% 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_472__3255014_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:15:12.809

% Could-be-implicit typings (7)
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__Ovname, type,
    vname : $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 (26)
thf(sy_c_Com_Ocom_OAss, type,
    ass : vname > (state > nat) > com).
thf(sy_c_Com_Ocom_OCond, type,
    cond : (state > $o) > com > com > com).
thf(sy_c_Com_Ocom_OSKIP, type,
    skip : com).
thf(sy_c_Com_Ocom_OSemi, type,
    semi : com > com > com).
thf(sy_c_Com_Ocom_OWhile, type,
    while : (state > $o) > com > com).
thf(sy_c_Fun_Ocomp_001_Eo_001_Eo_001t__Com__Ostate, type,
    comp_o_o_state : ($o > $o) > (state > $o) > state > $o).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    minus_1852999390iple_a : set_Ho137910533iple_a > set_Ho137910533iple_a > set_Ho137910533iple_a).
thf(sy_c_HOL_OEx1_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    ex1_Ho899980597iple_a : (hoare_1678595023iple_a > $o) > $o).
thf(sy_c_HOL_OThe_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    the_Ho1602500360iple_a : (hoare_1678595023iple_a > $o) > hoare_1678595023iple_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_Opeek__and_001tf__a, type,
    hoare_1795417776_and_a : (a > state > $o) > (state > $o) > a > state > $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_Natural_Oupdate, type,
    update : state > vname > nat > state).
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_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_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_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_Q, type,
    q : a > state > $o).
thf(sy_v_c, type,
    c : com).

% Relevant facts (92)
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_escape, axiom,
    ((![P : a > state > $o, G : set_Ho137910533iple_a, C : com, Q : a > state > $o]: ((![Z : a, S : state]: ((P @ Z @ S) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ (^[Za : a]: (^[S2 : state]: (S2 = S))) @ C @ (^[Z2 : a]: (Q @ Z))) @ bot_bo1298296729iple_a)))) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q) @ bot_bo1298296729iple_a)))))). % escape
thf(fact_2_conseq1, axiom,
    ((![G : set_Ho137910533iple_a, P2 : a > state > $o, C : com, Q : a > state > $o, P : a > state > $o]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P2 @ C @ Q) @ bot_bo1298296729iple_a)) => ((![Z : a, S : state]: ((P @ Z @ S) => (P2 @ Z @ S))) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q) @ bot_bo1298296729iple_a))))))). % conseq1
thf(fact_3_conseq2, axiom,
    ((![G : set_Ho137910533iple_a, P : a > state > $o, C : com, Q2 : 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]: ((Q2 @ Z @ S) => (Q @ Z @ S))) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q) @ bot_bo1298296729iple_a))))))). % conseq2
thf(fact_4_conseq12, axiom,
    ((![G : set_Ho137910533iple_a, P2 : a > state > $o, C : com, Q2 : a > state > $o, P : 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]: ((P @ Z @ S) => (![S3 : state]: ((![Z3 : a]: ((P2 @ Z3 @ S) => (Q2 @ Z3 @ S3))) => (Q @ Z @ S3))))) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q) @ bot_bo1298296729iple_a))))))). % conseq12
thf(fact_5__Cconstant_C, axiom,
