% 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_227__3251724_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:12:55.239

% 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__Com__Opname, type,
    pname : $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_OBODY, type,
    body : pname > 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_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    uminus922456654iple_a : set_Ho137910533iple_a > set_Ho137910533iple_a).
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 > $o).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    bot_bo1298296729iple_a : set_Ho137910533iple_a).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J_J, type,
    ord_le1048771374iple_a : ($o > set_Ho137910533iple_a) > ($o > set_Ho137910533iple_a) > $o).
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_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    order_929906668iple_a : (set_Ho137910533iple_a > $o) > set_Ho137910533iple_a).
thf(sy_c_Set_OCollect_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    collec1600235172iple_a : (hoare_1678595023iple_a > $o) > set_Ho137910533iple_a).
thf(sy_c_Set_Oinsert_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    insert1477804543iple_a : hoare_1678595023iple_a > set_Ho137910533iple_a > set_Ho137910533iple_a).
thf(sy_c_Set_Ois__singleton_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    is_sin1784037339iple_a : set_Ho137910533iple_a > $o).
thf(sy_c_Set_Othe__elem_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    the_el434698138iple_a : set_Ho137910533iple_a > hoare_1678595023iple_a).
thf(sy_c_member_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    member1332298086iple_a : hoare_1678595023iple_a > set_Ho137910533iple_a > $o).
thf(sy_v_G, type,
    g : set_Ho137910533iple_a).
thf(sy_v_P, type,
    p : a > state > $o).
thf(sy_v_Q, type,
    q : a > state > $o).
thf(sy_v_pn, type,
    pn : pname).

% Relevant facts (107)
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_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_2_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_3_insert__subset, axiom,
    ((![X : hoare_1678595023iple_a, A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ (insert1477804543iple_a @ X @ A) @ B) = (((member1332298086iple_a @ X @ B)) & ((ord_le1221261669iple_a @ A @ B))))))). % insert_subset
thf(fact_4_com_Oinject_I6_J, axiom,
    ((![X7 : pname, Y7 : pname]: (((body @ X7) = (body @ Y7)) = (X7 = Y7))))). % com.inject(6)
thf(fact_5_insertCI, axiom,
    ((![A2 : hoare_1678595023iple_a, B : set_Ho137910533iple_a, B2 : hoare_1678595023iple_a]: (((~ ((member1332298086iple_a @ A2 @ B))) => (A2 = B2)) => (member1332298086iple_a @ A2 @ (insert1477804543iple_a @ B2 @ B)))))). % insertCI
thf(fact_6_insert__iff, axiom,
    ((![A2 : hoare_1678595023iple_a, B2 : hoare_1678595023iple_a, A : set_Ho137910533iple_a]: ((member1332298086iple_a @ A2 @ (insert1477804543iple_a @ B2 @ A)) = (((A2 = B2)) | ((member1332298086iple_a @ A2 @ A))))))). % insert_iff
thf(fact_7_insert__absorb2, axiom,
    ((![X : hoare_1678595023iple_a, A : set_Ho137910533iple_a]: ((insert1477804543iple_a @ X @ (insert1477804543iple_a @ X @ A)) = (insert1477804543iple_a @ X @ A))))). % insert_absorb2
thf(fact_8_subsetI, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((![X4 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X4 @ A) => (member1332298086iple_a @ X4 @ B))) => (ord_le1221261669iple_a @ A @ B))))). % subsetI
