% 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_251__3251976_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:13:02.799

% 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 (25)
thf(sy_c_Com_Ocom_OCond, type,
    cond : (state > $o) > com > 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_Ocom_OWhile, type,
    while : (state > $o) > com > com).
thf(sy_c_Fun_Ocomp_001_Eo_001_Eo_001_Eo, type,
    comp_o_o_o : ($o > $o) > ($o > $o) > $o > $o).
thf(sy_c_Fun_Ocomp_001_Eo_001_Eo_001t__Com__Ostate, type,
    comp_o_o_state : ($o > $o) > (state > $o) > state > $o).
thf(sy_c_Fun_Ocomp_001t__Com__Ostate_001_Eo_001t__Com__Ostate, type,
    comp_state_o_state : (state > $o) > (state > state) > 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_Ohoare__valids_001tf__a, type,
    hoare_1775499016lids_a : set_Ho137910533iple_a > set_Ho137910533iple_a > $o).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Opeek__and_001tf__a, type,
    hoare_1795417776_and_a : (a > state > $o) > (state > $o) > a > state > $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_Oinsert_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    insert1477804543iple_a : hoare_1678595023iple_a > set_Ho137910533iple_a > set_Ho137910533iple_a).
thf(sy_c_Set_Ois__empty_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    is_emp901906557iple_a : set_Ho137910533iple_a > $o).
thf(sy_c_Set_Ois__singleton_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    is_sin1784037339iple_a : set_Ho137910533iple_a > $o).
thf(sy_c_Set_Opairwise_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    pairwi531237284iple_a : (hoare_1678595023iple_a > hoare_1678595023iple_a > $o) > set_Ho137910533iple_a > $o).
thf(sy_c_Set_Othe__elem_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    the_el434698138iple_a : set_Ho137910533iple_a > hoare_1678595023iple_a).
thf(sy_c_member_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    member1332298086iple_a : hoare_1678595023iple_a > set_Ho137910533iple_a > $o).
thf(sy_v_G, type,
    g : set_Ho137910533iple_a).
thf(sy_v_P, type,
    p : a > state > $o).
thf(sy_v_b, type,
    b : state > $o).
thf(sy_v_c, type,
    c : com).

% Relevant facts (96)
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_singletonI, axiom,
    ((![A : hoare_1678595023iple_a]: (member1332298086iple_a @ A @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a))))). % singletonI
thf(fact_4_com_Oinject_I5_J, axiom,
    ((![X61 : state > $o, X62 : com, Y61 : state > $o, Y62 : com]: (((while @ X61 @ X62) = (while @ Y61 @ Y62)) = (((X61 = Y61)) & ((X62 = Y62))))))). % com.inject(5)
thf(fact_5_Loop, axiom,
    ((![G : set_Ho137910533iple_a, P : a > state > $o, B : state > $o, C : com]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ (hoare_1795417776_and_a @ P @ B) @ C @ P) @ bot_bo1298296729iple_a)) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ (while @ B @ C) @ (hoare_1795417776_and_a @ P @ (comp_o_o_state @ (~) @ B))) @ bot_bo1298296729iple_a)))))). % Loop
thf(fact_6_comp__apply, axiom,
    ((comp_o_o_state = (^[F : $o > $o]: (^[G2 : state > $o]: (^[X : state]: (F @ (G2 @ X)))))))). % comp_apply
thf(fact_7_insertCI, axiom,
    ((![A : hoare_1678595023iple_a, B2 : set_Ho137910533iple_a, B : hoare_1678595023iple_a]: (((~ ((member1332298086iple_a @ A @ B2))) => (A = B)) => (member1332298086iple_a @ A @ (insert1477804543iple_a @ B @ B2)))))). % insertCI
thf(fact_8_insert__iff, axiom,
