% 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_497__3255328_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:15:21.859

% Could-be-implicit typings (5)
thf(ty_n_t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    set_Ho137910533iple_a : $tType).
thf(ty_n_t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    hoare_1678595023iple_a : $tType).
thf(ty_n_t__Com__Ostate, type,
    state : $tType).
thf(ty_n_t__Com__Ocom, type,
    com : $tType).
thf(ty_n_tf__a, type,
    a : $tType).

% Explicit typings (18)
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_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_Oimage_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    image_1577193952iple_a : (hoare_1678595023iple_a > hoare_1678595023iple_a) > set_Ho137910533iple_a > 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_t, type,
    t : hoare_1678595023iple_a).

% Relevant facts (86)
thf(fact_0_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_1_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_2_empty, axiom,
    ((![G : set_Ho137910533iple_a]: (hoare_129598474rivs_a @ G @ bot_bo1298296729iple_a)))). % empty
thf(fact_3_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_4_singletonI, axiom,
    ((![A : hoare_1678595023iple_a]: (member1332298086iple_a @ A @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a))))). % singletonI
thf(fact_5_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_6_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_7_insert__absorb2, axiom,
    ((![X : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((insert1477804543iple_a @ X @ (insert1477804543iple_a @ X @ A2)) = (insert1477804543iple_a @ X @ A2))))). % insert_absorb2
thf(fact_8_empty__iff, axiom,
    ((![C : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ C @ bot_bo1298296729iple_a)))))). % empty_iff
thf(fact_9_all__not__in__conv, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((![X2 : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ X2 @ A2)))) = (A2 = bot_bo1298296729iple_a))))). % all_not_in_conv
thf(fact_10_Collect__empty__eq, axiom,
    ((![P : hoare_1678595023iple_a > $o]: (((collec1600235172iple_a @ P) = bot_bo1298296729iple_a) = (![X2 : hoare_1678595023iple_a]: (~ ((P @ X2)))))))). % Collect_empty_eq
thf(fact_11_empty__Collect__eq, axiom,
    ((![P : hoare_1678595023iple_a > $o]: ((bot_bo1298296729iple_a = (collec1600235172iple_a @ P)) = (![X2 : hoare_1678595023iple_a]: (~ ((P @ X2)))))))). % empty_Collect_eq
thf(fact_12_singletonD, axiom,
    ((![B2 : hoare_1678595023iple_a, A : hoare_1678595023iple_a]: ((member1332298086iple_a @ B2 @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a)) => (B2 = A))))). % singletonD
thf(fact_13_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_14_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_15_bot__set__def, axiom,
    ((bot_bo1298296729iple_a = (collec1600235172iple_a @ bot_bo431311916le_a_o)))). % bot_set_def
thf(fact_16_ex__in__conv, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((?[X2 : hoare_1678595023iple_a]: (member1332298086iple_a @ X2 @ A2)) = (~ ((A2 = bot_bo1298296729iple_a))))))). % ex_in_conv
thf(fact_17_equals0I, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((![Y : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ Y @ A2)))) => (A2 = bot_bo1298296729iple_a))))). % equals0I
thf(fact_18_equals0D, axiom,
    ((![A2 : set_Ho137910533iple_a, A : hoare_1678595023iple_a]: ((A2 = bot_bo1298296729iple_a) => (~ ((member1332298086iple_a @ A @ A2))))))). % equals0D
thf(fact_19_emptyE, axiom,
