% 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_200__3251462_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:12:42.091

% Could-be-implicit typings (11)
thf(ty_n_t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J, type,
    set_Ho840737317_state : $tType).
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_It__Com__Ostate_J, type,
    hoare_958474565_state : $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__Oglb, type,
    glb : $tType).
thf(ty_n_t__Com__Ocom, type,
    com : $tType).
thf(ty_n_tf__a, type,
    a : $tType).

% Explicit typings (47)
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_OGlb, type,
    glb2 : glb > vname).
thf(sy_c_Com_Ovname_OLoc, type,
    loc2 : loc > vname).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J, type,
    minus_1628593292_state : set_Ho840737317_state > set_Ho840737317_state > set_Ho840737317_state).
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_It__Com__Ostate_J, type,
    ex1_Ho557811039_state : (hoare_958474565_state > $o) > $o).
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_It__Com__Ostate_J, type,
    the_Ho1908033100_state : (hoare_958474565_state > $o) > hoare_958474565_state).
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_OMGT, type,
    hoare_Mirabelle_MGT : com > hoare_958474565_state).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Ohoare__derivs_001t__Com__Ostate, type,
    hoare_604442164_state : set_Ho840737317_state > set_Ho840737317_state > $o).
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_001t__Com__Ostate, type,
    hoare_1659279548_state : (state > state > $o) > com > (state > state > $o) > hoare_958474565_state).
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_001t__Com__Ostate, type,
    hoare_364318704_state : nat > hoare_958474565_state > $o).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Otriple__valid_001tf__a, type,
    hoare_1926814542alid_a : nat > hoare_1678595023iple_a > $o).
thf(sy_c_Natural_Oevalc, type,
    evalc : com > state > state > $o).
thf(sy_c_Natural_Oevaln, type,
    evaln : com > state > nat > state > $o).
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_It__Com__Ostate_J_M_Eo_J, type,
    bot_bo1428770700tate_o : hoare_958474565_state > $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_It__Com__Ostate_J_J, type,
    bot_bo105666705_state : set_Ho840737317_state).
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_It__Com__Ostate_J, type,
    collec305460656_state : (hoare_958474565_state > $o) > set_Ho840737317_state).
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_It__Com__Ostate_J, type,
    insert776267541_state : hoare_958474565_state > set_Ho840737317_state > set_Ho840737317_state).
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_It__Com__Ostate_J, type,
    is_emp266830871_state : set_Ho840737317_state > $o).
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_It__Com__Ostate_J, type,
    is_sin536631353_state : set_Ho840737317_state > $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_It__Com__Ostate_J, type,
    pairwi17519280_state : (hoare_958474565_state > hoare_958474565_state > $o) > set_Ho840737317_state > $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_Oremove_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    remove586333994_state : hoare_958474565_state > set_Ho840737317_state > set_Ho840737317_state).
thf(sy_c_Set_Oremove_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    remove1512401578iple_a : hoare_1678595023iple_a > set_Ho137910533iple_a > set_Ho137910533iple_a).
thf(sy_c_Set_Othe__elem_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    the_el1300254266_state : set_Ho840737317_state > hoare_958474565_state).
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_It__Com__Ostate_J, type,
    member109514606_state : hoare_958474565_state > set_Ho840737317_state > $o).
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_b, type,
    b : state > $o).
thf(sy_v_c, type,
    c : com).

% Relevant facts (187)
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_triple_Oinject, axiom,
    ((![X1 : state > state > $o, X2 : com, X3 : state > state > $o, Y1 : state > state > $o, Y2 : com, Y3 : state > state > $o]: (((hoare_1659279548_state @ X1 @ X2 @ X3) = (hoare_1659279548_state @ Y1 @ Y2 @ Y3)) = (((X1 = Y1)) & ((((X2 = Y2)) & ((X3 = Y3))))))))). % triple.inject
thf(fact_2_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_3_escape, axiom,
    ((![P : state > state > $o, G : set_Ho840737317_state, C : com, Q : state > state > $o]: ((![Z : state, S : state]: ((P @ Z @ S) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ (^[Za : state]: (^[S2 : state]: (S2 = S))) @ C @ (^[Z2 : state]: (Q @ Z))) @ bot_bo105666705_state)))) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ C @ Q) @ bot_bo105666705_state)))))). % escape
thf(fact_4_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_5_conseq1, axiom,
