% 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_288__3252536_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:13:31.141

% Could-be-implicit typings (6)
thf(ty_n_t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    set_Ho137910533iple_a : $tType).
thf(ty_n_t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    hoare_1678595023iple_a : $tType).
thf(ty_n_t__Com__Ostate, type,
    state : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).
thf(ty_n_t__Com__Ocom, type,
    com : $tType).
thf(ty_n_tf__a, type,
    a : $tType).

% Explicit typings (23)
thf(sy_c_Com_Ocom_OSKIP, type,
    skip : com).
thf(sy_c_Com_Ocom_OSemi, type,
    semi : com > com > com).
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_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_Otriple_Otriple_001tf__a, type,
    hoare_719046530iple_a : (a > state > $o) > com > (a > state > $o) > hoare_1678595023iple_a).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Otriple__valid_001tf__a, type,
    hoare_1926814542alid_a : nat > hoare_1678595023iple_a > $o).
thf(sy_c_Nat_OSuc, type,
    suc : nat > nat).
thf(sy_c_Natural_Oevaln, type,
    evaln : com > state > nat > 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_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_Ga, type,
    ga : set_Ho137910533iple_a).
thf(sy_v_P, type,
    p : a > state > $o).
thf(sy_v_Q, type,
    q : a > state > $o).
thf(sy_v_c, type,
    c : com).

% Relevant facts (84)
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_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_6_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_7_cut, axiom,
    ((![G2 : set_Ho137910533iple_a, Ts : set_Ho137910533iple_a, G : set_Ho137910533iple_a]: ((hoare_129598474rivs_a @ G2 @ Ts) => ((hoare_129598474rivs_a @ G @ G2) => (hoare_129598474rivs_a @ G @ Ts)))))). % cut
thf(fact_8_empty, axiom,
    ((![G : set_Ho137910533iple_a]: (hoare_129598474rivs_a @ G @ bot_bo1298296729iple_a)))). % empty
thf(fact_9_conseq, axiom,
    ((![P2 : a > state > $o, G : set_Ho137910533iple_a, C : com, Q : a > state > $o]: ((![Z : a, S : state]: ((P2 @ Z @ S) => (?[P3 : a > state > $o, Q3 : a > state > $o]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P3 @ C @ Q3) @ bot_bo1298296729iple_a)) & (![S2 : state]: ((![Z2 : a]: ((P3 @ Z2 @ S) => (Q3 @ Z2 @ S2))) => (Q @ Z @ S2))))))) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P2 @ C @ Q) @ bot_bo1298296729iple_a)))))). % conseq
thf(fact_10_hoare__derivs_Oinsert, axiom,
    ((![G : set_Ho137910533iple_a, T : hoare_1678595023iple_a, Ts : set_Ho137910533iple_a]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ T @ bot_bo1298296729iple_a)) => ((hoare_129598474rivs_a @ G @ Ts) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ T @ Ts))))))). % hoare_derivs.insert
thf(fact_11_ball__empty, axiom,
    ((![P2 : hoare_1678595023iple_a > $o, X : hoare_1678595023iple_a]: ((member1332298086iple_a @ X @ bot_bo1298296729iple_a) => (P2 @ X))))). % ball_empty
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]: ((![X5 : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ X5 @ 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) = (![X5 : hoare_1678595023iple_a]: (~ ((P2 @ X5)))))))). % Collect_empty_eq
thf(fact_19_empty__Collect__eq, axiom,
    ((![P2 : hoare_1678595023iple_a > $o]: ((bot_bo1298296729iple_a = (collec1600235172iple_a @ P2)) = (![X5 : hoare_1678595023iple_a]: (~ ((P2 @ X5)))))))). % empty_Collect_eq
thf(fact_20_hoare__valids__def, axiom,
