% 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/FFT/prob_385__3226278_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:10:54.663

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

% Explicit typings (14)
thf(sy_c_Fields_Oinverse__class_Oinverse_001tf__a, type,
    inverse_inverse_a : a > a).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat, type,
    minus_minus_nat : nat > nat > nat).
thf(sy_c_Groups_Ominus__class_Ominus_001tf__a, type,
    minus_minus_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_001tf__a, type,
    zero_zero_a : a).
thf(sy_c_Nat_OSuc, type,
    suc : nat > nat).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat, type,
    ord_less_eq_nat : nat > nat > $o).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat, type,
    power_power_nat : nat > nat > nat).
thf(sy_c_Power_Opower__class_Opower_001tf__a, type,
    power_power_a : a > nat > a).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat, type,
    divide_divide_nat : nat > nat > nat).
thf(sy_c_Rings_Odivide__class_Odivide_001tf__a, type,
    divide_divide_a : a > a > a).
thf(sy_v_a, type,
    a2 : a).
thf(sy_v_x, type,
    x : nat).
thf(sy_v_y, type,
    y : nat).

% Relevant facts (137)
thf(fact_0_nz, axiom,
    ((~ ((a2 = zero_zero_a))))). % nz
thf(fact_1_diff__diff__cancel, axiom,
    ((![I : nat, N : nat]: ((ord_less_eq_nat @ I @ N) => ((minus_minus_nat @ N @ (minus_minus_nat @ N @ I)) = I))))). % diff_diff_cancel
thf(fact_2_inverse__divide, axiom,
    ((![A : a, B : a]: ((inverse_inverse_a @ (divide_divide_a @ A @ B)) = (divide_divide_a @ B @ A))))). % inverse_divide
thf(fact_3_diff__Suc__Suc, axiom,
    ((![M : nat, N : nat]: ((minus_minus_nat @ (suc @ M) @ (suc @ N)) = (minus_minus_nat @ M @ N))))). % diff_Suc_Suc
thf(fact_4_Suc__diff__diff, axiom,
    ((![M : nat, N : nat, K : nat]: ((minus_minus_nat @ (minus_minus_nat @ (suc @ M) @ N) @ (suc @ K)) = (minus_minus_nat @ (minus_minus_nat @ M @ N) @ K))))). % Suc_diff_diff
thf(fact_5_Suc__le__mono, axiom,
    ((![N : nat, M : nat]: ((ord_less_eq_nat @ (suc @ N) @ (suc @ M)) = (ord_less_eq_nat @ N @ M))))). % Suc_le_mono
thf(fact_6_Suc__diff__le, axiom,
    ((![N : nat, M : nat]: ((ord_less_eq_nat @ N @ M) => ((minus_minus_nat @ (suc @ M) @ N) = (suc @ (minus_minus_nat @ M @ N))))))). % Suc_diff_le
thf(fact_7_lift__Suc__mono__le, axiom,
    ((![F : nat > nat, N : nat, N2 : nat]: ((![N3 : nat]: (ord_less_eq_nat @ (F @ N3) @ (F @ (suc @ N3)))) => ((ord_less_eq_nat @ N @ N2) => (ord_less_eq_nat @ (F @ N) @ (F @ N2))))))). % lift_Suc_mono_le
thf(fact_8_lift__Suc__antimono__le, axiom,
    ((![F : nat > nat, N : nat, N2 : nat]: ((![N3 : nat]: (ord_less_eq_nat @ (F @ (suc @ N3)) @ (F @ N3))) => ((ord_less_eq_nat @ N @ N2) => (ord_less_eq_nat @ (F @ N2) @ (F @ N))))))). % lift_Suc_antimono_le
thf(fact_9_inverse__eq__iff__eq, axiom,
