero_poly_nat)))). % one_poly_eq_simps(1)
thf(fact_65_one__poly__eq__simps_I2_J, axiom,
    (((pCons_nat @ one_one_nat @ zero_zero_poly_nat) = one_one_poly_nat))). % one_poly_eq_simps(2)
thf(fact_66_lead__coeff__1, axiom,
    (((coeff_nat @ one_one_poly_nat @ (degree_nat @ one_one_poly_nat)) = one_one_nat))). % lead_coeff_1
thf(fact_67_pcompose__idR, axiom,
    ((![P3 : poly_nat]: ((pcompose_nat @ P3 @ (pCons_nat @ zero_zero_nat @ (pCons_nat @ one_one_nat @ zero_zero_poly_nat))) = P3)))). % pcompose_idR
thf(fact_68_one__reorient, axiom,
    ((![X3 : nat]: ((one_one_nat = X3) = (X3 = one_one_nat))))). % one_reorient
thf(fact_69_pCons__one, axiom,
    (((pCons_nat @ one_one_nat @ zero_zero_poly_nat) = one_one_poly_nat))). % pCons_one
thf(fact_70_zero__neq__one, axiom,
    ((~ ((zero_zero_nat = one_one_nat))))). % zero_neq_one
thf(fact_71_poly__reflect__poly__0, axiom,
    ((![P3 : poly_nat]: ((poly_nat2 @ (reflect_poly_nat @ P3) @ zero_zero_nat) = (coeff_nat @ P3 @ (degree_nat @ P3)))))). % poly_reflect_poly_0
thf(fact_72_map__poly__1, axiom,
    ((![F : nat > a]: ((map_poly_nat_a @ F @ one_one_poly_nat) = (pCons_a @ (F @ one_one_nat) @ zero_zero_poly_a))))). % map_poly_1
thf(fact_73_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_74_less__one, axiom,
    ((![N : nat]: ((ord_less_nat @ N @ one_one_nat) = (N = zero_zero_nat))))). % less_one
thf(fact_75_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_76_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_77_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_78_lessI, axiom,
    ((![N : nat]: (ord_less_nat @ N @ (suc @ N))))). % lessI
thf(fact_79_Suc__mono, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_nat @ (suc @ M) @ (suc @ N)))))). % Suc_mono
thf(fact_80_Suc__less__eq, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (suc @ M) @ (suc @ N)) = (ord_less_nat @ M @ N))))). % Suc_less_eq
thf(fact_81_map__poly__0, axiom,
    ((![F : a > a]: ((map_poly_a_a @ F @ zero_zero_poly_a) = zero_zero_poly_a)))). % map_poly_0
thf(fact_82_less__Suc0, axiom,
    ((![N : nat]: ((ord_less_nat @ N @ (suc @ zero_zero_nat)) = (N = zero_zero_nat))))). % less_Suc0
thf(fact_83_zero__less__Suc, axiom,
    ((![N : nat]: (ord_less_nat @ zero_zero_nat @ (suc @ N))))). % zero_less_Suc
thf(fact_84_poly__0, axiom,
    ((![X3 : nat]: ((poly_nat2 @ zero_zero_poly_nat @ X3) = zero_zero_nat)))). % poly_0
thf(fact_85_poly__1, axiom,
    ((![X3 : nat]: ((poly_nat2 @ one_one_poly_nat @ X3) = one_one_nat)))). % poly_1
thf(fact_86_map__poly__1_H, axiom,
    ((![F : nat > nat]: (((F @ one_one_nat) = one_one_nat) => ((map_poly_nat_nat @ F @ one_one_poly_nat) = one_one_poly_nat))))). % map_poly_1'
thf(fact_87_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P3 : poly_nat]: (((poly_nat2 @ (reflect_poly_nat @ P3) @ zero_zero_nat) = zero_zero_nat) = (P3 = zero_zero_poly_nat))))). % reflect_poly_at_0_eq_0_iff
thf(fact_88_nat__induct__non__zero, axiom,
    ((![N : nat, P2 : nat > $o]: ((ord_less_nat @ zero_zero_nat @ N) => ((P2 @ one_one_nat) => ((![N2 : nat]: ((ord_less_nat @ zero_zero_nat @ N2) => ((P2 @ N2) => (P2 @ (suc @ N2))))) => (P2 @ N))))))). % nat_induct_non_zero
thf(fact_89_zero__less__iff__neq__zero, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) = (~ ((N = zero_zero_nat))))))). % zero_less_iff_neq_zero
