TPTP Problem File: SWW470

TPTP Problem File: SWW470_2.p

%------------------------------------------------------------------------------
% File     : SWW470_2 : TPTP v8.2.0. Released v5.3.0.
% Domain   : Software Verification
% Problem  : Hoare's Logic with Procedures line 200, 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_l200 [Bla11]

% Status   : Theorem
% Rating   : 0.22 v8.2.0, 0.40 v8.1.0, 0.27 v7.5.0, 0.25 v7.4.0, 0.20 v7.3.0, 0.17 v7.1.0, 0.00 v6.1.0, 0.33 v6.0.0, 0.20 v5.5.0, 0.25 v5.4.0, 0.67 v5.3.0
% Syntax   : Number of formulae    : 1339 ( 281 unt; 537 typ;   0 def)
%            Number of atoms       : 1858 ( 723 equ)
%            Maximal formula atoms :    9 (   1 avg)
%            Number of connectives : 1325 ( 269   ~;  51   |; 110   &)
%                                         ( 145 <=>; 750  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   16 (   5 avg)
%            Maximal term depth    :   11 (   2 avg)
%            Number of types       :  177 ( 176 usr)
%            Number of type conns  :  433 ( 224   >; 209   *;   0   +;   0  <<)
%            Number of predicates  :    2 (   1 usr;   0 prp; 1-2 aty)
%            Number of functors    :  360 ( 360 usr; 137 con; 0-4 aty)
%            Number of variables   : 2162 (2110   !;  52   ?;2162   :)
% SPC      : TF0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2011-08-09 17:19:08
%------------------------------------------------------------------------------
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tff(sy_c_COMBI_000tc__Nat__Onat,type,
    cOMBI_nat: fun_nat_nat ).

tff(sy_c_COMBK_000tc__HOL__Obool_000tc__Com__Ostate,type,
    cOMBK_bool_state: fun_bo1936561970e_bool ).

tff(sy_c_COMBK_000tc__HOL__Obool_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It,type,
    cOMBK_329760839iple_a: fun_bo873159128a_bool ).

tff(sy_c_COMBK_000tc__HOL__Obool_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It_105,type,
    cOMBK_2086958736_state: fun_bo221168653e_bool ).

tff(sy_c_COMBK_000tc__HOL__Obool_000tc__Nat__Onat,type,
    cOMBK_bool_nat: fun_bo1570548802t_bool ).

tff(sy_c_COMBK_000tc__HOL__Obool_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    cOMBK_1994329625t_bool: fun_bo45372054l_bool ).

tff(sy_c_COMBK_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_000tc__Hoare,type,
    cOMBK_1156021774iple_a: fun_Ho115022426iple_a ).

tff(sy_c_COMBK_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_000tc__Hoare_106,type,
    cOMBK_875269129_state: fun_Ho316875795iple_a ).

tff(sy_c_COMBK_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_000tc__Nat__,type,
    cOMBK_830162040_a_nat: fun_Ho480463728iple_a ).

tff(sy_c_COMBK_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostate_J_0,type,
    cOMBK_718351789iple_a: fun_Ho1675781772_state ).

tff(sy_c_COMBK_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostate_J_0_107,type,
    cOMBK_477315498_state: fun_Ho784854669_state ).

tff(sy_c_COMBK_000tc__Nat__Onat_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It_,type,
    cOMBK_1122917412iple_a: fun_na1678609370_a_nat ).

tff(sy_c_COMBK_000tc__Nat__Onat_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc,type,
    cOMBK_809880179_state: fun_na913373671te_nat ).

tff(sy_c_COMBK_000tc__Nat__Onat_000tc__Nat__Onat,type,
    cOMBK_nat_nat: fun_nat_fun_nat_nat ).

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__Ostate_000tc__HOL__Obool_000tc__HOL__Obool,type,
    cOMBS_1378840469l_bool: fun_fu1047394976e_bool ).

tff(sy_c_COMBS_000tc__Com__Ostate_000tc__Nat__Onat_000tc__Com__Ostate,type,
    cOMBS_777315357_state: fun_fu227828410_state ).

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_COMBS_000tc__Nat__Onat_000tc__HOL__Obool_000tc__HOL__Obool,type,
    cOMBS_nat_bool_bool: fun_fu1901564496t_bool ).

tff(sy_c_COMBS_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obool_000tc__,type,
    cOMBS_1187019125l_bool: fun_fu898693662l_bool ).

tff(sy_c_Com_Ocom_OAss,type,
    ass: ( vname * fun_state_nat ) > com ).

tff(sy_c_Com_Ocom_OLocal,type,
    local: ( loc * fun_state_nat * 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_Ovname_OGlb,type,
    glb_1: glb > vname ).

tff(sy_c_Com_Ovname_OLoc,type,
    loc_1: loc > vname ).

tff(sy_c_Finite__Set_Ocomp__fun__commute_000tc__Hoare____Mirabelle____xlrqixeqwe__Ot,type,
    finite531755217iple_a: fun_Ho115022426iple_a > bool ).

tff(sy_c_Finite__Set_Ocomp__fun__idem_000tc__Hoare____Mirabelle____xlrqixeqwe__Otrip,type,
    finite701293802a_bool: fun_Ho611385006a_bool > bool ).

tff(sy_c_Finite__Set_Ocomp__fun__idem_000tc__Hoare____Mirabelle____xlrqixeqwe__Otrip_108,type,
    finite1936929544e_bool: fun_Ho1110608055e_bool > bool ).

tff(sy_c_Finite__Set_Ocomp__fun__idem_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc,type,
    finite1956789438t_bool: fun_na35521688t_bool > 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__Nat__Onat,type,
    finite_finite_nat: fun_fu425979586l_bool ).

tff(sy_c_Finite__Set_Ofinite_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    finite2012431853t_bool: fun_fu814369080l_bool ).

tff(sy_c_Finite__Set_Ofold1Set_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__,type,
    finite1715754352iple_a: ( fun_Ho115022426iple_a * fun_Ho287446294a_bool ) > fun_Ho287446294a_bool ).

tff(sy_c_Finite__Set_Ofold1Set_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc_,type,
    finite1016865319_state: ( fun_Ho784854669_state * fun_Ho1996104121e_bool ) > fun_Ho1996104121e_bool ).

tff(sy_c_Finite__Set_Ofold1Set_000tc__Nat__Onat,type,
    finite_fold1Set_nat: ( fun_nat_fun_nat_nat * fun_nat_bool ) > fun_nat_bool ).

tff(sy_c_Finite__Set_Ofold1_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J,type,
    finite1560330752iple_a: fun_Ho115022426iple_a > fun_fu1634330235iple_a ).

tff(sy_c_Finite__Set_Ofold1_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Co,type,
    finite206258583_state: fun_Ho784854669_state > fun_fu385705929_state ).

tff(sy_c_Finite__Set_Ofold1_000tc__Nat__Onat,type,
    finite_fold1_nat: fun_nat_fun_nat_nat > fun_fun_nat_bool_nat ).

tff(sy_c_Finite__Set_Ofold_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_,type,
    finite309585736iple_a: ( fun_Ho115022426iple_a * hoare_1544627872iple_a ) > fun_fu1634330235iple_a ).

tff(sy_c_Finite__Set_Ofold_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J__109,type,
    finite113656430a_bool: ( fun_Ho611385006a_bool * fun_Ho287446294a_bool ) > fun_fu410471825a_bool ).

tff(sy_c_Finite__Set_Ofold_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com,type,
    finite2136361956_state: ( fun_Ho784854669_state * hoare_1875481847_state ) > fun_fu385705929_state ).

tff(sy_c_Finite__Set_Ofold_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com_110,type,
    finite1131674252e_bool: ( fun_Ho1110608055e_bool * fun_Ho1996104121e_bool ) > fun_fu1296727421e_bool ).

tff(sy_c_Finite__Set_Ofold_000tc__Nat__Onat_000tc__Nat__Onat,type,
    finite_fold_nat_nat: ( fun_nat_fun_nat_nat * nat ) > fun_fun_nat_bool_nat ).

tff(sy_c_Finite__Set_Ofold_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Oboo,type,
    finite1578363458t_bool: ( fun_na35521688t_bool * fun_nat_bool ) > fun_fu821463397t_bool ).

tff(sy_c_Finite__Set_Ofold__graph_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_I,type,
    finite1341991622iple_a: ( fun_Ho115022426iple_a * hoare_1544627872iple_a * fun_Ho287446294a_bool ) > fun_Ho287446294a_bool ).

tff(sy_c_Finite__Set_Ofold__graph_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_I_111,type,
    finite1721989730_state: ( fun_Ho784854669_state * hoare_1875481847_state * fun_Ho1996104121e_bool ) > fun_Ho1996104121e_bool ).

tff(sy_c_Finite__Set_Ofold__graph_000tc__Nat__Onat_000tc__Nat__Onat,type,
    finite929467206at_nat: ( fun_nat_fun_nat_nat * nat * fun_nat_bool ) > fun_nat_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__112,type,
    finite1782051386_state: ( fun_Ho784854669_state * fun_fu385705929_state ) > bool ).

tff(sy_c_Finite__Set_Ofolding__one_000tc__Nat__Onat,type,
    finite988810631ne_nat: ( fun_nat_fun_nat_nat * fun_fun_nat_bool_nat ) > 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_113,type,
    finite1514162941_state: ( fun_Ho784854669_state * fun_fu385705929_state ) > bool ).

tff(sy_c_Finite__Set_Ofolding__one__idem_000tc__Nat__Onat,type,
    finite795500164em_nat: ( fun_nat_fun_nat_nat * fun_fun_nat_bool_nat ) > bool ).

tff(sy_c_Groups_Ominus__class_Ominus_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqw,type,
    minus_1054521797a_bool: fun_fu601883884a_bool ).

tff(sy_c_Groups_Ominus__class_Ominus_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqw_114,type,
    minus_589155914e_bool: fun_fu947547457e_bool ).

tff(sy_c_Groups_Ominus__class_Ominus_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    minus_2067140911t_bool: fun_fu915923926t_bool ).

tff(sy_c_Groups_Otimes__class_Otimes_000tc__Nat__Onat,type,
    times_times_nat: fun_nat_fun_nat_nat ).

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_HOL_OThe_000tc__Nat__Onat,type,
    the_nat: fun_fun_nat_bool_nat ).

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_Otriple_Otriple_000t__a,type,
    hoare_196563068iple_a: ( fun_a_fun_state_bool * com * fun_a_fun_state_bool ) > hoare_1544627872iple_a ).

tff(sy_c_Hoare__Mirabelle__xlrqixeqwe_Otriple_Otriple_000tc__Com__Ostate,type,
    hoare_13225861_state: ( fun_st1506752259e_bool * com * fun_st1506752259e_bool ) > hoare_1875481847_state ).

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_Lattices_Osemilattice__inf__class_Oinf_000tc__HOL__Obool,type,
    semila310582991f_bool: fun_bo1549164019l_bool ).

tff(sy_c_Lattices_Osemilattice__inf__class_Oinf_000tc__Nat__Onat,type,
    semila80283416nf_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Lattices_Osemilattice__inf__class_Oinf_000tc__fun_Itc__Hoare____Mirabelle__,type,
    semila1256126856a_bool: fun_fu601883884a_bool ).

tff(sy_c_Lattices_Osemilattice__inf__class_Oinf_000tc__fun_Itc__Hoare____Mirabelle___115,type,
    semila657954503e_bool: fun_fu947547457e_bool ).

tff(sy_c_Lattices_Osemilattice__inf__class_Oinf_000tc__fun_Itc__Nat__Onat_Mtc__HOL__,type,
    semila840543986t_bool: fun_fu915923926t_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__Nat__Onat,type,
    semila972727038up_nat: fun_nat_fun_nat_nat ).

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___116,type,
    semila1107643821e_bool: fun_fu947547457e_bool ).

tff(sy_c_Lattices_Osemilattice__sup__class_Osup_000tc__fun_Itc__Nat__Onat_Mtc__HOL__,type,
    semila465093516t_bool: fun_fu915923926t_bool ).

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

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

tff(sy_c_Natural_Ogetlocs,type,
    getlocs: ( state * loc ) > nat ).

tff(sy_c_Natural_Oupdate,type,
    update: fun_st1816159843_state ).

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__Hoare____Mirabelle____xlrqixeqwe,type,
    bot_bo1766443648a_bool: fun_Ho287446294a_bool ).

tff(sy_c_Orderings_Obot__class_Obot_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe_117,type,
    bot_bo1715400655e_bool: fun_Ho1996104121e_bool ).

tff(sy_c_Orderings_Obot__class_Obot_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    bot_bot_fun_nat_bool: fun_nat_bool ).

tff(sy_c_Orderings_Obot__class_Obot_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obo,type,
    bot_bo1701429464l_bool: fun_fu425979586l_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__HOL__Obool,type,
    ord_less_eq_bool: fun_bo1549164019l_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__Nat__Onat,type,
    ord_less_eq_nat: fun_nat_fun_nat_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__fun_Itc__Hoare____Mirabelle____xlrqi,type,
    ord_le345502796a_bool: fun_fu629000755l_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__fun_Itc__Hoare____Mirabelle____xlrqi_118,type,
    ord_le233469699e_bool: fun_fu1173762305l_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    ord_le1568362934t_bool: fun_fu1217155507l_bool ).

tff(sy_c_Partial__Function_Oflat__lub_000tc__Hoare____Mirabelle____xlrqixeqwe__Otrip,type,
    partia2044047638iple_a: hoare_1544627872iple_a > fun_fu1634330235iple_a ).

tff(sy_c_Partial__Function_Oflat__lub_000tc__Hoare____Mirabelle____xlrqixeqwe__Otrip_119,type,
    partia1423322881_state: hoare_1875481847_state > fun_fu385705929_state ).

tff(sy_c_Partial__Function_Oflat__lub_000tc__Nat__Onat,type,
    partial_flat_lub_nat: nat > fun_fun_nat_bool_nat ).

tff(sy_c_Set_OCollect_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J,type,
    collec445968053iple_a: fun_Ho287446294a_bool > fun_Ho287446294a_bool ).

tff(sy_c_Set_OCollect_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ost,type,
    collec1735317154_state: fun_Ho1996104121e_bool > fun_Ho1996104121e_bool ).

tff(sy_c_Set_OCollect_000tc__Nat__Onat,type,
    collect_nat: fun_nat_bool > fun_nat_bool ).

tff(sy_c_Set_OCollect_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    collect_fun_nat_bool: fun_fu425979586l_bool > fun_fu425979586l_bool ).

tff(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_000tc__,type,
    image_1784540805iple_a: ( fun_Ho1275796541iple_a * fun_Ho287446294a_bool ) > fun_Ho287446294a_bool ).

tff(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_000tc___120,type,
    image_456807122_state: ( fun_Ho1528294576_state * fun_Ho287446294a_bool ) > fun_Ho1996104121e_bool ).

tff(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_000tc___121,type,
    image_1667308271_a_nat: ( fun_Ho753038759_a_nat * fun_Ho287446294a_bool ) > fun_nat_bool ).

tff(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostat,type,
    image_299889782iple_a: ( fun_Ho1202345178iple_a * fun_Ho1996104121e_bool ) > fun_Ho287446294a_bool ).

tff(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostat_122,type,
    image_436941729_state: ( fun_Ho1455141843_state * fun_Ho1996104121e_bool ) > fun_Ho1996104121e_bool ).

tff(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostat_123,type,
    image_1722141408te_nat: ( fun_Ho1284378692te_nat * fun_Ho1996104121e_bool ) > fun_nat_bool ).

tff(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Hoare____Mirabelle____xlrqixeqwe__Otripl,type,
    image_1960063643iple_a: ( fun_na98436179iple_a * fun_nat_bool ) > fun_Ho287446294a_bool ).

tff(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Hoare____Mirabelle____xlrqixeqwe__Otripl_124,type,
    image_681421628_state: ( fun_na1612816666_state * fun_nat_bool ) > fun_Ho1996104121e_bool ).

tff(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Nat__Onat,type,
    image_nat_nat: ( fun_nat_nat * fun_nat_bool ) > fun_nat_bool ).

tff(sy_c_Set_Oinsert_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J,type,
    insert1051021594iple_a: fun_Ho611385006a_bool ).

tff(sy_c_Set_Oinsert_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Osta,type,
    insert694999549_state: fun_Ho1110608055e_bool ).

tff(sy_c_Set_Oinsert_000tc__Nat__Onat,type,
    insert_nat: fun_na35521688t_bool ).

tff(sy_c_Set_Oinsert_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    insert_fun_nat_bool: fun_fu1516990510l_bool ).

tff(sy_c_Set_Othe__elem_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J,type,
    the_el1614276927iple_a: fun_fu1634330235iple_a ).

tff(sy_c_Set_Othe__elem_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__O,type,
    the_el1031305112_state: fun_fu385705929_state ).

tff(sy_c_Set_Othe__elem_000tc__Nat__Onat,type,
    the_elem_nat: fun_fun_nat_bool_nat ).

tff(sy_c_fFalse,type,
    fFalse: bool ).

tff(sy_c_fNot,type,
    fNot: fun_bool_bool ).

tff(sy_c_fTrue,type,
    fTrue: bool ).

tff(sy_c_fconj,type,
    fconj: fun_bo1549164019l_bool ).

tff(sy_c_fdisj,type,
    fdisj: fun_bo1549164019l_bool ).

tff(sy_c_fequal_000tc__Com__Ostate,type,
    fequal_state: fun_st1506752259e_bool ).

tff(sy_c_fequal_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J,type,
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    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__Nat__Onat,type,
    member_nat: fun_na1436237685l_bool ).

tff(sy_c_member_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    member_fun_nat_bool: fun_fu1436348701l_bool ).

tff(sy_v_G,type,
    g: fun_Ho287446294a_bool ).

tff(sy_v_P,type,
    p: fun_a_fun_state_bool ).

tff(sy_v_b,type,
    b: fun_state_bool ).

tff(sy_v_c,type,
    c: com ).

%----Relevant facts (700)
tff(fact_0_empty,axiom,
    ! [Ga: fun_Ho287446294a_bool] : hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),bot_bo1766443648a_bool)) ).

tff(fact_1_empty,axiom,
    ! [Ga: fun_Ho1996104121e_bool] : hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),bot_bo1715400655e_bool)) ).

tff(fact_2_triple_Oinject,axiom,
    ! [Fun1_2: fun_st1506752259e_bool,Com_1: com,Fun2_2: fun_st1506752259e_bool,Fun1_1: fun_st1506752259e_bool,Com: com,Fun2_1: fun_st1506752259e_bool] :
      ( ( hoare_13225861_state(Fun1_2,Com_1,Fun2_2) = hoare_13225861_state(Fun1_1,Com,Fun2_1) )
    <=> ( ( Fun1_2 = Fun1_1 )
        & ( Com_1 = Com )
        & ( Fun2_2 = Fun2_1 ) ) ) ).

tff(fact_3_triple_Oinject,axiom,
    ! [Fun1_2: fun_a_fun_state_bool,Com_1: com,Fun2_2: fun_a_fun_state_bool,Fun1_1: fun_a_fun_state_bool,Com: com,Fun2_1: fun_a_fun_state_bool] :
      ( ( hoare_196563068iple_a(Fun1_2,Com_1,Fun2_2) = hoare_196563068iple_a(Fun1_1,Com,Fun2_1) )
    <=> ( ( Fun1_2 = Fun1_1 )
        & ( Com_1 = Com )
        & ( Fun2_2 = Fun2_1 ) ) ) ).

tff(fact_4_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_5_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_6_hoare__derivs_Oinsert,axiom,
    ! [Ts: fun_Ho287446294a_bool,Ga: fun_Ho287446294a_bool,T_1: hoare_1544627872iple_a] :
      ( hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,T_1),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_1),Ts))) ) ) ).

tff(fact_7_hoare__derivs_Oinsert,axiom,
    ! [Ts: fun_Ho1996104121e_bool,Ga: fun_Ho1996104121e_bool,T_1: hoare_1875481847_state] :
      ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,T_1),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_1),Ts))) ) ) ).

tff(fact_8_constant,axiom,
    ! [Ga: fun_Ho287446294a_bool,Pa: fun_a_fun_state_bool,Ca: com,Q_1: fun_a_fun_state_bool,C: bool] :
      ( ( hBOOL(C)
       => hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hoare_196563068iple_a(Pa,Ca,Q_1)),bot_bo1766443648a_bool))) )
     => hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hoare_196563068iple_a(hAPP_b540892988e_bool(hAPP_f1824947087e_bool(cOMBC_41962815e_bool,hAPP_f340725611e_bool(hAPP_f1006724181e_bool(cOMBB_1348041619bool_a,cOMBC_231445413l_bool),hAPP_f1509969235l_bool(hAPP_f1178339559l_bool(cOMBB_1355796797bool_a,hAPP_f1561913689l_bool(cOMBB_188601460_state,fconj)),Pa))),C),Ca,Q_1)),bot_bo1766443648a_bool))) ) ).

tff(fact_9_constant,axiom,
    ! [Ga: fun_Ho1996104121e_bool,Pa: fun_st1506752259e_bool,Ca: com,Q_1: fun_st1506752259e_bool,C: bool] :
      ( ( hBOOL(C)
       => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hoare_13225861_state(Pa,Ca,Q_1)),bot_bo1715400655e_bool))) )
     => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hoare_13225861_state(hAPP_b1095269219e_bool(hAPP_f701449317e_bool(cOMBC_471052088e_bool,hAPP_f873506917e_bool(hAPP_f1398071125e_bool(cOMBB_1757942702_state,cOMBC_231445413l_bool),hAPP_f1283379615l_bool(hAPP_f1345202233l_bool(cOMBB_1759179140_state,hAPP_f1561913689l_bool(cOMBB_188601460_state,fconj)),Pa))),C),Ca,Q_1)),bot_bo1715400655e_bool))) ) ).

tff(fact_10_escape,axiom,
    ! [Ga: fun_Ho287446294a_bool,Ca: com,Q_1: fun_a_fun_state_bool,Pa: fun_a_fun_state_bool] :
      ( ! [Z_7: x_a,S_2: state] :
          ( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Pa,Z_7),S_2))
         => hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hoare_196563068iple_a(hAPP_f762886889e_bool(cOMBK_1458035955bool_a,hAPP_s1806633685e_bool(hAPP_f817621513e_bool(cOMBC_2027030106e_bool,fequal_state),S_2)),Ca,hAPP_f762886889e_bool(cOMBK_1458035955bool_a,hAPP_a2036067514e_bool(Q_1,Z_7)))),bot_bo1766443648a_bool))) )
     => hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hoare_196563068iple_a(Pa,Ca,Q_1)),bot_bo1766443648a_bool))) ) ).

tff(fact_11_escape,axiom,
    ! [Ga: fun_Ho1996104121e_bool,Ca: com,Q_1: fun_st1506752259e_bool,Pa: fun_st1506752259e_bool] :
      ( ! [Z_7: state,S_2: state] :
          ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Pa,Z_7),S_2))
         => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hoare_13225861_state(hAPP_f1863945078e_bool(cOMBK_631994958_state,hAPP_s1806633685e_bool(hAPP_f817621513e_bool(cOMBC_2027030106e_bool,fequal_state),S_2)),Ca,hAPP_f1863945078e_bool(cOMBK_631994958_state,hAPP_s1806633685e_bool(Q_1,Z_7)))),bot_bo1715400655e_bool))) )
     => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hoare_13225861_state(Pa,Ca,Q_1)),bot_bo1715400655e_bool))) ) ).

tff(fact_12_conseq2,axiom,
    ! [Q_1: fun_a_fun_state_bool,Ga: fun_Ho287446294a_bool,Pa: fun_a_fun_state_bool,Ca: com,Q_3: fun_a_fun_state_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hoare_196563068iple_a(Pa,Ca,Q_3)),bot_bo1766443648a_bool)))
     => ( ! [Z_7: x_a,S_2: state] :
            ( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_3,Z_7),S_2))
           => hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_1,Z_7),S_2)) )
       => hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hoare_196563068iple_a(Pa,Ca,Q_1)),bot_bo1766443648a_bool))) ) ) ).

tff(fact_13_conseq2,axiom,
    ! [Q_1: fun_st1506752259e_bool,Ga: fun_Ho1996104121e_bool,Pa: fun_st1506752259e_bool,Ca: com,Q_3: fun_st1506752259e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hoare_13225861_state(Pa,Ca,Q_3)),bot_bo1715400655e_bool)))
     => ( ! [Z_7: state,S_2: state] :
            ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Q_3,Z_7),S_2))
           => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Q_1,Z_7),S_2)) )
       => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hoare_13225861_state(Pa,Ca,Q_1)),bot_bo1715400655e_bool))) ) ) ).

tff(fact_14_conseq1,axiom,
    ! [Pa: fun_a_fun_state_bool,Ga: fun_Ho287446294a_bool,P_2: fun_a_fun_state_bool,Ca: com,Q_1: fun_a_fun_state_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hoare_196563068iple_a(P_2,Ca,Q_1)),bot_bo1766443648a_bool)))
     => ( ! [Z_7: x_a,S_2: state] :
            ( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Pa,Z_7),S_2))
           => hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(P_2,Z_7),S_2)) )
       => hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hoare_196563068iple_a(Pa,Ca,Q_1)),bot_bo1766443648a_bool))) ) ) ).

tff(fact_15_conseq1,axiom,
    ! [Pa: fun_st1506752259e_bool,Ga: fun_Ho1996104121e_bool,P_2: fun_st1506752259e_bool,Ca: com,Q_1: fun_st1506752259e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hoare_13225861_state(P_2,Ca,Q_1)),bot_bo1715400655e_bool)))
     => ( ! [Z_7: state,S_2: state] :
            ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Pa,Z_7),S_2))
           => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(P_2,Z_7),S_2)) )
       => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hoare_13225861_state(Pa,Ca,Q_1)),bot_bo1715400655e_bool))) ) ) ).

tff(fact_16_conseq12,axiom,
    ! [Q_1: fun_a_fun_state_bool,Pa: fun_a_fun_state_bool,Ga: fun_Ho287446294a_bool,P_2: fun_a_fun_state_bool,Ca: com,Q_3: fun_a_fun_state_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hoare_196563068iple_a(P_2,Ca,Q_3)),bot_bo1766443648a_bool)))
     => ( ! [Z_7: x_a,S_2: state] :
            ( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Pa,Z_7),S_2))
           => ! [S_3: state] :
                ( ! [Z_8: x_a] :
                    ( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(P_2,Z_8),S_2))
                   => hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_3,Z_8),S_3)) )
               => hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_1,Z_7),S_3)) ) )
       => hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hoare_196563068iple_a(Pa,Ca,Q_1)),bot_bo1766443648a_bool))) ) ) ).

tff(fact_17_conseq12,axiom,
    ! [Q_1: fun_st1506752259e_bool,Pa: fun_st1506752259e_bool,Ga: fun_Ho1996104121e_bool,P_2: fun_st1506752259e_bool,Ca: com,Q_3: fun_st1506752259e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hoare_13225861_state(P_2,Ca,Q_3)),bot_bo1715400655e_bool)))
     => ( ! [Z_7: state,S_2: state] :
            ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Pa,Z_7),S_2))
           => ! [S_3: state] :
                ( ! [Z_8: state] :
                    ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(P_2,Z_8),S_2))
                   => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Q_3,Z_8),S_3)) )
               => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Q_1,Z_7),S_3)) ) )
       => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hoare_13225861_state(Pa,Ca,Q_1)),bot_bo1715400655e_bool))) ) ) ).

tff(fact_18_insertE,axiom,
    ! [A_1: hoare_1544627872iple_a,Ba: 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,Ba),A)))
     => ( ( A_1 != Ba )
       => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),A)) ) ) ).

tff(fact_19_insertE,axiom,
    ! [A_1: nat,Ba: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),A)))
     => ( ( A_1 != Ba )
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A)) ) ) ).

tff(fact_20_insertE,axiom,
    ! [A_1: hoare_1875481847_state,Ba: 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,Ba),A)))
     => ( ( A_1 != Ba )
       => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),A)) ) ) ).

tff(fact_21_insertCI,axiom,
    ! [Ba: hoare_1544627872iple_a,A_1: hoare_1544627872iple_a,B: fun_Ho287446294a_bool] :
      ( ( ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),B))
       => ( A_1 = Ba ) )
     => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,Ba),B))) ) ).

tff(fact_22_insertCI,axiom,
    ! [Ba: nat,A_1: nat,B: fun_nat_bool] :
      ( ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),B))
       => ( A_1 = Ba ) )
     => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),B))) ) ).

tff(fact_23_insertCI,axiom,
    ! [Ba: hoare_1875481847_state,A_1: hoare_1875481847_state,B: fun_Ho1996104121e_bool] :
      ( ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),B))
       => ( A_1 = Ba ) )
     => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,Ba),B))) ) ).

tff(fact_24_emptyE,axiom,
    ! [A_1: hoare_1544627872iple_a] : ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),bot_bo1766443648a_bool)) ).

tff(fact_25_emptyE,axiom,
    ! [A_1: hoare_1875481847_state] : ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),bot_bo1715400655e_bool)) ).

tff(fact_26_emptyE,axiom,
    ! [A_1: nat] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),bot_bot_fun_nat_bool)) ).

tff(fact_27_singleton__conv2,axiom,
    ! [A_1: hoare_1544627872iple_a] : collec445968053iple_a(hAPP_H1472549281a_bool(fequal1057774495iple_a,A_1)) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),bot_bo1766443648a_bool) ).

tff(fact_28_singleton__conv2,axiom,
    ! [A_1: nat] : collect_nat(hAPP_n1699378549t_bool(fequal_nat,A_1)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool) ).

tff(fact_29_singleton__conv2,axiom,
    ! [A_1: fun_nat_bool] : collect_fun_nat_bool(hAPP_f103356543l_bool(fequal_fun_nat_bool,A_1)) = hAPP_f1246832597l_bool(hAPP_f633452666l_bool(insert_fun_nat_bool,A_1),bot_bo1701429464l_bool) ).

tff(fact_30_singleton__conv2,axiom,
    ! [A_1: hoare_1875481847_state] : collec1735317154_state(hAPP_H117293375e_bool(fequal391417144_state,A_1)) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),bot_bo1715400655e_bool) ).

tff(fact_31_singleton__conv,axiom,
    ! [A_1: hoare_1544627872iple_a] : 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_32_singleton__conv,axiom,
    ! [A_1: nat] : collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),A_1)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool) ).

tff(fact_33_singleton__conv,axiom,
    ! [A_1: fun_nat_bool] : collect_fun_nat_bool(hAPP_f103356543l_bool(hAPP_f643944041l_bool(cOMBC_1693257480l_bool,fequal_fun_nat_bool),A_1)) = hAPP_f1246832597l_bool(hAPP_f633452666l_bool(insert_fun_nat_bool,A_1),bot_bo1701429464l_bool) ).

tff(fact_34_singleton__conv,axiom,
    ! [A_1: hoare_1875481847_state] : 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_35_Collect__conv__if2,axiom,
    ! [Pa: fun_Ho287446294a_bool,A_1: hoare_1544627872iple_a] :
      ( ( hBOOL(hAPP_H2006434664a_bool(Pa,A_1))
       => ( collec445968053iple_a(hAPP_f909437487a_bool(hAPP_f1099051562a_bool(cOMBS_523298619l_bool,hAPP_f224574947l_bool(hAPP_f11333901l_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))
       => ( collec445968053iple_a(hAPP_f909437487a_bool(hAPP_f1099051562a_bool(cOMBS_523298619l_bool,hAPP_f224574947l_bool(hAPP_f11333901l_bool(cOMBB_1960865690iple_a,fconj),hAPP_H1472549281a_bool(fequal1057774495iple_a,A_1))),Pa)) = bot_bo1766443648a_bool ) ) ) ).

