% 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_283__3252288_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:13:26.400

% 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__Set__Oset_It__Nat__Onat_J, type,
    set_nat : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (23)
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_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_Obot__class_Obot_001_062_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_M_Eo_J, type,
    bot_bo431311916le_a_o : hoare_1678595023iple_a > $o).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat, type,
    bot_bot_nat : nat).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    bot_bo1298296729iple_a : set_Ho137910533iple_a).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Nat__Onat_J, type,
    ord_less_eq_o_nat : ($o > nat) > ($o > nat) > $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__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    ord_le1221261669iple_a : set_Ho137910533iple_a > set_Ho137910533iple_a > $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_Orderings_Oorder__class_OGreatest_001t__Nat__Onat, type,
    order_Greatest_nat : (nat > $o) > nat).
thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Nat__Onat, type,
    order_1631207636at_nat : (nat > nat) > $o).
thf(sy_c_Set_OBall_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    ball_H465710501iple_a : set_Ho137910533iple_a > (hoare_1678595023iple_a > $o) > $o).
thf(sy_c_Set_OCollect_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    collec1600235172iple_a : (hoare_1678595023iple_a > $o) > set_Ho137910533iple_a).
thf(sy_c_Set_Obind_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    bind_H209762450iple_a : set_Ho137910533iple_a > (hoare_1678595023iple_a > set_Ho137910533iple_a) > set_Ho137910533iple_a).
thf(sy_c_Set_Odisjnt_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    disjnt925686856iple_a : set_Ho137910533iple_a > set_Ho137910533iple_a > $o).
thf(sy_c_Set_Oinsert_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    insert1477804543iple_a : hoare_1678595023iple_a > set_Ho137910533iple_a > set_Ho137910533iple_a).
thf(sy_c_Set_Ois__empty_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    is_emp901906557iple_a : set_Ho137910533iple_a > $o).
thf(sy_c_Set_Ois__singleton_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    is_sin1784037339iple_a : set_Ho137910533iple_a > $o).
thf(sy_c_Set_Opairwise_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    pairwi531237284iple_a : (hoare_1678595023iple_a > hoare_1678595023iple_a > $o) > set_Ho137910533iple_a > $o).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat, type,
    set_or1965240170an_nat : nat > set_nat).
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_Ga, type,
    ga : set_Ho137910533iple_a).

% Relevant facts (122)
thf(fact_0_ball__empty, axiom,
    ((![P : hoare_1678595023iple_a > $o, X : hoare_1678595023iple_a]: ((member1332298086iple_a @ X @ bot_bo1298296729iple_a) => (P @ X))))). % ball_empty
thf(fact_1_empty__iff, axiom,
    ((![C : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ C @ bot_bo1298296729iple_a)))))). % empty_iff
thf(fact_2_all__not__in__conv, axiom,
    ((![A : set_Ho137910533iple_a]: ((![X2 : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ X2 @ A)))) = (A = bot_bo1298296729iple_a))))). % all_not_in_conv
thf(fact_3_Collect__empty__eq, axiom,
    ((![P : hoare_1678595023iple_a > $o]: (((collec1600235172iple_a @ P) = bot_bo1298296729iple_a) = (![X2 : hoare_1678595023iple_a]: (~ ((P @ X2)))))))). % Collect_empty_eq
