TPTP Problem File: SWW471

TPTP Problem File: SWW471_2.p

%------------------------------------------------------------------------------
% File     : SWW471_2 : TPTP v8.2.0. Released v5.3.0.
% Domain   : Software Verification
% Problem  : Hoare's Logic with Procedures line 269, 500 axioms selected
% Version  : Especial.
% English  :

% Refs     : [BN10]  Boehme & Nipkow (2010), Sledgehammer: Judgement Day
%          : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source   : [Bla11]
% Names    : hoare_500_tff_l269 [Bla11]

% Status   : Theorem
% Rating   : 1.00 v5.5.0, 0.75 v5.4.0, 1.00 v5.3.0
% Syntax   : Number of formulae    : 1389 ( 360 unt; 588 typ;   0 def)
%            Number of atoms       : 1712 ( 739 equ)
%            Maximal formula atoms :    9 (   1 avg)
%            Number of connectives : 1181 ( 270   ~;  53   |;  77   &)
%                                         ( 150 <=>; 631  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   16 (   5 avg)
%            Maximal term depth    :   13 (   2 avg)
%            Number of types       :  198 ( 197 usr)
%            Number of type conns  :  485 ( 276   >; 209   *;   0   +;   0  <<)
%            Number of predicates  :    2 (   1 usr;   0 prp; 1-2 aty)
%            Number of functors    :  390 ( 390 usr; 115 con; 0-4 aty)
%            Number of variables   : 2171 (2136   !;  35   ?;2171   :)
% SPC      : TF0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2011-08-09 17:27:06
%------------------------------------------------------------------------------
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tff(sy_c_COMBB_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__O,type,
    cOMBB_1380679615_state: fun_fu689207471l_bool > fun_fu572183031e_bool ).

tff(sy_c_COMBC_000t__a_000tc__HOL__Obool_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool,type,
    cOMBC_41962815e_bool: fun_fu222103665e_bool ).

tff(sy_c_COMBC_000t__a_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc__fun_Itc_,type,
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tff(sy_c_COMBC_000tc__Com__Opname_000tc__Com__Opname_000tc__HOL__Obool,type,
    cOMBC_1149511130e_bool: fun_fu577621651e_bool ).

tff(sy_c_COMBC_000tc__Com__Opname_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_I,type,
    cOMBC_131447350a_bool: fun_fu1017342411e_bool ).

tff(sy_c_COMBC_000tc__Com__Opname_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_I_120,type,
    cOMBC_951029257e_bool: fun_fu1045989665e_bool ).

tff(sy_c_COMBC_000tc__Com__Opname_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000t,type,
    cOMBC_1058051404l_bool: fun_fu696558561e_bool ).

tff(sy_c_COMBC_000tc__Com__Opname_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000t_121,type,
    cOMBC_1123258281e_bool: fun_fu1018334113e_bool ).

tff(sy_c_COMBC_000tc__Com__Opname_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__,type,
    cOMBC_845146337a_bool: fun_fu362704033a_bool ).

tff(sy_c_COMBC_000tc__Com__Opname_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe___122,type,
    cOMBC_1520907207e_bool: fun_fu738531389e_bool ).

tff(sy_c_COMBC_000tc__Com__Ostate_000tc__Com__Ostate_000tc__HOL__Obool,type,
    cOMBC_2027030106e_bool: fun_fu402792811e_bool ).

tff(sy_c_COMBC_000tc__Com__Ostate_000tc__HOL__Obool_000tc__HOL__Obool,type,
    cOMBC_231445413l_bool: fun_fu1219323149e_bool ).

tff(sy_c_COMBC_000tc__Com__Ostate_000tc__HOL__Obool_000tc__fun_Itc__Com__Ostate_Mtc_,type,
    cOMBC_471052088e_bool: fun_fu13754701e_bool ).

tff(sy_c_COMBC_000tc__Com__Ostate_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000t,type,
    cOMBC_952831051e_bool: fun_fu1436120077e_bool ).

tff(sy_c_COMBC_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_000tc__Hoare,type,
    cOMBC_827727450a_bool: fun_fu36510979a_bool ).

tff(sy_c_COMBC_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_000tc__fun_I,type,
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tff(sy_c_COMBC_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_000tc__fun_I_123,type,
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tff(sy_c_COMBC_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostate_J_0,type,
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tff(sy_c_COMBC_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostate_J_0_124,type,
    cOMBC_1099403170l_bool: fun_fu1067265135e_bool ).

tff(sy_c_COMBC_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostate_J_0_125,type,
    cOMBC_483615668e_bool: fun_fu349737825e_bool ).

tff(sy_c_COMBI_000tc__Nat__Onat,type,
    cOMBI_nat: fun_nat_nat ).

tff(sy_c_COMBK_000tc__Com__Opname_000tc__Com__Opname,type,
    cOMBK_pname_pname: fun_pn1208287926_pname ).

tff(sy_c_COMBK_000tc__HOL__Obool_000tc__Com__Opname,type,
    cOMBK_bool_pname: fun_bo1031758332e_bool ).

tff(sy_c_COMBK_000tc__HOL__Obool_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It,type,
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tff(sy_c_COMBK_000tc__HOL__Obool_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It_126,type,
    cOMBK_2086958736_state: fun_bo221168653e_bool ).

tff(sy_c_COMBK_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_000tc__Com__,type,
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tff(sy_c_COMBK_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostate_J_0,type,
    cOMBK_1032401873_pname: fun_Ho299647920_state ).

tff(sy_c_COMBK_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000t__a,type,
    cOMBK_1458035955bool_a: fun_fu281355805e_bool ).

tff(sy_c_COMBK_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc__Com__Ostate,type,
    cOMBK_631994958_state: fun_fu1998731480e_bool ).

tff(sy_c_COMBS_000tc__Com__Opname_000tc__Com__Ocom_000tc__fun_Itc__fun_It__a_Mtc__fu,type,
    cOMBS_2128932974iple_a: fun_fu874046788iple_a ).

tff(sy_c_COMBS_000tc__Com__Opname_000tc__Com__Ocom_000tc__fun_Itc__fun_Itc__Com__Ost,type,
    cOMBS_1674056844_state: fun_fu993978052_state ).

tff(sy_c_COMBS_000tc__Com__Opname_000tc__HOL__Obool_000tc__HOL__Obool,type,
    cOMBS_568398431l_bool: fun_fu1576495102e_bool ).

tff(sy_c_COMBS_000tc__Com__Opname_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_I,type,
    cOMBS_459699575iple_a: fun_fu2136234884iple_a ).

tff(sy_c_COMBS_000tc__Com__Opname_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_I_127,type,
    cOMBS_1079312276iple_a: fun_fu1621911748iple_a ).

tff(sy_c_COMBS_000tc__Com__Opname_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_I_128,type,
    cOMBS_629630867_state: fun_fu1646366334_state ).

tff(sy_c_COMBS_000tc__Com__Opname_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_I_129,type,
    cOMBS_602905655_state: fun_fu1848601144_state ).

tff(sy_c_COMBS_000tc__Com__Opname_000tc__fun_It__a_Mtc__fun_Itc__Com__Ostate_Mtc__HO,type,
    cOMBS_438391419iple_a: fun_fu1240923524iple_a ).

tff(sy_c_COMBS_000tc__Com__Opname_000tc__fun_Itc__Com__Ostate_Mtc__fun_Itc__Com__Ost,type,
    cOMBS_475568287_state: fun_fu1956604286_state ).

tff(sy_c_COMBS_000tc__Com__Ostate_000tc__HOL__Obool_000tc__HOL__Obool,type,
    cOMBS_1378840469l_bool: fun_fu1047394976e_bool ).

tff(sy_c_COMBS_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_000tc__HOL__,type,
    cOMBS_523298619l_bool: fun_fu429465874a_bool ).

tff(sy_c_COMBS_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostate_J_0,type,
    cOMBS_896408300l_bool: fun_fu1856979029e_bool ).

tff(sy_c_Com_Obody,type,
    body_1: fun_pname_option_com ).

tff(sy_c_Com_Ocom_OBODY,type,
    body: fun_pname_com ).

tff(sy_c_Com_Ocom_OCond,type,
    cond: ( fun_state_bool * com * com ) > com ).

tff(sy_c_Com_Ocom_OSKIP,type,
    skip: com ).

tff(sy_c_Com_Ocom_OSemi,type,
    semi: ( com * com ) > com ).

tff(sy_c_Com_Ocom_OWhile,type,
    while: ( fun_state_bool * com ) > com ).

tff(sy_c_Com_Ocom_Ocom__size,type,
    com_size: com > nat ).

tff(sy_c_Finite__Set_Ofinite_000tc__Com__Opname,type,
    finite_finite_pname: fun_fu1430349052l_bool ).

tff(sy_c_Finite__Set_Ofinite_000tc__HOL__Obool,type,
    finite_finite_bool: fun_fu1255657237l_bool ).

tff(sy_c_Finite__Set_Ofinite_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_,type,
    finite1833138547iple_a: fun_fu1441721944l_bool ).

tff(sy_c_Finite__Set_Ofinite_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__C,type,
    finite1792194148_state: fun_fu689207471l_bool ).

tff(sy_c_Finite__Set_Ofinite_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
    finite595471783e_bool: fun_fu399576434l_bool ).

tff(sy_c_Finite__Set_Ofinite_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otrip,type,
    finite1237779075a_bool: fun_fu911114702l_bool ).

tff(sy_c_Finite__Set_Ofinite_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otrip_130,type,
    finite1559130124e_bool: fun_fu237807729l_bool ).

tff(sy_c_Finite__Set_Ofold__image_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__,type,
    finite642022125_pname: ( fun_fu601883884a_bool * fun_pn1843974899a_bool * fun_Ho287446294a_bool * fun_pname_bool ) > fun_Ho287446294a_bool ).

tff(sy_c_Finite__Set_Ofold__image_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe___131,type,
    finite750308382_pname: ( fun_fu947547457e_bool * fun_pn760644914e_bool * fun_Ho1996104121e_bool * fun_pname_bool ) > fun_Ho1996104121e_bool ).

tff(sy_c_Finite__Set_Ofolding__one_000tc__Com__Opname,type,
    finite1282449217_pname: ( fun_pn1208287926_pname * fun_fu1499449723_pname ) > bool ).

tff(sy_c_Finite__Set_Ofolding__one_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_,type,
    finite1847914781iple_a: ( fun_Ho115022426iple_a * fun_fu1634330235iple_a ) > bool ).

tff(sy_c_Finite__Set_Ofolding__one_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple__132,type,
    finite1782051386_state: ( fun_Ho784854669_state * fun_fu385705929_state ) > bool ).

tff(sy_c_Finite__Set_Ofolding__one__idem_000tc__Com__Opname,type,
    finite89670078_pname: ( fun_pn1208287926_pname * fun_fu1499449723_pname ) > bool ).

tff(sy_c_Finite__Set_Ofolding__one__idem_000tc__Hoare____Mirabelle____xlrqixeqwe__Ot,type,
    finite1127948314iple_a: ( fun_Ho115022426iple_a * fun_fu1634330235iple_a ) > bool ).

tff(sy_c_Finite__Set_Ofolding__one__idem_000tc__Hoare____Mirabelle____xlrqixeqwe__Ot_133,type,
    finite1514162941_state: ( fun_Ho784854669_state * fun_fu385705929_state ) > bool ).

tff(sy_c_Groups_Ominus__class_Ominus_000tc__Nat__Onat,type,
    minus_minus_nat: nat > fun_nat_nat ).

tff(sy_c_Groups_Ominus__class_Ominus_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
    minus_1015773161e_bool: fun_pname_bool > fun_fu410713561e_bool ).

tff(sy_c_Groups_Ominus__class_Ominus_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqw,type,
    minus_1054521797a_bool: fun_Ho287446294a_bool > fun_fu410471825a_bool ).

tff(sy_c_Groups_Ominus__class_Ominus_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqw_134,type,
    minus_589155914e_bool: fun_Ho1996104121e_bool > fun_fu1296727421e_bool ).

tff(sy_c_Groups_Oone__class_Oone_000tc__Nat__Onat,type,
    one_one_nat: nat ).

tff(sy_c_Groups_Oplus__class_Oplus_000tc__Nat__Onat,type,
    plus_plus_nat: ( nat * nat ) > nat ).

tff(sy_c_Groups_Ozero__class_Ozero_000tc__Nat__Onat,type,
    zero_zero_nat: nat ).

tff(sy_c_HOL_OThe_000tc__Com__Opname,type,
    the_pname: fun_fu1499449723_pname ).

tff(sy_c_HOL_OThe_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J,type,
    the_Ho924576593iple_a: fun_fu1634330235iple_a ).

tff(sy_c_HOL_OThe_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostate_,type,
    the_Ho1017792262_state: fun_fu385705929_state ).

tff(sy_c_Hoare__Mirabelle__xlrqixeqwe_OMGT,type,
    hoare_Mirabelle_MGT: com > hoare_1875481847_state ).

tff(sy_c_Hoare__Mirabelle__xlrqixeqwe_Ohoare__derivs_000t__a,type,
    hoare_1546678894rivs_a: fun_Ho287446294a_bool > fun_fu1441721944l_bool ).

tff(sy_c_Hoare__Mirabelle__xlrqixeqwe_Ohoare__derivs_000tc__Com__Ostate,type,
    hoare_2131502867_state: fun_Ho1996104121e_bool > fun_fu689207471l_bool ).

tff(sy_c_Hoare__Mirabelle__xlrqixeqwe_Ohoare__valids_000t__a,type,
    hoare_1884512240lids_a: fun_Ho287446294a_bool > fun_fu1441721944l_bool ).

tff(sy_c_Hoare__Mirabelle__xlrqixeqwe_Ohoare__valids_000tc__Com__Ostate,type,
    hoare_390430481_state: fun_Ho1996104121e_bool > fun_fu689207471l_bool ).

tff(sy_c_Hoare__Mirabelle__xlrqixeqwe_Otriple_Otriple_000t__a,type,
    hoare_196563068iple_a: fun_fu18778709iple_a ).

tff(sy_c_Hoare__Mirabelle__xlrqixeqwe_Otriple_Otriple_000tc__Com__Ostate,type,
    hoare_13225861_state: fun_fu404278912_state ).

tff(sy_c_Hoare__Mirabelle__xlrqixeqwe_Otriple_Otriple__size_000t__a,type,
    hoare_1225927005size_a: ( fun_a_nat * hoare_1544627872iple_a ) > nat ).

tff(sy_c_Hoare__Mirabelle__xlrqixeqwe_Otriple_Otriple__size_000tc__Com__Ostate,type,
    hoare_1677521252_state: ( fun_state_nat * hoare_1875481847_state ) > nat ).

tff(sy_c_Hoare__Mirabelle__xlrqixeqwe_Otriple__valid_000t__a,type,
    hoare_1500711466alid_a: nat > fun_Ho287446294a_bool ).

tff(sy_c_Hoare__Mirabelle__xlrqixeqwe_Otriple__valid_000tc__Com__Ostate,type,
    hoare_2065208407_state: nat > fun_Ho1996104121e_bool ).

tff(sy_c_If_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J,type,
    if_Hoa816906256iple_a: fun_bo170780846iple_a ).

tff(sy_c_If_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostate_J,type,
    if_Hoa1541320583_state: fun_bo840793679_state ).

tff(sy_c_Lattices_Osemilattice__inf__class_Oinf_000tc__HOL__Obool,type,
    semila310582991f_bool: bool > fun_bool_bool ).

tff(sy_c_Lattices_Osemilattice__inf__class_Oinf_000tc__fun_Itc__Com__Opname_Mtc__HOL,type,
    semila1635148844e_bool: fun_pname_bool > fun_fu410713561e_bool ).

tff(sy_c_Lattices_Osemilattice__inf__class_Oinf_000tc__fun_Itc__HOL__Obool_Mtc__HOL_,type,
    semila1551573549l_bool: fun_bool_bool > fun_fu701319549l_bool ).

tff(sy_c_Lattices_Osemilattice__inf__class_Oinf_000tc__fun_Itc__Hoare____Mirabelle__,type,
    semila1256126856a_bool: fun_Ho287446294a_bool > fun_fu410471825a_bool ).

tff(sy_c_Lattices_Osemilattice__inf__class_Oinf_000tc__fun_Itc__Hoare____Mirabelle___135,type,
    semila657954503e_bool: fun_Ho1996104121e_bool > fun_fu1296727421e_bool ).

tff(sy_c_Lattices_Osemilattice__inf__class_Oinf_000tc__fun_Itc__fun_Itc__Com__Opname,type,
    semila1130628874l_bool: fun_fu1430349052l_bool > fun_fu2087345469l_bool ).

tff(sy_c_Lattices_Osemilattice__inf__class_Oinf_000tc__fun_Itc__fun_Itc__Hoare____Mi,type,
    semila1122052966l_bool: fun_fu1441721944l_bool > fun_fu902983229l_bool ).

tff(sy_c_Lattices_Osemilattice__inf__class_Oinf_000tc__fun_Itc__fun_Itc__Hoare____Mi_136,type,
    semila1058800161l_bool: fun_fu689207471l_bool > fun_fu663891139l_bool ).

tff(sy_c_Lattices_Osemilattice__sup__class_Osup_000tc__HOL__Obool,type,
    semila1168014441p_bool: fun_bo1549164019l_bool ).

tff(sy_c_Lattices_Osemilattice__sup__class_Osup_000tc__fun_Itc__Com__Opname_Mtc__HOL,type,
    semila278973382e_bool: fun_fu2101097872e_bool ).

tff(sy_c_Lattices_Osemilattice__sup__class_Osup_000tc__fun_Itc__HOL__Obool_Mtc__HOL_,type,
    semila1866150931l_bool: fun_fu744228187l_bool ).

tff(sy_c_Lattices_Osemilattice__sup__class_Osup_000tc__fun_Itc__Hoare____Mirabelle__,type,
    semila2083752482a_bool: fun_fu601883884a_bool ).

tff(sy_c_Lattices_Osemilattice__sup__class_Osup_000tc__fun_Itc__Hoare____Mirabelle___137,type,
    semila1107643821e_bool: fun_fu947547457e_bool ).

tff(sy_c_Lattices_Osemilattice__sup__class_Osup_000tc__fun_Itc__fun_Itc__Com__Opname,type,
    semila1782091504l_bool: fun_fu1759027710l_bool ).

tff(sy_c_Lattices_Osemilattice__sup__class_Osup_000tc__fun_Itc__fun_Itc__Hoare____Mi,type,
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tff(sy_c_Lattices_Osemilattice__sup__class_Osup_000tc__fun_Itc__fun_Itc__Hoare____Mi_138,type,
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tff(sy_c_Nat_OSuc,type,
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tff(sy_c_Nat_Onat_Onat__case_000tc__Nat__Onat,type,
    nat_case_nat: ( nat * fun_nat_nat * nat ) > nat ).

tff(sy_c_Nat_Osize__class_Osize_000tc__Com__Ocom,type,
    size_size_com: com > nat ).

tff(sy_c_Nat_Osize__class_Osize_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It_,type,
    size_s493879222iple_a: hoare_1544627872iple_a > nat ).

tff(sy_c_Nat_Osize__class_Osize_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc,type,
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tff(sy_c_Natural_Oevalc,type,
    evalc: com > fun_st1506752259e_bool ).

tff(sy_c_Natural_Oevaln,type,
    evaln: ( com * state * nat ) > fun_state_bool ).

tff(sy_c_Option_Othe_000tc__Com__Ocom,type,
    the_com: fun_option_com_com ).

tff(sy_c_Orderings_Obot__class_Obot_000tc__HOL__Obool,type,
    bot_bot_bool: bool ).

tff(sy_c_Orderings_Obot__class_Obot_000tc__Nat__Onat,type,
    bot_bot_nat: nat ).

tff(sy_c_Orderings_Obot__class_Obot_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
    bot_bo844097828e_bool: fun_pname_bool ).

tff(sy_c_Orderings_Obot__class_Obot_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J,type,
    bot_bo806936373l_bool: fun_bool_bool ).

tff(sy_c_Orderings_Obot__class_Obot_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe,type,
    bot_bo1766443648a_bool: fun_Ho287446294a_bool ).

tff(sy_c_Orderings_Obot__class_Obot_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe_139,type,
    bot_bo1715400655e_bool: fun_Ho1996104121e_bool ).

tff(sy_c_Orderings_Obot__class_Obot_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__O,type,
    bot_bo1649642514l_bool: fun_fu1430349052l_bool ).

tff(sy_c_Orderings_Obot__class_Obot_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____x,type,
    bot_bo700474478l_bool: fun_fu1441721944l_bool ).

tff(sy_c_Orderings_Obot__class_Obot_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____x_140,type,
    bot_bo973479705l_bool: fun_fu689207471l_bool ).

tff(sy_c_Set_OCollect_000tc__Com__Opname,type,
    collect_pname: fun_fu410713561e_bool ).

tff(sy_c_Set_OCollect_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J,type,
    collec445968053iple_a: fun_fu410471825a_bool ).

tff(sy_c_Set_OCollect_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ost,type,
    collec1735317154_state: fun_fu1296727421e_bool ).

tff(sy_c_Set_Oimage_000tc__Com__Opname_000tc__Com__Opname,type,
    image_pname_pname: fun_pname_pname > fun_fu410713561e_bool ).

tff(sy_c_Set_Oimage_000tc__Com__Opname_000tc__Hoare____Mirabelle____xlrqixeqwe__Otri,type,
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    hAPP_f1876759039l_bool: ( fun_fu629000755l_bool * fun_Ho287446294a_bool ) > fun_fu1441721944l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_Mtc__220,type,
    hAPP_f439328184a_bool: ( fun_fu601883884a_bool * fun_Ho287446294a_bool ) > fun_fu410471825a_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_Mtc__221,type,
    hAPP_f372929895l_bool: ( fun_fu1642358345l_bool * fun_Ho287446294a_bool ) > fun_fu911114702l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_Mtc__222,type,
    hAPP_f709279716l_bool: ( fun_fu643565464l_bool * fun_Ho287446294a_bool ) > fun_fu902983229l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_Mtc__223,type,
    hAPP_f1099051562a_bool: ( fun_fu429465874a_bool * fun_Ho1937021500l_bool ) > fun_fu410471825a_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_Mtc__224,type,
    hAPP_f1726109544a_bool: ( fun_fu312520138a_bool * fun_Ho965678415a_bool ) > fun_Ho287446294a_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_Mtc__225,type,
    hAPP_f1878404769a_bool: ( fun_fu36510979a_bool * fun_Ho965678415a_bool ) > fun_Ho965678415a_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_Mtc__226,type,
    hAPP_f482270129a_bool: ( fun_fu1184349593a_bool * fun_Ho448013557l_bool ) > fun_fu410471825a_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_Mtc__227,type,
    hAPP_f1399402097a_bool: ( fun_fu1749506137a_bool * fun_Ho611385006a_bool ) > fun_fu605018442a_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Os,type,
    hAPP_f1408815105l_bool: ( fun_fu689207471l_bool * fun_Ho1996104121e_bool ) > bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Os_228,type,
    hAPP_f890249627_state: ( fun_fu385705929_state * fun_Ho1996104121e_bool ) > hoare_1875481847_state ).

tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Os_229,type,
    hAPP_f827623310e_bool: ( fun_fu1291396470e_bool * fun_Ho1996104121e_bool ) > fun_pn760644914e_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Os_230,type,
    hAPP_f1438912954a_bool: ( fun_fu1181985896a_bool * fun_Ho1996104121e_bool ) > fun_Ho287446294a_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Os_231,type,
    hAPP_f121055253e_bool: ( fun_fu1296727421e_bool * fun_Ho1996104121e_bool ) > fun_Ho1996104121e_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Os_232,type,
    hAPP_f932622461l_bool: ( fun_fu811519275l_bool * fun_Ho1996104121e_bool ) > fun_Ho1896286681l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Os_233,type,
    hAPP_f56162711e_bool: ( fun_fu775812037e_bool * fun_Ho1996104121e_bool ) > fun_Ho1695028467e_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Os_234,type,
    hAPP_f1873026899l_bool: ( fun_fu1173762305l_bool * fun_Ho1996104121e_bool ) > fun_fu689207471l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Os_235,type,
    hAPP_f1204759513e_bool: ( fun_fu947547457e_bool * fun_Ho1996104121e_bool ) > fun_fu1296727421e_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Os_236,type,
    hAPP_f640685261l_bool: ( fun_fu2083718837l_bool * fun_Ho1996104121e_bool ) > fun_fu237807729l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Os_237,type,
    hAPP_f872497439l_bool: ( fun_fu2107095047l_bool * fun_Ho1996104121e_bool ) > fun_fu663891139l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Os_238,type,
    hAPP_f1300953505e_bool: ( fun_fu1856979029e_bool * fun_Ho1896286681l_bool ) > fun_fu1296727421e_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Os_239,type,
    hAPP_f169892419e_bool: ( fun_fu572183031e_bool * fun_Ho1695028467e_bool ) > fun_Ho1996104121e_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Os_240,type,
    hAPP_f2101312745e_bool: ( fun_fu302189515e_bool * fun_Ho1695028467e_bool ) > fun_Ho1695028467e_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Os_241,type,
    hAPP_f1795624711e_bool: ( fun_fu1067265135e_bool * fun_Ho1085560331l_bool ) > fun_fu1296727421e_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Os_242,type,
    hAPP_f392546175e_bool: ( fun_fu349737825e_bool * fun_Ho1110608055e_bool ) > fun_fu775812037e_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__HOL__Obool,type,
    hAPP_f1935102916l_bool: ( fun_fu399576434l_bool * fun_fu1430349052l_bool ) > bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__HOL__Obool_243,type,
    hAPP_f559147733l_bool: ( fun_fu2087345469l_bool * fun_fu1430349052l_bool ) > fun_fu1430349052l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__HOL__Obool_244,type,
    hAPP_f230953622l_bool: ( fun_fu1759027710l_bool * fun_fu1430349052l_bool ) > fun_fu2087345469l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It_,type,
    hAPP_f1059025184l_bool: ( fun_fu911114702l_bool * fun_fu1441721944l_bool ) > bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__245,type,
    hAPP_f535450709l_bool: ( fun_fu902983229l_bool * fun_fu1441721944l_bool ) > fun_fu1441721944l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__246,type,
    hAPP_f468151866l_bool: ( fun_fu133562274l_bool * fun_fu1441721944l_bool ) > fun_fu902983229l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc,type,
    hAPP_f1699935247l_bool: ( fun_fu237807729l_bool * fun_fu689207471l_bool ) > bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc_247,type,
    hAPP_f900945761l_bool: ( fun_fu663891139l_bool * fun_fu689207471l_bool ) > fun_fu689207471l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc_248,type,
    hAPP_f19215249l_bool: ( fun_fu1415736773l_bool * fun_fu689207471l_bool ) > fun_fu663891139l_bool ).

tff(sy_c_hBOOL,type,
    hBOOL: bool > $o ).

tff(sy_c_member_000tc__Com__Opname,type,
    member_pname: fun_pn422929397l_bool ).

tff(sy_c_member_000tc__HOL__Obool,type,
    member_bool: fun_bo492988759l_bool ).

tff(sy_c_member_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J,type,
    member1891733107iple_a: fun_Ho448013557l_bool ).

tff(sy_c_member_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostate_J,type,
    member618553700_state: fun_Ho1085560331l_bool ).

tff(sy_c_member_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
    member799430823e_bool: fun_fu931343505l_bool ).

tff(sy_c_member_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_Mt,type,
    member1738976643a_bool: fun_fu1642358345l_bool ).

tff(sy_c_member_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__,type,
    member1062536460e_bool: fun_fu2083718837l_bool ).

tff(sy_v_G,type,
    g: fun_Ho287446294a_bool ).

tff(sy_v_P,type,
    p: fun_pn1683930517e_bool ).

tff(sy_v_Procs,type,
    procs: fun_pname_bool ).

tff(sy_v_Q,type,
    q: fun_pn1683930517e_bool ).

tff(sy_v_n,type,
    n: nat ).

%----Relevant facts (698)
tff(fact_0_triple_Oinject,axiom,
    ! [Fun1: fun_a_fun_state_bool,Com: com,Fun2: fun_a_fun_state_bool,Fun1_2: fun_a_fun_state_bool,Com_1: com,Fun2_2: fun_a_fun_state_bool] :
      ( ( hAPP_f328191961iple_a(hAPP_c1432218316iple_a(hAPP_f1102554477iple_a(hoare_196563068iple_a,Fun1),Com),Fun2) = hAPP_f328191961iple_a(hAPP_c1432218316iple_a(hAPP_f1102554477iple_a(hoare_196563068iple_a,Fun1_2),Com_1),Fun2_2) )
    <=> ( ( Fun1 = Fun1_2 )
        & ( Com = Com_1 )
        & ( Fun2 = Fun2_2 ) ) ) ).

tff(fact_1_triple_Oinject,axiom,
    ! [Fun1: fun_st1506752259e_bool,Com: com,Fun2: fun_st1506752259e_bool,Fun1_2: fun_st1506752259e_bool,Com_1: com,Fun2_2: fun_st1506752259e_bool] :
      ( ( hAPP_f1595846909_state(hAPP_c681126378_state(hAPP_f1930074316_state(hoare_13225861_state,Fun1),Com),Fun2) = hAPP_f1595846909_state(hAPP_c681126378_state(hAPP_f1930074316_state(hoare_13225861_state,Fun1_2),Com_1),Fun2_2) )
    <=> ( ( Fun1 = Fun1_2 )
        & ( Com = Com_1 )
        & ( Fun2 = Fun2_2 ) ) ) ).

tff(fact_2_hoare__valids__def,axiom,
    ! [Ga: fun_Ho1996104121e_bool,Ts: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hoare_390430481_state(Ga),Ts))
    <=> ! [N: nat] :
          ( ! [X: hoare_1875481847_state] :
              ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X),Ga))
             => hBOOL(hAPP_H1173941847e_bool(hoare_2065208407_state(N),X)) )
         => ! [X: hoare_1875481847_state] :
              ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X),Ts))
             => hBOOL(hAPP_H1173941847e_bool(hoare_2065208407_state(N),X)) ) ) ) ).

tff(fact_3_hoare__valids__def,axiom,
    ! [Ga: fun_Ho287446294a_bool,Ts: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hoare_1884512240lids_a(Ga),Ts))
    <=> ! [N: nat] :
          ( ! [X: hoare_1544627872iple_a] :
              ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X),Ga))
             => hBOOL(hAPP_H2006434664a_bool(hoare_1500711466alid_a(N),X)) )
         => ! [X: hoare_1544627872iple_a] :
              ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X),Ts))
             => hBOOL(hAPP_H2006434664a_bool(hoare_1500711466alid_a(N),X)) ) ) ) ).

tff(fact_4_hoare__derivs_OBody,axiom,
    ! [Ga: fun_Ho1996104121e_bool,Pa: fun_pn12888928e_bool,Qa: fun_pn12888928e_bool,Procsa: fun_pname_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,Ga),hAPP_f1291720380e_bool(image_1283223414_state(hAPP_f1568887991_state(hAPP_f891245078_state(cOMBS_475568287_state,hAPP_f2116202538_state(hAPP_f1339343196_state(cOMBS_1674056844_state,hAPP_f275933352_state(cOMBB_718493681_pname(hoare_13225861_state),Pa)),body)),Qa)),Procsa))),hAPP_f1291720380e_bool(image_1283223414_state(hAPP_f1568887991_state(hAPP_f891245078_state(cOMBS_475568287_state,hAPP_f2116202538_state(hAPP_f1339343196_state(cOMBS_1674056844_state,hAPP_f275933352_state(cOMBB_718493681_pname(hoare_13225861_state),Pa)),hAPP_f96342628me_com(cOMBB_923936821_pname(the_com),body_1))),Qa)),Procsa)))
     => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f1291720380e_bool(image_1283223414_state(hAPP_f1568887991_state(hAPP_f891245078_state(cOMBS_475568287_state,hAPP_f2116202538_state(hAPP_f1339343196_state(cOMBS_1674056844_state,hAPP_f275933352_state(cOMBB_718493681_pname(hoare_13225861_state),Pa)),body)),Qa)),Procsa))) ) ).

tff(fact_5_hoare__derivs_OBody,axiom,
    ! [Ga: fun_Ho287446294a_bool,Pa: fun_pn1683930517e_bool,Qa: fun_pn1683930517e_bool,Procsa: fun_pname_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,Ga),hAPP_f856094675a_bool(image_1832685281iple_a(hAPP_f104000281iple_a(hAPP_f2028456860iple_a(cOMBS_438391419iple_a,hAPP_f68608908iple_a(hAPP_f1015300700iple_a(cOMBS_2128932974iple_a,hAPP_f298026029iple_a(cOMBB_124251824_pname(hoare_196563068iple_a),Pa)),body)),Qa)),Procsa))),hAPP_f856094675a_bool(image_1832685281iple_a(hAPP_f104000281iple_a(hAPP_f2028456860iple_a(cOMBS_438391419iple_a,hAPP_f68608908iple_a(hAPP_f1015300700iple_a(cOMBS_2128932974iple_a,hAPP_f298026029iple_a(cOMBB_124251824_pname(hoare_196563068iple_a),Pa)),hAPP_f96342628me_com(cOMBB_923936821_pname(the_com),body_1))),Qa)),Procsa)))
     => hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f856094675a_bool(image_1832685281iple_a(hAPP_f104000281iple_a(hAPP_f2028456860iple_a(cOMBS_438391419iple_a,hAPP_f68608908iple_a(hAPP_f1015300700iple_a(cOMBS_2128932974iple_a,hAPP_f298026029iple_a(cOMBB_124251824_pname(hoare_196563068iple_a),Pa)),body)),Qa)),Procsa))) ) ).

tff(fact_6_UnE,axiom,
    ! [C: hoare_1875481847_state,A: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,C),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),B_1)))
     => ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,C),A))
       => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,C),B_1)) ) ) ).

tff(fact_7_UnE,axiom,
    ! [C: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,B_1: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B_1)))
     => ( ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),A))
       => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),B_1)) ) ) ).

tff(fact_8_UnE,axiom,
    ! [C: pname,A: fun_pname_bool,B_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),B_1)))
     => ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),A))
       => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),B_1)) ) ) ).

tff(fact_9_sup1E,axiom,
    ! [A: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool,X_2: hoare_1875481847_state] :
      ( hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),B_1),X_2))
     => ( ~ hBOOL(hAPP_H1173941847e_bool(A,X_2))
       => hBOOL(hAPP_H1173941847e_bool(B_1,X_2)) ) ) ).

tff(fact_10_sup1E,axiom,
    ! [A: fun_pname_bool,B_1: fun_pname_bool,X_2: pname] :
      ( hBOOL(hAPP_pname_bool(hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),B_1),X_2))
     => ( ~ hBOOL(hAPP_pname_bool(A,X_2))
       => hBOOL(hAPP_pname_bool(B_1,X_2)) ) ) ).

tff(fact_11_sup1E,axiom,
    ! [A: fun_Ho287446294a_bool,B_1: fun_Ho287446294a_bool,X_2: hoare_1544627872iple_a] :
      ( hBOOL(hAPP_H2006434664a_bool(hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B_1),X_2))
     => ( ~ hBOOL(hAPP_H2006434664a_bool(A,X_2))
       => hBOOL(hAPP_H2006434664a_bool(B_1,X_2)) ) ) ).

tff(fact_12_sup1CI,axiom,
    ! [A: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool,X_2: hoare_1875481847_state] :
      ( ( ~ hBOOL(hAPP_H1173941847e_bool(B_1,X_2))
       => hBOOL(hAPP_H1173941847e_bool(A,X_2)) )
     => hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),B_1),X_2)) ) ).

tff(fact_13_sup1CI,axiom,
    ! [A: fun_pname_bool,B_1: fun_pname_bool,X_2: pname] :
      ( ( ~ hBOOL(hAPP_pname_bool(B_1,X_2))
       => hBOOL(hAPP_pname_bool(A,X_2)) )
     => hBOOL(hAPP_pname_bool(hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),B_1),X_2)) ) ).

tff(fact_14_sup1CI,axiom,
    ! [A: fun_Ho287446294a_bool,B_1: fun_Ho287446294a_bool,X_2: hoare_1544627872iple_a] :
      ( ( ~ hBOOL(hAPP_H2006434664a_bool(B_1,X_2))
       => hBOOL(hAPP_H2006434664a_bool(A,X_2)) )
     => hBOOL(hAPP_H2006434664a_bool(hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B_1),X_2)) ) ).

tff(fact_15_UnCI,axiom,
    ! [A: fun_Ho1996104121e_bool,C: hoare_1875481847_state,B_1: fun_Ho1996104121e_bool] :
      ( ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,C),B_1))
       => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,C),A)) )
     => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,C),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),B_1))) ) ).

tff(fact_16_UnCI,axiom,
    ! [A: fun_Ho287446294a_bool,C: hoare_1544627872iple_a,B_1: fun_Ho287446294a_bool] :
      ( ( ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),B_1))
       => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),A)) )
     => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B_1))) ) ).

tff(fact_17_UnCI,axiom,
    ! [A: fun_pname_bool,C: pname,B_1: fun_pname_bool] :
      ( ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),B_1))
       => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),A)) )
     => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),B_1))) ) ).

tff(fact_18_image__eqI,axiom,
    ! [A: fun_pname_bool,B: hoare_1875481847_state,F_1: fun_pn664418900_state,X_2: pname] :
      ( ( B = hAPP_p702226982_state(F_1,X_2) )
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
       => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,B),hAPP_f1291720380e_bool(image_1283223414_state(F_1),A))) ) ) ).

tff(fact_19_image__eqI,axiom,
    ! [A: fun_Ho287446294a_bool,B: pname,F_1: fun_Ho1170546657_pname,X_2: hoare_1544627872iple_a] :
      ( ( B = hAPP_H326197753_pname(F_1,X_2) )
     => ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_2),A))
       => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,B),hAPP_f257603411e_bool(image_1717663913_pname(F_1),A))) ) ) ).

tff(fact_20_image__eqI,axiom,
    ! [A: fun_pname_bool,B: hoare_1544627872iple_a,F_1: fun_pn1711459225iple_a,X_2: pname] :
      ( ( B = hAPP_p441219121iple_a(F_1,X_2) )
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
       => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,B),hAPP_f856094675a_bool(image_1832685281iple_a(F_1),A))) ) ) ).

tff(fact_21_image__Un,axiom,
    ! [F_1: fun_pn664418900_state,A: fun_pname_bool,B_1: fun_pname_bool] : hAPP_f1291720380e_bool(image_1283223414_state(F_1),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),B_1)) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,hAPP_f1291720380e_bool(image_1283223414_state(F_1),A)),hAPP_f1291720380e_bool(image_1283223414_state(F_1),B_1)) ).

tff(fact_22_image__Un,axiom,
    ! [F_1: fun_Ho1528294576_state,A: fun_Ho287446294a_bool,B_1: fun_Ho287446294a_bool] : hAPP_f473063776e_bool(image_456807122_state(F_1),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B_1)) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,hAPP_f473063776e_bool(image_456807122_state(F_1),A)),hAPP_f473063776e_bool(image_456807122_state(F_1),B_1)) ).

