% 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_280__3252248_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:13:23.725

% Could-be-implicit typings (6)
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).
thf(ty_n_t__Com__Ocom, type,
    com : $tType).
thf(ty_n_tf__a, type,
    a : $tType).

% Explicit typings (20)
thf(sy_c_Com_Ocom_OSKIP, type,
    skip : com).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    uminus922456654iple_a : set_Ho137910533iple_a > set_Ho137910533iple_a).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Ohoare__derivs_001tf__a, type,
    hoare_129598474rivs_a : set_Ho137910533iple_a > set_Ho137910533iple_a > $o).
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_Otriple_001tf__a, type,
    hoare_719046530iple_a : (a > state > $o) > com > (a > state > $o) > hoare_1678595023iple_a).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Otriple__valid_001tf__a, type,
    hoare_1926814542alid_a : nat > hoare_1678595023iple_a > $o).
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__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_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_Oorder__class_OGreatest_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    order_929906668iple_a : (set_Ho137910533iple_a > $o) > set_Ho137910533iple_a).
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_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_Othe__elem_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    the_el434698138iple_a : set_Ho137910533iple_a > hoare_1678595023iple_a).
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_G, type,
    g : set_Ho137910533iple_a).
thf(sy_v_ts, type,
    ts : set_Ho137910533iple_a).

% Relevant facts (112)
thf(fact_0_cut, axiom,
    ((![G : set_Ho137910533iple_a, Ts : set_Ho137910533iple_a, G2 : set_Ho137910533iple_a]: ((hoare_129598474rivs_a @ G @ Ts) => ((hoare_129598474rivs_a @ G2 @ G) => (hoare_129598474rivs_a @ G2 @ Ts)))))). % cut
thf(fact_1_empty, axiom,
    ((![G2 : set_Ho137910533iple_a]: (hoare_129598474rivs_a @ G2 @ bot_bo1298296729iple_a)))). % empty
thf(fact_2_thin, axiom,
    ((![G : set_Ho137910533iple_a, Ts : set_Ho137910533iple_a, G2 : set_Ho137910533iple_a]: ((hoare_129598474rivs_a @ G @ Ts) => ((ord_le1221261669iple_a @ G @ G2) => (hoare_129598474rivs_a @ G2 @ Ts)))))). % thin
thf(fact_3_asm, axiom,
    ((![Ts : set_Ho137910533iple_a, G2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ Ts @ G2) => (hoare_129598474rivs_a @ G2 @ Ts))))). % asm
thf(fact_4_weaken, axiom,
    ((![G2 : set_Ho137910533iple_a, Ts2 : set_Ho137910533iple_a, Ts : set_Ho137910533iple_a]: ((hoare_129598474rivs_a @ G2 @ Ts2) => ((ord_le1221261669iple_a @ Ts @ Ts2) => (hoare_129598474rivs_a @ G2 @ Ts)))))). % weaken
thf(fact_5_single__stateE, axiom,
    ((hoare_405891322gleton => (![T : state]: (~ ((![S : state]: (S = T)))))))). % single_stateE
thf(fact_6_state__not__singleton__def, axiom,
    ((hoare_405891322gleton = (?[S2 : state]: (?[T2 : state]: (~ ((S2 = T2)))))))). % state_not_singleton_def
thf(fact_7_hoare__valids__def, axiom,
    ((hoare_1775499016lids_a = (^[G3 : set_Ho137910533iple_a]: (^[Ts3 : set_Ho137910533iple_a]: (![N : nat]: (((![X : hoare_1678595023iple_a]: (((member1332298086iple_a @ X @ G3)) => ((hoare_1926814542alid_a @ N @ X))))) => ((![X : hoare_1678595023iple_a]: (((member1332298086iple_a @ X @ Ts3)) => ((hoare_1926814542alid_a @ N @ X)))))))))))). % hoare_valids_def
thf(fact_8_derivs__insertD, axiom,
    ((![G2 : set_Ho137910533iple_a, T3 : hoare_1678595023iple_a, Ts : set_Ho137910533iple_a]: ((hoare_129598474rivs_a @ G2 @ (insert1477804543iple_a @ T3 @ Ts)) => ((hoare_129598474rivs_a @ G2 @ (insert1477804543iple_a @ T3 @ bot_bo1298296729iple_a)) & (hoare_129598474rivs_a @ G2 @ Ts)))))). % derivs_insertD
