% 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_273__3252170_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:13:19.149

% Could-be-implicit typings (9)
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__Option__Ooption_It__Com__Ocom_J, type,
    option_com : $tType).
thf(ty_n_t__Set__Oset_It__Com__Opname_J, type,
    set_pname : $tType).
thf(ty_n_t__Com__Ostate, type,
    state : $tType).
thf(ty_n_t__Com__Opname, type,
    pname : $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 (31)
thf(sy_c_Com_Obody, type,
    body : pname > option_com).
thf(sy_c_Com_Ocom_OBODY, type,
    body2 : pname > com).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
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_Lattices_Osup__class_Osup_001_062_It__Com__Opname_M_Eo_J, type,
    sup_sup_pname_o : (pname > $o) > (pname > $o) > pname > $o).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_M_Eo_J, type,
    sup_su335701908le_a_o : (hoare_1678595023iple_a > $o) > (hoare_1678595023iple_a > $o) > hoare_1678595023iple_a > $o).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Com__Opname_J, type,
    sup_sup_set_pname : set_pname > set_pname > set_pname).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    sup_su1000235569iple_a : set_Ho137910533iple_a > set_Ho137910533iple_a > set_Ho137910533iple_a).
thf(sy_c_Nat_OSuc, type,
    suc : nat > nat).
thf(sy_c_Option_Ooption_Othe_001t__Com__Ocom, type,
    the_com : option_com > com).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Com__Opname_J, type,
    bot_bot_set_pname : set_pname).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    bot_bo1298296729iple_a : set_Ho137910533iple_a).
thf(sy_c_Set_OBall_001t__Com__Opname, type,
    ball_pname : set_pname > (pname > $o) > $o).
thf(sy_c_Set_OCollect_001t__Com__Opname, type,
    collect_pname : (pname > $o) > set_pname).
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__Com__Opname_001t__Com__Opname, type,
    image_pname_pname : (pname > pname) > set_pname > set_pname).
thf(sy_c_Set_Oimage_001t__Com__Opname_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    image_2025567968iple_a : (pname > hoare_1678595023iple_a) > set_pname > 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__Com__Opname, type,
    insert_pname : pname > set_pname > set_pname).
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_member_001t__Com__Opname, type,
    member_pname : pname > set_pname > $o).
thf(sy_c_member_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    member1332298086iple_a : hoare_1678595023iple_a > set_Ho137910533iple_a > $o).
thf(sy_v_G, type,
    g : set_Ho137910533iple_a).
thf(sy_v_P, type,
    p : pname > a > state > $o).
thf(sy_v_Procs, type,
    procs : set_pname).
thf(sy_v_Q, type,
    q : pname > a > state > $o).
thf(sy_v_na, type,
    na : nat).
thf(sy_v_x, type,
    x : pname).

% Relevant facts (137)
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_triple__valid__Suc, axiom,
    ((![N : nat, T : hoare_1678595023iple_a]: ((hoare_1926814542alid_a @ (suc @ N) @ T) => (hoare_1926814542alid_a @ N @ T))))). % triple_valid_Suc
thf(fact_4_triples__valid__Suc, axiom,
    ((![Ts : set_Ho137910533iple_a, N : nat]: ((![X : hoare_1678595023iple_a]: ((member1332298086iple_a @ X @ Ts) => (hoare_1926814542alid_a @ (suc @ N) @ X))) => (![X4 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X4 @ Ts) => (hoare_1926814542alid_a @ N @ X4))))))). % triples_valid_Suc
thf(fact_5_Body__triple__valid__Suc, axiom,
    ((![N : nat, P : a > state > $o, Pn : pname, Q : a > state > $o]: ((hoare_1926814542alid_a @ N @ (hoare_719046530iple_a @ P @ (the_com @ (body @ Pn)) @ Q)) = (hoare_1926814542alid_a @ (suc @ N) @ (hoare_719046530iple_a @ P @ (body2 @ Pn) @ Q)))))). % Body_triple_valid_Suc
thf(fact_6_com_Oinject_I6_J, axiom,
    ((![X7 : pname, Y7 : pname]: (((body2 @ X7) = (body2 @ Y7)) = (X7 = Y7))))). % com.inject(6)
thf(fact_7_hoare__valids__def, axiom,
    ((hoare_1775499016lids_a = (^[G : set_Ho137910533iple_a]: (^[Ts2 : set_Ho137910533iple_a]: (![N2 : nat]: (((![X5 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X5 @ G)) => ((hoare_1926814542alid_a @ N2 @ X5))))) => ((![X5 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X5 @ Ts2)) => ((hoare_1926814542alid_a @ N2 @ X5)))))))))))). % hoare_valids_def
thf(fact_8_hoare__derivs_OBody, axiom,
