%------------------------------------------------------------------------------
% File     : SWW472_2 : TPTP v8.2.0. Released v5.3.0.
% Domain   : Software Verification
% Problem  : Hoare's Logic with Procedures line 327, 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_l327 [Bla11]

% Status   : Theorem
% Rating   : 1.00 v5.5.0, 0.75 v5.4.0, 1.00 v5.3.0
% Syntax   : Number of formulae    :  445 (  58 unt; 106 typ;   0 def)
%            Number of atoms       :  965 ( 192 equ)
%            Maximal formula atoms :    9 (   2 avg)
%            Number of connectives :  742 ( 116   ~;  28   |;  36   &)
%                                         (  84 <=>; 478  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   16 (   6 avg)
%            Maximal term depth    :    8 (   2 avg)
%            Number of types       :   27 (  26 usr)
%            Number of type conns  :   86 (  55   >;  31   *;   0   +;   0  <<)
%            Number of predicates  :    2 (   1 usr;   0 prp; 1-2 aty)
%            Number of functors    :   79 (  79 usr;  25 con; 0-3 aty)
%            Number of variables   :  972 ( 954   !;  18   ?; 972   :)
% SPC      : TF0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2011-08-09 17:35:04
%------------------------------------------------------------------------------
%----Should-be-implicit typings (26)
tff(ty_ty_tc__Com__Ocom,type,
    com: $tType ).

tff(ty_ty_tc__Com__Ostate,type,
    state: $tType ).

tff(ty_ty_tc__HOL__Obool,type,
    bool: $tType ).

tff(ty_ty_tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostate_J,type,
    hoare_1875481847_state: $tType ).

tff(ty_ty_tc__Nat__Onat,type,
    nat: $tType ).

tff(ty_ty_tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J,type,
    fun_state_bool: $tType ).

tff(ty_ty_tc__fun_Itc__Com__Ostate_Mtc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_J,type,
    fun_st1506752259e_bool: $tType ).

tff(ty_ty_tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J,type,
    fun_bool_bool: $tType ).

tff(ty_ty_tc__fun_Itc__HOL__Obool_Mtc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J_J,type,
    fun_bo1549164019l_bool: $tType ).

tff(ty_ty_tc__fun_Itc__HOL__Obool_Mtc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Ot,type,
    fun_bo221168653e_bool: $tType ).

tff(ty_ty_tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostate_J_,type,
    fun_Ho1996104121e_bool: $tType ).

tff(ty_ty_tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostate_J__001,type,
    fun_Ho1455141843_state: $tType ).

tff(ty_ty_tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostate_J__002,type,
    fun_Ho1284378692te_nat: $tType ).

tff(ty_ty_tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostate_J__003,type,
    fun_Ho1896286681l_bool: $tType ).

tff(ty_ty_tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostate_J__004,type,
    fun_Ho1695028467e_bool: $tType ).

tff(ty_ty_tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostate_J__005,type,
    fun_Ho784854669_state: $tType ).

tff(ty_ty_tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostate_J__006,type,
    fun_Ho1085560331l_bool: $tType ).

tff(ty_ty_tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostate_J__007,type,
    fun_Ho1110608055e_bool: $tType ).

tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    fun_nat_bool: $tType ).

tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J,type,
    fun_nat_nat: $tType ).

tff(ty_ty_tc__fun_Itc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__,type,
    fun_fu689207471l_bool: $tType ).

tff(ty_ty_tc__fun_Itc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com___008,type,
    fun_fu385705929_state: $tType ).

tff(ty_ty_tc__fun_Itc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com___009,type,
    fun_fu1296727421e_bool: $tType ).

tff(ty_ty_tc__fun_Itc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com___010,type,
    fun_fu811519275l_bool: $tType ).

tff(ty_ty_tc__fun_Itc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com___011,type,
    fun_fu1173762305l_bool: $tType ).

tff(ty_ty_tc__fun_Itc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com___012,type,
    fun_fu947547457e_bool: $tType ).

%----Explicit typings (80)
tff(sy_c_Big__Operators_Ocomm__monoid__add__class_Osetsum_000tc__Hoare____Mirabelle_,type,
    big_co583362288te_nat: ( fun_Ho1284378692te_nat * fun_Ho1996104121e_bool ) > nat ).

tff(sy_c_Big__Operators_Osemilattice__big_000tc__Hoare____Mirabelle____xlrqixeqwe__O,type,
    big_se163227553_state: ( fun_Ho784854669_state * fun_fu385705929_state ) > bool ).

tff(sy_c_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Hoare____Mirabelle____xlrq,type,
    cOMBB_1424001755_state: fun_bool_bool > fun_fu1296727421e_bool ).

tff(sy_c_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J_000tc_,type,
    cOMBB_242064253_state: fun_bo1549164019l_bool > fun_fu811519275l_bool ).

tff(sy_c_COMBC_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostate_J_0,type,
    cOMBC_1972244470e_bool: fun_Ho1695028467e_bool > fun_Ho1695028467e_bool ).

tff(sy_c_COMBC_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostate_J_0_013,type,
    cOMBC_1099403170l_bool: fun_Ho1085560331l_bool > fun_fu1296727421e_bool ).

tff(sy_c_COMBK_000tc__HOL__Obool_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_It,type,
    cOMBK_2086958736_state: fun_bo221168653e_bool ).

tff(sy_c_COMBS_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostate_J_0,type,
    cOMBS_896408300l_bool: fun_Ho1896286681l_bool > fun_fu1296727421e_bool ).

tff(sy_c_Com_Ocom_OSKIP,type,
    skip: com ).

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

tff(sy_c_Finite__Set_Ocomp__fun__commute_000tc__Hoare____Mirabelle____xlrqixeqwe__Ot,type,
    finite1496568813_state: fun_Ho784854669_state > bool ).

tff(sy_c_Finite__Set_Ocomp__fun__idem_000tc__Hoare____Mirabelle____xlrqixeqwe__Otrip,type,
    finite158617192_state: fun_Ho784854669_state > bool ).

tff(sy_c_Finite__Set_Ocomp__fun__idem_000tc__Hoare____Mirabelle____xlrqixeqwe__Otrip_014,type,
    finite1936929544e_bool: fun_Ho1110608055e_bool > 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_Ofold1Set_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc_,type,
    finite1016865319_state: fun_Ho784854669_state > fun_fu1296727421e_bool ).

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_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_015,type,
    finite1131674252e_bool: ( fun_Ho1110608055e_bool * fun_Ho1996104121e_bool ) > fun_fu1296727421e_bool ).

tff(sy_c_Finite__Set_Ofold__graph_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_I,type,
    finite1721989730_state: fun_Ho784854669_state > fun_Ho1110608055e_bool ).

tff(sy_c_Finite__Set_Ofolding__one_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_,type,
    finite1782051386_state: ( fun_Ho784854669_state * fun_fu385705929_state ) > bool ).

tff(sy_c_Finite__Set_Ofolding__one__idem_000tc__Hoare____Mirabelle____xlrqixeqwe__Ot,type,
    finite1514162941_state: ( fun_Ho784854669_state * fun_fu385705929_state ) > bool ).

tff(sy_c_Groups_Ominus__class_Ominus_000tc__HOL__Obool,type,
    minus_minus_bool: bool > fun_bool_bool ).

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

tff(sy_c_Groups_Ominus__class_Ominus_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqw,type,
    minus_589155914e_bool: fun_Ho1996104121e_bool > fun_fu1296727421e_bool ).

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

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

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

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

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_Orderings_Obot__class_Obot_000tc__HOL__Obool,type,
    bot_bot_bool: bool ).

tff(sy_c_Orderings_Obot__class_Obot_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe,type,
    bot_bo1715400655e_bool: fun_Ho1996104121e_bool ).

tff(sy_c_Orderings_Oord_Omax_000tc__HOL__Obool,type,
    max_bool: fun_bo1549164019l_bool > fun_bo1549164019l_bool ).

tff(sy_c_Orderings_Oord_Omax_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otrip,type,
    max_fu690965116e_bool: fun_fu1173762305l_bool > fun_fu947547457e_bool ).

tff(sy_c_Orderings_Oord_Omin_000tc__HOL__Obool,type,
    min_bool: fun_bo1549164019l_bool > fun_bo1549164019l_bool ).

tff(sy_c_Orderings_Oord_Omin_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otrip,type,
    min_fu1173241102e_bool: fun_fu1173762305l_bool > fun_fu947547457e_bool ).

tff(sy_c_Orderings_Oord__class_Oless_000tc__HOL__Obool,type,
    ord_less_bool: bool > fun_bool_bool ).

tff(sy_c_Orderings_Oord__class_Oless_000tc__Nat__Onat,type,
    ord_less_nat: nat > fun_nat_bool ).

tff(sy_c_Orderings_Oord__class_Oless_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqw,type,
    ord_le920868879e_bool: fun_Ho1996104121e_bool > fun_fu689207471l_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__fun_Itc__Hoare____Mirabelle____xlrqi,type,
    ord_le233469699e_bool: fun_fu1173762305l_bool ).

tff(sy_c_Orderings_Oord__class_Omax_000tc__HOL__Obool,type,
    ord_max_bool: fun_bo1549164019l_bool ).

tff(sy_c_Orderings_Oord__class_Omax_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe,type,
    ord_ma955633336e_bool: fun_fu947547457e_bool ).

tff(sy_c_Orderings_Oord__class_Omin_000tc__HOL__Obool,type,
    ord_min_bool: fun_bo1549164019l_bool ).

tff(sy_c_Orderings_Oord__class_Omin_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe,type,
    ord_mi1437909322e_bool: fun_fu947547457e_bool ).

tff(sy_c_Set_OCollect_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ost,type,
    collec1735317154_state: fun_fu1296727421e_bool ).

tff(sy_c_Set_Oinsert_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Osta,type,
    insert694999549_state: fun_Ho1110608055e_bool ).

tff(sy_c_Set_Othe__elem_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__O,type,
    the_el1031305112_state: fun_fu385705929_state ).

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__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostate_J,type,
    fequal391417144_state: fun_Ho1695028467e_bool ).

tff(sy_c_fimplies,type,
    fimplies: fun_bo1549164019l_bool ).

tff(sy_c_hAPP_000tc__Com__Ostate_000tc__HOL__Obool,type,
    hAPP_state_bool: ( fun_state_bool * state ) > bool ).

tff(sy_c_hAPP_000tc__Com__Ostate_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J,type,
    hAPP_s1806633685e_bool: ( fun_st1506752259e_bool * state ) > fun_state_bool ).

tff(sy_c_hAPP_000tc__HOL__Obool_000tc__HOL__Obool,type,
    hAPP_bool_bool: ( fun_bool_bool * bool ) > bool ).

tff(sy_c_hAPP_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J,type,
    hAPP_b589554111l_bool: ( fun_bo1549164019l_bool * bool ) > fun_bool_bool ).

tff(sy_c_hAPP_000tc__HOL__Obool_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Ot,type,
    hAPP_b1906038105e_bool: ( fun_bo221168653e_bool * bool ) > fun_Ho1996104121e_bool ).

tff(sy_c_hAPP_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostate_J_00,type,
    hAPP_H1173941847e_bool: ( fun_Ho1996104121e_bool * hoare_1875481847_state ) > bool ).

tff(sy_c_hAPP_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostate_J_00_016,type,
    hAPP_H350317809_state: ( fun_Ho1455141843_state * hoare_1875481847_state ) > hoare_1875481847_state ).

tff(sy_c_hAPP_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostate_J_00_017,type,
    hAPP_H1027528080te_nat: ( fun_Ho1284378692te_nat * hoare_1875481847_state ) > nat ).

tff(sy_c_hAPP_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostate_J_00_018,type,
    hAPP_H377399461l_bool: ( fun_Ho1896286681l_bool * hoare_1875481847_state ) > fun_bool_bool ).

tff(sy_c_hAPP_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostate_J_00_019,type,
    hAPP_H117293375e_bool: ( fun_Ho1695028467e_bool * hoare_1875481847_state ) > fun_Ho1996104121e_bool ).

tff(sy_c_hAPP_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostate_J_00_020,type,
    hAPP_H743221337_state: ( fun_Ho784854669_state * hoare_1875481847_state ) > fun_Ho1455141843_state ).

tff(sy_c_hAPP_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostate_J_00_021,type,
    hAPP_H810811817l_bool: ( fun_Ho1085560331l_bool * hoare_1875481847_state ) > fun_fu689207471l_bool ).

tff(sy_c_hAPP_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostate_J_00_022,type,
    hAPP_H1625489667e_bool: ( fun_Ho1110608055e_bool * hoare_1875481847_state ) > fun_fu1296727421e_bool ).

tff(sy_c_hAPP_000tc__Nat__Onat_000tc__HOL__Obool,type,
    hAPP_nat_bool: ( fun_nat_bool * nat ) > bool ).

tff(sy_c_hAPP_000tc__Nat__Onat_000tc__Nat__Onat,type,
    hAPP_nat_nat: ( fun_nat_nat * nat ) > nat ).

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

tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Os_023,type,
    hAPP_f890249627_state: ( fun_fu385705929_state * fun_Ho1996104121e_bool ) > hoare_1875481847_state ).

tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Os_024,type,
    hAPP_f121055253e_bool: ( fun_fu1296727421e_bool * fun_Ho1996104121e_bool ) > fun_Ho1996104121e_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Os_025,type,
    hAPP_f932622461l_bool: ( fun_fu811519275l_bool * fun_Ho1996104121e_bool ) > fun_Ho1896286681l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Os_026,type,
    hAPP_f1873026899l_bool: ( fun_fu1173762305l_bool * fun_Ho1996104121e_bool ) > fun_fu689207471l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Os_027,type,
    hAPP_f1204759513e_bool: ( fun_fu947547457e_bool * fun_Ho1996104121e_bool ) > fun_fu1296727421e_bool ).

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

tff(sy_c_member_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__Ostate_J,type,
    member618553700_state: fun_Ho1085560331l_bool ).

tff(sy_v_P,type,
    p: fun_st1506752259e_bool ).

tff(sy_v_Q,type,
    q: fun_st1506752259e_bool ).

tff(sy_v_c,type,
    c: com ).

