% 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_218__3251618_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:12:50.402

% Could-be-implicit typings (5)
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__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_Com_Ocom_OSemi, type,
    semi : com > com > 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_Otriple_Otriple_001tf__a, type,
    hoare_719046530iple_a : (a > state > $o) > com > (a > state > $o) > hoare_1678595023iple_a).
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_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_Ga, type,
    ga : set_Ho137910533iple_a).
thf(sy_v_P, type,
    p : a > state > $o).
thf(sy_v_Q, type,
    q : a > state > $o).
thf(sy_v_c, type,
    c : com).

% Relevant facts (111)
thf(fact_0_triple_Oinject, axiom,
    ((![X1 : a > state > $o, X2 : com, X3 : a > state > $o, Y1 : a > state > $o, Y2 : com, Y3 : a > state > $o]: (((hoare_719046530iple_a @ X1 @ X2 @ X3) = (hoare_719046530iple_a @ Y1 @ Y2 @ Y3)) = (((X1 = Y1)) & ((((X2 = Y2)) & ((X3 = Y3))))))))). % triple.inject
thf(fact_1_conseq1, axiom,
    ((![G : set_Ho137910533iple_a, P : a > state > $o, C : com, Q : a > state > $o, P2 : a > state > $o]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q) @ bot_bo1298296729iple_a)) => ((![Z : a, S : state]: ((P2 @ Z @ S) => (P @ Z @ S))) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P2 @ C @ Q) @ bot_bo1298296729iple_a))))))). % conseq1
thf(fact_2_conseq2, axiom,
    ((![G : set_Ho137910533iple_a, P2 : a > state > $o, C : com, Q2 : a > state > $o, Q : a > state > $o]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P2 @ C @ Q2) @ bot_bo1298296729iple_a)) => ((![Z : a, S : state]: ((Q2 @ Z @ S) => (Q @ Z @ S))) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P2 @ C @ Q) @ bot_bo1298296729iple_a))))))). % conseq2
thf(fact_3_conseq12, axiom,
    ((![G : set_Ho137910533iple_a, P : a > state > $o, C : com, Q2 : a > state > $o, P2 : a > state > $o, Q : a > state > $o]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q2) @ bot_bo1298296729iple_a)) => ((![Z : a, S : state]: ((P2 @ Z @ S) => (![S2 : state]: ((![Z2 : a]: ((P @ Z2 @ S) => (Q2 @ Z2 @ S2))) => (Q @ Z @ S2))))) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P2 @ C @ Q) @ bot_bo1298296729iple_a))))))). % conseq12
thf(fact_4_triple_Oinduct, axiom,
    ((![P2 : hoare_1678595023iple_a > $o, Triple : hoare_1678595023iple_a]: ((![X1a : a > state > $o, X2a : com, X3a : a > state > $o]: (P2 @ (hoare_719046530iple_a @ X1a @ X2a @ X3a))) => (P2 @ Triple))))). % triple.induct
thf(fact_5_triple_Oexhaust, axiom,
    ((![Y : hoare_1678595023iple_a]: (~ ((![X12 : a > state > $o, X22 : com, X32 : a > state > $o]: (~ ((Y = (hoare_719046530iple_a @ X12 @ X22 @ X32)))))))))). % triple.exhaust
thf(fact_6_asm, axiom,
    ((![Ts : set_Ho137910533iple_a, G : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ Ts @ G) => (hoare_129598474rivs_a @ G @ Ts))))). % asm
thf(fact_7_cut, axiom,
    ((![G2 : set_Ho137910533iple_a, Ts : set_Ho137910533iple_a, G : set_Ho137910533iple_a]: ((hoare_129598474rivs_a @ G2 @ Ts) => ((hoare_129598474rivs_a @ G @ G2) => (hoare_129598474rivs_a @ G @ Ts)))))). % cut
thf(fact_8_empty, axiom,
    ((![G : set_Ho137910533iple_a]: (hoare_129598474rivs_a @ G @ bot_bo1298296729iple_a)))). % empty
thf(fact_9_conseq, axiom,
    ((![P2 : a > state > $o, G : set_Ho137910533iple_a, C : com, Q : a > state > $o]: ((![Z : a, S : state]: ((P2 @ Z @ S) => (?[P3 : a > state > $o, Q3 : a > state > $o]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P3 @ C @ Q3) @ bot_bo1298296729iple_a)) & (![S2 : state]: ((![Z2 : a]: ((P3 @ Z2 @ S) => (Q3 @ Z2 @ S2))) => (Q @ Z @ S2))))))) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P2 @ C @ Q) @ bot_bo1298296729iple_a)))))). % conseq
thf(fact_10_hoare__derivs_Oinsert, axiom,
