% 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_87__5367722_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:26:55.003

% 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 (33)
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_001t__Polynomial__Opoly_Itf__a_J, type,
    fundam1343031620poly_a : poly_poly_a > poly_a > poly_poly_a).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Ooffset__poly_001tf__a, type,
    fundam1358810038poly_a : poly_a > a > poly_a).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat, type,
    plus_plus_nat : nat > nat > nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    plus_plus_poly_nat : poly_nat > poly_nat > poly_nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    plus_p1976640465poly_a : poly_poly_a > poly_poly_a > poly_poly_a).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Polynomial__Opoly_Itf__a_J, type,
    plus_plus_poly_a : poly_a > poly_a > poly_a).
thf(sy_c_Groups_Oplus__class_Oplus_001tf__a, type,
    plus_plus_a : a > a > a).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat, type,
    times_times_nat : nat > nat > nat).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    times_times_poly_nat : poly_nat > poly_nat > poly_nat).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    times_545135445poly_a : poly_poly_a > poly_poly_a > poly_poly_a).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_Itf__a_J, type,
    times_times_poly_a : poly_a > poly_a > poly_a).
thf(sy_c_Groups_Otimes__class_Otimes_001tf__a, type,
    times_times_a : a > 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_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_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_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_Opoly_001t__Nat__Onat, type,
    poly_nat2 : poly_nat > nat > nat).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_Itf__a_J, type,
    poly_poly_a2 : poly_poly_a > poly_a > poly_a).
thf(sy_c_Polynomial_Opoly_001tf__a, type,
    poly_a2 : poly_a > a > a).
thf(sy_c_Polynomial_Osmult_001t__Nat__Onat, type,
    smult_nat : nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_Osmult_001t__Polynomial__Opoly_Itf__a_J, type,
    smult_poly_a : poly_a > poly_poly_a > poly_poly_a).
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_v_h, type,
    h : a).
thf(sy_v_p, type,
    p : poly_a).

% Relevant facts (153)
thf(fact_0_offset__poly__0, axiom,
    ((![H : nat]: ((fundam170929432ly_nat @ zero_zero_poly_nat @ H) = zero_zero_poly_nat)))). % offset_poly_0
thf(fact_1_offset__poly__0, axiom,
    ((![H : a]: ((fundam1358810038poly_a @ zero_zero_poly_a @ H) = zero_zero_poly_a)))). % offset_poly_0
thf(fact_2_offset__poly__eq__0__iff, axiom,
    ((![P : poly_nat, H : nat]: (((fundam170929432ly_nat @ P @ H) = zero_zero_poly_nat) = (P = zero_zero_poly_nat))))). % offset_poly_eq_0_iff
thf(fact_3_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_4_offset__poly__single, axiom,
    ((![A : nat, H : nat]: ((fundam170929432ly_nat @ (pCons_nat @ A @ zero_zero_poly_nat) @ H) = (pCons_nat @ A @ zero_zero_poly_nat))))). % offset_poly_single
thf(fact_5_offset__poly__single, axiom,
    ((![A : a, H : a]: ((fundam1358810038poly_a @ (pCons_a @ A @ zero_zero_poly_a) @ H) = (pCons_a @ A @ zero_zero_poly_a))))). % offset_poly_single
thf(fact_6_poly__offset__poly, axiom,
    ((![P : poly_poly_a, H : poly_a, X : poly_a]: ((poly_poly_a2 @ (fundam1343031620poly_a @ P @ H) @ X) = (poly_poly_a2 @ P @ (plus_plus_poly_a @ H @ X)))))). % poly_offset_poly
thf(fact_7_poly__offset__poly, axiom,
    ((![P : poly_nat, H : nat, X : nat]: ((poly_nat2 @ (fundam170929432ly_nat @ P @ H) @ X) = (poly_nat2 @ P @ (plus_plus_nat @ H @ X)))))). % poly_offset_poly
