% 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_1137__5378588_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:36:05.493

% Could-be-implicit typings (6)
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J, type,
    poly_poly_nat : $tType).
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 (40)
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat, type,
    minus_minus_nat : nat > nat > nat).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    minus_154650241poly_a : poly_poly_a > poly_poly_a > poly_poly_a).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Polynomial__Opoly_Itf__a_J, type,
    minus_minus_poly_a : poly_a > poly_a > poly_a).
thf(sy_c_Groups_Ominus__class_Ominus_001tf__a, type,
    minus_minus_a : a > a > a).
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_It__Polynomial__Opoly_It__Nat__Onat_J_J, type,
    one_on1411366565ly_nat : poly_poly_nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    one_one_poly_poly_a : poly_poly_a).
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_It__Polynomial__Opoly_It__Nat__Onat_J_J, type,
    zero_z1059985641ly_nat : poly_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_Ocontent_001t__Nat__Onat, type,
    content_nat : poly_nat > 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_It__Nat__Onat_J, type,
    pCons_poly_nat : poly_nat > poly_poly_nat > poly_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_Opcompose_001tf__a, type,
    pcompose_a : poly_a > poly_a > poly_a).
thf(sy_c_Polynomial_Oprimitive__part_001t__Nat__Onat, type,
    primitive_part_nat : poly_nat > poly_nat).
thf(sy_c_Polynomial_Osmult_001t__Nat__Onat, type,
    smult_nat : nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_Osmult_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    smult_poly_nat : poly_nat > poly_poly_nat > poly_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_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_It__Polynomial__Opoly_It__Nat__Onat_J_J, type,
    dvd_dv944831366ly_nat : poly_poly_nat > poly_poly_nat > $o).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    dvd_dvd_poly_poly_a : poly_poly_a > poly_poly_a > $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_c_Rings_Onormalization__semidom__class_Onormalize_001t__Nat__Onat, type,
    normal728885956ze_nat : nat > nat).
thf(sy_v_a, type,
    a2 : a).
thf(sy_v_p, type,
    p : poly_a).
thf(sy_v_p_H, type,
    p2 : poly_a).
thf(sy_v_q, type,
    q : poly_a).
thf(sy_v_r, type,
    r : poly_a).

% Relevant facts (152)
thf(fact_0_pp_H, axiom,
    ((dvd_dvd_poly_a @ p @ p2))). % pp'
thf(fact_1_qrp_H, axiom,
    (((minus_minus_poly_a @ (smult_a @ a2 @ q) @ p2) = r))). % qrp'
thf(fact_2_dvd__refl, axiom,
    ((![A : poly_nat]: (dvd_dvd_poly_nat @ A @ A)))). % dvd_refl
thf(fact_3_dvd__refl, axiom,
    ((![A : poly_a]: (dvd_dvd_poly_a @ A @ A)))). % dvd_refl
thf(fact_4_dvd__refl, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ A)))). % dvd_refl
thf(fact_5_dvd__trans, axiom,
    ((![A : poly_nat, B : poly_nat, C : poly_nat]: ((dvd_dvd_poly_nat @ A @ B) => ((dvd_dvd_poly_nat @ B @ C) => (dvd_dvd_poly_nat @ A @ C)))))). % dvd_trans
thf(fact_6_dvd__trans, axiom,
    ((![A : poly_a, B : poly_a, C : poly_a]: ((dvd_dvd_poly_a @ A @ B) => ((dvd_dvd_poly_a @ B @ C) => (dvd_dvd_poly_a @ A @ C)))))). % dvd_trans