%----Relevant facts (315)
tff(fact_0_empty,axiom,
    ! [G_2: fun_Ho1996104121e_bool] : hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(G_2),bot_bo1715400655e_bool)) ).

tff(fact_1_triple_Oinject,axiom,
    ! [Fun1_2: fun_st1506752259e_bool,Com_2: com,Fun2_2: fun_st1506752259e_bool,Fun1_1: fun_st1506752259e_bool,Com_1: com,Fun2_1: fun_st1506752259e_bool] :
      ( ( hoare_13225861_state(Fun1_2,Com_2,Fun2_2) = hoare_13225861_state(Fun1_1,Com_1,Fun2_1) )
    <=> ( ( Fun1_2 = Fun1_1 )
        & ( Com_2 = Com_1 )
        & ( Fun2_2 = Fun2_1 ) ) ) ).

tff(fact_2_hoare__sound,axiom,
    ! [G_2: fun_Ho1996104121e_bool,Ts: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(G_2),Ts))
     => hBOOL(hAPP_f1408815105l_bool(hoare_390430481_state(G_2),Ts)) ) ).

tff(fact_3_cut,axiom,
    ! [G_2: 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(G_2),G_1))
       => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(G_2),Ts)) ) ) ).

tff(fact_4_hoare__derivs_Oinsert,axiom,
    ! [Ts: fun_Ho1996104121e_bool,G_2: fun_Ho1996104121e_bool,T: hoare_1875481847_state] :
      ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(G_2),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,T),bot_bo1715400655e_bool)))
     => ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(G_2),Ts))
       => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(G_2),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,T),Ts))) ) ) ).

tff(fact_5_derivs__insertD,axiom,
    ! [G_2: fun_Ho1996104121e_bool,T: hoare_1875481847_state,Ts: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(G_2),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,T),Ts)))
     => ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(G_2),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,T),bot_bo1715400655e_bool)))
        & hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(G_2),Ts)) ) ) ).

tff(fact_6_conseq2,axiom,
    ! [Qa: fun_st1506752259e_bool,G_2: fun_Ho1996104121e_bool,Pa: fun_st1506752259e_bool,Ca: com,Q_2: fun_st1506752259e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(G_2),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hoare_13225861_state(Pa,Ca,Q_2)),bot_bo1715400655e_bool)))
     => ( ! [Z_1: state,S: state] :
            ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Q_2,Z_1),S))
           => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Qa,Z_1),S)) )
       => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(G_2),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hoare_13225861_state(Pa,Ca,Qa)),bot_bo1715400655e_bool))) ) ) ).

tff(fact_7_conseq1,axiom,
    ! [Pa: fun_st1506752259e_bool,G_2: fun_Ho1996104121e_bool,P_2: fun_st1506752259e_bool,Ca: com,Qa: fun_st1506752259e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(G_2),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hoare_13225861_state(P_2,Ca,Qa)),bot_bo1715400655e_bool)))
     => ( ! [Z_1: state,S: state] :
            ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Pa,Z_1),S))
           => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(P_2,Z_1),S)) )
       => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(G_2),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hoare_13225861_state(Pa,Ca,Qa)),bot_bo1715400655e_bool))) ) ) ).

tff(fact_8_insertE,axiom,
    ! [A: hoare_1875481847_state,B: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,B),A_1)))
     => ( ( A != B )
       => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A),A_1)) ) ) ).

tff(fact_9_insertCI,axiom,
    ! [B: hoare_1875481847_state,A: hoare_1875481847_state,B_1: fun_Ho1996104121e_bool] :
      ( ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A),B_1))
       => ( A = B ) )
     => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,B),B_1))) ) ).

tff(fact_10_conseq12,axiom,
    ! [Qa: fun_st1506752259e_bool,Pa: fun_st1506752259e_bool,G_2: fun_Ho1996104121e_bool,P_2: fun_st1506752259e_bool,Ca: com,Q_2: fun_st1506752259e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(G_2),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hoare_13225861_state(P_2,Ca,Q_2)),bot_bo1715400655e_bool)))
     => ( ! [Z_1: state,S: state] :
            ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Pa,Z_1),S))
           => ! [S_1: state] :
                ( ! [Z_2: state] :
                    ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(P_2,Z_2),S))
                   => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Q_2,Z_2),S_1)) )
               => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Qa,Z_1),S_1)) ) )
       => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(G_2),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hoare_13225861_state(Pa,Ca,Qa)),bot_bo1715400655e_bool))) ) ) ).

tff(fact_11_emptyE,axiom,
    ! [A: hoare_1875481847_state] : ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A),bot_bo1715400655e_bool)) ).

tff(fact_12_empty__not__insert,axiom,
    ! [A: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool] : bot_bo1715400655e_bool != hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),A_1) ).

tff(fact_13_insert__not__empty,axiom,
    ! [A: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),A_1) != bot_bo1715400655e_bool ).

tff(fact_14_singleton__iff,axiom,
    ! [B: hoare_1875481847_state,A: hoare_1875481847_state] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,B),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),bot_bo1715400655e_bool)))
    <=> ( B = A ) ) ).

tff(fact_15_doubleton__eq__iff,axiom,
    ! [A: hoare_1875481847_state,B: hoare_1875481847_state,Ca: hoare_1875481847_state,D_1: hoare_1875481847_state] :
      ( ( hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,B),bot_bo1715400655e_bool)) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,Ca),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,D_1),bot_bo1715400655e_bool)) )
    <=> ( ( ( A = Ca )
          & ( B = D_1 ) )
        | ( ( A = D_1 )
          & ( B = Ca ) ) ) ) ).

tff(fact_16_equals0D,axiom,
    ! [A: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool] :
      ( ( A_1 = bot_bo1715400655e_bool )
     => ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A),A_1)) ) ).

tff(fact_17_Collect__empty__eq,axiom,
    ! [Pa: fun_Ho1996104121e_bool] :
      ( ( hAPP_f121055253e_bool(collec1735317154_state,Pa) = bot_bo1715400655e_bool )
    <=> ! [X_1: hoare_1875481847_state] : ~ hBOOL(hAPP_H1173941847e_bool(Pa,X_1)) ) ).

tff(fact_18_empty__iff,axiom,
    ! [Ca: hoare_1875481847_state] : ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,Ca),bot_bo1715400655e_bool)) ).

tff(fact_19_empty__Collect__eq,axiom,
    ! [Pa: fun_Ho1996104121e_bool] :
      ( ( bot_bo1715400655e_bool = hAPP_f121055253e_bool(collec1735317154_state,Pa) )
    <=> ! [X_1: hoare_1875481847_state] : ~ hBOOL(hAPP_H1173941847e_bool(Pa,X_1)) ) ).

tff(fact_20_ex__in__conv,axiom,
    ! [A_1: fun_Ho1996104121e_bool] :
      ( ? [X_1: hoare_1875481847_state] : hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),A_1))
    <=> ( A_1 != bot_bo1715400655e_bool ) ) ).

tff(fact_21_all__not__in__conv,axiom,
    ! [A_1: fun_Ho1996104121e_bool] :
      ( ! [X_1: hoare_1875481847_state] : ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),A_1))
    <=> ( A_1 = bot_bo1715400655e_bool ) ) ).

tff(fact_22_empty__def,axiom,
    bot_bo1715400655e_bool = hAPP_f121055253e_bool(collec1735317154_state,hAPP_b1906038105e_bool(cOMBK_2086958736_state,fFalse)) ).

tff(fact_23_insert__absorb,axiom,
    ! [A: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A),A_1))
     => ( hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),A_1) = A_1 ) ) ).

tff(fact_24_insertI2,axiom,
    ! [B: hoare_1875481847_state,A: hoare_1875481847_state,B_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A),B_1))
     => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,B),B_1))) ) ).

tff(fact_25_insert__ident,axiom,
    ! [B_1: fun_Ho1996104121e_bool,X_2: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool] :
      ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),A_1))
     => ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),B_1))
       => ( ( hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),A_1) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),B_1) )
        <=> ( A_1 = B_1 ) ) ) ) ).

tff(fact_26_insert__code,axiom,
    ! [Y_2: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool,X_2: hoare_1875481847_state] :
      ( hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,Y_2),A_1),X_2))
    <=> ( ( Y_2 = X_2 )
        | hBOOL(hAPP_H1173941847e_bool(A_1,X_2)) ) ) ).

tff(fact_27_insert__iff,axiom,
    ! [A: hoare_1875481847_state,B: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,B),A_1)))
    <=> ( ( A = B )
        | hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A),A_1)) ) ) ).

tff(fact_28_insert__commute,axiom,
    ! [X_2: hoare_1875481847_state,Y_2: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,Y_2),A_1)) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,Y_2),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),A_1)) ).

tff(fact_29_insert__absorb2,axiom,
    ! [X_2: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),A_1)) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),A_1) ).

tff(fact_30_insert__Collect,axiom,
    ! [A: hoare_1875481847_state,Pa: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),hAPP_f121055253e_bool(collec1735317154_state,Pa)) = hAPP_f121055253e_bool(collec1735317154_state,hAPP_f121055253e_bool(cOMBS_896408300l_bool(hAPP_f932622461l_bool(cOMBB_242064253_state(fimplies),hAPP_f121055253e_bool(cOMBB_1424001755_state(fNot),hAPP_H117293375e_bool(cOMBC_1972244470e_bool(fequal391417144_state),A)))),Pa)) ).

tff(fact_31_insert__compr,axiom,
    ! [A: hoare_1875481847_state,B_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),B_1) = hAPP_f121055253e_bool(collec1735317154_state,hAPP_f121055253e_bool(cOMBS_896408300l_bool(hAPP_f932622461l_bool(cOMBB_242064253_state(fdisj),hAPP_H117293375e_bool(cOMBC_1972244470e_bool(fequal391417144_state),A))),hAPP_f121055253e_bool(cOMBC_1099403170l_bool(member618553700_state),B_1))) ).

tff(fact_32_insertI1,axiom,
    ! [A: hoare_1875481847_state,B_1: fun_Ho1996104121e_bool] : hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),B_1))) ).

tff(fact_33_singleton__inject,axiom,
    ! [A: hoare_1875481847_state,B: hoare_1875481847_state] :
      ( ( hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),bot_bo1715400655e_bool) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,B),bot_bo1715400655e_bool) )
     => ( A = B ) ) ).

tff(fact_34_singletonE,axiom,
    ! [B: hoare_1875481847_state,A: hoare_1875481847_state] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,B),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),bot_bo1715400655e_bool)))
     => ( B = A ) ) ).

tff(fact_35_the__elem__eq,axiom,
    ! [X_2: hoare_1875481847_state] : hAPP_f890249627_state(the_el1031305112_state,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),bot_bo1715400655e_bool)) = X_2 ).

tff(fact_36_bot__apply,axiom,
    ! [X_2: hoare_1875481847_state] :
      ( hBOOL(hAPP_H1173941847e_bool(bot_bo1715400655e_bool,X_2))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_37_bot__fun__def,axiom,
    ! [X_1: hoare_1875481847_state] :
      ( hBOOL(hAPP_H1173941847e_bool(bot_bo1715400655e_bool,X_1))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_38_hoare__derivs_OSkip,axiom,
    ! [G_2: fun_Ho1996104121e_bool,Pa: fun_st1506752259e_bool] : hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(G_2),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hoare_13225861_state(Pa,skip,Pa)),bot_bo1715400655e_bool))) ).

tff(fact_39_Comp,axiom,
    ! [D_1: com,R_1: fun_st1506752259e_bool,G_2: fun_Ho1996104121e_bool,Pa: fun_st1506752259e_bool,Ca: com,Qa: fun_st1506752259e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(G_2),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hoare_13225861_state(Pa,Ca,Qa)),bot_bo1715400655e_bool)))
     => ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(G_2),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hoare_13225861_state(Qa,D_1,R_1)),bot_bo1715400655e_bool)))
       => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(G_2),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hoare_13225861_state(Pa,semi(Ca,D_1),R_1)),bot_bo1715400655e_bool))) ) ) ).

tff(fact_40_triple_Oexhaust,axiom,
    ! [Y_2: hoare_1875481847_state] :
      ~ ! [Fun1: fun_st1506752259e_bool,Com: com,Fun2: fun_st1506752259e_bool] : Y_2 != hoare_13225861_state(Fun1,Com,Fun2) ).

tff(fact_41_Set_Oset__insert,axiom,
    ! [X_2: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),A_1))
     => ~ ! [B_3: fun_Ho1996104121e_bool] :
            ( ( A_1 = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),B_3) )
           => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),B_3)) ) ) ).

tff(fact_42_mk__disjoint__insert,axiom,
    ! [A: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A),A_1))
     => ? [B_3: fun_Ho1996104121e_bool] :
          ( ( A_1 = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),B_3) )
          & ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A),B_3)) ) ) ).

tff(fact_43_equals0I,axiom,
    ! [A_1: fun_Ho1996104121e_bool] :
      ( ! [Y_1: hoare_1875481847_state] : ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,Y_1),A_1))
     => ( A_1 = bot_bo1715400655e_bool ) ) ).

tff(fact_44_conseq,axiom,
    ! [Qa: fun_st1506752259e_bool,G_2: fun_Ho1996104121e_bool,Ca: com,Pa: fun_st1506752259e_bool] :
      ( ! [Z_1: state,S: state] :
          ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Pa,Z_1),S))
         => ? [P_1: fun_st1506752259e_bool,Q_1: fun_st1506752259e_bool] :
              ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(G_2),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hoare_13225861_state(P_1,Ca,Q_1)),bot_bo1715400655e_bool)))
              & ! [S_1: state] :
                  ( ! [Z_2: state] :
                      ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(P_1,Z_2),S))
                     => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Q_1,Z_2),S_1)) )
                 => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Qa,Z_1),S_1)) ) ) )
     => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(G_2),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hoare_13225861_state(Pa,Ca,Qa)),bot_bo1715400655e_bool))) ) ).

tff(fact_45_nonempty__iff,axiom,
    ! [A_1: fun_Ho1996104121e_bool] :
      ( ( A_1 != bot_bo1715400655e_bool )
    <=> ? [X_1: hoare_1875481847_state,B_3: fun_Ho1996104121e_bool] :
          ( ( A_1 = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),B_3) )
          & ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),B_3)) ) ) ).

tff(fact_46_com_Osimps_I13_J,axiom,
    ! [Com1_2: com,Com2_2: com] : semi(Com1_2,Com2_2) != skip ).

tff(fact_47_com_Osimps_I12_J,axiom,
    ! [Com1_2: com,Com2_2: com] : skip != semi(Com1_2,Com2_2) ).

tff(fact_48_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_49_fold1Set__sing,axiom,
    ! [F: fun_Ho784854669_state,A: hoare_1875481847_state,B: hoare_1875481847_state] :
      ( hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(finite1016865319_state(F),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),bot_bo1715400655e_bool)),B))
    <=> ( A = B ) ) ).

tff(fact_50_folding__one_Osingleton,axiom,
    ! [X_2: hoare_1875481847_state,F: fun_Ho784854669_state,F_1: fun_fu385705929_state] :
      ( hBOOL(finite1782051386_state(F,F_1))
     => ( hAPP_f890249627_state(F_1,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),bot_bo1715400655e_bool)) = X_2 ) ) ).

tff(fact_51_bot__empty__eq,axiom,
    ! [X_1: hoare_1875481847_state] :
      ( hBOOL(hAPP_H1173941847e_bool(bot_bo1715400655e_bool,X_1))
    <=> hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),bot_bo1715400655e_bool)) ) ).

tff(fact_52_fold1__singleton,axiom,
    ! [F: fun_Ho784854669_state,A: hoare_1875481847_state] : hAPP_f890249627_state(finite206258583_state(F),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),bot_bo1715400655e_bool)) = A ).

tff(fact_53_fold1__singleton__def,axiom,
    ! [A: hoare_1875481847_state,G: fun_fu385705929_state,F: fun_Ho784854669_state] :
      ( ( G = finite206258583_state(F) )
     => ( hAPP_f890249627_state(G,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),bot_bo1715400655e_bool)) = A ) ) ).

tff(fact_54_empty__fold1SetE,axiom,
    ! [F: fun_Ho784854669_state,X_2: hoare_1875481847_state] : ~ hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(finite1016865319_state(F),bot_bo1715400655e_bool),X_2)) ).

