% 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_1126__5378478_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:36:02.664

% Could-be-implicit typings (2)
thf(ty_n_t__Polynomial__Opoly_Itf__a_J, type,
    poly_a : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (9)
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_Itf__a_J, type,
    zero_zero_poly_a : poly_a).
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_001tf__a, type,
    degree_a : poly_a > nat).
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_Itf__a_J, type,
    dvd_dvd_poly_a : poly_a > poly_a > $o).
thf(sy_v_p, type,
    p : poly_a).
thf(sy_v_q, type,
    q : poly_a).

% Relevant facts (133)
thf(fact_0_q0, axiom,
    ((~ ((q = zero_zero_poly_a))))). % q0
thf(fact_1_lq, axiom,
    ((~ ((p = zero_zero_poly_a))))). % lq
thf(fact_2_l, axiom,
    ((dvd_dvd_poly_a @ p @ q))). % l
thf(fact_3_lgpq, axiom,
    ((ord_less_nat @ (degree_a @ q) @ (degree_a @ p)))). % lgpq
thf(fact_4_order__refl, axiom,
    ((![X : nat]: (ord_less_eq_nat @ X @ X)))). % order_refl
thf(fact_5_le__refl, axiom,
    ((![N : nat]: (ord_less_eq_nat @ N @ N)))). % le_refl
thf(fact_6_le__trans, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_eq_nat @ I @ J) => ((ord_less_eq_nat @ J @ K) => (ord_less_eq_nat @ I @ K)))))). % le_trans
thf(fact_7_eq__imp__le, axiom,
    ((![M : nat, N : nat]: ((M = N) => (ord_less_eq_nat @ M @ N))))). % eq_imp_le
thf(fact_8_le__antisym, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ N) => ((ord_less_eq_nat @ N @ M) => (M = N)))))). % le_antisym
thf(fact_9_nat__le__linear, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ N) | (ord_less_eq_nat @ N @ M))))). % nat_le_linear
thf(fact_10_Nat_Oex__has__greatest__nat, axiom,
    ((![P : nat > $o, K : nat, B : nat]: ((P @ K) => ((![Y : nat]: ((P @ Y) => (ord_less_eq_nat @ Y @ B))) => (?[X2 : nat]: ((P @ X2) & (![Y2 : nat]: ((P @ Y2) => (ord_less_eq_nat @ Y2 @ X2)))))))))). % Nat.ex_has_greatest_nat
thf(fact_11_bounded__Max__nat, axiom,
    ((![P : nat > $o, X : nat, M2 : nat]: ((P @ X) => ((![X2 : nat]: ((P @ X2) => (ord_less_eq_nat @ X2 @ M2))) => (~ ((![M3 : nat]: ((P @ M3) => (~ ((![X3 : nat]: ((P @ X3) => (ord_less_eq_nat @ X3 @ M3)))))))))))))). % bounded_Max_nat
thf(fact_12_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_13_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_14_less__not__refl2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ N @ M) => (~ ((M = N))))))). % less_not_refl2
thf(fact_15_less__not__refl3, axiom,
    ((![S : nat, T : nat]: ((ord_less_nat @ S @ T) => (~ ((S = T))))))). % less_not_refl3
thf(fact_16_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_17_nat__less__induct, axiom,
    ((![P : nat > $o, N : nat]: ((![N2 : nat]: ((![M4 : nat]: ((ord_less_nat @ M4 @ N2) => (P @ M4))) => (P @ N2))) => (P @ N))))). % nat_less_induct
thf(fact_18_infinite__descent, axiom,
    ((![P : nat > $o, N : nat]: ((![N2 : nat]: ((~ ((P @ N2))) => (?[M4 : nat]: ((ord_less_nat @ M4 @ N2) & (~ ((P @ M4))))))) => (P @ N))))). % infinite_descent
thf(fact_19_linorder__neqE__nat, axiom,