    ((![G : set_Ho137910533iple_a, T : hoare_1678595023iple_a, Ts : set_Ho137910533iple_a]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ T @ bot_bo1298296729iple_a)) => ((hoare_129598474rivs_a @ G @ Ts) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ T @ Ts))))))). % hoare_derivs.insert
thf(fact_11_weaken, axiom,
    ((![G : set_Ho137910533iple_a, Ts2 : set_Ho137910533iple_a, Ts : set_Ho137910533iple_a]: ((hoare_129598474rivs_a @ G @ Ts2) => ((ord_le1221261669iple_a @ Ts @ Ts2) => (hoare_129598474rivs_a @ G @ Ts)))))). % weaken
thf(fact_12_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_13_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_14_insert__subset, axiom,
    ((![X : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ (insert1477804543iple_a @ X @ A2) @ B2) = (((member1332298086iple_a @ X @ B2)) & ((ord_le1221261669iple_a @ A2 @ B2))))))). % insert_subset
thf(fact_15_singletonI, axiom,
    ((![A : hoare_1678595023iple_a]: (member1332298086iple_a @ A @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a))))). % singletonI
thf(fact_16_subset__empty, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ bot_bo1298296729iple_a) = (A2 = bot_bo1298296729iple_a))))). % subset_empty
thf(fact_17_empty__subsetI, axiom,
    ((![A2 : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ bot_bo1298296729iple_a @ A2)))). % empty_subsetI
thf(fact_18_subset__singletonD, axiom,
    ((![A2 : set_Ho137910533iple_a, X : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A2 @ (insert1477804543iple_a @ X @ bot_bo1298296729iple_a)) => ((A2 = bot_bo1298296729iple_a) | (A2 = (insert1477804543iple_a @ X @ bot_bo1298296729iple_a))))))). % subset_singletonD
thf(fact_19_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_20_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_21_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_22_insert__absorb2, axiom,
    ((![X : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((insert1477804543iple_a @ X @ (insert1477804543iple_a @ X @ A2)) = (insert1477804543iple_a @ X @ A2))))). % insert_absorb2
thf(fact_23_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_24_empty__Collect__eq, axiom,
    ((![P2 : hoare_1678595023iple_a > $o]: ((bot_bo1298296729iple_a = (collec1600235172iple_a @ P2)) = (![X6 : hoare_1678595023iple_a]: (~ ((P2 @ X6)))))))). % empty_Collect_eq
thf(fact_25_Collect__empty__eq, axiom,
    ((![P2 : hoare_1678595023iple_a > $o]: (((collec1600235172iple_a @ P2) = bot_bo1298296729iple_a) = (![X6 : hoare_1678595023iple_a]: (~ ((P2 @ X6)))))))). % Collect_empty_eq
thf(fact_26_all__not__in__conv, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((![X6 : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ X6 @ A2)))) = (A2 = bot_bo1298296729iple_a))))). % all_not_in_conv
thf(fact_27_empty__iff, axiom,
    ((![C : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ C @ bot_bo1298296729iple_a)))))). % empty_iff
thf(fact_28_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_29_ex__in__conv, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((?[X6 : hoare_1678595023iple_a]: (member1332298086iple_a @ X6 @ A2)) = (~ ((A2 = bot_bo1298296729iple_a))))))). % ex_in_conv
thf(fact_30_equals0I, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((![Y4 : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ Y4 @ A2)))) => (A2 = bot_bo1298296729iple_a))))). % equals0I
thf(fact_31_equals0D, axiom,
    ((![A2 : set_Ho137910533iple_a, A : hoare_1678595023iple_a]: ((A2 = bot_bo1298296729iple_a) => (~ ((member1332298086iple_a @ A @ A2))))))). % equals0D
thf(fact_32_emptyE, axiom,
    ((![A : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ A @ bot_bo1298296729iple_a)))))). % emptyE
thf(fact_33_Collect__mono__iff, axiom,
    ((![P2 : hoare_1678595023iple_a > $o, Q : hoare_1678595023iple_a > $o]: ((ord_le1221261669iple_a @ (collec1600235172iple_a @ P2) @ (collec1600235172iple_a @ Q)) = (![X6 : hoare_1678595023iple_a]: (((P2 @ X6)) => ((Q @ X6)))))))). % Collect_mono_iff
thf(fact_34_set__eq__subset, axiom,
