%------------------------------------------------------------------------------
% File     : SWW471_30 : TPTP v9.0.0. Released v8.2.0.
% Domain   : Software Verification
% Problem  : Hoare's Logic with Procedures line 269, 1000 axioms selected
% Version  : SWW471_3 with the conjecture removed
% English  :

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

% Status   : Satisfiable
% Rating   : 1.00 v8.2.0
% Syntax   : Number of formulae    : 2355 ( 603 unt; 998 typ;   0 def)
%            Number of atoms       : 2848 (1168 equ)
%            Maximal formula atoms :    9 (   2 avg)
%            Number of connectives : 1910 ( 419   ~;  88   |; 137   &)
%                                         ( 258 <=>;1008  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   16 (   5 avg)
%            Maximal term depth    :   13 (   2 avg)
%            Number of types       :  348 ( 347 usr)
%            Number of type conns  :  813 ( 448   >; 365   *;   0   +;   0  <<)
%            Number of predicates  :    2 (   1 usr;   0 prp; 1-2 aty)
%            Number of functors    :  650 ( 650 usr; 203 con; 0-3 aty)
%            Number of variables   : 3584 (3514   !;  70   ?;3584   :)
% SPC      : TF0_SAT_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2011-08-09 17:31:07
%------------------------------------------------------------------------------
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tff(sy_c_COMBS_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com__Ostate_J_0,type,
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tff(sy_c_COMBS_000tc__Nat__Onat_000tc__HOL__Obool_000tc__HOL__Obool,type,
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tff(sy_c_COMBS_000tc__Nat__Onat_000tc__fun_It__a_Mtc__fun_Itc__Com__Ostate_Mtc__HOL_,type,
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tff(sy_c_COMBS_000tc__Nat__Onat_000tc__fun_Itc__Com__Ostate_Mtc__fun_Itc__Com__Ostat,type,
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tff(sy_c_COMBS_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_Mtc,type,
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tff(sy_c_COMBS_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_Mtc_245,type,
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tff(sy_c_Com_Obody,type,
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tff(sy_c_Com_Ocom_OBODY,type,
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tff(sy_c_Com_Ocom_OCond,type,
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tff(sy_c_Com_Ocom_OSKIP,type,
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tff(sy_c_Com_Ocom_OSemi,type,
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tff(sy_c_Com_Ocom_OWhile,type,
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tff(sy_c_Com_Ocom_Ocom__size,type,
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tff(sy_c_Finite__Set_Ocard_000tc__Nat__Onat,type,
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tff(sy_c_Finite__Set_Ofinite_000tc__Com__Opname,type,
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tff(sy_c_Finite__Set_Ofinite_000tc__HOL__Obool,type,
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tff(sy_c_Finite__Set_Ofinite_000tc__Nat__Onat,type,
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tff(sy_c_Finite__Set_Ofold__image_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy___249,type,
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tff(sy_c_Finite__Set_Ofold__image_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy___250,type,
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tff(sy_c_Finite__Set_Ofold__image_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy___251,type,
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tff(sy_c_Finite__Set_Ofold__image_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy___253,type,
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tff(sy_c_Finite__Set_Ofold__image_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc_,type,
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tff(sy_c_Finite__Set_Ofold__image_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdc,type,
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tff(sy_c_Finite__Set_Ofolding__one_000tc__Com__Opname,type,
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tff(sy_c_Finite__Set_Ofolding__one_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_,type,
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tff(sy_c_Finite__Set_Ofolding__one_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple__254,type,
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tff(sy_c_Finite__Set_Ofolding__one_000tc__Nat__Onat,type,
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tff(sy_c_Finite__Set_Ofolding__one__idem_000tc__Com__Opname,type,
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tff(sy_c_Finite__Set_Ofolding__one__idem_000tc__Hoare____Mirabelle____wdcyoqcnvy__Ot,type,
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tff(sy_c_Finite__Set_Ofolding__one__idem_000tc__Hoare____Mirabelle____wdcyoqcnvy__Ot_255,type,
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tff(sy_c_Finite__Set_Ofolding__one__idem_000tc__Nat__Onat,type,
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tff(sy_c_Finite__Set_Ofolding__one__idem_000tc__fun_Itc__Hoare____Mirabelle____wdcyo,type,
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tff(sy_c_Groups_Ominus__class_Ominus_000tc__HOL__Obool,type,
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tff(sy_c_Groups_Ominus__class_Ominus_000tc__Nat__Onat,type,
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tff(sy_c_Groups_Ominus__class_Ominus_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
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tff(sy_c_Groups_Ominus__class_Ominus_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnv,type,
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tff(sy_c_Groups_Ominus__class_Ominus_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnv_256,type,
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tff(sy_c_Groups_Ominus__class_Ominus_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
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tff(sy_c_Groups_Ominus__class_Ominus_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____,type,
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tff(sy_c_Groups_Oone__class_Oone_000tc__Nat__Onat,type,
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tff(sy_c_Groups_Oplus__class_Oplus_000tc__Nat__Onat,type,
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tff(sy_c_Groups_Otimes__class_Otimes_000tc__Nat__Onat,type,
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tff(sy_c_Groups_Ozero__class_Ozero_000tc__Nat__Onat,type,
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tff(sy_c_HOL_OThe_000tc__Com__Opname,type,
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tff(sy_c_HOL_OThe_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J,type,
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tff(sy_c_HOL_OThe_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com__Ostate_,type,
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tff(sy_c_HOL_OThe_000tc__Nat__Onat,type,
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tff(sy_c_HOL_OThe_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_,type,
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tff(sy_c_Hoare__Mirabelle__wdcyoqcnvy_OMGT,type,
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tff(sy_c_Hoare__Mirabelle__wdcyoqcnvy_Ohoare__derivs_000t__a,type,
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tff(sy_c_Hoare__Mirabelle__wdcyoqcnvy_Ohoare__derivs_000tc__Com__Ostate,type,
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tff(sy_c_Hoare__Mirabelle__wdcyoqcnvy_Ohoare__valids_000t__a,type,
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tff(sy_c_Hoare__Mirabelle__wdcyoqcnvy_Ohoare__valids_000tc__Com__Ostate,type,
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tff(sy_c_Hoare__Mirabelle__wdcyoqcnvy_Otriple_Otriple_000t__a,type,
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tff(sy_c_Hoare__Mirabelle__wdcyoqcnvy_Otriple_Otriple_000tc__Com__Ostate,type,
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tff(sy_c_Hoare__Mirabelle__wdcyoqcnvy_Otriple_Otriple__size_000t__a,type,
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tff(sy_c_Hoare__Mirabelle__wdcyoqcnvy_Otriple_Otriple__size_000tc__Com__Ostate,type,
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tff(sy_c_Hoare__Mirabelle__wdcyoqcnvy_Otriple__valid_000t__a,type,
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tff(sy_c_Hoare__Mirabelle__wdcyoqcnvy_Otriple__valid_000tc__Com__Ostate,type,
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tff(sy_c_Lattices_Osemilattice__inf__class_Oinf_000tc__HOL__Obool,type,
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tff(sy_c_Lattices_Osemilattice__inf__class_Oinf_000tc__Nat__Onat,type,
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tff(sy_c_Lattices_Osemilattice__inf__class_Oinf_000tc__fun_Itc__Com__Opname_Mtc__HOL,type,
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tff(sy_c_Lattices_Osemilattice__inf__class_Oinf_000tc__fun_Itc__Hoare____Mirabelle__,type,
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tff(sy_c_Lattices_Osemilattice__inf__class_Oinf_000tc__fun_Itc__Nat__Onat_Mtc__HOL__,type,
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tff(sy_c_Lattices_Osemilattice__inf__class_Oinf_000tc__fun_Itc__fun_Itc__Hoare____Mi,type,
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tff(sy_c_Lattices_Osemilattice__sup__class_Osup_000tc__HOL__Obool,type,
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tff(sy_c_Lattices_Osemilattice__sup__class_Osup_000tc__Nat__Onat,type,
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tff(sy_c_Lattices_Osemilattice__sup__class_Osup_000tc__fun_Itc__Com__Opname_Mtc__HOL,type,
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tff(sy_c_Lattices_Osemilattice__sup__class_Osup_000tc__fun_Itc__HOL__Obool_Mtc__HOL_,type,
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tff(sy_c_Lattices_Osemilattice__sup__class_Osup_000tc__fun_Itc__Hoare____Mirabelle__,type,
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tff(sy_c_Lattices_Osemilattice__sup__class_Osup_000tc__fun_Itc__Nat__Onat_Mtc__HOL__,type,
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tff(sy_c_Lattices_Osemilattice__sup__class_Osup_000tc__fun_Itc__fun_Itc__Com__Opname,type,
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tff(sy_c_Lattices_Osemilattice__sup__class_Osup_000tc__fun_Itc__fun_Itc__Hoare____Mi,type,
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tff(sy_c_Lattices_Osemilattice__sup__class_Osup_000tc__fun_Itc__fun_Itc__Hoare____Mi_259,type,
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tff(sy_c_Lattices_Osemilattice__sup__class_Osup_000tc__fun_Itc__fun_Itc__Nat__Onat_M,type,
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tff(sy_c_Lattices_Osemilattice__sup__class_Osup_000tc__fun_Itc__fun_Itc__fun_Itc__Ho,type,
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tff(sy_c_Nat_OSuc,type,
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tff(sy_c_Nat_Onat_Onat__case_000tc__HOL__Obool,type,
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tff(sy_c_Nat_Onat_Onat__case_000tc__Nat__Onat,type,
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tff(sy_c_Nat_Osize__class_Osize_000tc__Com__Ocom,type,
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tff(sy_c_Nat_Osize__class_Osize_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It_,type,
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tff(sy_c_Natural_Oevalc,type,
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tff(sy_c_Natural_Oevaln,type,
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tff(sy_c_Option_Othe_000tc__Com__Ocom,type,
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tff(sy_c_Orderings_Obot__class_Obot_000tc__HOL__Obool,type,
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tff(sy_c_Orderings_Obot__class_Obot_000tc__Nat__Onat,type,
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tff(sy_c_Orderings_Obot__class_Obot_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__O,type,
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tff(sy_c_Orderings_Obot__class_Obot_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____w,type,
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tff(sy_c_Orderings_Obot__class_Obot_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____w_261,type,
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    hAPP_f1684296317l_bool: ( fun_fu1976078123l_bool * fun_fu2026518849l_bool ) > fun_fu1788233809l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__438,type,
    hAPP_f1662919455l_bool: ( fun_fu862088525l_bool * fun_fu2026518849l_bool ) > fun_fu1789210355l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__439,type,
    hAPP_f1523760350l_bool: ( fun_fu474871180l_bool * fun_fu2026518849l_bool ) > fun_fu1000411314l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__440,type,
    hAPP_f1238945379l_bool: ( fun_fu1676769297l_bool * fun_fu2026518849l_bool ) > fun_fu1342882871l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__441,type,
    hAPP_f1976119377l_bool: ( fun_fu2076516793l_bool * fun_fu2026518849l_bool ) > fun_fu1354972029l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__442,type,
    hAPP_f1351571149l_bool: ( fun_fu945563829l_bool * fun_fu2026518849l_bool ) > fun_fu1999239161l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__443,type,
    hAPP_f674363047l_bool: ( fun_fu1035965839l_bool * fun_fu2026518849l_bool ) > fun_fu280784851l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__444,type,
    hAPP_f15901907a_bool: ( fun_fu1700257333a_bool * fun_fu253747899iple_a ) > fun_fu2051904533a_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__445,type,
    hAPP_f1800675941t_bool: ( fun_fu938226457t_bool * fun_fu328701628ol_nat ) > fun_fu2145798756t_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__446,type,
    hAPP_f375308812iple_a: ( fun_fu466416494iple_a * fun_fu1307637710e_bool ) > fun_fu253747899iple_a ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__447,type,
    hAPP_f42286368iple_a: ( fun_fu819106050iple_a * fun_fu1307637710e_bool ) > fun_fu28937807iple_a ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__448,type,
    hAPP_f1723480683iple_a: ( fun_fu63530143iple_a * fun_fu28937807iple_a ) > fun_fu12385085iple_a ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__449,type,
    hAPP_f995418987_state: ( fun_fu1098249503_state * fun_fu2018970971_state ) > fun_fu1759214921_state ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__450,type,
    hAPP_f1319084128_state: ( fun_fu2020579016_state * fun_fu857054695e_bool ) > fun_fu370532530_state ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__451,type,
    hAPP_f1181766445_state: ( fun_fu1699599323_state * fun_fu857054695e_bool ) > fun_fu2018970971_state ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__452,type,
    hAPP_f1121033001l_bool: ( fun_fu691339741l_bool * fun_fu1788233809l_bool ) > fun_fu1354972029l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__453,type,
    hAPP_f524575973l_bool: ( fun_fu2039753549l_bool * fun_fu2145798756t_bool ) > fun_na2131225204l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__454,type,
    hAPP_f2030579535iple_a: ( fun_fu516960899iple_a * fun_fu102784604iple_a ) > fun_fu466416494iple_a ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__455,type,
    hAPP_f1676710173_state: ( fun_fu797275345_state * fun_fu1800449896_state ) > fun_fu2020579016_state ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__456,type,
    hAPP_f1605242827l_bool: ( fun_fu604335999l_bool * fun_fu1789210355l_bool ) > fun_fu2026518849l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__457,type,
    hAPP_f246259945l_bool: ( fun_fu1236051915l_bool * fun_fu1789210355l_bool ) > fun_fu1789210355l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__458,type,
    hAPP_f86610733a_bool: ( fun_fu1595031695a_bool * fun_fu2051904533a_bool ) > fun_fu2051904533a_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__459,type,
    hAPP_f638520823l_bool: ( fun_fu1847882591l_bool * fun_fu330842395l_bool ) > fun_fu1354972029l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__460,type,
    hAPP_f2041002367l_bool: ( fun_fu1387427169l_bool * fun_fu25215279l_bool ) > fun_fu862088525l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc,type,
    hAPP_f1287348838l_bool: ( fun_fu1972704968l_bool * fun_fu1996308102l_bool ) > bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc_461,type,
    hAPP_f1180969799e_bool: ( fun_fu1671561851e_bool * fun_fu1996308102l_bool ) > fun_Ho320143120e_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc_462,type,
    hAPP_f85652879l_bool: ( fun_fu1858490225l_bool * fun_fu1996308102l_bool ) > fun_fu1996308102l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc_463,type,
    hAPP_f1175211816l_bool: ( fun_fu1392972380l_bool * fun_fu1996308102l_bool ) > fun_fu1858490225l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J,type,
    hAPP_f1637334154l_bool: ( fun_fu814369080l_bool * fun_fu425979586l_bool ) > bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_464,type,
    hAPP_f582319405t_bool: ( fun_fu496092379t_bool * fun_fu425979586l_bool ) > fun_nat_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_465,type,
    hAPP_f1246832597l_bool: ( fun_fu616551101l_bool * fun_fu425979586l_bool ) > fun_fu425979586l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_466,type,
    hAPP_f540020688l_bool: ( fun_fu891365688l_bool * fun_fu425979586l_bool ) > fun_fu616551101l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Ot,type,
    hAPP_f249932695l_bool: ( fun_fu1999239161l_bool * fun_fu1342882871l_bool ) > bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Ot_467,type,
    hAPP_f1453855623l_bool: ( fun_fu1258464827l_bool * fun_fu1342882871l_bool ) > fun_fu2026518849l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Ot_468,type,
    hAPP_f1122542193l_bool: ( fun_fu280784851l_bool * fun_fu1342882871l_bool ) > fun_fu1342882871l_bool ).

tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Ot_469,type,
    hAPP_f392660249l_bool: ( fun_fu1956796237l_bool * fun_fu1342882871l_bool ) > fun_fu280784851l_bool ).

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

tff(sy_c_member_000tc__Com__Opname,type,
    member_pname: fun_pn422929397l_bool ).

tff(sy_c_member_000tc__HOL__Obool,type,
    member_bool: fun_bo492988759l_bool ).

tff(sy_c_member_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J,type,
    member276146780iple_a: fun_Ho1910759733l_bool ).

tff(sy_c_member_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com__Ostate_J,type,
    member1253905979_state: fun_Ho177530809l_bool ).

tff(sy_c_member_000tc__Nat__Onat,type,
    member_nat: fun_na1436237685l_bool ).

tff(sy_c_member_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
    member799430823e_bool: fun_fu931343505l_bool ).

tff(sy_c_member_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_Mt,type,
    member186597612a_bool: fun_fu330842395l_bool ).

tff(sy_c_member_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com__,type,
    member1534059107e_bool: fun_fu1467933941l_bool ).

tff(sy_c_member_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
    member_fun_nat_bool: fun_fu1436348701l_bool ).

tff(sy_c_member_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_I,type,
    member991318164l_bool: fun_fu945563829l_bool ).

tff(sy_v_G,type,
    g: fun_Ho882550911a_bool ).

tff(sy_v_P,type,
    p: fun_pn1683930517e_bool ).

tff(sy_v_Procs,type,
    procs: fun_pname_bool ).

tff(sy_v_Q,type,
    q: fun_pn1683930517e_bool ).

tff(sy_v_n,type,
    n: nat ).

%----Relevant facts (1196)
tff(fact_0_triple_Oinject,axiom,
    ! [Fun1: fun_a_fun_state_bool,Com_1: com,Fun2: fun_a_fun_state_bool,Fun1_2: fun_a_fun_state_bool,Com: com,Fun2_2: fun_a_fun_state_bool] :
      ( ( hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,Fun1),Com_1),Fun2) = hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,Fun1_2),Com),Fun2_2) )
    <=> ( ( Fun1 = Fun1_2 )
        & ( Com_1 = Com )
        & ( Fun2 = Fun2_2 ) ) ) ).

tff(fact_1_triple_Oinject,axiom,
    ! [Fun1: fun_st1506752259e_bool,Com_1: com,Fun2: fun_st1506752259e_bool,Fun1_2: fun_st1506752259e_bool,Com: com,Fun2_2: fun_st1506752259e_bool] :
      ( ( hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,Fun1),Com_1),Fun2) = hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,Fun1_2),Com),Fun2_2) )
    <=> ( ( Fun1 = Fun1_2 )
        & ( Com_1 = Com )
        & ( Fun2 = Fun2_2 ) ) ) ).

tff(fact_2_hoare__valids__def,axiom,
    ! [Ga: fun_Ho320143120e_bool,Ts: fun_Ho320143120e_bool] :
      ( hBOOL(hAPP_f568432088l_bool(hoare_501514600_state(Ga),Ts))
    <=> ! [N: nat] :
          ( ! [X: hoare_363350478_state] :
              ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X),Ga))
             => hBOOL(hAPP_H1645464494e_bool(hAPP_n753050587e_bool(hoare_28808878_state,N),X)) )
         => ! [X: hoare_363350478_state] :
              ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X),Ts))
             => hBOOL(hAPP_H1645464494e_bool(hAPP_n753050587e_bool(hoare_28808878_state,N),X)) ) ) ) ).

tff(fact_3_hoare__valids__def,axiom,
    ! [Ga: fun_Ho882550911a_bool,Ts: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hoare_910298713lids_a(Ga),Ts))
    <=> ! [N: nat] :
          ( ! [X: hoare_2076525193iple_a] :
              ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X),Ga))
             => hBOOL(hAPP_H454055633a_bool(hAPP_n1335981428a_bool(hoare_526497939alid_a,N),X)) )
         => ! [X: hoare_2076525193iple_a] :
              ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X),Ts))
             => hBOOL(hAPP_H454055633a_bool(hAPP_n1335981428a_bool(hoare_526497939alid_a,N),X)) ) ) ) ).

tff(fact_4_hoare__derivs_OBody,axiom,
    ! [Ga: fun_Ho320143120e_bool,Pa: fun_pn12888928e_bool,Qa: fun_pn12888928e_bool,Procsa: fun_pname_bool] :
      ( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,Ga),hAPP_f1763243027e_bool(image_1918575693_state(hAPP_f19336718_state(hAPP_f720869462_state(cOMBS_1110920566_state,hAPP_f1146170881_state(hAPP_f1311198492_state(cOMBS_124505571_state,hAPP_f387017471_state(cOMBB_1212270426_pname(hoare_608330478_state),Pa)),body)),Qa)),Procsa))),hAPP_f1763243027e_bool(image_1918575693_state(hAPP_f19336718_state(hAPP_f720869462_state(cOMBS_1110920566_state,hAPP_f1146170881_state(hAPP_f1311198492_state(cOMBS_124505571_state,hAPP_f387017471_state(cOMBB_1212270426_pname(hoare_608330478_state),Pa)),hAPP_f96342628me_com(cOMBB_923936821_pname(the_com),body_1))),Qa)),Procsa)))
     => hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1763243027e_bool(image_1918575693_state(hAPP_f19336718_state(hAPP_f720869462_state(cOMBS_1110920566_state,hAPP_f1146170881_state(hAPP_f1311198492_state(cOMBS_124505571_state,hAPP_f387017471_state(cOMBB_1212270426_pname(hoare_608330478_state),Pa)),body)),Qa)),Procsa))) ) ).

tff(fact_5_hoare__derivs_OBody,axiom,
    ! [Ga: fun_Ho882550911a_bool,Pa: fun_pn1683930517e_bool,Qa: fun_pn1683930517e_bool,Procsa: fun_pname_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,Ga),hAPP_f1451199292a_bool(image_217098954iple_a(hAPP_f1246077570iple_a(hAPP_f345351132iple_a(cOMBS_970288740iple_a,hAPP_f804419189iple_a(hAPP_f900936988iple_a(cOMBS_1123526615iple_a,hAPP_f1471296150iple_a(cOMBB_1308294791_pname(hoare_1876769875iple_a),Pa)),body)),Qa)),Procsa))),hAPP_f1451199292a_bool(image_217098954iple_a(hAPP_f1246077570iple_a(hAPP_f345351132iple_a(cOMBS_970288740iple_a,hAPP_f804419189iple_a(hAPP_f900936988iple_a(cOMBS_1123526615iple_a,hAPP_f1471296150iple_a(cOMBB_1308294791_pname(hoare_1876769875iple_a),Pa)),hAPP_f96342628me_com(cOMBB_923936821_pname(the_com),body_1))),Qa)),Procsa)))
     => hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f1451199292a_bool(image_217098954iple_a(hAPP_f1246077570iple_a(hAPP_f345351132iple_a(cOMBS_970288740iple_a,hAPP_f804419189iple_a(hAPP_f900936988iple_a(cOMBS_1123526615iple_a,hAPP_f1471296150iple_a(cOMBB_1308294791_pname(hoare_1876769875iple_a),Pa)),body)),Qa)),Procsa))) ) ).

tff(fact_6_UnE,axiom,
    ! [C_2: nat,A_1: fun_nat_bool,B_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_1),B_2)))
     => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),A_1))
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),B_2)) ) ) ).

tff(fact_7_UnE,axiom,
    ! [C_2: fun_Ho882550911a_bool,A_1: fun_fu2026518849l_bool,B_2: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,C_2),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_1),B_2)))
     => ( ~ hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,C_2),A_1))
       => hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,C_2),B_2)) ) ) ).

tff(fact_8_UnE,axiom,
    ! [C_2: hoare_363350478_state,A_1: fun_Ho320143120e_bool,B_2: fun_Ho320143120e_bool] :
      ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,C_2),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_1),B_2)))
     => ( ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,C_2),A_1))
       => hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,C_2),B_2)) ) ) ).

tff(fact_9_UnE,axiom,
    ! [C_2: hoare_2076525193iple_a,A_1: fun_Ho882550911a_bool,B_2: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,C_2),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_1),B_2)))
     => ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,C_2),A_1))
       => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,C_2),B_2)) ) ) ).

tff(fact_10_UnE,axiom,
    ! [C_2: pname,A_1: fun_pname_bool,B_2: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),B_2)))
     => ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_2),A_1))
       => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_2),B_2)) ) ) ).

tff(fact_11_sup1E,axiom,
    ! [A_1: fun_nat_bool,B_2: fun_nat_bool,X_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_1),B_2),X_2))
     => ( ~ hBOOL(hAPP_nat_bool(A_1,X_2))
       => hBOOL(hAPP_nat_bool(B_2,X_2)) ) ) ).

tff(fact_12_sup1E,axiom,
    ! [A_1: fun_fu2026518849l_bool,B_2: fun_fu2026518849l_bool,X_2: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_1),B_2),X_2))
     => ( ~ hBOOL(hAPP_f500854239l_bool(A_1,X_2))
       => hBOOL(hAPP_f500854239l_bool(B_2,X_2)) ) ) ).

tff(fact_13_sup1E,axiom,
    ! [A_1: fun_Ho320143120e_bool,B_2: fun_Ho320143120e_bool,X_2: hoare_363350478_state] :
      ( hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_1),B_2),X_2))
     => ( ~ hBOOL(hAPP_H1645464494e_bool(A_1,X_2))
       => hBOOL(hAPP_H1645464494e_bool(B_2,X_2)) ) ) ).

tff(fact_14_sup1E,axiom,
    ! [A_1: fun_pname_bool,B_2: fun_pname_bool,X_2: pname] :
      ( hBOOL(hAPP_pname_bool(hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),B_2),X_2))
     => ( ~ hBOOL(hAPP_pname_bool(A_1,X_2))
       => hBOOL(hAPP_pname_bool(B_2,X_2)) ) ) ).

tff(fact_15_sup1E,axiom,
    ! [A_1: fun_Ho882550911a_bool,B_2: fun_Ho882550911a_bool,X_2: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_1),B_2),X_2))
     => ( ~ hBOOL(hAPP_H454055633a_bool(A_1,X_2))
       => hBOOL(hAPP_H454055633a_bool(B_2,X_2)) ) ) ).

tff(fact_16_sup1CI,axiom,
    ! [A_1: fun_nat_bool,B_2: fun_nat_bool,X_2: nat] :
      ( ( ~ hBOOL(hAPP_nat_bool(B_2,X_2))
       => hBOOL(hAPP_nat_bool(A_1,X_2)) )
     => hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_1),B_2),X_2)) ) ).

tff(fact_17_sup1CI,axiom,
    ! [A_1: fun_fu2026518849l_bool,B_2: fun_fu2026518849l_bool,X_2: fun_Ho882550911a_bool] :
      ( ( ~ hBOOL(hAPP_f500854239l_bool(B_2,X_2))
       => hBOOL(hAPP_f500854239l_bool(A_1,X_2)) )
     => hBOOL(hAPP_f500854239l_bool(hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_1),B_2),X_2)) ) ).

tff(fact_18_sup1CI,axiom,
    ! [A_1: fun_Ho320143120e_bool,B_2: fun_Ho320143120e_bool,X_2: hoare_363350478_state] :
      ( ( ~ hBOOL(hAPP_H1645464494e_bool(B_2,X_2))
       => hBOOL(hAPP_H1645464494e_bool(A_1,X_2)) )
     => hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_1),B_2),X_2)) ) ).

tff(fact_19_sup1CI,axiom,
    ! [A_1: fun_pname_bool,B_2: fun_pname_bool,X_2: pname] :
      ( ( ~ hBOOL(hAPP_pname_bool(B_2,X_2))
       => hBOOL(hAPP_pname_bool(A_1,X_2)) )
     => hBOOL(hAPP_pname_bool(hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),B_2),X_2)) ) ).

tff(fact_20_sup1CI,axiom,
    ! [A_1: fun_Ho882550911a_bool,B_2: fun_Ho882550911a_bool,X_2: hoare_2076525193iple_a] :
      ( ( ~ hBOOL(hAPP_H454055633a_bool(B_2,X_2))
       => hBOOL(hAPP_H454055633a_bool(A_1,X_2)) )
     => hBOOL(hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_1),B_2),X_2)) ) ).

tff(fact_21_UnCI,axiom,
    ! [A_1: fun_nat_bool,C_2: nat,B_2: fun_nat_bool] :
      ( ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),B_2))
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),A_1)) )
     => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_1),B_2))) ) ).

tff(fact_22_UnCI,axiom,
    ! [A_1: fun_fu2026518849l_bool,C_2: fun_Ho882550911a_bool,B_2: fun_fu2026518849l_bool] :
      ( ( ~ hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,C_2),B_2))
       => hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,C_2),A_1)) )
     => hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,C_2),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_1),B_2))) ) ).