tff(fact_55_fold1Set__nonempty,axiom,
    ! [F: fun_Ho784854669_state,A_1: fun_Ho1996104121e_bool,X_2: hoare_1875481847_state] :
      ( hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(finite1016865319_state(F),A_1),X_2))
     => ( A_1 != bot_bo1715400655e_bool ) ) ).

tff(fact_56_fold1Set_Ointros,axiom,
    ! [F: fun_Ho784854669_state,A: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool,X_2: hoare_1875481847_state] :
      ( hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(hAPP_H1625489667e_bool(finite1721989730_state(F),A),A_1),X_2))
     => ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A),A_1))
       => hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(finite1016865319_state(F),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),A_1)),X_2)) ) ) ).

tff(fact_57_folding__one_Oinsert,axiom,
    ! [X_2: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool,F: fun_Ho784854669_state,F_1: fun_fu385705929_state] :
      ( hBOOL(finite1782051386_state(F,F_1))
     => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1))
       => ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),A_1))
         => ( ( A_1 != bot_bo1715400655e_bool )
           => ( hAPP_f890249627_state(F_1,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),A_1)) = hAPP_H350317809_state(hAPP_H743221337_state(F,X_2),hAPP_f890249627_state(F_1,A_1)) ) ) ) ) ) ).

tff(fact_58_folding__one_Oeq__fold,axiom,
    ! [A_1: fun_Ho1996104121e_bool,F: fun_Ho784854669_state,F_1: fun_fu385705929_state] :
      ( hBOOL(finite1782051386_state(F,F_1))
     => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1))
       => ( hAPP_f890249627_state(F_1,A_1) = hAPP_f890249627_state(finite206258583_state(F),A_1) ) ) ) ).

tff(fact_59_folding__one_Oclosed,axiom,
    ! [A_1: fun_Ho1996104121e_bool,F: fun_Ho784854669_state,F_1: fun_fu385705929_state] :
      ( hBOOL(finite1782051386_state(F,F_1))
     => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1))
       => ( ( A_1 != bot_bo1715400655e_bool )
         => ( ! [X_1: hoare_1875481847_state,Y_1: hoare_1875481847_state] : hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,hAPP_H350317809_state(hAPP_H743221337_state(F,X_1),Y_1)),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,Y_1),bot_bo1715400655e_bool))))
           => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,hAPP_f890249627_state(F_1,A_1)),A_1)) ) ) ) ) ).

tff(fact_60_insert__fold1SetE,axiom,
    ! [F: fun_Ho784854669_state,A: hoare_1875481847_state,X_3: fun_Ho1996104121e_bool,X_2: hoare_1875481847_state] :
      ( hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(finite1016865319_state(F),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),X_3)),X_2))
     => ~ ! [A_3: hoare_1875481847_state,A_2: fun_Ho1996104121e_bool] :
            ( ( hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),X_3) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_3),A_2) )
           => ( hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(hAPP_H1625489667e_bool(finite1721989730_state(F),A_3),A_2),X_2))
             => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_3),A_2)) ) ) ) ).

tff(fact_61_subset__singletonD,axiom,
    ! [A_1: fun_Ho1996104121e_bool,X_2: hoare_1875481847_state] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),bot_bo1715400655e_bool)))
     => ( ( A_1 = bot_bo1715400655e_bool )
        | ( A_1 = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),bot_bo1715400655e_bool) ) ) ) ).

tff(fact_62_order__refl,axiom,
    ! [X_2: bool] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,X_2),X_2)) ).

tff(fact_63_order__refl,axiom,
    ! [X_2: fun_Ho1996104121e_bool] : hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,X_2),X_2)) ).

tff(fact_64_subsetD,axiom,
    ! [Ca: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A_1),B_1))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,Ca),A_1))
       => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,Ca),B_1)) ) ) ).

tff(fact_65_equalityI,axiom,
    ! [A_1: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A_1),B_1))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,B_1),A_1))
       => ( A_1 = B_1 ) ) ) ).

tff(fact_66_finite_OemptyI,axiom,
    hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,bot_bo1715400655e_bool)) ).

tff(fact_67_finite_OinsertI,axiom,
    ! [A: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1))
     => hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),A_1))) ) ).

tff(fact_68_empty__subsetI,axiom,
    ! [A_1: fun_Ho1996104121e_bool] : hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,bot_bo1715400655e_bool),A_1)) ).

tff(fact_69_rev__predicate1D,axiom,
    ! [Qa: fun_Ho1996104121e_bool,Pa: fun_Ho1996104121e_bool,X_2: hoare_1875481847_state] :
      ( hBOOL(hAPP_H1173941847e_bool(Pa,X_2))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Pa),Qa))
       => hBOOL(hAPP_H1173941847e_bool(Qa,X_2)) ) ) ).

tff(fact_70_predicate1D,axiom,
    ! [X_2: hoare_1875481847_state,Pa: fun_Ho1996104121e_bool,Qa: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Pa),Qa))
     => ( hBOOL(hAPP_H1173941847e_bool(Pa,X_2))
       => hBOOL(hAPP_H1173941847e_bool(Qa,X_2)) ) ) ).

tff(fact_71_mem__def,axiom,
    ! [X_2: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),A_1))
    <=> hBOOL(hAPP_H1173941847e_bool(A_1,X_2)) ) ).

tff(fact_72_Collect__def,axiom,
    ! [Pa: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(collec1735317154_state,Pa) = Pa ).

tff(fact_73_subset__refl,axiom,
    ! [A_1: fun_Ho1996104121e_bool] : hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A_1),A_1)) ).

tff(fact_74_set__eq__subset,axiom,
    ! [A_1: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( ( A_1 = B_1 )
    <=> ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A_1),B_1))
        & hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,B_1),A_1)) ) ) ).

tff(fact_75_equalityD1,axiom,
    ! [A_1: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( ( A_1 = B_1 )
     => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A_1),B_1)) ) ).

tff(fact_76_equalityD2,axiom,
    ! [A_1: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( ( A_1 = B_1 )
     => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,B_1),A_1)) ) ).

tff(fact_77_in__mono,axiom,
    ! [X_2: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A_1),B_1))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),A_1))
       => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),B_1)) ) ) ).

tff(fact_78_set__rev__mp,axiom,
    ! [B_1: fun_Ho1996104121e_bool,X_2: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),A_1))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A_1),B_1))
       => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),B_1)) ) ) ).

tff(fact_79_set__mp,axiom,
    ! [X_2: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A_1),B_1))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),A_1))
       => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),B_1)) ) ) ).

tff(fact_80_subset__trans,axiom,
    ! [C: fun_Ho1996104121e_bool,A_1: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A_1),B_1))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,B_1),C))
       => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A_1),C)) ) ) ).

tff(fact_81_equalityE,axiom,
    ! [A_1: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( ( A_1 = B_1 )
     => ~ ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A_1),B_1))
         => ~ hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,B_1),A_1)) ) ) ).

tff(fact_82_le__fun__def,axiom,
    ! [F: fun_Ho1996104121e_bool,G: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,F),G))
    <=> ! [X_1: hoare_1875481847_state] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_H1173941847e_bool(F,X_1)),hAPP_H1173941847e_bool(G,X_1))) ) ).

tff(fact_83_order__eq__iff,axiom,
    ! [Y_2: bool,X_2: bool] :
      ( ( hBOOL(X_2)
      <=> hBOOL(Y_2) )
    <=> ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,X_2),Y_2))
        & hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Y_2),X_2)) ) ) ).

tff(fact_84_order__eq__iff,axiom,
    ! [X_2: fun_Ho1996104121e_bool,Y_2: fun_Ho1996104121e_bool] :
      ( ( X_2 = Y_2 )
    <=> ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,X_2),Y_2))
        & hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Y_2),X_2)) ) ) ).

tff(fact_85_rev__finite__subset,axiom,
    ! [A_1: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,B_1))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A_1),B_1))
       => hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1)) ) ) ).

tff(fact_86_order__eq__refl,axiom,
    ! [Y_2: bool,X_2: bool] :
      ( ( hBOOL(X_2)
      <=> hBOOL(Y_2) )
     => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,X_2),Y_2)) ) ).

tff(fact_87_order__eq__refl,axiom,
    ! [X_2: fun_Ho1996104121e_bool,Y_2: fun_Ho1996104121e_bool] :
      ( ( X_2 = Y_2 )
     => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,X_2),Y_2)) ) ).

tff(fact_88_le__funD,axiom,
    ! [X_2: hoare_1875481847_state,F: fun_Ho1996104121e_bool,G: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,F),G))
     => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_H1173941847e_bool(F,X_2)),hAPP_H1173941847e_bool(G,X_2))) ) ).

tff(fact_89_order__antisym__conv,axiom,
    ! [Y_2: bool,X_2: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Y_2),X_2))
     => ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,X_2),Y_2))
      <=> ( hBOOL(X_2)
        <=> hBOOL(Y_2) ) ) ) ).

tff(fact_90_order__antisym__conv,axiom,
    ! [Y_2: fun_Ho1996104121e_bool,X_2: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Y_2),X_2))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,X_2),Y_2))
      <=> ( X_2 = Y_2 ) ) ) ).

tff(fact_91_finite__subset,axiom,
    ! [A_1: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A_1),B_1))
     => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,B_1))
       => hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1)) ) ) ).

tff(fact_92_ord__eq__le__trans,axiom,
    ! [Ca: bool,B: bool,A: bool] :
      ( ( hBOOL(A)
      <=> hBOOL(B) )
     => ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,B),Ca))
       => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,A),Ca)) ) ) ).

tff(fact_93_ord__eq__le__trans,axiom,
    ! [Ca: fun_Ho1996104121e_bool,A: fun_Ho1996104121e_bool,B: fun_Ho1996104121e_bool] :
      ( ( A = B )
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,B),Ca))
       => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A),Ca)) ) ) ).

tff(fact_94_xt1_I3_J,axiom,
    ! [Ca: bool,B: bool,A: bool] :
      ( ( hBOOL(A)
      <=> hBOOL(B) )
     => ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Ca),B))
       => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Ca),A)) ) ) ).

tff(fact_95_xt1_I3_J,axiom,
    ! [Ca: fun_Ho1996104121e_bool,A: fun_Ho1996104121e_bool,B: fun_Ho1996104121e_bool] :
      ( ( A = B )
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Ca),B))
       => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Ca),A)) ) ) ).

tff(fact_96_ord__le__eq__trans,axiom,
    ! [Ca: bool,A: bool,B: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,A),B))
     => ( ( hBOOL(B)
        <=> hBOOL(Ca) )
       => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,A),Ca)) ) ) ).

tff(fact_97_ord__le__eq__trans,axiom,
    ! [Ca: fun_Ho1996104121e_bool,A: fun_Ho1996104121e_bool,B: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A),B))
     => ( ( B = Ca )
       => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A),Ca)) ) ) ).

tff(fact_98_xt1_I4_J,axiom,
    ! [Ca: bool,B: bool,A: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,B),A))
     => ( ( hBOOL(B)
        <=> hBOOL(Ca) )
       => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Ca),A)) ) ) ).

tff(fact_99_xt1_I4_J,axiom,
    ! [Ca: fun_Ho1996104121e_bool,B: fun_Ho1996104121e_bool,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,B),A))
     => ( ( B = Ca )
       => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Ca),A)) ) ) ).

tff(fact_100_order__antisym,axiom,
    ! [X_2: bool,Y_2: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,X_2),Y_2))
     => ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Y_2),X_2))
       => ( hBOOL(X_2)
        <=> hBOOL(Y_2) ) ) ) ).

tff(fact_101_order__antisym,axiom,
    ! [X_2: fun_Ho1996104121e_bool,Y_2: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,X_2),Y_2))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Y_2),X_2))
       => ( X_2 = Y_2 ) ) ) ).

tff(fact_102_order__trans,axiom,
    ! [Z: bool,X_2: bool,Y_2: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,X_2),Y_2))
     => ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Y_2),Z))
       => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,X_2),Z)) ) ) ).

tff(fact_103_order__trans,axiom,
    ! [Z: fun_Ho1996104121e_bool,X_2: fun_Ho1996104121e_bool,Y_2: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,X_2),Y_2))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Y_2),Z))
       => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,X_2),Z)) ) ) ).

tff(fact_104_xt1_I5_J,axiom,
    ! [Y_2: bool,X_2: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Y_2),X_2))
     => ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,X_2),Y_2))
       => ( hBOOL(X_2)
        <=> hBOOL(Y_2) ) ) ) ).

tff(fact_105_xt1_I5_J,axiom,
    ! [Y_2: fun_Ho1996104121e_bool,X_2: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Y_2),X_2))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,X_2),Y_2))
       => ( X_2 = Y_2 ) ) ) ).

tff(fact_106_xt1_I6_J,axiom,
    ! [Z: bool,Y_2: bool,X_2: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Y_2),X_2))
     => ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Z),Y_2))
       => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Z),X_2)) ) ) ).

tff(fact_107_xt1_I6_J,axiom,
    ! [Z: fun_Ho1996104121e_bool,Y_2: fun_Ho1996104121e_bool,X_2: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Y_2),X_2))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Z),Y_2))
       => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Z),X_2)) ) ) ).

tff(fact_108_le__funE,axiom,
    ! [X_2: hoare_1875481847_state,F: fun_Ho1996104121e_bool,G: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,F),G))
     => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_H1173941847e_bool(F,X_2)),hAPP_H1173941847e_bool(G,X_2))) ) ).

tff(fact_109_bot__least,axiom,
    ! [A: fun_Ho1996104121e_bool] : hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,bot_bo1715400655e_bool),A)) ).

tff(fact_110_bot__least,axiom,
    ! [A: bool] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,bot_bot_bool),A)) ).

tff(fact_111_bot__unique,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_112_bot__unique,axiom,
    ! [A: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,A),bot_bot_bool))
    <=> ( hBOOL(A)
      <=> hBOOL(bot_bot_bool) ) ) ).

tff(fact_113_le__bot,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_114_le__bot,axiom,
    ! [A: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,A),bot_bot_bool))
     => ( hBOOL(A)
      <=> hBOOL(bot_bot_bool) ) ) ).

tff(fact_115_subset__empty,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_116_subset__insertI,axiom,
    ! [B_1: fun_Ho1996104121e_bool,A: hoare_1875481847_state] : hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,B_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),B_1))) ).

tff(fact_117_insert__subset,axiom,
    ! [X_2: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),A_1)),B_1))
    <=> ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),B_1))
        & hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A_1),B_1)) ) ) ).

tff(fact_118_subset__insert,axiom,
    ! [B_1: fun_Ho1996104121e_bool,X_2: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool] :
      ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),A_1))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),B_1)))
      <=> hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A_1),B_1)) ) ) ).

tff(fact_119_subset__insertI2,axiom,
    ! [B: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A_1),B_1))
     => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,B),B_1))) ) ).

tff(fact_120_insert__mono,axiom,
    ! [A: hoare_1875481847_state,C: fun_Ho1996104121e_bool,D: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,C),D))
     => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),C)),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),D))) ) ).

tff(fact_121_finite__insert,axiom,
    ! [A: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),A_1)))
    <=> hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1)) ) ).

tff(fact_122_asm,axiom,
    ! [Ts: fun_Ho1996104121e_bool,G_2: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Ts),G_2))
     => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(G_2),Ts)) ) ).