thf(fact_7_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_8_a0, axiom,
    ((~ ((a2 = zero_zero_a))))). % a0
thf(fact_9_content__dvd__contentI, axiom,
    ((![P : poly_nat, Q : poly_nat]: ((dvd_dvd_poly_nat @ P @ Q) => (dvd_dvd_nat @ (content_nat @ P) @ (content_nat @ Q)))))). % content_dvd_contentI
thf(fact_10_dvd__0__left__iff, axiom,
    ((![A : poly_a]: ((dvd_dvd_poly_a @ zero_zero_poly_a @ A) = (A = zero_zero_poly_a))))). % dvd_0_left_iff
thf(fact_11_dvd__0__left__iff, axiom,
    ((![A : poly_nat]: ((dvd_dvd_poly_nat @ zero_zero_poly_nat @ A) = (A = zero_zero_poly_nat))))). % dvd_0_left_iff
thf(fact_12_dvd__0__left__iff, axiom,
    ((![A : a]: ((dvd_dvd_a @ zero_zero_a @ A) = (A = zero_zero_a))))). % dvd_0_left_iff
thf(fact_13_dvd__0__left__iff, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) = (A = zero_zero_nat))))). % dvd_0_left_iff
thf(fact_14_dvd__0__right, axiom,
    ((![A : poly_a]: (dvd_dvd_poly_a @ A @ zero_zero_poly_a)))). % dvd_0_right
thf(fact_15_dvd__0__right, axiom,
    ((![A : poly_nat]: (dvd_dvd_poly_nat @ A @ zero_zero_poly_nat)))). % dvd_0_right
thf(fact_16_dvd__0__right, axiom,
    ((![A : a]: (dvd_dvd_a @ A @ zero_zero_a)))). % dvd_0_right
thf(fact_17_dvd__0__right, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ zero_zero_nat)))). % dvd_0_right
thf(fact_18_smult__0__left, axiom,
    ((![P : poly_a]: ((smult_a @ zero_zero_a @ P) = zero_zero_poly_a)))). % smult_0_left
thf(fact_19_smult__0__left, axiom,
    ((![P : poly_nat]: ((smult_nat @ zero_zero_nat @ P) = zero_zero_poly_nat)))). % smult_0_left
thf(fact_20_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_21_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_22_content__0, axiom,
    (((content_nat @ zero_zero_poly_nat) = zero_zero_nat))). % content_0
thf(fact_23_content__eq__zero__iff, axiom,
    ((![P : poly_nat]: (((content_nat @ P) = zero_zero_nat) = (P = zero_zero_poly_nat))))). % content_eq_zero_iff
thf(fact_24_content__times__primitive__part, axiom,
    ((![P : poly_nat]: ((smult_nat @ (content_nat @ P) @ (primitive_part_nat @ P)) = P)))). % content_times_primitive_part
thf(fact_25_smult__dvd, axiom,
    ((![P : poly_a, Q : poly_a, A : a]: ((dvd_dvd_poly_a @ P @ Q) => ((~ ((A = zero_zero_a))) => (dvd_dvd_poly_a @ (smult_a @ A @ P) @ Q)))))). % smult_dvd
thf(fact_26_dvd__smult__iff, axiom,
    ((![A : a, P : poly_a, Q : poly_a]: ((~ ((A = zero_zero_a))) => ((dvd_dvd_poly_a @ P @ (smult_a @ A @ Q)) = (dvd_dvd_poly_a @ P @ Q)))))). % dvd_smult_iff
thf(fact_27_smult__dvd__iff, axiom,
    ((![A : a, P : poly_a, Q : poly_a]: ((dvd_dvd_poly_a @ (smult_a @ A @ P) @ Q) = (((((A = zero_zero_a)) => ((Q = zero_zero_poly_a)))) & ((((~ ((A = zero_zero_a)))) => ((dvd_dvd_poly_a @ P @ Q))))))))). % smult_dvd_iff
thf(fact_28_smult__diff__left, axiom,
    ((![A : a, B : a, P : poly_a]: ((smult_a @ (minus_minus_a @ A @ B) @ P) = (minus_minus_poly_a @ (smult_a @ A @ P) @ (smult_a @ B @ P)))))). % smult_diff_left
thf(fact_29_smult__diff__left, axiom,
    ((![A : poly_a, B : poly_a, P : poly_poly_a]: ((smult_poly_a @ (minus_minus_poly_a @ A @ B) @ P) = (minus_154650241poly_a @ (smult_poly_a @ A @ P) @ (smult_poly_a @ B @ P)))))). % smult_diff_left
thf(fact_30_dvd__smult__cancel, axiom,
    ((![P : poly_a, A : a, Q : poly_a]: ((dvd_dvd_poly_a @ P @ (smult_a @ A @ Q)) => ((~ ((A = zero_zero_a))) => (dvd_dvd_poly_a @ P @ Q)))))). % dvd_smult_cancel