    ((![A : hoare_1678595023iple_a, B : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ A @ (insert1477804543iple_a @ B @ A2)) = (((A = B)) | ((member1332298086iple_a @ A @ A2))))))). % insert_iff
thf(fact_9_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_10_empty__iff, axiom,
    ((![C : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ C @ bot_bo1298296729iple_a)))))). % empty_iff
thf(fact_11_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_12_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_13_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_14_cut, axiom,
    ((![G3 : set_Ho137910533iple_a, Ts : set_Ho137910533iple_a, G : set_Ho137910533iple_a]: ((hoare_129598474rivs_a @ G3 @ Ts) => ((hoare_129598474rivs_a @ G @ G3) => (hoare_129598474rivs_a @ G @ Ts)))))). % cut
thf(fact_15_bot__set__def, axiom,
    ((bot_bo1298296729iple_a = (collec1600235172iple_a @ bot_bo431311916le_a_o)))). % bot_set_def
thf(fact_16_empty, axiom,
    ((![G : set_Ho137910533iple_a]: (hoare_129598474rivs_a @ G @ bot_bo1298296729iple_a)))). % empty
thf(fact_17_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_18_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_19_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_20_equals0I, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((![Y4 : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ Y4 @ A2)))) => (A2 = bot_bo1298296729iple_a))))). % equals0I
thf(fact_21_equals0D, axiom,
    ((![A2 : set_Ho137910533iple_a, A : hoare_1678595023iple_a]: ((A2 = bot_bo1298296729iple_a) => (~ ((member1332298086iple_a @ A @ A2))))))). % equals0D
thf(fact_22_emptyE, axiom,
    ((![A : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ A @ bot_bo1298296729iple_a)))))). % emptyE
thf(fact_23_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_24_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_25_insert__eq__iff, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B : hoare_1678595023iple_a, B2 : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ A @ A2))) => ((~ ((member1332298086iple_a @ B @ B2))) => (((insert1477804543iple_a @ A @ A2) = (insert1477804543iple_a @ B @ B2)) = (((((A = B)) => ((A2 = B2)))) & ((((~ ((A = B)))) => ((?[C2 : set_Ho137910533iple_a]: (((A2 = (insert1477804543iple_a @ B @ C2))) & ((((~ ((member1332298086iple_a @ B @ C2)))) & ((((B2 = (insert1477804543iple_a @ A @ C2))) & ((~ ((member1332298086iple_a @ A @ C2)))))))))))))))))))). % insert_eq_iff
thf(fact_26_insert__absorb, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ A @ A2) => ((insert1477804543iple_a @ A @ A2) = A2))))). % insert_absorb
thf(fact_27_insert__ident, axiom,
    ((![X4 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ X4 @ A2))) => ((~ ((member1332298086iple_a @ X4 @ B2))) => (((insert1477804543iple_a @ X4 @ A2) = (insert1477804543iple_a @ X4 @ B2)) = (A2 = B2))))))). % insert_ident
thf(fact_28_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_29_insertI2, axiom,
    ((![A : hoare_1678595023iple_a, B2 : set_Ho137910533iple_a, B : hoare_1678595023iple_a]: ((member1332298086iple_a @ A @ B2) => (member1332298086iple_a @ A @ (insert1477804543iple_a @ B @ B2)))))). % insertI2
thf(fact_30_insertI1, axiom,
    ((![A : hoare_1678595023iple_a, B2 : set_Ho137910533iple_a]: (member1332298086iple_a @ A @ (insert1477804543iple_a @ A @ B2))))). % insertI1
thf(fact_31_insertE, axiom,
    ((![A : hoare_1678595023iple_a, B : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ A @ (insert1477804543iple_a @ B @ A2)) => ((~ ((A = B))) => (member1332298086iple_a @ A @ A2)))))). % insertE