tff(fact_123_weaken,axiom,
    ! [Ts: fun_Ho1996104121e_bool,G_2: fun_Ho1996104121e_bool,Ts_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(G_2),Ts_1))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Ts),Ts_1))
       => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(G_2),Ts)) ) ) ).

tff(fact_124_thin,axiom,
    ! [G_2: 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(hAPP_f1873026899l_bool(ord_le233469699e_bool,G_1),G_2))
       => hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(G_2),Ts)) ) ) ).

tff(fact_125_fold__graph_OemptyI,axiom,
    ! [F: fun_Ho784854669_state,Z: hoare_1875481847_state] : hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(hAPP_H1625489667e_bool(finite1721989730_state(F),Z),bot_bo1715400655e_bool),Z)) ).

tff(fact_126_empty__fold__graphE,axiom,
    ! [F: fun_Ho784854669_state,Z: hoare_1875481847_state,X_2: hoare_1875481847_state] :
      ( hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(hAPP_H1625489667e_bool(finite1721989730_state(F),Z),bot_bo1715400655e_bool),X_2))
     => ( X_2 = Z ) ) ).

tff(fact_127_fold__graph_OinsertI,axiom,
    ! [F: fun_Ho784854669_state,Z: hoare_1875481847_state,Y_2: hoare_1875481847_state,X_2: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool] :
      ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),A_1))
     => ( hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(hAPP_H1625489667e_bool(finite1721989730_state(F),Z),A_1),Y_2))
       => hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(hAPP_H1625489667e_bool(finite1721989730_state(F),Z),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),A_1)),hAPP_H350317809_state(hAPP_H743221337_state(F,X_2),Y_2))) ) ) ).

tff(fact_128_finite__subset__induct,axiom,
    ! [Pa: fun_fu689207471l_bool,A_1: fun_Ho1996104121e_bool,F_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,F_1))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,F_1),A_1))
       => ( hBOOL(hAPP_f1408815105l_bool(Pa,bot_bo1715400655e_bool))
         => ( ! [A_3: 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_3),A_1))
                 => ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_3),F_2))
                   => ( hBOOL(hAPP_f1408815105l_bool(Pa,F_2))
                     => hBOOL(hAPP_f1408815105l_bool(Pa,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_3),F_2))) ) ) ) )
           => hBOOL(hAPP_f1408815105l_bool(Pa,F_1)) ) ) ) ) ).

tff(fact_129_finite__nonempty__imp__fold1Set,axiom,
    ! [F: fun_Ho784854669_state,A_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1))
     => ( ( A_1 != bot_bo1715400655e_bool )
       => ? [X1: hoare_1875481847_state] : hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(finite1016865319_state(F),A_1),X1)) ) ) ).

tff(fact_130_subsetI,axiom,
    ! [B_1: fun_Ho1996104121e_bool,A_1: fun_Ho1996104121e_bool] :
      ( ! [X_1: hoare_1875481847_state] :
          ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),A_1))
         => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),B_1)) )
     => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A_1),B_1)) ) ).

tff(fact_131_finite_Osimps,axiom,
    ! [A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A))
    <=> ( ( A = bot_bo1715400655e_bool )
        | ? [A_2: fun_Ho1996104121e_bool,A_3: hoare_1875481847_state] :
            ( ( A = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_3),A_2) )
            & hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_2)) ) ) ) ).

tff(fact_132_finite__induct,axiom,
    ! [Pa: fun_fu689207471l_bool,F_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,F_1))
     => ( hBOOL(hAPP_f1408815105l_bool(Pa,bot_bo1715400655e_bool))
       => ( ! [X_1: 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_1),F_2))
               => ( hBOOL(hAPP_f1408815105l_bool(Pa,F_2))
                 => hBOOL(hAPP_f1408815105l_bool(Pa,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),F_2))) ) ) )
         => hBOOL(hAPP_f1408815105l_bool(Pa,F_1)) ) ) ) ).

tff(fact_133_finite__imp__fold__graph,axiom,
    ! [F: fun_Ho784854669_state,Z: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1))
     => ? [X1: hoare_1875481847_state] : hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(hAPP_H1625489667e_bool(finite1721989730_state(F),Z),A_1),X1)) ) ).

tff(fact_134_le__funI,axiom,
    ! [F: fun_Ho1996104121e_bool,G: fun_Ho1996104121e_bool] :
      ( ! [X_1: hoare_1875481847_state] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_H1173941847e_bool(F,X_1)),hAPP_H1173941847e_bool(G,X_1)))
     => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,F),G)) ) ).

tff(fact_135_fold1Set_Osimps,axiom,
    ! [F: fun_Ho784854669_state,A1: fun_Ho1996104121e_bool,A2: hoare_1875481847_state] :
      ( hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(finite1016865319_state(F),A1),A2))
    <=> ? [A_3: hoare_1875481847_state,A_2: fun_Ho1996104121e_bool,X_1: hoare_1875481847_state] :
          ( ( A1 = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_3),A_2) )
          & ( A2 = X_1 )
          & hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(hAPP_H1625489667e_bool(finite1721989730_state(F),A_3),A_2),X_1))
          & ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_3),A_2)) ) ) ).

tff(fact_136_fold__graph_Osimps,axiom,
    ! [F: fun_Ho784854669_state,Z: hoare_1875481847_state,A1: fun_Ho1996104121e_bool,A2: hoare_1875481847_state] :
      ( hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(hAPP_H1625489667e_bool(finite1721989730_state(F),Z),A1),A2))
    <=> ( ( ( A1 = bot_bo1715400655e_bool )
          & ( A2 = Z ) )
        | ? [X_1: hoare_1875481847_state,A_2: fun_Ho1996104121e_bool,Y_1: hoare_1875481847_state] :
            ( ( A1 = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),A_2) )
            & ( A2 = hAPP_H350317809_state(hAPP_H743221337_state(F,X_1),Y_1) )
            & ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),A_2))
            & hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(hAPP_H1625489667e_bool(finite1721989730_state(F),Z),A_2),Y_1)) ) ) ) ).

tff(fact_137_folding__one__idem_Osubset__idem,axiom,
    ! [B_1: fun_Ho1996104121e_bool,A_1: fun_Ho1996104121e_bool,F: fun_Ho784854669_state,F_1: fun_fu385705929_state] :
      ( hBOOL(finite1514162941_state(F,F_1))
     => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1))
       => ( ( B_1 != bot_bo1715400655e_bool )
         => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,B_1),A_1))
           => ( hAPP_H350317809_state(hAPP_H743221337_state(F,hAPP_f890249627_state(F_1,B_1)),hAPP_f890249627_state(F_1,A_1)) = hAPP_f890249627_state(F_1,A_1) ) ) ) ) ) ).

tff(fact_138_folding__one__idem_Oinsert__idem,axiom,
    ! [X_2: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool,F: fun_Ho784854669_state,F_1: fun_fu385705929_state] :
      ( hBOOL(finite1514162941_state(F,F_1))
     => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1))
       => ( ( A_1 != bot_bo1715400655e_bool )
         => ( hAPP_f890249627_state(F_1,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),A_1)) = hAPP_H350317809_state(hAPP_H743221337_state(F,X_2),hAPP_f890249627_state(F_1,A_1)) ) ) ) ) ).

tff(fact_139_finite__ne__induct,axiom,
    ! [Pa: fun_fu689207471l_bool,F_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,F_1))
     => ( ( F_1 != bot_bo1715400655e_bool )
       => ( ! [X_1: hoare_1875481847_state] : hBOOL(hAPP_f1408815105l_bool(Pa,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),bot_bo1715400655e_bool)))
         => ( ! [X_1: 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_1),F_2))
                   => ( hBOOL(hAPP_f1408815105l_bool(Pa,F_2))
                     => hBOOL(hAPP_f1408815105l_bool(Pa,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_1),F_2))) ) ) ) )
           => hBOOL(hAPP_f1408815105l_bool(Pa,F_1)) ) ) ) ) ).

tff(fact_140_Collect__mono,axiom,
    ! [Qa: fun_Ho1996104121e_bool,Pa: fun_Ho1996104121e_bool] :
      ( ! [X_1: hoare_1875481847_state] :
          ( hBOOL(hAPP_H1173941847e_bool(Pa,X_1))
         => hBOOL(hAPP_H1173941847e_bool(Qa,X_1)) )
     => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,hAPP_f121055253e_bool(collec1735317154_state,Pa)),hAPP_f121055253e_bool(collec1735317154_state,Qa))) ) ).

tff(fact_141_folding__one__idem_Oidem,axiom,
    ! [X_2: hoare_1875481847_state,F: fun_Ho784854669_state,F_1: fun_fu385705929_state] :
      ( hBOOL(finite1514162941_state(F,F_1))
     => ( hAPP_H350317809_state(hAPP_H743221337_state(F,X_2),X_2) = X_2 ) ) ).

tff(fact_142_folding__one__idem_Oin__idem,axiom,
    ! [X_2: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool,F: fun_Ho784854669_state,F_1: fun_fu385705929_state] :
      ( hBOOL(finite1514162941_state(F,F_1))
     => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1))
       => ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),A_1))
         => ( hAPP_H350317809_state(hAPP_H743221337_state(F,X_2),hAPP_f890249627_state(F_1,A_1)) = hAPP_f890249627_state(F_1,A_1) ) ) ) ) ).

tff(fact_143_predicate1I,axiom,
    ! [Qa: fun_Ho1996104121e_bool,Pa: fun_Ho1996104121e_bool] :
      ( ! [X_1: hoare_1875481847_state] :
          ( hBOOL(hAPP_H1173941847e_bool(Pa,X_1))
         => hBOOL(hAPP_H1173941847e_bool(Qa,X_1)) )
     => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Pa),Qa)) ) ).

tff(fact_144_xt3,axiom,
    ! [Ca: bool,F: fun_fu689207471l_bool,B: fun_Ho1996104121e_bool,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,B),A))
     => ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Ca),hAPP_f1408815105l_bool(F,B)))
       => ( ! [X_1: fun_Ho1996104121e_bool,Y_1: fun_Ho1996104121e_bool] :
              ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Y_1),X_1))
             => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_f1408815105l_bool(F,Y_1)),hAPP_f1408815105l_bool(F,X_1))) )
         => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Ca),hAPP_f1408815105l_bool(F,A))) ) ) ) ).

tff(fact_145_xt3,axiom,
    ! [Ca: fun_Ho1996104121e_bool,F: fun_bo221168653e_bool,B: bool,A: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,B),A))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Ca),hAPP_b1906038105e_bool(F,B)))
       => ( ! [X_1: bool,Y_1: bool] :
              ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Y_1),X_1))
             => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,hAPP_b1906038105e_bool(F,Y_1)),hAPP_b1906038105e_bool(F,X_1))) )
         => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Ca),hAPP_b1906038105e_bool(F,A))) ) ) ) ).

tff(fact_146_xt1_I16_J,axiom,
    ! [Ca: bool,F: fun_bool_bool,B: bool,A: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,B),A))
     => ( ( hBOOL(hAPP_bool_bool(F,B))
        <=> hBOOL(Ca) )
       => ( ! [X_1: bool,Y_1: bool] :
              ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Y_1),X_1))
             => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_bool_bool(F,Y_1)),hAPP_bool_bool(F,X_1))) )
         => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Ca),hAPP_bool_bool(F,A))) ) ) ) ).

tff(fact_147_xt1_I16_J,axiom,
    ! [F: fun_fu1296727421e_bool,Ca: fun_Ho1996104121e_bool,B: fun_Ho1996104121e_bool,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,B),A))
     => ( ( hAPP_f121055253e_bool(F,B) = Ca )
       => ( ! [X_1: fun_Ho1996104121e_bool,Y_1: fun_Ho1996104121e_bool] :
              ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Y_1),X_1))
             => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,hAPP_f121055253e_bool(F,Y_1)),hAPP_f121055253e_bool(F,X_1))) )
         => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Ca),hAPP_f121055253e_bool(F,A))) ) ) ) ).

tff(fact_148_ord__le__eq__subst,axiom,
    ! [Ca: bool,F: fun_fu689207471l_bool,A: fun_Ho1996104121e_bool,B: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A),B))
     => ( ( hBOOL(hAPP_f1408815105l_bool(F,B))
        <=> hBOOL(Ca) )
       => ( ! [X_1: fun_Ho1996104121e_bool,Y_1: fun_Ho1996104121e_bool] :
              ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,X_1),Y_1))
             => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_f1408815105l_bool(F,X_1)),hAPP_f1408815105l_bool(F,Y_1))) )
         => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_f1408815105l_bool(F,A)),Ca)) ) ) ) ).

tff(fact_149_ord__le__eq__subst,axiom,
    ! [F: fun_bo221168653e_bool,Ca: fun_Ho1996104121e_bool,A: bool,B: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,A),B))
     => ( ( hAPP_b1906038105e_bool(F,B) = Ca )
       => ( ! [X_1: bool,Y_1: bool] :
              ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,X_1),Y_1))
             => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,hAPP_b1906038105e_bool(F,X_1)),hAPP_b1906038105e_bool(F,Y_1))) )
         => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,hAPP_b1906038105e_bool(F,A)),Ca)) ) ) ) ).

tff(fact_150_order__subst2,axiom,
    ! [F: fun_fu689207471l_bool,Ca: bool,A: fun_Ho1996104121e_bool,B: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A),B))
     => ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_f1408815105l_bool(F,B)),Ca))
       => ( ! [X_1: fun_Ho1996104121e_bool,Y_1: fun_Ho1996104121e_bool] :
              ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,X_1),Y_1))
             => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_f1408815105l_bool(F,X_1)),hAPP_f1408815105l_bool(F,Y_1))) )
         => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_f1408815105l_bool(F,A)),Ca)) ) ) ) ).

tff(fact_151_order__subst2,axiom,
    ! [F: fun_bo221168653e_bool,Ca: fun_Ho1996104121e_bool,A: bool,B: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,A),B))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,hAPP_b1906038105e_bool(F,B)),Ca))
       => ( ! [X_1: bool,Y_1: bool] :
              ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,X_1),Y_1))
             => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,hAPP_b1906038105e_bool(F,X_1)),hAPP_b1906038105e_bool(F,Y_1))) )
         => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,hAPP_b1906038105e_bool(F,A)),Ca)) ) ) ) ).

tff(fact_152_ord__eq__le__subst,axiom,
    ! [Ca: fun_Ho1996104121e_bool,F: fun_fu689207471l_bool,B: fun_Ho1996104121e_bool,A: bool] :
      ( ( hBOOL(A)
      <=> hBOOL(hAPP_f1408815105l_bool(F,B)) )
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,B),Ca))
       => ( ! [X_1: fun_Ho1996104121e_bool,Y_1: fun_Ho1996104121e_bool] :
              ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,X_1),Y_1))
             => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_f1408815105l_bool(F,X_1)),hAPP_f1408815105l_bool(F,Y_1))) )
         => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,A),hAPP_f1408815105l_bool(F,Ca))) ) ) ) ).

tff(fact_153_ord__eq__le__subst,axiom,
    ! [Ca: bool,A: fun_Ho1996104121e_bool,F: fun_bo221168653e_bool,B: bool] :
      ( ( A = hAPP_b1906038105e_bool(F,B) )
     => ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,B),Ca))
       => ( ! [X_1: bool,Y_1: bool] :
              ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,X_1),Y_1))
             => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,hAPP_b1906038105e_bool(F,X_1)),hAPP_b1906038105e_bool(F,Y_1))) )
         => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A),hAPP_b1906038105e_bool(F,Ca))) ) ) ) ).