    ((![G : set_Ho840737317_state, P2 : state > state > $o, C : com, Q : state > state > $o, P : state > state > $o]: ((hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P2 @ C @ Q) @ bot_bo105666705_state)) => ((![Z : state, S : state]: ((P @ Z @ S) => (P2 @ Z @ S))) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ C @ Q) @ bot_bo105666705_state))))))). % conseq1
thf(fact_6_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_7_conseq2, axiom,
    ((![G : set_Ho840737317_state, P : state > state > $o, C : com, Q2 : state > state > $o, Q : state > state > $o]: ((hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ C @ Q2) @ bot_bo105666705_state)) => ((![Z : state, S : state]: ((Q2 @ Z @ S) => (Q @ Z @ S))) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ C @ Q) @ bot_bo105666705_state))))))). % conseq2
thf(fact_8_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_9_conseq12, axiom,
    ((![G : set_Ho840737317_state, P2 : state > state > $o, C : com, Q2 : state > state > $o, P : state > state > $o, Q : state > state > $o]: ((hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P2 @ C @ Q2) @ bot_bo105666705_state)) => ((![Z : state, S : state]: ((P @ Z @ S) => (![S3 : state]: ((![Z3 : state]: ((P2 @ Z3 @ S) => (Q2 @ Z3 @ S3))) => (Q @ Z @ S3))))) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ C @ Q) @ bot_bo105666705_state))))))). % conseq12
thf(fact_10__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_11__Cconstant_C, axiom,
    ((![C2 : $o, G : set_Ho840737317_state, P : state > state > $o, C : com, Q : state > state > $o]: ((C2 => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ C @ Q) @ bot_bo105666705_state))) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ (^[Z4 : state]: (^[S4 : state]: (((P @ Z4 @ S4)) & (C2)))) @ C @ Q) @ bot_bo105666705_state)))))). % "constant"
thf(fact_12_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_13_triple_Oinduct, axiom,
    ((![P : hoare_958474565_state > $o, Triple : hoare_958474565_state]: ((![X1a : state > state > $o, X2a : com, X3a : state > state > $o]: (P @ (hoare_1659279548_state @ X1a @ X2a @ X3a))) => (P @ Triple))))). % triple.induct
thf(fact_14_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_15_triple_Oexhaust, axiom,
    ((![Y : hoare_958474565_state]: (~ ((![X12 : state > state > $o, X22 : com, X32 : state > state > $o]: (~ ((Y = (hoare_1659279548_state @ X12 @ X22 @ X32)))))))))). % triple.exhaust
thf(fact_16_cut, axiom,
    ((![G2 : set_Ho840737317_state, Ts : set_Ho840737317_state, G : set_Ho840737317_state]: ((hoare_604442164_state @ G2 @ Ts) => ((hoare_604442164_state @ G @ G2) => (hoare_604442164_state @ G @ Ts)))))). % cut
thf(fact_17_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_18_empty, axiom,
    ((![G : set_Ho840737317_state]: (hoare_604442164_state @ G @ bot_bo105666705_state)))). % empty
thf(fact_19_empty, axiom,
    ((![G : set_Ho137910533iple_a]: (hoare_129598474rivs_a @ G @ bot_bo1298296729iple_a)))). % empty
thf(fact_20_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_21_conseq, axiom,
    ((![P : state > state > $o, G : set_Ho840737317_state, C : com, Q : state > state > $o]: ((![Z : state, S : state]: ((P @ Z @ S) => (?[P3 : state > state > $o, Q3 : state > state > $o]: ((hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P3 @ C @ Q3) @ bot_bo105666705_state)) & (![S3 : state]: ((![Z3 : state]: ((P3 @ Z3 @ S) => (Q3 @ Z3 @ S3))) => (Q @ Z @ S3))))))) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ C @ Q) @ bot_bo105666705_state)))))). % conseq
thf(fact_22_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_23_hoare__derivs_Oinsert, axiom,
    ((![G : set_Ho840737317_state, T : hoare_958474565_state, Ts : set_Ho840737317_state]: ((hoare_604442164_state @ G @ (insert776267541_state @ T @ bot_bo105666705_state)) => ((hoare_604442164_state @ G @ Ts) => (hoare_604442164_state @ G @ (insert776267541_state @ T @ Ts))))))). % hoare_derivs.insert
thf(fact_24_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_25_singleton__conv, axiom,
    ((![A : hoare_958474565_state]: ((collec305460656_state @ (^[X : hoare_958474565_state]: (X = A))) = (insert776267541_state @ A @ bot_bo105666705_state))))). % singleton_conv
thf(fact_26_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_27_singleton__conv2, axiom,
    ((![A : hoare_958474565_state]: ((collec305460656_state @ ((^[Y4 : hoare_958474565_state]: (^[Z5 : hoare_958474565_state]: (Y4 = Z5))) @ A)) = (insert776267541_state @ A @ bot_bo105666705_state))))). % singleton_conv2
thf(fact_28_singletonI, axiom,
    ((![A : hoare_1678595023iple_a]: (member1332298086iple_a @ A @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a))))). % singletonI
thf(fact_29_singletonI, axiom,
    ((![A : hoare_958474565_state]: (member109514606_state @ A @ (insert776267541_state @ A @ bot_bo105666705_state))))). % singletonI
thf(fact_30_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_31_insertCI, axiom,