    ((![X : nat, Y3 : nat]: ((~ ((X = Y3))) => ((~ ((ord_less_nat @ X @ Y3))) => (ord_less_nat @ Y3 @ X)))))). % linorder_neqE_nat
thf(fact_20_ord__eq__less__subst, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_nat @ B @ C) => ((![X2 : nat, Y : nat]: ((ord_less_nat @ X2 @ Y) => (ord_less_nat @ (F @ X2) @ (F @ Y)))) => (ord_less_nat @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_21_ord__less__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_nat @ A @ B) => (((F @ B) = C) => ((![X2 : nat, Y : nat]: ((ord_less_nat @ X2 @ Y) => (ord_less_nat @ (F @ X2) @ (F @ Y)))) => (ord_less_nat @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_22_order__less__subst1, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((ord_less_nat @ A @ (F @ B)) => ((ord_less_nat @ B @ C) => ((![X2 : nat, Y : nat]: ((ord_less_nat @ X2 @ Y) => (ord_less_nat @ (F @ X2) @ (F @ Y)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_23_order__less__subst2, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ (F @ B) @ C) => ((![X2 : nat, Y : nat]: ((ord_less_nat @ X2 @ Y) => (ord_less_nat @ (F @ X2) @ (F @ Y)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_24_gt__ex, axiom,
    ((![X : nat]: (?[X_1 : nat]: (ord_less_nat @ X @ X_1))))). % gt_ex
thf(fact_25_neqE, axiom,
    ((![X : nat, Y3 : nat]: ((~ ((X = Y3))) => ((~ ((ord_less_nat @ X @ Y3))) => (ord_less_nat @ Y3 @ X)))))). % neqE
thf(fact_26_neq__iff, axiom,
    ((![X : nat, Y3 : nat]: ((~ ((X = Y3))) = (((ord_less_nat @ X @ Y3)) | ((ord_less_nat @ Y3 @ X))))))). % neq_iff
thf(fact_27_order_Oasym, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % order.asym
thf(fact_28_less__imp__neq, axiom,
    ((![X : nat, Y3 : nat]: ((ord_less_nat @ X @ Y3) => (~ ((X = Y3))))))). % less_imp_neq
thf(fact_29_less__asym, axiom,
    ((![X : nat, Y3 : nat]: ((ord_less_nat @ X @ Y3) => (~ ((ord_less_nat @ Y3 @ X))))))). % less_asym
thf(fact_30_less__asym_H, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % less_asym'
thf(fact_31_less__trans, axiom,
    ((![X : nat, Y3 : nat, Z : nat]: ((ord_less_nat @ X @ Y3) => ((ord_less_nat @ Y3 @ Z) => (ord_less_nat @ X @ Z)))))). % less_trans
thf(fact_32_less__linear, axiom,
    ((![X : nat, Y3 : nat]: ((ord_less_nat @ X @ Y3) | ((X = Y3) | (ord_less_nat @ Y3 @ X)))))). % less_linear
thf(fact_33_less__irrefl, axiom,
    ((![X : nat]: (~ ((ord_less_nat @ X @ X)))))). % less_irrefl
thf(fact_34_ord__eq__less__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((A = B) => ((ord_less_nat @ B @ C) => (ord_less_nat @ A @ C)))))). % ord_eq_less_trans
thf(fact_35_ord__less__eq__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((B = C) => (ord_less_nat @ A @ C)))))). % ord_less_eq_trans
thf(fact_36_dual__order_Oasym, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ B @ A) => (~ ((ord_less_nat @ A @ B))))))). % dual_order.asym
thf(fact_37_less__imp__not__eq, axiom,
    ((![X : nat, Y3 : nat]: ((ord_less_nat @ X @ Y3) => (~ ((X = Y3))))))). % less_imp_not_eq
thf(fact_38_less__not__sym, axiom,
    ((![X : nat, Y3 : nat]: ((ord_less_nat @ X @ Y3) => (~ ((ord_less_nat @ Y3 @ X))))))). % less_not_sym
thf(fact_39_less__induct, axiom,