    ((hoare_1775499016lids_a = (^[G3 : set_Ho137910533iple_a]: (^[Ts2 : set_Ho137910533iple_a]: (![N : nat]: (((![X5 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X5 @ G3)) => ((hoare_1926814542alid_a @ N @ X5))))) => ((![X5 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X5 @ Ts2)) => ((hoare_1926814542alid_a @ N @ X5)))))))))))). % hoare_valids_def
thf(fact_21_singletonD, axiom,
    ((![B2 : hoare_1678595023iple_a, A : hoare_1678595023iple_a]: ((member1332298086iple_a @ B2 @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a)) => (B2 = A))))). % singletonD
thf(fact_22_bot__set__def, axiom,
    ((bot_bo1298296729iple_a = (collec1600235172iple_a @ bot_bo431311916le_a_o)))). % bot_set_def
thf(fact_23_ex__in__conv, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((?[X5 : hoare_1678595023iple_a]: (member1332298086iple_a @ X5 @ A2)) = (~ ((A2 = bot_bo1298296729iple_a))))))). % ex_in_conv
thf(fact_24_equals0I, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((![Y4 : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ Y4 @ A2)))) => (A2 = bot_bo1298296729iple_a))))). % equals0I
thf(fact_25_equals0D, axiom,
    ((![A2 : set_Ho137910533iple_a, A : hoare_1678595023iple_a]: ((A2 = bot_bo1298296729iple_a) => (~ ((member1332298086iple_a @ A @ A2))))))). % equals0D
thf(fact_26_emptyE, axiom,
    ((![A : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ A @ bot_bo1298296729iple_a)))))). % emptyE
thf(fact_27_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_28_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_29_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_30_insert__absorb, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ A @ A2) => ((insert1477804543iple_a @ A @ A2) = A2))))). % insert_absorb
thf(fact_31_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_32_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_33_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_34_insertI1, axiom,
    ((![A : hoare_1678595023iple_a, B : set_Ho137910533iple_a]: (member1332298086iple_a @ A @ (insert1477804543iple_a @ A @ B))))). % insertI1
thf(fact_35_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_36_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_37_insert__not__empty, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: (~ (((insert1477804543iple_a @ A @ A2) = bot_bo1298296729iple_a)))))). % insert_not_empty
thf(fact_38_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_39_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_40_the__elem__eq, axiom,
    ((![X4 : hoare_1678595023iple_a]: ((the_el434698138iple_a @ (insert1477804543iple_a @ X4 @ bot_bo1298296729iple_a)) = X4)))). % the_elem_eq
thf(fact_41_is__singletonI, axiom,
    ((![X4 : hoare_1678595023iple_a]: (is_sin1784037339iple_a @ (insert1477804543iple_a @ X4 @ bot_bo1298296729iple_a))))). % is_singletonI
thf(fact_42_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_43_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_44_Set_Ois__empty__def, axiom,
    ((is_emp901906557iple_a = (^[A3 : set_Ho137910533iple_a]: (A3 = bot_bo1298296729iple_a))))). % Set.is_empty_def
thf(fact_45_is__singleton__def, axiom,
    ((is_sin1784037339iple_a = (^[A3 : set_Ho137910533iple_a]: (?[X5 : hoare_1678595023iple_a]: (A3 = (insert1477804543iple_a @ X5 @ bot_bo1298296729iple_a))))))). % is_singleton_def
thf(fact_46_is__singletonE, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((is_sin1784037339iple_a @ A2) => (~ ((![X6 : hoare_1678595023iple_a]: (~ ((A2 = (insert1477804543iple_a @ X6 @ bot_bo1298296729iple_a))))))))))). % is_singletonE
thf(fact_47_is__singleton__the__elem, axiom,
