% 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_507__3228788_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:11:44.583

% Could-be-implicit typings (3)
thf(ty_n_t__Num__Onum, type,
    num : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).
thf(ty_n_t__Int__Oint, type,
    int : $tType).

% Explicit typings (20)
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint, type,
    minus_minus_int : int > int > int).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat, type,
    minus_minus_nat : nat > nat > nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint, type,
    zero_zero_int : int).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Nat_OSuc, type,
    suc : nat > nat).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint, type,
    semiri2019852685at_int : nat > int).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat, type,
    semiri1382578993at_nat : nat > nat).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint, type,
    neg_numeral_dbl_int : int > int).
thf(sy_c_Num_Onum_OBit0, type,
    bit0 : num > num).
thf(sy_c_Num_Onum_OOne, type,
    one : num).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint, type,
    numeral_numeral_int : num > int).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat, type,
    numeral_numeral_nat : num > nat).
thf(sy_c_Num_Opow, type,
    pow : num > num > num).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint, type,
    ord_less_int : int > int > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum, type,
    ord_less_num : num > num > $o).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint, type,
    power_power_int : int > nat > int).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat, type,
    power_power_nat : nat > nat > nat).
thf(sy_v_i____, type,
    i : nat).
thf(sy_v_ka____, type,
    ka : nat).

% Relevant facts (178)
thf(fact_0_i, axiom,
    ((ord_less_nat @ i @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (suc @ ka))))). % i
thf(fact_1_False, axiom,
    ((~ ((ord_less_nat @ i @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ ka)))))). % False
thf(fact_2_power2__commute, axiom,
    ((![X : int, Y : int]: ((power_power_int @ (minus_minus_int @ X @ Y) @ (numeral_numeral_nat @ (bit0 @ one))) = (power_power_int @ (minus_minus_int @ Y @ X) @ (numeral_numeral_nat @ (bit0 @ one))))))). % power2_commute
thf(fact_3_semiring__norm_I85_J, axiom,
    ((![M : num]: (~ (((bit0 @ M) = one)))))). % semiring_norm(85)
thf(fact_4_semiring__norm_I83_J, axiom,
    ((![N : num]: (~ ((one = (bit0 @ N))))))). % semiring_norm(83)
thf(fact_5_numeral__less__iff, axiom,
    ((![M : num, N : num]: ((ord_less_nat @ (numeral_numeral_nat @ M) @ (numeral_numeral_nat @ N)) = (ord_less_num @ M @ N))))). % numeral_less_iff
thf(fact_6_numeral__less__iff, axiom,
    ((![M : num, N : num]: ((ord_less_int @ (numeral_numeral_int @ M) @ (numeral_numeral_int @ N)) = (ord_less_num @ M @ N))))). % numeral_less_iff
thf(fact_7_verit__eq__simplify_I8_J, axiom,
    ((![X2 : num, Y2 : num]: (((bit0 @ X2) = (bit0 @ Y2)) = (X2 = Y2))))). % verit_eq_simplify(8)
thf(fact_8_semiring__norm_I87_J, axiom,
    ((![M : num, N : num]: (((bit0 @ M) = (bit0 @ N)) = (M = N))))). % semiring_norm(87)
thf(fact_9_numeral__eq__iff, axiom,
    ((![M : num, N : num]: (((numeral_numeral_nat @ M) = (numeral_numeral_nat @ N)) = (M = N))))). % numeral_eq_iff
thf(fact_10_numeral__eq__iff, axiom,
    ((![M : num, N : num]: (((numeral_numeral_int @ M) = (numeral_numeral_int @ N)) = (M = N))))). % numeral_eq_iff
thf(fact_11_diff__less__mono2, axiom,
    ((![M : nat, N : nat, L : nat]: ((ord_less_nat @ M @ N) => ((ord_less_nat @ M @ L) => (ord_less_nat @ (minus_minus_nat @ L @ N) @ (minus_minus_nat @ L @ M))))))). % diff_less_mono2
thf(fact_12_less__imp__diff__less, axiom,