tff(fact_36_Collect__conv__if2,axiom,
    ! [Pa: fun_nat_bool,A_1: nat] :
      ( ( hBOOL(hAPP_nat_bool(Pa,A_1))
       => ( collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_n1699378549t_bool(fequal_nat,A_1))),Pa)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool) ) )
      & ( ~ hBOOL(hAPP_nat_bool(Pa,A_1))
       => ( collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_n1699378549t_bool(fequal_nat,A_1))),Pa)) = bot_bot_fun_nat_bool ) ) ) ).

tff(fact_37_Collect__conv__if2,axiom,
    ! [Pa: fun_fu425979586l_bool,A_1: fun_nat_bool] :
      ( ( hBOOL(hAPP_f54304608l_bool(Pa,A_1))
       => ( collect_fun_nat_bool(hAPP_f1246832597l_bool(hAPP_f1743029098l_bool(cOMBS_1187019125l_bool,hAPP_f1974927549l_bool(hAPP_f857404385l_bool(cOMBB_444170502t_bool,fconj),hAPP_f103356543l_bool(fequal_fun_nat_bool,A_1))),Pa)) = hAPP_f1246832597l_bool(hAPP_f633452666l_bool(insert_fun_nat_bool,A_1),bot_bo1701429464l_bool) ) )
      & ( ~ hBOOL(hAPP_f54304608l_bool(Pa,A_1))
       => ( collect_fun_nat_bool(hAPP_f1246832597l_bool(hAPP_f1743029098l_bool(cOMBS_1187019125l_bool,hAPP_f1974927549l_bool(hAPP_f857404385l_bool(cOMBB_444170502t_bool,fconj),hAPP_f103356543l_bool(fequal_fun_nat_bool,A_1))),Pa)) = bot_bo1701429464l_bool ) ) ) ).

tff(fact_38_Collect__conv__if2,axiom,
    ! [Pa: fun_Ho1996104121e_bool,A_1: hoare_1875481847_state] :
      ( ( hBOOL(hAPP_H1173941847e_bool(Pa,A_1))
       => ( collec1735317154_state(hAPP_f121055253e_bool(hAPP_f1300953505e_bool(cOMBS_896408300l_bool,hAPP_f932622461l_bool(hAPP_f1158435105l_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))
       => ( collec1735317154_state(hAPP_f121055253e_bool(hAPP_f1300953505e_bool(cOMBS_896408300l_bool,hAPP_f932622461l_bool(hAPP_f1158435105l_bool(cOMBB_242064253_state,fconj),hAPP_H117293375e_bool(fequal391417144_state,A_1))),Pa)) = bot_bo1715400655e_bool ) ) ) ).

tff(fact_39_Collect__conv__if,axiom,
    ! [Pa: fun_Ho287446294a_bool,A_1: hoare_1544627872iple_a] :
      ( ( hBOOL(hAPP_H2006434664a_bool(Pa,A_1))
       => ( collec445968053iple_a(hAPP_f909437487a_bool(hAPP_f1099051562a_bool(cOMBS_523298619l_bool,hAPP_f224574947l_bool(hAPP_f11333901l_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))
       => ( collec445968053iple_a(hAPP_f909437487a_bool(hAPP_f1099051562a_bool(cOMBS_523298619l_bool,hAPP_f224574947l_bool(hAPP_f11333901l_bool(cOMBB_1960865690iple_a,fconj),hAPP_H1472549281a_bool(hAPP_f1878404769a_bool(cOMBC_827727450a_bool,fequal1057774495iple_a),A_1))),Pa)) = bot_bo1766443648a_bool ) ) ) ).

tff(fact_40_Collect__conv__if,axiom,
    ! [Pa: fun_nat_bool,A_1: nat] :
      ( ( hBOOL(hAPP_nat_bool(Pa,A_1))
       => ( collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),A_1))),Pa)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool) ) )
      & ( ~ hBOOL(hAPP_nat_bool(Pa,A_1))
       => ( collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),A_1))),Pa)) = bot_bot_fun_nat_bool ) ) ) ).

tff(fact_41_Collect__conv__if,axiom,
    ! [Pa: fun_fu425979586l_bool,A_1: fun_nat_bool] :
      ( ( hBOOL(hAPP_f54304608l_bool(Pa,A_1))
       => ( collect_fun_nat_bool(hAPP_f1246832597l_bool(hAPP_f1743029098l_bool(cOMBS_1187019125l_bool,hAPP_f1974927549l_bool(hAPP_f857404385l_bool(cOMBB_444170502t_bool,fconj),hAPP_f103356543l_bool(hAPP_f643944041l_bool(cOMBC_1693257480l_bool,fequal_fun_nat_bool),A_1))),Pa)) = hAPP_f1246832597l_bool(hAPP_f633452666l_bool(insert_fun_nat_bool,A_1),bot_bo1701429464l_bool) ) )
      & ( ~ hBOOL(hAPP_f54304608l_bool(Pa,A_1))
       => ( collect_fun_nat_bool(hAPP_f1246832597l_bool(hAPP_f1743029098l_bool(cOMBS_1187019125l_bool,hAPP_f1974927549l_bool(hAPP_f857404385l_bool(cOMBB_444170502t_bool,fconj),hAPP_f103356543l_bool(hAPP_f643944041l_bool(cOMBC_1693257480l_bool,fequal_fun_nat_bool),A_1))),Pa)) = bot_bo1701429464l_bool ) ) ) ).

tff(fact_42_Collect__conv__if,axiom,
    ! [Pa: fun_Ho1996104121e_bool,A_1: hoare_1875481847_state] :
      ( ( hBOOL(hAPP_H1173941847e_bool(Pa,A_1))
       => ( collec1735317154_state(hAPP_f121055253e_bool(hAPP_f1300953505e_bool(cOMBS_896408300l_bool,hAPP_f932622461l_bool(hAPP_f1158435105l_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))
       => ( collec1735317154_state(hAPP_f121055253e_bool(hAPP_f1300953505e_bool(cOMBS_896408300l_bool,hAPP_f932622461l_bool(hAPP_f1158435105l_bool(cOMBB_242064253_state,fconj),hAPP_H117293375e_bool(hAPP_f2101312745e_bool(cOMBC_1972244470e_bool,fequal391417144_state),A_1))),Pa)) = bot_bo1715400655e_bool ) ) ) ).

tff(fact_43_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_44_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_45_equals0D,axiom,
    ! [A_1: nat,A: fun_nat_bool] :
      ( ( A = bot_bot_fun_nat_bool )
     => ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A)) ) ).

tff(fact_46_Collect__empty__eq,axiom,
    ! [Pa: fun_nat_bool] :
      ( ( collect_nat(Pa) = bot_bot_fun_nat_bool )
    <=> ! [X_2: nat] : ~ hBOOL(hAPP_nat_bool(Pa,X_2)) ) ).

tff(fact_47_Collect__empty__eq,axiom,
    ! [Pa: fun_Ho287446294a_bool] :
      ( ( collec445968053iple_a(Pa) = bot_bo1766443648a_bool )
    <=> ! [X_2: hoare_1544627872iple_a] : ~ hBOOL(hAPP_H2006434664a_bool(Pa,X_2)) ) ).

tff(fact_48_Collect__empty__eq,axiom,
    ! [Pa: fun_fu425979586l_bool] :
      ( ( collect_fun_nat_bool(Pa) = bot_bo1701429464l_bool )
    <=> ! [X_2: fun_nat_bool] : ~ hBOOL(hAPP_f54304608l_bool(Pa,X_2)) ) ).

tff(fact_49_Collect__empty__eq,axiom,
    ! [Pa: fun_Ho1996104121e_bool] :
      ( ( collec1735317154_state(Pa) = bot_bo1715400655e_bool )
    <=> ! [X_2: hoare_1875481847_state] : ~ hBOOL(hAPP_H1173941847e_bool(Pa,X_2)) ) ).

tff(fact_50_empty__iff,axiom,
    ! [Ca: hoare_1544627872iple_a] : ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),bot_bo1766443648a_bool)) ).

tff(fact_51_empty__iff,axiom,
    ! [Ca: hoare_1875481847_state] : ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,Ca),bot_bo1715400655e_bool)) ).

tff(fact_52_empty__iff,axiom,
    ! [Ca: nat] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),bot_bot_fun_nat_bool)) ).

tff(fact_53_empty__Collect__eq,axiom,
    ! [Pa: fun_nat_bool] :
      ( ( bot_bot_fun_nat_bool = collect_nat(Pa) )
    <=> ! [X_2: nat] : ~ hBOOL(hAPP_nat_bool(Pa,X_2)) ) ).

tff(fact_54_empty__Collect__eq,axiom,
    ! [Pa: fun_Ho287446294a_bool] :
      ( ( bot_bo1766443648a_bool = collec445968053iple_a(Pa) )
    <=> ! [X_2: hoare_1544627872iple_a] : ~ hBOOL(hAPP_H2006434664a_bool(Pa,X_2)) ) ).

tff(fact_55_empty__Collect__eq,axiom,
    ! [Pa: fun_fu425979586l_bool] :
      ( ( bot_bo1701429464l_bool = collect_fun_nat_bool(Pa) )
    <=> ! [X_2: fun_nat_bool] : ~ hBOOL(hAPP_f54304608l_bool(Pa,X_2)) ) ).

tff(fact_56_empty__Collect__eq,axiom,
    ! [Pa: fun_Ho1996104121e_bool] :
      ( ( bot_bo1715400655e_bool = collec1735317154_state(Pa) )
    <=> ! [X_2: hoare_1875481847_state] : ~ hBOOL(hAPP_H1173941847e_bool(Pa,X_2)) ) ).

tff(fact_57_ex__in__conv,axiom,
    ! [A: fun_Ho287446294a_bool] :
      ( ? [X_2: hoare_1544627872iple_a] : hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_2),A))
    <=> ( A != bot_bo1766443648a_bool ) ) ).

tff(fact_58_ex__in__conv,axiom,
    ! [A: fun_Ho1996104121e_bool] :
      ( ? [X_2: hoare_1875481847_state] : hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),A))
    <=> ( A != bot_bo1715400655e_bool ) ) ).

tff(fact_59_ex__in__conv,axiom,
    ! [A: fun_nat_bool] :
      ( ? [X_2: nat] : hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
    <=> ( A != bot_bot_fun_nat_bool ) ) ).

tff(fact_60_all__not__in__conv,axiom,
    ! [A: fun_Ho287446294a_bool] :
      ( ! [X_2: hoare_1544627872iple_a] : ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_2),A))
    <=> ( A = bot_bo1766443648a_bool ) ) ).

tff(fact_61_all__not__in__conv,axiom,
    ! [A: fun_Ho1996104121e_bool] :
      ( ! [X_2: hoare_1875481847_state] : ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),A))
    <=> ( A = bot_bo1715400655e_bool ) ) ).

tff(fact_62_all__not__in__conv,axiom,
    ! [A: fun_nat_bool] :
      ( ! [X_2: nat] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
    <=> ( A = bot_bot_fun_nat_bool ) ) ).

tff(fact_63_empty__def,axiom,
    bot_bot_fun_nat_bool = collect_nat(hAPP_b1013836512t_bool(cOMBK_bool_nat,fFalse)) ).

tff(fact_64_empty__def,axiom,
    bot_bo1766443648a_bool = collec445968053iple_a(hAPP_b704538998a_bool(cOMBK_329760839iple_a,fFalse)) ).

tff(fact_65_empty__def,axiom,
    bot_bo1701429464l_bool = collect_fun_nat_bool(hAPP_b1630757474l_bool(cOMBK_1994329625t_bool,fFalse)) ).

tff(fact_66_empty__def,axiom,
    bot_bo1715400655e_bool = collec1735317154_state(hAPP_b1906038105e_bool(cOMBK_2086958736_state,fFalse)) ).

tff(fact_67_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_68_insert__absorb,axiom,
    ! [A_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A))
     => ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),A) = A ) ) ).

tff(fact_69_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_70_insertI2,axiom,
    ! [Ba: hoare_1544627872iple_a,A_1: hoare_1544627872iple_a,B: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),B))
     => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,Ba),B))) ) ).

tff(fact_71_insertI2,axiom,
    ! [Ba: nat,A_1: nat,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),B))
     => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),B))) ) ).

tff(fact_72_insertI2,axiom,
    ! [Ba: hoare_1875481847_state,A_1: hoare_1875481847_state,B: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),B))
     => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,Ba),B))) ) ).

tff(fact_73_insert__ident,axiom,
    ! [B: fun_Ho287446294a_bool,X_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),A))
     => ( ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),B))
       => ( ( hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_1),A) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_1),B) )
        <=> ( A = B ) ) ) ) ).

tff(fact_74_insert__ident,axiom,
    ! [B: fun_nat_bool,X_1: nat,A: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B))
       => ( ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),B) )
        <=> ( A = B ) ) ) ) ).

tff(fact_75_insert__ident,axiom,
    ! [B: fun_Ho1996104121e_bool,X_1: hoare_1875481847_state,A: fun_Ho1996104121e_bool] :
      ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),A))
     => ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),B))
       => ( ( hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),A) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),B) )
        <=> ( A = B ) ) ) ) ).

tff(fact_76_insert__code,axiom,
    ! [Y_7: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,X_1: hoare_1544627872iple_a] :
      ( hBOOL(hAPP_H2006434664a_bool(hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,Y_7),A),X_1))
    <=> ( ( Y_7 = X_1 )
        | hBOOL(hAPP_H2006434664a_bool(A,X_1)) ) ) ).

tff(fact_77_insert__code,axiom,
    ! [Y_7: nat,A: fun_nat_bool,X_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Y_7),A),X_1))
    <=> ( ( Y_7 = X_1 )
        | hBOOL(hAPP_nat_bool(A,X_1)) ) ) ).

tff(fact_78_insert__code,axiom,
    ! [Y_7: hoare_1875481847_state,A: fun_Ho1996104121e_bool,X_1: hoare_1875481847_state] :
      ( hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,Y_7),A),X_1))
    <=> ( ( Y_7 = X_1 )
        | hBOOL(hAPP_H1173941847e_bool(A,X_1)) ) ) ).

tff(fact_79_insert__iff,axiom,
    ! [A_1: hoare_1544627872iple_a,Ba: 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,Ba),A)))
    <=> ( ( A_1 = Ba )
        | hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),A)) ) ) ).

tff(fact_80_insert__iff,axiom,
    ! [A_1: nat,Ba: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),A)))
    <=> ( ( A_1 = Ba )
        | hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A)) ) ) ).

tff(fact_81_insert__iff,axiom,
    ! [A_1: hoare_1875481847_state,Ba: 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,Ba),A)))
    <=> ( ( A_1 = Ba )
        | hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),A)) ) ) ).

tff(fact_82_insert__commute,axiom,
    ! [X_1: hoare_1544627872iple_a,Y_7: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_1),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,Y_7),A)) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,Y_7),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_1),A)) ).

tff(fact_83_insert__commute,axiom,
    ! [X_1: nat,Y_7: nat,A: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Y_7),A)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Y_7),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) ).

tff(fact_84_insert__commute,axiom,
    ! [X_1: hoare_1875481847_state,Y_7: hoare_1875481847_state,A: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,Y_7),A)) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,Y_7),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),A)) ).

tff(fact_85_insert__absorb2,axiom,
    ! [X_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_1),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_1),A)) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_1),A) ).

tff(fact_86_insert__absorb2,axiom,
    ! [X_1: nat,A: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A) ).

tff(fact_87_insert__absorb2,axiom,
    ! [X_1: hoare_1875481847_state,A: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),A)) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),A) ).

tff(fact_88_insert__Collect,axiom,
    ! [A_1: hoare_1544627872iple_a,Pa: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),collec445968053iple_a(Pa)) = collec445968053iple_a(hAPP_f909437487a_bool(hAPP_f1099051562a_bool(cOMBS_523298619l_bool,hAPP_f224574947l_bool(hAPP_f11333901l_bool(cOMBB_1960865690iple_a,fimplies),hAPP_f909437487a_bool(hAPP_f1524052231a_bool(cOMBB_1977449916iple_a,fNot),hAPP_H1472549281a_bool(hAPP_f1878404769a_bool(cOMBC_827727450a_bool,fequal1057774495iple_a),A_1)))),Pa)) ).

tff(fact_89_insert__Collect,axiom,
    ! [A_1: nat,Pa: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),collect_nat(Pa)) = collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fimplies),hAPP_f800510211t_bool(hAPP_f894608603t_bool(cOMBB_bool_bool_nat,fNot),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),A_1)))),Pa)) ).

tff(fact_90_insert__Collect,axiom,
    ! [A_1: fun_nat_bool,Pa: fun_fu425979586l_bool] : hAPP_f1246832597l_bool(hAPP_f633452666l_bool(insert_fun_nat_bool,A_1),collect_fun_nat_bool(Pa)) = collect_fun_nat_bool(hAPP_f1246832597l_bool(hAPP_f1743029098l_bool(cOMBS_1187019125l_bool,hAPP_f1974927549l_bool(hAPP_f857404385l_bool(cOMBB_444170502t_bool,fimplies),hAPP_f1246832597l_bool(hAPP_f1164891443l_bool(cOMBB_238756964t_bool,fNot),hAPP_f103356543l_bool(hAPP_f643944041l_bool(cOMBC_1693257480l_bool,fequal_fun_nat_bool),A_1)))),Pa)) ).

tff(fact_91_insert__Collect,axiom,
    ! [A_1: hoare_1875481847_state,Pa: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),collec1735317154_state(Pa)) = collec1735317154_state(hAPP_f121055253e_bool(hAPP_f1300953505e_bool(cOMBS_896408300l_bool,hAPP_f932622461l_bool(hAPP_f1158435105l_bool(cOMBB_242064253_state,fimplies),hAPP_f121055253e_bool(hAPP_f263911411e_bool(cOMBB_1424001755_state,fNot),hAPP_H117293375e_bool(hAPP_f2101312745e_bool(cOMBC_1972244470e_bool,fequal391417144_state),A_1)))),Pa)) ).

tff(fact_92_insert__compr,axiom,
    ! [A_1: hoare_1544627872iple_a,B: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),B) = collec445968053iple_a(hAPP_f909437487a_bool(hAPP_f1099051562a_bool(cOMBS_523298619l_bool,hAPP_f224574947l_bool(hAPP_f11333901l_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))) ).

tff(fact_93_insert__compr,axiom,
    ! [A_1: nat,B: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B) = collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fdisj),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),A_1))),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),B))) ).

tff(fact_94_insert__compr,axiom,
    ! [A_1: fun_nat_bool,B: fun_fu425979586l_bool] : hAPP_f1246832597l_bool(hAPP_f633452666l_bool(insert_fun_nat_bool,A_1),B) = collect_fun_nat_bool(hAPP_f1246832597l_bool(hAPP_f1743029098l_bool(cOMBS_1187019125l_bool,hAPP_f1974927549l_bool(hAPP_f857404385l_bool(cOMBB_444170502t_bool,fdisj),hAPP_f103356543l_bool(hAPP_f643944041l_bool(cOMBC_1693257480l_bool,fequal_fun_nat_bool),A_1))),hAPP_f1246832597l_bool(hAPP_f1443436725l_bool(cOMBC_1245412066l_bool,member_fun_nat_bool),B))) ).

tff(fact_95_insert__compr,axiom,
    ! [A_1: hoare_1875481847_state,B: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),B) = collec1735317154_state(hAPP_f121055253e_bool(hAPP_f1300953505e_bool(cOMBS_896408300l_bool,hAPP_f932622461l_bool(hAPP_f1158435105l_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))) ).

tff(fact_96_insertI1,axiom,
    ! [A_1: hoare_1544627872iple_a,B: 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))) ).

tff(fact_97_insertI1,axiom,
    ! [A_1: nat,B: fun_nat_bool] : hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B))) ).

tff(fact_98_insertI1,axiom,
    ! [A_1: hoare_1875481847_state,B: 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))) ).

tff(fact_99_insert__compr__raw,axiom,
    ! [X_2: hoare_1544627872iple_a,Xa: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_2),Xa) = collec445968053iple_a(hAPP_f909437487a_bool(hAPP_f1099051562a_bool(cOMBS_523298619l_bool,hAPP_f224574947l_bool(hAPP_f11333901l_bool(cOMBB_1960865690iple_a,fdisj),hAPP_H1472549281a_bool(hAPP_f1878404769a_bool(cOMBC_827727450a_bool,fequal1057774495iple_a),X_2))),hAPP_f909437487a_bool(hAPP_f482270129a_bool(cOMBC_1807206788l_bool,member1891733107iple_a),Xa))) ).

tff(fact_100_insert__compr__raw,axiom,
    ! [X_2: nat,Xa: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),Xa) = collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fdisj),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),X_2))),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),Xa))) ).

tff(fact_101_insert__compr__raw,axiom,
    ! [X_2: fun_nat_bool,Xa: fun_fu425979586l_bool] : hAPP_f1246832597l_bool(hAPP_f633452666l_bool(insert_fun_nat_bool,X_2),Xa) = collect_fun_nat_bool(hAPP_f1246832597l_bool(hAPP_f1743029098l_bool(cOMBS_1187019125l_bool,hAPP_f1974927549l_bool(hAPP_f857404385l_bool(cOMBB_444170502t_bool,fdisj),hAPP_f103356543l_bool(hAPP_f643944041l_bool(cOMBC_1693257480l_bool,fequal_fun_nat_bool),X_2))),hAPP_f1246832597l_bool(hAPP_f1443436725l_bool(cOMBC_1245412066l_bool,member_fun_nat_bool),Xa))) ).

tff(fact_102_insert__compr__raw,axiom,
    ! [X_2: hoare_1875481847_state,Xa: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),Xa) = collec1735317154_state(hAPP_f121055253e_bool(hAPP_f1300953505e_bool(cOMBS_896408300l_bool,hAPP_f932622461l_bool(hAPP_f1158435105l_bool(cOMBB_242064253_state,fdisj),hAPP_H117293375e_bool(hAPP_f2101312745e_bool(cOMBC_1972244470e_bool,fequal391417144_state),X_2))),hAPP_f121055253e_bool(hAPP_f1795624711e_bool(cOMBC_1099403170l_bool,member618553700_state),Xa))) ).

tff(fact_103_singleton__inject,axiom,
    ! [A_1: hoare_1544627872iple_a,Ba: hoare_1544627872iple_a] :
      ( ( hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),bot_bo1766443648a_bool) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,Ba),bot_bo1766443648a_bool) )
     => ( A_1 = Ba ) ) ).

tff(fact_104_singleton__inject,axiom,
    ! [A_1: nat,Ba: nat] :
      ( ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),bot_bot_fun_nat_bool) )
     => ( A_1 = Ba ) ) ).

tff(fact_105_singleton__inject,axiom,
    ! [A_1: hoare_1875481847_state,Ba: hoare_1875481847_state] :
      ( ( hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),bot_bo1715400655e_bool) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,Ba),bot_bo1715400655e_bool) )
     => ( A_1 = Ba ) ) ).

tff(fact_106_singletonE,axiom,
    ! [Ba: hoare_1544627872iple_a,A_1: hoare_1544627872iple_a] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ba),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),bot_bo1766443648a_bool)))
     => ( Ba = A_1 ) ) ).

tff(fact_107_singletonE,axiom,
    ! [Ba: nat,A_1: nat] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool)))
     => ( Ba = A_1 ) ) ).

tff(fact_108_singletonE,axiom,
    ! [Ba: hoare_1875481847_state,A_1: hoare_1875481847_state] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,Ba),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),bot_bo1715400655e_bool)))
     => ( Ba = A_1 ) ) ).

tff(fact_109_doubleton__eq__iff,axiom,
    ! [A_1: hoare_1544627872iple_a,Ba: hoare_1544627872iple_a,Ca: hoare_1544627872iple_a,D_3: hoare_1544627872iple_a] :
      ( ( hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,Ba),bot_bo1766443648a_bool)) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,Ca),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,D_3),bot_bo1766443648a_bool)) )
    <=> ( ( ( A_1 = Ca )
          & ( Ba = D_3 ) )
        | ( ( A_1 = D_3 )
          & ( Ba = Ca ) ) ) ) ).

tff(fact_110_doubleton__eq__iff,axiom,
    ! [A_1: nat,Ba: nat,Ca: nat,D_3: nat] :
      ( ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),bot_bot_fun_nat_bool)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ca),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,D_3),bot_bot_fun_nat_bool)) )
    <=> ( ( ( A_1 = Ca )
          & ( Ba = D_3 ) )
        | ( ( A_1 = D_3 )
          & ( Ba = Ca ) ) ) ) ).

tff(fact_111_doubleton__eq__iff,axiom,
    ! [A_1: hoare_1875481847_state,Ba: hoare_1875481847_state,Ca: hoare_1875481847_state,D_3: hoare_1875481847_state] :
      ( ( hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,Ba),bot_bo1715400655e_bool)) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,Ca),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,D_3),bot_bo1715400655e_bool)) )
    <=> ( ( ( A_1 = Ca )
          & ( Ba = D_3 ) )
        | ( ( A_1 = D_3 )
          & ( Ba = Ca ) ) ) ) ).

tff(fact_112_singleton__iff,axiom,
    ! [Ba: hoare_1544627872iple_a,A_1: hoare_1544627872iple_a] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ba),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),bot_bo1766443648a_bool)))
    <=> ( Ba = A_1 ) ) ).

tff(fact_113_singleton__iff,axiom,
    ! [Ba: nat,A_1: nat] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool)))
    <=> ( Ba = A_1 ) ) ).

tff(fact_114_singleton__iff,axiom,
    ! [Ba: hoare_1875481847_state,A_1: hoare_1875481847_state] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,Ba),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),bot_bo1715400655e_bool)))
    <=> ( Ba = A_1 ) ) ).

tff(fact_115_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_116_insert__not__empty,axiom,
    ! [A_1: nat,A: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),A) != bot_bot_fun_nat_bool ).

tff(fact_117_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_118_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_119_empty__not__insert,axiom,
    ! [A_1: nat,A: fun_nat_bool] : bot_bot_fun_nat_bool != hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),A) ).

tff(fact_120_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_121_the__elem__eq,axiom,
    ! [X_1: hoare_1544627872iple_a] : hAPP_f1853245639iple_a(the_el1614276927iple_a,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_1),bot_bo1766443648a_bool)) = X_1 ).

tff(fact_122_the__elem__eq,axiom,
    ! [X_1: nat] : hAPP_f22106695ol_nat(the_elem_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) = X_1 ).

tff(fact_123_the__elem__eq,axiom,
    ! [X_1: hoare_1875481847_state] : hAPP_f890249627_state(the_el1031305112_state,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),bot_bo1715400655e_bool)) = X_1 ).

tff(fact_124_bot__apply,axiom,
    ! [X_1: hoare_1544627872iple_a] :
      ( hBOOL(hAPP_H2006434664a_bool(bot_bo1766443648a_bool,X_1))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_125_bot__apply,axiom,
    ! [X_1: nat] :
      ( hBOOL(hAPP_nat_bool(bot_bot_fun_nat_bool,X_1))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_126_bot__apply,axiom,
    ! [X_1: hoare_1875481847_state] :
      ( hBOOL(hAPP_H1173941847e_bool(bot_bo1715400655e_bool,X_1))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_127_bot__fun__def,axiom,
    ! [X_2: hoare_1544627872iple_a] :
      ( hBOOL(hAPP_H2006434664a_bool(bot_bo1766443648a_bool,X_2))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_128_bot__fun__def,axiom,
    ! [X_2: nat] :
      ( hBOOL(hAPP_nat_bool(bot_bot_fun_nat_bool,X_2))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_129_bot__fun__def,axiom,
    ! [X_2: hoare_1875481847_state] :
      ( hBOOL(hAPP_H1173941847e_bool(bot_bo1715400655e_bool,X_2))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_130_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,hoare_196563068iple_a(Pa,skip,Pa)),bot_bo1766443648a_bool))) ).

tff(fact_131_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,hoare_13225861_state(Pa,skip,Pa)),bot_bo1715400655e_bool))) ).

tff(fact_132_Comp,axiom,
    ! [D_3: com,R_1: fun_a_fun_state_bool,Ga: fun_Ho287446294a_bool,Pa: fun_a_fun_state_bool,Ca: com,Q_1: fun_a_fun_state_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hoare_196563068iple_a(Pa,Ca,Q_1)),bot_bo1766443648a_bool)))
     => ( hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hoare_196563068iple_a(Q_1,D_3,R_1)),bot_bo1766443648a_bool)))
       => hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hoare_196563068iple_a(Pa,semi(Ca,D_3),R_1)),bot_bo1766443648a_bool))) ) ) ).

tff(fact_133_Comp,axiom,
    ! [D_3: com,R_1: fun_st1506752259e_bool,Ga: fun_Ho1996104121e_bool,Pa: fun_st1506752259e_bool,Ca: com,Q_1: fun_st1506752259e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hoare_13225861_state(Pa,Ca,Q_1)),bot_bo1715400655e_bool)))
     => ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hoare_13225861_state(Q_1,D_3,R_1)),bot_bo1715400655e_bool)))
       => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hoare_13225861_state(Pa,semi(Ca,D_3),R_1)),bot_bo1715400655e_bool))) ) ) ).

tff(fact_134_triple_Oexhaust,axiom,
    ! [Y_7: hoare_1875481847_state] :
      ~ ! [Fun1: fun_st1506752259e_bool,Com_2: com,Fun2: fun_st1506752259e_bool] : Y_7 != hoare_13225861_state(Fun1,Com_2,Fun2) ).

tff(fact_135_triple_Oexhaust,axiom,
    ! [Y_7: hoare_1544627872iple_a] :
      ~ ! [Fun1: fun_a_fun_state_bool,Com_2: com,Fun2: fun_a_fun_state_bool] : Y_7 != hoare_196563068iple_a(Fun1,Com_2,Fun2) ).

tff(fact_136_Set_Oset__insert,axiom,
    ! [X_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),A))
     => ~ ! [B_15: fun_Ho287446294a_bool] :
            ( ( A = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_1),B_15) )
           => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),B_15)) ) ) ).

tff(fact_137_Set_Oset__insert,axiom,
    ! [X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => ~ ! [B_15: fun_nat_bool] :
            ( ( A = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),B_15) )
           => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B_15)) ) ) ).

tff(fact_138_Set_Oset__insert,axiom,
    ! [X_1: hoare_1875481847_state,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),A))
     => ~ ! [B_15: fun_Ho1996104121e_bool] :
            ( ( A = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),B_15) )
           => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),B_15)) ) ) ).

tff(fact_139_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_15: fun_Ho287446294a_bool] :
          ( ( A = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),B_15) )
          & ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),B_15)) ) ) ).

tff(fact_140_mk__disjoint__insert,axiom,
    ! [A_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A))
     => ? [B_15: fun_nat_bool] :
          ( ( A = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B_15) )
          & ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),B_15)) ) ) ).

tff(fact_141_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_15: fun_Ho1996104121e_bool] :
          ( ( A = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),B_15) )
          & ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),B_15)) ) ) ).

tff(fact_142_equals0I,axiom,
    ! [A: fun_Ho287446294a_bool] :
      ( ! [Y_11: hoare_1544627872iple_a] : ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Y_11),A))
     => ( A = bot_bo1766443648a_bool ) ) ).