    ((![A : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ A @ bot_bo1298296729iple_a)))))). % emptyE
thf(fact_20_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_21_insert__commute, axiom,
    ((![X : hoare_1678595023iple_a, Y2 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((insert1477804543iple_a @ X @ (insert1477804543iple_a @ Y2 @ A2)) = (insert1477804543iple_a @ Y2 @ (insert1477804543iple_a @ X @ A2)))))). % insert_commute
thf(fact_22_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_23_insert__absorb, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ A @ A2) => ((insert1477804543iple_a @ A @ A2) = A2))))). % insert_absorb
thf(fact_24_insert__ident, axiom,
    ((![X : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ X @ A2))) => ((~ ((member1332298086iple_a @ X @ B))) => (((insert1477804543iple_a @ X @ A2) = (insert1477804543iple_a @ X @ B)) = (A2 = B))))))). % insert_ident
thf(fact_25_Set_Oset__insert, axiom,
    ((![X : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ X @ A2) => (~ ((![B3 : set_Ho137910533iple_a]: ((A2 = (insert1477804543iple_a @ X @ B3)) => (member1332298086iple_a @ X @ B3))))))))). % Set.set_insert
thf(fact_26_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_27_insertI1, axiom,
    ((![A : hoare_1678595023iple_a, B : set_Ho137910533iple_a]: (member1332298086iple_a @ A @ (insert1477804543iple_a @ A @ B))))). % insertI1
thf(fact_28_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_29_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_30_insert__not__empty, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: (~ (((insert1477804543iple_a @ A @ A2) = bot_bo1298296729iple_a)))))). % insert_not_empty
thf(fact_31_the__elem__eq, axiom,
    ((![X : hoare_1678595023iple_a]: ((the_el434698138iple_a @ (insert1477804543iple_a @ X @ bot_bo1298296729iple_a)) = X)))). % the_elem_eq
thf(fact_32_is__singletonI, axiom,
    ((![X : hoare_1678595023iple_a]: (is_sin1784037339iple_a @ (insert1477804543iple_a @ X @ bot_bo1298296729iple_a))))). % is_singletonI
thf(fact_33_Set_Ois__empty__def, axiom,
    ((is_emp901906557iple_a = (^[A3 : set_Ho137910533iple_a]: (A3 = bot_bo1298296729iple_a))))). % Set.is_empty_def
thf(fact_34_is__singleton__def, axiom,
    ((is_sin1784037339iple_a = (^[A3 : set_Ho137910533iple_a]: (?[X2 : hoare_1678595023iple_a]: (A3 = (insert1477804543iple_a @ X2 @ bot_bo1298296729iple_a))))))). % is_singleton_def
thf(fact_35_is__singletonE, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((is_sin1784037339iple_a @ A2) => (~ ((![X3 : hoare_1678595023iple_a]: (~ ((A2 = (insert1477804543iple_a @ X3 @ bot_bo1298296729iple_a))))))))))). % is_singletonE
thf(fact_36_conseq, axiom,
    ((![P : a > state > $o, G : set_Ho137910533iple_a, C : com, Q : a > state > $o]: ((![Z : a, S : state]: ((P @ Z @ S) => (?[P2 : a > state > $o, Q2 : a > state > $o]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P2 @ C @ Q2) @ bot_bo1298296729iple_a)) & (![S2 : state]: ((![Z2 : a]: ((P2 @ Z2 @ S) => (Q2 @ Z2 @ S2))) => (Q @ Z @ S2))))))) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q) @ bot_bo1298296729iple_a)))))). % conseq
thf(fact_37_conseq12, axiom,
    ((![G : set_Ho137910533iple_a, P3 : a > state > $o, C : com, Q3 : a > state > $o, P : a > state > $o, Q : a > state > $o]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P3 @ C @ Q3) @ bot_bo1298296729iple_a)) => ((![Z : a, S : state]: ((P @ Z @ S) => (![S2 : state]: ((![Z2 : a]: ((P3 @ Z2 @ S) => (Q3 @ Z2 @ S2))) => (Q @ Z @ S2))))) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q) @ bot_bo1298296729iple_a))))))). % conseq12