tff(fact_23_UnCI,axiom,
    ! [A_1: fun_Ho320143120e_bool,C_2: hoare_363350478_state,B_2: fun_Ho320143120e_bool] :
      ( ( ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,C_2),B_2))
       => hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,C_2),A_1)) )
     => hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,C_2),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_1),B_2))) ) ).

tff(fact_24_UnCI,axiom,
    ! [A_1: fun_Ho882550911a_bool,C_2: hoare_2076525193iple_a,B_2: fun_Ho882550911a_bool] :
      ( ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,C_2),B_2))
       => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,C_2),A_1)) )
     => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,C_2),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_1),B_2))) ) ).

tff(fact_25_UnCI,axiom,
    ! [A_1: fun_pname_bool,C_2: pname,B_2: fun_pname_bool] :
      ( ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_2),B_2))
       => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_2),A_1)) )
     => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),B_2))) ) ).

tff(fact_26_image__eqI,axiom,
    ! [A_1: fun_nat_bool,B_1: nat,F: fun_nat_nat,X_2: nat] :
      ( ( B_1 = hAPP_nat_nat(F,X_2) )
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A_1))
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,B_1),hAPP_f800510211t_bool(image_nat_nat(F),A_1))) ) ) ).

tff(fact_27_image__eqI,axiom,
    ! [A_1: fun_pname_bool,B_1: hoare_363350478_state,F: fun_pn1262351275_state,X_2: pname] :
      ( ( B_1 = hAPP_p1337579261_state(F,X_2) )
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A_1))
       => hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,B_1),hAPP_f1763243027e_bool(image_1918575693_state(F),A_1))) ) ) ).

tff(fact_28_image__eqI,axiom,
    ! [A_1: fun_pname_bool,B_1: nat,F: fun_pname_nat,X_2: pname] :
      ( ( B_1 = hAPP_pname_nat(F,X_2) )
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A_1))
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,B_1),hAPP_f1066163005t_bool(image_pname_nat(F),A_1))) ) ) ).

tff(fact_29_image__eqI,axiom,
    ! [A_1: fun_pname_bool,B_1: fun_Ho882550911a_bool,F: fun_pn1811694044a_bool,X_2: pname] :
      ( ( B_1 = hAPP_p1160247726a_bool(F,X_2) )
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A_1))
       => hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,B_1),hAPP_f1243652676l_bool(image_1433143038a_bool(F),A_1))) ) ) ).

tff(fact_30_image__eqI,axiom,
    ! [A_1: fun_nat_bool,B_1: pname,F: fun_nat_pname,X_2: nat] :
      ( ( B_1 = hAPP_nat_pname(F,X_2) )
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A_1))
       => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,B_1),hAPP_f654413245e_bool(image_nat_pname(F),A_1))) ) ) ).

tff(fact_31_image__eqI,axiom,
    ! [A_1: fun_fu2026518849l_bool,B_1: pname,F: fun_fu1817249398_pname,X_2: fun_Ho882550911a_bool] :
      ( ( B_1 = hAPP_f676353346_pname(F,X_2) )
     => ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,X_2),A_1))
       => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,B_1),hAPP_f1882202854e_bool(image_949248658_pname(F),A_1))) ) ) ).

tff(fact_32_image__eqI,axiom,
    ! [A_1: fun_Ho882550911a_bool,B_1: pname,F: fun_Ho333080248_pname,X_2: hoare_2076525193iple_a] :
      ( ( B_1 = hAPP_H1636214992_pname(F,X_2) )
     => ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_2),A_1))
       => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,B_1),hAPP_f213795644e_bool(image_880197504_pname(F),A_1))) ) ) ).

tff(fact_33_image__eqI,axiom,
    ! [A_1: fun_pname_bool,B_1: hoare_2076525193iple_a,F: fun_pn706052866iple_a,X_2: pname] :
      ( ( B_1 = hAPP_p973116442iple_a(F,X_2) )
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A_1))
       => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,B_1),hAPP_f1451199292a_bool(image_217098954iple_a(F),A_1))) ) ) ).

tff(fact_34_image__Un,axiom,
    ! [F: fun_nat_nat,A_1: fun_nat_bool,B_2: fun_nat_bool] : ( hAPP_f800510211t_bool(image_nat_nat(F),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_1),B_2)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,hAPP_f800510211t_bool(image_nat_nat(F),A_1)),hAPP_f800510211t_bool(image_nat_nat(F),B_2)) ) ).

tff(fact_35_image__Un,axiom,
    ! [F: fun_pn1262351275_state,A_1: fun_pname_bool,B_2: fun_pname_bool] : ( hAPP_f1763243027e_bool(image_1918575693_state(F),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),B_2)) = hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,hAPP_f1763243027e_bool(image_1918575693_state(F),A_1)),hAPP_f1763243027e_bool(image_1918575693_state(F),B_2)) ) ).

tff(fact_36_image__Un,axiom,
    ! [F: fun_Ho864122878_a_nat,A_1: fun_Ho882550911a_bool,B_2: fun_Ho882550911a_bool] : ( hAPP_f306451970t_bool(image_1778392390_a_nat(F),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_1),B_2)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,hAPP_f306451970t_bool(image_1778392390_a_nat(F),A_1)),hAPP_f306451970t_bool(image_1778392390_a_nat(F),B_2)) ) ).

tff(fact_37_image__Un,axiom,
    ! [F: fun_Ho943022881a_bool,A_1: fun_Ho882550911a_bool,B_2: fun_Ho882550911a_bool] : ( hAPP_f256727871l_bool(image_1537300803a_bool(F),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_1),B_2)) = hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,hAPP_f256727871l_bool(image_1537300803a_bool(F),A_1)),hAPP_f256727871l_bool(image_1537300803a_bool(F),B_2)) ) ).

tff(fact_38_image__Un,axiom,
    ! [F: fun_Ho1658235120_state,A_1: fun_Ho882550911a_bool,B_2: fun_Ho882550911a_bool] : ( hAPP_f225384718e_bool(image_624167570_state(F),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_1),B_2)) = hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,hAPP_f225384718e_bool(image_624167570_state(F),A_1)),hAPP_f225384718e_bool(image_624167570_state(F),B_2)) ) ).

tff(fact_39_image__Un,axiom,
    ! [F: fun_Ho333080248_pname,A_1: fun_Ho882550911a_bool,B_2: fun_Ho882550911a_bool] : ( hAPP_f213795644e_bool(image_880197504_pname(F),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_1),B_2)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f213795644e_bool(image_880197504_pname(F),A_1)),hAPP_f213795644e_bool(image_880197504_pname(F),B_2)) ) ).

tff(fact_40_image__Un,axiom,
    ! [F: fun_na1240513468iple_a,A_1: fun_nat_bool,B_2: fun_nat_bool] : ( hAPP_f1985266946a_bool(image_344477316iple_a(F),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_1),B_2)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,hAPP_f1985266946a_bool(image_344477316iple_a(F),A_1)),hAPP_f1985266946a_bool(image_344477316iple_a(F),B_2)) ) ).

tff(fact_41_image__Un,axiom,
    ! [F: fun_fu253747899iple_a,A_1: fun_fu2026518849l_bool,B_2: fun_fu2026518849l_bool] : ( hAPP_f1758315691a_bool(image_1469988567iple_a(F),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_1),B_2)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,hAPP_f1758315691a_bool(image_1469988567iple_a(F),A_1)),hAPP_f1758315691a_bool(image_1469988567iple_a(F),B_2)) ) ).

tff(fact_42_image__Un,axiom,
    ! [F: fun_Ho1876430636iple_a,A_1: fun_Ho320143120e_bool,B_2: fun_Ho320143120e_bool] : ( hAPP_f1314815866a_bool(image_363795272iple_a(F),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_1),B_2)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,hAPP_f1314815866a_bool(image_363795272iple_a(F),A_1)),hAPP_f1314815866a_bool(image_363795272iple_a(F),B_2)) ) ).

tff(fact_43_image__Un,axiom,
    ! [F: fun_pn706052866iple_a,A_1: fun_pname_bool,B_2: fun_pname_bool] : ( hAPP_f1451199292a_bool(image_217098954iple_a(F),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),B_2)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,hAPP_f1451199292a_bool(image_217098954iple_a(F),A_1)),hAPP_f1451199292a_bool(image_217098954iple_a(F),B_2)) ) ).

tff(fact_44_sup__fun__def,axiom,
    ! [F: fun_nat_bool,G: fun_nat_bool,X: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,F),G),X))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,hAPP_nat_bool(F,X)),hAPP_nat_bool(G,X))) ) ).

tff(fact_45_sup__fun__def,axiom,
    ! [F: fun_fu2026518849l_bool,G: fun_fu2026518849l_bool,X: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,F),G),X))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,hAPP_f500854239l_bool(F,X)),hAPP_f500854239l_bool(G,X))) ) ).

tff(fact_46_sup__fun__def,axiom,
    ! [F: fun_Ho320143120e_bool,G: fun_Ho320143120e_bool,X: hoare_363350478_state] :
      ( hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,F),G),X))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,hAPP_H1645464494e_bool(F,X)),hAPP_H1645464494e_bool(G,X))) ) ).

tff(fact_47_sup__fun__def,axiom,
    ! [F: fun_pname_bool,G: fun_pname_bool,X: pname] :
      ( hBOOL(hAPP_pname_bool(hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,F),G),X))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,hAPP_pname_bool(F,X)),hAPP_pname_bool(G,X))) ) ).

tff(fact_48_sup__fun__def,axiom,
    ! [F: fun_Ho882550911a_bool,G: fun_Ho882550911a_bool,X: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,F),G),X))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,hAPP_H454055633a_bool(F,X)),hAPP_H454055633a_bool(G,X))) ) ).

tff(fact_49_sup__apply,axiom,
    ! [F: fun_nat_bool,G: fun_nat_bool,X_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,F),G),X_2))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,hAPP_nat_bool(F,X_2)),hAPP_nat_bool(G,X_2))) ) ).

tff(fact_50_sup__apply,axiom,
    ! [F: fun_fu2026518849l_bool,G: fun_fu2026518849l_bool,X_2: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,F),G),X_2))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,hAPP_f500854239l_bool(F,X_2)),hAPP_f500854239l_bool(G,X_2))) ) ).

tff(fact_51_sup__apply,axiom,
    ! [F: fun_Ho320143120e_bool,G: fun_Ho320143120e_bool,X_2: hoare_363350478_state] :
      ( hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,F),G),X_2))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,hAPP_H1645464494e_bool(F,X_2)),hAPP_H1645464494e_bool(G,X_2))) ) ).

tff(fact_52_sup__apply,axiom,
    ! [F: fun_pname_bool,G: fun_pname_bool,X_2: pname] :
      ( hBOOL(hAPP_pname_bool(hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,F),G),X_2))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,hAPP_pname_bool(F,X_2)),hAPP_pname_bool(G,X_2))) ) ).

tff(fact_53_sup__apply,axiom,
    ! [F: fun_Ho882550911a_bool,G: fun_Ho882550911a_bool,X_2: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,F),G),X_2))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,hAPP_H454055633a_bool(F,X_2)),hAPP_H454055633a_bool(G,X_2))) ) ).

tff(fact_54_cut,axiom,
    ! [Ga: fun_Ho882550911a_bool,G_1: fun_Ho882550911a_bool,Ts: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(G_1),Ts))
     => ( hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),G_1))
       => hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),Ts)) ) ) ).

tff(fact_55_cut,axiom,
    ! [Ga: fun_Ho320143120e_bool,G_1: fun_Ho320143120e_bool,Ts: fun_Ho320143120e_bool] :
      ( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(G_1),Ts))
     => ( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),G_1))
       => hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),Ts)) ) ) ).

tff(fact_56_sup__assoc,axiom,
    ! [X_2: fun_nat_bool,Y_1: fun_nat_bool,Z_1: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,X_2),Y_1)),Z_1) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,X_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,Y_1),Z_1)) ) ).

tff(fact_57_sup__assoc,axiom,
    ! [X_20: nat,Y_17: nat,Z_9: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_20),Y_17)),Z_9) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_20),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,Y_17),Z_9)) ) ).

tff(fact_58_sup__assoc,axiom,
    ! [X_2: fun_fu2026518849l_bool,Y_1: fun_fu2026518849l_bool,Z_1: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,X_2),Y_1)),Z_1) = hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,X_2),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,Y_1),Z_1)) ) ).

tff(fact_59_sup__assoc,axiom,
    ! [X_2: fun_Ho320143120e_bool,Y_1: fun_Ho320143120e_bool,Z_1: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,X_2),Y_1)),Z_1) = hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,X_2),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,Y_1),Z_1)) ) ).

tff(fact_60_sup__assoc,axiom,
    ! [X_2: fun_pname_bool,Y_1: fun_pname_bool,Z_1: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),Y_1)),Z_1) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,Y_1),Z_1)) ) ).

tff(fact_61_sup__assoc,axiom,
    ! [X_2: bool,Y_1: bool,Z_1: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),Y_1)),Z_1))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,Y_1),Z_1))) ) ).

tff(fact_62_sup__assoc,axiom,
    ! [X_2: fun_Ho882550911a_bool,Y_1: fun_Ho882550911a_bool,Z_1: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,X_2),Y_1)),Z_1) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,X_2),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,Y_1),Z_1)) ) ).

tff(fact_63_inf__sup__aci_I6_J,axiom,
    ! [X_2: fun_nat_bool,Y_1: fun_nat_bool,Z_1: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,X_2),Y_1)),Z_1) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,X_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,Y_1),Z_1)) ) ).

tff(fact_64_inf__sup__aci_I6_J,axiom,
    ! [X_19: nat,Y_16: nat,Z_8: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_19),Y_16)),Z_8) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_19),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,Y_16),Z_8)) ) ).

tff(fact_65_inf__sup__aci_I6_J,axiom,
    ! [X_2: fun_fu2026518849l_bool,Y_1: fun_fu2026518849l_bool,Z_1: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,X_2),Y_1)),Z_1) = hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,X_2),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,Y_1),Z_1)) ) ).

tff(fact_66_inf__sup__aci_I6_J,axiom,
    ! [X_2: fun_Ho320143120e_bool,Y_1: fun_Ho320143120e_bool,Z_1: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,X_2),Y_1)),Z_1) = hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,X_2),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,Y_1),Z_1)) ) ).

tff(fact_67_inf__sup__aci_I6_J,axiom,
    ! [X_2: fun_pname_bool,Y_1: fun_pname_bool,Z_1: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),Y_1)),Z_1) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,Y_1),Z_1)) ) ).

tff(fact_68_inf__sup__aci_I6_J,axiom,
    ! [X_2: bool,Y_1: bool,Z_1: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),Y_1)),Z_1))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,Y_1),Z_1))) ) ).

tff(fact_69_inf__sup__aci_I6_J,axiom,
    ! [X_2: fun_Ho882550911a_bool,Y_1: fun_Ho882550911a_bool,Z_1: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,X_2),Y_1)),Z_1) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,X_2),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,Y_1),Z_1)) ) ).

tff(fact_70_sup_Oassoc,axiom,
    ! [A_2: fun_nat_bool,B_1: fun_nat_bool,C_2: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_2),B_1)),C_2) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,B_1),C_2)) ) ).

tff(fact_71_sup_Oassoc,axiom,
    ! [A_13: nat,B_10: nat,C_5: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,A_13),B_10)),C_5) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,A_13),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,B_10),C_5)) ) ).

tff(fact_72_sup_Oassoc,axiom,
    ! [A_2: fun_fu2026518849l_bool,B_1: fun_fu2026518849l_bool,C_2: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_2),B_1)),C_2) = hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_2),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,B_1),C_2)) ) ).

tff(fact_73_sup_Oassoc,axiom,
    ! [A_2: fun_Ho320143120e_bool,B_1: fun_Ho320143120e_bool,C_2: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_2),B_1)),C_2) = hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_2),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,B_1),C_2)) ) ).

tff(fact_74_sup_Oassoc,axiom,
    ! [A_2: fun_pname_bool,B_1: fun_pname_bool,C_2: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_2),B_1)),C_2) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,B_1),C_2)) ) ).

tff(fact_75_sup_Oassoc,axiom,
    ! [A_2: bool,B_1: bool,C_2: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,A_2),B_1)),C_2))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,A_2),hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,B_1),C_2))) ) ).

tff(fact_76_sup_Oassoc,axiom,
    ! [A_2: fun_Ho882550911a_bool,B_1: fun_Ho882550911a_bool,C_2: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_2),B_1)),C_2) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_2),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,B_1),C_2)) ) ).

tff(fact_77_sup__left__commute,axiom,
    ! [X_2: fun_nat_bool,Y_1: fun_nat_bool,Z_1: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,X_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,Y_1),Z_1)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,Y_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,X_2),Z_1)) ) ).

tff(fact_78_sup__left__commute,axiom,
    ! [X_18: nat,Y_15: nat,Z_7: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_18),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,Y_15),Z_7)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,Y_15),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_18),Z_7)) ) ).

tff(fact_79_sup__left__commute,axiom,
    ! [X_2: fun_fu2026518849l_bool,Y_1: fun_fu2026518849l_bool,Z_1: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,X_2),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,Y_1),Z_1)) = hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,Y_1),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,X_2),Z_1)) ) ).

tff(fact_80_sup__left__commute,axiom,
    ! [X_2: fun_Ho320143120e_bool,Y_1: fun_Ho320143120e_bool,Z_1: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,X_2),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,Y_1),Z_1)) = hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,Y_1),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,X_2),Z_1)) ) ).

tff(fact_81_sup__left__commute,axiom,
    ! [X_2: fun_pname_bool,Y_1: fun_pname_bool,Z_1: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,Y_1),Z_1)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,Y_1),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),Z_1)) ) ).

tff(fact_82_sup__left__commute,axiom,
    ! [X_2: bool,Y_1: bool,Z_1: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,Y_1),Z_1)))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,Y_1),hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),Z_1))) ) ).

tff(fact_83_sup__left__commute,axiom,
    ! [X_2: fun_Ho882550911a_bool,Y_1: fun_Ho882550911a_bool,Z_1: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,X_2),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,Y_1),Z_1)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,Y_1),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,X_2),Z_1)) ) ).

tff(fact_84_inf__sup__aci_I7_J,axiom,
    ! [X_2: fun_nat_bool,Y_1: fun_nat_bool,Z_1: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,X_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,Y_1),Z_1)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,Y_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,X_2),Z_1)) ) ).

tff(fact_85_inf__sup__aci_I7_J,axiom,
    ! [X_17: nat,Y_14: nat,Z_6: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,Y_14),Z_6)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,Y_14),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_17),Z_6)) ) ).

tff(fact_86_inf__sup__aci_I7_J,axiom,
    ! [X_2: fun_fu2026518849l_bool,Y_1: fun_fu2026518849l_bool,Z_1: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,X_2),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,Y_1),Z_1)) = hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,Y_1),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,X_2),Z_1)) ) ).

tff(fact_87_inf__sup__aci_I7_J,axiom,
    ! [X_2: fun_Ho320143120e_bool,Y_1: fun_Ho320143120e_bool,Z_1: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,X_2),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,Y_1),Z_1)) = hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,Y_1),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,X_2),Z_1)) ) ).

tff(fact_88_inf__sup__aci_I7_J,axiom,
    ! [X_2: fun_pname_bool,Y_1: fun_pname_bool,Z_1: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,Y_1),Z_1)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,Y_1),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),Z_1)) ) ).

tff(fact_89_inf__sup__aci_I7_J,axiom,
    ! [X_2: bool,Y_1: bool,Z_1: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,Y_1),Z_1)))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,Y_1),hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),Z_1))) ) ).

tff(fact_90_inf__sup__aci_I7_J,axiom,
    ! [X_2: fun_Ho882550911a_bool,Y_1: fun_Ho882550911a_bool,Z_1: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,X_2),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,Y_1),Z_1)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,Y_1),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,X_2),Z_1)) ) ).

tff(fact_91_sup_Oleft__commute,axiom,
    ! [B_1: fun_nat_bool,A_2: fun_nat_bool,C_2: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,B_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_2),C_2)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,B_1),C_2)) ) ).

tff(fact_92_sup_Oleft__commute,axiom,
    ! [B_9: nat,A_12: nat,C_4: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,B_9),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,A_12),C_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,A_12),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,B_9),C_4)) ) ).

tff(fact_93_sup_Oleft__commute,axiom,
    ! [B_1: fun_fu2026518849l_bool,A_2: fun_fu2026518849l_bool,C_2: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,B_1),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_2),C_2)) = hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_2),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,B_1),C_2)) ) ).

tff(fact_94_sup_Oleft__commute,axiom,
    ! [B_1: fun_Ho320143120e_bool,A_2: fun_Ho320143120e_bool,C_2: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,B_1),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_2),C_2)) = hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_2),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,B_1),C_2)) ) ).

tff(fact_95_sup_Oleft__commute,axiom,
    ! [B_1: fun_pname_bool,A_2: fun_pname_bool,C_2: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,B_1),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_2),C_2)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,B_1),C_2)) ) ).

tff(fact_96_sup_Oleft__commute,axiom,
    ! [B_1: bool,A_2: bool,C_2: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,B_1),hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,A_2),C_2)))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,A_2),hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,B_1),C_2))) ) ).

tff(fact_97_sup_Oleft__commute,axiom,
    ! [B_1: fun_Ho882550911a_bool,A_2: fun_Ho882550911a_bool,C_2: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,B_1),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_2),C_2)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_2),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,B_1),C_2)) ) ).

tff(fact_98_sup__left__idem,axiom,
    ! [X_2: fun_nat_bool,Y_1: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,X_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,X_2),Y_1)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,X_2),Y_1) ) ).

tff(fact_99_sup__left__idem,axiom,
    ! [X_16: nat,Y_13: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_16),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_16),Y_13)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_16),Y_13) ) ).

tff(fact_100_sup__left__idem,axiom,
    ! [X_2: fun_fu2026518849l_bool,Y_1: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,X_2),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,X_2),Y_1)) = hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,X_2),Y_1) ) ).

tff(fact_101_sup__left__idem,axiom,
    ! [X_2: fun_Ho320143120e_bool,Y_1: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,X_2),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,X_2),Y_1)) = hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,X_2),Y_1) ) ).

tff(fact_102_sup__left__idem,axiom,
    ! [X_2: fun_pname_bool,Y_1: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),Y_1)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),Y_1) ) ).

tff(fact_103_sup__left__idem,axiom,
    ! [X_2: bool,Y_1: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),Y_1)))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),Y_1)) ) ).

tff(fact_104_sup__left__idem,axiom,
    ! [X_2: fun_Ho882550911a_bool,Y_1: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,X_2),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,X_2),Y_1)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,X_2),Y_1) ) ).

tff(fact_105_inf__sup__aci_I8_J,axiom,
    ! [X_2: fun_nat_bool,Y_1: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,X_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,X_2),Y_1)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,X_2),Y_1) ) ).

tff(fact_106_inf__sup__aci_I8_J,axiom,
    ! [X_15: nat,Y_12: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_15),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_15),Y_12)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_15),Y_12) ) ).

tff(fact_107_inf__sup__aci_I8_J,axiom,
    ! [X_2: fun_fu2026518849l_bool,Y_1: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,X_2),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,X_2),Y_1)) = hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,X_2),Y_1) ) ).

tff(fact_108_inf__sup__aci_I8_J,axiom,
    ! [X_2: fun_Ho320143120e_bool,Y_1: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,X_2),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,X_2),Y_1)) = hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,X_2),Y_1) ) ).

tff(fact_109_inf__sup__aci_I8_J,axiom,
    ! [X_2: fun_pname_bool,Y_1: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),Y_1)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),Y_1) ) ).

tff(fact_110_inf__sup__aci_I8_J,axiom,
    ! [X_2: bool,Y_1: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),Y_1)))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),Y_1)) ) ).

tff(fact_111_inf__sup__aci_I8_J,axiom,
    ! [X_2: fun_Ho882550911a_bool,Y_1: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,X_2),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,X_2),Y_1)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,X_2),Y_1) ) ).

tff(fact_112_sup_Oleft__idem,axiom,
    ! [A_2: fun_nat_bool,B_1: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_2),B_1)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_2),B_1) ) ).

tff(fact_113_sup_Oleft__idem,axiom,
    ! [A_11: nat,B_8: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,A_11),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,A_11),B_8)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,A_11),B_8) ) ).

tff(fact_114_sup_Oleft__idem,axiom,
    ! [A_2: fun_fu2026518849l_bool,B_1: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_2),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_2),B_1)) = hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_2),B_1) ) ).

tff(fact_115_sup_Oleft__idem,axiom,
    ! [A_2: fun_Ho320143120e_bool,B_1: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_2),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_2),B_1)) = hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_2),B_1) ) ).

tff(fact_116_sup_Oleft__idem,axiom,
    ! [A_2: fun_pname_bool,B_1: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_2),B_1)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_2),B_1) ) ).

tff(fact_117_sup_Oleft__idem,axiom,
    ! [A_2: bool,B_1: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,A_2),hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,A_2),B_1)))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,A_2),B_1)) ) ).

tff(fact_118_sup_Oleft__idem,axiom,
    ! [A_2: fun_Ho882550911a_bool,B_1: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_2),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_2),B_1)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_2),B_1) ) ).

tff(fact_119_sup__commute,axiom,
    ! [X_2: fun_nat_bool,Y_1: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,X_2),Y_1) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,Y_1),X_2) ) ).

tff(fact_120_sup__commute,axiom,
    ! [X_14: nat,Y_11: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_14),Y_11) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,Y_11),X_14) ) ).

tff(fact_121_sup__commute,axiom,
    ! [X_2: fun_fu2026518849l_bool,Y_1: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,X_2),Y_1) = hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,Y_1),X_2) ) ).

tff(fact_122_sup__commute,axiom,
    ! [X_2: fun_Ho320143120e_bool,Y_1: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,X_2),Y_1) = hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,Y_1),X_2) ) ).

tff(fact_123_sup__commute,axiom,
    ! [X_2: fun_pname_bool,Y_1: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),Y_1) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,Y_1),X_2) ) ).

tff(fact_124_sup__commute,axiom,
    ! [X_2: bool,Y_1: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),Y_1))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,Y_1),X_2)) ) ).

tff(fact_125_sup__commute,axiom,
    ! [X_2: fun_Ho882550911a_bool,Y_1: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,X_2),Y_1) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,Y_1),X_2) ) ).

tff(fact_126_inf__sup__aci_I5_J,axiom,
    ! [X_2: fun_nat_bool,Y_1: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,X_2),Y_1) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,Y_1),X_2) ) ).

tff(fact_127_inf__sup__aci_I5_J,axiom,
    ! [X_13: nat,Y_10: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_13),Y_10) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,Y_10),X_13) ) ).

tff(fact_128_inf__sup__aci_I5_J,axiom,
    ! [X_2: fun_fu2026518849l_bool,Y_1: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,X_2),Y_1) = hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,Y_1),X_2) ) ).

tff(fact_129_inf__sup__aci_I5_J,axiom,
    ! [X_2: fun_Ho320143120e_bool,Y_1: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,X_2),Y_1) = hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,Y_1),X_2) ) ).

tff(fact_130_inf__sup__aci_I5_J,axiom,
    ! [X_2: fun_pname_bool,Y_1: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),Y_1) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,Y_1),X_2) ) ).

tff(fact_131_inf__sup__aci_I5_J,axiom,
    ! [X_2: bool,Y_1: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),Y_1))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,Y_1),X_2)) ) ).

tff(fact_132_inf__sup__aci_I5_J,axiom,
    ! [X_2: fun_Ho882550911a_bool,Y_1: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,X_2),Y_1) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,Y_1),X_2) ) ).

tff(fact_133_sup_Ocommute,axiom,
    ! [A_2: fun_nat_bool,B_1: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_2),B_1) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,B_1),A_2) ) ).

tff(fact_134_sup_Ocommute,axiom,
    ! [A_10: nat,B_7: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,A_10),B_7) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,B_7),A_10) ) ).