thf(fact_4_empty__Collect__eq, axiom,
    ((![P : hoare_1678595023iple_a > $o]: ((bot_bo1298296729iple_a = (collec1600235172iple_a @ P)) = (![X2 : hoare_1678595023iple_a]: (~ ((P @ X2)))))))). % empty_Collect_eq
thf(fact_5_hoare__valids__def, axiom,
    ((hoare_1775499016lids_a = (^[G : set_Ho137910533iple_a]: (^[Ts : set_Ho137910533iple_a]: (![N : nat]: (((![X2 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X2 @ G)) => ((hoare_1926814542alid_a @ N @ X2))))) => ((![X2 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X2 @ Ts)) => ((hoare_1926814542alid_a @ N @ X2)))))))))))). % hoare_valids_def
thf(fact_6_Ball__def, axiom,
    ((ball_H465710501iple_a = (^[A2 : set_Ho137910533iple_a]: (^[P2 : hoare_1678595023iple_a > $o]: (![X2 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X2 @ A2)) => ((P2 @ X2))))))))). % Ball_def
thf(fact_7_ball__reg, axiom,
    ((![R : set_Ho137910533iple_a, P : hoare_1678595023iple_a > $o, Q : hoare_1678595023iple_a > $o]: ((![X3 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X3 @ R) => ((P @ X3) => (Q @ X3)))) => ((![X3 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X3 @ R) => (P @ X3))) => (![X : hoare_1678595023iple_a]: ((member1332298086iple_a @ X @ R) => (Q @ X)))))))). % ball_reg
thf(fact_8_emptyE, axiom,
    ((![A3 : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ A3 @ bot_bo1298296729iple_a)))))). % emptyE
thf(fact_9_equals0D, axiom,
    ((![A : set_Ho137910533iple_a, A3 : hoare_1678595023iple_a]: ((A = bot_bo1298296729iple_a) => (~ ((member1332298086iple_a @ A3 @ A))))))). % equals0D
thf(fact_10_equals0I, axiom,
    ((![A : set_Ho137910533iple_a]: ((![Y : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ Y @ A)))) => (A = bot_bo1298296729iple_a))))). % equals0I
thf(fact_11_bot__set__def, axiom,
    ((bot_bo1298296729iple_a = (collec1600235172iple_a @ bot_bo431311916le_a_o)))). % bot_set_def
thf(fact_12_ex__in__conv, axiom,
    ((![A : set_Ho137910533iple_a]: ((?[X2 : hoare_1678595023iple_a]: (member1332298086iple_a @ X2 @ A)) = (~ ((A = bot_bo1298296729iple_a))))))). % ex_in_conv
thf(fact_13_Set_Ois__empty__def, axiom,
    ((is_emp901906557iple_a = (^[A2 : set_Ho137910533iple_a]: (A2 = bot_bo1298296729iple_a))))). % Set.is_empty_def
thf(fact_14_Collect__empty__eq__bot, axiom,
    ((![P : hoare_1678595023iple_a > $o]: (((collec1600235172iple_a @ P) = bot_bo1298296729iple_a) = (P = bot_bo431311916le_a_o))))). % Collect_empty_eq_bot
thf(fact_15_bot__empty__eq, axiom,
    ((bot_bo431311916le_a_o = (^[X2 : hoare_1678595023iple_a]: (member1332298086iple_a @ X2 @ bot_bo1298296729iple_a))))). % bot_empty_eq
thf(fact_16_triples__valid__Suc, axiom,
    ((![Ts2 : set_Ho137910533iple_a, N2 : nat]: ((![X3 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X3 @ Ts2) => (hoare_1926814542alid_a @ (suc @ N2) @ X3))) => (![X : hoare_1678595023iple_a]: ((member1332298086iple_a @ X @ Ts2) => (hoare_1926814542alid_a @ N2 @ X))))))). % triples_valid_Suc
thf(fact_17_is__singletonI_H, axiom,
    ((![A : set_Ho137910533iple_a]: ((~ ((A = bot_bo1298296729iple_a))) => ((![X3 : hoare_1678595023iple_a, Y : hoare_1678595023iple_a]: ((member1332298086iple_a @ X3 @ A) => ((member1332298086iple_a @ Y @ A) => (X3 = Y)))) => (is_sin1784037339iple_a @ A)))))). % is_singletonI'
thf(fact_18_empty__bind, axiom,
    ((![F : hoare_1678595023iple_a > set_Ho137910533iple_a]: ((bind_H209762450iple_a @ bot_bo1298296729iple_a @ F) = bot_bo1298296729iple_a)))). % empty_bind
thf(fact_19_disjnt__self__iff__empty, axiom,
    ((![S : set_Ho137910533iple_a]: ((disjnt925686856iple_a @ S @ S) = (S = bot_bo1298296729iple_a))))). % disjnt_self_iff_empty