tff(fact_23_image__Un,axiom,
    ! [F_1: fun_Ho1170546657_pname,A: fun_Ho287446294a_bool,B_1: fun_Ho287446294a_bool] : hAPP_f257603411e_bool(image_1717663913_pname(F_1),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B_1)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f257603411e_bool(image_1717663913_pname(F_1),A)),hAPP_f257603411e_bool(image_1717663913_pname(F_1),B_1)) ).

tff(fact_24_image__Un,axiom,
    ! [F_1: fun_Ho1202345178iple_a,A: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] : hAPP_f1438912954a_bool(image_299889782iple_a(F_1),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),B_1)) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,hAPP_f1438912954a_bool(image_299889782iple_a(F_1),A)),hAPP_f1438912954a_bool(image_299889782iple_a(F_1),B_1)) ).

tff(fact_25_image__Un,axiom,
    ! [F_1: fun_pn1711459225iple_a,A: fun_pname_bool,B_1: fun_pname_bool] : hAPP_f856094675a_bool(image_1832685281iple_a(F_1),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),B_1)) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,hAPP_f856094675a_bool(image_1832685281iple_a(F_1),A)),hAPP_f856094675a_bool(image_1832685281iple_a(F_1),B_1)) ).

tff(fact_26_sup__fun__def,axiom,
    ! [F_1: fun_Ho1996104121e_bool,G: fun_Ho1996104121e_bool,X: hoare_1875481847_state] :
      ( hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,F_1),G),X))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,hAPP_H1173941847e_bool(F_1,X)),hAPP_H1173941847e_bool(G,X))) ) ).

tff(fact_27_sup__fun__def,axiom,
    ! [F_1: fun_pname_bool,G: fun_pname_bool,X: pname] :
      ( hBOOL(hAPP_pname_bool(hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,F_1),G),X))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,hAPP_pname_bool(F_1,X)),hAPP_pname_bool(G,X))) ) ).

tff(fact_28_sup__fun__def,axiom,
    ! [F_1: fun_Ho287446294a_bool,G: fun_Ho287446294a_bool,X: hoare_1544627872iple_a] :
      ( hBOOL(hAPP_H2006434664a_bool(hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,F_1),G),X))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,hAPP_H2006434664a_bool(F_1,X)),hAPP_H2006434664a_bool(G,X))) ) ).

tff(fact_29_sup__apply,axiom,
    ! [F_1: fun_Ho1996104121e_bool,G: fun_Ho1996104121e_bool,X_2: hoare_1875481847_state] :
      ( hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,F_1),G),X_2))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,hAPP_H1173941847e_bool(F_1,X_2)),hAPP_H1173941847e_bool(G,X_2))) ) ).

tff(fact_30_sup__apply,axiom,
    ! [F_1: fun_pname_bool,G: fun_pname_bool,X_2: pname] :
      ( hBOOL(hAPP_pname_bool(hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,F_1),G),X_2))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,hAPP_pname_bool(F_1,X_2)),hAPP_pname_bool(G,X_2))) ) ).

tff(fact_31_sup__apply,axiom,
    ! [F_1: fun_Ho287446294a_bool,G: fun_Ho287446294a_bool,X_2: hoare_1544627872iple_a] :
      ( hBOOL(hAPP_H2006434664a_bool(hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,F_1),G),X_2))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,hAPP_H2006434664a_bool(F_1,X_2)),hAPP_H2006434664a_bool(G,X_2))) ) ).

tff(fact_32_cut,axiom,
    ! [Ga: fun_Ho1996104121e_bool,G_1: fun_Ho1996104121e_bool,Ts: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(G_1),Ts))
     => ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),G_1))
       => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),Ts)) ) ) ).

tff(fact_33_cut,axiom,
    ! [Ga: fun_Ho287446294a_bool,G_1: fun_Ho287446294a_bool,Ts: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(G_1),Ts))
     => ( hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),G_1))
       => hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),Ts)) ) ) ).

tff(fact_34_sup__assoc,axiom,
    ! [X_2: fun_Ho1996104121e_bool,Y_1: fun_Ho1996104121e_bool,Z_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X_2),Y_1)),Z_1) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X_2),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,Y_1),Z_1)) ).

tff(fact_35_sup__assoc,axiom,
    ! [X_2: fun_pname_bool,Y_1: fun_pname_bool,Z_1: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),Y_1)),Z_1) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,Y_1),Z_1)) ).

tff(fact_36_sup__assoc,axiom,
    ! [X_2: bool,Y_1: bool,Z_1: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),Y_1)),Z_1))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,Y_1),Z_1))) ) ).

tff(fact_37_sup__assoc,axiom,
    ! [X_2: fun_Ho287446294a_bool,Y_1: fun_Ho287446294a_bool,Z_1: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X_2),Y_1)),Z_1) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X_2),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,Y_1),Z_1)) ).

tff(fact_38_inf__sup__aci_I6_J,axiom,
    ! [X_2: fun_Ho1996104121e_bool,Y_1: fun_Ho1996104121e_bool,Z_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X_2),Y_1)),Z_1) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X_2),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,Y_1),Z_1)) ).

tff(fact_39_inf__sup__aci_I6_J,axiom,
    ! [X_2: fun_pname_bool,Y_1: fun_pname_bool,Z_1: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),Y_1)),Z_1) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,Y_1),Z_1)) ).

tff(fact_40_inf__sup__aci_I6_J,axiom,
    ! [X_2: bool,Y_1: bool,Z_1: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),Y_1)),Z_1))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,Y_1),Z_1))) ) ).

tff(fact_41_inf__sup__aci_I6_J,axiom,
    ! [X_2: fun_Ho287446294a_bool,Y_1: fun_Ho287446294a_bool,Z_1: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X_2),Y_1)),Z_1) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X_2),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,Y_1),Z_1)) ).

tff(fact_42_sup_Oassoc,axiom,
    ! [A_1: fun_Ho1996104121e_bool,B: fun_Ho1996104121e_bool,C: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A_1),B)),C) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A_1),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,B),C)) ).

tff(fact_43_sup_Oassoc,axiom,
    ! [A_1: fun_pname_bool,B: fun_pname_bool,C: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),B)),C) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,B),C)) ).

tff(fact_44_sup_Oassoc,axiom,
    ! [A_1: bool,B: bool,C: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,A_1),B)),C))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,A_1),hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,B),C))) ) ).

tff(fact_45_sup_Oassoc,axiom,
    ! [A_1: fun_Ho287446294a_bool,B: fun_Ho287446294a_bool,C: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A_1),B)),C) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A_1),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,B),C)) ).

tff(fact_46_sup__left__commute,axiom,
    ! [X_2: fun_Ho1996104121e_bool,Y_1: fun_Ho1996104121e_bool,Z_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X_2),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,Y_1),Z_1)) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,Y_1),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X_2),Z_1)) ).

tff(fact_47_sup__left__commute,axiom,
    ! [X_2: fun_pname_bool,Y_1: fun_pname_bool,Z_1: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,Y_1),Z_1)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,Y_1),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),Z_1)) ).

tff(fact_48_sup__left__commute,axiom,
    ! [X_2: bool,Y_1: bool,Z_1: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,Y_1),Z_1)))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,Y_1),hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),Z_1))) ) ).

tff(fact_49_sup__left__commute,axiom,
    ! [X_2: fun_Ho287446294a_bool,Y_1: fun_Ho287446294a_bool,Z_1: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X_2),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,Y_1),Z_1)) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,Y_1),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X_2),Z_1)) ).

tff(fact_50_inf__sup__aci_I7_J,axiom,
    ! [X_2: fun_Ho1996104121e_bool,Y_1: fun_Ho1996104121e_bool,Z_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X_2),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,Y_1),Z_1)) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,Y_1),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X_2),Z_1)) ).

tff(fact_51_inf__sup__aci_I7_J,axiom,
    ! [X_2: fun_pname_bool,Y_1: fun_pname_bool,Z_1: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,Y_1),Z_1)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,Y_1),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),Z_1)) ).

tff(fact_52_inf__sup__aci_I7_J,axiom,
    ! [X_2: bool,Y_1: bool,Z_1: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,Y_1),Z_1)))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,Y_1),hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),Z_1))) ) ).

tff(fact_53_inf__sup__aci_I7_J,axiom,
    ! [X_2: fun_Ho287446294a_bool,Y_1: fun_Ho287446294a_bool,Z_1: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X_2),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,Y_1),Z_1)) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,Y_1),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X_2),Z_1)) ).

tff(fact_54_sup_Oleft__commute,axiom,
    ! [B: fun_Ho1996104121e_bool,A_1: fun_Ho1996104121e_bool,C: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,B),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A_1),C)) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A_1),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,B),C)) ).

tff(fact_55_sup_Oleft__commute,axiom,
    ! [B: fun_pname_bool,A_1: fun_pname_bool,C: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,B),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),C)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,B),C)) ).

tff(fact_56_sup_Oleft__commute,axiom,
    ! [B: bool,A_1: bool,C: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,B),hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,A_1),C)))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,A_1),hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,B),C))) ) ).

tff(fact_57_sup_Oleft__commute,axiom,
    ! [B: fun_Ho287446294a_bool,A_1: fun_Ho287446294a_bool,C: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,B),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A_1),C)) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A_1),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,B),C)) ).

tff(fact_58_sup__left__idem,axiom,
    ! [X_2: fun_Ho1996104121e_bool,Y_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X_2),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X_2),Y_1)) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X_2),Y_1) ).

tff(fact_59_sup__left__idem,axiom,
    ! [X_2: fun_pname_bool,Y_1: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),Y_1)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),Y_1) ).

tff(fact_60_sup__left__idem,axiom,
    ! [X_2: bool,Y_1: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),Y_1)))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),Y_1)) ) ).

tff(fact_61_sup__left__idem,axiom,
    ! [X_2: fun_Ho287446294a_bool,Y_1: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X_2),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X_2),Y_1)) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X_2),Y_1) ).

tff(fact_62_inf__sup__aci_I8_J,axiom,
    ! [X_2: fun_Ho1996104121e_bool,Y_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X_2),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X_2),Y_1)) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X_2),Y_1) ).

tff(fact_63_inf__sup__aci_I8_J,axiom,
    ! [X_2: fun_pname_bool,Y_1: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),Y_1)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),Y_1) ).

tff(fact_64_inf__sup__aci_I8_J,axiom,
    ! [X_2: bool,Y_1: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),Y_1)))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),Y_1)) ) ).

tff(fact_65_inf__sup__aci_I8_J,axiom,
    ! [X_2: fun_Ho287446294a_bool,Y_1: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X_2),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X_2),Y_1)) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X_2),Y_1) ).

tff(fact_66_sup_Oleft__idem,axiom,
    ! [A_1: fun_Ho1996104121e_bool,B: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A_1),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A_1),B)) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A_1),B) ).

tff(fact_67_sup_Oleft__idem,axiom,
    ! [A_1: fun_pname_bool,B: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),B)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),B) ).

tff(fact_68_sup_Oleft__idem,axiom,
    ! [A_1: bool,B: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,A_1),hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,A_1),B)))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,A_1),B)) ) ).

tff(fact_69_sup_Oleft__idem,axiom,
    ! [A_1: fun_Ho287446294a_bool,B: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A_1),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A_1),B)) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A_1),B) ).

tff(fact_70_sup__commute,axiom,
    ! [X_2: fun_Ho1996104121e_bool,Y_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X_2),Y_1) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,Y_1),X_2) ).

tff(fact_71_sup__commute,axiom,
    ! [X_2: fun_pname_bool,Y_1: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),Y_1) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,Y_1),X_2) ).

tff(fact_72_sup__commute,axiom,
    ! [X_2: bool,Y_1: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),Y_1))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,Y_1),X_2)) ) ).

tff(fact_73_sup__commute,axiom,
    ! [X_2: fun_Ho287446294a_bool,Y_1: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X_2),Y_1) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,Y_1),X_2) ).

tff(fact_74_inf__sup__aci_I5_J,axiom,
    ! [X_2: fun_Ho1996104121e_bool,Y_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X_2),Y_1) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,Y_1),X_2) ).

tff(fact_75_inf__sup__aci_I5_J,axiom,
    ! [X_2: fun_pname_bool,Y_1: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),Y_1) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,Y_1),X_2) ).

tff(fact_76_inf__sup__aci_I5_J,axiom,
    ! [X_2: bool,Y_1: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),Y_1))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,Y_1),X_2)) ) ).

tff(fact_77_inf__sup__aci_I5_J,axiom,
    ! [X_2: fun_Ho287446294a_bool,Y_1: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X_2),Y_1) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,Y_1),X_2) ).

tff(fact_78_sup_Ocommute,axiom,
    ! [A_1: fun_Ho1996104121e_bool,B: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A_1),B) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,B),A_1) ).

tff(fact_79_sup_Ocommute,axiom,
    ! [A_1: fun_pname_bool,B: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),B) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,B),A_1) ).

tff(fact_80_sup_Ocommute,axiom,
    ! [A_1: bool,B: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,A_1),B))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,B),A_1)) ) ).

tff(fact_81_sup_Ocommute,axiom,
    ! [A_1: fun_Ho287446294a_bool,B: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A_1),B) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,B),A_1) ).

tff(fact_82_sup__idem,axiom,
    ! [X_2: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X_2),X_2) = X_2 ).

tff(fact_83_sup__idem,axiom,
    ! [X_2: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),X_2) = X_2 ).

tff(fact_84_sup__idem,axiom,
    ! [X_2: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),X_2))
    <=> hBOOL(X_2) ) ).

tff(fact_85_sup__idem,axiom,
    ! [X_2: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X_2),X_2) = X_2 ).

tff(fact_86_sup_Oidem,axiom,
    ! [A_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A_1),A_1) = A_1 ).

tff(fact_87_sup_Oidem,axiom,
    ! [A_1: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),A_1) = A_1 ).

tff(fact_88_sup_Oidem,axiom,
    ! [A_1: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,A_1),A_1))
    <=> hBOOL(A_1) ) ).

tff(fact_89_sup_Oidem,axiom,
    ! [A_1: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A_1),A_1) = A_1 ).

tff(fact_90_rev__image__eqI,axiom,
    ! [B: hoare_1875481847_state,F_1: fun_pn664418900_state,X_2: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
     => ( ( B = hAPP_p702226982_state(F_1,X_2) )
       => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,B),hAPP_f1291720380e_bool(image_1283223414_state(F_1),A))) ) ) ).

tff(fact_91_rev__image__eqI,axiom,
    ! [B: pname,F_1: fun_Ho1170546657_pname,X_2: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_2),A))
     => ( ( B = hAPP_H326197753_pname(F_1,X_2) )
       => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,B),hAPP_f257603411e_bool(image_1717663913_pname(F_1),A))) ) ) ).

tff(fact_92_rev__image__eqI,axiom,
    ! [B: hoare_1544627872iple_a,F_1: fun_pn1711459225iple_a,X_2: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
     => ( ( B = hAPP_p441219121iple_a(F_1,X_2) )
       => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,B),hAPP_f856094675a_bool(image_1832685281iple_a(F_1),A))) ) ) ).

tff(fact_93_imageI,axiom,
    ! [F_1: fun_pn664418900_state,X_2: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
     => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,hAPP_p702226982_state(F_1,X_2)),hAPP_f1291720380e_bool(image_1283223414_state(F_1),A))) ) ).

tff(fact_94_imageI,axiom,
    ! [F_1: fun_Ho1170546657_pname,X_2: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_2),A))
     => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,hAPP_H326197753_pname(F_1,X_2)),hAPP_f257603411e_bool(image_1717663913_pname(F_1),A))) ) ).

tff(fact_95_imageI,axiom,
    ! [F_1: fun_pn1711459225iple_a,X_2: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
     => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,hAPP_p441219121iple_a(F_1,X_2)),hAPP_f856094675a_bool(image_1832685281iple_a(F_1),A))) ) ).

tff(fact_96_image__iff,axiom,
    ! [Z_1: hoare_1875481847_state,F_1: fun_pn664418900_state,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,Z_1),hAPP_f1291720380e_bool(image_1283223414_state(F_1),A)))
    <=> ? [X: pname] :
          ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),A))
          & ( Z_1 = hAPP_p702226982_state(F_1,X) ) ) ) ).

tff(fact_97_image__iff,axiom,
    ! [Z_1: hoare_1544627872iple_a,F_1: fun_pn1711459225iple_a,A: fun_pname_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Z_1),hAPP_f856094675a_bool(image_1832685281iple_a(F_1),A)))
    <=> ? [X: pname] :
          ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),A))
          & ( Z_1 = hAPP_p441219121iple_a(F_1,X) ) ) ) ).

tff(fact_98_UnI2,axiom,
    ! [A: fun_Ho1996104121e_bool,C: hoare_1875481847_state,B_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,C),B_1))
     => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,C),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),B_1))) ) ).

tff(fact_99_UnI2,axiom,
    ! [A: fun_Ho287446294a_bool,C: hoare_1544627872iple_a,B_1: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),B_1))
     => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B_1))) ) ).

tff(fact_100_UnI2,axiom,
    ! [A: fun_pname_bool,C: pname,B_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),B_1))
     => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),B_1))) ) ).

tff(fact_101_UnI1,axiom,
    ! [B_1: fun_Ho1996104121e_bool,C: hoare_1875481847_state,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,C),A))
     => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,C),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),B_1))) ) ).

tff(fact_102_UnI1,axiom,
    ! [B_1: fun_Ho287446294a_bool,C: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),A))
     => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B_1))) ) ).

tff(fact_103_UnI1,axiom,
    ! [B_1: fun_pname_bool,C: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),A))
     => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),B_1))) ) ).

tff(fact_104_sup1I2,axiom,
    ! [A: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool,X_2: hoare_1875481847_state] :
      ( hBOOL(hAPP_H1173941847e_bool(B_1,X_2))
     => hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),B_1),X_2)) ) ).

tff(fact_105_sup1I2,axiom,
    ! [A: fun_pname_bool,B_1: fun_pname_bool,X_2: pname] :
      ( hBOOL(hAPP_pname_bool(B_1,X_2))
     => hBOOL(hAPP_pname_bool(hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),B_1),X_2)) ) ).

tff(fact_106_sup1I2,axiom,
    ! [A: fun_Ho287446294a_bool,B_1: fun_Ho287446294a_bool,X_2: hoare_1544627872iple_a] :
      ( hBOOL(hAPP_H2006434664a_bool(B_1,X_2))
     => hBOOL(hAPP_H2006434664a_bool(hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B_1),X_2)) ) ).

tff(fact_107_sup1I1,axiom,
    ! [B_1: fun_Ho1996104121e_bool,A: fun_Ho1996104121e_bool,X_2: hoare_1875481847_state] :
      ( hBOOL(hAPP_H1173941847e_bool(A,X_2))
     => hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),B_1),X_2)) ) ).

tff(fact_108_sup1I1,axiom,
    ! [B_1: fun_pname_bool,A: fun_pname_bool,X_2: pname] :
      ( hBOOL(hAPP_pname_bool(A,X_2))
     => hBOOL(hAPP_pname_bool(hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),B_1),X_2)) ) ).

tff(fact_109_sup1I1,axiom,
    ! [B_1: fun_Ho287446294a_bool,A: fun_Ho287446294a_bool,X_2: hoare_1544627872iple_a] :
      ( hBOOL(hAPP_H2006434664a_bool(A,X_2))
     => hBOOL(hAPP_H2006434664a_bool(hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B_1),X_2)) ) ).

tff(fact_110_ball__Un,axiom,
    ! [Pa: fun_Ho1996104121e_bool,A: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( ! [X: hoare_1875481847_state] :
          ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),B_1)))
         => hBOOL(hAPP_H1173941847e_bool(Pa,X)) )
    <=> ( ! [X: hoare_1875481847_state] :
            ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X),A))
           => hBOOL(hAPP_H1173941847e_bool(Pa,X)) )
        & ! [X: hoare_1875481847_state] :
            ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X),B_1))
           => hBOOL(hAPP_H1173941847e_bool(Pa,X)) ) ) ) ).

tff(fact_111_ball__Un,axiom,
    ! [Pa: fun_pname_bool,A: fun_pname_bool,B_1: fun_pname_bool] :
      ( ! [X: pname] :
          ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),B_1)))
         => hBOOL(hAPP_pname_bool(Pa,X)) )
    <=> ( ! [X: pname] :
            ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),A))
           => hBOOL(hAPP_pname_bool(Pa,X)) )
        & ! [X: pname] :
            ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),B_1))
           => hBOOL(hAPP_pname_bool(Pa,X)) ) ) ) ).

tff(fact_112_ball__Un,axiom,
    ! [Pa: fun_Ho287446294a_bool,A: fun_Ho287446294a_bool,B_1: fun_Ho287446294a_bool] :
      ( ! [X: hoare_1544627872iple_a] :
          ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B_1)))
         => hBOOL(hAPP_H2006434664a_bool(Pa,X)) )
    <=> ( ! [X: hoare_1544627872iple_a] :
            ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X),A))
           => hBOOL(hAPP_H2006434664a_bool(Pa,X)) )
        & ! [X: hoare_1544627872iple_a] :
            ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X),B_1))
           => hBOOL(hAPP_H2006434664a_bool(Pa,X)) ) ) ) ).

tff(fact_113_bex__Un,axiom,
    ! [Pa: fun_Ho1996104121e_bool,A: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( ? [X: hoare_1875481847_state] :
          ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),B_1)))
          & hBOOL(hAPP_H1173941847e_bool(Pa,X)) )
    <=> ( ? [X: hoare_1875481847_state] :
            ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X),A))
            & hBOOL(hAPP_H1173941847e_bool(Pa,X)) )
        | ? [X: hoare_1875481847_state] :
            ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X),B_1))
            & hBOOL(hAPP_H1173941847e_bool(Pa,X)) ) ) ) ).

tff(fact_114_bex__Un,axiom,
    ! [Pa: fun_pname_bool,A: fun_pname_bool,B_1: fun_pname_bool] :
      ( ? [X: pname] :
          ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),B_1)))
          & hBOOL(hAPP_pname_bool(Pa,X)) )
    <=> ( ? [X: pname] :
            ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),A))
            & hBOOL(hAPP_pname_bool(Pa,X)) )
        | ? [X: pname] :
            ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),B_1))
            & hBOOL(hAPP_pname_bool(Pa,X)) ) ) ) ).

tff(fact_115_bex__Un,axiom,
    ! [Pa: fun_Ho287446294a_bool,A: fun_Ho287446294a_bool,B_1: fun_Ho287446294a_bool] :
      ( ? [X: hoare_1544627872iple_a] :
          ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B_1)))
          & hBOOL(hAPP_H2006434664a_bool(Pa,X)) )
    <=> ( ? [X: hoare_1544627872iple_a] :
            ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X),A))
            & hBOOL(hAPP_H2006434664a_bool(Pa,X)) )
        | ? [X: hoare_1544627872iple_a] :
            ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X),B_1))
            & hBOOL(hAPP_H2006434664a_bool(Pa,X)) ) ) ) ).

tff(fact_116_Un__assoc,axiom,
    ! [A: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool,C_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),B_1)),C_1) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,B_1),C_1)) ).

tff(fact_117_Un__assoc,axiom,
    ! [A: fun_pname_bool,B_1: fun_pname_bool,C_1: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),B_1)),C_1) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,B_1),C_1)) ).

tff(fact_118_Un__assoc,axiom,
    ! [A: fun_Ho287446294a_bool,B_1: fun_Ho287446294a_bool,C_1: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B_1)),C_1) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,B_1),C_1)) ).

tff(fact_119_Un__iff,axiom,
    ! [C: hoare_1875481847_state,A: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,C),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),B_1)))
    <=> ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,C),A))
        | hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,C),B_1)) ) ) ).

tff(fact_120_Un__iff,axiom,
    ! [C: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,B_1: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B_1)))
    <=> ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),A))
        | hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),B_1)) ) ) ).

tff(fact_121_Un__iff,axiom,
    ! [C: pname,A: fun_pname_bool,B_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),B_1)))
    <=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),A))
        | hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),B_1)) ) ) ).

tff(fact_122_Un__left__commute,axiom,
    ! [A: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool,C_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,B_1),C_1)) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,B_1),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),C_1)) ).

tff(fact_123_Un__left__commute,axiom,
    ! [A: fun_pname_bool,B_1: fun_pname_bool,C_1: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,B_1),C_1)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,B_1),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),C_1)) ).

tff(fact_124_Un__left__commute,axiom,
    ! [A: fun_Ho287446294a_bool,B_1: fun_Ho287446294a_bool,C_1: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,B_1),C_1)) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,B_1),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),C_1)) ).

tff(fact_125_Un__left__absorb,axiom,
    ! [A: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),B_1)) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),B_1) ).

tff(fact_126_Un__left__absorb,axiom,
    ! [A: fun_pname_bool,B_1: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),B_1)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),B_1) ).

tff(fact_127_Un__left__absorb,axiom,
    ! [A: fun_Ho287446294a_bool,B_1: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B_1)) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B_1) ).

tff(fact_128_Un__commute,axiom,
    ! [A: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),B_1) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,B_1),A) ).

tff(fact_129_Un__commute,axiom,
    ! [A: fun_pname_bool,B_1: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),B_1) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,B_1),A) ).

tff(fact_130_Un__commute,axiom,
    ! [A: fun_Ho287446294a_bool,B_1: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B_1) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,B_1),A) ).

tff(fact_131_Un__def,axiom,
    ! [A: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),B_1) = hAPP_f121055253e_bool(collec1735317154_state,hAPP_f121055253e_bool(hAPP_f1300953505e_bool(cOMBS_896408300l_bool,hAPP_f932622461l_bool(cOMBB_242064253_state(fdisj),hAPP_f121055253e_bool(hAPP_f1795624711e_bool(cOMBC_1099403170l_bool,member618553700_state),A))),hAPP_f121055253e_bool(hAPP_f1795624711e_bool(cOMBC_1099403170l_bool,member618553700_state),B_1))) ).

tff(fact_132_Un__def,axiom,
    ! [A: fun_Ho287446294a_bool,B_1: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B_1) = hAPP_f909437487a_bool(collec445968053iple_a,hAPP_f909437487a_bool(hAPP_f1099051562a_bool(cOMBS_523298619l_bool,hAPP_f224574947l_bool(cOMBB_1960865690iple_a(fdisj),hAPP_f909437487a_bool(hAPP_f482270129a_bool(cOMBC_1807206788l_bool,member1891733107iple_a),A))),hAPP_f909437487a_bool(hAPP_f482270129a_bool(cOMBC_1807206788l_bool,member1891733107iple_a),B_1))) ).

tff(fact_133_Un__def,axiom,
    ! [A: fun_pname_bool,B_1: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),B_1) = hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,hAPP_f1402196763l_bool(cOMBB_675860798_pname(fdisj),hAPP_f759274231e_bool(hAPP_f661147897e_bool(cOMBC_1058051404l_bool,member_pname),A))),hAPP_f759274231e_bool(hAPP_f661147897e_bool(cOMBC_1058051404l_bool,member_pname),B_1))) ).

tff(fact_134_Un__absorb,axiom,
    ! [A: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),A) = A ).

tff(fact_135_Un__absorb,axiom,
    ! [A: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),A) = A ).

tff(fact_136_Un__absorb,axiom,
    ! [A: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),A) = A ).

tff(fact_137_image__image,axiom,
    ! [F_1: fun_Ho1528294576_state,G: fun_pn1711459225iple_a,A: fun_pname_bool] : hAPP_f473063776e_bool(image_456807122_state(F_1),hAPP_f856094675a_bool(image_1832685281iple_a(G),A)) = hAPP_f1291720380e_bool(image_1283223414_state(hAPP_f793302744_state(cOMBB_531782587_pname(F_1),G)),A) ).

tff(fact_138_image__image,axiom,
    ! [F_1: fun_Ho1202345178iple_a,G: fun_pn664418900_state,A: fun_pname_bool] : hAPP_f1438912954a_bool(image_299889782iple_a(F_1),hAPP_f1291720380e_bool(image_1283223414_state(G),A)) = hAPP_f856094675a_bool(image_1832685281iple_a(hAPP_f1780080898iple_a(cOMBB_1283441943_pname(F_1),G)),A) ).

tff(fact_139_sup__Un__eq,axiom,
    ! [R_1: fun_Ho1996104121e_bool,S_1: fun_Ho1996104121e_bool,X: hoare_1875481847_state] :
      ( hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,hAPP_f121055253e_bool(hAPP_f1795624711e_bool(cOMBC_1099403170l_bool,member618553700_state),R_1)),hAPP_f121055253e_bool(hAPP_f1795624711e_bool(cOMBC_1099403170l_bool,member618553700_state),S_1)),X))
    <=> hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,R_1),S_1))) ) ).

tff(fact_140_sup__Un__eq,axiom,
    ! [R_1: fun_Ho287446294a_bool,S_1: fun_Ho287446294a_bool,X: hoare_1544627872iple_a] :
      ( hBOOL(hAPP_H2006434664a_bool(hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,hAPP_f909437487a_bool(hAPP_f482270129a_bool(cOMBC_1807206788l_bool,member1891733107iple_a),R_1)),hAPP_f909437487a_bool(hAPP_f482270129a_bool(cOMBC_1807206788l_bool,member1891733107iple_a),S_1)),X))
    <=> hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,R_1),S_1))) ) ).

tff(fact_141_sup__Un__eq,axiom,
    ! [R_1: fun_pname_bool,S_1: fun_pname_bool,X: pname] :
      ( hBOOL(hAPP_pname_bool(hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f759274231e_bool(hAPP_f661147897e_bool(cOMBC_1058051404l_bool,member_pname),R_1)),hAPP_f759274231e_bool(hAPP_f661147897e_bool(cOMBC_1058051404l_bool,member_pname),S_1)),X))
    <=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,R_1),S_1))) ) ).

tff(fact_142_Collect__disj__eq,axiom,
    ! [Pa: fun_pname_bool,Qa: fun_pname_bool] : hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,hAPP_f1402196763l_bool(cOMBB_675860798_pname(fdisj),Pa)),Qa)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f759274231e_bool(collect_pname,Pa)),hAPP_f759274231e_bool(collect_pname,Qa)) ).

tff(fact_143_Collect__disj__eq,axiom,
    ! [Pa: fun_Ho1996104121e_bool,Qa: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(collec1735317154_state,hAPP_f121055253e_bool(hAPP_f1300953505e_bool(cOMBS_896408300l_bool,hAPP_f932622461l_bool(cOMBB_242064253_state(fdisj),Pa)),Qa)) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,hAPP_f121055253e_bool(collec1735317154_state,Pa)),hAPP_f121055253e_bool(collec1735317154_state,Qa)) ).

tff(fact_144_Collect__disj__eq,axiom,
    ! [Pa: fun_Ho287446294a_bool,Qa: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(collec445968053iple_a,hAPP_f909437487a_bool(hAPP_f1099051562a_bool(cOMBS_523298619l_bool,hAPP_f224574947l_bool(cOMBB_1960865690iple_a(fdisj),Pa)),Qa)) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,hAPP_f909437487a_bool(collec445968053iple_a,Pa)),hAPP_f909437487a_bool(collec445968053iple_a,Qa)) ).

tff(fact_145_imageE,axiom,
    ! [B: pname,F_1: fun_Ho1170546657_pname,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,B),hAPP_f257603411e_bool(image_1717663913_pname(F_1),A)))
     => ~ ! [X: hoare_1544627872iple_a] :
            ( ( B = hAPP_H326197753_pname(F_1,X) )
           => ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X),A)) ) ) ).

tff(fact_146_imageE,axiom,
    ! [B: hoare_1875481847_state,F_1: fun_pn664418900_state,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,B),hAPP_f1291720380e_bool(image_1283223414_state(F_1),A)))
     => ~ ! [X: pname] :
            ( ( B = hAPP_p702226982_state(F_1,X) )
           => ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),A)) ) ) ).

tff(fact_147_imageE,axiom,
    ! [B: hoare_1544627872iple_a,F_1: fun_pn1711459225iple_a,A: fun_pname_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,B),hAPP_f856094675a_bool(image_1832685281iple_a(F_1),A)))
     => ~ ! [X: pname] :
            ( ( B = hAPP_p441219121iple_a(F_1,X) )
           => ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),A)) ) ) ).

tff(fact_148_Body__triple__valid__Suc,axiom,
    ! [N_1: nat,Pa: fun_st1506752259e_bool,Pn_1: pname,Qa: fun_st1506752259e_bool] :
      ( hBOOL(hAPP_H1173941847e_bool(hoare_2065208407_state(N_1),hAPP_f1595846909_state(hAPP_c681126378_state(hAPP_f1930074316_state(hoare_13225861_state,Pa),hAPP_option_com_com(the_com,hAPP_p799580910on_com(body_1,Pn_1))),Qa)))
    <=> hBOOL(hAPP_H1173941847e_bool(hoare_2065208407_state(suc(N_1)),hAPP_f1595846909_state(hAPP_c681126378_state(hAPP_f1930074316_state(hoare_13225861_state,Pa),hAPP_pname_com(body,Pn_1)),Qa))) ) ).

tff(fact_149_Body__triple__valid__Suc,axiom,
    ! [N_1: nat,Pa: fun_a_fun_state_bool,Pn_1: pname,Qa: fun_a_fun_state_bool] :
      ( hBOOL(hAPP_H2006434664a_bool(hoare_1500711466alid_a(N_1),hAPP_f328191961iple_a(hAPP_c1432218316iple_a(hAPP_f1102554477iple_a(hoare_196563068iple_a,Pa),hAPP_option_com_com(the_com,hAPP_p799580910on_com(body_1,Pn_1))),Qa)))
    <=> hBOOL(hAPP_H2006434664a_bool(hoare_1500711466alid_a(suc(N_1)),hAPP_f328191961iple_a(hAPP_c1432218316iple_a(hAPP_f1102554477iple_a(hoare_196563068iple_a,Pa),hAPP_pname_com(body,Pn_1)),Qa))) ) ).

tff(fact_150_triple_Oexhaust,axiom,
    ! [Y_1: hoare_1544627872iple_a] :
      ~ ! [Fun1_1: fun_a_fun_state_bool,Com_2: com,Fun2_1: fun_a_fun_state_bool] : Y_1 != hAPP_f328191961iple_a(hAPP_c1432218316iple_a(hAPP_f1102554477iple_a(hoare_196563068iple_a,Fun1_1),Com_2),Fun2_1) ).

tff(fact_151_triple_Oexhaust,axiom,
    ! [Y_1: hoare_1875481847_state] :
      ~ ! [Fun1_1: fun_st1506752259e_bool,Com_2: com,Fun2_1: fun_st1506752259e_bool] : Y_1 != hAPP_f1595846909_state(hAPP_c681126378_state(hAPP_f1930074316_state(hoare_13225861_state,Fun1_1),Com_2),Fun2_1) ).

tff(fact_152_Body1,axiom,
    ! [Pn_1: pname,Ga: fun_Ho1996104121e_bool,Pa: fun_pn12888928e_bool,Qa: fun_pn12888928e_bool,Procsa: fun_pname_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,Ga),hAPP_f1291720380e_bool(image_1283223414_state(hAPP_f1568887991_state(hAPP_f891245078_state(cOMBS_475568287_state,hAPP_f2116202538_state(hAPP_f1339343196_state(cOMBS_1674056844_state,hAPP_f275933352_state(cOMBB_718493681_pname(hoare_13225861_state),Pa)),body)),Qa)),Procsa))),hAPP_f1291720380e_bool(image_1283223414_state(hAPP_f1568887991_state(hAPP_f891245078_state(cOMBS_475568287_state,hAPP_f2116202538_state(hAPP_f1339343196_state(cOMBS_1674056844_state,hAPP_f275933352_state(cOMBB_718493681_pname(hoare_13225861_state),Pa)),hAPP_f96342628me_com(cOMBB_923936821_pname(the_com),body_1))),Qa)),Procsa)))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,Pn_1),Procsa))
       => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hAPP_f1595846909_state(hAPP_c681126378_state(hAPP_f1930074316_state(hoare_13225861_state,hAPP_p1637813682e_bool(Pa,Pn_1)),hAPP_pname_com(body,Pn_1)),hAPP_p1637813682e_bool(Qa,Pn_1))),bot_bo1715400655e_bool))) ) ) ).

tff(fact_153_Body1,axiom,
    ! [Pn_1: pname,Ga: fun_Ho287446294a_bool,Pa: fun_pn1683930517e_bool,Qa: fun_pn1683930517e_bool,Procsa: fun_pname_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,Ga),hAPP_f856094675a_bool(image_1832685281iple_a(hAPP_f104000281iple_a(hAPP_f2028456860iple_a(cOMBS_438391419iple_a,hAPP_f68608908iple_a(hAPP_f1015300700iple_a(cOMBS_2128932974iple_a,hAPP_f298026029iple_a(cOMBB_124251824_pname(hoare_196563068iple_a),Pa)),body)),Qa)),Procsa))),hAPP_f856094675a_bool(image_1832685281iple_a(hAPP_f104000281iple_a(hAPP_f2028456860iple_a(cOMBS_438391419iple_a,hAPP_f68608908iple_a(hAPP_f1015300700iple_a(cOMBS_2128932974iple_a,hAPP_f298026029iple_a(cOMBB_124251824_pname(hoare_196563068iple_a),Pa)),hAPP_f96342628me_com(cOMBB_923936821_pname(the_com),body_1))),Qa)),Procsa)))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,Pn_1),Procsa))
       => hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hAPP_f328191961iple_a(hAPP_c1432218316iple_a(hAPP_f1102554477iple_a(hoare_196563068iple_a,hAPP_p635540397e_bool(Pa,Pn_1)),hAPP_pname_com(body,Pn_1)),hAPP_p635540397e_bool(Qa,Pn_1))),bot_bo1766443648a_bool))) ) ) ).

tff(fact_154_image__cong,axiom,
    ! [F_1: fun_pn664418900_state,G: fun_pn664418900_state,M_3: fun_pname_bool,N_3: fun_pname_bool] :
      ( ( M_3 = N_3 )
     => ( ! [X: pname] :
            ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),N_3))
           => ( hAPP_p702226982_state(F_1,X) = hAPP_p702226982_state(G,X) ) )
       => ( hAPP_f1291720380e_bool(image_1283223414_state(F_1),M_3) = hAPP_f1291720380e_bool(image_1283223414_state(G),N_3) ) ) ) ).

tff(fact_155_image__cong,axiom,
    ! [F_1: fun_pn1711459225iple_a,G: fun_pn1711459225iple_a,M_3: fun_pname_bool,N_3: fun_pname_bool] :
      ( ( M_3 = N_3 )
     => ( ! [X: pname] :
            ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),N_3))
           => ( hAPP_p441219121iple_a(F_1,X) = hAPP_p441219121iple_a(G,X) ) )
       => ( hAPP_f856094675a_bool(image_1832685281iple_a(F_1),M_3) = hAPP_f856094675a_bool(image_1832685281iple_a(G),N_3) ) ) ) ).

tff(fact_156_Body__triple__valid__0,axiom,
    ! [Pa: fun_st1506752259e_bool,Pn_1: pname,Qa: fun_st1506752259e_bool] : hBOOL(hAPP_H1173941847e_bool(hoare_2065208407_state(zero_zero_nat),hAPP_f1595846909_state(hAPP_c681126378_state(hAPP_f1930074316_state(hoare_13225861_state,Pa),hAPP_pname_com(body,Pn_1)),Qa))) ).

