% 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/Hoare/prob_149__3250838_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:12:22.591

% Could-be-implicit typings (4)
thf(ty_n_t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    set_Ho137910533iple_a : $tType).
thf(ty_n_t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    hoare_1678595023iple_a : $tType).
thf(ty_n_t__Com__Ostate, type,
    state : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (12)
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat, type,
    minus_minus_nat : nat > nat > nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat, type,
    plus_plus_nat : nat > nat > nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Ohoare__valids_001tf__a, type,
    hoare_1775499016lids_a : set_Ho137910533iple_a > set_Ho137910533iple_a > $o).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Ostate__not__singleton, type,
    hoare_405891322gleton : $o).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Otriple__valid_001tf__a, type,
    hoare_1926814542alid_a : nat > hoare_1678595023iple_a > $o).
thf(sy_c_Nat_OSuc, type,
    suc : nat > nat).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat, type,
    ord_less_eq_nat : nat > nat > $o).
thf(sy_c_member_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    member1332298086iple_a : hoare_1678595023iple_a > set_Ho137910533iple_a > $o).
thf(sy_v_n, type,
    n : nat).
thf(sy_v_ts, type,
    ts : set_Ho137910533iple_a).

% Relevant facts (127)
thf(fact_0_triple__valid__Suc, axiom,
    ((![N : nat, T : hoare_1678595023iple_a]: ((hoare_1926814542alid_a @ (suc @ N) @ T) => (hoare_1926814542alid_a @ N @ T))))). % triple_valid_Suc
thf(fact_1_hoare__valids__def, axiom,
    ((hoare_1775499016lids_a = (^[G : set_Ho137910533iple_a]: (^[Ts : set_Ho137910533iple_a]: (![N2 : nat]: (((![X : hoare_1678595023iple_a]: (((member1332298086iple_a @ X @ G)) => ((hoare_1926814542alid_a @ N2 @ X))))) => ((![X : hoare_1678595023iple_a]: (((member1332298086iple_a @ X @ Ts)) => ((hoare_1926814542alid_a @ N2 @ X)))))))))))). % hoare_valids_def
thf(fact_2_nat_Oinject, axiom,
    ((![X2 : nat, Y2 : nat]: (((suc @ X2) = (suc @ Y2)) = (X2 = Y2))))). % nat.inject
thf(fact_3_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_4_Suc__inject, axiom,
    ((![X3 : nat, Y : nat]: (((suc @ X3) = (suc @ Y)) => (X3 = Y))))). % Suc_inject
thf(fact_5_n__not__Suc__n, axiom,
    ((![N : nat]: (~ ((N = (suc @ N))))))). % n_not_Suc_n
thf(fact_6_single__stateE, axiom,
    ((hoare_405891322gleton => (![T2 : state]: (~ ((![S : state]: (S = T2)))))))). % single_stateE
thf(fact_7_state__not__singleton__def, axiom,
    ((hoare_405891322gleton = (?[S2 : state]: (?[T3 : state]: (~ ((S2 = T3)))))))). % state_not_singleton_def
thf(fact_8_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_9_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_10_Suc__mono, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_nat @ (suc @ M) @ (suc @ N)))))). % Suc_mono
thf(fact_11_lessI, axiom,
    ((![N : nat]: (ord_less_nat @ N @ (suc @ N))))). % lessI
thf(fact_12_add__Suc__right, axiom,
    ((![M : nat, N : nat]: ((plus_plus_nat @ M @ (suc @ N)) = (suc @ (plus_plus_nat @ M @ N)))))). % add_Suc_right
thf(fact_13_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_14_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_15_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_16_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_17_le0, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % le0
thf(fact_18_bot__nat__0_Oextremum, axiom,
    ((![A : nat]: (ord_less_eq_nat @ zero_zero_nat @ A)))). % bot_nat_0.extremum
thf(fact_19_add__is__0, axiom,
    ((![M : nat, N : nat]: (((plus_plus_nat @ M @ N) = zero_zero_nat) = (((M = zero_zero_nat)) & ((N = zero_zero_nat))))))). % add_is_0