thf(fact_8_poly__offset__poly, axiom,
    ((![P : poly_a, H : a, X : a]: ((poly_a2 @ (fundam1358810038poly_a @ P @ H) @ X) = (poly_a2 @ P @ (plus_plus_a @ H @ X)))))). % poly_offset_poly
thf(fact_9_offset__poly__pCons, axiom,
    ((![A : nat, P : poly_nat, H : nat]: ((fundam170929432ly_nat @ (pCons_nat @ A @ P) @ H) = (plus_plus_poly_nat @ (smult_nat @ H @ (fundam170929432ly_nat @ P @ H)) @ (pCons_nat @ A @ (fundam170929432ly_nat @ P @ H))))))). % offset_poly_pCons
thf(fact_10_offset__poly__pCons, axiom,
    ((![A : a, P : poly_a, H : a]: ((fundam1358810038poly_a @ (pCons_a @ A @ P) @ H) = (plus_plus_poly_a @ (smult_a @ H @ (fundam1358810038poly_a @ P @ H)) @ (pCons_a @ A @ (fundam1358810038poly_a @ P @ H))))))). % offset_poly_pCons
thf(fact_11_pCons__eq__iff, axiom,
    ((![A : a, P : poly_a, B : a, Q : poly_a]: (((pCons_a @ A @ P) = (pCons_a @ B @ Q)) = (((A = B)) & ((P = Q))))))). % pCons_eq_iff
thf(fact_12_smult__0__right, axiom,
    ((![A : a]: ((smult_a @ A @ zero_zero_poly_a) = zero_zero_poly_a)))). % smult_0_right
thf(fact_13_pCons__0__0, axiom,
    (((pCons_a @ zero_zero_a @ zero_zero_poly_a) = zero_zero_poly_a))). % pCons_0_0
thf(fact_14_pCons__0__0, axiom,
    (((pCons_nat @ zero_zero_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % pCons_0_0
thf(fact_15_pCons__0__0, axiom,
    (((pCons_poly_a @ zero_zero_poly_a @ zero_z2096148049poly_a) = zero_z2096148049poly_a))). % pCons_0_0
thf(fact_16_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_17_pCons__eq__0__iff, axiom,
    ((![A : poly_a, P : poly_poly_a]: (((pCons_poly_a @ A @ P) = zero_z2096148049poly_a) = (((A = zero_zero_poly_a)) & ((P = zero_z2096148049poly_a))))))). % pCons_eq_0_iff
thf(fact_18_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_19_smult__0__left, axiom,
    ((![P : poly_a]: ((smult_a @ zero_zero_a @ P) = zero_zero_poly_a)))). % smult_0_left
thf(fact_20_smult__0__left, axiom,
    ((![P : poly_nat]: ((smult_nat @ zero_zero_nat @ P) = zero_zero_poly_nat)))). % smult_0_left
thf(fact_21_smult__0__left, axiom,
    ((![P : poly_poly_a]: ((smult_poly_a @ zero_zero_poly_a @ P) = zero_z2096148049poly_a)))). % smult_0_left
thf(fact_22_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_23_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_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 : poly_a]: ((poly_poly_a2 @ zero_z2096148049poly_a @ X) = zero_zero_poly_a)))). % poly_0
thf(fact_26_poly__0, axiom,
    ((![X : a]: ((poly_a2 @ zero_zero_poly_a @ X) = zero_zero_a)))). % poly_0
thf(fact_27_add__pCons, axiom,
    ((![A : nat, P : poly_nat, B : nat, Q : poly_nat]: ((plus_plus_poly_nat @ (pCons_nat @ A @ P) @ (pCons_nat @ B @ Q)) = (pCons_nat @ (plus_plus_nat @ A @ B) @ (plus_plus_poly_nat @ P @ Q)))))). % add_pCons
thf(fact_28_add__pCons, axiom,
    ((![A : poly_a, P : poly_poly_a, B : poly_a, Q : poly_poly_a]: ((plus_p1976640465poly_a @ (pCons_poly_a @ A @ P) @ (pCons_poly_a @ B @ Q)) = (pCons_poly_a @ (plus_plus_poly_a @ A @ B) @ (plus_p1976640465poly_a @ P @ Q)))))). % add_pCons
thf(fact_29_add__pCons, axiom,
    ((![A : a, P : poly_a, B : a, Q : poly_a]: ((plus_plus_poly_a @ (pCons_a @ A @ P) @ (pCons_a @ B @ Q)) = (pCons_a @ (plus_plus_a @ A @ B) @ (plus_plus_poly_a @ P @ Q)))))). % add_pCons
thf(fact_30_poly__add, axiom,
    ((![P : poly_nat, Q : poly_nat, X : nat]: ((poly_nat2 @ (plus_plus_poly_nat @ P @ Q) @ X) = (plus_plus_nat @ (poly_nat2 @ P @ X) @ (poly_nat2 @ Q @ X)))))). % poly_add
thf(fact_31_poly__add, axiom,
    ((![P : poly_poly_a, Q : poly_poly_a, X : poly_a]: ((poly_poly_a2 @ (plus_p1976640465poly_a @ P @ Q) @ X) = (plus_plus_poly_a @ (poly_poly_a2 @ P @ X) @ (poly_poly_a2 @ Q @ X)))))). % poly_add
thf(fact_32_poly__add, axiom,