thf(fact_31_smult__diff__right, axiom,
    ((![A : a, P : poly_a, Q : poly_a]: ((smult_a @ A @ (minus_minus_poly_a @ P @ Q)) = (minus_minus_poly_a @ (smult_a @ A @ P) @ (smult_a @ A @ Q)))))). % smult_diff_right
thf(fact_32_dvd__diff, axiom,
    ((![X : poly_a, Y : poly_a, Z : poly_a]: ((dvd_dvd_poly_a @ X @ Y) => ((dvd_dvd_poly_a @ X @ Z) => (dvd_dvd_poly_a @ X @ (minus_minus_poly_a @ Y @ Z))))))). % dvd_diff
thf(fact_33_dvd__0__left, axiom,
    ((![A : poly_a]: ((dvd_dvd_poly_a @ zero_zero_poly_a @ A) => (A = zero_zero_poly_a))))). % dvd_0_left
thf(fact_34_dvd__0__left, axiom,
    ((![A : poly_nat]: ((dvd_dvd_poly_nat @ zero_zero_poly_nat @ A) => (A = zero_zero_poly_nat))))). % dvd_0_left
thf(fact_35_dvd__0__left, axiom,
    ((![A : a]: ((dvd_dvd_a @ zero_zero_a @ A) => (A = zero_zero_a))))). % dvd_0_left
thf(fact_36_dvd__0__left, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) => (A = zero_zero_nat))))). % dvd_0_left
thf(fact_37_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_38_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_39_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_40_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_41_diff__self, axiom,
    ((![A : a]: ((minus_minus_a @ A @ A) = zero_zero_a)))). % diff_self
thf(fact_42_diff__self, axiom,
    ((![A : poly_a]: ((minus_minus_poly_a @ A @ A) = zero_zero_poly_a)))). % diff_self
thf(fact_43_diff__0__right, axiom,
    ((![A : a]: ((minus_minus_a @ A @ zero_zero_a) = A)))). % diff_0_right
thf(fact_44_diff__0__right, axiom,
    ((![A : poly_a]: ((minus_minus_poly_a @ A @ zero_zero_poly_a) = A)))). % diff_0_right
thf(fact_45_zero__diff, axiom,
    ((![A : nat]: ((minus_minus_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % zero_diff
thf(fact_46_diff__zero, axiom,
    ((![A : a]: ((minus_minus_a @ A @ zero_zero_a) = A)))). % diff_zero
thf(fact_47_diff__zero, axiom,
    ((![A : nat]: ((minus_minus_nat @ A @ zero_zero_nat) = A)))). % diff_zero
thf(fact_48_diff__zero, axiom,
    ((![A : poly_a]: ((minus_minus_poly_a @ A @ zero_zero_poly_a) = A)))). % diff_zero
thf(fact_49_cancel__comm__monoid__add__class_Odiff__cancel, axiom,
    ((![A : a]: ((minus_minus_a @ A @ A) = zero_zero_a)))). % cancel_comm_monoid_add_class.diff_cancel
thf(fact_50_cancel__comm__monoid__add__class_Odiff__cancel, axiom,
    ((![A : nat]: ((minus_minus_nat @ A @ A) = zero_zero_nat)))). % cancel_comm_monoid_add_class.diff_cancel
thf(fact_51_cancel__comm__monoid__add__class_Odiff__cancel, axiom,
    ((![A : poly_a]: ((minus_minus_poly_a @ A @ A) = zero_zero_poly_a)))). % cancel_comm_monoid_add_class.diff_cancel
thf(fact_52_dvd__field__iff, axiom,
    ((dvd_dvd_a = (^[A2 : a]: (^[B2 : a]: (((A2 = zero_zero_a)) => ((B2 = zero_zero_a)))))))). % dvd_field_iff
thf(fact_53_eq__iff__diff__eq__0, axiom,
    (((^[Y2 : a]: (^[Z2 : a]: (Y2 = Z2))) = (^[A2 : a]: (^[B2 : a]: ((minus_minus_a @ A2 @ B2) = zero_zero_a)))))). % eq_iff_diff_eq_0
thf(fact_54_eq__iff__diff__eq__0, axiom,
    (((^[Y2 : poly_a]: (^[Z2 : poly_a]: (Y2 = Z2))) = (^[A2 : poly_a]: (^[B2 : poly_a]: ((minus_minus_poly_a @ A2 @ B2) = zero_zero_poly_a)))))). % eq_iff_diff_eq_0
thf(fact_55_smult__0__right, axiom,
    ((![A : a]: ((smult_a @ A @ zero_zero_poly_a) = zero_zero_poly_a)))). % smult_0_right
thf(fact_56_zero__reorient, axiom,