    ((![J : nat, K : nat, N : nat]: ((ord_less_nat @ J @ K) => (ord_less_nat @ (minus_minus_nat @ J @ N) @ K))))). % less_imp_diff_less
thf(fact_13_verit__eq__simplify_I10_J, axiom,
    ((![X2 : num]: (~ ((one = (bit0 @ X2))))))). % verit_eq_simplify(10)
thf(fact_14_nat_Oinject, axiom,
    ((![X2 : nat, Y2 : nat]: (((suc @ X2) = (suc @ Y2)) = (X2 = Y2))))). % nat.inject
thf(fact_15_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_16_Suc__less__eq, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (suc @ M) @ (suc @ N)) = (ord_less_nat @ M @ N))))). % Suc_less_eq
thf(fact_17_Suc__mono, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_nat @ (suc @ M) @ (suc @ N)))))). % Suc_mono
thf(fact_18_lessI, axiom,
    ((![N : nat]: (ord_less_nat @ N @ (suc @ N))))). % lessI
thf(fact_19_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_20_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_21_semiring__norm_I78_J, axiom,
    ((![M : num, N : num]: ((ord_less_num @ (bit0 @ M) @ (bit0 @ N)) = (ord_less_num @ M @ N))))). % semiring_norm(78)
thf(fact_22_semiring__norm_I75_J, axiom,
    ((![M : num]: (~ ((ord_less_num @ M @ one)))))). % semiring_norm(75)
thf(fact_23_semiring__norm_I76_J, axiom,
    ((![N : num]: (ord_less_num @ one @ (bit0 @ N))))). % semiring_norm(76)
thf(fact_24_Suc__inject, axiom,
    ((![X : nat, Y : nat]: (((suc @ X) = (suc @ Y)) => (X = Y))))). % Suc_inject
thf(fact_25_n__not__Suc__n, axiom,
    ((![N : nat]: (~ ((N = (suc @ N))))))). % n_not_Suc_n
thf(fact_26_not__less__less__Suc__eq, axiom,
    ((![N : nat, M : nat]: ((~ ((ord_less_nat @ N @ M))) => ((ord_less_nat @ N @ (suc @ M)) = (N = M)))))). % not_less_less_Suc_eq
thf(fact_27_strict__inc__induct, axiom,
    ((![I : nat, J : nat, P : nat > $o]: ((ord_less_nat @ I @ J) => ((![I2 : nat]: ((J = (suc @ I2)) => (P @ I2))) => ((![I2 : nat]: ((ord_less_nat @ I2 @ J) => ((P @ (suc @ I2)) => (P @ I2)))) => (P @ I))))))). % strict_inc_induct
thf(fact_28_less__Suc__induct, axiom,
    ((![I : nat, J : nat, P : nat > nat > $o]: ((ord_less_nat @ I @ J) => ((![I2 : nat]: (P @ I2 @ (suc @ I2))) => ((![I2 : nat, J2 : nat, K2 : nat]: ((ord_less_nat @ I2 @ J2) => ((ord_less_nat @ J2 @ K2) => ((P @ I2 @ J2) => ((P @ J2 @ K2) => (P @ I2 @ K2)))))) => (P @ I @ J))))))). % less_Suc_induct
thf(fact_29_less__trans__Suc, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_nat @ I @ J) => ((ord_less_nat @ J @ K) => (ord_less_nat @ (suc @ I) @ K)))))). % less_trans_Suc
thf(fact_30_Suc__less__SucD, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (suc @ M) @ (suc @ N)) => (ord_less_nat @ M @ N))))). % Suc_less_SucD
thf(fact_31_less__antisym, axiom,
    ((![N : nat, M : nat]: ((~ ((ord_less_nat @ N @ M))) => ((ord_less_nat @ N @ (suc @ M)) => (M = N)))))). % less_antisym
thf(fact_32_Suc__less__eq2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ (suc @ N) @ M) = (?[M2 : nat]: (((M = (suc @ M2))) & ((ord_less_nat @ N @ M2)))))))). % Suc_less_eq2
thf(fact_33_All__less__Suc, axiom,
    ((![N : nat, P : nat > $o]: ((![I3 : nat]: (((ord_less_nat @ I3 @ (suc @ N))) => ((P @ I3)))) = (((P @ N)) & ((![I3 : nat]: (((ord_less_nat @ I3 @ N)) => ((P @ I3)))))))))). % All_less_Suc
thf(fact_34_not__less__eq, axiom,
    ((![M : nat, N : nat]: ((~ ((ord_less_nat @ M @ N))) = (ord_less_nat @ N @ (suc @ M)))))). % not_less_eq
thf(fact_35_less__Suc__eq, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ (suc @ N)) = (((ord_less_nat @ M @ N)) | ((M = N))))))). % less_Suc_eq
thf(fact_36_Ex__less__Suc, axiom,
    ((![N : nat, P : nat > $o]: ((?[I3 : nat]: (((ord_less_nat @ I3 @ (suc @ N))) & ((P @ I3)))) = (((P @ N)) | ((?[I3 : nat]: (((ord_less_nat @ I3 @ N)) & ((P @ I3)))))))))). % Ex_less_Suc
thf(fact_37_less__SucI, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_nat @ M @ (suc @ N)))))). % less_SucI