    ((![G2 : set_Ho137910533iple_a, P : pname > a > state > $o, Q : pname > a > state > $o, Procs : set_pname]: ((hoare_129598474rivs_a @ (sup_su1000235569iple_a @ G2 @ (image_2025567968iple_a @ (^[P2 : pname]: (hoare_719046530iple_a @ (P @ P2) @ (body2 @ P2) @ (Q @ P2))) @ Procs)) @ (image_2025567968iple_a @ (^[P2 : pname]: (hoare_719046530iple_a @ (P @ P2) @ (the_com @ (body @ P2)) @ (Q @ P2))) @ Procs)) => (hoare_129598474rivs_a @ G2 @ (image_2025567968iple_a @ (^[P2 : pname]: (hoare_719046530iple_a @ (P @ P2) @ (body2 @ P2) @ (Q @ P2))) @ Procs)))))). % hoare_derivs.Body
thf(fact_9_UnCI, axiom,
    ((![C : pname, B : set_pname, A : set_pname]: (((~ ((member_pname @ C @ B))) => (member_pname @ C @ A)) => (member_pname @ C @ (sup_sup_set_pname @ A @ B)))))). % UnCI
thf(fact_10_UnCI, axiom,
    ((![C : hoare_1678595023iple_a, B : set_Ho137910533iple_a, A : set_Ho137910533iple_a]: (((~ ((member1332298086iple_a @ C @ B))) => (member1332298086iple_a @ C @ A)) => (member1332298086iple_a @ C @ (sup_su1000235569iple_a @ A @ B)))))). % UnCI
thf(fact_11_Un__iff, axiom,
    ((![C : pname, A : set_pname, B : set_pname]: ((member_pname @ C @ (sup_sup_set_pname @ A @ B)) = (((member_pname @ C @ A)) | ((member_pname @ C @ B))))))). % Un_iff
thf(fact_12_Un__iff, axiom,
    ((![C : hoare_1678595023iple_a, A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((member1332298086iple_a @ C @ (sup_su1000235569iple_a @ A @ B)) = (((member1332298086iple_a @ C @ A)) | ((member1332298086iple_a @ C @ B))))))). % Un_iff
thf(fact_13_sup_Oidem, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ A2 @ A2) = A2)))). % sup.idem
thf(fact_14_sup__idem, axiom,
    ((![X6 : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ X6 @ X6) = X6)))). % sup_idem
thf(fact_15_sup_Oleft__idem, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ A2 @ (sup_su1000235569iple_a @ A2 @ B2)) = (sup_su1000235569iple_a @ A2 @ B2))))). % sup.left_idem
thf(fact_16_sup__left__idem, axiom,
    ((![X6 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ X6 @ (sup_su1000235569iple_a @ X6 @ Y)) = (sup_su1000235569iple_a @ X6 @ Y))))). % sup_left_idem
thf(fact_17_sup_Oright__idem, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ (sup_su1000235569iple_a @ A2 @ B2) @ B2) = (sup_su1000235569iple_a @ A2 @ B2))))). % sup.right_idem
thf(fact_18_image__eqI, axiom,
    ((![B2 : hoare_1678595023iple_a, F : pname > hoare_1678595023iple_a, X6 : pname, A : set_pname]: ((B2 = (F @ X6)) => ((member_pname @ X6 @ A) => (member1332298086iple_a @ B2 @ (image_2025567968iple_a @ F @ A))))))). % image_eqI
thf(fact_19_image__eqI, axiom,
    ((![B2 : pname, F : pname > pname, X6 : pname, A : set_pname]: ((B2 = (F @ X6)) => ((member_pname @ X6 @ A) => (member_pname @ B2 @ (image_pname_pname @ F @ A))))))). % image_eqI
thf(fact_20_cut, axiom,
    ((![G3 : set_Ho137910533iple_a, Ts : set_Ho137910533iple_a, G2 : set_Ho137910533iple_a]: ((hoare_129598474rivs_a @ G3 @ Ts) => ((hoare_129598474rivs_a @ G2 @ G3) => (hoare_129598474rivs_a @ G2 @ Ts)))))). % cut
thf(fact_21_sup__set__def, axiom,
    ((sup_sup_set_pname = (^[A3 : set_pname]: (^[B3 : set_pname]: (collect_pname @ (sup_sup_pname_o @ (^[X5 : pname]: (member_pname @ X5 @ A3)) @ (^[X5 : pname]: (member_pname @ X5 @ B3))))))))). % sup_set_def
thf(fact_22_sup__set__def, axiom,
    ((sup_su1000235569iple_a = (^[A3 : set_Ho137910533iple_a]: (^[B3 : set_Ho137910533iple_a]: (collec1600235172iple_a @ (sup_su335701908le_a_o @ (^[X5 : hoare_1678595023iple_a]: (member1332298086iple_a @ X5 @ A3)) @ (^[X5 : hoare_1678595023iple_a]: (member1332298086iple_a @ X5 @ B3))))))))). % sup_set_def
thf(fact_23_rev__image__eqI, axiom,
    ((![X6 : pname, A : set_pname, B2 : hoare_1678595023iple_a, F : pname > hoare_1678595023iple_a]: ((member_pname @ X6 @ A) => ((B2 = (F @ X6)) => (member1332298086iple_a @ B2 @ (image_2025567968iple_a @ F @ A))))))). % rev_image_eqI
thf(fact_24_rev__image__eqI, axiom,
    ((![X6 : pname, A : set_pname, B2 : pname, F : pname > pname]: ((member_pname @ X6 @ A) => ((B2 = (F @ X6)) => (member_pname @ B2 @ (image_pname_pname @ F @ A))))))). % rev_image_eqI
thf(fact_25_ball__imageD, axiom,
    ((![F : pname > hoare_1678595023iple_a, A : set_pname, P : hoare_1678595023iple_a > $o]: ((![X : hoare_1678595023iple_a]: ((member1332298086iple_a @ X @ (image_2025567968iple_a @ F @ A)) => (P @ X))) => (![X4 : pname]: ((member_pname @ X4 @ A) => (P @ (F @ X4)))))))). % ball_imageD
thf(fact_26_image__cong, axiom,
    ((![M : set_pname, N3 : set_pname, F : pname > hoare_1678595023iple_a, G4 : pname > hoare_1678595023iple_a]: ((M = N3) => ((![X : pname]: ((member_pname @ X @ N3) => ((F @ X) = (G4 @ X)))) => ((image_2025567968iple_a @ F @ M) = (image_2025567968iple_a @ G4 @ N3))))))). % image_cong
thf(fact_27_bex__imageD, axiom,