    ((![C2 : $o, G : set_Ho137910533iple_a, P : a > state > $o, C : com, Q : a > state > $o]: ((C2 => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q) @ bot_bo1298296729iple_a))) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ (^[Z4 : a]: (^[S4 : state]: (((P @ Z4 @ S4)) & (C2)))) @ C @ Q) @ bot_bo1298296729iple_a)))))). % "constant"
thf(fact_6_triple_Oinduct, axiom,
    ((![P : hoare_1678595023iple_a > $o, Triple : hoare_1678595023iple_a]: ((![X1a : a > state > $o, X2a : com, X3a : a > state > $o]: (P @ (hoare_719046530iple_a @ X1a @ X2a @ X3a))) => (P @ Triple))))). % triple.induct
thf(fact_7_derivs__insertD, axiom,
    ((![G : set_Ho137910533iple_a, T : hoare_1678595023iple_a, Ts : set_Ho137910533iple_a]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ T @ Ts)) => ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ T @ bot_bo1298296729iple_a)) & (hoare_129598474rivs_a @ G @ Ts)))))). % derivs_insertD
thf(fact_8_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_9_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_10_empty, axiom,
    ((![G : set_Ho137910533iple_a]: (hoare_129598474rivs_a @ G @ bot_bo1298296729iple_a)))). % empty
thf(fact_11_conseq, axiom,
    ((![P : a > state > $o, G : set_Ho137910533iple_a, C : com, Q : a > state > $o]: ((![Z : a, S : state]: ((P @ 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)) & (![S3 : state]: ((![Z3 : a]: ((P3 @ Z3 @ S) => (Q3 @ Z3 @ S3))) => (Q @ Z @ S3))))))) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q) @ bot_bo1298296729iple_a)))))). % conseq
thf(fact_12_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_13_singleton__conv, axiom,
    ((![A : hoare_1678595023iple_a]: ((collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: (X = A))) = (insert1477804543iple_a @ A @ bot_bo1298296729iple_a))))). % singleton_conv
thf(fact_14_singleton__conv2, axiom,
    ((![A : hoare_1678595023iple_a]: ((collec1600235172iple_a @ ((^[Y4 : hoare_1678595023iple_a]: (^[Z5 : hoare_1678595023iple_a]: (Y4 = Z5))) @ A)) = (insert1477804543iple_a @ A @ bot_bo1298296729iple_a))))). % singleton_conv2
thf(fact_15_singletonI, axiom,
    ((![A : hoare_1678595023iple_a]: (member1332298086iple_a @ A @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a))))). % singletonI
thf(fact_16_insertCI, axiom,
    ((![A : hoare_1678595023iple_a, B : set_Ho137910533iple_a, B2 : hoare_1678595023iple_a]: (((~ ((member1332298086iple_a @ A @ B))) => (A = B2)) => (member1332298086iple_a @ A @ (insert1477804543iple_a @ B2 @ B)))))). % insertCI
thf(fact_17_insert__iff, axiom,
    ((![A : hoare_1678595023iple_a, B2 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ A @ (insert1477804543iple_a @ B2 @ A2)) = (((A = B2)) | ((member1332298086iple_a @ A @ A2))))))). % insert_iff
thf(fact_18_insert__absorb2, axiom,
    ((![X4 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((insert1477804543iple_a @ X4 @ (insert1477804543iple_a @ X4 @ A2)) = (insert1477804543iple_a @ X4 @ A2))))). % insert_absorb2
thf(fact_19_empty__iff, axiom,
    ((![C : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ C @ bot_bo1298296729iple_a)))))). % empty_iff
thf(fact_20_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_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_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_23_Collect__conv__if, axiom,