    ((![P : nat > $o, A : nat]: ((![X2 : nat]: ((![Y2 : nat]: ((ord_less_nat @ Y2 @ X2) => (P @ Y2))) => (P @ X2))) => (P @ A))))). % less_induct
thf(fact_40_antisym__conv3, axiom,
    ((![Y3 : nat, X : nat]: ((~ ((ord_less_nat @ Y3 @ X))) => ((~ ((ord_less_nat @ X @ Y3))) = (X = Y3)))))). % antisym_conv3
thf(fact_41_less__imp__not__eq2, axiom,
    ((![X : nat, Y3 : nat]: ((ord_less_nat @ X @ Y3) => (~ ((Y3 = X))))))). % less_imp_not_eq2
thf(fact_42_less__imp__triv, axiom,
    ((![X : nat, Y3 : nat, P : $o]: ((ord_less_nat @ X @ Y3) => ((ord_less_nat @ Y3 @ X) => P))))). % less_imp_triv
thf(fact_43_linorder__cases, axiom,
    ((![X : nat, Y3 : nat]: ((~ ((ord_less_nat @ X @ Y3))) => ((~ ((X = Y3))) => (ord_less_nat @ Y3 @ X)))))). % linorder_cases
thf(fact_44_dual__order_Oirrefl, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ A)))))). % dual_order.irrefl
thf(fact_45_order_Ostrict__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ B @ C) => (ord_less_nat @ A @ C)))))). % order.strict_trans
thf(fact_46_less__imp__not__less, axiom,
    ((![X : nat, Y3 : nat]: ((ord_less_nat @ X @ Y3) => (~ ((ord_less_nat @ Y3 @ X))))))). % less_imp_not_less
thf(fact_47_exists__least__iff, axiom,
    (((^[P2 : nat > $o]: (?[X4 : nat]: (P2 @ X4))) = (^[P3 : nat > $o]: (?[N3 : nat]: (((P3 @ N3)) & ((![M5 : nat]: (((ord_less_nat @ M5 @ N3)) => ((~ ((P3 @ M5))))))))))))). % exists_least_iff
thf(fact_48_linorder__less__wlog, axiom,
    ((![P : nat > nat > $o, A : nat, B : nat]: ((![A2 : nat, B2 : nat]: ((ord_less_nat @ A2 @ B2) => (P @ A2 @ B2))) => ((![A2 : nat]: (P @ A2 @ A2)) => ((![A2 : nat, B2 : nat]: ((P @ B2 @ A2) => (P @ A2 @ B2))) => (P @ A @ B))))))). % linorder_less_wlog
thf(fact_49_dual__order_Ostrict__trans, axiom,
    ((![B : nat, A : nat, C : nat]: ((ord_less_nat @ B @ A) => ((ord_less_nat @ C @ B) => (ord_less_nat @ C @ A)))))). % dual_order.strict_trans
thf(fact_50_not__less__iff__gr__or__eq, axiom,
    ((![X : nat, Y3 : nat]: ((~ ((ord_less_nat @ X @ Y3))) = (((ord_less_nat @ Y3 @ X)) | ((X = Y3))))))). % not_less_iff_gr_or_eq
thf(fact_51_order_Ostrict__implies__not__eq, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_52_dual__order_Ostrict__implies__not__eq, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ B @ A) => (~ ((A = B))))))). % dual_order.strict_implies_not_eq
thf(fact_53_order_Onot__eq__order__implies__strict, axiom,
    ((![A : nat, B : nat]: ((~ ((A = B))) => ((ord_less_eq_nat @ A @ B) => (ord_less_nat @ A @ B)))))). % order.not_eq_order_implies_strict
thf(fact_54_dual__order_Ostrict__implies__order, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ B @ A) => (ord_less_eq_nat @ B @ A))))). % dual_order.strict_implies_order
thf(fact_55_dual__order_Ostrict__iff__order, axiom,
    ((ord_less_nat = (^[B3 : nat]: (^[A3 : nat]: (((ord_less_eq_nat @ B3 @ A3)) & ((~ ((A3 = B3)))))))))). % dual_order.strict_iff_order
thf(fact_56_dual__order_Oorder__iff__strict, axiom,
    ((ord_less_eq_nat = (^[B3 : nat]: (^[A3 : nat]: (((ord_less_nat @ B3 @ A3)) | ((A3 = B3)))))))). % dual_order.order_iff_strict
thf(fact_57_order_Ostrict__implies__order, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (ord_less_eq_nat @ A @ B))))). % order.strict_implies_order