    ((![A : a, B : a]: (((inverse_inverse_a @ A) = (inverse_inverse_a @ B)) = (A = B))))). % inverse_eq_iff_eq
thf(fact_10_inverse__inverse__eq, axiom,
    ((![A : a]: ((inverse_inverse_a @ (inverse_inverse_a @ A)) = A)))). % inverse_inverse_eq
thf(fact_11_nat_Oinject, axiom,
    ((![X2 : nat, Y2 : nat]: (((suc @ X2) = (suc @ Y2)) = (X2 = Y2))))). % nat.inject
thf(fact_12_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_13_division__ring__divide__zero, axiom,
    ((![A : a]: ((divide_divide_a @ A @ zero_zero_a) = zero_zero_a)))). % division_ring_divide_zero
thf(fact_14_divide__cancel__right, axiom,
    ((![A : a, C : a, B : a]: (((divide_divide_a @ A @ C) = (divide_divide_a @ B @ C)) = (((C = zero_zero_a)) | ((A = B))))))). % divide_cancel_right
thf(fact_15_divide__cancel__left, axiom,
    ((![C : a, A : a, B : a]: (((divide_divide_a @ C @ A) = (divide_divide_a @ C @ B)) = (((C = zero_zero_a)) | ((A = B))))))). % divide_cancel_left
thf(fact_16_divide__eq__0__iff, axiom,
    ((![A : a, B : a]: (((divide_divide_a @ A @ B) = zero_zero_a) = (((A = zero_zero_a)) | ((B = zero_zero_a))))))). % divide_eq_0_iff
thf(fact_17_inverse__nonzero__iff__nonzero, axiom,
    ((![A : a]: (((inverse_inverse_a @ A) = zero_zero_a) = (A = zero_zero_a))))). % inverse_nonzero_iff_nonzero
thf(fact_18_inverse__zero, axiom,
    (((inverse_inverse_a @ zero_zero_a) = zero_zero_a))). % inverse_zero
thf(fact_19_nonzero__imp__inverse__nonzero, axiom,
    ((![A : a]: ((~ ((A = zero_zero_a))) => (~ (((inverse_inverse_a @ A) = zero_zero_a))))))). % nonzero_imp_inverse_nonzero
thf(fact_20_nonzero__inverse__inverse__eq, axiom,
    ((![A : a]: ((~ ((A = zero_zero_a))) => ((inverse_inverse_a @ (inverse_inverse_a @ A)) = A))))). % nonzero_inverse_inverse_eq
thf(fact_21_nonzero__inverse__eq__imp__eq, axiom,
    ((![A : a, B : a]: (((inverse_inverse_a @ A) = (inverse_inverse_a @ B)) => ((~ ((A = zero_zero_a))) => ((~ ((B = zero_zero_a))) => (A = B))))))). % nonzero_inverse_eq_imp_eq
thf(fact_22_inverse__zero__imp__zero, axiom,
    ((![A : a]: (((inverse_inverse_a @ A) = zero_zero_a) => (A = zero_zero_a))))). % inverse_zero_imp_zero
thf(fact_23_field__class_Ofield__inverse__zero, axiom,
    (((inverse_inverse_a @ zero_zero_a) = zero_zero_a))). % field_class.field_inverse_zero
thf(fact_24_n__not__Suc__n, axiom,
    ((![N : nat]: (~ ((N = (suc @ N))))))). % n_not_Suc_n
thf(fact_25_Suc__inject, axiom,
    ((![X : nat, Y : nat]: (((suc @ X) = (suc @ Y)) => (X = Y))))). % Suc_inject
thf(fact_26_Nat_Oex__has__greatest__nat, axiom,
    ((![P : nat > $o, K : nat, B : nat]: ((P @ K) => ((![Y3 : nat]: ((P @ Y3) => (ord_less_eq_nat @ Y3 @ B))) => (?[X3 : nat]: ((P @ X3) & (![Y4 : nat]: ((P @ Y4) => (ord_less_eq_nat @ Y4 @ X3)))))))))). % Nat.ex_has_greatest_nat
thf(fact_27_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_28_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_29_eq__imp__le, axiom,