thf(fact_20_pairwise__empty, axiom,
    ((![P : hoare_1678595023iple_a > hoare_1678595023iple_a > $o]: (pairwi531237284iple_a @ P @ bot_bo1298296729iple_a)))). % pairwise_empty
thf(fact_21_triple__valid__Suc, axiom,
    ((![N2 : nat, T : hoare_1678595023iple_a]: ((hoare_1926814542alid_a @ (suc @ N2) @ T) => (hoare_1926814542alid_a @ N2 @ T))))). % triple_valid_Suc
thf(fact_22_empty__subsetI, axiom,
    ((![A : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ bot_bo1298296729iple_a @ A)))). % empty_subsetI
thf(fact_23_order__refl, axiom,
    ((![X4 : nat]: (ord_less_eq_nat @ X4 @ X4)))). % order_refl
thf(fact_24_subsetI, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((![X3 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X3 @ A) => (member1332298086iple_a @ X3 @ B))) => (ord_le1221261669iple_a @ A @ B))))). % subsetI
thf(fact_25_subset__empty, axiom,
    ((![A : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ bot_bo1298296729iple_a) = (A = bot_bo1298296729iple_a))))). % subset_empty
thf(fact_26_in__mono, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, X4 : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A @ B) => ((member1332298086iple_a @ X4 @ A) => (member1332298086iple_a @ X4 @ B)))))). % in_mono
thf(fact_27_subsetD, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, C : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A @ B) => ((member1332298086iple_a @ C @ A) => (member1332298086iple_a @ C @ B)))))). % subsetD
thf(fact_28_pairwiseD, axiom,
    ((![R : hoare_1678595023iple_a > hoare_1678595023iple_a > $o, S : set_Ho137910533iple_a, X4 : hoare_1678595023iple_a, Y2 : hoare_1678595023iple_a]: ((pairwi531237284iple_a @ R @ S) => ((member1332298086iple_a @ X4 @ S) => ((member1332298086iple_a @ Y2 @ S) => ((~ ((X4 = Y2))) => (R @ X4 @ Y2)))))))). % pairwiseD
thf(fact_29_pairwiseI, axiom,
    ((![S : set_Ho137910533iple_a, R : hoare_1678595023iple_a > hoare_1678595023iple_a > $o]: ((![X3 : hoare_1678595023iple_a, Y : hoare_1678595023iple_a]: ((member1332298086iple_a @ X3 @ S) => ((member1332298086iple_a @ Y @ S) => ((~ ((X3 = Y))) => (R @ X3 @ Y))))) => (pairwi531237284iple_a @ R @ S))))). % pairwiseI
thf(fact_30_subset__eq, axiom,
    ((ord_le1221261669iple_a = (^[A2 : set_Ho137910533iple_a]: (^[B2 : set_Ho137910533iple_a]: (![X2 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X2 @ A2)) => ((member1332298086iple_a @ X2 @ B2))))))))). % subset_eq
thf(fact_31_disjnt__iff, axiom,
    ((disjnt925686856iple_a = (^[A2 : set_Ho137910533iple_a]: (^[B2 : set_Ho137910533iple_a]: (![X2 : hoare_1678595023iple_a]: (~ ((((member1332298086iple_a @ X2 @ A2)) & ((member1332298086iple_a @ X2 @ B2))))))))))). % disjnt_iff
thf(fact_32_subset__iff, axiom,
    ((ord_le1221261669iple_a = (^[A2 : set_Ho137910533iple_a]: (^[B2 : set_Ho137910533iple_a]: (![T2 : hoare_1678595023iple_a]: (((member1332298086iple_a @ T2 @ A2)) => ((member1332298086iple_a @ T2 @ B2))))))))). % subset_iff
thf(fact_33_Collect__mono, axiom,
    ((![P : hoare_1678595023iple_a > $o, Q : hoare_1678595023iple_a > $o]: ((![X3 : hoare_1678595023iple_a]: ((P @ X3) => (Q @ X3))) => (ord_le1221261669iple_a @ (collec1600235172iple_a @ P) @ (collec1600235172iple_a @ Q)))))). % Collect_mono
thf(fact_34_mem__Collect__eq, axiom,
    ((![A3 : hoare_1678595023iple_a, P : hoare_1678595023iple_a > $o]: ((member1332298086iple_a @ A3 @ (collec1600235172iple_a @ P)) = (P @ A3))))). % mem_Collect_eq
thf(fact_35_Collect__mem__eq, axiom,
    ((![A : set_Ho137910533iple_a]: ((collec1600235172iple_a @ (^[X2 : hoare_1678595023iple_a]: (member1332298086iple_a @ X2 @ A))) = A)))). % Collect_mem_eq