    ((![X : a]: ((zero_zero_a = X) = (X = zero_zero_a))))). % zero_reorient
thf(fact_57_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_58_cancel__ab__semigroup__add__class_Odiff__right__commute, axiom,
    ((![A : poly_a, C : poly_a, B : poly_a]: ((minus_minus_poly_a @ (minus_minus_poly_a @ A @ C) @ B) = (minus_minus_poly_a @ (minus_minus_poly_a @ A @ B) @ C))))). % cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_59_diff__eq__diff__eq, axiom,
    ((![A : poly_a, B : poly_a, C : poly_a, D : poly_a]: (((minus_minus_poly_a @ A @ B) = (minus_minus_poly_a @ C @ D)) => ((A = B) = (C = D)))))). % diff_eq_diff_eq
thf(fact_60_content__primitive__part, axiom,
    ((![P : poly_nat]: ((~ ((P = zero_zero_poly_nat))) => ((content_nat @ (primitive_part_nat @ P)) = one_one_nat))))). % content_primitive_part
thf(fact_61_const__poly__dvd__iff__dvd__content, axiom,
    ((![C : nat, P : poly_nat]: ((dvd_dvd_poly_nat @ (pCons_nat @ C @ zero_zero_poly_nat) @ P) = (dvd_dvd_nat @ C @ (content_nat @ P)))))). % const_poly_dvd_iff_dvd_content
thf(fact_62_normalize__0, axiom,
    (((normal728885956ze_nat @ zero_zero_nat) = zero_zero_nat))). % normalize_0
thf(fact_63_normalize__eq__0__iff, axiom,
    ((![A : nat]: (((normal728885956ze_nat @ A) = zero_zero_nat) = (A = zero_zero_nat))))). % normalize_eq_0_iff
thf(fact_64_normalize__dvd__iff, axiom,
    ((![A : nat, B : nat]: ((dvd_dvd_nat @ (normal728885956ze_nat @ A) @ B) = (dvd_dvd_nat @ A @ B))))). % normalize_dvd_iff
thf(fact_65_dvd__normalize__iff, axiom,
    ((![A : nat, B : nat]: ((dvd_dvd_nat @ A @ (normal728885956ze_nat @ B)) = (dvd_dvd_nat @ A @ B))))). % dvd_normalize_iff
thf(fact_66_smult__1__left, axiom,
    ((![P : poly_a]: ((smult_a @ one_one_a @ P) = P)))). % smult_1_left
thf(fact_67_content__1, axiom,
    (((content_nat @ one_one_poly_nat) = one_one_nat))). % content_1
thf(fact_68_normalize__content, axiom,
    ((![P : poly_nat]: ((normal728885956ze_nat @ (content_nat @ P)) = (content_nat @ P))))). % normalize_content
thf(fact_69_pCons__0__0, axiom,
    (((pCons_a @ zero_zero_a @ zero_zero_poly_a) = zero_zero_poly_a))). % pCons_0_0
thf(fact_70_pCons__0__0, axiom,
    (((pCons_nat @ zero_zero_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % pCons_0_0
thf(fact_71_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_72_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_73_is__unit__content__iff, axiom,
    ((![P : poly_nat]: ((dvd_dvd_nat @ (content_nat @ P) @ one_one_nat) = ((content_nat @ P) = one_one_nat))))). % is_unit_content_iff
thf(fact_74_smult__one, axiom,
    ((![C : a]: ((smult_a @ C @ one_one_poly_a) = (pCons_a @ C @ zero_zero_poly_a))))). % smult_one
thf(fact_75_diff__pCons, axiom,
    ((![A : poly_a, P : poly_poly_a, B : poly_a, Q : poly_poly_a]: ((minus_154650241poly_a @ (pCons_poly_a @ A @ P) @ (pCons_poly_a @ B @ Q)) = (pCons_poly_a @ (minus_minus_poly_a @ A @ B) @ (minus_154650241poly_a @ P @ Q)))))). % diff_pCons
thf(fact_76_diff__pCons, axiom,
    ((![A : a, P : poly_a, B : a, Q : poly_a]: ((minus_minus_poly_a @ (pCons_a @ A @ P) @ (pCons_a @ B @ Q)) = (pCons_a @ (minus_minus_a @ A @ B) @ (minus_minus_poly_a @ P @ Q)))))). % diff_pCons
thf(fact_77_const__poly__dvd__const__poly__iff, axiom,
    ((![A : poly_a, B : poly_a]: ((dvd_dvd_poly_poly_a @ (pCons_poly_a @ A @ zero_z2096148049poly_a) @ (pCons_poly_a @ B @ zero_z2096148049poly_a)) = (dvd_dvd_poly_a @ A @ B))))). % const_poly_dvd_const_poly_iff