thf(fact_9_subset__antisym, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ B) => ((ord_le1221261669iple_a @ B @ A) => (A = B)))))). % subset_antisym
thf(fact_10_order__refl, axiom,
    ((![X : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ X @ X)))). % order_refl
thf(fact_11_insert__mono, axiom,
    ((![C : set_Ho137910533iple_a, D : set_Ho137910533iple_a, A2 : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ C @ D) => (ord_le1221261669iple_a @ (insert1477804543iple_a @ A2 @ C) @ (insert1477804543iple_a @ A2 @ D)))))). % insert_mono
thf(fact_12_subset__insert, axiom,
    ((![X : hoare_1678595023iple_a, A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ X @ A))) => ((ord_le1221261669iple_a @ A @ (insert1477804543iple_a @ X @ B)) = (ord_le1221261669iple_a @ A @ B)))))). % subset_insert
thf(fact_13_subset__insertI, axiom,
    ((![B : set_Ho137910533iple_a, A2 : hoare_1678595023iple_a]: (ord_le1221261669iple_a @ B @ (insert1477804543iple_a @ A2 @ B))))). % subset_insertI
thf(fact_14_subset__insertI2, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, B2 : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A @ B) => (ord_le1221261669iple_a @ A @ (insert1477804543iple_a @ B2 @ B)))))). % subset_insertI2
thf(fact_15_dual__order_Oantisym, axiom,
    ((![B2 : set_Ho137910533iple_a, A2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ B2 @ A2) => ((ord_le1221261669iple_a @ A2 @ B2) => (A2 = B2)))))). % dual_order.antisym
thf(fact_16_dual__order_Oeq__iff, axiom,
    (((^[Y4 : set_Ho137910533iple_a]: (^[Z : set_Ho137910533iple_a]: (Y4 = Z))) = (^[A3 : set_Ho137910533iple_a]: (^[B3 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ B3 @ A3)) & ((ord_le1221261669iple_a @ A3 @ B3)))))))). % dual_order.eq_iff
thf(fact_17_dual__order_Otrans, axiom,
    ((![B2 : set_Ho137910533iple_a, A2 : set_Ho137910533iple_a, C2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ B2 @ A2) => ((ord_le1221261669iple_a @ C2 @ B2) => (ord_le1221261669iple_a @ C2 @ A2)))))). % dual_order.trans
thf(fact_18_dual__order_Orefl, axiom,
    ((![A2 : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ A2 @ A2)))). % dual_order.refl
thf(fact_19_order__trans, axiom,
    ((![X : set_Ho137910533iple_a, Y : set_Ho137910533iple_a, Z2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X @ Y) => ((ord_le1221261669iple_a @ Y @ Z2) => (ord_le1221261669iple_a @ X @ Z2)))))). % order_trans
thf(fact_20_order__class_Oorder_Oantisym, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ B2) => ((ord_le1221261669iple_a @ B2 @ A2) => (A2 = B2)))))). % order_class.order.antisym
thf(fact_21_ord__le__eq__trans, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, C2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ B2) => ((B2 = C2) => (ord_le1221261669iple_a @ A2 @ C2)))))). % ord_le_eq_trans
thf(fact_22_ord__eq__le__trans, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, C2 : set_Ho137910533iple_a]: ((A2 = B2) => ((ord_le1221261669iple_a @ B2 @ C2) => (ord_le1221261669iple_a @ A2 @ C2)))))). % ord_eq_le_trans
thf(fact_23_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y4 : set_Ho137910533iple_a]: (^[Z : set_Ho137910533iple_a]: (Y4 = Z))) = (^[A3 : set_Ho137910533iple_a]: (^[B3 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ A3 @ B3)) & ((ord_le1221261669iple_a @ B3 @ A3)))))))). % order_class.order.eq_iff