    ((![A : hoare_958474565_state, B : set_Ho840737317_state, B2 : hoare_958474565_state]: (((~ ((member109514606_state @ A @ B))) => (A = B2)) => (member109514606_state @ A @ (insert776267541_state @ B2 @ B)))))). % insertCI
thf(fact_32_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_33_insert__iff, axiom,
    ((![A : hoare_958474565_state, B2 : hoare_958474565_state, A2 : set_Ho840737317_state]: ((member109514606_state @ A @ (insert776267541_state @ B2 @ A2)) = (((A = B2)) | ((member109514606_state @ A @ A2))))))). % insert_iff
thf(fact_34_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_35_insert__absorb2, axiom,
    ((![X4 : hoare_958474565_state, A2 : set_Ho840737317_state]: ((insert776267541_state @ X4 @ (insert776267541_state @ X4 @ A2)) = (insert776267541_state @ X4 @ A2))))). % insert_absorb2
thf(fact_36_empty__iff, axiom,
    ((![C : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ C @ bot_bo1298296729iple_a)))))). % empty_iff
thf(fact_37_empty__iff, axiom,
    ((![C : hoare_958474565_state]: (~ ((member109514606_state @ C @ bot_bo105666705_state)))))). % empty_iff
thf(fact_38_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_39_all__not__in__conv, axiom,
    ((![A2 : set_Ho840737317_state]: ((![X : hoare_958474565_state]: (~ ((member109514606_state @ X @ A2)))) = (A2 = bot_bo105666705_state))))). % all_not_in_conv
thf(fact_40_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_41_Collect__empty__eq, axiom,
    ((![P : hoare_958474565_state > $o]: (((collec305460656_state @ P) = bot_bo105666705_state) = (![X : hoare_958474565_state]: (~ ((P @ X)))))))). % Collect_empty_eq
thf(fact_42_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_43_empty__Collect__eq, axiom,
    ((![P : hoare_958474565_state > $o]: ((bot_bo105666705_state = (collec305460656_state @ P)) = (![X : hoare_958474565_state]: (~ ((P @ X)))))))). % empty_Collect_eq
thf(fact_44_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_45_Collect__conv__if, axiom,
    ((![P : hoare_958474565_state > $o, A : hoare_958474565_state]: (((P @ A) => ((collec305460656_state @ (^[X : hoare_958474565_state]: (((X = A)) & ((P @ X))))) = (insert776267541_state @ A @ bot_bo105666705_state))) & ((~ ((P @ A))) => ((collec305460656_state @ (^[X : hoare_958474565_state]: (((X = A)) & ((P @ X))))) = bot_bo105666705_state)))))). % Collect_conv_if
thf(fact_46_bot__set__def, axiom,
    ((bot_bo1298296729iple_a = (collec1600235172iple_a @ bot_bo431311916le_a_o)))). % bot_set_def
thf(fact_47_bot__set__def, axiom,
    ((bot_bo105666705_state = (collec305460656_state @ bot_bo1428770700tate_o)))). % bot_set_def
thf(fact_48_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_49_ex__in__conv, axiom,
    ((![A2 : set_Ho840737317_state]: ((?[X : hoare_958474565_state]: (member109514606_state @ X @ A2)) = (~ ((A2 = bot_bo105666705_state))))))). % ex_in_conv
thf(fact_50_equals0I, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((![Y5 : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ Y5 @ A2)))) => (A2 = bot_bo1298296729iple_a))))). % equals0I
thf(fact_51_equals0I, axiom,
    ((![A2 : set_Ho840737317_state]: ((![Y5 : hoare_958474565_state]: (~ ((member109514606_state @ Y5 @ A2)))) => (A2 = bot_bo105666705_state))))). % equals0I
thf(fact_52_equals0D, axiom,
    ((![A2 : set_Ho137910533iple_a, A : hoare_1678595023iple_a]: ((A2 = bot_bo1298296729iple_a) => (~ ((member1332298086iple_a @ A @ A2))))))). % equals0D
thf(fact_53_equals0D, axiom,
    ((![A2 : set_Ho840737317_state, A : hoare_958474565_state]: ((A2 = bot_bo105666705_state) => (~ ((member109514606_state @ A @ A2))))))). % equals0D
thf(fact_54_emptyE, axiom,
    ((![A : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ A @ bot_bo1298296729iple_a)))))). % emptyE
thf(fact_55_emptyE, axiom,
    ((![A : hoare_958474565_state]: (~ ((member109514606_state @ A @ bot_bo105666705_state)))))). % emptyE
thf(fact_56_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_57_mk__disjoint__insert, axiom,
    ((![A : hoare_958474565_state, A2 : set_Ho840737317_state]: ((member109514606_state @ A @ A2) => (?[B3 : set_Ho840737317_state]: ((A2 = (insert776267541_state @ A @ B3)) & (~ ((member109514606_state @ A @ B3))))))))). % mk_disjoint_insert
thf(fact_58_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_59_insert__commute, axiom,
    ((![X4 : hoare_958474565_state, Y : hoare_958474565_state, A2 : set_Ho840737317_state]: ((insert776267541_state @ X4 @ (insert776267541_state @ Y @ A2)) = (insert776267541_state @ Y @ (insert776267541_state @ X4 @ A2)))))). % insert_commute
thf(fact_60_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_61_insert__eq__iff, axiom,
    ((![A : hoare_958474565_state, A2 : set_Ho840737317_state, B2 : hoare_958474565_state, B : set_Ho840737317_state]: ((~ ((member109514606_state @ A @ A2))) => ((~ ((member109514606_state @ B2 @ B))) => (((insert776267541_state @ A @ A2) = (insert776267541_state @ B2 @ B)) = (((((A = B2)) => ((A2 = B)))) & ((((~ ((A = B2)))) => ((?[C3 : set_Ho840737317_state]: (((A2 = (insert776267541_state @ B2 @ C3))) & ((((~ ((member109514606_state @ B2 @ C3)))) & ((((B = (insert776267541_state @ A @ C3))) & ((~ ((member109514606_state @ A @ C3)))))))))))))))))))). % insert_eq_iff