    ((![P : hoare_1678595023iple_a > $o, A : hoare_1678595023iple_a]: (((P @ A) => ((collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: (((X = A)) & ((P @ X))))) = (insert1477804543iple_a @ A @ bot_bo1298296729iple_a))) & ((~ ((P @ A))) => ((collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: (((X = A)) & ((P @ X))))) = bot_bo1298296729iple_a)))))). % Collect_conv_if
thf(fact_24_bot__set__def, axiom,
    ((bot_bo1298296729iple_a = (collec1600235172iple_a @ bot_bo431311916le_a_o)))). % bot_set_def
thf(fact_25_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_26_equals0I, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((![Y5 : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ Y5 @ A2)))) => (A2 = bot_bo1298296729iple_a))))). % equals0I
thf(fact_27_equals0D, axiom,
    ((![A2 : set_Ho137910533iple_a, A : hoare_1678595023iple_a]: ((A2 = bot_bo1298296729iple_a) => (~ ((member1332298086iple_a @ A @ A2))))))). % equals0D
thf(fact_28_emptyE, axiom,
    ((![A : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ A @ bot_bo1298296729iple_a)))))). % emptyE
thf(fact_29_mk__disjoint__insert, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ A @ A2) => (?[B3 : set_Ho137910533iple_a]: ((A2 = (insert1477804543iple_a @ A @ B3)) & (~ ((member1332298086iple_a @ A @ B3))))))))). % mk_disjoint_insert
thf(fact_30_insert__commute, axiom,
    ((![X4 : hoare_1678595023iple_a, Y : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((insert1477804543iple_a @ X4 @ (insert1477804543iple_a @ Y @ A2)) = (insert1477804543iple_a @ Y @ (insert1477804543iple_a @ X4 @ A2)))))). % insert_commute
thf(fact_31_insert__eq__iff, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B2 : hoare_1678595023iple_a, B : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ A @ A2))) => ((~ ((member1332298086iple_a @ B2 @ B))) => (((insert1477804543iple_a @ A @ A2) = (insert1477804543iple_a @ B2 @ B)) = (((((A = B2)) => ((A2 = B)))) & ((((~ ((A = B2)))) => ((?[C3 : set_Ho137910533iple_a]: (((A2 = (insert1477804543iple_a @ B2 @ C3))) & ((((~ ((member1332298086iple_a @ B2 @ C3)))) & ((((B = (insert1477804543iple_a @ A @ C3))) & ((~ ((member1332298086iple_a @ A @ C3)))))))))))))))))))). % insert_eq_iff
thf(fact_32_insert__absorb, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ A @ A2) => ((insert1477804543iple_a @ A @ A2) = A2))))). % insert_absorb
thf(fact_33_insert__ident, axiom,
    ((![X4 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ X4 @ A2))) => ((~ ((member1332298086iple_a @ X4 @ B))) => (((insert1477804543iple_a @ X4 @ A2) = (insert1477804543iple_a @ X4 @ B)) = (A2 = B))))))). % insert_ident
thf(fact_34_Set_Oset__insert, axiom,
    ((![X4 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ X4 @ A2) => (~ ((![B3 : set_Ho137910533iple_a]: ((A2 = (insert1477804543iple_a @ X4 @ B3)) => (member1332298086iple_a @ X4 @ B3))))))))). % Set.set_insert
thf(fact_35_insertI2, axiom,
    ((![A : hoare_1678595023iple_a, B : set_Ho137910533iple_a, B2 : hoare_1678595023iple_a]: ((member1332298086iple_a @ A @ B) => (member1332298086iple_a @ A @ (insert1477804543iple_a @ B2 @ B)))))). % insertI2
thf(fact_36_insertI1, axiom,
    ((![A : hoare_1678595023iple_a, B : set_Ho137910533iple_a]: (member1332298086iple_a @ A @ (insert1477804543iple_a @ A @ B))))). % insertI1
thf(fact_37_insertE, axiom,