thf(fact_38_less__SucE, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ (suc @ N)) => ((~ ((ord_less_nat @ M @ N))) => (M = N)))))). % less_SucE
thf(fact_39_Suc__lessI, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => ((~ (((suc @ M) = N))) => (ord_less_nat @ (suc @ M) @ N)))))). % Suc_lessI
thf(fact_40_Suc__lessE, axiom,
    ((![I : nat, K : nat]: ((ord_less_nat @ (suc @ I) @ K) => (~ ((![J2 : nat]: ((ord_less_nat @ I @ J2) => (~ ((K = (suc @ J2)))))))))))). % Suc_lessE
thf(fact_41_Suc__lessD, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (suc @ M) @ N) => (ord_less_nat @ M @ N))))). % Suc_lessD
thf(fact_42_Nat_OlessE, axiom,
    ((![I : nat, K : nat]: ((ord_less_nat @ I @ K) => ((~ ((K = (suc @ I)))) => (~ ((![J2 : nat]: ((ord_less_nat @ I @ J2) => (~ ((K = (suc @ J2))))))))))))). % Nat.lessE
thf(fact_43_zero__induct__lemma, axiom,
    ((![P : nat > $o, K : nat, I : nat]: ((P @ K) => ((![N2 : nat]: ((P @ (suc @ N2)) => (P @ N2))) => (P @ (minus_minus_nat @ K @ I))))))). % zero_induct_lemma
thf(fact_44_verit__comp__simplify1_I1_J, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ A)))))). % verit_comp_simplify1(1)
thf(fact_45_verit__comp__simplify1_I1_J, axiom,
    ((![A : num]: (~ ((ord_less_num @ A @ A)))))). % verit_comp_simplify1(1)
thf(fact_46_verit__comp__simplify1_I1_J, axiom,
    ((![A : int]: (~ ((ord_less_int @ A @ A)))))). % verit_comp_simplify1(1)
thf(fact_47_linorder__neqE__nat, axiom,
    ((![X : nat, Y : nat]: ((~ ((X = Y))) => ((~ ((ord_less_nat @ X @ Y))) => (ord_less_nat @ Y @ X)))))). % linorder_neqE_nat
thf(fact_48_infinite__descent, axiom,
    ((![P : nat > $o, N : nat]: ((![N2 : nat]: ((~ ((P @ N2))) => (?[M3 : nat]: ((ord_less_nat @ M3 @ N2) & (~ ((P @ M3))))))) => (P @ N))))). % infinite_descent
thf(fact_49_nat__less__induct, axiom,
    ((![P : nat > $o, N : nat]: ((![N2 : nat]: ((![M3 : nat]: ((ord_less_nat @ M3 @ N2) => (P @ M3))) => (P @ N2))) => (P @ N))))). % nat_less_induct
thf(fact_50_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_51_less__not__refl3, axiom,
    ((![S : nat, T : nat]: ((ord_less_nat @ S @ T) => (~ ((S = T))))))). % less_not_refl3
thf(fact_52_less__not__refl2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ N @ M) => (~ ((M = N))))))). % less_not_refl2
thf(fact_53_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_54_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_55_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_56_lift__Suc__mono__less__iff, axiom,
    ((![F : nat > nat, N : nat, M : nat]: ((![N2 : nat]: (ord_less_nat @ (F @ N2) @ (F @ (suc @ N2)))) => ((ord_less_nat @ (F @ N) @ (F @ M)) = (ord_less_nat @ N @ M)))))). % lift_Suc_mono_less_iff
thf(fact_57_lift__Suc__mono__less__iff, axiom,
    ((![F : nat > num, N : nat, M : nat]: ((![N2 : nat]: (ord_less_num @ (F @ N2) @ (F @ (suc @ N2)))) => ((ord_less_num @ (F @ N) @ (F @ M)) = (ord_less_nat @ N @ M)))))). % lift_Suc_mono_less_iff
thf(fact_58_lift__Suc__mono__less__iff, axiom,
    ((![F : nat > int, N : nat, M : nat]: ((![N2 : nat]: (ord_less_int @ (F @ N2) @ (F @ (suc @ N2)))) => ((ord_less_int @ (F @ N) @ (F @ M)) = (ord_less_nat @ N @ M)))))). % lift_Suc_mono_less_iff
thf(fact_59_lift__Suc__mono__less, axiom,
    ((![F : nat > nat, N : nat, N3 : nat]: ((![N2 : nat]: (ord_less_nat @ (F @ N2) @ (F @ (suc @ N2)))) => ((ord_less_nat @ N @ N3) => (ord_less_nat @ (F @ N) @ (F @ N3))))))). % lift_Suc_mono_less
thf(fact_60_lift__Suc__mono__less, axiom,
    ((![F : nat > num, N : nat, N3 : nat]: ((![N2 : nat]: (ord_less_num @ (F @ N2) @ (F @ (suc @ N2)))) => ((ord_less_nat @ N @ N3) => (ord_less_num @ (F @ N) @ (F @ N3))))))). % lift_Suc_mono_less
thf(fact_61_lift__Suc__mono__less, axiom,
    ((![F : nat > int, N : nat, N3 : nat]: ((![N2 : nat]: (ord_less_int @ (F @ N2) @ (F @ (suc @ N2)))) => ((ord_less_nat @ N @ N3) => (ord_less_int @ (F @ N) @ (F @ N3))))))). % lift_Suc_mono_less