tff(fact_157_Body__triple__valid__0,axiom,
    ! [Pa: fun_a_fun_state_bool,Pn_1: pname,Qa: fun_a_fun_state_bool] : hBOOL(hAPP_H2006434664a_bool(hoare_1500711466alid_a(zero_zero_nat),hAPP_f328191961iple_a(hAPP_c1432218316iple_a(hAPP_f1102554477iple_a(hoare_196563068iple_a,Pa),hAPP_pname_com(body,Pn_1)),Qa))) ).

tff(fact_158_com_Osimps_I6_J,axiom,
    ! [Pname_3: pname,Pname_2: pname] :
      ( ( hAPP_pname_com(body,Pname_3) = hAPP_pname_com(body,Pname_2) )
    <=> ( Pname_3 = Pname_2 ) ) ).

tff(fact_159_evalc_OBody,axiom,
    ! [Pn: pname,S0: state,S1_1: state] :
      ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(hAPP_option_com_com(the_com,hAPP_p799580910on_com(body_1,Pn))),S0),S1_1))
     => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(hAPP_pname_com(body,Pn)),S0),S1_1)) ) ).

tff(fact_160_evalc__elim__cases_I6_J,axiom,
    ! [P: pname,S_2: state,S1_1: state] :
      ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(hAPP_pname_com(body,P)),S_2),S1_1))
     => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(hAPP_option_com_com(the_com,hAPP_p799580910on_com(body_1,P))),S_2),S1_1)) ) ).

tff(fact_161_Sup__fin_Oidem,axiom,
    ! [X_2: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X_2),X_2) = X_2 ).

tff(fact_162_Sup__fin_Oidem,axiom,
    ! [X_2: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),X_2) = X_2 ).

tff(fact_163_Sup__fin_Oidem,axiom,
    ! [X_2: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),X_2))
    <=> hBOOL(X_2) ) ).

tff(fact_164_Sup__fin_Oidem,axiom,
    ! [X_2: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X_2),X_2) = X_2 ).

tff(fact_165_emptyE,axiom,
    ! [A_1: hoare_1544627872iple_a] : ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),bot_bo1766443648a_bool)) ).

tff(fact_166_emptyE,axiom,
    ! [A_1: hoare_1875481847_state] : ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),bot_bo1715400655e_bool)) ).

tff(fact_167_emptyE,axiom,
    ! [A_1: pname] : ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),bot_bo844097828e_bool)) ).

tff(fact_168_insertE,axiom,
    ! [A_1: hoare_1875481847_state,B: hoare_1875481847_state,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,B),A)))
     => ( ( A_1 != B )
       => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),A)) ) ) ).

tff(fact_169_insertE,axiom,
    ! [A_1: hoare_1544627872iple_a,B: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,B),A)))
     => ( ( A_1 != B )
       => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),A)) ) ) ).

tff(fact_170_insertE,axiom,
    ! [A_1: pname,B: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,B),A)))
     => ( ( A_1 != B )
       => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),A)) ) ) ).

tff(fact_171_insertCI,axiom,
    ! [B: hoare_1875481847_state,A_1: hoare_1875481847_state,B_1: fun_Ho1996104121e_bool] :
      ( ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),B_1))
       => ( A_1 = B ) )
     => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,B),B_1))) ) ).

tff(fact_172_insertCI,axiom,
    ! [B: hoare_1544627872iple_a,A_1: hoare_1544627872iple_a,B_1: fun_Ho287446294a_bool] :
      ( ( ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),B_1))
       => ( A_1 = B ) )
     => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,B),B_1))) ) ).

tff(fact_173_insertCI,axiom,
    ! [B: pname,A_1: pname,B_1: fun_pname_bool] :
      ( ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),B_1))
       => ( A_1 = B ) )
     => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,B),B_1))) ) ).

tff(fact_174_empty__not__insert,axiom,
    ! [A_1: hoare_1875481847_state,A: fun_Ho1996104121e_bool] : bot_bo1715400655e_bool != hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),A) ).

tff(fact_175_empty__not__insert,axiom,
    ! [A_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] : bot_bo1766443648a_bool != hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),A) ).

tff(fact_176_empty__not__insert,axiom,
    ! [A_1: pname,A: fun_pname_bool] : bot_bo844097828e_bool != hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),A) ).

tff(fact_177_insert__not__empty,axiom,
    ! [A_1: hoare_1875481847_state,A: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),A) != bot_bo1715400655e_bool ).

tff(fact_178_insert__not__empty,axiom,
    ! [A_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),A) != bot_bo1766443648a_bool ).

tff(fact_179_insert__not__empty,axiom,
    ! [A_1: pname,A: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),A) != bot_bo844097828e_bool ).

tff(fact_180_bot__empty__eq,axiom,
    ! [X: hoare_1544627872iple_a] :
      ( hBOOL(hAPP_H2006434664a_bool(bot_bo1766443648a_bool,X))
    <=> hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X),bot_bo1766443648a_bool)) ) ).

tff(fact_181_bot__empty__eq,axiom,
    ! [X: hoare_1875481847_state] :
      ( hBOOL(hAPP_H1173941847e_bool(bot_bo1715400655e_bool,X))
    <=> hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X),bot_bo1715400655e_bool)) ) ).

tff(fact_182_bot__empty__eq,axiom,
    ! [X: pname] :
      ( hBOOL(hAPP_pname_bool(bot_bo844097828e_bool,X))
    <=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),bot_bo844097828e_bool)) ) ).

tff(fact_183_empty__def,axiom,
    bot_bo1766443648a_bool = hAPP_f909437487a_bool(collec445968053iple_a,hAPP_b704538998a_bool(cOMBK_329760839iple_a,fFalse)) ).

tff(fact_184_empty__def,axiom,
    bot_bo844097828e_bool = hAPP_f759274231e_bool(collect_pname,hAPP_b76515610e_bool(cOMBK_bool_pname,fFalse)) ).

tff(fact_185_empty__def,axiom,
    bot_bo1715400655e_bool = hAPP_f121055253e_bool(collec1735317154_state,hAPP_b1906038105e_bool(cOMBK_2086958736_state,fFalse)) ).

tff(fact_186_insertI1,axiom,
    ! [A_1: hoare_1875481847_state,B_1: fun_Ho1996104121e_bool] : hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),B_1))) ).

tff(fact_187_insertI1,axiom,
    ! [A_1: hoare_1544627872iple_a,B_1: fun_Ho287446294a_bool] : hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),B_1))) ).

tff(fact_188_insertI1,axiom,
    ! [A_1: pname,B_1: fun_pname_bool] : hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),B_1))) ).

tff(fact_189_all__not__in__conv,axiom,
    ! [A: fun_Ho287446294a_bool] :
      ( ! [X: hoare_1544627872iple_a] : ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X),A))
    <=> ( A = bot_bo1766443648a_bool ) ) ).

tff(fact_190_all__not__in__conv,axiom,
    ! [A: fun_Ho1996104121e_bool] :
      ( ! [X: hoare_1875481847_state] : ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X),A))
    <=> ( A = bot_bo1715400655e_bool ) ) ).

tff(fact_191_all__not__in__conv,axiom,
    ! [A: fun_pname_bool] :
      ( ! [X: pname] : ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),A))
    <=> ( A = bot_bo844097828e_bool ) ) ).

tff(fact_192_singleton__conv2,axiom,
    ! [A_1: hoare_1875481847_state] : hAPP_f121055253e_bool(collec1735317154_state,hAPP_H117293375e_bool(fequal391417144_state,A_1)) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),bot_bo1715400655e_bool) ).

tff(fact_193_singleton__conv2,axiom,
    ! [A_1: hoare_1544627872iple_a] : hAPP_f909437487a_bool(collec445968053iple_a,hAPP_H1472549281a_bool(fequal1057774495iple_a,A_1)) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),bot_bo1766443648a_bool) ).

tff(fact_194_singleton__conv2,axiom,
    ! [A_1: pname] : hAPP_f759274231e_bool(collect_pname,hAPP_p61793385e_bool(fequal_pname,A_1)) = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),bot_bo844097828e_bool) ).

tff(fact_195_ex__in__conv,axiom,
    ! [A: fun_Ho287446294a_bool] :
      ( ? [X: hoare_1544627872iple_a] : hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X),A))
    <=> ( A != bot_bo1766443648a_bool ) ) ).

tff(fact_196_ex__in__conv,axiom,
    ! [A: fun_Ho1996104121e_bool] :
      ( ? [X: hoare_1875481847_state] : hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X),A))
    <=> ( A != bot_bo1715400655e_bool ) ) ).

tff(fact_197_ex__in__conv,axiom,
    ! [A: fun_pname_bool] :
      ( ? [X: pname] : hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),A))
    <=> ( A != bot_bo844097828e_bool ) ) ).

tff(fact_198_singleton__conv,axiom,
    ! [A_1: hoare_1875481847_state] : hAPP_f121055253e_bool(collec1735317154_state,hAPP_H117293375e_bool(hAPP_f2101312745e_bool(cOMBC_1972244470e_bool,fequal391417144_state),A_1)) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),bot_bo1715400655e_bool) ).

tff(fact_199_singleton__conv,axiom,
    ! [A_1: hoare_1544627872iple_a] : hAPP_f909437487a_bool(collec445968053iple_a,hAPP_H1472549281a_bool(hAPP_f1878404769a_bool(cOMBC_827727450a_bool,fequal1057774495iple_a),A_1)) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),bot_bo1766443648a_bool) ).

tff(fact_200_singleton__conv,axiom,
    ! [A_1: pname] : hAPP_f759274231e_bool(collect_pname,hAPP_p61793385e_bool(hAPP_f22061361e_bool(cOMBC_1149511130e_bool,fequal_pname),A_1)) = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),bot_bo844097828e_bool) ).

tff(fact_201_Collect__conv__if2,axiom,
    ! [Pa: fun_Ho1996104121e_bool,A_1: hoare_1875481847_state] :
      ( ( hBOOL(hAPP_H1173941847e_bool(Pa,A_1))
       => ( hAPP_f121055253e_bool(collec1735317154_state,hAPP_f121055253e_bool(hAPP_f1300953505e_bool(cOMBS_896408300l_bool,hAPP_f932622461l_bool(cOMBB_242064253_state(fconj),hAPP_H117293375e_bool(fequal391417144_state,A_1))),Pa)) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),bot_bo1715400655e_bool) ) )
      & ( ~ hBOOL(hAPP_H1173941847e_bool(Pa,A_1))
       => ( hAPP_f121055253e_bool(collec1735317154_state,hAPP_f121055253e_bool(hAPP_f1300953505e_bool(cOMBS_896408300l_bool,hAPP_f932622461l_bool(cOMBB_242064253_state(fconj),hAPP_H117293375e_bool(fequal391417144_state,A_1))),Pa)) = bot_bo1715400655e_bool ) ) ) ).

tff(fact_202_Collect__conv__if2,axiom,
    ! [Pa: fun_Ho287446294a_bool,A_1: hoare_1544627872iple_a] :
      ( ( hBOOL(hAPP_H2006434664a_bool(Pa,A_1))
       => ( hAPP_f909437487a_bool(collec445968053iple_a,hAPP_f909437487a_bool(hAPP_f1099051562a_bool(cOMBS_523298619l_bool,hAPP_f224574947l_bool(cOMBB_1960865690iple_a(fconj),hAPP_H1472549281a_bool(fequal1057774495iple_a,A_1))),Pa)) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),bot_bo1766443648a_bool) ) )
      & ( ~ hBOOL(hAPP_H2006434664a_bool(Pa,A_1))
       => ( hAPP_f909437487a_bool(collec445968053iple_a,hAPP_f909437487a_bool(hAPP_f1099051562a_bool(cOMBS_523298619l_bool,hAPP_f224574947l_bool(cOMBB_1960865690iple_a(fconj),hAPP_H1472549281a_bool(fequal1057774495iple_a,A_1))),Pa)) = bot_bo1766443648a_bool ) ) ) ).

tff(fact_203_Collect__conv__if2,axiom,
    ! [Pa: fun_pname_bool,A_1: pname] :
      ( ( hBOOL(hAPP_pname_bool(Pa,A_1))
       => ( hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,hAPP_f1402196763l_bool(cOMBB_675860798_pname(fconj),hAPP_p61793385e_bool(fequal_pname,A_1))),Pa)) = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),bot_bo844097828e_bool) ) )
      & ( ~ hBOOL(hAPP_pname_bool(Pa,A_1))
       => ( hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,hAPP_f1402196763l_bool(cOMBB_675860798_pname(fconj),hAPP_p61793385e_bool(fequal_pname,A_1))),Pa)) = bot_bo844097828e_bool ) ) ) ).

tff(fact_204_Collect__conv__if,axiom,
    ! [Pa: fun_Ho1996104121e_bool,A_1: hoare_1875481847_state] :
      ( ( hBOOL(hAPP_H1173941847e_bool(Pa,A_1))
       => ( hAPP_f121055253e_bool(collec1735317154_state,hAPP_f121055253e_bool(hAPP_f1300953505e_bool(cOMBS_896408300l_bool,hAPP_f932622461l_bool(cOMBB_242064253_state(fconj),hAPP_H117293375e_bool(hAPP_f2101312745e_bool(cOMBC_1972244470e_bool,fequal391417144_state),A_1))),Pa)) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),bot_bo1715400655e_bool) ) )
      & ( ~ hBOOL(hAPP_H1173941847e_bool(Pa,A_1))
       => ( hAPP_f121055253e_bool(collec1735317154_state,hAPP_f121055253e_bool(hAPP_f1300953505e_bool(cOMBS_896408300l_bool,hAPP_f932622461l_bool(cOMBB_242064253_state(fconj),hAPP_H117293375e_bool(hAPP_f2101312745e_bool(cOMBC_1972244470e_bool,fequal391417144_state),A_1))),Pa)) = bot_bo1715400655e_bool ) ) ) ).

tff(fact_205_Collect__conv__if,axiom,
    ! [Pa: fun_Ho287446294a_bool,A_1: hoare_1544627872iple_a] :
      ( ( hBOOL(hAPP_H2006434664a_bool(Pa,A_1))
       => ( hAPP_f909437487a_bool(collec445968053iple_a,hAPP_f909437487a_bool(hAPP_f1099051562a_bool(cOMBS_523298619l_bool,hAPP_f224574947l_bool(cOMBB_1960865690iple_a(fconj),hAPP_H1472549281a_bool(hAPP_f1878404769a_bool(cOMBC_827727450a_bool,fequal1057774495iple_a),A_1))),Pa)) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),bot_bo1766443648a_bool) ) )
      & ( ~ hBOOL(hAPP_H2006434664a_bool(Pa,A_1))
       => ( hAPP_f909437487a_bool(collec445968053iple_a,hAPP_f909437487a_bool(hAPP_f1099051562a_bool(cOMBS_523298619l_bool,hAPP_f224574947l_bool(cOMBB_1960865690iple_a(fconj),hAPP_H1472549281a_bool(hAPP_f1878404769a_bool(cOMBC_827727450a_bool,fequal1057774495iple_a),A_1))),Pa)) = bot_bo1766443648a_bool ) ) ) ).

tff(fact_206_Collect__conv__if,axiom,
    ! [Pa: fun_pname_bool,A_1: pname] :
      ( ( hBOOL(hAPP_pname_bool(Pa,A_1))
       => ( hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,hAPP_f1402196763l_bool(cOMBB_675860798_pname(fconj),hAPP_p61793385e_bool(hAPP_f22061361e_bool(cOMBC_1149511130e_bool,fequal_pname),A_1))),Pa)) = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),bot_bo844097828e_bool) ) )
      & ( ~ hBOOL(hAPP_pname_bool(Pa,A_1))
       => ( hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,hAPP_f1402196763l_bool(cOMBB_675860798_pname(fconj),hAPP_p61793385e_bool(hAPP_f22061361e_bool(cOMBC_1149511130e_bool,fequal_pname),A_1))),Pa)) = bot_bo844097828e_bool ) ) ) ).

tff(fact_207_empty__Collect__eq,axiom,
    ! [Pa: fun_Ho287446294a_bool] :
      ( ( bot_bo1766443648a_bool = hAPP_f909437487a_bool(collec445968053iple_a,Pa) )
    <=> ! [X: hoare_1544627872iple_a] : ~ hBOOL(hAPP_H2006434664a_bool(Pa,X)) ) ).

tff(fact_208_empty__Collect__eq,axiom,
    ! [Pa: fun_pname_bool] :
      ( ( bot_bo844097828e_bool = hAPP_f759274231e_bool(collect_pname,Pa) )
    <=> ! [X: pname] : ~ hBOOL(hAPP_pname_bool(Pa,X)) ) ).

tff(fact_209_empty__Collect__eq,axiom,
    ! [Pa: fun_Ho1996104121e_bool] :
      ( ( bot_bo1715400655e_bool = hAPP_f121055253e_bool(collec1735317154_state,Pa) )
    <=> ! [X: hoare_1875481847_state] : ~ hBOOL(hAPP_H1173941847e_bool(Pa,X)) ) ).

tff(fact_210_mem__def,axiom,
    ! [X_2: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_2),A))
    <=> hBOOL(hAPP_H2006434664a_bool(A,X_2)) ) ).

tff(fact_211_mem__def,axiom,
    ! [X_2: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
    <=> hBOOL(hAPP_pname_bool(A,X_2)) ) ).

tff(fact_212_Collect__def,axiom,
    ! [Pa: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(collec445968053iple_a,Pa) = Pa ).

tff(fact_213_Collect__def,axiom,
    ! [Pa: fun_pname_bool] : hAPP_f759274231e_bool(collect_pname,Pa) = Pa ).

tff(fact_214_empty__iff,axiom,
    ! [C: hoare_1544627872iple_a] : ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),bot_bo1766443648a_bool)) ).

tff(fact_215_empty__iff,axiom,
    ! [C: hoare_1875481847_state] : ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,C),bot_bo1715400655e_bool)) ).

tff(fact_216_empty__iff,axiom,
    ! [C: pname] : ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),bot_bo844097828e_bool)) ).

tff(fact_217_insert__compr,axiom,
    ! [A_1: hoare_1875481847_state,B_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),B_1) = hAPP_f121055253e_bool(collec1735317154_state,hAPP_f121055253e_bool(hAPP_f1300953505e_bool(cOMBS_896408300l_bool,hAPP_f932622461l_bool(cOMBB_242064253_state(fdisj),hAPP_H117293375e_bool(hAPP_f2101312745e_bool(cOMBC_1972244470e_bool,fequal391417144_state),A_1))),hAPP_f121055253e_bool(hAPP_f1795624711e_bool(cOMBC_1099403170l_bool,member618553700_state),B_1))) ).

tff(fact_218_insert__compr,axiom,
    ! [A_1: hoare_1544627872iple_a,B_1: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),B_1) = hAPP_f909437487a_bool(collec445968053iple_a,hAPP_f909437487a_bool(hAPP_f1099051562a_bool(cOMBS_523298619l_bool,hAPP_f224574947l_bool(cOMBB_1960865690iple_a(fdisj),hAPP_H1472549281a_bool(hAPP_f1878404769a_bool(cOMBC_827727450a_bool,fequal1057774495iple_a),A_1))),hAPP_f909437487a_bool(hAPP_f482270129a_bool(cOMBC_1807206788l_bool,member1891733107iple_a),B_1))) ).

tff(fact_219_insert__compr,axiom,
    ! [A_1: pname,B_1: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),B_1) = hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,hAPP_f1402196763l_bool(cOMBB_675860798_pname(fdisj),hAPP_p61793385e_bool(hAPP_f22061361e_bool(cOMBC_1149511130e_bool,fequal_pname),A_1))),hAPP_f759274231e_bool(hAPP_f661147897e_bool(cOMBC_1058051404l_bool,member_pname),B_1))) ).

tff(fact_220_insert__Collect,axiom,
    ! [A_1: hoare_1875481847_state,Pa: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),hAPP_f121055253e_bool(collec1735317154_state,Pa)) = hAPP_f121055253e_bool(collec1735317154_state,hAPP_f121055253e_bool(hAPP_f1300953505e_bool(cOMBS_896408300l_bool,hAPP_f932622461l_bool(cOMBB_242064253_state(fimplies),hAPP_f121055253e_bool(cOMBB_1424001755_state(fNot),hAPP_H117293375e_bool(hAPP_f2101312745e_bool(cOMBC_1972244470e_bool,fequal391417144_state),A_1)))),Pa)) ).

tff(fact_221_insert__Collect,axiom,
    ! [A_1: hoare_1544627872iple_a,Pa: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),hAPP_f909437487a_bool(collec445968053iple_a,Pa)) = hAPP_f909437487a_bool(collec445968053iple_a,hAPP_f909437487a_bool(hAPP_f1099051562a_bool(cOMBS_523298619l_bool,hAPP_f224574947l_bool(cOMBB_1960865690iple_a(fimplies),hAPP_f909437487a_bool(cOMBB_1977449916iple_a(fNot),hAPP_H1472549281a_bool(hAPP_f1878404769a_bool(cOMBC_827727450a_bool,fequal1057774495iple_a),A_1)))),Pa)) ).

tff(fact_222_insert__Collect,axiom,
    ! [A_1: pname,Pa: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),hAPP_f759274231e_bool(collect_pname,Pa)) = hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,hAPP_f1402196763l_bool(cOMBB_675860798_pname(fimplies),hAPP_f759274231e_bool(cOMBB_647938656_pname(fNot),hAPP_p61793385e_bool(hAPP_f22061361e_bool(cOMBC_1149511130e_bool,fequal_pname),A_1)))),Pa)) ).

tff(fact_223_singleton__iff,axiom,
    ! [B: hoare_1875481847_state,A_1: hoare_1875481847_state] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,B),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),bot_bo1715400655e_bool)))
    <=> ( B = A_1 ) ) ).

tff(fact_224_singleton__iff,axiom,
    ! [B: hoare_1544627872iple_a,A_1: hoare_1544627872iple_a] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,B),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),bot_bo1766443648a_bool)))
    <=> ( B = A_1 ) ) ).

tff(fact_225_singleton__iff,axiom,
    ! [B: pname,A_1: pname] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,B),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),bot_bo844097828e_bool)))
    <=> ( B = A_1 ) ) ).

tff(fact_226_insert__absorb2,axiom,
    ! [X_2: hoare_1875481847_state,A: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),A)) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),A) ).

tff(fact_227_insert__absorb2,axiom,
    ! [X_2: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_2),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_2),A)) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_2),A) ).

tff(fact_228_insert__absorb2,axiom,
    ! [X_2: pname,A: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),A)) = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),A) ).

tff(fact_229_insert__commute,axiom,
    ! [X_2: hoare_1875481847_state,Y_1: hoare_1875481847_state,A: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,Y_1),A)) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,Y_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),A)) ).

tff(fact_230_insert__commute,axiom,
    ! [X_2: hoare_1544627872iple_a,Y_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_2),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,Y_1),A)) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,Y_1),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_2),A)) ).

tff(fact_231_insert__commute,axiom,
    ! [X_2: pname,Y_1: pname,A: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,Y_1),A)) = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,Y_1),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),A)) ).

tff(fact_232_insert__iff,axiom,
    ! [A_1: hoare_1875481847_state,B: hoare_1875481847_state,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,B),A)))
    <=> ( ( A_1 = B )
        | hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),A)) ) ) ).

tff(fact_233_insert__iff,axiom,
    ! [A_1: hoare_1544627872iple_a,B: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,B),A)))
    <=> ( ( A_1 = B )
        | hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),A)) ) ) ).

tff(fact_234_insert__iff,axiom,
    ! [A_1: pname,B: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,B),A)))
    <=> ( ( A_1 = B )
        | hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),A)) ) ) ).

tff(fact_235_Collect__empty__eq,axiom,
    ! [Pa: fun_Ho287446294a_bool] :
      ( ( hAPP_f909437487a_bool(collec445968053iple_a,Pa) = bot_bo1766443648a_bool )
    <=> ! [X: hoare_1544627872iple_a] : ~ hBOOL(hAPP_H2006434664a_bool(Pa,X)) ) ).

tff(fact_236_Collect__empty__eq,axiom,
    ! [Pa: fun_pname_bool] :
      ( ( hAPP_f759274231e_bool(collect_pname,Pa) = bot_bo844097828e_bool )
    <=> ! [X: pname] : ~ hBOOL(hAPP_pname_bool(Pa,X)) ) ).

tff(fact_237_Collect__empty__eq,axiom,
    ! [Pa: fun_Ho1996104121e_bool] :
      ( ( hAPP_f121055253e_bool(collec1735317154_state,Pa) = bot_bo1715400655e_bool )
    <=> ! [X: hoare_1875481847_state] : ~ hBOOL(hAPP_H1173941847e_bool(Pa,X)) ) ).

tff(fact_238_doubleton__eq__iff,axiom,
    ! [A_1: hoare_1875481847_state,B: hoare_1875481847_state,C: hoare_1875481847_state,D: hoare_1875481847_state] :
      ( ( hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,B),bot_bo1715400655e_bool)) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,C),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,D),bot_bo1715400655e_bool)) )
    <=> ( ( ( A_1 = C )
          & ( B = D ) )
        | ( ( A_1 = D )
          & ( B = C ) ) ) ) ).

tff(fact_239_doubleton__eq__iff,axiom,
    ! [A_1: hoare_1544627872iple_a,B: hoare_1544627872iple_a,C: hoare_1544627872iple_a,D: hoare_1544627872iple_a] :
      ( ( hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,B),bot_bo1766443648a_bool)) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,C),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,D),bot_bo1766443648a_bool)) )
    <=> ( ( ( A_1 = C )
          & ( B = D ) )
        | ( ( A_1 = D )
          & ( B = C ) ) ) ) ).

tff(fact_240_doubleton__eq__iff,axiom,
    ! [A_1: pname,B: pname,C: pname,D: pname] :
      ( ( hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,B),bot_bo844097828e_bool)) = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,C),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,D),bot_bo844097828e_bool)) )
    <=> ( ( ( A_1 = C )
          & ( B = D ) )
        | ( ( A_1 = D )
          & ( B = C ) ) ) ) ).

tff(fact_241_insert__code,axiom,
    ! [Y_1: hoare_1875481847_state,A: fun_Ho1996104121e_bool,X_2: hoare_1875481847_state] :
      ( hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,Y_1),A),X_2))
    <=> ( ( Y_1 = X_2 )
        | hBOOL(hAPP_H1173941847e_bool(A,X_2)) ) ) ).

tff(fact_242_insert__code,axiom,
    ! [Y_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,X_2: hoare_1544627872iple_a] :
      ( hBOOL(hAPP_H2006434664a_bool(hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,Y_1),A),X_2))
    <=> ( ( Y_1 = X_2 )
        | hBOOL(hAPP_H2006434664a_bool(A,X_2)) ) ) ).

tff(fact_243_insert__code,axiom,
    ! [Y_1: pname,A: fun_pname_bool,X_2: pname] :
      ( hBOOL(hAPP_pname_bool(hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,Y_1),A),X_2))
    <=> ( ( Y_1 = X_2 )
        | hBOOL(hAPP_pname_bool(A,X_2)) ) ) ).

tff(fact_244_insert__ident,axiom,
    ! [B_1: fun_Ho1996104121e_bool,X_2: hoare_1875481847_state,A: fun_Ho1996104121e_bool] :
      ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),A))
     => ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),B_1))
       => ( ( hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),A) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),B_1) )
        <=> ( A = B_1 ) ) ) ) ).

tff(fact_245_insert__ident,axiom,
    ! [B_1: fun_Ho287446294a_bool,X_2: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_2),A))
     => ( ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_2),B_1))
       => ( ( hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_2),A) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_2),B_1) )
        <=> ( A = B_1 ) ) ) ) ).

tff(fact_246_insert__ident,axiom,
    ! [B_1: fun_pname_bool,X_2: pname,A: fun_pname_bool] :
      ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
     => ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),B_1))
       => ( ( hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),A) = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),B_1) )
        <=> ( A = B_1 ) ) ) ) ).

tff(fact_247_equals0D,axiom,
    ! [A_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( ( A = bot_bo1766443648a_bool )
     => ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),A)) ) ).

tff(fact_248_equals0D,axiom,
    ! [A_1: hoare_1875481847_state,A: fun_Ho1996104121e_bool] :
      ( ( A = bot_bo1715400655e_bool )
     => ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),A)) ) ).

tff(fact_249_equals0D,axiom,
    ! [A_1: pname,A: fun_pname_bool] :
      ( ( A = bot_bo844097828e_bool )
     => ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),A)) ) ).

tff(fact_250_insertI2,axiom,
    ! [B: hoare_1875481847_state,A_1: hoare_1875481847_state,B_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),B_1))
     => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,B),B_1))) ) ).

tff(fact_251_insertI2,axiom,
    ! [B: hoare_1544627872iple_a,A_1: hoare_1544627872iple_a,B_1: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),B_1))
     => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,B),B_1))) ) ).

tff(fact_252_insertI2,axiom,
    ! [B: pname,A_1: pname,B_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),B_1))
     => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,B),B_1))) ) ).

tff(fact_253_insert__absorb,axiom,
    ! [A_1: hoare_1875481847_state,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),A))
     => ( hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),A) = A ) ) ).

tff(fact_254_insert__absorb,axiom,
    ! [A_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),A))
     => ( hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),A) = A ) ) ).

tff(fact_255_insert__absorb,axiom,
    ! [A_1: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),A))
     => ( hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),A) = A ) ) ).

tff(fact_256_singletonE,axiom,
    ! [B: hoare_1875481847_state,A_1: hoare_1875481847_state] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,B),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),bot_bo1715400655e_bool)))
     => ( B = A_1 ) ) ).

tff(fact_257_singletonE,axiom,
    ! [B: hoare_1544627872iple_a,A_1: hoare_1544627872iple_a] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,B),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),bot_bo1766443648a_bool)))
     => ( B = A_1 ) ) ).

tff(fact_258_singletonE,axiom,
    ! [B: pname,A_1: pname] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,B),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),bot_bo844097828e_bool)))
     => ( B = A_1 ) ) ).

tff(fact_259_singleton__inject,axiom,
    ! [A_1: hoare_1875481847_state,B: hoare_1875481847_state] :
      ( ( hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),bot_bo1715400655e_bool) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,B),bot_bo1715400655e_bool) )
     => ( A_1 = B ) ) ).

tff(fact_260_singleton__inject,axiom,
    ! [A_1: hoare_1544627872iple_a,B: hoare_1544627872iple_a] :
      ( ( hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),bot_bo1766443648a_bool) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,B),bot_bo1766443648a_bool) )
     => ( A_1 = B ) ) ).

tff(fact_261_singleton__inject,axiom,
    ! [A_1: pname,B: pname] :
      ( ( hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),bot_bo844097828e_bool) = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,B),bot_bo844097828e_bool) )
     => ( A_1 = B ) ) ).

tff(fact_262_com__det,axiom,
    ! [U_1: state,C_2: com,S_2: state,T_1: state] :
      ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C_2),S_2),T_1))
     => ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C_2),S_2),U_1))
       => ( U_1 = T_1 ) ) ) ).

tff(fact_263_insert__is__Un,axiom,
    ! [A_1: hoare_1875481847_state,A: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),A) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),bot_bo1715400655e_bool)),A) ).

tff(fact_264_insert__is__Un,axiom,
    ! [A_1: pname,A: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),A) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),bot_bo844097828e_bool)),A) ).

tff(fact_265_insert__is__Un,axiom,
    ! [A_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),A) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),bot_bo1766443648a_bool)),A) ).

tff(fact_266_insert__compr__raw,axiom,
    ! [X: hoare_1875481847_state,Xa: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X),Xa) = hAPP_f121055253e_bool(collec1735317154_state,hAPP_f121055253e_bool(hAPP_f1300953505e_bool(cOMBS_896408300l_bool,hAPP_f932622461l_bool(cOMBB_242064253_state(fdisj),hAPP_H117293375e_bool(hAPP_f2101312745e_bool(cOMBC_1972244470e_bool,fequal391417144_state),X))),hAPP_f121055253e_bool(hAPP_f1795624711e_bool(cOMBC_1099403170l_bool,member618553700_state),Xa))) ).

tff(fact_267_insert__compr__raw,axiom,
    ! [X: hoare_1544627872iple_a,Xa: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X),Xa) = hAPP_f909437487a_bool(collec445968053iple_a,hAPP_f909437487a_bool(hAPP_f1099051562a_bool(cOMBS_523298619l_bool,hAPP_f224574947l_bool(cOMBB_1960865690iple_a(fdisj),hAPP_H1472549281a_bool(hAPP_f1878404769a_bool(cOMBC_827727450a_bool,fequal1057774495iple_a),X))),hAPP_f909437487a_bool(hAPP_f482270129a_bool(cOMBC_1807206788l_bool,member1891733107iple_a),Xa))) ).

tff(fact_268_insert__compr__raw,axiom,
    ! [X: pname,Xa: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X),Xa) = hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,hAPP_f1402196763l_bool(cOMBB_675860798_pname(fdisj),hAPP_p61793385e_bool(hAPP_f22061361e_bool(cOMBC_1149511130e_bool,fequal_pname),X))),hAPP_f759274231e_bool(hAPP_f661147897e_bool(cOMBC_1058051404l_bool,member_pname),Xa))) ).

tff(fact_269_derivs__insertD,axiom,
    ! [Ga: fun_Ho1996104121e_bool,T: hoare_1875481847_state,Ts: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,T),Ts)))
     => ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,T),bot_bo1715400655e_bool)))
        & hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),Ts)) ) ) ).

tff(fact_270_derivs__insertD,axiom,
    ! [Ga: fun_Ho287446294a_bool,T: hoare_1544627872iple_a,Ts: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,T),Ts)))
     => ( hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,T),bot_bo1766443648a_bool)))
        & hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),Ts)) ) ) ).

tff(fact_271_hoare__derivs_Oinsert,axiom,
    ! [Ts: fun_Ho1996104121e_bool,Ga: fun_Ho1996104121e_bool,T: hoare_1875481847_state] :
      ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,T),bot_bo1715400655e_bool)))
     => ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),Ts))
       => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,T),Ts))) ) ) ).

tff(fact_272_hoare__derivs_Oinsert,axiom,
    ! [Ts: fun_Ho287446294a_bool,Ga: fun_Ho287446294a_bool,T: hoare_1544627872iple_a] :
      ( hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,T),bot_bo1766443648a_bool)))
     => ( hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),Ts))
       => hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,T),Ts))) ) ) ).

tff(fact_273_image__constant__conv,axiom,
    ! [C: hoare_1875481847_state,A: fun_pname_bool] :
      ( ( ( A = bot_bo844097828e_bool )
       => ( hAPP_f1291720380e_bool(image_1283223414_state(hAPP_H939818062_state(cOMBK_1032401873_pname,C)),A) = bot_bo1715400655e_bool ) )
      & ( ( A != bot_bo844097828e_bool )
       => ( hAPP_f1291720380e_bool(image_1283223414_state(hAPP_H939818062_state(cOMBK_1032401873_pname,C)),A) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,C),bot_bo1715400655e_bool) ) ) ) ).

tff(fact_274_image__constant__conv,axiom,
    ! [C: hoare_1544627872iple_a,A: fun_pname_bool] :
      ( ( ( A = bot_bo844097828e_bool )
       => ( hAPP_f856094675a_bool(image_1832685281iple_a(hAPP_H79278542iple_a(cOMBK_997027250_pname,C)),A) = bot_bo1766443648a_bool ) )
      & ( ( A != bot_bo844097828e_bool )
       => ( hAPP_f856094675a_bool(image_1832685281iple_a(hAPP_H79278542iple_a(cOMBK_997027250_pname,C)),A) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,C),bot_bo1766443648a_bool) ) ) ) ).

tff(fact_275_image__constant,axiom,
    ! [C: hoare_1875481847_state,X_2: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
     => ( hAPP_f1291720380e_bool(image_1283223414_state(hAPP_H939818062_state(cOMBK_1032401873_pname,C)),A) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,C),bot_bo1715400655e_bool) ) ) ).

tff(fact_276_image__constant,axiom,
    ! [C: pname,X_2: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
     => ( hAPP_f759274231e_bool(image_pname_pname(hAPP_p1170154830_pname(cOMBK_pname_pname,C)),A) = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,C),bot_bo844097828e_bool) ) ) ).

tff(fact_277_image__constant,axiom,
    ! [C: hoare_1544627872iple_a,X_2: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
     => ( hAPP_f856094675a_bool(image_1832685281iple_a(hAPP_H79278542iple_a(cOMBK_997027250_pname,C)),A) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,C),bot_bo1766443648a_bool) ) ) ).

tff(fact_278_image__insert,axiom,
    ! [F_1: fun_pn664418900_state,A_1: pname,B_1: fun_pname_bool] : hAPP_f1291720380e_bool(image_1283223414_state(F_1),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),B_1)) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hAPP_p702226982_state(F_1,A_1)),hAPP_f1291720380e_bool(image_1283223414_state(F_1),B_1)) ).

tff(fact_279_image__insert,axiom,
    ! [F_1: fun_pn1711459225iple_a,A_1: pname,B_1: fun_pname_bool] : hAPP_f856094675a_bool(image_1832685281iple_a(F_1),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),B_1)) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hAPP_p441219121iple_a(F_1,A_1)),hAPP_f856094675a_bool(image_1832685281iple_a(F_1),B_1)) ).

tff(fact_280_insert__image,axiom,
    ! [F_1: fun_pn664418900_state,X_2: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
     => ( hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hAPP_p702226982_state(F_1,X_2)),hAPP_f1291720380e_bool(image_1283223414_state(F_1),A)) = hAPP_f1291720380e_bool(image_1283223414_state(F_1),A) ) ) ).

tff(fact_281_insert__image,axiom,
    ! [F_1: fun_pname_pname,X_2: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
     => ( hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,hAPP_pname_pname(F_1,X_2)),hAPP_f759274231e_bool(image_pname_pname(F_1),A)) = hAPP_f759274231e_bool(image_pname_pname(F_1),A) ) ) ).

tff(fact_282_insert__image,axiom,
    ! [F_1: fun_pn1711459225iple_a,X_2: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
     => ( hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hAPP_p441219121iple_a(F_1,X_2)),hAPP_f856094675a_bool(image_1832685281iple_a(F_1),A)) = hAPP_f856094675a_bool(image_1832685281iple_a(F_1),A) ) ) ).

tff(fact_283_Un__insert__right,axiom,
    ! [A: fun_Ho1996104121e_bool,A_1: hoare_1875481847_state,B_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),B_1)) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),B_1)) ).

tff(fact_284_Un__insert__right,axiom,
    ! [A: fun_pname_bool,A_1: pname,B_1: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),B_1)) = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),B_1)) ).

tff(fact_285_Un__insert__right,axiom,
    ! [A: fun_Ho287446294a_bool,A_1: hoare_1544627872iple_a,B_1: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),B_1)) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B_1)) ).

tff(fact_286_Un__insert__left,axiom,
    ! [A_1: hoare_1875481847_state,B_1: fun_Ho1996104121e_bool,C_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),B_1)),C_1) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,B_1),C_1)) ).

tff(fact_287_Un__insert__left,axiom,
    ! [A_1: pname,B_1: fun_pname_bool,C_1: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),B_1)),C_1) = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,B_1),C_1)) ).

tff(fact_288_Un__insert__left,axiom,
    ! [A_1: hoare_1544627872iple_a,B_1: fun_Ho287446294a_bool,C_1: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),B_1)),C_1) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,B_1),C_1)) ).

tff(fact_289_empty__is__image,axiom,
    ! [F_1: fun_pn664418900_state,A: fun_pname_bool] :
      ( ( bot_bo1715400655e_bool = hAPP_f1291720380e_bool(image_1283223414_state(F_1),A) )
    <=> ( A = bot_bo844097828e_bool ) ) ).