thf(fact_38_conseq2, axiom,
    ((![G : set_Ho137910533iple_a, P : a > state > $o, C : com, Q3 : a > state > $o, Q : a > state > $o]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q3) @ bot_bo1298296729iple_a)) => ((![Z : a, S : state]: ((Q3 @ Z @ S) => (Q @ Z @ S))) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q) @ bot_bo1298296729iple_a))))))). % conseq2
thf(fact_39_conseq1, axiom,
    ((![G : set_Ho137910533iple_a, P3 : a > state > $o, C : com, Q : a > state > $o, P : a > state > $o]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P3 @ C @ Q) @ bot_bo1298296729iple_a)) => ((![Z : a, S : state]: ((P @ Z @ S) => (P3 @ Z @ S))) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q) @ bot_bo1298296729iple_a))))))). % conseq1
thf(fact_40_mem__Collect__eq, axiom,
    ((![A : hoare_1678595023iple_a, P : hoare_1678595023iple_a > $o]: ((member1332298086iple_a @ A @ (collec1600235172iple_a @ P)) = (P @ A))))). % mem_Collect_eq
thf(fact_41_Collect__mem__eq, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((collec1600235172iple_a @ (^[X2 : hoare_1678595023iple_a]: (member1332298086iple_a @ X2 @ A2))) = A2)))). % Collect_mem_eq
thf(fact_42_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_43_is__singletonI_H, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((~ ((A2 = bot_bo1298296729iple_a))) => ((![X3 : hoare_1678595023iple_a, Y : hoare_1678595023iple_a]: ((member1332298086iple_a @ X3 @ A2) => ((member1332298086iple_a @ Y @ A2) => (X3 = Y)))) => (is_sin1784037339iple_a @ A2)))))). % is_singletonI'
thf(fact_44_bot__empty__eq, axiom,
    ((bot_bo431311916le_a_o = (^[X2 : hoare_1678595023iple_a]: (member1332298086iple_a @ X2 @ bot_bo1298296729iple_a))))). % bot_empty_eq
thf(fact_45_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_46_hoare__SkipI, axiom,
    ((![P : a > state > $o, Q : a > state > $o, G : set_Ho137910533iple_a]: ((![Z : a, S : state]: ((P @ Z @ S) => (Q @ Z @ S))) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ skip @ Q) @ bot_bo1298296729iple_a)))))). % hoare_SkipI
thf(fact_47_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_48_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_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_hoare__sound, axiom,
    ((![G : set_Ho137910533iple_a, Ts : set_Ho137910533iple_a]: ((hoare_129598474rivs_a @ G @ Ts) => (hoare_1775499016lids_a @ G @ Ts))))). % hoare_sound
thf(fact_52_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_53_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_54_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_55_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_56_image__eqI, axiom,
    ((![B2 : hoare_1678595023iple_a, F : hoare_1678595023iple_a > hoare_1678595023iple_a, X : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((B2 = (F @ X)) => ((member1332298086iple_a @ X @ A2) => (member1332298086iple_a @ B2 @ (image_1577193952iple_a @ F @ A2))))))). % image_eqI
thf(fact_57_subsetI, axiom,
    ((![A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((![X3 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X3 @ A2) => (member1332298086iple_a @ X3 @ B))) => (ord_le1221261669iple_a @ A2 @ B))))). % subsetI
thf(fact_58_Diff__iff, axiom,
    ((![C : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((member1332298086iple_a @ C @ (minus_1852999390iple_a @ A2 @ B)) = (((member1332298086iple_a @ C @ A2)) & ((~ ((member1332298086iple_a @ C @ B))))))))). % Diff_iff
thf(fact_59_DiffI, axiom,