thf(fact_20_Nat_Oadd__0__right, axiom,
    ((![M : nat]: ((plus_plus_nat @ M @ zero_zero_nat) = M)))). % Nat.add_0_right
thf(fact_21_nat__add__left__cancel__less, axiom,
    ((![K : nat, M : nat, N : nat]: ((ord_less_nat @ (plus_plus_nat @ K @ M) @ (plus_plus_nat @ K @ N)) = (ord_less_nat @ M @ N))))). % nat_add_left_cancel_less
thf(fact_22_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_23_diff__0__eq__0, axiom,
    ((![N : nat]: ((minus_minus_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % diff_0_eq_0
thf(fact_24_diff__self__eq__0, axiom,
    ((![M : nat]: ((minus_minus_nat @ M @ M) = zero_zero_nat)))). % diff_self_eq_0
thf(fact_25_nat__add__left__cancel__le, axiom,
    ((![K : nat, M : nat, N : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ K @ M) @ (plus_plus_nat @ K @ N)) = (ord_less_eq_nat @ M @ N))))). % nat_add_left_cancel_le
thf(fact_26_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_27_diff__diff__left, axiom,
    ((![I : nat, J : nat, K : nat]: ((minus_minus_nat @ (minus_minus_nat @ I @ J) @ K) = (minus_minus_nat @ I @ (plus_plus_nat @ J @ K)))))). % diff_diff_left
thf(fact_28_zero__less__Suc, axiom,
    ((![N : nat]: (ord_less_nat @ zero_zero_nat @ (suc @ N))))). % zero_less_Suc
thf(fact_29_less__Suc0, axiom,
    ((![N : nat]: ((ord_less_nat @ N @ (suc @ zero_zero_nat)) = (N = zero_zero_nat))))). % less_Suc0
thf(fact_30_add__gr__0, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (plus_plus_nat @ M @ N)) = (((ord_less_nat @ zero_zero_nat @ M)) | ((ord_less_nat @ zero_zero_nat @ N))))))). % add_gr_0
thf(fact_31_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_32_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_33_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_34_Nat_Oadd__diff__assoc, axiom,
    ((![K : nat, J : nat, I : nat]: ((ord_less_eq_nat @ K @ J) => ((plus_plus_nat @ I @ (minus_minus_nat @ J @ K)) = (minus_minus_nat @ (plus_plus_nat @ I @ J) @ K)))))). % Nat.add_diff_assoc
thf(fact_35_Nat_Oadd__diff__assoc2, axiom,
    ((![K : nat, J : nat, I : nat]: ((ord_less_eq_nat @ K @ J) => ((plus_plus_nat @ (minus_minus_nat @ J @ K) @ I) = (minus_minus_nat @ (plus_plus_nat @ J @ I) @ K)))))). % Nat.add_diff_assoc2
thf(fact_36_Nat_Odiff__diff__right, axiom,
    ((![K : nat, J : nat, I : nat]: ((ord_less_eq_nat @ K @ J) => ((minus_minus_nat @ I @ (minus_minus_nat @ J @ K)) = (minus_minus_nat @ (plus_plus_nat @ I @ K) @ J)))))). % Nat.diff_diff_right
thf(fact_37_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_38_diff__Suc__diff__eq1, axiom,
    ((![K : nat, J : nat, I : nat]: ((ord_less_eq_nat @ K @ J) => ((minus_minus_nat @ I @ (suc @ (minus_minus_nat @ J @ K))) = (minus_minus_nat @ (plus_plus_nat @ I @ K) @ (suc @ J))))))). % diff_Suc_diff_eq1
thf(fact_39_diff__Suc__diff__eq2, axiom,
    ((![K : nat, J : nat, I : nat]: ((ord_less_eq_nat @ K @ J) => ((minus_minus_nat @ (suc @ (minus_minus_nat @ J @ K)) @ I) = (minus_minus_nat @ (suc @ J) @ (plus_plus_nat @ K @ I))))))). % diff_Suc_diff_eq2
thf(fact_40_less__eq__nat_Osimps_I1_J, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % less_eq_nat.simps(1)
thf(fact_41_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I
thf(fact_42_Suc__leI, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_eq_nat @ (suc @ M) @ N))))). % Suc_leI
thf(fact_43_add__leE, axiom,
    ((![M : nat, K : nat, N : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ M @ K) @ N) => (~ (((ord_less_eq_nat @ M @ N) => (~ ((ord_less_eq_nat @ K @ N)))))))))). % add_leE
thf(fact_44_le__0__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_0_eq
thf(fact_45_le__add1, axiom,
    ((![N : nat, M : nat]: (ord_less_eq_nat @ N @ (plus_plus_nat @ N @ M))))). % le_add1
thf(fact_46_le__add2, axiom,