tff(fact_143_equals0I,axiom,
    ! [A: fun_Ho1996104121e_bool] :
      ( ! [Y_11: hoare_1875481847_state] : ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,Y_11),A))
     => ( A = bot_bo1715400655e_bool ) ) ).

tff(fact_144_equals0I,axiom,
    ! [A: fun_nat_bool] :
      ( ! [Y_11: nat] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Y_11),A))
     => ( A = bot_bot_fun_nat_bool ) ) ).

tff(fact_145_conseq,axiom,
    ! [Q_1: fun_a_fun_state_bool,Ga: fun_Ho287446294a_bool,Ca: com,Pa: fun_a_fun_state_bool] :
      ( ! [Z_7: x_a,S_2: state] :
          ( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Pa,Z_7),S_2))
         => ? [P_1: fun_a_fun_state_bool,Q_2: fun_a_fun_state_bool] :
              ( hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hoare_196563068iple_a(P_1,Ca,Q_2)),bot_bo1766443648a_bool)))
              & ! [S_3: state] :
                  ( ! [Z_8: x_a] :
                      ( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(P_1,Z_8),S_2))
                     => hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_2,Z_8),S_3)) )
                 => hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_1,Z_7),S_3)) ) ) )
     => hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hoare_196563068iple_a(Pa,Ca,Q_1)),bot_bo1766443648a_bool))) ) ).

tff(fact_146_conseq,axiom,
    ! [Q_1: fun_st1506752259e_bool,Ga: fun_Ho1996104121e_bool,Ca: com,Pa: fun_st1506752259e_bool] :
      ( ! [Z_7: state,S_2: state] :
          ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Pa,Z_7),S_2))
         => ? [P_1: fun_st1506752259e_bool,Q_2: fun_st1506752259e_bool] :
              ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hoare_13225861_state(P_1,Ca,Q_2)),bot_bo1715400655e_bool)))
              & ! [S_3: state] :
                  ( ! [Z_8: state] :
                      ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(P_1,Z_8),S_2))
                     => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Q_2,Z_8),S_3)) )
                 => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Q_1,Z_7),S_3)) ) ) )
     => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hoare_13225861_state(Pa,Ca,Q_1)),bot_bo1715400655e_bool))) ) ).

tff(fact_147_com_Osimps_I13_J,axiom,
    ! [Com1_2: com,Com2_2: com] : semi(Com1_2,Com2_2) != skip ).

tff(fact_148_com_Osimps_I12_J,axiom,
    ! [Com1_2: com,Com2_2: com] : skip != semi(Com1_2,Com2_2) ).

tff(fact_149_the__elem__def,axiom,
    ! [X_35: fun_Ho287446294a_bool] : hAPP_f1853245639iple_a(the_el1614276927iple_a,X_35) = hAPP_f1853245639iple_a(the_Ho924576593iple_a,hAPP_f1726109544a_bool(hAPP_f1948079815a_bool(cOMBB_1474127043iple_a,hAPP_f1876759039l_bool(fequal1843628375a_bool,X_35)),hAPP_f693853672a_bool(hAPP_f1399402097a_bool(cOMBC_894585277a_bool,insert1051021594iple_a),bot_bo1766443648a_bool))) ).

tff(fact_150_the__elem__def,axiom,
    ! [X_35: fun_nat_bool] : hAPP_f22106695ol_nat(the_elem_nat,X_35) = hAPP_f22106695ol_nat(the_nat,hAPP_f158894502t_bool(hAPP_f1777703707t_bool(cOMBB_955900739ol_nat,hAPP_f103356543l_bool(fequal_fun_nat_bool,X_35)),hAPP_f1570313510t_bool(hAPP_f688831301t_bool(cOMBC_178881787t_bool,insert_nat),bot_bot_fun_nat_bool))) ).

tff(fact_151_the__elem__def,axiom,
    ! [X_35: fun_Ho1996104121e_bool] : hAPP_f890249627_state(the_el1031305112_state,X_35) = hAPP_f890249627_state(the_Ho1017792262_state,hAPP_f169892419e_bool(hAPP_f1983876009e_bool(cOMBB_1380679615_state,hAPP_f1873026899l_bool(fequal259518520e_bool,X_35)),hAPP_f56162711e_bool(hAPP_f392546175e_bool(cOMBC_483615668e_bool,insert694999549_state),bot_bo1715400655e_bool))) ).

tff(fact_152_com_Osimps_I3_J,axiom,
    ! [Com1_1: com,Com2_1: com,Com1: com,Com2: com] :
      ( ( semi(Com1_1,Com2_1) = semi(Com1,Com2) )
    <=> ( ( Com1_1 = Com1 )
        & ( Com2_1 = Com2 ) ) ) ).

tff(fact_153_nonempty__iff,axiom,
    ! [A: fun_Ho287446294a_bool] :
      ( ( A != bot_bo1766443648a_bool )
    <=> ? [X_2: hoare_1544627872iple_a,B_15: fun_Ho287446294a_bool] :
          ( ( A = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_2),B_15) )
          & ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_2),B_15)) ) ) ).

tff(fact_154_nonempty__iff,axiom,
    ! [A: fun_nat_bool] :
      ( ( A != bot_bot_fun_nat_bool )
    <=> ? [X_2: nat,B_15: fun_nat_bool] :
          ( ( A = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),B_15) )
          & ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),B_15)) ) ) ).

tff(fact_155_nonempty__iff,axiom,
    ! [A: fun_Ho1996104121e_bool] :
      ( ( A != bot_bo1715400655e_bool )
    <=> ? [X_2: hoare_1875481847_state,B_15: fun_Ho1996104121e_bool] :
          ( ( A = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),B_15) )
          & ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),B_15)) ) ) ).

tff(fact_156_bot__empty__eq,axiom,
    ! [X_2: hoare_1544627872iple_a] :
      ( hBOOL(hAPP_H2006434664a_bool(bot_bo1766443648a_bool,X_2))
    <=> hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_2),bot_bo1766443648a_bool)) ) ).

tff(fact_157_bot__empty__eq,axiom,
    ! [X_2: hoare_1875481847_state] :
      ( hBOOL(hAPP_H1173941847e_bool(bot_bo1715400655e_bool,X_2))
    <=> hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),bot_bo1715400655e_bool)) ) ).

tff(fact_158_bot__empty__eq,axiom,
    ! [X_2: nat] :
      ( hBOOL(hAPP_nat_bool(bot_bot_fun_nat_bool,X_2))
    <=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),bot_bot_fun_nat_bool)) ) ).

tff(fact_159_Ass,axiom,
    ! [Ga: fun_Ho287446294a_bool,Pa: fun_a_fun_state_bool,X_35: vname,A_1: fun_state_nat] : hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hoare_196563068iple_a(hAPP_f100967412e_bool(hAPP_f1460451647e_bool(cOMBC_524597097e_bool,hAPP_f1406200875e_bool(hAPP_f635443597e_bool(cOMBB_2139825703bool_a,cOMBB_844853809_state),Pa)),hAPP_f871651461_state(hAPP_f851239890_state(cOMBS_777315357_state,hAPP_v365393659_state(hAPP_f1542232213_state(cOMBC_1193272608_state,update),X_35)),A_1)),ass(X_35,A_1),Pa)),bot_bo1766443648a_bool))) ).

tff(fact_160_Ass,axiom,
    ! [Ga: fun_Ho1996104121e_bool,Pa: fun_st1506752259e_bool,X_35: vname,A_1: fun_state_nat] : hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hoare_13225861_state(hAPP_f1838002347e_bool(hAPP_f1706273077e_bool(cOMBC_867582640e_bool,hAPP_f289738463e_bool(hAPP_f1151843515e_bool(cOMBB_1941618714_state,cOMBB_844853809_state),Pa)),hAPP_f871651461_state(hAPP_f851239890_state(cOMBS_777315357_state,hAPP_v365393659_state(hAPP_f1542232213_state(cOMBC_1193272608_state,update),X_35)),A_1)),ass(X_35,A_1),Pa)),bot_bo1715400655e_bool))) ).

tff(fact_161_image__constant__conv,axiom,
    ! [Ca: hoare_1544627872iple_a,A: fun_nat_bool] :
      ( ( ( A = bot_bot_fun_nat_bool )
       => ( image_1960063643iple_a(hAPP_H325842184iple_a(cOMBK_830162040_a_nat,Ca),A) = bot_bo1766443648a_bool ) )
      & ( ( A != bot_bot_fun_nat_bool )
       => ( image_1960063643iple_a(hAPP_H325842184iple_a(cOMBK_830162040_a_nat,Ca),A) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,Ca),bot_bo1766443648a_bool) ) ) ) ).

tff(fact_162_image__constant__conv,axiom,
    ! [Ca: hoare_1544627872iple_a,A: fun_Ho1996104121e_bool] :
      ( ( ( A = bot_bo1715400655e_bool )
       => ( image_299889782iple_a(hAPP_H180680677iple_a(cOMBK_875269129_state,Ca),A) = bot_bo1766443648a_bool ) )
      & ( ( A != bot_bo1715400655e_bool )
       => ( image_299889782iple_a(hAPP_H180680677iple_a(cOMBK_875269129_state,Ca),A) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,Ca),bot_bo1766443648a_bool) ) ) ) ).

tff(fact_163_image__constant__conv,axiom,
    ! [Ca: nat,A: fun_Ho287446294a_bool] :
      ( ( ( A = bot_bo1766443648a_bool )
       => ( image_1667308271_a_nat(hAPP_n1125129074_a_nat(cOMBK_1122917412iple_a,Ca),A) = bot_bot_fun_nat_bool ) )
      & ( ( A != bot_bo1766443648a_bool )
       => ( image_1667308271_a_nat(hAPP_n1125129074_a_nat(cOMBK_1122917412iple_a,Ca),A) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ca),bot_bot_fun_nat_bool) ) ) ) ).

tff(fact_164_image__constant__conv,axiom,
    ! [Ca: hoare_1875481847_state,A: fun_Ho287446294a_bool] :
      ( ( ( A = bot_bo1766443648a_bool )
       => ( image_456807122_state(hAPP_H296470442_state(cOMBK_718351789iple_a,Ca),A) = bot_bo1715400655e_bool ) )
      & ( ( A != bot_bo1766443648a_bool )
       => ( image_456807122_state(hAPP_H296470442_state(cOMBK_718351789iple_a,Ca),A) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,Ca),bot_bo1715400655e_bool) ) ) ) ).

tff(fact_165_image__constant__conv,axiom,
    ! [Ca: nat,A: fun_nat_bool] :
      ( ( ( A = bot_bot_fun_nat_bool )
       => ( image_nat_nat(hAPP_nat_fun_nat_nat(cOMBK_nat_nat,Ca),A) = bot_bot_fun_nat_bool ) )
      & ( ( A != bot_bot_fun_nat_bool )
       => ( image_nat_nat(hAPP_nat_fun_nat_nat(cOMBK_nat_nat,Ca),A) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ca),bot_bot_fun_nat_bool) ) ) ) ).

tff(fact_166_image__constant__conv,axiom,
    ! [Ca: hoare_1875481847_state,A: fun_Ho1996104121e_bool] :
      ( ( ( A = bot_bo1715400655e_bool )
       => ( image_436941729_state(hAPP_H743221337_state(cOMBK_477315498_state,Ca),A) = bot_bo1715400655e_bool ) )
      & ( ( A != bot_bo1715400655e_bool )
       => ( image_436941729_state(hAPP_H743221337_state(cOMBK_477315498_state,Ca),A) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,Ca),bot_bo1715400655e_bool) ) ) ) ).

tff(fact_167_image__constant__conv,axiom,
    ! [Ca: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( ( ( A = bot_bo1766443648a_bool )
       => ( image_1784540805iple_a(hAPP_H1988213490iple_a(cOMBK_1156021774iple_a,Ca),A) = bot_bo1766443648a_bool ) )
      & ( ( A != bot_bo1766443648a_bool )
       => ( image_1784540805iple_a(hAPP_H1988213490iple_a(cOMBK_1156021774iple_a,Ca),A) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,Ca),bot_bo1766443648a_bool) ) ) ) ).

tff(fact_168_image__constant,axiom,
    ! [Ca: hoare_1544627872iple_a,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => ( image_1960063643iple_a(hAPP_H325842184iple_a(cOMBK_830162040_a_nat,Ca),A) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,Ca),bot_bo1766443648a_bool) ) ) ).

tff(fact_169_image__constant,axiom,
    ! [Ca: hoare_1544627872iple_a,X_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),A))
     => ( image_1784540805iple_a(hAPP_H1988213490iple_a(cOMBK_1156021774iple_a,Ca),A) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,Ca),bot_bo1766443648a_bool) ) ) ).

tff(fact_170_image__constant,axiom,
    ! [Ca: hoare_1544627872iple_a,X_1: hoare_1875481847_state,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),A))
     => ( image_299889782iple_a(hAPP_H180680677iple_a(cOMBK_875269129_state,Ca),A) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,Ca),bot_bo1766443648a_bool) ) ) ).

tff(fact_171_image__constant,axiom,
    ! [Ca: nat,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => ( image_nat_nat(hAPP_nat_fun_nat_nat(cOMBK_nat_nat,Ca),A) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ca),bot_bot_fun_nat_bool) ) ) ).

tff(fact_172_image__constant,axiom,
    ! [Ca: nat,X_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),A))
     => ( image_1667308271_a_nat(hAPP_n1125129074_a_nat(cOMBK_1122917412iple_a,Ca),A) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ca),bot_bot_fun_nat_bool) ) ) ).

tff(fact_173_image__constant,axiom,
    ! [Ca: nat,X_1: hoare_1875481847_state,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),A))
     => ( image_1722141408te_nat(hAPP_n747866297te_nat(cOMBK_809880179_state,Ca),A) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ca),bot_bot_fun_nat_bool) ) ) ).

tff(fact_174_image__constant,axiom,
    ! [Ca: hoare_1875481847_state,X_1: hoare_1875481847_state,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),A))
     => ( image_436941729_state(hAPP_H743221337_state(cOMBK_477315498_state,Ca),A) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,Ca),bot_bo1715400655e_bool) ) ) ).

tff(fact_175_image__constant,axiom,
    ! [Ca: hoare_1875481847_state,X_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),A))
     => ( image_456807122_state(hAPP_H296470442_state(cOMBK_718351789iple_a,Ca),A) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,Ca),bot_bo1715400655e_bool) ) ) ).

tff(fact_176_image__eqI,axiom,
    ! [A: fun_Ho1996104121e_bool,Ba: hoare_1875481847_state,F_1: fun_Ho1455141843_state,X_1: hoare_1875481847_state] :
      ( ( Ba = hAPP_H350317809_state(F_1,X_1) )
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),A))
       => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,Ba),image_436941729_state(F_1,A))) ) ) ).

tff(fact_177_image__eqI,axiom,
    ! [A: fun_nat_bool,Ba: hoare_1875481847_state,F_1: fun_na1612816666_state,X_1: nat] :
      ( ( Ba = hAPP_n2134291948_state(F_1,X_1) )
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,Ba),image_681421628_state(F_1,A))) ) ) ).

tff(fact_178_image__eqI,axiom,
    ! [A: fun_Ho287446294a_bool,Ba: hoare_1875481847_state,F_1: fun_Ho1528294576_state,X_1: hoare_1544627872iple_a] :
      ( ( Ba = hAPP_H1928101250_state(F_1,X_1) )
     => ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),A))
       => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,Ba),image_456807122_state(F_1,A))) ) ) ).

tff(fact_179_image__eqI,axiom,
    ! [A: fun_nat_bool,Ba: nat,F_1: fun_nat_nat,X_1: nat] :
      ( ( Ba = hAPP_nat_nat(F_1,X_1) )
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),image_nat_nat(F_1,A))) ) ) ).

tff(fact_180_image__eqI,axiom,
    ! [A: fun_Ho287446294a_bool,Ba: nat,F_1: fun_Ho753038759_a_nat,X_1: hoare_1544627872iple_a] :
      ( ( Ba = hAPP_H63220799_a_nat(F_1,X_1) )
     => ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),A))
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),image_1667308271_a_nat(F_1,A))) ) ) ).

tff(fact_181_image__eqI,axiom,
    ! [A: fun_Ho1996104121e_bool,Ba: nat,F_1: fun_Ho1284378692te_nat,X_1: hoare_1875481847_state] :
      ( ( Ba = hAPP_H1027528080te_nat(F_1,X_1) )
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),A))
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),image_1722141408te_nat(F_1,A))) ) ) ).

tff(fact_182_image__eqI,axiom,
    ! [A: fun_Ho287446294a_bool,Ba: hoare_1544627872iple_a,F_1: fun_Ho1275796541iple_a,X_1: hoare_1544627872iple_a] :
      ( ( Ba = hAPP_H9461717iple_a(F_1,X_1) )
     => ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),A))
       => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ba),image_1784540805iple_a(F_1,A))) ) ) ).

tff(fact_183_image__eqI,axiom,
    ! [A: fun_Ho1996104121e_bool,Ba: hoare_1544627872iple_a,F_1: fun_Ho1202345178iple_a,X_1: hoare_1875481847_state] :
      ( ( Ba = hAPP_H1771183910iple_a(F_1,X_1) )
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),A))
       => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ba),image_299889782iple_a(F_1,A))) ) ) ).

tff(fact_184_image__eqI,axiom,
    ! [A: fun_nat_bool,Ba: hoare_1544627872iple_a,F_1: fun_na98436179iple_a,X_1: nat] :
      ( ( Ba = hAPP_n355976171iple_a(F_1,X_1) )
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ba),image_1960063643iple_a(F_1,A))) ) ) ).

tff(fact_185_image__ident,axiom,
    ! [Y_27: fun_nat_bool] : image_nat_nat(cOMBI_nat,Y_27) = Y_27 ).

tff(fact_186_image__ident,axiom,
    ! [Y_27: fun_Ho1996104121e_bool] : image_436941729_state(cOMBI_227521594_state,Y_27) = Y_27 ).

tff(fact_187_image__ident,axiom,
    ! [Y_27: fun_Ho287446294a_bool] : image_1784540805iple_a(cOMBI_1396158237iple_a,Y_27) = Y_27 ).

tff(fact_188_com_Osimps_I1_J,axiom,
    ! [Vname: vname,Fun_1: fun_state_nat,Vname_1: vname,Fun: fun_state_nat] :
      ( ( ass(Vname,Fun_1) = ass(Vname_1,Fun) )
    <=> ( ( Vname = Vname_1 )
        & ( Fun_1 = Fun ) ) ) ).

tff(fact_189_rev__image__eqI,axiom,
    ! [Ba: hoare_1875481847_state,F_1: fun_Ho1455141843_state,X_1: hoare_1875481847_state,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),A))
     => ( ( Ba = hAPP_H350317809_state(F_1,X_1) )
       => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,Ba),image_436941729_state(F_1,A))) ) ) ).

tff(fact_190_rev__image__eqI,axiom,
    ! [Ba: hoare_1875481847_state,F_1: fun_na1612816666_state,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => ( ( Ba = hAPP_n2134291948_state(F_1,X_1) )
       => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,Ba),image_681421628_state(F_1,A))) ) ) ).

tff(fact_191_rev__image__eqI,axiom,
    ! [Ba: hoare_1875481847_state,F_1: fun_Ho1528294576_state,X_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),A))
     => ( ( Ba = hAPP_H1928101250_state(F_1,X_1) )
       => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,Ba),image_456807122_state(F_1,A))) ) ) ).

tff(fact_192_rev__image__eqI,axiom,
    ! [Ba: nat,F_1: fun_nat_nat,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => ( ( Ba = hAPP_nat_nat(F_1,X_1) )
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),image_nat_nat(F_1,A))) ) ) ).

tff(fact_193_rev__image__eqI,axiom,
    ! [Ba: nat,F_1: fun_Ho753038759_a_nat,X_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),A))
     => ( ( Ba = hAPP_H63220799_a_nat(F_1,X_1) )
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),image_1667308271_a_nat(F_1,A))) ) ) ).

tff(fact_194_rev__image__eqI,axiom,
    ! [Ba: nat,F_1: fun_Ho1284378692te_nat,X_1: hoare_1875481847_state,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),A))
     => ( ( Ba = hAPP_H1027528080te_nat(F_1,X_1) )
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),image_1722141408te_nat(F_1,A))) ) ) ).

tff(fact_195_rev__image__eqI,axiom,
    ! [Ba: hoare_1544627872iple_a,F_1: fun_Ho1275796541iple_a,X_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),A))
     => ( ( Ba = hAPP_H9461717iple_a(F_1,X_1) )
       => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ba),image_1784540805iple_a(F_1,A))) ) ) ).

tff(fact_196_rev__image__eqI,axiom,
    ! [Ba: hoare_1544627872iple_a,F_1: fun_Ho1202345178iple_a,X_1: hoare_1875481847_state,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),A))
     => ( ( Ba = hAPP_H1771183910iple_a(F_1,X_1) )
       => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ba),image_299889782iple_a(F_1,A))) ) ) ).

tff(fact_197_rev__image__eqI,axiom,
    ! [Ba: hoare_1544627872iple_a,F_1: fun_na98436179iple_a,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => ( ( Ba = hAPP_n355976171iple_a(F_1,X_1) )
       => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ba),image_1960063643iple_a(F_1,A))) ) ) ).

tff(fact_198_imageI,axiom,
    ! [F_1: fun_Ho1455141843_state,X_1: hoare_1875481847_state,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),A))
     => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,hAPP_H350317809_state(F_1,X_1)),image_436941729_state(F_1,A))) ) ).

tff(fact_199_imageI,axiom,
    ! [F_1: fun_na1612816666_state,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,hAPP_n2134291948_state(F_1,X_1)),image_681421628_state(F_1,A))) ) ).

tff(fact_200_imageI,axiom,
    ! [F_1: fun_Ho1528294576_state,X_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),A))
     => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,hAPP_H1928101250_state(F_1,X_1)),image_456807122_state(F_1,A))) ) ).

tff(fact_201_imageI,axiom,
    ! [F_1: fun_nat_nat,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_nat_nat(F_1,X_1)),image_nat_nat(F_1,A))) ) ).

tff(fact_202_imageI,axiom,
    ! [F_1: fun_Ho753038759_a_nat,X_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),A))
     => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_H63220799_a_nat(F_1,X_1)),image_1667308271_a_nat(F_1,A))) ) ).

tff(fact_203_imageI,axiom,
    ! [F_1: fun_Ho1284378692te_nat,X_1: hoare_1875481847_state,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),A))
     => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_H1027528080te_nat(F_1,X_1)),image_1722141408te_nat(F_1,A))) ) ).

tff(fact_204_imageI,axiom,
    ! [F_1: fun_Ho1275796541iple_a,X_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),A))
     => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,hAPP_H9461717iple_a(F_1,X_1)),image_1784540805iple_a(F_1,A))) ) ).

tff(fact_205_imageI,axiom,
    ! [F_1: fun_Ho1202345178iple_a,X_1: hoare_1875481847_state,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),A))
     => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,hAPP_H1771183910iple_a(F_1,X_1)),image_299889782iple_a(F_1,A))) ) ).

tff(fact_206_imageI,axiom,
    ! [F_1: fun_na98436179iple_a,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,hAPP_n355976171iple_a(F_1,X_1)),image_1960063643iple_a(F_1,A))) ) ).

tff(fact_207_image__iff,axiom,
    ! [Z_2: hoare_1875481847_state,F_1: fun_Ho1455141843_state,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,Z_2),image_436941729_state(F_1,A)))
    <=> ? [X_2: hoare_1875481847_state] :
          ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),A))
          & ( Z_2 = hAPP_H350317809_state(F_1,X_2) ) ) ) ).

tff(fact_208_image__iff,axiom,
    ! [Z_2: hoare_1875481847_state,F_1: fun_Ho1528294576_state,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,Z_2),image_456807122_state(F_1,A)))
    <=> ? [X_2: hoare_1544627872iple_a] :
          ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_2),A))
          & ( Z_2 = hAPP_H1928101250_state(F_1,X_2) ) ) ) ).

tff(fact_209_image__iff,axiom,
    ! [Z_2: nat,F_1: fun_nat_nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Z_2),image_nat_nat(F_1,A)))
    <=> ? [X_2: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
          & ( Z_2 = hAPP_nat_nat(F_1,X_2) ) ) ) ).

tff(fact_210_image__iff,axiom,
    ! [Z_2: nat,F_1: fun_Ho753038759_a_nat,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Z_2),image_1667308271_a_nat(F_1,A)))
    <=> ? [X_2: hoare_1544627872iple_a] :
          ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_2),A))
          & ( Z_2 = hAPP_H63220799_a_nat(F_1,X_2) ) ) ) ).

tff(fact_211_image__iff,axiom,
    ! [Z_2: hoare_1544627872iple_a,F_1: fun_Ho1275796541iple_a,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Z_2),image_1784540805iple_a(F_1,A)))
    <=> ? [X_2: hoare_1544627872iple_a] :
          ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_2),A))
          & ( Z_2 = hAPP_H9461717iple_a(F_1,X_2) ) ) ) ).

tff(fact_212_image__iff,axiom,
    ! [Z_2: hoare_1544627872iple_a,F_1: fun_Ho1202345178iple_a,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Z_2),image_299889782iple_a(F_1,A)))
    <=> ? [X_2: hoare_1875481847_state] :
          ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),A))
          & ( Z_2 = hAPP_H1771183910iple_a(F_1,X_2) ) ) ) ).

tff(fact_213_image__iff,axiom,
    ! [Z_2: hoare_1544627872iple_a,F_1: fun_na98436179iple_a,A: fun_nat_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Z_2),image_1960063643iple_a(F_1,A)))
    <=> ? [X_2: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
          & ( Z_2 = hAPP_n355976171iple_a(F_1,X_2) ) ) ) ).

tff(fact_214_com_Osimps_I24_J,axiom,
    ! [Vname: vname,Fun_1: fun_state_nat,Com1: com,Com2: com] : ass(Vname,Fun_1) != semi(Com1,Com2) ).

tff(fact_215_com_Osimps_I25_J,axiom,
    ! [Com1: com,Com2: com,Vname: vname,Fun_1: fun_state_nat] : semi(Com1,Com2) != ass(Vname,Fun_1) ).

tff(fact_216_com_Osimps_I8_J,axiom,
    ! [Vname_1: vname,Fun: fun_state_nat] : skip != ass(Vname_1,Fun) ).

tff(fact_217_com_Osimps_I9_J,axiom,
    ! [Vname_1: vname,Fun: fun_state_nat] : ass(Vname_1,Fun) != skip ).

tff(fact_218_image__is__empty,axiom,
    ! [F_1: fun_Ho1202345178iple_a,A: fun_Ho1996104121e_bool] :
      ( ( image_299889782iple_a(F_1,A) = bot_bo1766443648a_bool )
    <=> ( A = bot_bo1715400655e_bool ) ) ).

tff(fact_219_image__is__empty,axiom,
    ! [F_1: fun_na98436179iple_a,A: fun_nat_bool] :
      ( ( image_1960063643iple_a(F_1,A) = bot_bo1766443648a_bool )
    <=> ( A = bot_bot_fun_nat_bool ) ) ).

tff(fact_220_image__is__empty,axiom,
    ! [F_1: fun_Ho1528294576_state,A: fun_Ho287446294a_bool] :
      ( ( image_456807122_state(F_1,A) = bot_bo1715400655e_bool )
    <=> ( A = bot_bo1766443648a_bool ) ) ).

tff(fact_221_image__is__empty,axiom,
    ! [F_1: fun_Ho753038759_a_nat,A: fun_Ho287446294a_bool] :
      ( ( image_1667308271_a_nat(F_1,A) = bot_bot_fun_nat_bool )
    <=> ( A = bot_bo1766443648a_bool ) ) ).

tff(fact_222_image__empty,axiom,
    ! [F_1: fun_Ho1528294576_state] : image_456807122_state(F_1,bot_bo1766443648a_bool) = bot_bo1715400655e_bool ).

tff(fact_223_image__empty,axiom,
    ! [F_1: fun_Ho753038759_a_nat] : image_1667308271_a_nat(F_1,bot_bo1766443648a_bool) = bot_bot_fun_nat_bool ).

tff(fact_224_image__empty,axiom,
    ! [F_1: fun_Ho1202345178iple_a] : image_299889782iple_a(F_1,bot_bo1715400655e_bool) = bot_bo1766443648a_bool ).

tff(fact_225_image__empty,axiom,
    ! [F_1: fun_na98436179iple_a] : image_1960063643iple_a(F_1,bot_bot_fun_nat_bool) = bot_bo1766443648a_bool ).

tff(fact_226_empty__is__image,axiom,
    ! [F_1: fun_Ho1202345178iple_a,A: fun_Ho1996104121e_bool] :
      ( ( bot_bo1766443648a_bool = image_299889782iple_a(F_1,A) )
    <=> ( A = bot_bo1715400655e_bool ) ) ).

tff(fact_227_empty__is__image,axiom,
    ! [F_1: fun_na98436179iple_a,A: fun_nat_bool] :
      ( ( bot_bo1766443648a_bool = image_1960063643iple_a(F_1,A) )
    <=> ( A = bot_bot_fun_nat_bool ) ) ).

tff(fact_228_empty__is__image,axiom,
    ! [F_1: fun_Ho1528294576_state,A: fun_Ho287446294a_bool] :
      ( ( bot_bo1715400655e_bool = image_456807122_state(F_1,A) )
    <=> ( A = bot_bo1766443648a_bool ) ) ).

tff(fact_229_empty__is__image,axiom,
    ! [F_1: fun_Ho753038759_a_nat,A: fun_Ho287446294a_bool] :
      ( ( bot_bot_fun_nat_bool = image_1667308271_a_nat(F_1,A) )
    <=> ( A = bot_bo1766443648a_bool ) ) ).

tff(fact_230_mem__def,axiom,
    ! [X_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),A))
    <=> hBOOL(hAPP_H2006434664a_bool(A,X_1)) ) ).

tff(fact_231_mem__def,axiom,
    ! [X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
    <=> hBOOL(hAPP_nat_bool(A,X_1)) ) ).

tff(fact_232_Collect__def,axiom,
    ! [Pa: fun_Ho287446294a_bool] : collec445968053iple_a(Pa) = Pa ).

tff(fact_233_Collect__def,axiom,
    ! [Pa: fun_nat_bool] : collect_nat(Pa) = Pa ).

tff(fact_234_insert__image,axiom,
    ! [F_1: fun_Ho1275796541iple_a,X_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),A))
     => ( hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hAPP_H9461717iple_a(F_1,X_1)),image_1784540805iple_a(F_1,A)) = image_1784540805iple_a(F_1,A) ) ) ).

tff(fact_235_insert__image,axiom,
    ! [F_1: fun_na98436179iple_a,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => ( hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hAPP_n355976171iple_a(F_1,X_1)),image_1960063643iple_a(F_1,A)) = image_1960063643iple_a(F_1,A) ) ) ).

tff(fact_236_image__insert,axiom,
    ! [F_1: fun_Ho1528294576_state,A_1: hoare_1544627872iple_a,B: fun_Ho287446294a_bool] : image_456807122_state(F_1,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),B)) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hAPP_H1928101250_state(F_1,A_1)),image_456807122_state(F_1,B)) ).

tff(fact_237_image__insert,axiom,
    ! [F_1: fun_Ho753038759_a_nat,A_1: hoare_1544627872iple_a,B: fun_Ho287446294a_bool] : image_1667308271_a_nat(F_1,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),B)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,hAPP_H63220799_a_nat(F_1,A_1)),image_1667308271_a_nat(F_1,B)) ).