thf(fact_9_hoare__derivs_Oinsert, axiom,
    ((![G2 : set_Ho137910533iple_a, T3 : hoare_1678595023iple_a, Ts : set_Ho137910533iple_a]: ((hoare_129598474rivs_a @ G2 @ (insert1477804543iple_a @ T3 @ bot_bo1298296729iple_a)) => ((hoare_129598474rivs_a @ G2 @ Ts) => (hoare_129598474rivs_a @ G2 @ (insert1477804543iple_a @ T3 @ Ts))))))). % hoare_derivs.insert
thf(fact_10_singleton__insert__inj__eq, axiom,
    ((![B : hoare_1678595023iple_a, A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: (((insert1477804543iple_a @ B @ bot_bo1298296729iple_a) = (insert1477804543iple_a @ A @ A2)) = (((A = B)) & ((ord_le1221261669iple_a @ A2 @ (insert1477804543iple_a @ B @ bot_bo1298296729iple_a)))))))). % singleton_insert_inj_eq
thf(fact_11_singleton__insert__inj__eq_H, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B : hoare_1678595023iple_a]: (((insert1477804543iple_a @ A @ A2) = (insert1477804543iple_a @ B @ bot_bo1298296729iple_a)) = (((A = B)) & ((ord_le1221261669iple_a @ A2 @ (insert1477804543iple_a @ B @ bot_bo1298296729iple_a)))))))). % singleton_insert_inj_eq'
thf(fact_12_insert__subset, axiom,
    ((![X2 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ (insert1477804543iple_a @ X2 @ A2) @ B2) = (((member1332298086iple_a @ X2 @ B2)) & ((ord_le1221261669iple_a @ A2 @ B2))))))). % insert_subset
thf(fact_13_ball__empty, axiom,
    ((![P : hoare_1678595023iple_a > $o, X3 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X3 @ bot_bo1298296729iple_a) => (P @ X3))))). % ball_empty
thf(fact_14_singletonI, axiom,
    ((![A : hoare_1678595023iple_a]: (member1332298086iple_a @ A @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a))))). % singletonI
thf(fact_15_subset__empty, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ bot_bo1298296729iple_a) = (A2 = bot_bo1298296729iple_a))))). % subset_empty
thf(fact_16_empty__subsetI, axiom,
    ((![A2 : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ bot_bo1298296729iple_a @ A2)))). % empty_subsetI
thf(fact_17_subset__singletonD, axiom,
    ((![A2 : set_Ho137910533iple_a, X2 : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A2 @ (insert1477804543iple_a @ X2 @ bot_bo1298296729iple_a)) => ((A2 = bot_bo1298296729iple_a) | (A2 = (insert1477804543iple_a @ X2 @ bot_bo1298296729iple_a))))))). % subset_singletonD
thf(fact_18_subset__singleton__iff, axiom,
    ((![X4 : set_Ho137910533iple_a, A : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ X4 @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a)) = (((X4 = bot_bo1298296729iple_a)) | ((X4 = (insert1477804543iple_a @ A @ bot_bo1298296729iple_a)))))))). % subset_singleton_iff
thf(fact_19_insertCI, axiom,
    ((![A : hoare_1678595023iple_a, B2 : set_Ho137910533iple_a, B : hoare_1678595023iple_a]: (((~ ((member1332298086iple_a @ A @ B2))) => (A = B)) => (member1332298086iple_a @ A @ (insert1477804543iple_a @ B @ B2)))))). % insertCI
thf(fact_20_empty__Collect__eq, axiom,
    ((![P : hoare_1678595023iple_a > $o]: ((bot_bo1298296729iple_a = (collec1600235172iple_a @ P)) = (![X : hoare_1678595023iple_a]: (~ ((P @ X)))))))). % empty_Collect_eq
thf(fact_21_Collect__empty__eq, axiom,
    ((![P : hoare_1678595023iple_a > $o]: (((collec1600235172iple_a @ P) = bot_bo1298296729iple_a) = (![X : hoare_1678595023iple_a]: (~ ((P @ X)))))))). % Collect_empty_eq
thf(fact_22_all__not__in__conv, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((![X : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ X @ A2)))) = (A2 = bot_bo1298296729iple_a))))). % all_not_in_conv
