% 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_1127__5378512_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:36:04.042

% 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 (21)
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Ooffset__poly_001tf__a, type,
    fundam1358810038poly_a : poly_a > a > poly_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_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_Ocontent_001t__Nat__Onat, type,
    content_nat : poly_nat > nat).
thf(sy_c_Polynomial_Odegree_001tf__a, type,
    degree_a : poly_a > nat).
thf(sy_c_Polynomial_Odivide__poly__main_001t__Polynomial__Opoly_Itf__a_J, type,
    divide518463757poly_a : poly_a > poly_poly_a > poly_poly_a > poly_poly_a > nat > nat > poly_poly_a).
thf(sy_c_Polynomial_Odivide__poly__main_001tf__a, type,
    divide_poly_main_a : a > poly_a > poly_a > poly_a > nat > nat > 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_Opseudo__mod_001tf__a, type,
    pseudo_mod_a : poly_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_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 (138)
thf(fact_0_lq, axiom,
    ((~ ((p = zero_zero_poly_a))))). % lq
thf(fact_1__092_060open_062degree_Ap_A_092_060le_062_Adegree_Aq_092_060close_062, axiom,
    ((ord_less_eq_nat @ (degree_a @ p) @ (degree_a @ q)))). % \<open>degree p \<le> degree q\<close>
thf(fact_2_lgpq, axiom,
    ((ord_less_nat @ (degree_a @ q) @ (degree_a @ p)))). % lgpq
thf(fact_3_q0, axiom,
    ((~ ((q = zero_zero_poly_a))))). % q0
thf(fact_4_l, axiom,
    ((dvd_dvd_poly_a @ p @ q))). % l
thf(fact_5_zero__reorient, axiom,
    ((![X : poly_a]: ((zero_zero_poly_a = X) = (X = zero_zero_poly_a))))). % zero_reorient
thf(fact_6_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_7_le__zero__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_zero_eq
thf(fact_8_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_9_offset__poly__0, axiom,
    ((![H : a]: ((fundam1358810038poly_a @ zero_zero_poly_a @ H) = zero_zero_poly_a)))). % offset_poly_0
thf(fact_10_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_11_divide__poly__main__0, axiom,
    ((![R : poly_a, D : poly_a, Dr : nat, N : nat]: ((divide_poly_main_a @ zero_zero_a @ zero_zero_poly_a @ R @ D @ Dr @ N) = zero_zero_poly_a)))). % divide_poly_main_0
thf(fact_12_divide__poly__main__0, axiom,
    ((![R : poly_poly_a, D : poly_poly_a, Dr : nat, N : nat]: ((divide518463757poly_a @ zero_zero_poly_a @ zero_z2096148049poly_a @ R @ D @ Dr @ N) = zero_z2096148049poly_a)))). % divide_poly_main_0
thf(fact_13_mpoly__base__conv_I1_J, axiom,
    ((![X : a]: (zero_zero_a = (poly_a2 @ zero_zero_poly_a @ X))))). % mpoly_base_conv(1)
thf(fact_14_mpoly__base__conv_I1_J, axiom,
    ((![X : poly_a]: (zero_zero_poly_a = (poly_poly_a2 @ zero_z2096148049poly_a @ X))))). % mpoly_base_conv(1)
thf(fact_15_content__0, axiom,
    (((content_nat @ zero_zero_poly_nat) = zero_zero_nat))). % content_0
thf(fact_16_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_17_degree__0, axiom,
    (((degree_a @ zero_zero_poly_a) = zero_zero_nat))). % degree_0
thf(fact_18_poly__0, axiom,
    ((![X : poly_a]: ((poly_poly_a2 @ zero_z2096148049poly_a @ X) = zero_zero_poly_a)))). % poly_0
thf(fact_19_poly__0, axiom,
    ((![X : nat]: ((poly_nat2 @ zero_zero_poly_nat @ X) = zero_zero_nat)))). % poly_0
thf(fact_20_poly__0, axiom,
    ((![X : a]: ((poly_a2 @ zero_zero_poly_a @ X) = zero_zero_a)))). % poly_0
thf(fact_21_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_22_degree__offset__poly, axiom,
    ((![P : poly_a, H : a]: ((degree_a @ (fundam1358810038poly_a @ P @ H)) = (degree_a @ P))))). % degree_offset_poly
thf(fact_23_dvd__imp__degree__le, axiom,