    (((^[Y5 : set_Ho137910533iple_a]: (^[Z3 : set_Ho137910533iple_a]: (Y5 = Z3))) = (^[A3 : set_Ho137910533iple_a]: (^[B3 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ A3 @ B3)) & ((ord_le1221261669iple_a @ B3 @ A3)))))))). % set_eq_subset
thf(fact_35_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_36_Collect__mono, axiom,
    ((![P2 : hoare_1678595023iple_a > $o, Q : hoare_1678595023iple_a > $o]: ((![X5 : hoare_1678595023iple_a]: ((P2 @ X5) => (Q @ X5))) => (ord_le1221261669iple_a @ (collec1600235172iple_a @ P2) @ (collec1600235172iple_a @ Q)))))). % Collect_mono
thf(fact_37_subset__refl, axiom,
    ((![A2 : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ A2 @ A2)))). % subset_refl
thf(fact_38_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_39_equalityD2, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((A2 = B2) => (ord_le1221261669iple_a @ B2 @ A2))))). % equalityD2
thf(fact_40_equalityD1, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((A2 = B2) => (ord_le1221261669iple_a @ A2 @ B2))))). % equalityD1
thf(fact_41_subset__eq, axiom,
    ((ord_le1221261669iple_a = (^[A3 : set_Ho137910533iple_a]: (^[B3 : set_Ho137910533iple_a]: (![X6 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X6 @ A3)) => ((member1332298086iple_a @ X6 @ B3))))))))). % subset_eq
thf(fact_42_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_43_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_44_in__mono, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, X : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A2 @ B2) => ((member1332298086iple_a @ X @ A2) => (member1332298086iple_a @ X @ B2)))))). % in_mono
thf(fact_45_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_46_insert__commute, axiom,
    ((![X : hoare_1678595023iple_a, Y : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((insert1477804543iple_a @ X @ (insert1477804543iple_a @ Y @ A2)) = (insert1477804543iple_a @ Y @ (insert1477804543iple_a @ X @ A2)))))). % insert_commute
thf(fact_47_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_48_insert__absorb, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ A @ A2) => ((insert1477804543iple_a @ A @ A2) = A2))))). % insert_absorb
thf(fact_49_insert__ident, axiom,
    ((![X : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ X @ A2))) => ((~ ((member1332298086iple_a @ X @ B2))) => (((insert1477804543iple_a @ X @ A2) = (insert1477804543iple_a @ X @ B2)) = (A2 = B2))))))). % insert_ident
thf(fact_50_Set_Oset__insert, axiom,
    ((![X : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ X @ A2) => (~ ((![B4 : set_Ho137910533iple_a]: ((A2 = (insert1477804543iple_a @ X @ B4)) => (member1332298086iple_a @ X @ B4))))))))). % Set.set_insert
thf(fact_51_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_52_insertI1, axiom,
    ((![A : hoare_1678595023iple_a, B2 : set_Ho137910533iple_a]: (member1332298086iple_a @ A @ (insert1477804543iple_a @ A @ B2))))). % insertI1
thf(fact_53_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_54_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_55_insert__not__empty, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: (~ (((insert1477804543iple_a @ A @ A2) = bot_bo1298296729iple_a)))))). % insert_not_empty
thf(fact_56_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_57_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_58_singletonD, axiom,
    ((![B : hoare_1678595023iple_a, A : hoare_1678595023iple_a]: ((member1332298086iple_a @ B @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a)) => (B = A))))). % singletonD
thf(fact_59_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_60_subset__insertI, axiom,
    ((![B2 : set_Ho137910533iple_a, A : hoare_1678595023iple_a]: (ord_le1221261669iple_a @ B2 @ (insert1477804543iple_a @ A @ B2))))). % subset_insertI
thf(fact_61_subset__insert, axiom,
    ((![X : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ X @ A2))) => ((ord_le1221261669iple_a @ A2 @ (insert1477804543iple_a @ X @ B2)) = (ord_le1221261669iple_a @ A2 @ B2)))))). % subset_insert