    ((![M : nat, N : nat]: ((M = N) => (ord_less_eq_nat @ M @ N))))). % eq_imp_le
thf(fact_30_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_31_le__refl, axiom,
    ((![N : nat]: (ord_less_eq_nat @ N @ N)))). % le_refl
thf(fact_32_diff__commute, axiom,
    ((![I : nat, J : nat, K : nat]: ((minus_minus_nat @ (minus_minus_nat @ I @ J) @ K) = (minus_minus_nat @ (minus_minus_nat @ I @ K) @ J))))). % diff_commute
thf(fact_33_inverse__eq__imp__eq, axiom,
    ((![A : a, B : a]: (((inverse_inverse_a @ A) = (inverse_inverse_a @ B)) => (A = B))))). % inverse_eq_imp_eq
thf(fact_34_diff__divide__distrib, axiom,
    ((![A : a, B : a, C : a]: ((divide_divide_a @ (minus_minus_a @ A @ B) @ C) = (minus_minus_a @ (divide_divide_a @ A @ C) @ (divide_divide_a @ B @ C)))))). % diff_divide_distrib
thf(fact_35_transitive__stepwise__le, axiom,
    ((![M : nat, N : nat, R : nat > nat > $o]: ((ord_less_eq_nat @ M @ N) => ((![X3 : nat]: (R @ X3 @ X3)) => ((![X3 : nat, Y3 : nat, Z : nat]: ((R @ X3 @ Y3) => ((R @ Y3 @ Z) => (R @ X3 @ Z)))) => ((![N3 : nat]: (R @ N3 @ (suc @ N3))) => (R @ M @ N)))))))). % transitive_stepwise_le
thf(fact_36_nat__induct__at__least, axiom,
    ((![M : nat, N : nat, P : nat > $o]: ((ord_less_eq_nat @ M @ N) => ((P @ M) => ((![N3 : nat]: ((ord_less_eq_nat @ M @ N3) => ((P @ N3) => (P @ (suc @ N3))))) => (P @ N))))))). % nat_induct_at_least
thf(fact_37_full__nat__induct, axiom,
    ((![P : nat > $o, N : nat]: ((![N3 : nat]: ((![M2 : nat]: ((ord_less_eq_nat @ (suc @ M2) @ N3) => (P @ M2))) => (P @ N3))) => (P @ N))))). % full_nat_induct
thf(fact_38_not__less__eq__eq, axiom,
    ((![M : nat, N : nat]: ((~ ((ord_less_eq_nat @ M @ N))) = (ord_less_eq_nat @ (suc @ N) @ M))))). % not_less_eq_eq
thf(fact_39_Suc__n__not__le__n, axiom,
    ((![N : nat]: (~ ((ord_less_eq_nat @ (suc @ N) @ N)))))). % Suc_n_not_le_n
thf(fact_40_le__Suc__eq, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ (suc @ N)) = (((ord_less_eq_nat @ M @ N)) | ((M = (suc @ N)))))))). % le_Suc_eq
thf(fact_41_Suc__le__D, axiom,
    ((![N : nat, M3 : nat]: ((ord_less_eq_nat @ (suc @ N) @ M3) => (?[M4 : nat]: (M3 = (suc @ M4))))))). % Suc_le_D
thf(fact_42_le__SucI, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ N) => (ord_less_eq_nat @ M @ (suc @ N)))))). % le_SucI
thf(fact_43_le__SucE, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ (suc @ N)) => ((~ ((ord_less_eq_nat @ M @ N))) => (M = (suc @ N))))))). % le_SucE
thf(fact_44_Suc__leD, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ (suc @ M) @ N) => (ord_less_eq_nat @ M @ N))))). % Suc_leD
thf(fact_45_zero__induct__lemma, axiom,
    ((![P : nat > $o, K : nat, I : nat]: ((P @ K) => ((![N3 : nat]: ((P @ (suc @ N3)) => (P @ N3))) => (P @ (minus_minus_nat @ K @ I))))))). % zero_induct_lemma
thf(fact_46_diff__le__mono2, axiom,