    ((![P : poly_a, Q : poly_a]: ((dvd_dvd_poly_a @ P @ Q) => ((~ ((Q = zero_zero_poly_a))) => (ord_less_eq_nat @ (degree_a @ P) @ (degree_a @ Q))))))). % dvd_imp_degree_le
thf(fact_24_divides__degree, axiom,
    ((![P : poly_a, Q : poly_a]: ((dvd_dvd_poly_a @ P @ Q) => ((ord_less_eq_nat @ (degree_a @ P) @ (degree_a @ Q)) | (Q = zero_zero_poly_a)))))). % divides_degree
thf(fact_25_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_26_gr__implies__not__zero, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_27_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_28_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_29_zero__le, axiom,
    ((![X : nat]: (ord_less_eq_nat @ zero_zero_nat @ X)))). % zero_le
thf(fact_30_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_31_dvd__0__left__iff, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) = (A = zero_zero_nat))))). % dvd_0_left_iff
thf(fact_32_dvd__0__right, axiom,
    ((![A : poly_a]: (dvd_dvd_poly_a @ A @ zero_zero_poly_a)))). % dvd_0_right
thf(fact_33_dvd__0__right, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ zero_zero_nat)))). % dvd_0_right
thf(fact_34_le0, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % le0
thf(fact_35_bot__nat__0_Oextremum, axiom,
    ((![A : nat]: (ord_less_eq_nat @ zero_zero_nat @ A)))). % bot_nat_0.extremum
thf(fact_36_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_37_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_38_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_39_order__refl, axiom,
    ((![X : nat]: (ord_less_eq_nat @ X @ X)))). % order_refl
thf(fact_40_pseudo__mod_I2_J, axiom,
    ((![G : poly_a, F : poly_a]: ((~ ((G = zero_zero_poly_a))) => (((pseudo_mod_a @ F @ G) = zero_zero_poly_a) | (ord_less_nat @ (degree_a @ (pseudo_mod_a @ F @ G)) @ (degree_a @ G))))))). % pseudo_mod(2)
thf(fact_41_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_42_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_43_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_44_dual__order_Oeq__iff, axiom,
    (((^[Y : nat]: (^[Z : nat]: (Y = Z))) = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_eq_nat @ B2 @ A2)) & ((ord_less_eq_nat @ A2 @ B2)))))))). % dual_order.eq_iff
thf(fact_45_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_46_linorder__wlog, axiom,
    ((![P2 : nat > nat > $o, A : nat, B : nat]: ((![A3 : nat, B3 : nat]: ((ord_less_eq_nat @ A3 @ B3) => (P2 @ A3 @ B3))) => ((![A3 : nat, B3 : nat]: ((P2 @ B3 @ A3) => (P2 @ A3 @ B3))) => (P2 @ A @ B)))))). % linorder_wlog
thf(fact_47_dual__order_Orefl, axiom,
    ((![A : nat]: (ord_less_eq_nat @ A @ A)))). % dual_order.refl
thf(fact_48_order__trans, axiom,
    ((![X : nat, Y2 : nat, Z2 : nat]: ((ord_less_eq_nat @ X @ Y2) => ((ord_less_eq_nat @ Y2 @ Z2) => (ord_less_eq_nat @ X @ Z2)))))). % order_trans
thf(fact_49_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_50_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_51_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_52_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y : nat]: (^[Z : nat]: (Y = Z))) = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_eq_nat @ A2 @ B2)) & ((ord_less_eq_nat @ B2 @ A2)))))))). % order_class.order.eq_iff
thf(fact_53_antisym__conv, axiom,
    ((![Y2 : nat, X : nat]: ((ord_less_eq_nat @ Y2 @ X) => ((ord_less_eq_nat @ X @ Y2) = (X = Y2)))))). % antisym_conv
thf(fact_54_le__cases3, axiom,
    ((![X : nat, Y2 : nat, Z2 : nat]: (((ord_less_eq_nat @ X @ Y2) => (~ ((ord_less_eq_nat @ Y2 @ Z2)))) => (((ord_less_eq_nat @ Y2 @ X) => (~ ((ord_less_eq_nat @ X @ Z2)))) => (((ord_less_eq_nat @ X @ Z2) => (~ ((ord_less_eq_nat @ Z2 @ Y2)))) => (((ord_less_eq_nat @ Z2 @ Y2) => (~ ((ord_less_eq_nat @ Y2 @ X)))) => (((ord_less_eq_nat @ Y2 @ Z2) => (~ ((ord_less_eq_nat @ Z2 @ X)))) => (~ (((ord_less_eq_nat @ Z2 @ X) => (~ ((ord_less_eq_nat @ X @ Y2)))))))))))))). % le_cases3
