% 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/StrongNorm/prob_136__5210622_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:37:42.671

% Could-be-implicit typings (6)
thf(ty_n_t__Set__Oset_It__Nat__Onat_J, type,
    set_nat : $tType).
thf(ty_n_t__Set__Oset_It__Int__Oint_J, type,
    set_int : $tType).
thf(ty_n_t__LambdaType__Otype, type,
    type : $tType).
thf(ty_n_t__Lambda__OdB, type,
    dB : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).
thf(ty_n_t__Int__Oint, type,
    int : $tType).

% Explicit typings (37)
thf(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size_001t__Int__Oint, type,
    euclid1863447361ze_int : int > nat).
thf(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size_001t__Nat__Onat, type,
    euclid1226173669ze_nat : nat > nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint, type,
    one_one_int : int).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat, type,
    one_one_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_If_001t__Nat__Onat, type,
    if_nat : $o > nat > nat > nat).
thf(sy_c_InductTermi_OIT, type,
    it : dB > $o).
thf(sy_c_LambdaType_Oshift_001t__LambdaType__Otype, type,
    shift_type : (nat > type) > nat > type > nat > type).
thf(sy_c_LambdaType_Otyping, type,
    typing : (nat > type) > dB > type > $o).
thf(sy_c_Lambda_Olift, type,
    lift : dB > nat > dB).
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_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__Set__Oset_It__Int__Oint_J, type,
    ord_less_set_int : set_int > set_int > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J, type,
    ord_less_set_nat : set_nat > set_nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint, type,
    ord_less_eq_int : int > int > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat, type,
    ord_less_eq_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J, type,
    ord_less_eq_set_int : set_int > set_int > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J, type,
    ord_less_eq_set_nat : set_nat > set_nat > $o).
thf(sy_c_Parity_Osemiring__bit__shifts__class_Otake__bit_001t__Int__Oint, type,
    semiri1605039314it_int : nat > int > int).
thf(sy_c_Parity_Osemiring__bit__shifts__class_Otake__bit_001t__Nat__Onat, type,
    semiri967765622it_nat : nat > nat > nat).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint, type,
    dvd_dvd_int : int > int > $o).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat, type,
    dvd_dvd_nat : nat > nat > $o).
thf(sy_c_Set_OCollect_001t__Int__Oint, type,
    collect_int : (int > $o) > set_int).
thf(sy_c_Set_OCollect_001t__Nat__Onat, type,
    collect_nat : (nat > $o) > set_nat).
thf(sy_v_T_H_H____, type,
    t : type).
thf(sy_v_T____, type,
    t2 : type).
thf(sy_v_a____, type,
    a : dB).
thf(sy_v_e1____, type,
    e1 : nat > type).
thf(sy_v_e____, type,
    e : nat > type).
thf(sy_v_i____, type,
    i : nat).
thf(sy_v_n____, type,
    n : nat).
thf(sy_v_t____, type,
    t3 : dB).
thf(sy_v_u1____, type,
    u1 : dB).
thf(sy_v_u____, type,
    u : dB).

% Relevant facts (193)
thf(fact_0__092_060open_062IT_At_092_060close_062, axiom,
    ((it @ t3))). % \<open>IT t\<close>
thf(fact_1_Var_Oprems_I2_J, axiom,
    ((it @ u1))). % Var.prems(2)
thf(fact_2_uIT, axiom,
    ((it @ u))). % uIT
thf(fact_3_lift__IT, axiom,
    ((![T : dB, I : nat]: ((it @ T) => (it @ (lift @ T @ I)))))). % lift_IT
thf(fact_4_zero__natural_Orsp, axiom,
    ((zero_zero_nat = zero_zero_nat))). % zero_natural.rsp
thf(fact_5_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_6_zero__reorient, axiom,
    ((![X : int]: ((zero_zero_int = X) = (X = zero_zero_int))))). % zero_reorient