    ((![C : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((member1332298086iple_a @ C @ A2) => ((~ ((member1332298086iple_a @ C @ B))) => (member1332298086iple_a @ C @ (minus_1852999390iple_a @ A2 @ B))))))). % DiffI
thf(fact_60_image__is__empty, axiom,
    ((![F : hoare_1678595023iple_a > hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: (((image_1577193952iple_a @ F @ A2) = bot_bo1298296729iple_a) = (A2 = bot_bo1298296729iple_a))))). % image_is_empty
thf(fact_61_empty__is__image, axiom,
    ((![F : hoare_1678595023iple_a > hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((bot_bo1298296729iple_a = (image_1577193952iple_a @ F @ A2)) = (A2 = bot_bo1298296729iple_a))))). % empty_is_image
thf(fact_62_image__empty, axiom,
    ((![F : hoare_1678595023iple_a > hoare_1678595023iple_a]: ((image_1577193952iple_a @ F @ bot_bo1298296729iple_a) = bot_bo1298296729iple_a)))). % image_empty
thf(fact_63_insert__image, axiom,
    ((![X : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, F : hoare_1678595023iple_a > hoare_1678595023iple_a]: ((member1332298086iple_a @ X @ A2) => ((insert1477804543iple_a @ (F @ X) @ (image_1577193952iple_a @ F @ A2)) = (image_1577193952iple_a @ F @ A2)))))). % insert_image
thf(fact_64_image__insert, axiom,
    ((![F : hoare_1678595023iple_a > hoare_1678595023iple_a, A : hoare_1678595023iple_a, B : set_Ho137910533iple_a]: ((image_1577193952iple_a @ F @ (insert1477804543iple_a @ A @ B)) = (insert1477804543iple_a @ (F @ A) @ (image_1577193952iple_a @ F @ B)))))). % image_insert
thf(fact_65_empty__subsetI, axiom,
    ((![A2 : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ bot_bo1298296729iple_a @ A2)))). % empty_subsetI
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_insert__subset, axiom,
    ((![X : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ (insert1477804543iple_a @ X @ A2) @ B) = (((member1332298086iple_a @ X @ B)) & ((ord_le1221261669iple_a @ A2 @ B))))))). % insert_subset
thf(fact_68_Diff__cancel, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((minus_1852999390iple_a @ A2 @ A2) = bot_bo1298296729iple_a)))). % Diff_cancel
thf(fact_69_empty__Diff, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((minus_1852999390iple_a @ bot_bo1298296729iple_a @ A2) = bot_bo1298296729iple_a)))). % empty_Diff
thf(fact_70_Diff__empty, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((minus_1852999390iple_a @ A2 @ bot_bo1298296729iple_a) = A2)))). % Diff_empty
thf(fact_71_insert__Diff1, axiom,
    ((![X : hoare_1678595023iple_a, B : set_Ho137910533iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ X @ B) => ((minus_1852999390iple_a @ (insert1477804543iple_a @ X @ A2) @ B) = (minus_1852999390iple_a @ A2 @ B)))))). % insert_Diff1
thf(fact_72_Diff__insert0, axiom,
    ((![X : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ X @ A2))) => ((minus_1852999390iple_a @ A2 @ (insert1477804543iple_a @ X @ B)) = (minus_1852999390iple_a @ A2 @ B)))))). % Diff_insert0
thf(fact_73_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_74_pairwise__alt, axiom,
    ((pairwi531237284iple_a = (^[R2 : hoare_1678595023iple_a > hoare_1678595023iple_a > $o]: (^[S3 : set_Ho137910533iple_a]: (![X2 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X2 @ S3)) => ((![Y3 : hoare_1678595023iple_a]: (((member1332298086iple_a @ Y3 @ (minus_1852999390iple_a @ S3 @ (insert1477804543iple_a @ X2 @ bot_bo1298296729iple_a)))) => ((R2 @ X2 @ Y3)))))))))))). % pairwise_alt
thf(fact_75_insert__Diff__if, axiom,