thf(fact_24_antisym__conv, axiom,
    ((![Y : set_Ho137910533iple_a, X : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ Y @ X) => ((ord_le1221261669iple_a @ X @ Y) = (X = Y)))))). % antisym_conv
thf(fact_25_order_Otrans, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, C2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ B2) => ((ord_le1221261669iple_a @ B2 @ C2) => (ord_le1221261669iple_a @ A2 @ C2)))))). % order.trans
thf(fact_26_eq__refl, axiom,
    ((![X : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((X = Y) => (ord_le1221261669iple_a @ X @ Y))))). % eq_refl
thf(fact_27_antisym, axiom,
    ((![X : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X @ Y) => ((ord_le1221261669iple_a @ Y @ X) => (X = Y)))))). % antisym
thf(fact_28_eq__iff, axiom,
    (((^[Y4 : set_Ho137910533iple_a]: (^[Z : set_Ho137910533iple_a]: (Y4 = Z))) = (^[X5 : set_Ho137910533iple_a]: (^[Y5 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ X5 @ Y5)) & ((ord_le1221261669iple_a @ Y5 @ X5)))))))). % eq_iff
thf(fact_29_ord__le__eq__subst, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, F : set_Ho137910533iple_a > set_Ho137910533iple_a, C2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ B2) => (((F @ B2) = C2) => ((![X4 : set_Ho137910533iple_a, Y6 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X4 @ Y6) => (ord_le1221261669iple_a @ (F @ X4) @ (F @ Y6)))) => (ord_le1221261669iple_a @ (F @ A2) @ C2))))))). % ord_le_eq_subst
thf(fact_30_ord__eq__le__subst, axiom,
    ((![A2 : set_Ho137910533iple_a, F : set_Ho137910533iple_a > set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, C2 : set_Ho137910533iple_a]: ((A2 = (F @ B2)) => ((ord_le1221261669iple_a @ B2 @ C2) => ((![X4 : set_Ho137910533iple_a, Y6 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X4 @ Y6) => (ord_le1221261669iple_a @ (F @ X4) @ (F @ Y6)))) => (ord_le1221261669iple_a @ A2 @ (F @ C2)))))))). % ord_eq_le_subst
thf(fact_31_order__subst2, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, F : set_Ho137910533iple_a > set_Ho137910533iple_a, C2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ B2) => ((ord_le1221261669iple_a @ (F @ B2) @ C2) => ((![X4 : set_Ho137910533iple_a, Y6 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X4 @ Y6) => (ord_le1221261669iple_a @ (F @ X4) @ (F @ Y6)))) => (ord_le1221261669iple_a @ (F @ A2) @ C2))))))). % order_subst2
thf(fact_32_order__subst1, axiom,
    ((![A2 : set_Ho137910533iple_a, F : set_Ho137910533iple_a > set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, C2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ (F @ B2)) => ((ord_le1221261669iple_a @ B2 @ C2) => ((![X4 : set_Ho137910533iple_a, Y6 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X4 @ Y6) => (ord_le1221261669iple_a @ (F @ X4) @ (F @ Y6)))) => (ord_le1221261669iple_a @ A2 @ (F @ C2)))))))). % order_subst1
thf(fact_33_Collect__mono__iff, axiom,
    ((![P : hoare_1678595023iple_a > $o, Q : hoare_1678595023iple_a > $o]: ((ord_le1221261669iple_a @ (collec1600235172iple_a @ P) @ (collec1600235172iple_a @ Q)) = (![X5 : hoare_1678595023iple_a]: (((P @ X5)) => ((Q @ X5)))))))). % Collect_mono_iff
thf(fact_34_set__eq__subset, axiom,
    (((^[Y4 : set_Ho137910533iple_a]: (^[Z : set_Ho137910533iple_a]: (Y4 = Z))) = (^[A4 : set_Ho137910533iple_a]: (^[B4 : set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ A4 @ B4)) & ((ord_le1221261669iple_a @ B4 @ A4)))))))). % set_eq_subset