tff(fact_135_sup_Ocommute,axiom,
    ! [A_2: fun_fu2026518849l_bool,B_1: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_2),B_1) = hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,B_1),A_2) ) ).

tff(fact_136_sup_Ocommute,axiom,
    ! [A_2: fun_Ho320143120e_bool,B_1: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_2),B_1) = hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,B_1),A_2) ) ).

tff(fact_137_sup_Ocommute,axiom,
    ! [A_2: fun_pname_bool,B_1: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_2),B_1) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,B_1),A_2) ) ).

tff(fact_138_sup_Ocommute,axiom,
    ! [A_2: bool,B_1: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,A_2),B_1))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,B_1),A_2)) ) ).

tff(fact_139_sup_Ocommute,axiom,
    ! [A_2: fun_Ho882550911a_bool,B_1: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_2),B_1) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,B_1),A_2) ) ).

tff(fact_140_sup__idem,axiom,
    ! [X_2: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,X_2),X_2) = X_2 ) ).

tff(fact_141_sup__idem,axiom,
    ! [X_12: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_12),X_12) = X_12 ) ).

tff(fact_142_sup__idem,axiom,
    ! [X_2: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,X_2),X_2) = X_2 ) ).

tff(fact_143_sup__idem,axiom,
    ! [X_2: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,X_2),X_2) = X_2 ) ).

tff(fact_144_sup__idem,axiom,
    ! [X_2: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),X_2) = X_2 ) ).

tff(fact_145_sup__idem,axiom,
    ! [X_2: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),X_2))
    <=> hBOOL(X_2) ) ).

tff(fact_146_sup__idem,axiom,
    ! [X_2: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,X_2),X_2) = X_2 ) ).

tff(fact_147_sup_Oidem,axiom,
    ! [A_2: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_2),A_2) = A_2 ) ).

tff(fact_148_sup_Oidem,axiom,
    ! [A_9: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,A_9),A_9) = A_9 ) ).

tff(fact_149_sup_Oidem,axiom,
    ! [A_2: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_2),A_2) = A_2 ) ).

tff(fact_150_sup_Oidem,axiom,
    ! [A_2: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_2),A_2) = A_2 ) ).

tff(fact_151_sup_Oidem,axiom,
    ! [A_2: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_2),A_2) = A_2 ) ).

tff(fact_152_sup_Oidem,axiom,
    ! [A_2: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,A_2),A_2))
    <=> hBOOL(A_2) ) ).

tff(fact_153_sup_Oidem,axiom,
    ! [A_2: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_2),A_2) = A_2 ) ).

tff(fact_154_rev__image__eqI,axiom,
    ! [B_1: nat,F: fun_nat_nat,X_2: nat,A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A_1))
     => ( ( B_1 = hAPP_nat_nat(F,X_2) )
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,B_1),hAPP_f800510211t_bool(image_nat_nat(F),A_1))) ) ) ).

tff(fact_155_rev__image__eqI,axiom,
    ! [B_1: hoare_363350478_state,F: fun_pn1262351275_state,X_2: pname,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A_1))
     => ( ( B_1 = hAPP_p1337579261_state(F,X_2) )
       => hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,B_1),hAPP_f1763243027e_bool(image_1918575693_state(F),A_1))) ) ) ).

tff(fact_156_rev__image__eqI,axiom,
    ! [B_1: nat,F: fun_pname_nat,X_2: pname,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A_1))
     => ( ( B_1 = hAPP_pname_nat(F,X_2) )
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,B_1),hAPP_f1066163005t_bool(image_pname_nat(F),A_1))) ) ) ).

tff(fact_157_rev__image__eqI,axiom,
    ! [B_1: fun_Ho882550911a_bool,F: fun_pn1811694044a_bool,X_2: pname,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A_1))
     => ( ( B_1 = hAPP_p1160247726a_bool(F,X_2) )
       => hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,B_1),hAPP_f1243652676l_bool(image_1433143038a_bool(F),A_1))) ) ) ).

tff(fact_158_rev__image__eqI,axiom,
    ! [B_1: pname,F: fun_nat_pname,X_2: nat,A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A_1))
     => ( ( B_1 = hAPP_nat_pname(F,X_2) )
       => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,B_1),hAPP_f654413245e_bool(image_nat_pname(F),A_1))) ) ) ).

tff(fact_159_rev__image__eqI,axiom,
    ! [B_1: pname,F: fun_fu1817249398_pname,X_2: fun_Ho882550911a_bool,A_1: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,X_2),A_1))
     => ( ( B_1 = hAPP_f676353346_pname(F,X_2) )
       => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,B_1),hAPP_f1882202854e_bool(image_949248658_pname(F),A_1))) ) ) ).

tff(fact_160_rev__image__eqI,axiom,
    ! [B_1: pname,F: fun_Ho333080248_pname,X_2: hoare_2076525193iple_a,A_1: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_2),A_1))
     => ( ( B_1 = hAPP_H1636214992_pname(F,X_2) )
       => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,B_1),hAPP_f213795644e_bool(image_880197504_pname(F),A_1))) ) ) ).

tff(fact_161_rev__image__eqI,axiom,
    ! [B_1: hoare_2076525193iple_a,F: fun_pn706052866iple_a,X_2: pname,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A_1))
     => ( ( B_1 = hAPP_p973116442iple_a(F,X_2) )
       => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,B_1),hAPP_f1451199292a_bool(image_217098954iple_a(F),A_1))) ) ) ).

tff(fact_162_imageI,axiom,
    ! [F: fun_nat_nat,X_2: nat,A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A_1))
     => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_nat_nat(F,X_2)),hAPP_f800510211t_bool(image_nat_nat(F),A_1))) ) ).

tff(fact_163_imageI,axiom,
    ! [F: fun_pn1262351275_state,X_2: pname,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A_1))
     => hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,hAPP_p1337579261_state(F,X_2)),hAPP_f1763243027e_bool(image_1918575693_state(F),A_1))) ) ).

tff(fact_164_imageI,axiom,
    ! [F: fun_pname_nat,X_2: pname,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A_1))
     => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_pname_nat(F,X_2)),hAPP_f1066163005t_bool(image_pname_nat(F),A_1))) ) ).

tff(fact_165_imageI,axiom,
    ! [F: fun_pn1811694044a_bool,X_2: pname,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A_1))
     => hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,hAPP_p1160247726a_bool(F,X_2)),hAPP_f1243652676l_bool(image_1433143038a_bool(F),A_1))) ) ).

tff(fact_166_imageI,axiom,
    ! [F: fun_nat_pname,X_2: nat,A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A_1))
     => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,hAPP_nat_pname(F,X_2)),hAPP_f654413245e_bool(image_nat_pname(F),A_1))) ) ).

tff(fact_167_imageI,axiom,
    ! [F: fun_fu1817249398_pname,X_2: fun_Ho882550911a_bool,A_1: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,X_2),A_1))
     => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,hAPP_f676353346_pname(F,X_2)),hAPP_f1882202854e_bool(image_949248658_pname(F),A_1))) ) ).

tff(fact_168_imageI,axiom,
    ! [F: fun_Ho333080248_pname,X_2: hoare_2076525193iple_a,A_1: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_2),A_1))
     => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,hAPP_H1636214992_pname(F,X_2)),hAPP_f213795644e_bool(image_880197504_pname(F),A_1))) ) ).

tff(fact_169_imageI,axiom,
    ! [F: fun_pn706052866iple_a,X_2: pname,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A_1))
     => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,hAPP_p973116442iple_a(F,X_2)),hAPP_f1451199292a_bool(image_217098954iple_a(F),A_1))) ) ).

tff(fact_170_image__iff,axiom,
    ! [Z_1: nat,F: fun_nat_nat,A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Z_1),hAPP_f800510211t_bool(image_nat_nat(F),A_1)))
    <=> ? [X: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),A_1))
          & ( Z_1 = hAPP_nat_nat(F,X) ) ) ) ).

tff(fact_171_image__iff,axiom,
    ! [Z_1: hoare_363350478_state,F: fun_pn1262351275_state,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,Z_1),hAPP_f1763243027e_bool(image_1918575693_state(F),A_1)))
    <=> ? [X: pname] :
          ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),A_1))
          & ( Z_1 = hAPP_p1337579261_state(F,X) ) ) ) ).

tff(fact_172_image__iff,axiom,
    ! [Z_1: hoare_2076525193iple_a,F: fun_pn706052866iple_a,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,Z_1),hAPP_f1451199292a_bool(image_217098954iple_a(F),A_1)))
    <=> ? [X: pname] :
          ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),A_1))
          & ( Z_1 = hAPP_p973116442iple_a(F,X) ) ) ) ).

tff(fact_173_UnI2,axiom,
    ! [A_1: fun_nat_bool,C_2: nat,B_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),B_2))
     => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_1),B_2))) ) ).

tff(fact_174_UnI2,axiom,
    ! [A_1: fun_fu2026518849l_bool,C_2: fun_Ho882550911a_bool,B_2: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,C_2),B_2))
     => hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,C_2),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_1),B_2))) ) ).

tff(fact_175_UnI2,axiom,
    ! [A_1: fun_Ho320143120e_bool,C_2: hoare_363350478_state,B_2: fun_Ho320143120e_bool] :
      ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,C_2),B_2))
     => hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,C_2),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_1),B_2))) ) ).

tff(fact_176_UnI2,axiom,
    ! [A_1: fun_Ho882550911a_bool,C_2: hoare_2076525193iple_a,B_2: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,C_2),B_2))
     => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,C_2),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_1),B_2))) ) ).

tff(fact_177_UnI2,axiom,
    ! [A_1: fun_pname_bool,C_2: pname,B_2: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_2),B_2))
     => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),B_2))) ) ).

tff(fact_178_UnI1,axiom,
    ! [B_2: fun_nat_bool,C_2: nat,A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),A_1))
     => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_1),B_2))) ) ).

tff(fact_179_UnI1,axiom,
    ! [B_2: fun_fu2026518849l_bool,C_2: fun_Ho882550911a_bool,A_1: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,C_2),A_1))
     => hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,C_2),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_1),B_2))) ) ).

tff(fact_180_UnI1,axiom,
    ! [B_2: fun_Ho320143120e_bool,C_2: hoare_363350478_state,A_1: fun_Ho320143120e_bool] :
      ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,C_2),A_1))
     => hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,C_2),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_1),B_2))) ) ).

tff(fact_181_UnI1,axiom,
    ! [B_2: fun_Ho882550911a_bool,C_2: hoare_2076525193iple_a,A_1: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,C_2),A_1))
     => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,C_2),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_1),B_2))) ) ).

tff(fact_182_UnI1,axiom,
    ! [B_2: fun_pname_bool,C_2: pname,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_2),A_1))
     => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),B_2))) ) ).

tff(fact_183_sup1I2,axiom,
    ! [A_1: fun_nat_bool,B_2: fun_nat_bool,X_2: nat] :
      ( hBOOL(hAPP_nat_bool(B_2,X_2))
     => hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_1),B_2),X_2)) ) ).

tff(fact_184_sup1I2,axiom,
    ! [A_1: fun_fu2026518849l_bool,B_2: fun_fu2026518849l_bool,X_2: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(B_2,X_2))
     => hBOOL(hAPP_f500854239l_bool(hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_1),B_2),X_2)) ) ).

tff(fact_185_sup1I2,axiom,
    ! [A_1: fun_Ho320143120e_bool,B_2: fun_Ho320143120e_bool,X_2: hoare_363350478_state] :
      ( hBOOL(hAPP_H1645464494e_bool(B_2,X_2))
     => hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_1),B_2),X_2)) ) ).

tff(fact_186_sup1I2,axiom,
    ! [A_1: fun_pname_bool,B_2: fun_pname_bool,X_2: pname] :
      ( hBOOL(hAPP_pname_bool(B_2,X_2))
     => hBOOL(hAPP_pname_bool(hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),B_2),X_2)) ) ).

tff(fact_187_sup1I2,axiom,
    ! [A_1: fun_Ho882550911a_bool,B_2: fun_Ho882550911a_bool,X_2: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_H454055633a_bool(B_2,X_2))
     => hBOOL(hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_1),B_2),X_2)) ) ).

tff(fact_188_sup1I1,axiom,
    ! [B_2: fun_nat_bool,A_1: fun_nat_bool,X_2: nat] :
      ( hBOOL(hAPP_nat_bool(A_1,X_2))
     => hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_1),B_2),X_2)) ) ).

tff(fact_189_sup1I1,axiom,
    ! [B_2: fun_fu2026518849l_bool,A_1: fun_fu2026518849l_bool,X_2: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(A_1,X_2))
     => hBOOL(hAPP_f500854239l_bool(hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_1),B_2),X_2)) ) ).

tff(fact_190_sup1I1,axiom,
    ! [B_2: fun_Ho320143120e_bool,A_1: fun_Ho320143120e_bool,X_2: hoare_363350478_state] :
      ( hBOOL(hAPP_H1645464494e_bool(A_1,X_2))
     => hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_1),B_2),X_2)) ) ).

tff(fact_191_sup1I1,axiom,
    ! [B_2: fun_pname_bool,A_1: fun_pname_bool,X_2: pname] :
      ( hBOOL(hAPP_pname_bool(A_1,X_2))
     => hBOOL(hAPP_pname_bool(hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),B_2),X_2)) ) ).

tff(fact_192_sup1I1,axiom,
    ! [B_2: fun_Ho882550911a_bool,A_1: fun_Ho882550911a_bool,X_2: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_H454055633a_bool(A_1,X_2))
     => hBOOL(hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_1),B_2),X_2)) ) ).

tff(fact_193_ball__Un,axiom,
    ! [Pa: fun_nat_bool,A_1: fun_nat_bool,B_2: fun_nat_bool] :
      ( ! [X: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_1),B_2)))
         => hBOOL(hAPP_nat_bool(Pa,X)) )
    <=> ( ! [X: nat] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),A_1))
           => hBOOL(hAPP_nat_bool(Pa,X)) )
        & ! [X: nat] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),B_2))
           => hBOOL(hAPP_nat_bool(Pa,X)) ) ) ) ).

tff(fact_194_ball__Un,axiom,
    ! [Pa: fun_fu2026518849l_bool,A_1: fun_fu2026518849l_bool,B_2: fun_fu2026518849l_bool] :
      ( ! [X: fun_Ho882550911a_bool] :
          ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,X),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_1),B_2)))
         => hBOOL(hAPP_f500854239l_bool(Pa,X)) )
    <=> ( ! [X: fun_Ho882550911a_bool] :
            ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,X),A_1))
           => hBOOL(hAPP_f500854239l_bool(Pa,X)) )
        & ! [X: fun_Ho882550911a_bool] :
            ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,X),B_2))
           => hBOOL(hAPP_f500854239l_bool(Pa,X)) ) ) ) ).

tff(fact_195_ball__Un,axiom,
    ! [Pa: fun_Ho320143120e_bool,A_1: fun_Ho320143120e_bool,B_2: fun_Ho320143120e_bool] :
      ( ! [X: hoare_363350478_state] :
          ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_1),B_2)))
         => hBOOL(hAPP_H1645464494e_bool(Pa,X)) )
    <=> ( ! [X: hoare_363350478_state] :
            ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X),A_1))
           => hBOOL(hAPP_H1645464494e_bool(Pa,X)) )
        & ! [X: hoare_363350478_state] :
            ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X),B_2))
           => hBOOL(hAPP_H1645464494e_bool(Pa,X)) ) ) ) ).

tff(fact_196_ball__Un,axiom,
    ! [Pa: fun_pname_bool,A_1: fun_pname_bool,B_2: fun_pname_bool] :
      ( ! [X: pname] :
          ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),B_2)))
         => hBOOL(hAPP_pname_bool(Pa,X)) )
    <=> ( ! [X: pname] :
            ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),A_1))
           => hBOOL(hAPP_pname_bool(Pa,X)) )
        & ! [X: pname] :
            ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),B_2))
           => hBOOL(hAPP_pname_bool(Pa,X)) ) ) ) ).

tff(fact_197_ball__Un,axiom,
    ! [Pa: fun_Ho882550911a_bool,A_1: fun_Ho882550911a_bool,B_2: fun_Ho882550911a_bool] :
      ( ! [X: hoare_2076525193iple_a] :
          ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_1),B_2)))
         => hBOOL(hAPP_H454055633a_bool(Pa,X)) )
    <=> ( ! [X: hoare_2076525193iple_a] :
            ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X),A_1))
           => hBOOL(hAPP_H454055633a_bool(Pa,X)) )
        & ! [X: hoare_2076525193iple_a] :
            ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X),B_2))
           => hBOOL(hAPP_H454055633a_bool(Pa,X)) ) ) ) ).

tff(fact_198_bex__Un,axiom,
    ! [Pa: fun_nat_bool,A_1: fun_nat_bool,B_2: fun_nat_bool] :
      ( ? [X: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_1),B_2)))
          & hBOOL(hAPP_nat_bool(Pa,X)) )
    <=> ( ? [X: nat] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),A_1))
            & hBOOL(hAPP_nat_bool(Pa,X)) )
        | ? [X: nat] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),B_2))
            & hBOOL(hAPP_nat_bool(Pa,X)) ) ) ) ).

tff(fact_199_bex__Un,axiom,
    ! [Pa: fun_fu2026518849l_bool,A_1: fun_fu2026518849l_bool,B_2: fun_fu2026518849l_bool] :
      ( ? [X: fun_Ho882550911a_bool] :
          ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,X),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_1),B_2)))
          & hBOOL(hAPP_f500854239l_bool(Pa,X)) )
    <=> ( ? [X: fun_Ho882550911a_bool] :
            ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,X),A_1))
            & hBOOL(hAPP_f500854239l_bool(Pa,X)) )
        | ? [X: fun_Ho882550911a_bool] :
            ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,X),B_2))
            & hBOOL(hAPP_f500854239l_bool(Pa,X)) ) ) ) ).

tff(fact_200_bex__Un,axiom,
    ! [Pa: fun_Ho320143120e_bool,A_1: fun_Ho320143120e_bool,B_2: fun_Ho320143120e_bool] :
      ( ? [X: hoare_363350478_state] :
          ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_1),B_2)))
          & hBOOL(hAPP_H1645464494e_bool(Pa,X)) )
    <=> ( ? [X: hoare_363350478_state] :
            ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X),A_1))
            & hBOOL(hAPP_H1645464494e_bool(Pa,X)) )
        | ? [X: hoare_363350478_state] :
            ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X),B_2))
            & hBOOL(hAPP_H1645464494e_bool(Pa,X)) ) ) ) ).

tff(fact_201_bex__Un,axiom,
    ! [Pa: fun_pname_bool,A_1: fun_pname_bool,B_2: fun_pname_bool] :
      ( ? [X: pname] :
          ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),B_2)))
          & hBOOL(hAPP_pname_bool(Pa,X)) )
    <=> ( ? [X: pname] :
            ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),A_1))
            & hBOOL(hAPP_pname_bool(Pa,X)) )
        | ? [X: pname] :
            ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),B_2))
            & hBOOL(hAPP_pname_bool(Pa,X)) ) ) ) ).

tff(fact_202_bex__Un,axiom,
    ! [Pa: fun_Ho882550911a_bool,A_1: fun_Ho882550911a_bool,B_2: fun_Ho882550911a_bool] :
      ( ? [X: hoare_2076525193iple_a] :
          ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_1),B_2)))
          & hBOOL(hAPP_H454055633a_bool(Pa,X)) )
    <=> ( ? [X: hoare_2076525193iple_a] :
            ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X),A_1))
            & hBOOL(hAPP_H454055633a_bool(Pa,X)) )
        | ? [X: hoare_2076525193iple_a] :
            ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X),B_2))
            & hBOOL(hAPP_H454055633a_bool(Pa,X)) ) ) ) ).

tff(fact_203_Un__assoc,axiom,
    ! [A_1: fun_nat_bool,B_2: fun_nat_bool,C_1: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_1),B_2)),C_1) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,B_2),C_1)) ) ).

tff(fact_204_Un__assoc,axiom,
    ! [A_1: fun_fu2026518849l_bool,B_2: fun_fu2026518849l_bool,C_1: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_1),B_2)),C_1) = hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_1),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,B_2),C_1)) ) ).

tff(fact_205_Un__assoc,axiom,
    ! [A_1: fun_Ho320143120e_bool,B_2: fun_Ho320143120e_bool,C_1: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_1),B_2)),C_1) = hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_1),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,B_2),C_1)) ) ).

tff(fact_206_Un__assoc,axiom,
    ! [A_1: fun_pname_bool,B_2: fun_pname_bool,C_1: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),B_2)),C_1) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,B_2),C_1)) ) ).

tff(fact_207_Un__assoc,axiom,
    ! [A_1: fun_Ho882550911a_bool,B_2: fun_Ho882550911a_bool,C_1: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_1),B_2)),C_1) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_1),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,B_2),C_1)) ) ).

tff(fact_208_Un__iff,axiom,
    ! [C_2: nat,A_1: fun_nat_bool,B_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_1),B_2)))
    <=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),A_1))
        | hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),B_2)) ) ) ).

tff(fact_209_Un__iff,axiom,
    ! [C_2: fun_Ho882550911a_bool,A_1: fun_fu2026518849l_bool,B_2: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,C_2),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_1),B_2)))
    <=> ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,C_2),A_1))
        | hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,C_2),B_2)) ) ) ).

tff(fact_210_Un__iff,axiom,
    ! [C_2: hoare_363350478_state,A_1: fun_Ho320143120e_bool,B_2: fun_Ho320143120e_bool] :
      ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,C_2),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_1),B_2)))
    <=> ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,C_2),A_1))
        | hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,C_2),B_2)) ) ) ).

tff(fact_211_Un__iff,axiom,
    ! [C_2: hoare_2076525193iple_a,A_1: fun_Ho882550911a_bool,B_2: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,C_2),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_1),B_2)))
    <=> ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,C_2),A_1))
        | hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,C_2),B_2)) ) ) ).

tff(fact_212_Un__iff,axiom,
    ! [C_2: pname,A_1: fun_pname_bool,B_2: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),B_2)))
    <=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_2),A_1))
        | hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_2),B_2)) ) ) ).

tff(fact_213_Un__left__commute,axiom,
    ! [A_1: fun_nat_bool,B_2: fun_nat_bool,C_1: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,B_2),C_1)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,B_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_1),C_1)) ) ).

tff(fact_214_Un__left__commute,axiom,
    ! [A_1: fun_fu2026518849l_bool,B_2: fun_fu2026518849l_bool,C_1: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_1),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,B_2),C_1)) = hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,B_2),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_1),C_1)) ) ).

tff(fact_215_Un__left__commute,axiom,
    ! [A_1: fun_Ho320143120e_bool,B_2: fun_Ho320143120e_bool,C_1: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_1),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,B_2),C_1)) = hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,B_2),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_1),C_1)) ) ).

tff(fact_216_Un__left__commute,axiom,
    ! [A_1: fun_pname_bool,B_2: fun_pname_bool,C_1: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,B_2),C_1)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,B_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),C_1)) ) ).

tff(fact_217_Un__left__commute,axiom,
    ! [A_1: fun_Ho882550911a_bool,B_2: fun_Ho882550911a_bool,C_1: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_1),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,B_2),C_1)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,B_2),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_1),C_1)) ) ).

tff(fact_218_Un__left__absorb,axiom,
    ! [A_1: fun_nat_bool,B_2: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_1),B_2)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_1),B_2) ) ).

tff(fact_219_Un__left__absorb,axiom,
    ! [A_1: fun_fu2026518849l_bool,B_2: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_1),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_1),B_2)) = hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_1),B_2) ) ).

tff(fact_220_Un__left__absorb,axiom,
    ! [A_1: fun_Ho320143120e_bool,B_2: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_1),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_1),B_2)) = hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_1),B_2) ) ).

tff(fact_221_Un__left__absorb,axiom,
    ! [A_1: fun_pname_bool,B_2: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),B_2)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),B_2) ) ).

tff(fact_222_Un__left__absorb,axiom,
    ! [A_1: fun_Ho882550911a_bool,B_2: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_1),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_1),B_2)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_1),B_2) ) ).

tff(fact_223_Un__commute,axiom,
    ! [A_1: fun_nat_bool,B_2: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_1),B_2) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,B_2),A_1) ) ).

tff(fact_224_Un__commute,axiom,
    ! [A_1: fun_fu2026518849l_bool,B_2: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_1),B_2) = hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,B_2),A_1) ) ).

tff(fact_225_Un__commute,axiom,
    ! [A_1: fun_Ho320143120e_bool,B_2: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_1),B_2) = hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,B_2),A_1) ) ).

tff(fact_226_Un__commute,axiom,
    ! [A_1: fun_pname_bool,B_2: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),B_2) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,B_2),A_1) ) ).

tff(fact_227_Un__commute,axiom,
    ! [A_1: fun_Ho882550911a_bool,B_2: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_1),B_2) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,B_2),A_1) ) ).

tff(fact_228_Un__def,axiom,
    ! [A_1: fun_fu2026518849l_bool,B_2: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_1),B_2) = hAPP_f354682389l_bool(collec416005674a_bool,hAPP_f354682389l_bool(hAPP_f1121033001l_bool(cOMBS_204124788l_bool,hAPP_f1684296317l_bool(cOMBB_120690437a_bool(fdisj),hAPP_f354682389l_bool(hAPP_f638520823l_bool(cOMBC_1528738210l_bool,member186597612a_bool),A_1))),hAPP_f354682389l_bool(hAPP_f638520823l_bool(cOMBC_1528738210l_bool,member186597612a_bool),B_2))) ) ).

tff(fact_229_Un__def,axiom,
    ! [A_1: fun_Ho320143120e_bool,B_2: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_1),B_2) = hAPP_f1749453397e_bool(collec223185785_state,hAPP_f1749453397e_bool(hAPP_f1612626488e_bool(cOMBS_56025283l_bool,hAPP_f1422032445l_bool(cOMBB_877416532_state(fdisj),hAPP_f1749453397e_bool(hAPP_f1072682137e_bool(cOMBC_1415895650l_bool,member1253905979_state),A_1))),hAPP_f1749453397e_bool(hAPP_f1072682137e_bool(cOMBC_1415895650l_bool,member1253905979_state),B_2))) ) ).

tff(fact_230_Un__def,axiom,
    ! [A_1: fun_Ho882550911a_bool,B_2: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_1),B_2) = hAPP_f642208769a_bool(collec977865374iple_a,hAPP_f642208769a_bool(hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,hAPP_f268025233l_bool(cOMBB_345279363iple_a(fdisj),hAPP_f642208769a_bool(hAPP_f325481667a_bool(cOMBC_1529670358l_bool,member276146780iple_a),A_1))),hAPP_f642208769a_bool(hAPP_f325481667a_bool(cOMBC_1529670358l_bool,member276146780iple_a),B_2))) ) ).

tff(fact_231_Un__def,axiom,
    ! [A_1: fun_nat_bool,B_2: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_1),B_2) = hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(cOMBB_1015721476ol_nat(fdisj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),A_1))),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),B_2))) ) ).

tff(fact_232_Un__def,axiom,
    ! [A_1: fun_pname_bool,B_2: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),B_2) = hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,hAPP_f1402196763l_bool(cOMBB_675860798_pname(fdisj),hAPP_f759274231e_bool(hAPP_f661147897e_bool(cOMBC_1058051404l_bool,member_pname),A_1))),hAPP_f759274231e_bool(hAPP_f661147897e_bool(cOMBC_1058051404l_bool,member_pname),B_2))) ) ).

tff(fact_233_Un__absorb,axiom,
    ! [A_1: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_1),A_1) = A_1 ) ).

tff(fact_234_Un__absorb,axiom,
    ! [A_1: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_1),A_1) = A_1 ) ).

tff(fact_235_Un__absorb,axiom,
    ! [A_1: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_1),A_1) = A_1 ) ).

tff(fact_236_Un__absorb,axiom,
    ! [A_1: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),A_1) = A_1 ) ).

tff(fact_237_Un__absorb,axiom,
    ! [A_1: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_1),A_1) = A_1 ) ).

tff(fact_238_image__ident,axiom,
    ! [Y_9: fun_nat_bool] : ( hAPP_f800510211t_bool(image_nat_nat(cOMBI_nat),Y_9) = Y_9 ) ).