thf(fact_58_dual__order_Ostrict__trans2, axiom,
    ((![B : nat, A : nat, C : nat]: ((ord_less_nat @ B @ A) => ((ord_less_eq_nat @ C @ B) => (ord_less_nat @ C @ A)))))). % dual_order.strict_trans2
thf(fact_59_dual__order_Ostrict__trans1, axiom,
    ((![B : nat, A : nat, C : nat]: ((ord_less_eq_nat @ B @ A) => ((ord_less_nat @ C @ B) => (ord_less_nat @ C @ A)))))). % dual_order.strict_trans1
thf(fact_60_order_Ostrict__iff__order, axiom,
    ((ord_less_nat = (^[A3 : nat]: (^[B3 : nat]: (((ord_less_eq_nat @ A3 @ B3)) & ((~ ((A3 = B3)))))))))). % order.strict_iff_order
thf(fact_61_order_Oorder__iff__strict, axiom,
    ((ord_less_eq_nat = (^[A3 : nat]: (^[B3 : nat]: (((ord_less_nat @ A3 @ B3)) | ((A3 = B3)))))))). % order.order_iff_strict
thf(fact_62_order_Ostrict__trans2, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_eq_nat @ B @ C) => (ord_less_nat @ A @ C)))))). % order.strict_trans2
thf(fact_63_order_Ostrict__trans1, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_nat @ B @ C) => (ord_less_nat @ A @ C)))))). % order.strict_trans1
thf(fact_64_not__le__imp__less, axiom,
    ((![Y3 : nat, X : nat]: ((~ ((ord_less_eq_nat @ Y3 @ X))) => (ord_less_nat @ X @ Y3))))). % not_le_imp_less
thf(fact_65_less__le__not__le, axiom,
    ((ord_less_nat = (^[X5 : nat]: (^[Y4 : nat]: (((ord_less_eq_nat @ X5 @ Y4)) & ((~ ((ord_less_eq_nat @ Y4 @ X5)))))))))). % less_le_not_le
thf(fact_66_le__imp__less__or__eq, axiom,
    ((![X : nat, Y3 : nat]: ((ord_less_eq_nat @ X @ Y3) => ((ord_less_nat @ X @ Y3) | (X = Y3)))))). % le_imp_less_or_eq
thf(fact_67_le__less__linear, axiom,
    ((![X : nat, Y3 : nat]: ((ord_less_eq_nat @ X @ Y3) | (ord_less_nat @ Y3 @ X))))). % le_less_linear
thf(fact_68_less__le__trans, axiom,
    ((![X : nat, Y3 : nat, Z : nat]: ((ord_less_nat @ X @ Y3) => ((ord_less_eq_nat @ Y3 @ Z) => (ord_less_nat @ X @ Z)))))). % less_le_trans
thf(fact_69_le__less__trans, axiom,
    ((![X : nat, Y3 : nat, Z : nat]: ((ord_less_eq_nat @ X @ Y3) => ((ord_less_nat @ Y3 @ Z) => (ord_less_nat @ X @ Z)))))). % le_less_trans
thf(fact_70_less__imp__le, axiom,
    ((![X : nat, Y3 : nat]: ((ord_less_nat @ X @ Y3) => (ord_less_eq_nat @ X @ Y3))))). % less_imp_le
thf(fact_71_antisym__conv2, axiom,
    ((![X : nat, Y3 : nat]: ((ord_less_eq_nat @ X @ Y3) => ((~ ((ord_less_nat @ X @ Y3))) = (X = Y3)))))). % antisym_conv2
thf(fact_72_antisym__conv1, axiom,
    ((![X : nat, Y3 : nat]: ((~ ((ord_less_nat @ X @ Y3))) => ((ord_less_eq_nat @ X @ Y3) = (X = Y3)))))). % antisym_conv1
thf(fact_73_le__neq__trans, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ A @ B) => ((~ ((A = B))) => (ord_less_nat @ A @ B)))))). % le_neq_trans
thf(fact_74_not__less, axiom,
    ((![X : nat, Y3 : nat]: ((~ ((ord_less_nat @ X @ Y3))) = (ord_less_eq_nat @ Y3 @ X))))). % not_less
thf(fact_75_not__le, axiom,
    ((![X : nat, Y3 : nat]: ((~ ((ord_less_eq_nat @ X @ Y3))) = (ord_less_nat @ Y3 @ X))))). % not_le
thf(fact_76_order__less__le__subst2, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_eq_nat @ (F @ B) @ C) => ((![X2 : nat, Y : nat]: ((ord_less_nat @ X2 @ Y) => (ord_less_nat @ (F @ X2) @ (F @ Y)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_less_le_subst2