    ((![M : nat, N : nat, L : nat]: ((ord_less_eq_nat @ M @ N) => (ord_less_eq_nat @ (minus_minus_nat @ L @ N) @ (minus_minus_nat @ L @ M)))))). % diff_le_mono2
thf(fact_47_le__diff__iff_H, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_eq_nat @ A @ C) => ((ord_less_eq_nat @ B @ C) => ((ord_less_eq_nat @ (minus_minus_nat @ C @ A) @ (minus_minus_nat @ C @ B)) = (ord_less_eq_nat @ B @ A))))))). % le_diff_iff'
thf(fact_48_diff__le__self, axiom,
    ((![M : nat, N : nat]: (ord_less_eq_nat @ (minus_minus_nat @ M @ N) @ M)))). % diff_le_self
thf(fact_49_diff__le__mono, axiom,
    ((![M : nat, N : nat, L : nat]: ((ord_less_eq_nat @ M @ N) => (ord_less_eq_nat @ (minus_minus_nat @ M @ L) @ (minus_minus_nat @ N @ L)))))). % diff_le_mono
thf(fact_50_Nat_Odiff__diff__eq, axiom,
    ((![K : nat, M : nat, N : nat]: ((ord_less_eq_nat @ K @ M) => ((ord_less_eq_nat @ K @ N) => ((minus_minus_nat @ (minus_minus_nat @ M @ K) @ (minus_minus_nat @ N @ K)) = (minus_minus_nat @ M @ N))))))). % Nat.diff_diff_eq
thf(fact_51_le__diff__iff, axiom,
    ((![K : nat, M : nat, N : nat]: ((ord_less_eq_nat @ K @ M) => ((ord_less_eq_nat @ K @ N) => ((ord_less_eq_nat @ (minus_minus_nat @ M @ K) @ (minus_minus_nat @ N @ K)) = (ord_less_eq_nat @ M @ N))))))). % le_diff_iff
thf(fact_52_eq__diff__iff, axiom,
    ((![K : nat, M : nat, N : nat]: ((ord_less_eq_nat @ K @ M) => ((ord_less_eq_nat @ K @ N) => (((minus_minus_nat @ M @ K) = (minus_minus_nat @ N @ K)) = (M = N))))))). % eq_diff_iff
thf(fact_53_power__0__Suc, axiom,
    ((![N : nat]: ((power_power_a @ zero_zero_a @ (suc @ N)) = zero_zero_a)))). % power_0_Suc
thf(fact_54_power__0__Suc, axiom,
    ((![N : nat]: ((power_power_nat @ zero_zero_nat @ (suc @ N)) = zero_zero_nat)))). % power_0_Suc
thf(fact_55_power__diff, axiom,
    ((![A : a, N : nat, M : nat]: ((~ ((A = zero_zero_a))) => ((ord_less_eq_nat @ N @ M) => ((power_power_a @ A @ (minus_minus_nat @ M @ N)) = (divide_divide_a @ (power_power_a @ A @ M) @ (power_power_a @ A @ N)))))))). % power_diff
thf(fact_56_power__diff, axiom,
    ((![A : nat, N : nat, M : nat]: ((~ ((A = zero_zero_nat))) => ((ord_less_eq_nat @ N @ M) => ((power_power_nat @ A @ (minus_minus_nat @ M @ N)) = (divide_divide_nat @ (power_power_nat @ A @ M) @ (power_power_nat @ A @ N)))))))). % power_diff
thf(fact_57_bits__div__by__0, axiom,
    ((![A : nat]: ((divide_divide_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % bits_div_by_0
thf(fact_58_bits__div__0, axiom,
    ((![A : nat]: ((divide_divide_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % bits_div_0
thf(fact_59_div__by__0, axiom,
    ((![A : a]: ((divide_divide_a @ A @ zero_zero_a) = zero_zero_a)))). % div_by_0
thf(fact_60_div__by__0, axiom,
    ((![A : nat]: ((divide_divide_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % div_by_0
thf(fact_61_div__0, axiom,