tff(fact_238_image__insert,axiom,
    ! [F_1: fun_Ho1202345178iple_a,A_1: hoare_1875481847_state,B: fun_Ho1996104121e_bool] : image_299889782iple_a(F_1,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),B)) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hAPP_H1771183910iple_a(F_1,A_1)),image_299889782iple_a(F_1,B)) ).

tff(fact_239_image__insert,axiom,
    ! [F_1: fun_na98436179iple_a,A_1: nat,B: fun_nat_bool] : image_1960063643iple_a(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B)) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hAPP_n355976171iple_a(F_1,A_1)),image_1960063643iple_a(F_1,B)) ).

tff(fact_240_the__sym__eq__trivial,axiom,
    ! [X_1: hoare_1544627872iple_a] : hAPP_f1853245639iple_a(the_Ho924576593iple_a,hAPP_H1472549281a_bool(fequal1057774495iple_a,X_1)) = X_1 ).

tff(fact_241_the__eq__trivial,axiom,
    ! [A_1: hoare_1544627872iple_a] : hAPP_f1853245639iple_a(the_Ho924576593iple_a,hAPP_H1472549281a_bool(hAPP_f1878404769a_bool(cOMBC_827727450a_bool,fequal1057774495iple_a),A_1)) = A_1 ).

tff(fact_242_If__def,axiom,
    ! [X_1: hoare_1544627872iple_a,Y_7: hoare_1544627872iple_a,Pa: bool] :
      ( ( hBOOL(Pa)
       => ( X_1 = hAPP_f1853245639iple_a(the_Ho924576593iple_a,hAPP_f909437487a_bool(hAPP_f1099051562a_bool(cOMBS_523298619l_bool,hAPP_f224574947l_bool(hAPP_f11333901l_bool(cOMBB_1960865690iple_a,fconj),hAPP_f909437487a_bool(hAPP_f1524052231a_bool(cOMBB_1977449916iple_a,hAPP_b589554111l_bool(fimplies,Pa)),hAPP_H1472549281a_bool(hAPP_f1878404769a_bool(cOMBC_827727450a_bool,fequal1057774495iple_a),X_1)))),hAPP_f909437487a_bool(hAPP_f1524052231a_bool(cOMBB_1977449916iple_a,hAPP_b589554111l_bool(fimplies,hAPP_bool_bool(fNot,Pa))),hAPP_H1472549281a_bool(hAPP_f1878404769a_bool(cOMBC_827727450a_bool,fequal1057774495iple_a),Y_7)))) ) )
      & ( ~ hBOOL(Pa)
       => ( Y_7 = hAPP_f1853245639iple_a(the_Ho924576593iple_a,hAPP_f909437487a_bool(hAPP_f1099051562a_bool(cOMBS_523298619l_bool,hAPP_f224574947l_bool(hAPP_f11333901l_bool(cOMBB_1960865690iple_a,fconj),hAPP_f909437487a_bool(hAPP_f1524052231a_bool(cOMBB_1977449916iple_a,hAPP_b589554111l_bool(fimplies,Pa)),hAPP_H1472549281a_bool(hAPP_f1878404769a_bool(cOMBC_827727450a_bool,fequal1057774495iple_a),X_1)))),hAPP_f909437487a_bool(hAPP_f1524052231a_bool(cOMBB_1977449916iple_a,hAPP_b589554111l_bool(fimplies,hAPP_bool_bool(fNot,Pa))),hAPP_H1472549281a_bool(hAPP_f1878404769a_bool(cOMBC_827727450a_bool,fequal1057774495iple_a),Y_7)))) ) ) ) ).

tff(fact_243_fold1Set__sing,axiom,
    ! [F_1: fun_Ho784854669_state,A_1: hoare_1875481847_state,Ba: hoare_1875481847_state] :
      ( hBOOL(hAPP_H1173941847e_bool(finite1016865319_state(F_1,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),bot_bo1715400655e_bool)),Ba))
    <=> ( A_1 = Ba ) ) ).

tff(fact_244_fold1Set__sing,axiom,
    ! [F_1: fun_nat_fun_nat_nat,A_1: nat,Ba: nat] :
      ( hBOOL(hAPP_nat_bool(finite_fold1Set_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool)),Ba))
    <=> ( A_1 = Ba ) ) ).

tff(fact_245_fold1Set__sing,axiom,
    ! [F_1: fun_Ho115022426iple_a,A_1: hoare_1544627872iple_a,Ba: hoare_1544627872iple_a] :
      ( hBOOL(hAPP_H2006434664a_bool(finite1715754352iple_a(F_1,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),bot_bo1766443648a_bool)),Ba))
    <=> ( A_1 = Ba ) ) ).

tff(fact_246_the__equality,axiom,
    ! [Pa: fun_Ho287446294a_bool,A_1: hoare_1544627872iple_a] :
      ( hBOOL(hAPP_H2006434664a_bool(Pa,A_1))
     => ( ! [X_2: hoare_1544627872iple_a] :
            ( hBOOL(hAPP_H2006434664a_bool(Pa,X_2))
           => ( X_2 = A_1 ) )
       => ( hAPP_f1853245639iple_a(the_Ho924576593iple_a,Pa) = A_1 ) ) ) ).

tff(fact_247_folding__one_Osingleton,axiom,
    ! [X_1: 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_1),bot_bo1715400655e_bool)) = X_1 ) ) ).

tff(fact_248_folding__one_Osingleton,axiom,
    ! [X_1: nat,F_1: fun_nat_fun_nat_nat,F: fun_fun_nat_bool_nat] :
      ( hBOOL(finite988810631ne_nat(F_1,F))
     => ( hAPP_f22106695ol_nat(F,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) = X_1 ) ) ).

tff(fact_249_folding__one_Osingleton,axiom,
    ! [X_1: 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_1),bot_bo1766443648a_bool)) = X_1 ) ) ).

tff(fact_250_hoare__derivs_OLocal,axiom,
    ! [A_1: fun_state_nat,Ga: fun_Ho1996104121e_bool,Pa: fun_st1506752259e_bool,Ca: com,Q_1: fun_st1506752259e_bool,X_35: loc,S_5: state] :
      ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hoare_13225861_state(Pa,Ca,hAPP_f1838002347e_bool(hAPP_f1706273077e_bool(cOMBC_867582640e_bool,hAPP_f289738463e_bool(hAPP_f1151843515e_bool(cOMBB_1941618714_state,cOMBB_844853809_state),Q_1)),hAPP_n1547241352_state(hAPP_f1848060885_state(cOMBC_1777403949_state,hAPP_v365393659_state(hAPP_f1542232213_state(cOMBC_1193272608_state,update),loc_1(X_35))),getlocs(S_5,X_35))))),bot_bo1715400655e_bool)))
     => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hoare_13225861_state(hAPP_f817621513e_bool(hAPP_f512427579e_bool(cOMBB_572666224_state,hAPP_f644196280e_bool(cOMBS_1378840469l_bool,hAPP_f1259673775l_bool(hAPP_f1561913689l_bool(cOMBB_188601460_state,fconj),hAPP_s1806633685e_bool(fequal_state,S_5)))),hAPP_f1838002347e_bool(hAPP_f1706273077e_bool(cOMBC_867582640e_bool,hAPP_f289738463e_bool(hAPP_f1151843515e_bool(cOMBB_1941618714_state,cOMBB_844853809_state),Pa)),hAPP_f871651461_state(hAPP_f851239890_state(cOMBS_777315357_state,hAPP_v365393659_state(hAPP_f1542232213_state(cOMBC_1193272608_state,update),loc_1(X_35))),A_1))),local(X_35,A_1,Ca),Q_1)),bot_bo1715400655e_bool))) ) ).

tff(fact_251_hoare__derivs_OLocal,axiom,
    ! [A_1: fun_state_nat,Ga: fun_Ho287446294a_bool,Pa: fun_a_fun_state_bool,Ca: com,Q_1: fun_a_fun_state_bool,X_35: loc,S_5: state] :
      ( hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hoare_196563068iple_a(Pa,Ca,hAPP_f100967412e_bool(hAPP_f1460451647e_bool(cOMBC_524597097e_bool,hAPP_f1406200875e_bool(hAPP_f635443597e_bool(cOMBB_2139825703bool_a,cOMBB_844853809_state),Q_1)),hAPP_n1547241352_state(hAPP_f1848060885_state(cOMBC_1777403949_state,hAPP_v365393659_state(hAPP_f1542232213_state(cOMBC_1193272608_state,update),loc_1(X_35))),getlocs(S_5,X_35))))),bot_bo1766443648a_bool)))
     => hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hoare_196563068iple_a(hAPP_f769584981e_bool(hAPP_f1159960589e_bool(cOMBB_237455441bool_a,hAPP_f644196280e_bool(cOMBS_1378840469l_bool,hAPP_f1259673775l_bool(hAPP_f1561913689l_bool(cOMBB_188601460_state,fconj),hAPP_s1806633685e_bool(fequal_state,S_5)))),hAPP_f100967412e_bool(hAPP_f1460451647e_bool(cOMBC_524597097e_bool,hAPP_f1406200875e_bool(hAPP_f635443597e_bool(cOMBB_2139825703bool_a,cOMBB_844853809_state),Pa)),hAPP_f871651461_state(hAPP_f851239890_state(cOMBS_777315357_state,hAPP_v365393659_state(hAPP_f1542232213_state(cOMBC_1193272608_state,update),loc_1(X_35))),A_1))),local(X_35,A_1,Ca),Q_1)),bot_bo1766443648a_bool))) ) ).

tff(fact_252_vname_Osimps_I2_J,axiom,
    ! [Loc_3: loc,Loc_2: loc] :
      ( ( loc_1(Loc_3) = loc_1(Loc_2) )
    <=> ( Loc_3 = Loc_2 ) ) ).

tff(fact_253_com_Osimps_I2_J,axiom,
    ! [Loc_3: loc,Fun_1: fun_state_nat,Com_1: com,Loc_2: loc,Fun: fun_state_nat,Com: com] :
      ( ( local(Loc_3,Fun_1,Com_1) = local(Loc_2,Fun,Com) )
    <=> ( ( Loc_3 = Loc_2 )
        & ( Fun_1 = Fun )
        & ( Com_1 = Com ) ) ) ).

tff(fact_254_com_Osimps_I34_J,axiom,
    ! [Loc_3: loc,Fun_1: fun_state_nat,Com_1: com,Com1: com,Com2: com] : local(Loc_3,Fun_1,Com_1) != semi(Com1,Com2) ).

tff(fact_255_com_Osimps_I35_J,axiom,
    ! [Com1: com,Com2: com,Loc_3: loc,Fun_1: fun_state_nat,Com_1: com] : semi(Com1,Com2) != local(Loc_3,Fun_1,Com_1) ).

tff(fact_256_com_Osimps_I23_J,axiom,
    ! [Loc_2: loc,Fun: fun_state_nat,Com: com,Vname: vname,Fun_1: fun_state_nat] : local(Loc_2,Fun,Com) != ass(Vname,Fun_1) ).

tff(fact_257_com_Osimps_I22_J,axiom,
    ! [Vname: vname,Fun_1: fun_state_nat,Loc_2: loc,Fun: fun_state_nat,Com: com] : ass(Vname,Fun_1) != local(Loc_2,Fun,Com) ).

tff(fact_258_com_Osimps_I11_J,axiom,
    ! [Loc_2: loc,Fun: fun_state_nat,Com: com] : local(Loc_2,Fun,Com) != skip ).

tff(fact_259_com_Osimps_I10_J,axiom,
    ! [Loc_2: loc,Fun: fun_state_nat,Com: com] : skip != local(Loc_2,Fun,Com) ).

tff(fact_260_empty__fold1SetE,axiom,
    ! [F_1: fun_Ho784854669_state,X_1: hoare_1875481847_state] : ~ hBOOL(hAPP_H1173941847e_bool(finite1016865319_state(F_1,bot_bo1715400655e_bool),X_1)) ).

tff(fact_261_empty__fold1SetE,axiom,
    ! [F_1: fun_nat_fun_nat_nat,X_1: nat] : ~ hBOOL(hAPP_nat_bool(finite_fold1Set_nat(F_1,bot_bot_fun_nat_bool),X_1)) ).

tff(fact_262_empty__fold1SetE,axiom,
    ! [F_1: fun_Ho115022426iple_a,X_1: hoare_1544627872iple_a] : ~ hBOOL(hAPP_H2006434664a_bool(finite1715754352iple_a(F_1,bot_bo1766443648a_bool),X_1)) ).

tff(fact_263_fold1Set__nonempty,axiom,
    ! [F_1: fun_Ho784854669_state,A: fun_Ho1996104121e_bool,X_1: hoare_1875481847_state] :
      ( hBOOL(hAPP_H1173941847e_bool(finite1016865319_state(F_1,A),X_1))
     => ( A != bot_bo1715400655e_bool ) ) ).

tff(fact_264_fold1Set__nonempty,axiom,
    ! [F_1: fun_nat_fun_nat_nat,A: fun_nat_bool,X_1: nat] :
      ( hBOOL(hAPP_nat_bool(finite_fold1Set_nat(F_1,A),X_1))
     => ( A != bot_bot_fun_nat_bool ) ) ).

tff(fact_265_fold1Set__nonempty,axiom,
    ! [F_1: fun_Ho115022426iple_a,A: fun_Ho287446294a_bool,X_1: hoare_1544627872iple_a] :
      ( hBOOL(hAPP_H2006434664a_bool(finite1715754352iple_a(F_1,A),X_1))
     => ( A != bot_bo1766443648a_bool ) ) ).

tff(fact_266_theI,axiom,
    ! [Pa: fun_Ho287446294a_bool,A_1: hoare_1544627872iple_a] :
      ( hBOOL(hAPP_H2006434664a_bool(Pa,A_1))
     => ( ! [X_2: hoare_1544627872iple_a] :
            ( hBOOL(hAPP_H2006434664a_bool(Pa,X_2))
           => ( X_2 = A_1 ) )
       => hBOOL(hAPP_H2006434664a_bool(Pa,hAPP_f1853245639iple_a(the_Ho924576593iple_a,Pa))) ) ) ).

tff(fact_267_the1__equality,axiom,
    ! [A_1: hoare_1544627872iple_a,Pa: fun_Ho287446294a_bool] :
      ( ? [X_2: hoare_1544627872iple_a] :
          ( hBOOL(hAPP_H2006434664a_bool(Pa,X_2))
          & ! [Y_11: hoare_1544627872iple_a] :
              ( hBOOL(hAPP_H2006434664a_bool(Pa,Y_11))
             => ( Y_11 = X_2 ) ) )
     => ( hBOOL(hAPP_H2006434664a_bool(Pa,A_1))
       => ( hAPP_f1853245639iple_a(the_Ho924576593iple_a,Pa) = A_1 ) ) ) ).

tff(fact_268_theI_H,axiom,
    ! [Pa: fun_Ho287446294a_bool] :
      ( ? [X_2: hoare_1544627872iple_a] :
          ( hBOOL(hAPP_H2006434664a_bool(Pa,X_2))
          & ! [Y_11: hoare_1544627872iple_a] :
              ( hBOOL(hAPP_H2006434664a_bool(Pa,Y_11))
             => ( Y_11 = X_2 ) ) )
     => hBOOL(hAPP_H2006434664a_bool(Pa,hAPP_f1853245639iple_a(the_Ho924576593iple_a,Pa))) ) ).

tff(fact_269_evalc_OLocal,axiom,
    ! [Ca: com,S0_1: state,Y_27: loc,A_1: fun_state_nat,S1_2: state] :
      ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(Ca),hAPP_nat_state(hAPP_v594194232_state(hAPP_s712361723_state(update,S0_1),loc_1(Y_27)),hAPP_state_nat(A_1,S0_1))),S1_2))
     => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(local(Y_27,A_1,Ca)),S0_1),hAPP_nat_state(hAPP_v594194232_state(hAPP_s712361723_state(update,S1_2),loc_1(Y_27)),getlocs(S0_1,Y_27)))) ) ).

tff(fact_270_evaln_OLocal,axiom,
    ! [Ca: com,S0_1: state,Y_27: loc,A_1: fun_state_nat,N_3: nat,S1_2: state] :
      ( hBOOL(evaln(Ca,hAPP_nat_state(hAPP_v594194232_state(hAPP_s712361723_state(update,S0_1),loc_1(Y_27)),hAPP_state_nat(A_1,S0_1)),N_3,S1_2))
     => hBOOL(evaln(local(Y_27,A_1,Ca),S0_1,N_3,hAPP_nat_state(hAPP_v594194232_state(hAPP_s712361723_state(update,S1_2),loc_1(Y_27)),getlocs(S0_1,Y_27)))) ) ).

tff(fact_271_fold1Set_Ointros,axiom,
    ! [F_1: fun_Ho784854669_state,A_1: hoare_1875481847_state,A: fun_Ho1996104121e_bool,X_1: hoare_1875481847_state] :
      ( hBOOL(hAPP_H1173941847e_bool(finite1721989730_state(F_1,A_1,A),X_1))
     => ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),A))
       => hBOOL(hAPP_H1173941847e_bool(finite1016865319_state(F_1,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),A)),X_1)) ) ) ).

tff(fact_272_fold1Set_Ointros,axiom,
    ! [F_1: fun_nat_fun_nat_nat,A_1: nat,A: fun_nat_bool,X_1: nat] :
      ( hBOOL(hAPP_nat_bool(finite929467206at_nat(F_1,A_1,A),X_1))
     => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A))
       => hBOOL(hAPP_nat_bool(finite_fold1Set_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),A)),X_1)) ) ) ).

tff(fact_273_fold1Set_Ointros,axiom,
    ! [F_1: fun_Ho115022426iple_a,A_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,X_1: hoare_1544627872iple_a] :
      ( hBOOL(hAPP_H2006434664a_bool(finite1341991622iple_a(F_1,A_1,A),X_1))
     => ( ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),A))
       => hBOOL(hAPP_H2006434664a_bool(finite1715754352iple_a(F_1,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),A)),X_1)) ) ) ).

tff(fact_274_evaln_OSemi,axiom,
    ! [C1: com,S2: state,C0: com,S0: state,N_2: nat,S1: state] :
      ( hBOOL(evaln(C0,S0,N_2,S1))
     => ( hBOOL(evaln(C1,S1,N_2,S2))
       => hBOOL(evaln(semi(C0,C1),S0,N_2,S2)) ) ) ).

tff(fact_275_evaln_OSkip,axiom,
    ! [S_1: state,N_2: nat] : hBOOL(evaln(skip,S_1,N_2,S_1)) ).

tff(fact_276_evaln__elim__cases_I1_J,axiom,
    ! [S_1: state,N_2: nat,T: state] :
      ( hBOOL(evaln(skip,S_1,N_2,T))
     => ( T = S_1 ) ) ).

tff(fact_277_evalc_OSemi,axiom,
    ! [C1: com,S2: state,C0: com,S0: state,S1: state] :
      ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C0),S0),S1))
     => ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C1),S1),S2))
       => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(semi(C0,C1)),S0),S2)) ) ) ).

tff(fact_278_evalc_OSkip,axiom,
    ! [S_1: state] : hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(skip),S_1),S_1)) ).

tff(fact_279_evalc__elim__cases_I1_J,axiom,
    ! [S_1: state,T: state] :
      ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(skip),S_1),T))
     => ( T = S_1 ) ) ).

tff(fact_280_evaln_OAssign,axiom,
    ! [X_35: vname,A_1: fun_state_nat,S_4: state,N_3: nat] : hBOOL(evaln(ass(X_35,A_1),S_4,N_3,hAPP_nat_state(hAPP_v594194232_state(hAPP_s712361723_state(update,S_4),X_35),hAPP_state_nat(A_1,S_4)))) ).

tff(fact_281_evaln__elim__cases_I2_J,axiom,
    ! [X_35: vname,A_1: fun_state_nat,S_4: state,N_3: nat,T_1: state] :
      ( hBOOL(evaln(ass(X_35,A_1),S_4,N_3,T_1))
     => ( T_1 = hAPP_nat_state(hAPP_v594194232_state(hAPP_s712361723_state(update,S_4),X_35),hAPP_state_nat(A_1,S_4)) ) ) ).

tff(fact_282_evalc_OAssign,axiom,
    ! [X_35: vname,A_1: fun_state_nat,S_4: state] : hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(ass(X_35,A_1)),S_4),hAPP_nat_state(hAPP_v594194232_state(hAPP_s712361723_state(update,S_4),X_35),hAPP_state_nat(A_1,S_4)))) ).

tff(fact_283_evalc__elim__cases_I2_J,axiom,
    ! [X_35: vname,A_1: fun_state_nat,S_4: state,T_1: state] :
      ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(ass(X_35,A_1)),S_4),T_1))
     => ( T_1 = hAPP_nat_state(hAPP_v594194232_state(hAPP_s712361723_state(update,S_4),X_35),hAPP_state_nat(A_1,S_4)) ) ) ).

tff(fact_284_eval__eq,axiom,
    ! [Ca: com,S_4: state,T_1: state] :
      ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(Ca),S_4),T_1))
    <=> ? [N_1: nat] : hBOOL(evaln(Ca,S_4,N_1,T_1)) ) ).

tff(fact_285_com__det,axiom,
    ! [U_1: state,C_2: com,S_1: state,T: state] :
      ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C_2),S_1),T))
     => ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C_2),S_1),U_1))
       => ( U_1 = T ) ) ) ).

tff(fact_286_evaln__evalc,axiom,
    ! [C_2: com,S_1: state,N_2: nat,T: state] :
      ( hBOOL(evaln(C_2,S_1,N_2,T))
     => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C_2),S_1),T)) ) ).

tff(fact_287_empty__fold__graphE,axiom,
    ! [F_1: fun_Ho115022426iple_a,Z_2: hoare_1544627872iple_a,X_1: hoare_1544627872iple_a] :
      ( hBOOL(hAPP_H2006434664a_bool(finite1341991622iple_a(F_1,Z_2,bot_bo1766443648a_bool),X_1))
     => ( X_1 = Z_2 ) ) ).

tff(fact_288_fold__graph_OemptyI,axiom,
    ! [F_1: fun_Ho115022426iple_a,Z_2: hoare_1544627872iple_a] : hBOOL(hAPP_H2006434664a_bool(finite1341991622iple_a(F_1,Z_2,bot_bo1766443648a_bool),Z_2)) ).

tff(fact_289_fold__graph_OinsertI,axiom,
    ! [F_1: fun_Ho115022426iple_a,Z_2: hoare_1544627872iple_a,Y_7: hoare_1544627872iple_a,X_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),A))
     => ( hBOOL(hAPP_H2006434664a_bool(finite1341991622iple_a(F_1,Z_2,A),Y_7))
       => hBOOL(hAPP_H2006434664a_bool(finite1341991622iple_a(F_1,Z_2,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_1),A)),hAPP_H9461717iple_a(hAPP_H1988213490iple_a(F_1,X_1),Y_7))) ) ) ).

tff(fact_290_evalc__elim__cases_I3_J,axiom,
    ! [Y_27: loc,A_1: fun_state_nat,Ca: com,S_4: state,T_1: state] :
      ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(local(Y_27,A_1,Ca)),S_4),T_1))
     => ~ ! [S1_1: state] :
            ( ( T_1 = hAPP_nat_state(hAPP_v594194232_state(hAPP_s712361723_state(update,S1_1),loc_1(Y_27)),getlocs(S_4,Y_27)) )
           => ~ hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(Ca),hAPP_nat_state(hAPP_v594194232_state(hAPP_s712361723_state(update,S_4),loc_1(Y_27)),hAPP_state_nat(A_1,S_4))),S1_1)) ) ) ).

tff(fact_291_evaln__elim__cases_I3_J,axiom,
    ! [Y_27: loc,A_1: fun_state_nat,Ca: com,S_4: state,N_3: nat,T_1: state] :
      ( hBOOL(evaln(local(Y_27,A_1,Ca),S_4,N_3,T_1))
     => ~ ! [S1_1: state] :
            ( ( T_1 = hAPP_nat_state(hAPP_v594194232_state(hAPP_s712361723_state(update,S1_1),loc_1(Y_27)),getlocs(S_4,Y_27)) )
           => ~ hBOOL(evaln(Ca,hAPP_nat_state(hAPP_v594194232_state(hAPP_s712361723_state(update,S_4),loc_1(Y_27)),hAPP_state_nat(A_1,S_4)),N_3,S1_1)) ) ) ).

tff(fact_292_evalc__elim__cases_I4_J,axiom,
    ! [C1: com,C2: com,S_1: state,T: state] :
      ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(semi(C1,C2)),S_1),T))
     => ~ ! [S1_1: state] :
            ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C1),S_1),S1_1))
           => ~ hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C2),S1_1),T)) ) ) ).

tff(fact_293_evaln__elim__cases_I4_J,axiom,
    ! [C1: com,C2: com,S_1: state,N_2: nat,T: state] :
      ( hBOOL(evaln(semi(C1,C2),S_1,N_2,T))
     => ~ ! [S1_1: state] :
            ( hBOOL(evaln(C1,S_1,N_2,S1_1))
           => ~ hBOOL(evaln(C2,S1_1,N_2,T)) ) ) ).

tff(fact_294_insert__fold1SetE,axiom,
    ! [F_1: fun_nat_fun_nat_nat,A_1: nat,X_35: fun_nat_bool,X_1: nat] :
      ( hBOOL(hAPP_nat_bool(finite_fold1Set_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),X_35)),X_1))
     => ~ ! [A_7: nat,A_19: fun_nat_bool] :
            ( ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),X_35) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_7),A_19) )
           => ( hBOOL(hAPP_nat_bool(finite929467206at_nat(F_1,A_7,A_19),X_1))
             => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_7),A_19)) ) ) ) ).

tff(fact_295_insert__fold1SetE,axiom,
    ! [F_1: fun_Ho784854669_state,A_1: hoare_1875481847_state,X_35: fun_Ho1996104121e_bool,X_1: hoare_1875481847_state] :
      ( hBOOL(hAPP_H1173941847e_bool(finite1016865319_state(F_1,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),X_35)),X_1))
     => ~ ! [A_7: hoare_1875481847_state,A_19: fun_Ho1996104121e_bool] :
            ( ( hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),X_35) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_7),A_19) )
           => ( hBOOL(hAPP_H1173941847e_bool(finite1721989730_state(F_1,A_7,A_19),X_1))
             => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_7),A_19)) ) ) ) ).

tff(fact_296_insert__fold1SetE,axiom,
    ! [F_1: fun_Ho115022426iple_a,A_1: hoare_1544627872iple_a,X_35: fun_Ho287446294a_bool,X_1: hoare_1544627872iple_a] :
      ( hBOOL(hAPP_H2006434664a_bool(finite1715754352iple_a(F_1,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),X_35)),X_1))
     => ~ ! [A_7: hoare_1544627872iple_a,A_19: fun_Ho287446294a_bool] :
            ( ( hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),X_35) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_7),A_19) )
           => ( hBOOL(hAPP_H2006434664a_bool(finite1341991622iple_a(F_1,A_7,A_19),X_1))
             => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_7),A_19)) ) ) ) ).

tff(fact_297_MGT__def,axiom,
    ! [Ca: com] : hoare_Mirabelle_MGT(Ca) = hoare_13225861_state(fequal_state,Ca,evalc(Ca)) ).

tff(fact_298_evalc__evaln,axiom,
    ! [C_2: com,S_1: state,T: state] :
      ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C_2),S_1),T))
     => ? [N_1: nat] : hBOOL(evaln(C_2,S_1,N_1,T)) ) ).

tff(fact_299_fold1Set_Osimps,axiom,
    ! [F_1: fun_nat_fun_nat_nat,A1: fun_nat_bool,A2: nat] :
      ( hBOOL(hAPP_nat_bool(finite_fold1Set_nat(F_1,A1),A2))
    <=> ? [A_7: nat,A_19: fun_nat_bool,X_2: nat] :
          ( ( A1 = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_7),A_19) )
          & ( A2 = X_2 )
          & hBOOL(hAPP_nat_bool(finite929467206at_nat(F_1,A_7,A_19),X_2))
          & ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_7),A_19)) ) ) ).

tff(fact_300_fold1Set_Osimps,axiom,
    ! [F_1: fun_Ho784854669_state,A1: fun_Ho1996104121e_bool,A2: hoare_1875481847_state] :
      ( hBOOL(hAPP_H1173941847e_bool(finite1016865319_state(F_1,A1),A2))
    <=> ? [A_7: hoare_1875481847_state,A_19: fun_Ho1996104121e_bool,X_2: hoare_1875481847_state] :
          ( ( A1 = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_7),A_19) )
          & ( A2 = X_2 )
          & hBOOL(hAPP_H1173941847e_bool(finite1721989730_state(F_1,A_7,A_19),X_2))
          & ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_7),A_19)) ) ) ).

tff(fact_301_fold1Set_Osimps,axiom,
    ! [F_1: fun_Ho115022426iple_a,A1: fun_Ho287446294a_bool,A2: hoare_1544627872iple_a] :
      ( hBOOL(hAPP_H2006434664a_bool(finite1715754352iple_a(F_1,A1),A2))
    <=> ? [A_7: hoare_1544627872iple_a,A_19: fun_Ho287446294a_bool,X_2: hoare_1544627872iple_a] :
          ( ( A1 = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_7),A_19) )
          & ( A2 = X_2 )
          & hBOOL(hAPP_H2006434664a_bool(finite1341991622iple_a(F_1,A_7,A_19),X_2))
          & ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_7),A_19)) ) ) ).

tff(fact_302_fold__graph_Osimps,axiom,
    ! [F_1: fun_Ho115022426iple_a,Z_2: hoare_1544627872iple_a,A1: fun_Ho287446294a_bool,A2: hoare_1544627872iple_a] :
      ( hBOOL(hAPP_H2006434664a_bool(finite1341991622iple_a(F_1,Z_2,A1),A2))
    <=> ( ( ( A1 = bot_bo1766443648a_bool )
          & ( A2 = Z_2 ) )
        | ? [X_2: hoare_1544627872iple_a,A_19: fun_Ho287446294a_bool,Y_11: hoare_1544627872iple_a] :
            ( ( A1 = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_2),A_19) )
            & ( A2 = hAPP_H9461717iple_a(hAPP_H1988213490iple_a(F_1,X_2),Y_11) )
            & ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_2),A_19))
            & hBOOL(hAPP_H2006434664a_bool(finite1341991622iple_a(F_1,Z_2,A_19),Y_11)) ) ) ) ).

tff(fact_303_evaln__max2,axiom,
    ! [C2: com,S2: state,N2: nat,T2: state,C1: com,S1: state,N1: nat,T1: state] :
      ( hBOOL(evaln(C1,S1,N1,T1))
     => ( hBOOL(evaln(C2,S2,N2,T2))
       => ? [N_1: nat] :
            ( hBOOL(evaln(C1,S1,N_1,T1))
            & hBOOL(evaln(C2,S2,N_1,T2)) ) ) ) ).

tff(fact_304_triple__valid__def2,axiom,
    ! [N_3: nat,Pa: fun_st1506752259e_bool,Ca: com,Q_1: fun_st1506752259e_bool] :
      ( hBOOL(hAPP_H1173941847e_bool(hoare_2065208407_state(N_3),hoare_13225861_state(Pa,Ca,Q_1)))
    <=> ! [Z_7: state,S_2: state] :
          ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Pa,Z_7),S_2))
         => ! [S_3: state] :
              ( hBOOL(evaln(Ca,S_2,N_3,S_3))
             => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Q_1,Z_7),S_3)) ) ) ) ).

