% 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_189__3251312_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:12:36.333

% Could-be-implicit typings (8)
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__Oloc, type,
    loc : $tType).
thf(ty_n_t__Com__Ocom, type,
    com : $tType).
thf(ty_n_tf__a, type,
    a : $tType).

% Explicit typings (25)
thf(sy_c_Com_Ocom_OAss, type,
    ass : vname > (state > nat) > com).
thf(sy_c_Com_Ocom_OLocal, type,
    local : loc > (state > nat) > 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_Ovname_OLoc, type,
    loc2 : loc > vname).
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_Otriple_Otriple_001tf__a, type,
    hoare_719046530iple_a : (a > state > $o) > com > (a > state > $o) > hoare_1678595023iple_a).
thf(sy_c_Natural_Ogetlocs, type,
    getlocs : state > loc > nat).
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_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_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_P, type,
    p : a > state > $o).
thf(sy_v_Q, type,
    q : a > state > $o).
thf(sy_v_c, type,
    c : com).

% Relevant facts (81)
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_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_7_empty, axiom,
    ((![G : set_Ho137910533iple_a]: (hoare_129598474rivs_a @ G @ bot_bo1298296729iple_a)))). % empty
thf(fact_8_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_9_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_10_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_11_singleton__conv2, axiom,
    ((![A : hoare_1678595023iple_a]: ((collec1600235172iple_a @ ((^[Y4 : hoare_1678595023iple_a]: (^[Z3 : hoare_1678595023iple_a]: (Y4 = Z3))) @ A)) = (insert1477804543iple_a @ A @ bot_bo1298296729iple_a))))). % singleton_conv2
thf(fact_12_singletonI, axiom,
    ((![A : hoare_1678595023iple_a]: (member1332298086iple_a @ A @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a))))). % singletonI
thf(fact_13_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_14_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_15_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_16_empty__iff, axiom,
    ((![C : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ C @ bot_bo1298296729iple_a)))))). % empty_iff
thf(fact_17_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_18_Collect__empty__eq, axiom,
    ((![P2 : hoare_1678595023iple_a > $o]: (((collec1600235172iple_a @ P2) = bot_bo1298296729iple_a) = (![X : hoare_1678595023iple_a]: (~ ((P2 @ X)))))))). % Collect_empty_eq
thf(fact_19_empty__Collect__eq, axiom,
    ((![P2 : hoare_1678595023iple_a > $o]: ((bot_bo1298296729iple_a = (collec1600235172iple_a @ P2)) = (![X : hoare_1678595023iple_a]: (~ ((P2 @ X)))))))). % empty_Collect_eq
thf(fact_20_Collect__conv__if, axiom,
    ((![P2 : hoare_1678595023iple_a > $o, A : hoare_1678595023iple_a]: (((P2 @ A) => ((collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: (((X = A)) & ((P2 @ X))))) = (insert1477804543iple_a @ A @ bot_bo1298296729iple_a))) & ((~ ((P2 @ A))) => ((collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: (((X = A)) & ((P2 @ X))))) = bot_bo1298296729iple_a)))))). % Collect_conv_if
thf(fact_21_bot__set__def, axiom,
    ((bot_bo1298296729iple_a = (collec1600235172iple_a @ bot_bo431311916le_a_o)))). % bot_set_def
thf(fact_22_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_23_equals0I, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((![Y5 : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ Y5 @ A2)))) => (A2 = bot_bo1298296729iple_a))))). % equals0I
thf(fact_24_equals0D, axiom,
    ((![A2 : set_Ho137910533iple_a, A : hoare_1678595023iple_a]: ((A2 = bot_bo1298296729iple_a) => (~ ((member1332298086iple_a @ A @ A2))))))). % equals0D
thf(fact_25_emptyE, axiom,
    ((![A : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ A @ bot_bo1298296729iple_a)))))). % emptyE
thf(fact_26_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_27_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_28_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)))) => ((?[C2 : set_Ho137910533iple_a]: (((A2 = (insert1477804543iple_a @ B2 @ C2))) & ((((~ ((member1332298086iple_a @ B2 @ C2)))) & ((((B = (insert1477804543iple_a @ A @ C2))) & ((~ ((member1332298086iple_a @ A @ C2)))))))))))))))))))). % insert_eq_iff