thf(fact_23_empty__iff, axiom,
    ((![C : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ C @ bot_bo1298296729iple_a)))))). % empty_iff
thf(fact_24_subset__antisym, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ B2) => ((ord_le1221261669iple_a @ B2 @ A2) => (A2 = B2)))))). % subset_antisym
thf(fact_25_subsetI, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((![X5 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X5 @ A2) => (member1332298086iple_a @ X5 @ B2))) => (ord_le1221261669iple_a @ A2 @ B2))))). % subsetI
thf(fact_26_insert__absorb2, axiom,
    ((![X2 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((insert1477804543iple_a @ X2 @ (insert1477804543iple_a @ X2 @ A2)) = (insert1477804543iple_a @ X2 @ A2))))). % insert_absorb2
thf(fact_27_insert__iff, axiom,
    ((![A : hoare_1678595023iple_a, B : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ A @ (insert1477804543iple_a @ B @ A2)) = (((A = B)) | ((member1332298086iple_a @ A @ A2))))))). % insert_iff
thf(fact_28_ex__in__conv, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((?[X : hoare_1678595023iple_a]: (member1332298086iple_a @ X @ A2)) = (~ ((A2 = bot_bo1298296729iple_a))))))). % ex_in_conv
thf(fact_29_equals0I, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((![Y : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ Y @ A2)))) => (A2 = bot_bo1298296729iple_a))))). % equals0I
thf(fact_30_equals0D, axiom,
    ((![A2 : set_Ho137910533iple_a, A : hoare_1678595023iple_a]: ((A2 = bot_bo1298296729iple_a) => (~ ((member1332298086iple_a @ A @ A2))))))). % equals0D
thf(fact_31_emptyE, axiom,
    ((![A : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ A @ bot_bo1298296729iple_a)))))). % emptyE
thf(fact_32_Collect__mono__iff, axiom,
    ((![P : hoare_1678595023iple_a > $o, Q : hoare_1678595023iple_a > $o]: ((ord_le1221261669iple_a @ (collec1600235172iple_a @ P) @ (collec1600235172iple_a @ Q)) = (![X : hoare_1678595023iple_a]: (((P @ X)) => ((Q @ X)))))))). % Collect_mono_iff
thf(fact_33_set__eq__subset, axiom,
    (((^[Y2 : set_Ho137910533iple_a]: (^[Z : set_Ho137910533iple_a]: (Y2 = Z))) = (^[A3 : set_Ho137910533iple_a]: (^[B3 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ A3 @ B3)) & ((ord_le1221261669iple_a @ B3 @ A3)))))))). % set_eq_subset
thf(fact_34_subset__trans, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, C2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ B2) => ((ord_le1221261669iple_a @ B2 @ C2) => (ord_le1221261669iple_a @ A2 @ C2)))))). % subset_trans
thf(fact_35_Collect__mono, axiom,
    ((![P : hoare_1678595023iple_a > $o, Q : hoare_1678595023iple_a > $o]: ((![X5 : hoare_1678595023iple_a]: ((P @ X5) => (Q @ X5))) => (ord_le1221261669iple_a @ (collec1600235172iple_a @ P) @ (collec1600235172iple_a @ Q)))))). % Collect_mono
thf(fact_36_subset__refl, axiom,
    ((![A2 : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ A2 @ A2)))). % subset_refl
thf(fact_37_subset__iff, axiom,
    ((ord_le1221261669iple_a = (^[A3 : set_Ho137910533iple_a]: (^[B3 : set_Ho137910533iple_a]: (![T2 : hoare_1678595023iple_a]: (((member1332298086iple_a @ T2 @ A3)) => ((member1332298086iple_a @ T2 @ B3))))))))). % subset_iff
thf(fact_38_equalityD2, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((A2 = B2) => (ord_le1221261669iple_a @ B2 @ A2))))). % equalityD2
thf(fact_39_equalityD1, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((A2 = B2) => (ord_le1221261669iple_a @ A2 @ B2))))). % equalityD1
thf(fact_40_subset__eq, axiom,
    ((ord_le1221261669iple_a = (^[A3 : set_Ho137910533iple_a]: (^[B3 : set_Ho137910533iple_a]: (![X : hoare_1678595023iple_a]: (((member1332298086iple_a @ X @ A3)) => ((member1332298086iple_a @ X @ B3))))))))). % subset_eq