tff(fact_305_triple__valid__def2,axiom,
    ! [N_3: nat,Pa: fun_a_fun_state_bool,Ca: com,Q_1: fun_a_fun_state_bool] :
      ( hBOOL(hAPP_H2006434664a_bool(hoare_1500711466alid_a(N_3),hoare_196563068iple_a(Pa,Ca,Q_1)))
    <=> ! [Z_7: x_a,S_2: state] :
          ( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Pa,Z_7),S_2))
         => ! [S_3: state] :
              ( hBOOL(evaln(Ca,S_2,N_3,S_3))
             => hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_1,Z_7),S_3)) ) ) ) ).

tff(fact_306_folding__one_Oinsert,axiom,
    ! [X_1: nat,A: fun_nat_bool,F_1: fun_nat_fun_nat_nat,F: fun_fun_nat_bool_nat] :
      ( hBOOL(finite988810631ne_nat(F_1,F))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => ( ( A != bot_bot_fun_nat_bool )
           => ( hAPP_f22106695ol_nat(F,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F_1,X_1),hAPP_f22106695ol_nat(F,A)) ) ) ) ) ) ).

tff(fact_307_folding__one_Oinsert,axiom,
    ! [X_1: 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_1),A))
         => ( ( A != bot_bo1715400655e_bool )
           => ( hAPP_f890249627_state(F,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),A)) = hAPP_H350317809_state(hAPP_H743221337_state(F_1,X_1),hAPP_f890249627_state(F,A)) ) ) ) ) ) ).

tff(fact_308_folding__one_Oinsert,axiom,
    ! [X_1: 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_1),A))
         => ( ( A != bot_bo1766443648a_bool )
           => ( hAPP_f1853245639iple_a(F,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_1),A)) = hAPP_H9461717iple_a(hAPP_H1988213490iple_a(F_1,X_1),hAPP_f1853245639iple_a(F,A)) ) ) ) ) ) ).

tff(fact_309_finite__Collect__conjI,axiom,
    ! [Q_1: fun_Ho287446294a_bool,Pa: fun_Ho287446294a_bool] :
      ( ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,collec445968053iple_a(Pa)))
        | hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,collec445968053iple_a(Q_1))) )
     => hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,collec445968053iple_a(hAPP_f909437487a_bool(hAPP_f1099051562a_bool(cOMBS_523298619l_bool,hAPP_f224574947l_bool(hAPP_f11333901l_bool(cOMBB_1960865690iple_a,fconj),Pa)),Q_1)))) ) ).

tff(fact_310_finite__Collect__conjI,axiom,
    ! [Q_1: fun_nat_bool,Pa: fun_nat_bool] :
      ( ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(Pa)))
        | hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(Q_1))) )
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),Pa)),Q_1)))) ) ).

tff(fact_311_finite_OemptyI,axiom,
    hBOOL(hAPP_f54304608l_bool(finite_finite_nat,bot_bot_fun_nat_bool)) ).

tff(fact_312_finite_OemptyI,axiom,
    hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,bot_bo1715400655e_bool)) ).

tff(fact_313_finite_OemptyI,axiom,
    hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,bot_bo1766443648a_bool)) ).

tff(fact_314_finite_OinsertI,axiom,
    ! [A_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),A))) ) ).

tff(fact_315_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_316_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_317_finite__Collect__disjI,axiom,
    ! [Pa: fun_Ho287446294a_bool,Q_1: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,collec445968053iple_a(hAPP_f909437487a_bool(hAPP_f1099051562a_bool(cOMBS_523298619l_bool,hAPP_f224574947l_bool(hAPP_f11333901l_bool(cOMBB_1960865690iple_a,fdisj),Pa)),Q_1))))
    <=> ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,collec445968053iple_a(Pa)))
        & hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,collec445968053iple_a(Q_1))) ) ) ).

tff(fact_318_finite__Collect__disjI,axiom,
    ! [Pa: fun_nat_bool,Q_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fdisj),Pa)),Q_1))))
    <=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(Pa)))
        & hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(Q_1))) ) ) ).

tff(fact_319_vname_Osimps_I1_J,axiom,
    ! [Glb_3: glb,Glb_2: glb] :
      ( ( glb_1(Glb_3) = glb_1(Glb_2) )
    <=> ( Glb_3 = Glb_2 ) ) ).

tff(fact_320_finite__insert,axiom,
    ! [A_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),A)))
    <=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) ) ).

tff(fact_321_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_322_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_323_vname_Osimps_I4_J,axiom,
    ! [Loc_1: loc,Glb_1: glb] : loc_1(Loc_1) != glb_1(Glb_1) ).

tff(fact_324_vname_Osimps_I3_J,axiom,
    ! [Glb_1: glb,Loc_1: loc] : glb_1(Glb_1) != loc_1(Loc_1) ).

tff(fact_325_folding__one_Oclosed,axiom,
    ! [A: fun_nat_bool,F_1: fun_nat_fun_nat_nat,F: fun_fun_nat_bool_nat] :
      ( hBOOL(finite988810631ne_nat(F_1,F))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( ( A != bot_bot_fun_nat_bool )
         => ( ! [X_2: nat,Y_11: nat] : hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(F_1,X_2),Y_11)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Y_11),bot_bot_fun_nat_bool))))
           => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_f22106695ol_nat(F,A)),A)) ) ) ) ) ).

tff(fact_326_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_2: hoare_1875481847_state,Y_11: hoare_1875481847_state] : hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,hAPP_H350317809_state(hAPP_H743221337_state(F_1,X_2),Y_11)),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,Y_11),bot_bo1715400655e_bool))))
           => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,hAPP_f890249627_state(F,A)),A)) ) ) ) ) ).

tff(fact_327_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_2: hoare_1544627872iple_a,Y_11: hoare_1544627872iple_a] : hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,hAPP_H9461717iple_a(hAPP_H1988213490iple_a(F_1,X_2),Y_11)),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_2),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,Y_11),bot_bo1766443648a_bool))))
           => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,hAPP_f1853245639iple_a(F,A)),A)) ) ) ) ) ).

tff(fact_328_finite__nonempty__imp__fold1Set,axiom,
    ! [F_1: fun_nat_fun_nat_nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( ( A != bot_bot_fun_nat_bool )
       => ? [X1: nat] : hBOOL(hAPP_nat_bool(finite_fold1Set_nat(F_1,A),X1)) ) ) ).

tff(fact_329_finite__nonempty__imp__fold1Set,axiom,
    ! [F_1: fun_Ho115022426iple_a,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,A))
     => ( ( A != bot_bo1766443648a_bool )
       => ? [X1: hoare_1544627872iple_a] : hBOOL(hAPP_H2006434664a_bool(finite1715754352iple_a(F_1,A),X1)) ) ) ).

tff(fact_330_finite__nonempty__imp__fold1Set,axiom,
    ! [F_1: fun_Ho784854669_state,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A))
     => ( ( A != bot_bo1715400655e_bool )
       => ? [X1: hoare_1875481847_state] : hBOOL(hAPP_H1173941847e_bool(finite1016865319_state(F_1,A),X1)) ) ) ).

tff(fact_331_finite__induct,axiom,
    ! [Pa: fun_fu425979586l_bool,F: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F))
     => ( hBOOL(hAPP_f54304608l_bool(Pa,bot_bot_fun_nat_bool))
       => ( ! [X_2: nat,F_2: fun_nat_bool] :
              ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_2))
             => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),F_2))
               => ( hBOOL(hAPP_f54304608l_bool(Pa,F_2))
                 => hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),F_2))) ) ) )
         => hBOOL(hAPP_f54304608l_bool(Pa,F)) ) ) ) ).

tff(fact_332_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_2: 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_2),F_2))
               => ( hBOOL(hAPP_f1408815105l_bool(Pa,F_2))
                 => hBOOL(hAPP_f1408815105l_bool(Pa,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),F_2))) ) ) )
         => hBOOL(hAPP_f1408815105l_bool(Pa,F)) ) ) ) ).

tff(fact_333_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_2: 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_2),F_2))
               => ( hBOOL(hAPP_f2063540982l_bool(Pa,F_2))
                 => hBOOL(hAPP_f2063540982l_bool(Pa,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_2),F_2))) ) ) )
         => hBOOL(hAPP_f2063540982l_bool(Pa,F)) ) ) ) ).

tff(fact_334_finite_Osimps,axiom,
    ! [A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
    <=> ( ( A_1 = bot_bot_fun_nat_bool )
        | ? [A_19: fun_nat_bool,A_7: nat] :
            ( ( A_1 = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_7),A_19) )
            & hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_19)) ) ) ) ).

tff(fact_335_finite_Osimps,axiom,
    ! [A_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1))
    <=> ( ( A_1 = bot_bo1715400655e_bool )
        | ? [A_19: fun_Ho1996104121e_bool,A_7: hoare_1875481847_state] :
            ( ( A_1 = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_7),A_19) )
            & hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_19)) ) ) ) ).

tff(fact_336_finite_Osimps,axiom,
    ! [A_1: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,A_1))
    <=> ( ( A_1 = bot_bo1766443648a_bool )
        | ? [A_19: fun_Ho287446294a_bool,A_7: hoare_1544627872iple_a] :
            ( ( A_1 = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_7),A_19) )
            & hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,A_19)) ) ) ) ).

tff(fact_337_finite__imp__fold__graph,axiom,
    ! [F_1: fun_Ho115022426iple_a,Z_2: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,A))
     => ? [X1: hoare_1544627872iple_a] : hBOOL(hAPP_H2006434664a_bool(finite1341991622iple_a(F_1,Z_2,A),X1)) ) ).

tff(fact_338_folding__one__idem_Oinsert__idem,axiom,
    ! [X_1: nat,A: fun_nat_bool,F_1: fun_nat_fun_nat_nat,F: fun_fun_nat_bool_nat] :
      ( hBOOL(finite795500164em_nat(F_1,F))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( ( A != bot_bot_fun_nat_bool )
         => ( hAPP_f22106695ol_nat(F,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F_1,X_1),hAPP_f22106695ol_nat(F,A)) ) ) ) ) ).

tff(fact_339_folding__one__idem_Oinsert__idem,axiom,
    ! [X_1: 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_1),A)) = hAPP_H350317809_state(hAPP_H743221337_state(F_1,X_1),hAPP_f890249627_state(F,A)) ) ) ) ) ).

tff(fact_340_folding__one__idem_Oinsert__idem,axiom,
    ! [X_1: 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_1),A)) = hAPP_H9461717iple_a(hAPP_H1988213490iple_a(F_1,X_1),hAPP_f1853245639iple_a(F,A)) ) ) ) ) ).

tff(fact_341_finite__ne__induct,axiom,
    ! [Pa: fun_fu425979586l_bool,F: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F))
     => ( ( F != bot_bot_fun_nat_bool )
       => ( ! [X_2: nat] : hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),bot_bot_fun_nat_bool)))
         => ( ! [X_2: nat,F_2: fun_nat_bool] :
                ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_2))
               => ( ( F_2 != bot_bot_fun_nat_bool )
                 => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),F_2))
                   => ( hBOOL(hAPP_f54304608l_bool(Pa,F_2))
                     => hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),F_2))) ) ) ) )
           => hBOOL(hAPP_f54304608l_bool(Pa,F)) ) ) ) ) ).

tff(fact_342_finite__ne__induct,axiom,
    ! [Pa: fun_fu689207471l_bool,F: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,F))
     => ( ( F != bot_bo1715400655e_bool )
       => ( ! [X_2: hoare_1875481847_state] : hBOOL(hAPP_f1408815105l_bool(Pa,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),bot_bo1715400655e_bool)))
         => ( ! [X_2: 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_2),F_2))
                   => ( hBOOL(hAPP_f1408815105l_bool(Pa,F_2))
                     => hBOOL(hAPP_f1408815105l_bool(Pa,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),F_2))) ) ) ) )
           => hBOOL(hAPP_f1408815105l_bool(Pa,F)) ) ) ) ) ).

tff(fact_343_finite__ne__induct,axiom,
    ! [Pa: fun_fu1441721944l_bool,F: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,F))
     => ( ( F != bot_bo1766443648a_bool )
       => ( ! [X_2: hoare_1544627872iple_a] : hBOOL(hAPP_f2063540982l_bool(Pa,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_2),bot_bo1766443648a_bool)))
         => ( ! [X_2: 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_2),F_2))
                   => ( hBOOL(hAPP_f2063540982l_bool(Pa,F_2))
                     => hBOOL(hAPP_f2063540982l_bool(Pa,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_2),F_2))) ) ) ) )
           => hBOOL(hAPP_f2063540982l_bool(Pa,F)) ) ) ) ) ).

tff(fact_344_vname_Oexhaust,axiom,
    ! [Y: vname] :
      ( ! [Glb: glb] : Y != glb_1(Glb)
     => ~ ! [Loc: loc] : Y != loc_1(Loc) ) ).

tff(fact_345_folding__one_Oremove,axiom,
    ! [X_1: nat,A: fun_nat_bool,F_1: fun_nat_fun_nat_nat,F: fun_fun_nat_bool_nat] :
      ( hBOOL(finite988810631ne_nat(F_1,F))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => ( ( ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) = bot_bot_fun_nat_bool )
             => ( hAPP_f22106695ol_nat(F,A) = X_1 ) )
            & ( ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) != bot_bot_fun_nat_bool )
             => ( hAPP_f22106695ol_nat(F,A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F_1,X_1),hAPP_f22106695ol_nat(F,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ) ) ) ) ).

tff(fact_346_folding__one_Oremove,axiom,
    ! [X_1: 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_1),A))
         => ( ( ( hAPP_f121055253e_bool(hAPP_f1204759513e_bool(minus_589155914e_bool,A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),bot_bo1715400655e_bool)) = bot_bo1715400655e_bool )
             => ( hAPP_f890249627_state(F,A) = X_1 ) )
            & ( ( hAPP_f121055253e_bool(hAPP_f1204759513e_bool(minus_589155914e_bool,A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),bot_bo1715400655e_bool)) != bot_bo1715400655e_bool )
             => ( hAPP_f890249627_state(F,A) = hAPP_H350317809_state(hAPP_H743221337_state(F_1,X_1),hAPP_f890249627_state(F,hAPP_f121055253e_bool(hAPP_f1204759513e_bool(minus_589155914e_bool,A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),bot_bo1715400655e_bool)))) ) ) ) ) ) ) ).

tff(fact_347_folding__one_Oremove,axiom,
    ! [X_1: 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_1),A))
         => ( ( ( hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,A),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_1),bot_bo1766443648a_bool)) = bot_bo1766443648a_bool )
             => ( hAPP_f1853245639iple_a(F,A) = X_1 ) )
            & ( ( hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,A),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_1),bot_bo1766443648a_bool)) != bot_bo1766443648a_bool )
             => ( hAPP_f1853245639iple_a(F,A) = hAPP_H9461717iple_a(hAPP_H1988213490iple_a(F_1,X_1),hAPP_f1853245639iple_a(F,hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,A),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_1),bot_bo1766443648a_bool)))) ) ) ) ) ) ) ).

tff(fact_348_folding__one_Oinsert__remove,axiom,
    ! [X_1: nat,A: fun_nat_bool,F_1: fun_nat_fun_nat_nat,F: fun_fun_nat_bool_nat] :
      ( hBOOL(finite988810631ne_nat(F_1,F))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( ( ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) = bot_bot_fun_nat_bool )
           => ( hAPP_f22106695ol_nat(F,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = X_1 ) )
          & ( ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) != bot_bot_fun_nat_bool )
           => ( hAPP_f22106695ol_nat(F,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F_1,X_1),hAPP_f22106695ol_nat(F,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ) ) ) ).

tff(fact_349_folding__one_Oinsert__remove,axiom,
    ! [X_1: 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))
       => ( ( ( hAPP_f121055253e_bool(hAPP_f1204759513e_bool(minus_589155914e_bool,A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),bot_bo1715400655e_bool)) = bot_bo1715400655e_bool )
           => ( hAPP_f890249627_state(F,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),A)) = X_1 ) )
          & ( ( hAPP_f121055253e_bool(hAPP_f1204759513e_bool(minus_589155914e_bool,A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),bot_bo1715400655e_bool)) != bot_bo1715400655e_bool )
           => ( hAPP_f890249627_state(F,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),A)) = hAPP_H350317809_state(hAPP_H743221337_state(F_1,X_1),hAPP_f890249627_state(F,hAPP_f121055253e_bool(hAPP_f1204759513e_bool(minus_589155914e_bool,A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),bot_bo1715400655e_bool)))) ) ) ) ) ) ).

tff(fact_350_folding__one_Oinsert__remove,axiom,
    ! [X_1: 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))
       => ( ( ( hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,A),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_1),bot_bo1766443648a_bool)) = bot_bo1766443648a_bool )
           => ( hAPP_f1853245639iple_a(F,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_1),A)) = X_1 ) )
          & ( ( hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,A),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_1),bot_bo1766443648a_bool)) != bot_bo1766443648a_bool )
           => ( hAPP_f1853245639iple_a(F,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_1),A)) = hAPP_H9461717iple_a(hAPP_H1988213490iple_a(F_1,X_1),hAPP_f1853245639iple_a(F,hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,A),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_1),bot_bo1766443648a_bool)))) ) ) ) ) ) ).

tff(fact_351_DiffE,axiom,
    ! [Ca: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,B: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,A),B)))
     => ~ ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),A))
         => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),B)) ) ) ).

tff(fact_352_DiffE,axiom,
    ! [Ca: nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))
     => ~ ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A))
         => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B)) ) ) ).

tff(fact_353_DiffI,axiom,
    ! [B: fun_Ho287446294a_bool,Ca: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),A))
     => ( ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),B))
       => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,A),B))) ) ) ).

tff(fact_354_DiffI,axiom,
    ! [B: fun_nat_bool,Ca: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A))
     => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B))
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B))) ) ) ).

tff(fact_355_finite__Diff,axiom,
    ! [B: fun_nat_bool,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B))) ) ).

tff(fact_356_DiffD2,axiom,
    ! [Ca: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,B: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,A),B)))
     => ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),B)) ) ).

tff(fact_357_DiffD2,axiom,
    ! [Ca: nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))
     => ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B)) ) ).

tff(fact_358_DiffD1,axiom,
    ! [Ca: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,B: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,A),B)))
     => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),A)) ) ).

tff(fact_359_DiffD1,axiom,
    ! [Ca: nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))
     => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A)) ) ).

tff(fact_360_Diff__iff,axiom,
    ! [Ca: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,B: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,A),B)))
    <=> ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),A))
        & ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),B)) ) ) ).

tff(fact_361_Diff__iff,axiom,
    ! [Ca: nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))
    <=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A))
        & ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B)) ) ) ).

tff(fact_362_set__diff__eq,axiom,
    ! [A: fun_Ho287446294a_bool,B: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,A),B) = collec445968053iple_a(hAPP_f909437487a_bool(hAPP_f1099051562a_bool(cOMBS_523298619l_bool,hAPP_f224574947l_bool(hAPP_f11333901l_bool(cOMBB_1960865690iple_a,fconj),hAPP_f909437487a_bool(hAPP_f482270129a_bool(cOMBC_1807206788l_bool,member1891733107iple_a),A))),hAPP_f909437487a_bool(hAPP_f1524052231a_bool(cOMBB_1977449916iple_a,fNot),hAPP_f909437487a_bool(hAPP_f482270129a_bool(cOMBC_1807206788l_bool,member1891733107iple_a),B)))) ).

tff(fact_363_set__diff__eq,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B) = collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),A))),hAPP_f800510211t_bool(hAPP_f894608603t_bool(cOMBB_bool_bool_nat,fNot),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),B)))) ).

tff(fact_364_Diff__cancel,axiom,
    ! [A: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),A) = bot_bot_fun_nat_bool ).

tff(fact_365_Diff__cancel,axiom,
    ! [A: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(minus_589155914e_bool,A),A) = bot_bo1715400655e_bool ).

tff(fact_366_Diff__cancel,axiom,
    ! [A: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,A),A) = bot_bo1766443648a_bool ).

tff(fact_367_Diff__empty,axiom,
    ! [A: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),bot_bot_fun_nat_bool) = A ).

tff(fact_368_Diff__empty,axiom,
    ! [A: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(minus_589155914e_bool,A),bot_bo1715400655e_bool) = A ).

tff(fact_369_Diff__empty,axiom,
    ! [A: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,A),bot_bo1766443648a_bool) = A ).

tff(fact_370_empty__Diff,axiom,
    ! [A: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,bot_bot_fun_nat_bool),A) = bot_bot_fun_nat_bool ).

tff(fact_371_empty__Diff,axiom,
    ! [A: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(minus_589155914e_bool,bot_bo1715400655e_bool),A) = bot_bo1715400655e_bool ).

tff(fact_372_empty__Diff,axiom,
    ! [A: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,bot_bo1766443648a_bool),A) = bot_bo1766443648a_bool ).

tff(fact_373_finite__Diff2,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))
      <=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) ) ) ).

tff(fact_374_insert__Diff1,axiom,
    ! [A: fun_nat_bool,X_1: nat,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B))
     => ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)),B) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B) ) ) ).

tff(fact_375_insert__Diff1,axiom,
    ! [A: fun_Ho1996104121e_bool,X_1: hoare_1875481847_state,B: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),B))
     => ( hAPP_f121055253e_bool(hAPP_f1204759513e_bool(minus_589155914e_bool,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),A)),B) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(minus_589155914e_bool,A),B) ) ) ).

tff(fact_376_insert__Diff1,axiom,
    ! [A: fun_Ho287446294a_bool,X_1: hoare_1544627872iple_a,B: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),B))
     => ( hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_1),A)),B) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,A),B) ) ) ).

tff(fact_377_insert__Diff__if,axiom,
    ! [A: fun_nat_bool,X_1: nat,B: fun_nat_bool] :
      ( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B))
       => ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)),B) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B) ) )
      & ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B))
       => ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)),B) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)) ) ) ) ).

tff(fact_378_insert__Diff__if,axiom,
    ! [A: fun_Ho1996104121e_bool,X_1: hoare_1875481847_state,B: fun_Ho1996104121e_bool] :
      ( ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),B))
       => ( hAPP_f121055253e_bool(hAPP_f1204759513e_bool(minus_589155914e_bool,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),A)),B) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(minus_589155914e_bool,A),B) ) )
      & ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),B))
       => ( hAPP_f121055253e_bool(hAPP_f1204759513e_bool(minus_589155914e_bool,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),A)),B) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(minus_589155914e_bool,A),B)) ) ) ) ).

tff(fact_379_insert__Diff__if,axiom,
    ! [A: fun_Ho287446294a_bool,X_1: hoare_1544627872iple_a,B: fun_Ho287446294a_bool] :
      ( ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),B))
       => ( hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_1),A)),B) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,A),B) ) )
      & ( ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),B))
       => ( hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_1),A)),B) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_1),hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,A),B)) ) ) ) ).

tff(fact_380_insert__Diff,axiom,
    ! [A_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A))
     => ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool))) = A ) ) ).

tff(fact_381_insert__Diff,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),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(minus_589155914e_bool,A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),bot_bo1715400655e_bool))) = A ) ) ).

tff(fact_382_insert__Diff,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),hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,A),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),bot_bo1766443648a_bool))) = A ) ) ).

tff(fact_383_Diff__insert__absorb,axiom,
    ! [X_1: nat,A: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) = A ) ) ).

tff(fact_384_Diff__insert__absorb,axiom,
    ! [X_1: hoare_1875481847_state,A: fun_Ho1996104121e_bool] :
      ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),A))
     => ( hAPP_f121055253e_bool(hAPP_f1204759513e_bool(minus_589155914e_bool,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),A)),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),bot_bo1715400655e_bool)) = A ) ) ).

tff(fact_385_Diff__insert__absorb,axiom,
    ! [X_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),A))
     => ( hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_1),A)),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_1),bot_bo1766443648a_bool)) = A ) ) ).

tff(fact_386_insert__Diff__single,axiom,
    ! [A_1: nat,A: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool))) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),A) ).

tff(fact_387_insert__Diff__single,axiom,
    ! [A_1: hoare_1875481847_state,A: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(minus_589155914e_bool,A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),bot_bo1715400655e_bool))) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),A) ).

tff(fact_388_insert__Diff__single,axiom,
    ! [A_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,A),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),bot_bo1766443648a_bool))) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),A) ).

tff(fact_389_Diff__insert2,axiom,
    ! [A: fun_nat_bool,A_1: nat,B: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool))),B) ).

tff(fact_390_Diff__insert2,axiom,
    ! [A: fun_Ho1996104121e_bool,A_1: hoare_1875481847_state,B: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(minus_589155914e_bool,A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),B)) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(minus_589155914e_bool,hAPP_f121055253e_bool(hAPP_f1204759513e_bool(minus_589155914e_bool,A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),bot_bo1715400655e_bool))),B) ).

tff(fact_391_Diff__insert2,axiom,
    ! [A: fun_Ho287446294a_bool,A_1: hoare_1544627872iple_a,B: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,A),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),B)) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,A),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),bot_bo1766443648a_bool))),B) ).

tff(fact_392_Diff__insert,axiom,
    ! [A: fun_nat_bool,A_1: nat,B: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool)) ).

tff(fact_393_Diff__insert,axiom,
    ! [A: fun_Ho1996104121e_bool,A_1: hoare_1875481847_state,B: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(minus_589155914e_bool,A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),B)) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(minus_589155914e_bool,hAPP_f121055253e_bool(hAPP_f1204759513e_bool(minus_589155914e_bool,A),B)),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),bot_bo1715400655e_bool)) ).

tff(fact_394_Diff__insert,axiom,
    ! [A: fun_Ho287446294a_bool,A_1: hoare_1544627872iple_a,B: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,A),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),B)) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,A),B)),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),bot_bo1766443648a_bool)) ).

tff(fact_395_finite__Diff__insert,axiom,
    ! [A: fun_nat_bool,A_1: nat,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B))))
    <=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B))) ) ).

tff(fact_396_finite__Diff__insert,axiom,
    ! [A: fun_Ho1996104121e_bool,A_1: hoare_1875481847_state,B: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,hAPP_f121055253e_bool(hAPP_f1204759513e_bool(minus_589155914e_bool,A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),B))))
    <=> hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,hAPP_f121055253e_bool(hAPP_f1204759513e_bool(minus_589155914e_bool,A),B))) ) ).

tff(fact_397_finite__Diff__insert,axiom,
    ! [A: fun_Ho287446294a_bool,A_1: hoare_1544627872iple_a,B: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,A),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),B))))
    <=> hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,A),B))) ) ).

tff(fact_398_folding__one__idem_Oin__idem,axiom,
    ! [X_1: 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_1),A))
         => ( hAPP_H9461717iple_a(hAPP_H1988213490iple_a(F_1,X_1),hAPP_f1853245639iple_a(F,A)) = hAPP_f1853245639iple_a(F,A) ) ) ) ) ).

tff(fact_399_folding__one__idem_Oin__idem,axiom,
    ! [X_1: nat,A: fun_nat_bool,F_1: fun_nat_fun_nat_nat,F: fun_fun_nat_bool_nat] :
      ( hBOOL(finite795500164em_nat(F_1,F))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(F_1,X_1),hAPP_f22106695ol_nat(F,A)) = hAPP_f22106695ol_nat(F,A) ) ) ) ) ).

tff(fact_400_folding__one__idem_Ohom__commute,axiom,
    ! [N: fun_nat_bool,H: fun_nat_nat,F_1: fun_nat_fun_nat_nat,F: fun_fun_nat_bool_nat] :
      ( hBOOL(finite795500164em_nat(F_1,F))
     => ( ! [X_2: nat,Y_11: nat] : hAPP_nat_nat(H,hAPP_nat_nat(hAPP_nat_fun_nat_nat(F_1,X_2),Y_11)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F_1,hAPP_nat_nat(H,X_2)),hAPP_nat_nat(H,Y_11))
       => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N))
         => ( ( N != bot_bot_fun_nat_bool )
           => ( hAPP_nat_nat(H,hAPP_f22106695ol_nat(F,N)) = hAPP_f22106695ol_nat(F,image_nat_nat(H,N)) ) ) ) ) ) ).

tff(fact_401_folding__one__idem_Ohom__commute,axiom,
    ! [N: fun_Ho1996104121e_bool,H: fun_Ho1455141843_state,F_1: fun_Ho784854669_state,F: fun_fu385705929_state] :
      ( hBOOL(finite1514162941_state(F_1,F))
     => ( ! [X_2: hoare_1875481847_state,Y_11: hoare_1875481847_state] : hAPP_H350317809_state(H,hAPP_H350317809_state(hAPP_H743221337_state(F_1,X_2),Y_11)) = hAPP_H350317809_state(hAPP_H743221337_state(F_1,hAPP_H350317809_state(H,X_2)),hAPP_H350317809_state(H,Y_11))
       => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,N))
         => ( ( N != bot_bo1715400655e_bool )
           => ( hAPP_H350317809_state(H,hAPP_f890249627_state(F,N)) = hAPP_f890249627_state(F,image_436941729_state(H,N)) ) ) ) ) ) ).

tff(fact_402_folding__one__idem_Ohom__commute,axiom,
    ! [N: fun_Ho287446294a_bool,H: fun_Ho1275796541iple_a,F_1: fun_Ho115022426iple_a,F: fun_fu1634330235iple_a] :
      ( hBOOL(finite1127948314iple_a(F_1,F))
     => ( ! [X_2: hoare_1544627872iple_a,Y_11: hoare_1544627872iple_a] : hAPP_H9461717iple_a(H,hAPP_H9461717iple_a(hAPP_H1988213490iple_a(F_1,X_2),Y_11)) = hAPP_H9461717iple_a(hAPP_H1988213490iple_a(F_1,hAPP_H9461717iple_a(H,X_2)),hAPP_H9461717iple_a(H,Y_11))
       => ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,N))
         => ( ( N != bot_bo1766443648a_bool )
           => ( hAPP_H9461717iple_a(H,hAPP_f1853245639iple_a(F,N)) = hAPP_f1853245639iple_a(F,image_1784540805iple_a(H,N)) ) ) ) ) ) ).

tff(fact_403_finite__empty__induct,axiom,
    ! [Pa: fun_fu425979586l_bool,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f54304608l_bool(Pa,A))
       => ( ! [A_7: nat,A_19: fun_nat_bool] :
              ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_19))
             => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_7),A_19))
               => ( hBOOL(hAPP_f54304608l_bool(Pa,A_19))
                 => hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_19),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_7),bot_bot_fun_nat_bool)))) ) ) )
         => hBOOL(hAPP_f54304608l_bool(Pa,bot_bot_fun_nat_bool)) ) ) ) ).

tff(fact_404_finite__empty__induct,axiom,
    ! [Pa: fun_fu689207471l_bool,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A))
     => ( hBOOL(hAPP_f1408815105l_bool(Pa,A))
       => ( ! [A_7: hoare_1875481847_state,A_19: fun_Ho1996104121e_bool] :
              ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_19))
             => ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_7),A_19))
               => ( hBOOL(hAPP_f1408815105l_bool(Pa,A_19))
                 => hBOOL(hAPP_f1408815105l_bool(Pa,hAPP_f121055253e_bool(hAPP_f1204759513e_bool(minus_589155914e_bool,A_19),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_7),bot_bo1715400655e_bool)))) ) ) )
         => hBOOL(hAPP_f1408815105l_bool(Pa,bot_bo1715400655e_bool)) ) ) ) ).