thf(fact_36_Collect__cong, axiom,
    ((![P : hoare_1678595023iple_a > $o, Q : hoare_1678595023iple_a > $o]: ((![X3 : hoare_1678595023iple_a]: ((P @ X3) = (Q @ X3))) => ((collec1600235172iple_a @ P) = (collec1600235172iple_a @ Q)))))). % Collect_cong
thf(fact_37_Collect__mono__iff, axiom,
    ((![P : hoare_1678595023iple_a > $o, Q : hoare_1678595023iple_a > $o]: ((ord_le1221261669iple_a @ (collec1600235172iple_a @ P) @ (collec1600235172iple_a @ Q)) = (![X2 : hoare_1678595023iple_a]: (((P @ X2)) => ((Q @ X2)))))))). % Collect_mono_iff
thf(fact_38_order__subst1, axiom,
    ((![A3 : nat, F : nat > nat, B3 : nat, C : nat]: ((ord_less_eq_nat @ A3 @ (F @ B3)) => ((ord_less_eq_nat @ B3 @ C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_nat @ A3 @ (F @ C)))))))). % order_subst1
thf(fact_39_order__subst2, axiom,
    ((![A3 : nat, B3 : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A3 @ B3) => ((ord_less_eq_nat @ (F @ B3) @ C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_nat @ (F @ A3) @ C))))))). % order_subst2
thf(fact_40_ord__eq__le__subst, axiom,
    ((![A3 : nat, F : nat > nat, B3 : nat, C : nat]: ((A3 = (F @ B3)) => ((ord_less_eq_nat @ B3 @ C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_nat @ A3 @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_41_ord__le__eq__subst, axiom,
    ((![A3 : nat, B3 : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A3 @ B3) => (((F @ B3) = C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_nat @ (F @ A3) @ C))))))). % ord_le_eq_subst
thf(fact_42_eq__iff, axiom,
    (((^[Y3 : nat]: (^[Z : nat]: (Y3 = Z))) = (^[X2 : nat]: (^[Y4 : nat]: (((ord_less_eq_nat @ X2 @ Y4)) & ((ord_less_eq_nat @ Y4 @ X2)))))))). % eq_iff
thf(fact_43_antisym, axiom,
    ((![X4 : nat, Y2 : nat]: ((ord_less_eq_nat @ X4 @ Y2) => ((ord_less_eq_nat @ Y2 @ X4) => (X4 = Y2)))))). % antisym
thf(fact_44_linear, axiom,
    ((![X4 : nat, Y2 : nat]: ((ord_less_eq_nat @ X4 @ Y2) | (ord_less_eq_nat @ Y2 @ X4))))). % linear
thf(fact_45_eq__refl, axiom,
    ((![X4 : nat, Y2 : nat]: ((X4 = Y2) => (ord_less_eq_nat @ X4 @ Y2))))). % eq_refl
thf(fact_46_le__cases, axiom,
    ((![X4 : nat, Y2 : nat]: ((~ ((ord_less_eq_nat @ X4 @ Y2))) => (ord_less_eq_nat @ Y2 @ X4))))). % le_cases
thf(fact_47_order_Otrans, axiom,
    ((![A3 : nat, B3 : nat, C : nat]: ((ord_less_eq_nat @ A3 @ B3) => ((ord_less_eq_nat @ B3 @ C) => (ord_less_eq_nat @ A3 @ C)))))). % order.trans
thf(fact_48_le__cases3, axiom,
    ((![X4 : nat, Y2 : nat, Z2 : nat]: (((ord_less_eq_nat @ X4 @ Y2) => (~ ((ord_less_eq_nat @ Y2 @ Z2)))) => (((ord_less_eq_nat @ Y2 @ X4) => (~ ((ord_less_eq_nat @ X4 @ Z2)))) => (((ord_less_eq_nat @ X4 @ Z2) => (~ ((ord_less_eq_nat @ Z2 @ Y2)))) => (((ord_less_eq_nat @ Z2 @ Y2) => (~ ((ord_less_eq_nat @ Y2 @ X4)))) => (((ord_less_eq_nat @ Y2 @ Z2) => (~ ((ord_less_eq_nat @ Z2 @ X4)))) => (~ (((ord_less_eq_nat @ Z2 @ X4) => (~ ((ord_less_eq_nat @ X4 @ Y2)))))))))))))). % le_cases3
thf(fact_49_antisym__conv, axiom,
    ((![Y2 : nat, X4 : nat]: ((ord_less_eq_nat @ Y2 @ X4) => ((ord_less_eq_nat @ X4 @ Y2) = (X4 = Y2)))))). % antisym_conv
thf(fact_50_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y3 : nat]: (^[Z : nat]: (Y3 = Z))) = (^[A4 : nat]: (^[B4 : nat]: (((ord_less_eq_nat @ A4 @ B4)) & ((ord_less_eq_nat @ B4 @ A4)))))))). % order_class.order.eq_iff
thf(fact_51_ord__eq__le__trans, axiom,
    ((![A3 : nat, B3 : nat, C : nat]: ((A3 = B3) => ((ord_less_eq_nat @ B3 @ C) => (ord_less_eq_nat @ A3 @ C)))))). % ord_eq_le_trans
thf(fact_52_ord__le__eq__trans, axiom,