thf(fact_62_insert__absorb, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ A @ A2) => ((insert1477804543iple_a @ A @ A2) = A2))))). % insert_absorb
thf(fact_63_insert__absorb, axiom,
    ((![A : hoare_958474565_state, A2 : set_Ho840737317_state]: ((member109514606_state @ A @ A2) => ((insert776267541_state @ A @ A2) = A2))))). % insert_absorb
thf(fact_64_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_65_insert__ident, axiom,
    ((![X4 : hoare_958474565_state, A2 : set_Ho840737317_state, B : set_Ho840737317_state]: ((~ ((member109514606_state @ X4 @ A2))) => ((~ ((member109514606_state @ X4 @ B))) => (((insert776267541_state @ X4 @ A2) = (insert776267541_state @ X4 @ B)) = (A2 = B))))))). % insert_ident
thf(fact_66_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_67_Set_Oset__insert, axiom,
    ((![X4 : hoare_958474565_state, A2 : set_Ho840737317_state]: ((member109514606_state @ X4 @ A2) => (~ ((![B3 : set_Ho840737317_state]: ((A2 = (insert776267541_state @ X4 @ B3)) => (member109514606_state @ X4 @ B3))))))))). % Set.set_insert
thf(fact_68_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_69_insertI2, axiom,
    ((![A : hoare_958474565_state, B : set_Ho840737317_state, B2 : hoare_958474565_state]: ((member109514606_state @ A @ B) => (member109514606_state @ A @ (insert776267541_state @ B2 @ B)))))). % insertI2
thf(fact_70_insertI1, axiom,
    ((![A : hoare_1678595023iple_a, B : set_Ho137910533iple_a]: (member1332298086iple_a @ A @ (insert1477804543iple_a @ A @ B))))). % insertI1
thf(fact_71_insertI1, axiom,
    ((![A : hoare_958474565_state, B : set_Ho840737317_state]: (member109514606_state @ A @ (insert776267541_state @ A @ B))))). % insertI1
thf(fact_72_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_73_insertE, axiom,
    ((![A : hoare_958474565_state, B2 : hoare_958474565_state, A2 : set_Ho840737317_state]: ((member109514606_state @ A @ (insert776267541_state @ B2 @ A2)) => ((~ ((A = B2))) => (member109514606_state @ A @ A2)))))). % insertE
thf(fact_74_empty__def, axiom,
    ((bot_bo1298296729iple_a = (collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: $false))))). % empty_def
thf(fact_75_empty__def, axiom,
    ((bot_bo105666705_state = (collec305460656_state @ (^[X : hoare_958474565_state]: $false))))). % empty_def
thf(fact_76_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_77_insert__Collect, axiom,
    ((![A : hoare_958474565_state, P : hoare_958474565_state > $o]: ((insert776267541_state @ A @ (collec305460656_state @ P)) = (collec305460656_state @ (^[U : hoare_958474565_state]: (((~ ((U = A)))) => ((P @ U))))))))). % insert_Collect
thf(fact_78_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_79_insert__compr, axiom,
    ((insert776267541_state = (^[A3 : hoare_958474565_state]: (^[B4 : set_Ho840737317_state]: (collec305460656_state @ (^[X : hoare_958474565_state]: (((X = A3)) | ((member109514606_state @ X @ B4)))))))))). % insert_compr
thf(fact_80_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_81_singleton__inject, axiom,
    ((![A : hoare_958474565_state, B2 : hoare_958474565_state]: (((insert776267541_state @ A @ bot_bo105666705_state) = (insert776267541_state @ B2 @ bot_bo105666705_state)) => (A = B2))))). % singleton_inject
thf(fact_82_insert__not__empty, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: (~ (((insert1477804543iple_a @ A @ A2) = bot_bo1298296729iple_a)))))). % insert_not_empty
thf(fact_83_insert__not__empty, axiom,
    ((![A : hoare_958474565_state, A2 : set_Ho840737317_state]: (~ (((insert776267541_state @ A @ A2) = bot_bo105666705_state)))))). % insert_not_empty
thf(fact_84_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_85_doubleton__eq__iff, axiom,
    ((![A : hoare_958474565_state, B2 : hoare_958474565_state, C : hoare_958474565_state, D : hoare_958474565_state]: (((insert776267541_state @ A @ (insert776267541_state @ B2 @ bot_bo105666705_state)) = (insert776267541_state @ C @ (insert776267541_state @ D @ bot_bo105666705_state))) = (((((A = C)) & ((B2 = D)))) | ((((A = D)) & ((B2 = C))))))))). % doubleton_eq_iff
thf(fact_86_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_87_singleton__iff, axiom,
    ((![B2 : hoare_958474565_state, A : hoare_958474565_state]: ((member109514606_state @ B2 @ (insert776267541_state @ A @ bot_bo105666705_state)) = (B2 = A))))). % singleton_iff
thf(fact_88_singletonD, axiom,
    ((![B2 : hoare_1678595023iple_a, A : hoare_1678595023iple_a]: ((member1332298086iple_a @ B2 @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a)) => (B2 = A))))). % singletonD
thf(fact_89_singletonD, axiom,