tff(fact_154_xt2,axiom,
    ! [Ca: bool,F: fun_bo221168653e_bool,B: bool,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,hAPP_b1906038105e_bool(F,B)),A))
     => ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Ca),B))
       => ( ! [X_1: bool,Y_1: bool] :
              ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Y_1),X_1))
             => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,hAPP_b1906038105e_bool(F,Y_1)),hAPP_b1906038105e_bool(F,X_1))) )
         => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,hAPP_b1906038105e_bool(F,Ca)),A)) ) ) ) ).

tff(fact_155_xt2,axiom,
    ! [Ca: fun_Ho1996104121e_bool,F: fun_fu689207471l_bool,B: fun_Ho1996104121e_bool,A: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_f1408815105l_bool(F,B)),A))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Ca),B))
       => ( ! [X_1: fun_Ho1996104121e_bool,Y_1: fun_Ho1996104121e_bool] :
              ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Y_1),X_1))
             => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_f1408815105l_bool(F,Y_1)),hAPP_f1408815105l_bool(F,X_1))) )
         => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_f1408815105l_bool(F,Ca)),A)) ) ) ) ).

tff(fact_156_xt1_I15_J,axiom,
    ! [Ca: bool,F: fun_bool_bool,B: bool,A: bool] :
      ( ( hBOOL(A)
      <=> hBOOL(hAPP_bool_bool(F,B)) )
     => ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Ca),B))
       => ( ! [X_1: bool,Y_1: bool] :
              ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Y_1),X_1))
             => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_bool_bool(F,Y_1)),hAPP_bool_bool(F,X_1))) )
         => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_bool_bool(F,Ca)),A)) ) ) ) ).

tff(fact_157_xt1_I15_J,axiom,
    ! [Ca: fun_Ho1996104121e_bool,A: fun_Ho1996104121e_bool,F: fun_fu1296727421e_bool,B: fun_Ho1996104121e_bool] :
      ( ( A = hAPP_f121055253e_bool(F,B) )
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Ca),B))
       => ( ! [X_1: fun_Ho1996104121e_bool,Y_1: fun_Ho1996104121e_bool] :
              ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Y_1),X_1))
             => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,hAPP_f121055253e_bool(F,Y_1)),hAPP_f121055253e_bool(F,X_1))) )
         => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,hAPP_f121055253e_bool(F,Ca)),A)) ) ) ) ).

tff(fact_158_order__subst1,axiom,
    ! [Ca: bool,A: fun_Ho1996104121e_bool,F: fun_bo221168653e_bool,B: bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A),hAPP_b1906038105e_bool(F,B)))
     => ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,B),Ca))
       => ( ! [X_1: bool,Y_1: bool] :
              ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,X_1),Y_1))
             => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,hAPP_b1906038105e_bool(F,X_1)),hAPP_b1906038105e_bool(F,Y_1))) )
         => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A),hAPP_b1906038105e_bool(F,Ca))) ) ) ) ).

tff(fact_159_order__subst1,axiom,
    ! [Ca: fun_Ho1996104121e_bool,A: bool,F: fun_fu689207471l_bool,B: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,A),hAPP_f1408815105l_bool(F,B)))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,B),Ca))
       => ( ! [X_1: fun_Ho1996104121e_bool,Y_1: fun_Ho1996104121e_bool] :
              ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,X_1),Y_1))
             => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_f1408815105l_bool(F,X_1)),hAPP_f1408815105l_bool(F,Y_1))) )
         => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,A),hAPP_f1408815105l_bool(F,Ca))) ) ) ) ).

tff(fact_160_semilattice__big_OF__eq,axiom,
    ! [A_1: fun_Ho1996104121e_bool,F: fun_Ho784854669_state,F_1: fun_fu385705929_state] :
      ( hBOOL(big_se163227553_state(F,F_1))
     => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1))
       => ( hAPP_f890249627_state(F_1,A_1) = hAPP_f890249627_state(finite206258583_state(F),A_1) ) ) ) ).

tff(fact_161_folding__one_Oremove,axiom,
    ! [X_2: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool,F: fun_Ho784854669_state,F_1: fun_fu385705929_state] :
      ( hBOOL(finite1782051386_state(F,F_1))
     => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1))
       => ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),A_1))
         => ( ( ( hAPP_f121055253e_bool(minus_589155914e_bool(A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),bot_bo1715400655e_bool)) = bot_bo1715400655e_bool )
             => ( hAPP_f890249627_state(F_1,A_1) = X_2 ) )
            & ( ( hAPP_f121055253e_bool(minus_589155914e_bool(A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),bot_bo1715400655e_bool)) != bot_bo1715400655e_bool )
             => ( hAPP_f890249627_state(F_1,A_1) = hAPP_H350317809_state(hAPP_H743221337_state(F,X_2),hAPP_f890249627_state(F_1,hAPP_f121055253e_bool(minus_589155914e_bool(A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),bot_bo1715400655e_bool)))) ) ) ) ) ) ) ).

tff(fact_162_folding__one_Oinsert__remove,axiom,
    ! [X_2: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool,F: fun_Ho784854669_state,F_1: fun_fu385705929_state] :
      ( hBOOL(finite1782051386_state(F,F_1))
     => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1))
       => ( ( ( hAPP_f121055253e_bool(minus_589155914e_bool(A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),bot_bo1715400655e_bool)) = bot_bo1715400655e_bool )
           => ( hAPP_f890249627_state(F_1,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),A_1)) = X_2 ) )
          & ( ( hAPP_f121055253e_bool(minus_589155914e_bool(A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),bot_bo1715400655e_bool)) != bot_bo1715400655e_bool )
           => ( hAPP_f890249627_state(F_1,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),A_1)) = hAPP_H350317809_state(hAPP_H743221337_state(F,X_2),hAPP_f890249627_state(F_1,hAPP_f121055253e_bool(minus_589155914e_bool(A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),bot_bo1715400655e_bool)))) ) ) ) ) ) ).

tff(fact_163_DiffI,axiom,
    ! [B_1: fun_Ho1996104121e_bool,Ca: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,Ca),A_1))
     => ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,Ca),B_1))
       => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,Ca),hAPP_f121055253e_bool(minus_589155914e_bool(A_1),B_1))) ) ) ).

tff(fact_164_DiffE,axiom,
    ! [Ca: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,Ca),hAPP_f121055253e_bool(minus_589155914e_bool(A_1),B_1)))
     => ~ ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,Ca),A_1))
         => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,Ca),B_1)) ) ) ).

tff(fact_165_finite__Diff,axiom,
    ! [B_1: fun_Ho1996104121e_bool,A_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1))
     => hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,hAPP_f121055253e_bool(minus_589155914e_bool(A_1),B_1))) ) ).

tff(fact_166_set__diff__eq,axiom,
    ! [A_1: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(minus_589155914e_bool(A_1),B_1) = hAPP_f121055253e_bool(collec1735317154_state,hAPP_f121055253e_bool(cOMBS_896408300l_bool(hAPP_f932622461l_bool(cOMBB_242064253_state(fconj),hAPP_f121055253e_bool(cOMBC_1099403170l_bool(member618553700_state),A_1))),hAPP_f121055253e_bool(cOMBB_1424001755_state(fNot),hAPP_f121055253e_bool(cOMBC_1099403170l_bool(member618553700_state),B_1)))) ).

tff(fact_167_Diff__iff,axiom,
    ! [Ca: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,Ca),hAPP_f121055253e_bool(minus_589155914e_bool(A_1),B_1)))
    <=> ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,Ca),A_1))
        & ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,Ca),B_1)) ) ) ).

tff(fact_168_Diff__idemp,axiom,
    ! [A_1: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(minus_589155914e_bool(hAPP_f121055253e_bool(minus_589155914e_bool(A_1),B_1)),B_1) = hAPP_f121055253e_bool(minus_589155914e_bool(A_1),B_1) ).

tff(fact_169_DiffD1,axiom,
    ! [Ca: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,Ca),hAPP_f121055253e_bool(minus_589155914e_bool(A_1),B_1)))
     => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,Ca),A_1)) ) ).

tff(fact_170_DiffD2,axiom,
    ! [Ca: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,Ca),hAPP_f121055253e_bool(minus_589155914e_bool(A_1),B_1)))
     => ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,Ca),B_1)) ) ).

tff(fact_171_empty__Diff,axiom,
    ! [A_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(minus_589155914e_bool(bot_bo1715400655e_bool),A_1) = bot_bo1715400655e_bool ).

tff(fact_172_Diff__empty,axiom,
    ! [A_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(minus_589155914e_bool(A_1),bot_bo1715400655e_bool) = A_1 ).

tff(fact_173_Diff__cancel,axiom,
    ! [A_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(minus_589155914e_bool(A_1),A_1) = bot_bo1715400655e_bool ).

tff(fact_174_finite__Diff2,axiom,
    ! [A_1: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,B_1))
     => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,hAPP_f121055253e_bool(minus_589155914e_bool(A_1),B_1)))
      <=> hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1)) ) ) ).

tff(fact_175_insert__Diff__if,axiom,
    ! [A_1: fun_Ho1996104121e_bool,X_2: hoare_1875481847_state,B_1: fun_Ho1996104121e_bool] :
      ( ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),B_1))
       => ( hAPP_f121055253e_bool(minus_589155914e_bool(hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),A_1)),B_1) = hAPP_f121055253e_bool(minus_589155914e_bool(A_1),B_1) ) )
      & ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),B_1))
       => ( hAPP_f121055253e_bool(minus_589155914e_bool(hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),A_1)),B_1) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),hAPP_f121055253e_bool(minus_589155914e_bool(A_1),B_1)) ) ) ) ).

tff(fact_176_insert__Diff1,axiom,
    ! [A_1: fun_Ho1996104121e_bool,X_2: hoare_1875481847_state,B_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),B_1))
     => ( hAPP_f121055253e_bool(minus_589155914e_bool(hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),A_1)),B_1) = hAPP_f121055253e_bool(minus_589155914e_bool(A_1),B_1) ) ) ).

tff(fact_177_Diff__subset,axiom,
    ! [A_1: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] : hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,hAPP_f121055253e_bool(minus_589155914e_bool(A_1),B_1)),A_1)) ).

tff(fact_178_Diff__mono,axiom,
    ! [D: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool,A_1: fun_Ho1996104121e_bool,C: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A_1),C))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,D),B_1))
       => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,hAPP_f121055253e_bool(minus_589155914e_bool(A_1),B_1)),hAPP_f121055253e_bool(minus_589155914e_bool(C),D))) ) ) ).

tff(fact_179_double__diff,axiom,
    ! [C: fun_Ho1996104121e_bool,A_1: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A_1),B_1))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,B_1),C))
       => ( hAPP_f121055253e_bool(minus_589155914e_bool(B_1),hAPP_f121055253e_bool(minus_589155914e_bool(C),A_1)) = A_1 ) ) ) ).

tff(fact_180_Diff__insert,axiom,
    ! [A_1: fun_Ho1996104121e_bool,A: hoare_1875481847_state,B_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(minus_589155914e_bool(A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),B_1)) = hAPP_f121055253e_bool(minus_589155914e_bool(hAPP_f121055253e_bool(minus_589155914e_bool(A_1),B_1)),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),bot_bo1715400655e_bool)) ).

tff(fact_181_Diff__insert2,axiom,
    ! [A_1: fun_Ho1996104121e_bool,A: hoare_1875481847_state,B_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(minus_589155914e_bool(A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),B_1)) = hAPP_f121055253e_bool(minus_589155914e_bool(hAPP_f121055253e_bool(minus_589155914e_bool(A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),bot_bo1715400655e_bool))),B_1) ).

tff(fact_182_insert__Diff__single,axiom,
    ! [A: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool] : hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),hAPP_f121055253e_bool(minus_589155914e_bool(A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),bot_bo1715400655e_bool))) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),A_1) ).

tff(fact_183_Diff__insert__absorb,axiom,
    ! [X_2: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool] :
      ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),A_1))
     => ( hAPP_f121055253e_bool(minus_589155914e_bool(hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),A_1)),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),bot_bo1715400655e_bool)) = A_1 ) ) ).

tff(fact_184_insert__Diff,axiom,
    ! [A: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A),A_1))
     => ( hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),hAPP_f121055253e_bool(minus_589155914e_bool(A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),bot_bo1715400655e_bool))) = A_1 ) ) ).

tff(fact_185_finite__Diff__insert,axiom,
    ! [A_1: fun_Ho1996104121e_bool,A: hoare_1875481847_state,B_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,hAPP_f121055253e_bool(minus_589155914e_bool(A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),B_1))))
    <=> hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,hAPP_f121055253e_bool(minus_589155914e_bool(A_1),B_1))) ) ).

tff(fact_186_subset__insert__iff,axiom,
    ! [A_1: fun_Ho1996104121e_bool,X_2: hoare_1875481847_state,B_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),B_1)))
    <=> ( ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),A_1))
         => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,hAPP_f121055253e_bool(minus_589155914e_bool(A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),bot_bo1715400655e_bool))),B_1)) )
        & ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),A_1))
         => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A_1),B_1)) ) ) ) ).

tff(fact_187_diff__single__insert,axiom,
    ! [A_1: fun_Ho1996104121e_bool,X_2: hoare_1875481847_state,B_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,hAPP_f121055253e_bool(minus_589155914e_bool(A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),bot_bo1715400655e_bool))),B_1))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),A_1))
       => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),B_1))) ) ) ).

tff(fact_188_finite__empty__induct,axiom,
    ! [Pa: fun_fu689207471l_bool,A_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1))
     => ( hBOOL(hAPP_f1408815105l_bool(Pa,A_1))
       => ( ! [A_3: hoare_1875481847_state,A_2: fun_Ho1996104121e_bool] :
              ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_2))
             => ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A_3),A_2))
               => ( hBOOL(hAPP_f1408815105l_bool(Pa,A_2))
                 => hBOOL(hAPP_f1408815105l_bool(Pa,hAPP_f121055253e_bool(minus_589155914e_bool(A_2),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A_3),bot_bo1715400655e_bool)))) ) ) )
         => hBOOL(hAPP_f1408815105l_bool(Pa,bot_bo1715400655e_bool)) ) ) ) ).

tff(fact_189_minus__apply,axiom,
    ! [A_1: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool,X_2: hoare_1875481847_state] :
      ( hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(minus_589155914e_bool(A_1),B_1),X_2))
    <=> hBOOL(hAPP_bool_bool(minus_minus_bool(hAPP_H1173941847e_bool(A_1,X_2)),hAPP_H1173941847e_bool(B_1,X_2))) ) ).

tff(fact_190_fun__diff__def,axiom,
    ! [A_1: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool,X_1: hoare_1875481847_state] :
      ( hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(minus_589155914e_bool(A_1),B_1),X_1))
    <=> hBOOL(hAPP_bool_bool(minus_minus_bool(hAPP_H1173941847e_bool(A_1,X_1)),hAPP_H1173941847e_bool(B_1,X_1))) ) ).