    ((![A3 : nat, B3 : nat, C : nat]: ((ord_less_eq_nat @ A3 @ B3) => ((B3 = C) => (ord_less_eq_nat @ A3 @ C)))))). % ord_le_eq_trans
thf(fact_53_order__class_Oorder_Oantisym, axiom,
    ((![A3 : nat, B3 : nat]: ((ord_less_eq_nat @ A3 @ B3) => ((ord_less_eq_nat @ B3 @ A3) => (A3 = B3)))))). % order_class.order.antisym
thf(fact_54_order__trans, axiom,
    ((![X4 : nat, Y2 : nat, Z2 : nat]: ((ord_less_eq_nat @ X4 @ Y2) => ((ord_less_eq_nat @ Y2 @ Z2) => (ord_less_eq_nat @ X4 @ Z2)))))). % order_trans
thf(fact_55_dual__order_Orefl, axiom,
    ((![A3 : nat]: (ord_less_eq_nat @ A3 @ A3)))). % dual_order.refl
thf(fact_56_linorder__wlog, axiom,
    ((![P : nat > nat > $o, A3 : nat, B3 : nat]: ((![A5 : nat, B5 : nat]: ((ord_less_eq_nat @ A5 @ B5) => (P @ A5 @ B5))) => ((![A5 : nat, B5 : nat]: ((P @ B5 @ A5) => (P @ A5 @ B5))) => (P @ A3 @ B3)))))). % linorder_wlog
thf(fact_57_dual__order_Otrans, axiom,
    ((![B3 : nat, A3 : nat, C : nat]: ((ord_less_eq_nat @ B3 @ A3) => ((ord_less_eq_nat @ C @ B3) => (ord_less_eq_nat @ C @ A3)))))). % dual_order.trans
thf(fact_58_dual__order_Oeq__iff, axiom,
    (((^[Y3 : nat]: (^[Z : nat]: (Y3 = Z))) = (^[A4 : nat]: (^[B4 : nat]: (((ord_less_eq_nat @ B4 @ A4)) & ((ord_less_eq_nat @ A4 @ B4)))))))). % dual_order.eq_iff
thf(fact_59_dual__order_Oantisym, axiom,
    ((![B3 : nat, A3 : nat]: ((ord_less_eq_nat @ B3 @ A3) => ((ord_less_eq_nat @ A3 @ B3) => (A3 = B3)))))). % dual_order.antisym
thf(fact_60_bot_Oextremum, axiom,
    ((![A3 : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ bot_bo1298296729iple_a @ A3)))). % bot.extremum
thf(fact_61_bot_Oextremum, axiom,
    ((![A3 : nat]: (ord_less_eq_nat @ bot_bot_nat @ A3)))). % bot.extremum
thf(fact_62_bot_Oextremum__unique, axiom,
    ((![A3 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A3 @ bot_bo1298296729iple_a) = (A3 = bot_bo1298296729iple_a))))). % bot.extremum_unique
thf(fact_63_bot_Oextremum__unique, axiom,
    ((![A3 : nat]: ((ord_less_eq_nat @ A3 @ bot_bot_nat) = (A3 = bot_bot_nat))))). % bot.extremum_unique
thf(fact_64_bot_Oextremum__uniqueI, axiom,
    ((![A3 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A3 @ bot_bo1298296729iple_a) => (A3 = bot_bo1298296729iple_a))))). % bot.extremum_uniqueI
thf(fact_65_bot_Oextremum__uniqueI, axiom,
    ((![A3 : nat]: ((ord_less_eq_nat @ A3 @ bot_bot_nat) => (A3 = bot_bot_nat))))). % bot.extremum_uniqueI
thf(fact_66_disjnt__empty1, axiom,
    ((![A : set_Ho137910533iple_a]: (disjnt925686856iple_a @ bot_bo1298296729iple_a @ A)))). % disjnt_empty1
thf(fact_67_disjnt__empty2, axiom,
    ((![A : set_Ho137910533iple_a]: (disjnt925686856iple_a @ A @ bot_bo1298296729iple_a)))). % disjnt_empty2
thf(fact_68_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_69_nat_Oinject, axiom,
    ((![X22 : nat, Y22 : nat]: (((suc @ X22) = (suc @ Y22)) = (X22 = Y22))))). % nat.inject
thf(fact_70_Ball__Collect, axiom,
    ((ball_H465710501iple_a = (^[A2 : set_Ho137910533iple_a]: (^[P2 : hoare_1678595023iple_a > $o]: (ord_le1221261669iple_a @ A2 @ (collec1600235172iple_a @ P2))))))). % Ball_Collect
thf(fact_71_subset__emptyI, axiom,
    ((![A : set_Ho137910533iple_a]: ((![X3 : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ X3 @ A)))) => (ord_le1221261669iple_a @ A @ bot_bo1298296729iple_a))))). % subset_emptyI
thf(fact_72_lift__Suc__antimono__le, axiom,
    ((![F : nat > nat, N2 : nat, N3 : nat]: ((![N4 : nat]: (ord_less_eq_nat @ (F @ (suc @ N4)) @ (F @ N4))) => ((ord_less_eq_nat @ N2 @ N3) => (ord_less_eq_nat @ (F @ N3) @ (F @ N2))))))). % lift_Suc_antimono_le
thf(fact_73_lift__Suc__mono__le, axiom,