    ((![B2 : hoare_958474565_state, A : hoare_958474565_state]: ((member109514606_state @ B2 @ (insert776267541_state @ A @ bot_bo105666705_state)) => (B2 = A))))). % singletonD
thf(fact_90_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_91_Collect__conv__if2, axiom,
    ((![P : hoare_958474565_state > $o, A : hoare_958474565_state]: (((P @ A) => ((collec305460656_state @ (^[X : hoare_958474565_state]: (((A = X)) & ((P @ X))))) = (insert776267541_state @ A @ bot_bo105666705_state))) & ((~ ((P @ A))) => ((collec305460656_state @ (^[X : hoare_958474565_state]: (((A = X)) & ((P @ X))))) = bot_bo105666705_state)))))). % Collect_conv_if2
thf(fact_92_the__elem__eq, axiom,
    ((![X4 : hoare_1678595023iple_a]: ((the_el434698138iple_a @ (insert1477804543iple_a @ X4 @ bot_bo1298296729iple_a)) = X4)))). % the_elem_eq
thf(fact_93_the__elem__eq, axiom,
    ((![X4 : hoare_958474565_state]: ((the_el1300254266_state @ (insert776267541_state @ X4 @ bot_bo105666705_state)) = X4)))). % the_elem_eq
thf(fact_94_is__singletonI, axiom,
    ((![X4 : hoare_1678595023iple_a]: (is_sin1784037339iple_a @ (insert1477804543iple_a @ X4 @ bot_bo1298296729iple_a))))). % is_singletonI
thf(fact_95_is__singletonI, axiom,
    ((![X4 : hoare_958474565_state]: (is_sin536631353_state @ (insert776267541_state @ X4 @ bot_bo105666705_state))))). % is_singletonI
thf(fact_96_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_97_hoare__derivs_OSkip, axiom,
    ((![G : set_Ho840737317_state, P : state > state > $o]: (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ skip @ P) @ bot_bo105666705_state))))). % hoare_derivs.Skip
thf(fact_98_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_99_Comp, axiom,
    ((![G : set_Ho840737317_state, P : state > state > $o, C : com, Q : state > state > $o, D : com, R : state > state > $o]: ((hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ C @ Q) @ bot_bo105666705_state)) => ((hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ Q @ D @ R) @ bot_bo105666705_state)) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ (semi @ C @ D) @ R) @ bot_bo105666705_state))))))). % Comp
thf(fact_100_Set_Ois__empty__def, axiom,
    ((is_emp901906557iple_a = (^[A4 : set_Ho137910533iple_a]: (A4 = bot_bo1298296729iple_a))))). % Set.is_empty_def
thf(fact_101_Set_Ois__empty__def, axiom,
    ((is_emp266830871_state = (^[A4 : set_Ho840737317_state]: (A4 = bot_bo105666705_state))))). % Set.is_empty_def
thf(fact_102_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_103_is__singleton__def, axiom,
    ((is_sin536631353_state = (^[A4 : set_Ho840737317_state]: (?[X : hoare_958474565_state]: (A4 = (insert776267541_state @ X @ bot_bo105666705_state))))))). % is_singleton_def
thf(fact_104_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_105_is__singletonE, axiom,
    ((![A2 : set_Ho840737317_state]: ((is_sin536631353_state @ A2) => (~ ((![X5 : hoare_958474565_state]: (~ ((A2 = (insert776267541_state @ X5 @ bot_bo105666705_state))))))))))). % is_singletonE
thf(fact_106_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_107_is__singleton__the__elem, axiom,
    ((is_sin536631353_state = (^[A4 : set_Ho840737317_state]: (A4 = (insert776267541_state @ (the_el1300254266_state @ A4) @ bot_bo105666705_state)))))). % is_singleton_the_elem
thf(fact_108_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_109_is__singletonI_H, axiom,
    ((![A2 : set_Ho840737317_state]: ((~ ((A2 = bot_bo105666705_state))) => ((![X5 : hoare_958474565_state, Y5 : hoare_958474565_state]: ((member109514606_state @ X5 @ A2) => ((member109514606_state @ Y5 @ A2) => (X5 = Y5)))) => (is_sin536631353_state @ A2)))))). % is_singletonI'
thf(fact_110_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_111_com_Odistinct_I5_J, axiom,
    ((![X41 : com, X42 : com]: (~ ((skip = (semi @ X41 @ X42))))))). % com.distinct(5)
thf(fact_112_bot__empty__eq, axiom,
    ((bot_bo431311916le_a_o = (^[X : hoare_1678595023iple_a]: (member1332298086iple_a @ X @ bot_bo1298296729iple_a))))). % bot_empty_eq
thf(fact_113_bot__empty__eq, axiom,
    ((bot_bo1428770700tate_o = (^[X : hoare_958474565_state]: (member109514606_state @ X @ bot_bo105666705_state))))). % bot_empty_eq
thf(fact_114_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_115_Collect__empty__eq__bot, axiom,
    ((![P : hoare_958474565_state > $o]: (((collec305460656_state @ P) = bot_bo105666705_state) = (P = bot_bo1428770700tate_o))))). % Collect_empty_eq_bot
thf(fact_116_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_117_the__elem__def, axiom,
    ((the_el1300254266_state = (^[X6 : set_Ho840737317_state]: (the_Ho1908033100_state @ (^[X : hoare_958474565_state]: (X6 = (insert776267541_state @ X @ bot_bo105666705_state)))))))). % the_elem_def
thf(fact_118_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_119_Ass, axiom,
    ((![G : set_Ho840737317_state, P : state > state > $o, X7 : vname, A : state > nat]: (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ (^[Z4 : state]: (^[S4 : state]: (P @ Z4 @ (update @ S4 @ X7 @ (A @ S4))))) @ (ass @ X7 @ A) @ P) @ bot_bo105666705_state))))). % Ass