tff(fact_405_finite__empty__induct,axiom,
    ! [Pa: fun_fu1441721944l_bool,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,A))
     => ( hBOOL(hAPP_f2063540982l_bool(Pa,A))
       => ( ! [A_7: hoare_1544627872iple_a,A_19: fun_Ho287446294a_bool] :
              ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,A_19))
             => ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_7),A_19))
               => ( hBOOL(hAPP_f2063540982l_bool(Pa,A_19))
                 => hBOOL(hAPP_f2063540982l_bool(Pa,hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,A_19),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_7),bot_bo1766443648a_bool)))) ) ) )
         => hBOOL(hAPP_f2063540982l_bool(Pa,bot_bo1766443648a_bool)) ) ) ) ).

tff(fact_406_comp__fun__idem__remove,axiom,
    hBOOL(finite1956789438t_bool(hAPP_f10625010t_bool(hAPP_f1115950719t_bool(cOMBB_1654519265ol_nat,hAPP_f1311642927t_bool(cOMBC_646315179t_bool,minus_2067140911t_bool)),hAPP_f1570313510t_bool(hAPP_f688831301t_bool(cOMBC_178881787t_bool,insert_nat),bot_bot_fun_nat_bool)))) ).

tff(fact_407_comp__fun__idem__remove,axiom,
    hBOOL(finite1936929544e_bool(hAPP_f1821865133e_bool(hAPP_f967770723e_bool(cOMBB_107478263_state,hAPP_f961873429e_bool(cOMBC_1168545212e_bool,minus_589155914e_bool)),hAPP_f56162711e_bool(hAPP_f392546175e_bool(cOMBC_483615668e_bool,insert694999549_state),bot_bo1715400655e_bool)))) ).

tff(fact_408_comp__fun__idem__remove,axiom,
    hBOOL(finite701293802a_bool(hAPP_f915134428a_bool(hAPP_f1987684735a_bool(cOMBB_958433249iple_a,hAPP_f1118475355a_bool(cOMBC_1385723501a_bool,minus_1054521797a_bool)),hAPP_f693853672a_bool(hAPP_f1399402097a_bool(cOMBC_894585277a_bool,insert1051021594iple_a),bot_bo1766443648a_bool)))) ).

tff(fact_409_comp__fun__commute_Ofold__graph__insertE__aux,axiom,
    ! [A_1: hoare_1544627872iple_a,Z_2: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,Y_7: hoare_1544627872iple_a,F_1: fun_Ho115022426iple_a] :
      ( hBOOL(finite531755217iple_a(F_1))
     => ( hBOOL(hAPP_H2006434664a_bool(finite1341991622iple_a(F_1,Z_2,A),Y_7))
       => ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),A))
         => ? [Y_26: hoare_1544627872iple_a] :
              ( ( Y_7 = hAPP_H9461717iple_a(hAPP_H1988213490iple_a(F_1,A_1),Y_26) )
              & hBOOL(hAPP_H2006434664a_bool(finite1341991622iple_a(F_1,Z_2,hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,A),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),bot_bo1766443648a_bool))),Y_26)) ) ) ) ) ).

tff(fact_410_fold__graph__permute__diff,axiom,
    ! [A_1: nat,Ba: nat,A: fun_nat_bool,X_1: nat] :
      ( hBOOL(hAPP_nat_bool(finite929467206at_nat(times_times_nat,Ba,A),X_1))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A))
       => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),A))
         => hBOOL(hAPP_nat_bool(finite929467206at_nat(times_times_nat,A_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool)))),X_1)) ) ) ) ).

tff(fact_411_comp__fun__idem__insert,axiom,
    hBOOL(finite1956789438t_bool(insert_nat)) ).

tff(fact_412_comp__fun__idem__insert,axiom,
    hBOOL(finite1936929544e_bool(insert694999549_state)) ).

tff(fact_413_comp__fun__idem__insert,axiom,
    hBOOL(finite701293802a_bool(insert1051021594iple_a)) ).

tff(fact_414_comp__fun__commute_Ofold__graph__determ,axiom,
    ! [Y_7: hoare_1544627872iple_a,Z_2: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,X_1: hoare_1544627872iple_a,F_1: fun_Ho115022426iple_a] :
      ( hBOOL(finite531755217iple_a(F_1))
     => ( hBOOL(hAPP_H2006434664a_bool(finite1341991622iple_a(F_1,Z_2,A),X_1))
       => ( hBOOL(hAPP_H2006434664a_bool(finite1341991622iple_a(F_1,Z_2,A),Y_7))
         => ( Y_7 = X_1 ) ) ) ) ).

tff(fact_415_fold__graph__insert__swap,axiom,
    ! [Z_2: nat,Ba: nat,A: fun_nat_bool,Y_7: nat] :
      ( hBOOL(hAPP_nat_bool(finite929467206at_nat(times_times_nat,Ba,A),Y_7))
     => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ba),A))
       => hBOOL(hAPP_nat_bool(finite929467206at_nat(times_times_nat,Z_2,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),A)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Z_2),Y_7))) ) ) ).

tff(fact_416_comp__fun__commute_Ofold__graph__insertE,axiom,
    ! [Z_2: hoare_1544627872iple_a,X_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,V: hoare_1544627872iple_a,F_1: fun_Ho115022426iple_a] :
      ( hBOOL(finite531755217iple_a(F_1))
     => ( hBOOL(hAPP_H2006434664a_bool(finite1341991622iple_a(F_1,Z_2,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_1),A)),V))
       => ( ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),A))
         => ~ ! [Y_11: hoare_1544627872iple_a] :
                ( ( V = hAPP_H9461717iple_a(hAPP_H1988213490iple_a(F_1,X_1),Y_11) )
               => ~ hBOOL(hAPP_H2006434664a_bool(finite1341991622iple_a(F_1,Z_2,A),Y_11)) ) ) ) ) ).

tff(fact_417_fold1__insert,axiom,
    ! [X_1: nat,A: fun_nat_bool] :
      ( ( A != bot_bot_fun_nat_bool )
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => ( hAPP_f22106695ol_nat(finite_fold1_nat(times_times_nat),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_1),hAPP_f22106695ol_nat(finite_fold1_nat(times_times_nat),A)) ) ) ) ) ).

tff(fact_418_fold1__eq__fold,axiom,
    ! [A_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A))
       => ( hAPP_f22106695ol_nat(finite_fold1_nat(times_times_nat),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),A)) = hAPP_f22106695ol_nat(finite_fold_nat_nat(times_times_nat,A_1),A) ) ) ) ).

tff(fact_419_fold1__singleton,axiom,
    ! [F_1: fun_nat_fun_nat_nat,A_1: nat] : hAPP_f22106695ol_nat(finite_fold1_nat(F_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool)) = A_1 ).

tff(fact_420_fold1__singleton,axiom,
    ! [F_1: fun_Ho784854669_state,A_1: hoare_1875481847_state] : hAPP_f890249627_state(finite206258583_state(F_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),bot_bo1715400655e_bool)) = A_1 ).

tff(fact_421_fold1__singleton,axiom,
    ! [F_1: fun_Ho115022426iple_a,A_1: hoare_1544627872iple_a] : hAPP_f1853245639iple_a(finite1560330752iple_a(F_1),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),bot_bo1766443648a_bool)) = A_1 ).

tff(fact_422_fold1__singleton__def,axiom,
    ! [A_1: nat,G: fun_fun_nat_bool_nat,F_1: fun_nat_fun_nat_nat] :
      ( ( G = finite_fold1_nat(F_1) )
     => ( hAPP_f22106695ol_nat(G,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool)) = A_1 ) ) ).

tff(fact_423_fold1__singleton__def,axiom,
    ! [A_1: hoare_1875481847_state,G: fun_fu385705929_state,F_1: fun_Ho784854669_state] :
      ( ( G = finite206258583_state(F_1) )
     => ( hAPP_f890249627_state(G,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),bot_bo1715400655e_bool)) = A_1 ) ) ).

tff(fact_424_fold1__singleton__def,axiom,
    ! [A_1: hoare_1544627872iple_a,G: fun_fu1634330235iple_a,F_1: fun_Ho115022426iple_a] :
      ( ( G = finite1560330752iple_a(F_1) )
     => ( hAPP_f1853245639iple_a(G,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),bot_bo1766443648a_bool)) = A_1 ) ) ).

tff(fact_425_comp__fun__commute_Ofold__equality,axiom,
    ! [Z_2: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,Y_7: hoare_1544627872iple_a,F_1: fun_Ho115022426iple_a] :
      ( hBOOL(finite531755217iple_a(F_1))
     => ( hBOOL(hAPP_H2006434664a_bool(finite1341991622iple_a(F_1,Z_2,A),Y_7))
       => ( hAPP_f1853245639iple_a(finite309585736iple_a(F_1,Z_2),A) = Y_7 ) ) ) ).

tff(fact_426_fold__def,axiom,
    ! [F_1: fun_Ho115022426iple_a,Z_2: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] : hAPP_f1853245639iple_a(finite309585736iple_a(F_1,Z_2),A) = hAPP_f1853245639iple_a(the_Ho924576593iple_a,finite1341991622iple_a(F_1,Z_2,A)) ).

tff(fact_427_folding__one_Oeq__fold,axiom,
    ! [A: fun_nat_bool,F_1: fun_nat_fun_nat_nat,F: fun_fun_nat_bool_nat] :
      ( hBOOL(finite988810631ne_nat(F_1,F))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( hAPP_f22106695ol_nat(F,A) = hAPP_f22106695ol_nat(finite_fold1_nat(F_1),A) ) ) ) ).

tff(fact_428_folding__one_Oeq__fold,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))
       => ( hAPP_f1853245639iple_a(F,A) = hAPP_f1853245639iple_a(finite1560330752iple_a(F_1),A) ) ) ) ).

tff(fact_429_folding__one_Oeq__fold_H,axiom,
    ! [X_1: nat,A: fun_nat_bool,F_1: fun_nat_fun_nat_nat,F: fun_fun_nat_bool_nat] :
      ( hBOOL(finite988810631ne_nat(F_1,F))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => ( hAPP_f22106695ol_nat(F,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_f22106695ol_nat(finite_fold_nat_nat(F_1,X_1),A) ) ) ) ) ).

tff(fact_430_folding__one_Oeq__fold_H,axiom,
    ! [X_1: 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_1),A))
         => ( hAPP_f890249627_state(F,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),A)) = hAPP_f890249627_state(finite2136361956_state(F_1,X_1),A) ) ) ) ) ).

tff(fact_431_folding__one_Oeq__fold_H,axiom,
    ! [X_1: 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_1),A))
         => ( hAPP_f1853245639iple_a(F,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_1),A)) = hAPP_f1853245639iple_a(finite309585736iple_a(F_1,X_1),A) ) ) ) ) ).

tff(fact_432_folding__one__idem_Oeq__fold__idem_H,axiom,
    ! [A_1: nat,A: fun_nat_bool,F_1: fun_nat_fun_nat_nat,F: fun_fun_nat_bool_nat] :
      ( hBOOL(finite795500164em_nat(F_1,F))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( hAPP_f22106695ol_nat(F,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),A)) = hAPP_f22106695ol_nat(finite_fold_nat_nat(F_1,A_1),A) ) ) ) ).

tff(fact_433_folding__one__idem_Oeq__fold__idem_H,axiom,
    ! [A_1: 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))
       => ( hAPP_f890249627_state(F,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),A)) = hAPP_f890249627_state(finite2136361956_state(F_1,A_1),A) ) ) ) ).

tff(fact_434_folding__one__idem_Oeq__fold__idem_H,axiom,
    ! [A_1: 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))
       => ( hAPP_f1853245639iple_a(F,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),A)) = hAPP_f1853245639iple_a(finite309585736iple_a(F_1,A_1),A) ) ) ) ).

tff(fact_435_comp__fun__commute_Ofold__graph__fold,axiom,
    ! [Z_2: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,F_1: fun_Ho115022426iple_a] :
      ( hBOOL(finite531755217iple_a(F_1))
     => ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,A))
       => hBOOL(hAPP_H2006434664a_bool(finite1341991622iple_a(F_1,Z_2,A),hAPP_f1853245639iple_a(finite309585736iple_a(F_1,Z_2),A))) ) ) ).

tff(fact_436_fold1__def,axiom,
    ! [F_1: fun_Ho115022426iple_a,A: fun_Ho287446294a_bool] : hAPP_f1853245639iple_a(finite1560330752iple_a(F_1),A) = hAPP_f1853245639iple_a(the_Ho924576593iple_a,finite1715754352iple_a(F_1,A)) ).

tff(fact_437_minus__fold__remove,axiom,
    ! [B: fun_nat_bool,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,B),A) = hAPP_f800510211t_bool(finite1578363458t_bool(hAPP_f10625010t_bool(hAPP_f1115950719t_bool(cOMBB_1654519265ol_nat,hAPP_f1311642927t_bool(cOMBC_646315179t_bool,minus_2067140911t_bool)),hAPP_f1570313510t_bool(hAPP_f688831301t_bool(cOMBC_178881787t_bool,insert_nat),bot_bot_fun_nat_bool)),B),A) ) ) ).

tff(fact_438_minus__fold__remove,axiom,
    ! [B: fun_Ho1996104121e_bool,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A))
     => ( hAPP_f121055253e_bool(hAPP_f1204759513e_bool(minus_589155914e_bool,B),A) = hAPP_f121055253e_bool(finite1131674252e_bool(hAPP_f1821865133e_bool(hAPP_f967770723e_bool(cOMBB_107478263_state,hAPP_f961873429e_bool(cOMBC_1168545212e_bool,minus_589155914e_bool)),hAPP_f56162711e_bool(hAPP_f392546175e_bool(cOMBC_483615668e_bool,insert694999549_state),bot_bo1715400655e_bool)),B),A) ) ) ).

tff(fact_439_minus__fold__remove,axiom,
    ! [B: fun_Ho287446294a_bool,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,A))
     => ( hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,B),A) = hAPP_f909437487a_bool(finite113656430a_bool(hAPP_f915134428a_bool(hAPP_f1987684735a_bool(cOMBB_958433249iple_a,hAPP_f1118475355a_bool(cOMBC_1385723501a_bool,minus_1054521797a_bool)),hAPP_f693853672a_bool(hAPP_f1399402097a_bool(cOMBC_894585277a_bool,insert1051021594iple_a),bot_bo1766443648a_bool)),B),A) ) ) ).

tff(fact_440_fold1__in,axiom,
    ! [A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( ( A != bot_bot_fun_nat_bool )
       => ( ! [X_2: nat,Y_11: nat] : hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_2),Y_11)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Y_11),bot_bot_fun_nat_bool))))
         => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_f22106695ol_nat(finite_fold1_nat(times_times_nat),A)),A)) ) ) ) ).

tff(fact_441_semilattice__big_OF__eq,axiom,
    ! [A: fun_nat_bool,F_1: fun_nat_fun_nat_nat,F: fun_fun_nat_bool_nat] :
      ( hBOOL(big_se275732192ig_nat(F_1,F))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( hAPP_f22106695ol_nat(F,A) = hAPP_f22106695ol_nat(finite_fold1_nat(F_1),A) ) ) ) ).

tff(fact_442_folding__one__idem_Osubset__idem,axiom,
    ! [B: fun_nat_bool,A: fun_nat_bool,F_1: fun_nat_fun_nat_nat,F: fun_fun_nat_bool_nat] :
      ( hBOOL(finite795500164em_nat(F_1,F))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( ( B != bot_bot_fun_nat_bool )
         => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A))
           => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(F_1,hAPP_f22106695ol_nat(F,B)),hAPP_f22106695ol_nat(F,A)) = hAPP_f22106695ol_nat(F,A) ) ) ) ) ) ).

tff(fact_443_folding__one__idem_Osubset__idem,axiom,
    ! [B: 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))
       => ( ( B != bot_bo1715400655e_bool )
         => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,B),A))
           => ( hAPP_H350317809_state(hAPP_H743221337_state(F_1,hAPP_f890249627_state(F,B)),hAPP_f890249627_state(F,A)) = hAPP_f890249627_state(F,A) ) ) ) ) ) ).

tff(fact_444_folding__one__idem_Osubset__idem,axiom,
    ! [B: 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))
       => ( ( B != bot_bo1766443648a_bool )
         => ( hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(ord_le345502796a_bool,B),A))
           => ( hAPP_H9461717iple_a(hAPP_H1988213490iple_a(F_1,hAPP_f1853245639iple_a(F,B)),hAPP_f1853245639iple_a(F,A)) = hAPP_f1853245639iple_a(F,A) ) ) ) ) ) ).

tff(fact_445_order__refl,axiom,
    ! [X_34: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_34),X_34)) ).

tff(fact_446_subsetD,axiom,
    ! [Ca: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,B: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(ord_le345502796a_bool,A),B))
     => ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),A))
       => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),B)) ) ) ).

tff(fact_447_subsetD,axiom,
    ! [Ca: nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A))
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B)) ) ) ).

tff(fact_448_UnCI,axiom,
    ! [A: fun_Ho287446294a_bool,Ca: hoare_1544627872iple_a,B: fun_Ho287446294a_bool] :
      ( ( ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),B))
       => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),A)) )
     => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B))) ) ).

tff(fact_449_UnCI,axiom,
    ! [A: fun_nat_bool,Ca: nat,B: fun_nat_bool] :
      ( ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B))
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A)) )
     => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A),B))) ) ).

tff(fact_450_UnE,axiom,
    ! [Ca: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,B: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B)))
     => ( ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),A))
       => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),B)) ) ) ).

tff(fact_451_UnE,axiom,
    ! [Ca: nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A),B)))
     => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A))
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B)) ) ) ).

tff(fact_452_empty__subsetI,axiom,
    ! [A: fun_nat_bool] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,bot_bot_fun_nat_bool),A)) ).

tff(fact_453_empty__subsetI,axiom,
    ! [A: fun_Ho1996104121e_bool] : hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,bot_bo1715400655e_bool),A)) ).

tff(fact_454_empty__subsetI,axiom,
    ! [A: fun_Ho287446294a_bool] : hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(ord_le345502796a_bool,bot_bo1766443648a_bool),A)) ).

tff(fact_455_finite__Collect__subsets,axiom,
    ! [A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(hAPP_f103356543l_bool(hAPP_f643944041l_bool(cOMBC_1693257480l_bool,ord_le1568362934t_bool),A)))) ) ).

tff(fact_456_sup__le__fold__sup,axiom,
    ! [Ba: nat,A_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,A_1),Ba)),hAPP_f22106695ol_nat(finite_fold_nat_nat(semila972727038up_nat,Ba),A))) ) ) ).

tff(fact_457_subset__empty,axiom,
    ! [A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),bot_bot_fun_nat_bool))
    <=> ( A = bot_bot_fun_nat_bool ) ) ).

tff(fact_458_subset__empty,axiom,
    ! [A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A),bot_bo1715400655e_bool))
    <=> ( A = bot_bo1715400655e_bool ) ) ).

tff(fact_459_subset__empty,axiom,
    ! [A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(ord_le345502796a_bool,A),bot_bo1766443648a_bool))
    <=> ( A = bot_bo1766443648a_bool ) ) ).

tff(fact_460_rev__finite__subset,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
       => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) ) ) ).

tff(fact_461_finite__subset,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
       => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) ) ) ).

tff(fact_462_subset__insertI,axiom,
    ! [B: fun_nat_bool,A_1: nat] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B))) ).

tff(fact_463_subset__insertI,axiom,
    ! [B: fun_Ho1996104121e_bool,A_1: hoare_1875481847_state] : hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,B),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),B))) ).

tff(fact_464_subset__insertI,axiom,
    ! [B: fun_Ho287446294a_bool,A_1: hoare_1544627872iple_a] : hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(ord_le345502796a_bool,B),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),B))) ).

tff(fact_465_insert__subset,axiom,
    ! [X_1: nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)),B))
    <=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B))
        & hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) ) ) ).

tff(fact_466_insert__subset,axiom,
    ! [X_1: hoare_1875481847_state,A: fun_Ho1996104121e_bool,B: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),A)),B))
    <=> ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),B))
        & hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A),B)) ) ) ).

tff(fact_467_insert__subset,axiom,
    ! [X_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,B: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(ord_le345502796a_bool,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_1),A)),B))
    <=> ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),B))
        & hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(ord_le345502796a_bool,A),B)) ) ) ).

tff(fact_468_subset__insert,axiom,
    ! [B: fun_nat_bool,X_1: nat,A: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),B)))
      <=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) ) ) ).

tff(fact_469_subset__insert,axiom,
    ! [B: fun_Ho1996104121e_bool,X_1: hoare_1875481847_state,A: fun_Ho1996104121e_bool] :
      ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),A))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),B)))
      <=> hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A),B)) ) ) ).

tff(fact_470_subset__insert,axiom,
    ! [B: fun_Ho287446294a_bool,X_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),A))
     => ( hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(ord_le345502796a_bool,A),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_1),B)))
      <=> hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(ord_le345502796a_bool,A),B)) ) ) ).

tff(fact_471_subset__insertI2,axiom,
    ! [Ba: nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),B))) ) ).

tff(fact_472_subset__insertI2,axiom,
    ! [Ba: hoare_1875481847_state,A: fun_Ho1996104121e_bool,B: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A),B))
     => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,Ba),B))) ) ).

tff(fact_473_subset__insertI2,axiom,
    ! [Ba: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,B: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(ord_le345502796a_bool,A),B))
     => hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(ord_le345502796a_bool,A),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,Ba),B))) ) ).

tff(fact_474_insert__mono,axiom,
    ! [A_1: nat,C: fun_nat_bool,D_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C),D_2))
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),C)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),D_2))) ) ).

tff(fact_475_insert__mono,axiom,
    ! [A_1: hoare_1875481847_state,C: fun_Ho1996104121e_bool,D_2: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,C),D_2))
     => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),C)),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),D_2))) ) ).

tff(fact_476_insert__mono,axiom,
    ! [A_1: hoare_1544627872iple_a,C: fun_Ho287446294a_bool,D_2: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(ord_le345502796a_bool,C),D_2))
     => hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(ord_le345502796a_bool,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),C)),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),D_2))) ) ).

tff(fact_477_le__supE,axiom,
    ! [A_18: nat,B_14: nat,X_33: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,A_18),B_14)),X_33))
     => ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_18),X_33))
         => ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_14),X_33)) ) ) ).

tff(fact_478_sup__mono,axiom,
    ! [B_13: nat,D_1: nat,A_17: nat,C_7: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_17),C_7))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_13),D_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,A_17),B_13)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,C_7),D_1))) ) ) ).

tff(fact_479_sup__least,axiom,
    ! [Z_6: nat,Y_25: nat,X_32: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_25),X_32))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_6),X_32))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,Y_25),Z_6)),X_32)) ) ) ).

tff(fact_480_le__supI,axiom,
    ! [B_12: nat,A_16: nat,X_31: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_16),X_31))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_12),X_31))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,A_16),B_12)),X_31)) ) ) ).

tff(fact_481_sup__absorb1,axiom,
    ! [Y_24: nat,X_30: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_24),X_30))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_30),Y_24) = X_30 ) ) ).

tff(fact_482_sup__absorb2,axiom,
    ! [X_29: nat,Y_23: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_29),Y_23))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_29),Y_23) = Y_23 ) ) ).

tff(fact_483_le__supI2,axiom,
    ! [A_15: nat,X_28: nat,B_11: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_28),B_11))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_28),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,A_15),B_11))) ) ).

tff(fact_484_le__supI1,axiom,
    ! [B_10: nat,X_27: nat,A_14: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_27),A_14))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_27),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,A_14),B_10))) ) ).

tff(fact_485_le__sup__iff,axiom,
    ! [X_1: nat,Y_7: nat,Z_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_1),Y_7)),Z_2))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Z_2))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_7),Z_2)) ) ) ).

tff(fact_486_le__iff__sup,axiom,
    ! [X_1: nat,Y_7: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_7))
    <=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_1),Y_7) = Y_7 ) ) ).

tff(fact_487_sup__ge2,axiom,
    ! [Y_22: nat,X_26: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_22),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_26),Y_22))) ).

tff(fact_488_inf__sup__ord_I4_J,axiom,
    ! [Y_21: nat,X_25: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_21),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_25),Y_21))) ).

tff(fact_489_sup__ge1,axiom,
    ! [X_24: nat,Y_20: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_24),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_24),Y_20))) ).

tff(fact_490_inf__sup__ord_I3_J,axiom,
    ! [X_23: nat,Y_19: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_23),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_23),Y_19))) ).

tff(fact_491_sup__eq__bot__iff,axiom,
    ! [X_1: bool,Y_7: bool] :
      ( ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_1),Y_7))
      <=> hBOOL(bot_bot_bool) )
    <=> ( ( hBOOL(X_1)
        <=> hBOOL(bot_bot_bool) )
        & ( hBOOL(Y_7)
        <=> hBOOL(bot_bot_bool) ) ) ) ).

tff(fact_492_sup__eq__bot__iff,axiom,
    ! [X_1: fun_nat_bool,Y_7: fun_nat_bool] :
      ( ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,X_1),Y_7) = bot_bot_fun_nat_bool )
    <=> ( ( X_1 = bot_bot_fun_nat_bool )
        & ( Y_7 = bot_bot_fun_nat_bool ) ) ) ).

tff(fact_493_sup__eq__bot__iff,axiom,
    ! [X_1: fun_Ho1996104121e_bool,Y_7: fun_Ho1996104121e_bool] :
      ( ( hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X_1),Y_7) = bot_bo1715400655e_bool )
    <=> ( ( X_1 = bot_bo1715400655e_bool )
        & ( Y_7 = bot_bo1715400655e_bool ) ) ) ).

tff(fact_494_sup__eq__bot__iff,axiom,
    ! [X_1: fun_Ho287446294a_bool,Y_7: fun_Ho287446294a_bool] :
      ( ( hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X_1),Y_7) = bot_bo1766443648a_bool )
    <=> ( ( X_1 = bot_bo1766443648a_bool )
        & ( Y_7 = bot_bo1766443648a_bool ) ) ) ).

tff(fact_495_sup__bot__right,axiom,
    ! [X_1: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_1),bot_bot_bool))
    <=> hBOOL(X_1) ) ).

tff(fact_496_sup__bot__right,axiom,
    ! [X_1: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,X_1),bot_bot_fun_nat_bool) = X_1 ).

tff(fact_497_sup__bot__right,axiom,
    ! [X_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,X_1),bot_bo1715400655e_bool) = X_1 ).

tff(fact_498_sup__bot__right,axiom,
    ! [X_1: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,X_1),bot_bo1766443648a_bool) = X_1 ).

tff(fact_499_sup__bot__left,axiom,
    ! [X_1: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,bot_bot_bool),X_1))
    <=> hBOOL(X_1) ) ).

tff(fact_500_sup__bot__left,axiom,
    ! [X_1: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,bot_bot_fun_nat_bool),X_1) = X_1 ).

tff(fact_501_sup__bot__left,axiom,
    ! [X_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,bot_bo1715400655e_bool),X_1) = X_1 ).

tff(fact_502_sup__bot__left,axiom,
    ! [X_1: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,bot_bo1766443648a_bool),X_1) = X_1 ).

tff(fact_503_Un__empty__left,axiom,
    ! [B: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,bot_bot_fun_nat_bool),B) = B ).

tff(fact_504_Un__empty__left,axiom,
    ! [B: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,bot_bo1715400655e_bool),B) = B ).

tff(fact_505_Un__empty__left,axiom,
    ! [B: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,bot_bo1766443648a_bool),B) = B ).

tff(fact_506_Un__empty__right,axiom,
    ! [A: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A),bot_bot_fun_nat_bool) = A ).

tff(fact_507_Un__empty__right,axiom,
    ! [A: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),bot_bo1715400655e_bool) = A ).

tff(fact_508_Un__empty__right,axiom,
    ! [A: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),bot_bo1766443648a_bool) = A ).

tff(fact_509_Un__empty,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] :
      ( ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A),B) = bot_bot_fun_nat_bool )
    <=> ( ( A = bot_bot_fun_nat_bool )
        & ( B = bot_bot_fun_nat_bool ) ) ) ).

tff(fact_510_Un__empty,axiom,
    ! [A: fun_Ho1996104121e_bool,B: fun_Ho1996104121e_bool] :
      ( ( hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),B) = bot_bo1715400655e_bool )
    <=> ( ( A = bot_bo1715400655e_bool )
        & ( B = bot_bo1715400655e_bool ) ) ) ).

tff(fact_511_Un__empty,axiom,
    ! [A: fun_Ho287446294a_bool,B: fun_Ho287446294a_bool] :
      ( ( hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B) = bot_bo1766443648a_bool )
    <=> ( ( A = bot_bo1766443648a_bool )
        & ( B = bot_bo1766443648a_bool ) ) ) ).

tff(fact_512_finite__Un,axiom,
    ! [F: fun_nat_bool,Ga: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,F),Ga)))
    <=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F))
        & hBOOL(hAPP_f54304608l_bool(finite_finite_nat,Ga)) ) ) ).

tff(fact_513_finite__UnI,axiom,
    ! [Ga: fun_nat_bool,F: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,Ga))
       => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,F),Ga))) ) ) ).

tff(fact_514_linorder__le__cases,axiom,
    ! [X_22: nat,Y_18: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_22),Y_18))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_18),X_22)) ) ).

tff(fact_515_xt1_I6_J,axiom,
    ! [Z_5: nat,Y_17: nat,X_21: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_17),X_21))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_5),Y_17))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_5),X_21)) ) ) ).

tff(fact_516_xt1_I5_J,axiom,
    ! [Y_16: nat,X_20: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_16),X_20))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_20),Y_16))
       => ( X_20 = Y_16 ) ) ) ).

tff(fact_517_order__trans,axiom,
    ! [Z_4: nat,X_19: nat,Y_15: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_19),Y_15))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_15),Z_4))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_19),Z_4)) ) ) ).

tff(fact_518_order__antisym,axiom,
    ! [X_18: nat,Y_14: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_18),Y_14))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_14),X_18))
       => ( X_18 = Y_14 ) ) ) ).

tff(fact_519_xt1_I4_J,axiom,
    ! [C_6: nat,B_9: nat,A_13: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_9),A_13))
     => ( ( B_9 = C_6 )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_6),A_13)) ) ) ).

tff(fact_520_ord__le__eq__trans,axiom,
    ! [C_5: nat,A_12: nat,B_8: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_12),B_8))
     => ( ( B_8 = C_5 )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_12),C_5)) ) ) ).

tff(fact_521_xt1_I3_J,axiom,
    ! [C_4: nat,A_11: nat,B_7: nat] :
      ( ( A_11 = B_7 )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_4),B_7))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_4),A_11)) ) ) ).

tff(fact_522_ord__eq__le__trans,axiom,
    ! [C_3: nat,A_10: nat,B_6: nat] :
      ( ( A_10 = B_6 )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_6),C_3))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_10),C_3)) ) ) ).

tff(fact_523_order__antisym__conv,axiom,
    ! [Y_7: nat,X_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_7),X_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_7))
      <=> ( X_1 = Y_7 ) ) ) ).

tff(fact_524_order__eq__refl,axiom,
    ! [X_17: nat,Y_13: nat] :
      ( ( X_17 = Y_13 )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_17),Y_13)) ) ).

tff(fact_525_order__eq__iff,axiom,
    ! [X_1: nat,Y_7: nat] :
      ( ( X_1 = Y_7 )
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_7))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_7),X_1)) ) ) ).

tff(fact_526_linorder__linear,axiom,
    ! [X_16: nat,Y_12: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_16),Y_12))
      | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_12),X_16)) ) ).