thf(fact_55_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_56_le__cases, axiom,
    ((![X : nat, Y2 : nat]: ((~ ((ord_less_eq_nat @ X @ Y2))) => (ord_less_eq_nat @ Y2 @ X))))). % le_cases
thf(fact_57_eq__refl, axiom,
    ((![X : nat, Y2 : nat]: ((X = Y2) => (ord_less_eq_nat @ X @ Y2))))). % eq_refl
thf(fact_58_linear, axiom,
    ((![X : nat, Y2 : nat]: ((ord_less_eq_nat @ X @ Y2) | (ord_less_eq_nat @ Y2 @ X))))). % linear
thf(fact_59_antisym, axiom,
    ((![X : nat, Y2 : nat]: ((ord_less_eq_nat @ X @ Y2) => ((ord_less_eq_nat @ Y2 @ X) => (X = Y2)))))). % antisym
thf(fact_60_eq__iff, axiom,
    (((^[Y : nat]: (^[Z : nat]: (Y = Z))) = (^[X2 : nat]: (^[Y3 : nat]: (((ord_less_eq_nat @ X2 @ Y3)) & ((ord_less_eq_nat @ Y3 @ X2)))))))). % eq_iff
thf(fact_61_ord__le__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B) => (((F @ B) = C) => ((![X3 : nat, Y4 : nat]: ((ord_less_eq_nat @ X3 @ Y4) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y4)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_62_ord__eq__le__subst, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X3 : nat, Y4 : nat]: ((ord_less_eq_nat @ X3 @ Y4) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y4)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_63_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) => ((![X3 : nat, Y4 : nat]: ((ord_less_eq_nat @ X3 @ Y4) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y4)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % order_subst2
thf(fact_64_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) => ((![X3 : nat, Y4 : nat]: ((ord_less_eq_nat @ X3 @ Y4) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y4)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % order_subst1
thf(fact_65_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_66_order_Ostrict__implies__not__eq, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_67_not__less__iff__gr__or__eq, axiom,
    ((![X : nat, Y2 : nat]: ((~ ((ord_less_nat @ X @ Y2))) = (((ord_less_nat @ Y2 @ X)) | ((X = Y2))))))). % not_less_iff_gr_or_eq
thf(fact_68_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_69_linorder__less__wlog, axiom,
    ((![P2 : nat > nat > $o, A : nat, B : nat]: ((![A3 : nat, B3 : nat]: ((ord_less_nat @ A3 @ B3) => (P2 @ A3 @ B3))) => ((![A3 : nat]: (P2 @ A3 @ A3)) => ((![A3 : nat, B3 : nat]: ((P2 @ B3 @ A3) => (P2 @ A3 @ B3))) => (P2 @ A @ B))))))). % linorder_less_wlog
thf(fact_70_exists__least__iff, axiom,
    (((^[P3 : nat > $o]: (?[X4 : nat]: (P3 @ X4))) = (^[P4 : nat > $o]: (?[N2 : nat]: (((P4 @ N2)) & ((![M2 : nat]: (((ord_less_nat @ M2 @ N2)) => ((~ ((P4 @ M2))))))))))))). % exists_least_iff
thf(fact_71_less__imp__not__less, axiom,
    ((![X : nat, Y2 : nat]: ((ord_less_nat @ X @ Y2) => (~ ((ord_less_nat @ Y2 @ X))))))). % less_imp_not_less
thf(fact_72_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_73_dual__order_Oirrefl, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ A)))))). % dual_order.irrefl
thf(fact_74_linorder__cases, axiom,
    ((![X : nat, Y2 : nat]: ((~ ((ord_less_nat @ X @ Y2))) => ((~ ((X = Y2))) => (ord_less_nat @ Y2 @ X)))))). % linorder_cases
thf(fact_75_less__imp__triv, axiom,
    ((![X : nat, Y2 : nat, P2 : $o]: ((ord_less_nat @ X @ Y2) => ((ord_less_nat @ Y2 @ X) => P2))))). % less_imp_triv
thf(fact_76_less__imp__not__eq2, axiom,
    ((![X : nat, Y2 : nat]: ((ord_less_nat @ X @ Y2) => (~ ((Y2 = X))))))). % less_imp_not_eq2
thf(fact_77_antisym__conv3, axiom,
    ((![Y2 : nat, X : nat]: ((~ ((ord_less_nat @ Y2 @ X))) => ((~ ((ord_less_nat @ X @ Y2))) = (X = Y2)))))). % antisym_conv3