    ((![A : a]: ((divide_divide_a @ zero_zero_a @ A) = zero_zero_a)))). % div_0
thf(fact_62_div__0, axiom,
    ((![A : nat]: ((divide_divide_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % div_0
thf(fact_63_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_64_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_65_diff__zero, axiom,
    ((![A : a]: ((minus_minus_a @ A @ zero_zero_a) = A)))). % diff_zero
thf(fact_66_diff__zero, axiom,
    ((![A : nat]: ((minus_minus_nat @ A @ zero_zero_nat) = A)))). % diff_zero
thf(fact_67_zero__diff, axiom,
    ((![A : nat]: ((minus_minus_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % zero_diff
thf(fact_68_power__Suc__0, axiom,
    ((![N : nat]: ((power_power_nat @ (suc @ zero_zero_nat) @ N) = (suc @ zero_zero_nat))))). % power_Suc_0
thf(fact_69_nat__power__eq__Suc__0__iff, axiom,
    ((![X : nat, M : nat]: (((power_power_nat @ X @ M) = (suc @ zero_zero_nat)) = (((M = zero_zero_nat)) | ((X = (suc @ zero_zero_nat)))))))). % nat_power_eq_Suc_0_iff
thf(fact_70_le0, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % le0
thf(fact_71_bot__nat__0_Oextremum, axiom,
    ((![A : nat]: (ord_less_eq_nat @ zero_zero_nat @ A)))). % bot_nat_0.extremum
thf(fact_72_diff__0__eq__0, axiom,
    ((![N : nat]: ((minus_minus_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % diff_0_eq_0
thf(fact_73_diff__self__eq__0, axiom,
    ((![M : nat]: ((minus_minus_nat @ M @ M) = zero_zero_nat)))). % diff_self_eq_0
thf(fact_74_le__zero__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_zero_eq
thf(fact_75_diff__self, axiom,
    ((![A : a]: ((minus_minus_a @ A @ A) = zero_zero_a)))). % diff_self
thf(fact_76_diff__0__right, axiom,
    ((![A : a]: ((minus_minus_a @ A @ zero_zero_a) = A)))). % diff_0_right
thf(fact_77_power__Suc0__right, axiom,
    ((![A : a]: ((power_power_a @ A @ (suc @ zero_zero_nat)) = A)))). % power_Suc0_right
thf(fact_78_power__Suc0__right, axiom,
    ((![A : nat]: ((power_power_nat @ A @ (suc @ zero_zero_nat)) = A)))). % power_Suc0_right
thf(fact_79_diff__is__0__eq, axiom,
    ((![M : nat, N : nat]: (((minus_minus_nat @ M @ N) = zero_zero_nat) = (ord_less_eq_nat @ M @ N))))). % diff_is_0_eq
thf(fact_80_diff__is__0__eq_H, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ N) => ((minus_minus_nat @ M @ N) = zero_zero_nat))))). % diff_is_0_eq'
thf(fact_81_nat__one__le__power, axiom,
    ((![I : nat, N : nat]: ((ord_less_eq_nat @ (suc @ zero_zero_nat) @ I) => (ord_less_eq_nat @ (suc @ zero_zero_nat) @ (power_power_nat @ I @ N)))))). % nat_one_le_power
thf(fact_82_nat_Odistinct_I1_J, axiom,
    ((![X2 : nat]: (~ ((zero_zero_nat = (suc @ X2))))))). % nat.distinct(1)
thf(fact_83_old_Onat_Odistinct_I2_J, axiom,
    ((![Nat2 : nat]: (~ (((suc @ Nat2) = zero_zero_nat)))))). % old.nat.distinct(2)
thf(fact_84_old_Onat_Odistinct_I1_J, axiom,
    ((![Nat2 : nat]: (~ ((zero_zero_nat = (suc @ Nat2))))))). % old.nat.distinct(1)