    ((![P : poly_a, Q : poly_a, X : a]: ((poly_a2 @ (plus_plus_poly_a @ P @ Q) @ X) = (plus_plus_a @ (poly_a2 @ P @ X) @ (poly_a2 @ Q @ X)))))). % poly_add
thf(fact_33_pCons__cases, axiom,
    ((![P : poly_a]: (~ ((![A2 : a, Q2 : poly_a]: (~ ((P = (pCons_a @ A2 @ Q2)))))))))). % pCons_cases
thf(fact_34_pCons__induct, axiom,
    ((![P2 : poly_nat > $o, P : poly_nat]: ((P2 @ zero_zero_poly_nat) => ((![A2 : nat, P3 : poly_nat]: (((~ ((A2 = zero_zero_nat))) | (~ ((P3 = zero_zero_poly_nat)))) => ((P2 @ P3) => (P2 @ (pCons_nat @ A2 @ P3))))) => (P2 @ P)))))). % pCons_induct
thf(fact_35_pCons__induct, axiom,
    ((![P2 : poly_poly_a > $o, P : poly_poly_a]: ((P2 @ zero_z2096148049poly_a) => ((![A2 : poly_a, P3 : poly_poly_a]: (((~ ((A2 = zero_zero_poly_a))) | (~ ((P3 = zero_z2096148049poly_a)))) => ((P2 @ P3) => (P2 @ (pCons_poly_a @ A2 @ P3))))) => (P2 @ P)))))). % pCons_induct
thf(fact_36_pCons__induct, axiom,
    ((![P2 : poly_a > $o, P : poly_a]: ((P2 @ zero_zero_poly_a) => ((![A2 : a, P3 : poly_a]: (((~ ((A2 = zero_zero_a))) | (~ ((P3 = zero_zero_poly_a)))) => ((P2 @ P3) => (P2 @ (pCons_a @ A2 @ P3))))) => (P2 @ P)))))). % pCons_induct
thf(fact_37_poly__induct2, axiom,
    ((![P2 : poly_a > poly_a > $o, P : poly_a, Q : poly_a]: ((P2 @ zero_zero_poly_a @ zero_zero_poly_a) => ((![A2 : a, P3 : poly_a, B2 : a, Q2 : poly_a]: ((P2 @ P3 @ Q2) => (P2 @ (pCons_a @ A2 @ P3) @ (pCons_a @ B2 @ Q2)))) => (P2 @ P @ Q)))))). % poly_induct2
thf(fact_38_smult__add__left, axiom,
    ((![A : nat, B : nat, P : poly_nat]: ((smult_nat @ (plus_plus_nat @ A @ B) @ P) = (plus_plus_poly_nat @ (smult_nat @ A @ P) @ (smult_nat @ B @ P)))))). % smult_add_left
thf(fact_39_smult__add__left, axiom,
    ((![A : a, B : a, P : poly_a]: ((smult_a @ (plus_plus_a @ A @ B) @ P) = (plus_plus_poly_a @ (smult_a @ A @ P) @ (smult_a @ B @ P)))))). % smult_add_left
thf(fact_40_smult__add__left, axiom,
    ((![A : poly_a, B : poly_a, P : poly_poly_a]: ((smult_poly_a @ (plus_plus_poly_a @ A @ B) @ P) = (plus_p1976640465poly_a @ (smult_poly_a @ A @ P) @ (smult_poly_a @ B @ P)))))). % smult_add_left
thf(fact_41_smult__add__right, axiom,
    ((![A : a, P : poly_a, Q : poly_a]: ((smult_a @ A @ (plus_plus_poly_a @ P @ Q)) = (plus_plus_poly_a @ (smult_a @ A @ P) @ (smult_a @ A @ Q)))))). % smult_add_right
thf(fact_42_synthetic__div__unique__lemma, axiom,
    ((![C : a, P : poly_a, A : a]: (((smult_a @ C @ P) = (pCons_a @ A @ P)) => (P = zero_zero_poly_a))))). % synthetic_div_unique_lemma
thf(fact_43_offset__poly__eq__0__lemma, axiom,
    ((![C : a, P : poly_a, A : a]: (((plus_plus_poly_a @ (smult_a @ C @ P) @ (pCons_a @ A @ P)) = zero_zero_poly_a) => (P = zero_zero_poly_a))))). % offset_poly_eq_0_lemma
thf(fact_44_add_Oleft__neutral, axiom,
    ((![A : a]: ((plus_plus_a @ zero_zero_a @ A) = A)))). % add.left_neutral
thf(fact_45_add_Oleft__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % add.left_neutral
thf(fact_46_add_Oleft__neutral, axiom,
    ((![A : poly_a]: ((plus_plus_poly_a @ zero_zero_poly_a @ A) = A)))). % add.left_neutral
thf(fact_47_add__right__cancel, axiom,
    ((![B : nat, A : nat, C : nat]: (((plus_plus_nat @ B @ A) = (plus_plus_nat @ C @ A)) = (B = C))))). % add_right_cancel
thf(fact_48_add__left__cancel, axiom,
    ((![A : nat, B : nat, C : nat]: (((plus_plus_nat @ A @ B) = (plus_plus_nat @ A @ C)) = (B = C))))). % add_left_cancel
thf(fact_49_zero__eq__add__iff__both__eq__0, axiom,
    ((![X : nat, Y : nat]: ((zero_zero_nat = (plus_plus_nat @ X @ Y)) = (((X = zero_zero_nat)) & ((Y = zero_zero_nat))))))). % zero_eq_add_iff_both_eq_0
thf(fact_50_add__eq__0__iff__both__eq__0, axiom,
    ((![X : nat, Y : nat]: (((plus_plus_nat @ X @ Y) = zero_zero_nat) = (((X = zero_zero_nat)) & ((Y = zero_zero_nat))))))). % add_eq_0_iff_both_eq_0