thf(fact_41_equalityE, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((A2 = B2) => (~ (((ord_le1221261669iple_a @ A2 @ B2) => (~ ((ord_le1221261669iple_a @ B2 @ A2)))))))))). % equalityE
thf(fact_42_subsetD, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, C : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A2 @ B2) => ((member1332298086iple_a @ C @ A2) => (member1332298086iple_a @ C @ B2)))))). % subsetD
thf(fact_43_in__mono, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, X2 : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A2 @ B2) => ((member1332298086iple_a @ X2 @ A2) => (member1332298086iple_a @ X2 @ B2)))))). % in_mono
thf(fact_44_mk__disjoint__insert, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ A @ A2) => (?[B4 : set_Ho137910533iple_a]: ((A2 = (insert1477804543iple_a @ A @ B4)) & (~ ((member1332298086iple_a @ A @ B4))))))))). % mk_disjoint_insert
thf(fact_45_insert__commute, axiom,
    ((![X2 : hoare_1678595023iple_a, Y3 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((insert1477804543iple_a @ X2 @ (insert1477804543iple_a @ Y3 @ A2)) = (insert1477804543iple_a @ Y3 @ (insert1477804543iple_a @ X2 @ A2)))))). % insert_commute
thf(fact_46_insert__eq__iff, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B : hoare_1678595023iple_a, B2 : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ A @ A2))) => ((~ ((member1332298086iple_a @ B @ B2))) => (((insert1477804543iple_a @ A @ A2) = (insert1477804543iple_a @ B @ B2)) = (((((A = B)) => ((A2 = B2)))) & ((((~ ((A = B)))) => ((?[C3 : set_Ho137910533iple_a]: (((A2 = (insert1477804543iple_a @ B @ C3))) & ((((~ ((member1332298086iple_a @ B @ C3)))) & ((((B2 = (insert1477804543iple_a @ A @ C3))) & ((~ ((member1332298086iple_a @ A @ C3)))))))))))))))))))). % insert_eq_iff
thf(fact_47_insert__absorb, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ A @ A2) => ((insert1477804543iple_a @ A @ A2) = A2))))). % insert_absorb
thf(fact_48_insert__ident, axiom,
    ((![X2 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ X2 @ A2))) => ((~ ((member1332298086iple_a @ X2 @ B2))) => (((insert1477804543iple_a @ X2 @ A2) = (insert1477804543iple_a @ X2 @ B2)) = (A2 = B2))))))). % insert_ident
thf(fact_49_Set_Oset__insert, axiom,
    ((![X2 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ X2 @ A2) => (~ ((![B4 : set_Ho137910533iple_a]: ((A2 = (insert1477804543iple_a @ X2 @ B4)) => (member1332298086iple_a @ X2 @ B4))))))))). % Set.set_insert
thf(fact_50_insertI2, axiom,
    ((![A : hoare_1678595023iple_a, B2 : set_Ho137910533iple_a, B : hoare_1678595023iple_a]: ((member1332298086iple_a @ A @ B2) => (member1332298086iple_a @ A @ (insert1477804543iple_a @ B @ B2)))))). % insertI2
thf(fact_51_insertI1, axiom,
    ((![A : hoare_1678595023iple_a, B2 : set_Ho137910533iple_a]: (member1332298086iple_a @ A @ (insert1477804543iple_a @ A @ B2))))). % insertI1
thf(fact_52_insertE, axiom,
    ((![A : hoare_1678595023iple_a, B : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ A @ (insert1477804543iple_a @ B @ A2)) => ((~ ((A = B))) => (member1332298086iple_a @ A @ A2)))))). % insertE
thf(fact_53_singleton__inject, axiom,
    ((![A : hoare_1678595023iple_a, B : hoare_1678595023iple_a]: (((insert1477804543iple_a @ A @ bot_bo1298296729iple_a) = (insert1477804543iple_a @ B @ bot_bo1298296729iple_a)) => (A = B))))). % singleton_inject
thf(fact_54_insert__not__empty, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: (~ (((insert1477804543iple_a @ A @ A2) = bot_bo1298296729iple_a)))))). % insert_not_empty
thf(fact_55_doubleton__eq__iff, axiom,