    ((![F : nat > nat, N2 : nat, N3 : nat]: ((![N4 : nat]: (ord_less_eq_nat @ (F @ N4) @ (F @ (suc @ N4)))) => ((ord_less_eq_nat @ N2 @ N3) => (ord_less_eq_nat @ (F @ N2) @ (F @ N3))))))). % lift_Suc_mono_le
thf(fact_74_GreatestI2__order, axiom,
    ((![P : nat > $o, X4 : nat, Q : nat > $o]: ((P @ X4) => ((![Y : nat]: ((P @ Y) => (ord_less_eq_nat @ Y @ X4))) => ((![X3 : nat]: ((P @ X3) => ((![Y5 : nat]: ((P @ Y5) => (ord_less_eq_nat @ Y5 @ X3))) => (Q @ X3)))) => (Q @ (order_Greatest_nat @ P)))))))). % GreatestI2_order
thf(fact_75_Suc__le__mono, axiom,
    ((![N2 : nat, M : nat]: ((ord_less_eq_nat @ (suc @ N2) @ (suc @ M)) = (ord_less_eq_nat @ N2 @ M))))). % Suc_le_mono
thf(fact_76_transitive__stepwise__le, axiom,
    ((![M : nat, N2 : nat, R : nat > nat > $o]: ((ord_less_eq_nat @ M @ N2) => ((![X3 : nat]: (R @ X3 @ X3)) => ((![X3 : nat, Y : nat, Z3 : nat]: ((R @ X3 @ Y) => ((R @ Y @ Z3) => (R @ X3 @ Z3)))) => ((![N4 : nat]: (R @ N4 @ (suc @ N4))) => (R @ M @ N2)))))))). % transitive_stepwise_le
thf(fact_77_nat__induct__at__least, axiom,
    ((![M : nat, N2 : nat, P : nat > $o]: ((ord_less_eq_nat @ M @ N2) => ((P @ M) => ((![N4 : nat]: ((ord_less_eq_nat @ M @ N4) => ((P @ N4) => (P @ (suc @ N4))))) => (P @ N2))))))). % nat_induct_at_least
thf(fact_78_Nat_Oex__has__greatest__nat, axiom,
    ((![P : nat > $o, K : nat, B3 : nat]: ((P @ K) => ((![Y : nat]: ((P @ Y) => (ord_less_eq_nat @ Y @ B3))) => (?[X3 : nat]: ((P @ X3) & (![Y5 : nat]: ((P @ Y5) => (ord_less_eq_nat @ Y5 @ X3)))))))))). % Nat.ex_has_greatest_nat
thf(fact_79_GreatestI__ex__nat, axiom,
    ((![P : nat > $o, B3 : nat]: ((?[X_1 : nat]: (P @ X_1)) => ((![Y : nat]: ((P @ Y) => (ord_less_eq_nat @ Y @ B3))) => (P @ (order_Greatest_nat @ P))))))). % GreatestI_ex_nat
thf(fact_80_full__nat__induct, axiom,
    ((![P : nat > $o, N2 : nat]: ((![N4 : nat]: ((![M2 : nat]: ((ord_less_eq_nat @ (suc @ M2) @ N4) => (P @ M2))) => (P @ N4))) => (P @ N2))))). % full_nat_induct
thf(fact_81_Greatest__le__nat, axiom,
    ((![P : nat > $o, K : nat, B3 : nat]: ((P @ K) => ((![Y : nat]: ((P @ Y) => (ord_less_eq_nat @ Y @ B3))) => (ord_less_eq_nat @ K @ (order_Greatest_nat @ P))))))). % Greatest_le_nat
thf(fact_82_not__less__eq__eq, axiom,
    ((![M : nat, N2 : nat]: ((~ ((ord_less_eq_nat @ M @ N2))) = (ord_less_eq_nat @ (suc @ N2) @ M))))). % not_less_eq_eq
thf(fact_83_Suc__n__not__le__n, axiom,
    ((![N2 : nat]: (~ ((ord_less_eq_nat @ (suc @ N2) @ N2)))))). % Suc_n_not_le_n
thf(fact_84_nat__le__linear, axiom,
    ((![M : nat, N2 : nat]: ((ord_less_eq_nat @ M @ N2) | (ord_less_eq_nat @ N2 @ M))))). % nat_le_linear
thf(fact_85_GreatestI__nat, axiom,
    ((![P : nat > $o, K : nat, B3 : nat]: ((P @ K) => ((![Y : nat]: ((P @ Y) => (ord_less_eq_nat @ Y @ B3))) => (P @ (order_Greatest_nat @ P))))))). % GreatestI_nat
thf(fact_86_le__antisym, axiom,
    ((![M : nat, N2 : nat]: ((ord_less_eq_nat @ M @ N2) => ((ord_less_eq_nat @ N2 @ M) => (M = N2)))))). % le_antisym
thf(fact_87_le__Suc__eq, axiom,
    ((![M : nat, N2 : nat]: ((ord_less_eq_nat @ M @ (suc @ N2)) = (((ord_less_eq_nat @ M @ N2)) | ((M = (suc @ N2)))))))). % le_Suc_eq
thf(fact_88_eq__imp__le, axiom,
    ((![M : nat, N2 : nat]: ((M = N2) => (ord_less_eq_nat @ M @ N2))))). % eq_imp_le
thf(fact_89_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_90_Suc__le__D, axiom,
    ((![N2 : nat, M3 : nat]: ((ord_less_eq_nat @ (suc @ N2) @ M3) => (?[M4 : nat]: (M3 = (suc @ M4))))))). % Suc_le_D
thf(fact_91_le__refl, axiom,
    ((![N2 : nat]: (ord_less_eq_nat @ N2 @ N2)))). % le_refl
thf(fact_92_le__SucI, axiom,