tff(fact_527_Collect__disj__eq,axiom,
    ! [Pa: fun_Ho287446294a_bool,Q_1: fun_Ho287446294a_bool] : collec445968053iple_a(hAPP_f909437487a_bool(hAPP_f1099051562a_bool(cOMBS_523298619l_bool,hAPP_f224574947l_bool(hAPP_f11333901l_bool(cOMBB_1960865690iple_a,fdisj),Pa)),Q_1)) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,collec445968053iple_a(Pa)),collec445968053iple_a(Q_1)) ).

tff(fact_528_Collect__disj__eq,axiom,
    ! [Pa: fun_nat_bool,Q_1: fun_nat_bool] : collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fdisj),Pa)),Q_1)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,collect_nat(Pa)),collect_nat(Q_1)) ).

tff(fact_529_set__mp,axiom,
    ! [X_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,B: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(ord_le345502796a_bool,A),B))
     => ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),A))
       => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),B)) ) ) ).

tff(fact_530_set__mp,axiom,
    ! [X_1: nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B)) ) ) ).

tff(fact_531_set__rev__mp,axiom,
    ! [B: fun_Ho287446294a_bool,X_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),A))
     => ( hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(ord_le345502796a_bool,A),B))
       => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),B)) ) ) ).

tff(fact_532_set__rev__mp,axiom,
    ! [B: fun_nat_bool,X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B)) ) ) ).

tff(fact_533_in__mono,axiom,
    ! [X_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,B: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(ord_le345502796a_bool,A),B))
     => ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),A))
       => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),B)) ) ) ).

tff(fact_534_in__mono,axiom,
    ! [X_1: nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B)) ) ) ).

tff(fact_535_UnI2,axiom,
    ! [A: fun_Ho287446294a_bool,Ca: hoare_1544627872iple_a,B: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),B))
     => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B))) ) ).

tff(fact_536_UnI2,axiom,
    ! [A: fun_nat_bool,Ca: nat,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B))
     => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A),B))) ) ).

tff(fact_537_UnI1,axiom,
    ! [B: fun_Ho287446294a_bool,Ca: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),A))
     => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B))) ) ).

tff(fact_538_UnI1,axiom,
    ! [B: fun_nat_bool,Ca: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A))
     => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A),B))) ) ).

tff(fact_539_Un__iff,axiom,
    ! [Ca: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,B: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B)))
    <=> ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),A))
        | hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),B)) ) ) ).

tff(fact_540_Un__iff,axiom,
    ! [Ca: nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A),B)))
    <=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A))
        | hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B)) ) ) ).

tff(fact_541_Un__def,axiom,
    ! [A: fun_Ho287446294a_bool,B: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B) = collec445968053iple_a(hAPP_f909437487a_bool(hAPP_f1099051562a_bool(cOMBS_523298619l_bool,hAPP_f224574947l_bool(hAPP_f11333901l_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))) ).

tff(fact_542_Un__def,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A),B) = collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fdisj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),A))),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),B))) ).

tff(fact_543_pred__subset__eq,axiom,
    ! [R_1: fun_Ho287446294a_bool,S: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(ord_le345502796a_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)))
    <=> hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(ord_le345502796a_bool,R_1),S)) ) ).

tff(fact_544_pred__subset__eq,axiom,
    ! [R_1: fun_nat_bool,S: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),R_1)),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),S)))
    <=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,R_1),S)) ) ).

tff(fact_545_sup__Un__eq,axiom,
    ! [R_1: fun_Ho287446294a_bool,S: fun_Ho287446294a_bool,X_2: 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)),X_2))
    <=> hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_2),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,R_1),S))) ) ).

tff(fact_546_sup__Un__eq,axiom,
    ! [R_1: fun_nat_bool,S: fun_nat_bool,X_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),R_1)),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),S)),X_2))
    <=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,R_1),S))) ) ).

tff(fact_547_bot__least,axiom,
    ! [A_1: bool] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,bot_bot_bool),A_1)) ).

tff(fact_548_bot__least,axiom,
    ! [A_1: fun_nat_bool] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,bot_bot_fun_nat_bool),A_1)) ).

tff(fact_549_bot__least,axiom,
    ! [A_1: fun_Ho1996104121e_bool] : hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,bot_bo1715400655e_bool),A_1)) ).

tff(fact_550_bot__least,axiom,
    ! [A_1: fun_Ho287446294a_bool] : hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(ord_le345502796a_bool,bot_bo1766443648a_bool),A_1)) ).

tff(fact_551_bot__least,axiom,
    ! [A_9: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,bot_bot_nat),A_9)) ).

tff(fact_552_bot__unique,axiom,
    ! [A_1: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,A_1),bot_bot_bool))
    <=> ( hBOOL(A_1)
      <=> hBOOL(bot_bot_bool) ) ) ).

tff(fact_553_bot__unique,axiom,
    ! [A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),bot_bot_fun_nat_bool))
    <=> ( A_1 = bot_bot_fun_nat_bool ) ) ).

tff(fact_554_bot__unique,axiom,
    ! [A_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A_1),bot_bo1715400655e_bool))
    <=> ( A_1 = bot_bo1715400655e_bool ) ) ).

tff(fact_555_bot__unique,axiom,
    ! [A_1: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(ord_le345502796a_bool,A_1),bot_bo1766443648a_bool))
    <=> ( A_1 = bot_bo1766443648a_bool ) ) ).

tff(fact_556_bot__unique,axiom,
    ! [A_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_1),bot_bot_nat))
    <=> ( A_1 = bot_bot_nat ) ) ).

tff(fact_557_le__bot,axiom,
    ! [A_1: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,A_1),bot_bot_bool))
     => ( hBOOL(A_1)
      <=> hBOOL(bot_bot_bool) ) ) ).

tff(fact_558_le__bot,axiom,
    ! [A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),bot_bot_fun_nat_bool))
     => ( A_1 = bot_bot_fun_nat_bool ) ) ).

tff(fact_559_le__bot,axiom,
    ! [A_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A_1),bot_bo1715400655e_bool))
     => ( A_1 = bot_bo1715400655e_bool ) ) ).

tff(fact_560_le__bot,axiom,
    ! [A_1: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(ord_le345502796a_bool,A_1),bot_bo1766443648a_bool))
     => ( A_1 = bot_bo1766443648a_bool ) ) ).

tff(fact_561_le__bot,axiom,
    ! [A_8: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_8),bot_bot_nat))
     => ( A_8 = bot_bot_nat ) ) ).

tff(fact_562_Un__insert__right,axiom,
    ! [A: fun_nat_bool,A_1: nat,B: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A),B)) ).

tff(fact_563_Un__insert__right,axiom,
    ! [A: fun_Ho1996104121e_bool,A_1: hoare_1875481847_state,B: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),B)) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),B)) ).

tff(fact_564_Un__insert__right,axiom,
    ! [A: fun_Ho287446294a_bool,A_1: hoare_1544627872iple_a,B: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),B)) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B)) ).

tff(fact_565_Un__insert__left,axiom,
    ! [A_1: nat,B: fun_nat_bool,C: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B)),C) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,B),C)) ).

tff(fact_566_Un__insert__left,axiom,
    ! [A_1: hoare_1875481847_state,B: fun_Ho1996104121e_bool,C: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),B)),C) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,B),C)) ).

tff(fact_567_Un__insert__left,axiom,
    ! [A_1: hoare_1544627872iple_a,B: fun_Ho287446294a_bool,C: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),B)),C) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,B),C)) ).

tff(fact_568_weaken,axiom,
    ! [Ts: fun_Ho1996104121e_bool,Ga: fun_Ho1996104121e_bool,Ts_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),Ts_1))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Ts),Ts_1))
       => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),Ts)) ) ) ).

tff(fact_569_weaken,axiom,
    ! [Ts: fun_Ho287446294a_bool,Ga: fun_Ho287446294a_bool,Ts_1: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),Ts_1))
     => ( hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(ord_le345502796a_bool,Ts),Ts_1))
       => hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),Ts)) ) ) ).

tff(fact_570_asm,axiom,
    ! [Ts: fun_Ho1996104121e_bool,Ga: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Ts),Ga))
     => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(Ga),Ts)) ) ).

tff(fact_571_asm,axiom,
    ! [Ts: fun_Ho287446294a_bool,Ga: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(ord_le345502796a_bool,Ts),Ga))
     => hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(Ga),Ts)) ) ).

tff(fact_572_insert__def,axiom,
    ! [A_1: nat,B: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),A_1))),B) ).

tff(fact_573_insert__def,axiom,
    ! [A_1: hoare_1875481847_state,B: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),B) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,collec1735317154_state(hAPP_H117293375e_bool(hAPP_f2101312745e_bool(cOMBC_1972244470e_bool,fequal391417144_state),A_1))),B) ).

tff(fact_574_insert__def,axiom,
    ! [A_1: hoare_1544627872iple_a,B: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),B) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,collec445968053iple_a(hAPP_H1472549281a_bool(hAPP_f1878404769a_bool(cOMBC_827727450a_bool,fequal1057774495iple_a),A_1))),B) ).

tff(fact_575_insert__is__Un,axiom,
    ! [A_1: nat,A: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),A) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),bot_bot_fun_nat_bool)),A) ).

tff(fact_576_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_577_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_578_fold__sup__insert,axiom,
    ! [Ba: nat,A_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hAPP_f22106695ol_nat(finite_fold_nat_nat(semila972727038up_nat,Ba),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,A_1),hAPP_f22106695ol_nat(finite_fold_nat_nat(semila972727038up_nat,Ba),A)) ) ) ).

tff(fact_579_union__fold__insert,axiom,
    ! [B: fun_nat_bool,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A),B) = hAPP_f800510211t_bool(finite1578363458t_bool(insert_nat,B),A) ) ) ).

tff(fact_580_union__fold__insert,axiom,
    ! [B: fun_Ho1996104121e_bool,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A))
     => ( hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),B) = hAPP_f121055253e_bool(finite1131674252e_bool(insert694999549_state,B),A) ) ) ).

tff(fact_581_union__fold__insert,axiom,
    ! [B: fun_Ho287446294a_bool,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,A))
     => ( hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B) = hAPP_f909437487a_bool(finite113656430a_bool(insert1051021594iple_a,B),A) ) ) ).

tff(fact_582_subset__singletonD,axiom,
    ! [A: fun_nat_bool,X_1: nat] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))
     => ( ( A = bot_bot_fun_nat_bool )
        | ( A = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool) ) ) ) ).

tff(fact_583_subset__singletonD,axiom,
    ! [A: fun_Ho1996104121e_bool,X_1: hoare_1875481847_state] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),bot_bo1715400655e_bool)))
     => ( ( A = bot_bo1715400655e_bool )
        | ( A = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),bot_bo1715400655e_bool) ) ) ) ).

tff(fact_584_subset__singletonD,axiom,
    ! [A: fun_Ho287446294a_bool,X_1: hoare_1544627872iple_a] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(ord_le345502796a_bool,A),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_1),bot_bo1766443648a_bool)))
     => ( ( A = bot_bo1766443648a_bool )
        | ( A = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_1),bot_bo1766443648a_bool) ) ) ) ).

tff(fact_585_diff__single__insert,axiom,
    ! [A: fun_nat_bool,X_1: nat,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool))),B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),B))) ) ) ).

tff(fact_586_diff__single__insert,axiom,
    ! [A: fun_Ho1996104121e_bool,X_1: hoare_1875481847_state,B: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,hAPP_f121055253e_bool(hAPP_f1204759513e_bool(minus_589155914e_bool,A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),bot_bo1715400655e_bool))),B))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),A))
       => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),B))) ) ) ).

tff(fact_587_diff__single__insert,axiom,
    ! [A: fun_Ho287446294a_bool,X_1: hoare_1544627872iple_a,B: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(ord_le345502796a_bool,hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,A),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_1),bot_bo1766443648a_bool))),B))
     => ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),A))
       => hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(ord_le345502796a_bool,A),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_1),B))) ) ) ).

tff(fact_588_subset__insert__iff,axiom,
    ! [A: fun_nat_bool,X_1: nat,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),B)))
    <=> ( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool))),B)) )
        & ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) ) ) ) ).

tff(fact_589_subset__insert__iff,axiom,
    ! [A: fun_Ho1996104121e_bool,X_1: hoare_1875481847_state,B: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),B)))
    <=> ( ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),A))
         => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,hAPP_f121055253e_bool(hAPP_f1204759513e_bool(minus_589155914e_bool,A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),bot_bo1715400655e_bool))),B)) )
        & ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),A))
         => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A),B)) ) ) ) ).

tff(fact_590_subset__insert__iff,axiom,
    ! [A: fun_Ho287446294a_bool,X_1: hoare_1544627872iple_a,B: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(ord_le345502796a_bool,A),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_1),B)))
    <=> ( ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),A))
         => hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(ord_le345502796a_bool,hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,A),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,X_1),bot_bo1766443648a_bool))),B)) )
        & ( ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),A))
         => hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(ord_le345502796a_bool,A),B)) ) ) ) ).

tff(fact_591_folding__one__idem_Ounion__idem,axiom,
    ! [B: fun_nat_bool,A: fun_nat_bool,F_1: fun_nat_fun_nat_nat,F: fun_fun_nat_bool_nat] :
      ( hBOOL(finite795500164em_nat(F_1,F))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( ( A != bot_bot_fun_nat_bool )
         => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
           => ( ( B != bot_bot_fun_nat_bool )
             => ( hAPP_f22106695ol_nat(F,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A),B)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F_1,hAPP_f22106695ol_nat(F,A)),hAPP_f22106695ol_nat(F,B)) ) ) ) ) ) ) ).

tff(fact_592_folding__one__idem_Ounion__idem,axiom,
    ! [B: 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))
           => ( ( B != bot_bo1715400655e_bool )
             => ( hAPP_f890249627_state(F,hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila1107643821e_bool,A),B)) = hAPP_H350317809_state(hAPP_H743221337_state(F_1,hAPP_f890249627_state(F,A)),hAPP_f890249627_state(F,B)) ) ) ) ) ) ) ).

tff(fact_593_folding__one__idem_Ounion__idem,axiom,
    ! [B: 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))
           => ( ( B != bot_bo1766443648a_bool )
             => ( hAPP_f1853245639iple_a(F,hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila2083752482a_bool,A),B)) = hAPP_H9461717iple_a(hAPP_H1988213490iple_a(F_1,hAPP_f1853245639iple_a(F,A)),hAPP_f1853245639iple_a(F,B)) ) ) ) ) ) ) ).

tff(fact_594_fold__sup__le__sup,axiom,
    ! [Ca: nat,Ba: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( ! [X_2: nat] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
           => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),Ba)) )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_fold_nat_nat(semila972727038up_nat,Ca),A)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,Ba),Ca))) ) ) ).

tff(fact_595_finite__subset__induct,axiom,
    ! [Pa: fun_fu425979586l_bool,A: fun_nat_bool,F: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F),A))
       => ( hBOOL(hAPP_f54304608l_bool(Pa,bot_bot_fun_nat_bool))
         => ( ! [A_7: nat,F_2: fun_nat_bool] :
                ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_2))
               => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_7),A))
                 => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_7),F_2))
                   => ( hBOOL(hAPP_f54304608l_bool(Pa,F_2))
                     => hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_7),F_2))) ) ) ) )
           => hBOOL(hAPP_f54304608l_bool(Pa,F)) ) ) ) ) ).

tff(fact_596_finite__subset__induct,axiom,
    ! [Pa: fun_fu689207471l_bool,A: fun_Ho1996104121e_bool,F: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,F))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,F),A))
       => ( hBOOL(hAPP_f1408815105l_bool(Pa,bot_bo1715400655e_bool))
         => ( ! [A_7: 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,A_7),A))
                 => ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_7),F_2))
                   => ( hBOOL(hAPP_f1408815105l_bool(Pa,F_2))
                     => hBOOL(hAPP_f1408815105l_bool(Pa,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_7),F_2))) ) ) ) )
           => hBOOL(hAPP_f1408815105l_bool(Pa,F)) ) ) ) ) ).

tff(fact_597_finite__subset__induct,axiom,
    ! [Pa: fun_fu1441721944l_bool,A: fun_Ho287446294a_bool,F: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(finite1833138547iple_a,F))
     => ( hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(ord_le345502796a_bool,F),A))
       => ( hBOOL(hAPP_f2063540982l_bool(Pa,bot_bo1766443648a_bool))
         => ( ! [A_7: 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,A_7),A))
                 => ( ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_7),F_2))
                   => ( hBOOL(hAPP_f2063540982l_bool(Pa,F_2))
                     => hBOOL(hAPP_f2063540982l_bool(Pa,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_7),F_2))) ) ) ) )
           => hBOOL(hAPP_f2063540982l_bool(Pa,F)) ) ) ) ) ).

tff(fact_598_subsetI,axiom,
    ! [B: fun_Ho287446294a_bool,A: fun_Ho287446294a_bool] :
      ( ! [X_2: hoare_1544627872iple_a] :
          ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_2),A))
         => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_2),B)) )
     => hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(ord_le345502796a_bool,A),B)) ) ).

tff(fact_599_subsetI,axiom,
    ! [B: fun_nat_bool,A: fun_nat_bool] :
      ( ! [X_2: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
         => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),B)) )
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) ) ).

tff(fact_600_evaln__nonstrict,axiom,
    ! [M_1: nat,C_2: com,S_1: state,N_2: nat,T: state] :
      ( hBOOL(evaln(C_2,S_1,N_2,T))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),M_1))
       => hBOOL(evaln(C_2,S_1,M_1,T)) ) ) ).

tff(fact_601_finite__Collect__le__nat,axiom,
    ! [K: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_eq_nat),K)))) ).

tff(fact_602_flat__lub__def,axiom,
    ! [A: fun_Ho287446294a_bool,Ba: hoare_1544627872iple_a] :
      ( ( hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(ord_le345502796a_bool,A),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,Ba),bot_bo1766443648a_bool)))
       => ( hAPP_f1853245639iple_a(partia2044047638iple_a(Ba),A) = Ba ) )
      & ( ~ hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(ord_le345502796a_bool,A),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,Ba),bot_bo1766443648a_bool)))
       => ( hAPP_f1853245639iple_a(partia2044047638iple_a(Ba),A) = hAPP_f1853245639iple_a(the_Ho924576593iple_a,hAPP_f909437487a_bool(hAPP_f482270129a_bool(cOMBC_1807206788l_bool,member1891733107iple_a),hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,A),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,Ba),bot_bo1766443648a_bool)))) ) ) ) ).

tff(fact_603_flat__lub__def,axiom,
    ! [A: fun_nat_bool,Ba: nat] :
      ( ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),bot_bot_fun_nat_bool)))
       => ( hAPP_f22106695ol_nat(partial_flat_lub_nat(Ba),A) = Ba ) )
      & ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),bot_bot_fun_nat_bool)))
       => ( hAPP_f22106695ol_nat(partial_flat_lub_nat(Ba),A) = hAPP_f22106695ol_nat(the_nat,hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Ba),bot_bot_fun_nat_bool)))) ) ) ) ).

tff(fact_604_flat__lub__def,axiom,
    ! [A: fun_Ho1996104121e_bool,Ba: hoare_1875481847_state] :
      ( ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,Ba),bot_bo1715400655e_bool)))
       => ( hAPP_f890249627_state(partia1423322881_state(Ba),A) = Ba ) )
      & ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,Ba),bot_bo1715400655e_bool)))
       => ( hAPP_f890249627_state(partia1423322881_state(Ba),A) = hAPP_f890249627_state(the_Ho1017792262_state,hAPP_f121055253e_bool(hAPP_f1795624711e_bool(cOMBC_1099403170l_bool,member618553700_state),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(minus_589155914e_bool,A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,Ba),bot_bo1715400655e_bool)))) ) ) ) ).

tff(fact_605_finite__nat__set__iff__bounded__le,axiom,
    ! [N: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N))
    <=> ? [M: nat] :
        ! [X_2: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),N))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),M)) ) ) ).

tff(fact_606_finite__less__ub,axiom,
    ! [U: nat,F_1: fun_nat_nat] :
      ( ! [N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),hAPP_nat_nat(F_1,N_1)))
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,ord_less_eq_nat),F_1)),U)))) ) ).

tff(fact_607_Sup__fin_Oremove,axiom,
    ! [X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => ( ( ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) = bot_bot_fun_nat_bool )
           => ( hAPP_f22106695ol_nat(big_la43341705in_nat,A) = X_1 ) )
          & ( ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) != bot_bot_fun_nat_bool )
           => ( hAPP_f22106695ol_nat(big_la43341705in_nat,A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_1),hAPP_f22106695ol_nat(big_la43341705in_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ) ) ) ).

tff(fact_608_Sup__fin_Osingleton,axiom,
    ! [X_1: nat] : hAPP_f22106695ol_nat(big_la43341705in_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) = X_1 ).

tff(fact_609_Sup__fin_Oin__idem,axiom,
    ! [X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_1),hAPP_f22106695ol_nat(big_la43341705in_nat,A)) = hAPP_f22106695ol_nat(big_la43341705in_nat,A) ) ) ) ).

tff(fact_610_Sup__fin_OF__eq,axiom,
    ! [A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hAPP_f22106695ol_nat(big_la43341705in_nat,A) = hAPP_f22106695ol_nat(finite_fold1_nat(semila972727038up_nat),A) ) ) ).

tff(fact_611_Sup__fin_Oinsert__idem,axiom,
    ! [X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( ( A != bot_bot_fun_nat_bool )
       => ( hAPP_f22106695ol_nat(big_la43341705in_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_1),hAPP_f22106695ol_nat(big_la43341705in_nat,A)) ) ) ) ).

tff(fact_612_Sup__fin_Oinsert,axiom,
    ! [X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => ( ( A != bot_bot_fun_nat_bool )
         => ( hAPP_f22106695ol_nat(big_la43341705in_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_1),hAPP_f22106695ol_nat(big_la43341705in_nat,A)) ) ) ) ) ).

tff(fact_613_Sup__fin_Osubset__idem,axiom,
    ! [B: fun_nat_bool,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( ( B != bot_bot_fun_nat_bool )
       => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A))
         => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,hAPP_f22106695ol_nat(big_la43341705in_nat,B)),hAPP_f22106695ol_nat(big_la43341705in_nat,A)) = hAPP_f22106695ol_nat(big_la43341705in_nat,A) ) ) ) ) ).

tff(fact_614_Sup__fin_Ounion__idem,axiom,
    ! [B: fun_nat_bool,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( ( A != bot_bot_fun_nat_bool )
       => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
         => ( ( B != bot_bot_fun_nat_bool )
           => ( hAPP_f22106695ol_nat(big_la43341705in_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A),B)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,hAPP_f22106695ol_nat(big_la43341705in_nat,A)),hAPP_f22106695ol_nat(big_la43341705in_nat,B)) ) ) ) ) ) ).

tff(fact_615_Sup__fin_Oeq__fold__idem_H,axiom,
    ! [A_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hAPP_f22106695ol_nat(big_la43341705in_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),A)) = hAPP_f22106695ol_nat(finite_fold_nat_nat(semila972727038up_nat,A_1),A) ) ) ).

tff(fact_616_Sup__fin_Oeq__fold_H,axiom,
    ! [X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => ( hAPP_f22106695ol_nat(big_la43341705in_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_f22106695ol_nat(finite_fold_nat_nat(semila972727038up_nat,X_1),A) ) ) ) ).

tff(fact_617_Sup__fin_Oinsert__remove,axiom,
    ! [X_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( ( ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) = bot_bot_fun_nat_bool )
         => ( hAPP_f22106695ol_nat(big_la43341705in_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = X_1 ) )
        & ( ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) != bot_bot_fun_nat_bool )
         => ( hAPP_f22106695ol_nat(big_la43341705in_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_1),hAPP_f22106695ol_nat(big_la43341705in_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ) ) ).

tff(fact_618_Sup__fin_Ohom__commute,axiom,
    ! [N: fun_nat_bool,H: fun_nat_nat] :
      ( ! [X_2: nat,Y_11: nat] : hAPP_nat_nat(H,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_2),Y_11)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,hAPP_nat_nat(H,X_2)),hAPP_nat_nat(H,Y_11))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N))
       => ( ( N != bot_bot_fun_nat_bool )
         => ( hAPP_nat_nat(H,hAPP_f22106695ol_nat(big_la43341705in_nat,N)) = hAPP_f22106695ol_nat(big_la43341705in_nat,image_nat_nat(H,N)) ) ) ) ) ).

tff(fact_619_Sup__fin_Oclosed,axiom,
    ! [A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( ( A != bot_bot_fun_nat_bool )
       => ( ! [X_2: nat,Y_11: nat] : hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_2),Y_11)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Y_11),bot_bot_fun_nat_bool))))
         => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_f22106695ol_nat(big_la43341705in_nat,A)),A)) ) ) ) ).

tff(fact_620_Sup__fin_Ounion__disjoint,axiom,
    ! [B: fun_nat_bool,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( ( A != bot_bot_fun_nat_bool )
       => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
         => ( ( B != bot_bot_fun_nat_bool )
           => ( ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),B) = bot_bot_fun_nat_bool )
             => ( hAPP_f22106695ol_nat(big_la43341705in_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A),B)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,hAPP_f22106695ol_nat(big_la43341705in_nat,A)),hAPP_f22106695ol_nat(big_la43341705in_nat,B)) ) ) ) ) ) ) ).

tff(fact_621_IntI,axiom,
    ! [B: fun_Ho287446294a_bool,Ca: hoare_1544627872iple_a,A: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),A))
     => ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),B))
       => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila1256126856a_bool,A),B))) ) ) ).

tff(fact_622_IntI,axiom,
    ! [B: fun_nat_bool,Ca: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B))
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),B))) ) ) ).

tff(fact_623_IntE,axiom,
    ! [Ca: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,B: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila1256126856a_bool,A),B)))
     => ~ ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),A))
         => ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),B)) ) ) ).

tff(fact_624_IntE,axiom,
    ! [Ca: nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),B)))
     => ~ ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A))
         => ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B)) ) ) ).

tff(fact_625_finite__Int,axiom,
    ! [Ga: fun_nat_bool,F: fun_nat_bool] :
      ( ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F))
        | hBOOL(hAPP_f54304608l_bool(finite_finite_nat,Ga)) )
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,F),Ga))) ) ).

tff(fact_626_le__infE,axiom,
    ! [X_15: nat,A_6: nat,B_5: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_15),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,A_6),B_5)))
     => ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_15),A_6))
         => ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_15),B_5)) ) ) ).

tff(fact_627_inf__mono,axiom,
    ! [B_4: nat,D: nat,A_5: nat,C_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_5),C_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_4),D))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,A_5),B_4)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,C_1),D))) ) ) ).

tff(fact_628_inf__greatest,axiom,
    ! [Z_3: nat,X_14: nat,Y_10: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_14),Y_10))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_14),Z_3))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_14),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,Y_10),Z_3))) ) ) ).

tff(fact_629_le__infI,axiom,
    ! [B_3: nat,X_13: nat,A_4: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_13),A_4))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_13),B_3))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_13),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,A_4),B_3))) ) ) ).

tff(fact_630_inf__absorb2,axiom,
    ! [Y_9: nat,X_12: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_9),X_12))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,X_12),Y_9) = Y_9 ) ) ).

tff(fact_631_inf__absorb1,axiom,
    ! [X_11: nat,Y_8: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_11),Y_8))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,X_11),Y_8) = X_11 ) ) ).

tff(fact_632_le__infI2,axiom,
    ! [A_3: nat,B_2: nat,X_10: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_2),X_10))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,A_3),B_2)),X_10)) ) ).

tff(fact_633_le__infI1,axiom,
    ! [B_1: nat,A_2: nat,X_9: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_2),X_9))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,A_2),B_1)),X_9)) ) ).

tff(fact_634_le__inf__iff,axiom,
    ! [X_1: nat,Y_7: nat,Z_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,Y_7),Z_2)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_7))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Z_2)) ) ) ).

tff(fact_635_le__iff__inf,axiom,
    ! [X_1: nat,Y_7: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_7))
    <=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,X_1),Y_7) = X_1 ) ) ).

tff(fact_636_inf__le2,axiom,
    ! [X_8: nat,Y_6: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,X_8),Y_6)),Y_6)) ).

tff(fact_637_inf__sup__ord_I2_J,axiom,
    ! [X_7: nat,Y_5: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,X_7),Y_5)),Y_5)) ).

tff(fact_638_inf__le1,axiom,
    ! [X_6: nat,Y_4: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,X_6),Y_4)),X_6)) ).

tff(fact_639_inf__sup__ord_I1_J,axiom,
    ! [X_5: nat,Y_3: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,X_5),Y_3)),X_5)) ).

tff(fact_640_distrib__sup__le,axiom,
    ! [X_4: nat,Y_2: nat,Z_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_4),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,Y_2),Z_1))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_4),Y_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_4),Z_1)))) ).

tff(fact_641_distrib__inf__le,axiom,
    ! [X_3: nat,Y_1: nat,Z: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,X_3),Y_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,X_3),Z))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,X_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,Y_1),Z)))) ).

tff(fact_642_Int__insert__left__if1,axiom,
    ! [B: fun_nat_bool,A_1: nat,C: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),C))
     => ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B)),C) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,B),C)) ) ) ).

tff(fact_643_Int__insert__left__if1,axiom,
    ! [B: fun_Ho1996104121e_bool,A_1: hoare_1875481847_state,C: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),C))
     => ( hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila657954503e_bool,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),B)),C) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila657954503e_bool,B),C)) ) ) ).

tff(fact_644_Int__insert__left__if1,axiom,
    ! [B: fun_Ho287446294a_bool,A_1: hoare_1544627872iple_a,C: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),C))
     => ( hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila1256126856a_bool,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),B)),C) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila1256126856a_bool,B),C)) ) ) ).

tff(fact_645_Int__insert__right__if1,axiom,
    ! [B: fun_nat_bool,A_1: nat,A: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A))
     => ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),B)) ) ) ).

tff(fact_646_Int__insert__right__if1,axiom,
    ! [B: 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(hAPP_f1204759513e_bool(semila657954503e_bool,A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),B)) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila657954503e_bool,A),B)) ) ) ).

tff(fact_647_Int__insert__right__if1,axiom,
    ! [B: 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(hAPP_f439328184a_bool(semila1256126856a_bool,A),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),B)) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila1256126856a_bool,A),B)) ) ) ).

tff(fact_648_Int__insert__left__if0,axiom,
    ! [B: fun_nat_bool,A_1: nat,C: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),C))
     => ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B)),C) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,B),C) ) ) ).

tff(fact_649_Int__insert__left__if0,axiom,
    ! [B: fun_Ho1996104121e_bool,A_1: hoare_1875481847_state,C: fun_Ho1996104121e_bool] :
      ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),C))
     => ( hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila657954503e_bool,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),B)),C) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila657954503e_bool,B),C) ) ) ).

tff(fact_650_Int__insert__left__if0,axiom,
    ! [B: fun_Ho287446294a_bool,A_1: hoare_1544627872iple_a,C: fun_Ho287446294a_bool] :
      ( ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),C))
     => ( hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila1256126856a_bool,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),B)),C) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila1256126856a_bool,B),C) ) ) ).

tff(fact_651_Int__insert__right__if0,axiom,
    ! [B: fun_nat_bool,A_1: nat,A: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A))
     => ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),B) ) ) ).

tff(fact_652_Int__insert__right__if0,axiom,
    ! [B: 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(hAPP_f1204759513e_bool(semila657954503e_bool,A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),B)) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila657954503e_bool,A),B) ) ) ).