    ((![A : hoare_1678595023iple_a, B : hoare_1678595023iple_a, C : hoare_1678595023iple_a, D : hoare_1678595023iple_a]: (((insert1477804543iple_a @ A @ (insert1477804543iple_a @ B @ bot_bo1298296729iple_a)) = (insert1477804543iple_a @ C @ (insert1477804543iple_a @ D @ bot_bo1298296729iple_a))) = (((((A = C)) & ((B = D)))) | ((((A = D)) & ((B = C))))))))). % doubleton_eq_iff
thf(fact_56_singleton__iff, axiom,
    ((![B : hoare_1678595023iple_a, A : hoare_1678595023iple_a]: ((member1332298086iple_a @ B @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a)) = (B = A))))). % singleton_iff
thf(fact_57_singletonD, axiom,
    ((![B : hoare_1678595023iple_a, A : hoare_1678595023iple_a]: ((member1332298086iple_a @ B @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a)) => (B = A))))). % singletonD
thf(fact_58_subset__insertI2, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, B : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A2 @ B2) => (ord_le1221261669iple_a @ A2 @ (insert1477804543iple_a @ B @ B2)))))). % subset_insertI2
thf(fact_59_subset__insertI, axiom,
    ((![B2 : set_Ho137910533iple_a, A : hoare_1678595023iple_a]: (ord_le1221261669iple_a @ B2 @ (insert1477804543iple_a @ A @ B2))))). % subset_insertI
thf(fact_60_subset__insert, axiom,
    ((![X2 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ X2 @ A2))) => ((ord_le1221261669iple_a @ A2 @ (insert1477804543iple_a @ X2 @ B2)) = (ord_le1221261669iple_a @ A2 @ B2)))))). % subset_insert
thf(fact_61_insert__mono, axiom,
    ((![C2 : set_Ho137910533iple_a, D2 : set_Ho137910533iple_a, A : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ C2 @ D2) => (ord_le1221261669iple_a @ (insert1477804543iple_a @ A @ C2) @ (insert1477804543iple_a @ A @ D2)))))). % insert_mono
thf(fact_62_the__elem__eq, axiom,
    ((![X2 : hoare_1678595023iple_a]: ((the_el434698138iple_a @ (insert1477804543iple_a @ X2 @ bot_bo1298296729iple_a)) = X2)))). % the_elem_eq
thf(fact_63_order__refl, axiom,
    ((![X2 : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ X2 @ X2)))). % order_refl
thf(fact_64_is__singletonI, axiom,
    ((![X2 : hoare_1678595023iple_a]: (is_sin1784037339iple_a @ (insert1477804543iple_a @ X2 @ bot_bo1298296729iple_a))))). % is_singletonI
thf(fact_65_Ball__Collect, axiom,
    ((ball_H465710501iple_a = (^[A3 : set_Ho137910533iple_a]: (^[P2 : hoare_1678595023iple_a > $o]: (ord_le1221261669iple_a @ A3 @ (collec1600235172iple_a @ P2))))))). % Ball_Collect
thf(fact_66_insert__subsetI, axiom,
    ((![X2 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, X4 : set_Ho137910533iple_a]: ((member1332298086iple_a @ X2 @ A2) => ((ord_le1221261669iple_a @ X4 @ A2) => (ord_le1221261669iple_a @ (insert1477804543iple_a @ X2 @ X4) @ A2)))))). % insert_subsetI
thf(fact_67_subset__emptyI, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((![X5 : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ X5 @ A2)))) => (ord_le1221261669iple_a @ A2 @ bot_bo1298296729iple_a))))). % subset_emptyI
thf(fact_68_bot_Oextremum__uniqueI, axiom,
    ((![A : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ bot_bo1298296729iple_a) => (A = bot_bo1298296729iple_a))))). % bot.extremum_uniqueI
thf(fact_69_bot_Oextremum__unique, axiom,
    ((![A : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ bot_bo1298296729iple_a) = (A = bot_bo1298296729iple_a))))). % bot.extremum_unique
thf(fact_70_bot__set__def, axiom,
    ((bot_bo1298296729iple_a = (collec1600235172iple_a @ bot_bo431311916le_a_o)))). % bot_set_def
thf(fact_71_is__singleton__the__elem, axiom,
    ((is_sin1784037339iple_a = (^[A3 : set_Ho137910533iple_a]: (A3 = (insert1477804543iple_a @ (the_el434698138iple_a @ A3) @ bot_bo1298296729iple_a)))))). % is_singleton_the_elem