thf(fact_62_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_63_the__elem__eq, axiom,
    ((![X : hoare_1678595023iple_a]: ((the_el434698138iple_a @ (insert1477804543iple_a @ X @ bot_bo1298296729iple_a)) = X)))). % the_elem_eq
thf(fact_64_order__refl, axiom,
    ((![X : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ X @ X)))). % order_refl
thf(fact_65_is__singletonI, axiom,
    ((![X : hoare_1678595023iple_a]: (is_sin1784037339iple_a @ (insert1477804543iple_a @ X @ bot_bo1298296729iple_a))))). % is_singletonI
thf(fact_66_insert__subsetI, axiom,
    ((![X : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, X4 : set_Ho137910533iple_a]: ((member1332298086iple_a @ X @ A2) => ((ord_le1221261669iple_a @ X4 @ A2) => (ord_le1221261669iple_a @ (insert1477804543iple_a @ X @ 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_hoare__derivs_OSkip, axiom,
    ((![G : set_Ho137910533iple_a, P2 : a > state > $o]: (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P2 @ skip @ P2) @ bot_bo1298296729iple_a))))). % hoare_derivs.Skip
thf(fact_69_bot_Oextremum__uniqueI, axiom,
    ((![A : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ bot_bo1298296729iple_a) => (A = bot_bo1298296729iple_a))))). % bot.extremum_uniqueI
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, Y4 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X5 @ A2) => ((member1332298086iple_a @ Y4 @ A2) => (X5 = Y4)))) => (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, Y4 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X5 @ Y4) => (ord_le1221261669iple_a @ (F @ X5) @ (F @ Y4)))) => (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, Y4 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X5 @ Y4) => (ord_le1221261669iple_a @ (F @ X5) @ (F @ Y4)))) => (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, Y4 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X5 @ Y4) => (ord_le1221261669iple_a @ (F @ X5) @ (F @ Y4)))) => (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, Y4 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X5 @ Y4) => (ord_le1221261669iple_a @ (F @ X5) @ (F @ Y4)))) => (ord_le1221261669iple_a @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_77_eq__iff, axiom,
    (((^[Y5 : set_Ho137910533iple_a]: (^[Z3 : set_Ho137910533iple_a]: (Y5 = Z3))) = (^[X6 : set_Ho137910533iple_a]: (^[Y6 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ X6 @ Y6)) & ((ord_le1221261669iple_a @ Y6 @ X6)))))))). % eq_iff
thf(fact_78_antisym, axiom,
    ((![X : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X @ Y) => ((ord_le1221261669iple_a @ Y @ X) => (X = Y)))))). % antisym
thf(fact_79_eq__refl, axiom,
    ((![X : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((X = Y) => (ord_le1221261669iple_a @ X @ Y))))). % 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,
    ((![Y : set_Ho137910533iple_a, X : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ Y @ X) => ((ord_le1221261669iple_a @ X @ Y) = (X = Y)))))). % antisym_conv
thf(fact_82_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y5 : set_Ho137910533iple_a]: (^[Z3 : set_Ho137910533iple_a]: (Y5 = Z3))) = (^[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,
    ((![X : set_Ho137910533iple_a, Y : set_Ho137910533iple_a, Z4 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X @ Y) => ((ord_le1221261669iple_a @ Y @ Z4) => (ord_le1221261669iple_a @ X @ Z4)))))). % 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,
    (((^[Y5 : set_Ho137910533iple_a]: (^[Z3 : set_Ho137910533iple_a]: (Y5 = Z3))) = (^[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]: (?[X6 : hoare_1678595023iple_a]: (A3 = (insert1477804543iple_a @ X6 @ 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_bot_Oextremum__unique, axiom,
    ((![A : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ bot_bo1298296729iple_a) = (A = bot_bo1298296729iple_a))))). % bot.extremum_unique