thf(fact_35_subset__trans, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ B) => ((ord_le1221261669iple_a @ B @ C) => (ord_le1221261669iple_a @ A @ C)))))). % subset_trans
thf(fact_36_Collect__mono, axiom,
    ((![P : hoare_1678595023iple_a > $o, Q : hoare_1678595023iple_a > $o]: ((![X4 : hoare_1678595023iple_a]: ((P @ X4) => (Q @ X4))) => (ord_le1221261669iple_a @ (collec1600235172iple_a @ P) @ (collec1600235172iple_a @ Q)))))). % Collect_mono
thf(fact_37_mem__Collect__eq, axiom,
    ((![A2 : hoare_1678595023iple_a, P : hoare_1678595023iple_a > $o]: ((member1332298086iple_a @ A2 @ (collec1600235172iple_a @ P)) = (P @ A2))))). % mem_Collect_eq
thf(fact_38_Collect__mem__eq, axiom,
    ((![A : set_Ho137910533iple_a]: ((collec1600235172iple_a @ (^[X5 : hoare_1678595023iple_a]: (member1332298086iple_a @ X5 @ A))) = A)))). % Collect_mem_eq
thf(fact_39_subset__refl, axiom,
    ((![A : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ A @ A)))). % subset_refl
thf(fact_40_subset__iff, axiom,
    ((ord_le1221261669iple_a = (^[A4 : set_Ho137910533iple_a]: (^[B4 : set_Ho137910533iple_a]: (![T : hoare_1678595023iple_a]: (((member1332298086iple_a @ T @ A4)) => ((member1332298086iple_a @ T @ B4))))))))). % subset_iff
thf(fact_41_equalityD2, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((A = B) => (ord_le1221261669iple_a @ B @ A))))). % equalityD2
thf(fact_42_equalityD1, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((A = B) => (ord_le1221261669iple_a @ A @ B))))). % equalityD1
thf(fact_43_subset__eq, axiom,
    ((ord_le1221261669iple_a = (^[A4 : set_Ho137910533iple_a]: (^[B4 : set_Ho137910533iple_a]: (![X5 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X5 @ A4)) => ((member1332298086iple_a @ X5 @ B4))))))))). % subset_eq
thf(fact_44_equalityE, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((A = B) => (~ (((ord_le1221261669iple_a @ A @ B) => (~ ((ord_le1221261669iple_a @ B @ A)))))))))). % equalityE
thf(fact_45_subsetD, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, C2 : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A @ B) => ((member1332298086iple_a @ C2 @ A) => (member1332298086iple_a @ C2 @ B)))))). % subsetD
thf(fact_46_in__mono, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, X : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A @ B) => ((member1332298086iple_a @ X @ A) => (member1332298086iple_a @ X @ B)))))). % in_mono
thf(fact_47_mk__disjoint__insert, axiom,
    ((![A2 : hoare_1678595023iple_a, A : set_Ho137910533iple_a]: ((member1332298086iple_a @ A2 @ A) => (?[B5 : set_Ho137910533iple_a]: ((A = (insert1477804543iple_a @ A2 @ B5)) & (~ ((member1332298086iple_a @ A2 @ B5))))))))). % mk_disjoint_insert
thf(fact_48_insert__commute, axiom,
    ((![X : hoare_1678595023iple_a, Y : hoare_1678595023iple_a, A : set_Ho137910533iple_a]: ((insert1477804543iple_a @ X @ (insert1477804543iple_a @ Y @ A)) = (insert1477804543iple_a @ Y @ (insert1477804543iple_a @ X @ A)))))). % insert_commute
thf(fact_49_insert__eq__iff, axiom,
    ((![A2 : hoare_1678595023iple_a, A : set_Ho137910533iple_a, B2 : hoare_1678595023iple_a, B : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ A2 @ A))) => ((~ ((member1332298086iple_a @ B2 @ B))) => (((insert1477804543iple_a @ A2 @ A) = (insert1477804543iple_a @ B2 @ B)) = (((((A2 = B2)) => ((A = B)))) & ((((~ ((A2 = B2)))) => ((?[C3 : set_Ho137910533iple_a]: (((A = (insert1477804543iple_a @ B2 @ C3))) & ((((~ ((member1332298086iple_a @ B2 @ C3)))) & ((((B = (insert1477804543iple_a @ A2 @ C3))) & ((~ ((member1332298086iple_a @ A2 @ C3)))))))))))))))))))). % insert_eq_iff