tff(fact_239_image__image,axiom,
    ! [F: fun_Ho1658235120_state,G: fun_pn706052866iple_a,A_1: fun_pname_bool] : ( hAPP_f225384718e_bool(image_624167570_state(F),hAPP_f1451199292a_bool(image_217098954iple_a(G),A_1)) = hAPP_f1763243027e_bool(image_1918575693_state(hAPP_f1641181080_state(cOMBB_911045627_pname(F),G)),A_1) ) ).

tff(fact_240_image__image,axiom,
    ! [F: fun_Ho1876430636iple_a,G: fun_pn1262351275_state,A_1: fun_pname_bool] : ( hAPP_f1314815866a_bool(image_363795272iple_a(F),hAPP_f1763243027e_bool(image_1918575693_state(G),A_1)) = hAPP_f1451199292a_bool(image_217098954iple_a(hAPP_f1024620500iple_a(cOMBB_1765583813_pname(F),G)),A_1) ) ).

tff(fact_241_sup__Un__eq,axiom,
    ! [R_1: fun_nat_bool,S_2: fun_nat_bool,X: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),R_1)),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),S_2)),X))
    <=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,R_1),S_2))) ) ).

tff(fact_242_sup__Un__eq,axiom,
    ! [R_1: fun_fu2026518849l_bool,S_2: fun_fu2026518849l_bool,X: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,hAPP_f354682389l_bool(hAPP_f638520823l_bool(cOMBC_1528738210l_bool,member186597612a_bool),R_1)),hAPP_f354682389l_bool(hAPP_f638520823l_bool(cOMBC_1528738210l_bool,member186597612a_bool),S_2)),X))
    <=> hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,X),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,R_1),S_2))) ) ).

tff(fact_243_sup__Un__eq,axiom,
    ! [R_1: fun_Ho320143120e_bool,S_2: fun_Ho320143120e_bool,X: hoare_363350478_state] :
      ( hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,hAPP_f1749453397e_bool(hAPP_f1072682137e_bool(cOMBC_1415895650l_bool,member1253905979_state),R_1)),hAPP_f1749453397e_bool(hAPP_f1072682137e_bool(cOMBC_1415895650l_bool,member1253905979_state),S_2)),X))
    <=> hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,R_1),S_2))) ) ).

tff(fact_244_sup__Un__eq,axiom,
    ! [R_1: fun_Ho882550911a_bool,S_2: fun_Ho882550911a_bool,X: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,hAPP_f642208769a_bool(hAPP_f325481667a_bool(cOMBC_1529670358l_bool,member276146780iple_a),R_1)),hAPP_f642208769a_bool(hAPP_f325481667a_bool(cOMBC_1529670358l_bool,member276146780iple_a),S_2)),X))
    <=> hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,R_1),S_2))) ) ).

tff(fact_245_sup__Un__eq,axiom,
    ! [R_1: fun_pname_bool,S_2: fun_pname_bool,X: pname] :
      ( hBOOL(hAPP_pname_bool(hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f759274231e_bool(hAPP_f661147897e_bool(cOMBC_1058051404l_bool,member_pname),R_1)),hAPP_f759274231e_bool(hAPP_f661147897e_bool(cOMBC_1058051404l_bool,member_pname),S_2)),X))
    <=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,R_1),S_2))) ) ).

tff(fact_246_Collect__disj__eq,axiom,
    ! [Pa: fun_pname_bool,Qa: fun_pname_bool] : ( hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,hAPP_f1402196763l_bool(cOMBB_675860798_pname(fdisj),Pa)),Qa)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f759274231e_bool(collect_pname,Pa)),hAPP_f759274231e_bool(collect_pname,Qa)) ) ).

tff(fact_247_Collect__disj__eq,axiom,
    ! [Pa: fun_fu2026518849l_bool,Qa: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(collec416005674a_bool,hAPP_f354682389l_bool(hAPP_f1121033001l_bool(cOMBS_204124788l_bool,hAPP_f1684296317l_bool(cOMBB_120690437a_bool(fdisj),Pa)),Qa)) = hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,hAPP_f354682389l_bool(collec416005674a_bool,Pa)),hAPP_f354682389l_bool(collec416005674a_bool,Qa)) ) ).

tff(fact_248_Collect__disj__eq,axiom,
    ! [Pa: fun_Ho320143120e_bool,Qa: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(collec223185785_state,hAPP_f1749453397e_bool(hAPP_f1612626488e_bool(cOMBS_56025283l_bool,hAPP_f1422032445l_bool(cOMBB_877416532_state(fdisj),Pa)),Qa)) = hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,hAPP_f1749453397e_bool(collec223185785_state,Pa)),hAPP_f1749453397e_bool(collec223185785_state,Qa)) ) ).

tff(fact_249_Collect__disj__eq,axiom,
    ! [Pa: fun_Ho882550911a_bool,Qa: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(collec977865374iple_a,hAPP_f642208769a_bool(hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,hAPP_f268025233l_bool(cOMBB_345279363iple_a(fdisj),Pa)),Qa)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,hAPP_f642208769a_bool(collec977865374iple_a,Pa)),hAPP_f642208769a_bool(collec977865374iple_a,Qa)) ) ).

tff(fact_250_Collect__disj__eq,axiom,
    ! [Pa: fun_nat_bool,Qa: fun_nat_bool] : ( hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(cOMBB_1015721476ol_nat(fdisj),Pa)),Qa)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,hAPP_f800510211t_bool(collect_nat,Pa)),hAPP_f800510211t_bool(collect_nat,Qa)) ) ).

tff(fact_251_imageE,axiom,
    ! [B_1: nat,F: fun_nat_nat,A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,B_1),hAPP_f800510211t_bool(image_nat_nat(F),A_1)))
     => ~ ! [X: nat] :
            ( ( B_1 = hAPP_nat_nat(F,X) )
           => ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),A_1)) ) ) ).

tff(fact_252_imageE,axiom,
    ! [B_1: pname,F: fun_nat_pname,A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,B_1),hAPP_f654413245e_bool(image_nat_pname(F),A_1)))
     => ~ ! [X: nat] :
            ( ( B_1 = hAPP_nat_pname(F,X) )
           => ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),A_1)) ) ) ).

tff(fact_253_imageE,axiom,
    ! [B_1: pname,F: fun_fu1817249398_pname,A_1: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,B_1),hAPP_f1882202854e_bool(image_949248658_pname(F),A_1)))
     => ~ ! [X: fun_Ho882550911a_bool] :
            ( ( B_1 = hAPP_f676353346_pname(F,X) )
           => ~ hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,X),A_1)) ) ) ).

tff(fact_254_imageE,axiom,
    ! [B_1: pname,F: fun_Ho333080248_pname,A_1: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,B_1),hAPP_f213795644e_bool(image_880197504_pname(F),A_1)))
     => ~ ! [X: hoare_2076525193iple_a] :
            ( ( B_1 = hAPP_H1636214992_pname(F,X) )
           => ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X),A_1)) ) ) ).

tff(fact_255_imageE,axiom,
    ! [B_1: hoare_363350478_state,F: fun_pn1262351275_state,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,B_1),hAPP_f1763243027e_bool(image_1918575693_state(F),A_1)))
     => ~ ! [X: pname] :
            ( ( B_1 = hAPP_p1337579261_state(F,X) )
           => ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),A_1)) ) ) ).

tff(fact_256_imageE,axiom,
    ! [B_1: nat,F: fun_pname_nat,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,B_1),hAPP_f1066163005t_bool(image_pname_nat(F),A_1)))
     => ~ ! [X: pname] :
            ( ( B_1 = hAPP_pname_nat(F,X) )
           => ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),A_1)) ) ) ).

tff(fact_257_imageE,axiom,
    ! [B_1: fun_Ho882550911a_bool,F: fun_pn1811694044a_bool,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,B_1),hAPP_f1243652676l_bool(image_1433143038a_bool(F),A_1)))
     => ~ ! [X: pname] :
            ( ( B_1 = hAPP_p1160247726a_bool(F,X) )
           => ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),A_1)) ) ) ).

tff(fact_258_imageE,axiom,
    ! [B_1: hoare_2076525193iple_a,F: fun_pn706052866iple_a,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,B_1),hAPP_f1451199292a_bool(image_217098954iple_a(F),A_1)))
     => ~ ! [X: pname] :
            ( ( B_1 = hAPP_p973116442iple_a(F,X) )
           => ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),A_1)) ) ) ).

tff(fact_259_Body__triple__valid__Suc,axiom,
    ! [N_1: nat,Pa: fun_st1506752259e_bool,Pn_1: pname,Qa: fun_st1506752259e_bool] :
      ( hBOOL(hAPP_H1645464494e_bool(hAPP_n753050587e_bool(hoare_28808878_state,N_1),hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,Pa),hAPP_option_com_com(the_com,hAPP_p799580910on_com(body_1,Pn_1))),Qa)))
    <=> hBOOL(hAPP_H1645464494e_bool(hAPP_n753050587e_bool(hoare_28808878_state,hAPP_nat_nat(suc,N_1)),hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,Pa),hAPP_pname_com(body,Pn_1)),Qa))) ) ).

tff(fact_260_Body__triple__valid__Suc,axiom,
    ! [N_1: nat,Pa: fun_a_fun_state_bool,Pn_1: pname,Qa: fun_a_fun_state_bool] :
      ( hBOOL(hAPP_H454055633a_bool(hAPP_n1335981428a_bool(hoare_526497939alid_a,N_1),hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,Pa),hAPP_option_com_com(the_com,hAPP_p799580910on_com(body_1,Pn_1))),Qa)))
    <=> hBOOL(hAPP_H454055633a_bool(hAPP_n1335981428a_bool(hoare_526497939alid_a,hAPP_nat_nat(suc,N_1)),hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,Pa),hAPP_pname_com(body,Pn_1)),Qa))) ) ).

tff(fact_261_triple_Oexhaust,axiom,
    ! [Y_1: hoare_2076525193iple_a] :
      ~ ! [Fun1_1: fun_a_fun_state_bool,Com_2: com,Fun2_1: fun_a_fun_state_bool] : ( Y_1 != hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,Fun1_1),Com_2),Fun2_1) ) ).

tff(fact_262_triple_Oexhaust,axiom,
    ! [Y_1: hoare_363350478_state] :
      ~ ! [Fun1_1: fun_st1506752259e_bool,Com_2: com,Fun2_1: fun_st1506752259e_bool] : ( Y_1 != hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,Fun1_1),Com_2),Fun2_1) ) ).

tff(fact_263_Body1,axiom,
    ! [Pn_1: pname,Ga: fun_Ho320143120e_bool,Pa: fun_pn12888928e_bool,Qa: fun_pn12888928e_bool,Procsa: fun_pname_bool] :
      ( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,Ga),hAPP_f1763243027e_bool(image_1918575693_state(hAPP_f19336718_state(hAPP_f720869462_state(cOMBS_1110920566_state,hAPP_f1146170881_state(hAPP_f1311198492_state(cOMBS_124505571_state,hAPP_f387017471_state(cOMBB_1212270426_pname(hoare_608330478_state),Pa)),body)),Qa)),Procsa))),hAPP_f1763243027e_bool(image_1918575693_state(hAPP_f19336718_state(hAPP_f720869462_state(cOMBS_1110920566_state,hAPP_f1146170881_state(hAPP_f1311198492_state(cOMBS_124505571_state,hAPP_f387017471_state(cOMBB_1212270426_pname(hoare_608330478_state),Pa)),hAPP_f96342628me_com(cOMBB_923936821_pname(the_com),body_1))),Qa)),Procsa)))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,Pn_1),Procsa))
       => hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,hAPP_p1637813682e_bool(Pa,Pn_1)),hAPP_pname_com(body,Pn_1)),hAPP_p1637813682e_bool(Qa,Pn_1))),bot_bo39439654e_bool))) ) ) ).

tff(fact_264_Body1,axiom,
    ! [Pn_1: pname,Ga: fun_Ho882550911a_bool,Pa: fun_pn1683930517e_bool,Qa: fun_pn1683930517e_bool,Procsa: fun_pname_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,Ga),hAPP_f1451199292a_bool(image_217098954iple_a(hAPP_f1246077570iple_a(hAPP_f345351132iple_a(cOMBS_970288740iple_a,hAPP_f804419189iple_a(hAPP_f900936988iple_a(cOMBS_1123526615iple_a,hAPP_f1471296150iple_a(cOMBB_1308294791_pname(hoare_1876769875iple_a),Pa)),body)),Qa)),Procsa))),hAPP_f1451199292a_bool(image_217098954iple_a(hAPP_f1246077570iple_a(hAPP_f345351132iple_a(cOMBS_970288740iple_a,hAPP_f804419189iple_a(hAPP_f900936988iple_a(cOMBS_1123526615iple_a,hAPP_f1471296150iple_a(cOMBB_1308294791_pname(hoare_1876769875iple_a),Pa)),hAPP_f96342628me_com(cOMBB_923936821_pname(the_com),body_1))),Qa)),Procsa)))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,Pn_1),Procsa))
       => hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,hAPP_p635540397e_bool(Pa,Pn_1)),hAPP_pname_com(body,Pn_1)),hAPP_p635540397e_bool(Qa,Pn_1))),bot_bo214064617a_bool))) ) ) ).

tff(fact_265_image__cong,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,M_5: fun_nat_bool,N_3: fun_nat_bool] :
      ( ( M_5 = N_3 )
     => ( ! [X: nat] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),N_3))
           => ( hAPP_nat_nat(F,X) = hAPP_nat_nat(G,X) ) )
       => ( hAPP_f800510211t_bool(image_nat_nat(F),M_5) = hAPP_f800510211t_bool(image_nat_nat(G),N_3) ) ) ) ).

tff(fact_266_image__cong,axiom,
    ! [F: fun_pn1262351275_state,G: fun_pn1262351275_state,M_5: fun_pname_bool,N_3: fun_pname_bool] :
      ( ( M_5 = N_3 )
     => ( ! [X: pname] :
            ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),N_3))
           => ( hAPP_p1337579261_state(F,X) = hAPP_p1337579261_state(G,X) ) )
       => ( hAPP_f1763243027e_bool(image_1918575693_state(F),M_5) = hAPP_f1763243027e_bool(image_1918575693_state(G),N_3) ) ) ) ).

tff(fact_267_image__cong,axiom,
    ! [F: fun_pn706052866iple_a,G: fun_pn706052866iple_a,M_5: fun_pname_bool,N_3: fun_pname_bool] :
      ( ( M_5 = N_3 )
     => ( ! [X: pname] :
            ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),N_3))
           => ( hAPP_p973116442iple_a(F,X) = hAPP_p973116442iple_a(G,X) ) )
       => ( hAPP_f1451199292a_bool(image_217098954iple_a(F),M_5) = hAPP_f1451199292a_bool(image_217098954iple_a(G),N_3) ) ) ) ).

tff(fact_268_Body__triple__valid__0,axiom,
    ! [Pa: fun_st1506752259e_bool,Pn_1: pname,Qa: fun_st1506752259e_bool] : hBOOL(hAPP_H1645464494e_bool(hAPP_n753050587e_bool(hoare_28808878_state,zero_zero_nat),hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,Pa),hAPP_pname_com(body,Pn_1)),Qa))) ).

tff(fact_269_Body__triple__valid__0,axiom,
    ! [Pa: fun_a_fun_state_bool,Pn_1: pname,Qa: fun_a_fun_state_bool] : hBOOL(hAPP_H454055633a_bool(hAPP_n1335981428a_bool(hoare_526497939alid_a,zero_zero_nat),hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,Pa),hAPP_pname_com(body,Pn_1)),Qa))) ).

tff(fact_270_com_Osimps_I6_J,axiom,
    ! [Pname_3: pname,Pname: pname] :
      ( ( hAPP_pname_com(body,Pname_3) = hAPP_pname_com(body,Pname) )
    <=> ( Pname_3 = Pname ) ) ).

tff(fact_271_evalc_OBody,axiom,
    ! [Pn: pname,S0: state,S1_2: state] :
      ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(hAPP_option_com_com(the_com,hAPP_p799580910on_com(body_1,Pn))),S0),S1_2))
     => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(hAPP_pname_com(body,Pn)),S0),S1_2)) ) ).

tff(fact_272_evalc__elim__cases_I6_J,axiom,
    ! [P: pname,S: state,S1_2: state] :
      ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(hAPP_pname_com(body,P)),S),S1_2))
     => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(hAPP_option_com_com(the_com,hAPP_p799580910on_com(body_1,P))),S),S1_2)) ) ).

tff(fact_273_Sup__fin_Oidem,axiom,
    ! [X_2: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,X_2),X_2) = X_2 ) ).

tff(fact_274_Sup__fin_Oidem,axiom,
    ! [X_11: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_11),X_11) = X_11 ) ).

tff(fact_275_Sup__fin_Oidem,axiom,
    ! [X_2: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,X_2),X_2) = X_2 ) ).

tff(fact_276_Sup__fin_Oidem,axiom,
    ! [X_2: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,X_2),X_2) = X_2 ) ).

tff(fact_277_Sup__fin_Oidem,axiom,
    ! [X_2: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),X_2) = X_2 ) ).

tff(fact_278_Sup__fin_Oidem,axiom,
    ! [X_2: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),X_2))
    <=> hBOOL(X_2) ) ).

tff(fact_279_Sup__fin_Oidem,axiom,
    ! [X_2: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,X_2),X_2) = X_2 ) ).

tff(fact_280_triples__valid__Suc,axiom,
    ! [N_1: nat,Ts: fun_Ho320143120e_bool] :
      ( ! [X: hoare_363350478_state] :
          ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X),Ts))
         => hBOOL(hAPP_H1645464494e_bool(hAPP_n753050587e_bool(hoare_28808878_state,hAPP_nat_nat(suc,N_1)),X)) )
     => ! [X: hoare_363350478_state] :
          ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X),Ts))
         => hBOOL(hAPP_H1645464494e_bool(hAPP_n753050587e_bool(hoare_28808878_state,N_1),X)) ) ) ).

tff(fact_281_triples__valid__Suc,axiom,
    ! [N_1: nat,Ts: fun_Ho882550911a_bool] :
      ( ! [X: hoare_2076525193iple_a] :
          ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X),Ts))
         => hBOOL(hAPP_H454055633a_bool(hAPP_n1335981428a_bool(hoare_526497939alid_a,hAPP_nat_nat(suc,N_1)),X)) )
     => ! [X: hoare_2076525193iple_a] :
          ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X),Ts))
         => hBOOL(hAPP_H454055633a_bool(hAPP_n1335981428a_bool(hoare_526497939alid_a,N_1),X)) ) ) ).

tff(fact_282_emptyE,axiom,
    ! [A_2: nat] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),bot_bot_fun_nat_bool)) ).

tff(fact_283_emptyE,axiom,
    ! [A_2: fun_Ho882550911a_bool] : ~ hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,A_2),bot_bo1285271383l_bool)) ).

tff(fact_284_emptyE,axiom,
    ! [A_2: hoare_2076525193iple_a] : ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_2),bot_bo214064617a_bool)) ).

tff(fact_285_emptyE,axiom,
    ! [A_2: hoare_363350478_state] : ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_2),bot_bo39439654e_bool)) ).

tff(fact_286_emptyE,axiom,
    ! [A_2: pname] : ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),bot_bo844097828e_bool)) ).

tff(fact_287_insertE,axiom,
    ! [A_2: nat,B_1: nat,A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,B_1),A_1)))
     => ( ( A_2 != B_1 )
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),A_1)) ) ) ).

tff(fact_288_insertE,axiom,
    ! [A_2: fun_Ho882550911a_bool,B_1: fun_Ho882550911a_bool,A_1: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,A_2),hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,B_1),A_1)))
     => ( ( A_2 != B_1 )
       => hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,A_2),A_1)) ) ) ).

tff(fact_289_insertE,axiom,
    ! [A_2: hoare_2076525193iple_a,B_1: hoare_2076525193iple_a,A_1: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_2),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,B_1),A_1)))
     => ( ( A_2 != B_1 )
       => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_2),A_1)) ) ) ).

tff(fact_290_insertE,axiom,
    ! [A_2: hoare_363350478_state,B_1: hoare_363350478_state,A_1: fun_Ho320143120e_bool] :
      ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,B_1),A_1)))
     => ( ( A_2 != B_1 )
       => hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_2),A_1)) ) ) ).

tff(fact_291_insertE,axiom,
    ! [A_2: pname,B_1: pname,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,B_1),A_1)))
     => ( ( A_2 != B_1 )
       => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),A_1)) ) ) ).

tff(fact_292_insertCI,axiom,
    ! [B_1: nat,A_2: nat,B_2: fun_nat_bool] :
      ( ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),B_2))
       => ( A_2 = B_1 ) )
     => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,B_1),B_2))) ) ).

tff(fact_293_insertCI,axiom,
    ! [B_1: fun_Ho882550911a_bool,A_2: fun_Ho882550911a_bool,B_2: fun_fu2026518849l_bool] :
      ( ( ~ hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,A_2),B_2))
       => ( A_2 = B_1 ) )
     => hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,A_2),hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,B_1),B_2))) ) ).

tff(fact_294_insertCI,axiom,
    ! [B_1: hoare_2076525193iple_a,A_2: hoare_2076525193iple_a,B_2: fun_Ho882550911a_bool] :
      ( ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_2),B_2))
       => ( A_2 = B_1 ) )
     => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_2),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,B_1),B_2))) ) ).

tff(fact_295_insertCI,axiom,
    ! [B_1: hoare_363350478_state,A_2: hoare_363350478_state,B_2: fun_Ho320143120e_bool] :
      ( ( ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_2),B_2))
       => ( A_2 = B_1 ) )
     => hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,B_1),B_2))) ) ).

tff(fact_296_insertCI,axiom,
    ! [B_1: pname,A_2: pname,B_2: fun_pname_bool] :
      ( ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),B_2))
       => ( A_2 = B_1 ) )
     => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,B_1),B_2))) ) ).

tff(fact_297_empty__not__insert,axiom,
    ! [A_2: nat,A_1: fun_nat_bool] : ( bot_bot_fun_nat_bool != hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),A_1) ) ).

tff(fact_298_empty__not__insert,axiom,
    ! [A_2: fun_Ho882550911a_bool,A_1: fun_fu2026518849l_bool] : ( bot_bo1285271383l_bool != hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,A_2),A_1) ) ).

tff(fact_299_empty__not__insert,axiom,
    ! [A_2: pname,A_1: fun_pname_bool] : ( bot_bo844097828e_bool != hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),A_1) ) ).

tff(fact_300_empty__not__insert,axiom,
    ! [A_2: hoare_2076525193iple_a,A_1: fun_Ho882550911a_bool] : ( bot_bo214064617a_bool != hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_2),A_1) ) ).

tff(fact_301_empty__not__insert,axiom,
    ! [A_2: hoare_363350478_state,A_1: fun_Ho320143120e_bool] : ( bot_bo39439654e_bool != hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_2),A_1) ) ).

tff(fact_302_insert__not__empty,axiom,
    ! [A_2: nat,A_1: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),A_1) != bot_bot_fun_nat_bool ) ).

tff(fact_303_insert__not__empty,axiom,
    ! [A_2: fun_Ho882550911a_bool,A_1: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,A_2),A_1) != bot_bo1285271383l_bool ) ).

tff(fact_304_insert__not__empty,axiom,
    ! [A_2: pname,A_1: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),A_1) != bot_bo844097828e_bool ) ).

tff(fact_305_insert__not__empty,axiom,
    ! [A_2: hoare_2076525193iple_a,A_1: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_2),A_1) != bot_bo214064617a_bool ) ).

tff(fact_306_insert__not__empty,axiom,
    ! [A_2: hoare_363350478_state,A_1: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_2),A_1) != bot_bo39439654e_bool ) ).

tff(fact_307_bot__empty__eq,axiom,
    ! [X: nat] :
      ( hBOOL(hAPP_nat_bool(bot_bot_fun_nat_bool,X))
    <=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),bot_bot_fun_nat_bool)) ) ).

tff(fact_308_bot__empty__eq,axiom,
    ! [X: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(bot_bo1285271383l_bool,X))
    <=> hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,X),bot_bo1285271383l_bool)) ) ).

tff(fact_309_bot__empty__eq,axiom,
    ! [X: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_H454055633a_bool(bot_bo214064617a_bool,X))
    <=> hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X),bot_bo214064617a_bool)) ) ).

tff(fact_310_bot__empty__eq,axiom,
    ! [X: hoare_363350478_state] :
      ( hBOOL(hAPP_H1645464494e_bool(bot_bo39439654e_bool,X))
    <=> hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X),bot_bo39439654e_bool)) ) ).

tff(fact_311_bot__empty__eq,axiom,
    ! [X: pname] :
      ( hBOOL(hAPP_pname_bool(bot_bo844097828e_bool,X))
    <=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),bot_bo844097828e_bool)) ) ).

tff(fact_312_empty__def,axiom,
    bot_bo844097828e_bool = hAPP_f759274231e_bool(collect_pname,hAPP_b76515610e_bool(cOMBK_bool_pname,fFalse)) ).

tff(fact_313_empty__def,axiom,
    bot_bo214064617a_bool = hAPP_f642208769a_bool(collec977865374iple_a,hAPP_b1299643615a_bool(cOMBK_861658160iple_a,fFalse)) ).

tff(fact_314_empty__def,axiom,
    bot_bo1285271383l_bool = hAPP_f354682389l_bool(collec416005674a_bool,hAPP_b1036872929l_bool(cOMBK_1968875032a_bool,fFalse)) ).

tff(fact_315_empty__def,axiom,
    bot_bo39439654e_bool = hAPP_f1749453397e_bool(collec223185785_state,hAPP_b230077104e_bool(cOMBK_574827367_state,fFalse)) ).

tff(fact_316_empty__def,axiom,
    bot_bot_fun_nat_bool = hAPP_f800510211t_bool(collect_nat,hAPP_b1013836512t_bool(cOMBK_bool_nat,fFalse)) ).

tff(fact_317_insertI1,axiom,
    ! [A_2: nat,B_2: fun_nat_bool] : hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),B_2))) ).

tff(fact_318_insertI1,axiom,
    ! [A_2: fun_Ho882550911a_bool,B_2: fun_fu2026518849l_bool] : hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,A_2),hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,A_2),B_2))) ).

tff(fact_319_insertI1,axiom,
    ! [A_2: hoare_2076525193iple_a,B_2: fun_Ho882550911a_bool] : hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_2),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_2),B_2))) ).

tff(fact_320_insertI1,axiom,
    ! [A_2: hoare_363350478_state,B_2: fun_Ho320143120e_bool] : hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_2),B_2))) ).

tff(fact_321_insertI1,axiom,
    ! [A_2: pname,B_2: fun_pname_bool] : hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),B_2))) ).

tff(fact_322_all__not__in__conv,axiom,
    ! [A_1: fun_nat_bool] :
      ( ! [X: nat] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),A_1))
    <=> ( A_1 = bot_bot_fun_nat_bool ) ) ).

tff(fact_323_all__not__in__conv,axiom,
    ! [A_1: fun_fu2026518849l_bool] :
      ( ! [X: fun_Ho882550911a_bool] : ~ hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,X),A_1))
    <=> ( A_1 = bot_bo1285271383l_bool ) ) ).

tff(fact_324_all__not__in__conv,axiom,
    ! [A_1: fun_Ho882550911a_bool] :
      ( ! [X: hoare_2076525193iple_a] : ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X),A_1))
    <=> ( A_1 = bot_bo214064617a_bool ) ) ).

tff(fact_325_all__not__in__conv,axiom,
    ! [A_1: fun_Ho320143120e_bool] :
      ( ! [X: hoare_363350478_state] : ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X),A_1))
    <=> ( A_1 = bot_bo39439654e_bool ) ) ).

tff(fact_326_all__not__in__conv,axiom,
    ! [A_1: fun_pname_bool] :
      ( ! [X: pname] : ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),A_1))
    <=> ( A_1 = bot_bo844097828e_bool ) ) ).

tff(fact_327_singleton__conv2,axiom,
    ! [A_2: fun_Ho882550911a_bool] : ( hAPP_f354682389l_bool(collec416005674a_bool,hAPP_f256727871l_bool(fequal291249344a_bool,A_2)) = hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,A_2),bot_bo1285271383l_bool) ) ).