tff(fact_191_comp__fun__commute_Ofold__graph__insertE__aux,axiom,
    ! [A: hoare_1875481847_state,Z: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool,Y_2: hoare_1875481847_state,F: fun_Ho784854669_state] :
      ( hBOOL(finite1496568813_state(F))
     => ( hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(hAPP_H1625489667e_bool(finite1721989730_state(F),Z),A_1),Y_2))
       => ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A),A_1))
         => ? [Y_3: hoare_1875481847_state] :
              ( ( Y_2 = hAPP_H350317809_state(hAPP_H743221337_state(F,A),Y_3) )
              & hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(hAPP_H1625489667e_bool(finite1721989730_state(F),Z),hAPP_f121055253e_bool(minus_589155914e_bool(A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),bot_bo1715400655e_bool))),Y_3)) ) ) ) ) ).

tff(fact_192_comp__fun__commute_Ofun__left__comm,axiom,
    ! [X_2: hoare_1875481847_state,Y_2: hoare_1875481847_state,Z: hoare_1875481847_state,F: fun_Ho784854669_state] :
      ( hBOOL(finite1496568813_state(F))
     => ( hAPP_H350317809_state(hAPP_H743221337_state(F,X_2),hAPP_H350317809_state(hAPP_H743221337_state(F,Y_2),Z)) = hAPP_H350317809_state(hAPP_H743221337_state(F,Y_2),hAPP_H350317809_state(hAPP_H743221337_state(F,X_2),Z)) ) ) ).

tff(fact_193_comp__fun__commute_Ofold__graph__determ,axiom,
    ! [Y_2: hoare_1875481847_state,Z: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool,X_2: hoare_1875481847_state,F: fun_Ho784854669_state] :
      ( hBOOL(finite1496568813_state(F))
     => ( hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(hAPP_H1625489667e_bool(finite1721989730_state(F),Z),A_1),X_2))
       => ( hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(hAPP_H1625489667e_bool(finite1721989730_state(F),Z),A_1),Y_2))
         => ( Y_2 = X_2 ) ) ) ) ).

tff(fact_194_comp__fun__commute_Ofold__graph__insertE,axiom,
    ! [Z: hoare_1875481847_state,X_2: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool,V: hoare_1875481847_state,F: fun_Ho784854669_state] :
      ( hBOOL(finite1496568813_state(F))
     => ( hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(hAPP_H1625489667e_bool(finite1721989730_state(F),Z),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),A_1)),V))
       => ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),A_1))
         => ~ ! [Y_1: hoare_1875481847_state] :
                ( ( V = hAPP_H350317809_state(hAPP_H743221337_state(F,X_2),Y_1) )
               => ~ hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(hAPP_H1625489667e_bool(finite1721989730_state(F),Z),A_1),Y_1)) ) ) ) ) ).

tff(fact_195_min__leastR,axiom,
    ! [X_2: bool,Least: bool] :
      ( ! [X_1: bool] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Least),X_1))
     => ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_min_bool,X_2),Least))
      <=> hBOOL(Least) ) ) ).

tff(fact_196_min__leastR,axiom,
    ! [X_2: fun_Ho1996104121e_bool,Least: fun_Ho1996104121e_bool] :
      ( ! [X_1: fun_Ho1996104121e_bool] : hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Least),X_1))
     => ( hAPP_f121055253e_bool(hAPP_f1204759513e_bool(ord_mi1437909322e_bool,X_2),Least) = Least ) ) ).

tff(fact_197_min__leastL,axiom,
    ! [X_2: bool,Least: bool] :
      ( ! [X_1: bool] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Least),X_1))
     => ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_min_bool,Least),X_2))
      <=> hBOOL(Least) ) ) ).

tff(fact_198_min__leastL,axiom,
    ! [X_2: fun_Ho1996104121e_bool,Least: fun_Ho1996104121e_bool] :
      ( ! [X_1: fun_Ho1996104121e_bool] : hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Least),X_1))
     => ( hAPP_f121055253e_bool(hAPP_f1204759513e_bool(ord_mi1437909322e_bool,Least),X_2) = Least ) ) ).

tff(fact_199_min__ord__min,axiom,
    ord_min_bool = min_bool(ord_less_eq_bool) ).

tff(fact_200_min__ord__min,axiom,
    ord_mi1437909322e_bool = min_fu1173241102e_bool(ord_le233469699e_bool) ).

tff(fact_201_comp__fun__commute_Ofold__insert__remove,axiom,
    ! [Z: hoare_1875481847_state,X_2: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool,F: fun_Ho784854669_state] :
      ( hBOOL(finite1496568813_state(F))
     => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1))
       => ( hAPP_f890249627_state(finite2136361956_state(F,Z),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),A_1)) = hAPP_H350317809_state(hAPP_H743221337_state(F,X_2),hAPP_f890249627_state(finite2136361956_state(F,Z),hAPP_f121055253e_bool(minus_589155914e_bool(A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),bot_bo1715400655e_bool)))) ) ) ) ).

tff(fact_202_comp__fun__commute_Ofold__rec,axiom,
    ! [Z: hoare_1875481847_state,X_2: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool,F: fun_Ho784854669_state] :
      ( hBOOL(finite1496568813_state(F))
     => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1))
       => ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),A_1))
         => ( hAPP_f890249627_state(finite2136361956_state(F,Z),A_1) = hAPP_H350317809_state(hAPP_H743221337_state(F,X_2),hAPP_f890249627_state(finite2136361956_state(F,Z),hAPP_f121055253e_bool(minus_589155914e_bool(A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),bot_bo1715400655e_bool)))) ) ) ) ) ).

tff(fact_203_fold__empty,axiom,
    ! [F: fun_Ho784854669_state,Z: hoare_1875481847_state] : hAPP_f890249627_state(finite2136361956_state(F,Z),bot_bo1715400655e_bool) = Z ).

tff(fact_204_comp__fun__commute_Ofold__fun__comm,axiom,
    ! [X_2: hoare_1875481847_state,Z: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool,F: fun_Ho784854669_state] :
      ( hBOOL(finite1496568813_state(F))
     => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1))
       => ( hAPP_H350317809_state(hAPP_H743221337_state(F,X_2),hAPP_f890249627_state(finite2136361956_state(F,Z),A_1)) = hAPP_f890249627_state(finite2136361956_state(F,hAPP_H350317809_state(hAPP_H743221337_state(F,X_2),Z)),A_1) ) ) ) ).

tff(fact_205_comp__fun__commute_Ofold__equality,axiom,
    ! [Z: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool,Y_2: hoare_1875481847_state,F: fun_Ho784854669_state] :
      ( hBOOL(finite1496568813_state(F))
     => ( hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(hAPP_H1625489667e_bool(finite1721989730_state(F),Z),A_1),Y_2))
       => ( hAPP_f890249627_state(finite2136361956_state(F,Z),A_1) = Y_2 ) ) ) ).

tff(fact_206_comp__fun__commute_Ofold__insert2,axiom,
    ! [Z: hoare_1875481847_state,X_2: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool,F: fun_Ho784854669_state] :
      ( hBOOL(finite1496568813_state(F))
     => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1))
       => ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),A_1))
         => ( hAPP_f890249627_state(finite2136361956_state(F,Z),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),A_1)) = hAPP_f890249627_state(finite2136361956_state(F,hAPP_H350317809_state(hAPP_H743221337_state(F,X_2),Z)),A_1) ) ) ) ) ).

tff(fact_207_comp__fun__commute_Ofold__insert,axiom,
    ! [Z: hoare_1875481847_state,X_2: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool,F: fun_Ho784854669_state] :
      ( hBOOL(finite1496568813_state(F))
     => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1))
       => ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),A_1))
         => ( hAPP_f890249627_state(finite2136361956_state(F,Z),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),A_1)) = hAPP_H350317809_state(hAPP_H743221337_state(F,X_2),hAPP_f890249627_state(finite2136361956_state(F,Z),A_1)) ) ) ) ) ).

tff(fact_208_folding__one_Oeq__fold_H,axiom,
    ! [X_2: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool,F: fun_Ho784854669_state,F_1: fun_fu385705929_state] :
      ( hBOOL(finite1782051386_state(F,F_1))
     => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1))
       => ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),A_1))
         => ( hAPP_f890249627_state(F_1,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),A_1)) = hAPP_f890249627_state(finite2136361956_state(F,X_2),A_1) ) ) ) ) ).

tff(fact_209_folding__one__idem_Oeq__fold__idem_H,axiom,
    ! [A: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool,F: fun_Ho784854669_state,F_1: fun_fu385705929_state] :
      ( hBOOL(finite1514162941_state(F,F_1))
     => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1))
       => ( hAPP_f890249627_state(F_1,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),A_1)) = hAPP_f890249627_state(finite2136361956_state(F,A),A_1) ) ) ) ).

tff(fact_210_comp__fun__commute_Ofold__graph__fold,axiom,
    ! [Z: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool,F: fun_Ho784854669_state] :
      ( hBOOL(finite1496568813_state(F))
     => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1))
       => hBOOL(hAPP_H1173941847e_bool(hAPP_f121055253e_bool(hAPP_H1625489667e_bool(finite1721989730_state(F),Z),A_1),hAPP_f890249627_state(finite2136361956_state(F,Z),A_1))) ) ) ).

tff(fact_211_comp__fun__idem_Ofold__insert__idem,axiom,
    ! [Z: hoare_1875481847_state,X_2: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool,F: fun_Ho784854669_state] :
      ( hBOOL(finite158617192_state(F))
     => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1))
       => ( hAPP_f890249627_state(finite2136361956_state(F,Z),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),A_1)) = hAPP_H350317809_state(hAPP_H743221337_state(F,X_2),hAPP_f890249627_state(finite2136361956_state(F,Z),A_1)) ) ) ) ).

tff(fact_212_comp__fun__idem_Ofold__insert__idem,axiom,
    ! [Z: fun_Ho1996104121e_bool,X_2: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool,F: fun_Ho1110608055e_bool] :
      ( hBOOL(finite1936929544e_bool(F))
     => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1))
       => ( hAPP_f121055253e_bool(finite1131674252e_bool(F,Z),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),A_1)) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(F,X_2),hAPP_f121055253e_bool(finite1131674252e_bool(F,Z),A_1)) ) ) ) ).

tff(fact_213_comp__fun__idem_Ofold__insert__idem2,axiom,
    ! [Z: hoare_1875481847_state,X_2: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool,F: fun_Ho784854669_state] :
      ( hBOOL(finite158617192_state(F))
     => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1))
       => ( hAPP_f890249627_state(finite2136361956_state(F,Z),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),A_1)) = hAPP_f890249627_state(finite2136361956_state(F,hAPP_H350317809_state(hAPP_H743221337_state(F,X_2),Z)),A_1) ) ) ) ).

tff(fact_214_comp__fun__idem_Ofold__insert__idem2,axiom,
    ! [Z: fun_Ho1996104121e_bool,X_2: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool,F: fun_Ho1110608055e_bool] :
      ( hBOOL(finite1936929544e_bool(F))
     => ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1))
       => ( hAPP_f121055253e_bool(finite1131674252e_bool(F,Z),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),A_1)) = hAPP_f121055253e_bool(finite1131674252e_bool(F,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(F,X_2),Z)),A_1) ) ) ) ).

tff(fact_215_comp__fun__idem_Ofun__left__idem,axiom,
    ! [X_2: hoare_1875481847_state,Z: fun_Ho1996104121e_bool,F: fun_Ho1110608055e_bool] :
      ( hBOOL(finite1936929544e_bool(F))
     => ( hAPP_f121055253e_bool(hAPP_H1625489667e_bool(F,X_2),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(F,X_2),Z)) = hAPP_f121055253e_bool(hAPP_H1625489667e_bool(F,X_2),Z) ) ) ).

tff(fact_216_comp__fun__idem__insert,axiom,
    hBOOL(finite1936929544e_bool(insert694999549_state)) ).

tff(fact_217_setsum__diff1__nat,axiom,
    ! [F: fun_Ho1284378692te_nat,A: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool] :
      ( ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A),A_1))
       => ( big_co583362288te_nat(F,hAPP_f121055253e_bool(minus_589155914e_bool(A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),bot_bo1715400655e_bool))) = hAPP_nat_nat(minus_minus_nat(big_co583362288te_nat(F,A_1)),hAPP_H1027528080te_nat(F,A)) ) )
      & ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A),A_1))
       => ( big_co583362288te_nat(F,hAPP_f121055253e_bool(minus_589155914e_bool(A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),bot_bo1715400655e_bool))) = big_co583362288te_nat(F,A_1) ) ) ) ).

tff(fact_218_setsum__diff__nat,axiom,
    ! [F: fun_Ho1284378692te_nat,A_1: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,B_1))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,B_1),A_1))
       => ( big_co583362288te_nat(F,hAPP_f121055253e_bool(minus_589155914e_bool(A_1),B_1)) = hAPP_nat_nat(minus_minus_nat(big_co583362288te_nat(F,A_1)),big_co583362288te_nat(F,B_1)) ) ) ) ).

tff(fact_219_setsum_Ocong,axiom,
    ! [G: fun_Ho1284378692te_nat,H: fun_Ho1284378692te_nat,A_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1))
     => ( ! [X_1: hoare_1875481847_state] :
            ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),A_1))
           => ( hAPP_H1027528080te_nat(G,X_1) = hAPP_H1027528080te_nat(H,X_1) ) )
       => ( big_co583362288te_nat(G,A_1) = big_co583362288te_nat(H,A_1) ) ) ) ).

tff(fact_220_max__leastR,axiom,
    ! [X_2: bool,Least: bool] :
      ( ! [X_1: bool] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Least),X_1))
     => ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_max_bool,X_2),Least))
      <=> hBOOL(X_2) ) ) ).

tff(fact_221_max__leastR,axiom,
    ! [X_2: fun_Ho1996104121e_bool,Least: fun_Ho1996104121e_bool] :
      ( ! [X_1: fun_Ho1996104121e_bool] : hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Least),X_1))
     => ( hAPP_f121055253e_bool(hAPP_f1204759513e_bool(ord_ma955633336e_bool,X_2),Least) = X_2 ) ) ).

tff(fact_222_max__leastL,axiom,
    ! [X_2: bool,Least: bool] :
      ( ! [X_1: bool] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Least),X_1))
     => ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_max_bool,Least),X_2))
      <=> hBOOL(X_2) ) ) ).

tff(fact_223_max__leastL,axiom,
    ! [X_2: fun_Ho1996104121e_bool,Least: fun_Ho1996104121e_bool] :
      ( ! [X_1: fun_Ho1996104121e_bool] : hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Least),X_1))
     => ( hAPP_f121055253e_bool(hAPP_f1204759513e_bool(ord_ma955633336e_bool,Least),X_2) = X_2 ) ) ).

tff(fact_224_max__ord__max,axiom,
    ord_max_bool = max_bool(ord_less_eq_bool) ).

tff(fact_225_max__ord__max,axiom,
    ord_ma955633336e_bool = max_fu690965116e_bool(ord_le233469699e_bool) ).

tff(fact_226_setsum_Oremove,axiom,
    ! [G: fun_Ho1284378692te_nat,X_2: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),A_1))
       => ( big_co583362288te_nat(G,A_1) = plus_plus_nat(hAPP_H1027528080te_nat(G,X_2),big_co583362288te_nat(G,hAPP_f121055253e_bool(minus_589155914e_bool(A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),bot_bo1715400655e_bool)))) ) ) ) ).