    ((![N : nat, M : nat]: (ord_less_eq_nat @ N @ (plus_plus_nat @ M @ N))))). % le_add2
thf(fact_47_le__refl, axiom,
    ((![N : nat]: (ord_less_eq_nat @ N @ N)))). % le_refl
thf(fact_48_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_49_add__leD1, axiom,
    ((![M : nat, K : nat, N : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ M @ K) @ N) => (ord_less_eq_nat @ M @ N))))). % add_leD1
thf(fact_50_add__leD2, axiom,
    ((![M : nat, K : nat, N : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ M @ K) @ N) => (ord_less_eq_nat @ K @ N))))). % add_leD2
thf(fact_51_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_52_Suc__le__eq, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ (suc @ M) @ N) = (ord_less_nat @ M @ N))))). % Suc_le_eq
thf(fact_53_diff__less, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_nat @ zero_zero_nat @ M) => (ord_less_nat @ (minus_minus_nat @ M @ N) @ M)))))). % diff_less
thf(fact_54_eq__imp__le, axiom,
    ((![M : nat, N : nat]: ((M = N) => (ord_less_eq_nat @ M @ N))))). % eq_imp_le
thf(fact_55_le__Suc__ex, axiom,
    ((![K : nat, L : nat]: ((ord_less_eq_nat @ K @ L) => (?[N3 : nat]: (L = (plus_plus_nat @ K @ N3))))))). % le_Suc_ex
thf(fact_56_less__natE, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((![Q : nat]: (~ ((N = (suc @ (plus_plus_nat @ M @ Q)))))))))))). % less_natE
thf(fact_57_not__less0, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less0
thf(fact_58_add__lessD1, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_nat @ (plus_plus_nat @ I @ J) @ K) => (ord_less_nat @ I @ K))))). % add_lessD1
thf(fact_59_dec__induct, axiom,
    ((![I : nat, J : nat, P : nat > $o]: ((ord_less_eq_nat @ I @ J) => ((P @ I) => ((![N3 : nat]: ((ord_less_eq_nat @ I @ N3) => ((ord_less_nat @ N3 @ J) => ((P @ N3) => (P @ (suc @ N3)))))) => (P @ J))))))). % dec_induct
thf(fact_60_diff__add__0, axiom,
    ((![N : nat, M : nat]: ((minus_minus_nat @ N @ (plus_plus_nat @ N @ M)) = zero_zero_nat)))). % diff_add_0
thf(fact_61_inc__induct, axiom,
    ((![I : nat, J : nat, P : nat > $o]: ((ord_less_eq_nat @ I @ J) => ((P @ J) => ((![N3 : nat]: ((ord_less_eq_nat @ I @ N3) => ((ord_less_nat @ N3 @ J) => ((P @ (suc @ N3)) => (P @ N3))))) => (P @ I))))))). % inc_induct
thf(fact_62_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_63_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE
thf(fact_64_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_65_add__le__mono, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: ((ord_less_eq_nat @ I @ J) => ((ord_less_eq_nat @ K @ L) => (ord_less_eq_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L))))))). % add_le_mono
thf(fact_66_Nat_Odiff__cancel, axiom,
    ((![K : nat, M : nat, N : nat]: ((minus_minus_nat @ (plus_plus_nat @ K @ M) @ (plus_plus_nat @ K @ N)) = (minus_minus_nat @ M @ N))))). % Nat.diff_cancel
thf(fact_67_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_68_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_69_nat__less__le, axiom,
    ((ord_less_nat = (^[M2 : nat]: (^[N2 : nat]: (((ord_less_eq_nat @ M2 @ N2)) & ((~ ((M2 = N2)))))))))). % nat_less_le
thf(fact_70_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_71_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_72_Suc__le__lessD, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ (suc @ M) @ N) => (ord_less_nat @ M @ N))))). % Suc_le_lessD
thf(fact_73_add__le__mono1, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_eq_nat @ I @ J) => (ord_less_eq_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ K)))))). % add_le_mono1
thf(fact_74_diff__cancel2, axiom,
    ((![M : nat, K : nat, N : nat]: ((minus_minus_nat @ (plus_plus_nat @ M @ K) @ (plus_plus_nat @ N @ K)) = (minus_minus_nat @ M @ N))))). % diff_cancel2