thf(fact_7_size__0, axiom,
    (((euclid1226173669ze_nat @ zero_zero_nat) = zero_zero_nat))). % size_0
thf(fact_8_size__0, axiom,
    (((euclid1863447361ze_int @ zero_zero_int) = zero_zero_nat))). % size_0
thf(fact_9_euclidean__size__eq__0__iff, axiom,
    ((![B : nat]: (((euclid1226173669ze_nat @ B) = zero_zero_nat) = (B = zero_zero_nat))))). % euclidean_size_eq_0_iff
thf(fact_10_euclidean__size__eq__0__iff, axiom,
    ((![B : int]: (((euclid1863447361ze_int @ B) = zero_zero_nat) = (B = zero_zero_int))))). % euclidean_size_eq_0_iff
thf(fact_11_take__bit__0, axiom,
    ((![A : int]: ((semiri1605039314it_int @ zero_zero_nat @ A) = zero_zero_int)))). % take_bit_0
thf(fact_12_take__bit__0, axiom,
    ((![A : nat]: ((semiri967765622it_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % take_bit_0
thf(fact_13_uT, axiom,
    ((typing @ e @ u @ t2))). % uT
thf(fact_14_of__nat__0, axiom,
    (((semiri1382578993at_nat @ zero_zero_nat) = zero_zero_nat))). % of_nat_0
thf(fact_15_of__nat__0, axiom,
    (((semiri2019852685at_int @ zero_zero_nat) = zero_zero_int))). % of_nat_0
thf(fact_16_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_17_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_18_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_19_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_20_of__nat__eq__iff, axiom,
    ((![M : nat, N : nat]: (((semiri2019852685at_int @ M) = (semiri2019852685at_int @ N)) = (M = N))))). % of_nat_eq_iff
thf(fact_21_take__bit__of__0, axiom,
    ((![N : nat]: ((semiri1605039314it_int @ N @ zero_zero_int) = zero_zero_int)))). % take_bit_of_0
thf(fact_22_take__bit__of__0, axiom,
    ((![N : nat]: ((semiri967765622it_nat @ N @ zero_zero_nat) = zero_zero_nat)))). % take_bit_of_0
thf(fact_23_euclidean__size__of__nat, axiom,
    ((![N : nat]: ((euclid1863447361ze_int @ (semiri2019852685at_int @ N)) = N)))). % euclidean_size_of_nat
thf(fact_24_euclidean__size__of__nat, axiom,
    ((![N : nat]: ((euclid1226173669ze_nat @ (semiri1382578993at_nat @ N)) = N)))). % euclidean_size_of_nat
thf(fact_25_take__bit__of__nat, axiom,
    ((![N : nat, M : nat]: ((semiri1605039314it_int @ N @ (semiri2019852685at_int @ M)) = (semiri2019852685at_int @ (semiri967765622it_nat @ N @ M)))))). % take_bit_of_nat
thf(fact_26_take__bit__of__nat, axiom,
    ((![N : nat, M : nat]: ((semiri967765622it_nat @ N @ (semiri1382578993at_nat @ M)) = (semiri1382578993at_nat @ (semiri967765622it_nat @ N @ M)))))). % take_bit_of_nat
thf(fact_27_Var_Oprems_I3_J, axiom,
    ((typing @ e1 @ u1 @ t2))). % Var.prems(3)
thf(fact_28_take__bit__of__1__eq__0__iff, axiom,
    ((![N : nat]: (((semiri1605039314it_int @ N @ one_one_int) = zero_zero_int) = (N = zero_zero_nat))))). % take_bit_of_1_eq_0_iff
thf(fact_29_take__bit__of__1__eq__0__iff, axiom,
    ((![N : nat]: (((semiri967765622it_nat @ N @ one_one_nat) = zero_zero_nat) = (N = zero_zero_nat))))). % take_bit_of_1_eq_0_iff
thf(fact_30_euclidean__size__greater__0__iff, axiom,
    ((![B : int]: ((ord_less_nat @ zero_zero_nat @ (euclid1863447361ze_int @ B)) = (~ ((B = zero_zero_int))))))). % euclidean_size_greater_0_iff
thf(fact_31_euclidean__size__greater__0__iff, axiom,
    ((![B : nat]: ((ord_less_nat @ zero_zero_nat @ (euclid1226173669ze_nat @ B)) = (~ ((B = zero_zero_nat))))))). % euclidean_size_greater_0_iff
thf(fact_32_of__nat__le__0__iff, axiom,
    ((![M : nat]: ((ord_less_eq_nat @ (semiri1382578993at_nat @ M) @ zero_zero_nat) = (M = zero_zero_nat))))). % of_nat_le_0_iff
thf(fact_33_of__nat__le__0__iff, axiom,
    ((![M : nat]: ((ord_less_eq_int @ (semiri2019852685at_int @ M) @ zero_zero_int) = (M = zero_zero_nat))))). % of_nat_le_0_iff