    ((![A : hoare_1678595023iple_a, B2 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ A @ (insert1477804543iple_a @ B2 @ A2)) => ((~ ((A = B2))) => (member1332298086iple_a @ A @ A2)))))). % insertE
thf(fact_38_empty__def, axiom,
    ((bot_bo1298296729iple_a = (collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: $false))))). % empty_def
thf(fact_39_insert__Collect, axiom,
    ((![A : hoare_1678595023iple_a, P : hoare_1678595023iple_a > $o]: ((insert1477804543iple_a @ A @ (collec1600235172iple_a @ P)) = (collec1600235172iple_a @ (^[U : hoare_1678595023iple_a]: (((~ ((U = A)))) => ((P @ U))))))))). % insert_Collect
thf(fact_40_insert__compr, axiom,
    ((insert1477804543iple_a = (^[A3 : hoare_1678595023iple_a]: (^[B4 : set_Ho137910533iple_a]: (collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: (((X = A3)) | ((member1332298086iple_a @ X @ B4)))))))))). % insert_compr
thf(fact_41_singleton__inject, axiom,
    ((![A : hoare_1678595023iple_a, B2 : hoare_1678595023iple_a]: (((insert1477804543iple_a @ A @ bot_bo1298296729iple_a) = (insert1477804543iple_a @ B2 @ bot_bo1298296729iple_a)) => (A = B2))))). % singleton_inject
thf(fact_42_insert__not__empty, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: (~ (((insert1477804543iple_a @ A @ A2) = bot_bo1298296729iple_a)))))). % insert_not_empty
thf(fact_43_doubleton__eq__iff, axiom,
    ((![A : hoare_1678595023iple_a, B2 : hoare_1678595023iple_a, C : hoare_1678595023iple_a, D : hoare_1678595023iple_a]: (((insert1477804543iple_a @ A @ (insert1477804543iple_a @ B2 @ bot_bo1298296729iple_a)) = (insert1477804543iple_a @ C @ (insert1477804543iple_a @ D @ bot_bo1298296729iple_a))) = (((((A = C)) & ((B2 = D)))) | ((((A = D)) & ((B2 = C))))))))). % doubleton_eq_iff
thf(fact_44_singleton__iff, axiom,
    ((![B2 : hoare_1678595023iple_a, A : hoare_1678595023iple_a]: ((member1332298086iple_a @ B2 @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a)) = (B2 = A))))). % singleton_iff
thf(fact_45_singletonD, axiom,
    ((![B2 : hoare_1678595023iple_a, A : hoare_1678595023iple_a]: ((member1332298086iple_a @ B2 @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a)) => (B2 = A))))). % singletonD
thf(fact_46_Collect__conv__if2, axiom,
    ((![P : hoare_1678595023iple_a > $o, A : hoare_1678595023iple_a]: (((P @ A) => ((collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: (((A = X)) & ((P @ X))))) = (insert1477804543iple_a @ A @ bot_bo1298296729iple_a))) & ((~ ((P @ A))) => ((collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: (((A = X)) & ((P @ X))))) = bot_bo1298296729iple_a)))))). % Collect_conv_if2
thf(fact_47_the__elem__eq, axiom,
    ((![X4 : hoare_1678595023iple_a]: ((the_el434698138iple_a @ (insert1477804543iple_a @ X4 @ bot_bo1298296729iple_a)) = X4)))). % the_elem_eq
thf(fact_48_is__singletonI, axiom,
    ((![X4 : hoare_1678595023iple_a]: (is_sin1784037339iple_a @ (insert1477804543iple_a @ X4 @ bot_bo1298296729iple_a))))). % is_singletonI
thf(fact_49_hoare__derivs_OSkip, axiom,
    ((![G : set_Ho137910533iple_a, P : a > state > $o]: (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ skip @ P) @ bot_bo1298296729iple_a))))). % hoare_derivs.Skip
thf(fact_50_Comp, axiom,
    ((![G : set_Ho137910533iple_a, P : a > state > $o, C : com, Q : a > state > $o, D : com, R : a > state > $o]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ 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 @ P @ (semi @ C @ D) @ R) @ bot_bo1298296729iple_a))))))). % Comp