tff(fact_653_Int__insert__right__if0,axiom,
    ! [B: 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(hAPP_f439328184a_bool(semila1256126856a_bool,A),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),B)) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila1256126856a_bool,A),B) ) ) ).

tff(fact_654_insert__inter__insert,axiom,
    ! [A_1: nat,A: fun_nat_bool,B: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),A)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),B)) ).

tff(fact_655_insert__inter__insert,axiom,
    ! [A_1: hoare_1875481847_state,A: fun_Ho1996104121e_bool,B: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_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)) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila657954503e_bool,A),B)) ).

tff(fact_656_insert__inter__insert,axiom,
    ! [A_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,B: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_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)) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila1256126856a_bool,A),B)) ).

tff(fact_657_Int__insert__left,axiom,
    ! [B: fun_nat_bool,A_1: nat,C: fun_nat_bool] :
      ( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),C))
       => ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B)),C) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,B),C)) ) )
      & ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),C))
       => ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B)),C) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,B),C) ) ) ) ).

tff(fact_658_Int__insert__left,axiom,
    ! [B: fun_Ho1996104121e_bool,A_1: hoare_1875481847_state,C: fun_Ho1996104121e_bool] :
      ( ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),C))
       => ( hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila657954503e_bool,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),B)),C) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila657954503e_bool,B),C)) ) )
      & ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),C))
       => ( hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila657954503e_bool,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),B)),C) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila657954503e_bool,B),C) ) ) ) ).

tff(fact_659_Int__insert__left,axiom,
    ! [B: fun_Ho287446294a_bool,A_1: hoare_1544627872iple_a,C: fun_Ho287446294a_bool] :
      ( ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),C))
       => ( hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila1256126856a_bool,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),B)),C) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila1256126856a_bool,B),C)) ) )
      & ( ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),C))
       => ( hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila1256126856a_bool,hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),B)),C) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila1256126856a_bool,B),C) ) ) ) ).

tff(fact_660_Int__insert__right,axiom,
    ! [B: fun_nat_bool,A_1: nat,A: fun_nat_bool] :
      ( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A))
       => ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),B)) ) )
      & ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A))
       => ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_1),B)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),B) ) ) ) ).

tff(fact_661_Int__insert__right,axiom,
    ! [B: 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(hAPP_f1204759513e_bool(semila657954503e_bool,A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),B)) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila657954503e_bool,A),B)) ) )
      & ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_1),A))
       => ( hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila657954503e_bool,A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_1),B)) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila657954503e_bool,A),B) ) ) ) ).

tff(fact_662_Int__insert__right,axiom,
    ! [B: 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(hAPP_f439328184a_bool(semila1256126856a_bool,A),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),B)) = hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila1256126856a_bool,A),B)) ) )
      & ( ~ hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,A_1),A))
       => ( hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila1256126856a_bool,A),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,A_1),B)) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila1256126856a_bool,A),B) ) ) ) ).

tff(fact_663_inf__Int__eq,axiom,
    ! [R_1: fun_Ho287446294a_bool,S: fun_Ho287446294a_bool,X_2: hoare_1544627872iple_a] :
      ( hBOOL(hAPP_H2006434664a_bool(hAPP_f909437487a_bool(hAPP_f439328184a_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)),X_2))
    <=> hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_2),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila1256126856a_bool,R_1),S))) ) ).

tff(fact_664_inf__Int__eq,axiom,
    ! [R_1: fun_nat_bool,S: fun_nat_bool,X_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),R_1)),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),S)),X_2))
    <=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,R_1),S))) ) ).

tff(fact_665_Collect__conj__eq,axiom,
    ! [Pa: fun_Ho287446294a_bool,Q_1: fun_Ho287446294a_bool] : collec445968053iple_a(hAPP_f909437487a_bool(hAPP_f1099051562a_bool(cOMBS_523298619l_bool,hAPP_f224574947l_bool(hAPP_f11333901l_bool(cOMBB_1960865690iple_a,fconj),Pa)),Q_1)) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila1256126856a_bool,collec445968053iple_a(Pa)),collec445968053iple_a(Q_1)) ).

tff(fact_666_Collect__conj__eq,axiom,
    ! [Pa: fun_nat_bool,Q_1: fun_nat_bool] : collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),Pa)),Q_1)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,collect_nat(Pa)),collect_nat(Q_1)) ).

tff(fact_667_Int__Collect,axiom,
    ! [X_1: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,Pa: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila1256126856a_bool,A),collec445968053iple_a(Pa))))
    <=> ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_1),A))
        & hBOOL(hAPP_H2006434664a_bool(Pa,X_1)) ) ) ).

tff(fact_668_Int__Collect,axiom,
    ! [X_1: nat,A: fun_nat_bool,Pa: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),collect_nat(Pa))))
    <=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
        & hBOOL(hAPP_nat_bool(Pa,X_1)) ) ) ).

tff(fact_669_Int__def,axiom,
    ! [A: fun_Ho287446294a_bool,B: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila1256126856a_bool,A),B) = collec445968053iple_a(hAPP_f909437487a_bool(hAPP_f1099051562a_bool(cOMBS_523298619l_bool,hAPP_f224574947l_bool(hAPP_f11333901l_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))) ).

tff(fact_670_Int__def,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),B) = collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),A))),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),B))) ).

tff(fact_671_Int__iff,axiom,
    ! [Ca: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,B: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila1256126856a_bool,A),B)))
    <=> ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),A))
        & hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),B)) ) ) ).

tff(fact_672_Int__iff,axiom,
    ! [Ca: nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),B)))
    <=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A))
        & hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B)) ) ) ).

tff(fact_673_IntD1,axiom,
    ! [Ca: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,B: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila1256126856a_bool,A),B)))
     => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),A)) ) ).

tff(fact_674_IntD1,axiom,
    ! [Ca: nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),B)))
     => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),A)) ) ).

tff(fact_675_IntD2,axiom,
    ! [Ca: hoare_1544627872iple_a,A: fun_Ho287446294a_bool,B: fun_Ho287446294a_bool] :
      ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila1256126856a_bool,A),B)))
     => hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Ca),B)) ) ).

tff(fact_676_IntD2,axiom,
    ! [Ca: nat,A: fun_nat_bool,B: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),B)))
     => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Ca),B)) ) ).

tff(fact_677_disjoint__iff__not__equal,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] :
      ( ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),B) = bot_bot_fun_nat_bool )
    <=> ! [X_2: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
         => ! [Xa: nat] :
              ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Xa),B))
             => ( X_2 != Xa ) ) ) ) ).

tff(fact_678_disjoint__iff__not__equal,axiom,
    ! [A: fun_Ho1996104121e_bool,B: fun_Ho1996104121e_bool] :
      ( ( hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila657954503e_bool,A),B) = bot_bo1715400655e_bool )
    <=> ! [X_2: hoare_1875481847_state] :
          ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),A))
         => ! [Xa: hoare_1875481847_state] :
              ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,Xa),B))
             => ( X_2 != Xa ) ) ) ) ).

tff(fact_679_disjoint__iff__not__equal,axiom,
    ! [A: fun_Ho287446294a_bool,B: fun_Ho287446294a_bool] :
      ( ( hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila1256126856a_bool,A),B) = bot_bo1766443648a_bool )
    <=> ! [X_2: hoare_1544627872iple_a] :
          ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,X_2),A))
         => ! [Xa: hoare_1544627872iple_a] :
              ( hBOOL(hAPP_f2063540982l_bool(hAPP_H736103437l_bool(member1891733107iple_a,Xa),B))
             => ( X_2 != Xa ) ) ) ) ).

tff(fact_680_Int__empty__right,axiom,
    ! [A: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),bot_bot_fun_nat_bool) = bot_bot_fun_nat_bool ).

tff(fact_681_Int__empty__right,axiom,
    ! [A: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila657954503e_bool,A),bot_bo1715400655e_bool) = bot_bo1715400655e_bool ).

tff(fact_682_Int__empty__right,axiom,
    ! [A: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila1256126856a_bool,A),bot_bo1766443648a_bool) = bot_bo1766443648a_bool ).

tff(fact_683_Int__empty__left,axiom,
    ! [B: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,bot_bot_fun_nat_bool),B) = bot_bot_fun_nat_bool ).

tff(fact_684_Int__empty__left,axiom,
    ! [B: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila657954503e_bool,bot_bo1715400655e_bool),B) = bot_bo1715400655e_bool ).

tff(fact_685_Int__empty__left,axiom,
    ! [B: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila1256126856a_bool,bot_bo1766443648a_bool),B) = bot_bo1766443648a_bool ).

tff(fact_686_inf__bot__left,axiom,
    ! [X_1: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila310582991f_bool,bot_bot_bool),X_1))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_687_inf__bot__left,axiom,
    ! [X_1: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,bot_bot_fun_nat_bool),X_1) = bot_bot_fun_nat_bool ).

tff(fact_688_inf__bot__left,axiom,
    ! [X_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila657954503e_bool,bot_bo1715400655e_bool),X_1) = bot_bo1715400655e_bool ).

tff(fact_689_inf__bot__left,axiom,
    ! [X_1: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila1256126856a_bool,bot_bo1766443648a_bool),X_1) = bot_bo1766443648a_bool ).

tff(fact_690_inf__bot__right,axiom,
    ! [X_1: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila310582991f_bool,X_1),bot_bot_bool))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_691_inf__bot__right,axiom,
    ! [X_1: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,X_1),bot_bot_fun_nat_bool) = bot_bot_fun_nat_bool ).

tff(fact_692_inf__bot__right,axiom,
    ! [X_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila657954503e_bool,X_1),bot_bo1715400655e_bool) = bot_bo1715400655e_bool ).

tff(fact_693_inf__bot__right,axiom,
    ! [X_1: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila1256126856a_bool,X_1),bot_bo1766443648a_bool) = bot_bo1766443648a_bool ).

tff(fact_694_Diff__triv,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] :
      ( ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),B) = bot_bot_fun_nat_bool )
     => ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B) = A ) ) ).

tff(fact_695_Diff__triv,axiom,
    ! [A: fun_Ho1996104121e_bool,B: fun_Ho1996104121e_bool] :
      ( ( hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila657954503e_bool,A),B) = bot_bo1715400655e_bool )
     => ( hAPP_f121055253e_bool(hAPP_f1204759513e_bool(minus_589155914e_bool,A),B) = A ) ) ).

tff(fact_696_Diff__triv,axiom,
    ! [A: fun_Ho287446294a_bool,B: fun_Ho287446294a_bool] :
      ( ( hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila1256126856a_bool,A),B) = bot_bo1766443648a_bool )
     => ( hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,A),B) = A ) ) ).

tff(fact_697_Diff__disjoint,axiom,
    ! [A: fun_nat_bool,B: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,B),A)) = bot_bot_fun_nat_bool ).

tff(fact_698_Diff__disjoint,axiom,
    ! [A: fun_Ho1996104121e_bool,B: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(semila657954503e_bool,A),hAPP_f121055253e_bool(hAPP_f1204759513e_bool(minus_589155914e_bool,B),A)) = bot_bo1715400655e_bool ).

tff(fact_699_Diff__disjoint,axiom,
    ! [A: fun_Ho287446294a_bool,B: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(semila1256126856a_bool,A),hAPP_f909437487a_bool(hAPP_f439328184a_bool(minus_1054521797a_bool,B),A)) = bot_bo1766443648a_bool ).

%----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__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X),Y)) ) ).

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

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

tff(help_COMBK_1_1_COMBK_000tc__Nat__Onat_000tc__Nat__Onat_U,axiom,
    ! [P: nat,Q: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(cOMBK_nat_nat,P),Q) = P ).

tff(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Nat__Onat_U,axiom,
    ! [P: bool,Q: nat] : hAPP_nat_bool(hAPP_b1013836512t_bool(cOMBK_bool_nat,P),Q) = P ).

tff(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Com__Ostate_U,axiom,
    ! [P: bool,Q: state] : hAPP_state_bool(hAPP_b2019457360e_bool(cOMBK_bool_state,P),Q) = P ).

tff(help_fequal_1_1_fequal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_T,axiom,
    ! [X: fun_nat_bool,Y: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(fequal_fun_nat_bool,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_T,axiom,
    ! [X: fun_nat_bool,Y: fun_nat_bool] :
      ( ( X != Y )
      | hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(fequal_fun_nat_bool,X),Y)) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Nat__Onat_000tc__HOL__Obool_U,axiom,
    ! [P: fun_nat_fun_nat_bool,Q: nat,R: nat] : hAPP_nat_bool(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,P),Q),R) = hAPP_nat_bool(hAPP_n1699378549t_bool(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Nat__Onat_U,axiom,
    ! [P: fun_bool_bool,Q: fun_nat_bool,R: nat] : hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f894608603t_bool(cOMBB_bool_bool_nat,P),Q),R) = hAPP_bool_bool(P,hAPP_nat_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
    ! [P: fun_na1469252690l_bool,Q: fun_nat_bool,R: nat] : hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,P),Q),R) = hAPP_bool_bool(hAPP_n1006566506l_bool(P,R),hAPP_nat_bool(Q,R)) ).

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_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(hAPP_f2073279419e_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_COMBC_1_1_COMBC_000tc__Com__Ostate_000tc__Nat__Onat_000tc__Com__Ostate_U,axiom,
    ! [P: fun_st2072884720_state,Q: nat,R: state] : hAPP_state_state(hAPP_n1547241352_state(hAPP_f1848060885_state(cOMBC_1777403949_state,P),Q),R) = hAPP_nat_state(hAPP_s1892499976_state(P,R),Q) ).

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_COMBS_1_1_COMBS_000tc__Com__Ostate_000tc__Nat__Onat_000tc__Com__Ostate_U,axiom,
    ! [P: fun_st2072884720_state,Q: fun_state_nat,R: state] : hAPP_state_state(hAPP_f871651461_state(hAPP_f851239890_state(cOMBS_777315357_state,P),Q),R) = hAPP_nat_state(hAPP_s1892499976_state(P,R),hAPP_state_nat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__Com__Ostate_000tc__HOL__Obool_000tc__Com__Ostate_U,axiom,
    ! [P: fun_state_bool,Q: fun_state_state,R: state] : hAPP_state_bool(hAPP_f746301080e_bool(hAPP_f162060345e_bool(cOMBB_844853809_state,P),Q),R) = hAPP_state_bool(P,hAPP_state_state(Q,R)) ).

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_COMBI_1_1_COMBI_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_U,axiom,
    ! [P: hoare_1544627872iple_a] : hAPP_H9461717iple_a(cOMBI_1396158237iple_a,P) = P ).

tff(help_fequal_1_1_fequal_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_,axiom,
    ! [X: hoare_1544627872iple_a,Y: hoare_1544627872iple_a] :
      ( ~ hBOOL(hAPP_H2006434664a_bool(hAPP_H1472549281a_bool(fequal1057774495iple_a,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_,axiom,
    ! [X: hoare_1544627872iple_a,Y: hoare_1544627872iple_a] :
      ( ( X != Y )
      | hBOOL(hAPP_H2006434664a_bool(hAPP_H1472549281a_bool(fequal1057774495iple_a,X),Y)) ) ).

tff(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool,axiom,
    ! [P: bool,Q: fun_nat_bool] : hAPP_f54304608l_bool(hAPP_b1630757474l_bool(cOMBK_1994329625t_bool,P),Q) = P ).

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_COMBI_1_1_COMBI_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__,axiom,
    ! [P: hoare_1875481847_state] : hAPP_H350317809_state(cOMBI_227521594_state,P) = P ).

tff(help_fequal_1_1_fequal_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com,axiom,
    ! [X: hoare_1875481847_state,Y: hoare_1875481847_state] :
      ( ~ hBOOL(hAPP_H1173941847e_bool(hAPP_H117293375e_bool(fequal391417144_state,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com,axiom,
    ! [X: hoare_1875481847_state,Y: hoare_1875481847_state] :
      ( ( X != Y )
      | hBOOL(hAPP_H1173941847e_bool(hAPP_H117293375e_bool(fequal391417144_state,X),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__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_00,axiom,
    ! [P: hoare_1544627872iple_a,Q: nat] : hAPP_n355976171iple_a(hAPP_H325842184iple_a(cOMBK_830162040_a_nat,P),Q) = P ).

tff(help_COMBK_1_1_COMBK_000tc__Nat__Onat_000tc__Hoare____Mirabelle____xlrqixeqwe__O,axiom,
    ! [P: nat,Q: hoare_1544627872iple_a] : hAPP_H63220799_a_nat(hAPP_n1125129074_a_nat(cOMBK_1122917412iple_a,P),Q) = P ).

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__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_,axiom,
    ! [P: fun_nat_fun_nat_bool,Q: fun_nat_nat,R: nat] : hAPP_n1699378549t_bool(hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,P),Q),R) = hAPP_n1699378549t_bool(P,hAPP_nat_nat(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obo,axiom,
    ! [P: fun_fu425979586l_bool,Q: fun_nat_fun_nat_bool,R: nat] : hAPP_nat_bool(hAPP_f158894502t_bool(hAPP_f1777703707t_bool(cOMBB_955900739ol_nat,P),Q),R) = hAPP_f54304608l_bool(P,hAPP_n1699378549t_bool(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_,axiom,
    ! [P: fun_na1436237685l_bool,Q: fun_nat_bool,R: nat] : hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,P),Q),R) = hAPP_f54304608l_bool(hAPP_n215258509l_bool(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__fun_Itc__Nat__On,axiom,
    ! [P: fun_bool_bool,Q: fun_fu425979586l_bool,R: fun_nat_bool] : hAPP_f54304608l_bool(hAPP_f1246832597l_bool(hAPP_f1164891443l_bool(cOMBB_238756964t_bool,P),Q),R) = hAPP_bool_bool(P,hAPP_f54304608l_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo,axiom,
    ! [P: fun_bo1549164019l_bool,Q: fun_nat_bool,R: nat] : hAPP_n1006566506l_bool(hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,P),Q),R) = hAPP_b589554111l_bool(P,hAPP_nat_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obo,axiom,
    ! [P: fun_fu1016514960l_bool,Q: fun_fu425979586l_bool,R: fun_nat_bool] : hAPP_f54304608l_bool(hAPP_f1246832597l_bool(hAPP_f1743029098l_bool(cOMBS_1187019125l_bool,P),Q),R) = hAPP_bool_bool(hAPP_f1748468828l_bool(P,R),hAPP_f54304608l_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_210,axiom,
    ! [P: fun_bo1549164019l_bool,Q: fun_state_bool,R: state] : hAPP_s58564346l_bool(hAPP_f1259673775l_bool(hAPP_f1561913689l_bool(cOMBB_188601460_state,P),Q),R) = hAPP_b589554111l_bool(P,hAPP_state_bool(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Com__Ostate_000tc__Com__Ovname_000tc__fun_Itc__Nat__,axiom,
    ! [P: fun_st1816159843_state,Q: vname,R: state] : hAPP_s1892499976_state(hAPP_v365393659_state(hAPP_f1542232213_state(cOMBC_1193272608_state,P),Q),R) = hAPP_v594194232_state(hAPP_s712361723_state(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_COMBK_1_1_COMBK_000tc__Nat__Onat_000tc__Hoare____Mirabelle____xlrqixeqwe__O_211,axiom,
    ! [P: nat,Q: hoare_1875481847_state] : hAPP_H1027528080te_nat(hAPP_n747866297te_nat(cOMBK_809880179_state,P),Q) = P ).

tff(help_fequal_1_1_fequal_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple,axiom,
    ! [X: fun_Ho287446294a_bool,Y: fun_Ho287446294a_bool] :
      ( ~ hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(fequal1843628375a_bool,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple,axiom,
    ! [X: fun_Ho287446294a_bool,Y: fun_Ho287446294a_bool] :
      ( ( X != Y )
      | hBOOL(hAPP_f2063540982l_bool(hAPP_f1876759039l_bool(fequal1843628375a_bool,X),Y)) ) ).

tff(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Hoare____Mirabelle____xlrqixeqwe___212,axiom,
    ! [P: bool,Q: hoare_1875481847_state] : hAPP_H1173941847e_bool(hAPP_b1906038105e_bool(cOMBK_2086958736_state,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(hAPP_f1524052231a_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_fequal_1_1_fequal_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_213,axiom,
    ! [X: fun_Ho1996104121e_bool,Y: fun_Ho1996104121e_bool] :
      ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(fequal259518520e_bool,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_214,axiom,
    ! [X: fun_Ho1996104121e_bool,Y: fun_Ho1996104121e_bool] :
      ( ( X != Y )
      | hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(fequal259518520e_bool,X),Y)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc__fun_It,axiom,
    ! [P: fun_fu373216837e_bool,Q: fun_a_fun_state_bool,R: x_a] : hAPP_a2036067514e_bool(hAPP_f769584981e_bool(hAPP_f1159960589e_bool(cOMBB_237455441bool_a,P),Q),R) = hAPP_f1759915619e_bool(P,hAPP_a2036067514e_bool(Q,R)) ).

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_COMBC_1_1_COMBC_000t__a_000tc__fun_Itc__Com__Ostate_Mtc__Com__Ostate_J_000t,axiom,
    ! [P: fun_a_340690546e_bool,Q: fun_state_state,R: x_a] : hAPP_a2036067514e_bool(hAPP_f100967412e_bool(hAPP_f1460451647e_bool(cOMBC_524597097e_bool,P),Q),R) = hAPP_f746301080e_bool(hAPP_a1224971408e_bool(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Hoare____Mirabel_215,axiom,
    ! [P: fun_bool_bool,Q: fun_Ho1996104121e_bool,R: hoare_1875481847_state] : hAPP_H1173941847e_bool(hAPP_f121055253e_bool(hAPP_f263911411e_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__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool__216,axiom,
    ! [P: fun_na35521688t_bool,Q: fun_nat_bool,R: nat] : hAPP_n1699378549t_bool(hAPP_f1570313510t_bool(hAPP_f688831301t_bool(cOMBC_178881787t_bool,P),Q),R) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc_,axiom,
    ! [P: fun_fu1217155507l_bool,Q: fun_nat_bool,R: fun_nat_bool] : hAPP_f54304608l_bool(hAPP_f103356543l_bool(hAPP_f643944041l_bool(cOMBC_1693257480l_bool,P),Q),R) = hAPP_f54304608l_bool(hAPP_f103356543l_bool(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_217,axiom,
    ! [P: fun_bo1549164019l_bool,Q: fun_fu425979586l_bool,R: fun_nat_bool] : hAPP_f1748468828l_bool(hAPP_f1974927549l_bool(hAPP_f857404385l_bool(cOMBB_444170502t_bool,P),Q),R) = hAPP_b589554111l_bool(P,hAPP_f54304608l_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc__fun_It_218,axiom,
    ! [P: fun_fu373216837e_bool,Q: fun_st1506752259e_bool,R: state] : hAPP_s1806633685e_bool(hAPP_f817621513e_bool(hAPP_f512427579e_bool(cOMBB_572666224_state,P),Q),R) = hAPP_f1759915619e_bool(P,hAPP_s1806633685e_bool(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Com__Ostate_000tc__fun_Itc__Com__Ostate_Mtc__Com__Os,axiom,
    ! [P: fun_st623980505e_bool,Q: fun_state_state,R: state] : hAPP_s1806633685e_bool(hAPP_f1838002347e_bool(hAPP_f1706273077e_bool(cOMBC_867582640e_bool,P),Q),R) = hAPP_f746301080e_bool(hAPP_s286259371e_bool(P,R),Q) ).

tff(help_COMBK_1_1_COMBK_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_00_219,axiom,
    ! [P: hoare_1544627872iple_a,Q: hoare_1544627872iple_a] : hAPP_H9461717iple_a(hAPP_H1988213490iple_a(cOMBK_1156021774iple_a,P),Q) = P ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_220,axiom,
    ! [P: fun_bo1549164019l_bool,Q: fun_Ho287446294a_bool,R: hoare_1544627872iple_a] : hAPP_H320345684l_bool(hAPP_f224574947l_bool(hAPP_f11333901l_bool(cOMBB_1960865690iple_a,P),Q),R) = hAPP_b589554111l_bool(P,hAPP_H2006434664a_bool(Q,R)) ).

tff(help_COMBK_1_1_COMBK_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_00_221,axiom,
    ! [P: hoare_1544627872iple_a,Q: hoare_1875481847_state] : hAPP_H1771183910iple_a(hAPP_H180680677iple_a(cOMBK_875269129_state,P),Q) = P ).

tff(help_COMBK_1_1_COMBK_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__,axiom,
    ! [P: hoare_1875481847_state,Q: hoare_1544627872iple_a] : hAPP_H1928101250_state(hAPP_H296470442_state(cOMBK_718351789iple_a,P),Q) = P ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc__fun_It_222,axiom,
    ! [P: fun_fu1658206819l_bool,Q: fun_a_fun_state_bool,R: x_a] : hAPP_a849909144l_bool(hAPP_f1509969235l_bool(hAPP_f1178339559l_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_223,axiom,
    ! [P: fun_bo1549164019l_bool,Q: fun_Ho1996104121e_bool,R: hoare_1875481847_state] : hAPP_H377399461l_bool(hAPP_f932622461l_bool(hAPP_f1158435105l_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_COMBC_1_1_COMBC_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__224,axiom,
    ! [P: fun_fu915923926t_bool,Q: fun_nat_bool,R: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(hAPP_f1311642927t_bool(cOMBC_646315179t_bool,P),Q),R) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__225,axiom,
    ! [P: fun_fu1436348701l_bool,Q: fun_fu425979586l_bool,R: fun_nat_bool] : hAPP_f54304608l_bool(hAPP_f1246832597l_bool(hAPP_f1443436725l_bool(cOMBC_1245412066l_bool,P),Q),R) = hAPP_f1637334154l_bool(hAPP_f1951378235l_bool(P,R),Q) ).

tff(help_COMBK_1_1_COMBK_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com___226,axiom,
    ! [P: hoare_1875481847_state,Q: hoare_1875481847_state] : hAPP_H350317809_state(hAPP_H743221337_state(cOMBK_477315498_state,P),Q) = P ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc__fun_It_227,axiom,
    ! [P: fun_fu1658206819l_bool,Q: fun_st1506752259e_bool,R: state] : hAPP_s2001034685l_bool(hAPP_f1283379615l_bool(hAPP_f1345202233l_bool(cOMBB_1759179140_state,P),Q),R) = hAPP_f1259673775l_bool(P,hAPP_s1806633685e_bool(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(hAPP_f1006724181e_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_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc__fun_It_228,axiom,
    ! [P: fun_fu2127278875e_bool,Q: fun_a_fun_state_bool,R: x_a] : hAPP_a1224971408e_bool(hAPP_f1406200875e_bool(hAPP_f635443597e_bool(cOMBB_2139825703bool_a,P),Q),R) = hAPP_f162060345e_bool(P,hAPP_a2036067514e_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc_,axiom,
    ! [P: fun_fu915923926t_bool,Q: fun_nat_fun_nat_bool,R: nat] : hAPP_n1512601776t_bool(hAPP_f10625010t_bool(hAPP_f1115950719t_bool(cOMBB_1654519265ol_nat,P),Q),R) = hAPP_f1730770594t_bool(P,hAPP_n1699378549t_bool(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(hAPP_f1948079815a_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_229,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__fun_Itc__Com__Ostate_Mtc__fun_Itc__HOL__Obool_Mtc__H_230,axiom,
    ! [P: fun_fu1219323149e_bool,Q: fun_st2116860965l_bool,R: state] : hAPP_s1874344717e_bool(hAPP_f873506917e_bool(hAPP_f1398071125e_bool(cOMBB_1757942702_state,P),Q),R) = hAPP_f167292325e_bool(P,hAPP_s2001034685l_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc__fun_It_231,axiom,
    ! [P: fun_fu2127278875e_bool,Q: fun_st1506752259e_bool,R: state] : hAPP_s286259371e_bool(hAPP_f289738463e_bool(hAPP_f1151843515e_bool(cOMBB_1941618714_state,P),Q),R) = hAPP_f162060345e_bool(P,hAPP_s1806633685e_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_I_232,axiom,
    ! [P: fun_fu689207471l_bool,Q: fun_Ho1695028467e_bool,R: hoare_1875481847_state] : hAPP_H1173941847e_bool(hAPP_f169892419e_bool(hAPP_f1983876009e_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___233,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__fun_Itc__Com__Ostate_Mtc__fun_Itc__HOL__Obool_Mtc__H_234,axiom,
    ! [P: fun_fu1047394976e_bool,Q: fun_a_1632297036l_bool,R: x_a] : hAPP_a1200519163e_bool(hAPP_f963367678e_bool(hAPP_f375255701e_bool(cOMBB_145932198bool_a,P),Q),R) = hAPP_f644196280e_bool(P,hAPP_a849909144l_bool(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It__a_J_00_235,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_COMBC_1_1_COMBC_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_I,axiom,
    ! [P: fun_fu601883884a_bool,Q: fun_Ho287446294a_bool,R: fun_Ho287446294a_bool] : hAPP_f909437487a_bool(hAPP_f439328184a_bool(hAPP_f1118475355a_bool(cOMBC_1385723501a_bool,P),Q),R) = hAPP_f909437487a_bool(hAPP_f439328184a_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com___236,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_COMBC_1_1_COMBC_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_I_237,axiom,
    ! [P: fun_fu947547457e_bool,Q: fun_Ho1996104121e_bool,R: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_f1204759513e_bool(hAPP_f961873429e_bool(cOMBC_1168545212e_bool,P),Q),R) = hAPP_f121055253e_bool(hAPP_f1204759513e_bool(P,R),Q) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_I_238,axiom,
    ! [P: fun_fu601883884a_bool,Q: fun_Ho965678415a_bool,R: hoare_1544627872iple_a] : hAPP_H762155206a_bool(hAPP_f915134428a_bool(hAPP_f1987684735a_bool(cOMBB_958433249iple_a,P),Q),R) = hAPP_f439328184a_bool(P,hAPP_H1472549281a_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_I_239,axiom,
    ! [P: fun_fu947547457e_bool,Q: fun_Ho1695028467e_bool,R: hoare_1875481847_state] : hAPP_H1625489667e_bool(hAPP_f1821865133e_bool(hAPP_f967770723e_bool(cOMBB_107478263_state,P),Q),R) = hAPP_f1204759513e_bool(P,hAPP_H117293375e_bool(Q,R)) ).

%----Conjectures (1)
tff(conj_0,conjecture,
    hBOOL(hAPP_f2063540982l_bool(hoare_1546678894rivs_a(g),hAPP_f909437487a_bool(hAPP_H762155206a_bool(insert1051021594iple_a,hoare_196563068iple_a(hAPP_f762886889e_bool(cOMBK_1458035955bool_a,hAPP_b2019457360e_bool(cOMBK_bool_state,fFalse)),c,hAPP_f762886889e_bool(hAPP_f1261923407e_bool(cOMBC_892787026e_bool,hAPP_f963367678e_bool(hAPP_f375255701e_bool(cOMBB_145932198bool_a,cOMBS_1378840469l_bool),hAPP_f1509969235l_bool(hAPP_f1178339559l_bool(cOMBB_1355796797bool_a,hAPP_f1561913689l_bool(cOMBB_188601460_state,fconj)),p))),hAPP_f1759915619e_bool(hAPP_f2073279419e_bool(cOMBB_160679318_state,fNot),b)))),bot_bo1766443648a_bool))) ).

%------------------------------------------------------------------------------