    ((![F : pname > hoare_1678595023iple_a, A : set_pname, P : hoare_1678595023iple_a > $o]: ((?[X4 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X4 @ (image_2025567968iple_a @ F @ A)) & (P @ X4))) => (?[X : pname]: ((member_pname @ X @ A) & (P @ (F @ X)))))))). % bex_imageD
thf(fact_28_image__iff, axiom,
    ((![Z : hoare_1678595023iple_a, F : pname > hoare_1678595023iple_a, A : set_pname]: ((member1332298086iple_a @ Z @ (image_2025567968iple_a @ F @ A)) = (?[X5 : pname]: (((member_pname @ X5 @ A)) & ((Z = (F @ X5))))))))). % image_iff
thf(fact_29_imageI, axiom,
    ((![X6 : pname, A : set_pname, F : pname > hoare_1678595023iple_a]: ((member_pname @ X6 @ A) => (member1332298086iple_a @ (F @ X6) @ (image_2025567968iple_a @ F @ A)))))). % imageI
thf(fact_30_imageI, axiom,
    ((![X6 : pname, A : set_pname, F : pname > pname]: ((member_pname @ X6 @ A) => (member_pname @ (F @ X6) @ (image_pname_pname @ F @ A)))))). % imageI
thf(fact_31_sup__left__commute, axiom,
    ((![X6 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a, Z : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ X6 @ (sup_su1000235569iple_a @ Y @ Z)) = (sup_su1000235569iple_a @ Y @ (sup_su1000235569iple_a @ X6 @ Z)))))). % sup_left_commute
thf(fact_32_sup_Oleft__commute, axiom,
    ((![B2 : set_Ho137910533iple_a, A2 : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ B2 @ (sup_su1000235569iple_a @ A2 @ C)) = (sup_su1000235569iple_a @ A2 @ (sup_su1000235569iple_a @ B2 @ C)))))). % sup.left_commute
thf(fact_33_sup__commute, axiom,
    ((sup_su1000235569iple_a = (^[X5 : set_Ho137910533iple_a]: (^[Y4 : set_Ho137910533iple_a]: (sup_su1000235569iple_a @ Y4 @ X5)))))). % sup_commute
thf(fact_34_sup_Ocommute, axiom,
    ((sup_su1000235569iple_a = (^[A4 : set_Ho137910533iple_a]: (^[B4 : set_Ho137910533iple_a]: (sup_su1000235569iple_a @ B4 @ A4)))))). % sup.commute
thf(fact_35_sup__assoc, axiom,
    ((![X6 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a, Z : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ (sup_su1000235569iple_a @ X6 @ Y) @ Z) = (sup_su1000235569iple_a @ X6 @ (sup_su1000235569iple_a @ Y @ Z)))))). % sup_assoc
thf(fact_36_sup_Oassoc, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, C : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ (sup_su1000235569iple_a @ A2 @ B2) @ C) = (sup_su1000235569iple_a @ A2 @ (sup_su1000235569iple_a @ B2 @ C)))))). % sup.assoc
thf(fact_37_boolean__algebra__cancel_Osup2, axiom,
    ((![B : set_Ho137910533iple_a, K : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a, A2 : set_Ho137910533iple_a]: ((B = (sup_su1000235569iple_a @ K @ B2)) => ((sup_su1000235569iple_a @ A2 @ B) = (sup_su1000235569iple_a @ K @ (sup_su1000235569iple_a @ A2 @ B2))))))). % boolean_algebra_cancel.sup2
thf(fact_38_boolean__algebra__cancel_Osup1, axiom,
    ((![A : set_Ho137910533iple_a, K : set_Ho137910533iple_a, A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((A = (sup_su1000235569iple_a @ K @ A2)) => ((sup_su1000235569iple_a @ A @ B2) = (sup_su1000235569iple_a @ K @ (sup_su1000235569iple_a @ A2 @ B2))))))). % boolean_algebra_cancel.sup1
thf(fact_39_inf__sup__aci_I5_J, axiom,
    ((sup_su1000235569iple_a = (^[X5 : set_Ho137910533iple_a]: (^[Y4 : set_Ho137910533iple_a]: (sup_su1000235569iple_a @ Y4 @ X5)))))). % inf_sup_aci(5)
thf(fact_40_inf__sup__aci_I6_J, axiom,
    ((![X6 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a, Z : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ (sup_su1000235569iple_a @ X6 @ Y) @ Z) = (sup_su1000235569iple_a @ X6 @ (sup_su1000235569iple_a @ Y @ Z)))))). % inf_sup_aci(6)
thf(fact_41_inf__sup__aci_I7_J, axiom,
    ((![X6 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a, Z : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ X6 @ (sup_su1000235569iple_a @ Y @ Z)) = (sup_su1000235569iple_a @ Y @ (sup_su1000235569iple_a @ X6 @ Z)))))). % inf_sup_aci(7)
thf(fact_42_inf__sup__aci_I8_J, axiom,
    ((![X6 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ X6 @ (sup_su1000235569iple_a @ X6 @ Y)) = (sup_su1000235569iple_a @ X6 @ Y))))). % inf_sup_aci(8)
thf(fact_43_Ball__def, axiom,
    ((ball_pname = (^[A3 : set_pname]: (^[P3 : pname > $o]: (![X5 : pname]: (((member_pname @ X5 @ A3)) => ((P3 @ X5))))))))). % Ball_def
thf(fact_44_Un__left__commute, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, C2 : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ A @ (sup_su1000235569iple_a @ B @ C2)) = (sup_su1000235569iple_a @ B @ (sup_su1000235569iple_a @ A @ C2)))))). % Un_left_commute
thf(fact_45_Un__left__absorb, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ A @ (sup_su1000235569iple_a @ A @ B)) = (sup_su1000235569iple_a @ A @ B))))). % Un_left_absorb