thf(fact_62_diff__less__Suc, axiom,
    ((![M : nat, N : nat]: (ord_less_nat @ (minus_minus_nat @ M @ N) @ (suc @ M))))). % diff_less_Suc
thf(fact_63_Suc__diff__Suc, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ N @ M) => ((suc @ (minus_minus_nat @ M @ (suc @ N))) = (minus_minus_nat @ M @ N)))))). % Suc_diff_Suc
thf(fact_64_power__numeral, axiom,
    ((![K : num, L : num]: ((power_power_nat @ (numeral_numeral_nat @ K) @ (numeral_numeral_nat @ L)) = (numeral_numeral_nat @ (pow @ K @ L)))))). % power_numeral
thf(fact_65_power__numeral, axiom,
    ((![K : num, L : num]: ((power_power_int @ (numeral_numeral_int @ K) @ (numeral_numeral_nat @ L)) = (numeral_numeral_int @ (pow @ K @ L)))))). % power_numeral
thf(fact_66_of__nat__less__numeral__power__cancel__iff, axiom,
    ((![X : nat, I : num, N : nat]: ((ord_less_nat @ (semiri1382578993at_nat @ X) @ (power_power_nat @ (numeral_numeral_nat @ I) @ N)) = (ord_less_nat @ X @ (power_power_nat @ (numeral_numeral_nat @ I) @ N)))))). % of_nat_less_numeral_power_cancel_iff
thf(fact_67_of__nat__less__numeral__power__cancel__iff, axiom,
    ((![X : nat, I : num, N : nat]: ((ord_less_int @ (semiri2019852685at_int @ X) @ (power_power_int @ (numeral_numeral_int @ I) @ N)) = (ord_less_nat @ X @ (power_power_nat @ (numeral_numeral_nat @ I) @ N)))))). % of_nat_less_numeral_power_cancel_iff
thf(fact_68_numeral__power__less__of__nat__cancel__iff, axiom,
    ((![I : num, N : nat, X : nat]: ((ord_less_nat @ (power_power_nat @ (numeral_numeral_nat @ I) @ N) @ (semiri1382578993at_nat @ X)) = (ord_less_nat @ (power_power_nat @ (numeral_numeral_nat @ I) @ N) @ X))))). % numeral_power_less_of_nat_cancel_iff
thf(fact_69_numeral__power__less__of__nat__cancel__iff, axiom,
    ((![I : num, N : nat, X : nat]: ((ord_less_int @ (power_power_int @ (numeral_numeral_int @ I) @ N) @ (semiri2019852685at_int @ X)) = (ord_less_nat @ (power_power_nat @ (numeral_numeral_nat @ I) @ N) @ X))))). % numeral_power_less_of_nat_cancel_iff
thf(fact_70_diff__strict__right__mono, axiom,
    ((![A : int, B : int, C : int]: ((ord_less_int @ A @ B) => (ord_less_int @ (minus_minus_int @ A @ C) @ (minus_minus_int @ B @ C)))))). % diff_strict_right_mono
thf(fact_71_diff__strict__left__mono, axiom,
    ((![B : int, A : int, C : int]: ((ord_less_int @ B @ A) => (ord_less_int @ (minus_minus_int @ C @ A) @ (minus_minus_int @ C @ B)))))). % diff_strict_left_mono
thf(fact_72_diff__eq__diff__less, axiom,
    ((![A : int, B : int, C : int, D : int]: (((minus_minus_int @ A @ B) = (minus_minus_int @ C @ D)) => ((ord_less_int @ A @ B) = (ord_less_int @ C @ D)))))). % diff_eq_diff_less
thf(fact_73_diff__strict__mono, axiom,
    ((![A : int, B : int, D : int, C : int]: ((ord_less_int @ A @ B) => ((ord_less_int @ D @ C) => (ord_less_int @ (minus_minus_int @ A @ C) @ (minus_minus_int @ B @ D))))))). % diff_strict_mono
thf(fact_74_zero__less__power2, axiom,
    ((![A : int]: ((ord_less_int @ zero_zero_int @ (power_power_int @ A @ (numeral_numeral_nat @ (bit0 @ one)))) = (~ ((A = zero_zero_int))))))). % zero_less_power2
thf(fact_75_dbl__simps_I5_J, axiom,
    ((![K : num]: ((neg_numeral_dbl_int @ (numeral_numeral_int @ K)) = (numeral_numeral_int @ (bit0 @ K)))))). % dbl_simps(5)
thf(fact_76_less__2__cases__iff, axiom,
    ((![N : nat]: ((ord_less_nat @ N @ (numeral_numeral_nat @ (bit0 @ one))) = (((N = zero_zero_nat)) | ((N = (suc @ zero_zero_nat)))))))). % less_2_cases_iff
thf(fact_77_of__nat__eq__iff, axiom,
    ((![M : nat, N : nat]: (((semiri2019852685at_int @ M) = (semiri2019852685at_int @ N)) = (M = N))))). % of_nat_eq_iff
thf(fact_78_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_79_diff__self, axiom,
    ((![A : int]: ((minus_minus_int @ A @ A) = zero_zero_int)))). % diff_self
thf(fact_80_diff__0__right, axiom,