thf(fact_34_int__ops_I1_J, axiom,
    (((semiri2019852685at_int @ zero_zero_nat) = zero_zero_int))). % int_ops(1)
thf(fact_35_argT, axiom,
    ((typing @ (shift_type @ e @ i @ t2) @ a @ t))). % argT
thf(fact_36_dvd__euclidean__size__eq__imp__dvd, axiom,
    ((![A : int, B : int]: ((~ ((A = zero_zero_int))) => (((euclid1863447361ze_int @ A) = (euclid1863447361ze_int @ B)) => ((dvd_dvd_int @ B @ A) => (dvd_dvd_int @ A @ B))))))). % dvd_euclidean_size_eq_imp_dvd
thf(fact_37_dvd__euclidean__size__eq__imp__dvd, axiom,
    ((![A : nat, B : nat]: ((~ ((A = zero_zero_nat))) => (((euclid1226173669ze_nat @ A) = (euclid1226173669ze_nat @ B)) => ((dvd_dvd_nat @ B @ A) => (dvd_dvd_nat @ A @ B))))))). % dvd_euclidean_size_eq_imp_dvd
thf(fact_38_True, axiom,
    ((n = i))). % True
thf(fact_39_le0, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % le0
thf(fact_40_bot__nat__0_Oextremum, axiom,
    ((![A : nat]: (ord_less_eq_nat @ zero_zero_nat @ A)))). % bot_nat_0.extremum
thf(fact_41_le__zero__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_zero_eq
thf(fact_42_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_43_of__nat__le__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)) = (ord_less_eq_nat @ M @ N))))). % of_nat_le_iff
thf(fact_44_of__nat__le__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)) = (ord_less_eq_nat @ M @ N))))). % of_nat_le_iff
thf(fact_45_of__nat__eq__1__iff, axiom,
    ((![N : nat]: (((semiri1382578993at_nat @ N) = one_one_nat) = (N = one_one_nat))))). % of_nat_eq_1_iff
thf(fact_46_of__nat__eq__1__iff, axiom,
    ((![N : nat]: (((semiri2019852685at_int @ N) = one_one_int) = (N = one_one_nat))))). % of_nat_eq_1_iff
thf(fact_47_of__nat__1__eq__iff, axiom,
    ((![N : nat]: ((one_one_nat = (semiri1382578993at_nat @ N)) = (N = one_one_nat))))). % of_nat_1_eq_iff
thf(fact_48_of__nat__1__eq__iff, axiom,
    ((![N : nat]: ((one_one_int = (semiri2019852685at_int @ N)) = (N = one_one_nat))))). % of_nat_1_eq_iff
thf(fact_49_of__nat__1, axiom,
    (((semiri1382578993at_nat @ one_one_nat) = one_one_nat))). % of_nat_1
thf(fact_50_of__nat__1, axiom,
    (((semiri2019852685at_int @ one_one_nat) = one_one_int))). % of_nat_1
thf(fact_51_less__one, axiom,
    ((![N : nat]: ((ord_less_nat @ N @ one_one_nat) = (N = zero_zero_nat))))). % less_one
thf(fact_52_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_53_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_54_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_55_euclidean__size__1, axiom,
    (((euclid1863447361ze_int @ one_one_int) = one_one_nat))). % euclidean_size_1
thf(fact_56_euclidean__size__1, axiom,
    (((euclid1226173669ze_nat @ one_one_nat) = one_one_nat))). % euclidean_size_1
thf(fact_57_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_58_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_59_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_60_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_61_int__ops_I2_J, axiom,
    (((semiri2019852685at_int @ one_one_nat) = one_one_int))). % int_ops(2)
thf(fact_62_nat__int__comparison_I3_J, axiom,
    ((ord_less_eq_nat = (^[A2 : nat]: (^[B2 : nat]: (ord_less_eq_int @ (semiri2019852685at_int @ A2) @ (semiri2019852685at_int @ B2))))))). % nat_int_comparison(3)
thf(fact_63_nat__int__comparison_I1_J, axiom,
    (((^[Y : nat]: (^[Z : nat]: (Y = Z))) = (^[A2 : nat]: (^[B2 : nat]: ((semiri2019852685at_int @ A2) = (semiri2019852685at_int @ B2))))))). % nat_int_comparison(1)
thf(fact_64_int__if, axiom,
    ((![P : $o, A : nat, B : nat]: ((P => ((semiri2019852685at_int @ (if_nat @ P @ A @ B)) = (semiri2019852685at_int @ A))) & ((~ (P)) => ((semiri2019852685at_int @ (if_nat @ P @ A @ B)) = (semiri2019852685at_int @ B))))))). % int_if