    ((is_sin1784037339iple_a = (^[A3 : set_Ho137910533iple_a]: (A3 = (insert1477804543iple_a @ (the_el434698138iple_a @ A3) @ bot_bo1298296729iple_a)))))). % is_singleton_the_elem
thf(fact_48_is__singletonI_H, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((~ ((A2 = bot_bo1298296729iple_a))) => ((![X6 : hoare_1678595023iple_a, Y4 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X6 @ A2) => ((member1332298086iple_a @ Y4 @ A2) => (X6 = Y4)))) => (is_sin1784037339iple_a @ A2)))))). % is_singletonI'
thf(fact_49_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_50_com_Odistinct_I5_J, axiom,
    ((![X41 : com, X42 : com]: (~ ((skip = (semi @ X41 @ X42))))))). % com.distinct(5)
thf(fact_51_bot__empty__eq, axiom,
    ((bot_bo431311916le_a_o = (^[X5 : hoare_1678595023iple_a]: (member1332298086iple_a @ X5 @ bot_bo1298296729iple_a))))). % bot_empty_eq
thf(fact_52_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_53_triples__valid__Suc, axiom,
    ((![Ts : set_Ho137910533iple_a, N2 : nat]: ((![X6 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X6 @ Ts) => (hoare_1926814542alid_a @ (suc @ N2) @ X6))) => (![X : hoare_1678595023iple_a]: ((member1332298086iple_a @ X @ Ts) => (hoare_1926814542alid_a @ N2 @ X))))))). % triples_valid_Suc
thf(fact_54_triple__valid__Suc, axiom,
    ((![N2 : nat, T : hoare_1678595023iple_a]: ((hoare_1926814542alid_a @ (suc @ N2) @ T) => (hoare_1926814542alid_a @ N2 @ T))))). % triple_valid_Suc
thf(fact_55_nat_Oinject, axiom,
    ((![X2 : nat, Y2 : nat]: (((suc @ X2) = (suc @ Y2)) = (X2 = Y2))))). % nat.inject
thf(fact_56_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_57_Suc__inject, axiom,
    ((![X4 : nat, Y : nat]: (((suc @ X4) = (suc @ Y)) => (X4 = Y))))). % Suc_inject
thf(fact_58_n__not__Suc__n, axiom,
    ((![N2 : nat]: (~ ((N2 = (suc @ N2))))))). % n_not_Suc_n
thf(fact_59_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_60_pairwise__empty, axiom,
    ((![P2 : hoare_1678595023iple_a > hoare_1678595023iple_a > $o]: (pairwi531237284iple_a @ P2 @ bot_bo1298296729iple_a)))). % pairwise_empty
thf(fact_61_pairwise__insert, axiom,
    ((![R2 : hoare_1678595023iple_a > hoare_1678595023iple_a > $o, X4 : hoare_1678595023iple_a, S3 : set_Ho137910533iple_a]: ((pairwi531237284iple_a @ R2 @ (insert1477804543iple_a @ X4 @ S3)) = (((![Y5 : hoare_1678595023iple_a]: (((((member1332298086iple_a @ Y5 @ S3)) & ((~ ((Y5 = X4)))))) => ((((R2 @ X4 @ Y5)) & ((R2 @ Y5 @ X4))))))) & ((pairwi531237284iple_a @ R2 @ S3))))))). % pairwise_insert
thf(fact_62_pairwise__alt, axiom,
    ((pairwi531237284iple_a = (^[R3 : hoare_1678595023iple_a > hoare_1678595023iple_a > $o]: (^[S4 : set_Ho137910533iple_a]: (![X5 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X5 @ S4)) => ((![Y5 : hoare_1678595023iple_a]: (((member1332298086iple_a @ Y5 @ (minus_1852999390iple_a @ S4 @ (insert1477804543iple_a @ X5 @ bot_bo1298296729iple_a)))) => ((R3 @ X5 @ Y5)))))))))))). % pairwise_alt
thf(fact_63_triple__valid__def2, axiom,
    ((![N2 : nat, P2 : a > state > $o, C : com, Q : a > state > $o]: ((hoare_1926814542alid_a @ N2 @ (hoare_719046530iple_a @ P2 @ C @ Q)) = (![Z3 : a]: (![S5 : state]: (((P2 @ Z3 @ S5)) => ((![S6 : state]: (((evaln @ C @ S5 @ N2 @ S6)) => ((Q @ Z3 @ S6)))))))))))). % triple_valid_def2
thf(fact_64_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_65_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_66_subset__empty, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ bot_bo1298296729iple_a) = (A2 = bot_bo1298296729iple_a))))). % subset_empty
thf(fact_67_empty__subsetI, axiom,
    ((![A2 : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ bot_bo1298296729iple_a @ A2)))). % empty_subsetI