    ((![A : int]: ((minus_minus_int @ A @ zero_zero_int) = A)))). % diff_0_right
thf(fact_81_zero__diff, axiom,
    ((![A : nat]: ((minus_minus_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % zero_diff
thf(fact_82_diff__zero, axiom,
    ((![A : nat]: ((minus_minus_nat @ A @ zero_zero_nat) = A)))). % diff_zero
thf(fact_83_diff__zero, axiom,
    ((![A : int]: ((minus_minus_int @ A @ zero_zero_int) = A)))). % diff_zero
thf(fact_84_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_85_cancel__comm__monoid__add__class_Odiff__cancel, axiom,
    ((![A : int]: ((minus_minus_int @ A @ A) = zero_zero_int)))). % cancel_comm_monoid_add_class.diff_cancel
thf(fact_86_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_87_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_88_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_89_diff__0__eq__0, axiom,
    ((![N : nat]: ((minus_minus_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % diff_0_eq_0
thf(fact_90_diff__self__eq__0, axiom,
    ((![M : nat]: ((minus_minus_nat @ M @ M) = zero_zero_nat)))). % diff_self_eq_0
thf(fact_91_dbl__simps_I2_J, axiom,
    (((neg_numeral_dbl_int @ zero_zero_int) = zero_zero_int))). % dbl_simps(2)
thf(fact_92_diff__gt__0__iff__gt, axiom,
    ((![A : int, B : int]: ((ord_less_int @ zero_zero_int @ (minus_minus_int @ A @ B)) = (ord_less_int @ B @ A))))). % diff_gt_0_iff_gt
thf(fact_93_power__0__Suc, axiom,
    ((![N : nat]: ((power_power_nat @ zero_zero_nat @ (suc @ N)) = zero_zero_nat)))). % power_0_Suc
thf(fact_94_power__0__Suc, axiom,
    ((![N : nat]: ((power_power_int @ zero_zero_int @ (suc @ N)) = zero_zero_int)))). % power_0_Suc
thf(fact_95_power__zero__numeral, axiom,
    ((![K : num]: ((power_power_nat @ zero_zero_nat @ (numeral_numeral_nat @ K)) = zero_zero_nat)))). % power_zero_numeral
thf(fact_96_power__zero__numeral, axiom,
    ((![K : num]: ((power_power_int @ zero_zero_int @ (numeral_numeral_nat @ K)) = zero_zero_int)))). % power_zero_numeral
thf(fact_97_of__nat__0, axiom,
    (((semiri1382578993at_nat @ zero_zero_nat) = zero_zero_nat))). % of_nat_0
thf(fact_98_of__nat__0, axiom,
    (((semiri2019852685at_int @ zero_zero_nat) = zero_zero_int))). % of_nat_0
thf(fact_99_of__nat__0__eq__iff, axiom,
    ((![N : nat]: ((zero_zero_nat = (semiri1382578993at_nat @ N)) = (zero_zero_nat = N))))). % of_nat_0_eq_iff
thf(fact_100_of__nat__0__eq__iff, axiom,
    ((![N : nat]: ((zero_zero_int = (semiri2019852685at_int @ N)) = (zero_zero_nat = N))))). % of_nat_0_eq_iff
thf(fact_101_of__nat__eq__0__iff, axiom,
    ((![M : nat]: (((semiri1382578993at_nat @ M) = zero_zero_nat) = (M = zero_zero_nat))))). % of_nat_eq_0_iff
thf(fact_102_of__nat__eq__0__iff, axiom,
    ((![M : nat]: (((semiri2019852685at_int @ M) = zero_zero_int) = (M = zero_zero_nat))))). % of_nat_eq_0_iff
thf(fact_103_of__nat__less__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)) = (ord_less_nat @ M @ N))))). % of_nat_less_iff
thf(fact_104_of__nat__less__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)) = (ord_less_nat @ M @ N))))). % of_nat_less_iff
thf(fact_105_power__Suc0__right, axiom,
    ((![A : nat]: ((power_power_nat @ A @ (suc @ zero_zero_nat)) = A)))). % power_Suc0_right
thf(fact_106_power__Suc0__right, axiom,
    ((![A : int]: ((power_power_int @ A @ (suc @ zero_zero_nat)) = A)))). % power_Suc0_right
thf(fact_107_of__nat__numeral, axiom,
    ((![N : num]: ((semiri1382578993at_nat @ (numeral_numeral_nat @ N)) = (numeral_numeral_nat @ N))))). % of_nat_numeral
thf(fact_108_of__nat__numeral, axiom,
    ((![N : num]: ((semiri2019852685at_int @ (numeral_numeral_nat @ N)) = (numeral_numeral_int @ N))))). % of_nat_numeral
thf(fact_109_less__Suc0, axiom,
    ((![N : nat]: ((ord_less_nat @ N @ (suc @ zero_zero_nat)) = (N = zero_zero_nat))))). % less_Suc0
thf(fact_110_zero__less__Suc, axiom,