tff(fact_227_setsum__diff1_H,axiom,
    ! [F: fun_Ho1284378692te_nat,A: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A),A_1))
       => ( big_co583362288te_nat(F,A_1) = plus_plus_nat(hAPP_H1027528080te_nat(F,A),big_co583362288te_nat(F,hAPP_f121055253e_bool(minus_589155914e_bool(A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),bot_bo1715400655e_bool)))) ) ) ) ).

tff(fact_228_setsum_Oinsert,axiom,
    ! [G: fun_Ho1284378692te_nat,X_2: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1))
     => ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),A_1))
       => ( big_co583362288te_nat(G,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),A_1)) = plus_plus_nat(hAPP_H1027528080te_nat(G,X_2),big_co583362288te_nat(G,A_1)) ) ) ) ).

tff(fact_229_setsum__insert,axiom,
    ! [F: fun_Ho1284378692te_nat,A: hoare_1875481847_state,F_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,F_1))
     => ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,A),F_1))
       => ( big_co583362288te_nat(F,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,A),F_1)) = plus_plus_nat(hAPP_H1027528080te_nat(F,A),big_co583362288te_nat(F,F_1)) ) ) ) ).

tff(fact_230_setsum_Oinsert__remove,axiom,
    ! [G: fun_Ho1284378692te_nat,X_2: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1))
     => ( big_co583362288te_nat(G,hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),A_1)) = plus_plus_nat(hAPP_H1027528080te_nat(G,X_2),big_co583362288te_nat(G,hAPP_f121055253e_bool(minus_589155914e_bool(A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),bot_bo1715400655e_bool)))) ) ) ).

tff(fact_231_setsum__cong2,axiom,
    ! [F: fun_Ho1284378692te_nat,G: fun_Ho1284378692te_nat,A_1: fun_Ho1996104121e_bool] :
      ( ! [X_1: hoare_1875481847_state] :
          ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),A_1))
         => ( hAPP_H1027528080te_nat(F,X_1) = hAPP_H1027528080te_nat(G,X_1) ) )
     => ( big_co583362288te_nat(F,A_1) = big_co583362288te_nat(G,A_1) ) ) ).

tff(fact_232_setsum__cong,axiom,
    ! [F: fun_Ho1284378692te_nat,G: fun_Ho1284378692te_nat,A_1: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( ( A_1 = B_1 )
     => ( ! [X_1: hoare_1875481847_state] :
            ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),B_1))
           => ( hAPP_H1027528080te_nat(F,X_1) = hAPP_H1027528080te_nat(G,X_1) ) )
       => ( big_co583362288te_nat(F,A_1) = big_co583362288te_nat(G,B_1) ) ) ) ).

tff(fact_233_setsum_OF__cong,axiom,
    ! [H: fun_Ho1284378692te_nat,G: fun_Ho1284378692te_nat,A_1: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( ( A_1 = B_1 )
     => ( ! [X_1: hoare_1875481847_state] :
            ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),B_1))
           => ( hAPP_H1027528080te_nat(H,X_1) = hAPP_H1027528080te_nat(G,X_1) ) )
       => ( big_co583362288te_nat(H,A_1) = big_co583362288te_nat(G,B_1) ) ) ) ).

tff(fact_234_not__less__bot,axiom,
    ! [A: fun_Ho1996104121e_bool] : ~ hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(A),bot_bo1715400655e_bool)) ).

tff(fact_235_not__less__bot,axiom,
    ! [A: bool] : ~ hBOOL(hAPP_bool_bool(ord_less_bool(A),bot_bot_bool)) ).

tff(fact_236_bot__less,axiom,
    ! [A: fun_Ho1996104121e_bool] :
      ( ( A != bot_bo1715400655e_bool )
    <=> hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(bot_bo1715400655e_bool),A)) ) ).

tff(fact_237_bot__less,axiom,
    ! [A: bool] :
      ( ~ ( hBOOL(A)
        <=> hBOOL(bot_bot_bool) )
    <=> hBOOL(hAPP_bool_bool(ord_less_bool(bot_bot_bool),A)) ) ).

tff(fact_238_not__psubset__empty,axiom,
    ! [A_1: fun_Ho1996104121e_bool] : ~ hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(A_1),bot_bo1715400655e_bool)) ).

tff(fact_239_psubset__eq,axiom,
    ! [A_1: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(A_1),B_1))
    <=> ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A_1),B_1))
        & ( A_1 != B_1 ) ) ) ).

tff(fact_240_subset__iff__psubset__eq,axiom,
    ! [A_1: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A_1),B_1))
    <=> ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(A_1),B_1))
        | ( A_1 = B_1 ) ) ) ).

tff(fact_241_psubset__imp__subset,axiom,
    ! [A_1: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(A_1),B_1))
     => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A_1),B_1)) ) ).

tff(fact_242_psubset__subset__trans,axiom,
    ! [C: fun_Ho1996104121e_bool,A_1: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(A_1),B_1))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,B_1),C))
       => hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(A_1),C)) ) ) ).

tff(fact_243_subset__psubset__trans,axiom,
    ! [C: fun_Ho1996104121e_bool,A_1: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A_1),B_1))
     => ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(B_1),C))
       => hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(A_1),C)) ) ) ).

tff(fact_244_order__less__asym,axiom,
    ! [X_2: fun_Ho1996104121e_bool,Y_2: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(X_2),Y_2))
     => ~ hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(Y_2),X_2)) ) ).

tff(fact_245_order__less__asym,axiom,
    ! [X_2: bool,Y_2: bool] :
      ( hBOOL(hAPP_bool_bool(ord_less_bool(X_2),Y_2))
     => ~ hBOOL(hAPP_bool_bool(ord_less_bool(Y_2),X_2)) ) ).

tff(fact_246_xt1_I10_J,axiom,
    ! [Z: fun_Ho1996104121e_bool,Y_2: fun_Ho1996104121e_bool,X_2: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(Y_2),X_2))
     => ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(Z),Y_2))
       => hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(Z),X_2)) ) ) ).

tff(fact_247_xt1_I10_J,axiom,
    ! [Z: bool,Y_2: bool,X_2: bool] :
      ( hBOOL(hAPP_bool_bool(ord_less_bool(Y_2),X_2))
     => ( hBOOL(hAPP_bool_bool(ord_less_bool(Z),Y_2))
       => hBOOL(hAPP_bool_bool(ord_less_bool(Z),X_2)) ) ) ).

tff(fact_248_order__less__trans,axiom,
    ! [Z: fun_Ho1996104121e_bool,X_2: fun_Ho1996104121e_bool,Y_2: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(X_2),Y_2))
     => ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(Y_2),Z))
       => hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(X_2),Z)) ) ) ).

tff(fact_249_order__less__trans,axiom,
    ! [Z: bool,X_2: bool,Y_2: bool] :
      ( hBOOL(hAPP_bool_bool(ord_less_bool(X_2),Y_2))
     => ( hBOOL(hAPP_bool_bool(ord_less_bool(Y_2),Z))
       => hBOOL(hAPP_bool_bool(ord_less_bool(X_2),Z)) ) ) ).

tff(fact_250_xt1_I2_J,axiom,
    ! [Ca: fun_Ho1996104121e_bool,B: fun_Ho1996104121e_bool,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(B),A))
     => ( ( B = Ca )
       => hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(Ca),A)) ) ) ).

tff(fact_251_xt1_I2_J,axiom,
    ! [Ca: bool,B: bool,A: bool] :
      ( hBOOL(hAPP_bool_bool(ord_less_bool(B),A))
     => ( ( hBOOL(B)
        <=> hBOOL(Ca) )
       => hBOOL(hAPP_bool_bool(ord_less_bool(Ca),A)) ) ) ).

tff(fact_252_ord__less__eq__trans,axiom,
    ! [Ca: fun_Ho1996104121e_bool,A: fun_Ho1996104121e_bool,B: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(A),B))
     => ( ( B = Ca )
       => hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(A),Ca)) ) ) ).

tff(fact_253_ord__less__eq__trans,axiom,
    ! [Ca: bool,A: bool,B: bool] :
      ( hBOOL(hAPP_bool_bool(ord_less_bool(A),B))
     => ( ( hBOOL(B)
        <=> hBOOL(Ca) )
       => hBOOL(hAPP_bool_bool(ord_less_bool(A),Ca)) ) ) ).

tff(fact_254_xt1_I1_J,axiom,
    ! [Ca: fun_Ho1996104121e_bool,A: fun_Ho1996104121e_bool,B: fun_Ho1996104121e_bool] :
      ( ( A = B )
     => ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(Ca),B))
       => hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(Ca),A)) ) ) ).

tff(fact_255_xt1_I1_J,axiom,
    ! [Ca: bool,B: bool,A: bool] :
      ( ( hBOOL(A)
      <=> hBOOL(B) )
     => ( hBOOL(hAPP_bool_bool(ord_less_bool(Ca),B))
       => hBOOL(hAPP_bool_bool(ord_less_bool(Ca),A)) ) ) ).

tff(fact_256_ord__eq__less__trans,axiom,
    ! [Ca: fun_Ho1996104121e_bool,A: fun_Ho1996104121e_bool,B: fun_Ho1996104121e_bool] :
      ( ( A = B )
     => ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(B),Ca))
       => hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(A),Ca)) ) ) ).

tff(fact_257_ord__eq__less__trans,axiom,
    ! [Ca: bool,B: bool,A: bool] :
      ( ( hBOOL(A)
      <=> hBOOL(B) )
     => ( hBOOL(hAPP_bool_bool(ord_less_bool(B),Ca))
       => hBOOL(hAPP_bool_bool(ord_less_bool(A),Ca)) ) ) ).

tff(fact_258_xt1_I9_J,axiom,
    ! [B: fun_Ho1996104121e_bool,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(B),A))
     => ~ hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(A),B)) ) ).

tff(fact_259_xt1_I9_J,axiom,
    ! [B: bool,A: bool] :
      ( hBOOL(hAPP_bool_bool(ord_less_bool(B),A))
     => ~ hBOOL(hAPP_bool_bool(ord_less_bool(A),B)) ) ).

tff(fact_260_order__less__asym_H,axiom,
    ! [A: fun_Ho1996104121e_bool,B: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(A),B))
     => ~ hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(B),A)) ) ).

tff(fact_261_order__less__asym_H,axiom,
    ! [A: bool,B: bool] :
      ( hBOOL(hAPP_bool_bool(ord_less_bool(A),B))
     => ~ hBOOL(hAPP_bool_bool(ord_less_bool(B),A)) ) ).

tff(fact_262_order__less__imp__triv,axiom,
    ! [Pa: bool,X_2: fun_Ho1996104121e_bool,Y_2: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(X_2),Y_2))
     => ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(Y_2),X_2))
       => hBOOL(Pa) ) ) ).

tff(fact_263_order__less__imp__triv,axiom,
    ! [Pa: bool,X_2: bool,Y_2: bool] :
      ( hBOOL(hAPP_bool_bool(ord_less_bool(X_2),Y_2))
     => ( hBOOL(hAPP_bool_bool(ord_less_bool(Y_2),X_2))
       => hBOOL(Pa) ) ) ).

tff(fact_264_order__less__imp__not__eq2,axiom,
    ! [X_2: fun_Ho1996104121e_bool,Y_2: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(X_2),Y_2))
     => ( Y_2 != X_2 ) ) ).

tff(fact_265_order__less__imp__not__eq2,axiom,
    ! [X_2: bool,Y_2: bool] :
      ( hBOOL(hAPP_bool_bool(ord_less_bool(X_2),Y_2))
     => ( hBOOL(Y_2)
      <=> ~ hBOOL(X_2) ) ) ).

tff(fact_266_order__less__imp__not__eq,axiom,
    ! [X_2: fun_Ho1996104121e_bool,Y_2: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(X_2),Y_2))
     => ( X_2 != Y_2 ) ) ).

tff(fact_267_order__less__imp__not__eq,axiom,
    ! [X_2: bool,Y_2: bool] :
      ( hBOOL(hAPP_bool_bool(ord_less_bool(X_2),Y_2))
     => ( hBOOL(X_2)
      <=> ~ hBOOL(Y_2) ) ) ).

tff(fact_268_order__less__imp__not__less,axiom,
    ! [X_2: fun_Ho1996104121e_bool,Y_2: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(X_2),Y_2))
     => ~ hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(Y_2),X_2)) ) ).

tff(fact_269_order__less__imp__not__less,axiom,
    ! [X_2: bool,Y_2: bool] :
      ( hBOOL(hAPP_bool_bool(ord_less_bool(X_2),Y_2))
     => ~ hBOOL(hAPP_bool_bool(ord_less_bool(Y_2),X_2)) ) ).

tff(fact_270_order__less__not__sym,axiom,
    ! [X_2: fun_Ho1996104121e_bool,Y_2: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(X_2),Y_2))
     => ~ hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(Y_2),X_2)) ) ).

tff(fact_271_order__less__not__sym,axiom,
    ! [X_2: bool,Y_2: bool] :
      ( hBOOL(hAPP_bool_bool(ord_less_bool(X_2),Y_2))
     => ~ hBOOL(hAPP_bool_bool(ord_less_bool(Y_2),X_2)) ) ).

tff(fact_272_less__imp__neq,axiom,
    ! [X_2: fun_Ho1996104121e_bool,Y_2: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(X_2),Y_2))
     => ( X_2 != Y_2 ) ) ).

tff(fact_273_less__imp__neq,axiom,
    ! [X_2: bool,Y_2: bool] :
      ( hBOOL(hAPP_bool_bool(ord_less_bool(X_2),Y_2))
     => ~ ( hBOOL(X_2)
        <=> hBOOL(Y_2) ) ) ).

tff(fact_274_order__less__irrefl,axiom,
    ! [X_2: fun_Ho1996104121e_bool] : ~ hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(X_2),X_2)) ).

tff(fact_275_order__less__irrefl,axiom,
    ! [X_2: bool] : ~ hBOOL(hAPP_bool_bool(ord_less_bool(X_2),X_2)) ).

tff(fact_276_order__less__le,axiom,
    ! [X_2: bool,Y_2: bool] :
      ( hBOOL(hAPP_bool_bool(ord_less_bool(X_2),Y_2))
    <=> ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,X_2),Y_2))
        & ~ ( hBOOL(X_2)
          <=> hBOOL(Y_2) ) ) ) ).

tff(fact_277_order__less__le,axiom,
    ! [X_2: fun_Ho1996104121e_bool,Y_2: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(X_2),Y_2))
    <=> ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,X_2),Y_2))
        & ( X_2 != Y_2 ) ) ) ).

tff(fact_278_less__le__not__le,axiom,
    ! [X_2: bool,Y_2: bool] :
      ( hBOOL(hAPP_bool_bool(ord_less_bool(X_2),Y_2))
    <=> ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,X_2),Y_2))
        & ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Y_2),X_2)) ) ) ).

tff(fact_279_less__le__not__le,axiom,
    ! [X_2: fun_Ho1996104121e_bool,Y_2: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(X_2),Y_2))
    <=> ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,X_2),Y_2))
        & ~ hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Y_2),X_2)) ) ) ).

tff(fact_280_order__le__less,axiom,
    ! [X_2: bool,Y_2: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,X_2),Y_2))
    <=> ( hBOOL(hAPP_bool_bool(ord_less_bool(X_2),Y_2))
        | ( hBOOL(X_2)
        <=> hBOOL(Y_2) ) ) ) ).