thf(fact_65_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_66_nat__less__le, axiom,
    ((ord_less_nat = (^[M2 : nat]: (^[N2 : nat]: (((ord_less_eq_nat @ M2 @ N2)) & ((~ ((M2 = N2)))))))))). % nat_less_le
thf(fact_67_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_68_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_69_less__not__refl2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ N @ M) => (~ ((M = N))))))). % less_not_refl2
thf(fact_70_less__not__refl3, axiom,
    ((![S : nat, T : nat]: ((ord_less_nat @ S @ T) => (~ ((S = T))))))). % less_not_refl3
thf(fact_71_nat__leq__as__int, axiom,
    ((ord_less_eq_nat = (^[A2 : nat]: (^[B2 : nat]: (ord_less_eq_int @ (semiri2019852685at_int @ A2) @ (semiri2019852685at_int @ B2))))))). % nat_leq_as_int
thf(fact_72_one__reorient, axiom,
    ((![X : nat]: ((one_one_nat = X) = (X = one_one_nat))))). % one_reorient
thf(fact_73_one__reorient, axiom,
    ((![X : int]: ((one_one_int = X) = (X = one_one_int))))). % one_reorient
thf(fact_74_ex__least__nat__le, axiom,
    ((![P : nat > $o, N : nat]: ((P @ N) => ((~ ((P @ zero_zero_nat))) => (?[K2 : nat]: ((ord_less_eq_nat @ K2 @ N) & ((![I2 : nat]: ((ord_less_nat @ I2 @ K2) => (~ ((P @ I2))))) & (P @ K2))))))))). % ex_least_nat_le
thf(fact_75_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_76_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_77_nat__less__induct, axiom,
    ((![P : nat > $o, N : nat]: ((![N3 : nat]: ((![M3 : nat]: ((ord_less_nat @ M3 @ N3) => (P @ M3))) => (P @ N3))) => (P @ N))))). % nat_less_induct
thf(fact_78_infinite__descent, axiom,
    ((![P : nat > $o, N : nat]: ((![N3 : nat]: ((~ ((P @ N3))) => (?[M3 : nat]: ((ord_less_nat @ M3 @ N3) & (~ ((P @ M3))))))) => (P @ N))))). % infinite_descent
thf(fact_79_le__eq__less__or__eq, axiom,
    ((ord_less_eq_nat = (^[M2 : nat]: (^[N2 : nat]: (((ord_less_nat @ M2 @ N2)) | ((M2 = N2)))))))). % le_eq_less_or_eq
thf(fact_80_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_81_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_82_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_83_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_84_less__mono__imp__le__mono, axiom,
    ((![F : nat > nat, I : nat, J : nat]: ((![I3 : nat, J2 : nat]: ((ord_less_nat @ I3 @ J2) => (ord_less_nat @ (F @ I3) @ (F @ J2)))) => ((ord_less_eq_nat @ I @ J) => (ord_less_eq_nat @ (F @ I) @ (F @ J))))))). % less_mono_imp_le_mono
thf(fact_85_take__bit__nonnegative, axiom,
    ((![N : nat, K : int]: (ord_less_eq_int @ zero_zero_int @ (semiri1605039314it_int @ N @ K))))). % take_bit_nonnegative
thf(fact_86_zero__integer_Orsp, axiom,
    ((zero_zero_int = zero_zero_int))). % zero_integer.rsp
thf(fact_87_take__bit__nat__less__eq__self, axiom,
    ((![N : nat, M : nat]: (ord_less_eq_nat @ (semiri967765622it_nat @ N @ M) @ M)))). % take_bit_nat_less_eq_self
thf(fact_88_of__nat__mono, axiom,
    ((![I : nat, J : nat]: ((ord_less_eq_nat @ I @ J) => (ord_less_eq_nat @ (semiri1382578993at_nat @ I) @ (semiri1382578993at_nat @ J)))))). % of_nat_mono
thf(fact_89_of__nat__mono, axiom,
    ((![I : nat, J : nat]: ((ord_less_eq_nat @ I @ J) => (ord_less_eq_int @ (semiri2019852685at_int @ I) @ (semiri2019852685at_int @ J)))))). % of_nat_mono
thf(fact_90_less__imp__of__nat__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)))))). % less_imp_of_nat_less
thf(fact_91_less__imp__of__nat__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)))))). % less_imp_of_nat_less
thf(fact_92_of__nat__less__imp__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)) => (ord_less_nat @ M @ N))))). % of_nat_less_imp_less