    ((![N : nat]: (ord_less_nat @ zero_zero_nat @ (suc @ N))))). % zero_less_Suc
thf(fact_111_zero__less__diff, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ zero_zero_nat @ (minus_minus_nat @ N @ M)) = (ord_less_nat @ M @ N))))). % zero_less_diff
thf(fact_112_power__Suc__0, axiom,
    ((![N : nat]: ((power_power_nat @ (suc @ zero_zero_nat) @ N) = (suc @ zero_zero_nat))))). % power_Suc_0
thf(fact_113_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_114_of__nat__power, axiom,
    ((![M : nat, N : nat]: ((semiri1382578993at_nat @ (power_power_nat @ M @ N)) = (power_power_nat @ (semiri1382578993at_nat @ M) @ N))))). % of_nat_power
thf(fact_115_of__nat__power, axiom,
    ((![M : nat, N : nat]: ((semiri2019852685at_int @ (power_power_nat @ M @ N)) = (power_power_int @ (semiri2019852685at_int @ M) @ N))))). % of_nat_power
thf(fact_116_of__nat__eq__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: (((power_power_nat @ (semiri1382578993at_nat @ B) @ W) = (semiri1382578993at_nat @ X)) = ((power_power_nat @ B @ W) = X))))). % of_nat_eq_of_nat_power_cancel_iff
thf(fact_117_of__nat__eq__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: (((power_power_int @ (semiri2019852685at_int @ B) @ W) = (semiri2019852685at_int @ X)) = ((power_power_nat @ B @ W) = X))))). % of_nat_eq_of_nat_power_cancel_iff
thf(fact_118_of__nat__power__eq__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: (((semiri1382578993at_nat @ X) = (power_power_nat @ (semiri1382578993at_nat @ B) @ W)) = (X = (power_power_nat @ B @ W)))))). % of_nat_power_eq_of_nat_cancel_iff
thf(fact_119_of__nat__power__eq__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: (((semiri2019852685at_int @ X) = (power_power_int @ (semiri2019852685at_int @ B) @ W)) = (X = (power_power_nat @ B @ W)))))). % of_nat_power_eq_of_nat_cancel_iff
thf(fact_120_nat__zero__less__power__iff, axiom,
    ((![X : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (power_power_nat @ X @ N)) = (((ord_less_nat @ zero_zero_nat @ X)) | ((N = zero_zero_nat))))))). % nat_zero_less_power_iff
thf(fact_121_power__eq__0__iff, axiom,
    ((![A : nat, N : nat]: (((power_power_nat @ A @ N) = zero_zero_nat) = (((A = zero_zero_nat)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_122_power__eq__0__iff, axiom,
    ((![A : int, N : nat]: (((power_power_int @ A @ N) = zero_zero_int) = (((A = zero_zero_int)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_123_Suc__pred, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((suc @ (minus_minus_nat @ N @ (suc @ zero_zero_nat))) = N))))). % Suc_pred
thf(fact_124_zero__eq__power2, axiom,
    ((![A : nat]: (((power_power_nat @ A @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_nat) = (A = zero_zero_nat))))). % zero_eq_power2
thf(fact_125_zero__eq__power2, axiom,
    ((![A : int]: (((power_power_int @ A @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_int) = (A = zero_zero_int))))). % zero_eq_power2
thf(fact_126_of__nat__0__less__iff, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ (semiri1382578993at_nat @ N)) = (ord_less_nat @ zero_zero_nat @ N))))). % of_nat_0_less_iff
thf(fact_127_of__nat__0__less__iff, axiom,
    ((![N : nat]: ((ord_less_int @ zero_zero_int @ (semiri2019852685at_int @ N)) = (ord_less_nat @ zero_zero_nat @ N))))). % of_nat_0_less_iff
thf(fact_128_of__nat__less__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: ((ord_less_nat @ (power_power_nat @ (semiri1382578993at_nat @ B) @ W) @ (semiri1382578993at_nat @ X)) = (ord_less_nat @ (power_power_nat @ B @ W) @ X))))). % of_nat_less_of_nat_power_cancel_iff
thf(fact_129_of__nat__less__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: ((ord_less_int @ (power_power_int @ (semiri2019852685at_int @ B) @ W) @ (semiri2019852685at_int @ X)) = (ord_less_nat @ (power_power_nat @ B @ W) @ X))))). % of_nat_less_of_nat_power_cancel_iff