thf(fact_78_const__poly__dvd__const__poly__iff, axiom,
    ((![A : poly_nat, B : poly_nat]: ((dvd_dv944831366ly_nat @ (pCons_poly_nat @ A @ zero_z1059985641ly_nat) @ (pCons_poly_nat @ B @ zero_z1059985641ly_nat)) = (dvd_dvd_poly_nat @ A @ B))))). % const_poly_dvd_const_poly_iff
thf(fact_79_const__poly__dvd__const__poly__iff, axiom,
    ((![A : a, B : a]: ((dvd_dvd_poly_a @ (pCons_a @ A @ zero_zero_poly_a) @ (pCons_a @ B @ zero_zero_poly_a)) = (dvd_dvd_a @ A @ B))))). % const_poly_dvd_const_poly_iff
thf(fact_80_const__poly__dvd__const__poly__iff, axiom,
    ((![A : nat, B : nat]: ((dvd_dvd_poly_nat @ (pCons_nat @ A @ zero_zero_poly_nat) @ (pCons_nat @ B @ zero_zero_poly_nat)) = (dvd_dvd_nat @ A @ B))))). % const_poly_dvd_const_poly_iff
thf(fact_81_content__const, axiom,
    ((![C : nat]: ((content_nat @ (pCons_nat @ C @ zero_zero_poly_nat)) = (normal728885956ze_nat @ C))))). % content_const
thf(fact_82_normalize__idem__imp__is__unit__iff, axiom,
    ((![A : nat]: (((normal728885956ze_nat @ A) = A) => ((dvd_dvd_nat @ A @ one_one_nat) = (A = one_one_nat)))))). % normalize_idem_imp_is_unit_iff
thf(fact_83_is__unit__normalize, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ A @ one_one_nat) => ((normal728885956ze_nat @ A) = one_one_nat))))). % is_unit_normalize
thf(fact_84_normalize__1__iff, axiom,
    ((![A : nat]: (((normal728885956ze_nat @ A) = one_one_nat) = (dvd_dvd_nat @ A @ one_one_nat))))). % normalize_1_iff
thf(fact_85_associated__unit, axiom,
    ((![A : nat, B : nat]: (((normal728885956ze_nat @ A) = (normal728885956ze_nat @ B)) => ((dvd_dvd_nat @ A @ one_one_nat) => (dvd_dvd_nat @ B @ one_one_nat)))))). % associated_unit
thf(fact_86_is__unit__const__poly__iff, axiom,
    ((![C : poly_a]: ((dvd_dvd_poly_poly_a @ (pCons_poly_a @ C @ zero_z2096148049poly_a) @ one_one_poly_poly_a) = (dvd_dvd_poly_a @ C @ one_one_poly_a))))). % is_unit_const_poly_iff
thf(fact_87_is__unit__const__poly__iff, axiom,
    ((![C : poly_nat]: ((dvd_dv944831366ly_nat @ (pCons_poly_nat @ C @ zero_z1059985641ly_nat) @ one_on1411366565ly_nat) = (dvd_dvd_poly_nat @ C @ one_one_poly_nat))))). % is_unit_const_poly_iff
thf(fact_88_is__unit__const__poly__iff, axiom,
    ((![C : a]: ((dvd_dvd_poly_a @ (pCons_a @ C @ zero_zero_poly_a) @ one_one_poly_a) = (dvd_dvd_a @ C @ one_one_a))))). % is_unit_const_poly_iff
thf(fact_89_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_90_is__unit__poly__iff, axiom,
    ((![P : poly_poly_a]: ((dvd_dvd_poly_poly_a @ P @ one_one_poly_poly_a) = (?[C2 : poly_a]: (((P = (pCons_poly_a @ C2 @ zero_z2096148049poly_a))) & ((dvd_dvd_poly_a @ C2 @ one_one_poly_a)))))))). % is_unit_poly_iff
thf(fact_91_is__unit__poly__iff, axiom,
    ((![P : poly_poly_nat]: ((dvd_dv944831366ly_nat @ P @ one_on1411366565ly_nat) = (?[C2 : poly_nat]: (((P = (pCons_poly_nat @ C2 @ zero_z1059985641ly_nat))) & ((dvd_dvd_poly_nat @ C2 @ one_one_poly_nat)))))))). % is_unit_poly_iff
thf(fact_92_is__unit__poly__iff, axiom,
    ((![P : poly_a]: ((dvd_dvd_poly_a @ P @ one_one_poly_a) = (?[C2 : a]: (((P = (pCons_a @ C2 @ zero_zero_poly_a))) & ((dvd_dvd_a @ C2 @ one_one_a)))))))). % is_unit_poly_iff
thf(fact_93_is__unit__poly__iff, axiom,