thf(fact_50_insert__absorb, axiom,
    ((![A2 : hoare_1678595023iple_a, A : set_Ho137910533iple_a]: ((member1332298086iple_a @ A2 @ A) => ((insert1477804543iple_a @ A2 @ A) = A))))). % insert_absorb
thf(fact_51_insert__ident, axiom,
    ((![X : hoare_1678595023iple_a, A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ X @ A))) => ((~ ((member1332298086iple_a @ X @ B))) => (((insert1477804543iple_a @ X @ A) = (insert1477804543iple_a @ X @ B)) = (A = B))))))). % insert_ident
thf(fact_52_Set_Oset__insert, axiom,
    ((![X : hoare_1678595023iple_a, A : set_Ho137910533iple_a]: ((member1332298086iple_a @ X @ A) => (~ ((![B5 : set_Ho137910533iple_a]: ((A = (insert1477804543iple_a @ X @ B5)) => (member1332298086iple_a @ X @ B5))))))))). % Set.set_insert
thf(fact_53_insertI2, axiom,
    ((![A2 : hoare_1678595023iple_a, B : set_Ho137910533iple_a, B2 : hoare_1678595023iple_a]: ((member1332298086iple_a @ A2 @ B) => (member1332298086iple_a @ A2 @ (insert1477804543iple_a @ B2 @ B)))))). % insertI2
thf(fact_54_insertI1, axiom,
    ((![A2 : hoare_1678595023iple_a, B : set_Ho137910533iple_a]: (member1332298086iple_a @ A2 @ (insert1477804543iple_a @ A2 @ B))))). % insertI1
thf(fact_55_insertE, axiom,
    ((![A2 : hoare_1678595023iple_a, B2 : hoare_1678595023iple_a, A : set_Ho137910533iple_a]: ((member1332298086iple_a @ A2 @ (insert1477804543iple_a @ B2 @ A)) => ((~ ((A2 = B2))) => (member1332298086iple_a @ A2 @ A)))))). % insertE
thf(fact_56_insert__subsetI, axiom,
    ((![X : hoare_1678595023iple_a, A : set_Ho137910533iple_a, X6 : set_Ho137910533iple_a]: ((member1332298086iple_a @ X @ A) => ((ord_le1221261669iple_a @ X6 @ A) => (ord_le1221261669iple_a @ (insert1477804543iple_a @ X @ X6) @ A)))))). % insert_subsetI
thf(fact_57_Greatest__equality, axiom,
    ((![P : set_Ho137910533iple_a > $o, X : set_Ho137910533iple_a]: ((P @ X) => ((![Y6 : set_Ho137910533iple_a]: ((P @ Y6) => (ord_le1221261669iple_a @ Y6 @ X))) => ((order_929906668iple_a @ P) = X)))))). % Greatest_equality
thf(fact_58_GreatestI2__order, axiom,
    ((![P : set_Ho137910533iple_a > $o, X : set_Ho137910533iple_a, Q : set_Ho137910533iple_a > $o]: ((P @ X) => ((![Y6 : set_Ho137910533iple_a]: ((P @ Y6) => (ord_le1221261669iple_a @ Y6 @ X))) => ((![X4 : set_Ho137910533iple_a]: ((P @ X4) => ((![Y8 : set_Ho137910533iple_a]: ((P @ Y8) => (ord_le1221261669iple_a @ Y8 @ X4))) => (Q @ X4)))) => (Q @ (order_929906668iple_a @ P)))))))). % GreatestI2_order
thf(fact_59_le__rel__bool__arg__iff, axiom,
    ((ord_le1048771374iple_a = (^[X8 : $o > set_Ho137910533iple_a]: (^[Y9 : $o > set_Ho137910533iple_a]: (((ord_le1221261669iple_a @ (X8 @ $false) @ (Y9 @ $false))) & ((ord_le1221261669iple_a @ (X8 @ $true) @ (Y9 @ $true))))))))). % le_rel_bool_arg_iff
thf(fact_60_singleton__insert__inj__eq_H, axiom,
    ((![A2 : hoare_1678595023iple_a, A : set_Ho137910533iple_a, B2 : hoare_1678595023iple_a]: (((insert1477804543iple_a @ A2 @ A) = (insert1477804543iple_a @ B2 @ bot_bo1298296729iple_a)) = (((A2 = B2)) & ((ord_le1221261669iple_a @ A @ (insert1477804543iple_a @ B2 @ bot_bo1298296729iple_a)))))))). % singleton_insert_inj_eq'
thf(fact_61_singleton__insert__inj__eq, axiom,