thf(fact_46_Un__commute, axiom,
    ((sup_su1000235569iple_a = (^[A3 : set_Ho137910533iple_a]: (^[B3 : set_Ho137910533iple_a]: (sup_su1000235569iple_a @ B3 @ A3)))))). % Un_commute
thf(fact_47_Un__absorb, axiom,
    ((![A : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ A @ A) = A)))). % Un_absorb
thf(fact_48_mem__Collect__eq, axiom,
    ((![A2 : pname, P : pname > $o]: ((member_pname @ A2 @ (collect_pname @ P)) = (P @ A2))))). % mem_Collect_eq
thf(fact_49_Collect__mem__eq, axiom,
    ((![A : set_pname]: ((collect_pname @ (^[X5 : pname]: (member_pname @ X5 @ A))) = A)))). % Collect_mem_eq
thf(fact_50_Un__assoc, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, C2 : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ (sup_su1000235569iple_a @ A @ B) @ C2) = (sup_su1000235569iple_a @ A @ (sup_su1000235569iple_a @ B @ C2)))))). % Un_assoc
thf(fact_51_ball__Un, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, P : hoare_1678595023iple_a > $o]: ((![X5 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X5 @ (sup_su1000235569iple_a @ A @ B))) => ((P @ X5)))) = (((![X5 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X5 @ A)) => ((P @ X5))))) & ((![X5 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X5 @ B)) => ((P @ X5)))))))))). % ball_Un
thf(fact_52_bex__Un, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, P : hoare_1678595023iple_a > $o]: ((?[X5 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X5 @ (sup_su1000235569iple_a @ A @ B))) & ((P @ X5)))) = (((?[X5 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X5 @ A)) & ((P @ X5))))) | ((?[X5 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X5 @ B)) & ((P @ X5)))))))))). % bex_Un
thf(fact_53_UnI2, axiom,
    ((![C : pname, B : set_pname, A : set_pname]: ((member_pname @ C @ B) => (member_pname @ C @ (sup_sup_set_pname @ A @ B)))))). % UnI2
thf(fact_54_UnI2, axiom,
    ((![C : hoare_1678595023iple_a, B : set_Ho137910533iple_a, A : set_Ho137910533iple_a]: ((member1332298086iple_a @ C @ B) => (member1332298086iple_a @ C @ (sup_su1000235569iple_a @ A @ B)))))). % UnI2
thf(fact_55_UnI1, axiom,
    ((![C : pname, A : set_pname, B : set_pname]: ((member_pname @ C @ A) => (member_pname @ C @ (sup_sup_set_pname @ A @ B)))))). % UnI1
thf(fact_56_UnI1, axiom,
    ((![C : hoare_1678595023iple_a, A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((member1332298086iple_a @ C @ A) => (member1332298086iple_a @ C @ (sup_su1000235569iple_a @ A @ B)))))). % UnI1
thf(fact_57_UnE, axiom,
    ((![C : pname, A : set_pname, B : set_pname]: ((member_pname @ C @ (sup_sup_set_pname @ A @ B)) => ((~ ((member_pname @ C @ A))) => (member_pname @ C @ B)))))). % UnE
thf(fact_58_UnE, axiom,
    ((![C : hoare_1678595023iple_a, A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((member1332298086iple_a @ C @ (sup_su1000235569iple_a @ A @ B)) => ((~ ((member1332298086iple_a @ C @ A))) => (member1332298086iple_a @ C @ B)))))). % UnE
thf(fact_59_Compr__image__eq, axiom,
    ((![F : pname > hoare_1678595023iple_a, A : set_pname, P : hoare_1678595023iple_a > $o]: ((collec1600235172iple_a @ (^[X5 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X5 @ (image_2025567968iple_a @ F @ A))) & ((P @ X5))))) = (image_2025567968iple_a @ F @ (collect_pname @ (^[X5 : pname]: (((member_pname @ X5 @ A)) & ((P @ (F @ X5))))))))))). % Compr_image_eq
thf(fact_60_Compr__image__eq, axiom,
    ((![F : pname > pname, A : set_pname, P : pname > $o]: ((collect_pname @ (^[X5 : pname]: (((member_pname @ X5 @ (image_pname_pname @ F @ A))) & ((P @ X5))))) = (image_pname_pname @ F @ (collect_pname @ (^[X5 : pname]: (((member_pname @ X5 @ A)) & ((P @ (F @ X5))))))))))). % Compr_image_eq
thf(fact_61_image__image, axiom,
    ((![F : hoare_1678595023iple_a > hoare_1678595023iple_a, G4 : pname > hoare_1678595023iple_a, A : set_pname]: ((image_1577193952iple_a @ F @ (image_2025567968iple_a @ G4 @ A)) = (image_2025567968iple_a @ (^[X5 : pname]: (F @ (G4 @ X5))) @ A))))). % image_image
thf(fact_62_image__image, axiom,
    ((![F : pname > hoare_1678595023iple_a, G4 : pname > pname, A : set_pname]: ((image_2025567968iple_a @ F @ (image_pname_pname @ G4 @ A)) = (image_2025567968iple_a @ (^[X5 : pname]: (F @ (G4 @ X5))) @ A))))). % image_image
thf(fact_63_imageE, axiom,
    ((![B2 : hoare_1678595023iple_a, F : pname > hoare_1678595023iple_a, A : set_pname]: ((member1332298086iple_a @ B2 @ (image_2025567968iple_a @ F @ A)) => (~ ((![X : pname]: ((B2 = (F @ X)) => (~ ((member_pname @ X @ A))))))))))). % imageE