thf(fact_68_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_69_Diff__empty, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((minus_1852999390iple_a @ A2 @ bot_bo1298296729iple_a) = A2)))). % Diff_empty
thf(fact_70_empty__Diff, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((minus_1852999390iple_a @ bot_bo1298296729iple_a @ A2) = bot_bo1298296729iple_a)))). % empty_Diff
thf(fact_71_Diff__cancel, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((minus_1852999390iple_a @ A2 @ A2) = bot_bo1298296729iple_a)))). % Diff_cancel
thf(fact_72_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_73_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_74_Diff__eq__empty__iff, axiom,
    ((![A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: (((minus_1852999390iple_a @ A2 @ B) = bot_bo1298296729iple_a) = (ord_le1221261669iple_a @ A2 @ B))))). % Diff_eq_empty_iff
thf(fact_75_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_76_weaken, axiom,
    ((![G : set_Ho137910533iple_a, Ts3 : set_Ho137910533iple_a, Ts : set_Ho137910533iple_a]: ((hoare_129598474rivs_a @ G @ Ts3) => ((ord_le1221261669iple_a @ Ts @ Ts3) => (hoare_129598474rivs_a @ G @ Ts)))))). % weaken
thf(fact_77_asm, axiom,
    ((![Ts : set_Ho137910533iple_a, G : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ Ts @ G) => (hoare_129598474rivs_a @ G @ Ts))))). % asm
thf(fact_78_thin, axiom,
    ((![G2 : set_Ho137910533iple_a, Ts : set_Ho137910533iple_a, G : set_Ho137910533iple_a]: ((hoare_129598474rivs_a @ G2 @ Ts) => ((ord_le1221261669iple_a @ G2 @ G) => (hoare_129598474rivs_a @ G @ Ts)))))). % thin
thf(fact_79_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_80_bot_Oextremum__uniqueI, axiom,
    ((![A : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ bot_bo1298296729iple_a) => (A = bot_bo1298296729iple_a))))). % bot.extremum_uniqueI
thf(fact_81_bot_Oextremum__unique, axiom,
    ((![A : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ bot_bo1298296729iple_a) = (A = bot_bo1298296729iple_a))))). % bot.extremum_unique
thf(fact_82_bot_Oextremum, axiom,
    ((![A : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ bot_bo1298296729iple_a @ A)))). % bot.extremum
thf(fact_83_subset__Diff__insert, axiom,
    ((![A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a, X4 : hoare_1678595023iple_a, C3 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ (minus_1852999390iple_a @ B @ (insert1477804543iple_a @ X4 @ C3))) = (((ord_le1221261669iple_a @ A2 @ (minus_1852999390iple_a @ B @ C3))) & ((~ ((member1332298086iple_a @ X4 @ A2))))))))). % subset_Diff_insert

% Conjectures (2)
thf(conj_0, hypothesis,
    ((![Z4 : a, S7 : state]: ((p @ Z4 @ S7) => (?[P4 : a > state > $o, Q4 : a > state > $o]: (((hoare_129598474rivs_a @ ga @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P4 @ c @ Q4) @ bot_bo1298296729iple_a)) & (![N3 : nat]: ((![X6 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X6 @ ga) => (hoare_1926814542alid_a @ N3 @ X6))) => (![X : hoare_1678595023iple_a]: ((member1332298086iple_a @ X @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P4 @ c @ Q4) @ bot_bo1298296729iple_a)) => (hoare_1926814542alid_a @ N3 @ X)))))) & (![S8 : state]: ((![Z5 : a]: ((P4 @ Z5 @ S7) => (Q4 @ Z5 @ S8))) => (q @ Z4 @ S8))))))))).
thf(conj_1, conjecture,
    ((![N4 : nat]: ((?[X : hoare_1678595023iple_a]: ((member1332298086iple_a @ X @ ga) & (~ ((hoare_1926814542alid_a @ N4 @ X))))) | (![X6 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X6 @ (insert1477804543iple_a @ (hoare_719046530iple_a @ p @ c @ q) @ bot_bo1298296729iple_a)) => (hoare_1926814542alid_a @ N4 @ X6))))))).