    ((![B2 : hoare_1678595023iple_a, A2 : hoare_1678595023iple_a, A : set_Ho137910533iple_a]: (((insert1477804543iple_a @ B2 @ bot_bo1298296729iple_a) = (insert1477804543iple_a @ A2 @ A)) = (((A2 = B2)) & ((ord_le1221261669iple_a @ A @ (insert1477804543iple_a @ B2 @ bot_bo1298296729iple_a)))))))). % singleton_insert_inj_eq
thf(fact_62_subset__singleton__iff, axiom,
    ((![X6 : set_Ho137910533iple_a, A2 : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ X6 @ (insert1477804543iple_a @ A2 @ bot_bo1298296729iple_a)) = (((X6 = bot_bo1298296729iple_a)) | ((X6 = (insert1477804543iple_a @ A2 @ bot_bo1298296729iple_a)))))))). % subset_singleton_iff
thf(fact_63_empty__iff, axiom,
    ((![C2 : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ C2 @ bot_bo1298296729iple_a)))))). % empty_iff
thf(fact_64_all__not__in__conv, axiom,
    ((![A : set_Ho137910533iple_a]: ((![X5 : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ X5 @ A)))) = (A = bot_bo1298296729iple_a))))). % all_not_in_conv
thf(fact_65_subset__empty, axiom,
    ((![A : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ bot_bo1298296729iple_a) = (A = bot_bo1298296729iple_a))))). % subset_empty
thf(fact_66_empty__subsetI, axiom,
    ((![A : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ bot_bo1298296729iple_a @ A)))). % empty_subsetI
thf(fact_67_singletonI, axiom,
    ((![A2 : hoare_1678595023iple_a]: (member1332298086iple_a @ A2 @ (insert1477804543iple_a @ A2 @ bot_bo1298296729iple_a))))). % singletonI
thf(fact_68_emptyE, axiom,
    ((![A2 : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ A2 @ bot_bo1298296729iple_a)))))). % emptyE
thf(fact_69_equals0D, axiom,
    ((![A : set_Ho137910533iple_a, A2 : hoare_1678595023iple_a]: ((A = bot_bo1298296729iple_a) => (~ ((member1332298086iple_a @ A2 @ A))))))). % equals0D
thf(fact_70_equals0I, axiom,
    ((![A : set_Ho137910533iple_a]: ((![Y6 : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ Y6 @ A)))) => (A = bot_bo1298296729iple_a))))). % equals0I
thf(fact_71_ex__in__conv, axiom,
    ((![A : set_Ho137910533iple_a]: ((?[X5 : hoare_1678595023iple_a]: (member1332298086iple_a @ X5 @ A)) = (~ ((A = bot_bo1298296729iple_a))))))). % ex_in_conv
thf(fact_72_subset__emptyI, axiom,
    ((![A : set_Ho137910533iple_a]: ((![X4 : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ X4 @ A)))) => (ord_le1221261669iple_a @ A @ bot_bo1298296729iple_a))))). % subset_emptyI
thf(fact_73_bot_Oextremum, axiom,
    ((![A2 : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ bot_bo1298296729iple_a @ A2)))). % bot.extremum
thf(fact_74_bot_Oextremum__unique, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ bot_bo1298296729iple_a) = (A2 = bot_bo1298296729iple_a))))). % bot.extremum_unique
thf(fact_75_bot_Oextremum__uniqueI, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ bot_bo1298296729iple_a) => (A2 = bot_bo1298296729iple_a))))). % bot.extremum_uniqueI
thf(fact_76_singletonD, axiom,
    ((![B2 : hoare_1678595023iple_a, A2 : hoare_1678595023iple_a]: ((member1332298086iple_a @ B2 @ (insert1477804543iple_a @ A2 @ bot_bo1298296729iple_a)) => (B2 = A2))))). % singletonD
thf(fact_77_singleton__iff, axiom,
    ((![B2 : hoare_1678595023iple_a, A2 : hoare_1678595023iple_a]: ((member1332298086iple_a @ B2 @ (insert1477804543iple_a @ A2 @ bot_bo1298296729iple_a)) = (B2 = A2))))). % singleton_iff
thf(fact_78_doubleton__eq__iff, axiom,