thf(fact_64_imageE, axiom,
    ((![B2 : pname, F : pname > pname, A : set_pname]: ((member_pname @ B2 @ (image_pname_pname @ F @ A)) => (~ ((![X : pname]: ((B2 = (F @ X)) => (~ ((member_pname @ X @ A))))))))))). % imageE
thf(fact_65_Collect__disj__eq, axiom,
    ((![P : hoare_1678595023iple_a > $o, Q : hoare_1678595023iple_a > $o]: ((collec1600235172iple_a @ (^[X5 : hoare_1678595023iple_a]: (((P @ X5)) | ((Q @ X5))))) = (sup_su1000235569iple_a @ (collec1600235172iple_a @ P) @ (collec1600235172iple_a @ Q)))))). % Collect_disj_eq
thf(fact_66_Un__def, axiom,
    ((sup_sup_set_pname = (^[A3 : set_pname]: (^[B3 : set_pname]: (collect_pname @ (^[X5 : pname]: (((member_pname @ X5 @ A3)) | ((member_pname @ X5 @ B3)))))))))). % Un_def
thf(fact_67_Un__def, axiom,
    ((sup_su1000235569iple_a = (^[A3 : set_Ho137910533iple_a]: (^[B3 : set_Ho137910533iple_a]: (collec1600235172iple_a @ (^[X5 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X5 @ A3)) | ((member1332298086iple_a @ X5 @ B3)))))))))). % Un_def
thf(fact_68_image__Un, axiom,
    ((![F : pname > hoare_1678595023iple_a, A : set_pname, B : set_pname]: ((image_2025567968iple_a @ F @ (sup_sup_set_pname @ A @ B)) = (sup_su1000235569iple_a @ (image_2025567968iple_a @ F @ A) @ (image_2025567968iple_a @ F @ B)))))). % image_Un
thf(fact_69_image__Un, axiom,
    ((![F : hoare_1678595023iple_a > hoare_1678595023iple_a, A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((image_1577193952iple_a @ F @ (sup_su1000235569iple_a @ A @ B)) = (sup_su1000235569iple_a @ (image_1577193952iple_a @ F @ A) @ (image_1577193952iple_a @ F @ B)))))). % image_Un
thf(fact_70_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_71_nat_Oinject, axiom,
    ((![X2 : nat, Y2 : nat]: (((suc @ X2) = (suc @ Y2)) = (X2 = Y2))))). % nat.inject
thf(fact_72_Body1, axiom,
    ((![G2 : set_Ho137910533iple_a, P : pname > a > state > $o, Q : pname > a > state > $o, Procs : set_pname, Pn : pname]: ((hoare_129598474rivs_a @ (sup_su1000235569iple_a @ G2 @ (image_2025567968iple_a @ (^[P2 : pname]: (hoare_719046530iple_a @ (P @ P2) @ (body2 @ P2) @ (Q @ P2))) @ Procs)) @ (image_2025567968iple_a @ (^[P2 : pname]: (hoare_719046530iple_a @ (P @ P2) @ (the_com @ (body @ P2)) @ (Q @ P2))) @ Procs)) => ((member_pname @ Pn @ Procs) => (hoare_129598474rivs_a @ G2 @ (insert1477804543iple_a @ (hoare_719046530iple_a @ (P @ Pn) @ (body2 @ Pn) @ (Q @ Pn)) @ bot_bo1298296729iple_a))))))). % Body1
thf(fact_73_sup__Un__eq, axiom,
    ((![R : set_pname, S : set_pname]: ((sup_sup_pname_o @ (^[X5 : pname]: (member_pname @ X5 @ R)) @ (^[X5 : pname]: (member_pname @ X5 @ S))) = (^[X5 : pname]: (member_pname @ X5 @ (sup_sup_set_pname @ R @ S))))))). % sup_Un_eq
thf(fact_74_sup__Un__eq, axiom,
    ((![R : set_Ho137910533iple_a, S : set_Ho137910533iple_a]: ((sup_su335701908le_a_o @ (^[X5 : hoare_1678595023iple_a]: (member1332298086iple_a @ X5 @ R)) @ (^[X5 : hoare_1678595023iple_a]: (member1332298086iple_a @ X5 @ S))) = (^[X5 : hoare_1678595023iple_a]: (member1332298086iple_a @ X5 @ (sup_su1000235569iple_a @ R @ S))))))). % sup_Un_eq
thf(fact_75_Body__triple__valid__0, axiom,
    ((![P : a > state > $o, Pn : pname, Q : a > state > $o]: (hoare_1926814542alid_a @ zero_zero_nat @ (hoare_719046530iple_a @ P @ (body2 @ Pn) @ Q))))). % Body_triple_valid_0
thf(fact_76_empty__iff, axiom,
    ((![C : pname]: (~ ((member_pname @ C @ bot_bot_set_pname)))))). % empty_iff
thf(fact_77_all__not__in__conv, axiom,
    ((![A : set_pname]: ((![X5 : pname]: (~ ((member_pname @ X5 @ A)))) = (A = bot_bot_set_pname))))). % all_not_in_conv
thf(fact_78_insertCI, axiom,
    ((![A2 : pname, B : set_pname, B2 : pname]: (((~ ((member_pname @ A2 @ B))) => (A2 = B2)) => (member_pname @ A2 @ (insert_pname @ B2 @ B)))))). % insertCI
thf(fact_79_insert__iff, axiom,
    ((![A2 : pname, B2 : pname, A : set_pname]: ((member_pname @ A2 @ (insert_pname @ B2 @ A)) = (((A2 = B2)) | ((member_pname @ A2 @ A))))))). % insert_iff
thf(fact_80_image__empty, axiom,
    ((![F : pname > hoare_1678595023iple_a]: ((image_2025567968iple_a @ F @ bot_bot_set_pname) = bot_bo1298296729iple_a)))). % image_empty
thf(fact_81_empty__is__image, axiom,
    ((![F : pname > hoare_1678595023iple_a, A : set_pname]: ((bot_bo1298296729iple_a = (image_2025567968iple_a @ F @ A)) = (A = bot_bot_set_pname))))). % empty_is_image
thf(fact_82_image__is__empty, axiom,
    ((![F : pname > hoare_1678595023iple_a, A : set_pname]: (((image_2025567968iple_a @ F @ A) = bot_bo1298296729iple_a) = (A = bot_bot_set_pname))))). % image_is_empty
thf(fact_83_image__insert, axiom,