thf(fact_77_order__less__le__subst1, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((ord_less_nat @ A @ (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X2 : nat, Y : nat]: ((ord_less_eq_nat @ X2 @ Y) => (ord_less_eq_nat @ (F @ X2) @ (F @ Y)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_less_le_subst1
thf(fact_78_order__le__less__subst2, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_nat @ (F @ B) @ C) => ((![X2 : nat, Y : nat]: ((ord_less_eq_nat @ X2 @ Y) => (ord_less_eq_nat @ (F @ X2) @ (F @ Y)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_le_less_subst2
thf(fact_79_order__le__less__subst1, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ (F @ B)) => ((ord_less_nat @ B @ C) => ((![X2 : nat, Y : nat]: ((ord_less_nat @ X2 @ Y) => (ord_less_nat @ (F @ X2) @ (F @ Y)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_le_less_subst1
thf(fact_80_less__le, axiom,
    ((ord_less_nat = (^[X5 : nat]: (^[Y4 : nat]: (((ord_less_eq_nat @ X5 @ Y4)) & ((~ ((X5 = Y4)))))))))). % less_le
thf(fact_81_le__less, axiom,
    ((ord_less_eq_nat = (^[X5 : nat]: (^[Y4 : nat]: (((ord_less_nat @ X5 @ Y4)) | ((X5 = Y4)))))))). % le_less
thf(fact_82_leI, axiom,
    ((![X : nat, Y3 : nat]: ((~ ((ord_less_nat @ X @ Y3))) => (ord_less_eq_nat @ Y3 @ X))))). % leI
thf(fact_83_leD, axiom,
    ((![Y3 : nat, X : nat]: ((ord_less_eq_nat @ Y3 @ X) => (~ ((ord_less_nat @ X @ Y3))))))). % leD
thf(fact_84_less__mono__imp__le__mono, axiom,
    ((![F : nat > nat, I : nat, J : nat]: ((![I2 : nat, J2 : nat]: ((ord_less_nat @ I2 @ J2) => (ord_less_nat @ (F @ I2) @ (F @ J2)))) => ((ord_less_eq_nat @ I @ J) => (ord_less_eq_nat @ (F @ I) @ (F @ J))))))). % less_mono_imp_le_mono
thf(fact_85_le__neq__implies__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ N) => ((~ ((M = N))) => (ord_less_nat @ M @ N)))))). % le_neq_implies_less
thf(fact_86_less__or__eq__imp__le, axiom,
    ((![M : nat, N : nat]: (((ord_less_nat @ M @ N) | (M = N)) => (ord_less_eq_nat @ M @ N))))). % less_or_eq_imp_le
thf(fact_87_le__eq__less__or__eq, axiom,
    ((ord_less_eq_nat = (^[M5 : nat]: (^[N3 : nat]: (((ord_less_nat @ M5 @ N3)) | ((M5 = N3)))))))). % le_eq_less_or_eq
thf(fact_88_less__imp__le__nat, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_eq_nat @ M @ N))))). % less_imp_le_nat
thf(fact_89_nat__less__le, axiom,
    ((ord_less_nat = (^[M5 : nat]: (^[N3 : nat]: (((ord_less_eq_nat @ M5 @ N3)) & ((~ ((M5 = N3)))))))))). % nat_less_le
thf(fact_90_dual__order_Oantisym, axiom,
    ((![B : nat, A : nat]: ((ord_less_eq_nat @ B @ A) => ((ord_less_eq_nat @ A @ B) => (A = B)))))). % dual_order.antisym
thf(fact_91_dual__order_Oeq__iff, axiom,
    (((^[Y5 : nat]: (^[Z2 : nat]: (Y5 = Z2))) = (^[A3 : nat]: (^[B3 : nat]: (((ord_less_eq_nat @ B3 @ A3)) & ((ord_less_eq_nat @ A3 @ B3)))))))). % dual_order.eq_iff
thf(fact_92_dual__order_Otrans, axiom,
    ((![B : nat, A : nat, C : nat]: ((ord_less_eq_nat @ B @ A) => ((ord_less_eq_nat @ C @ B) => (ord_less_eq_nat @ C @ A)))))). % dual_order.trans
thf(fact_93_linorder__wlog, axiom,