thf(fact_95_Comp, axiom,
    ((![G : set_Ho137910533iple_a, P2 : a > state > $o, C : com, Q : a > state > $o, D : com, R : a > state > $o]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P2 @ C @ Q) @ bot_bo1298296729iple_a)) => ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ Q @ D @ R) @ bot_bo1298296729iple_a)) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P2 @ (semi @ C @ D) @ R) @ bot_bo1298296729iple_a))))))). % Comp
thf(fact_96_Set_Ois__empty__def, axiom,
    ((is_emp901906557iple_a = (^[A3 : set_Ho137910533iple_a]: (A3 = bot_bo1298296729iple_a))))). % Set.is_empty_def
thf(fact_97_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_98_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_99_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_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,
    ((![X : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ (uminus922456654iple_a @ X) @ (uminus922456654iple_a @ Y)) = (ord_le1221261669iple_a @ Y @ X))))). % compl_le_compl_iff
thf(fact_102_compl__mono, axiom,
    ((![X : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X @ Y) => (ord_le1221261669iple_a @ (uminus922456654iple_a @ Y) @ (uminus922456654iple_a @ X)))))). % compl_mono
thf(fact_103_compl__le__swap1, axiom,
    ((![Y : set_Ho137910533iple_a, X : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ Y @ (uminus922456654iple_a @ X)) => (ord_le1221261669iple_a @ X @ (uminus922456654iple_a @ Y)))))). % compl_le_swap1
thf(fact_104_compl__le__swap2, axiom,
    ((![Y : set_Ho137910533iple_a, X : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ (uminus922456654iple_a @ Y) @ X) => (ord_le1221261669iple_a @ (uminus922456654iple_a @ X) @ Y))))). % compl_le_swap2
thf(fact_105_com_Oinject_I3_J, axiom,
    ((![X41 : com, X42 : com, Y41 : com, Y42 : com]: (((semi @ X41 @ X42) = (semi @ Y41 @ Y42)) = (((X41 = Y41)) & ((X42 = Y42))))))). % com.inject(3)
thf(fact_106_com_Odistinct_I5_J, axiom,
    ((![X41 : com, X42 : com]: (~ ((skip = (semi @ X41 @ X42))))))). % com.distinct(5)
thf(fact_107_bot__empty__eq, axiom,
    ((bot_bo431311916le_a_o = (^[X6 : hoare_1678595023iple_a]: (member1332298086iple_a @ X6 @ bot_bo1298296729iple_a))))). % bot_empty_eq
thf(fact_108_Collect__empty__eq__bot, axiom,
    ((![P2 : hoare_1678595023iple_a > $o]: (((collec1600235172iple_a @ P2) = bot_bo1298296729iple_a) = (P2 = bot_bo431311916le_a_o))))). % Collect_empty_eq_bot
thf(fact_109_GreatestI2__order, axiom,
    ((![P2 : set_Ho137910533iple_a > $o, X : set_Ho137910533iple_a, Q : set_Ho137910533iple_a > $o]: ((P2 @ X) => ((![Y4 : set_Ho137910533iple_a]: ((P2 @ Y4) => (ord_le1221261669iple_a @ Y4 @ X))) => ((![X5 : set_Ho137910533iple_a]: ((P2 @ X5) => ((![Y7 : set_Ho137910533iple_a]: ((P2 @ Y7) => (ord_le1221261669iple_a @ Y7 @ X5))) => (Q @ X5)))) => (Q @ (order_929906668iple_a @ P2)))))))). % GreatestI2_order
thf(fact_110_Greatest__equality, axiom,
    ((![P2 : set_Ho137910533iple_a > $o, X : set_Ho137910533iple_a]: ((P2 @ X) => ((![Y4 : set_Ho137910533iple_a]: ((P2 @ Y4) => (ord_le1221261669iple_a @ Y4 @ X))) => ((order_929906668iple_a @ P2) = X)))))). % Greatest_equality

% Conjectures (3)
thf(conj_0, hypothesis,
    ((![Z5 : a, S3 : state]: ((p @ Z5 @ S3) => (?[P4 : a > state > $o, Q4 : a > state > $o]: (((hoare_129598474rivs_a @ g @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P4 @ c @ Q4) @ bot_bo1298296729iple_a)) & (![G3 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ g @ G3) => (hoare_129598474rivs_a @ G3 @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P4 @ c @ Q4) @ bot_bo1298296729iple_a))))) & (![S4 : state]: ((![Z6 : a]: ((P4 @ Z6 @ S3) => (Q4 @ Z6 @ S4))) => (q @ Z5 @ S4))))))))).
thf(conj_1, hypothesis,
    ((ord_le1221261669iple_a @ g @ ga))).
thf(conj_2, conjecture,
    ((hoare_129598474rivs_a @ ga @ (insert1477804543iple_a @ (hoare_719046530iple_a @ p @ c @ q) @ bot_bo1298296729iple_a)))).