thf(fact_51_LoopF, axiom,
    ((![G : set_Ho137910533iple_a, P : a > state > $o, B2 : state > $o, C : com]: (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ (^[Z4 : a]: (^[S4 : state]: (((P @ Z4 @ S4)) & ((~ ((B2 @ S4))))))) @ (while @ B2 @ C) @ P) @ bot_bo1298296729iple_a))))). % LoopF
thf(fact_52_Set_Ois__empty__def, axiom,
    ((is_emp901906557iple_a = (^[A4 : set_Ho137910533iple_a]: (A4 = bot_bo1298296729iple_a))))). % Set.is_empty_def
thf(fact_53_is__singleton__def, axiom,
    ((is_sin1784037339iple_a = (^[A4 : set_Ho137910533iple_a]: (?[X : hoare_1678595023iple_a]: (A4 = (insert1477804543iple_a @ X @ bot_bo1298296729iple_a))))))). % is_singleton_def
thf(fact_54_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_55_is__singleton__the__elem, axiom,
    ((is_sin1784037339iple_a = (^[A4 : set_Ho137910533iple_a]: (A4 = (insert1477804543iple_a @ (the_el434698138iple_a @ A4) @ bot_bo1298296729iple_a)))))). % is_singleton_the_elem
thf(fact_56_is__singletonI_H, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((~ ((A2 = bot_bo1298296729iple_a))) => ((![X5 : hoare_1678595023iple_a, Y5 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X5 @ A2) => ((member1332298086iple_a @ Y5 @ A2) => (X5 = Y5)))) => (is_sin1784037339iple_a @ A2)))))). % is_singletonI'
thf(fact_57_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_58_com_Oinject_I5_J, axiom,
    ((![X61 : state > $o, X62 : com, Y61 : state > $o, Y62 : com]: (((while @ X61 @ X62) = (while @ Y61 @ Y62)) = (((X61 = Y61)) & ((X62 = Y62))))))). % com.inject(5)
thf(fact_59_com_Odistinct_I5_J, axiom,
    ((![X41 : com, X42 : com]: (~ ((skip = (semi @ X41 @ X42))))))). % com.distinct(5)
thf(fact_60_com_Odistinct_I9_J, axiom,
    ((![X61 : state > $o, X62 : com]: (~ ((skip = (while @ X61 @ X62))))))). % com.distinct(9)
thf(fact_61_com_Odistinct_I39_J, axiom,
    ((![X41 : com, X42 : com, X61 : state > $o, X62 : com]: (~ (((semi @ X41 @ X42) = (while @ X61 @ X62))))))). % com.distinct(39)
thf(fact_62_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_63_bot__empty__eq, axiom,
    ((bot_bo431311916le_a_o = (^[X : hoare_1678595023iple_a]: (member1332298086iple_a @ X @ bot_bo1298296729iple_a))))). % bot_empty_eq
thf(fact_64_the__elem__def, axiom,
    ((the_el434698138iple_a = (^[X6 : set_Ho137910533iple_a]: (the_Ho1602500360iple_a @ (^[X : hoare_1678595023iple_a]: (X6 = (insert1477804543iple_a @ X @ bot_bo1298296729iple_a)))))))). % the_elem_def
thf(fact_65_Loop, axiom,
    ((![G : set_Ho137910533iple_a, P : a > state > $o, B2 : state > $o, C : com]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ (hoare_1795417776_and_a @ P @ B2) @ C @ P) @ bot_bo1298296729iple_a)) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ (while @ B2 @ C) @ (hoare_1795417776_and_a @ P @ (comp_o_o_state @ (~) @ B2))) @ bot_bo1298296729iple_a)))))). % Loop
thf(fact_66_Loop__sound__lemma, axiom,
    ((![G : set_Ho137910533iple_a, P : a > state > $o, B2 : state > $o, C : com]: ((hoare_1775499016lids_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ (hoare_1795417776_and_a @ P @ B2) @ C @ P) @ bot_bo1298296729iple_a)) => (hoare_1775499016lids_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ (while @ B2 @ C) @ (hoare_1795417776_and_a @ P @ (comp_o_o_state @ (~) @ B2))) @ bot_bo1298296729iple_a)))))). % Loop_sound_lemma