    ((![P : nat > nat > $o, A : nat, B : nat]: ((![A2 : nat, B2 : nat]: ((ord_less_eq_nat @ A2 @ B2) => (P @ A2 @ B2))) => ((![A2 : nat, B2 : nat]: ((P @ B2 @ A2) => (P @ A2 @ B2))) => (P @ A @ B)))))). % linorder_wlog
thf(fact_94_dual__order_Orefl, axiom,
    ((![A : nat]: (ord_less_eq_nat @ A @ A)))). % dual_order.refl
thf(fact_95_order__trans, axiom,
    ((![X : nat, Y3 : nat, Z : nat]: ((ord_less_eq_nat @ X @ Y3) => ((ord_less_eq_nat @ Y3 @ Z) => (ord_less_eq_nat @ X @ Z)))))). % order_trans
thf(fact_96_order__class_Oorder_Oantisym, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ B @ A) => (A = B)))))). % order_class.order.antisym
thf(fact_97_ord__le__eq__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((B = C) => (ord_less_eq_nat @ A @ C)))))). % ord_le_eq_trans
thf(fact_98_ord__eq__le__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((A = B) => ((ord_less_eq_nat @ B @ C) => (ord_less_eq_nat @ A @ C)))))). % ord_eq_le_trans
thf(fact_99_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y5 : nat]: (^[Z2 : nat]: (Y5 = Z2))) = (^[A3 : nat]: (^[B3 : nat]: (((ord_less_eq_nat @ A3 @ B3)) & ((ord_less_eq_nat @ B3 @ A3)))))))). % order_class.order.eq_iff
thf(fact_100_antisym__conv, axiom,
    ((![Y3 : nat, X : nat]: ((ord_less_eq_nat @ Y3 @ X) => ((ord_less_eq_nat @ X @ Y3) = (X = Y3)))))). % antisym_conv
thf(fact_101_le__cases3, axiom,
    ((![X : nat, Y3 : nat, Z : nat]: (((ord_less_eq_nat @ X @ Y3) => (~ ((ord_less_eq_nat @ Y3 @ Z)))) => (((ord_less_eq_nat @ Y3 @ X) => (~ ((ord_less_eq_nat @ X @ Z)))) => (((ord_less_eq_nat @ X @ Z) => (~ ((ord_less_eq_nat @ Z @ Y3)))) => (((ord_less_eq_nat @ Z @ Y3) => (~ ((ord_less_eq_nat @ Y3 @ X)))) => (((ord_less_eq_nat @ Y3 @ Z) => (~ ((ord_less_eq_nat @ Z @ X)))) => (~ (((ord_less_eq_nat @ Z @ X) => (~ ((ord_less_eq_nat @ X @ Y3)))))))))))))). % le_cases3
thf(fact_102_order_Otrans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ B @ C) => (ord_less_eq_nat @ A @ C)))))). % order.trans
thf(fact_103_le__cases, axiom,
    ((![X : nat, Y3 : nat]: ((~ ((ord_less_eq_nat @ X @ Y3))) => (ord_less_eq_nat @ Y3 @ X))))). % le_cases
thf(fact_104_eq__refl, axiom,
    ((![X : nat, Y3 : nat]: ((X = Y3) => (ord_less_eq_nat @ X @ Y3))))). % eq_refl
thf(fact_105_linear, axiom,
    ((![X : nat, Y3 : nat]: ((ord_less_eq_nat @ X @ Y3) | (ord_less_eq_nat @ Y3 @ X))))). % linear
thf(fact_106_antisym, axiom,
    ((![X : nat, Y3 : nat]: ((ord_less_eq_nat @ X @ Y3) => ((ord_less_eq_nat @ Y3 @ X) => (X = Y3)))))). % antisym
thf(fact_107_eq__iff, axiom,
    (((^[Y5 : nat]: (^[Z2 : nat]: (Y5 = Z2))) = (^[X5 : nat]: (^[Y4 : nat]: (((ord_less_eq_nat @ X5 @ Y4)) & ((ord_less_eq_nat @ Y4 @ X5)))))))). % eq_iff
thf(fact_108_ord__le__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B) => (((F @ B) = C) => ((![X2 : nat, Y : nat]: ((ord_less_eq_nat @ X2 @ Y) => (ord_less_eq_nat @ (F @ X2) @ (F @ Y)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_109_ord__eq__le__subst, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X2 : nat, Y : nat]: ((ord_less_eq_nat @ X2 @ Y) => (ord_less_eq_nat @ (F @ X2) @ (F @ Y)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_110_order__subst2, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ (F @ B) @ C) => ((![X2 : nat, Y : nat]: ((ord_less_eq_nat @ X2 @ Y) => (ord_less_eq_nat @ (F @ X2) @ (F @ Y)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % order_subst2