tff(fact_290_empty__is__image,axiom,
    ! [F_1: fun_pn1711459225iple_a,A: fun_pname_bool] :
      ( ( bot_bo1766443648a_bool = hAPP_f856094675a_bool(image_1832685281iple_a(F_1),A) )
    <=> ( A = bot_bo844097828e_bool ) ) ).

tff(fact_291_image__empty,axiom,
    ! [F_1: fun_pn664418900_state] : hAPP_f1291720380e_bool(image_1283223414_state(F_1),bot_bo844097828e_bool) = bot_bo1715400655e_bool ).

tff(fact_292_image__empty,axiom,
    ! [F_1: fun_pn1711459225iple_a] : hAPP_f856094675a_bool(image_1832685281iple_a(F_1),bot_bo844097828e_bool) = bot_bo1766443648a_bool ).

tff(fact_293_image__is__empty,axiom,
    ! [F_1: fun_pn664418900_state,A: fun_pname_bool] :
      ( ( hAPP_f1291720380e_bool(image_1283223414_state(F_1),A) = bot_bo1715400655e_bool )
    <=> ( A = bot_bo844097828e_bool ) ) ).

tff(fact_294_image__is__empty,axiom,
    ! [F_1: fun_pn1711459225iple_a,A: fun_pname_bool] :
      ( ( hAPP_f856094675a_bool(image_1832685281iple_a(F_1),A) = bot_bo1766443648a_bool )
    <=> ( A = bot_bo844097828e_bool ) ) ).

tff(fact_295_ball__empty,axiom,
    ! [Pa: fun_Ho1996104121e_bool,X: hoare_1875481847_state] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X),bot_bo1715400655e_bool))
     => hBOOL(hAPP_H1173941847e_bool(Pa,X)) ) ).

tff(fact_296_ball__empty,axiom,
    ! [Pa: fun_Ho287446294a_bool,X: hoare_1544627872iple_a] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X),bot_bo1766443648a_bool))
     => hBOOL(hAPP_H2006434664a_bool(Pa,X)) ) ).

tff(fact_297_ball__empty,axiom,
    ! [Pa: fun_pname_bool,X: pname] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),bot_bo844097828e_bool))
     => hBOOL(hAPP_pname_bool(Pa,X)) ) ).

tff(fact_298_Un__empty__left,axiom,
    ! [B_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,bot_bo1715400655e_bool),B_1) = B_1 ).

tff(fact_299_Un__empty__left,axiom,
    ! [B_1: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,bot_bo844097828e_bool),B_1) = B_1 ).

tff(fact_300_Un__empty__left,axiom,
    ! [B_1: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,bot_bo1766443648a_bool),B_1) = B_1 ).

tff(fact_301_Un__empty__right,axiom,
    ! [A: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),bot_bo1715400655e_bool) = A ).

tff(fact_302_Un__empty__right,axiom,
    ! [A: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),bot_bo844097828e_bool) = A ).

tff(fact_303_Un__empty__right,axiom,
    ! [A: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),bot_bo1766443648a_bool) = A ).

tff(fact_304_Un__empty,axiom,
    ! [A: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( ( hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),B_1) = bot_bo1715400655e_bool )
    <=> ( ( A = bot_bo1715400655e_bool )
        & ( B_1 = bot_bo1715400655e_bool ) ) ) ).

tff(fact_305_Un__empty,axiom,
    ! [A: fun_pname_bool,B_1: fun_pname_bool] :
      ( ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),B_1) = bot_bo844097828e_bool )
    <=> ( ( A = bot_bo844097828e_bool )
        & ( B_1 = bot_bo844097828e_bool ) ) ) ).

tff(fact_306_Un__empty,axiom,
    ! [A: fun_Ho287446294a_bool,B_1: fun_Ho287446294a_bool] :
      ( ( hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B_1) = bot_bo1766443648a_bool )
    <=> ( ( A = bot_bo1766443648a_bool )
        & ( B_1 = bot_bo1766443648a_bool ) ) ) ).

tff(fact_307_constant,axiom,
    ! [Ga: fun_Ho287446294a_bool,Pa: fun_a_fun_state_bool,C: com,Qa: fun_a_fun_state_bool,C_1: bool] :
      ( ( hBOOL(C_1)
       => hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hAPP_f328191961iple_a(hAPP_c1432218316iple_a(hAPP_f1102554477iple_a(hoare_196563068iple_a,Pa),C),Qa)),bot_bo1766443648a_bool))) )
     => hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hAPP_f328191961iple_a(hAPP_c1432218316iple_a(hAPP_f1102554477iple_a(hoare_196563068iple_a,hAPP_b540892988e_bool(hAPP_f1824947087e_bool(cOMBC_41962815e_bool,hAPP_f340725611e_bool(cOMBB_1348041619bool_a(cOMBC_231445413l_bool),hAPP_f1509969235l_bool(cOMBB_1355796797bool_a(cOMBB_188601460_state(fconj)),Pa))),C_1)),C),Qa)),bot_bo1766443648a_bool))) ) ).

tff(fact_308_constant,axiom,
    ! [Ga: fun_Ho1996104121e_bool,Pa: fun_st1506752259e_bool,C: com,Qa: fun_st1506752259e_bool,C_1: bool] :
      ( ( hBOOL(C_1)
       => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hAPP_f1595846909_state(hAPP_c681126378_state(hAPP_f1930074316_state(hoare_13225861_state,Pa),C),Qa)),bot_bo1715400655e_bool))) )
     => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hAPP_f1595846909_state(hAPP_c681126378_state(hAPP_f1930074316_state(hoare_13225861_state,hAPP_b1095269219e_bool(hAPP_f701449317e_bool(cOMBC_471052088e_bool,hAPP_f873506917e_bool(cOMBB_1757942702_state(cOMBC_231445413l_bool),hAPP_f1283379615l_bool(cOMBB_1759179140_state(cOMBB_188601460_state(fconj)),Pa))),C_1)),C),Qa)),bot_bo1715400655e_bool))) ) ).

tff(fact_309_empty,axiom,
    ! [Ga: fun_Ho1996104121e_bool] : hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),bot_bo1715400655e_bool)) ).

tff(fact_310_empty,axiom,
    ! [Ga: fun_Ho287446294a_bool] : hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),bot_bo1766443648a_bool)) ).

tff(fact_311_sup__bot__left,axiom,
    ! [X_2: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,bot_bo1715400655e_bool),X_2) = X_2 ).

tff(fact_312_sup__bot__left,axiom,
    ! [X_2: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,bot_bo844097828e_bool),X_2) = X_2 ).

tff(fact_313_sup__bot__left,axiom,
    ! [X_2: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,bot_bot_bool),X_2))
    <=> hBOOL(X_2) ) ).

tff(fact_314_sup__bot__left,axiom,
    ! [X_2: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,bot_bo1766443648a_bool),X_2) = X_2 ).

tff(fact_315_sup__bot__right,axiom,
    ! [X_2: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X_2),bot_bo1715400655e_bool) = X_2 ).

tff(fact_316_sup__bot__right,axiom,
    ! [X_2: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),bot_bo844097828e_bool) = X_2 ).

tff(fact_317_sup__bot__right,axiom,
    ! [X_2: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),bot_bot_bool))
    <=> hBOOL(X_2) ) ).

tff(fact_318_sup__bot__right,axiom,
    ! [X_2: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X_2),bot_bo1766443648a_bool) = X_2 ).

tff(fact_319_sup__eq__bot__iff,axiom,
    ! [X_2: fun_Ho1996104121e_bool,Y_1: fun_Ho1996104121e_bool] :
      ( ( hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X_2),Y_1) = bot_bo1715400655e_bool )
    <=> ( ( X_2 = bot_bo1715400655e_bool )
        & ( Y_1 = bot_bo1715400655e_bool ) ) ) ).

tff(fact_320_sup__eq__bot__iff,axiom,
    ! [X_2: fun_pname_bool,Y_1: fun_pname_bool] :
      ( ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),Y_1) = bot_bo844097828e_bool )
    <=> ( ( X_2 = bot_bo844097828e_bool )
        & ( Y_1 = bot_bo844097828e_bool ) ) ) ).

tff(fact_321_sup__eq__bot__iff,axiom,
    ! [X_2: bool,Y_1: bool] :
      ( ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),Y_1))
      <=> hBOOL(bot_bot_bool) )
    <=> ( ( hBOOL(X_2)
        <=> hBOOL(bot_bot_bool) )
        & ( hBOOL(Y_1)
        <=> hBOOL(bot_bot_bool) ) ) ) ).

tff(fact_322_sup__eq__bot__iff,axiom,
    ! [X_2: fun_Ho287446294a_bool,Y_1: fun_Ho287446294a_bool] :
      ( ( hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X_2),Y_1) = bot_bo1766443648a_bool )
    <=> ( ( X_2 = bot_bo1766443648a_bool )
        & ( Y_1 = bot_bo1766443648a_bool ) ) ) ).

tff(fact_323_triple__valid__Suc,axiom,
    ! [N_4: nat,T_2: hoare_1875481847_state] :
      ( hBOOL(hAPP_H1173941847e_bool(hoare_2065208407_state(suc(N_4)),T_2))
     => hBOOL(hAPP_H1173941847e_bool(hoare_2065208407_state(N_4),T_2)) ) ).

tff(fact_324_triple__valid__Suc,axiom,
    ! [N_4: nat,T_2: hoare_1544627872iple_a] :
      ( hBOOL(hAPP_H2006434664a_bool(hoare_1500711466alid_a(suc(N_4)),T_2))
     => hBOOL(hAPP_H2006434664a_bool(hoare_1500711466alid_a(N_4),T_2)) ) ).

tff(fact_325_insert__def,axiom,
    ! [A_1: hoare_1875481847_state,B_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),B_1) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,hAPP_f121055253e_bool(collec1735317154_state,hAPP_H117293375e_bool(hAPP_f2101312745e_bool(cOMBC_1972244470e_bool,fequal391417144_state),A_1))),B_1) ).

tff(fact_326_insert__def,axiom,
    ! [A_1: pname,B_1: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),B_1) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f759274231e_bool(collect_pname,hAPP_p61793385e_bool(hAPP_f22061361e_bool(cOMBC_1149511130e_bool,fequal_pname),A_1))),B_1) ).

tff(fact_327_insert__def,axiom,
    ! [A_1: hoare_1544627872iple_a,B_1: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),B_1) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,hAPP_f909437487a_bool(collec445968053iple_a,hAPP_H1472549281a_bool(hAPP_f1878404769a_bool(cOMBC_827727450a_bool,fequal1057774495iple_a),A_1))),B_1) ).

tff(fact_328_weak__Body,axiom,
    ! [Ga: fun_Ho287446294a_bool,Pa: fun_a_fun_state_bool,Pn_1: pname,Qa: fun_a_fun_state_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hAPP_f328191961iple_a(hAPP_c1432218316iple_a(hAPP_f1102554477iple_a(hoare_196563068iple_a,Pa),hAPP_option_com_com(the_com,hAPP_p799580910on_com(body_1,Pn_1))),Qa)),bot_bo1766443648a_bool)))
     => hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hAPP_f328191961iple_a(hAPP_c1432218316iple_a(hAPP_f1102554477iple_a(hoare_196563068iple_a,Pa),hAPP_pname_com(body,Pn_1)),Qa)),bot_bo1766443648a_bool))) ) ).

tff(fact_329_weak__Body,axiom,
    ! [Ga: fun_Ho1996104121e_bool,Pa: fun_st1506752259e_bool,Pn_1: pname,Qa: fun_st1506752259e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hAPP_f1595846909_state(hAPP_c681126378_state(hAPP_f1930074316_state(hoare_13225861_state,Pa),hAPP_option_com_com(the_com,hAPP_p799580910on_com(body_1,Pn_1))),Qa)),bot_bo1715400655e_bool)))
     => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hAPP_f1595846909_state(hAPP_c681126378_state(hAPP_f1930074316_state(hoare_13225861_state,Pa),hAPP_pname_com(body,Pn_1)),Qa)),bot_bo1715400655e_bool))) ) ).

tff(fact_330_BodyN,axiom,
    ! [Pa: fun_a_fun_state_bool,Pn_1: pname,Qa: fun_a_fun_state_bool,Ga: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hAPP_f328191961iple_a(hAPP_c1432218316iple_a(hAPP_f1102554477iple_a(hoare_196563068iple_a,Pa),hAPP_pname_com(body,Pn_1)),Qa)),Ga)),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hAPP_f328191961iple_a(hAPP_c1432218316iple_a(hAPP_f1102554477iple_a(hoare_196563068iple_a,Pa),hAPP_option_com_com(the_com,hAPP_p799580910on_com(body_1,Pn_1))),Qa)),bot_bo1766443648a_bool)))
     => hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hAPP_f328191961iple_a(hAPP_c1432218316iple_a(hAPP_f1102554477iple_a(hoare_196563068iple_a,Pa),hAPP_pname_com(body,Pn_1)),Qa)),bot_bo1766443648a_bool))) ) ).

tff(fact_331_BodyN,axiom,
    ! [Pa: fun_st1506752259e_bool,Pn_1: pname,Qa: fun_st1506752259e_bool,Ga: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hAPP_f1595846909_state(hAPP_c681126378_state(hAPP_f1930074316_state(hoare_13225861_state,Pa),hAPP_pname_com(body,Pn_1)),Qa)),Ga)),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hAPP_f1595846909_state(hAPP_c681126378_state(hAPP_f1930074316_state(hoare_13225861_state,Pa),hAPP_option_com_com(the_com,hAPP_p799580910on_com(body_1,Pn_1))),Qa)),bot_bo1715400655e_bool)))
     => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hAPP_f1595846909_state(hAPP_c681126378_state(hAPP_f1930074316_state(hoare_13225861_state,Pa),hAPP_pname_com(body,Pn_1)),Qa)),bot_bo1715400655e_bool))) ) ).

tff(fact_332_triples__valid__Suc,axiom,
    ! [N_1: nat,Ts: fun_Ho1996104121e_bool] :
      ( ! [X: hoare_1875481847_state] :
          ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X),Ts))
         => hBOOL(hAPP_H1173941847e_bool(hoare_2065208407_state(suc(N_1)),X)) )
     => ! [X: hoare_1875481847_state] :
          ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X),Ts))
         => hBOOL(hAPP_H1173941847e_bool(hoare_2065208407_state(N_1),X)) ) ) ).

tff(fact_333_triples__valid__Suc,axiom,
    ! [N_1: nat,Ts: fun_Ho287446294a_bool] :
      ( ! [X: hoare_1544627872iple_a] :
          ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X),Ts))
         => hBOOL(hAPP_H2006434664a_bool(hoare_1500711466alid_a(suc(N_1)),X)) )
     => ! [X: hoare_1544627872iple_a] :
          ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X),Ts))
         => hBOOL(hAPP_H2006434664a_bool(hoare_1500711466alid_a(N_1),X)) ) ) ).

tff(fact_334_escape,axiom,
    ! [Ga: fun_Ho287446294a_bool,C: com,Qa: fun_a_fun_state_bool,Pa: fun_a_fun_state_bool] :
      ( ! [Z_3: x_a,S_3: state] :
          ( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Pa,Z_3),S_3))
         => hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hAPP_f328191961iple_a(hAPP_c1432218316iple_a(hAPP_f1102554477iple_a(hoare_196563068iple_a,hAPP_f762886889e_bool(cOMBK_1458035955bool_a,hAPP_s1806633685e_bool(hAPP_f817621513e_bool(cOMBC_2027030106e_bool,fequal_state),S_3))),C),hAPP_f762886889e_bool(cOMBK_1458035955bool_a,hAPP_a2036067514e_bool(Qa,Z_3)))),bot_bo1766443648a_bool))) )
     => hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hAPP_f328191961iple_a(hAPP_c1432218316iple_a(hAPP_f1102554477iple_a(hoare_196563068iple_a,Pa),C),Qa)),bot_bo1766443648a_bool))) ) ).

tff(fact_335_escape,axiom,
    ! [Ga: fun_Ho1996104121e_bool,C: com,Qa: fun_st1506752259e_bool,Pa: fun_st1506752259e_bool] :
      ( ! [Z_3: state,S_3: state] :
          ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Pa,Z_3),S_3))
         => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hAPP_f1595846909_state(hAPP_c681126378_state(hAPP_f1930074316_state(hoare_13225861_state,hAPP_f1863945078e_bool(cOMBK_631994958_state,hAPP_s1806633685e_bool(hAPP_f817621513e_bool(cOMBC_2027030106e_bool,fequal_state),S_3))),C),hAPP_f1863945078e_bool(cOMBK_631994958_state,hAPP_s1806633685e_bool(Qa,Z_3)))),bot_bo1715400655e_bool))) )
     => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hAPP_f1595846909_state(hAPP_c681126378_state(hAPP_f1930074316_state(hoare_13225861_state,Pa),C),Qa)),bot_bo1715400655e_bool))) ) ).

tff(fact_336_conseq1,axiom,
    ! [Pa: fun_a_fun_state_bool,Ga: fun_Ho287446294a_bool,P_1: fun_a_fun_state_bool,C: com,Qa: fun_a_fun_state_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hAPP_f328191961iple_a(hAPP_c1432218316iple_a(hAPP_f1102554477iple_a(hoare_196563068iple_a,P_1),C),Qa)),bot_bo1766443648a_bool)))
     => ( ! [Z_3: x_a,S_3: state] :
            ( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Pa,Z_3),S_3))
           => hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(P_1,Z_3),S_3)) )
       => hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hAPP_f328191961iple_a(hAPP_c1432218316iple_a(hAPP_f1102554477iple_a(hoare_196563068iple_a,Pa),C),Qa)),bot_bo1766443648a_bool))) ) ) ).

tff(fact_337_conseq1,axiom,
    ! [Pa: fun_st1506752259e_bool,Ga: fun_Ho1996104121e_bool,P_1: fun_st1506752259e_bool,C: com,Qa: fun_st1506752259e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hAPP_f1595846909_state(hAPP_c681126378_state(hAPP_f1930074316_state(hoare_13225861_state,P_1),C),Qa)),bot_bo1715400655e_bool)))
     => ( ! [Z_3: state,S_3: state] :
            ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Pa,Z_3),S_3))
           => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(P_1,Z_3),S_3)) )
       => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hAPP_f1595846909_state(hAPP_c681126378_state(hAPP_f1930074316_state(hoare_13225861_state,Pa),C),Qa)),bot_bo1715400655e_bool))) ) ) ).

tff(fact_338_conseq2,axiom,
    ! [Qa: fun_a_fun_state_bool,Ga: fun_Ho287446294a_bool,Pa: fun_a_fun_state_bool,C: com,Q_1: fun_a_fun_state_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hAPP_f328191961iple_a(hAPP_c1432218316iple_a(hAPP_f1102554477iple_a(hoare_196563068iple_a,Pa),C),Q_1)),bot_bo1766443648a_bool)))
     => ( ! [Z_3: x_a,S_3: state] :
            ( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_1,Z_3),S_3))
           => hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Qa,Z_3),S_3)) )
       => hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hAPP_f328191961iple_a(hAPP_c1432218316iple_a(hAPP_f1102554477iple_a(hoare_196563068iple_a,Pa),C),Qa)),bot_bo1766443648a_bool))) ) ) ).

tff(fact_339_conseq2,axiom,
    ! [Qa: fun_st1506752259e_bool,Ga: fun_Ho1996104121e_bool,Pa: fun_st1506752259e_bool,C: com,Q_1: fun_st1506752259e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hAPP_f1595846909_state(hAPP_c681126378_state(hAPP_f1930074316_state(hoare_13225861_state,Pa),C),Q_1)),bot_bo1715400655e_bool)))
     => ( ! [Z_3: state,S_3: state] :
            ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Q_1,Z_3),S_3))
           => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Qa,Z_3),S_3)) )
       => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hAPP_f1595846909_state(hAPP_c681126378_state(hAPP_f1930074316_state(hoare_13225861_state,Pa),C),Qa)),bot_bo1715400655e_bool))) ) ) ).

tff(fact_340_triple_Osize_I1_J,axiom,
    ! [Fa: fun_state_nat,Fun1: fun_st1506752259e_bool,Com: com,Fun2: fun_st1506752259e_bool] : hoare_1677521252_state(Fa,hAPP_f1595846909_state(hAPP_c681126378_state(hAPP_f1930074316_state(hoare_13225861_state,Fun1),Com),Fun2)) = zero_zero_nat ).

tff(fact_341_triple_Osize_I1_J,axiom,
    ! [Fa: fun_a_nat,Fun1: fun_a_fun_state_bool,Com: com,Fun2: fun_a_fun_state_bool] : hoare_1225927005size_a(Fa,hAPP_f328191961iple_a(hAPP_c1432218316iple_a(hAPP_f1102554477iple_a(hoare_196563068iple_a,Fun1),Com),Fun2)) = zero_zero_nat ).

tff(fact_342_MGT__def,axiom,
    ! [C: com] : hoare_Mirabelle_MGT(C) = hAPP_f1595846909_state(hAPP_c681126378_state(hAPP_f1930074316_state(hoare_13225861_state,fequal_state),C),evalc(C)) ).

tff(fact_343_triple_Osize_I2_J,axiom,
    ! [Fun1: fun_st1506752259e_bool,Com: com,Fun2: fun_st1506752259e_bool] : size_s1353586785_state(hAPP_f1595846909_state(hAPP_c681126378_state(hAPP_f1930074316_state(hoare_13225861_state,Fun1),Com),Fun2)) = zero_zero_nat ).

tff(fact_344_triple_Osize_I2_J,axiom,
    ! [Fun1: fun_a_fun_state_bool,Com: com,Fun2: fun_a_fun_state_bool] : size_s493879222iple_a(hAPP_f328191961iple_a(hAPP_c1432218316iple_a(hAPP_f1102554477iple_a(hoare_196563068iple_a,Fun1),Com),Fun2)) = zero_zero_nat ).

tff(fact_345_conseq12,axiom,
    ! [Qa: fun_st1506752259e_bool,Pa: fun_st1506752259e_bool,Ga: fun_Ho1996104121e_bool,P_1: fun_st1506752259e_bool,C: com,Q_1: fun_st1506752259e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hAPP_f1595846909_state(hAPP_c681126378_state(hAPP_f1930074316_state(hoare_13225861_state,P_1),C),Q_1)),bot_bo1715400655e_bool)))
     => ( ! [Z_3: state,S_3: state] :
            ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Pa,Z_3),S_3))
           => ! [S_4: state] :
                ( ! [Z_4: state] :
                    ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(P_1,Z_4),S_3))
                   => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Q_1,Z_4),S_4)) )
               => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Qa,Z_3),S_4)) ) )
       => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hAPP_f1595846909_state(hAPP_c681126378_state(hAPP_f1930074316_state(hoare_13225861_state,Pa),C),Qa)),bot_bo1715400655e_bool))) ) ) ).

tff(fact_346_conseq12,axiom,
    ! [Qa: fun_a_fun_state_bool,Pa: fun_a_fun_state_bool,Ga: fun_Ho287446294a_bool,P_1: fun_a_fun_state_bool,C: com,Q_1: fun_a_fun_state_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hAPP_f328191961iple_a(hAPP_c1432218316iple_a(hAPP_f1102554477iple_a(hoare_196563068iple_a,P_1),C),Q_1)),bot_bo1766443648a_bool)))
     => ( ! [Z_3: x_a,S_3: state] :
            ( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Pa,Z_3),S_3))
           => ! [S_4: state] :
                ( ! [Z_4: x_a] :
                    ( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(P_1,Z_4),S_3))
                   => hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_1,Z_4),S_4)) )
               => hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Qa,Z_3),S_4)) ) )
       => hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hAPP_f328191961iple_a(hAPP_c1432218316iple_a(hAPP_f1102554477iple_a(hoare_196563068iple_a,Pa),C),Qa)),bot_bo1766443648a_bool))) ) ) ).

tff(fact_347_the__elem__eq,axiom,
    ! [X_2: hoare_1875481847_state] : hAPP_f890249627_state(the_el1031305112_state,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),bot_bo1715400655e_bool)) = X_2 ).

tff(fact_348_the__elem__eq,axiom,
    ! [X_2: hoare_1544627872iple_a] : hAPP_f1853245639iple_a(the_el1614276927iple_a,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_2),bot_bo1766443648a_bool)) = X_2 ).

tff(fact_349_the__elem__eq,axiom,
    ! [X_2: pname] : hAPP_f1297739591_pname(the_elem_pname,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),bot_bo844097828e_bool)) = X_2 ).

tff(fact_350_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero_nat != suc(M) ).

tff(fact_351_nat_Osimps_I2_J,axiom,
    ! [Nat_4: nat] : zero_zero_nat != suc(Nat_4) ).

tff(fact_352_Suc__not__Zero,axiom,
    ! [M: nat] : suc(M) != zero_zero_nat ).

tff(fact_353_nat_Osimps_I3_J,axiom,
    ! [Nat_3: nat] : suc(Nat_3) != zero_zero_nat ).

tff(fact_354_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero_nat != suc(M) ).

tff(fact_355_Suc__neq__Zero,axiom,
    ! [M: nat] : suc(M) != zero_zero_nat ).

tff(fact_356_bot__fun__def,axiom,
    ! [X: pname] :
      ( hBOOL(hAPP_pname_bool(bot_bo844097828e_bool,X))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_357_bot__fun__def,axiom,
    ! [X: hoare_1544627872iple_a] :
      ( hBOOL(hAPP_H2006434664a_bool(bot_bo1766443648a_bool,X))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_358_bot__fun__def,axiom,
    ! [X: hoare_1875481847_state] :
      ( hBOOL(hAPP_H1173941847e_bool(bot_bo1715400655e_bool,X))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_359_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

tff(fact_360_Suc__inject,axiom,
    ! [X_1: nat,Y: nat] :
      ( ( suc(X_1) = suc(Y) )
     => ( X_1 = Y ) ) ).

tff(fact_361_nat_Oinject,axiom,
    ! [Nat_2: nat,Nat_1: nat] :
      ( ( suc(Nat_2) = suc(Nat_1) )
    <=> ( Nat_2 = Nat_1 ) ) ).

tff(fact_362_Suc__n__not__n,axiom,
    ! [N_2: nat] : suc(N_2) != N_2 ).

tff(fact_363_n__not__Suc__n,axiom,
    ! [N_2: nat] : N_2 != suc(N_2) ).

tff(fact_364_bot__apply,axiom,
    ! [X_2: pname] :
      ( hBOOL(hAPP_pname_bool(bot_bo844097828e_bool,X_2))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_365_bot__apply,axiom,
    ! [X_2: hoare_1544627872iple_a] :
      ( hBOOL(hAPP_H2006434664a_bool(bot_bo1766443648a_bool,X_2))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_366_bot__apply,axiom,
    ! [X_2: hoare_1875481847_state] :
      ( hBOOL(hAPP_H1173941847e_bool(bot_bo1715400655e_bool,X_2))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_367_nat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat: nat] : Y != suc(Nat) ) ).

tff(fact_368_zero__induct,axiom,
    ! [Pa: fun_nat_bool,K: nat] :
      ( hBOOL(hAPP_nat_bool(Pa,K))
     => ( ! [N: nat] :
            ( hBOOL(hAPP_nat_bool(Pa,suc(N)))
           => hBOOL(hAPP_nat_bool(Pa,N)) )
       => hBOOL(hAPP_nat_bool(Pa,zero_zero_nat)) ) ) ).

tff(fact_369_nat__induct,axiom,
    ! [N_1: nat,Pa: fun_nat_bool] :
      ( hBOOL(hAPP_nat_bool(Pa,zero_zero_nat))
     => ( ! [N: nat] :
            ( hBOOL(hAPP_nat_bool(Pa,N))
           => hBOOL(hAPP_nat_bool(Pa,suc(N))) )
       => hBOOL(hAPP_nat_bool(Pa,N_1)) ) ) ).

tff(fact_370_not0__implies__Suc,axiom,
    ! [N_2: nat] :
      ( ( N_2 != zero_zero_nat )
     => ? [M_2: nat] : N_2 = suc(M_2) ) ).

tff(fact_371_evaln_OBody,axiom,
    ! [Pn: pname,S0: state,N_2: nat,S1_1: state] :
      ( hBOOL(hAPP_state_bool(evaln(hAPP_option_com_com(the_com,hAPP_p799580910on_com(body_1,Pn)),S0,N_2),S1_1))
     => hBOOL(hAPP_state_bool(evaln(hAPP_pname_com(body,Pn),S0,suc(N_2)),S1_1)) ) ).

tff(fact_372_hoare__derivs_OSkip,axiom,
    ! [Ga: fun_Ho1996104121e_bool,Pa: fun_st1506752259e_bool] : hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hAPP_f1595846909_state(hAPP_c681126378_state(hAPP_f1930074316_state(hoare_13225861_state,Pa),skip),Pa)),bot_bo1715400655e_bool))) ).

tff(fact_373_hoare__derivs_OSkip,axiom,
    ! [Ga: fun_Ho287446294a_bool,Pa: fun_a_fun_state_bool] : hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hAPP_f328191961iple_a(hAPP_c1432218316iple_a(hAPP_f1102554477iple_a(hoare_196563068iple_a,Pa),skip),Pa)),bot_bo1766443648a_bool))) ).

tff(fact_374_evaln__elim__cases_I1_J,axiom,
    ! [S_2: state,N_2: nat,T_1: state] :
      ( hBOOL(hAPP_state_bool(evaln(skip,S_2,N_2),T_1))
     => ( T_1 = S_2 ) ) ).

tff(fact_375_evaln_OSkip,axiom,
    ! [S_2: state,N_2: nat] : hBOOL(hAPP_state_bool(evaln(skip,S_2,N_2),S_2)) ).

tff(fact_376_evalc_OSkip,axiom,
    ! [S_2: state] : hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(skip),S_2),S_2)) ).

tff(fact_377_evalc__elim__cases_I1_J,axiom,
    ! [S_2: state,T_1: state] :
      ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(skip),S_2),T_1))
     => ( T_1 = S_2 ) ) ).

tff(fact_378_evaln__Suc,axiom,
    ! [C_2: com,S_2: state,N_2: nat,S_5: state] :
      ( hBOOL(hAPP_state_bool(evaln(C_2,S_2,N_2),S_5))
     => hBOOL(hAPP_state_bool(evaln(C_2,S_2,suc(N_2)),S_5)) ) ).

tff(fact_379_eval__eq,axiom,
    ! [C: com,S: state,T: state] :
      ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C),S),T))
    <=> ? [N: nat] : hBOOL(hAPP_state_bool(evaln(C,S,N),T)) ) ).

tff(fact_380_evaln__evalc,axiom,
    ! [C_2: com,S_2: state,N_2: nat,T_1: state] :
      ( hBOOL(hAPP_state_bool(evaln(C_2,S_2,N_2),T_1))
     => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C_2),S_2),T_1)) ) ).

tff(fact_381_com_Osimps_I19_J,axiom,
    ! [Pname_1: pname] : hAPP_pname_com(body,Pname_1) != skip ).

tff(fact_382_com_Osimps_I18_J,axiom,
    ! [Pname_1: pname] : skip != hAPP_pname_com(body,Pname_1) ).

tff(fact_383_triple__valid__def2,axiom,
    ! [N_1: nat,Pa: fun_st1506752259e_bool,C: com,Qa: fun_st1506752259e_bool] :
      ( hBOOL(hAPP_H1173941847e_bool(hoare_2065208407_state(N_1),hAPP_f1595846909_state(hAPP_c681126378_state(hAPP_f1930074316_state(hoare_13225861_state,Pa),C),Qa)))
    <=> ! [Z_3: state,S_3: state] :
          ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Pa,Z_3),S_3))
         => ! [S_4: state] :
              ( hBOOL(hAPP_state_bool(evaln(C,S_3,N_1),S_4))
             => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Qa,Z_3),S_4)) ) ) ) ).

tff(fact_384_triple__valid__def2,axiom,
    ! [N_1: nat,Pa: fun_a_fun_state_bool,C: com,Qa: fun_a_fun_state_bool] :
      ( hBOOL(hAPP_H2006434664a_bool(hoare_1500711466alid_a(N_1),hAPP_f328191961iple_a(hAPP_c1432218316iple_a(hAPP_f1102554477iple_a(hoare_196563068iple_a,Pa),C),Qa)))
    <=> ! [Z_3: x_a,S_3: state] :
          ( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Pa,Z_3),S_3))
         => ! [S_4: state] :
              ( hBOOL(hAPP_state_bool(evaln(C,S_3,N_1),S_4))
             => hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Qa,Z_3),S_4)) ) ) ) ).

tff(fact_385_evaln__elim__cases_I6_J,axiom,
    ! [P: pname,S_2: state,N_2: nat,S1_1: state] :
      ( hBOOL(hAPP_state_bool(evaln(hAPP_pname_com(body,P),S_2,N_2),S1_1))
     => ~ ! [N: nat] :
            ( ( N_2 = suc(N) )
           => ~ hBOOL(hAPP_state_bool(evaln(hAPP_option_com_com(the_com,hAPP_p799580910on_com(body_1,P)),S_2,N),S1_1)) ) ) ).

tff(fact_386_evalc__evaln,axiom,
    ! [C_2: com,S_2: state,T_1: state] :
      ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C_2),S_2),T_1))
     => ? [N: nat] : hBOOL(hAPP_state_bool(evaln(C_2,S_2,N),T_1)) ) ).

tff(fact_387_LoopF,axiom,
    ! [Ga: fun_Ho1996104121e_bool,Pa: fun_st1506752259e_bool,B: fun_state_bool,C: com] : hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hAPP_f1595846909_state(hAPP_c681126378_state(hAPP_f1930074316_state(hoare_13225861_state,hAPP_f1863945078e_bool(hAPP_f915354021e_bool(cOMBC_952831051e_bool,hAPP_f1138284024e_bool(cOMBB_20296667_state(cOMBS_1378840469l_bool),hAPP_f1283379615l_bool(cOMBB_1759179140_state(cOMBB_188601460_state(fconj)),Pa))),hAPP_f1759915619e_bool(cOMBB_160679318_state(fNot),B))),while(B,C)),Pa)),bot_bo1715400655e_bool))) ).

tff(fact_388_LoopF,axiom,
    ! [Ga: fun_Ho287446294a_bool,Pa: fun_a_fun_state_bool,B: fun_state_bool,C: com] : hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hAPP_f328191961iple_a(hAPP_c1432218316iple_a(hAPP_f1102554477iple_a(hoare_196563068iple_a,hAPP_f762886889e_bool(hAPP_f1261923407e_bool(cOMBC_892787026e_bool,hAPP_f963367678e_bool(cOMBB_145932198bool_a(cOMBS_1378840469l_bool),hAPP_f1509969235l_bool(cOMBB_1355796797bool_a(cOMBB_188601460_state(fconj)),Pa))),hAPP_f1759915619e_bool(cOMBB_160679318_state(fNot),B))),while(B,C)),Pa)),bot_bo1766443648a_bool))) ).

tff(fact_389_Comp,axiom,
    ! [D: com,R_1: fun_st1506752259e_bool,Ga: fun_Ho1996104121e_bool,Pa: fun_st1506752259e_bool,C: com,Qa: fun_st1506752259e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hAPP_f1595846909_state(hAPP_c681126378_state(hAPP_f1930074316_state(hoare_13225861_state,Pa),C),Qa)),bot_bo1715400655e_bool)))
     => ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hAPP_f1595846909_state(hAPP_c681126378_state(hAPP_f1930074316_state(hoare_13225861_state,Qa),D),R_1)),bot_bo1715400655e_bool)))
       => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hAPP_f1595846909_state(hAPP_c681126378_state(hAPP_f1930074316_state(hoare_13225861_state,Pa),semi(C,D)),R_1)),bot_bo1715400655e_bool))) ) ) ).

tff(fact_390_Comp,axiom,
    ! [D: com,R_1: fun_a_fun_state_bool,Ga: fun_Ho287446294a_bool,Pa: fun_a_fun_state_bool,C: com,Qa: fun_a_fun_state_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hAPP_f328191961iple_a(hAPP_c1432218316iple_a(hAPP_f1102554477iple_a(hoare_196563068iple_a,Pa),C),Qa)),bot_bo1766443648a_bool)))
     => ( hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hAPP_f328191961iple_a(hAPP_c1432218316iple_a(hAPP_f1102554477iple_a(hoare_196563068iple_a,Qa),D),R_1)),bot_bo1766443648a_bool)))
       => hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hAPP_f328191961iple_a(hAPP_c1432218316iple_a(hAPP_f1102554477iple_a(hoare_196563068iple_a,Pa),semi(C,D)),R_1)),bot_bo1766443648a_bool))) ) ) ).

tff(fact_391_the__elem__def,axiom,
    ! [X_3: fun_Ho1996104121e_bool] : hAPP_f890249627_state(the_el1031305112_state,X_3) = hAPP_f890249627_state(the_Ho1017792262_state,hAPP_f169892419e_bool(cOMBB_1380679615_state(hAPP_f1873026899l_bool(fequal259518520e_bool,X_3)),hAPP_f56162711e_bool(hAPP_f392546175e_bool(cOMBC_483615668e_bool,insert694999549_state),bot_bo1715400655e_bool))) ).

tff(fact_392_the__elem__def,axiom,
    ! [X_3: fun_Ho287446294a_bool] : hAPP_f1853245639iple_a(the_el1614276927iple_a,X_3) = hAPP_f1853245639iple_a(the_Ho924576593iple_a,hAPP_f1726109544a_bool(cOMBB_1474127043iple_a(hAPP_f1876759039l_bool(fequal1843628375a_bool,X_3)),hAPP_f693853672a_bool(hAPP_f1399402097a_bool(cOMBC_894585277a_bool,insert1051021594iple_a),bot_bo1766443648a_bool))) ).

tff(fact_393_the__elem__def,axiom,
    ! [X_3: fun_pname_bool] : hAPP_f1297739591_pname(the_elem_pname,X_3) = hAPP_f1297739591_pname(the_pname,hAPP_f42430548e_bool(cOMBB_530759491_pname(hAPP_f434788991l_bool(fequal533582459e_bool,X_3)),hAPP_f180827860e_bool(hAPP_f698292281e_bool(cOMBC_1123258281e_bool,insert_pname),bot_bo844097828e_bool))) ).

tff(fact_394_finite__pointwise,axiom,
    ! [Pa: fun_pn12888928e_bool,Qa: fun_pn12888928e_bool,Ga: fun_Ho1996104121e_bool,P_1: fun_pn12888928e_bool,C0: fun_pname_com,Q_1: fun_pn12888928e_bool,U: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,U))
     => ( ! [P_2: pname] :
            ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hAPP_f1595846909_state(hAPP_c681126378_state(hAPP_f1930074316_state(hoare_13225861_state,hAPP_p1637813682e_bool(P_1,P_2)),hAPP_pname_com(C0,P_2)),hAPP_p1637813682e_bool(Q_1,P_2))),bot_bo1715400655e_bool)))
           => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hAPP_f1595846909_state(hAPP_c681126378_state(hAPP_f1930074316_state(hoare_13225861_state,hAPP_p1637813682e_bool(Pa,P_2)),hAPP_pname_com(C0,P_2)),hAPP_p1637813682e_bool(Qa,P_2))),bot_bo1715400655e_bool))) )
       => ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f1291720380e_bool(image_1283223414_state(hAPP_f1568887991_state(hAPP_f891245078_state(cOMBS_475568287_state,hAPP_f2116202538_state(hAPP_f1339343196_state(cOMBS_1674056844_state,hAPP_f275933352_state(cOMBB_718493681_pname(hoare_13225861_state),P_1)),C0)),Q_1)),U)))
         => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f1291720380e_bool(image_1283223414_state(hAPP_f1568887991_state(hAPP_f891245078_state(cOMBS_475568287_state,hAPP_f2116202538_state(hAPP_f1339343196_state(cOMBS_1674056844_state,hAPP_f275933352_state(cOMBB_718493681_pname(hoare_13225861_state),Pa)),C0)),Qa)),U))) ) ) ) ).