thf(fact_85_nat_OdiscI, axiom,
    ((![Nat : nat, X2 : nat]: ((Nat = (suc @ X2)) => (~ ((Nat = zero_zero_nat))))))). % nat.discI
thf(fact_86_nat__induct, axiom,
    ((![P : nat > $o, N : nat]: ((P @ zero_zero_nat) => ((![N3 : nat]: ((P @ N3) => (P @ (suc @ N3)))) => (P @ N)))))). % nat_induct
thf(fact_87_diff__induct, axiom,
    ((![P : nat > nat > $o, M : nat, N : nat]: ((![X3 : nat]: (P @ X3 @ zero_zero_nat)) => ((![Y3 : nat]: (P @ zero_zero_nat @ (suc @ Y3))) => ((![X3 : nat, Y3 : nat]: ((P @ X3 @ Y3) => (P @ (suc @ X3) @ (suc @ Y3)))) => (P @ M @ N))))))). % diff_induct
thf(fact_88_zero__induct, axiom,
    ((![P : nat > $o, K : nat]: ((P @ K) => ((![N3 : nat]: ((P @ (suc @ N3)) => (P @ N3))) => (P @ zero_zero_nat)))))). % zero_induct
thf(fact_89_Suc__neq__Zero, axiom,
    ((![M : nat]: (~ (((suc @ M) = zero_zero_nat)))))). % Suc_neq_Zero
thf(fact_90_Zero__neq__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_neq_Suc
thf(fact_91_Zero__not__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_not_Suc
thf(fact_92_old_Onat_Oexhaust, axiom,
    ((![Y : nat]: ((~ ((Y = zero_zero_nat))) => (~ ((![Nat3 : nat]: (~ ((Y = (suc @ Nat3))))))))))). % old.nat.exhaust
thf(fact_93_old_Onat_Oinducts, axiom,
    ((![P : nat > $o, Nat : nat]: ((P @ zero_zero_nat) => ((![Nat3 : nat]: ((P @ Nat3) => (P @ (suc @ Nat3)))) => (P @ Nat)))))). % old.nat.inducts
thf(fact_94_not0__implies__Suc, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (?[M4 : nat]: (N = (suc @ M4))))))). % not0_implies_Suc
thf(fact_95_less__eq__nat_Osimps_I1_J, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % less_eq_nat.simps(1)
thf(fact_96_le__0__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_0_eq
thf(fact_97_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_98_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_99_diffs0__imp__equal, axiom,
    ((![M : nat, N : nat]: (((minus_minus_nat @ M @ N) = zero_zero_nat) => (((minus_minus_nat @ N @ M) = zero_zero_nat) => (M = N)))))). % diffs0_imp_equal
thf(fact_100_minus__nat_Odiff__0, axiom,
    ((![M : nat]: ((minus_minus_nat @ M @ zero_zero_nat) = M)))). % minus_nat.diff_0
thf(fact_101_zero__reorient, axiom,
    ((![X : a]: ((zero_zero_a = X) = (X = zero_zero_a))))). % zero_reorient
thf(fact_102_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_103_diff__right__commute, axiom,
    ((![A : nat, C : nat, B : nat]: ((minus_minus_nat @ (minus_minus_nat @ A @ C) @ B) = (minus_minus_nat @ (minus_minus_nat @ A @ B) @ C))))). % diff_right_commute
thf(fact_104_zero__le, axiom,
    ((![X : nat]: (ord_less_eq_nat @ zero_zero_nat @ X)))). % zero_le
thf(fact_105_eq__iff__diff__eq__0, axiom,
    (((^[Y5 : a]: (^[Z2 : a]: (Y5 = Z2))) = (^[A2 : a]: (^[B2 : a]: ((minus_minus_a @ A2 @ B2) = zero_zero_a)))))). % eq_iff_diff_eq_0
thf(fact_106_power__not__zero, axiom,