thf(fact_90_gr__implies__not__zero, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_91_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_92_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_93_bot__nat__0_Oextremum__strict, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_94_infinite__descent0, axiom,
    ((![P2 : nat > $o, N : nat]: ((P2 @ zero_zero_nat) => ((![N2 : nat]: ((ord_less_nat @ zero_zero_nat @ N2) => ((~ ((P2 @ N2))) => (?[M3 : nat]: ((ord_less_nat @ M3 @ N2) & (~ ((P2 @ M3)))))))) => (P2 @ N)))))). % infinite_descent0
thf(fact_95_gr__implies__not0, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not0
thf(fact_96_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE
thf(fact_97_not__less0, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less0
thf(fact_98_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_99_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I
thf(fact_100_not__less__less__Suc__eq, axiom,
    ((![N : nat, M : nat]: ((~ ((ord_less_nat @ N @ M))) => ((ord_less_nat @ N @ (suc @ M)) = (N = M)))))). % not_less_less_Suc_eq
thf(fact_101_strict__inc__induct, axiom,
    ((![I : nat, J : nat, P2 : nat > $o]: ((ord_less_nat @ I @ J) => ((![I2 : nat]: ((J = (suc @ I2)) => (P2 @ I2))) => ((![I2 : nat]: ((ord_less_nat @ I2 @ J) => ((P2 @ (suc @ I2)) => (P2 @ I2)))) => (P2 @ I))))))). % strict_inc_induct
thf(fact_102_less__Suc__induct, axiom,
    ((![I : nat, J : nat, P2 : nat > nat > $o]: ((ord_less_nat @ I @ J) => ((![I2 : nat]: (P2 @ I2 @ (suc @ I2))) => ((![I2 : nat, J2 : nat, K2 : nat]: ((ord_less_nat @ I2 @ J2) => ((ord_less_nat @ J2 @ K2) => ((P2 @ I2 @ J2) => ((P2 @ J2 @ K2) => (P2 @ I2 @ K2)))))) => (P2 @ I @ J))))))). % less_Suc_induct
thf(fact_103_less__trans__Suc, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_nat @ I @ J) => ((ord_less_nat @ J @ K) => (ord_less_nat @ (suc @ I) @ K)))))). % less_trans_Suc
thf(fact_104_Suc__less__SucD, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (suc @ M) @ (suc @ N)) => (ord_less_nat @ M @ N))))). % Suc_less_SucD
thf(fact_105_less__antisym, axiom,
    ((![N : nat, M : nat]: ((~ ((ord_less_nat @ N @ M))) => ((ord_less_nat @ N @ (suc @ M)) => (M = N)))))). % less_antisym
thf(fact_106_Suc__less__eq2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ (suc @ N) @ M) = (?[M4 : nat]: (((M = (suc @ M4))) & ((ord_less_nat @ N @ M4)))))))). % Suc_less_eq2
thf(fact_107_All__less__Suc, axiom,
    ((![N : nat, P2 : nat > $o]: ((![I3 : nat]: (((ord_less_nat @ I3 @ (suc @ N))) => ((P2 @ I3)))) = (((P2 @ N)) & ((![I3 : nat]: (((ord_less_nat @ I3 @ N)) => ((P2 @ I3)))))))))). % All_less_Suc
thf(fact_108_not__less__eq, axiom,
    ((![M : nat, N : nat]: ((~ ((ord_less_nat @ M @ N))) = (ord_less_nat @ N @ (suc @ M)))))). % not_less_eq
thf(fact_109_less__Suc__eq, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ (suc @ N)) = (((ord_less_nat @ M @ N)) | ((M = N))))))). % less_Suc_eq
thf(fact_110_Ex__less__Suc, axiom,
    ((![N : nat, P2 : nat > $o]: ((?[I3 : nat]: (((ord_less_nat @ I3 @ (suc @ N))) & ((P2 @ I3)))) = (((P2 @ N)) | ((?[I3 : nat]: (((ord_less_nat @ I3 @ N)) & ((P2 @ I3)))))))))). % Ex_less_Suc