thf(fact_51_add__cancel__right__right, axiom,
    ((![A : nat, B : nat]: ((A = (plus_plus_nat @ A @ B)) = (B = zero_zero_nat))))). % add_cancel_right_right
thf(fact_52_add__cancel__right__left, axiom,
    ((![A : nat, B : nat]: ((A = (plus_plus_nat @ B @ A)) = (B = zero_zero_nat))))). % add_cancel_right_left
thf(fact_53_add__cancel__left__right, axiom,
    ((![A : nat, B : nat]: (((plus_plus_nat @ A @ B) = A) = (B = zero_zero_nat))))). % add_cancel_left_right
thf(fact_54_add__cancel__left__left, axiom,
    ((![B : nat, A : nat]: (((plus_plus_nat @ B @ A) = A) = (B = zero_zero_nat))))). % add_cancel_left_left
thf(fact_55_add_Oright__neutral, axiom,
    ((![A : a]: ((plus_plus_a @ A @ zero_zero_a) = A)))). % add.right_neutral
thf(fact_56_add_Oright__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.right_neutral
thf(fact_57_add_Oright__neutral, axiom,
    ((![A : poly_a]: ((plus_plus_poly_a @ A @ zero_zero_poly_a) = A)))). % add.right_neutral
thf(fact_58_degree__0, axiom,
    (((degree_a @ zero_zero_poly_a) = zero_zero_nat))). % degree_0
thf(fact_59_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_60_zero__reorient, axiom,
    ((![X : poly_a]: ((zero_zero_poly_a = X) = (X = zero_zero_poly_a))))). % zero_reorient
thf(fact_61_add__right__imp__eq, axiom,
    ((![B : nat, A : nat, C : nat]: (((plus_plus_nat @ B @ A) = (plus_plus_nat @ C @ A)) => (B = C))))). % add_right_imp_eq
thf(fact_62_add__left__imp__eq, axiom,
    ((![A : nat, B : nat, C : nat]: (((plus_plus_nat @ A @ B) = (plus_plus_nat @ A @ C)) => (B = C))))). % add_left_imp_eq
thf(fact_63_add_Oleft__commute, axiom,
    ((![B : nat, A : nat, C : nat]: ((plus_plus_nat @ B @ (plus_plus_nat @ A @ C)) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % add.left_commute
thf(fact_64_add_Oleft__commute, axiom,
    ((![B : a, A : a, C : a]: ((plus_plus_a @ B @ (plus_plus_a @ A @ C)) = (plus_plus_a @ A @ (plus_plus_a @ B @ C)))))). % add.left_commute
thf(fact_65_add_Oleft__commute, axiom,
    ((![B : poly_a, A : poly_a, C : poly_a]: ((plus_plus_poly_a @ B @ (plus_plus_poly_a @ A @ C)) = (plus_plus_poly_a @ A @ (plus_plus_poly_a @ B @ C)))))). % add.left_commute
thf(fact_66_add_Ocommute, axiom,
    ((plus_plus_nat = (^[A3 : nat]: (^[B3 : nat]: (plus_plus_nat @ B3 @ A3)))))). % add.commute
thf(fact_67_add_Ocommute, axiom,
    ((plus_plus_a = (^[A3 : a]: (^[B3 : a]: (plus_plus_a @ B3 @ A3)))))). % add.commute
thf(fact_68_add_Ocommute, axiom,
    ((plus_plus_poly_a = (^[A3 : poly_a]: (^[B3 : poly_a]: (plus_plus_poly_a @ B3 @ A3)))))). % add.commute
thf(fact_69_add_Oassoc, axiom,
    ((![A : nat, B : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % add.assoc
thf(fact_70_add_Oassoc, axiom,
    ((![A : a, B : a, C : a]: ((plus_plus_a @ (plus_plus_a @ A @ B) @ C) = (plus_plus_a @ A @ (plus_plus_a @ B @ C)))))). % add.assoc
thf(fact_71_add_Oassoc, axiom,
    ((![A : poly_a, B : poly_a, C : poly_a]: ((plus_plus_poly_a @ (plus_plus_poly_a @ A @ B) @ C) = (plus_plus_poly_a @ A @ (plus_plus_poly_a @ B @ C)))))). % add.assoc
thf(fact_72_group__cancel_Oadd2, axiom,
    ((![B4 : nat, K : nat, B : nat, A : nat]: ((B4 = (plus_plus_nat @ K @ B)) => ((plus_plus_nat @ A @ B4) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B))))))). % group_cancel.add2
thf(fact_73_group__cancel_Oadd2, axiom,
    ((![B4 : a, K : a, B : a, A : a]: ((B4 = (plus_plus_a @ K @ B)) => ((plus_plus_a @ A @ B4) = (plus_plus_a @ K @ (plus_plus_a @ A @ B))))))). % group_cancel.add2
thf(fact_74_group__cancel_Oadd2, axiom,
    ((![B4 : poly_a, K : poly_a, B : poly_a, A : poly_a]: ((B4 = (plus_plus_poly_a @ K @ B)) => ((plus_plus_poly_a @ A @ B4) = (plus_plus_poly_a @ K @ (plus_plus_poly_a @ A @ B))))))). % group_cancel.add2