    ((![P : poly_nat]: ((dvd_dvd_poly_nat @ P @ one_one_poly_nat) = (?[C2 : nat]: (((P = (pCons_nat @ C2 @ zero_zero_poly_nat))) & ((dvd_dvd_nat @ C2 @ one_one_nat)))))))). % is_unit_poly_iff
thf(fact_94_is__unit__polyE, axiom,
    ((![P : poly_poly_a]: ((dvd_dvd_poly_poly_a @ P @ one_one_poly_poly_a) => (~ ((![C3 : poly_a]: ((P = (pCons_poly_a @ C3 @ zero_z2096148049poly_a)) => (~ ((dvd_dvd_poly_a @ C3 @ one_one_poly_a))))))))))). % is_unit_polyE
thf(fact_95_is__unit__polyE, axiom,
    ((![P : poly_poly_nat]: ((dvd_dv944831366ly_nat @ P @ one_on1411366565ly_nat) => (~ ((![C3 : poly_nat]: ((P = (pCons_poly_nat @ C3 @ zero_z1059985641ly_nat)) => (~ ((dvd_dvd_poly_nat @ C3 @ one_one_poly_nat))))))))))). % is_unit_polyE
thf(fact_96_is__unit__polyE, axiom,
    ((![P : poly_a]: ((dvd_dvd_poly_a @ P @ one_one_poly_a) => (~ ((![C3 : a]: ((P = (pCons_a @ C3 @ zero_zero_poly_a)) => (~ ((dvd_dvd_a @ C3 @ one_one_a))))))))))). % is_unit_polyE
thf(fact_97_is__unit__polyE, axiom,
    ((![P : poly_nat]: ((dvd_dvd_poly_nat @ P @ one_one_poly_nat) => (~ ((![C3 : nat]: ((P = (pCons_nat @ C3 @ zero_zero_poly_nat)) => (~ ((dvd_dvd_nat @ C3 @ one_one_nat))))))))))). % is_unit_polyE
thf(fact_98_associated__iff__dvd, axiom,
    ((![A : nat, B : nat]: (((normal728885956ze_nat @ A) = (normal728885956ze_nat @ B)) = (((dvd_dvd_nat @ A @ B)) & ((dvd_dvd_nat @ B @ A))))))). % associated_iff_dvd
thf(fact_99_associated__eqI, axiom,
    ((![A : nat, B : nat]: ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ B @ A) => (((normal728885956ze_nat @ A) = A) => (((normal728885956ze_nat @ B) = B) => (A = B)))))))). % associated_eqI
thf(fact_100_associatedD2, axiom,
    ((![A : nat, B : nat]: (((normal728885956ze_nat @ A) = (normal728885956ze_nat @ B)) => (dvd_dvd_nat @ B @ A))))). % associatedD2
thf(fact_101_associatedD1, axiom,
    ((![A : nat, B : nat]: (((normal728885956ze_nat @ A) = (normal728885956ze_nat @ B)) => (dvd_dvd_nat @ A @ B))))). % associatedD1
thf(fact_102_associatedI, axiom,
    ((![A : nat, B : nat]: ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ B @ A) => ((normal728885956ze_nat @ A) = (normal728885956ze_nat @ B))))))). % associatedI
thf(fact_103_zero__neq__one, axiom,
    ((~ ((zero_zero_a = one_one_a))))). % zero_neq_one
thf(fact_104_zero__neq__one, axiom,
    ((~ ((zero_zero_nat = one_one_nat))))). % zero_neq_one
thf(fact_105_dvd__unit__imp__unit, axiom,
    ((![A : poly_a, B : poly_a]: ((dvd_dvd_poly_a @ A @ B) => ((dvd_dvd_poly_a @ B @ one_one_poly_a) => (dvd_dvd_poly_a @ A @ one_one_poly_a)))))). % dvd_unit_imp_unit
thf(fact_106_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_107_unit__imp__dvd, axiom,
    ((![B : poly_a, A : poly_a]: ((dvd_dvd_poly_a @ B @ one_one_poly_a) => (dvd_dvd_poly_a @ B @ A))))). % unit_imp_dvd
thf(fact_108_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_109_one__dvd, axiom,
    ((![A : poly_a]: (dvd_dvd_poly_a @ one_one_poly_a @ A)))). % one_dvd
thf(fact_110_one__dvd, axiom,
    ((![A : nat]: (dvd_dvd_nat @ one_one_nat @ A)))). % one_dvd
thf(fact_111_one__dvd, axiom,
    ((![A : poly_nat]: (dvd_dvd_poly_nat @ one_one_poly_nat @ A)))). % one_dvd
thf(fact_112_is__unit__triv, axiom,
    ((![A : a]: ((~ ((A = zero_zero_a))) => (dvd_dvd_poly_a @ (pCons_a @ A @ zero_zero_poly_a) @ one_one_poly_a))))). % is_unit_triv
thf(fact_113_is__unit__pCons__iff, axiom,
    ((![A : a, P : poly_a]: ((dvd_dvd_poly_a @ (pCons_a @ A @ P) @ one_one_poly_a) = (((P = zero_zero_poly_a)) & ((~ ((A = zero_zero_a))))))))). % is_unit_pCons_iff