    ((![F : pname > hoare_1678595023iple_a, A2 : pname, B : set_pname]: ((image_2025567968iple_a @ F @ (insert_pname @ A2 @ B)) = (insert1477804543iple_a @ (F @ A2) @ (image_2025567968iple_a @ F @ B)))))). % image_insert
thf(fact_84_insert__image, axiom,
    ((![X6 : pname, A : set_pname, F : pname > hoare_1678595023iple_a]: ((member_pname @ X6 @ A) => ((insert1477804543iple_a @ (F @ X6) @ (image_2025567968iple_a @ F @ A)) = (image_2025567968iple_a @ F @ A)))))). % insert_image
thf(fact_85_sup__bot__left, axiom,
    ((![X6 : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ bot_bo1298296729iple_a @ X6) = X6)))). % sup_bot_left
thf(fact_86_sup__bot__right, axiom,
    ((![X6 : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ X6 @ bot_bo1298296729iple_a) = X6)))). % sup_bot_right
thf(fact_87_bot__eq__sup__iff, axiom,
    ((![X6 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: ((bot_bo1298296729iple_a = (sup_su1000235569iple_a @ X6 @ Y)) = (((X6 = bot_bo1298296729iple_a)) & ((Y = bot_bo1298296729iple_a))))))). % bot_eq_sup_iff
thf(fact_88_sup__eq__bot__iff, axiom,
    ((![X6 : set_Ho137910533iple_a, Y : set_Ho137910533iple_a]: (((sup_su1000235569iple_a @ X6 @ Y) = bot_bo1298296729iple_a) = (((X6 = bot_bo1298296729iple_a)) & ((Y = bot_bo1298296729iple_a))))))). % sup_eq_bot_iff
thf(fact_89_sup__bot_Oeq__neutr__iff, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: (((sup_su1000235569iple_a @ A2 @ B2) = bot_bo1298296729iple_a) = (((A2 = bot_bo1298296729iple_a)) & ((B2 = bot_bo1298296729iple_a))))))). % sup_bot.eq_neutr_iff
thf(fact_90_sup__bot_Oleft__neutral, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ bot_bo1298296729iple_a @ A2) = A2)))). % sup_bot.left_neutral
thf(fact_91_sup__bot_Oneutr__eq__iff, axiom,
    ((![A2 : set_Ho137910533iple_a, B2 : set_Ho137910533iple_a]: ((bot_bo1298296729iple_a = (sup_su1000235569iple_a @ A2 @ B2)) = (((A2 = bot_bo1298296729iple_a)) & ((B2 = bot_bo1298296729iple_a))))))). % sup_bot.neutr_eq_iff
thf(fact_92_sup__bot_Oright__neutral, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ A2 @ bot_bo1298296729iple_a) = A2)))). % sup_bot.right_neutral
thf(fact_93_singletonI, axiom,
    ((![A2 : pname]: (member_pname @ A2 @ (insert_pname @ A2 @ bot_bot_set_pname))))). % singletonI
thf(fact_94_Un__empty, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: (((sup_su1000235569iple_a @ A @ B) = bot_bo1298296729iple_a) = (((A = bot_bo1298296729iple_a)) & ((B = bot_bo1298296729iple_a))))))). % Un_empty
thf(fact_95_Un__insert__left, axiom,
    ((![A2 : hoare_1678595023iple_a, B : set_Ho137910533iple_a, C2 : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ (insert1477804543iple_a @ A2 @ B) @ C2) = (insert1477804543iple_a @ A2 @ (sup_su1000235569iple_a @ B @ C2)))))). % Un_insert_left
thf(fact_96_Un__insert__right, axiom,
    ((![A : set_Ho137910533iple_a, A2 : hoare_1678595023iple_a, B : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ A @ (insert1477804543iple_a @ A2 @ B)) = (insert1477804543iple_a @ A2 @ (sup_su1000235569iple_a @ A @ B)))))). % Un_insert_right
thf(fact_97_emptyE, axiom,
    ((![A2 : pname]: (~ ((member_pname @ A2 @ bot_bot_set_pname)))))). % emptyE
thf(fact_98_insertE, axiom,
    ((![A2 : pname, B2 : pname, A : set_pname]: ((member_pname @ A2 @ (insert_pname @ B2 @ A)) => ((~ ((A2 = B2))) => (member_pname @ A2 @ A)))))). % insertE
thf(fact_99_equals0D, axiom,
    ((![A : set_pname, A2 : pname]: ((A = bot_bot_set_pname) => (~ ((member_pname @ A2 @ A))))))). % equals0D
thf(fact_100_equals0I, axiom,
    ((![A : set_pname]: ((![Y5 : pname]: (~ ((member_pname @ Y5 @ A)))) => (A = bot_bot_set_pname))))). % equals0I
thf(fact_101_insertI1, axiom,
    ((![A2 : pname, B : set_pname]: (member_pname @ A2 @ (insert_pname @ A2 @ B))))). % insertI1
thf(fact_102_insertI2, axiom,
    ((![A2 : pname, B : set_pname, B2 : pname]: ((member_pname @ A2 @ B) => (member_pname @ A2 @ (insert_pname @ B2 @ B)))))). % insertI2
thf(fact_103_ex__in__conv, axiom,
    ((![A : set_pname]: ((?[X5 : pname]: (member_pname @ X5 @ A)) = (~ ((A = bot_bot_set_pname))))))). % ex_in_conv
thf(fact_104_Set_Oset__insert, axiom,
    ((![X6 : pname, A : set_pname]: ((member_pname @ X6 @ A) => (~ ((![B5 : set_pname]: ((A = (insert_pname @ X6 @ B5)) => (member_pname @ X6 @ B5))))))))). % Set.set_insert
thf(fact_105_singletonD, axiom,
    ((![B2 : pname, A2 : pname]: ((member_pname @ B2 @ (insert_pname @ A2 @ bot_bot_set_pname)) => (B2 = A2))))). % singletonD
thf(fact_106_insert__compr, axiom,
    ((insert_pname = (^[A4 : pname]: (^[B3 : set_pname]: (collect_pname @ (^[X5 : pname]: (((X5 = A4)) | ((member_pname @ X5 @ B3)))))))))). % insert_compr