thf(fact_75_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_76_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_77_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_78_diff__le__self, axiom,
    ((![M : nat, N : nat]: (ord_less_eq_nat @ (minus_minus_nat @ M @ N) @ M)))). % diff_le_self
thf(fact_79_le__diff__conv, axiom,
    ((![J : nat, K : nat, I : nat]: ((ord_less_eq_nat @ (minus_minus_nat @ J @ K) @ I) = (ord_less_eq_nat @ J @ (plus_plus_nat @ I @ K)))))). % le_diff_conv
thf(fact_80_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_81_add__less__mono, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: ((ord_less_nat @ I @ J) => ((ord_less_nat @ K @ L) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L))))))). % add_less_mono
thf(fact_82_diff__Suc__less, axiom,
    ((![N : nat, I : nat]: ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_nat @ (minus_minus_nat @ N @ (suc @ I)) @ N))))). % diff_Suc_less
thf(fact_83_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_84_diff__less__Suc, axiom,
    ((![M : nat, N : nat]: (ord_less_nat @ (minus_minus_nat @ M @ N) @ (suc @ M))))). % diff_less_Suc
thf(fact_85_Nat_Ole__diff__conv2, axiom,
    ((![K : nat, J : nat, I : nat]: ((ord_less_eq_nat @ K @ J) => ((ord_less_eq_nat @ I @ (minus_minus_nat @ J @ K)) = (ord_less_eq_nat @ (plus_plus_nat @ I @ K) @ J)))))). % Nat.le_diff_conv2
thf(fact_86_less__add__Suc1, axiom,
    ((![I : nat, M : nat]: (ord_less_nat @ I @ (suc @ (plus_plus_nat @ I @ M)))))). % less_add_Suc1
thf(fact_87_less__add__Suc2, axiom,
    ((![I : nat, M : nat]: (ord_less_nat @ I @ (suc @ (plus_plus_nat @ M @ I)))))). % less_add_Suc2
thf(fact_88_less__diff__iff, axiom,
    ((![K : nat, M : nat, N : nat]: ((ord_less_eq_nat @ K @ M) => ((ord_less_eq_nat @ K @ N) => ((ord_less_nat @ (minus_minus_nat @ M @ K) @ (minus_minus_nat @ N @ K)) = (ord_less_nat @ M @ N))))))). % less_diff_iff
thf(fact_89_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_90_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_91_not__add__less1, axiom,
    ((![I : nat, J : nat]: (~ ((ord_less_nat @ (plus_plus_nat @ I @ J) @ I)))))). % not_add_less1
thf(fact_92_not__add__less2, axiom,
    ((![J : nat, I : nat]: (~ ((ord_less_nat @ (plus_plus_nat @ J @ I) @ I)))))). % not_add_less2
thf(fact_93_trans__le__add1, axiom,
    ((![I : nat, J : nat, M : nat]: ((ord_less_eq_nat @ I @ J) => (ord_less_eq_nat @ I @ (plus_plus_nat @ J @ M)))))). % trans_le_add1
thf(fact_94_trans__le__add2, axiom,
    ((![I : nat, J : nat, M : nat]: ((ord_less_eq_nat @ I @ J) => (ord_less_eq_nat @ I @ (plus_plus_nat @ M @ J)))))). % trans_le_add2
thf(fact_95_add__less__mono1, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_nat @ I @ J) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ K)))))). % add_less_mono1
thf(fact_96_Nat_Odiff__add__assoc, axiom,
    ((![K : nat, J : nat, I : nat]: ((ord_less_eq_nat @ K @ J) => ((minus_minus_nat @ (plus_plus_nat @ I @ J) @ K) = (plus_plus_nat @ I @ (minus_minus_nat @ J @ K))))))). % Nat.diff_add_assoc
thf(fact_97_diff__less__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_eq_nat @ C @ A) => (ord_less_nat @ (minus_minus_nat @ A @ C) @ (minus_minus_nat @ B @ C))))))). % diff_less_mono
thf(fact_98_le__less__Suc__eq, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ N) => ((ord_less_nat @ N @ (suc @ M)) = (N = M)))))). % le_less_Suc_eq
thf(fact_99_less__Suc__eq__le, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ (suc @ N)) = (ord_less_eq_nat @ M @ N))))). % less_Suc_eq_le
thf(fact_100_less__diff__conv, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_nat @ I @ (minus_minus_nat @ J @ K)) = (ord_less_nat @ (plus_plus_nat @ I @ K) @ J))))). % less_diff_conv
thf(fact_101_less__eq__Suc__le, axiom,
    ((ord_less_nat = (^[N2 : nat]: (ord_less_eq_nat @ (suc @ N2)))))). % less_eq_Suc_le