    ((![M : nat, N2 : nat]: ((ord_less_eq_nat @ M @ N2) => (ord_less_eq_nat @ M @ (suc @ N2)))))). % le_SucI
thf(fact_93_le__SucE, axiom,
    ((![M : nat, N2 : nat]: ((ord_less_eq_nat @ M @ (suc @ N2)) => ((~ ((ord_less_eq_nat @ M @ N2))) => (M = (suc @ N2))))))). % le_SucE
thf(fact_94_Suc__leD, axiom,
    ((![M : nat, N2 : nat]: ((ord_less_eq_nat @ (suc @ M) @ N2) => (ord_less_eq_nat @ M @ N2))))). % Suc_leD
thf(fact_95_Suc__inject, axiom,
    ((![X4 : nat, Y2 : nat]: (((suc @ X4) = (suc @ Y2)) => (X4 = Y2))))). % Suc_inject
thf(fact_96_n__not__Suc__n, axiom,
    ((![N2 : nat]: (~ ((N2 = (suc @ N2))))))). % n_not_Suc_n
thf(fact_97_Greatest__equality, axiom,
    ((![P : nat > $o, X4 : nat]: ((P @ X4) => ((![Y : nat]: ((P @ Y) => (ord_less_eq_nat @ Y @ X4))) => ((order_Greatest_nat @ P) = X4)))))). % Greatest_equality
thf(fact_98_Suc__le__D__lemma, axiom,
    ((![N2 : nat, M3 : nat, P : nat > $o]: ((ord_less_eq_nat @ (suc @ N2) @ M3) => ((![M4 : nat]: ((ord_less_eq_nat @ N2 @ M4) => (P @ (suc @ M4)))) => (P @ M3)))))). % Suc_le_D_lemma
thf(fact_99_antimono__iff__le__Suc, axiom,
    ((order_1631207636at_nat = (^[F2 : nat > nat]: (![N : nat]: (ord_less_eq_nat @ (F2 @ (suc @ N)) @ (F2 @ N))))))). % antimono_iff_le_Suc
thf(fact_100_antimonoD, axiom,
    ((![F : nat > nat, X4 : nat, Y2 : nat]: ((order_1631207636at_nat @ F) => ((ord_less_eq_nat @ X4 @ Y2) => (ord_less_eq_nat @ (F @ Y2) @ (F @ X4))))))). % antimonoD
thf(fact_101_antimonoE, axiom,
    ((![F : nat > nat, X4 : nat, Y2 : nat]: ((order_1631207636at_nat @ F) => ((ord_less_eq_nat @ X4 @ Y2) => (ord_less_eq_nat @ (F @ Y2) @ (F @ X4))))))). % antimonoE
thf(fact_102_antimonoI, axiom,
    ((![F : nat > nat]: ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_nat @ (F @ Y) @ (F @ X3)))) => (order_1631207636at_nat @ F))))). % antimonoI
thf(fact_103_antimono__def, axiom,
    ((order_1631207636at_nat = (^[F2 : nat > nat]: (![X2 : nat]: (![Y4 : nat]: (((ord_less_eq_nat @ X2 @ Y4)) => ((ord_less_eq_nat @ (F2 @ Y4) @ (F2 @ X2)))))))))). % antimono_def
thf(fact_104_bounded__Max__nat, axiom,
    ((![P : nat > $o, X4 : nat, M5 : nat]: ((P @ X4) => ((![X3 : nat]: ((P @ X3) => (ord_less_eq_nat @ X3 @ M5))) => (~ ((![M4 : nat]: ((P @ M4) => (~ ((![X : nat]: ((P @ X) => (ord_less_eq_nat @ X @ M4)))))))))))))). % bounded_Max_nat
thf(fact_105_greaterThan__subset__iff, axiom,
    ((![X4 : nat, Y2 : nat]: ((ord_less_eq_set_nat @ (set_or1965240170an_nat @ X4) @ (set_or1965240170an_nat @ Y2)) = (ord_less_eq_nat @ Y2 @ X4))))). % greaterThan_subset_iff
thf(fact_106_verit__la__disequality, axiom,
    ((![A3 : nat, B3 : nat]: ((A3 = B3) | ((~ ((ord_less_eq_nat @ A3 @ B3))) | (~ ((ord_less_eq_nat @ B3 @ A3)))))))). % verit_la_disequality
thf(fact_107_le__rel__bool__arg__iff, axiom,
    ((ord_less_eq_o_nat = (^[X5 : $o > nat]: (^[Y6 : $o > nat]: (((ord_less_eq_nat @ (X5 @ $false) @ (Y6 @ $false))) & ((ord_less_eq_nat @ (X5 @ $true) @ (Y6 @ $true))))))))). % le_rel_bool_arg_iff
thf(fact_108_is__singletonI, axiom,
    ((![X4 : hoare_1678595023iple_a]: (is_sin1784037339iple_a @ (insert1477804543iple_a @ X4 @ bot_bo1298296729iple_a))))). % is_singletonI
thf(fact_109_insertCI, axiom,
    ((![A3 : hoare_1678595023iple_a, B : set_Ho137910533iple_a, B3 : hoare_1678595023iple_a]: (((~ ((member1332298086iple_a @ A3 @ B))) => (A3 = B3)) => (member1332298086iple_a @ A3 @ (insert1477804543iple_a @ B3 @ B)))))). % insertCI
thf(fact_110_insert__iff, axiom,
    ((![A3 : hoare_1678595023iple_a, B3 : hoare_1678595023iple_a, A : set_Ho137910533iple_a]: ((member1332298086iple_a @ A3 @ (insert1477804543iple_a @ B3 @ A)) = (((A3 = B3)) | ((member1332298086iple_a @ A3 @ A))))))). % insert_iff