thf(fact_107_insert__ident, axiom,
    ((![X6 : pname, A : set_pname, B : set_pname]: ((~ ((member_pname @ X6 @ A))) => ((~ ((member_pname @ X6 @ B))) => (((insert_pname @ X6 @ A) = (insert_pname @ X6 @ B)) = (A = B))))))). % insert_ident
thf(fact_108_insert__is__Un, axiom,
    ((insert1477804543iple_a = (^[A4 : hoare_1678595023iple_a]: (sup_su1000235569iple_a @ (insert1477804543iple_a @ A4 @ bot_bo1298296729iple_a)))))). % insert_is_Un
thf(fact_109_insert__absorb, axiom,
    ((![A2 : pname, A : set_pname]: ((member_pname @ A2 @ A) => ((insert_pname @ A2 @ A) = A))))). % insert_absorb
thf(fact_110_insert__eq__iff, axiom,
    ((![A2 : pname, A : set_pname, B2 : pname, B : set_pname]: ((~ ((member_pname @ A2 @ A))) => ((~ ((member_pname @ B2 @ B))) => (((insert_pname @ A2 @ A) = (insert_pname @ B2 @ B)) = (((((A2 = B2)) => ((A = B)))) & ((((~ ((A2 = B2)))) => ((?[C3 : set_pname]: (((A = (insert_pname @ B2 @ C3))) & ((((~ ((member_pname @ B2 @ C3)))) & ((((B = (insert_pname @ A2 @ C3))) & ((~ ((member_pname @ A2 @ C3)))))))))))))))))))). % insert_eq_iff
thf(fact_111_singleton__iff, axiom,
    ((![B2 : pname, A2 : pname]: ((member_pname @ B2 @ (insert_pname @ A2 @ bot_bot_set_pname)) = (B2 = A2))))). % singleton_iff
thf(fact_112_Un__singleton__iff, axiom,
    ((![A : set_Ho137910533iple_a, B : set_Ho137910533iple_a, X6 : hoare_1678595023iple_a]: (((sup_su1000235569iple_a @ A @ B) = (insert1477804543iple_a @ X6 @ bot_bo1298296729iple_a)) = (((((A = bot_bo1298296729iple_a)) & ((B = (insert1477804543iple_a @ X6 @ bot_bo1298296729iple_a))))) | ((((((A = (insert1477804543iple_a @ X6 @ bot_bo1298296729iple_a))) & ((B = bot_bo1298296729iple_a)))) | ((((A = (insert1477804543iple_a @ X6 @ bot_bo1298296729iple_a))) & ((B = (insert1477804543iple_a @ X6 @ bot_bo1298296729iple_a)))))))))))). % Un_singleton_iff
thf(fact_113_singleton__Un__iff, axiom,
    ((![X6 : hoare_1678595023iple_a, A : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: (((insert1477804543iple_a @ X6 @ bot_bo1298296729iple_a) = (sup_su1000235569iple_a @ A @ B)) = (((((A = bot_bo1298296729iple_a)) & ((B = (insert1477804543iple_a @ X6 @ bot_bo1298296729iple_a))))) | ((((((A = (insert1477804543iple_a @ X6 @ bot_bo1298296729iple_a))) & ((B = bot_bo1298296729iple_a)))) | ((((A = (insert1477804543iple_a @ X6 @ bot_bo1298296729iple_a))) & ((B = (insert1477804543iple_a @ X6 @ bot_bo1298296729iple_a)))))))))))). % singleton_Un_iff
thf(fact_114_mk__disjoint__insert, axiom,
    ((![A2 : pname, A : set_pname]: ((member_pname @ A2 @ A) => (?[B5 : set_pname]: ((A = (insert_pname @ A2 @ B5)) & (~ ((member_pname @ A2 @ B5))))))))). % mk_disjoint_insert
thf(fact_115_hoare__derivs_Oinsert, axiom,
    ((![G2 : set_Ho137910533iple_a, T : hoare_1678595023iple_a, Ts : set_Ho137910533iple_a]: ((hoare_129598474rivs_a @ G2 @ (insert1477804543iple_a @ T @ bot_bo1298296729iple_a)) => ((hoare_129598474rivs_a @ G2 @ Ts) => (hoare_129598474rivs_a @ G2 @ (insert1477804543iple_a @ T @ Ts))))))). % hoare_derivs.insert
thf(fact_116_derivs__insertD, axiom,
    ((![G2 : set_Ho137910533iple_a, T : hoare_1678595023iple_a, Ts : set_Ho137910533iple_a]: ((hoare_129598474rivs_a @ G2 @ (insert1477804543iple_a @ T @ Ts)) => ((hoare_129598474rivs_a @ G2 @ (insert1477804543iple_a @ T @ bot_bo1298296729iple_a)) & (hoare_129598474rivs_a @ G2 @ Ts)))))). % derivs_insertD
thf(fact_117_nat_Odistinct_I1_J, axiom,
    ((![X2 : nat]: (~ ((zero_zero_nat = (suc @ X2))))))). % nat.distinct(1)
thf(fact_118_old_Onat_Odistinct_I2_J, axiom,
    ((![Nat2 : nat]: (~ (((suc @ Nat2) = zero_zero_nat)))))). % old.nat.distinct(2)
thf(fact_119_old_Onat_Odistinct_I1_J, axiom,
    ((![Nat2 : nat]: (~ ((zero_zero_nat = (suc @ Nat2))))))). % old.nat.distinct(1)
thf(fact_120_nat_OdiscI, axiom,
    ((![Nat : nat, X2 : nat]: ((Nat = (suc @ X2)) => (~ ((Nat = zero_zero_nat))))))). % nat.discI
thf(fact_121_nat__induct, axiom,
    ((![P : nat > $o, N : nat]: ((P @ zero_zero_nat) => ((![N4 : nat]: ((P @ N4) => (P @ (suc @ N4)))) => (P @ N)))))). % nat_induct
thf(fact_122_diff__induct, axiom,
    ((![P : nat > nat > $o, M2 : nat, N : nat]: ((![X : nat]: (P @ X @ zero_zero_nat)) => ((![Y5 : nat]: (P @ zero_zero_nat @ (suc @ Y5))) => ((![X : nat, Y5 : nat]: ((P @ X @ Y5) => (P @ (suc @ X) @ (suc @ Y5)))) => (P @ M2 @ N))))))). % diff_induct
thf(fact_123_zero__induct, axiom,
    ((![P : nat > $o, K : nat]: ((P @ K) => ((![N4 : nat]: ((P @ (suc @ N4)) => (P @ N4))) => (P @ zero_zero_nat)))))). % zero_induct
thf(fact_124_Suc__neq__Zero, axiom,
    ((![M2 : nat]: (~ (((suc @ M2) = zero_zero_nat)))))). % Suc_neq_Zero