    ((![A2 : hoare_1678595023iple_a, B2 : hoare_1678595023iple_a, C2 : hoare_1678595023iple_a, D2 : hoare_1678595023iple_a]: (((insert1477804543iple_a @ A2 @ (insert1477804543iple_a @ B2 @ bot_bo1298296729iple_a)) = (insert1477804543iple_a @ C2 @ (insert1477804543iple_a @ D2 @ bot_bo1298296729iple_a))) = (((((A2 = C2)) & ((B2 = D2)))) | ((((A2 = D2)) & ((B2 = C2))))))))). % doubleton_eq_iff
thf(fact_79_insert__not__empty, axiom,
    ((![A2 : hoare_1678595023iple_a, A : set_Ho137910533iple_a]: (~ (((insert1477804543iple_a @ A2 @ A) = bot_bo1298296729iple_a)))))). % insert_not_empty
thf(fact_80_singleton__inject, axiom,
    ((![A2 : hoare_1678595023iple_a, B2 : hoare_1678595023iple_a]: (((insert1477804543iple_a @ A2 @ bot_bo1298296729iple_a) = (insert1477804543iple_a @ B2 @ bot_bo1298296729iple_a)) => (A2 = B2))))). % singleton_inject
thf(fact_81_subset__singletonD, axiom,
    ((![A : set_Ho137910533iple_a, X : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A @ (insert1477804543iple_a @ X @ bot_bo1298296729iple_a)) => ((A = bot_bo1298296729iple_a) | (A = (insert1477804543iple_a @ X @ bot_bo1298296729iple_a))))))). % subset_singletonD
thf(fact_82_the__elem__eq, axiom,
    ((![X : hoare_1678595023iple_a]: ((the_el434698138iple_a @ (insert1477804543iple_a @ X @ bot_bo1298296729iple_a)) = X)))). % the_elem_eq
thf(fact_83_is__singletonI, axiom,
    ((![X : hoare_1678595023iple_a]: (is_sin1784037339iple_a @ (insert1477804543iple_a @ X @ bot_bo1298296729iple_a))))). % is_singletonI
thf(fact_84_is__singletonE, axiom,
    ((![A : set_Ho137910533iple_a]: ((is_sin1784037339iple_a @ A) => (~ ((![X4 : hoare_1678595023iple_a]: (~ ((A = (insert1477804543iple_a @ X4 @ bot_bo1298296729iple_a))))))))))). % is_singletonE
thf(fact_85_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_86_is__singletonI_H, axiom,
    ((![A : set_Ho137910533iple_a]: ((~ ((A = bot_bo1298296729iple_a))) => ((![X4 : hoare_1678595023iple_a, Y6 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X4 @ A) => ((member1332298086iple_a @ Y6 @ A) => (X4 = Y6)))) => (is_sin1784037339iple_a @ A)))))). % is_singletonI'
thf(fact_87_is__singleton__def, axiom,
    ((is_sin1784037339iple_a = (^[A4 : set_Ho137910533iple_a]: (?[X5 : hoare_1678595023iple_a]: (A4 = (insert1477804543iple_a @ X5 @ bot_bo1298296729iple_a))))))). % is_singleton_def
thf(fact_88_subset__Compl__singleton, axiom,
    ((![A : set_Ho137910533iple_a, B2 : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A @ (uminus922456654iple_a @ (insert1477804543iple_a @ B2 @ bot_bo1298296729iple_a))) = (~ ((member1332298086iple_a @ B2 @ A))))))). % subset_Compl_singleton
thf(fact_89_bot__empty__eq, axiom,
    ((bot_bo431311916le_a_o = (^[X5 : hoare_1678595023iple_a]: (member1332298086iple_a @ X5 @ bot_bo1298296729iple_a))))). % bot_empty_eq
thf(fact_90_ComplI, axiom,
    ((![C2 : hoare_1678595023iple_a, A : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ C2 @ A))) => (member1332298086iple_a @ C2 @ (uminus922456654iple_a @ A)))))). % ComplI
thf(fact_91_Compl__iff, axiom,
    ((![C2 : hoare_1678595023iple_a, A : set_Ho137910533iple_a]: ((member1332298086iple_a @ C2 @ (uminus922456654iple_a @ A)) = (~ ((member1332298086iple_a @ C2 @ A))))))). % Compl_iff
thf(fact_92_Compl__subset__Compl__iff, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ (uminus922456654iple_a @ A) @ (uminus922456654iple_a @ B)) = (ord_le1221261669iple_a @ B @ A))))). % Compl_subset_Compl_iff
thf(fact_93_Compl__anti__mono, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ B) => (ord_le1221261669iple_a @ (uminus922456654iple_a @ B) @ (uminus922456654iple_a @ A)))))). % Compl_anti_mono