thf(fact_93_of__nat__less__imp__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)) => (ord_less_nat @ M @ N))))). % of_nat_less_imp_less
thf(fact_94_of__nat__dvd__iff, axiom,
    ((![M : nat, N : nat]: ((dvd_dvd_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)) = (dvd_dvd_nat @ M @ N))))). % of_nat_dvd_iff
thf(fact_95_of__nat__dvd__iff, axiom,
    ((![M : nat, N : nat]: ((dvd_dvd_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)) = (dvd_dvd_nat @ M @ N))))). % of_nat_dvd_iff
thf(fact_96_verit__la__disequality, axiom,
    ((![A : nat, B : nat]: ((A = B) | ((~ ((ord_less_eq_nat @ A @ B))) | (~ ((ord_less_eq_nat @ B @ A)))))))). % verit_la_disequality
thf(fact_97_verit__la__disequality, axiom,
    ((![A : int, B : int]: ((A = B) | ((~ ((ord_less_eq_int @ A @ B))) | (~ ((ord_less_eq_int @ B @ A)))))))). % verit_la_disequality
thf(fact_98_verit__comp__simplify1_I1_J, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ A)))))). % verit_comp_simplify1(1)
thf(fact_99_verit__comp__simplify1_I1_J, axiom,
    ((![A : int]: (~ ((ord_less_int @ A @ A)))))). % verit_comp_simplify1(1)
thf(fact_100_verit__comp__simplify1_I3_J, axiom,
    ((![B3 : nat, A3 : nat]: ((~ ((ord_less_eq_nat @ B3 @ A3))) = (ord_less_nat @ A3 @ B3))))). % verit_comp_simplify1(3)
thf(fact_101_verit__comp__simplify1_I3_J, axiom,
    ((![B3 : int, A3 : int]: ((~ ((ord_less_eq_int @ B3 @ A3))) = (ord_less_int @ A3 @ B3))))). % verit_comp_simplify1(3)
thf(fact_102_euclidean__size__unit, axiom,
    ((![A : int]: ((dvd_dvd_int @ A @ one_one_int) => ((euclid1863447361ze_int @ A) = (euclid1863447361ze_int @ one_one_int)))))). % euclidean_size_unit
thf(fact_103_euclidean__size__unit, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ A @ one_one_nat) => ((euclid1226173669ze_nat @ A) = (euclid1226173669ze_nat @ one_one_nat)))))). % euclidean_size_unit
thf(fact_104_dvd__proper__imp__size__less, axiom,
    ((![A : int, B : int]: ((dvd_dvd_int @ A @ B) => ((~ ((dvd_dvd_int @ B @ A))) => ((~ ((B = zero_zero_int))) => (ord_less_nat @ (euclid1863447361ze_int @ A) @ (euclid1863447361ze_int @ B)))))))). % dvd_proper_imp_size_less
thf(fact_105_dvd__proper__imp__size__less, axiom,
    ((![A : nat, B : nat]: ((dvd_dvd_nat @ A @ B) => ((~ ((dvd_dvd_nat @ B @ A))) => ((~ ((B = zero_zero_nat))) => (ord_less_nat @ (euclid1226173669ze_nat @ A) @ (euclid1226173669ze_nat @ B)))))))). % dvd_proper_imp_size_less
thf(fact_106_dvd__imp__size__le, axiom,
    ((![A : int, B : int]: ((dvd_dvd_int @ A @ B) => ((~ ((B = zero_zero_int))) => (ord_less_eq_nat @ (euclid1863447361ze_int @ A) @ (euclid1863447361ze_int @ B))))))). % dvd_imp_size_le
thf(fact_107_dvd__imp__size__le, axiom,
    ((![A : nat, B : nat]: ((dvd_dvd_nat @ A @ B) => ((~ ((B = zero_zero_nat))) => (ord_less_eq_nat @ (euclid1226173669ze_nat @ A) @ (euclid1226173669ze_nat @ B))))))). % dvd_imp_size_le
thf(fact_108_unit__iff__euclidean__size, axiom,
    ((![A : int]: ((dvd_dvd_int @ A @ one_one_int) = ((((euclid1863447361ze_int @ A) = (euclid1863447361ze_int @ one_one_int))) & ((~ ((A = zero_zero_int))))))))). % unit_iff_euclidean_size
thf(fact_109_unit__iff__euclidean__size, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ A @ one_one_nat) = ((((euclid1226173669ze_nat @ A) = (euclid1226173669ze_nat @ one_one_nat))) & ((~ ((A = zero_zero_nat))))))))). % unit_iff_euclidean_size
thf(fact_110_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_111_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_112_gr__implies__not__zero, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_113_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_114_zero__le, axiom,