thf(fact_67_hoare__sound, axiom,
    ((![G : set_Ho137910533iple_a, Ts : set_Ho137910533iple_a]: ((hoare_129598474rivs_a @ G @ Ts) => (hoare_1775499016lids_a @ G @ Ts))))). % hoare_sound
thf(fact_68_hoare__derivs_OIf, axiom,
    ((![G : set_Ho137910533iple_a, P : a > state > $o, B2 : state > $o, C : com, Q : a > state > $o, D : com]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ (hoare_1795417776_and_a @ P @ B2) @ C @ Q) @ bot_bo1298296729iple_a)) => ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ (hoare_1795417776_and_a @ P @ (comp_o_o_state @ (~) @ B2)) @ D @ Q) @ bot_bo1298296729iple_a)) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ (cond @ B2 @ C @ D) @ Q) @ bot_bo1298296729iple_a))))))). % hoare_derivs.If
thf(fact_69_com_Oinject_I4_J, axiom,
    ((![X51 : state > $o, X52 : com, X53 : com, Y51 : state > $o, Y52 : com, Y53 : com]: (((cond @ X51 @ X52 @ X53) = (cond @ Y51 @ Y52 @ Y53)) = (((X51 = Y51)) & ((((X52 = Y52)) & ((X53 = Y53))))))))). % com.inject(4)
thf(fact_70_com_Odistinct_I45_J, axiom,
    ((![X51 : state > $o, X52 : com, X53 : com, X61 : state > $o, X62 : com]: (~ (((cond @ X51 @ X52 @ X53) = (while @ X61 @ X62))))))). % com.distinct(45)
thf(fact_71_com_Odistinct_I37_J, axiom,
    ((![X41 : com, X42 : com, X51 : state > $o, X52 : com, X53 : com]: (~ (((semi @ X41 @ X42) = (cond @ X51 @ X52 @ X53))))))). % com.distinct(37)
thf(fact_72_com_Odistinct_I7_J, axiom,
    ((![X51 : state > $o, X52 : com, X53 : com]: (~ ((skip = (cond @ X51 @ X52 @ X53))))))). % com.distinct(7)
thf(fact_73_Ass, axiom,
    ((![G : set_Ho137910533iple_a, P : a > state > $o, X7 : vname, A : state > nat]: (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ (^[Z4 : a]: (^[S4 : state]: (P @ Z4 @ (update @ S4 @ X7 @ (A @ S4))))) @ (ass @ X7 @ A) @ P) @ bot_bo1298296729iple_a))))). % Ass
thf(fact_74_com_Oinject_I1_J, axiom,
    ((![X21 : vname, X222 : state > nat, Y21 : vname, Y22 : state > nat]: (((ass @ X21 @ X222) = (ass @ Y21 @ Y22)) = (((X21 = Y21)) & ((X222 = Y22))))))). % com.inject(1)
thf(fact_75_com_Odistinct_I21_J, axiom,
    ((![X21 : vname, X222 : state > nat, X61 : state > $o, X62 : com]: (~ (((ass @ X21 @ X222) = (while @ X61 @ X62))))))). % com.distinct(21)
thf(fact_76_com_Odistinct_I19_J, axiom,
    ((![X21 : vname, X222 : state > nat, X51 : state > $o, X52 : com, X53 : com]: (~ (((ass @ X21 @ X222) = (cond @ X51 @ X52 @ X53))))))). % com.distinct(19)
thf(fact_77_com_Odistinct_I17_J, axiom,
    ((![X21 : vname, X222 : state > nat, X41 : com, X42 : com]: (~ (((ass @ X21 @ X222) = (semi @ X41 @ X42))))))). % com.distinct(17)
thf(fact_78_com_Odistinct_I1_J, axiom,
    ((![X21 : vname, X222 : state > nat]: (~ ((skip = (ass @ X21 @ X222))))))). % com.distinct(1)
thf(fact_79_pairwise__singleton, axiom,
    ((![P : hoare_1678595023iple_a > hoare_1678595023iple_a > $o, A2 : hoare_1678595023iple_a]: (pairwi531237284iple_a @ P @ (insert1477804543iple_a @ A2 @ bot_bo1298296729iple_a))))). % pairwise_singleton
thf(fact_80_pairwise__insert, axiom,
    ((![R2 : hoare_1678595023iple_a > hoare_1678595023iple_a > $o, X4 : hoare_1678595023iple_a, S5 : set_Ho137910533iple_a]: ((pairwi531237284iple_a @ R2 @ (insert1477804543iple_a @ X4 @ S5)) = (((![Y6 : hoare_1678595023iple_a]: (((((member1332298086iple_a @ Y6 @ S5)) & ((~ ((Y6 = X4)))))) => ((((R2 @ X4 @ Y6)) & ((R2 @ Y6 @ X4))))))) & ((pairwi531237284iple_a @ R2 @ S5))))))). % pairwise_insert