thf(fact_120_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_121_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_122_com_Odistinct_I1_J, axiom,
    ((![X21 : vname, X222 : state > nat]: (~ ((skip = (ass @ X21 @ X222))))))). % com.distinct(1)
thf(fact_123_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_124_pairwise__singleton, axiom,
    ((![P : hoare_958474565_state > hoare_958474565_state > $o, A2 : hoare_958474565_state]: (pairwi17519280_state @ P @ (insert776267541_state @ A2 @ bot_bo105666705_state))))). % pairwise_singleton
thf(fact_125_hoare__derivs_OLocal, axiom,
    ((![G : set_Ho137910533iple_a, P : a > state > $o, C : com, Q : a > state > $o, X7 : loc, S5 : state, A : state > nat]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ (^[Z4 : a]: (^[S4 : state]: (Q @ Z4 @ (update @ S4 @ (loc2 @ X7) @ (getlocs @ S5 @ X7)))))) @ bot_bo1298296729iple_a)) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ (^[Z4 : a]: (^[S4 : state]: (((S5 = S4)) & ((P @ Z4 @ (update @ S4 @ (loc2 @ X7) @ (A @ S4))))))) @ (local @ X7 @ A @ C) @ Q) @ bot_bo1298296729iple_a)))))). % hoare_derivs.Local
thf(fact_126_hoare__derivs_OLocal, axiom,
    ((![G : set_Ho840737317_state, P : state > state > $o, C : com, Q : state > state > $o, X7 : loc, S5 : state, A : state > nat]: ((hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ C @ (^[Z4 : state]: (^[S4 : state]: (Q @ Z4 @ (update @ S4 @ (loc2 @ X7) @ (getlocs @ S5 @ X7)))))) @ bot_bo105666705_state)) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ (^[Z4 : state]: (^[S4 : state]: (((S5 = S4)) & ((P @ Z4 @ (update @ S4 @ (loc2 @ X7) @ (A @ S4))))))) @ (local @ X7 @ A @ C) @ Q) @ bot_bo105666705_state)))))). % hoare_derivs.Local
thf(fact_127_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_128_Nitpick_OEx1__unfold, axiom,
    ((ex1_Ho557811039_state = (^[P4 : hoare_958474565_state > $o]: (?[X : hoare_958474565_state]: ((collec305460656_state @ P4) = (insert776267541_state @ X @ bot_bo105666705_state))))))). % Nitpick.Ex1_unfold
thf(fact_129_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_130_vname_Oinject_I2_J, axiom,
    ((![X2 : loc, Y2 : loc]: (((loc2 @ X2) = (loc2 @ Y2)) = (X2 = Y2))))). % vname.inject(2)
thf(fact_131_pairwise__empty, axiom,
    ((![P : hoare_1678595023iple_a > hoare_1678595023iple_a > $o]: (pairwi531237284iple_a @ P @ bot_bo1298296729iple_a)))). % pairwise_empty
thf(fact_132_pairwise__empty, axiom,
    ((![P : hoare_958474565_state > hoare_958474565_state > $o]: (pairwi17519280_state @ P @ bot_bo105666705_state)))). % pairwise_empty
thf(fact_133_pairwise__insert, axiom,
    ((![R2 : hoare_1678595023iple_a > hoare_1678595023iple_a > $o, X4 : hoare_1678595023iple_a, S6 : set_Ho137910533iple_a]: ((pairwi531237284iple_a @ R2 @ (insert1477804543iple_a @ X4 @ S6)) = (((![Y6 : hoare_1678595023iple_a]: (((((member1332298086iple_a @ Y6 @ S6)) & ((~ ((Y6 = X4)))))) => ((((R2 @ X4 @ Y6)) & ((R2 @ Y6 @ X4))))))) & ((pairwi531237284iple_a @ R2 @ S6))))))). % pairwise_insert
thf(fact_134_pairwise__insert, axiom,
    ((![R2 : hoare_958474565_state > hoare_958474565_state > $o, X4 : hoare_958474565_state, S6 : set_Ho840737317_state]: ((pairwi17519280_state @ R2 @ (insert776267541_state @ X4 @ S6)) = (((![Y6 : hoare_958474565_state]: (((((member109514606_state @ Y6 @ S6)) & ((~ ((Y6 = X4)))))) => ((((R2 @ X4 @ Y6)) & ((R2 @ Y6 @ X4))))))) & ((pairwi17519280_state @ R2 @ S6))))))). % pairwise_insert
thf(fact_135_com_Odistinct_I27_J, axiom,
    ((![X31 : loc, X322 : state > nat, X33 : com, X41 : com, X42 : com]: (~ (((local @ X31 @ X322 @ X33) = (semi @ X41 @ X42))))))). % com.distinct(27)
thf(fact_136_com_Odistinct_I15_J, axiom,
    ((![X21 : vname, X222 : state > nat, X31 : loc, X322 : state > nat, X33 : com]: (~ (((ass @ X21 @ X222) = (local @ X31 @ X322 @ X33))))))). % com.distinct(15)
thf(fact_137_com_Odistinct_I3_J, axiom,
    ((![X31 : loc, X322 : state > nat, X33 : com]: (~ ((skip = (local @ X31 @ X322 @ X33))))))). % com.distinct(3)
thf(fact_138_evalc__elim__cases_I3_J, axiom,
    ((![Y7 : loc, A : state > nat, C : com, S6 : state, T : state]: ((evalc @ (local @ Y7 @ A @ C) @ S6 @ T) => (~ ((![S1 : state]: ((T = (update @ S1 @ (loc2 @ Y7) @ (getlocs @ S6 @ Y7))) => (~ ((evalc @ C @ (update @ S6 @ (loc2 @ Y7) @ (A @ S6)) @ S1))))))))))). % evalc_elim_cases(3)
thf(fact_139_evalc_OLocal, axiom,
    ((![C : com, S0 : state, Y7 : loc, A : state > nat, S12 : state]: ((evalc @ C @ (update @ S0 @ (loc2 @ Y7) @ (A @ S0)) @ S12) => (evalc @ (local @ Y7 @ A @ C) @ S0 @ (update @ S12 @ (loc2 @ Y7) @ (getlocs @ S0 @ Y7))))))). % evalc.Local
thf(fact_140_evaln__elim__cases_I3_J, axiom,
    ((![Y7 : loc, A : state > nat, C : com, S6 : state, N : nat, T : state]: ((evaln @ (local @ Y7 @ A @ C) @ S6 @ N @ T) => (~ ((![S1 : state]: ((T = (update @ S1 @ (loc2 @ Y7) @ (getlocs @ S6 @ Y7))) => (~ ((evaln @ C @ (update @ S6 @ (loc2 @ Y7) @ (A @ S6)) @ N @ S1))))))))))). % evaln_elim_cases(3)