tff(fact_395_finite__pointwise,axiom,
    ! [Pa: fun_pn1683930517e_bool,Qa: fun_pn1683930517e_bool,Ga: fun_Ho287446294a_bool,P_1: fun_pn1683930517e_bool,C0: fun_pname_com,Q_1: fun_pn1683930517e_bool,U: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,U))
     => ( ! [P_2: pname] :
            ( hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hAPP_f328191961iple_a(hAPP_c1432218316iple_a(hAPP_f1102554477iple_a(hoare_196563068iple_a,hAPP_p635540397e_bool(P_1,P_2)),hAPP_pname_com(C0,P_2)),hAPP_p635540397e_bool(Q_1,P_2))),bot_bo1766443648a_bool)))
           => hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hAPP_f328191961iple_a(hAPP_c1432218316iple_a(hAPP_f1102554477iple_a(hoare_196563068iple_a,hAPP_p635540397e_bool(Pa,P_2)),hAPP_pname_com(C0,P_2)),hAPP_p635540397e_bool(Qa,P_2))),bot_bo1766443648a_bool))) )
       => ( hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f856094675a_bool(image_1832685281iple_a(hAPP_f104000281iple_a(hAPP_f2028456860iple_a(cOMBS_438391419iple_a,hAPP_f68608908iple_a(hAPP_f1015300700iple_a(cOMBS_2128932974iple_a,hAPP_f298026029iple_a(cOMBB_124251824_pname(hoare_196563068iple_a),P_1)),C0)),Q_1)),U)))
         => hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f856094675a_bool(image_1832685281iple_a(hAPP_f104000281iple_a(hAPP_f2028456860iple_a(cOMBS_438391419iple_a,hAPP_f68608908iple_a(hAPP_f1015300700iple_a(cOMBS_2128932974iple_a,hAPP_f298026029iple_a(cOMBB_124251824_pname(hoare_196563068iple_a),Pa)),C0)),Qa)),U))) ) ) ) ).

tff(fact_396_evaln_OWhileFalse,axiom,
    ! [C: com,N_1: nat,B: fun_state_bool,S: state] :
      ( ~ hBOOL(hAPP_state_bool(B,S))
     => hBOOL(hAPP_state_bool(evaln(while(B,C),S,N_1),S)) ) ).

tff(fact_397_evaln_OWhileTrue,axiom,
    ! [S2_1: state,C: com,N_1: nat,S1: state,B: fun_state_bool,S0_1: state] :
      ( hBOOL(hAPP_state_bool(B,S0_1))
     => ( hBOOL(hAPP_state_bool(evaln(C,S0_1,N_1),S1))
       => ( hBOOL(hAPP_state_bool(evaln(while(B,C),S1,N_1),S2_1))
         => hBOOL(hAPP_state_bool(evaln(while(B,C),S0_1,N_1),S2_1)) ) ) ) ).

tff(fact_398_evalc_OWhileTrue,axiom,
    ! [S2_1: state,C: com,S1: state,B: fun_state_bool,S0_1: state] :
      ( hBOOL(hAPP_state_bool(B,S0_1))
     => ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C),S0_1),S1))
       => ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(while(B,C)),S1),S2_1))
         => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(while(B,C)),S0_1),S2_1)) ) ) ) ).

tff(fact_399_evalc_OWhileFalse,axiom,
    ! [C: com,B: fun_state_bool,S: state] :
      ( ~ hBOOL(hAPP_state_bool(B,S))
     => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(while(B,C)),S),S)) ) ).

tff(fact_400_evaln_OSemi,axiom,
    ! [C1_1: com,S2: state,C0_1: com,S0: state,N_2: nat,S1_1: state] :
      ( hBOOL(hAPP_state_bool(evaln(C0_1,S0,N_2),S1_1))
     => ( hBOOL(hAPP_state_bool(evaln(C1_1,S1_1,N_2),S2))
       => hBOOL(hAPP_state_bool(evaln(semi(C0_1,C1_1),S0,N_2),S2)) ) ) ).

tff(fact_401_evalc_OSemi,axiom,
    ! [C1_1: com,S2: state,C0_1: com,S0: state,S1_1: state] :
      ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C0_1),S0),S1_1))
     => ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C1_1),S1_1),S2))
       => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(semi(C0_1,C1_1)),S0),S2)) ) ) ).

tff(fact_402_com_Osimps_I46_J,axiom,
    ! [Com1: com,Com2: com,Fun_1: fun_state_bool,Com_1: com] : semi(Com1,Com2) != while(Fun_1,Com_1) ).

tff(fact_403_com_Osimps_I47_J,axiom,
    ! [Fun_1: fun_state_bool,Com_1: com,Com1: com,Com2: com] : while(Fun_1,Com_1) != semi(Com1,Com2) ).

tff(fact_404_com_Osimps_I3_J,axiom,
    ! [Com1: com,Com2: com,Com1_3: com,Com2_3: com] :
      ( ( semi(Com1,Com2) = semi(Com1_3,Com2_3) )
    <=> ( ( Com1 = Com1_3 )
        & ( Com2 = Com2_3 ) ) ) ).

tff(fact_405_com_Osimps_I5_J,axiom,
    ! [Fun: fun_state_bool,Com: com,Fun_1: fun_state_bool,Com_1: com] :
      ( ( while(Fun,Com) = while(Fun_1,Com_1) )
    <=> ( ( Fun = Fun_1 )
        & ( Com = Com_1 ) ) ) ).

tff(fact_406_com_Osimps_I59_J,axiom,
    ! [Pname_2: pname,Fun: fun_state_bool,Com: com] : hAPP_pname_com(body,Pname_2) != while(Fun,Com) ).

tff(fact_407_com_Osimps_I58_J,axiom,
    ! [Fun: fun_state_bool,Com: com,Pname_2: pname] : while(Fun,Com) != hAPP_pname_com(body,Pname_2) ).

tff(fact_408_com_Osimps_I16_J,axiom,
    ! [Fun_1: fun_state_bool,Com_1: com] : skip != while(Fun_1,Com_1) ).

tff(fact_409_com_Osimps_I17_J,axiom,
    ! [Fun_1: fun_state_bool,Com_1: com] : while(Fun_1,Com_1) != skip ).

tff(fact_410_com_Osimps_I49_J,axiom,
    ! [Pname_1: pname,Com1_1: com,Com2_1: com] : hAPP_pname_com(body,Pname_1) != semi(Com1_1,Com2_1) ).

tff(fact_411_com_Osimps_I48_J,axiom,
    ! [Com1_1: com,Com2_1: com,Pname_1: pname] : semi(Com1_1,Com2_1) != hAPP_pname_com(body,Pname_1) ).

tff(fact_412_com_Osimps_I12_J,axiom,
    ! [Com1_2: com,Com2_2: com] : skip != semi(Com1_2,Com2_2) ).

tff(fact_413_com_Osimps_I13_J,axiom,
    ! [Com1_2: com,Com2_2: com] : semi(Com1_2,Com2_2) != skip ).

tff(fact_414_evalc__elim__cases_I4_J,axiom,
    ! [C1_1: com,C2_1: com,S_2: state,T_1: state] :
      ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(semi(C1_1,C2_1)),S_2),T_1))
     => ~ ! [S1_2: state] :
            ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C1_1),S_2),S1_2))
           => ~ hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C2_1),S1_2),T_1)) ) ) ).

tff(fact_415_evaln__elim__cases_I4_J,axiom,
    ! [C1_1: com,C2_1: com,S_2: state,N_2: nat,T_1: state] :
      ( hBOOL(hAPP_state_bool(evaln(semi(C1_1,C2_1),S_2,N_2),T_1))
     => ~ ! [S1_2: state] :
            ( hBOOL(hAPP_state_bool(evaln(C1_1,S_2,N_2),S1_2))
           => ~ hBOOL(hAPP_state_bool(evaln(C2_1,S1_2,N_2),T_1)) ) ) ).

tff(fact_416_finite__imageI,axiom,
    ! [H: fun_pn664418900_state,F: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F))
     => hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,hAPP_f1291720380e_bool(image_1283223414_state(H),F))) ) ).

tff(fact_417_finite__imageI,axiom,
    ! [H: fun_pn1711459225iple_a,F: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F))
     => hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,hAPP_f856094675a_bool(image_1832685281iple_a(H),F))) ) ).

tff(fact_418_finite_OinsertI,axiom,
    ! [A_1: hoare_1875481847_state,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A))
     => hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),A))) ) ).

tff(fact_419_finite_OinsertI,axiom,
    ! [A_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,A))
     => hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),A))) ) ).

tff(fact_420_finite_OinsertI,axiom,
    ! [A_1: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),A))) ) ).

tff(fact_421_finite_OemptyI,axiom,
    hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,bot_bo844097828e_bool)) ).

tff(fact_422_finite_OemptyI,axiom,
    hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,bot_bo1766443648a_bool)) ).

tff(fact_423_finite_OemptyI,axiom,
    hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,bot_bo1715400655e_bool)) ).

tff(fact_424_finite__Collect__conjI,axiom,
    ! [Qa: fun_Ho287446294a_bool,Pa: fun_Ho287446294a_bool] :
      ( ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,hAPP_f909437487a_bool(collec445968053iple_a,Pa)))
        | hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,hAPP_f909437487a_bool(collec445968053iple_a,Qa))) )
     => hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,hAPP_f909437487a_bool(collec445968053iple_a,hAPP_f909437487a_bool(hAPP_f1099051562a_bool(cOMBS_523298619l_bool,hAPP_f224574947l_bool(cOMBB_1960865690iple_a(fconj),Pa)),Qa)))) ) ).

tff(fact_425_finite__Collect__conjI,axiom,
    ! [Qa: fun_pname_bool,Pa: fun_pname_bool] :
      ( ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(collect_pname,Pa)))
        | hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(collect_pname,Qa))) )
     => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,hAPP_f1402196763l_bool(cOMBB_675860798_pname(fconj),Pa)),Qa)))) ) ).

tff(fact_426_finite__Collect__disjI,axiom,
    ! [Pa: fun_Ho287446294a_bool,Qa: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,hAPP_f909437487a_bool(collec445968053iple_a,hAPP_f909437487a_bool(hAPP_f1099051562a_bool(cOMBS_523298619l_bool,hAPP_f224574947l_bool(cOMBB_1960865690iple_a(fdisj),Pa)),Qa))))
    <=> ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,hAPP_f909437487a_bool(collec445968053iple_a,Pa)))
        & hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,hAPP_f909437487a_bool(collec445968053iple_a,Qa))) ) ) ).

tff(fact_427_finite__Collect__disjI,axiom,
    ! [Pa: fun_pname_bool,Qa: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,hAPP_f1402196763l_bool(cOMBB_675860798_pname(fdisj),Pa)),Qa))))
    <=> ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(collect_pname,Pa)))
        & hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(collect_pname,Qa))) ) ) ).

tff(fact_428_finite__insert,axiom,
    ! [A_1: hoare_1875481847_state,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),A)))
    <=> hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A)) ) ).

tff(fact_429_finite__insert,axiom,
    ! [A_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),A)))
    <=> hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,A)) ) ).

tff(fact_430_finite__insert,axiom,
    ! [A_1: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),A)))
    <=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A)) ) ).

tff(fact_431_finite__Un,axiom,
    ! [F: fun_pname_bool,Ga: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,F),Ga)))
    <=> ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F))
        & hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,Ga)) ) ) ).

tff(fact_432_finite__Un,axiom,
    ! [F: fun_Ho1996104121e_bool,Ga: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,F),Ga)))
    <=> ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,F))
        & hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,Ga)) ) ) ).

tff(fact_433_finite__Un,axiom,
    ! [F: fun_Ho287446294a_bool,Ga: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,F),Ga)))
    <=> ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,F))
        & hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,Ga)) ) ) ).

tff(fact_434_finite__UnI,axiom,
    ! [Ga: fun_pname_bool,F: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F))
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,Ga))
       => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,F),Ga))) ) ) ).

tff(fact_435_finite__UnI,axiom,
    ! [Ga: fun_Ho1996104121e_bool,F: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,F))
     => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,Ga))
       => hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,F),Ga))) ) ) ).

tff(fact_436_finite__UnI,axiom,
    ! [Ga: fun_Ho287446294a_bool,F: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,F))
     => ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,Ga))
       => hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,F),Ga))) ) ) ).

tff(fact_437_finite_Osimps,axiom,
    ! [A_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1))
    <=> ( ( A_1 = bot_bo1715400655e_bool )
        | ? [A_3: fun_Ho1996104121e_bool,A_2: hoare_1875481847_state] :
            ( ( A_1 = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_2),A_3) )
            & hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_3)) ) ) ) ).

tff(fact_438_finite_Osimps,axiom,
    ! [A_1: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,A_1))
    <=> ( ( A_1 = bot_bo1766443648a_bool )
        | ? [A_3: fun_Ho287446294a_bool,A_2: hoare_1544627872iple_a] :
            ( ( A_1 = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_2),A_3) )
            & hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,A_3)) ) ) ) ).

tff(fact_439_finite_Osimps,axiom,
    ! [A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
    <=> ( ( A_1 = bot_bo844097828e_bool )
        | ? [A_3: fun_pname_bool,A_2: pname] :
            ( ( A_1 = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),A_3) )
            & hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_3)) ) ) ) ).

tff(fact_440_finite__induct,axiom,
    ! [Pa: fun_fu689207471l_bool,F: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,F))
     => ( hBOOL(hAPP_f1408815105l_bool(Pa,bot_bo1715400655e_bool))
       => ( ! [X: hoare_1875481847_state,F_2: fun_Ho1996104121e_bool] :
              ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,F_2))
             => ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X),F_2))
               => ( hBOOL(hAPP_f1408815105l_bool(Pa,F_2))
                 => hBOOL(hAPP_f1408815105l_bool(Pa,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X),F_2))) ) ) )
         => hBOOL(hAPP_f1408815105l_bool(Pa,F)) ) ) ) ).

tff(fact_441_finite__induct,axiom,
    ! [Pa: fun_fu1441721944l_bool,F: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,F))
     => ( hBOOL(hAPP_f2063540982l_bool(Pa,bot_bo1766443648a_bool))
       => ( ! [X: hoare_1544627872iple_a,F_2: fun_Ho287446294a_bool] :
              ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,F_2))
             => ( ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X),F_2))
               => ( hBOOL(hAPP_f2063540982l_bool(Pa,F_2))
                 => hBOOL(hAPP_f2063540982l_bool(Pa,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X),F_2))) ) ) )
         => hBOOL(hAPP_f2063540982l_bool(Pa,F)) ) ) ) ).

tff(fact_442_finite__induct,axiom,
    ! [Pa: fun_fu1430349052l_bool,F: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F))
     => ( hBOOL(hAPP_f1664156314l_bool(Pa,bot_bo844097828e_bool))
       => ( ! [X: pname,F_2: fun_pname_bool] :
              ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_2))
             => ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),F_2))
               => ( hBOOL(hAPP_f1664156314l_bool(Pa,F_2))
                 => hBOOL(hAPP_f1664156314l_bool(Pa,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X),F_2))) ) ) )
         => hBOOL(hAPP_f1664156314l_bool(Pa,F)) ) ) ) ).

tff(fact_443_pigeonhole__infinite,axiom,
    ! [F_1: fun_pn664418900_state,A: fun_pname_bool] :
      ( ~ hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,hAPP_f1291720380e_bool(image_1283223414_state(F_1),A)))
       => ? [X: pname] :
            ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),A))
            & ~ hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,hAPP_f1402196763l_bool(cOMBB_675860798_pname(fconj),hAPP_f759274231e_bool(hAPP_f661147897e_bool(cOMBC_1058051404l_bool,member_pname),A))),hAPP_H785152948e_bool(hAPP_f1564627769e_bool(cOMBC_951029257e_bool,hAPP_f1601283355e_bool(cOMBB_2037342654_pname(fequal391417144_state),F_1)),hAPP_p702226982_state(F_1,X)))))) ) ) ) ).

tff(fact_444_pigeonhole__infinite,axiom,
    ! [F_1: fun_pn1711459225iple_a,A: fun_pname_bool] :
      ( ~ hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,hAPP_f856094675a_bool(image_1832685281iple_a(F_1),A)))
       => ? [X: pname] :
            ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),A))
            & ~ hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,hAPP_f1402196763l_bool(cOMBB_675860798_pname(fconj),hAPP_f759274231e_bool(hAPP_f661147897e_bool(cOMBC_1058051404l_bool,member_pname),A))),hAPP_H1178433989e_bool(hAPP_f565099497e_bool(cOMBC_131447350a_bool,hAPP_f1707302775a_bool(cOMBB_1584663260_pname(fequal1057774495iple_a),F_1)),hAPP_p441219121iple_a(F_1,X)))))) ) ) ) ).

tff(fact_445_evalc__WHILE__case,axiom,
    ! [B: fun_state_bool,C: com,S: state,T: state] :
      ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(while(B,C)),S),T))
     => ( ( ( T = S )
         => hBOOL(hAPP_state_bool(B,S)) )
       => ~ ( hBOOL(hAPP_state_bool(B,S))
           => ! [S1_2: state] :
                ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C),S),S1_2))
               => ~ hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(while(B,C)),S1_2),T)) ) ) ) ) ).

tff(fact_446_evaln__WHILE__case,axiom,
    ! [B: fun_state_bool,C: com,S: state,N_1: nat,T: state] :
      ( hBOOL(hAPP_state_bool(evaln(while(B,C),S,N_1),T))
     => ( ( ( T = S )
         => hBOOL(hAPP_state_bool(B,S)) )
       => ~ ( hBOOL(hAPP_state_bool(B,S))
           => ! [S1_2: state] :
                ( hBOOL(hAPP_state_bool(evaln(C,S,N_1),S1_2))
               => ~ hBOOL(hAPP_state_bool(evaln(while(B,C),S1_2,N_1),T)) ) ) ) ) ).

tff(fact_447_nonempty__iff,axiom,
    ! [A: fun_Ho1996104121e_bool] :
      ( ( A != bot_bo1715400655e_bool )
    <=> ? [X: hoare_1875481847_state,B_2: fun_Ho1996104121e_bool] :
          ( ( A = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X),B_2) )
          & ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X),B_2)) ) ) ).

tff(fact_448_nonempty__iff,axiom,
    ! [A: fun_Ho287446294a_bool] :
      ( ( A != bot_bo1766443648a_bool )
    <=> ? [X: hoare_1544627872iple_a,B_2: fun_Ho287446294a_bool] :
          ( ( A = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X),B_2) )
          & ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X),B_2)) ) ) ).

tff(fact_449_nonempty__iff,axiom,
    ! [A: fun_pname_bool] :
      ( ( A != bot_bo844097828e_bool )
    <=> ? [X: pname,B_2: fun_pname_bool] :
          ( ( A = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X),B_2) )
          & ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),B_2)) ) ) ).

tff(fact_450_folding__one__idem_Ounion__idem,axiom,
    ! [B_1: fun_pname_bool,A: fun_pname_bool,F_1: fun_pn1208287926_pname,F: fun_fu1499449723_pname] :
      ( hBOOL(finite89670078_pname(F_1,F))
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
       => ( ( A != bot_bo844097828e_bool )
         => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B_1))
           => ( ( B_1 != bot_bo844097828e_bool )
             => ( hAPP_f1297739591_pname(F,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),B_1)) = hAPP_pname_pname(hAPP_p1170154830_pname(F_1,hAPP_f1297739591_pname(F,A)),hAPP_f1297739591_pname(F,B_1)) ) ) ) ) ) ) ).

tff(fact_451_folding__one__idem_Ounion__idem,axiom,
    ! [B_1: fun_Ho287446294a_bool,A: fun_Ho287446294a_bool,F_1: fun_Ho115022426iple_a,F: fun_fu1634330235iple_a] :
      ( hBOOL(finite1127948314iple_a(F_1,F))
     => ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,A))
       => ( ( A != bot_bo1766443648a_bool )
         => ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,B_1))
           => ( ( B_1 != bot_bo1766443648a_bool )
             => ( hAPP_f1853245639iple_a(F,hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B_1)) = hAPP_H9461717iple_a(hAPP_H1988213490iple_a(F_1,hAPP_f1853245639iple_a(F,A)),hAPP_f1853245639iple_a(F,B_1)) ) ) ) ) ) ) ).

tff(fact_452_folding__one__idem_Ounion__idem,axiom,
    ! [B_1: fun_Ho1996104121e_bool,A: fun_Ho1996104121e_bool,F_1: fun_Ho784854669_state,F: fun_fu385705929_state] :
      ( hBOOL(finite1514162941_state(F_1,F))
     => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A))
       => ( ( A != bot_bo1715400655e_bool )
         => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,B_1))
           => ( ( B_1 != bot_bo1715400655e_bool )
             => ( hAPP_f890249627_state(F,hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),B_1)) = hAPP_H350317809_state(hAPP_H743221337_state(F_1,hAPP_f890249627_state(F,A)),hAPP_f890249627_state(F,B_1)) ) ) ) ) ) ) ).

tff(fact_453_folding__one__idem_Oinsert__idem,axiom,
    ! [X_2: hoare_1875481847_state,A: fun_Ho1996104121e_bool,F_1: fun_Ho784854669_state,F: fun_fu385705929_state] :
      ( hBOOL(finite1514162941_state(F_1,F))
     => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A))
       => ( ( A != bot_bo1715400655e_bool )
         => ( hAPP_f890249627_state(F,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),A)) = hAPP_H350317809_state(hAPP_H743221337_state(F_1,X_2),hAPP_f890249627_state(F,A)) ) ) ) ) ).

tff(fact_454_folding__one__idem_Oinsert__idem,axiom,
    ! [X_2: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,F_1: fun_Ho115022426iple_a,F: fun_fu1634330235iple_a] :
      ( hBOOL(finite1127948314iple_a(F_1,F))
     => ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,A))
       => ( ( A != bot_bo1766443648a_bool )
         => ( hAPP_f1853245639iple_a(F,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_2),A)) = hAPP_H9461717iple_a(hAPP_H1988213490iple_a(F_1,X_2),hAPP_f1853245639iple_a(F,A)) ) ) ) ) ).

tff(fact_455_folding__one__idem_Oinsert__idem,axiom,
    ! [X_2: pname,A: fun_pname_bool,F_1: fun_pn1208287926_pname,F: fun_fu1499449723_pname] :
      ( hBOOL(finite89670078_pname(F_1,F))
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
       => ( ( A != bot_bo844097828e_bool )
         => ( hAPP_f1297739591_pname(F,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),A)) = hAPP_pname_pname(hAPP_p1170154830_pname(F_1,X_2),hAPP_f1297739591_pname(F,A)) ) ) ) ) ).

tff(fact_456_image__eq__fold__image,axiom,
    ! [F_1: fun_pn664418900_state,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( hAPP_f1291720380e_bool(image_1283223414_state(F_1),A) = finite750308382_pname(semila1107643821e_bool,hAPP_f827623310e_bool(hAPP_f1497196629e_bool(cOMBC_1520907207e_bool,hAPP_f1023805663e_bool(cOMBB_261907706_pname(insert694999549_state),F_1)),bot_bo1715400655e_bool),bot_bo1715400655e_bool,A) ) ) ).

tff(fact_457_image__eq__fold__image,axiom,
    ! [F_1: fun_pn1711459225iple_a,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( hAPP_f856094675a_bool(image_1832685281iple_a(F_1),A) = finite642022125_pname(semila2083752482a_bool,hAPP_f1932171532a_bool(hAPP_f1035004473a_bool(cOMBC_845146337a_bool,hAPP_f616158854a_bool(cOMBB_1386904087_pname(insert1051021594iple_a),F_1)),bot_bo1766443648a_bool),bot_bo1766443648a_bool,A) ) ) ).

tff(fact_458_finite__ne__induct,axiom,
    ! [Pa: fun_fu689207471l_bool,F: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,F))
     => ( ( F != bot_bo1715400655e_bool )
       => ( ! [X: hoare_1875481847_state] : hBOOL(hAPP_f1408815105l_bool(Pa,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X),bot_bo1715400655e_bool)))
         => ( ! [X: hoare_1875481847_state,F_2: fun_Ho1996104121e_bool] :
                ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,F_2))
               => ( ( F_2 != bot_bo1715400655e_bool )
                 => ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X),F_2))
                   => ( hBOOL(hAPP_f1408815105l_bool(Pa,F_2))
                     => hBOOL(hAPP_f1408815105l_bool(Pa,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X),F_2))) ) ) ) )
           => hBOOL(hAPP_f1408815105l_bool(Pa,F)) ) ) ) ) ).

tff(fact_459_finite__ne__induct,axiom,
    ! [Pa: fun_fu1441721944l_bool,F: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,F))
     => ( ( F != bot_bo1766443648a_bool )
       => ( ! [X: hoare_1544627872iple_a] : hBOOL(hAPP_f2063540982l_bool(Pa,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X),bot_bo1766443648a_bool)))
         => ( ! [X: hoare_1544627872iple_a,F_2: fun_Ho287446294a_bool] :
                ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,F_2))
               => ( ( F_2 != bot_bo1766443648a_bool )
                 => ( ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X),F_2))
                   => ( hBOOL(hAPP_f2063540982l_bool(Pa,F_2))
                     => hBOOL(hAPP_f2063540982l_bool(Pa,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X),F_2))) ) ) ) )
           => hBOOL(hAPP_f2063540982l_bool(Pa,F)) ) ) ) ) ).

tff(fact_460_finite__ne__induct,axiom,
    ! [Pa: fun_fu1430349052l_bool,F: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F))
     => ( ( F != bot_bo844097828e_bool )
       => ( ! [X: pname] : hBOOL(hAPP_f1664156314l_bool(Pa,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X),bot_bo844097828e_bool)))
         => ( ! [X: pname,F_2: fun_pname_bool] :
                ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_2))
               => ( ( F_2 != bot_bo844097828e_bool )
                 => ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),F_2))
                   => ( hBOOL(hAPP_f1664156314l_bool(Pa,F_2))
                     => hBOOL(hAPP_f1664156314l_bool(Pa,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X),F_2))) ) ) ) )
           => hBOOL(hAPP_f1664156314l_bool(Pa,F)) ) ) ) ) ).

tff(fact_461_folding__one__idem_Oin__idem,axiom,
    ! [X_2: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,F_1: fun_Ho115022426iple_a,F: fun_fu1634330235iple_a] :
      ( hBOOL(finite1127948314iple_a(F_1,F))
     => ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,A))
       => ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_2),A))
         => ( hAPP_H9461717iple_a(hAPP_H1988213490iple_a(F_1,X_2),hAPP_f1853245639iple_a(F,A)) = hAPP_f1853245639iple_a(F,A) ) ) ) ) ).

tff(fact_462_folding__one__idem_Oin__idem,axiom,
    ! [X_2: pname,A: fun_pname_bool,F_1: fun_pn1208287926_pname,F: fun_fu1499449723_pname] :
      ( hBOOL(finite89670078_pname(F_1,F))
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
       => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
         => ( hAPP_pname_pname(hAPP_p1170154830_pname(F_1,X_2),hAPP_f1297739591_pname(F,A)) = hAPP_f1297739591_pname(F,A) ) ) ) ) ).

tff(fact_463_folding__one__idem_Ohom__commute,axiom,
    ! [N_3: fun_pname_bool,H: fun_pname_pname,F_1: fun_pn1208287926_pname,F: fun_fu1499449723_pname] :
      ( hBOOL(finite89670078_pname(F_1,F))
     => ( ! [X: pname,Y_2: pname] : hAPP_pname_pname(H,hAPP_pname_pname(hAPP_p1170154830_pname(F_1,X),Y_2)) = hAPP_pname_pname(hAPP_p1170154830_pname(F_1,hAPP_pname_pname(H,X)),hAPP_pname_pname(H,Y_2))
       => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,N_3))
         => ( ( N_3 != bot_bo844097828e_bool )
           => ( hAPP_pname_pname(H,hAPP_f1297739591_pname(F,N_3)) = hAPP_f1297739591_pname(F,hAPP_f759274231e_bool(image_pname_pname(H),N_3)) ) ) ) ) ) ).

tff(fact_464_folding__one__idem_Ohom__commute,axiom,
    ! [N_3: fun_Ho287446294a_bool,H: fun_Ho1275796541iple_a,F_1: fun_Ho115022426iple_a,F: fun_fu1634330235iple_a] :
      ( hBOOL(finite1127948314iple_a(F_1,F))
     => ( ! [X: hoare_1544627872iple_a,Y_2: hoare_1544627872iple_a] : hAPP_H9461717iple_a(H,hAPP_H9461717iple_a(hAPP_H1988213490iple_a(F_1,X),Y_2)) = hAPP_H9461717iple_a(hAPP_H1988213490iple_a(F_1,hAPP_H9461717iple_a(H,X)),hAPP_H9461717iple_a(H,Y_2))
       => ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,N_3))
         => ( ( N_3 != bot_bo1766443648a_bool )
           => ( hAPP_H9461717iple_a(H,hAPP_f1853245639iple_a(F,N_3)) = hAPP_f1853245639iple_a(F,hAPP_f909437487a_bool(image_1784540805iple_a(H),N_3)) ) ) ) ) ) ).

tff(fact_465_folding__one__idem_Ohom__commute,axiom,
    ! [N_3: fun_Ho1996104121e_bool,H: fun_Ho1455141843_state,F_1: fun_Ho784854669_state,F: fun_fu385705929_state] :
      ( hBOOL(finite1514162941_state(F_1,F))
     => ( ! [X: hoare_1875481847_state,Y_2: hoare_1875481847_state] : hAPP_H350317809_state(H,hAPP_H350317809_state(hAPP_H743221337_state(F_1,X),Y_2)) = hAPP_H350317809_state(hAPP_H743221337_state(F_1,hAPP_H350317809_state(H,X)),hAPP_H350317809_state(H,Y_2))
       => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,N_3))
         => ( ( N_3 != bot_bo1715400655e_bool )
           => ( hAPP_H350317809_state(H,hAPP_f890249627_state(F,N_3)) = hAPP_f890249627_state(F,hAPP_f121055253e_bool(image_436941729_state(H),N_3)) ) ) ) ) ) ).

tff(fact_466_folding__one_Oinsert,axiom,
    ! [X_2: hoare_1875481847_state,A: fun_Ho1996104121e_bool,F_1: fun_Ho784854669_state,F: fun_fu385705929_state] :
      ( hBOOL(finite1782051386_state(F_1,F))
     => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A))
       => ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),A))
         => ( ( A != bot_bo1715400655e_bool )
           => ( hAPP_f890249627_state(F,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),A)) = hAPP_H350317809_state(hAPP_H743221337_state(F_1,X_2),hAPP_f890249627_state(F,A)) ) ) ) ) ) ).

tff(fact_467_folding__one_Oinsert,axiom,
    ! [X_2: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,F_1: fun_Ho115022426iple_a,F: fun_fu1634330235iple_a] :
      ( hBOOL(finite1847914781iple_a(F_1,F))
     => ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,A))
       => ( ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_2),A))
         => ( ( A != bot_bo1766443648a_bool )
           => ( hAPP_f1853245639iple_a(F,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_2),A)) = hAPP_H9461717iple_a(hAPP_H1988213490iple_a(F_1,X_2),hAPP_f1853245639iple_a(F,A)) ) ) ) ) ) ).

tff(fact_468_folding__one_Oinsert,axiom,
    ! [X_2: pname,A: fun_pname_bool,F_1: fun_pn1208287926_pname,F: fun_fu1499449723_pname] :
      ( hBOOL(finite1282449217_pname(F_1,F))
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
       => ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
         => ( ( A != bot_bo844097828e_bool )
           => ( hAPP_f1297739591_pname(F,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),A)) = hAPP_pname_pname(hAPP_p1170154830_pname(F_1,X_2),hAPP_f1297739591_pname(F,A)) ) ) ) ) ) ).

tff(fact_469_folding__one_Osingleton,axiom,
    ! [X_2: hoare_1875481847_state,F_1: fun_Ho784854669_state,F: fun_fu385705929_state] :
      ( hBOOL(finite1782051386_state(F_1,F))
     => ( hAPP_f890249627_state(F,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),bot_bo1715400655e_bool)) = X_2 ) ) ).

tff(fact_470_folding__one_Osingleton,axiom,
    ! [X_2: hoare_1544627872iple_a,F_1: fun_Ho115022426iple_a,F: fun_fu1634330235iple_a] :
      ( hBOOL(finite1847914781iple_a(F_1,F))
     => ( hAPP_f1853245639iple_a(F,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_2),bot_bo1766443648a_bool)) = X_2 ) ) ).

tff(fact_471_folding__one_Osingleton,axiom,
    ! [X_2: pname,F_1: fun_pn1208287926_pname,F: fun_fu1499449723_pname] :
      ( hBOOL(finite1282449217_pname(F_1,F))
     => ( hAPP_f1297739591_pname(F,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),bot_bo844097828e_bool)) = X_2 ) ) ).

tff(fact_472_folding__one_Oclosed,axiom,
    ! [A: fun_Ho1996104121e_bool,F_1: fun_Ho784854669_state,F: fun_fu385705929_state] :
      ( hBOOL(finite1782051386_state(F_1,F))
     => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A))
       => ( ( A != bot_bo1715400655e_bool )
         => ( ! [X: hoare_1875481847_state,Y_2: hoare_1875481847_state] : hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,hAPP_H350317809_state(hAPP_H743221337_state(F_1,X),Y_2)),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,Y_2),bot_bo1715400655e_bool))))
           => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,hAPP_f890249627_state(F,A)),A)) ) ) ) ) ).

tff(fact_473_folding__one_Oclosed,axiom,
    ! [A: fun_Ho287446294a_bool,F_1: fun_Ho115022426iple_a,F: fun_fu1634330235iple_a] :
      ( hBOOL(finite1847914781iple_a(F_1,F))
     => ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,A))
       => ( ( A != bot_bo1766443648a_bool )
         => ( ! [X: hoare_1544627872iple_a,Y_2: hoare_1544627872iple_a] : hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,hAPP_H9461717iple_a(hAPP_H1988213490iple_a(F_1,X),Y_2)),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,Y_2),bot_bo1766443648a_bool))))
           => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,hAPP_f1853245639iple_a(F,A)),A)) ) ) ) ) ).

tff(fact_474_folding__one_Oclosed,axiom,
    ! [A: fun_pname_bool,F_1: fun_pn1208287926_pname,F: fun_fu1499449723_pname] :
      ( hBOOL(finite1282449217_pname(F_1,F))
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
       => ( ( A != bot_bo844097828e_bool )
         => ( ! [X: pname,Y_2: pname] : hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,hAPP_pname_pname(hAPP_p1170154830_pname(F_1,X),Y_2)),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,Y_2),bot_bo844097828e_bool))))
           => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,hAPP_f1297739591_pname(F,A)),A)) ) ) ) ) ).

tff(fact_475_evaln__max2,axiom,
    ! [C2_1: com,S2: state,N2: nat,T2: state,C1_1: com,S1_1: state,N1: nat,T1: state] :
      ( hBOOL(hAPP_state_bool(evaln(C1_1,S1_1,N1),T1))
     => ( hBOOL(hAPP_state_bool(evaln(C2_1,S2,N2),T2))
       => ? [N: nat] :
            ( hBOOL(hAPP_state_bool(evaln(C1_1,S1_1,N),T1))
            & hBOOL(hAPP_state_bool(evaln(C2_1,S2,N),T2)) ) ) ) ).

tff(fact_476_mk__disjoint__insert,axiom,
    ! [A_1: hoare_1875481847_state,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),A))
     => ? [B_2: fun_Ho1996104121e_bool] :
          ( ( A = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),B_2) )
          & ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),B_2)) ) ) ).

tff(fact_477_mk__disjoint__insert,axiom,
    ! [A_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),A))
     => ? [B_2: fun_Ho287446294a_bool] :
          ( ( A = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),B_2) )
          & ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),B_2)) ) ) ).

tff(fact_478_mk__disjoint__insert,axiom,
    ! [A_1: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),A))
     => ? [B_2: fun_pname_bool] :
          ( ( A = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),B_2) )
          & ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),B_2)) ) ) ).

tff(fact_479_Set_Oset__insert,axiom,
    ! [X_2: hoare_1875481847_state,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),A))
     => ~ ! [B_2: fun_Ho1996104121e_bool] :
            ( ( A = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),B_2) )
           => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),B_2)) ) ) ).

tff(fact_480_Set_Oset__insert,axiom,
    ! [X_2: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_2),A))
     => ~ ! [B_2: fun_Ho287446294a_bool] :
            ( ( A = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_2),B_2) )
           => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_2),B_2)) ) ) ).

tff(fact_481_Set_Oset__insert,axiom,
    ! [X_2: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
     => ~ ! [B_2: fun_pname_bool] :
            ( ( A = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),B_2) )
           => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),B_2)) ) ) ).

tff(fact_482_equals0I,axiom,
    ! [A: fun_Ho287446294a_bool] :
      ( ! [Y_2: hoare_1544627872iple_a] : ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Y_2),A))
     => ( A = bot_bo1766443648a_bool ) ) ).

tff(fact_483_equals0I,axiom,
    ! [A: fun_pname_bool] :
      ( ! [Y_2: pname] : ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,Y_2),A))
     => ( A = bot_bo844097828e_bool ) ) ).

tff(fact_484_equals0I,axiom,
    ! [A: fun_Ho1996104121e_bool] :
      ( ! [Y_2: hoare_1875481847_state] : ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,Y_2),A))
     => ( A = bot_bo1715400655e_bool ) ) ).

tff(fact_485_Sup__fin_Ounion__idem,axiom,
    ! [B_1: fun_bool_bool,A: fun_bool_bool] :
      ( hBOOL(hAPP_f387058535l_bool(finite_finite_bool,A))
     => ( ( A != bot_bo806936373l_bool )
       => ( hBOOL(hAPP_f387058535l_bool(finite_finite_bool,B_1))
         => ( ( B_1 != bot_bo806936373l_bool )
           => ( hBOOL(big_la1480321694n_bool(hAPP_f961197973l_bool(hAPP_f1172769267l_bool(semila1866150931l_bool,A),B_1)))
            <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,big_la1480321694n_bool(A)),big_la1480321694n_bool(B_1))) ) ) ) ) ) ).

tff(fact_486_Sup__fin_Ounion__idem,axiom,
    ! [B_1: fun_fu1430349052l_bool,A: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
     => ( ( A != bot_bo1649642514l_bool )
       => ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B_1))
         => ( ( B_1 != bot_bo1649642514l_bool )
           => ( big_la841148155e_bool(hAPP_f559147733l_bool(hAPP_f230953622l_bool(semila1782091504l_bool,A),B_1)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,big_la841148155e_bool(A)),big_la841148155e_bool(B_1)) ) ) ) ) ) ).