thf(fact_72_is__singletonI_H, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((~ ((A2 = bot_bo1298296729iple_a))) => ((![X5 : hoare_1678595023iple_a, Y : hoare_1678595023iple_a]: ((member1332298086iple_a @ X5 @ A2) => ((member1332298086iple_a @ Y @ A2) => (X5 = Y)))) => (is_sin1784037339iple_a @ A2)))))). % is_singletonI'
thf(fact_73_order__subst1, axiom,
    ((![A : set_Ho137910533iple_a, F : set_Ho137910533iple_a > set_Ho137910533iple_a, B : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ (F @ B)) => ((ord_le1221261669iple_a @ B @ C) => ((![X5 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X5 @ Y) => (ord_le1221261669iple_a @ (F @ X5) @ (F @ Y)))) => (ord_le1221261669iple_a @ A @ (F @ C)))))))). % order_subst1
thf(fact_74_order__subst2, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, F : set_Ho137910533iple_a > set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ B) => ((ord_le1221261669iple_a @ (F @ B) @ C) => ((![X5 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X5 @ Y) => (ord_le1221261669iple_a @ (F @ X5) @ (F @ Y)))) => (ord_le1221261669iple_a @ (F @ A) @ C))))))). % order_subst2
thf(fact_75_ord__eq__le__subst, axiom,
    ((![A : set_Ho137910533iple_a, F : set_Ho137910533iple_a > set_Ho137910533iple_a, B : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((A = (F @ B)) => ((ord_le1221261669iple_a @ B @ C) => ((![X5 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X5 @ Y) => (ord_le1221261669iple_a @ (F @ X5) @ (F @ Y)))) => (ord_le1221261669iple_a @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_76_ord__le__eq__subst, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, F : set_Ho137910533iple_a > set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ B) => (((F @ B) = C) => ((![X5 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X5 @ Y) => (ord_le1221261669iple_a @ (F @ X5) @ (F @ Y)))) => (ord_le1221261669iple_a @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_77_eq__iff, axiom,
    (((^[Y2 : set_Ho137910533iple_a]: (^[Z : set_Ho137910533iple_a]: (Y2 = Z))) = (^[X : set_Ho137910533iple_a]: (^[Y4 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ X @ Y4)) & ((ord_le1221261669iple_a @ Y4 @ X)))))))). % eq_iff
thf(fact_78_antisym, axiom,
    ((![X2 : set_Ho137910533iple_a, Y3 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X2 @ Y3) => ((ord_le1221261669iple_a @ Y3 @ X2) => (X2 = Y3)))))). % antisym
thf(fact_79_eq__refl, axiom,
    ((![X2 : set_Ho137910533iple_a, Y3 : set_Ho137910533iple_a]: ((X2 = Y3) => (ord_le1221261669iple_a @ X2 @ Y3))))). % eq_refl
thf(fact_80_order_Otrans, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ B) => ((ord_le1221261669iple_a @ B @ C) => (ord_le1221261669iple_a @ A @ C)))))). % order.trans
thf(fact_81_antisym__conv, axiom,
    ((![Y3 : set_Ho137910533iple_a, X2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ Y3 @ X2) => ((ord_le1221261669iple_a @ X2 @ Y3) = (X2 = Y3)))))). % antisym_conv
thf(fact_82_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y2 : set_Ho137910533iple_a]: (^[Z : set_Ho137910533iple_a]: (Y2 = Z))) = (^[A4 : set_Ho137910533iple_a]: (^[B5 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ A4 @ B5)) & ((ord_le1221261669iple_a @ B5 @ A4)))))))). % order_class.order.eq_iff
thf(fact_83_ord__eq__le__trans, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((A = B) => ((ord_le1221261669iple_a @ B @ C) => (ord_le1221261669iple_a @ A @ C)))))). % ord_eq_le_trans
thf(fact_84_ord__le__eq__trans, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ B) => ((B = C) => (ord_le1221261669iple_a @ A @ C)))))). % ord_le_eq_trans