tff(fact_328_singleton__conv2,axiom,
    ! [A_2: pname] : ( hAPP_f759274231e_bool(collect_pname,hAPP_p61793385e_bool(fequal_pname,A_2)) = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),bot_bo844097828e_bool) ) ).

tff(fact_329_singleton__conv2,axiom,
    ! [A_2: hoare_2076525193iple_a] : ( hAPP_f642208769a_bool(collec977865374iple_a,hAPP_H2077279219a_bool(fequal1589671816iple_a,A_2)) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_2),bot_bo214064617a_bool) ) ).

tff(fact_330_singleton__conv2,axiom,
    ! [A_2: hoare_363350478_state] : ( hAPP_f1749453397e_bool(collec223185785_state,hAPP_H705653759e_bool(fequal1026769423_state,A_2)) = hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_2),bot_bo39439654e_bool) ) ).

tff(fact_331_singleton__conv2,axiom,
    ! [A_2: nat] : ( hAPP_f800510211t_bool(collect_nat,hAPP_n1699378549t_bool(fequal_nat,A_2)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),bot_bot_fun_nat_bool) ) ).

tff(fact_332_ex__in__conv,axiom,
    ! [A_1: fun_nat_bool] :
      ( ? [X: nat] : hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),A_1))
    <=> ( A_1 != bot_bot_fun_nat_bool ) ) ).

tff(fact_333_ex__in__conv,axiom,
    ! [A_1: fun_fu2026518849l_bool] :
      ( ? [X: fun_Ho882550911a_bool] : hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,X),A_1))
    <=> ( A_1 != bot_bo1285271383l_bool ) ) ).

tff(fact_334_ex__in__conv,axiom,
    ! [A_1: fun_Ho882550911a_bool] :
      ( ? [X: hoare_2076525193iple_a] : hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X),A_1))
    <=> ( A_1 != bot_bo214064617a_bool ) ) ).

tff(fact_335_ex__in__conv,axiom,
    ! [A_1: fun_Ho320143120e_bool] :
      ( ? [X: hoare_363350478_state] : hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X),A_1))
    <=> ( A_1 != bot_bo39439654e_bool ) ) ).

tff(fact_336_ex__in__conv,axiom,
    ! [A_1: fun_pname_bool] :
      ( ? [X: pname] : hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),A_1))
    <=> ( A_1 != bot_bo844097828e_bool ) ) ).

tff(fact_337_singleton__conv,axiom,
    ! [A_2: fun_Ho882550911a_bool] : ( hAPP_f354682389l_bool(collec416005674a_bool,hAPP_f256727871l_bool(hAPP_f246259945l_bool(cOMBC_1515890438l_bool,fequal291249344a_bool),A_2)) = hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,A_2),bot_bo1285271383l_bool) ) ).

tff(fact_338_singleton__conv,axiom,
    ! [A_2: pname] : ( hAPP_f759274231e_bool(collect_pname,hAPP_p61793385e_bool(hAPP_f22061361e_bool(cOMBC_1149511130e_bool,fequal_pname),A_2)) = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),bot_bo844097828e_bool) ) ).

tff(fact_339_singleton__conv,axiom,
    ! [A_2: hoare_2076525193iple_a] : ( hAPP_f642208769a_bool(collec977865374iple_a,hAPP_H2077279219a_bool(hAPP_f877684549a_bool(cOMBC_1047656090a_bool,fequal1589671816iple_a),A_2)) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_2),bot_bo214064617a_bool) ) ).

tff(fact_340_singleton__conv,axiom,
    ! [A_2: hoare_363350478_state] : ( hAPP_f1749453397e_bool(collec223185785_state,hAPP_H705653759e_bool(hAPP_f114072425e_bool(cOMBC_13083556e_bool,fequal1026769423_state),A_2)) = hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_2),bot_bo39439654e_bool) ) ).

tff(fact_341_singleton__conv,axiom,
    ! [A_2: nat] : ( hAPP_f800510211t_bool(collect_nat,hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),A_2)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),bot_bot_fun_nat_bool) ) ).

tff(fact_342_Collect__conv__if2,axiom,
    ! [Pa: fun_fu2026518849l_bool,A_2: fun_Ho882550911a_bool] :
      ( ( hBOOL(hAPP_f500854239l_bool(Pa,A_2))
       => ( hAPP_f354682389l_bool(collec416005674a_bool,hAPP_f354682389l_bool(hAPP_f1121033001l_bool(cOMBS_204124788l_bool,hAPP_f1684296317l_bool(cOMBB_120690437a_bool(fconj),hAPP_f256727871l_bool(fequal291249344a_bool,A_2))),Pa)) = hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,A_2),bot_bo1285271383l_bool) ) )
      & ( ~ hBOOL(hAPP_f500854239l_bool(Pa,A_2))
       => ( hAPP_f354682389l_bool(collec416005674a_bool,hAPP_f354682389l_bool(hAPP_f1121033001l_bool(cOMBS_204124788l_bool,hAPP_f1684296317l_bool(cOMBB_120690437a_bool(fconj),hAPP_f256727871l_bool(fequal291249344a_bool,A_2))),Pa)) = bot_bo1285271383l_bool ) ) ) ).

tff(fact_343_Collect__conv__if2,axiom,
    ! [Pa: fun_pname_bool,A_2: pname] :
      ( ( hBOOL(hAPP_pname_bool(Pa,A_2))
       => ( hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,hAPP_f1402196763l_bool(cOMBB_675860798_pname(fconj),hAPP_p61793385e_bool(fequal_pname,A_2))),Pa)) = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),bot_bo844097828e_bool) ) )
      & ( ~ hBOOL(hAPP_pname_bool(Pa,A_2))
       => ( hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,hAPP_f1402196763l_bool(cOMBB_675860798_pname(fconj),hAPP_p61793385e_bool(fequal_pname,A_2))),Pa)) = bot_bo844097828e_bool ) ) ) ).

tff(fact_344_Collect__conv__if2,axiom,
    ! [Pa: fun_Ho882550911a_bool,A_2: hoare_2076525193iple_a] :
      ( ( hBOOL(hAPP_H454055633a_bool(Pa,A_2))
       => ( hAPP_f642208769a_bool(collec977865374iple_a,hAPP_f642208769a_bool(hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,hAPP_f268025233l_bool(cOMBB_345279363iple_a(fconj),hAPP_H2077279219a_bool(fequal1589671816iple_a,A_2))),Pa)) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_2),bot_bo214064617a_bool) ) )
      & ( ~ hBOOL(hAPP_H454055633a_bool(Pa,A_2))
       => ( hAPP_f642208769a_bool(collec977865374iple_a,hAPP_f642208769a_bool(hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,hAPP_f268025233l_bool(cOMBB_345279363iple_a(fconj),hAPP_H2077279219a_bool(fequal1589671816iple_a,A_2))),Pa)) = bot_bo214064617a_bool ) ) ) ).

tff(fact_345_Collect__conv__if2,axiom,
    ! [Pa: fun_Ho320143120e_bool,A_2: hoare_363350478_state] :
      ( ( hBOOL(hAPP_H1645464494e_bool(Pa,A_2))
       => ( hAPP_f1749453397e_bool(collec223185785_state,hAPP_f1749453397e_bool(hAPP_f1612626488e_bool(cOMBS_56025283l_bool,hAPP_f1422032445l_bool(cOMBB_877416532_state(fconj),hAPP_H705653759e_bool(fequal1026769423_state,A_2))),Pa)) = hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_2),bot_bo39439654e_bool) ) )
      & ( ~ hBOOL(hAPP_H1645464494e_bool(Pa,A_2))
       => ( hAPP_f1749453397e_bool(collec223185785_state,hAPP_f1749453397e_bool(hAPP_f1612626488e_bool(cOMBS_56025283l_bool,hAPP_f1422032445l_bool(cOMBB_877416532_state(fconj),hAPP_H705653759e_bool(fequal1026769423_state,A_2))),Pa)) = bot_bo39439654e_bool ) ) ) ).

tff(fact_346_Collect__conv__if2,axiom,
    ! [Pa: fun_nat_bool,A_2: nat] :
      ( ( hBOOL(hAPP_nat_bool(Pa,A_2))
       => ( hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(cOMBB_1015721476ol_nat(fconj),hAPP_n1699378549t_bool(fequal_nat,A_2))),Pa)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),bot_bot_fun_nat_bool) ) )
      & ( ~ hBOOL(hAPP_nat_bool(Pa,A_2))
       => ( hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(cOMBB_1015721476ol_nat(fconj),hAPP_n1699378549t_bool(fequal_nat,A_2))),Pa)) = bot_bot_fun_nat_bool ) ) ) ).

tff(fact_347_Collect__conv__if,axiom,
    ! [Pa: fun_fu2026518849l_bool,A_2: fun_Ho882550911a_bool] :
      ( ( hBOOL(hAPP_f500854239l_bool(Pa,A_2))
       => ( hAPP_f354682389l_bool(collec416005674a_bool,hAPP_f354682389l_bool(hAPP_f1121033001l_bool(cOMBS_204124788l_bool,hAPP_f1684296317l_bool(cOMBB_120690437a_bool(fconj),hAPP_f256727871l_bool(hAPP_f246259945l_bool(cOMBC_1515890438l_bool,fequal291249344a_bool),A_2))),Pa)) = hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,A_2),bot_bo1285271383l_bool) ) )
      & ( ~ hBOOL(hAPP_f500854239l_bool(Pa,A_2))
       => ( hAPP_f354682389l_bool(collec416005674a_bool,hAPP_f354682389l_bool(hAPP_f1121033001l_bool(cOMBS_204124788l_bool,hAPP_f1684296317l_bool(cOMBB_120690437a_bool(fconj),hAPP_f256727871l_bool(hAPP_f246259945l_bool(cOMBC_1515890438l_bool,fequal291249344a_bool),A_2))),Pa)) = bot_bo1285271383l_bool ) ) ) ).

tff(fact_348_Collect__conv__if,axiom,
    ! [Pa: fun_pname_bool,A_2: pname] :
      ( ( hBOOL(hAPP_pname_bool(Pa,A_2))
       => ( hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,hAPP_f1402196763l_bool(cOMBB_675860798_pname(fconj),hAPP_p61793385e_bool(hAPP_f22061361e_bool(cOMBC_1149511130e_bool,fequal_pname),A_2))),Pa)) = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),bot_bo844097828e_bool) ) )
      & ( ~ hBOOL(hAPP_pname_bool(Pa,A_2))
       => ( hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,hAPP_f1402196763l_bool(cOMBB_675860798_pname(fconj),hAPP_p61793385e_bool(hAPP_f22061361e_bool(cOMBC_1149511130e_bool,fequal_pname),A_2))),Pa)) = bot_bo844097828e_bool ) ) ) ).

tff(fact_349_Collect__conv__if,axiom,
    ! [Pa: fun_Ho882550911a_bool,A_2: hoare_2076525193iple_a] :
      ( ( hBOOL(hAPP_H454055633a_bool(Pa,A_2))
       => ( hAPP_f642208769a_bool(collec977865374iple_a,hAPP_f642208769a_bool(hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,hAPP_f268025233l_bool(cOMBB_345279363iple_a(fconj),hAPP_H2077279219a_bool(hAPP_f877684549a_bool(cOMBC_1047656090a_bool,fequal1589671816iple_a),A_2))),Pa)) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_2),bot_bo214064617a_bool) ) )
      & ( ~ hBOOL(hAPP_H454055633a_bool(Pa,A_2))
       => ( hAPP_f642208769a_bool(collec977865374iple_a,hAPP_f642208769a_bool(hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,hAPP_f268025233l_bool(cOMBB_345279363iple_a(fconj),hAPP_H2077279219a_bool(hAPP_f877684549a_bool(cOMBC_1047656090a_bool,fequal1589671816iple_a),A_2))),Pa)) = bot_bo214064617a_bool ) ) ) ).

tff(fact_350_Collect__conv__if,axiom,
    ! [Pa: fun_Ho320143120e_bool,A_2: hoare_363350478_state] :
      ( ( hBOOL(hAPP_H1645464494e_bool(Pa,A_2))
       => ( hAPP_f1749453397e_bool(collec223185785_state,hAPP_f1749453397e_bool(hAPP_f1612626488e_bool(cOMBS_56025283l_bool,hAPP_f1422032445l_bool(cOMBB_877416532_state(fconj),hAPP_H705653759e_bool(hAPP_f114072425e_bool(cOMBC_13083556e_bool,fequal1026769423_state),A_2))),Pa)) = hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_2),bot_bo39439654e_bool) ) )
      & ( ~ hBOOL(hAPP_H1645464494e_bool(Pa,A_2))
       => ( hAPP_f1749453397e_bool(collec223185785_state,hAPP_f1749453397e_bool(hAPP_f1612626488e_bool(cOMBS_56025283l_bool,hAPP_f1422032445l_bool(cOMBB_877416532_state(fconj),hAPP_H705653759e_bool(hAPP_f114072425e_bool(cOMBC_13083556e_bool,fequal1026769423_state),A_2))),Pa)) = bot_bo39439654e_bool ) ) ) ).

tff(fact_351_Collect__conv__if,axiom,
    ! [Pa: fun_nat_bool,A_2: nat] :
      ( ( hBOOL(hAPP_nat_bool(Pa,A_2))
       => ( hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(cOMBB_1015721476ol_nat(fconj),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),A_2))),Pa)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),bot_bot_fun_nat_bool) ) )
      & ( ~ hBOOL(hAPP_nat_bool(Pa,A_2))
       => ( hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(cOMBB_1015721476ol_nat(fconj),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),A_2))),Pa)) = bot_bot_fun_nat_bool ) ) ) ).

tff(fact_352_mem__def,axiom,
    ! [X_2: nat,A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A_1))
    <=> hBOOL(hAPP_nat_bool(A_1,X_2)) ) ).

tff(fact_353_mem__def,axiom,
    ! [X_2: fun_Ho882550911a_bool,A_1: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,X_2),A_1))
    <=> hBOOL(hAPP_f500854239l_bool(A_1,X_2)) ) ).

tff(fact_354_mem__def,axiom,
    ! [X_2: hoare_2076525193iple_a,A_1: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_2),A_1))
    <=> hBOOL(hAPP_H454055633a_bool(A_1,X_2)) ) ).

tff(fact_355_mem__def,axiom,
    ! [X_2: pname,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A_1))
    <=> hBOOL(hAPP_pname_bool(A_1,X_2)) ) ).

tff(fact_356_Collect__def,axiom,
    ! [Pa: fun_pname_bool] : ( hAPP_f759274231e_bool(collect_pname,Pa) = Pa ) ).

tff(fact_357_Collect__def,axiom,
    ! [Pa: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(collec977865374iple_a,Pa) = Pa ) ).

tff(fact_358_Collect__def,axiom,
    ! [Pa: fun_nat_bool] : ( hAPP_f800510211t_bool(collect_nat,Pa) = Pa ) ).

tff(fact_359_empty__Collect__eq,axiom,
    ! [Pa: fun_pname_bool] :
      ( ( bot_bo844097828e_bool = hAPP_f759274231e_bool(collect_pname,Pa) )
    <=> ! [X: pname] : ~ hBOOL(hAPP_pname_bool(Pa,X)) ) ).

tff(fact_360_empty__Collect__eq,axiom,
    ! [Pa: fun_Ho882550911a_bool] :
      ( ( bot_bo214064617a_bool = hAPP_f642208769a_bool(collec977865374iple_a,Pa) )
    <=> ! [X: hoare_2076525193iple_a] : ~ hBOOL(hAPP_H454055633a_bool(Pa,X)) ) ).

tff(fact_361_empty__Collect__eq,axiom,
    ! [Pa: fun_fu2026518849l_bool] :
      ( ( bot_bo1285271383l_bool = hAPP_f354682389l_bool(collec416005674a_bool,Pa) )
    <=> ! [X: fun_Ho882550911a_bool] : ~ hBOOL(hAPP_f500854239l_bool(Pa,X)) ) ).

tff(fact_362_empty__Collect__eq,axiom,
    ! [Pa: fun_Ho320143120e_bool] :
      ( ( bot_bo39439654e_bool = hAPP_f1749453397e_bool(collec223185785_state,Pa) )
    <=> ! [X: hoare_363350478_state] : ~ hBOOL(hAPP_H1645464494e_bool(Pa,X)) ) ).

tff(fact_363_empty__Collect__eq,axiom,
    ! [Pa: fun_nat_bool] :
      ( ( bot_bot_fun_nat_bool = hAPP_f800510211t_bool(collect_nat,Pa) )
    <=> ! [X: nat] : ~ hBOOL(hAPP_nat_bool(Pa,X)) ) ).

tff(fact_364_empty__iff,axiom,
    ! [C_2: nat] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),bot_bot_fun_nat_bool)) ).

tff(fact_365_empty__iff,axiom,
    ! [C_2: fun_Ho882550911a_bool] : ~ hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,C_2),bot_bo1285271383l_bool)) ).

tff(fact_366_empty__iff,axiom,
    ! [C_2: hoare_2076525193iple_a] : ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,C_2),bot_bo214064617a_bool)) ).

tff(fact_367_empty__iff,axiom,
    ! [C_2: hoare_363350478_state] : ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,C_2),bot_bo39439654e_bool)) ).

tff(fact_368_empty__iff,axiom,
    ! [C_2: pname] : ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_2),bot_bo844097828e_bool)) ).

tff(fact_369_insert__compr,axiom,
    ! [A_2: fun_Ho882550911a_bool,B_2: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,A_2),B_2) = hAPP_f354682389l_bool(collec416005674a_bool,hAPP_f354682389l_bool(hAPP_f1121033001l_bool(cOMBS_204124788l_bool,hAPP_f1684296317l_bool(cOMBB_120690437a_bool(fdisj),hAPP_f256727871l_bool(hAPP_f246259945l_bool(cOMBC_1515890438l_bool,fequal291249344a_bool),A_2))),hAPP_f354682389l_bool(hAPP_f638520823l_bool(cOMBC_1528738210l_bool,member186597612a_bool),B_2))) ) ).

tff(fact_370_insert__compr,axiom,
    ! [A_2: hoare_2076525193iple_a,B_2: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_2),B_2) = hAPP_f642208769a_bool(collec977865374iple_a,hAPP_f642208769a_bool(hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,hAPP_f268025233l_bool(cOMBB_345279363iple_a(fdisj),hAPP_H2077279219a_bool(hAPP_f877684549a_bool(cOMBC_1047656090a_bool,fequal1589671816iple_a),A_2))),hAPP_f642208769a_bool(hAPP_f325481667a_bool(cOMBC_1529670358l_bool,member276146780iple_a),B_2))) ) ).

tff(fact_371_insert__compr,axiom,
    ! [A_2: hoare_363350478_state,B_2: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_2),B_2) = hAPP_f1749453397e_bool(collec223185785_state,hAPP_f1749453397e_bool(hAPP_f1612626488e_bool(cOMBS_56025283l_bool,hAPP_f1422032445l_bool(cOMBB_877416532_state(fdisj),hAPP_H705653759e_bool(hAPP_f114072425e_bool(cOMBC_13083556e_bool,fequal1026769423_state),A_2))),hAPP_f1749453397e_bool(hAPP_f1072682137e_bool(cOMBC_1415895650l_bool,member1253905979_state),B_2))) ) ).

tff(fact_372_insert__compr,axiom,
    ! [A_2: nat,B_2: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),B_2) = hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(cOMBB_1015721476ol_nat(fdisj),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),A_2))),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),B_2))) ) ).

tff(fact_373_insert__compr,axiom,
    ! [A_2: pname,B_2: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),B_2) = hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,hAPP_f1402196763l_bool(cOMBB_675860798_pname(fdisj),hAPP_p61793385e_bool(hAPP_f22061361e_bool(cOMBC_1149511130e_bool,fequal_pname),A_2))),hAPP_f759274231e_bool(hAPP_f661147897e_bool(cOMBC_1058051404l_bool,member_pname),B_2))) ) ).

tff(fact_374_insert__Collect,axiom,
    ! [A_2: fun_Ho882550911a_bool,Pa: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,A_2),hAPP_f354682389l_bool(collec416005674a_bool,Pa)) = hAPP_f354682389l_bool(collec416005674a_bool,hAPP_f354682389l_bool(hAPP_f1121033001l_bool(cOMBS_204124788l_bool,hAPP_f1684296317l_bool(cOMBB_120690437a_bool(fimplies),hAPP_f354682389l_bool(cOMBB_1757941347a_bool(fNot),hAPP_f256727871l_bool(hAPP_f246259945l_bool(cOMBC_1515890438l_bool,fequal291249344a_bool),A_2)))),Pa)) ) ).

tff(fact_375_insert__Collect,axiom,
    ! [A_2: pname,Pa: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),hAPP_f759274231e_bool(collect_pname,Pa)) = hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,hAPP_f1402196763l_bool(cOMBB_675860798_pname(fimplies),hAPP_f759274231e_bool(cOMBB_647938656_pname(fNot),hAPP_p61793385e_bool(hAPP_f22061361e_bool(cOMBC_1149511130e_bool,fequal_pname),A_2)))),Pa)) ) ).

tff(fact_376_insert__Collect,axiom,
    ! [A_2: hoare_2076525193iple_a,Pa: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_2),hAPP_f642208769a_bool(collec977865374iple_a,Pa)) = hAPP_f642208769a_bool(collec977865374iple_a,hAPP_f642208769a_bool(hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,hAPP_f268025233l_bool(cOMBB_345279363iple_a(fimplies),hAPP_f642208769a_bool(cOMBB_361863589iple_a(fNot),hAPP_H2077279219a_bool(hAPP_f877684549a_bool(cOMBC_1047656090a_bool,fequal1589671816iple_a),A_2)))),Pa)) ) ).

tff(fact_377_insert__Collect,axiom,
    ! [A_2: hoare_363350478_state,Pa: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_2),hAPP_f1749453397e_bool(collec223185785_state,Pa)) = hAPP_f1749453397e_bool(collec223185785_state,hAPP_f1749453397e_bool(hAPP_f1612626488e_bool(cOMBS_56025283l_bool,hAPP_f1422032445l_bool(cOMBB_877416532_state(fimplies),hAPP_f1749453397e_bool(cOMBB_2059354034_state(fNot),hAPP_H705653759e_bool(hAPP_f114072425e_bool(cOMBC_13083556e_bool,fequal1026769423_state),A_2)))),Pa)) ) ).

tff(fact_378_insert__Collect,axiom,
    ! [A_2: nat,Pa: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),hAPP_f800510211t_bool(collect_nat,Pa)) = hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(cOMBB_1015721476ol_nat(fimplies),hAPP_f800510211t_bool(cOMBB_bool_bool_nat(fNot),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),A_2)))),Pa)) ) ).

tff(fact_379_singleton__iff,axiom,
    ! [B_1: nat,A_2: nat] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,B_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),bot_bot_fun_nat_bool)))
    <=> ( B_1 = A_2 ) ) ).

tff(fact_380_singleton__iff,axiom,
    ! [B_1: fun_Ho882550911a_bool,A_2: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,B_1),hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,A_2),bot_bo1285271383l_bool)))
    <=> ( B_1 = A_2 ) ) ).

tff(fact_381_singleton__iff,axiom,
    ! [B_1: hoare_2076525193iple_a,A_2: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,B_1),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_2),bot_bo214064617a_bool)))
    <=> ( B_1 = A_2 ) ) ).

tff(fact_382_singleton__iff,axiom,
    ! [B_1: hoare_363350478_state,A_2: hoare_363350478_state] :
      ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,B_1),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_2),bot_bo39439654e_bool)))
    <=> ( B_1 = A_2 ) ) ).

tff(fact_383_singleton__iff,axiom,
    ! [B_1: pname,A_2: pname] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,B_1),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),bot_bo844097828e_bool)))
    <=> ( B_1 = A_2 ) ) ).

tff(fact_384_insert__absorb2,axiom,
    ! [X_2: nat,A_1: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),A_1)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),A_1) ) ).

tff(fact_385_insert__absorb2,axiom,
    ! [X_2: fun_Ho882550911a_bool,A_1: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,X_2),hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,X_2),A_1)) = hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,X_2),A_1) ) ).

tff(fact_386_insert__absorb2,axiom,
    ! [X_2: pname,A_1: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),A_1)) = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),A_1) ) ).

tff(fact_387_insert__absorb2,axiom,
    ! [X_2: hoare_2076525193iple_a,A_1: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_2),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_2),A_1)) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_2),A_1) ) ).

tff(fact_388_insert__absorb2,axiom,
    ! [X_2: hoare_363350478_state,A_1: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_2),A_1)) = hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_2),A_1) ) ).

tff(fact_389_insert__commute,axiom,
    ! [X_2: nat,Y_1: nat,A_1: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Y_1),A_1)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Y_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),A_1)) ) ).

tff(fact_390_insert__commute,axiom,
    ! [X_2: fun_Ho882550911a_bool,Y_1: fun_Ho882550911a_bool,A_1: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,X_2),hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,Y_1),A_1)) = hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,Y_1),hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,X_2),A_1)) ) ).

tff(fact_391_insert__commute,axiom,
    ! [X_2: pname,Y_1: pname,A_1: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,Y_1),A_1)) = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,Y_1),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),A_1)) ) ).

tff(fact_392_insert__commute,axiom,
    ! [X_2: hoare_2076525193iple_a,Y_1: hoare_2076525193iple_a,A_1: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_2),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,Y_1),A_1)) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,Y_1),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_2),A_1)) ) ).

tff(fact_393_insert__commute,axiom,
    ! [X_2: hoare_363350478_state,Y_1: hoare_363350478_state,A_1: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,Y_1),A_1)) = hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,Y_1),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_2),A_1)) ) ).

tff(fact_394_insert__iff,axiom,
    ! [A_2: nat,B_1: nat,A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,B_1),A_1)))
    <=> ( ( A_2 = B_1 )
        | hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),A_1)) ) ) ).

tff(fact_395_insert__iff,axiom,
    ! [A_2: fun_Ho882550911a_bool,B_1: fun_Ho882550911a_bool,A_1: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,A_2),hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,B_1),A_1)))
    <=> ( ( A_2 = B_1 )
        | hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,A_2),A_1)) ) ) ).

tff(fact_396_insert__iff,axiom,
    ! [A_2: hoare_2076525193iple_a,B_1: hoare_2076525193iple_a,A_1: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_2),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,B_1),A_1)))
    <=> ( ( A_2 = B_1 )
        | hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_2),A_1)) ) ) ).

tff(fact_397_insert__iff,axiom,
    ! [A_2: hoare_363350478_state,B_1: hoare_363350478_state,A_1: fun_Ho320143120e_bool] :
      ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,B_1),A_1)))
    <=> ( ( A_2 = B_1 )
        | hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_2),A_1)) ) ) ).

tff(fact_398_insert__iff,axiom,
    ! [A_2: pname,B_1: pname,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,B_1),A_1)))
    <=> ( ( A_2 = B_1 )
        | hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),A_1)) ) ) ).

tff(fact_399_Collect__empty__eq,axiom,
    ! [Pa: fun_pname_bool] :
      ( ( hAPP_f759274231e_bool(collect_pname,Pa) = bot_bo844097828e_bool )
    <=> ! [X: pname] : ~ hBOOL(hAPP_pname_bool(Pa,X)) ) ).

tff(fact_400_Collect__empty__eq,axiom,
    ! [Pa: fun_Ho882550911a_bool] :
      ( ( hAPP_f642208769a_bool(collec977865374iple_a,Pa) = bot_bo214064617a_bool )
    <=> ! [X: hoare_2076525193iple_a] : ~ hBOOL(hAPP_H454055633a_bool(Pa,X)) ) ).

tff(fact_401_Collect__empty__eq,axiom,
    ! [Pa: fun_fu2026518849l_bool] :
      ( ( hAPP_f354682389l_bool(collec416005674a_bool,Pa) = bot_bo1285271383l_bool )
    <=> ! [X: fun_Ho882550911a_bool] : ~ hBOOL(hAPP_f500854239l_bool(Pa,X)) ) ).