thf(fact_32_comp__eq__dest__lhs, axiom,
    ((![A : $o > $o, B : state > $o, C : state > $o, V : state]: (((comp_o_o_state @ A @ B) = C) => ((A @ (B @ V)) = (C @ V)))))). % comp_eq_dest_lhs
thf(fact_33_comp__eq__elim, axiom,
    ((![A : $o > $o, B : state > $o, C : $o > $o, D : state > $o]: (((comp_o_o_state @ A @ B) = (comp_o_o_state @ C @ D)) => (![V2 : state]: ((A @ (B @ V2)) = (C @ (D @ V2)))))))). % comp_eq_elim
thf(fact_34_comp__eq__dest, axiom,
    ((![A : $o > $o, B : state > $o, C : $o > $o, D : state > $o, V : state]: (((comp_o_o_state @ A @ B) = (comp_o_o_state @ C @ D)) => ((A @ (B @ V)) = (C @ (D @ V))))))). % comp_eq_dest
thf(fact_35_comp__assoc, axiom,
    ((![F2 : $o > $o, G4 : state > $o, H : state > state]: ((comp_state_o_state @ (comp_o_o_state @ F2 @ G4) @ H) = (comp_o_o_state @ F2 @ (comp_state_o_state @ G4 @ H)))))). % comp_assoc
thf(fact_36_comp__assoc, axiom,
    ((![F2 : $o > $o, G4 : $o > $o, H : state > $o]: ((comp_o_o_state @ (comp_o_o_o @ F2 @ G4) @ H) = (comp_o_o_state @ F2 @ (comp_o_o_state @ G4 @ H)))))). % comp_assoc
thf(fact_37_comp__def, axiom,
    ((comp_o_o_state = (^[F : $o > $o]: (^[G2 : state > $o]: (^[X : state]: (F @ (G2 @ X)))))))). % comp_def
thf(fact_38_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_39_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_40_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_41_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_42_singleton__inject, axiom,
    ((![A : hoare_1678595023iple_a, B : hoare_1678595023iple_a]: (((insert1477804543iple_a @ A @ bot_bo1298296729iple_a) = (insert1477804543iple_a @ B @ bot_bo1298296729iple_a)) => (A = B))))). % singleton_inject
thf(fact_43_insert__not__empty, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: (~ (((insert1477804543iple_a @ A @ A2) = bot_bo1298296729iple_a)))))). % insert_not_empty
thf(fact_44_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_45_Collect__mem__eq, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: (member1332298086iple_a @ X @ A2))) = A2)))). % Collect_mem_eq
thf(fact_46_doubleton__eq__iff, axiom,
    ((![A : hoare_1678595023iple_a, B : hoare_1678595023iple_a, C : hoare_1678595023iple_a, D : hoare_1678595023iple_a]: (((insert1477804543iple_a @ A @ (insert1477804543iple_a @ B @ bot_bo1298296729iple_a)) = (insert1477804543iple_a @ C @ (insert1477804543iple_a @ D @ bot_bo1298296729iple_a))) = (((((A = C)) & ((B = D)))) | ((((A = D)) & ((B = C))))))))). % doubleton_eq_iff
thf(fact_47_singleton__iff, axiom,
    ((![B : hoare_1678595023iple_a, A : hoare_1678595023iple_a]: ((member1332298086iple_a @ B @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a)) = (B = A))))). % singleton_iff
thf(fact_48_singletonD, axiom,
    ((![B : hoare_1678595023iple_a, A : hoare_1678595023iple_a]: ((member1332298086iple_a @ B @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a)) => (B = A))))). % singletonD
thf(fact_49_hoare__derivs_OIf, axiom,
    ((![G : set_Ho137910533iple_a, P : a > state > $o, B : state > $o, C : com, Q : a > state > $o, D : com]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ (hoare_1795417776_and_a @ P @ B) @ C @ Q) @ bot_bo1298296729iple_a)) => ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ (hoare_1795417776_and_a @ P @ (comp_o_o_state @ (~) @ B)) @ D @ Q) @ bot_bo1298296729iple_a)) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ (cond @ B @ C @ D) @ Q) @ bot_bo1298296729iple_a))))))). % hoare_derivs.If