thf(fact_102_less__not__refl2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ N @ M) => (~ ((M = N))))))). % less_not_refl2
thf(fact_103_less__not__refl3, axiom,
    ((![S3 : nat, T : nat]: ((ord_less_nat @ S3 @ T) => (~ ((S3 = T))))))). % less_not_refl3
thf(fact_104_nat__diff__split, axiom,
    ((![P : nat > $o, A : nat, B : nat]: ((P @ (minus_minus_nat @ A @ B)) = (((((ord_less_nat @ A @ B)) => ((P @ zero_zero_nat)))) & ((![D : nat]: (((A = (plus_plus_nat @ B @ D))) => ((P @ D)))))))))). % nat_diff_split
thf(fact_105_nat__le__iff__add, axiom,
    ((ord_less_eq_nat = (^[M2 : nat]: (^[N2 : nat]: (?[K2 : nat]: (N2 = (plus_plus_nat @ M2 @ K2)))))))). % nat_le_iff_add
thf(fact_106_plus__nat_Oadd__0, axiom,
    ((![N : nat]: ((plus_plus_nat @ zero_zero_nat @ N) = N)))). % plus_nat.add_0
thf(fact_107_Nat_Odiff__add__assoc2, axiom,
    ((![K : nat, J : nat, I : nat]: ((ord_less_eq_nat @ K @ J) => ((minus_minus_nat @ (plus_plus_nat @ J @ I) @ K) = (plus_plus_nat @ (minus_minus_nat @ J @ K) @ I)))))). % Nat.diff_add_assoc2
thf(fact_108_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_109_ex__least__nat__le, axiom,
    ((![P : nat > $o, N : nat]: ((P @ N) => ((~ ((P @ zero_zero_nat))) => (?[K3 : nat]: ((ord_less_eq_nat @ K3 @ N) & ((![I2 : nat]: ((ord_less_nat @ I2 @ K3) => (~ ((P @ I2))))) & (P @ K3))))))))). % ex_least_nat_le
thf(fact_110_gr__implies__not0, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not0
thf(fact_111_le__imp__less__Suc, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ N) => (ord_less_nat @ M @ (suc @ N)))))). % le_imp_less_Suc
thf(fact_112_less__diff__conv2, axiom,
    ((![K : nat, J : nat, I : nat]: ((ord_less_eq_nat @ K @ J) => ((ord_less_nat @ (minus_minus_nat @ J @ K) @ I) = (ord_less_nat @ J @ (plus_plus_nat @ I @ K))))))). % less_diff_conv2
thf(fact_113_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_114_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_115_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_116_trans__less__add1, axiom,
    ((![I : nat, J : nat, M : nat]: ((ord_less_nat @ I @ J) => (ord_less_nat @ I @ (plus_plus_nat @ J @ M)))))). % trans_less_add1
thf(fact_117_trans__less__add2, axiom,
    ((![I : nat, J : nat, M : nat]: ((ord_less_nat @ I @ J) => (ord_less_nat @ I @ (plus_plus_nat @ M @ J)))))). % trans_less_add2
thf(fact_118_add__eq__self__zero, axiom,
    ((![M : nat, N : nat]: (((plus_plus_nat @ M @ N) = M) => (N = zero_zero_nat))))). % add_eq_self_zero
thf(fact_119_diff__add__inverse, axiom,
    ((![N : nat, M : nat]: ((minus_minus_nat @ (plus_plus_nat @ N @ M) @ N) = M)))). % diff_add_inverse
thf(fact_120_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_121_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_122_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_123_less__add__eq__less, axiom,
    ((![K : nat, L : nat, M : nat, N : nat]: ((ord_less_nat @ K @ L) => (((plus_plus_nat @ M @ L) = (plus_plus_nat @ K @ N)) => (ord_less_nat @ M @ N)))))). % less_add_eq_less
thf(fact_124_less__iff__Suc__add, axiom,
    ((ord_less_nat = (^[M2 : nat]: (^[N2 : nat]: (?[K2 : nat]: (N2 = (suc @ (plus_plus_nat @ M2 @ K2))))))))). % less_iff_Suc_add
thf(fact_125_less__imp__Suc__add, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (?[K3 : nat]: (N = (suc @ (plus_plus_nat @ M @ K3)))))))). % less_imp_Suc_add
thf(fact_126_minus__nat_Odiff__0, axiom,
    ((![M : nat]: ((minus_minus_nat @ M @ zero_zero_nat) = M)))). % minus_nat.diff_0

% Conjectures (2)
thf(conj_0, hypothesis,
    ((![X4 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X4 @ ts) => (hoare_1926814542alid_a @ (suc @ n) @ X4))))).
thf(conj_1, conjecture,
    ((![X5 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X5 @ ts) => (hoare_1926814542alid_a @ n @ X5))))).