tff(fact_487_Sup__fin_Ounion__idem,axiom,
    ! [B_1: fun_fu689207471l_bool,A: fun_fu689207471l_bool] :
      ( hBOOL(hAPP_f1699935247l_bool(finite1559130124e_bool,A))
     => ( ( A != bot_bo973479705l_bool )
       => ( hBOOL(hAPP_f1699935247l_bool(finite1559130124e_bool,B_1))
         => ( ( B_1 != bot_bo973479705l_bool )
           => ( big_la152959928e_bool(hAPP_f900945761l_bool(hAPP_f19215249l_bool(semila1703230395l_bool,A),B_1)) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,big_la152959928e_bool(A)),big_la152959928e_bool(B_1)) ) ) ) ) ) ).

tff(fact_488_Sup__fin_Ounion__idem,axiom,
    ! [B_1: fun_fu1441721944l_bool,A: fun_fu1441721944l_bool] :
      ( hBOOL(hAPP_f1059025184l_bool(finite1237779075a_bool,A))
     => ( ( A != bot_bo700474478l_bool )
       => ( hBOOL(hAPP_f1059025184l_bool(finite1237779075a_bool,B_1))
         => ( ( B_1 != bot_bo700474478l_bool )
           => ( big_la430824919a_bool(hAPP_f535450709l_bool(hAPP_f468151866l_bool(semila1202735692l_bool,A),B_1)) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,big_la430824919a_bool(A)),big_la430824919a_bool(B_1)) ) ) ) ) ) ).

tff(fact_489_Sup__fin_Oin__idem,axiom,
    ! [X_2: bool,A: fun_bool_bool] :
      ( hBOOL(hAPP_f387058535l_bool(finite_finite_bool,A))
     => ( hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,X_2),A))
       => ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),big_la1480321694n_bool(A)))
        <=> hBOOL(big_la1480321694n_bool(A)) ) ) ) ).

tff(fact_490_Sup__fin_Oin__idem,axiom,
    ! [X_2: fun_pname_bool,A: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
     => ( hBOOL(hAPP_f1935102916l_bool(hAPP_f556039215l_bool(member799430823e_bool,X_2),A))
       => ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),big_la841148155e_bool(A)) = big_la841148155e_bool(A) ) ) ) ).

tff(fact_491_Sup__fin_Oin__idem,axiom,
    ! [X_2: fun_Ho1996104121e_bool,A: fun_fu689207471l_bool] :
      ( hBOOL(hAPP_f1699935247l_bool(finite1559130124e_bool,A))
     => ( hBOOL(hAPP_f1699935247l_bool(hAPP_f640685261l_bool(member1062536460e_bool,X_2),A))
       => ( hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X_2),big_la152959928e_bool(A)) = big_la152959928e_bool(A) ) ) ) ).

tff(fact_492_Sup__fin_Oin__idem,axiom,
    ! [X_2: fun_Ho287446294a_bool,A: fun_fu1441721944l_bool] :
      ( hBOOL(hAPP_f1059025184l_bool(finite1237779075a_bool,A))
     => ( hBOOL(hAPP_f1059025184l_bool(hAPP_f372929895l_bool(member1738976643a_bool,X_2),A))
       => ( hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X_2),big_la430824919a_bool(A)) = big_la430824919a_bool(A) ) ) ) ).

tff(fact_493_Sup__fin_Oinsert,axiom,
    ! [X_2: bool,A: fun_bool_bool] :
      ( hBOOL(hAPP_f387058535l_bool(finite_finite_bool,A))
     => ( ~ hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,X_2),A))
       => ( ( A != bot_bo806936373l_bool )
         => ( hBOOL(big_la1480321694n_bool(hAPP_f961197973l_bool(hAPP_b496459037l_bool(insert_bool,X_2),A)))
          <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),big_la1480321694n_bool(A))) ) ) ) ) ).

tff(fact_494_Sup__fin_Oinsert,axiom,
    ! [X_2: fun_pname_bool,A: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
     => ( ~ hBOOL(hAPP_f1935102916l_bool(hAPP_f556039215l_bool(member799430823e_bool,X_2),A))
       => ( ( A != bot_bo1649642514l_bool )
         => ( big_la841148155e_bool(hAPP_f559147733l_bool(hAPP_f1320879424l_bool(insert1325755072e_bool,X_2),A)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),big_la841148155e_bool(A)) ) ) ) ) ).

tff(fact_495_Sup__fin_Oinsert,axiom,
    ! [X_2: fun_Ho1996104121e_bool,A: fun_fu689207471l_bool] :
      ( hBOOL(hAPP_f1699935247l_bool(finite1559130124e_bool,A))
     => ( ~ hBOOL(hAPP_f1699935247l_bool(hAPP_f640685261l_bool(member1062536460e_bool,X_2),A))
       => ( ( A != bot_bo973479705l_bool )
         => ( big_la152959928e_bool(hAPP_f900945761l_bool(hAPP_f872497439l_bool(insert504309043e_bool,X_2),A)) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X_2),big_la152959928e_bool(A)) ) ) ) ) ).

tff(fact_496_Sup__fin_Oinsert,axiom,
    ! [X_2: fun_Ho287446294a_bool,A: fun_fu1441721944l_bool] :
      ( hBOOL(hAPP_f1059025184l_bool(finite1237779075a_bool,A))
     => ( ~ hBOOL(hAPP_f1059025184l_bool(hAPP_f372929895l_bool(member1738976643a_bool,X_2),A))
       => ( ( A != bot_bo700474478l_bool )
         => ( big_la430824919a_bool(hAPP_f535450709l_bool(hAPP_f709279716l_bool(insert243155484a_bool,X_2),A)) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X_2),big_la430824919a_bool(A)) ) ) ) ) ).

tff(fact_497_Sup__fin_Oinsert__idem,axiom,
    ! [X_2: bool,A: fun_bool_bool] :
      ( hBOOL(hAPP_f387058535l_bool(finite_finite_bool,A))
     => ( ( A != bot_bo806936373l_bool )
       => ( hBOOL(big_la1480321694n_bool(hAPP_f961197973l_bool(hAPP_b496459037l_bool(insert_bool,X_2),A)))
        <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),big_la1480321694n_bool(A))) ) ) ) ).

tff(fact_498_Sup__fin_Oinsert__idem,axiom,
    ! [X_2: fun_pname_bool,A: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
     => ( ( A != bot_bo1649642514l_bool )
       => ( big_la841148155e_bool(hAPP_f559147733l_bool(hAPP_f1320879424l_bool(insert1325755072e_bool,X_2),A)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),big_la841148155e_bool(A)) ) ) ) ).

tff(fact_499_Sup__fin_Oinsert__idem,axiom,
    ! [X_2: fun_Ho1996104121e_bool,A: fun_fu689207471l_bool] :
      ( hBOOL(hAPP_f1699935247l_bool(finite1559130124e_bool,A))
     => ( ( A != bot_bo973479705l_bool )
       => ( big_la152959928e_bool(hAPP_f900945761l_bool(hAPP_f872497439l_bool(insert504309043e_bool,X_2),A)) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X_2),big_la152959928e_bool(A)) ) ) ) ).

tff(fact_500_Sup__fin_Oinsert__idem,axiom,
    ! [X_2: fun_Ho287446294a_bool,A: fun_fu1441721944l_bool] :
      ( hBOOL(hAPP_f1059025184l_bool(finite1237779075a_bool,A))
     => ( ( A != bot_bo700474478l_bool )
       => ( big_la430824919a_bool(hAPP_f535450709l_bool(hAPP_f709279716l_bool(insert243155484a_bool,X_2),A)) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X_2),big_la430824919a_bool(A)) ) ) ) ).

tff(fact_501_Sup__fin_Ohom__commute,axiom,
    ! [N_3: fun_bool_bool,H: fun_bool_bool] :
      ( ! [X: bool,Y_2: bool] :
          ( hBOOL(hAPP_bool_bool(H,hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X),Y_2)))
        <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,hAPP_bool_bool(H,X)),hAPP_bool_bool(H,Y_2))) )
     => ( hBOOL(hAPP_f387058535l_bool(finite_finite_bool,N_3))
       => ( ( N_3 != bot_bo806936373l_bool )
         => ( hBOOL(hAPP_bool_bool(H,big_la1480321694n_bool(N_3)))
          <=> hBOOL(big_la1480321694n_bool(hAPP_f961197973l_bool(image_bool_bool(H),N_3))) ) ) ) ) ).

tff(fact_502_Sup__fin_Ohom__commute,axiom,
    ! [N_3: fun_fu1430349052l_bool,H: fun_fu410713561e_bool] :
      ( ! [X: fun_pname_bool,Y_2: fun_pname_bool] : hAPP_f759274231e_bool(H,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X),Y_2)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f759274231e_bool(H,X)),hAPP_f759274231e_bool(H,Y_2))
     => ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,N_3))
       => ( ( N_3 != bot_bo1649642514l_bool )
         => ( hAPP_f759274231e_bool(H,big_la841148155e_bool(N_3)) = big_la841148155e_bool(hAPP_f559147733l_bool(image_2063528359e_bool(H),N_3)) ) ) ) ) ).

tff(fact_503_Sup__fin_Ohom__commute,axiom,
    ! [N_3: fun_fu689207471l_bool,H: fun_fu1296727421e_bool] :
      ( ! [X: fun_Ho1996104121e_bool,Y_2: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(H,hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X),Y_2)) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,hAPP_f121055253e_bool(H,X)),hAPP_f121055253e_bool(H,Y_2))
     => ( hBOOL(hAPP_f1699935247l_bool(finite1559130124e_bool,N_3))
       => ( ( N_3 != bot_bo973479705l_bool )
         => ( hAPP_f121055253e_bool(H,big_la152959928e_bool(N_3)) = big_la152959928e_bool(hAPP_f900945761l_bool(image_1852024517e_bool(H),N_3)) ) ) ) ) ).

tff(fact_504_Sup__fin_Ohom__commute,axiom,
    ! [N_3: fun_fu1441721944l_bool,H: fun_fu410471825a_bool] :
      ( ! [X: fun_Ho287446294a_bool,Y_2: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(H,hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X),Y_2)) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,hAPP_f909437487a_bool(H,X)),hAPP_f909437487a_bool(H,Y_2))
     => ( hBOOL(hAPP_f1059025184l_bool(finite1237779075a_bool,N_3))
       => ( ( N_3 != bot_bo700474478l_bool )
         => ( hAPP_f909437487a_bool(H,big_la430824919a_bool(N_3)) = big_la430824919a_bool(hAPP_f535450709l_bool(image_1585000671a_bool(H),N_3)) ) ) ) ) ).

tff(fact_505_Sup__fin_Oclosed,axiom,
    ! [A: fun_bool_bool] :
      ( hBOOL(hAPP_f387058535l_bool(finite_finite_bool,A))
     => ( ( A != bot_bo806936373l_bool )
       => ( ! [X: bool,Y_2: bool] : hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X),Y_2)),hAPP_f961197973l_bool(hAPP_b496459037l_bool(insert_bool,X),hAPP_f961197973l_bool(hAPP_b496459037l_bool(insert_bool,Y_2),bot_bo806936373l_bool))))
         => hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,big_la1480321694n_bool(A)),A)) ) ) ) ).

tff(fact_506_Sup__fin_Oclosed,axiom,
    ! [A: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
     => ( ( A != bot_bo1649642514l_bool )
       => ( ! [X: fun_pname_bool,Y_2: fun_pname_bool] : hBOOL(hAPP_f1935102916l_bool(hAPP_f556039215l_bool(member799430823e_bool,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X),Y_2)),hAPP_f559147733l_bool(hAPP_f1320879424l_bool(insert1325755072e_bool,X),hAPP_f559147733l_bool(hAPP_f1320879424l_bool(insert1325755072e_bool,Y_2),bot_bo1649642514l_bool))))
         => hBOOL(hAPP_f1935102916l_bool(hAPP_f556039215l_bool(member799430823e_bool,big_la841148155e_bool(A)),A)) ) ) ) ).

tff(fact_507_Sup__fin_Oclosed,axiom,
    ! [A: fun_fu689207471l_bool] :
      ( hBOOL(hAPP_f1699935247l_bool(finite1559130124e_bool,A))
     => ( ( A != bot_bo973479705l_bool )
       => ( ! [X: fun_Ho1996104121e_bool,Y_2: fun_Ho1996104121e_bool] : hBOOL(hAPP_f1699935247l_bool(hAPP_f640685261l_bool(member1062536460e_bool,hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X),Y_2)),hAPP_f900945761l_bool(hAPP_f872497439l_bool(insert504309043e_bool,X),hAPP_f900945761l_bool(hAPP_f872497439l_bool(insert504309043e_bool,Y_2),bot_bo973479705l_bool))))
         => hBOOL(hAPP_f1699935247l_bool(hAPP_f640685261l_bool(member1062536460e_bool,big_la152959928e_bool(A)),A)) ) ) ) ).

tff(fact_508_Sup__fin_Oclosed,axiom,
    ! [A: fun_fu1441721944l_bool] :
      ( hBOOL(hAPP_f1059025184l_bool(finite1237779075a_bool,A))
     => ( ( A != bot_bo700474478l_bool )
       => ( ! [X: fun_Ho287446294a_bool,Y_2: fun_Ho287446294a_bool] : hBOOL(hAPP_f1059025184l_bool(hAPP_f372929895l_bool(member1738976643a_bool,hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X),Y_2)),hAPP_f535450709l_bool(hAPP_f709279716l_bool(insert243155484a_bool,X),hAPP_f535450709l_bool(hAPP_f709279716l_bool(insert243155484a_bool,Y_2),bot_bo700474478l_bool))))
         => hBOOL(hAPP_f1059025184l_bool(hAPP_f372929895l_bool(member1738976643a_bool,big_la430824919a_bool(A)),A)) ) ) ) ).

tff(fact_509_Sup__fin_Ounion__inter,axiom,
    ! [B_1: fun_bool_bool,A: fun_bool_bool] :
      ( hBOOL(hAPP_f387058535l_bool(finite_finite_bool,A))
     => ( hBOOL(hAPP_f387058535l_bool(finite_finite_bool,B_1))
       => ( ( hAPP_f961197973l_bool(semila1551573549l_bool(A),B_1) != bot_bo806936373l_bool )
         => ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,big_la1480321694n_bool(hAPP_f961197973l_bool(hAPP_f1172769267l_bool(semila1866150931l_bool,A),B_1))),big_la1480321694n_bool(hAPP_f961197973l_bool(semila1551573549l_bool(A),B_1))))
          <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,big_la1480321694n_bool(A)),big_la1480321694n_bool(B_1))) ) ) ) ) ).

tff(fact_510_Sup__fin_Ounion__inter,axiom,
    ! [B_1: fun_fu1430349052l_bool,A: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
     => ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B_1))
       => ( ( hAPP_f559147733l_bool(semila1130628874l_bool(A),B_1) != bot_bo1649642514l_bool )
         => ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,big_la841148155e_bool(hAPP_f559147733l_bool(hAPP_f230953622l_bool(semila1782091504l_bool,A),B_1))),big_la841148155e_bool(hAPP_f559147733l_bool(semila1130628874l_bool(A),B_1))) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,big_la841148155e_bool(A)),big_la841148155e_bool(B_1)) ) ) ) ) ).

tff(fact_511_Sup__fin_Ounion__inter,axiom,
    ! [B_1: fun_fu689207471l_bool,A: fun_fu689207471l_bool] :
      ( hBOOL(hAPP_f1699935247l_bool(finite1559130124e_bool,A))
     => ( hBOOL(hAPP_f1699935247l_bool(finite1559130124e_bool,B_1))
       => ( ( hAPP_f900945761l_bool(semila1058800161l_bool(A),B_1) != bot_bo973479705l_bool )
         => ( hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,big_la152959928e_bool(hAPP_f900945761l_bool(hAPP_f19215249l_bool(semila1703230395l_bool,A),B_1))),big_la152959928e_bool(hAPP_f900945761l_bool(semila1058800161l_bool(A),B_1))) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,big_la152959928e_bool(A)),big_la152959928e_bool(B_1)) ) ) ) ) ).

tff(fact_512_Sup__fin_Ounion__inter,axiom,
    ! [B_1: fun_fu1441721944l_bool,A: fun_fu1441721944l_bool] :
      ( hBOOL(hAPP_f1059025184l_bool(finite1237779075a_bool,A))
     => ( hBOOL(hAPP_f1059025184l_bool(finite1237779075a_bool,B_1))
       => ( ( hAPP_f535450709l_bool(semila1122052966l_bool(A),B_1) != bot_bo700474478l_bool )
         => ( hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,big_la430824919a_bool(hAPP_f535450709l_bool(hAPP_f468151866l_bool(semila1202735692l_bool,A),B_1))),big_la430824919a_bool(hAPP_f535450709l_bool(semila1122052966l_bool(A),B_1))) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,big_la430824919a_bool(A)),big_la430824919a_bool(B_1)) ) ) ) ) ).

tff(fact_513_Sup__fin_Ounion__disjoint,axiom,
    ! [B_1: fun_bool_bool,A: fun_bool_bool] :
      ( hBOOL(hAPP_f387058535l_bool(finite_finite_bool,A))
     => ( ( A != bot_bo806936373l_bool )
       => ( hBOOL(hAPP_f387058535l_bool(finite_finite_bool,B_1))
         => ( ( B_1 != bot_bo806936373l_bool )
           => ( ( hAPP_f961197973l_bool(semila1551573549l_bool(A),B_1) = bot_bo806936373l_bool )
             => ( hBOOL(big_la1480321694n_bool(hAPP_f961197973l_bool(hAPP_f1172769267l_bool(semila1866150931l_bool,A),B_1)))
              <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,big_la1480321694n_bool(A)),big_la1480321694n_bool(B_1))) ) ) ) ) ) ) ).

tff(fact_514_Sup__fin_Ounion__disjoint,axiom,
    ! [B_1: fun_fu1430349052l_bool,A: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
     => ( ( A != bot_bo1649642514l_bool )
       => ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B_1))
         => ( ( B_1 != bot_bo1649642514l_bool )
           => ( ( hAPP_f559147733l_bool(semila1130628874l_bool(A),B_1) = bot_bo1649642514l_bool )
             => ( big_la841148155e_bool(hAPP_f559147733l_bool(hAPP_f230953622l_bool(semila1782091504l_bool,A),B_1)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,big_la841148155e_bool(A)),big_la841148155e_bool(B_1)) ) ) ) ) ) ) ).

tff(fact_515_Sup__fin_Ounion__disjoint,axiom,
    ! [B_1: fun_fu689207471l_bool,A: fun_fu689207471l_bool] :
      ( hBOOL(hAPP_f1699935247l_bool(finite1559130124e_bool,A))
     => ( ( A != bot_bo973479705l_bool )
       => ( hBOOL(hAPP_f1699935247l_bool(finite1559130124e_bool,B_1))
         => ( ( B_1 != bot_bo973479705l_bool )
           => ( ( hAPP_f900945761l_bool(semila1058800161l_bool(A),B_1) = bot_bo973479705l_bool )
             => ( big_la152959928e_bool(hAPP_f900945761l_bool(hAPP_f19215249l_bool(semila1703230395l_bool,A),B_1)) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,big_la152959928e_bool(A)),big_la152959928e_bool(B_1)) ) ) ) ) ) ) ).

tff(fact_516_Sup__fin_Ounion__disjoint,axiom,
    ! [B_1: fun_fu1441721944l_bool,A: fun_fu1441721944l_bool] :
      ( hBOOL(hAPP_f1059025184l_bool(finite1237779075a_bool,A))
     => ( ( A != bot_bo700474478l_bool )
       => ( hBOOL(hAPP_f1059025184l_bool(finite1237779075a_bool,B_1))
         => ( ( B_1 != bot_bo700474478l_bool )
           => ( ( hAPP_f535450709l_bool(semila1122052966l_bool(A),B_1) = bot_bo700474478l_bool )
             => ( big_la430824919a_bool(hAPP_f535450709l_bool(hAPP_f468151866l_bool(semila1202735692l_bool,A),B_1)) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,big_la430824919a_bool(A)),big_la430824919a_bool(B_1)) ) ) ) ) ) ) ).

tff(fact_517_IntI,axiom,
    ! [B_1: fun_Ho287446294a_bool,C: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),A))
     => ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),B_1))
       => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),hAPP_f909437487a_bool(semila1256126856a_bool(A),B_1))) ) ) ).

tff(fact_518_IntI,axiom,
    ! [B_1: fun_pname_bool,C: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),A))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),B_1))
       => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),hAPP_f759274231e_bool(semila1635148844e_bool(A),B_1))) ) ) ).

tff(fact_519_IntE,axiom,
    ! [C: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,B_1: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),hAPP_f909437487a_bool(semila1256126856a_bool(A),B_1)))
     => ~ ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),A))
         => ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),B_1)) ) ) ).

tff(fact_520_IntE,axiom,
    ! [C: pname,A: fun_pname_bool,B_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),hAPP_f759274231e_bool(semila1635148844e_bool(A),B_1)))
     => ~ ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),A))
         => ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),B_1)) ) ) ).

tff(fact_521_inf__bot__right,axiom,
    ! [X_2: fun_pname_bool] : hAPP_f759274231e_bool(semila1635148844e_bool(X_2),bot_bo844097828e_bool) = bot_bo844097828e_bool ).

tff(fact_522_inf__bot__right,axiom,
    ! [X_2: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(semila1256126856a_bool(X_2),bot_bo1766443648a_bool) = bot_bo1766443648a_bool ).

tff(fact_523_inf__bot__right,axiom,
    ! [X_2: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(semila657954503e_bool(X_2),bot_bo1715400655e_bool) = bot_bo1715400655e_bool ).

tff(fact_524_inf__bot__left,axiom,
    ! [X_2: fun_pname_bool] : hAPP_f759274231e_bool(semila1635148844e_bool(bot_bo844097828e_bool),X_2) = bot_bo844097828e_bool ).

tff(fact_525_inf__bot__left,axiom,
    ! [X_2: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(semila1256126856a_bool(bot_bo1766443648a_bool),X_2) = bot_bo1766443648a_bool ).

tff(fact_526_inf__bot__left,axiom,
    ! [X_2: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(semila657954503e_bool(bot_bo1715400655e_bool),X_2) = bot_bo1715400655e_bool ).

tff(fact_527_sup__inf__distrib2,axiom,
    ! [Y_1: bool,Z_1: bool,X_2: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,hAPP_bool_bool(semila310582991f_bool(Y_1),Z_1)),X_2))
    <=> hBOOL(hAPP_bool_bool(semila310582991f_bool(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,Y_1),X_2)),hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,Z_1),X_2))) ) ).

tff(fact_528_sup__inf__distrib2,axiom,
    ! [Y_1: fun_pname_bool,Z_1: fun_pname_bool,X_2: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f759274231e_bool(semila1635148844e_bool(Y_1),Z_1)),X_2) = hAPP_f759274231e_bool(semila1635148844e_bool(hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,Y_1),X_2)),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,Z_1),X_2)) ).

tff(fact_529_sup__inf__distrib2,axiom,
    ! [Y_1: fun_Ho1996104121e_bool,Z_1: fun_Ho1996104121e_bool,X_2: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,hAPP_f121055253e_bool(semila657954503e_bool(Y_1),Z_1)),X_2) = hAPP_f121055253e_bool(semila657954503e_bool(hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,Y_1),X_2)),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,Z_1),X_2)) ).

tff(fact_530_sup__inf__distrib2,axiom,
    ! [Y_1: fun_Ho287446294a_bool,Z_1: fun_Ho287446294a_bool,X_2: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,hAPP_f909437487a_bool(semila1256126856a_bool(Y_1),Z_1)),X_2) = hAPP_f909437487a_bool(semila1256126856a_bool(hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,Y_1),X_2)),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,Z_1),X_2)) ).

tff(fact_531_inf__sup__distrib2,axiom,
    ! [Y_1: bool,Z_1: bool,X_2: bool] :
      ( hBOOL(hAPP_bool_bool(semila310582991f_bool(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,Y_1),Z_1)),X_2))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,hAPP_bool_bool(semila310582991f_bool(Y_1),X_2)),hAPP_bool_bool(semila310582991f_bool(Z_1),X_2))) ) ).

tff(fact_532_inf__sup__distrib2,axiom,
    ! [Y_1: fun_pname_bool,Z_1: fun_pname_bool,X_2: fun_pname_bool] : hAPP_f759274231e_bool(semila1635148844e_bool(hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,Y_1),Z_1)),X_2) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f759274231e_bool(semila1635148844e_bool(Y_1),X_2)),hAPP_f759274231e_bool(semila1635148844e_bool(Z_1),X_2)) ).

tff(fact_533_inf__sup__distrib2,axiom,
    ! [Y_1: fun_Ho1996104121e_bool,Z_1: fun_Ho1996104121e_bool,X_2: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(semila657954503e_bool(hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,Y_1),Z_1)),X_2) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,hAPP_f121055253e_bool(semila657954503e_bool(Y_1),X_2)),hAPP_f121055253e_bool(semila657954503e_bool(Z_1),X_2)) ).

tff(fact_534_inf__sup__distrib2,axiom,
    ! [Y_1: fun_Ho287446294a_bool,Z_1: fun_Ho287446294a_bool,X_2: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(semila1256126856a_bool(hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,Y_1),Z_1)),X_2) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,hAPP_f909437487a_bool(semila1256126856a_bool(Y_1),X_2)),hAPP_f909437487a_bool(semila1256126856a_bool(Z_1),X_2)) ).

tff(fact_535_sup__inf__distrib1,axiom,
    ! [X_2: bool,Y_1: bool,Z_1: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),hAPP_bool_bool(semila310582991f_bool(Y_1),Z_1)))
    <=> hBOOL(hAPP_bool_bool(semila310582991f_bool(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),Y_1)),hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),Z_1))) ) ).

tff(fact_536_sup__inf__distrib1,axiom,
    ! [X_2: fun_pname_bool,Y_1: fun_pname_bool,Z_1: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),hAPP_f759274231e_bool(semila1635148844e_bool(Y_1),Z_1)) = hAPP_f759274231e_bool(semila1635148844e_bool(hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),Y_1)),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),Z_1)) ).

tff(fact_537_sup__inf__distrib1,axiom,
    ! [X_2: fun_Ho1996104121e_bool,Y_1: fun_Ho1996104121e_bool,Z_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X_2),hAPP_f121055253e_bool(semila657954503e_bool(Y_1),Z_1)) = hAPP_f121055253e_bool(semila657954503e_bool(hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X_2),Y_1)),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X_2),Z_1)) ).

tff(fact_538_sup__inf__distrib1,axiom,
    ! [X_2: fun_Ho287446294a_bool,Y_1: fun_Ho287446294a_bool,Z_1: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X_2),hAPP_f909437487a_bool(semila1256126856a_bool(Y_1),Z_1)) = hAPP_f909437487a_bool(semila1256126856a_bool(hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X_2),Y_1)),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X_2),Z_1)) ).

tff(fact_539_inf__sup__distrib1,axiom,
    ! [X_2: bool,Y_1: bool,Z_1: bool] :
      ( hBOOL(hAPP_bool_bool(semila310582991f_bool(X_2),hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,Y_1),Z_1)))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,hAPP_bool_bool(semila310582991f_bool(X_2),Y_1)),hAPP_bool_bool(semila310582991f_bool(X_2),Z_1))) ) ).

tff(fact_540_inf__sup__distrib1,axiom,
    ! [X_2: fun_pname_bool,Y_1: fun_pname_bool,Z_1: fun_pname_bool] : hAPP_f759274231e_bool(semila1635148844e_bool(X_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,Y_1),Z_1)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f759274231e_bool(semila1635148844e_bool(X_2),Y_1)),hAPP_f759274231e_bool(semila1635148844e_bool(X_2),Z_1)) ).

tff(fact_541_inf__sup__distrib1,axiom,
    ! [X_2: fun_Ho1996104121e_bool,Y_1: fun_Ho1996104121e_bool,Z_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(semila657954503e_bool(X_2),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,Y_1),Z_1)) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,hAPP_f121055253e_bool(semila657954503e_bool(X_2),Y_1)),hAPP_f121055253e_bool(semila657954503e_bool(X_2),Z_1)) ).

tff(fact_542_inf__sup__distrib1,axiom,
    ! [X_2: fun_Ho287446294a_bool,Y_1: fun_Ho287446294a_bool,Z_1: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(semila1256126856a_bool(X_2),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,Y_1),Z_1)) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,hAPP_f909437487a_bool(semila1256126856a_bool(X_2),Y_1)),hAPP_f909437487a_bool(semila1256126856a_bool(X_2),Z_1)) ).

tff(fact_543_sup__inf__absorb,axiom,
    ! [X_2: bool,Y_1: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),hAPP_bool_bool(semila310582991f_bool(X_2),Y_1)))
    <=> hBOOL(X_2) ) ).

tff(fact_544_sup__inf__absorb,axiom,
    ! [X_2: fun_pname_bool,Y_1: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),hAPP_f759274231e_bool(semila1635148844e_bool(X_2),Y_1)) = X_2 ).

tff(fact_545_sup__inf__absorb,axiom,
    ! [X_2: fun_Ho1996104121e_bool,Y_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X_2),hAPP_f121055253e_bool(semila657954503e_bool(X_2),Y_1)) = X_2 ).

tff(fact_546_sup__inf__absorb,axiom,
    ! [X_2: fun_Ho287446294a_bool,Y_1: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X_2),hAPP_f909437487a_bool(semila1256126856a_bool(X_2),Y_1)) = X_2 ).

tff(fact_547_inf__sup__absorb,axiom,
    ! [X_2: bool,Y_1: bool] :
      ( hBOOL(hAPP_bool_bool(semila310582991f_bool(X_2),hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),Y_1)))
    <=> hBOOL(X_2) ) ).

tff(fact_548_inf__sup__absorb,axiom,
    ! [X_2: fun_pname_bool,Y_1: fun_pname_bool] : hAPP_f759274231e_bool(semila1635148844e_bool(X_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),Y_1)) = X_2 ).

tff(fact_549_inf__sup__absorb,axiom,
    ! [X_2: fun_Ho1996104121e_bool,Y_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(semila657954503e_bool(X_2),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X_2),Y_1)) = X_2 ).

tff(fact_550_inf__sup__absorb,axiom,
    ! [X_2: fun_Ho287446294a_bool,Y_1: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(semila1256126856a_bool(X_2),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X_2),Y_1)) = X_2 ).

tff(fact_551_disjoint__iff__not__equal,axiom,
    ! [A: fun_pname_bool,B_1: fun_pname_bool] :
      ( ( hAPP_f759274231e_bool(semila1635148844e_bool(A),B_1) = bot_bo844097828e_bool )
    <=> ! [X: pname] :
          ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),A))
         => ! [Xa: pname] :
              ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,Xa),B_1))
             => ( X != Xa ) ) ) ) ).

tff(fact_552_disjoint__iff__not__equal,axiom,
    ! [A: fun_Ho287446294a_bool,B_1: fun_Ho287446294a_bool] :
      ( ( hAPP_f909437487a_bool(semila1256126856a_bool(A),B_1) = bot_bo1766443648a_bool )
    <=> ! [X: hoare_1544627872iple_a] :
          ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X),A))
         => ! [Xa: hoare_1544627872iple_a] :
              ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Xa),B_1))
             => ( X != Xa ) ) ) ) ).

tff(fact_553_disjoint__iff__not__equal,axiom,
    ! [A: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( ( hAPP_f121055253e_bool(semila657954503e_bool(A),B_1) = bot_bo1715400655e_bool )
    <=> ! [X: hoare_1875481847_state] :
          ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X),A))
         => ! [Xa: hoare_1875481847_state] :
              ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,Xa),B_1))
             => ( X != Xa ) ) ) ) ).

tff(fact_554_Int__empty__right,axiom,
    ! [A: fun_pname_bool] : hAPP_f759274231e_bool(semila1635148844e_bool(A),bot_bo844097828e_bool) = bot_bo844097828e_bool ).

tff(fact_555_Int__empty__right,axiom,
    ! [A: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(semila1256126856a_bool(A),bot_bo1766443648a_bool) = bot_bo1766443648a_bool ).

tff(fact_556_Int__empty__right,axiom,
    ! [A: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(semila657954503e_bool(A),bot_bo1715400655e_bool) = bot_bo1715400655e_bool ).

tff(fact_557_Int__empty__left,axiom,
    ! [B_1: fun_pname_bool] : hAPP_f759274231e_bool(semila1635148844e_bool(bot_bo844097828e_bool),B_1) = bot_bo844097828e_bool ).

tff(fact_558_Int__empty__left,axiom,
    ! [B_1: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(semila1256126856a_bool(bot_bo1766443648a_bool),B_1) = bot_bo1766443648a_bool ).

tff(fact_559_Int__empty__left,axiom,
    ! [B_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(semila657954503e_bool(bot_bo1715400655e_bool),B_1) = bot_bo1715400655e_bool ).

tff(fact_560_Int__def,axiom,
    ! [A: fun_Ho287446294a_bool,B_1: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(semila1256126856a_bool(A),B_1) = hAPP_f909437487a_bool(collec445968053iple_a,hAPP_f909437487a_bool(hAPP_f1099051562a_bool(cOMBS_523298619l_bool,hAPP_f224574947l_bool(cOMBB_1960865690iple_a(fconj),hAPP_f909437487a_bool(hAPP_f482270129a_bool(cOMBC_1807206788l_bool,member1891733107iple_a),A))),hAPP_f909437487a_bool(hAPP_f482270129a_bool(cOMBC_1807206788l_bool,member1891733107iple_a),B_1))) ).

tff(fact_561_Int__def,axiom,
    ! [A: fun_pname_bool,B_1: fun_pname_bool] : hAPP_f759274231e_bool(semila1635148844e_bool(A),B_1) = hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,hAPP_f1402196763l_bool(cOMBB_675860798_pname(fconj),hAPP_f759274231e_bool(hAPP_f661147897e_bool(cOMBC_1058051404l_bool,member_pname),A))),hAPP_f759274231e_bool(hAPP_f661147897e_bool(cOMBC_1058051404l_bool,member_pname),B_1))) ).

tff(fact_562_Int__iff,axiom,
    ! [C: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,B_1: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),hAPP_f909437487a_bool(semila1256126856a_bool(A),B_1)))
    <=> ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),A))
        & hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),B_1)) ) ) ).

tff(fact_563_Int__iff,axiom,
    ! [C: pname,A: fun_pname_bool,B_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),hAPP_f759274231e_bool(semila1635148844e_bool(A),B_1)))
    <=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),A))
        & hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),B_1)) ) ) ).

tff(fact_564_IntD1,axiom,
    ! [C: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,B_1: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),hAPP_f909437487a_bool(semila1256126856a_bool(A),B_1)))
     => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),A)) ) ).

tff(fact_565_IntD1,axiom,
    ! [C: pname,A: fun_pname_bool,B_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),hAPP_f759274231e_bool(semila1635148844e_bool(A),B_1)))
     => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),A)) ) ).

tff(fact_566_IntD2,axiom,
    ! [C: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,B_1: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),hAPP_f909437487a_bool(semila1256126856a_bool(A),B_1)))
     => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),B_1)) ) ).

tff(fact_567_IntD2,axiom,
    ! [C: pname,A: fun_pname_bool,B_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),hAPP_f759274231e_bool(semila1635148844e_bool(A),B_1)))
     => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),B_1)) ) ).

tff(fact_568_Collect__conj__eq,axiom,
    ! [Pa: fun_Ho287446294a_bool,Qa: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(collec445968053iple_a,hAPP_f909437487a_bool(hAPP_f1099051562a_bool(cOMBS_523298619l_bool,hAPP_f224574947l_bool(cOMBB_1960865690iple_a(fconj),Pa)),Qa)) = hAPP_f909437487a_bool(semila1256126856a_bool(hAPP_f909437487a_bool(collec445968053iple_a,Pa)),hAPP_f909437487a_bool(collec445968053iple_a,Qa)) ).

tff(fact_569_Collect__conj__eq,axiom,
    ! [Pa: fun_pname_bool,Qa: fun_pname_bool] : hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,hAPP_f1402196763l_bool(cOMBB_675860798_pname(fconj),Pa)),Qa)) = hAPP_f759274231e_bool(semila1635148844e_bool(hAPP_f759274231e_bool(collect_pname,Pa)),hAPP_f759274231e_bool(collect_pname,Qa)) ).

tff(fact_570_Int__Collect,axiom,
    ! [X_2: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,Pa: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_2),hAPP_f909437487a_bool(semila1256126856a_bool(A),hAPP_f909437487a_bool(collec445968053iple_a,Pa))))
    <=> ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_2),A))
        & hBOOL(hAPP_H2006434664a_bool(Pa,X_2)) ) ) ).

tff(fact_571_Int__Collect,axiom,
    ! [X_2: pname,A: fun_pname_bool,Pa: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),hAPP_f759274231e_bool(semila1635148844e_bool(A),hAPP_f759274231e_bool(collect_pname,Pa))))
    <=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
        & hBOOL(hAPP_pname_bool(Pa,X_2)) ) ) ).

tff(fact_572_inf__Int__eq,axiom,
    ! [R_1: fun_Ho287446294a_bool,S_1: fun_Ho287446294a_bool,X: hoare_1544627872iple_a] :
      ( hBOOL(hAPP_H2006434664a_bool(hAPP_f909437487a_bool(semila1256126856a_bool(hAPP_f909437487a_bool(hAPP_f482270129a_bool(cOMBC_1807206788l_bool,member1891733107iple_a),R_1)),hAPP_f909437487a_bool(hAPP_f482270129a_bool(cOMBC_1807206788l_bool,member1891733107iple_a),S_1)),X))
    <=> hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X),hAPP_f909437487a_bool(semila1256126856a_bool(R_1),S_1))) ) ).

tff(fact_573_inf__Int__eq,axiom,
    ! [R_1: fun_pname_bool,S_1: fun_pname_bool,X: pname] :
      ( hBOOL(hAPP_pname_bool(hAPP_f759274231e_bool(semila1635148844e_bool(hAPP_f759274231e_bool(hAPP_f661147897e_bool(cOMBC_1058051404l_bool,member_pname),R_1)),hAPP_f759274231e_bool(hAPP_f661147897e_bool(cOMBC_1058051404l_bool,member_pname),S_1)),X))
    <=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),hAPP_f759274231e_bool(semila1635148844e_bool(R_1),S_1))) ) ).

tff(fact_574_Un__Int__crazy,axiom,
    ! [A: fun_pname_bool,B_1: fun_pname_bool,C_1: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f759274231e_bool(semila1635148844e_bool(A),B_1)),hAPP_f759274231e_bool(semila1635148844e_bool(B_1),C_1))),hAPP_f759274231e_bool(semila1635148844e_bool(C_1),A)) = hAPP_f759274231e_bool(semila1635148844e_bool(hAPP_f759274231e_bool(semila1635148844e_bool(hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),B_1)),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,B_1),C_1))),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,C_1),A)) ).

tff(fact_575_Un__Int__crazy,axiom,
    ! [A: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool,C_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,hAPP_f121055253e_bool(semila657954503e_bool(A),B_1)),hAPP_f121055253e_bool(semila657954503e_bool(B_1),C_1))),hAPP_f121055253e_bool(semila657954503e_bool(C_1),A)) = hAPP_f121055253e_bool(semila657954503e_bool(hAPP_f121055253e_bool(semila657954503e_bool(hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),B_1)),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,B_1),C_1))),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,C_1),A)) ).