tff(fact_402_Collect__empty__eq,axiom,
    ! [Pa: fun_Ho320143120e_bool] :
      ( ( hAPP_f1749453397e_bool(collec223185785_state,Pa) = bot_bo39439654e_bool )
    <=> ! [X: hoare_363350478_state] : ~ hBOOL(hAPP_H1645464494e_bool(Pa,X)) ) ).

tff(fact_403_Collect__empty__eq,axiom,
    ! [Pa: fun_nat_bool] :
      ( ( hAPP_f800510211t_bool(collect_nat,Pa) = bot_bot_fun_nat_bool )
    <=> ! [X: nat] : ~ hBOOL(hAPP_nat_bool(Pa,X)) ) ).

tff(fact_404_doubleton__eq__iff,axiom,
    ! [A_2: nat,B_1: nat,C_2: nat,D: nat] :
      ( ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,B_1),bot_bot_fun_nat_bool)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,C_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,D),bot_bot_fun_nat_bool)) )
    <=> ( ( ( A_2 = C_2 )
          & ( B_1 = D ) )
        | ( ( A_2 = D )
          & ( B_1 = C_2 ) ) ) ) ).

tff(fact_405_doubleton__eq__iff,axiom,
    ! [A_2: fun_Ho882550911a_bool,B_1: fun_Ho882550911a_bool,C_2: fun_Ho882550911a_bool,D: fun_Ho882550911a_bool] :
      ( ( hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,A_2),hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,B_1),bot_bo1285271383l_bool)) = hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,C_2),hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,D),bot_bo1285271383l_bool)) )
    <=> ( ( ( A_2 = C_2 )
          & ( B_1 = D ) )
        | ( ( A_2 = D )
          & ( B_1 = C_2 ) ) ) ) ).

tff(fact_406_doubleton__eq__iff,axiom,
    ! [A_2: pname,B_1: pname,C_2: pname,D: pname] :
      ( ( hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,B_1),bot_bo844097828e_bool)) = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,C_2),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,D),bot_bo844097828e_bool)) )
    <=> ( ( ( A_2 = C_2 )
          & ( B_1 = D ) )
        | ( ( A_2 = D )
          & ( B_1 = C_2 ) ) ) ) ).

tff(fact_407_doubleton__eq__iff,axiom,
    ! [A_2: hoare_2076525193iple_a,B_1: hoare_2076525193iple_a,C_2: hoare_2076525193iple_a,D: hoare_2076525193iple_a] :
      ( ( hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_2),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,B_1),bot_bo214064617a_bool)) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,C_2),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,D),bot_bo214064617a_bool)) )
    <=> ( ( ( A_2 = C_2 )
          & ( B_1 = D ) )
        | ( ( A_2 = D )
          & ( B_1 = C_2 ) ) ) ) ).

tff(fact_408_doubleton__eq__iff,axiom,
    ! [A_2: hoare_363350478_state,B_1: hoare_363350478_state,C_2: hoare_363350478_state,D: hoare_363350478_state] :
      ( ( hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,B_1),bot_bo39439654e_bool)) = hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,C_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,D),bot_bo39439654e_bool)) )
    <=> ( ( ( A_2 = C_2 )
          & ( B_1 = D ) )
        | ( ( A_2 = D )
          & ( B_1 = C_2 ) ) ) ) ).

tff(fact_409_insert__code,axiom,
    ! [Y_1: nat,A_1: fun_nat_bool,X_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Y_1),A_1),X_2))
    <=> ( ( Y_1 = X_2 )
        | hBOOL(hAPP_nat_bool(A_1,X_2)) ) ) ).

tff(fact_410_insert__code,axiom,
    ! [Y_1: fun_Ho882550911a_bool,A_1: fun_fu2026518849l_bool,X_2: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,Y_1),A_1),X_2))
    <=> ( ( Y_1 = X_2 )
        | hBOOL(hAPP_f500854239l_bool(A_1,X_2)) ) ) ).

tff(fact_411_insert__code,axiom,
    ! [Y_1: pname,A_1: fun_pname_bool,X_2: pname] :
      ( hBOOL(hAPP_pname_bool(hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,Y_1),A_1),X_2))
    <=> ( ( Y_1 = X_2 )
        | hBOOL(hAPP_pname_bool(A_1,X_2)) ) ) ).

tff(fact_412_insert__code,axiom,
    ! [Y_1: hoare_2076525193iple_a,A_1: fun_Ho882550911a_bool,X_2: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,Y_1),A_1),X_2))
    <=> ( ( Y_1 = X_2 )
        | hBOOL(hAPP_H454055633a_bool(A_1,X_2)) ) ) ).

tff(fact_413_insert__code,axiom,
    ! [Y_1: hoare_363350478_state,A_1: fun_Ho320143120e_bool,X_2: hoare_363350478_state] :
      ( hBOOL(hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,Y_1),A_1),X_2))
    <=> ( ( Y_1 = X_2 )
        | hBOOL(hAPP_H1645464494e_bool(A_1,X_2)) ) ) ).

tff(fact_414_insert__ident,axiom,
    ! [B_2: fun_nat_bool,X_2: nat,A_1: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A_1))
     => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),B_2))
       => ( ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),A_1) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),B_2) )
        <=> ( A_1 = B_2 ) ) ) ) ).

tff(fact_415_insert__ident,axiom,
    ! [B_2: fun_fu2026518849l_bool,X_2: fun_Ho882550911a_bool,A_1: fun_fu2026518849l_bool] :
      ( ~ hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,X_2),A_1))
     => ( ~ hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,X_2),B_2))
       => ( ( hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,X_2),A_1) = hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,X_2),B_2) )
        <=> ( A_1 = B_2 ) ) ) ) ).

tff(fact_416_insert__ident,axiom,
    ! [B_2: fun_Ho882550911a_bool,X_2: hoare_2076525193iple_a,A_1: fun_Ho882550911a_bool] :
      ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_2),A_1))
     => ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_2),B_2))
       => ( ( hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_2),A_1) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_2),B_2) )
        <=> ( A_1 = B_2 ) ) ) ) ).

tff(fact_417_insert__ident,axiom,
    ! [B_2: fun_Ho320143120e_bool,X_2: hoare_363350478_state,A_1: fun_Ho320143120e_bool] :
      ( ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_2),A_1))
     => ( ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_2),B_2))
       => ( ( hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_2),A_1) = hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_2),B_2) )
        <=> ( A_1 = B_2 ) ) ) ) ).

tff(fact_418_insert__ident,axiom,
    ! [B_2: fun_pname_bool,X_2: pname,A_1: fun_pname_bool] :
      ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A_1))
     => ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),B_2))
       => ( ( hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),A_1) = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),B_2) )
        <=> ( A_1 = B_2 ) ) ) ) ).

tff(fact_419_equals0D,axiom,
    ! [A_2: nat,A_1: fun_nat_bool] :
      ( ( A_1 = bot_bot_fun_nat_bool )
     => ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),A_1)) ) ).

tff(fact_420_equals0D,axiom,
    ! [A_2: fun_Ho882550911a_bool,A_1: fun_fu2026518849l_bool] :
      ( ( A_1 = bot_bo1285271383l_bool )
     => ~ hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,A_2),A_1)) ) ).

tff(fact_421_equals0D,axiom,
    ! [A_2: hoare_2076525193iple_a,A_1: fun_Ho882550911a_bool] :
      ( ( A_1 = bot_bo214064617a_bool )
     => ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_2),A_1)) ) ).

tff(fact_422_equals0D,axiom,
    ! [A_2: hoare_363350478_state,A_1: fun_Ho320143120e_bool] :
      ( ( A_1 = bot_bo39439654e_bool )
     => ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_2),A_1)) ) ).

tff(fact_423_equals0D,axiom,
    ! [A_2: pname,A_1: fun_pname_bool] :
      ( ( A_1 = bot_bo844097828e_bool )
     => ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),A_1)) ) ).

tff(fact_424_insertI2,axiom,
    ! [B_1: nat,A_2: nat,B_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),B_2))
     => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,B_1),B_2))) ) ).

tff(fact_425_insertI2,axiom,
    ! [B_1: fun_Ho882550911a_bool,A_2: fun_Ho882550911a_bool,B_2: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,A_2),B_2))
     => hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,A_2),hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,B_1),B_2))) ) ).

tff(fact_426_insertI2,axiom,
    ! [B_1: hoare_2076525193iple_a,A_2: hoare_2076525193iple_a,B_2: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_2),B_2))
     => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_2),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,B_1),B_2))) ) ).

tff(fact_427_insertI2,axiom,
    ! [B_1: hoare_363350478_state,A_2: hoare_363350478_state,B_2: fun_Ho320143120e_bool] :
      ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_2),B_2))
     => hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_2),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,B_1),B_2))) ) ).

tff(fact_428_insertI2,axiom,
    ! [B_1: pname,A_2: pname,B_2: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),B_2))
     => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,B_1),B_2))) ) ).

tff(fact_429_insert__absorb,axiom,
    ! [A_2: nat,A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),A_1))
     => ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),A_1) = A_1 ) ) ).

tff(fact_430_insert__absorb,axiom,
    ! [A_2: fun_Ho882550911a_bool,A_1: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,A_2),A_1))
     => ( hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,A_2),A_1) = A_1 ) ) ).

tff(fact_431_insert__absorb,axiom,
    ! [A_2: hoare_2076525193iple_a,A_1: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_2),A_1))
     => ( hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_2),A_1) = A_1 ) ) ).

tff(fact_432_insert__absorb,axiom,
    ! [A_2: hoare_363350478_state,A_1: fun_Ho320143120e_bool] :
      ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_2),A_1))
     => ( hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_2),A_1) = A_1 ) ) ).

tff(fact_433_insert__absorb,axiom,
    ! [A_2: pname,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),A_1))
     => ( hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),A_1) = A_1 ) ) ).

tff(fact_434_singletonE,axiom,
    ! [B_1: nat,A_2: nat] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,B_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),bot_bot_fun_nat_bool)))
     => ( B_1 = A_2 ) ) ).

tff(fact_435_singletonE,axiom,
    ! [B_1: fun_Ho882550911a_bool,A_2: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,B_1),hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,A_2),bot_bo1285271383l_bool)))
     => ( B_1 = A_2 ) ) ).

tff(fact_436_singletonE,axiom,
    ! [B_1: hoare_2076525193iple_a,A_2: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,B_1),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_2),bot_bo214064617a_bool)))
     => ( B_1 = A_2 ) ) ).

tff(fact_437_singletonE,axiom,
    ! [B_1: hoare_363350478_state,A_2: hoare_363350478_state] :
      ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,B_1),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_2),bot_bo39439654e_bool)))
     => ( B_1 = A_2 ) ) ).

tff(fact_438_singletonE,axiom,
    ! [B_1: pname,A_2: pname] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,B_1),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),bot_bo844097828e_bool)))
     => ( B_1 = A_2 ) ) ).

tff(fact_439_singleton__inject,axiom,
    ! [A_2: nat,B_1: nat] :
      ( ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),bot_bot_fun_nat_bool) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,B_1),bot_bot_fun_nat_bool) )
     => ( A_2 = B_1 ) ) ).

tff(fact_440_singleton__inject,axiom,
    ! [A_2: fun_Ho882550911a_bool,B_1: fun_Ho882550911a_bool] :
      ( ( hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,A_2),bot_bo1285271383l_bool) = hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,B_1),bot_bo1285271383l_bool) )
     => ( A_2 = B_1 ) ) ).

tff(fact_441_singleton__inject,axiom,
    ! [A_2: pname,B_1: pname] :
      ( ( hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),bot_bo844097828e_bool) = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,B_1),bot_bo844097828e_bool) )
     => ( A_2 = B_1 ) ) ).

tff(fact_442_singleton__inject,axiom,
    ! [A_2: hoare_2076525193iple_a,B_1: hoare_2076525193iple_a] :
      ( ( hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_2),bot_bo214064617a_bool) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,B_1),bot_bo214064617a_bool) )
     => ( A_2 = B_1 ) ) ).

tff(fact_443_singleton__inject,axiom,
    ! [A_2: hoare_363350478_state,B_1: hoare_363350478_state] :
      ( ( hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_2),bot_bo39439654e_bool) = hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,B_1),bot_bo39439654e_bool) )
     => ( A_2 = B_1 ) ) ).

tff(fact_444_com__det,axiom,
    ! [U_1: state,C: com,S: state,T: state] :
      ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C),S),T))
     => ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C),S),U_1))
       => ( U_1 = T ) ) ) ).

tff(fact_445_insert__is__Un,axiom,
    ! [A_2: nat,A_1: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),A_1) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),bot_bot_fun_nat_bool)),A_1) ) ).

tff(fact_446_insert__is__Un,axiom,
    ! [A_2: fun_Ho882550911a_bool,A_1: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,A_2),A_1) = hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,A_2),bot_bo1285271383l_bool)),A_1) ) ).

tff(fact_447_insert__is__Un,axiom,
    ! [A_2: pname,A_1: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),A_1) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),bot_bo844097828e_bool)),A_1) ) ).

tff(fact_448_insert__is__Un,axiom,
    ! [A_2: hoare_363350478_state,A_1: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_2),A_1) = hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_2),bot_bo39439654e_bool)),A_1) ) ).

tff(fact_449_insert__is__Un,axiom,
    ! [A_2: hoare_2076525193iple_a,A_1: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_2),A_1) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_2),bot_bo214064617a_bool)),A_1) ) ).

tff(fact_450_insert__compr__raw,axiom,
    ! [X: fun_Ho882550911a_bool,Xa: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,X),Xa) = hAPP_f354682389l_bool(collec416005674a_bool,hAPP_f354682389l_bool(hAPP_f1121033001l_bool(cOMBS_204124788l_bool,hAPP_f1684296317l_bool(cOMBB_120690437a_bool(fdisj),hAPP_f256727871l_bool(hAPP_f246259945l_bool(cOMBC_1515890438l_bool,fequal291249344a_bool),X))),hAPP_f354682389l_bool(hAPP_f638520823l_bool(cOMBC_1528738210l_bool,member186597612a_bool),Xa))) ) ).

tff(fact_451_insert__compr__raw,axiom,
    ! [X: hoare_2076525193iple_a,Xa: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X),Xa) = hAPP_f642208769a_bool(collec977865374iple_a,hAPP_f642208769a_bool(hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,hAPP_f268025233l_bool(cOMBB_345279363iple_a(fdisj),hAPP_H2077279219a_bool(hAPP_f877684549a_bool(cOMBC_1047656090a_bool,fequal1589671816iple_a),X))),hAPP_f642208769a_bool(hAPP_f325481667a_bool(cOMBC_1529670358l_bool,member276146780iple_a),Xa))) ) ).

tff(fact_452_insert__compr__raw,axiom,
    ! [X: hoare_363350478_state,Xa: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X),Xa) = hAPP_f1749453397e_bool(collec223185785_state,hAPP_f1749453397e_bool(hAPP_f1612626488e_bool(cOMBS_56025283l_bool,hAPP_f1422032445l_bool(cOMBB_877416532_state(fdisj),hAPP_H705653759e_bool(hAPP_f114072425e_bool(cOMBC_13083556e_bool,fequal1026769423_state),X))),hAPP_f1749453397e_bool(hAPP_f1072682137e_bool(cOMBC_1415895650l_bool,member1253905979_state),Xa))) ) ).

tff(fact_453_insert__compr__raw,axiom,
    ! [X: nat,Xa: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X),Xa) = hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(cOMBB_1015721476ol_nat(fdisj),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),X))),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),Xa))) ) ).

tff(fact_454_insert__compr__raw,axiom,
    ! [X: pname,Xa: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X),Xa) = hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,hAPP_f1402196763l_bool(cOMBB_675860798_pname(fdisj),hAPP_p61793385e_bool(hAPP_f22061361e_bool(cOMBC_1149511130e_bool,fequal_pname),X))),hAPP_f759274231e_bool(hAPP_f661147897e_bool(cOMBC_1058051404l_bool,member_pname),Xa))) ) ).

tff(fact_455_derivs__insertD,axiom,
    ! [Ga: fun_Ho882550911a_bool,T_1: hoare_2076525193iple_a,Ts: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,T_1),Ts)))
     => ( hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,T_1),bot_bo214064617a_bool)))
        & hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),Ts)) ) ) ).

tff(fact_456_derivs__insertD,axiom,
    ! [Ga: fun_Ho320143120e_bool,T_1: hoare_363350478_state,Ts: fun_Ho320143120e_bool] :
      ( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,T_1),Ts)))
     => ( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,T_1),bot_bo39439654e_bool)))
        & hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),Ts)) ) ) ).

tff(fact_457_hoare__derivs_Oinsert,axiom,
    ! [Ts: fun_Ho882550911a_bool,Ga: fun_Ho882550911a_bool,T_1: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,T_1),bot_bo214064617a_bool)))
     => ( hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),Ts))
       => hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,T_1),Ts))) ) ) ).

tff(fact_458_hoare__derivs_Oinsert,axiom,
    ! [Ts: fun_Ho320143120e_bool,Ga: fun_Ho320143120e_bool,T_1: hoare_363350478_state] :
      ( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,T_1),bot_bo39439654e_bool)))
     => ( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),Ts))
       => hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,T_1),Ts))) ) ) ).

tff(fact_459_image__constant__conv,axiom,
    ! [C_2: nat,A_1: fun_nat_bool] :
      ( ( ( A_1 = bot_bot_fun_nat_bool )
       => ( hAPP_f800510211t_bool(image_nat_nat(hAPP_nat_fun_nat_nat(cOMBK_nat_nat,C_2)),A_1) = bot_bot_fun_nat_bool ) )
      & ( ( A_1 != bot_bot_fun_nat_bool )
       => ( hAPP_f800510211t_bool(image_nat_nat(hAPP_nat_fun_nat_nat(cOMBK_nat_nat,C_2)),A_1) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,C_2),bot_bot_fun_nat_bool) ) ) ) ).

tff(fact_460_image__constant__conv,axiom,
    ! [C_2: hoare_363350478_state,A_1: fun_pname_bool] :
      ( ( ( A_1 = bot_bo844097828e_bool )
       => ( hAPP_f1763243027e_bool(image_1918575693_state(hAPP_H1607320188_state(cOMBK_92056634_pname,C_2)),A_1) = bot_bo39439654e_bool ) )
      & ( ( A_1 != bot_bo844097828e_bool )
       => ( hAPP_f1763243027e_bool(image_1918575693_state(hAPP_H1607320188_state(cOMBK_92056634_pname,C_2)),A_1) = hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,C_2),bot_bo39439654e_bool) ) ) ) ).

tff(fact_461_image__constant__conv,axiom,
    ! [C_2: hoare_2076525193iple_a,A_1: fun_pname_bool] :
      ( ( ( A_1 = bot_bo844097828e_bool )
       => ( hAPP_f1451199292a_bool(image_217098954iple_a(hAPP_H1272595598iple_a(cOMBK_159560841_pname,C_2)),A_1) = bot_bo214064617a_bool ) )
      & ( ( A_1 != bot_bo844097828e_bool )
       => ( hAPP_f1451199292a_bool(image_217098954iple_a(hAPP_H1272595598iple_a(cOMBK_159560841_pname,C_2)),A_1) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,C_2),bot_bo214064617a_bool) ) ) ) ).

tff(fact_462_image__constant,axiom,
    ! [C_2: nat,X_2: nat,A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A_1))
     => ( hAPP_f800510211t_bool(image_nat_nat(hAPP_nat_fun_nat_nat(cOMBK_nat_nat,C_2)),A_1) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,C_2),bot_bot_fun_nat_bool) ) ) ).

tff(fact_463_image__constant,axiom,
    ! [C_2: hoare_363350478_state,X_2: pname,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A_1))
     => ( hAPP_f1763243027e_bool(image_1918575693_state(hAPP_H1607320188_state(cOMBK_92056634_pname,C_2)),A_1) = hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,C_2),bot_bo39439654e_bool) ) ) ).

tff(fact_464_image__constant,axiom,
    ! [C_2: nat,X_2: pname,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A_1))
     => ( hAPP_f1066163005t_bool(image_pname_nat(hAPP_n362732366me_nat(cOMBK_nat_pname,C_2)),A_1) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,C_2),bot_bot_fun_nat_bool) ) ) ).

tff(fact_465_image__constant,axiom,
    ! [C_2: fun_Ho882550911a_bool,X_2: pname,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A_1))
     => ( hAPP_f1243652676l_bool(image_1433143038a_bool(hAPP_f406822750a_bool(cOMBK_579861065_pname,C_2)),A_1) = hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,C_2),bot_bo1285271383l_bool) ) ) ).

tff(fact_466_image__constant,axiom,
    ! [C_2: pname,X_2: pname,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A_1))
     => ( hAPP_f759274231e_bool(image_pname_pname(hAPP_p1170154830_pname(cOMBK_pname_pname,C_2)),A_1) = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,C_2),bot_bo844097828e_bool) ) ) ).

tff(fact_467_image__constant,axiom,
    ! [C_2: hoare_2076525193iple_a,X_2: pname,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A_1))
     => ( hAPP_f1451199292a_bool(image_217098954iple_a(hAPP_H1272595598iple_a(cOMBK_159560841_pname,C_2)),A_1) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,C_2),bot_bo214064617a_bool) ) ) ).

tff(fact_468_image__insert,axiom,
    ! [F: fun_nat_nat,A_2: nat,B_2: fun_nat_bool] : ( hAPP_f800510211t_bool(image_nat_nat(F),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),B_2)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,hAPP_nat_nat(F,A_2)),hAPP_f800510211t_bool(image_nat_nat(F),B_2)) ) ).

tff(fact_469_image__insert,axiom,
    ! [F: fun_pn1262351275_state,A_2: pname,B_2: fun_pname_bool] : ( hAPP_f1763243027e_bool(image_1918575693_state(F),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),B_2)) = hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hAPP_p1337579261_state(F,A_2)),hAPP_f1763243027e_bool(image_1918575693_state(F),B_2)) ) ).

tff(fact_470_image__insert,axiom,
    ! [F: fun_pn706052866iple_a,A_2: pname,B_2: fun_pname_bool] : ( hAPP_f1451199292a_bool(image_217098954iple_a(F),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),B_2)) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hAPP_p973116442iple_a(F,A_2)),hAPP_f1451199292a_bool(image_217098954iple_a(F),B_2)) ) ).

tff(fact_471_insert__image,axiom,
    ! [F: fun_nat_nat,X_2: nat,A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A_1))
     => ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,hAPP_nat_nat(F,X_2)),hAPP_f800510211t_bool(image_nat_nat(F),A_1)) = hAPP_f800510211t_bool(image_nat_nat(F),A_1) ) ) ).

tff(fact_472_insert__image,axiom,
    ! [F: fun_pn1262351275_state,X_2: pname,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A_1))
     => ( hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hAPP_p1337579261_state(F,X_2)),hAPP_f1763243027e_bool(image_1918575693_state(F),A_1)) = hAPP_f1763243027e_bool(image_1918575693_state(F),A_1) ) ) ).

tff(fact_473_insert__image,axiom,
    ! [F: fun_pname_nat,X_2: pname,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A_1))
     => ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,hAPP_pname_nat(F,X_2)),hAPP_f1066163005t_bool(image_pname_nat(F),A_1)) = hAPP_f1066163005t_bool(image_pname_nat(F),A_1) ) ) ).

tff(fact_474_insert__image,axiom,
    ! [F: fun_pn1811694044a_bool,X_2: pname,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A_1))
     => ( hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,hAPP_p1160247726a_bool(F,X_2)),hAPP_f1243652676l_bool(image_1433143038a_bool(F),A_1)) = hAPP_f1243652676l_bool(image_1433143038a_bool(F),A_1) ) ) ).

tff(fact_475_insert__image,axiom,
    ! [F: fun_pname_pname,X_2: pname,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A_1))
     => ( hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,hAPP_pname_pname(F,X_2)),hAPP_f759274231e_bool(image_pname_pname(F),A_1)) = hAPP_f759274231e_bool(image_pname_pname(F),A_1) ) ) ).

tff(fact_476_insert__image,axiom,
    ! [F: fun_pn706052866iple_a,X_2: pname,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A_1))
     => ( hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hAPP_p973116442iple_a(F,X_2)),hAPP_f1451199292a_bool(image_217098954iple_a(F),A_1)) = hAPP_f1451199292a_bool(image_217098954iple_a(F),A_1) ) ) ).

tff(fact_477_Un__insert__right,axiom,
    ! [A_1: fun_nat_bool,A_2: nat,B_2: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),B_2)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_1),B_2)) ) ).

tff(fact_478_Un__insert__right,axiom,
    ! [A_1: fun_fu2026518849l_bool,A_2: fun_Ho882550911a_bool,B_2: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_1),hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,A_2),B_2)) = hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,A_2),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_1),B_2)) ) ).

tff(fact_479_Un__insert__right,axiom,
    ! [A_1: fun_pname_bool,A_2: pname,B_2: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),B_2)) = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),B_2)) ) ).

tff(fact_480_Un__insert__right,axiom,
    ! [A_1: fun_Ho320143120e_bool,A_2: hoare_363350478_state,B_2: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_1),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_2),B_2)) = hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_2),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_1),B_2)) ) ).

tff(fact_481_Un__insert__right,axiom,
    ! [A_1: fun_Ho882550911a_bool,A_2: hoare_2076525193iple_a,B_2: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_1),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_2),B_2)) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_2),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_1),B_2)) ) ).

tff(fact_482_Un__insert__left,axiom,
    ! [A_2: nat,B_2: fun_nat_bool,C_1: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),B_2)),C_1) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,B_2),C_1)) ) ).

tff(fact_483_Un__insert__left,axiom,
    ! [A_2: fun_Ho882550911a_bool,B_2: fun_fu2026518849l_bool,C_1: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,A_2),B_2)),C_1) = hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,A_2),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,B_2),C_1)) ) ).

tff(fact_484_Un__insert__left,axiom,
    ! [A_2: pname,B_2: fun_pname_bool,C_1: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),B_2)),C_1) = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,B_2),C_1)) ) ).

tff(fact_485_Un__insert__left,axiom,
    ! [A_2: hoare_363350478_state,B_2: fun_Ho320143120e_bool,C_1: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_2),B_2)),C_1) = hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_2),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,B_2),C_1)) ) ).

tff(fact_486_Un__insert__left,axiom,
    ! [A_2: hoare_2076525193iple_a,B_2: fun_Ho882550911a_bool,C_1: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_2),B_2)),C_1) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_2),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,B_2),C_1)) ) ).

tff(fact_487_empty__is__image,axiom,
    ! [F: fun_nat_nat,A_1: fun_nat_bool] :
      ( ( bot_bot_fun_nat_bool = hAPP_f800510211t_bool(image_nat_nat(F),A_1) )
    <=> ( A_1 = bot_bot_fun_nat_bool ) ) ).

tff(fact_488_empty__is__image,axiom,
    ! [F: fun_pn1262351275_state,A_1: fun_pname_bool] :
      ( ( bot_bo39439654e_bool = hAPP_f1763243027e_bool(image_1918575693_state(F),A_1) )
    <=> ( A_1 = bot_bo844097828e_bool ) ) ).

tff(fact_489_empty__is__image,axiom,
    ! [F: fun_pn706052866iple_a,A_1: fun_pname_bool] :
      ( ( bot_bo214064617a_bool = hAPP_f1451199292a_bool(image_217098954iple_a(F),A_1) )
    <=> ( A_1 = bot_bo844097828e_bool ) ) ).

tff(fact_490_image__empty,axiom,
    ! [F: fun_nat_nat] : ( hAPP_f800510211t_bool(image_nat_nat(F),bot_bot_fun_nat_bool) = bot_bot_fun_nat_bool ) ).

tff(fact_491_image__empty,axiom,
    ! [F: fun_pn1262351275_state] : ( hAPP_f1763243027e_bool(image_1918575693_state(F),bot_bo844097828e_bool) = bot_bo39439654e_bool ) ).

tff(fact_492_image__empty,axiom,
    ! [F: fun_pn706052866iple_a] : ( hAPP_f1451199292a_bool(image_217098954iple_a(F),bot_bo844097828e_bool) = bot_bo214064617a_bool ) ).