    ((![X : nat]: (ord_less_eq_nat @ zero_zero_nat @ X)))). % zero_le
thf(fact_115_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I
thf(fact_116_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_117_not__less0, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less0
thf(fact_118_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE
thf(fact_119_gr__implies__not0, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not0
thf(fact_120_infinite__descent0, axiom,
    ((![P : nat > $o, N : nat]: ((P @ zero_zero_nat) => ((![N3 : nat]: ((ord_less_nat @ zero_zero_nat @ N3) => ((~ ((P @ N3))) => (?[M3 : nat]: ((ord_less_nat @ M3 @ N3) & (~ ((P @ M3)))))))) => (P @ N)))))). % infinite_descent0
thf(fact_121_bot__nat__0_Oextremum__strict, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_122_less__eq__nat_Osimps_I1_J, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % less_eq_nat.simps(1)
thf(fact_123_le__0__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_0_eq
thf(fact_124_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_125_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_126_take__bit__tightened, axiom,
    ((![N : nat, A : int, B : int, M : nat]: (((semiri1605039314it_int @ N @ A) = (semiri1605039314it_int @ N @ B)) => ((ord_less_eq_nat @ M @ N) => ((semiri1605039314it_int @ M @ A) = (semiri1605039314it_int @ M @ B))))))). % take_bit_tightened
thf(fact_127_take__bit__tightened, axiom,
    ((![N : nat, A : nat, B : nat, M : nat]: (((semiri967765622it_nat @ N @ A) = (semiri967765622it_nat @ N @ B)) => ((ord_less_eq_nat @ M @ N) => ((semiri967765622it_nat @ M @ A) = (semiri967765622it_nat @ M @ B))))))). % take_bit_tightened
thf(fact_128_of__nat__less__0__iff, axiom,
    ((![M : nat]: (~ ((ord_less_nat @ (semiri1382578993at_nat @ M) @ zero_zero_nat)))))). % of_nat_less_0_iff
thf(fact_129_of__nat__less__0__iff, axiom,
    ((![M : nat]: (~ ((ord_less_int @ (semiri2019852685at_int @ M) @ zero_zero_int)))))). % of_nat_less_0_iff
thf(fact_130_of__nat__0__le__iff, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ (semiri1382578993at_nat @ N))))). % of_nat_0_le_iff
thf(fact_131_of__nat__0__le__iff, axiom,
    ((![N : nat]: (ord_less_eq_int @ zero_zero_int @ (semiri2019852685at_int @ N))))). % of_nat_0_le_iff
thf(fact_132_lift__type, axiom,
    ((![E : nat > type, T : dB, T2 : type, I : nat, U : type]: ((typing @ E @ T @ T2) => (typing @ (shift_type @ E @ I @ U) @ (lift @ T @ I) @ T2))))). % lift_type
thf(fact_133_shift__gt, axiom,
    ((![J : nat, I : nat, E : nat > type, T2 : type]: ((ord_less_nat @ J @ I) => ((shift_type @ E @ I @ T2 @ J) = (E @ J)))))). % shift_gt
thf(fact_134_dvd__0__right, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ zero_zero_nat)))). % dvd_0_right
thf(fact_135_dvd__0__right, axiom,
    ((![A : int]: (dvd_dvd_int @ A @ zero_zero_int)))). % dvd_0_right
thf(fact_136_dvd__0__left__iff, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) = (A = zero_zero_nat))))). % dvd_0_left_iff
thf(fact_137_dvd__0__left__iff, axiom,
    ((![A : int]: ((dvd_dvd_int @ zero_zero_int @ A) = (A = zero_zero_int))))). % dvd_0_left_iff
thf(fact_138_shift__eq, axiom,
    ((![I : nat, J : nat, E : nat > type, T2 : type]: ((I = J) => ((shift_type @ E @ I @ T2 @ J) = T2))))). % shift_eq
thf(fact_139_not__is__unit__0, axiom,
    ((~ ((dvd_dvd_nat @ zero_zero_nat @ one_one_nat))))). % not_is_unit_0
thf(fact_140_not__is__unit__0, axiom,
    ((~ ((dvd_dvd_int @ zero_zero_int @ one_one_int))))). % not_is_unit_0
thf(fact_141_less__numeral__extra_I1_J, axiom,
    ((ord_less_nat @ zero_zero_nat @ one_one_nat))). % less_numeral_extra(1)
thf(fact_142_less__numeral__extra_I1_J, axiom,