thf(fact_141_evaln_OSkip, axiom,
    ((![S6 : state, N : nat]: (evaln @ skip @ S6 @ N @ S6)))). % evaln.Skip
thf(fact_142_evaln_OSemi, axiom,
    ((![C0 : com, S0 : state, N : nat, S12 : state, C1 : com, S22 : state]: ((evaln @ C0 @ S0 @ N @ S12) => ((evaln @ C1 @ S12 @ N @ S22) => (evaln @ (semi @ C0 @ C1) @ S0 @ N @ S22)))))). % evaln.Semi
thf(fact_143_evalc_OSkip, axiom,
    ((![S6 : state]: (evalc @ skip @ S6 @ S6)))). % evalc.Skip
thf(fact_144_evalc_OSemi, axiom,
    ((![C0 : com, S0 : state, S12 : state, C1 : com, S22 : state]: ((evalc @ C0 @ S0 @ S12) => ((evalc @ C1 @ S12 @ S22) => (evalc @ (semi @ C0 @ C1) @ S0 @ S22)))))). % evalc.Semi
thf(fact_145_evaln__elim__cases_I1_J, axiom,
    ((![S6 : state, N : nat, T : state]: ((evaln @ skip @ S6 @ N @ T) => (T = S6))))). % evaln_elim_cases(1)
thf(fact_146_evaln__elim__cases_I4_J, axiom,
    ((![C1 : com, C22 : com, S6 : state, N : nat, T : state]: ((evaln @ (semi @ C1 @ C22) @ S6 @ N @ T) => (~ ((![S1 : state]: ((evaln @ C1 @ S6 @ N @ S1) => (~ ((evaln @ C22 @ S1 @ N @ T))))))))))). % evaln_elim_cases(4)
thf(fact_147_evalc__elim__cases_I1_J, axiom,
    ((![S6 : state, T : state]: ((evalc @ skip @ S6 @ T) => (T = S6))))). % evalc_elim_cases(1)
thf(fact_148_evalc__elim__cases_I4_J, axiom,
    ((![C1 : com, C22 : com, S6 : state, T : state]: ((evalc @ (semi @ C1 @ C22) @ S6 @ T) => (~ ((![S1 : state]: ((evalc @ C1 @ S6 @ S1) => (~ ((evalc @ C22 @ S1 @ T))))))))))). % evalc_elim_cases(4)
thf(fact_149_evaln__elim__cases_I2_J, axiom,
    ((![X7 : vname, A : state > nat, S6 : state, N : nat, T : state]: ((evaln @ (ass @ X7 @ A) @ S6 @ N @ T) => (T = (update @ S6 @ X7 @ (A @ S6))))))). % evaln_elim_cases(2)
thf(fact_150_evaln_OAssign, axiom,
    ((![X7 : vname, A : state > nat, S6 : state, N : nat]: (evaln @ (ass @ X7 @ A) @ S6 @ N @ (update @ S6 @ X7 @ (A @ S6)))))). % evaln.Assign
thf(fact_151_evalc__elim__cases_I2_J, axiom,
    ((![X7 : vname, A : state > nat, S6 : state, T : state]: ((evalc @ (ass @ X7 @ A) @ S6 @ T) => (T = (update @ S6 @ X7 @ (A @ S6))))))). % evalc_elim_cases(2)
thf(fact_152_evalc_OAssign, axiom,
    ((![X7 : vname, A : state > nat, S6 : state]: (evalc @ (ass @ X7 @ A) @ S6 @ (update @ S6 @ X7 @ (A @ S6)))))). % evalc.Assign
thf(fact_153_evaln_OLocal, axiom,
    ((![C : com, S0 : state, Y7 : loc, A : state > nat, N : nat, S12 : state]: ((evaln @ C @ (update @ S0 @ (loc2 @ Y7) @ (A @ S0)) @ N @ S12) => (evaln @ (local @ Y7 @ A @ C) @ S0 @ N @ (update @ S12 @ (loc2 @ Y7) @ (getlocs @ S0 @ Y7))))))). % evaln.Local
thf(fact_154_MGT__def, axiom,
    ((hoare_Mirabelle_MGT = (^[C4 : com]: (hoare_1659279548_state @ (^[Y4 : state]: (^[Z5 : state]: (Y4 = Z5))) @ C4 @ (evalc @ C4)))))). % MGT_def
thf(fact_155_triple__valid__def2, axiom,
    ((![N : nat, P : a > state > $o, C : com, Q : a > state > $o]: ((hoare_1926814542alid_a @ N @ (hoare_719046530iple_a @ P @ C @ Q)) = (![Z4 : a]: (![S4 : state]: (((P @ Z4 @ S4)) => ((![S2 : state]: (((evaln @ C @ S4 @ N @ S2)) => ((Q @ Z4 @ S2)))))))))))). % triple_valid_def2
thf(fact_156_triple__valid__def2, axiom,
    ((![N : nat, P : state > state > $o, C : com, Q : state > state > $o]: ((hoare_364318704_state @ N @ (hoare_1659279548_state @ P @ C @ Q)) = (![Z4 : state]: (![S4 : state]: (((P @ Z4 @ S4)) => ((![S2 : state]: (((evaln @ C @ S4 @ N @ S2)) => ((Q @ Z4 @ S2)))))))))))). % triple_valid_def2
thf(fact_157_pairwise__alt, axiom,
    ((pairwi531237284iple_a = (^[R3 : hoare_1678595023iple_a > hoare_1678595023iple_a > $o]: (^[S7 : set_Ho137910533iple_a]: (![X : hoare_1678595023iple_a]: (((member1332298086iple_a @ X @ S7)) => ((![Y6 : hoare_1678595023iple_a]: (((member1332298086iple_a @ Y6 @ (minus_1852999390iple_a @ S7 @ (insert1477804543iple_a @ X @ bot_bo1298296729iple_a)))) => ((R3 @ X @ Y6)))))))))))). % pairwise_alt
thf(fact_158_pairwise__alt, axiom,
    ((pairwi17519280_state = (^[R3 : hoare_958474565_state > hoare_958474565_state > $o]: (^[S7 : set_Ho840737317_state]: (![X : hoare_958474565_state]: (((member109514606_state @ X @ S7)) => ((![Y6 : hoare_958474565_state]: (((member109514606_state @ Y6 @ (minus_1628593292_state @ S7 @ (insert776267541_state @ X @ bot_bo105666705_state)))) => ((R3 @ X @ Y6)))))))))))). % pairwise_alt
thf(fact_159_vname_Oinject_I1_J, axiom,
    ((![X1 : glb, Y1 : glb]: (((glb2 @ X1) = (glb2 @ Y1)) = (X1 = Y1))))). % vname.inject(1)
thf(fact_160_Diff__empty, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((minus_1852999390iple_a @ A2 @ bot_bo1298296729iple_a) = A2)))). % Diff_empty
thf(fact_161_Diff__empty, axiom,
    ((![A2 : set_Ho840737317_state]: ((minus_1628593292_state @ A2 @ bot_bo105666705_state) = A2)))). % Diff_empty
thf(fact_162_empty__Diff, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((minus_1852999390iple_a @ bot_bo1298296729iple_a @ A2) = bot_bo1298296729iple_a)))). % empty_Diff
thf(fact_163_empty__Diff, axiom,