thf(fact_50_the__elem__eq, axiom,
    ((![X4 : hoare_1678595023iple_a]: ((the_el434698138iple_a @ (insert1477804543iple_a @ X4 @ bot_bo1298296729iple_a)) = X4)))). % the_elem_eq
thf(fact_51_is__singletonI, axiom,
    ((![X4 : hoare_1678595023iple_a]: (is_sin1784037339iple_a @ (insert1477804543iple_a @ X4 @ bot_bo1298296729iple_a))))). % is_singletonI
thf(fact_52_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_53_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_54_Set_Ois__empty__def, axiom,
    ((is_emp901906557iple_a = (^[A3 : set_Ho137910533iple_a]: (A3 = bot_bo1298296729iple_a))))). % Set.is_empty_def
thf(fact_55_is__singleton__def, axiom,
    ((is_sin1784037339iple_a = (^[A3 : set_Ho137910533iple_a]: (?[X : hoare_1678595023iple_a]: (A3 = (insert1477804543iple_a @ X @ bot_bo1298296729iple_a))))))). % is_singleton_def
thf(fact_56_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_57_rewriteR__comp__comp2, axiom,
    ((![G4 : state > $o, H : state > state, R1 : $o > $o, R2 : state > $o, F2 : $o > $o, L : $o > $o]: (((comp_state_o_state @ G4 @ H) = (comp_o_o_state @ R1 @ R2)) => (((comp_o_o_o @ F2 @ R1) = L) => ((comp_state_o_state @ (comp_o_o_state @ F2 @ G4) @ H) = (comp_o_o_state @ L @ R2))))))). % rewriteR_comp_comp2
thf(fact_58_rewriteR__comp__comp2, axiom,
    ((![G4 : $o > $o, H : state > $o, R1 : state > $o, R2 : state > state, F2 : $o > $o, L : state > $o]: (((comp_o_o_state @ G4 @ H) = (comp_state_o_state @ R1 @ R2)) => (((comp_o_o_state @ F2 @ R1) = L) => ((comp_o_o_state @ (comp_o_o_o @ F2 @ G4) @ H) = (comp_state_o_state @ L @ R2))))))). % rewriteR_comp_comp2
thf(fact_59_rewriteR__comp__comp2, axiom,
    ((![G4 : $o > $o, H : state > $o, R1 : $o > $o, R2 : state > $o, F2 : $o > $o, L : $o > $o]: (((comp_o_o_state @ G4 @ H) = (comp_o_o_state @ R1 @ R2)) => (((comp_o_o_o @ F2 @ R1) = L) => ((comp_o_o_state @ (comp_o_o_o @ F2 @ G4) @ H) = (comp_o_o_state @ L @ R2))))))). % rewriteR_comp_comp2
thf(fact_60_com_Oinject_I4_J, axiom,
    ((![X51 : state > $o, X52 : com, X53 : com, Y51 : state > $o, Y52 : com, Y53 : com]: (((cond @ X51 @ X52 @ X53) = (cond @ Y51 @ Y52 @ Y53)) = (((X51 = Y51)) & ((((X52 = Y52)) & ((X53 = Y53))))))))). % com.inject(4)
thf(fact_61_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_62_com_Odistinct_I37_J, axiom,
    ((![X41 : com, X42 : com, X51 : state > $o, X52 : com, X53 : com]: (~ (((semi @ X41 @ X42) = (cond @ X51 @ X52 @ X53))))))). % com.distinct(37)
thf(fact_63_com_Odistinct_I7_J, axiom,
    ((![X51 : state > $o, X52 : com, X53 : com]: (~ ((skip = (cond @ X51 @ X52 @ X53))))))). % com.distinct(7)
thf(fact_64_com_Odistinct_I5_J, axiom,
    ((![X41 : com, X42 : com]: (~ ((skip = (semi @ X41 @ X42))))))). % com.distinct(5)
thf(fact_65_com_Odistinct_I45_J, axiom,
    ((![X51 : state > $o, X52 : com, X53 : com, X61 : state > $o, X62 : com]: (~ (((cond @ X51 @ X52 @ X53) = (while @ X61 @ X62))))))). % com.distinct(45)
thf(fact_66_com_Odistinct_I39_J, axiom,
    ((![X41 : com, X42 : com, X61 : state > $o, X62 : com]: (~ (((semi @ X41 @ X42) = (while @ X61 @ X62))))))). % com.distinct(39)
thf(fact_67_com_Odistinct_I9_J, axiom,
    ((![X61 : state > $o, X62 : com]: (~ ((skip = (while @ X61 @ X62))))))). % com.distinct(9)