tff(fact_576_Un__Int__crazy,axiom,
    ! [A: fun_Ho287446294a_bool,B_1: fun_Ho287446294a_bool,C_1: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,hAPP_f909437487a_bool(semila1256126856a_bool(A),B_1)),hAPP_f909437487a_bool(semila1256126856a_bool(B_1),C_1))),hAPP_f909437487a_bool(semila1256126856a_bool(C_1),A)) = hAPP_f909437487a_bool(semila1256126856a_bool(hAPP_f909437487a_bool(semila1256126856a_bool(hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B_1)),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,B_1),C_1))),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,C_1),A)) ).

tff(fact_577_Un__Int__distrib2,axiom,
    ! [B_1: fun_pname_bool,C_1: fun_pname_bool,A: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f759274231e_bool(semila1635148844e_bool(B_1),C_1)),A) = hAPP_f759274231e_bool(semila1635148844e_bool(hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,B_1),A)),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,C_1),A)) ).

tff(fact_578_Un__Int__distrib2,axiom,
    ! [B_1: fun_Ho1996104121e_bool,C_1: fun_Ho1996104121e_bool,A: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,hAPP_f121055253e_bool(semila657954503e_bool(B_1),C_1)),A) = hAPP_f121055253e_bool(semila657954503e_bool(hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,B_1),A)),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,C_1),A)) ).

tff(fact_579_Un__Int__distrib2,axiom,
    ! [B_1: fun_Ho287446294a_bool,C_1: fun_Ho287446294a_bool,A: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,hAPP_f909437487a_bool(semila1256126856a_bool(B_1),C_1)),A) = hAPP_f909437487a_bool(semila1256126856a_bool(hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,B_1),A)),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,C_1),A)) ).

tff(fact_580_Int__Un__distrib2,axiom,
    ! [B_1: fun_pname_bool,C_1: fun_pname_bool,A: fun_pname_bool] : hAPP_f759274231e_bool(semila1635148844e_bool(hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,B_1),C_1)),A) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f759274231e_bool(semila1635148844e_bool(B_1),A)),hAPP_f759274231e_bool(semila1635148844e_bool(C_1),A)) ).

tff(fact_581_Int__Un__distrib2,axiom,
    ! [B_1: fun_Ho1996104121e_bool,C_1: fun_Ho1996104121e_bool,A: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(semila657954503e_bool(hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,B_1),C_1)),A) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,hAPP_f121055253e_bool(semila657954503e_bool(B_1),A)),hAPP_f121055253e_bool(semila657954503e_bool(C_1),A)) ).

tff(fact_582_Int__Un__distrib2,axiom,
    ! [B_1: fun_Ho287446294a_bool,C_1: fun_Ho287446294a_bool,A: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(semila1256126856a_bool(hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,B_1),C_1)),A) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,hAPP_f909437487a_bool(semila1256126856a_bool(B_1),A)),hAPP_f909437487a_bool(semila1256126856a_bool(C_1),A)) ).

tff(fact_583_Un__Int__distrib,axiom,
    ! [A: fun_pname_bool,B_1: fun_pname_bool,C_1: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),hAPP_f759274231e_bool(semila1635148844e_bool(B_1),C_1)) = hAPP_f759274231e_bool(semila1635148844e_bool(hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),B_1)),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),C_1)) ).

tff(fact_584_Un__Int__distrib,axiom,
    ! [A: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool,C_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),hAPP_f121055253e_bool(semila657954503e_bool(B_1),C_1)) = hAPP_f121055253e_bool(semila657954503e_bool(hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),B_1)),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),C_1)) ).

tff(fact_585_Un__Int__distrib,axiom,
    ! [A: fun_Ho287446294a_bool,B_1: fun_Ho287446294a_bool,C_1: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),hAPP_f909437487a_bool(semila1256126856a_bool(B_1),C_1)) = hAPP_f909437487a_bool(semila1256126856a_bool(hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B_1)),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),C_1)) ).

tff(fact_586_Int__Un__distrib,axiom,
    ! [A: fun_pname_bool,B_1: fun_pname_bool,C_1: fun_pname_bool] : hAPP_f759274231e_bool(semila1635148844e_bool(A),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,B_1),C_1)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f759274231e_bool(semila1635148844e_bool(A),B_1)),hAPP_f759274231e_bool(semila1635148844e_bool(A),C_1)) ).

tff(fact_587_Int__Un__distrib,axiom,
    ! [A: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool,C_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(semila657954503e_bool(A),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,B_1),C_1)) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,hAPP_f121055253e_bool(semila657954503e_bool(A),B_1)),hAPP_f121055253e_bool(semila657954503e_bool(A),C_1)) ).

tff(fact_588_Int__Un__distrib,axiom,
    ! [A: fun_Ho287446294a_bool,B_1: fun_Ho287446294a_bool,C_1: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(semila1256126856a_bool(A),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,B_1),C_1)) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,hAPP_f909437487a_bool(semila1256126856a_bool(A),B_1)),hAPP_f909437487a_bool(semila1256126856a_bool(A),C_1)) ).

tff(fact_589_Int__insert__left__if1,axiom,
    ! [B_1: fun_Ho1996104121e_bool,A_1: hoare_1875481847_state,C_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),C_1))
     => ( hAPP_f121055253e_bool(semila657954503e_bool(hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),B_1)),C_1) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),hAPP_f121055253e_bool(semila657954503e_bool(B_1),C_1)) ) ) ).

tff(fact_590_Int__insert__left__if1,axiom,
    ! [B_1: fun_Ho287446294a_bool,A_1: hoare_1544627872iple_a,C_1: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),C_1))
     => ( hAPP_f909437487a_bool(semila1256126856a_bool(hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),B_1)),C_1) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),hAPP_f909437487a_bool(semila1256126856a_bool(B_1),C_1)) ) ) ).

tff(fact_591_Int__insert__left__if1,axiom,
    ! [B_1: fun_pname_bool,A_1: pname,C_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),C_1))
     => ( hAPP_f759274231e_bool(semila1635148844e_bool(hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),B_1)),C_1) = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),hAPP_f759274231e_bool(semila1635148844e_bool(B_1),C_1)) ) ) ).

tff(fact_592_Int__insert__right__if1,axiom,
    ! [B_1: fun_Ho1996104121e_bool,A_1: hoare_1875481847_state,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),A))
     => ( hAPP_f121055253e_bool(semila657954503e_bool(A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),B_1)) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),hAPP_f121055253e_bool(semila657954503e_bool(A),B_1)) ) ) ).

tff(fact_593_Int__insert__right__if1,axiom,
    ! [B_1: fun_Ho287446294a_bool,A_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),A))
     => ( hAPP_f909437487a_bool(semila1256126856a_bool(A),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),B_1)) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),hAPP_f909437487a_bool(semila1256126856a_bool(A),B_1)) ) ) ).

tff(fact_594_Int__insert__right__if1,axiom,
    ! [B_1: fun_pname_bool,A_1: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),A))
     => ( hAPP_f759274231e_bool(semila1635148844e_bool(A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),B_1)) = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),hAPP_f759274231e_bool(semila1635148844e_bool(A),B_1)) ) ) ).

tff(fact_595_Int__insert__left__if0,axiom,
    ! [B_1: fun_Ho1996104121e_bool,A_1: hoare_1875481847_state,C_1: fun_Ho1996104121e_bool] :
      ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),C_1))
     => ( hAPP_f121055253e_bool(semila657954503e_bool(hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),B_1)),C_1) = hAPP_f121055253e_bool(semila657954503e_bool(B_1),C_1) ) ) ).

tff(fact_596_Int__insert__left__if0,axiom,
    ! [B_1: fun_Ho287446294a_bool,A_1: hoare_1544627872iple_a,C_1: fun_Ho287446294a_bool] :
      ( ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),C_1))
     => ( hAPP_f909437487a_bool(semila1256126856a_bool(hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),B_1)),C_1) = hAPP_f909437487a_bool(semila1256126856a_bool(B_1),C_1) ) ) ).

tff(fact_597_Int__insert__left__if0,axiom,
    ! [B_1: fun_pname_bool,A_1: pname,C_1: fun_pname_bool] :
      ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),C_1))
     => ( hAPP_f759274231e_bool(semila1635148844e_bool(hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),B_1)),C_1) = hAPP_f759274231e_bool(semila1635148844e_bool(B_1),C_1) ) ) ).

tff(fact_598_Int__insert__right__if0,axiom,
    ! [B_1: fun_Ho1996104121e_bool,A_1: hoare_1875481847_state,A: fun_Ho1996104121e_bool] :
      ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),A))
     => ( hAPP_f121055253e_bool(semila657954503e_bool(A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),B_1)) = hAPP_f121055253e_bool(semila657954503e_bool(A),B_1) ) ) ).

tff(fact_599_Int__insert__right__if0,axiom,
    ! [B_1: fun_Ho287446294a_bool,A_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),A))
     => ( hAPP_f909437487a_bool(semila1256126856a_bool(A),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),B_1)) = hAPP_f909437487a_bool(semila1256126856a_bool(A),B_1) ) ) ).

tff(fact_600_Int__insert__right__if0,axiom,
    ! [B_1: fun_pname_bool,A_1: pname,A: fun_pname_bool] :
      ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),A))
     => ( hAPP_f759274231e_bool(semila1635148844e_bool(A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),B_1)) = hAPP_f759274231e_bool(semila1635148844e_bool(A),B_1) ) ) ).

tff(fact_601_insert__inter__insert,axiom,
    ! [A_1: hoare_1875481847_state,A: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(semila657954503e_bool(hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),A)),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),B_1)) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),hAPP_f121055253e_bool(semila657954503e_bool(A),B_1)) ).

tff(fact_602_insert__inter__insert,axiom,
    ! [A_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,B_1: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(semila1256126856a_bool(hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),A)),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),B_1)) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),hAPP_f909437487a_bool(semila1256126856a_bool(A),B_1)) ).

tff(fact_603_insert__inter__insert,axiom,
    ! [A_1: pname,A: fun_pname_bool,B_1: fun_pname_bool] : hAPP_f759274231e_bool(semila1635148844e_bool(hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),A)),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),B_1)) = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),hAPP_f759274231e_bool(semila1635148844e_bool(A),B_1)) ).

tff(fact_604_Int__insert__left,axiom,
    ! [B_1: fun_Ho1996104121e_bool,A_1: hoare_1875481847_state,C_1: fun_Ho1996104121e_bool] :
      ( ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),C_1))
       => ( hAPP_f121055253e_bool(semila657954503e_bool(hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),B_1)),C_1) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),hAPP_f121055253e_bool(semila657954503e_bool(B_1),C_1)) ) )
      & ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),C_1))
       => ( hAPP_f121055253e_bool(semila657954503e_bool(hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),B_1)),C_1) = hAPP_f121055253e_bool(semila657954503e_bool(B_1),C_1) ) ) ) ).

tff(fact_605_Int__insert__left,axiom,
    ! [B_1: fun_Ho287446294a_bool,A_1: hoare_1544627872iple_a,C_1: fun_Ho287446294a_bool] :
      ( ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),C_1))
       => ( hAPP_f909437487a_bool(semila1256126856a_bool(hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),B_1)),C_1) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),hAPP_f909437487a_bool(semila1256126856a_bool(B_1),C_1)) ) )
      & ( ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),C_1))
       => ( hAPP_f909437487a_bool(semila1256126856a_bool(hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),B_1)),C_1) = hAPP_f909437487a_bool(semila1256126856a_bool(B_1),C_1) ) ) ) ).

tff(fact_606_Int__insert__left,axiom,
    ! [B_1: fun_pname_bool,A_1: pname,C_1: fun_pname_bool] :
      ( ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),C_1))
       => ( hAPP_f759274231e_bool(semila1635148844e_bool(hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),B_1)),C_1) = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),hAPP_f759274231e_bool(semila1635148844e_bool(B_1),C_1)) ) )
      & ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),C_1))
       => ( hAPP_f759274231e_bool(semila1635148844e_bool(hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),B_1)),C_1) = hAPP_f759274231e_bool(semila1635148844e_bool(B_1),C_1) ) ) ) ).

tff(fact_607_Int__insert__right,axiom,
    ! [B_1: fun_Ho1996104121e_bool,A_1: hoare_1875481847_state,A: fun_Ho1996104121e_bool] :
      ( ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),A))
       => ( hAPP_f121055253e_bool(semila657954503e_bool(A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),B_1)) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),hAPP_f121055253e_bool(semila657954503e_bool(A),B_1)) ) )
      & ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),A))
       => ( hAPP_f121055253e_bool(semila657954503e_bool(A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),B_1)) = hAPP_f121055253e_bool(semila657954503e_bool(A),B_1) ) ) ) ).

tff(fact_608_Int__insert__right,axiom,
    ! [B_1: fun_Ho287446294a_bool,A_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),A))
       => ( hAPP_f909437487a_bool(semila1256126856a_bool(A),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),B_1)) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),hAPP_f909437487a_bool(semila1256126856a_bool(A),B_1)) ) )
      & ( ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),A))
       => ( hAPP_f909437487a_bool(semila1256126856a_bool(A),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),B_1)) = hAPP_f909437487a_bool(semila1256126856a_bool(A),B_1) ) ) ) ).

tff(fact_609_Int__insert__right,axiom,
    ! [B_1: fun_pname_bool,A_1: pname,A: fun_pname_bool] :
      ( ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),A))
       => ( hAPP_f759274231e_bool(semila1635148844e_bool(A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),B_1)) = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),hAPP_f759274231e_bool(semila1635148844e_bool(A),B_1)) ) )
      & ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),A))
       => ( hAPP_f759274231e_bool(semila1635148844e_bool(A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_1),B_1)) = hAPP_f759274231e_bool(semila1635148844e_bool(A),B_1) ) ) ) ).

tff(fact_610_if__image__distrib,axiom,
    ! [Pa: fun_pname_bool,F_1: fun_pn664418900_state,G: fun_pn664418900_state,S_1: fun_pname_bool] : hAPP_f1291720380e_bool(image_1283223414_state(hAPP_f1981552043_state(hAPP_f1875279510_state(cOMBS_629630867_state,hAPP_f1087244981_state(hAPP_f294490838_state(cOMBS_602905655_state,hAPP_f980169511_state(cOMBB_1989814576_pname(if_Hoa1541320583_state),Pa)),F_1)),G)),S_1) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,hAPP_f1291720380e_bool(image_1283223414_state(F_1),hAPP_f759274231e_bool(semila1635148844e_bool(S_1),hAPP_f759274231e_bool(collect_pname,Pa)))),hAPP_f1291720380e_bool(image_1283223414_state(G),hAPP_f759274231e_bool(semila1635148844e_bool(S_1),hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(cOMBB_647938656_pname(fNot),Pa))))) ).

tff(fact_611_if__image__distrib,axiom,
    ! [Pa: fun_pname_bool,F_1: fun_pn1711459225iple_a,G: fun_pn1711459225iple_a,S_1: fun_pname_bool] : hAPP_f856094675a_bool(image_1832685281iple_a(hAPP_f687424533iple_a(hAPP_f16446748iple_a(cOMBS_459699575iple_a,hAPP_f638932530iple_a(hAPP_f176122972iple_a(cOMBS_1079312276iple_a,hAPP_f645774806iple_a(cOMBB_225107267_pname(if_Hoa816906256iple_a),Pa)),F_1)),G)),S_1) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,hAPP_f856094675a_bool(image_1832685281iple_a(F_1),hAPP_f759274231e_bool(semila1635148844e_bool(S_1),hAPP_f759274231e_bool(collect_pname,Pa)))),hAPP_f856094675a_bool(image_1832685281iple_a(G),hAPP_f759274231e_bool(semila1635148844e_bool(S_1),hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(cOMBB_647938656_pname(fNot),Pa))))) ).

tff(fact_612_folding__one_Ounion__inter,axiom,
    ! [B_1: fun_pname_bool,A: fun_pname_bool,F_1: fun_pn1208287926_pname,F: fun_fu1499449723_pname] :
      ( hBOOL(finite1282449217_pname(F_1,F))
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
       => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B_1))
         => ( ( hAPP_f759274231e_bool(semila1635148844e_bool(A),B_1) != bot_bo844097828e_bool )
           => ( hAPP_pname_pname(hAPP_p1170154830_pname(F_1,hAPP_f1297739591_pname(F,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),B_1))),hAPP_f1297739591_pname(F,hAPP_f759274231e_bool(semila1635148844e_bool(A),B_1))) = hAPP_pname_pname(hAPP_p1170154830_pname(F_1,hAPP_f1297739591_pname(F,A)),hAPP_f1297739591_pname(F,B_1)) ) ) ) ) ) ).

tff(fact_613_folding__one_Ounion__inter,axiom,
    ! [B_1: fun_Ho287446294a_bool,A: fun_Ho287446294a_bool,F_1: fun_Ho115022426iple_a,F: fun_fu1634330235iple_a] :
      ( hBOOL(finite1847914781iple_a(F_1,F))
     => ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,A))
       => ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,B_1))
         => ( ( hAPP_f909437487a_bool(semila1256126856a_bool(A),B_1) != bot_bo1766443648a_bool )
           => ( hAPP_H9461717iple_a(hAPP_H1988213490iple_a(F_1,hAPP_f1853245639iple_a(F,hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B_1))),hAPP_f1853245639iple_a(F,hAPP_f909437487a_bool(semila1256126856a_bool(A),B_1))) = hAPP_H9461717iple_a(hAPP_H1988213490iple_a(F_1,hAPP_f1853245639iple_a(F,A)),hAPP_f1853245639iple_a(F,B_1)) ) ) ) ) ) ).

tff(fact_614_folding__one_Ounion__inter,axiom,
    ! [B_1: fun_Ho1996104121e_bool,A: fun_Ho1996104121e_bool,F_1: fun_Ho784854669_state,F: fun_fu385705929_state] :
      ( hBOOL(finite1782051386_state(F_1,F))
     => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A))
       => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,B_1))
         => ( ( hAPP_f121055253e_bool(semila657954503e_bool(A),B_1) != bot_bo1715400655e_bool )
           => ( hAPP_H350317809_state(hAPP_H743221337_state(F_1,hAPP_f890249627_state(F,hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),B_1))),hAPP_f890249627_state(F,hAPP_f121055253e_bool(semila657954503e_bool(A),B_1))) = hAPP_H350317809_state(hAPP_H743221337_state(F_1,hAPP_f890249627_state(F,A)),hAPP_f890249627_state(F,B_1)) ) ) ) ) ) ).

tff(fact_615_folding__one_Ounion__disjoint,axiom,
    ! [B_1: fun_pname_bool,A: fun_pname_bool,F_1: fun_pn1208287926_pname,F: fun_fu1499449723_pname] :
      ( hBOOL(finite1282449217_pname(F_1,F))
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
       => ( ( A != bot_bo844097828e_bool )
         => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B_1))
           => ( ( B_1 != bot_bo844097828e_bool )
             => ( ( hAPP_f759274231e_bool(semila1635148844e_bool(A),B_1) = bot_bo844097828e_bool )
               => ( hAPP_f1297739591_pname(F,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),B_1)) = hAPP_pname_pname(hAPP_p1170154830_pname(F_1,hAPP_f1297739591_pname(F,A)),hAPP_f1297739591_pname(F,B_1)) ) ) ) ) ) ) ) ).

tff(fact_616_folding__one_Ounion__disjoint,axiom,
    ! [B_1: fun_Ho287446294a_bool,A: fun_Ho287446294a_bool,F_1: fun_Ho115022426iple_a,F: fun_fu1634330235iple_a] :
      ( hBOOL(finite1847914781iple_a(F_1,F))
     => ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,A))
       => ( ( A != bot_bo1766443648a_bool )
         => ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,B_1))
           => ( ( B_1 != bot_bo1766443648a_bool )
             => ( ( hAPP_f909437487a_bool(semila1256126856a_bool(A),B_1) = bot_bo1766443648a_bool )
               => ( hAPP_f1853245639iple_a(F,hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B_1)) = hAPP_H9461717iple_a(hAPP_H1988213490iple_a(F_1,hAPP_f1853245639iple_a(F,A)),hAPP_f1853245639iple_a(F,B_1)) ) ) ) ) ) ) ) ).

tff(fact_617_folding__one_Ounion__disjoint,axiom,
    ! [B_1: fun_Ho1996104121e_bool,A: fun_Ho1996104121e_bool,F_1: fun_Ho784854669_state,F: fun_fu385705929_state] :
      ( hBOOL(finite1782051386_state(F_1,F))
     => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A))
       => ( ( A != bot_bo1715400655e_bool )
         => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,B_1))
           => ( ( B_1 != bot_bo1715400655e_bool )
             => ( ( hAPP_f121055253e_bool(semila657954503e_bool(A),B_1) = bot_bo1715400655e_bool )
               => ( hAPP_f890249627_state(F,hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),B_1)) = hAPP_H350317809_state(hAPP_H743221337_state(F_1,hAPP_f890249627_state(F,A)),hAPP_f890249627_state(F,B_1)) ) ) ) ) ) ) ) ).

tff(fact_618_distrib__imp2,axiom,
    ! [X_2: bool,Y_1: bool,Z_1: bool] :
      ( ! [X: bool,Y_2: bool,Z_2: bool] :
          ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X),hAPP_bool_bool(semila310582991f_bool(Y_2),Z_2)))
        <=> hBOOL(hAPP_bool_bool(semila310582991f_bool(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X),Y_2)),hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X),Z_2))) )
     => ( hBOOL(hAPP_bool_bool(semila310582991f_bool(X_2),hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,Y_1),Z_1)))
      <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,hAPP_bool_bool(semila310582991f_bool(X_2),Y_1)),hAPP_bool_bool(semila310582991f_bool(X_2),Z_1))) ) ) ).

tff(fact_619_distrib__imp2,axiom,
    ! [X_2: fun_pname_bool,Y_1: fun_pname_bool,Z_1: fun_pname_bool] :
      ( ! [X: fun_pname_bool,Y_2: fun_pname_bool,Z_2: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X),hAPP_f759274231e_bool(semila1635148844e_bool(Y_2),Z_2)) = hAPP_f759274231e_bool(semila1635148844e_bool(hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X),Y_2)),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X),Z_2))
     => ( hAPP_f759274231e_bool(semila1635148844e_bool(X_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,Y_1),Z_1)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f759274231e_bool(semila1635148844e_bool(X_2),Y_1)),hAPP_f759274231e_bool(semila1635148844e_bool(X_2),Z_1)) ) ) ).

tff(fact_620_distrib__imp2,axiom,
    ! [X_2: fun_Ho1996104121e_bool,Y_1: fun_Ho1996104121e_bool,Z_1: fun_Ho1996104121e_bool] :
      ( ! [X: fun_Ho1996104121e_bool,Y_2: fun_Ho1996104121e_bool,Z_2: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X),hAPP_f121055253e_bool(semila657954503e_bool(Y_2),Z_2)) = hAPP_f121055253e_bool(semila657954503e_bool(hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X),Y_2)),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X),Z_2))
     => ( hAPP_f121055253e_bool(semila657954503e_bool(X_2),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,Y_1),Z_1)) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,hAPP_f121055253e_bool(semila657954503e_bool(X_2),Y_1)),hAPP_f121055253e_bool(semila657954503e_bool(X_2),Z_1)) ) ) ).

tff(fact_621_distrib__imp2,axiom,
    ! [X_2: fun_Ho287446294a_bool,Y_1: fun_Ho287446294a_bool,Z_1: fun_Ho287446294a_bool] :
      ( ! [X: fun_Ho287446294a_bool,Y_2: fun_Ho287446294a_bool,Z_2: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X),hAPP_f909437487a_bool(semila1256126856a_bool(Y_2),Z_2)) = hAPP_f909437487a_bool(semila1256126856a_bool(hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X),Y_2)),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X),Z_2))
     => ( hAPP_f909437487a_bool(semila1256126856a_bool(X_2),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,Y_1),Z_1)) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,hAPP_f909437487a_bool(semila1256126856a_bool(X_2),Y_1)),hAPP_f909437487a_bool(semila1256126856a_bool(X_2),Z_1)) ) ) ).

tff(fact_622_distrib__imp1,axiom,
    ! [X_2: bool,Y_1: bool,Z_1: bool] :
      ( ! [X: bool,Y_2: bool,Z_2: bool] :
          ( hBOOL(hAPP_bool_bool(semila310582991f_bool(X),hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,Y_2),Z_2)))
        <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,hAPP_bool_bool(semila310582991f_bool(X),Y_2)),hAPP_bool_bool(semila310582991f_bool(X),Z_2))) )
     => ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),hAPP_bool_bool(semila310582991f_bool(Y_1),Z_1)))
      <=> hBOOL(hAPP_bool_bool(semila310582991f_bool(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),Y_1)),hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),Z_1))) ) ) ).

tff(fact_623_distrib__imp1,axiom,
    ! [X_2: fun_pname_bool,Y_1: fun_pname_bool,Z_1: fun_pname_bool] :
      ( ! [X: fun_pname_bool,Y_2: fun_pname_bool,Z_2: fun_pname_bool] : hAPP_f759274231e_bool(semila1635148844e_bool(X),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,Y_2),Z_2)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f759274231e_bool(semila1635148844e_bool(X),Y_2)),hAPP_f759274231e_bool(semila1635148844e_bool(X),Z_2))
     => ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),hAPP_f759274231e_bool(semila1635148844e_bool(Y_1),Z_1)) = hAPP_f759274231e_bool(semila1635148844e_bool(hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),Y_1)),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),Z_1)) ) ) ).

tff(fact_624_distrib__imp1,axiom,
    ! [X_2: fun_Ho1996104121e_bool,Y_1: fun_Ho1996104121e_bool,Z_1: fun_Ho1996104121e_bool] :
      ( ! [X: fun_Ho1996104121e_bool,Y_2: fun_Ho1996104121e_bool,Z_2: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(semila657954503e_bool(X),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,Y_2),Z_2)) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,hAPP_f121055253e_bool(semila657954503e_bool(X),Y_2)),hAPP_f121055253e_bool(semila657954503e_bool(X),Z_2))
     => ( hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X_2),hAPP_f121055253e_bool(semila657954503e_bool(Y_1),Z_1)) = hAPP_f121055253e_bool(semila657954503e_bool(hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X_2),Y_1)),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X_2),Z_1)) ) ) ).

tff(fact_625_distrib__imp1,axiom,
    ! [X_2: fun_Ho287446294a_bool,Y_1: fun_Ho287446294a_bool,Z_1: fun_Ho287446294a_bool] :
      ( ! [X: fun_Ho287446294a_bool,Y_2: fun_Ho287446294a_bool,Z_2: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(semila1256126856a_bool(X),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,Y_2),Z_2)) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,hAPP_f909437487a_bool(semila1256126856a_bool(X),Y_2)),hAPP_f909437487a_bool(semila1256126856a_bool(X),Z_2))
     => ( hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X_2),hAPP_f909437487a_bool(semila1256126856a_bool(Y_1),Z_1)) = hAPP_f909437487a_bool(semila1256126856a_bool(hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X_2),Y_1)),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X_2),Z_1)) ) ) ).

tff(fact_626_sup__Inf__absorb,axiom,
    ! [A_1: bool,A: fun_bool_bool] :
      ( hBOOL(hAPP_f387058535l_bool(finite_finite_bool,A))
     => ( hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,A_1),A))
       => ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,A_1),big_la472677547n_bool(A)))
        <=> hBOOL(A_1) ) ) ) ).

tff(fact_627_sup__Inf__absorb,axiom,
    ! [A_1: fun_pname_bool,A: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
     => ( hBOOL(hAPP_f1935102916l_bool(hAPP_f556039215l_bool(member799430823e_bool,A_1),A))
       => ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),big_la28065288e_bool(A)) = A_1 ) ) ) ).

tff(fact_628_sup__Inf__absorb,axiom,
    ! [A_1: fun_Ho1996104121e_bool,A: fun_fu689207471l_bool] :
      ( hBOOL(hAPP_f1699935247l_bool(finite1559130124e_bool,A))
     => ( hBOOL(hAPP_f1699935247l_bool(hAPP_f640685261l_bool(member1062536460e_bool,A_1),A))
       => ( hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A_1),big_la1168147435e_bool(A)) = A_1 ) ) ) ).

tff(fact_629_sup__Inf__absorb,axiom,
    ! [A_1: fun_Ho287446294a_bool,A: fun_fu1441721944l_bool] :
      ( hBOOL(hAPP_f1059025184l_bool(finite1237779075a_bool,A))
     => ( hBOOL(hAPP_f1059025184l_bool(hAPP_f372929895l_bool(member1738976643a_bool,A_1),A))
       => ( hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A_1),big_la1749641572a_bool(A)) = A_1 ) ) ) ).

tff(fact_630_DiffE,axiom,
    ! [C: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,B_1: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),hAPP_f909437487a_bool(minus_1054521797a_bool(A),B_1)))
     => ~ ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),A))
         => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),B_1)) ) ) ).

tff(fact_631_DiffE,axiom,
    ! [C: pname,A: fun_pname_bool,B_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),hAPP_f759274231e_bool(minus_1015773161e_bool(A),B_1)))
     => ~ ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),A))
         => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),B_1)) ) ) ).

tff(fact_632_DiffI,axiom,
    ! [B_1: fun_Ho287446294a_bool,C: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),A))
     => ( ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),B_1))
       => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),hAPP_f909437487a_bool(minus_1054521797a_bool(A),B_1))) ) ) ).

tff(fact_633_DiffI,axiom,
    ! [B_1: fun_pname_bool,C: pname,A: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),A))
     => ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),B_1))
       => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),hAPP_f759274231e_bool(minus_1015773161e_bool(A),B_1))) ) ) ).

tff(fact_634_Un__Diff__cancel,axiom,
    ! [A: fun_pname_bool,B_1: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),hAPP_f759274231e_bool(minus_1015773161e_bool(B_1),A)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),B_1) ).

tff(fact_635_Un__Diff__cancel,axiom,
    ! [A: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),hAPP_f121055253e_bool(minus_589155914e_bool(B_1),A)) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),B_1) ).

tff(fact_636_Un__Diff__cancel,axiom,
    ! [A: fun_Ho287446294a_bool,B_1: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),hAPP_f909437487a_bool(minus_1054521797a_bool(B_1),A)) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B_1) ).

tff(fact_637_Un__Diff__cancel2,axiom,
    ! [B_1: fun_pname_bool,A: fun_pname_bool] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f759274231e_bool(minus_1015773161e_bool(B_1),A)),A) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,B_1),A) ).

tff(fact_638_Un__Diff__cancel2,axiom,
    ! [B_1: fun_Ho1996104121e_bool,A: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,hAPP_f121055253e_bool(minus_589155914e_bool(B_1),A)),A) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,B_1),A) ).

tff(fact_639_Un__Diff__cancel2,axiom,
    ! [B_1: fun_Ho287446294a_bool,A: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,hAPP_f909437487a_bool(minus_1054521797a_bool(B_1),A)),A) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,B_1),A) ).

tff(fact_640_Un__Diff,axiom,
    ! [A: fun_pname_bool,B_1: fun_pname_bool,C_1: fun_pname_bool] : hAPP_f759274231e_bool(minus_1015773161e_bool(hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A),B_1)),C_1) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f759274231e_bool(minus_1015773161e_bool(A),C_1)),hAPP_f759274231e_bool(minus_1015773161e_bool(B_1),C_1)) ).

tff(fact_641_Un__Diff,axiom,
    ! [A: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool,C_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(minus_589155914e_bool(hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),B_1)),C_1) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,hAPP_f121055253e_bool(minus_589155914e_bool(A),C_1)),hAPP_f121055253e_bool(minus_589155914e_bool(B_1),C_1)) ).

tff(fact_642_Un__Diff,axiom,
    ! [A: fun_Ho287446294a_bool,B_1: fun_Ho287446294a_bool,C_1: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(minus_1054521797a_bool(hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B_1)),C_1) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,hAPP_f909437487a_bool(minus_1054521797a_bool(A),C_1)),hAPP_f909437487a_bool(minus_1054521797a_bool(B_1),C_1)) ).

tff(fact_643_DiffD2,axiom,
    ! [C: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,B_1: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),hAPP_f909437487a_bool(minus_1054521797a_bool(A),B_1)))
     => ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),B_1)) ) ).

tff(fact_644_DiffD2,axiom,
    ! [C: pname,A: fun_pname_bool,B_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),hAPP_f759274231e_bool(minus_1015773161e_bool(A),B_1)))
     => ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),B_1)) ) ).

tff(fact_645_DiffD1,axiom,
    ! [C: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,B_1: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),hAPP_f909437487a_bool(minus_1054521797a_bool(A),B_1)))
     => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,C),A)) ) ).

tff(fact_646_DiffD1,axiom,
    ! [C: pname,A: fun_pname_bool,B_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),hAPP_f759274231e_bool(minus_1015773161e_bool(A),B_1)))
     => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),A)) ) ).

tff(fact_647_Diff__iff,axiom,
    ! [C: pname,A: fun_pname_bool,B_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),hAPP_f759274231e_bool(minus_1015773161e_bool(A),B_1)))
    <=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),A))
        & ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),B_1)) ) ) ).

tff(fact_648_diff__Suc__Suc,axiom,
    ! [M: nat,N_2: nat] : hAPP_nat_nat(minus_minus_nat(suc(M)),suc(N_2)) = hAPP_nat_nat(minus_minus_nat(M),N_2) ).

tff(fact_649_Suc__diff__diff,axiom,
    ! [M: nat,N_2: nat,K_1: nat] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(suc(M)),N_2)),suc(K_1)) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(M),N_2)),K_1) ).

tff(fact_650_diff__0__eq__0,axiom,
    ! [N_2: nat] : hAPP_nat_nat(minus_minus_nat(zero_zero_nat),N_2) = zero_zero_nat ).

tff(fact_651_minus__nat_Odiff__0,axiom,
    ! [M: nat] : hAPP_nat_nat(minus_minus_nat(M),zero_zero_nat) = M ).

tff(fact_652_diff__self__eq__0,axiom,
    ! [M: nat] : hAPP_nat_nat(minus_minus_nat(M),M) = zero_zero_nat ).

tff(fact_653_diffs0__imp__equal,axiom,
    ! [M: nat,N_2: nat] :
      ( ( hAPP_nat_nat(minus_minus_nat(M),N_2) = zero_zero_nat )
     => ( ( hAPP_nat_nat(minus_minus_nat(N_2),M) = zero_zero_nat )
       => ( M = N_2 ) ) ) ).

tff(fact_654_zero__induct__lemma,axiom,
    ! [I_1: nat,Pa: fun_nat_bool,K: nat] :
      ( hBOOL(hAPP_nat_bool(Pa,K))
     => ( ! [N: nat] :
            ( hBOOL(hAPP_nat_bool(Pa,suc(N)))
           => hBOOL(hAPP_nat_bool(Pa,N)) )
       => hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(minus_minus_nat(K),I_1))) ) ) ).

tff(fact_655_diff__commute,axiom,
    ! [I: nat,J: nat,K_1: nat] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(I),J)),K_1) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(I),K_1)),J) ).

tff(fact_656_diff__Suc,axiom,
    ! [M_1: nat,N_1: nat] : hAPP_nat_nat(minus_minus_nat(M_1),suc(N_1)) = nat_case_nat(zero_zero_nat,cOMBI_nat,hAPP_nat_nat(minus_minus_nat(M_1),N_1)) ).

tff(fact_657_diff__Suc__1,axiom,
    ! [N_2: nat] : hAPP_nat_nat(minus_minus_nat(suc(N_2)),one_one_nat) = N_2 ).

tff(fact_658_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N_2: nat] : hAPP_nat_nat(minus_minus_nat(M),suc(N_2)) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(M),one_one_nat)),N_2) ).

tff(fact_659_One__nat__def,axiom,
    one_one_nat = suc(zero_zero_nat) ).

tff(fact_660_Suc__eq__plus1,axiom,
    ! [N_2: nat] : suc(N_2) = plus_plus_nat(N_2,one_one_nat) ).

tff(fact_661_Suc__eq__plus1__left,axiom,
    ! [N_2: nat] : suc(N_2) = plus_plus_nat(one_one_nat,N_2) ).

tff(fact_662_diff__cancel2,axiom,
    ! [M: nat,K_1: nat,N_2: nat] : hAPP_nat_nat(minus_minus_nat(plus_plus_nat(M,K_1)),plus_plus_nat(N_2,K_1)) = hAPP_nat_nat(minus_minus_nat(M),N_2) ).

tff(fact_663_diff__cancel,axiom,
    ! [K_1: nat,M: nat,N_2: nat] : hAPP_nat_nat(minus_minus_nat(plus_plus_nat(K_1,M)),plus_plus_nat(K_1,N_2)) = hAPP_nat_nat(minus_minus_nat(M),N_2) ).

tff(fact_664_diff__diff__left,axiom,
    ! [I: nat,J: nat,K_1: nat] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(I),J)),K_1) = hAPP_nat_nat(minus_minus_nat(I),plus_plus_nat(J,K_1)) ).

tff(fact_665_diff__add__inverse,axiom,
    ! [N_2: nat,M: nat] : hAPP_nat_nat(minus_minus_nat(plus_plus_nat(N_2,M)),N_2) = M ).

tff(fact_666_diff__add__inverse2,axiom,
    ! [M: nat,N_2: nat] : hAPP_nat_nat(minus_minus_nat(plus_plus_nat(M,N_2)),N_2) = M ).

tff(fact_667_diff__add__0,axiom,
    ! [N_2: nat,M: nat] : hAPP_nat_nat(minus_minus_nat(N_2),plus_plus_nat(N_2,M)) = zero_zero_nat ).

tff(fact_668_nat__add__commute,axiom,
    ! [M: nat,N_2: nat] : plus_plus_nat(M,N_2) = plus_plus_nat(N_2,M) ).

tff(fact_669_nat__add__left__commute,axiom,
    ! [X_1: nat,Y: nat,Z: nat] : plus_plus_nat(X_1,plus_plus_nat(Y,Z)) = plus_plus_nat(Y,plus_plus_nat(X_1,Z)) ).

tff(fact_670_nat__add__assoc,axiom,
    ! [M: nat,N_2: nat,K_1: nat] : plus_plus_nat(plus_plus_nat(M,N_2),K_1) = plus_plus_nat(M,plus_plus_nat(N_2,K_1)) ).

tff(fact_671_nat__add__left__cancel,axiom,
    ! [K: nat,M_1: nat,N_1: nat] :
      ( ( plus_plus_nat(K,M_1) = plus_plus_nat(K,N_1) )
    <=> ( M_1 = N_1 ) ) ).

tff(fact_672_nat__add__right__cancel,axiom,
    ! [M_1: nat,K: nat,N_1: nat] :
      ( ( plus_plus_nat(M_1,K) = plus_plus_nat(N_1,K) )
    <=> ( M_1 = N_1 ) ) ).

tff(fact_673_add__Suc__shift,axiom,
    ! [M: nat,N_2: nat] : plus_plus_nat(suc(M),N_2) = plus_plus_nat(M,suc(N_2)) ).

tff(fact_674_add__Suc,axiom,
    ! [M: nat,N_2: nat] : plus_plus_nat(suc(M),N_2) = suc(plus_plus_nat(M,N_2)) ).