thf(fact_111_order__subst1, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X2 : nat, Y : nat]: ((ord_less_eq_nat @ X2 @ Y) => (ord_less_eq_nat @ (F @ X2) @ (F @ Y)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % order_subst1
thf(fact_112_dvd__0__right, axiom,
    ((![A : poly_a]: (dvd_dvd_poly_a @ A @ zero_zero_poly_a)))). % dvd_0_right
thf(fact_113_dvd__0__right, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ zero_zero_nat)))). % dvd_0_right
thf(fact_114_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_115_dvd__0__left__iff, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) = (A = zero_zero_nat))))). % dvd_0_left_iff
thf(fact_116_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_117_le__zero__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_zero_eq
thf(fact_118_dvd__imp__degree__le, axiom,
    ((![P4 : poly_a, Q : poly_a]: ((dvd_dvd_poly_a @ P4 @ Q) => ((~ ((Q = zero_zero_poly_a))) => (ord_less_eq_nat @ (degree_a @ P4) @ (degree_a @ Q))))))). % dvd_imp_degree_le
thf(fact_119_divides__degree, axiom,
    ((![P4 : poly_a, Q : poly_a]: ((dvd_dvd_poly_a @ P4 @ Q) => ((ord_less_eq_nat @ (degree_a @ P4) @ (degree_a @ Q)) | (Q = zero_zero_poly_a)))))). % divides_degree
thf(fact_120_nat__descend__induct, axiom,
    ((![N : nat, P : nat > $o, M : nat]: ((![K2 : nat]: ((ord_less_nat @ N @ K2) => (P @ K2))) => ((![K2 : nat]: ((ord_less_eq_nat @ K2 @ N) => ((![I3 : nat]: ((ord_less_nat @ K2 @ I3) => (P @ I3))) => (P @ K2)))) => (P @ M)))))). % nat_descend_induct
thf(fact_121_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_122_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_123_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_124_le0, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % le0
thf(fact_125_bot__nat__0_Oextremum, axiom,
    ((![A : nat]: (ord_less_eq_nat @ zero_zero_nat @ A)))). % bot_nat_0.extremum
thf(fact_126_degree__0, axiom,
    (((degree_a @ zero_zero_poly_a) = zero_zero_nat))). % degree_0
thf(fact_127_nat__dvd__not__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ M) => ((ord_less_nat @ M @ N) => (~ ((dvd_dvd_nat @ N @ M)))))))). % nat_dvd_not_less
thf(fact_128_dvd__imp__le, axiom,
    ((![K : nat, N : nat]: ((dvd_dvd_nat @ K @ N) => ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_eq_nat @ K @ N)))))). % dvd_imp_le
thf(fact_129_bot__nat__0_Oextremum__strict, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_130_infinite__descent0, axiom,
    ((![P : nat > $o, N : nat]: ((P @ zero_zero_nat) => ((![N2 : nat]: ((ord_less_nat @ zero_zero_nat @ N2) => ((~ ((P @ N2))) => (?[M4 : nat]: ((ord_less_nat @ M4 @ N2) & (~ ((P @ M4)))))))) => (P @ N)))))). % infinite_descent0
thf(fact_131_gr__implies__not0, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not0
thf(fact_132_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE

% Conjectures (1)
thf(conj_0, conjecture,
    ((ord_less_eq_nat @ (degree_a @ p) @ (degree_a @ q)))).