    ((ord_less_int @ zero_zero_int @ one_one_int))). % less_numeral_extra(1)
thf(fact_143_zero__less__one, axiom,
    ((ord_less_nat @ zero_zero_nat @ one_one_nat))). % zero_less_one
thf(fact_144_zero__less__one, axiom,
    ((ord_less_int @ zero_zero_int @ one_one_int))). % zero_less_one
thf(fact_145_nat__dvd__1__iff__1, axiom,
    ((![M : nat]: ((dvd_dvd_nat @ M @ one_one_nat) = (M = one_one_nat))))). % nat_dvd_1_iff_1
thf(fact_146_le__refl, axiom,
    ((![N : nat]: (ord_less_eq_nat @ N @ N)))). % le_refl
thf(fact_147_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_148_eq__imp__le, axiom,
    ((![M : nat, N : nat]: ((M = N) => (ord_less_eq_nat @ M @ N))))). % eq_imp_le
thf(fact_149_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_150_dvd__antisym, axiom,
    ((![M : nat, N : nat]: ((dvd_dvd_nat @ M @ N) => ((dvd_dvd_nat @ N @ M) => (M = N)))))). % dvd_antisym
thf(fact_151_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_152_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))) => (?[X2 : nat]: ((P @ X2) & (![Y4 : nat]: ((P @ Y4) => (ord_less_eq_nat @ Y4 @ X2)))))))))). % Nat.ex_has_greatest_nat
thf(fact_153_one__integer_Orsp, axiom,
    ((one_one_int = one_one_int))). % one_integer.rsp
thf(fact_154_one__natural_Orsp, axiom,
    ((one_one_nat = one_one_nat))). % one_natural.rsp
thf(fact_155_verit__la__generic, axiom,
    ((![A : int, X : int]: ((ord_less_eq_int @ A @ X) | ((A = X) | (ord_less_eq_int @ X @ A)))))). % verit_la_generic
thf(fact_156_strict__subset__divisors__dvd, axiom,
    ((![A : nat, B : nat]: ((ord_less_set_nat @ (collect_nat @ (^[C : nat]: (dvd_dvd_nat @ C @ A))) @ (collect_nat @ (^[C : nat]: (dvd_dvd_nat @ C @ B)))) = (((dvd_dvd_nat @ A @ B)) & ((~ ((dvd_dvd_nat @ B @ A))))))))). % strict_subset_divisors_dvd
thf(fact_157_strict__subset__divisors__dvd, axiom,
    ((![A : int, B : int]: ((ord_less_set_int @ (collect_int @ (^[C : int]: (dvd_dvd_int @ C @ A))) @ (collect_int @ (^[C : int]: (dvd_dvd_int @ C @ B)))) = (((dvd_dvd_int @ A @ B)) & ((~ ((dvd_dvd_int @ B @ A))))))))). % strict_subset_divisors_dvd
thf(fact_158_not__take__bit__negative, axiom,
    ((![N : nat, K : int]: (~ ((ord_less_int @ (semiri1605039314it_int @ N @ K) @ zero_zero_int)))))). % not_take_bit_negative
thf(fact_159_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_160_nat__less__as__int, axiom,
    ((ord_less_nat = (^[A2 : nat]: (^[B2 : nat]: (ord_less_int @ (semiri2019852685at_int @ A2) @ (semiri2019852685at_int @ B2))))))). % nat_less_as_int
thf(fact_161_linorder__neqE__linordered__idom, axiom,
    ((![X : int, Y2 : int]: ((~ ((X = Y2))) => ((~ ((ord_less_int @ X @ Y2))) => (ord_less_int @ Y2 @ X)))))). % linorder_neqE_linordered_idom
thf(fact_162_dvd__refl, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ A)))). % dvd_refl
thf(fact_163_dvd__refl, axiom,
    ((![A : int]: (dvd_dvd_int @ A @ A)))). % dvd_refl
thf(fact_164_dvd__trans, axiom,
    ((![A : nat, B : nat, C2 : nat]: ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ B @ C2) => (dvd_dvd_nat @ A @ C2)))))). % dvd_trans
thf(fact_165_dvd__trans, axiom,
    ((![A : int, B : int, C2 : int]: ((dvd_dvd_int @ A @ B) => ((dvd_dvd_int @ B @ C2) => (dvd_dvd_int @ A @ C2)))))). % dvd_trans
thf(fact_166_subset__divisors__dvd, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_set_nat @ (collect_nat @ (^[C : nat]: (dvd_dvd_nat @ C @ A))) @ (collect_nat @ (^[C : nat]: (dvd_dvd_nat @ C @ B)))) = (dvd_dvd_nat @ A @ B))))). % subset_divisors_dvd
thf(fact_167_subset__divisors__dvd, axiom,
    ((![A : int, B : int]: ((ord_less_eq_set_int @ (collect_int @ (^[C : int]: (dvd_dvd_int @ C @ A))) @ (collect_int @ (^[C : int]: (dvd_dvd_int @ C @ B)))) = (dvd_dvd_int @ A @ B))))). % subset_divisors_dvd