thf(fact_85_order__class_Oorder_Oantisym, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ B) => ((ord_le1221261669iple_a @ B @ A) => (A = B)))))). % order_class.order.antisym
thf(fact_86_order__trans, axiom,
    ((![X2 : set_Ho137910533iple_a, Y3 : set_Ho137910533iple_a, Z2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X2 @ Y3) => ((ord_le1221261669iple_a @ Y3 @ Z2) => (ord_le1221261669iple_a @ X2 @ Z2)))))). % order_trans
thf(fact_87_dual__order_Orefl, axiom,
    ((![A : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ A @ A)))). % dual_order.refl
thf(fact_88_dual__order_Otrans, axiom,
    ((![B : set_Ho137910533iple_a, A : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ B @ A) => ((ord_le1221261669iple_a @ C @ B) => (ord_le1221261669iple_a @ C @ A)))))). % dual_order.trans
thf(fact_89_dual__order_Oeq__iff, axiom,
    (((^[Y2 : set_Ho137910533iple_a]: (^[Z : set_Ho137910533iple_a]: (Y2 = Z))) = (^[A4 : set_Ho137910533iple_a]: (^[B5 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ B5 @ A4)) & ((ord_le1221261669iple_a @ A4 @ B5)))))))). % dual_order.eq_iff
thf(fact_90_dual__order_Oantisym, axiom,
    ((![B : set_Ho137910533iple_a, A : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ B @ A) => ((ord_le1221261669iple_a @ A @ B) => (A = B)))))). % dual_order.antisym
thf(fact_91_is__singletonE, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((is_sin1784037339iple_a @ A2) => (~ ((![X5 : hoare_1678595023iple_a]: (~ ((A2 = (insert1477804543iple_a @ X5 @ bot_bo1298296729iple_a))))))))))). % is_singletonE
thf(fact_92_is__singleton__def, axiom,
    ((is_sin1784037339iple_a = (^[A3 : set_Ho137910533iple_a]: (?[X : hoare_1678595023iple_a]: (A3 = (insert1477804543iple_a @ X @ bot_bo1298296729iple_a))))))). % is_singleton_def
thf(fact_93_bot_Oextremum, axiom,
    ((![A : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ bot_bo1298296729iple_a @ A)))). % bot.extremum
thf(fact_94_Set_Ois__empty__def, axiom,
    ((is_emp901906557iple_a = (^[A3 : set_Ho137910533iple_a]: (A3 = bot_bo1298296729iple_a))))). % Set.is_empty_def
thf(fact_95_subset__Compl__singleton, axiom,
    ((![A2 : set_Ho137910533iple_a, B : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A2 @ (uminus922456654iple_a @ (insert1477804543iple_a @ B @ bot_bo1298296729iple_a))) = (~ ((member1332298086iple_a @ B @ A2))))))). % subset_Compl_singleton
thf(fact_96_GreatestI2__order, axiom,
    ((![P : set_Ho137910533iple_a > $o, X2 : set_Ho137910533iple_a, Q : set_Ho137910533iple_a > $o]: ((P @ X2) => ((![Y : set_Ho137910533iple_a]: ((P @ Y) => (ord_le1221261669iple_a @ Y @ X2))) => ((![X5 : set_Ho137910533iple_a]: ((P @ X5) => ((![Y5 : set_Ho137910533iple_a]: ((P @ Y5) => (ord_le1221261669iple_a @ Y5 @ X5))) => (Q @ X5)))) => (Q @ (order_929906668iple_a @ P)))))))). % GreatestI2_order
thf(fact_97_Greatest__equality, axiom,
    ((![P : set_Ho137910533iple_a > $o, X2 : set_Ho137910533iple_a]: ((P @ X2) => ((![Y : set_Ho137910533iple_a]: ((P @ Y) => (ord_le1221261669iple_a @ Y @ X2))) => ((order_929906668iple_a @ P) = X2)))))). % Greatest_equality
thf(fact_98_Compl__anti__mono, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ B2) => (ord_le1221261669iple_a @ (uminus922456654iple_a @ B2) @ (uminus922456654iple_a @ A2)))))). % Compl_anti_mono
thf(fact_99_Compl__subset__Compl__iff, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ (uminus922456654iple_a @ A2) @ (uminus922456654iple_a @ B2)) = (ord_le1221261669iple_a @ B2 @ A2))))). % Compl_subset_Compl_iff
thf(fact_100_subset__Compl__self__eq, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ (uminus922456654iple_a @ A2)) = (A2 = bot_bo1298296729iple_a))))). % subset_Compl_self_eq