thf(fact_29_insert__absorb, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ A @ A2) => ((insert1477804543iple_a @ A @ A2) = A2))))). % insert_absorb
thf(fact_30_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_31_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_32_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_33_insertI1, axiom,
    ((![A : hoare_1678595023iple_a, B : set_Ho137910533iple_a]: (member1332298086iple_a @ A @ (insert1477804543iple_a @ A @ B))))). % insertI1
thf(fact_34_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_35_empty__def, axiom,
    ((bot_bo1298296729iple_a = (collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: $false))))). % empty_def
thf(fact_36_insert__Collect, axiom,
    ((![A : hoare_1678595023iple_a, P2 : hoare_1678595023iple_a > $o]: ((insert1477804543iple_a @ A @ (collec1600235172iple_a @ P2)) = (collec1600235172iple_a @ (^[U : hoare_1678595023iple_a]: (((~ ((U = A)))) => ((P2 @ U))))))))). % insert_Collect
thf(fact_37_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_38_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_39_insert__not__empty, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: (~ (((insert1477804543iple_a @ A @ A2) = bot_bo1298296729iple_a)))))). % insert_not_empty
thf(fact_40_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_41_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_42_singletonD, axiom,
    ((![B2 : hoare_1678595023iple_a, A : hoare_1678595023iple_a]: ((member1332298086iple_a @ B2 @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a)) => (B2 = A))))). % singletonD
thf(fact_43_Collect__conv__if2, axiom,
    ((![P2 : hoare_1678595023iple_a > $o, A : hoare_1678595023iple_a]: (((P2 @ A) => ((collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: (((A = X)) & ((P2 @ X))))) = (insert1477804543iple_a @ A @ bot_bo1298296729iple_a))) & ((~ ((P2 @ A))) => ((collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: (((A = X)) & ((P2 @ X))))) = bot_bo1298296729iple_a)))))). % Collect_conv_if2
thf(fact_44_the__elem__eq, axiom,
    ((![X4 : hoare_1678595023iple_a]: ((the_el434698138iple_a @ (insert1477804543iple_a @ X4 @ bot_bo1298296729iple_a)) = X4)))). % the_elem_eq
thf(fact_45_is__singletonI, axiom,
    ((![X4 : hoare_1678595023iple_a]: (is_sin1784037339iple_a @ (insert1477804543iple_a @ X4 @ bot_bo1298296729iple_a))))). % is_singletonI
thf(fact_46_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_47_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_48_Set_Ois__empty__def, axiom,
    ((is_emp901906557iple_a = (^[A4 : set_Ho137910533iple_a]: (A4 = bot_bo1298296729iple_a))))). % Set.is_empty_def
thf(fact_49_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_50_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_51_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_52_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_53_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_54_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_55_com_Odistinct_I5_J, axiom,
    ((![X41 : com, X42 : com]: (~ ((skip = (semi @ X41 @ X42))))))). % com.distinct(5)
thf(fact_56_bot__empty__eq, axiom,
    ((bot_bo431311916le_a_o = (^[X : hoare_1678595023iple_a]: (member1332298086iple_a @ X @ bot_bo1298296729iple_a))))). % bot_empty_eq
thf(fact_57_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_58_Ass, axiom,
    ((![G : set_Ho137910533iple_a, P2 : a > state > $o, X7 : vname, A : state > nat]: (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ (^[Z4 : a]: (^[S3 : state]: (P2 @ Z4 @ (update @ S3 @ X7 @ (A @ S3))))) @ (ass @ X7 @ A) @ P2) @ bot_bo1298296729iple_a))))). % Ass
thf(fact_59_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_60_com_Odistinct_I1_J, axiom,
    ((![X21 : vname, X222 : state > nat]: (~ ((skip = (ass @ X21 @ X222))))))). % com.distinct(1)
thf(fact_61_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_62_pairwise__singleton, axiom,
    ((![P2 : hoare_1678595023iple_a > hoare_1678595023iple_a > $o, A2 : hoare_1678595023iple_a]: (pairwi531237284iple_a @ P2 @ (insert1477804543iple_a @ A2 @ bot_bo1298296729iple_a))))). % pairwise_singleton
thf(fact_63_hoare__derivs_OLocal, axiom,