thf(fact_94_ComplD, axiom,
    ((![C2 : hoare_1678595023iple_a, A : set_Ho137910533iple_a]: ((member1332298086iple_a @ C2 @ (uminus922456654iple_a @ A)) => (~ ((member1332298086iple_a @ C2 @ A))))))). % ComplD
thf(fact_95_subset__Compl__self__eq, axiom,
    ((![A : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ (uminus922456654iple_a @ A)) = (A = bot_bo1298296729iple_a))))). % subset_Compl_self_eq
thf(fact_96_compl__le__compl__iff, axiom,
    ((![X : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ (uminus922456654iple_a @ X) @ (uminus922456654iple_a @ Y)) = (ord_le1221261669iple_a @ Y @ X))))). % compl_le_compl_iff
thf(fact_97_compl__mono, axiom,
    ((![X : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ X @ Y) => (ord_le1221261669iple_a @ (uminus922456654iple_a @ Y) @ (uminus922456654iple_a @ X)))))). % compl_mono
thf(fact_98_compl__le__swap1, axiom,
    ((![Y : set_Ho137910533iple_a, X : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ Y @ (uminus922456654iple_a @ X)) => (ord_le1221261669iple_a @ X @ (uminus922456654iple_a @ Y)))))). % compl_le_swap1
thf(fact_99_compl__le__swap2, axiom,
    ((![Y : set_Ho137910533iple_a, X : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ (uminus922456654iple_a @ Y) @ X) => (ord_le1221261669iple_a @ (uminus922456654iple_a @ X) @ Y))))). % compl_le_swap2
thf(fact_100_Compl__insert, axiom,
    ((![X : hoare_1678595023iple_a, A : set_Ho137910533iple_a]: ((uminus922456654iple_a @ (insert1477804543iple_a @ X @ A)) = (minus_1852999390iple_a @ (uminus922456654iple_a @ A) @ (insert1477804543iple_a @ X @ bot_bo1298296729iple_a)))))). % Compl_insert
thf(fact_101_DiffI, axiom,
    ((![C2 : hoare_1678595023iple_a, A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((member1332298086iple_a @ C2 @ A) => ((~ ((member1332298086iple_a @ C2 @ B))) => (member1332298086iple_a @ C2 @ (minus_1852999390iple_a @ A @ B))))))). % DiffI
thf(fact_102_Diff__iff, axiom,
    ((![C2 : hoare_1678595023iple_a, A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((member1332298086iple_a @ C2 @ (minus_1852999390iple_a @ A @ B)) = (((member1332298086iple_a @ C2 @ A)) & ((~ ((member1332298086iple_a @ C2 @ B))))))))). % Diff_iff
thf(fact_103_insert__Diff1, axiom,
    ((![X : hoare_1678595023iple_a, B : set_Ho137910533iple_a, A : set_Ho137910533iple_a]: ((member1332298086iple_a @ X @ B) => ((minus_1852999390iple_a @ (insert1477804543iple_a @ X @ A) @ B) = (minus_1852999390iple_a @ A @ B)))))). % insert_Diff1
thf(fact_104_Diff__insert0, axiom,
    ((![X : hoare_1678595023iple_a, A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ X @ A))) => ((minus_1852999390iple_a @ A @ (insert1477804543iple_a @ X @ B)) = (minus_1852999390iple_a @ A @ B)))))). % Diff_insert0
thf(fact_105_Diff__eq__empty__iff, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: (((minus_1852999390iple_a @ A @ B) = bot_bo1298296729iple_a) = (ord_le1221261669iple_a @ A @ B))))). % Diff_eq_empty_iff
thf(fact_106_insert__Diff__single, axiom,
    ((![A2 : hoare_1678595023iple_a, A : set_Ho137910533iple_a]: ((insert1477804543iple_a @ A2 @ (minus_1852999390iple_a @ A @ (insert1477804543iple_a @ A2 @ bot_bo1298296729iple_a))) = (insert1477804543iple_a @ A2 @ A))))). % insert_Diff_single

% Conjectures (1)
thf(conj_0, conjecture,
    ((ord_le1221261669iple_a @ g @ (insert1477804543iple_a @ (hoare_719046530iple_a @ p @ (body @ pn) @ q) @ g)))).