thf(fact_75_group__cancel_Oadd1, axiom,
    ((![A4 : nat, K : nat, A : nat, B : nat]: ((A4 = (plus_plus_nat @ K @ A)) => ((plus_plus_nat @ A4 @ B) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B))))))). % group_cancel.add1
thf(fact_76_group__cancel_Oadd1, axiom,
    ((![A4 : a, K : a, A : a, B : a]: ((A4 = (plus_plus_a @ K @ A)) => ((plus_plus_a @ A4 @ B) = (plus_plus_a @ K @ (plus_plus_a @ A @ B))))))). % group_cancel.add1
thf(fact_77_group__cancel_Oadd1, axiom,
    ((![A4 : poly_a, K : poly_a, A : poly_a, B : poly_a]: ((A4 = (plus_plus_poly_a @ K @ A)) => ((plus_plus_poly_a @ A4 @ B) = (plus_plus_poly_a @ K @ (plus_plus_poly_a @ A @ B))))))). % group_cancel.add1
thf(fact_78_add__mono__thms__linordered__semiring_I4_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((I = J) & (K = L)) => ((plus_plus_nat @ I @ K) = (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_semiring(4)
thf(fact_79_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A : nat, B : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_80_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A : a, B : a, C : a]: ((plus_plus_a @ (plus_plus_a @ A @ B) @ C) = (plus_plus_a @ A @ (plus_plus_a @ B @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_81_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A : poly_a, B : poly_a, C : poly_a]: ((plus_plus_poly_a @ (plus_plus_poly_a @ A @ B) @ C) = (plus_plus_poly_a @ A @ (plus_plus_poly_a @ B @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_82_degree__eq__zeroE, axiom,
    ((![P : poly_a]: (((degree_a @ P) = zero_zero_nat) => (~ ((![A2 : a]: (~ ((P = (pCons_a @ A2 @ zero_zero_poly_a))))))))))). % degree_eq_zeroE
thf(fact_83_degree__pCons__0, axiom,
    ((![A : a]: ((degree_a @ (pCons_a @ A @ zero_zero_poly_a)) = zero_zero_nat)))). % degree_pCons_0
thf(fact_84_add_Ocomm__neutral, axiom,
    ((![A : a]: ((plus_plus_a @ A @ zero_zero_a) = A)))). % add.comm_neutral
thf(fact_85_add_Ocomm__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.comm_neutral
thf(fact_86_add_Ocomm__neutral, axiom,
    ((![A : poly_a]: ((plus_plus_poly_a @ A @ zero_zero_poly_a) = A)))). % add.comm_neutral
thf(fact_87_comm__monoid__add__class_Oadd__0, axiom,
    ((![A : a]: ((plus_plus_a @ zero_zero_a @ A) = A)))). % comm_monoid_add_class.add_0
thf(fact_88_comm__monoid__add__class_Oadd__0, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % comm_monoid_add_class.add_0
thf(fact_89_comm__monoid__add__class_Oadd__0, axiom,
    ((![A : poly_a]: ((plus_plus_poly_a @ zero_zero_poly_a @ A) = A)))). % comm_monoid_add_class.add_0
thf(fact_90_add__is__0, axiom,
    ((![M : nat, N : nat]: (((plus_plus_nat @ M @ N) = zero_zero_nat) = (((M = zero_zero_nat)) & ((N = zero_zero_nat))))))). % add_is_0
thf(fact_91_Nat_Oadd__0__right, axiom,
    ((![M : nat]: ((plus_plus_nat @ M @ zero_zero_nat) = M)))). % Nat.add_0_right
thf(fact_92_add__eq__self__zero, axiom,
    ((![M : nat, N : nat]: (((plus_plus_nat @ M @ N) = M) => (N = zero_zero_nat))))). % add_eq_self_zero
thf(fact_93_plus__nat_Oadd__0, axiom,
    ((![N : nat]: ((plus_plus_nat @ zero_zero_nat @ N) = N)))). % plus_nat.add_0
thf(fact_94_verit__sum__simplify, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % verit_sum_simplify
thf(fact_95_synthetic__div__unique, axiom,
    ((![P : poly_nat, C : nat, Q : poly_nat, R : nat]: (((plus_plus_poly_nat @ P @ (smult_nat @ C @ Q)) = (pCons_nat @ R @ Q)) => ((R = (poly_nat2 @ P @ C)) & (Q = (synthetic_div_nat @ P @ C))))))). % synthetic_div_unique
thf(fact_96_synthetic__div__unique, axiom,
    ((![P : poly_a, C : a, Q : poly_a, R : a]: (((plus_plus_poly_a @ P @ (smult_a @ C @ Q)) = (pCons_a @ R @ Q)) => ((R = (poly_a2 @ P @ C)) & (Q = (synthetic_div_a @ P @ C))))))). % synthetic_div_unique