    ((![X : hoare_1678595023iple_a, B : set_Ho137910533iple_a, A2 : set_Ho137910533iple_a]: (((member1332298086iple_a @ X @ B) => ((minus_1852999390iple_a @ (insert1477804543iple_a @ X @ A2) @ B) = (minus_1852999390iple_a @ A2 @ B))) & ((~ ((member1332298086iple_a @ X @ B))) => ((minus_1852999390iple_a @ (insert1477804543iple_a @ X @ A2) @ B) = (insert1477804543iple_a @ X @ (minus_1852999390iple_a @ A2 @ B)))))))). % insert_Diff_if
thf(fact_76_subset__Diff__insert, axiom,
    ((![A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a, X : hoare_1678595023iple_a, C3 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ (minus_1852999390iple_a @ B @ (insert1477804543iple_a @ X @ C3))) = (((ord_le1221261669iple_a @ A2 @ (minus_1852999390iple_a @ B @ C3))) & ((~ ((member1332298086iple_a @ X @ A2))))))))). % subset_Diff_insert
thf(fact_77_rev__image__eqI, axiom,
    ((![X : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B2 : hoare_1678595023iple_a, F : hoare_1678595023iple_a > hoare_1678595023iple_a]: ((member1332298086iple_a @ X @ A2) => ((B2 = (F @ X)) => (member1332298086iple_a @ B2 @ (image_1577193952iple_a @ F @ A2))))))). % rev_image_eqI
thf(fact_78_image__subsetI, axiom,
    ((![A2 : set_Ho137910533iple_a, F : hoare_1678595023iple_a > hoare_1678595023iple_a, B : set_Ho137910533iple_a]: ((![X3 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X3 @ A2) => (member1332298086iple_a @ (F @ X3) @ B))) => (ord_le1221261669iple_a @ (image_1577193952iple_a @ F @ A2) @ B))))). % image_subsetI
thf(fact_79_subset__iff, axiom,
    ((ord_le1221261669iple_a = (^[A3 : set_Ho137910533iple_a]: (^[B4 : set_Ho137910533iple_a]: (![T2 : hoare_1678595023iple_a]: (((member1332298086iple_a @ T2 @ A3)) => ((member1332298086iple_a @ T2 @ B4))))))))). % subset_iff
thf(fact_80_subset__eq, axiom,
    ((ord_le1221261669iple_a = (^[A3 : set_Ho137910533iple_a]: (^[B4 : set_Ho137910533iple_a]: (![X2 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X2 @ A3)) => ((member1332298086iple_a @ X2 @ B4))))))))). % subset_eq
thf(fact_81_pairwiseI, axiom,
    ((![S4 : set_Ho137910533iple_a, R : hoare_1678595023iple_a > hoare_1678595023iple_a > $o]: ((![X3 : hoare_1678595023iple_a, Y : hoare_1678595023iple_a]: ((member1332298086iple_a @ X3 @ S4) => ((member1332298086iple_a @ Y @ S4) => ((~ ((X3 = Y))) => (R @ X3 @ Y))))) => (pairwi531237284iple_a @ R @ S4))))). % pairwiseI
thf(fact_82_pairwiseD, axiom,
    ((![R : hoare_1678595023iple_a > hoare_1678595023iple_a > $o, S4 : set_Ho137910533iple_a, X : hoare_1678595023iple_a, Y2 : hoare_1678595023iple_a]: ((pairwi531237284iple_a @ R @ S4) => ((member1332298086iple_a @ X @ S4) => ((member1332298086iple_a @ Y2 @ S4) => ((~ ((X = Y2))) => (R @ X @ Y2)))))))). % pairwiseD
thf(fact_83_subsetD, axiom,
    ((![A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a, C : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A2 @ B) => ((member1332298086iple_a @ C @ A2) => (member1332298086iple_a @ C @ B)))))). % subsetD
thf(fact_84_in__mono, axiom,
    ((![A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a, X : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A2 @ B) => ((member1332298086iple_a @ X @ A2) => (member1332298086iple_a @ X @ B)))))). % in_mono
thf(fact_85_imageI, axiom,
    ((![X : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, F : hoare_1678595023iple_a > hoare_1678595023iple_a]: ((member1332298086iple_a @ X @ A2) => (member1332298086iple_a @ (F @ X) @ (image_1577193952iple_a @ F @ A2)))))). % imageI

% Conjectures (1)
thf(conj_0, conjecture,
    ((hoare_129598474rivs_a @ (insert1477804543iple_a @ t @ bot_bo1298296729iple_a) @ (insert1477804543iple_a @ t @ bot_bo1298296729iple_a)))).