thf(fact_78_less__induct, axiom,
    ((![P2 : nat > $o, A : nat]: ((![X3 : nat]: ((![Y5 : nat]: ((ord_less_nat @ Y5 @ X3) => (P2 @ Y5))) => (P2 @ X3))) => (P2 @ A))))). % less_induct
thf(fact_79_less__not__sym, axiom,
    ((![X : nat, Y2 : nat]: ((ord_less_nat @ X @ Y2) => (~ ((ord_less_nat @ Y2 @ X))))))). % less_not_sym
thf(fact_80_less__imp__not__eq, axiom,
    ((![X : nat, Y2 : nat]: ((ord_less_nat @ X @ Y2) => (~ ((X = Y2))))))). % less_imp_not_eq
thf(fact_81_dual__order_Oasym, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ B @ A) => (~ ((ord_less_nat @ A @ B))))))). % dual_order.asym
thf(fact_82_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_83_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_84_less__irrefl, axiom,
    ((![X : nat]: (~ ((ord_less_nat @ X @ X)))))). % less_irrefl
thf(fact_85_less__linear, axiom,
    ((![X : nat, Y2 : nat]: ((ord_less_nat @ X @ Y2) | ((X = Y2) | (ord_less_nat @ Y2 @ X)))))). % less_linear
thf(fact_86_less__trans, axiom,
    ((![X : nat, Y2 : nat, Z2 : nat]: ((ord_less_nat @ X @ Y2) => ((ord_less_nat @ Y2 @ Z2) => (ord_less_nat @ X @ Z2)))))). % less_trans
thf(fact_87_less__asym_H, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % less_asym'
thf(fact_88_less__asym, axiom,
    ((![X : nat, Y2 : nat]: ((ord_less_nat @ X @ Y2) => (~ ((ord_less_nat @ Y2 @ X))))))). % less_asym
thf(fact_89_less__imp__neq, axiom,
    ((![X : nat, Y2 : nat]: ((ord_less_nat @ X @ Y2) => (~ ((X = Y2))))))). % less_imp_neq
thf(fact_90_order_Oasym, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % order.asym
thf(fact_91_neq__iff, axiom,
    ((![X : nat, Y2 : nat]: ((~ ((X = Y2))) = (((ord_less_nat @ X @ Y2)) | ((ord_less_nat @ Y2 @ X))))))). % neq_iff
thf(fact_92_neqE, axiom,
    ((![X : nat, Y2 : nat]: ((~ ((X = Y2))) => ((~ ((ord_less_nat @ X @ Y2))) => (ord_less_nat @ Y2 @ X)))))). % neqE
thf(fact_93_gt__ex, axiom,
    ((![X : nat]: (?[X_1 : nat]: (ord_less_nat @ X @ X_1))))). % gt_ex
thf(fact_94_order__less__subst2, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ (F @ B) @ C) => ((![X3 : nat, Y4 : nat]: ((ord_less_nat @ X3 @ Y4) => (ord_less_nat @ (F @ X3) @ (F @ Y4)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_95_order__less__subst1, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((ord_less_nat @ A @ (F @ B)) => ((ord_less_nat @ B @ C) => ((![X3 : nat, Y4 : nat]: ((ord_less_nat @ X3 @ Y4) => (ord_less_nat @ (F @ X3) @ (F @ Y4)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_96_ord__less__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_nat @ A @ B) => (((F @ B) = C) => ((![X3 : nat, Y4 : nat]: ((ord_less_nat @ X3 @ Y4) => (ord_less_nat @ (F @ X3) @ (F @ Y4)))) => (ord_less_nat @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_97_ord__eq__less__subst, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_nat @ B @ C) => ((![X3 : nat, Y4 : nat]: ((ord_less_nat @ X3 @ Y4) => (ord_less_nat @ (F @ X3) @ (F @ Y4)))) => (ord_less_nat @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_98_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_99_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_100_dvd__refl, axiom,
    ((![A : poly_a]: (dvd_dvd_poly_a @ A @ A)))). % dvd_refl
thf(fact_101_dvd__refl, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ A)))). % dvd_refl
thf(fact_102_bot__nat__0_Oextremum__strict, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_103_linorder__neqE__nat, axiom,
    ((![X : nat, Y2 : nat]: ((~ ((X = Y2))) => ((~ ((ord_less_nat @ X @ Y2))) => (ord_less_nat @ Y2 @ X)))))). % linorder_neqE_nat
thf(fact_104_infinite__descent0, axiom,
    ((![P2 : nat > $o, N : nat]: ((P2 @ zero_zero_nat) => ((![N3 : nat]: ((ord_less_nat @ zero_zero_nat @ N3) => ((~ ((P2 @ N3))) => (?[M3 : nat]: ((ord_less_nat @ M3 @ N3) & (~ ((P2 @ M3)))))))) => (P2 @ N)))))). % infinite_descent0