thf(fact_168_le__numeral__extra_I3_J, axiom,
    ((ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat))). % le_numeral_extra(3)
thf(fact_169_le__numeral__extra_I3_J, axiom,
    ((ord_less_eq_int @ zero_zero_int @ zero_zero_int))). % le_numeral_extra(3)
thf(fact_170_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_nat @ zero_zero_nat @ zero_zero_nat))))). % less_numeral_extra(3)
thf(fact_171_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_int @ zero_zero_int @ zero_zero_int))))). % less_numeral_extra(3)
thf(fact_172_le__numeral__extra_I4_J, axiom,
    ((ord_less_eq_nat @ one_one_nat @ one_one_nat))). % le_numeral_extra(4)
thf(fact_173_le__numeral__extra_I4_J, axiom,
    ((ord_less_eq_int @ one_one_int @ one_one_int))). % le_numeral_extra(4)
thf(fact_174_zero__neq__one, axiom,
    ((~ ((zero_zero_nat = one_one_nat))))). % zero_neq_one
thf(fact_175_zero__neq__one, axiom,
    ((~ ((zero_zero_int = one_one_int))))). % zero_neq_one
thf(fact_176_less__numeral__extra_I4_J, axiom,
    ((~ ((ord_less_nat @ one_one_nat @ one_one_nat))))). % less_numeral_extra(4)
thf(fact_177_less__numeral__extra_I4_J, axiom,
    ((~ ((ord_less_int @ one_one_int @ one_one_int))))). % less_numeral_extra(4)
thf(fact_178_dvd__0__left, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) => (A = zero_zero_nat))))). % dvd_0_left
thf(fact_179_dvd__0__left, axiom,
    ((![A : int]: ((dvd_dvd_int @ zero_zero_int @ A) => (A = zero_zero_int))))). % dvd_0_left
thf(fact_180_one__dvd, axiom,
    ((![A : nat]: (dvd_dvd_nat @ one_one_nat @ A)))). % one_dvd
thf(fact_181_one__dvd, axiom,
    ((![A : int]: (dvd_dvd_int @ one_one_int @ A)))). % one_dvd
thf(fact_182_unit__imp__dvd, axiom,
    ((![B : nat, A : nat]: ((dvd_dvd_nat @ B @ one_one_nat) => (dvd_dvd_nat @ B @ A))))). % unit_imp_dvd
thf(fact_183_unit__imp__dvd, axiom,
    ((![B : int, A : int]: ((dvd_dvd_int @ B @ one_one_int) => (dvd_dvd_int @ B @ A))))). % unit_imp_dvd
thf(fact_184_dvd__unit__imp__unit, axiom,
    ((![A : nat, B : nat]: ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ B @ one_one_nat) => (dvd_dvd_nat @ A @ one_one_nat)))))). % dvd_unit_imp_unit
thf(fact_185_dvd__unit__imp__unit, axiom,
    ((![A : int, B : int]: ((dvd_dvd_int @ A @ B) => ((dvd_dvd_int @ B @ one_one_int) => (dvd_dvd_int @ A @ one_one_int)))))). % dvd_unit_imp_unit
thf(fact_186_not__one__le__zero, axiom,
    ((~ ((ord_less_eq_nat @ one_one_nat @ zero_zero_nat))))). % not_one_le_zero
thf(fact_187_not__one__le__zero, axiom,
    ((~ ((ord_less_eq_int @ one_one_int @ zero_zero_int))))). % not_one_le_zero
thf(fact_188_zero__le__one, axiom,
    ((ord_less_eq_nat @ zero_zero_nat @ one_one_nat))). % zero_le_one
thf(fact_189_zero__le__one, axiom,
    ((ord_less_eq_int @ zero_zero_int @ one_one_int))). % zero_le_one
thf(fact_190_not__one__less__zero, axiom,
    ((~ ((ord_less_nat @ one_one_nat @ zero_zero_nat))))). % not_one_less_zero
thf(fact_191_not__one__less__zero, axiom,
    ((~ ((ord_less_int @ one_one_int @ zero_zero_int))))). % not_one_less_zero
thf(fact_192_int__dvd__int__iff, axiom,
    ((![M : nat, N : nat]: ((dvd_dvd_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)) = (dvd_dvd_nat @ M @ N))))). % int_dvd_int_iff

% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T, axiom,
    ((![P : $o]: ((P = $true) | (P = $false))))).
thf(help_If_2_1_If_001t__Nat__Onat_T, axiom,
    ((![X : nat, Y2 : nat]: ((if_nat @ $false @ X @ Y2) = Y2)))).
thf(help_If_1_1_If_001t__Nat__Onat_T, axiom,
    ((![X : nat, Y2 : nat]: ((if_nat @ $true @ X @ Y2) = X)))).

% Conjectures (1)
thf(conj_0, conjecture,
    ((it @ (lift @ u @ zero_zero_nat)))).