tff(fact_281_order__le__less,axiom,
    ! [X_2: fun_Ho1996104121e_bool,Y_2: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,X_2),Y_2))
    <=> ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(X_2),Y_2))
        | ( X_2 = Y_2 ) ) ) ).

tff(fact_282_order__neq__le__trans,axiom,
    ! [B: bool,A: bool] :
      ( ~ ( hBOOL(A)
        <=> hBOOL(B) )
     => ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,A),B))
       => hBOOL(hAPP_bool_bool(ord_less_bool(A),B)) ) ) ).

tff(fact_283_order__neq__le__trans,axiom,
    ! [A: fun_Ho1996104121e_bool,B: fun_Ho1996104121e_bool] :
      ( ( A != B )
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A),B))
       => hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(A),B)) ) ) ).

tff(fact_284_xt1_I12_J,axiom,
    ! [B: bool,A: bool] :
      ( ~ ( hBOOL(A)
        <=> hBOOL(B) )
     => ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,B),A))
       => hBOOL(hAPP_bool_bool(ord_less_bool(B),A)) ) ) ).

tff(fact_285_xt1_I12_J,axiom,
    ! [A: fun_Ho1996104121e_bool,B: fun_Ho1996104121e_bool] :
      ( ( A != B )
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,B),A))
       => hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(B),A)) ) ) ).

tff(fact_286_order__less__imp__le,axiom,
    ! [X_2: bool,Y_2: bool] :
      ( hBOOL(hAPP_bool_bool(ord_less_bool(X_2),Y_2))
     => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,X_2),Y_2)) ) ).

tff(fact_287_order__less__imp__le,axiom,
    ! [X_2: fun_Ho1996104121e_bool,Y_2: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(X_2),Y_2))
     => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,X_2),Y_2)) ) ).

tff(fact_288_order__le__imp__less__or__eq,axiom,
    ! [X_2: bool,Y_2: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,X_2),Y_2))
     => ( hBOOL(hAPP_bool_bool(ord_less_bool(X_2),Y_2))
        | ( hBOOL(X_2)
        <=> hBOOL(Y_2) ) ) ) ).

tff(fact_289_order__le__imp__less__or__eq,axiom,
    ! [X_2: fun_Ho1996104121e_bool,Y_2: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,X_2),Y_2))
     => ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(X_2),Y_2))
        | ( X_2 = Y_2 ) ) ) ).

tff(fact_290_order__le__neq__trans,axiom,
    ! [A: bool,B: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,A),B))
     => ( ~ ( hBOOL(A)
          <=> hBOOL(B) )
       => hBOOL(hAPP_bool_bool(ord_less_bool(A),B)) ) ) ).

tff(fact_291_order__le__neq__trans,axiom,
    ! [A: fun_Ho1996104121e_bool,B: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A),B))
     => ( ( A != B )
       => hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(A),B)) ) ) ).

tff(fact_292_xt1_I11_J,axiom,
    ! [B: bool,A: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,B),A))
     => ( ~ ( hBOOL(A)
          <=> hBOOL(B) )
       => hBOOL(hAPP_bool_bool(ord_less_bool(B),A)) ) ) ).

tff(fact_293_xt1_I11_J,axiom,
    ! [B: fun_Ho1996104121e_bool,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,B),A))
     => ( ( A != B )
       => hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(B),A)) ) ) ).

tff(fact_294_order__less__le__trans,axiom,
    ! [Z: bool,X_2: bool,Y_2: bool] :
      ( hBOOL(hAPP_bool_bool(ord_less_bool(X_2),Y_2))
     => ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Y_2),Z))
       => hBOOL(hAPP_bool_bool(ord_less_bool(X_2),Z)) ) ) ).

tff(fact_295_order__less__le__trans,axiom,
    ! [Z: fun_Ho1996104121e_bool,X_2: fun_Ho1996104121e_bool,Y_2: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(X_2),Y_2))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Y_2),Z))
       => hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(X_2),Z)) ) ) ).

tff(fact_296_xt1_I7_J,axiom,
    ! [Z: bool,Y_2: bool,X_2: bool] :
      ( hBOOL(hAPP_bool_bool(ord_less_bool(Y_2),X_2))
     => ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Z),Y_2))
       => hBOOL(hAPP_bool_bool(ord_less_bool(Z),X_2)) ) ) ).

tff(fact_297_xt1_I7_J,axiom,
    ! [Z: fun_Ho1996104121e_bool,Y_2: fun_Ho1996104121e_bool,X_2: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(Y_2),X_2))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Z),Y_2))
       => hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(Z),X_2)) ) ) ).

tff(fact_298_order__le__less__trans,axiom,
    ! [Z: bool,X_2: bool,Y_2: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,X_2),Y_2))
     => ( hBOOL(hAPP_bool_bool(ord_less_bool(Y_2),Z))
       => hBOOL(hAPP_bool_bool(ord_less_bool(X_2),Z)) ) ) ).

tff(fact_299_order__le__less__trans,axiom,
    ! [Z: fun_Ho1996104121e_bool,X_2: fun_Ho1996104121e_bool,Y_2: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,X_2),Y_2))
     => ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(Y_2),Z))
       => hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(X_2),Z)) ) ) ).

tff(fact_300_xt1_I8_J,axiom,
    ! [Z: bool,Y_2: bool,X_2: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Y_2),X_2))
     => ( hBOOL(hAPP_bool_bool(ord_less_bool(Z),Y_2))
       => hBOOL(hAPP_bool_bool(ord_less_bool(Z),X_2)) ) ) ).

tff(fact_301_xt1_I8_J,axiom,
    ! [Z: fun_Ho1996104121e_bool,Y_2: fun_Ho1996104121e_bool,X_2: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Y_2),X_2))
     => ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(Z),Y_2))
       => hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(Z),X_2)) ) ) ).

tff(fact_302_less__fun__def,axiom,
    ! [F: fun_Ho1996104121e_bool,G: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(F),G))
    <=> ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,F),G))
        & ~ hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,G),F)) ) ) ).

tff(fact_303_psubset__insert__iff,axiom,
    ! [A_1: fun_Ho1996104121e_bool,X_2: hoare_1875481847_state,B_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),B_1)))
    <=> ( ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),B_1))
         => hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(A_1),B_1)) )
        & ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),B_1))
         => ( ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),A_1))
             => hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(hAPP_f121055253e_bool(minus_589155914e_bool(A_1),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,X_2),bot_bo1715400655e_bool))),B_1)) )
            & ( ~ hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_2),A_1))
             => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,A_1),B_1)) ) ) ) ) ) ).

tff(fact_304_setsum__strict__mono,axiom,
    ! [F: fun_Ho1284378692te_nat,G: fun_Ho1284378692te_nat,A_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(finite1792194148_state,A_1))
     => ( ( A_1 != bot_bo1715400655e_bool )
       => ( ! [X_1: hoare_1875481847_state] :
              ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,X_1),A_1))
             => hBOOL(hAPP_nat_bool(ord_less_nat(hAPP_H1027528080te_nat(F,X_1)),hAPP_H1027528080te_nat(G,X_1))) )
         => hBOOL(hAPP_nat_bool(ord_less_nat(big_co583362288te_nat(F,A_1)),big_co583362288te_nat(G,A_1))) ) ) ) ).

tff(fact_305_psubset__trans,axiom,
    ! [C: fun_Ho1996104121e_bool,A_1: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(A_1),B_1))
     => ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(B_1),C))
       => hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(A_1),C)) ) ) ).

tff(fact_306_psubsetD,axiom,
    ! [Ca: hoare_1875481847_state,A_1: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(A_1),B_1))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,Ca),A_1))
       => hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,Ca),B_1)) ) ) ).

tff(fact_307_psubset__imp__ex__mem,axiom,
    ! [A_1: fun_Ho1996104121e_bool,B_1: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(A_1),B_1))
     => ? [B_2: hoare_1875481847_state] : hBOOL(hAPP_f1408815105l_bool(hAPP_H810811817l_bool(member618553700_state,B_2),hAPP_f121055253e_bool(minus_589155914e_bool(B_1),A_1))) ) ).

tff(fact_308_xt6,axiom,
    ! [Ca: fun_Ho1996104121e_bool,F: fun_fu1296727421e_bool,B: fun_Ho1996104121e_bool,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,hAPP_f121055253e_bool(F,B)),A))
     => ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(Ca),B))
       => ( ! [X_1: fun_Ho1996104121e_bool,Y_1: fun_Ho1996104121e_bool] :
              ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(Y_1),X_1))
             => hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(hAPP_f121055253e_bool(F,Y_1)),hAPP_f121055253e_bool(F,X_1))) )
         => hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(hAPP_f121055253e_bool(F,Ca)),A)) ) ) ) ).

tff(fact_309_xt6,axiom,
    ! [Ca: bool,F: fun_bo221168653e_bool,B: bool,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,hAPP_b1906038105e_bool(F,B)),A))
     => ( hBOOL(hAPP_bool_bool(ord_less_bool(Ca),B))
       => ( ! [X_1: bool,Y_1: bool] :
              ( hBOOL(hAPP_bool_bool(ord_less_bool(Y_1),X_1))
             => hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(hAPP_b1906038105e_bool(F,Y_1)),hAPP_b1906038105e_bool(F,X_1))) )
         => hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(hAPP_b1906038105e_bool(F,Ca)),A)) ) ) ) ).

tff(fact_310_xt5,axiom,
    ! [Ca: fun_Ho1996104121e_bool,F: fun_fu1296727421e_bool,B: fun_Ho1996104121e_bool,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(B),A))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Ca),hAPP_f121055253e_bool(F,B)))
       => ( ! [X_1: fun_Ho1996104121e_bool,Y_1: fun_Ho1996104121e_bool] :
              ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(Y_1),X_1))
             => hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(hAPP_f121055253e_bool(F,Y_1)),hAPP_f121055253e_bool(F,X_1))) )
         => hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(Ca),hAPP_f121055253e_bool(F,A))) ) ) ) ).

tff(fact_311_xt5,axiom,
    ! [Ca: fun_Ho1996104121e_bool,F: fun_bo221168653e_bool,B: bool,A: bool] :
      ( hBOOL(hAPP_bool_bool(ord_less_bool(B),A))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Ca),hAPP_b1906038105e_bool(F,B)))
       => ( ! [X_1: bool,Y_1: bool] :
              ( hBOOL(hAPP_bool_bool(ord_less_bool(Y_1),X_1))
             => hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(hAPP_b1906038105e_bool(F,Y_1)),hAPP_b1906038105e_bool(F,X_1))) )
         => hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(Ca),hAPP_b1906038105e_bool(F,A))) ) ) ) ).

tff(fact_312_xt4,axiom,
    ! [Ca: fun_Ho1996104121e_bool,F: fun_fu1296727421e_bool,B: fun_Ho1996104121e_bool,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(hAPP_f121055253e_bool(F,B)),A))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Ca),B))
       => ( ! [X_1: fun_Ho1996104121e_bool,Y_1: fun_Ho1996104121e_bool] :
              ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Y_1),X_1))
             => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,hAPP_f121055253e_bool(F,Y_1)),hAPP_f121055253e_bool(F,X_1))) )
         => hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(hAPP_f121055253e_bool(F,Ca)),A)) ) ) ) ).

tff(fact_313_xt4,axiom,
    ! [Ca: fun_Ho1996104121e_bool,F: fun_fu689207471l_bool,B: fun_Ho1996104121e_bool,A: bool] :
      ( hBOOL(hAPP_bool_bool(ord_less_bool(hAPP_f1408815105l_bool(F,B)),A))
     => ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Ca),B))
       => ( ! [X_1: fun_Ho1996104121e_bool,Y_1: fun_Ho1996104121e_bool] :
              ( hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,Y_1),X_1))
             => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_f1408815105l_bool(F,Y_1)),hAPP_f1408815105l_bool(F,X_1))) )
         => hBOOL(hAPP_bool_bool(ord_less_bool(hAPP_f1408815105l_bool(F,Ca)),A)) ) ) ) ).

tff(fact_314_xt4,axiom,
    ! [Ca: bool,F: fun_bo221168653e_bool,B: bool,A: fun_Ho1996104121e_bool] :
      ( hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(hAPP_b1906038105e_bool(F,B)),A))
     => ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Ca),B))
       => ( ! [X_1: bool,Y_1: bool] :
              ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Y_1),X_1))
             => hBOOL(hAPP_f1408815105l_bool(hAPP_f1873026899l_bool(ord_le233469699e_bool,hAPP_b1906038105e_bool(F,Y_1)),hAPP_b1906038105e_bool(F,X_1))) )
         => hBOOL(hAPP_f1408815105l_bool(ord_le920868879e_bool(hAPP_b1906038105e_bool(F,Ca)),A)) ) ) ) ).

%----Helper facts (21)
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_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_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Hoare____Mirabelle____xlrqixeqwe__,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_Ho1996104121e_bool,R: hoare_1875481847_state] : hAPP_H1173941847e_bool(hAPP_f121055253e_bool(cOMBB_1424001755_state(P),Q),R) = hAPP_bool_bool(P,hAPP_H1173941847e_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__,axiom,
    ! [P: fun_Ho1896286681l_bool,Q: fun_Ho1996104121e_bool,R: hoare_1875481847_state] : hAPP_H1173941847e_bool(hAPP_f121055253e_bool(cOMBS_896408300l_bool(P),Q),R) = hAPP_bool_bool(hAPP_H377399461l_bool(P,R),hAPP_H1173941847e_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_Ho1996104121e_bool,R: hoare_1875481847_state] : hAPP_H377399461l_bool(hAPP_f932622461l_bool(cOMBB_242064253_state(P),Q),R) = hAPP_b589554111l_bool(P,hAPP_H1173941847e_bool(Q,R)) ).

tff(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com__,axiom,
    ! [P: fun_Ho1695028467e_bool,Q: hoare_1875481847_state,R: hoare_1875481847_state] : hAPP_H1173941847e_bool(hAPP_H117293375e_bool(cOMBC_1972244470e_bool(P),Q),R) = hAPP_H1173941847e_bool(hAPP_H117293375e_bool(P,R),Q) ).

tff(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____xlrqixeqwe__Otriple_Itc__Com___028,axiom,
    ! [P: fun_Ho1085560331l_bool,Q: fun_Ho1996104121e_bool,R: hoare_1875481847_state] : hAPP_H1173941847e_bool(hAPP_f121055253e_bool(cOMBC_1099403170l_bool(P),Q),R) = hAPP_f1408815105l_bool(hAPP_H810811817l_bool(P,R),Q) ).

%----Conjectures (3)
tff(conj_0,hypothesis,
    hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(bot_bo1715400655e_bool),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hoare_Mirabelle_MGT(c)),bot_bo1715400655e_bool))) ).

tff(conj_1,hypothesis,
    hBOOL(hAPP_f1408815105l_bool(hoare_390430481_state(bot_bo1715400655e_bool),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hoare_13225861_state(p,c,q)),bot_bo1715400655e_bool))) ).

tff(conj_2,conjecture,
    hBOOL(hAPP_f1408815105l_bool(hoare_2131502867_state(bot_bo1715400655e_bool),hAPP_f121055253e_bool(hAPP_H1625489667e_bool(insert694999549_state,hoare_13225861_state(p,c,q)),bot_bo1715400655e_bool))) ).

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