thf(fact_97_synthetic__div__correct, axiom,
    ((![P : poly_nat, C : nat]: ((plus_plus_poly_nat @ P @ (smult_nat @ C @ (synthetic_div_nat @ P @ C))) = (pCons_nat @ (poly_nat2 @ P @ C) @ (synthetic_div_nat @ P @ C)))))). % synthetic_div_correct
thf(fact_98_synthetic__div__correct, axiom,
    ((![P : poly_a, C : a]: ((plus_plus_poly_a @ P @ (smult_a @ C @ (synthetic_div_a @ P @ C))) = (pCons_a @ (poly_a2 @ P @ C) @ (synthetic_div_a @ P @ C)))))). % synthetic_div_correct
thf(fact_99_Euclid__induct, axiom,
    ((![P2 : nat > nat > $o, A : nat, B : nat]: ((![A2 : nat, B2 : nat]: ((P2 @ A2 @ B2) = (P2 @ B2 @ A2))) => ((![A2 : nat]: (P2 @ A2 @ zero_zero_nat)) => ((![A2 : nat, B2 : nat]: ((P2 @ A2 @ B2) => (P2 @ A2 @ (plus_plus_nat @ A2 @ B2)))) => (P2 @ A @ B))))))). % Euclid_induct
thf(fact_100_synthetic__div__0, axiom,
    ((![C : a]: ((synthetic_div_a @ zero_zero_poly_a @ C) = zero_zero_poly_a)))). % synthetic_div_0
thf(fact_101_synthetic__div__pCons, axiom,
    ((![A : nat, P : poly_nat, C : nat]: ((synthetic_div_nat @ (pCons_nat @ A @ P) @ C) = (pCons_nat @ (poly_nat2 @ P @ C) @ (synthetic_div_nat @ P @ C)))))). % synthetic_div_pCons
thf(fact_102_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_103_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_104_add__0__iff, axiom,
    ((![B : nat, A : nat]: ((B = (plus_plus_nat @ B @ A)) = (A = zero_zero_nat))))). % add_0_iff
thf(fact_105_mult__pCons__right, axiom,
    ((![P : poly_a, A : a, Q : poly_a]: ((times_times_poly_a @ P @ (pCons_a @ A @ Q)) = (plus_plus_poly_a @ (smult_a @ A @ P) @ (pCons_a @ zero_zero_a @ (times_times_poly_a @ P @ Q))))))). % mult_pCons_right
thf(fact_106_mult__pCons__right, axiom,
    ((![P : poly_nat, A : nat, Q : poly_nat]: ((times_times_poly_nat @ P @ (pCons_nat @ A @ Q)) = (plus_plus_poly_nat @ (smult_nat @ A @ P) @ (pCons_nat @ zero_zero_nat @ (times_times_poly_nat @ P @ Q))))))). % mult_pCons_right
thf(fact_107_mult__pCons__right, axiom,
    ((![P : poly_poly_a, A : poly_a, Q : poly_poly_a]: ((times_545135445poly_a @ P @ (pCons_poly_a @ A @ Q)) = (plus_p1976640465poly_a @ (smult_poly_a @ A @ P) @ (pCons_poly_a @ zero_zero_poly_a @ (times_545135445poly_a @ P @ Q))))))). % mult_pCons_right
thf(fact_108_smult__smult, axiom,
    ((![A : a, B : a, P : poly_a]: ((smult_a @ A @ (smult_a @ B @ P)) = (smult_a @ (times_times_a @ A @ B) @ P))))). % smult_smult
thf(fact_109_smult__smult, axiom,
    ((![A : nat, B : nat, P : poly_nat]: ((smult_nat @ A @ (smult_nat @ B @ P)) = (smult_nat @ (times_times_nat @ A @ B) @ P))))). % smult_smult
thf(fact_110_mult__smult__right, axiom,
    ((![P : poly_a, A : a, Q : poly_a]: ((times_times_poly_a @ P @ (smult_a @ A @ Q)) = (smult_a @ A @ (times_times_poly_a @ P @ Q)))))). % mult_smult_right
thf(fact_111_mult__smult__left, axiom,
    ((![A : a, P : poly_a, Q : poly_a]: ((times_times_poly_a @ (smult_a @ A @ P) @ Q) = (smult_a @ A @ (times_times_poly_a @ P @ Q)))))). % mult_smult_left
thf(fact_112_smult__pCons, axiom,
    ((![A : a, B : a, P : poly_a]: ((smult_a @ A @ (pCons_a @ B @ P)) = (pCons_a @ (times_times_a @ A @ B) @ (smult_a @ A @ P)))))). % smult_pCons
thf(fact_113_smult__pCons, axiom,
    ((![A : nat, B : nat, P : poly_nat]: ((smult_nat @ A @ (pCons_nat @ B @ P)) = (pCons_nat @ (times_times_nat @ A @ B) @ (smult_nat @ A @ P)))))). % smult_pCons
thf(fact_114_poly__smult, axiom,
    ((![A : a, P : poly_a, X : a]: ((poly_a2 @ (smult_a @ A @ P) @ X) = (times_times_a @ A @ (poly_a2 @ P @ X)))))). % poly_smult
thf(fact_115_poly__smult, axiom,
    ((![A : nat, P : poly_nat, X : nat]: ((poly_nat2 @ (smult_nat @ A @ P) @ X) = (times_times_nat @ A @ (poly_nat2 @ P @ X)))))). % poly_smult