thf(fact_111_less__SucI, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_nat @ M @ (suc @ N)))))). % less_SucI
thf(fact_112_less__SucE, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ (suc @ N)) => ((~ ((ord_less_nat @ M @ N))) => (M = N)))))). % less_SucE
thf(fact_113_Suc__lessI, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => ((~ (((suc @ M) = N))) => (ord_less_nat @ (suc @ M) @ N)))))). % Suc_lessI
thf(fact_114_Suc__lessE, axiom,
    ((![I : nat, K : nat]: ((ord_less_nat @ (suc @ I) @ K) => (~ ((![J2 : nat]: ((ord_less_nat @ I @ J2) => (~ ((K = (suc @ J2)))))))))))). % Suc_lessE
thf(fact_115_Suc__lessD, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (suc @ M) @ N) => (ord_less_nat @ M @ N))))). % Suc_lessD
thf(fact_116_Nat_OlessE, axiom,
    ((![I : nat, K : nat]: ((ord_less_nat @ I @ K) => ((~ ((K = (suc @ I)))) => (~ ((![J2 : nat]: ((ord_less_nat @ I @ J2) => (~ ((K = (suc @ J2))))))))))))). % Nat.lessE
thf(fact_117_nat__neq__iff, axiom,
    ((![M : nat, N : nat]: ((~ ((M = N))) = (((ord_less_nat @ M @ N)) | ((ord_less_nat @ N @ M))))))). % nat_neq_iff
thf(fact_118_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_119_less__not__refl2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ N @ M) => (~ ((M = N))))))). % less_not_refl2
thf(fact_120_less__not__refl3, axiom,
    ((![S : nat, T : nat]: ((ord_less_nat @ S @ T) => (~ ((S = T))))))). % less_not_refl3
thf(fact_121_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_122_nat__less__induct, axiom,
    ((![P2 : nat > $o, N : nat]: ((![N2 : nat]: ((![M3 : nat]: ((ord_less_nat @ M3 @ N2) => (P2 @ M3))) => (P2 @ N2))) => (P2 @ N))))). % nat_less_induct
thf(fact_123_infinite__descent, axiom,
    ((![P2 : nat > $o, N : nat]: ((![N2 : nat]: ((~ ((P2 @ N2))) => (?[M3 : nat]: ((ord_less_nat @ M3 @ N2) & (~ ((P2 @ M3))))))) => (P2 @ N))))). % infinite_descent
thf(fact_124_linorder__neqE__nat, axiom,
    ((![X3 : nat, Y3 : nat]: ((~ ((X3 = Y3))) => ((~ ((ord_less_nat @ X3 @ Y3))) => (ord_less_nat @ Y3 @ X3)))))). % linorder_neqE_nat
thf(fact_125_lift__Suc__mono__less, axiom,
    ((![F : nat > nat, N : nat, N3 : nat]: ((![N2 : nat]: (ord_less_nat @ (F @ N2) @ (F @ (suc @ N2)))) => ((ord_less_nat @ N @ N3) => (ord_less_nat @ (F @ N) @ (F @ N3))))))). % lift_Suc_mono_less
thf(fact_126_lift__Suc__mono__less__iff, axiom,
    ((![F : nat > nat, N : nat, M : nat]: ((![N2 : nat]: (ord_less_nat @ (F @ N2) @ (F @ (suc @ N2)))) => ((ord_less_nat @ (F @ N) @ (F @ M)) = (ord_less_nat @ N @ M)))))). % lift_Suc_mono_less_iff
thf(fact_127_zero__less__one, axiom,
    ((ord_less_nat @ zero_zero_nat @ one_one_nat))). % zero_less_one
thf(fact_128_not__one__less__zero, axiom,
    ((~ ((ord_less_nat @ one_one_nat @ zero_zero_nat))))). % not_one_less_zero

% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T, axiom,
    ((![P2 : $o]: ((P2 = $true) | (P2 = $false))))).
thf(help_If_2_1_If_001t__Nat__Onat_T, axiom,
    ((![X3 : nat, Y3 : nat]: ((if_nat @ $false @ X3 @ Y3) = Y3)))).
thf(help_If_1_1_If_001t__Nat__Onat_T, axiom,
    ((![X3 : nat, Y3 : nat]: ((if_nat @ $true @ X3 @ Y3) = X3)))).

% Conjectures (1)
thf(conj_0, conjecture,
    ((~ (((((~ ((p = zero_zero_poly_a)))) => (((suc @ (degree_a @ p)) = zero_zero_nat))) = (~ ((p = zero_zero_poly_a)))))))).