thf(fact_101_compl__le__compl__iff, axiom,
    ((![X2 : set_Ho137910533iple_a, Y3 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ (uminus922456654iple_a @ X2) @ (uminus922456654iple_a @ Y3)) = (ord_le1221261669iple_a @ Y3 @ X2))))). % compl_le_compl_iff
thf(fact_102_compl__mono, axiom,
    ((![X2 : set_Ho137910533iple_a, Y3 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X2 @ Y3) => (ord_le1221261669iple_a @ (uminus922456654iple_a @ Y3) @ (uminus922456654iple_a @ X2)))))). % compl_mono
thf(fact_103_compl__le__swap1, axiom,
    ((![Y3 : set_Ho137910533iple_a, X2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ Y3 @ (uminus922456654iple_a @ X2)) => (ord_le1221261669iple_a @ X2 @ (uminus922456654iple_a @ Y3)))))). % compl_le_swap1
thf(fact_104_compl__le__swap2, axiom,
    ((![Y3 : set_Ho137910533iple_a, X2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ (uminus922456654iple_a @ Y3) @ X2) => (ord_le1221261669iple_a @ (uminus922456654iple_a @ X2) @ Y3))))). % compl_le_swap2
thf(fact_105_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_106_bot__empty__eq, axiom,
    ((bot_bo431311916le_a_o = (^[X : hoare_1678595023iple_a]: (member1332298086iple_a @ X @ bot_bo1298296729iple_a))))). % bot_empty_eq
thf(fact_107_conseq, axiom,
    ((![P : a > state > $o, G2 : set_Ho137910533iple_a, C : com, Q : a > state > $o]: ((![Z3 : a, S : state]: ((P @ Z3 @ S) => (?[P3 : a > state > $o, Q2 : a > state > $o]: ((hoare_129598474rivs_a @ G2 @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P3 @ C @ Q2) @ bot_bo1298296729iple_a)) & (![S3 : state]: ((![Z4 : a]: ((P3 @ Z4 @ S) => (Q2 @ Z4 @ S3))) => (Q @ Z3 @ S3))))))) => (hoare_129598474rivs_a @ G2 @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q) @ bot_bo1298296729iple_a)))))). % conseq
thf(fact_108_conseq1, axiom,
    ((![G2 : set_Ho137910533iple_a, P4 : a > state > $o, C : com, Q : a > state > $o, P : a > state > $o]: ((hoare_129598474rivs_a @ G2 @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P4 @ C @ Q) @ bot_bo1298296729iple_a)) => ((![Z3 : a, S : state]: ((P @ Z3 @ S) => (P4 @ Z3 @ S))) => (hoare_129598474rivs_a @ G2 @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q) @ bot_bo1298296729iple_a))))))). % conseq1
thf(fact_109_conseq2, axiom,
    ((![G2 : set_Ho137910533iple_a, P : a > state > $o, C : com, Q3 : a > state > $o, Q : a > state > $o]: ((hoare_129598474rivs_a @ G2 @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q3) @ bot_bo1298296729iple_a)) => ((![Z3 : a, S : state]: ((Q3 @ Z3 @ S) => (Q @ Z3 @ S))) => (hoare_129598474rivs_a @ G2 @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q) @ bot_bo1298296729iple_a))))))). % conseq2
thf(fact_110_conseq12, axiom,
    ((![G2 : set_Ho137910533iple_a, P4 : a > state > $o, C : com, Q3 : a > state > $o, P : a > state > $o, Q : a > state > $o]: ((hoare_129598474rivs_a @ G2 @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P4 @ C @ Q3) @ bot_bo1298296729iple_a)) => ((![Z3 : a, S : state]: ((P @ Z3 @ S) => (![S3 : state]: ((![Z4 : a]: ((P4 @ Z4 @ S) => (Q3 @ Z4 @ S3))) => (Q @ Z3 @ S3))))) => (hoare_129598474rivs_a @ G2 @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q) @ bot_bo1298296729iple_a))))))). % conseq12
thf(fact_111_hoare__derivs_OSkip, axiom,
    ((![G2 : set_Ho137910533iple_a, P : a > state > $o]: (hoare_129598474rivs_a @ G2 @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ skip @ P) @ bot_bo1298296729iple_a))))). % hoare_derivs.Skip

% Conjectures (2)
thf(conj_0, hypothesis,
    ((hoare_129598474rivs_a @ g @ ts))).
thf(conj_1, conjecture,
    ((hoare_1775499016lids_a @ g @ ts))).