thf(fact_130_of__nat__power__less__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: ((ord_less_nat @ (semiri1382578993at_nat @ X) @ (power_power_nat @ (semiri1382578993at_nat @ B) @ W)) = (ord_less_nat @ X @ (power_power_nat @ B @ W)))))). % of_nat_power_less_of_nat_cancel_iff
thf(fact_131_of__nat__power__less__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: ((ord_less_int @ (semiri2019852685at_int @ X) @ (power_power_int @ (semiri2019852685at_int @ B) @ W)) = (ord_less_nat @ X @ (power_power_nat @ B @ W)))))). % of_nat_power_less_of_nat_cancel_iff
thf(fact_132_numeral__power__eq__of__nat__cancel__iff, axiom,
    ((![X : num, N : nat, Y : nat]: (((power_power_nat @ (numeral_numeral_nat @ X) @ N) = (semiri1382578993at_nat @ Y)) = ((power_power_nat @ (numeral_numeral_nat @ X) @ N) = Y))))). % numeral_power_eq_of_nat_cancel_iff
thf(fact_133_numeral__power__eq__of__nat__cancel__iff, axiom,
    ((![X : num, N : nat, Y : nat]: (((power_power_int @ (numeral_numeral_int @ X) @ N) = (semiri2019852685at_int @ Y)) = ((power_power_nat @ (numeral_numeral_nat @ X) @ N) = Y))))). % numeral_power_eq_of_nat_cancel_iff
thf(fact_134_real__of__nat__eq__numeral__power__cancel__iff, axiom,
    ((![Y : nat, X : num, N : nat]: (((semiri1382578993at_nat @ Y) = (power_power_nat @ (numeral_numeral_nat @ X) @ N)) = (Y = (power_power_nat @ (numeral_numeral_nat @ X) @ N)))))). % real_of_nat_eq_numeral_power_cancel_iff
thf(fact_135_real__of__nat__eq__numeral__power__cancel__iff, axiom,
    ((![Y : nat, X : num, N : nat]: (((semiri2019852685at_int @ Y) = (power_power_int @ (numeral_numeral_int @ X) @ N)) = (Y = (power_power_nat @ (numeral_numeral_nat @ X) @ N)))))). % real_of_nat_eq_numeral_power_cancel_iff
thf(fact_136_of__nat__zero__less__power__iff, axiom,
    ((![X : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (power_power_nat @ (semiri1382578993at_nat @ X) @ N)) = (((ord_less_nat @ zero_zero_nat @ X)) | ((N = zero_zero_nat))))))). % of_nat_zero_less_power_iff
thf(fact_137_of__nat__zero__less__power__iff, axiom,
    ((![X : nat, N : nat]: ((ord_less_int @ zero_zero_int @ (power_power_int @ (semiri2019852685at_int @ X) @ N)) = (((ord_less_nat @ zero_zero_nat @ X)) | ((N = zero_zero_nat))))))). % of_nat_zero_less_power_iff
thf(fact_138_int__ops_I6_J, axiom,
    ((![A : nat, B : nat]: (((ord_less_int @ (semiri2019852685at_int @ A) @ (semiri2019852685at_int @ B)) => ((semiri2019852685at_int @ (minus_minus_nat @ A @ B)) = zero_zero_int)) & ((~ ((ord_less_int @ (semiri2019852685at_int @ A) @ (semiri2019852685at_int @ B)))) => ((semiri2019852685at_int @ (minus_minus_nat @ A @ B)) = (minus_minus_int @ (semiri2019852685at_int @ A) @ (semiri2019852685at_int @ B)))))))). % int_ops(6)
thf(fact_139_int__ops_I1_J, axiom,
    (((semiri2019852685at_int @ zero_zero_nat) = zero_zero_int))). % int_ops(1)
thf(fact_140_of__nat__neq__0, axiom,
    ((![N : nat]: (~ (((semiri1382578993at_nat @ (suc @ N)) = zero_zero_nat)))))). % of_nat_neq_0
thf(fact_141_of__nat__neq__0, axiom,
    ((![N : nat]: (~ (((semiri2019852685at_int @ (suc @ N)) = zero_zero_int)))))). % of_nat_neq_0
thf(fact_142_of__nat__less__0__iff, axiom,
    ((![M : nat]: (~ ((ord_less_nat @ (semiri1382578993at_nat @ M) @ zero_zero_nat)))))). % of_nat_less_0_iff
thf(fact_143_of__nat__less__0__iff, axiom,
    ((![M : nat]: (~ ((ord_less_int @ (semiri2019852685at_int @ M) @ zero_zero_int)))))). % of_nat_less_0_iff
thf(fact_144_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_145_zero__reorient, axiom,
    ((![X : int]: ((zero_zero_int = X) = (X = zero_zero_int))))). % zero_reorient
thf(fact_146_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_147_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_148_gr__implies__not__zero, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_149_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_150_eq__iff__diff__eq__0, axiom,