thf(fact_125_Zero__neq__Suc, axiom,
    ((![M2 : nat]: (~ ((zero_zero_nat = (suc @ M2))))))). % Zero_neq_Suc
thf(fact_126_Zero__not__Suc, axiom,
    ((![M2 : nat]: (~ ((zero_zero_nat = (suc @ M2))))))). % Zero_not_Suc
thf(fact_127_old_Onat_Oexhaust, axiom,
    ((![Y : nat]: ((~ ((Y = zero_zero_nat))) => (~ ((![Nat3 : nat]: (~ ((Y = (suc @ Nat3))))))))))). % old.nat.exhaust
thf(fact_128_old_Onat_Oinducts, axiom,
    ((![P : nat > $o, Nat : nat]: ((P @ zero_zero_nat) => ((![Nat3 : nat]: ((P @ Nat3) => (P @ (suc @ Nat3)))) => (P @ Nat)))))). % old.nat.inducts
thf(fact_129_not0__implies__Suc, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (?[M3 : nat]: (N = (suc @ M3))))))). % not0_implies_Suc
thf(fact_130_image__constant, axiom,
    ((![X6 : pname, A : set_pname, C : hoare_1678595023iple_a]: ((member_pname @ X6 @ A) => ((image_2025567968iple_a @ (^[X5 : pname]: C) @ A) = (insert1477804543iple_a @ C @ bot_bo1298296729iple_a)))))). % image_constant
thf(fact_131_image__constant__conv, axiom,
    ((![A : set_pname, C : hoare_1678595023iple_a]: (((A = bot_bot_set_pname) => ((image_2025567968iple_a @ (^[X5 : pname]: C) @ A) = bot_bo1298296729iple_a)) & ((~ ((A = bot_bot_set_pname))) => ((image_2025567968iple_a @ (^[X5 : pname]: C) @ A) = (insert1477804543iple_a @ C @ bot_bo1298296729iple_a))))))). % image_constant_conv
thf(fact_132_conseq, axiom,
    ((![P : a > state > $o, G2 : set_Ho137910533iple_a, C : com, Q : a > state > $o]: ((![Z2 : a, S2 : state]: ((P @ Z2 @ S2) => (?[P4 : a > state > $o, Q2 : a > state > $o]: ((hoare_129598474rivs_a @ G2 @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P4 @ C @ Q2) @ bot_bo1298296729iple_a)) & (![S3 : state]: ((![Z3 : a]: ((P4 @ Z3 @ S2) => (Q2 @ Z3 @ S3))) => (Q @ Z2 @ S3))))))) => (hoare_129598474rivs_a @ G2 @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q) @ bot_bo1298296729iple_a)))))). % conseq
thf(fact_133_conseq12, axiom,
    ((![G2 : set_Ho137910533iple_a, P5 : a > state > $o, C : com, Q3 : a > state > $o, P : a > state > $o, Q : a > state > $o]: ((hoare_129598474rivs_a @ G2 @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P5 @ C @ Q3) @ bot_bo1298296729iple_a)) => ((![Z2 : a, S2 : state]: ((P @ Z2 @ S2) => (![S3 : state]: ((![Z3 : a]: ((P5 @ Z3 @ S2) => (Q3 @ Z3 @ S3))) => (Q @ Z2 @ S3))))) => (hoare_129598474rivs_a @ G2 @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q) @ bot_bo1298296729iple_a))))))). % conseq12
thf(fact_134_conseq2, axiom,
    ((![G2 : set_Ho137910533iple_a, P : a > state > $o, C : com, Q3 : a > state > $o, Q : a > state > $o]: ((hoare_129598474rivs_a @ G2 @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q3) @ bot_bo1298296729iple_a)) => ((![Z2 : a, S2 : state]: ((Q3 @ Z2 @ S2) => (Q @ Z2 @ S2))) => (hoare_129598474rivs_a @ G2 @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q) @ bot_bo1298296729iple_a))))))). % conseq2
thf(fact_135_conseq1, axiom,
    ((![G2 : set_Ho137910533iple_a, P5 : a > state > $o, C : com, Q : a > state > $o, P : a > state > $o]: ((hoare_129598474rivs_a @ G2 @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P5 @ C @ Q) @ bot_bo1298296729iple_a)) => ((![Z2 : a, S2 : state]: ((P @ Z2 @ S2) => (P5 @ Z2 @ S2))) => (hoare_129598474rivs_a @ G2 @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q) @ bot_bo1298296729iple_a))))))). % conseq1
thf(fact_136_Un__empty__left, axiom,
    ((![B : set_Ho137910533iple_a]: ((sup_su1000235569iple_a @ bot_bo1298296729iple_a @ B) = B)))). % Un_empty_left

% Conjectures (5)
thf(conj_0, hypothesis,
    ((![N5 : nat]: ((![X : hoare_1678595023iple_a]: ((member1332298086iple_a @ X @ (sup_su1000235569iple_a @ g @ (image_2025567968iple_a @ (^[Pn2 : pname]: (hoare_719046530iple_a @ (p @ Pn2) @ (body2 @ Pn2) @ (q @ Pn2))) @ procs))) => (hoare_1926814542alid_a @ N5 @ X))) => (![X4 : pname]: ((member_pname @ X4 @ procs) => (hoare_1926814542alid_a @ N5 @ (hoare_719046530iple_a @ (p @ X4) @ (the_com @ (body @ X4)) @ (q @ X4))))))))).
thf(conj_1, hypothesis,
    (((![X : hoare_1678595023iple_a]: ((member1332298086iple_a @ X @ g) => (hoare_1926814542alid_a @ na @ X))) => (![X4 : pname]: ((member_pname @ X4 @ procs) => (hoare_1926814542alid_a @ na @ (hoare_719046530iple_a @ (p @ X4) @ (body2 @ X4) @ (q @ X4)))))))).
thf(conj_2, hypothesis,
    ((member_pname @ x @ procs))).
thf(conj_3, hypothesis,
    ((![X4 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X4 @ g) => (hoare_1926814542alid_a @ na @ X4))))).
thf(conj_4, conjecture,
    ((hoare_1926814542alid_a @ (suc @ na) @ (hoare_719046530iple_a @ (p @ x) @ (body2 @ x) @ (q @ x))))).