thf(fact_111_singletonI, axiom,
    ((![A3 : hoare_1678595023iple_a]: (member1332298086iple_a @ A3 @ (insert1477804543iple_a @ A3 @ bot_bo1298296729iple_a))))). % singletonI
thf(fact_112_insert__subset, axiom,
    ((![X4 : hoare_1678595023iple_a, A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ (insert1477804543iple_a @ X4 @ A) @ B) = (((member1332298086iple_a @ X4 @ B)) & ((ord_le1221261669iple_a @ A @ B))))))). % insert_subset
thf(fact_113_disjnt__insert2, axiom,
    ((![Y7 : set_Ho137910533iple_a, A3 : hoare_1678595023iple_a, X6 : set_Ho137910533iple_a]: ((disjnt925686856iple_a @ Y7 @ (insert1477804543iple_a @ A3 @ X6)) = (((~ ((member1332298086iple_a @ A3 @ Y7)))) & ((disjnt925686856iple_a @ Y7 @ X6))))))). % disjnt_insert2
thf(fact_114_disjnt__insert1, axiom,
    ((![A3 : hoare_1678595023iple_a, X6 : set_Ho137910533iple_a, Y7 : set_Ho137910533iple_a]: ((disjnt925686856iple_a @ (insert1477804543iple_a @ A3 @ X6) @ Y7) = (((~ ((member1332298086iple_a @ A3 @ Y7)))) & ((disjnt925686856iple_a @ X6 @ Y7))))))). % disjnt_insert1
thf(fact_115_singleton__insert__inj__eq_H, axiom,
    ((![A3 : hoare_1678595023iple_a, A : set_Ho137910533iple_a, B3 : hoare_1678595023iple_a]: (((insert1477804543iple_a @ A3 @ A) = (insert1477804543iple_a @ B3 @ bot_bo1298296729iple_a)) = (((A3 = B3)) & ((ord_le1221261669iple_a @ A @ (insert1477804543iple_a @ B3 @ bot_bo1298296729iple_a)))))))). % singleton_insert_inj_eq'
thf(fact_116_singleton__insert__inj__eq, axiom,
    ((![B3 : hoare_1678595023iple_a, A3 : hoare_1678595023iple_a, A : set_Ho137910533iple_a]: (((insert1477804543iple_a @ B3 @ bot_bo1298296729iple_a) = (insert1477804543iple_a @ A3 @ A)) = (((A3 = B3)) & ((ord_le1221261669iple_a @ A @ (insert1477804543iple_a @ B3 @ bot_bo1298296729iple_a)))))))). % singleton_insert_inj_eq
thf(fact_117_subset__singletonD, axiom,
    ((![A : set_Ho137910533iple_a, X4 : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A @ (insert1477804543iple_a @ X4 @ bot_bo1298296729iple_a)) => ((A = bot_bo1298296729iple_a) | (A = (insert1477804543iple_a @ X4 @ bot_bo1298296729iple_a))))))). % subset_singletonD
thf(fact_118_subset__singleton__iff, axiom,
    ((![X6 : set_Ho137910533iple_a, A3 : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ X6 @ (insert1477804543iple_a @ A3 @ bot_bo1298296729iple_a)) = (((X6 = bot_bo1298296729iple_a)) | ((X6 = (insert1477804543iple_a @ A3 @ bot_bo1298296729iple_a)))))))). % subset_singleton_iff
thf(fact_119_subset__insert, axiom,
    ((![X4 : hoare_1678595023iple_a, A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ X4 @ A))) => ((ord_le1221261669iple_a @ A @ (insert1477804543iple_a @ X4 @ B)) = (ord_le1221261669iple_a @ A @ B)))))). % subset_insert
thf(fact_120_insert__subsetI, axiom,
    ((![X4 : hoare_1678595023iple_a, A : set_Ho137910533iple_a, X6 : set_Ho137910533iple_a]: ((member1332298086iple_a @ X4 @ A) => ((ord_le1221261669iple_a @ X6 @ A) => (ord_le1221261669iple_a @ (insert1477804543iple_a @ X4 @ X6) @ A)))))). % insert_subsetI
thf(fact_121_singleton__inject, axiom,
    ((![A3 : hoare_1678595023iple_a, B3 : hoare_1678595023iple_a]: (((insert1477804543iple_a @ A3 @ bot_bo1298296729iple_a) = (insert1477804543iple_a @ B3 @ bot_bo1298296729iple_a)) => (A3 = B3))))). % singleton_inject

% Conjectures (1)
thf(conj_0, conjecture,
    ((![N4 : nat]: ((?[X : hoare_1678595023iple_a]: ((member1332298086iple_a @ X @ ga) & (~ ((hoare_1926814542alid_a @ N4 @ X))))) | (![X3 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X3 @ bot_bo1298296729iple_a) => (hoare_1926814542alid_a @ N4 @ X3))))))).