    (((^[Y3 : int]: (^[Z : int]: (Y3 = Z))) = (^[A2 : int]: (^[B2 : int]: ((minus_minus_int @ A2 @ B2) = zero_zero_int)))))). % eq_iff_diff_eq_0
thf(fact_151_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_power_nat @ zero_zero_nat @ N) = zero_zero_nat))))). % zero_power
thf(fact_152_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_power_int @ zero_zero_int @ N) = zero_zero_int))))). % zero_power
thf(fact_153_nat__int__comparison_I2_J, axiom,
    ((ord_less_nat = (^[A2 : nat]: (^[B2 : nat]: (ord_less_int @ (semiri2019852685at_int @ A2) @ (semiri2019852685at_int @ B2))))))). % nat_int_comparison(2)
thf(fact_154_int__ops_I3_J, axiom,
    ((![N : num]: ((semiri2019852685at_int @ (numeral_numeral_nat @ N)) = (numeral_numeral_int @ N))))). % int_ops(3)
thf(fact_155_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_nat @ zero_zero_nat @ zero_zero_nat))))). % less_numeral_extra(3)
thf(fact_156_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_int @ zero_zero_int @ zero_zero_int))))). % less_numeral_extra(3)
thf(fact_157_zero__neq__numeral, axiom,
    ((![N : num]: (~ ((zero_zero_nat = (numeral_numeral_nat @ N))))))). % zero_neq_numeral
thf(fact_158_zero__neq__numeral, axiom,
    ((![N : num]: (~ ((zero_zero_int = (numeral_numeral_int @ N))))))). % zero_neq_numeral
thf(fact_159_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_160_power__not__zero, axiom,
    ((![A : int, N : nat]: ((~ ((A = zero_zero_int))) => (~ (((power_power_int @ A @ N) = zero_zero_int))))))). % power_not_zero
thf(fact_161_not0__implies__Suc, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (?[M4 : nat]: (N = (suc @ M4))))))). % not0_implies_Suc
thf(fact_162_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_163_old_Onat_Oexhaust, axiom,
    ((![Y : nat]: ((~ ((Y = zero_zero_nat))) => (~ ((![Nat3 : nat]: (~ ((Y = (suc @ Nat3))))))))))). % old.nat.exhaust
thf(fact_164_Zero__not__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_not_Suc
thf(fact_165_Zero__neq__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_neq_Suc
thf(fact_166_Suc__neq__Zero, axiom,
    ((![M : nat]: (~ (((suc @ M) = zero_zero_nat)))))). % Suc_neq_Zero
thf(fact_167_zero__induct, axiom,
    ((![P : nat > $o, K : nat]: ((P @ K) => ((![N2 : nat]: ((P @ (suc @ N2)) => (P @ N2))) => (P @ zero_zero_nat)))))). % zero_induct
thf(fact_168_diff__induct, axiom,
    ((![P : nat > nat > $o, M : nat, N : nat]: ((![X3 : nat]: (P @ X3 @ zero_zero_nat)) => ((![Y4 : nat]: (P @ zero_zero_nat @ (suc @ Y4))) => ((![X3 : nat, Y4 : nat]: ((P @ X3 @ Y4) => (P @ (suc @ X3) @ (suc @ Y4)))) => (P @ M @ N))))))). % diff_induct
thf(fact_169_nat__induct, axiom,
    ((![P : nat > $o, N : nat]: ((P @ zero_zero_nat) => ((![N2 : nat]: ((P @ N2) => (P @ (suc @ N2)))) => (P @ N)))))). % nat_induct
thf(fact_170_nat_OdiscI, axiom,
    ((![Nat : nat, X2 : nat]: ((Nat = (suc @ X2)) => (~ ((Nat = zero_zero_nat))))))). % nat.discI
thf(fact_171_old_Onat_Odistinct_I1_J, axiom,
    ((![Nat2 : nat]: (~ ((zero_zero_nat = (suc @ Nat2))))))). % old.nat.distinct(1)
thf(fact_172_old_Onat_Odistinct_I2_J, axiom,
    ((![Nat2 : nat]: (~ (((suc @ Nat2) = zero_zero_nat)))))). % old.nat.distinct(2)
thf(fact_173_nat_Odistinct_I1_J, axiom,
    ((![X2 : nat]: (~ ((zero_zero_nat = (suc @ X2))))))). % nat.distinct(1)
thf(fact_174_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I
thf(fact_175_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_176_not__less0, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less0
thf(fact_177_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE

% Conjectures (1)
thf(conj_0, conjecture,
    ((ord_less_nat @ (minus_minus_nat @ i @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ ka)) @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ ka)))).