tff(fact_675_add__Suc__right,axiom,
    ! [M: nat,N_2: nat] : plus_plus_nat(M,suc(N_2)) = suc(plus_plus_nat(M,N_2)) ).

tff(fact_676_one__is__add,axiom,
    ! [M_1: nat,N_1: nat] :
      ( ( suc(zero_zero_nat) = plus_plus_nat(M_1,N_1) )
    <=> ( ( ( M_1 = suc(zero_zero_nat) )
          & ( N_1 = zero_zero_nat ) )
        | ( ( M_1 = zero_zero_nat )
          & ( N_1 = suc(zero_zero_nat) ) ) ) ) ).

tff(fact_677_add__is__1,axiom,
    ! [M_1: nat,N_1: nat] :
      ( ( plus_plus_nat(M_1,N_1) = suc(zero_zero_nat) )
    <=> ( ( ( M_1 = suc(zero_zero_nat) )
          & ( N_1 = zero_zero_nat ) )
        | ( ( M_1 = zero_zero_nat )
          & ( N_1 = suc(zero_zero_nat) ) ) ) ) ).

tff(fact_678_plus__nat_Oadd__0,axiom,
    ! [N_2: nat] : plus_plus_nat(zero_zero_nat,N_2) = N_2 ).

tff(fact_679_Nat_Oadd__0__right,axiom,
    ! [M: nat] : plus_plus_nat(M,zero_zero_nat) = M ).

tff(fact_680_add__is__0,axiom,
    ! [M_1: nat,N_1: nat] :
      ( ( plus_plus_nat(M_1,N_1) = zero_zero_nat )
    <=> ( ( M_1 = zero_zero_nat )
        & ( N_1 = zero_zero_nat ) ) ) ).

tff(fact_681_add__eq__self__zero,axiom,
    ! [M: nat,N_2: nat] :
      ( ( plus_plus_nat(M,N_2) = M )
     => ( N_2 = zero_zero_nat ) ) ).

tff(fact_682_add__eq__if,axiom,
    ! [N_2: nat,M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( plus_plus_nat(M,N_2) = N_2 ) )
      & ( ( M != zero_zero_nat )
       => ( plus_plus_nat(M,N_2) = suc(plus_plus_nat(hAPP_nat_nat(minus_minus_nat(M),one_one_nat),N_2)) ) ) ) ).

tff(fact_683_com_Osize_I4_J,axiom,
    ! [Com1_1: com,Com2_1: com] : com_size(semi(Com1_1,Com2_1)) = plus_plus_nat(plus_plus_nat(com_size(Com1_1),com_size(Com2_1)),suc(zero_zero_nat)) ).

tff(fact_684_com_Osize_I7_J,axiom,
    ! [Pname: pname] : com_size(hAPP_pname_com(body,Pname)) = zero_zero_nat ).

tff(fact_685_com_Osize_I1_J,axiom,
    com_size(skip) = zero_zero_nat ).

tff(fact_686_com_Osize_I6_J,axiom,
    ! [Fun: fun_state_bool,Com: com] : com_size(while(Fun,Com)) = plus_plus_nat(com_size(Com),suc(zero_zero_nat)) ).

tff(fact_687_com_Osize_I12_J,axiom,
    ! [Com1_1: com,Com2_1: com] : size_size_com(semi(Com1_1,Com2_1)) = plus_plus_nat(plus_plus_nat(size_size_com(Com1_1),size_size_com(Com2_1)),suc(zero_zero_nat)) ).

tff(fact_688_com_Osize_I15_J,axiom,
    ! [Pname: pname] : size_size_com(hAPP_pname_com(body,Pname)) = zero_zero_nat ).

tff(fact_689_com_Osize_I9_J,axiom,
    size_size_com(skip) = zero_zero_nat ).

tff(fact_690_com_Osize_I14_J,axiom,
    ! [Fun: fun_state_bool,Com: com] : size_size_com(while(Fun,Com)) = plus_plus_nat(size_size_com(Com),suc(zero_zero_nat)) ).

tff(fact_691_com_Osize_I13_J,axiom,
    ! [Fun: fun_state_bool,Com1: com,Com2: com] : size_size_com(cond(Fun,Com1,Com2)) = plus_plus_nat(plus_plus_nat(size_size_com(Com1),size_size_com(Com2)),suc(zero_zero_nat)) ).

tff(fact_692_evaln_OIfFalse,axiom,
    ! [C0: com,C1: com,N_1: nat,S1: state,B: fun_state_bool,S: state] :
      ( ~ hBOOL(hAPP_state_bool(B,S))
     => ( hBOOL(hAPP_state_bool(evaln(C1,S,N_1),S1))
       => hBOOL(hAPP_state_bool(evaln(cond(B,C0,C1),S,N_1),S1)) ) ) ).

tff(fact_693_evaln_OIfTrue,axiom,
    ! [C1: com,C0: com,N_1: nat,S1: state,B: fun_state_bool,S: state] :
      ( hBOOL(hAPP_state_bool(B,S))
     => ( hBOOL(hAPP_state_bool(evaln(C0,S,N_1),S1))
       => hBOOL(hAPP_state_bool(evaln(cond(B,C0,C1),S,N_1),S1)) ) ) ).

tff(fact_694_evaln__elim__cases_I5_J,axiom,
    ! [B: fun_state_bool,C1: com,C2: com,S: state,N_1: nat,T: state] :
      ( hBOOL(hAPP_state_bool(evaln(cond(B,C1,C2),S,N_1),T))
     => ( ( hBOOL(hAPP_state_bool(B,S))
         => ~ hBOOL(hAPP_state_bool(evaln(C1,S,N_1),T)) )
       => ~ ( ~ hBOOL(hAPP_state_bool(B,S))
           => ~ hBOOL(hAPP_state_bool(evaln(C2,S,N_1),T)) ) ) ) ).

tff(fact_695_evalc__elim__cases_I5_J,axiom,
    ! [B: fun_state_bool,C1: com,C2: com,S: state,T: state] :
      ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(cond(B,C1,C2)),S),T))
     => ( ( hBOOL(hAPP_state_bool(B,S))
         => ~ hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C1),S),T)) )
       => ~ ( ~ hBOOL(hAPP_state_bool(B,S))
           => ~ hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C2),S),T)) ) ) ) ).

tff(fact_696_evalc_OIfTrue,axiom,
    ! [C1: com,C0: com,S1: state,B: fun_state_bool,S: state] :
      ( hBOOL(hAPP_state_bool(B,S))
     => ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C0),S),S1))
       => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(cond(B,C0,C1)),S),S1)) ) ) ).

tff(fact_697_evalc_OIfFalse,axiom,
    ! [C0: com,C1: com,S1: state,B: fun_state_bool,S: state] :
      ( ~ hBOOL(hAPP_state_bool(B,S))
     => ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C1),S),S1))
       => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(cond(B,C0,C1)),S),S1)) ) ) ).

%----Helper facts (101)
tff(help_fNot_1_1_U,axiom,
    ! [P: bool] :
      ( ~ hBOOL(hAPP_bool_bool(fNot,P))
      | ~ hBOOL(P) ) ).

tff(help_fNot_2_1_U,axiom,
    ! [P: bool] :
      ( hBOOL(P)
      | hBOOL(hAPP_bool_bool(fNot,P)) ) ).

tff(help_fconj_1_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ hBOOL(P)
      | ~ hBOOL(Q)
      | hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q)) ) ).

tff(help_fconj_2_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q))
      | hBOOL(P) ) ).

tff(help_fconj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q))
      | hBOOL(Q) ) ).

tff(help_fdisj_1_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ hBOOL(P)
      | hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fdisj,P),Q)) ) ).

tff(help_fdisj_2_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ hBOOL(Q)
      | hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fdisj,P),Q)) ) ).

tff(help_fdisj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fdisj,P),Q))
      | hBOOL(P)
      | hBOOL(Q) ) ).

tff(help_fFalse_1_1_U,axiom,
    ~ hBOOL(fFalse) ).

tff(help_fFalse_1_1_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_fimplies_1_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( hBOOL(P)
      | hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fimplies,P),Q)) ) ).

tff(help_fimplies_2_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ hBOOL(Q)
      | hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fimplies,P),Q)) ) ).

tff(help_fimplies_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fimplies,P),Q))
      | ~ hBOOL(P)
      | hBOOL(Q) ) ).

tff(help_COMBI_1_1_COMBI_000tc__Nat__Onat_U,axiom,
    ! [P: nat] : hAPP_nat_nat(cOMBI_nat,P) = P ).

tff(help_fequal_1_1_fequal_000tc__Com__Opname_T,axiom,
    ! [X_1: pname,Y: pname] :
      ( ~ hBOOL(hAPP_pname_bool(hAPP_p61793385e_bool(fequal_pname,X_1),Y))
      | ( X_1 = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__Com__Opname_T,axiom,
    ! [X_1: pname,Y: pname] :
      ( ( X_1 != Y )
      | hBOOL(hAPP_pname_bool(hAPP_p61793385e_bool(fequal_pname,X_1),Y)) ) ).

tff(help_fequal_1_1_fequal_000tc__Com__Ostate_T,axiom,
    ! [X_1: state,Y: state] :
      ( ~ hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(fequal_state,X_1),Y))
      | ( X_1 = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__Com__Ostate_T,axiom,
    ! [X_1: state,Y: state] :
      ( ( X_1 != Y )
      | hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(fequal_state,X_1),Y)) ) ).

tff(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Com__Opname_U,axiom,
    ! [P: bool,Q: pname] : hAPP_pname_bool(hAPP_b76515610e_bool(cOMBK_bool_pname,P),Q) = P ).

tff(help_COMBK_1_1_COMBK_000tc__Com__Opname_000tc__Com__Opname_U,axiom,
    ! [P: pname,Q: pname] : hAPP_pname_pname(hAPP_p1170154830_pname(cOMBK_pname_pname,P),Q) = P ).

tff(help_fequal_1_1_fequal_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_T,axiom,
    ! [X_1: fun_pname_bool,Y: fun_pname_bool] :
      ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(fequal533582459e_bool,X_1),Y))
      | ( X_1 = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_T,axiom,
    ! [X_1: fun_pname_bool,Y: fun_pname_bool] :
      ( ( X_1 != Y )
      | hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(fequal533582459e_bool,X_1),Y)) ) ).

tff(help_COMBK_1_1_COMBK_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000t__a_U,axiom,
    ! [P: fun_state_bool,Q: x_a] : hAPP_a2036067514e_bool(hAPP_f762886889e_bool(cOMBK_1458035955bool_a,P),Q) = P ).

tff(help_If_1_1_If_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_T,axiom,
    ! [X_1: hoare_1544627872iple_a,Y: hoare_1544627872iple_a] : hAPP_H9461717iple_a(hAPP_H1988213490iple_a(hAPP_b1309515258iple_a(if_Hoa816906256iple_a,fTrue),X_1),Y) = X_1 ).

tff(help_If_2_1_If_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_T,axiom,
    ! [X_1: hoare_1544627872iple_a,Y: hoare_1544627872iple_a] : hAPP_H9461717iple_a(hAPP_H1988213490iple_a(hAPP_b1309515258iple_a(if_Hoa816906256iple_a,fFalse),X_1),Y) = Y ).

tff(help_If_3_1_If_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Com__Opname_U,axiom,
    ! [P: fun_bool_bool,Q: fun_pname_bool,R: pname] : hAPP_pname_bool(hAPP_f759274231e_bool(cOMBB_647938656_pname(P),Q),R) = hAPP_bool_bool(P,hAPP_pname_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Com__Ostate_U,axiom,
    ! [P: fun_bool_bool,Q: fun_state_bool,R: state] : hAPP_state_bool(hAPP_f1759915619e_bool(cOMBB_160679318_state(P),Q),R) = hAPP_bool_bool(P,hAPP_state_bool(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Com__Ostate_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
    ! [P: fun_st2063251938l_bool,Q: bool,R: state] : hAPP_state_bool(hAPP_b2019457360e_bool(hAPP_f167292325e_bool(cOMBC_231445413l_bool,P),Q),R) = hAPP_bool_bool(hAPP_s58564346l_bool(P,R),Q) ).

tff(help_COMBS_1_1_COMBS_000tc__Com__Opname_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
    ! [P: fun_pn250273176l_bool,Q: fun_pname_bool,R: pname] : hAPP_pname_bool(hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,P),Q),R) = hAPP_bool_bool(hAPP_p393069232l_bool(P,R),hAPP_pname_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__Com__Ostate_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
    ! [P: fun_st2063251938l_bool,Q: fun_state_bool,R: state] : hAPP_state_bool(hAPP_f1759915619e_bool(hAPP_f644196280e_bool(cOMBS_1378840469l_bool,P),Q),R) = hAPP_bool_bool(hAPP_s58564346l_bool(P,R),hAPP_state_bool(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Com__Opname_000tc__Com__Opname_000tc__HOL__Obool_U,axiom,
    ! [P: fun_pn800050071e_bool,Q: pname,R: pname] : hAPP_pname_bool(hAPP_p61793385e_bool(hAPP_f22061361e_bool(cOMBC_1149511130e_bool,P),Q),R) = hAPP_pname_bool(hAPP_p61793385e_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Com__Ostate_000tc__Com__Ostate_000tc__HOL__Obool_U,axiom,
    ! [P: fun_st1506752259e_bool,Q: state,R: state] : hAPP_state_bool(hAPP_s1806633685e_bool(hAPP_f817621513e_bool(cOMBC_2027030106e_bool,P),Q),R) = hAPP_state_bool(hAPP_s1806633685e_bool(P,R),Q) ).

tff(help_fequal_1_1_fequal_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_,axiom,
    ! [X_1: hoare_1544627872iple_a,Y: hoare_1544627872iple_a] :
      ( ~ hBOOL(hAPP_H2006434664a_bool(hAPP_H1472549281a_bool(fequal1057774495iple_a,X_1),Y))
      | ( X_1 = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_,axiom,
    ! [X_1: hoare_1544627872iple_a,Y: hoare_1544627872iple_a] :
      ( ( X_1 != Y )
      | hBOOL(hAPP_H2006434664a_bool(hAPP_H1472549281a_bool(fequal1057774495iple_a,X_1),Y)) ) ).

tff(help_COMBK_1_1_COMBK_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc__Com__O,axiom,
    ! [P: fun_state_bool,Q: state] : hAPP_s1806633685e_bool(hAPP_f1863945078e_bool(cOMBK_631994958_state,P),Q) = P ).

tff(help_If_1_1_If_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostate,axiom,
    ! [X_1: hoare_1875481847_state,Y: hoare_1875481847_state] : hAPP_H350317809_state(hAPP_H743221337_state(hAPP_b1347637997_state(if_Hoa1541320583_state,fTrue),X_1),Y) = X_1 ).

tff(help_If_2_1_If_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostate,axiom,
    ! [X_1: hoare_1875481847_state,Y: hoare_1875481847_state] : hAPP_H350317809_state(hAPP_H743221337_state(hAPP_b1347637997_state(if_Hoa1541320583_state,fFalse),X_1),Y) = Y ).

tff(help_If_3_1_If_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostate,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_fequal_1_1_fequal_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com,axiom,
    ! [X_1: hoare_1875481847_state,Y: hoare_1875481847_state] :
      ( ~ hBOOL(hAPP_H1173941847e_bool(hAPP_H117293375e_bool(fequal391417144_state,X_1),Y))
      | ( X_1 = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com,axiom,
    ! [X_1: hoare_1875481847_state,Y: hoare_1875481847_state] :
      ( ( X_1 != Y )
      | hBOOL(hAPP_H1173941847e_bool(hAPP_H117293375e_bool(fequal391417144_state,X_1),Y)) ) ).

tff(help_COMBC_1_1_COMBC_000t__a_000tc__HOL__Obool_000tc__fun_Itc__Com__Ostate_Mtc__,axiom,
    ! [P: fun_a_998512028e_bool,Q: bool,R: x_a] : hAPP_a2036067514e_bool(hAPP_b540892988e_bool(hAPP_f1824947087e_bool(cOMBC_41962815e_bool,P),Q),R) = hAPP_b2019457360e_bool(hAPP_a723219176e_bool(P,R),Q) ).

tff(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Hoare____Mirabelle____xlrqixeqwe__,axiom,
    ! [P: bool,Q: hoare_1544627872iple_a] : hAPP_H2006434664a_bool(hAPP_b704538998a_bool(cOMBK_329760839iple_a,P),Q) = P ).

tff(help_COMBB_1_1_COMBB_000tc__Option__Ooption_Itc__Com__Ocom_J_000tc__Com__Ocom_00,axiom,
    ! [P: fun_option_com_com,Q: fun_pname_option_com,R: pname] : hAPP_pname_com(hAPP_f96342628me_com(cOMBB_923936821_pname(P),Q),R) = hAPP_option_com_com(P,hAPP_p799580910on_com(Q,R)) ).

tff(help_COMBK_1_1_COMBK_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_00,axiom,
    ! [P: hoare_1544627872iple_a,Q: pname] : hAPP_p441219121iple_a(hAPP_H79278542iple_a(cOMBK_997027250_pname,P),Q) = P ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo,axiom,
    ! [P: fun_bo1549164019l_bool,Q: fun_pname_bool,R: pname] : hAPP_p393069232l_bool(hAPP_f1402196763l_bool(cOMBB_675860798_pname(P),Q),R) = hAPP_b589554111l_bool(P,hAPP_pname_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_249,axiom,
    ! [P: fun_bo1549164019l_bool,Q: fun_state_bool,R: state] : hAPP_s58564346l_bool(hAPP_f1259673775l_bool(cOMBB_188601460_state(P),Q),R) = hAPP_b589554111l_bool(P,hAPP_state_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__HOL__O,axiom,
    ! [P: fun_fu1430349052l_bool,Q: fun_pn800050071e_bool,R: pname] : hAPP_pname_bool(hAPP_f42430548e_bool(cOMBB_530759491_pname(P),Q),R) = hAPP_f1664156314l_bool(P,hAPP_p61793385e_bool(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Com__Opname_000tc__fun_Itc__Com__Opname_Mtc__HOL__Ob,axiom,
    ! [P: fun_pn422929397l_bool,Q: fun_pname_bool,R: pname] : hAPP_pname_bool(hAPP_f759274231e_bool(hAPP_f661147897e_bool(cOMBC_1058051404l_bool,P),Q),R) = hAPP_f1664156314l_bool(hAPP_p338031245l_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Com__Ostate_000tc__HOL__Obool_000tc__fun_Itc__Com__O,axiom,
    ! [P: fun_st1483075957e_bool,Q: bool,R: state] : hAPP_s1806633685e_bool(hAPP_b1095269219e_bool(hAPP_f701449317e_bool(cOMBC_471052088e_bool,P),Q),R) = hAPP_b2019457360e_bool(hAPP_s1874344717e_bool(P,R),Q) ).

tff(help_fequal_1_1_fequal_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple,axiom,
    ! [X_1: fun_Ho287446294a_bool,Y: fun_Ho287446294a_bool] :
      ( ~ hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(fequal1843628375a_bool,X_1),Y))
      | ( X_1 = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple,axiom,
    ! [X_1: fun_Ho287446294a_bool,Y: fun_Ho287446294a_bool] :
      ( ( X_1 != Y )
      | hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(fequal1843628375a_bool,X_1),Y)) ) ).

tff(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Hoare____Mirabelle____xlrqixeqwe___250,axiom,
    ! [P: bool,Q: hoare_1875481847_state] : hAPP_H1173941847e_bool(hAPP_b1906038105e_bool(cOMBK_2086958736_state,P),Q) = P ).

tff(help_COMBK_1_1_COMBK_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__,axiom,
    ! [P: hoare_1875481847_state,Q: pname] : hAPP_p702226982_state(hAPP_H939818062_state(cOMBK_1032401873_pname,P),Q) = P ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Hoare____Mirabel,axiom,
    ! [P: fun_bool_bool,Q: fun_Ho287446294a_bool,R: hoare_1544627872iple_a] : hAPP_H2006434664a_bool(hAPP_f909437487a_bool(cOMBB_1977449916iple_a(P),Q),R) = hAPP_bool_bool(P,hAPP_H2006434664a_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_00,axiom,
    ! [P: fun_Ho1937021500l_bool,Q: fun_Ho287446294a_bool,R: hoare_1544627872iple_a] : hAPP_H2006434664a_bool(hAPP_f909437487a_bool(hAPP_f1099051562a_bool(cOMBS_523298619l_bool,P),Q),R) = hAPP_bool_bool(hAPP_H320345684l_bool(P,R),hAPP_H2006434664a_bool(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Com__Opname_000tc__Hoare____Mirabelle____xlrqixeqwe_,axiom,
    ! [P: fun_pn1843974899a_bool,Q: hoare_1544627872iple_a,R: pname] : hAPP_pname_bool(hAPP_H1178433989e_bool(hAPP_f565099497e_bool(cOMBC_131447350a_bool,P),Q),R) = hAPP_H2006434664a_bool(hAPP_p565143109a_bool(P,R),Q) ).

tff(help_fequal_1_1_fequal_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_251,axiom,
    ! [X_1: fun_Ho1996104121e_bool,Y: fun_Ho1996104121e_bool] :
      ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(fequal259518520e_bool,X_1),Y))
      | ( X_1 = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_252,axiom,
    ! [X_1: fun_Ho1996104121e_bool,Y: fun_Ho1996104121e_bool] :
      ( ( X_1 != Y )
      | hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(fequal259518520e_bool,X_1),Y)) ) ).

tff(help_COMBC_1_1_COMBC_000t__a_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc,axiom,
    ! [P: fun_a_2117018159e_bool,Q: fun_state_bool,R: x_a] : hAPP_a2036067514e_bool(hAPP_f762886889e_bool(hAPP_f1261923407e_bool(cOMBC_892787026e_bool,P),Q),R) = hAPP_f1759915619e_bool(hAPP_a1200519163e_bool(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Hoare____Mirabel_253,axiom,
    ! [P: fun_bool_bool,Q: fun_Ho1996104121e_bool,R: hoare_1875481847_state] : hAPP_H1173941847e_bool(hAPP_f121055253e_bool(cOMBB_1424001755_state(P),Q),R) = hAPP_bool_bool(P,hAPP_H1173941847e_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__,axiom,
    ! [P: fun_Ho1896286681l_bool,Q: fun_Ho1996104121e_bool,R: hoare_1875481847_state] : hAPP_H1173941847e_bool(hAPP_f121055253e_bool(hAPP_f1300953505e_bool(cOMBS_896408300l_bool,P),Q),R) = hAPP_bool_bool(hAPP_H377399461l_bool(P,R),hAPP_H1173941847e_bool(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Com__Opname_000tc__Hoare____Mirabelle____xlrqixeqwe__254,axiom,
    ! [P: fun_pn760644914e_bool,Q: hoare_1875481847_state,R: pname] : hAPP_pname_bool(hAPP_H785152948e_bool(hAPP_f1564627769e_bool(cOMBC_951029257e_bool,P),Q),R) = hAPP_H1173941847e_bool(hAPP_p1537966538e_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Com__Opname_000tc__fun_Itc__Com__Opname_Mtc__HOL__Ob_255,axiom,
    ! [P: fun_pn479984722e_bool,Q: fun_pname_bool,R: pname] : hAPP_p61793385e_bool(hAPP_f180827860e_bool(hAPP_f698292281e_bool(cOMBC_1123258281e_bool,P),Q),R) = hAPP_f759274231e_bool(hAPP_p905327722e_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Com__Ostate_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Ob,axiom,
    ! [P: fun_st277393160e_bool,Q: fun_state_bool,R: state] : hAPP_s1806633685e_bool(hAPP_f1863945078e_bool(hAPP_f915354021e_bool(cOMBC_952831051e_bool,P),Q),R) = hAPP_f1759915619e_bool(hAPP_s1226857760e_bool(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_256,axiom,
    ! [P: fun_bo1549164019l_bool,Q: fun_Ho287446294a_bool,R: hoare_1544627872iple_a] : hAPP_H320345684l_bool(hAPP_f224574947l_bool(cOMBB_1960865690iple_a(P),Q),R) = hAPP_b589554111l_bool(P,hAPP_H2006434664a_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc__fun_It,axiom,
    ! [P: fun_fu1658206819l_bool,Q: fun_a_fun_state_bool,R: x_a] : hAPP_a849909144l_bool(hAPP_f1509969235l_bool(cOMBB_1355796797bool_a(P),Q),R) = hAPP_f1259673775l_bool(P,hAPP_a2036067514e_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_257,axiom,
    ! [P: fun_bo1549164019l_bool,Q: fun_Ho1996104121e_bool,R: hoare_1875481847_state] : hAPP_H377399461l_bool(hAPP_f932622461l_bool(cOMBB_242064253_state(P),Q),R) = hAPP_b589554111l_bool(P,hAPP_H1173941847e_bool(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_00,axiom,
    ! [P: fun_Ho965678415a_bool,Q: hoare_1544627872iple_a,R: hoare_1544627872iple_a] : hAPP_H2006434664a_bool(hAPP_H1472549281a_bool(hAPP_f1878404769a_bool(cOMBC_827727450a_bool,P),Q),R) = hAPP_H2006434664a_bool(hAPP_H1472549281a_bool(P,R),Q) ).

tff(help_COMBS_1_1_COMBS_000tc__Com__Opname_000tc__Hoare____Mirabelle____xlrqixeqwe_,axiom,
    ! [P: fun_pn1936399286iple_a,Q: fun_pn1711459225iple_a,R: pname] : hAPP_p441219121iple_a(hAPP_f687424533iple_a(hAPP_f16446748iple_a(cOMBS_459699575iple_a,P),Q),R) = hAPP_H9461717iple_a(hAPP_p1688315214iple_a(P,R),hAPP_p441219121iple_a(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc__fun_It_258,axiom,
    ! [P: fun_fu1658206819l_bool,Q: fun_st1506752259e_bool,R: state] : hAPP_s2001034685l_bool(hAPP_f1283379615l_bool(cOMBB_1759179140_state(P),Q),R) = hAPP_f1259673775l_bool(P,hAPP_s1806633685e_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__Com__Opname_000tc__fun_It__a_Mtc__fun_Itc__Com__Osta,axiom,
    ! [P: fun_pn1496034746iple_a,Q: fun_pn1683930517e_bool,R: pname] : hAPP_p441219121iple_a(hAPP_f104000281iple_a(hAPP_f2028456860iple_a(cOMBS_438391419iple_a,P),Q),R) = hAPP_f328191961iple_a(hAPP_p369566930iple_a(P,R),hAPP_p635540397e_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_00,axiom,
    ! [P: fun_Ho1528294576_state,Q: fun_pn1711459225iple_a,R: pname] : hAPP_p702226982_state(hAPP_f793302744_state(cOMBB_531782587_pname(P),Q),R) = hAPP_H1928101250_state(P,hAPP_p441219121iple_a(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__,axiom,
    ! [P: fun_Ho1202345178iple_a,Q: fun_pn664418900_state,R: pname] : hAPP_p441219121iple_a(hAPP_f1780080898iple_a(cOMBB_1283441943_pname(P),Q),R) = hAPP_H1771183910iple_a(P,hAPP_p702226982_state(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__fun_Itc__HOL__Obool_Mtc__H,axiom,
    ! [P: fun_fu1219323149e_bool,Q: fun_a_1632297036l_bool,R: x_a] : hAPP_a723219176e_bool(hAPP_f340725611e_bool(cOMBB_1348041619bool_a(P),Q),R) = hAPP_f167292325e_bool(P,hAPP_a849909144l_bool(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__,axiom,
    ! [P: fun_Ho1695028467e_bool,Q: hoare_1875481847_state,R: hoare_1875481847_state] : hAPP_H1173941847e_bool(hAPP_H117293375e_bool(hAPP_f2101312745e_bool(cOMBC_1972244470e_bool,P),Q),R) = hAPP_H1173941847e_bool(hAPP_H117293375e_bool(P,R),Q) ).

tff(help_COMBS_1_1_COMBS_000tc__Com__Opname_000tc__Hoare____Mirabelle____xlrqixeqwe__259,axiom,
    ! [P: fun_pn1776597836_state,Q: fun_pn664418900_state,R: pname] : hAPP_p702226982_state(hAPP_f1981552043_state(hAPP_f1875279510_state(cOMBS_629630867_state,P),Q),R) = hAPP_H350317809_state(hAPP_p1279250276_state(P,R),hAPP_p702226982_state(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_I,axiom,
    ! [P: fun_fu1441721944l_bool,Q: fun_Ho965678415a_bool,R: hoare_1544627872iple_a] : hAPP_H2006434664a_bool(hAPP_f1726109544a_bool(cOMBB_1474127043iple_a(P),Q),R) = hAPP_f2063540982l_bool(P,hAPP_H1472549281a_bool(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_00_260,axiom,
    ! [P: fun_Ho448013557l_bool,Q: fun_Ho287446294a_bool,R: hoare_1544627872iple_a] : hAPP_H2006434664a_bool(hAPP_f909437487a_bool(hAPP_f482270129a_bool(cOMBC_1807206788l_bool,P),Q),R) = hAPP_f2063540982l_bool(hAPP_H736103437l_bool(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_00_261,axiom,
    ! [P: fun_Ho965678415a_bool,Q: fun_pn1711459225iple_a,R: pname] : hAPP_p565143109a_bool(hAPP_f1707302775a_bool(cOMBB_1584663260_pname(P),Q),R) = hAPP_H1472549281a_bool(P,hAPP_p441219121iple_a(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__Com__Opname_000tc__fun_Itc__Com__Ostate_Mtc__fun_Itc,axiom,
    ! [P: fun_pn1610849624_state,Q: fun_pn12888928e_bool,R: pname] : hAPP_p702226982_state(hAPP_f1568887991_state(hAPP_f891245078_state(cOMBS_475568287_state,P),Q),R) = hAPP_f1595846909_state(hAPP_p1985340144_state(P,R),hAPP_p1637813682e_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__fun_Itc__HOL__Obool_Mtc__H_262,axiom,
    ! [P: fun_fu1219323149e_bool,Q: fun_st2116860965l_bool,R: state] : hAPP_s1874344717e_bool(hAPP_f873506917e_bool(cOMBB_1757942702_state(P),Q),R) = hAPP_f167292325e_bool(P,hAPP_s2001034685l_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__Com__Opname_000tc__Com__Ocom_000tc__fun_Itc__fun_It_,axiom,
    ! [P: fun_pn236922029iple_a,Q: fun_pname_com,R: pname] : hAPP_p369566930iple_a(hAPP_f68608908iple_a(hAPP_f1015300700iple_a(cOMBS_2128932974iple_a,P),Q),R) = hAPP_c1432218316iple_a(hAPP_p558195141iple_a(P,R),hAPP_pname_com(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_I_263,axiom,
    ! [P: fun_fu689207471l_bool,Q: fun_Ho1695028467e_bool,R: hoare_1875481847_state] : hAPP_H1173941847e_bool(hAPP_f169892419e_bool(cOMBB_1380679615_state(P),Q),R) = hAPP_f1408815105l_bool(P,hAPP_H117293375e_bool(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com___264,axiom,
    ! [P: fun_Ho1085560331l_bool,Q: fun_Ho1996104121e_bool,R: hoare_1875481847_state] : hAPP_H1173941847e_bool(hAPP_f121055253e_bool(hAPP_f1795624711e_bool(cOMBC_1099403170l_bool,P),Q),R) = hAPP_f1408815105l_bool(hAPP_H810811817l_bool(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com___265,axiom,
    ! [P: fun_Ho1695028467e_bool,Q: fun_pn664418900_state,R: pname] : hAPP_p1537966538e_bool(hAPP_f1601283355e_bool(cOMBB_2037342654_pname(P),Q),R) = hAPP_H117293375e_bool(P,hAPP_p702226982_state(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__fun_Itc__HOL__Obool_Mtc__H_266,axiom,
    ! [P: fun_fu1047394976e_bool,Q: fun_a_1632297036l_bool,R: x_a] : hAPP_a1200519163e_bool(hAPP_f963367678e_bool(cOMBB_145932198bool_a(P),Q),R) = hAPP_f644196280e_bool(P,hAPP_a849909144l_bool(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Com__Opname_000tc__fun_Itc__Hoare____Mirabelle____xl,axiom,
    ! [P: fun_pn660268554a_bool,Q: fun_Ho287446294a_bool,R: pname] : hAPP_p565143109a_bool(hAPP_f1932171532a_bool(hAPP_f1035004473a_bool(cOMBC_845146337a_bool,P),Q),R) = hAPP_f909437487a_bool(hAPP_p1544413474a_bool(P,R),Q) ).

tff(help_COMBS_1_1_COMBS_000tc__Com__Opname_000tc__Com__Ocom_000tc__fun_Itc__fun_Itc,axiom,
    ! [P: fun_pn92346699_state,Q: fun_pname_com,R: pname] : hAPP_p1985340144_state(hAPP_f2116202538_state(hAPP_f1339343196_state(cOMBS_1674056844_state,P),Q),R) = hAPP_c681126378_state(hAPP_p2012996835_state(P,R),hAPP_pname_com(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__fun_Itc__HOL__Obool_Mtc__H_267,axiom,
    ! [P: fun_fu1047394976e_bool,Q: fun_st2116860965l_bool,R: state] : hAPP_s1226857760e_bool(hAPP_f1138284024e_bool(cOMBB_20296667_state(P),Q),R) = hAPP_f644196280e_bool(P,hAPP_s2001034685l_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__Com__Opname_000tc__Hoare____Mirabelle____xlrqixeqwe__268,axiom,
    ! [P: fun_pn796963923iple_a,Q: fun_pn1711459225iple_a,R: pname] : hAPP_p1688315214iple_a(hAPP_f638932530iple_a(hAPP_f176122972iple_a(cOMBS_1079312276iple_a,P),Q),R) = hAPP_H1988213490iple_a(hAPP_p1244732139iple_a(P,R),hAPP_p441219121iple_a(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Com__Opname_000tc__fun_Itc__Hoare____Mirabelle____xl_269,axiom,
    ! [P: fun_pn558093046e_bool,Q: fun_Ho1996104121e_bool,R: pname] : hAPP_p1537966538e_bool(hAPP_f827623310e_bool(hAPP_f1497196629e_bool(cOMBC_1520907207e_bool,P),Q),R) = hAPP_f121055253e_bool(hAPP_p908911502e_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_00_270,axiom,
    ! [P: fun_Ho611385006a_bool,Q: fun_Ho287446294a_bool,R: hoare_1544627872iple_a] : hAPP_H1472549281a_bool(hAPP_f693853672a_bool(hAPP_f1399402097a_bool(cOMBC_894585277a_bool,P),Q),R) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__Hoare____Mirabelle____xlr,axiom,
    ! [P: fun_bo170780846iple_a,Q: fun_pname_bool,R: pname] : hAPP_p1244732139iple_a(hAPP_f645774806iple_a(cOMBB_225107267_pname(P),Q),R) = hAPP_b1309515258iple_a(P,hAPP_pname_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__Com__Opname_000tc__Hoare____Mirabelle____xlrqixeqwe__271,axiom,
    ! [P: fun_pn1243268074_state,Q: fun_pn664418900_state,R: pname] : hAPP_p1279250276_state(hAPP_f1087244981_state(hAPP_f294490838_state(cOMBS_602905655_state,P),Q),R) = hAPP_H743221337_state(hAPP_p1513527868_state(P,R),hAPP_p702226982_state(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_It__a_Mtc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_,axiom,
    ! [P: fun_fu18778709iple_a,Q: fun_pn1683930517e_bool,R: pname] : hAPP_p558195141iple_a(hAPP_f298026029iple_a(cOMBB_124251824_pname(P),Q),R) = hAPP_f1102554477iple_a(P,hAPP_p635540397e_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_00_272,axiom,
    ! [P: fun_Ho611385006a_bool,Q: fun_pn1711459225iple_a,R: pname] : hAPP_p1544413474a_bool(hAPP_f616158854a_bool(cOMBB_1386904087_pname(P),Q),R) = hAPP_H762155206a_bool(P,hAPP_p441219121iple_a(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com___273,axiom,
    ! [P: fun_Ho1110608055e_bool,Q: fun_Ho1996104121e_bool,R: hoare_1875481847_state] : hAPP_H117293375e_bool(hAPP_f56162711e_bool(hAPP_f392546175e_bool(cOMBC_483615668e_bool,P),Q),R) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__Hoare____Mirabelle____xlr_274,axiom,
    ! [P: fun_bo840793679_state,Q: fun_pname_bool,R: pname] : hAPP_p1513527868_state(hAPP_f980169511_state(cOMBB_1989814576_pname(P),Q),R) = hAPP_b1347637997_state(P,hAPP_pname_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__fun_Itc__Com__Ostate_Mtc__,axiom,
    ! [P: fun_fu404278912_state,Q: fun_pn12888928e_bool,R: pname] : hAPP_p2012996835_state(hAPP_f275933352_state(cOMBB_718493681_pname(P),Q),R) = hAPP_f1930074316_state(P,hAPP_p1637813682e_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com___275,axiom,
    ! [P: fun_Ho1110608055e_bool,Q: fun_pn664418900_state,R: pname] : hAPP_p908911502e_bool(hAPP_f1023805663e_bool(cOMBB_261907706_pname(P),Q),R) = hAPP_H1625489667e_bool(P,hAPP_p702226982_state(Q,R)) ).

%----Conjectures (2)
tff(conj_0,hypothesis,
    ! [N: nat] :
      ( ! [X: hoare_1544627872iple_a] :
          ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,g),hAPP_f856094675a_bool(image_1832685281iple_a(hAPP_f104000281iple_a(hAPP_f2028456860iple_a(cOMBS_438391419iple_a,hAPP_f68608908iple_a(hAPP_f1015300700iple_a(cOMBS_2128932974iple_a,hAPP_f298026029iple_a(cOMBB_124251824_pname(hoare_196563068iple_a),p)),body)),q)),procs))))
         => hBOOL(hAPP_H2006434664a_bool(hoare_1500711466alid_a(N),X)) )
     => ! [X: hoare_1544627872iple_a] :
          ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X),hAPP_f856094675a_bool(image_1832685281iple_a(hAPP_f104000281iple_a(hAPP_f2028456860iple_a(cOMBS_438391419iple_a,hAPP_f68608908iple_a(hAPP_f1015300700iple_a(cOMBS_2128932974iple_a,hAPP_f298026029iple_a(cOMBB_124251824_pname(hoare_196563068iple_a),p)),hAPP_f96342628me_com(cOMBB_923936821_pname(the_com),body_1))),q)),procs)))
         => hBOOL(hAPP_H2006434664a_bool(hoare_1500711466alid_a(N),X)) ) ) ).

tff(conj_1,conjecture,
    ( ! [X: hoare_1544627872iple_a] :
        ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X),g))
       => hBOOL(hAPP_H2006434664a_bool(hoare_1500711466alid_a(n),X)) )
   => ! [X: hoare_1544627872iple_a] :
        ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X),hAPP_f856094675a_bool(image_1832685281iple_a(hAPP_f104000281iple_a(hAPP_f2028456860iple_a(cOMBS_438391419iple_a,hAPP_f68608908iple_a(hAPP_f1015300700iple_a(cOMBS_2128932974iple_a,hAPP_f298026029iple_a(cOMBB_124251824_pname(hoare_196563068iple_a),p)),body)),q)),procs)))
       => hBOOL(hAPP_H2006434664a_bool(hoare_1500711466alid_a(n),X)) ) ) ).

%------------------------------------------------------------------------------