thf(fact_105_infinite__descent, axiom,
    ((![P2 : nat > $o, N : nat]: ((![N3 : nat]: ((~ ((P2 @ N3))) => (?[M3 : nat]: ((ord_less_nat @ M3 @ N3) & (~ ((P2 @ M3))))))) => (P2 @ N))))). % infinite_descent
thf(fact_106_nat__less__induct, axiom,
    ((![P2 : nat > $o, N : nat]: ((![N3 : nat]: ((![M3 : nat]: ((ord_less_nat @ M3 @ N3) => (P2 @ M3))) => (P2 @ N3))) => (P2 @ N))))). % nat_less_induct
thf(fact_107_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_108_gr__implies__not0, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not0
thf(fact_109_less__not__refl3, axiom,
    ((![S : nat, T : nat]: ((ord_less_nat @ S @ T) => (~ ((S = T))))))). % less_not_refl3
thf(fact_110_less__not__refl2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ N @ M) => (~ ((M = N))))))). % less_not_refl2
thf(fact_111_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_112_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_113_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE
thf(fact_114_not__less0, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less0
thf(fact_115_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_116_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I
thf(fact_117_bot__nat__0_Oextremum__uniqueI, axiom,
    ((![A : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) => (A = zero_zero_nat))))). % bot_nat_0.extremum_uniqueI
thf(fact_118_bot__nat__0_Oextremum__unique, axiom,
    ((![A : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) = (A = zero_zero_nat))))). % bot_nat_0.extremum_unique
thf(fact_119_Nat_Oex__has__greatest__nat, axiom,
    ((![P2 : nat > $o, K : nat, B : nat]: ((P2 @ K) => ((![Y4 : nat]: ((P2 @ Y4) => (ord_less_eq_nat @ Y4 @ B))) => (?[X3 : nat]: ((P2 @ X3) & (![Y5 : nat]: ((P2 @ Y5) => (ord_less_eq_nat @ Y5 @ X3)))))))))). % Nat.ex_has_greatest_nat
thf(fact_120_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_121_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_122_eq__imp__le, axiom,
    ((![M : nat, N : nat]: ((M = N) => (ord_less_eq_nat @ M @ N))))). % eq_imp_le
thf(fact_123_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_124_le__refl, axiom,
    ((![N : nat]: (ord_less_eq_nat @ N @ N)))). % le_refl
thf(fact_125_le__0__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_0_eq
thf(fact_126_less__eq__nat_Osimps_I1_J, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % less_eq_nat.simps(1)
thf(fact_127_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_128_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_129_dual__order_Ostrict__iff__order, axiom,
    ((ord_less_nat = (^[B2 : nat]: (^[A2 : nat]: (((ord_less_eq_nat @ B2 @ A2)) & ((~ ((A2 = B2)))))))))). % dual_order.strict_iff_order
thf(fact_130_dual__order_Oorder__iff__strict, axiom,
    ((ord_less_eq_nat = (^[B2 : nat]: (^[A2 : nat]: (((ord_less_nat @ B2 @ A2)) | ((A2 = B2)))))))). % dual_order.order_iff_strict
thf(fact_131_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_132_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_133_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_134_order_Ostrict__iff__order, axiom,
    ((ord_less_nat = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_eq_nat @ A2 @ B2)) & ((~ ((A2 = B2)))))))))). % order.strict_iff_order
thf(fact_135_order_Oorder__iff__strict, axiom,
    ((ord_less_eq_nat = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_nat @ A2 @ B2)) | ((A2 = B2)))))))). % order.order_iff_strict
thf(fact_136_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_137_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

% Conjectures (1)
thf(conj_0, conjecture,
    ((q = zero_zero_poly_a))).