    ((![A : a, N : nat]: ((~ ((A = zero_zero_a))) => (~ (((power_power_a @ A @ N) = zero_zero_a))))))). % power_not_zero
thf(fact_107_power__not__zero, axiom,
    ((![A : nat, N : nat]: ((~ ((A = zero_zero_nat))) => (~ (((power_power_nat @ A @ N) = zero_zero_nat))))))). % power_not_zero
thf(fact_108_power__divide, axiom,
    ((![A : a, B : a, N : nat]: ((power_power_a @ (divide_divide_a @ A @ B) @ N) = (divide_divide_a @ (power_power_a @ A @ N) @ (power_power_a @ B @ N)))))). % power_divide
thf(fact_109_power__inverse, axiom,
    ((![A : a, N : nat]: ((power_power_a @ (inverse_inverse_a @ A) @ N) = (inverse_inverse_a @ (power_power_a @ A @ N)))))). % power_inverse
thf(fact_110_power__mono, axiom,
    ((![A : nat, B : nat, N : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ zero_zero_nat @ A) => (ord_less_eq_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N))))))). % power_mono
thf(fact_111_zero__le__power, axiom,
    ((![A : nat, N : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A) => (ord_less_eq_nat @ zero_zero_nat @ (power_power_nat @ A @ N)))))). % zero_le_power
thf(fact_112_power__inject__base, axiom,
    ((![A : nat, N : nat, B : nat]: (((power_power_nat @ A @ (suc @ N)) = (power_power_nat @ B @ (suc @ N))) => ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ zero_zero_nat @ B) => (A = B))))))). % power_inject_base
thf(fact_113_power__le__imp__le__base, axiom,
    ((![A : nat, N : nat, B : nat]: ((ord_less_eq_nat @ (power_power_nat @ A @ (suc @ N)) @ (power_power_nat @ B @ (suc @ N))) => ((ord_less_eq_nat @ zero_zero_nat @ B) => (ord_less_eq_nat @ A @ B)))))). % power_le_imp_le_base
thf(fact_114_div__by__Suc__0, axiom,
    ((![M : nat]: ((divide_divide_nat @ M @ (suc @ zero_zero_nat)) = M)))). % div_by_Suc_0
thf(fact_115_order__refl, axiom,
    ((![X : nat]: (ord_less_eq_nat @ X @ X)))). % order_refl
thf(fact_116_Suc__div__le__mono, axiom,
    ((![M : nat, N : nat]: (ord_less_eq_nat @ (divide_divide_nat @ M @ N) @ (divide_divide_nat @ (suc @ M) @ N))))). % Suc_div_le_mono
thf(fact_117_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_118_dual__order_Oeq__iff, axiom,
    (((^[Y5 : nat]: (^[Z2 : nat]: (Y5 = Z2))) = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_eq_nat @ B2 @ A2)) & ((ord_less_eq_nat @ A2 @ B2)))))))). % dual_order.eq_iff
thf(fact_119_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_120_linorder__wlog, axiom,
    ((![P : nat > nat > $o, A : nat, B : nat]: ((![A3 : nat, B3 : nat]: ((ord_less_eq_nat @ A3 @ B3) => (P @ A3 @ B3))) => ((![A3 : nat, B3 : nat]: ((P @ B3 @ A3) => (P @ A3 @ B3))) => (P @ A @ B)))))). % linorder_wlog
thf(fact_121_dual__order_Orefl, axiom,
    ((![A : nat]: (ord_less_eq_nat @ A @ A)))). % dual_order.refl
thf(fact_122_order__trans, axiom,