thf(fact_81_pairwise__empty, axiom,
    ((![P : hoare_1678595023iple_a > hoare_1678595023iple_a > $o]: (pairwi531237284iple_a @ P @ bot_bo1298296729iple_a)))). % pairwise_empty
thf(fact_82_pairwise__alt, axiom,
    ((pairwi531237284iple_a = (^[R3 : hoare_1678595023iple_a > hoare_1678595023iple_a > $o]: (^[S6 : set_Ho137910533iple_a]: (![X : hoare_1678595023iple_a]: (((member1332298086iple_a @ X @ S6)) => ((![Y6 : hoare_1678595023iple_a]: (((member1332298086iple_a @ Y6 @ (minus_1852999390iple_a @ S6 @ (insert1477804543iple_a @ X @ bot_bo1298296729iple_a)))) => ((R3 @ X @ Y6)))))))))))). % pairwise_alt
thf(fact_83_Nitpick_OEx1__unfold, axiom,
    ((ex1_Ho899980597iple_a = (^[P4 : hoare_1678595023iple_a > $o]: (?[X : hoare_1678595023iple_a]: ((collec1600235172iple_a @ P4) = (insert1477804543iple_a @ X @ bot_bo1298296729iple_a))))))). % Nitpick.Ex1_unfold
thf(fact_84_Diff__empty, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((minus_1852999390iple_a @ A2 @ bot_bo1298296729iple_a) = A2)))). % Diff_empty
thf(fact_85_empty__Diff, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((minus_1852999390iple_a @ bot_bo1298296729iple_a @ A2) = bot_bo1298296729iple_a)))). % empty_Diff
thf(fact_86_Diff__cancel, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((minus_1852999390iple_a @ A2 @ A2) = bot_bo1298296729iple_a)))). % Diff_cancel
thf(fact_87_Diff__insert0, axiom,
    ((![X4 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ X4 @ A2))) => ((minus_1852999390iple_a @ A2 @ (insert1477804543iple_a @ X4 @ B)) = (minus_1852999390iple_a @ A2 @ B)))))). % Diff_insert0
thf(fact_88_insert__Diff1, axiom,
    ((![X4 : hoare_1678595023iple_a, B : set_Ho137910533iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ X4 @ B) => ((minus_1852999390iple_a @ (insert1477804543iple_a @ X4 @ A2) @ B) = (minus_1852999390iple_a @ A2 @ B)))))). % insert_Diff1
thf(fact_89_insert__Diff__single, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((insert1477804543iple_a @ A @ (minus_1852999390iple_a @ A2 @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a))) = (insert1477804543iple_a @ A @ A2))))). % insert_Diff_single
thf(fact_90_insert__Diff__if, axiom,
    ((![X4 : hoare_1678595023iple_a, B : set_Ho137910533iple_a, A2 : set_Ho137910533iple_a]: (((member1332298086iple_a @ X4 @ B) => ((minus_1852999390iple_a @ (insert1477804543iple_a @ X4 @ A2) @ B) = (minus_1852999390iple_a @ A2 @ B))) & ((~ ((member1332298086iple_a @ X4 @ B))) => ((minus_1852999390iple_a @ (insert1477804543iple_a @ X4 @ A2) @ B) = (insert1477804543iple_a @ X4 @ (minus_1852999390iple_a @ A2 @ B)))))))). % insert_Diff_if
thf(fact_91_Diff__insert, axiom,
    ((![A2 : set_Ho137910533iple_a, A : hoare_1678595023iple_a, B : set_Ho137910533iple_a]: ((minus_1852999390iple_a @ A2 @ (insert1477804543iple_a @ A @ B)) = (minus_1852999390iple_a @ (minus_1852999390iple_a @ A2 @ B) @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a)))))). % Diff_insert

% Conjectures (1)
thf(conj_0, conjecture,
    ((hoare_129598474rivs_a @ g @ (insert1477804543iple_a @ (hoare_719046530iple_a @ (^[Z4 : a]: (^[S4 : state]: $false)) @ c @ q) @ bot_bo1298296729iple_a)))).