tff(fact_493_image__is__empty,axiom,
    ! [F: fun_nat_nat,A_1: fun_nat_bool] :
      ( ( hAPP_f800510211t_bool(image_nat_nat(F),A_1) = bot_bot_fun_nat_bool )
    <=> ( A_1 = bot_bot_fun_nat_bool ) ) ).

tff(fact_494_image__is__empty,axiom,
    ! [F: fun_pn1262351275_state,A_1: fun_pname_bool] :
      ( ( hAPP_f1763243027e_bool(image_1918575693_state(F),A_1) = bot_bo39439654e_bool )
    <=> ( A_1 = bot_bo844097828e_bool ) ) ).

tff(fact_495_image__is__empty,axiom,
    ! [F: fun_pn706052866iple_a,A_1: fun_pname_bool] :
      ( ( hAPP_f1451199292a_bool(image_217098954iple_a(F),A_1) = bot_bo214064617a_bool )
    <=> ( A_1 = bot_bo844097828e_bool ) ) ).

tff(fact_496_ball__empty,axiom,
    ! [Pa: fun_nat_bool,X: nat] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),bot_bot_fun_nat_bool))
     => hBOOL(hAPP_nat_bool(Pa,X)) ) ).

tff(fact_497_ball__empty,axiom,
    ! [Pa: fun_Ho882550911a_bool,X: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X),bot_bo214064617a_bool))
     => hBOOL(hAPP_H454055633a_bool(Pa,X)) ) ).

tff(fact_498_ball__empty,axiom,
    ! [Pa: fun_fu2026518849l_bool,X: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,X),bot_bo1285271383l_bool))
     => hBOOL(hAPP_f500854239l_bool(Pa,X)) ) ).

tff(fact_499_ball__empty,axiom,
    ! [Pa: fun_pname_bool,X: pname] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),bot_bo844097828e_bool))
     => hBOOL(hAPP_pname_bool(Pa,X)) ) ).

tff(fact_500_ball__empty,axiom,
    ! [Pa: fun_Ho320143120e_bool,X: hoare_363350478_state] :
      ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X),bot_bo39439654e_bool))
     => hBOOL(hAPP_H1645464494e_bool(Pa,X)) ) ).

tff(fact_501_Un__empty__left,axiom,
    ! [B_2: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,bot_bot_fun_nat_bool),B_2) = B_2 ) ).

tff(fact_502_Un__empty__left,axiom,
    ! [B_2: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,bot_bo1285271383l_bool),B_2) = B_2 ) ).

tff(fact_503_Un__empty__left,axiom,
    ! [B_2: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,bot_bo844097828e_bool),B_2) = B_2 ) ).

tff(fact_504_Un__empty__left,axiom,
    ! [B_2: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,bot_bo39439654e_bool),B_2) = B_2 ) ).

tff(fact_505_Un__empty__left,axiom,
    ! [B_2: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,bot_bo214064617a_bool),B_2) = B_2 ) ).

tff(fact_506_Un__empty__right,axiom,
    ! [A_1: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_1),bot_bot_fun_nat_bool) = A_1 ) ).

tff(fact_507_Un__empty__right,axiom,
    ! [A_1: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_1),bot_bo1285271383l_bool) = A_1 ) ).

tff(fact_508_Un__empty__right,axiom,
    ! [A_1: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),bot_bo844097828e_bool) = A_1 ) ).

tff(fact_509_Un__empty__right,axiom,
    ! [A_1: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_1),bot_bo39439654e_bool) = A_1 ) ).

tff(fact_510_Un__empty__right,axiom,
    ! [A_1: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_1),bot_bo214064617a_bool) = A_1 ) ).

tff(fact_511_Un__empty,axiom,
    ! [A_1: fun_nat_bool,B_2: fun_nat_bool] :
      ( ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_1),B_2) = bot_bot_fun_nat_bool )
    <=> ( ( A_1 = bot_bot_fun_nat_bool )
        & ( B_2 = bot_bot_fun_nat_bool ) ) ) ).

tff(fact_512_Un__empty,axiom,
    ! [A_1: fun_fu2026518849l_bool,B_2: fun_fu2026518849l_bool] :
      ( ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_1),B_2) = bot_bo1285271383l_bool )
    <=> ( ( A_1 = bot_bo1285271383l_bool )
        & ( B_2 = bot_bo1285271383l_bool ) ) ) ).

tff(fact_513_Un__empty,axiom,
    ! [A_1: fun_pname_bool,B_2: fun_pname_bool] :
      ( ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),B_2) = bot_bo844097828e_bool )
    <=> ( ( A_1 = bot_bo844097828e_bool )
        & ( B_2 = bot_bo844097828e_bool ) ) ) ).

tff(fact_514_Un__empty,axiom,
    ! [A_1: fun_Ho320143120e_bool,B_2: fun_Ho320143120e_bool] :
      ( ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_1),B_2) = bot_bo39439654e_bool )
    <=> ( ( A_1 = bot_bo39439654e_bool )
        & ( B_2 = bot_bo39439654e_bool ) ) ) ).

tff(fact_515_Un__empty,axiom,
    ! [A_1: fun_Ho882550911a_bool,B_2: fun_Ho882550911a_bool] :
      ( ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_1),B_2) = bot_bo214064617a_bool )
    <=> ( ( A_1 = bot_bo214064617a_bool )
        & ( B_2 = bot_bo214064617a_bool ) ) ) ).

tff(fact_516_constant,axiom,
    ! [Ga: fun_Ho882550911a_bool,Pa: fun_a_fun_state_bool,C_2: com,Qa: fun_a_fun_state_bool,C_1: bool] :
      ( ( hBOOL(C_1)
       => hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,Pa),C_2),Qa)),bot_bo214064617a_bool))) )
     => hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,hAPP_b540892988e_bool(hAPP_f1824947087e_bool(cOMBC_41962815e_bool,hAPP_f340725611e_bool(cOMBB_1348041619bool_a(cOMBC_231445413l_bool),hAPP_f1509969235l_bool(cOMBB_1355796797bool_a(cOMBB_188601460_state(fconj)),Pa))),C_1)),C_2),Qa)),bot_bo214064617a_bool))) ) ).

tff(fact_517_constant,axiom,
    ! [Ga: fun_Ho320143120e_bool,Pa: fun_st1506752259e_bool,C_2: com,Qa: fun_st1506752259e_bool,C_1: bool] :
      ( ( hBOOL(C_1)
       => hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,Pa),C_2),Qa)),bot_bo39439654e_bool))) )
     => hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,hAPP_b1095269219e_bool(hAPP_f701449317e_bool(cOMBC_471052088e_bool,hAPP_f873506917e_bool(cOMBB_1757942702_state(cOMBC_231445413l_bool),hAPP_f1283379615l_bool(cOMBB_1759179140_state(cOMBB_188601460_state(fconj)),Pa))),C_1)),C_2),Qa)),bot_bo39439654e_bool))) ) ).

tff(fact_518_empty,axiom,
    ! [Ga: fun_Ho882550911a_bool] : hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),bot_bo214064617a_bool)) ).

tff(fact_519_empty,axiom,
    ! [Ga: fun_Ho320143120e_bool] : hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),bot_bo39439654e_bool)) ).

tff(fact_520_sup__bot__left,axiom,
    ! [X_2: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,bot_bot_fun_nat_bool),X_2) = X_2 ) ).

tff(fact_521_sup__bot__left,axiom,
    ! [X_2: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,bot_bo1285271383l_bool),X_2) = X_2 ) ).

tff(fact_522_sup__bot__left,axiom,
    ! [X_2: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,bot_bo844097828e_bool),X_2) = X_2 ) ).

tff(fact_523_sup__bot__left,axiom,
    ! [X_2: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,bot_bo39439654e_bool),X_2) = X_2 ) ).

tff(fact_524_sup__bot__left,axiom,
    ! [X_2: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,bot_bot_bool),X_2))
    <=> hBOOL(X_2) ) ).

tff(fact_525_sup__bot__left,axiom,
    ! [X_2: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,bot_bo214064617a_bool),X_2) = X_2 ) ).

tff(fact_526_sup__bot__right,axiom,
    ! [X_2: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,X_2),bot_bot_fun_nat_bool) = X_2 ) ).

tff(fact_527_sup__bot__right,axiom,
    ! [X_2: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,X_2),bot_bo1285271383l_bool) = X_2 ) ).

tff(fact_528_sup__bot__right,axiom,
    ! [X_2: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),bot_bo844097828e_bool) = X_2 ) ).

tff(fact_529_sup__bot__right,axiom,
    ! [X_2: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,X_2),bot_bo39439654e_bool) = X_2 ) ).

tff(fact_530_sup__bot__right,axiom,
    ! [X_2: bool] :
      ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),bot_bot_bool))
    <=> hBOOL(X_2) ) ).

tff(fact_531_sup__bot__right,axiom,
    ! [X_2: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,X_2),bot_bo214064617a_bool) = X_2 ) ).

tff(fact_532_sup__eq__bot__iff,axiom,
    ! [X_2: fun_nat_bool,Y_1: fun_nat_bool] :
      ( ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,X_2),Y_1) = bot_bot_fun_nat_bool )
    <=> ( ( X_2 = bot_bot_fun_nat_bool )
        & ( Y_1 = bot_bot_fun_nat_bool ) ) ) ).

tff(fact_533_sup__eq__bot__iff,axiom,
    ! [X_2: fun_fu2026518849l_bool,Y_1: fun_fu2026518849l_bool] :
      ( ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,X_2),Y_1) = bot_bo1285271383l_bool )
    <=> ( ( X_2 = bot_bo1285271383l_bool )
        & ( Y_1 = bot_bo1285271383l_bool ) ) ) ).

tff(fact_534_sup__eq__bot__iff,axiom,
    ! [X_2: fun_pname_bool,Y_1: fun_pname_bool] :
      ( ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),Y_1) = bot_bo844097828e_bool )
    <=> ( ( X_2 = bot_bo844097828e_bool )
        & ( Y_1 = bot_bo844097828e_bool ) ) ) ).

tff(fact_535_sup__eq__bot__iff,axiom,
    ! [X_2: fun_Ho320143120e_bool,Y_1: fun_Ho320143120e_bool] :
      ( ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,X_2),Y_1) = bot_bo39439654e_bool )
    <=> ( ( X_2 = bot_bo39439654e_bool )
        & ( Y_1 = bot_bo39439654e_bool ) ) ) ).

tff(fact_536_sup__eq__bot__iff,axiom,
    ! [X_2: bool,Y_1: bool] :
      ( ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),Y_1))
      <=> hBOOL(bot_bot_bool) )
    <=> ( ( hBOOL(X_2)
        <=> hBOOL(bot_bot_bool) )
        & ( hBOOL(Y_1)
        <=> hBOOL(bot_bot_bool) ) ) ) ).

tff(fact_537_sup__eq__bot__iff,axiom,
    ! [X_2: fun_Ho882550911a_bool,Y_1: fun_Ho882550911a_bool] :
      ( ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,X_2),Y_1) = bot_bo214064617a_bool )
    <=> ( ( X_2 = bot_bo214064617a_bool )
        & ( Y_1 = bot_bo214064617a_bool ) ) ) ).

tff(fact_538_triple__valid__Suc,axiom,
    ! [N_4: nat,T_2: hoare_363350478_state] :
      ( hBOOL(hAPP_H1645464494e_bool(hAPP_n753050587e_bool(hoare_28808878_state,hAPP_nat_nat(suc,N_4)),T_2))
     => hBOOL(hAPP_H1645464494e_bool(hAPP_n753050587e_bool(hoare_28808878_state,N_4),T_2)) ) ).

tff(fact_539_triple__valid__Suc,axiom,
    ! [N_4: nat,T_2: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_H454055633a_bool(hAPP_n1335981428a_bool(hoare_526497939alid_a,hAPP_nat_nat(suc,N_4)),T_2))
     => hBOOL(hAPP_H454055633a_bool(hAPP_n1335981428a_bool(hoare_526497939alid_a,N_4),T_2)) ) ).

tff(fact_540_insert__def,axiom,
    ! [A_2: fun_Ho882550911a_bool,B_2: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,A_2),B_2) = hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,hAPP_f354682389l_bool(collec416005674a_bool,hAPP_f256727871l_bool(hAPP_f246259945l_bool(cOMBC_1515890438l_bool,fequal291249344a_bool),A_2))),B_2) ) ).

tff(fact_541_insert__def,axiom,
    ! [A_2: pname,B_2: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),B_2) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f759274231e_bool(collect_pname,hAPP_p61793385e_bool(hAPP_f22061361e_bool(cOMBC_1149511130e_bool,fequal_pname),A_2))),B_2) ) ).

tff(fact_542_insert__def,axiom,
    ! [A_2: hoare_363350478_state,B_2: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_2),B_2) = hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,hAPP_f1749453397e_bool(collec223185785_state,hAPP_H705653759e_bool(hAPP_f114072425e_bool(cOMBC_13083556e_bool,fequal1026769423_state),A_2))),B_2) ) ).

tff(fact_543_insert__def,axiom,
    ! [A_2: hoare_2076525193iple_a,B_2: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_2),B_2) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,hAPP_f642208769a_bool(collec977865374iple_a,hAPP_H2077279219a_bool(hAPP_f877684549a_bool(cOMBC_1047656090a_bool,fequal1589671816iple_a),A_2))),B_2) ) ).

tff(fact_544_insert__def,axiom,
    ! [A_2: nat,B_2: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),B_2) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,hAPP_f800510211t_bool(collect_nat,hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,fequal_nat),A_2))),B_2) ) ).

tff(fact_545_weak__Body,axiom,
    ! [Ga: fun_Ho882550911a_bool,Pa: fun_a_fun_state_bool,Pn_1: pname,Qa: fun_a_fun_state_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,Pa),hAPP_option_com_com(the_com,hAPP_p799580910on_com(body_1,Pn_1))),Qa)),bot_bo214064617a_bool)))
     => hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,Pa),hAPP_pname_com(body,Pn_1)),Qa)),bot_bo214064617a_bool))) ) ).

tff(fact_546_weak__Body,axiom,
    ! [Ga: fun_Ho320143120e_bool,Pa: fun_st1506752259e_bool,Pn_1: pname,Qa: fun_st1506752259e_bool] :
      ( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,Pa),hAPP_option_com_com(the_com,hAPP_p799580910on_com(body_1,Pn_1))),Qa)),bot_bo39439654e_bool)))
     => hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,Pa),hAPP_pname_com(body,Pn_1)),Qa)),bot_bo39439654e_bool))) ) ).

tff(fact_547_BodyN,axiom,
    ! [Pa: fun_a_fun_state_bool,Pn_1: pname,Qa: fun_a_fun_state_bool,Ga: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,Pa),hAPP_pname_com(body,Pn_1)),Qa)),Ga)),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,Pa),hAPP_option_com_com(the_com,hAPP_p799580910on_com(body_1,Pn_1))),Qa)),bot_bo214064617a_bool)))
     => hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,Pa),hAPP_pname_com(body,Pn_1)),Qa)),bot_bo214064617a_bool))) ) ).

tff(fact_548_BodyN,axiom,
    ! [Pa: fun_st1506752259e_bool,Pn_1: pname,Qa: fun_st1506752259e_bool,Ga: fun_Ho320143120e_bool] :
      ( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,Pa),hAPP_pname_com(body,Pn_1)),Qa)),Ga)),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,Pa),hAPP_option_com_com(the_com,hAPP_p799580910on_com(body_1,Pn_1))),Qa)),bot_bo39439654e_bool)))
     => hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,Pa),hAPP_pname_com(body,Pn_1)),Qa)),bot_bo39439654e_bool))) ) ).

tff(fact_549_escape,axiom,
    ! [Ga: fun_Ho882550911a_bool,C_2: com,Qa: fun_a_fun_state_bool,Pa: fun_a_fun_state_bool] :
      ( ! [Z_4: x_a,S_3: state] :
          ( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Pa,Z_4),S_3))
         => hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,hAPP_f762886889e_bool(cOMBK_1458035955bool_a,hAPP_s1806633685e_bool(hAPP_f817621513e_bool(cOMBC_2027030106e_bool,fequal_state),S_3))),C_2),hAPP_f762886889e_bool(cOMBK_1458035955bool_a,hAPP_a2036067514e_bool(Qa,Z_4)))),bot_bo214064617a_bool))) )
     => hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,Pa),C_2),Qa)),bot_bo214064617a_bool))) ) ).

tff(fact_550_escape,axiom,
    ! [Ga: fun_Ho320143120e_bool,C_2: com,Qa: fun_st1506752259e_bool,Pa: fun_st1506752259e_bool] :
      ( ! [Z_4: state,S_3: state] :
          ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Pa,Z_4),S_3))
         => hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,hAPP_f1863945078e_bool(cOMBK_631994958_state,hAPP_s1806633685e_bool(hAPP_f817621513e_bool(cOMBC_2027030106e_bool,fequal_state),S_3))),C_2),hAPP_f1863945078e_bool(cOMBK_631994958_state,hAPP_s1806633685e_bool(Qa,Z_4)))),bot_bo39439654e_bool))) )
     => hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,Pa),C_2),Qa)),bot_bo39439654e_bool))) ) ).

tff(fact_551_conseq1,axiom,
    ! [Pa: fun_a_fun_state_bool,Ga: fun_Ho882550911a_bool,P_1: fun_a_fun_state_bool,C_2: com,Qa: fun_a_fun_state_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,P_1),C_2),Qa)),bot_bo214064617a_bool)))
     => ( ! [Z_4: x_a,S_3: state] :
            ( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Pa,Z_4),S_3))
           => hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(P_1,Z_4),S_3)) )
       => hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,Pa),C_2),Qa)),bot_bo214064617a_bool))) ) ) ).

tff(fact_552_conseq1,axiom,
    ! [Pa: fun_st1506752259e_bool,Ga: fun_Ho320143120e_bool,P_1: fun_st1506752259e_bool,C_2: com,Qa: fun_st1506752259e_bool] :
      ( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,P_1),C_2),Qa)),bot_bo39439654e_bool)))
     => ( ! [Z_4: state,S_3: state] :
            ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Pa,Z_4),S_3))
           => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(P_1,Z_4),S_3)) )
       => hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,Pa),C_2),Qa)),bot_bo39439654e_bool))) ) ) ).

tff(fact_553_conseq2,axiom,
    ! [Qa: fun_a_fun_state_bool,Ga: fun_Ho882550911a_bool,Pa: fun_a_fun_state_bool,C_2: com,Q_1: fun_a_fun_state_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,Pa),C_2),Q_1)),bot_bo214064617a_bool)))
     => ( ! [Z_4: x_a,S_3: state] :
            ( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_1,Z_4),S_3))
           => hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Qa,Z_4),S_3)) )
       => hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,Pa),C_2),Qa)),bot_bo214064617a_bool))) ) ) ).

tff(fact_554_conseq2,axiom,
    ! [Qa: fun_st1506752259e_bool,Ga: fun_Ho320143120e_bool,Pa: fun_st1506752259e_bool,C_2: com,Q_1: fun_st1506752259e_bool] :
      ( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,Pa),C_2),Q_1)),bot_bo39439654e_bool)))
     => ( ! [Z_4: state,S_3: state] :
            ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Q_1,Z_4),S_3))
           => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Qa,Z_4),S_3)) )
       => hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,Pa),C_2),Qa)),bot_bo39439654e_bool))) ) ) ).

tff(fact_555_triple_Osize_I1_J,axiom,
    ! [Fa: fun_a_nat,Fun1: fun_a_fun_state_bool,Com_1: com,Fun2: fun_a_fun_state_bool] : ( hAPP_H174304918_a_nat(hoare_1823859380size_a(Fa),hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,Fun1),Com_1),Fun2)) = zero_zero_nat ) ).

tff(fact_556_triple_Osize_I1_J,axiom,
    ! [Fa: fun_state_nat,Fun1: fun_st1506752259e_bool,Com_1: com,Fun2: fun_st1506752259e_bool] : ( hAPP_H124007033te_nat(hoare_1376048973_state(Fa),hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,Fun1),Com_1),Fun2)) = zero_zero_nat ) ).

tff(fact_557_MGT__def,axiom,
    ! [C_2: com] : ( hoare_Mirabelle_MGT(C_2) = hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,fequal_state),C_2),evalc(C_2)) ) ).

tff(fact_558_triple_Osize_I2_J,axiom,
    ! [Fun1: fun_a_fun_state_bool,Com_1: com,Fun2: fun_a_fun_state_bool] : ( hAPP_H174304918_a_nat(size_s1025776543iple_a,hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,Fun1),Com_1),Fun2)) = zero_zero_nat ) ).

tff(fact_559_triple_Osize_I2_J,axiom,
    ! [Fun1: fun_st1506752259e_bool,Com_1: com,Fun2: fun_st1506752259e_bool] : ( hAPP_H124007033te_nat(size_s1988939064_state,hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,Fun1),Com_1),Fun2)) = zero_zero_nat ) ).

tff(fact_560_conseq12,axiom,
    ! [Qa: fun_a_fun_state_bool,Pa: fun_a_fun_state_bool,Ga: fun_Ho882550911a_bool,P_1: fun_a_fun_state_bool,C_2: com,Q_1: fun_a_fun_state_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,P_1),C_2),Q_1)),bot_bo214064617a_bool)))
     => ( ! [Z_4: x_a,S_3: state] :
            ( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Pa,Z_4),S_3))
           => ! [S_4: state] :
                ( ! [Z_5: x_a] :
                    ( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(P_1,Z_5),S_3))
                   => hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_1,Z_5),S_4)) )
               => hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Qa,Z_4),S_4)) ) )
       => hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,Pa),C_2),Qa)),bot_bo214064617a_bool))) ) ) ).

tff(fact_561_conseq12,axiom,
    ! [Qa: fun_st1506752259e_bool,Pa: fun_st1506752259e_bool,Ga: fun_Ho320143120e_bool,P_1: fun_st1506752259e_bool,C_2: com,Q_1: fun_st1506752259e_bool] :
      ( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,P_1),C_2),Q_1)),bot_bo39439654e_bool)))
     => ( ! [Z_4: state,S_3: state] :
            ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Pa,Z_4),S_3))
           => ! [S_4: state] :
                ( ! [Z_5: state] :
                    ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(P_1,Z_5),S_3))
                   => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Q_1,Z_5),S_4)) )
               => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Qa,Z_4),S_4)) ) )
       => hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,Pa),C_2),Qa)),bot_bo39439654e_bool))) ) ) ).

tff(fact_562_the__elem__eq,axiom,
    ! [X_2: nat] : ( hAPP_f22106695ol_nat(the_elem_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),bot_bot_fun_nat_bool)) = X_2 ) ).

tff(fact_563_the__elem__eq,axiom,
    ! [X_2: fun_Ho882550911a_bool] : ( hAPP_f1758315691a_bool(the_el396789536a_bool,hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,X_2),bot_bo1285271383l_bool)) = X_2 ) ).

tff(fact_564_the__elem__eq,axiom,
    ! [X_2: pname] : ( hAPP_f1297739591_pname(the_elem_pname,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),bot_bo844097828e_bool)) = X_2 ) ).

tff(fact_565_the__elem__eq,axiom,
    ! [X_2: hoare_2076525193iple_a] : ( hAPP_f2009966983iple_a(the_el2146174248iple_a,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_2),bot_bo214064617a_bool)) = X_2 ) ).

tff(fact_566_the__elem__eq,axiom,
    ! [X_2: hoare_363350478_state] : ( hAPP_f1242401993_state(the_el1666657391_state,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_2),bot_bo39439654e_bool)) = X_2 ) ).

tff(fact_567_Suc__neq__Zero,axiom,
    ! [M_2: nat] : ( hAPP_nat_nat(suc,M_2) != zero_zero_nat ) ).

tff(fact_568_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

tff(fact_569_n__not__Suc__n,axiom,
    ! [N_2: nat] : ( N_2 != hAPP_nat_nat(suc,N_2) ) ).

tff(fact_570_Suc__n__not__n,axiom,
    ! [N_2: nat] : ( hAPP_nat_nat(suc,N_2) != N_2 ) ).

tff(fact_571_nat_Oinject,axiom,
    ! [Nat_4: nat,Nat_3: nat] :
      ( ( hAPP_nat_nat(suc,Nat_4) = hAPP_nat_nat(suc,Nat_3) )
    <=> ( Nat_4 = Nat_3 ) ) ).

tff(fact_572_Suc__inject,axiom,
    ! [X_1: nat,Y: nat] :
      ( ( hAPP_nat_nat(suc,X_1) = hAPP_nat_nat(suc,Y) )
     => ( X_1 = Y ) ) ).

tff(fact_573_Zero__not__Suc,axiom,
    ! [M_2: nat] : ( zero_zero_nat != hAPP_nat_nat(suc,M_2) ) ).

tff(fact_574_nat_Osimps_I2_J,axiom,
    ! [Nat_2: nat] : ( zero_zero_nat != hAPP_nat_nat(suc,Nat_2) ) ).

tff(fact_575_Suc__not__Zero,axiom,
    ! [M_2: nat] : ( hAPP_nat_nat(suc,M_2) != zero_zero_nat ) ).

tff(fact_576_nat_Osimps_I3_J,axiom,
    ! [Nat_1: nat] : ( hAPP_nat_nat(suc,Nat_1) != zero_zero_nat ) ).

tff(fact_577_Zero__neq__Suc,axiom,
    ! [M_2: nat] : ( zero_zero_nat != hAPP_nat_nat(suc,M_2) ) ).

tff(fact_578_not0__implies__Suc,axiom,
    ! [N_2: nat] :
      ( ( N_2 != zero_zero_nat )
     => ? [M_1: nat] : ( N_2 = hAPP_nat_nat(suc,M_1) ) ) ).

tff(fact_579_nat__induct,axiom,
    ! [N_1: nat,Pa: fun_nat_bool] :
      ( hBOOL(hAPP_nat_bool(Pa,zero_zero_nat))
     => ( ! [N: nat] :
            ( hBOOL(hAPP_nat_bool(Pa,N))
           => hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(suc,N))) )
       => hBOOL(hAPP_nat_bool(Pa,N_1)) ) ) ).

tff(fact_580_zero__induct,axiom,
    ! [Pa: fun_nat_bool,K_1: nat] :
      ( hBOOL(hAPP_nat_bool(Pa,K_1))
     => ( ! [N: nat] :
            ( hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(suc,N)))
           => hBOOL(hAPP_nat_bool(Pa,N)) )
       => hBOOL(hAPP_nat_bool(Pa,zero_zero_nat)) ) ) ).

tff(fact_581_nat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat: nat] : ( Y != hAPP_nat_nat(suc,Nat) ) ) ).

tff(fact_582_bot__fun__def,axiom,
    ! [X: nat] :
      ( hBOOL(hAPP_nat_bool(bot_bot_fun_nat_bool,X))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_583_bot__fun__def,axiom,
    ! [X: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_H454055633a_bool(bot_bo214064617a_bool,X))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_584_bot__fun__def,axiom,
    ! [X: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(bot_bo1285271383l_bool,X))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_585_bot__fun__def,axiom,
    ! [X: pname] :
      ( hBOOL(hAPP_pname_bool(bot_bo844097828e_bool,X))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_586_bot__fun__def,axiom,
    ! [X: hoare_363350478_state] :
      ( hBOOL(hAPP_H1645464494e_bool(bot_bo39439654e_bool,X))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_587_bot__apply,axiom,
    ! [X_2: nat] :
      ( hBOOL(hAPP_nat_bool(bot_bot_fun_nat_bool,X_2))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_588_bot__apply,axiom,
    ! [X_2: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_H454055633a_bool(bot_bo214064617a_bool,X_2))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_589_bot__apply,axiom,
    ! [X_2: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(bot_bo1285271383l_bool,X_2))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_590_bot__apply,axiom,
    ! [X_2: pname] :
      ( hBOOL(hAPP_pname_bool(bot_bo844097828e_bool,X_2))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_591_bot__apply,axiom,
    ! [X_2: hoare_363350478_state] :
      ( hBOOL(hAPP_H1645464494e_bool(bot_bo39439654e_bool,X_2))
    <=> hBOOL(bot_bot_bool) ) ).