thf(fact_114_is__unit__smult__iff, axiom,
    ((![C : poly_a, P : poly_poly_a]: ((dvd_dvd_poly_poly_a @ (smult_poly_a @ C @ P) @ one_one_poly_poly_a) = (((dvd_dvd_poly_a @ C @ one_one_poly_a)) & ((dvd_dvd_poly_poly_a @ P @ one_one_poly_poly_a))))))). % is_unit_smult_iff
thf(fact_115_is__unit__smult__iff, axiom,
    ((![C : poly_nat, P : poly_poly_nat]: ((dvd_dv944831366ly_nat @ (smult_poly_nat @ C @ P) @ one_on1411366565ly_nat) = (((dvd_dvd_poly_nat @ C @ one_one_poly_nat)) & ((dvd_dv944831366ly_nat @ P @ one_on1411366565ly_nat))))))). % is_unit_smult_iff
thf(fact_116_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_117_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_118_pCons__induct, axiom,
    ((![P2 : poly_a > $o, P : poly_a]: ((P2 @ zero_zero_poly_a) => ((![A3 : a, P3 : poly_a]: (((~ ((A3 = zero_zero_a))) | (~ ((P3 = zero_zero_poly_a)))) => ((P2 @ P3) => (P2 @ (pCons_a @ A3 @ P3))))) => (P2 @ P)))))). % pCons_induct
thf(fact_119_pCons__induct, axiom,
    ((![P2 : poly_nat > $o, P : poly_nat]: ((P2 @ zero_zero_poly_nat) => ((![A3 : nat, P3 : poly_nat]: (((~ ((A3 = zero_zero_nat))) | (~ ((P3 = zero_zero_poly_nat)))) => ((P2 @ P3) => (P2 @ (pCons_nat @ A3 @ P3))))) => (P2 @ P)))))). % pCons_induct
thf(fact_120_poly__divides__pad__rule, axiom,
    ((![P : poly_a, Q : poly_a]: ((dvd_dvd_poly_a @ P @ Q) => (dvd_dvd_poly_a @ P @ (pCons_a @ zero_zero_a @ Q)))))). % poly_divides_pad_rule
thf(fact_121_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_122_not__is__unit__0, axiom,
    ((~ ((dvd_dvd_poly_a @ zero_zero_poly_a @ one_one_poly_a))))). % not_is_unit_0
thf(fact_123_not__is__unit__0, axiom,
    ((~ ((dvd_dvd_nat @ zero_zero_nat @ one_one_nat))))). % not_is_unit_0
thf(fact_124_content__decompose, axiom,
    ((![P : poly_nat]: (~ ((![P4 : poly_nat]: ((P = (smult_nat @ (content_nat @ P) @ P4)) => (~ (((content_nat @ P4) = one_one_nat)))))))))). % content_decompose
thf(fact_125_primitive__part__prim, axiom,
    ((![P : poly_nat]: (((content_nat @ P) = one_one_nat) => ((primitive_part_nat @ P) = P))))). % primitive_part_prim
thf(fact_126_content__dvd, axiom,
    ((![P : poly_nat]: (dvd_dvd_poly_nat @ (pCons_nat @ (content_nat @ P) @ zero_zero_poly_nat) @ P)))). % content_dvd
thf(fact_127_diff__numeral__special_I9_J, axiom,
    (((minus_minus_a @ one_one_a @ one_one_a) = zero_zero_a))). % diff_numeral_special(9)
thf(fact_128_diff__numeral__special_I9_J, axiom,
    (((minus_minus_poly_a @ one_one_poly_a @ one_one_poly_a) = zero_zero_poly_a))). % diff_numeral_special(9)
thf(fact_129_lcm_Onormalize__bottom, axiom,
    (((normal728885956ze_nat @ zero_zero_nat) = zero_zero_nat))). % lcm.normalize_bottom
thf(fact_130_pcompose__idR, axiom,
    ((![P : poly_a]: ((pcompose_a @ P @ (pCons_a @ zero_zero_a @ (pCons_a @ one_one_a @ zero_zero_poly_a))) = P)))). % pcompose_idR
thf(fact_131_pcompose__idR, axiom,
    ((![P : poly_nat]: ((pcompose_nat @ P @ (pCons_nat @ zero_zero_nat @ (pCons_nat @ one_one_nat @ zero_zero_poly_nat))) = P)))). % pcompose_idR
thf(fact_132_gcd__nat_Oextremum__uniqueI, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) => (A = zero_zero_nat))))). % gcd_nat.extremum_uniqueI
thf(fact_133_gcd__nat_Onot__eq__extremum, axiom,