    ((![X : nat, Y : nat, Z3 : nat]: ((ord_less_eq_nat @ X @ Y) => ((ord_less_eq_nat @ Y @ Z3) => (ord_less_eq_nat @ X @ Z3)))))). % order_trans
thf(fact_123_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_124_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_125_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_126_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y5 : nat]: (^[Z2 : nat]: (Y5 = Z2))) = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_eq_nat @ A2 @ B2)) & ((ord_less_eq_nat @ B2 @ A2)))))))). % order_class.order.eq_iff
thf(fact_127_antisym__conv, axiom,
    ((![Y : nat, X : nat]: ((ord_less_eq_nat @ Y @ X) => ((ord_less_eq_nat @ X @ Y) = (X = Y)))))). % antisym_conv
thf(fact_128_le__cases3, axiom,
    ((![X : nat, Y : nat, Z3 : nat]: (((ord_less_eq_nat @ X @ Y) => (~ ((ord_less_eq_nat @ Y @ Z3)))) => (((ord_less_eq_nat @ Y @ X) => (~ ((ord_less_eq_nat @ X @ Z3)))) => (((ord_less_eq_nat @ X @ Z3) => (~ ((ord_less_eq_nat @ Z3 @ Y)))) => (((ord_less_eq_nat @ Z3 @ Y) => (~ ((ord_less_eq_nat @ Y @ X)))) => (((ord_less_eq_nat @ Y @ Z3) => (~ ((ord_less_eq_nat @ Z3 @ X)))) => (~ (((ord_less_eq_nat @ Z3 @ X) => (~ ((ord_less_eq_nat @ X @ Y)))))))))))))). % le_cases3
thf(fact_129_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_130_le__cases, axiom,
    ((![X : nat, Y : nat]: ((~ ((ord_less_eq_nat @ X @ Y))) => (ord_less_eq_nat @ Y @ X))))). % le_cases
thf(fact_131_eq__refl, axiom,
    ((![X : nat, Y : nat]: ((X = Y) => (ord_less_eq_nat @ X @ Y))))). % eq_refl
thf(fact_132_linear, axiom,
    ((![X : nat, Y : nat]: ((ord_less_eq_nat @ X @ Y) | (ord_less_eq_nat @ Y @ X))))). % linear
thf(fact_133_antisym, axiom,
    ((![X : nat, Y : nat]: ((ord_less_eq_nat @ X @ Y) => ((ord_less_eq_nat @ Y @ X) => (X = Y)))))). % antisym
thf(fact_134_eq__iff, axiom,
    (((^[Y5 : nat]: (^[Z2 : nat]: (Y5 = Z2))) = (^[X4 : nat]: (^[Y6 : nat]: (((ord_less_eq_nat @ X4 @ Y6)) & ((ord_less_eq_nat @ Y6 @ X4)))))))). % eq_iff
thf(fact_135_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, Y3 : nat]: ((ord_less_eq_nat @ X3 @ Y3) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y3)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_136_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, Y3 : nat]: ((ord_less_eq_nat @ X3 @ Y3) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y3)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % ord_eq_le_subst

% Conjectures (3)
thf(conj_0, hypothesis,
    (((ord_less_eq_nat @ y @ x) => ((power_power_a @ (inverse_inverse_a @ a2) @ (minus_minus_nat @ x @ y)) = (divide_divide_a @ (power_power_a @ a2 @ y) @ (power_power_a @ a2 @ x)))))).
thf(conj_1, hypothesis,
    ((ord_less_eq_nat @ (suc @ y) @ (suc @ x)))).
thf(conj_2, conjecture,
    (((power_power_a @ (inverse_inverse_a @ a2) @ (minus_minus_nat @ (suc @ x) @ (suc @ y))) = (divide_divide_a @ (power_power_a @ a2 @ (suc @ y)) @ (power_power_a @ a2 @ (suc @ x)))))).