thf(fact_116_poly__mult, axiom,
    ((![P : poly_a, Q : poly_a, X : a]: ((poly_a2 @ (times_times_poly_a @ P @ Q) @ X) = (times_times_a @ (poly_a2 @ P @ X) @ (poly_a2 @ Q @ X)))))). % poly_mult
thf(fact_117_poly__mult, axiom,
    ((![P : poly_nat, Q : poly_nat, X : nat]: ((poly_nat2 @ (times_times_poly_nat @ P @ Q) @ X) = (times_times_nat @ (poly_nat2 @ P @ X) @ (poly_nat2 @ Q @ X)))))). % poly_mult
thf(fact_118_poly__pCons, axiom,
    ((![A : a, P : poly_a, X : a]: ((poly_a2 @ (pCons_a @ A @ P) @ X) = (plus_plus_a @ A @ (times_times_a @ X @ (poly_a2 @ P @ X))))))). % poly_pCons
thf(fact_119_poly__pCons, axiom,
    ((![A : poly_a, P : poly_poly_a, X : poly_a]: ((poly_poly_a2 @ (pCons_poly_a @ A @ P) @ X) = (plus_plus_poly_a @ A @ (times_times_poly_a @ X @ (poly_poly_a2 @ P @ X))))))). % poly_pCons
thf(fact_120_poly__pCons, axiom,
    ((![A : nat, P : poly_nat, X : nat]: ((poly_nat2 @ (pCons_nat @ A @ P) @ X) = (plus_plus_nat @ A @ (times_times_nat @ X @ (poly_nat2 @ P @ X))))))). % poly_pCons
thf(fact_121_mult__pCons__left, axiom,
    ((![A : a, P : poly_a, Q : poly_a]: ((times_times_poly_a @ (pCons_a @ A @ P) @ Q) = (plus_plus_poly_a @ (smult_a @ A @ Q) @ (pCons_a @ zero_zero_a @ (times_times_poly_a @ P @ Q))))))). % mult_pCons_left
thf(fact_122_mult__pCons__left, axiom,
    ((![A : nat, P : poly_nat, Q : poly_nat]: ((times_times_poly_nat @ (pCons_nat @ A @ P) @ Q) = (plus_plus_poly_nat @ (smult_nat @ A @ Q) @ (pCons_nat @ zero_zero_nat @ (times_times_poly_nat @ P @ Q))))))). % mult_pCons_left
thf(fact_123_mult__pCons__left, axiom,
    ((![A : poly_a, P : poly_poly_a, Q : poly_poly_a]: ((times_545135445poly_a @ (pCons_poly_a @ A @ P) @ Q) = (plus_p1976640465poly_a @ (smult_poly_a @ A @ Q) @ (pCons_poly_a @ zero_zero_poly_a @ (times_545135445poly_a @ P @ Q))))))). % mult_pCons_left
thf(fact_124_mult_Oleft__commute, axiom,
    ((![B : nat, A : nat, C : nat]: ((times_times_nat @ B @ (times_times_nat @ A @ C)) = (times_times_nat @ A @ (times_times_nat @ B @ C)))))). % mult.left_commute
thf(fact_125_mult_Ocommute, axiom,
    ((times_times_nat = (^[A3 : nat]: (^[B3 : nat]: (times_times_nat @ B3 @ A3)))))). % mult.commute
thf(fact_126_mult_Oassoc, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ (times_times_nat @ A @ B) @ C) = (times_times_nat @ A @ (times_times_nat @ B @ C)))))). % mult.assoc
thf(fact_127_ab__semigroup__mult__class_Omult__ac_I1_J, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ (times_times_nat @ A @ B) @ C) = (times_times_nat @ A @ (times_times_nat @ B @ C)))))). % ab_semigroup_mult_class.mult_ac(1)
thf(fact_128_mult__poly__add__left, axiom,
    ((![P : poly_a, Q : poly_a, R : poly_a]: ((times_times_poly_a @ (plus_plus_poly_a @ P @ Q) @ R) = (plus_plus_poly_a @ (times_times_poly_a @ P @ R) @ (times_times_poly_a @ Q @ R)))))). % mult_poly_add_left
thf(fact_129_add__scale__eq__noteq, axiom,
    ((![R : nat, A : nat, B : nat, C : nat, D : nat]: ((~ ((R = zero_zero_nat))) => (((A = B) & (~ ((C = D)))) => (~ (((plus_plus_nat @ A @ (times_times_nat @ R @ C)) = (plus_plus_nat @ B @ (times_times_nat @ R @ D)))))))))). % add_scale_eq_noteq
thf(fact_130_mult__poly__0__right, axiom,
    ((![P : poly_a]: ((times_times_poly_a @ P @ zero_zero_poly_a) = zero_zero_poly_a)))). % mult_poly_0_right
thf(fact_131_mult__poly__0__left, axiom,
    ((![Q : poly_a]: ((times_times_poly_a @ zero_zero_poly_a @ Q) = zero_zero_poly_a)))). % mult_poly_0_left
thf(fact_132_crossproduct__eq, axiom,
    ((![W : nat, Y : nat, X : nat, Z : nat]: (((plus_plus_nat @ (times_times_nat @ W @ Y) @ (times_times_nat @ X @ Z)) = (plus_plus_nat @ (times_times_nat @ W @ Z) @ (times_times_nat @ X @ Y))) = (((W = X)) | ((Y = Z))))))). % crossproduct_eq