    ((![A : nat]: ((~ ((A = zero_zero_nat))) = (((dvd_dvd_nat @ A @ zero_zero_nat)) & ((~ ((A = zero_zero_nat))))))))). % gcd_nat.not_eq_extremum
thf(fact_134_gcd__nat_Oextremum__unique, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) = (A = zero_zero_nat))))). % gcd_nat.extremum_unique
thf(fact_135_gcd__nat_Oextremum__strict, axiom,
    ((![A : nat]: (~ (((dvd_dvd_nat @ zero_zero_nat @ A) & (~ ((zero_zero_nat = A))))))))). % gcd_nat.extremum_strict
thf(fact_136_gcd__nat_Oextremum, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ zero_zero_nat)))). % gcd_nat.extremum
thf(fact_137_pcompose__smult, axiom,
    ((![A : a, P : poly_a, R : poly_a]: ((pcompose_a @ (smult_a @ A @ P) @ R) = (smult_a @ A @ (pcompose_a @ P @ R)))))). % pcompose_smult
thf(fact_138_pcompose__diff, axiom,
    ((![P : poly_a, Q : poly_a, R : poly_a]: ((pcompose_a @ (minus_minus_poly_a @ P @ Q) @ R) = (minus_minus_poly_a @ (pcompose_a @ P @ R) @ (pcompose_a @ Q @ R)))))). % pcompose_diff
thf(fact_139_gcd__nat_Oasym, axiom,
    ((![A : nat, B : nat]: (((dvd_dvd_nat @ A @ B) & (~ ((A = B)))) => (~ (((dvd_dvd_nat @ B @ A) & (~ ((B = A)))))))))). % gcd_nat.asym
thf(fact_140_gcd__nat_Orefl, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ A)))). % gcd_nat.refl
thf(fact_141_gcd__nat_Otrans, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ B @ C) => (dvd_dvd_nat @ A @ C)))))). % gcd_nat.trans
thf(fact_142_gcd__nat_Oeq__iff, axiom,
    (((^[Y2 : nat]: (^[Z2 : nat]: (Y2 = Z2))) = (^[A2 : nat]: (^[B2 : nat]: (((dvd_dvd_nat @ A2 @ B2)) & ((dvd_dvd_nat @ B2 @ A2)))))))). % gcd_nat.eq_iff
thf(fact_143_gcd__nat_Oirrefl, axiom,
    ((![A : nat]: (~ (((dvd_dvd_nat @ A @ A) & (~ ((A = A))))))))). % gcd_nat.irrefl
thf(fact_144_gcd__nat_Oantisym, axiom,
    ((![A : nat, B : nat]: ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ B @ A) => (A = B)))))). % gcd_nat.antisym
thf(fact_145_gcd__nat_Ostrict__trans, axiom,
    ((![A : nat, B : nat, C : nat]: (((dvd_dvd_nat @ A @ B) & (~ ((A = B)))) => (((dvd_dvd_nat @ B @ C) & (~ ((B = C)))) => ((dvd_dvd_nat @ A @ C) & (~ ((A = C))))))))). % gcd_nat.strict_trans
thf(fact_146_gcd__nat_Ostrict__trans1, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ A @ B) => (((dvd_dvd_nat @ B @ C) & (~ ((B = C)))) => ((dvd_dvd_nat @ A @ C) & (~ ((A = C))))))))). % gcd_nat.strict_trans1
thf(fact_147_gcd__nat_Ostrict__trans2, axiom,
    ((![A : nat, B : nat, C : nat]: (((dvd_dvd_nat @ A @ B) & (~ ((A = B)))) => ((dvd_dvd_nat @ B @ C) => ((dvd_dvd_nat @ A @ C) & (~ ((A = C))))))))). % gcd_nat.strict_trans2
thf(fact_148_gcd__nat_Oorder__iff__strict, axiom,
    ((dvd_dvd_nat = (^[A2 : nat]: (^[B2 : nat]: (((((dvd_dvd_nat @ A2 @ B2)) & ((~ ((A2 = B2)))))) | ((A2 = B2)))))))). % gcd_nat.order_iff_strict
thf(fact_149_gcd__nat_Ostrict__iff__order, axiom,
    ((![A : nat, B : nat]: ((((dvd_dvd_nat @ A @ B)) & ((~ ((A = B))))) = (((dvd_dvd_nat @ A @ B)) & ((~ ((A = B))))))))). % gcd_nat.strict_iff_order
thf(fact_150_gcd__nat_Ostrict__implies__order, axiom,
    ((![A : nat, B : nat]: (((dvd_dvd_nat @ A @ B) & (~ ((A = B)))) => (dvd_dvd_nat @ A @ B))))). % gcd_nat.strict_implies_order
thf(fact_151_gcd__nat_Ostrict__implies__not__eq, axiom,
    ((![A : nat, B : nat]: (((dvd_dvd_nat @ A @ B) & (~ ((A = B)))) => (~ ((A = B))))))). % gcd_nat.strict_implies_not_eq

% Conjectures (1)
thf(conj_0, conjecture,
    (((dvd_dvd_poly_a @ p @ q) = (dvd_dvd_poly_a @ p @ r)))).