    ((![A2 : set_Ho840737317_state]: ((minus_1628593292_state @ bot_bo105666705_state @ A2) = bot_bo105666705_state)))). % empty_Diff
thf(fact_164_Diff__cancel, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((minus_1852999390iple_a @ A2 @ A2) = bot_bo1298296729iple_a)))). % Diff_cancel
thf(fact_165_Diff__cancel, axiom,
    ((![A2 : set_Ho840737317_state]: ((minus_1628593292_state @ A2 @ A2) = bot_bo105666705_state)))). % Diff_cancel
thf(fact_166_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_167_Diff__insert0, axiom,
    ((![X4 : hoare_958474565_state, A2 : set_Ho840737317_state, B : set_Ho840737317_state]: ((~ ((member109514606_state @ X4 @ A2))) => ((minus_1628593292_state @ A2 @ (insert776267541_state @ X4 @ B)) = (minus_1628593292_state @ A2 @ B)))))). % Diff_insert0
thf(fact_168_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_169_insert__Diff1, axiom,
    ((![X4 : hoare_958474565_state, B : set_Ho840737317_state, A2 : set_Ho840737317_state]: ((member109514606_state @ X4 @ B) => ((minus_1628593292_state @ (insert776267541_state @ X4 @ A2) @ B) = (minus_1628593292_state @ A2 @ B)))))). % insert_Diff1
thf(fact_170_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_171_insert__Diff__single, axiom,
    ((![A : hoare_958474565_state, A2 : set_Ho840737317_state]: ((insert776267541_state @ A @ (minus_1628593292_state @ A2 @ (insert776267541_state @ A @ bot_bo105666705_state))) = (insert776267541_state @ A @ A2))))). % insert_Diff_single
thf(fact_172_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
thf(fact_173_Diff__insert, axiom,
    ((![A2 : set_Ho840737317_state, A : hoare_958474565_state, B : set_Ho840737317_state]: ((minus_1628593292_state @ A2 @ (insert776267541_state @ A @ B)) = (minus_1628593292_state @ (minus_1628593292_state @ A2 @ B) @ (insert776267541_state @ A @ bot_bo105666705_state)))))). % Diff_insert
thf(fact_174_insert__Diff, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ A @ A2) => ((insert1477804543iple_a @ A @ (minus_1852999390iple_a @ A2 @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a))) = A2))))). % insert_Diff
thf(fact_175_insert__Diff, axiom,
    ((![A : hoare_958474565_state, A2 : set_Ho840737317_state]: ((member109514606_state @ A @ A2) => ((insert776267541_state @ A @ (minus_1628593292_state @ A2 @ (insert776267541_state @ A @ bot_bo105666705_state))) = A2))))). % insert_Diff
thf(fact_176_Diff__insert2, 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 @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a)) @ B))))). % Diff_insert2
thf(fact_177_Diff__insert2, axiom,
    ((![A2 : set_Ho840737317_state, A : hoare_958474565_state, B : set_Ho840737317_state]: ((minus_1628593292_state @ A2 @ (insert776267541_state @ A @ B)) = (minus_1628593292_state @ (minus_1628593292_state @ A2 @ (insert776267541_state @ A @ bot_bo105666705_state)) @ B))))). % Diff_insert2
thf(fact_178_Diff__insert__absorb, axiom,
    ((![X4 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ X4 @ A2))) => ((minus_1852999390iple_a @ (insert1477804543iple_a @ X4 @ A2) @ (insert1477804543iple_a @ X4 @ bot_bo1298296729iple_a)) = A2))))). % Diff_insert_absorb
thf(fact_179_Diff__insert__absorb, axiom,
    ((![X4 : hoare_958474565_state, A2 : set_Ho840737317_state]: ((~ ((member109514606_state @ X4 @ A2))) => ((minus_1628593292_state @ (insert776267541_state @ X4 @ A2) @ (insert776267541_state @ X4 @ bot_bo105666705_state)) = A2))))). % Diff_insert_absorb
thf(fact_180_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_181_insert__Diff__if, axiom,
    ((![X4 : hoare_958474565_state, B : set_Ho840737317_state, A2 : set_Ho840737317_state]: (((member109514606_state @ X4 @ B) => ((minus_1628593292_state @ (insert776267541_state @ X4 @ A2) @ B) = (minus_1628593292_state @ A2 @ B))) & ((~ ((member109514606_state @ X4 @ B))) => ((minus_1628593292_state @ (insert776267541_state @ X4 @ A2) @ B) = (insert776267541_state @ X4 @ (minus_1628593292_state @ A2 @ B)))))))). % insert_Diff_if
thf(fact_182_vname_Oexhaust, axiom,
    ((![Y : vname]: ((![X12 : glb]: (~ ((Y = (glb2 @ X12))))) => (~ ((![X22 : loc]: (~ ((Y = (loc2 @ X22))))))))))). % vname.exhaust
thf(fact_183_vname_Oinduct, axiom,
    ((![P : vname > $o, Vname : vname]: ((![X5 : glb]: (P @ (glb2 @ X5))) => ((![X5 : loc]: (P @ (loc2 @ X5))) => (P @ Vname)))))). % vname.induct
thf(fact_184_vname_Odistinct_I1_J, axiom,
    ((![X1 : glb, X2 : loc]: (~ (((glb2 @ X1) = (loc2 @ X2))))))). % vname.distinct(1)
thf(fact_185_remove__def, axiom,
    ((remove1512401578iple_a = (^[X : hoare_1678595023iple_a]: (^[A4 : set_Ho137910533iple_a]: (minus_1852999390iple_a @ A4 @ (insert1477804543iple_a @ X @ bot_bo1298296729iple_a))))))). % remove_def
thf(fact_186_remove__def, axiom,
    ((remove586333994_state = (^[X : hoare_958474565_state]: (^[A4 : set_Ho840737317_state]: (minus_1628593292_state @ A4 @ (insert776267541_state @ X @ bot_bo105666705_state))))))). % remove_def

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