thf(fact_68_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_69_is__singletonI_H, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((~ ((A2 = bot_bo1298296729iple_a))) => ((![X5 : hoare_1678595023iple_a, Y4 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X5 @ A2) => ((member1332298086iple_a @ Y4 @ A2) => (X5 = Y4)))) => (is_sin1784037339iple_a @ A2)))))). % is_singletonI'
thf(fact_70_rewriteL__comp__comp, axiom,
    ((![F2 : $o > $o, G4 : $o > $o, L : $o > $o, H : state > $o]: (((comp_o_o_o @ F2 @ G4) = L) => ((comp_o_o_state @ F2 @ (comp_o_o_state @ G4 @ H)) = (comp_o_o_state @ L @ H)))))). % rewriteL_comp_comp
thf(fact_71_rewriteL__comp__comp, axiom,
    ((![F2 : $o > $o, G4 : state > $o, L : state > $o, H : state > state]: (((comp_o_o_state @ F2 @ G4) = L) => ((comp_o_o_state @ F2 @ (comp_state_o_state @ G4 @ H)) = (comp_state_o_state @ L @ H)))))). % rewriteL_comp_comp
thf(fact_72_rewriteR__comp__comp, axiom,
    ((![G4 : state > $o, H : state > state, R3 : state > $o, F2 : $o > $o]: (((comp_state_o_state @ G4 @ H) = R3) => ((comp_state_o_state @ (comp_o_o_state @ F2 @ G4) @ H) = (comp_o_o_state @ F2 @ R3)))))). % rewriteR_comp_comp
thf(fact_73_rewriteR__comp__comp, axiom,
    ((![G4 : $o > $o, H : state > $o, R3 : state > $o, F2 : $o > $o]: (((comp_o_o_state @ G4 @ H) = R3) => ((comp_o_o_state @ (comp_o_o_o @ F2 @ G4) @ H) = (comp_o_o_state @ F2 @ R3)))))). % rewriteR_comp_comp
thf(fact_74_rewriteL__comp__comp2, axiom,
    ((![F2 : $o > $o, G4 : $o > $o, L1 : $o > $o, L2 : $o > $o, H : state > $o, R3 : state > $o]: (((comp_o_o_o @ F2 @ G4) = (comp_o_o_o @ L1 @ L2)) => (((comp_o_o_state @ L2 @ H) = R3) => ((comp_o_o_state @ F2 @ (comp_o_o_state @ G4 @ H)) = (comp_o_o_state @ L1 @ R3))))))). % rewriteL_comp_comp2
thf(fact_75_rewriteL__comp__comp2, axiom,
    ((![F2 : $o > $o, G4 : state > $o, L1 : $o > $o, L2 : state > $o, H : state > state, R3 : state > $o]: (((comp_o_o_state @ F2 @ G4) = (comp_o_o_state @ L1 @ L2)) => (((comp_state_o_state @ L2 @ H) = R3) => ((comp_o_o_state @ F2 @ (comp_state_o_state @ G4 @ H)) = (comp_o_o_state @ L1 @ R3))))))). % rewriteL_comp_comp2
thf(fact_76_bot__empty__eq, axiom,
    ((bot_bo431311916le_a_o = (^[X : hoare_1678595023iple_a]: (member1332298086iple_a @ X @ bot_bo1298296729iple_a))))). % bot_empty_eq
thf(fact_77_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_78_comp__cong, axiom,
    ((![F2 : $o > $o, G4 : state > $o, X4 : state, F3 : $o > $o, G5 : state > $o, X6 : state]: (((F2 @ (G4 @ X4)) = (F3 @ (G5 @ X6))) => ((comp_o_o_state @ F2 @ G4 @ X4) = (comp_o_o_state @ F3 @ G5 @ X6)))))). % comp_cong
thf(fact_79_comp__apply__eq, axiom,
    ((![F2 : $o > $o, G4 : state > $o, X4 : state, H : $o > $o, K : state > $o]: (((F2 @ (G4 @ X4)) = (H @ (K @ X4))) => ((comp_o_o_state @ F2 @ G4 @ X4) = (comp_o_o_state @ H @ K @ X4)))))). % comp_apply_eq
thf(fact_80_fun_Omap__comp, axiom,
    ((![G4 : $o > $o, F2 : state > $o, V : state > state]: ((comp_o_o_state @ G4 @ (comp_state_o_state @ F2 @ V)) = (comp_state_o_state @ (comp_o_o_state @ G4 @ F2) @ V))))). % fun.map_comp
thf(fact_81_fun_Omap__comp, axiom,
    ((![G4 : $o > $o, F2 : $o > $o, V : state > $o]: ((comp_o_o_state @ G4 @ (comp_o_o_state @ F2 @ V)) = (comp_o_o_state @ (comp_o_o_o @ G4 @ F2) @ V))))). % fun.map_comp
thf(fact_82_type__copy__map__cong0, axiom,
    ((![M : state > $o, G4 : state > state, X4 : state, N : $o > $o, H : state > $o, F2 : $o > $o]: (((M @ (G4 @ X4)) = (N @ (H @ X4))) => ((comp_state_o_state @ (comp_o_o_state @ F2 @ M) @ G4 @ X4) = (comp_o_o_state @ (comp_o_o_o @ F2 @ N) @ H @ X4)))))). % type_copy_map_cong0
thf(fact_83_type__copy__map__cong0, axiom,