    ((![G : set_Ho137910533iple_a, P2 : a > state > $o, C : com, Q : a > state > $o, X7 : loc, S4 : state, A : state > nat]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P2 @ C @ (^[Z4 : a]: (^[S3 : state]: (Q @ Z4 @ (update @ S3 @ (loc2 @ X7) @ (getlocs @ S4 @ X7)))))) @ bot_bo1298296729iple_a)) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ (^[Z4 : a]: (^[S3 : state]: (((S4 = S3)) & ((P2 @ Z4 @ (update @ S3 @ (loc2 @ X7) @ (A @ S3))))))) @ (local @ X7 @ A @ C) @ Q) @ bot_bo1298296729iple_a)))))). % hoare_derivs.Local
thf(fact_64_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_65_singleton__insert__inj__eq_H, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B2 : hoare_1678595023iple_a]: (((insert1477804543iple_a @ A @ A2) = (insert1477804543iple_a @ B2 @ bot_bo1298296729iple_a)) = (((A = B2)) & ((ord_le1221261669iple_a @ A2 @ (insert1477804543iple_a @ B2 @ bot_bo1298296729iple_a)))))))). % singleton_insert_inj_eq'
thf(fact_66_com_Oinject_I2_J, axiom,
    ((![X31 : loc, X322 : state > nat, X33 : com, Y31 : loc, Y32 : state > nat, Y33 : com]: (((local @ X31 @ X322 @ X33) = (local @ Y31 @ Y32 @ Y33)) = (((X31 = Y31)) & ((((X322 = Y32)) & ((X33 = Y33))))))))). % com.inject(2)
thf(fact_67_vname_Oinject_I2_J, axiom,
    ((![X2 : loc, Y2 : loc]: (((loc2 @ X2) = (loc2 @ Y2)) = (X2 = Y2))))). % vname.inject(2)
thf(fact_68_empty__subsetI, axiom,
    ((![A2 : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ bot_bo1298296729iple_a @ A2)))). % empty_subsetI
thf(fact_69_subset__empty, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ bot_bo1298296729iple_a) = (A2 = bot_bo1298296729iple_a))))). % subset_empty
thf(fact_70_insert__subset, axiom,
    ((![X4 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ (insert1477804543iple_a @ X4 @ A2) @ B) = (((member1332298086iple_a @ X4 @ B)) & ((ord_le1221261669iple_a @ A2 @ B))))))). % insert_subset
thf(fact_71_singleton__insert__inj__eq, axiom,
    ((![B2 : hoare_1678595023iple_a, A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: (((insert1477804543iple_a @ B2 @ bot_bo1298296729iple_a) = (insert1477804543iple_a @ A @ A2)) = (((A = B2)) & ((ord_le1221261669iple_a @ A2 @ (insert1477804543iple_a @ B2 @ bot_bo1298296729iple_a)))))))). % singleton_insert_inj_eq
thf(fact_72_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_73_asm, axiom,
    ((![Ts : set_Ho137910533iple_a, G : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ Ts @ G) => (hoare_129598474rivs_a @ G @ Ts))))). % asm
thf(fact_74_bot_Oextremum__uniqueI, axiom,
    ((![A : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ bot_bo1298296729iple_a) => (A = bot_bo1298296729iple_a))))). % bot.extremum_uniqueI
thf(fact_75_bot_Oextremum__unique, axiom,
    ((![A : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ bot_bo1298296729iple_a) = (A = bot_bo1298296729iple_a))))). % bot.extremum_unique
thf(fact_76_bot_Oextremum, axiom,
    ((![A : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ bot_bo1298296729iple_a @ A)))). % bot.extremum
thf(fact_77_subset__insertI2, axiom,
    ((![A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a, B2 : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A2 @ B) => (ord_le1221261669iple_a @ A2 @ (insert1477804543iple_a @ B2 @ B)))))). % subset_insertI2
thf(fact_78_subset__insertI, axiom,
    ((![B : set_Ho137910533iple_a, A : hoare_1678595023iple_a]: (ord_le1221261669iple_a @ B @ (insert1477804543iple_a @ A @ B))))). % subset_insertI
thf(fact_79_subset__insert, axiom,
    ((![X4 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ X4 @ A2))) => ((ord_le1221261669iple_a @ A2 @ (insert1477804543iple_a @ X4 @ B)) = (ord_le1221261669iple_a @ A2 @ B)))))). % subset_insert
thf(fact_80_insert__mono, axiom,
    ((![C3 : set_Ho137910533iple_a, D2 : set_Ho137910533iple_a, A : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ C3 @ D2) => (ord_le1221261669iple_a @ (insert1477804543iple_a @ A @ C3) @ (insert1477804543iple_a @ A @ D2)))))). % insert_mono

% Conjectures (2)
thf(conj_0, hypothesis,
    ((![Z5 : a, S5 : state]: ((p @ Z5 @ S5) => (hoare_129598474rivs_a @ g @ (insert1477804543iple_a @ (hoare_719046530iple_a @ (^[Za : a]: (^[S6 : state]: (S6 = S5))) @ c @ (^[Z6 : a]: (q @ Z5))) @ bot_bo1298296729iple_a)))))).
thf(conj_1, conjecture,
    ((![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)) & (![S2 : state]: ((?[Z2 : a]: ((P3 @ Z2 @ S) & (~ ((Q3 @ Z2 @ S2))))) | (q @ Z @ S2))))))))).