thf(fact_133_crossproduct__noteq, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((((~ ((A = B)))) & ((~ ((C = D))))) = (~ (((plus_plus_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ D)) = (plus_plus_nat @ (times_times_nat @ A @ D) @ (times_times_nat @ B @ C))))))))). % crossproduct_noteq
thf(fact_134_mult__cancel2, axiom,
    ((![M : nat, K : nat, N : nat]: (((times_times_nat @ M @ K) = (times_times_nat @ N @ K)) = (((M = N)) | ((K = zero_zero_nat))))))). % mult_cancel2
thf(fact_135_mult__cancel1, axiom,
    ((![K : nat, M : nat, N : nat]: (((times_times_nat @ K @ M) = (times_times_nat @ K @ N)) = (((M = N)) | ((K = zero_zero_nat))))))). % mult_cancel1
thf(fact_136_mult__0__right, axiom,
    ((![M : nat]: ((times_times_nat @ M @ zero_zero_nat) = zero_zero_nat)))). % mult_0_right
thf(fact_137_mult__is__0, axiom,
    ((![M : nat, N : nat]: (((times_times_nat @ M @ N) = zero_zero_nat) = (((M = zero_zero_nat)) | ((N = zero_zero_nat))))))). % mult_is_0
thf(fact_138_mult__zero__left, axiom,
    ((![A : poly_a]: ((times_times_poly_a @ zero_zero_poly_a @ A) = zero_zero_poly_a)))). % mult_zero_left
thf(fact_139_mult__zero__left, axiom,
    ((![A : nat]: ((times_times_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % mult_zero_left
thf(fact_140_mult__zero__right, axiom,
    ((![A : poly_a]: ((times_times_poly_a @ A @ zero_zero_poly_a) = zero_zero_poly_a)))). % mult_zero_right
thf(fact_141_mult__zero__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % mult_zero_right
thf(fact_142_mult__eq__0__iff, axiom,
    ((![A : nat, B : nat]: (((times_times_nat @ A @ B) = zero_zero_nat) = (((A = zero_zero_nat)) | ((B = zero_zero_nat))))))). % mult_eq_0_iff
thf(fact_143_mult__cancel__left, axiom,
    ((![C : nat, A : nat, B : nat]: (((times_times_nat @ C @ A) = (times_times_nat @ C @ B)) = (((C = zero_zero_nat)) | ((A = B))))))). % mult_cancel_left
thf(fact_144_mult__cancel__right, axiom,
    ((![A : nat, C : nat, B : nat]: (((times_times_nat @ A @ C) = (times_times_nat @ B @ C)) = (((C = zero_zero_nat)) | ((A = B))))))). % mult_cancel_right
thf(fact_145_add__mult__distrib, axiom,
    ((![M : nat, N : nat, K : nat]: ((times_times_nat @ (plus_plus_nat @ M @ N) @ K) = (plus_plus_nat @ (times_times_nat @ M @ K) @ (times_times_nat @ N @ K)))))). % add_mult_distrib
thf(fact_146_add__mult__distrib2, axiom,
    ((![K : nat, M : nat, N : nat]: ((times_times_nat @ K @ (plus_plus_nat @ M @ N)) = (plus_plus_nat @ (times_times_nat @ K @ M) @ (times_times_nat @ K @ N)))))). % add_mult_distrib2
thf(fact_147_mult__0, axiom,
    ((![N : nat]: ((times_times_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % mult_0
thf(fact_148_mult__not__zero, axiom,
    ((![A : poly_a, B : poly_a]: ((~ (((times_times_poly_a @ A @ B) = zero_zero_poly_a))) => ((~ ((A = zero_zero_poly_a))) & (~ ((B = zero_zero_poly_a)))))))). % mult_not_zero
thf(fact_149_mult__not__zero, axiom,
    ((![A : nat, B : nat]: ((~ (((times_times_nat @ A @ B) = zero_zero_nat))) => ((~ ((A = zero_zero_nat))) & (~ ((B = zero_zero_nat)))))))). % mult_not_zero
thf(fact_150_divisors__zero, axiom,
    ((![A : nat, B : nat]: (((times_times_nat @ A @ B) = zero_zero_nat) => ((A = zero_zero_nat) | (B = zero_zero_nat)))))). % divisors_zero
thf(fact_151_no__zero__divisors, axiom,
    ((![A : nat, B : nat]: ((~ ((A = zero_zero_nat))) => ((~ ((B = zero_zero_nat))) => (~ (((times_times_nat @ A @ B) = zero_zero_nat)))))))). % no_zero_divisors
thf(fact_152_mult__left__cancel, axiom,
    ((![C : nat, A : nat, B : nat]: ((~ ((C = zero_zero_nat))) => (((times_times_nat @ C @ A) = (times_times_nat @ C @ B)) = (A = B)))))). % mult_left_cancel

% Conjectures (1)
thf(conj_0, conjecture,
    (((degree_a @ (fundam1358810038poly_a @ p @ h)) = (degree_a @ p)))).