    ((![M : $o > $o, G4 : state > $o, X4 : state, N : state > $o, H : state > state, F2 : $o > $o]: (((M @ (G4 @ X4)) = (N @ (H @ X4))) => ((comp_o_o_state @ (comp_o_o_o @ F2 @ M) @ G4 @ X4) = (comp_state_o_state @ (comp_o_o_state @ F2 @ N) @ H @ X4)))))). % type_copy_map_cong0
thf(fact_84_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_85_singleton__insert__inj__eq, axiom,
    ((![B : hoare_1678595023iple_a, A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: (((insert1477804543iple_a @ B @ bot_bo1298296729iple_a) = (insert1477804543iple_a @ A @ A2)) = (((A = B)) & ((ord_le1221261669iple_a @ A2 @ (insert1477804543iple_a @ B @ bot_bo1298296729iple_a)))))))). % singleton_insert_inj_eq
thf(fact_86_subsetI, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((![X5 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X5 @ A2) => (member1332298086iple_a @ X5 @ B2))) => (ord_le1221261669iple_a @ A2 @ B2))))). % subsetI
thf(fact_87_subset__empty, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ bot_bo1298296729iple_a) = (A2 = bot_bo1298296729iple_a))))). % subset_empty
thf(fact_88_empty__subsetI, axiom,
    ((![A2 : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ bot_bo1298296729iple_a @ A2)))). % empty_subsetI
thf(fact_89_insert__subset, axiom,
    ((![X4 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ (insert1477804543iple_a @ X4 @ A2) @ B2) = (((member1332298086iple_a @ X4 @ B2)) & ((ord_le1221261669iple_a @ A2 @ B2))))))). % insert_subset
thf(fact_90_singleton__insert__inj__eq_H, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B : hoare_1678595023iple_a]: (((insert1477804543iple_a @ A @ A2) = (insert1477804543iple_a @ B @ bot_bo1298296729iple_a)) = (((A = B)) & ((ord_le1221261669iple_a @ A2 @ (insert1477804543iple_a @ B @ bot_bo1298296729iple_a)))))))). % singleton_insert_inj_eq'
thf(fact_91_pairwise__empty, axiom,
    ((![P : hoare_1678595023iple_a > hoare_1678595023iple_a > $o]: (pairwi531237284iple_a @ P @ bot_bo1298296729iple_a)))). % pairwise_empty
thf(fact_92_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_93_subset__eq, axiom,
    ((ord_le1221261669iple_a = (^[A3 : set_Ho137910533iple_a]: (^[B4 : set_Ho137910533iple_a]: (![X : hoare_1678595023iple_a]: (((member1332298086iple_a @ X @ A3)) => ((member1332298086iple_a @ X @ B4))))))))). % subset_eq
thf(fact_94_pairwiseI, axiom,
    ((![S3 : set_Ho137910533iple_a, R : hoare_1678595023iple_a > hoare_1678595023iple_a > $o]: ((![X5 : hoare_1678595023iple_a, Y4 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X5 @ S3) => ((member1332298086iple_a @ Y4 @ S3) => ((~ ((X5 = Y4))) => (R @ X5 @ Y4))))) => (pairwi531237284iple_a @ R @ S3))))). % pairwiseI
thf(fact_95_pairwiseD, axiom,
    ((![R : hoare_1678595023iple_a > hoare_1678595023iple_a > $o, S3 : set_Ho137910533iple_a, X4 : hoare_1678595023iple_a, Y : hoare_1678595023iple_a]: ((pairwi531237284iple_a @ R @ S3) => ((member1332298086iple_a @ X4 @ S3) => ((member1332298086iple_a @ Y @ S3) => ((~ ((X4 = Y))) => (R @ X4 @ Y)))))))). % pairwiseD

% Conjectures (2)
thf(conj_0, hypothesis,
    ((hoare_1775499016lids_a @ g @ (insert1477804543iple_a @ (hoare_719046530iple_a @ (hoare_1795417776_and_a @ p @ b) @ c @ p) @ bot_bo1298296729iple_a)))).
thf(conj_1, conjecture,
    ((hoare_1775499016lids_a @ g @ (insert1477804543iple_a @ (hoare_719046530iple_a @ p @ (while @ b @ c) @ (hoare_1795417776_and_a @ p @ (comp_o_o_state @ (~) @ b))) @ bot_bo1298296729iple_a)))).