tff(fact_592_evaln_OBody,axiom,
    ! [Pn: pname,S0: state,N_2: nat,S1_2: state] :
      ( hBOOL(hAPP_state_bool(evaln(hAPP_option_com_com(the_com,hAPP_p799580910on_com(body_1,Pn)),S0,N_2),S1_2))
     => hBOOL(hAPP_state_bool(evaln(hAPP_pname_com(body,Pn),S0,hAPP_nat_nat(suc,N_2)),S1_2)) ) ).

tff(fact_593_hoare__derivs_OSkip,axiom,
    ! [Ga: fun_Ho882550911a_bool,Pa: fun_a_fun_state_bool] : hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,Pa),skip),Pa)),bot_bo214064617a_bool))) ).

tff(fact_594_hoare__derivs_OSkip,axiom,
    ! [Ga: fun_Ho320143120e_bool,Pa: fun_st1506752259e_bool] : hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,Pa),skip),Pa)),bot_bo39439654e_bool))) ).

tff(fact_595_LoopF,axiom,
    ! [Ga: fun_Ho882550911a_bool,Pa: fun_a_fun_state_bool,B_1: fun_state_bool,C_2: com] : hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,hAPP_f762886889e_bool(hAPP_f1261923407e_bool(cOMBC_892787026e_bool,hAPP_f963367678e_bool(cOMBB_145932198bool_a(cOMBS_1378840469l_bool),hAPP_f1509969235l_bool(cOMBB_1355796797bool_a(cOMBB_188601460_state(fconj)),Pa))),hAPP_f1759915619e_bool(cOMBB_160679318_state(fNot),B_1))),while(B_1,C_2)),Pa)),bot_bo214064617a_bool))) ).

tff(fact_596_LoopF,axiom,
    ! [Ga: fun_Ho320143120e_bool,Pa: fun_st1506752259e_bool,B_1: fun_state_bool,C_2: com] : hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,hAPP_f1863945078e_bool(hAPP_f915354021e_bool(cOMBC_952831051e_bool,hAPP_f1138284024e_bool(cOMBB_20296667_state(cOMBS_1378840469l_bool),hAPP_f1283379615l_bool(cOMBB_1759179140_state(cOMBB_188601460_state(fconj)),Pa))),hAPP_f1759915619e_bool(cOMBB_160679318_state(fNot),B_1))),while(B_1,C_2)),Pa)),bot_bo39439654e_bool))) ).

tff(fact_597_evaln_OWhileFalse,axiom,
    ! [C_2: com,N_1: nat,B_1: fun_state_bool,S_1: state] :
      ( ~ hBOOL(hAPP_state_bool(B_1,S_1))
     => hBOOL(hAPP_state_bool(evaln(while(B_1,C_2),S_1,N_1),S_1)) ) ).

tff(fact_598_evaln_OWhileTrue,axiom,
    ! [S2_1: state,C_2: com,N_1: nat,S1: state,B_1: fun_state_bool,S0_1: state] :
      ( hBOOL(hAPP_state_bool(B_1,S0_1))
     => ( hBOOL(hAPP_state_bool(evaln(C_2,S0_1,N_1),S1))
       => ( hBOOL(hAPP_state_bool(evaln(while(B_1,C_2),S1,N_1),S2_1))
         => hBOOL(hAPP_state_bool(evaln(while(B_1,C_2),S0_1,N_1),S2_1)) ) ) ) ).

tff(fact_599_evalc_OWhileTrue,axiom,
    ! [S2_1: state,C_2: com,S1: state,B_1: fun_state_bool,S0_1: state] :
      ( hBOOL(hAPP_state_bool(B_1,S0_1))
     => ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C_2),S0_1),S1))
       => ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(while(B_1,C_2)),S1),S2_1))
         => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(while(B_1,C_2)),S0_1),S2_1)) ) ) ) ).

tff(fact_600_evalc_OWhileFalse,axiom,
    ! [C_2: com,B_1: fun_state_bool,S_1: state] :
      ( ~ hBOOL(hAPP_state_bool(B_1,S_1))
     => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(while(B_1,C_2)),S_1),S_1)) ) ).

tff(fact_601_evaln_OSkip,axiom,
    ! [S: state,N_2: nat] : hBOOL(hAPP_state_bool(evaln(skip,S,N_2),S)) ).

tff(fact_602_evaln__elim__cases_I1_J,axiom,
    ! [S: state,N_2: nat,T: state] :
      ( hBOOL(hAPP_state_bool(evaln(skip,S,N_2),T))
     => ( T = S ) ) ).

tff(fact_603_evalc__elim__cases_I1_J,axiom,
    ! [S: state,T: state] :
      ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(skip),S),T))
     => ( T = S ) ) ).

tff(fact_604_evalc_OSkip,axiom,
    ! [S: state] : hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(skip),S),S)) ).

tff(fact_605_com_Osimps_I16_J,axiom,
    ! [Fun: fun_state_bool,Com: com] : ( skip != while(Fun,Com) ) ).

tff(fact_606_com_Osimps_I17_J,axiom,
    ! [Fun: fun_state_bool,Com: com] : ( while(Fun,Com) != skip ) ).

tff(fact_607_com_Osimps_I5_J,axiom,
    ! [Fun_1: fun_state_bool,Com_1: com,Fun: fun_state_bool,Com: com] :
      ( ( while(Fun_1,Com_1) = while(Fun,Com) )
    <=> ( ( Fun_1 = Fun )
        & ( Com_1 = Com ) ) ) ).

tff(fact_608_evaln__Suc,axiom,
    ! [C: com,S: state,N_2: nat,S_5: state] :
      ( hBOOL(hAPP_state_bool(evaln(C,S,N_2),S_5))
     => hBOOL(hAPP_state_bool(evaln(C,S,hAPP_nat_nat(suc,N_2)),S_5)) ) ).

tff(fact_609_evaln__evalc,axiom,
    ! [C: com,S: state,N_2: nat,T: state] :
      ( hBOOL(hAPP_state_bool(evaln(C,S,N_2),T))
     => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C),S),T)) ) ).

tff(fact_610_eval__eq,axiom,
    ! [C_2: com,S_1: state,T_1: state] :
      ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C_2),S_1),T_1))
    <=> ? [N: nat] : hBOOL(hAPP_state_bool(evaln(C_2,S_1,N),T_1)) ) ).

tff(fact_611_com_Osimps_I59_J,axiom,
    ! [Pname: pname,Fun_1: fun_state_bool,Com_1: com] : ( hAPP_pname_com(body,Pname) != while(Fun_1,Com_1) ) ).

tff(fact_612_com_Osimps_I58_J,axiom,
    ! [Fun_1: fun_state_bool,Com_1: com,Pname: pname] : ( while(Fun_1,Com_1) != hAPP_pname_com(body,Pname) ) ).

tff(fact_613_com_Osimps_I18_J,axiom,
    ! [Pname_2: pname] : ( skip != hAPP_pname_com(body,Pname_2) ) ).

tff(fact_614_com_Osimps_I19_J,axiom,
    ! [Pname_2: pname] : ( hAPP_pname_com(body,Pname_2) != skip ) ).

tff(fact_615_triple__valid__def2,axiom,
    ! [N_1: nat,Pa: fun_st1506752259e_bool,C_2: com,Qa: fun_st1506752259e_bool] :
      ( hBOOL(hAPP_H1645464494e_bool(hAPP_n753050587e_bool(hoare_28808878_state,N_1),hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,Pa),C_2),Qa)))
    <=> ! [Z_4: state,S_3: state] :
          ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Pa,Z_4),S_3))
         => ! [S_4: state] :
              ( hBOOL(hAPP_state_bool(evaln(C_2,S_3,N_1),S_4))
             => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(Qa,Z_4),S_4)) ) ) ) ).

tff(fact_616_triple__valid__def2,axiom,
    ! [N_1: nat,Pa: fun_a_fun_state_bool,C_2: com,Qa: fun_a_fun_state_bool] :
      ( hBOOL(hAPP_H454055633a_bool(hAPP_n1335981428a_bool(hoare_526497939alid_a,N_1),hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,Pa),C_2),Qa)))
    <=> ! [Z_4: x_a,S_3: state] :
          ( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Pa,Z_4),S_3))
         => ! [S_4: state] :
              ( hBOOL(hAPP_state_bool(evaln(C_2,S_3,N_1),S_4))
             => hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Qa,Z_4),S_4)) ) ) ) ).

tff(fact_617_evaln__elim__cases_I6_J,axiom,
    ! [P: pname,S: state,N_2: nat,S1_2: state] :
      ( hBOOL(hAPP_state_bool(evaln(hAPP_pname_com(body,P),S,N_2),S1_2))
     => ~ ! [N: nat] :
            ( ( N_2 = hAPP_nat_nat(suc,N) )
           => ~ hBOOL(hAPP_state_bool(evaln(hAPP_option_com_com(the_com,hAPP_p799580910on_com(body_1,P)),S,N),S1_2)) ) ) ).

tff(fact_618_evalc__WHILE__case,axiom,
    ! [B_1: fun_state_bool,C_2: com,S_1: state,T_1: state] :
      ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(while(B_1,C_2)),S_1),T_1))
     => ( ( ( T_1 = S_1 )
         => hBOOL(hAPP_state_bool(B_1,S_1)) )
       => ~ ( hBOOL(hAPP_state_bool(B_1,S_1))
           => ! [S1_1: state] :
                ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C_2),S_1),S1_1))
               => ~ hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(while(B_1,C_2)),S1_1),T_1)) ) ) ) ) ).

tff(fact_619_evaln__WHILE__case,axiom,
    ! [B_1: fun_state_bool,C_2: com,S_1: state,N_1: nat,T_1: state] :
      ( hBOOL(hAPP_state_bool(evaln(while(B_1,C_2),S_1,N_1),T_1))
     => ( ( ( T_1 = S_1 )
         => hBOOL(hAPP_state_bool(B_1,S_1)) )
       => ~ ( hBOOL(hAPP_state_bool(B_1,S_1))
           => ! [S1_1: state] :
                ( hBOOL(hAPP_state_bool(evaln(C_2,S_1,N_1),S1_1))
               => ~ hBOOL(hAPP_state_bool(evaln(while(B_1,C_2),S1_1,N_1),T_1)) ) ) ) ) ).

tff(fact_620_evalc__evaln,axiom,
    ! [C: com,S: state,T: state] :
      ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C),S),T))
     => ? [N: nat] : hBOOL(hAPP_state_bool(evaln(C,S,N),T)) ) ).

tff(fact_621_Comp,axiom,
    ! [D: com,R_1: fun_a_fun_state_bool,Ga: fun_Ho882550911a_bool,Pa: fun_a_fun_state_bool,C_2: com,Qa: fun_a_fun_state_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,Pa),C_2),Qa)),bot_bo214064617a_bool)))
     => ( hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,Qa),D),R_1)),bot_bo214064617a_bool)))
       => hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,Pa),semi(C_2,D)),R_1)),bot_bo214064617a_bool))) ) ) ).

tff(fact_622_Comp,axiom,
    ! [D: com,R_1: fun_st1506752259e_bool,Ga: fun_Ho320143120e_bool,Pa: fun_st1506752259e_bool,C_2: com,Qa: fun_st1506752259e_bool] :
      ( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,Pa),C_2),Qa)),bot_bo39439654e_bool)))
     => ( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,Qa),D),R_1)),bot_bo39439654e_bool)))
       => hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,Pa),semi(C_2,D)),R_1)),bot_bo39439654e_bool))) ) ) ).

tff(fact_623_the__elem__def,axiom,
    ! [X_10: fun_nat_bool] : ( hAPP_f22106695ol_nat(the_elem_nat,X_10) = hAPP_f22106695ol_nat(the_nat,hAPP_f158894502t_bool(cOMBB_955900739ol_nat(hAPP_f103356543l_bool(fequal_fun_nat_bool,X_10)),hAPP_f1570313510t_bool(hAPP_f688831301t_bool(cOMBC_178881787t_bool,insert_nat),bot_bot_fun_nat_bool))) ) ).

tff(fact_624_the__elem__def,axiom,
    ! [X_10: fun_fu2026518849l_bool] : ( hAPP_f1758315691a_bool(the_el396789536a_bool,X_10) = hAPP_f1758315691a_bool(the_fu972267150a_bool,hAPP_f1605242827l_bool(cOMBB_4714191a_bool(hAPP_f1238945379l_bool(fequal261524416l_bool,X_10)),hAPP_f1662919455l_bool(hAPP_f2041002367l_bool(cOMBC_1141206332l_bool,insert838260101a_bool),bot_bo1285271383l_bool))) ) ).

tff(fact_625_the__elem__def,axiom,
    ! [X_10: fun_pname_bool] : ( hAPP_f1297739591_pname(the_elem_pname,X_10) = hAPP_f1297739591_pname(the_pname,hAPP_f42430548e_bool(cOMBB_530759491_pname(hAPP_f434788991l_bool(fequal533582459e_bool,X_10)),hAPP_f180827860e_bool(hAPP_f698292281e_bool(cOMBC_1123258281e_bool,insert_pname),bot_bo844097828e_bool))) ) ).

tff(fact_626_the__elem__def,axiom,
    ! [X_10: fun_Ho882550911a_bool] : ( hAPP_f2009966983iple_a(the_el2146174248iple_a,X_10) = hAPP_f2009966983iple_a(the_Ho1456473914iple_a,hAPP_f338738851a_bool(cOMBB_1676209667iple_a(hAPP_f256727871l_bool(fequal291249344a_bool,X_10)),hAPP_f219434403a_bool(hAPP_f1824299331a_bool(cOMBC_175592824a_bool,insert1582918915iple_a),bot_bo214064617a_bool))) ) ).

tff(fact_627_the__elem__def,axiom,
    ! [X_10: fun_Ho320143120e_bool] : ( hAPP_f1242401993_state(the_el1666657391_state,X_10) = hAPP_f1242401993_state(the_Ho1653144541_state,hAPP_f265222234e_bool(cOMBB_1356223341_state(hAPP_f726209793l_bool(fequal731041167e_bool,X_10)),hAPP_f596150574e_bool(hAPP_f961492031e_bool(cOMBC_1774708299e_bool,insert1330351828_state),bot_bo39439654e_bool))) ) ).

tff(fact_628_finite__pointwise,axiom,
    ! [Pa: fun_Ho1146995216e_bool,Qa: fun_Ho1146995216e_bool,Ga: fun_Ho882550911a_bool,P_1: fun_Ho1146995216e_bool,C0: fun_Ho1421867454_a_com,Q_1: fun_Ho1146995216e_bool,U_2: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,U_2))
     => ( ! [P_2: hoare_2076525193iple_a] :
            ( hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,hAPP_H1347006248e_bool(P_1,P_2)),hAPP_H45551702_a_com(C0,P_2)),hAPP_H1347006248e_bool(Q_1,P_2))),bot_bo214064617a_bool)))
           => hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,hAPP_H1347006248e_bool(Pa,P_2)),hAPP_H45551702_a_com(C0,P_2)),hAPP_H1347006248e_bool(Qa,P_2))),bot_bo214064617a_bool))) )
       => ( hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(image_1416816325iple_a(hAPP_f1255182146iple_a(hAPP_f2049934177iple_a(cOMBS_159040809iple_a,hAPP_f1031856565iple_a(hAPP_f1391221537iple_a(cOMBS_164138524iple_a,hAPP_f1151385430iple_a(cOMBB_782889804iple_a(hoare_1876769875iple_a),P_1)),C0)),Q_1)),U_2)))
         => hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(image_1416816325iple_a(hAPP_f1255182146iple_a(hAPP_f2049934177iple_a(cOMBS_159040809iple_a,hAPP_f1031856565iple_a(hAPP_f1391221537iple_a(cOMBS_164138524iple_a,hAPP_f1151385430iple_a(cOMBB_782889804iple_a(hoare_1876769875iple_a),Pa)),C0)),Qa)),U_2))) ) ) ) ).

tff(fact_629_finite__pointwise,axiom,
    ! [Pa: fun_fu1307637710e_bool,Qa: fun_fu1307637710e_bool,Ga: fun_Ho882550911a_bool,P_1: fun_fu1307637710e_bool,C0: fun_fu886446204ol_com,Q_1: fun_fu1307637710e_bool,U_2: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,U_2))
     => ( ! [P_2: fun_Ho882550911a_bool] :
            ( hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,hAPP_f369538458e_bool(P_1,P_2)),hAPP_f61887944ol_com(C0,P_2)),hAPP_f369538458e_bool(Q_1,P_2))),bot_bo214064617a_bool)))
           => hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,hAPP_f369538458e_bool(Pa,P_2)),hAPP_f61887944ol_com(C0,P_2)),hAPP_f369538458e_bool(Qa,P_2))),bot_bo214064617a_bool))) )
       => ( hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f1758315691a_bool(image_1469988567iple_a(hAPP_f375308812iple_a(hAPP_f2030579535iple_a(cOMBS_736671097iple_a,hAPP_f1229970779iple_a(hAPP_f1723480683iple_a(cOMBS_282424940iple_a,hAPP_f42286368iple_a(cOMBB_1339444220a_bool(hoare_1876769875iple_a),P_1)),C0)),Q_1)),U_2)))
         => hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f1758315691a_bool(image_1469988567iple_a(hAPP_f375308812iple_a(hAPP_f2030579535iple_a(cOMBS_736671097iple_a,hAPP_f1229970779iple_a(hAPP_f1723480683iple_a(cOMBS_282424940iple_a,hAPP_f42286368iple_a(cOMBB_1339444220a_bool(hoare_1876769875iple_a),Pa)),C0)),Qa)),U_2))) ) ) ) ).

tff(fact_630_finite__pointwise,axiom,
    ! [Pa: fun_pn12888928e_bool,Qa: fun_pn12888928e_bool,Ga: fun_Ho320143120e_bool,P_1: fun_pn12888928e_bool,C0: fun_pname_com,Q_1: fun_pn12888928e_bool,U_2: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,U_2))
     => ( ! [P_2: pname] :
            ( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,hAPP_p1637813682e_bool(P_1,P_2)),hAPP_pname_com(C0,P_2)),hAPP_p1637813682e_bool(Q_1,P_2))),bot_bo39439654e_bool)))
           => hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,hAPP_p1637813682e_bool(Pa,P_2)),hAPP_pname_com(C0,P_2)),hAPP_p1637813682e_bool(Qa,P_2))),bot_bo39439654e_bool))) )
       => ( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1763243027e_bool(image_1918575693_state(hAPP_f19336718_state(hAPP_f720869462_state(cOMBS_1110920566_state,hAPP_f1146170881_state(hAPP_f1311198492_state(cOMBS_124505571_state,hAPP_f387017471_state(cOMBB_1212270426_pname(hoare_608330478_state),P_1)),C0)),Q_1)),U_2)))
         => hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1763243027e_bool(image_1918575693_state(hAPP_f19336718_state(hAPP_f720869462_state(cOMBS_1110920566_state,hAPP_f1146170881_state(hAPP_f1311198492_state(cOMBS_124505571_state,hAPP_f387017471_state(cOMBB_1212270426_pname(hoare_608330478_state),Pa)),C0)),Qa)),U_2))) ) ) ) ).

tff(fact_631_finite__pointwise,axiom,
    ! [Pa: fun_Ho1217407269e_bool,Qa: fun_Ho1217407269e_bool,Ga: fun_Ho320143120e_bool,P_1: fun_Ho1217407269e_bool,C0: fun_Ho1421867454_a_com,Q_1: fun_Ho1217407269e_bool,U_2: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,U_2))
     => ( ! [P_2: hoare_2076525193iple_a] :
            ( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,hAPP_H166769783e_bool(P_1,P_2)),hAPP_H45551702_a_com(C0,P_2)),hAPP_H166769783e_bool(Q_1,P_2))),bot_bo39439654e_bool)))
           => hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,hAPP_H166769783e_bool(Pa,P_2)),hAPP_H45551702_a_com(C0,P_2)),hAPP_H166769783e_bool(Qa,P_2))),bot_bo39439654e_bool))) )
       => ( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f225384718e_bool(image_624167570_state(hAPP_f1907710744_state(hAPP_f1734768933_state(cOMBS_8231099_state,hAPP_f595441985_state(hAPP_f646848545_state(cOMBS_672164008_state,hAPP_f1793699445_state(cOMBB_1475458975iple_a(hoare_608330478_state),P_1)),C0)),Q_1)),U_2)))
         => hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f225384718e_bool(image_624167570_state(hAPP_f1907710744_state(hAPP_f1734768933_state(cOMBS_8231099_state,hAPP_f595441985_state(hAPP_f646848545_state(cOMBS_672164008_state,hAPP_f1793699445_state(cOMBB_1475458975iple_a(hoare_608330478_state),Pa)),C0)),Qa)),U_2))) ) ) ) ).

tff(fact_632_finite__pointwise,axiom,
    ! [Pa: fun_fu857054695e_bool,Qa: fun_fu857054695e_bool,Ga: fun_Ho320143120e_bool,P_1: fun_fu857054695e_bool,C0: fun_fu886446204ol_com,Q_1: fun_fu857054695e_bool,U_2: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,U_2))
     => ( ! [P_2: fun_Ho882550911a_bool] :
            ( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,hAPP_f1470859141e_bool(P_1,P_2)),hAPP_f61887944ol_com(C0,P_2)),hAPP_f1470859141e_bool(Q_1,P_2))),bot_bo39439654e_bool)))
           => hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,hAPP_f1470859141e_bool(Pa,P_2)),hAPP_f61887944ol_com(C0,P_2)),hAPP_f1470859141e_bool(Qa,P_2))),bot_bo39439654e_bool))) )
       => ( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1609309156e_bool(image_1856701440_state(hAPP_f1319084128_state(hAPP_f1676710173_state(cOMBS_1608831243_state,hAPP_f1517110247_state(hAPP_f995418987_state(cOMBS_1312987384_state,hAPP_f1181766445_state(cOMBB_498853865a_bool(hoare_608330478_state),P_1)),C0)),Q_1)),U_2)))
         => hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1609309156e_bool(image_1856701440_state(hAPP_f1319084128_state(hAPP_f1676710173_state(cOMBS_1608831243_state,hAPP_f1517110247_state(hAPP_f995418987_state(cOMBS_1312987384_state,hAPP_f1181766445_state(cOMBB_498853865a_bool(hoare_608330478_state),Pa)),C0)),Qa)),U_2))) ) ) ) ).

tff(fact_633_finite__pointwise,axiom,
    ! [Pa: fun_na186072911e_bool,Qa: fun_na186072911e_bool,Ga: fun_Ho882550911a_bool,P_1: fun_na186072911e_bool,C0: fun_nat_com,Q_1: fun_na186072911e_bool,U_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,U_2))
     => ( ! [P_2: nat] :
            ( hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,hAPP_n1063102567e_bool(P_1,P_2)),hAPP_nat_com(C0,P_2)),hAPP_n1063102567e_bool(Q_1,P_2))),bot_bo214064617a_bool)))
           => hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,hAPP_n1063102567e_bool(Pa,P_2)),hAPP_nat_com(C0,P_2)),hAPP_n1063102567e_bool(Qa,P_2))),bot_bo214064617a_bool))) )
       => ( hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f1985266946a_bool(image_344477316iple_a(hAPP_f1101337346iple_a(hAPP_f165988002iple_a(cOMBS_2043478954iple_a,hAPP_f198456565iple_a(hAPP_f603309282iple_a(cOMBS_803657245iple_a,hAPP_f391471894iple_a(cOMBB_1836265421_a_nat(hoare_1876769875iple_a),P_1)),C0)),Q_1)),U_2)))
         => hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f1985266946a_bool(image_344477316iple_a(hAPP_f1101337346iple_a(hAPP_f165988002iple_a(cOMBS_2043478954iple_a,hAPP_f198456565iple_a(hAPP_f603309282iple_a(cOMBS_803657245iple_a,hAPP_f391471894iple_a(cOMBB_1836265421_a_nat(hoare_1876769875iple_a),Pa)),C0)),Qa)),U_2))) ) ) ) ).

tff(fact_634_finite__pointwise,axiom,
    ! [Pa: fun_na1057458470e_bool,Qa: fun_na1057458470e_bool,Ga: fun_Ho320143120e_bool,P_1: fun_na1057458470e_bool,C0: fun_nat_com,Q_1: fun_na1057458470e_bool,U_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,U_2))
     => ( ! [P_2: nat] :
            ( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,hAPP_n1252169848e_bool(P_1,P_2)),hAPP_nat_com(C0,P_2)),hAPP_n1252169848e_bool(Q_1,P_2))),bot_bo39439654e_bool)))
           => hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,hAPP_f83715540_state(hAPP_c1279058753_state(hAPP_f960042659_state(hoare_608330478_state,hAPP_n1252169848e_bool(Pa,P_2)),hAPP_nat_com(C0,P_2)),hAPP_n1252169848e_bool(Qa,P_2))),bot_bo39439654e_bool))) )
       => ( hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1778645709e_bool(image_1316773907_state(hAPP_f1886686490_state(hAPP_f77533864_state(cOMBS_976781756_state,hAPP_f1844653185_state(hAPP_f1469189858_state(cOMBS_1435179817_state,hAPP_f15168243_state(cOMBB_186380832te_nat(hoare_608330478_state),P_1)),C0)),Q_1)),U_2)))
         => hBOOL(hAPP_f568432088l_bool(hoare_95103338_state(Ga),hAPP_f1778645709e_bool(image_1316773907_state(hAPP_f1886686490_state(hAPP_f77533864_state(cOMBS_976781756_state,hAPP_f1844653185_state(hAPP_f1469189858_state(cOMBS_1435179817_state,hAPP_f15168243_state(cOMBB_186380832te_nat(hoare_608330478_state),Pa)),C0)),Qa)),U_2))) ) ) ) ).

tff(fact_635_finite__pointwise,axiom,
    ! [Pa: fun_pn1683930517e_bool,Qa: fun_pn1683930517e_bool,Ga: fun_Ho882550911a_bool,P_1: fun_pn1683930517e_bool,C0: fun_pname_com,Q_1: fun_pn1683930517e_bool,U_2: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,U_2))
     => ( ! [P_2: pname] :
            ( hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,hAPP_p635540397e_bool(P_1,P_2)),hAPP_pname_com(C0,P_2)),hAPP_p635540397e_bool(Q_1,P_2))),bot_bo214064617a_bool)))
           => hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,hAPP_f860089282iple_a(hAPP_c426811957iple_a(hAPP_f1838364758iple_a(hoare_1876769875iple_a,hAPP_p635540397e_bool(Pa,P_2)),hAPP_pname_com(C0,P_2)),hAPP_p635540397e_bool(Qa,P_2))),bot_bo214064617a_bool))) )
       => ( hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f1451199292a_bool(image_217098954iple_a(hAPP_f1246077570iple_a(hAPP_f345351132iple_a(cOMBS_970288740iple_a,hAPP_f804419189iple_a(hAPP_f900936988iple_a(cOMBS_1123526615iple_a,hAPP_f1471296150iple_a(cOMBB_1308294791_pname(hoare_1876769875iple_a),P_1)),C0)),Q_1)),U_2)))
         => hBOOL(hAPP_f500854239l_bool(hoare_572465367rivs_a(Ga),hAPP_f1451199292a_bool(image_217098954iple_a(hAPP_f1246077570iple_a(hAPP_f345351132iple_a(cOMBS_970288740iple_a,hAPP_f804419189iple_a(hAPP_f900936988iple_a(cOMBS_1123526615iple_a,hAPP_f1471296150iple_a(cOMBB_1308294791_pname(hoare_1876769875iple_a),Pa)),C0)),Qa)),U_2))) ) ) ) ).

tff(fact_636_evaln__max2,axiom,
    ! [C2_1: com,S2: state,N2: nat,T2: state,C1_1: com,S1_2: state,N1: nat,T1: state] :
      ( hBOOL(hAPP_state_bool(evaln(C1_1,S1_2,N1),T1))
     => ( hBOOL(hAPP_state_bool(evaln(C2_1,S2,N2),T2))
       => ? [N: nat] :
            ( hBOOL(hAPP_state_bool(evaln(C1_1,S1_2,N),T1))
            & hBOOL(hAPP_state_bool(evaln(C2_1,S2,N),T2)) ) ) ) ).

tff(fact_637_mk__disjoint__insert,axiom,
    ! [A_2: nat,A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),A_1))
     => ? [B_6: fun_nat_bool] :
          ( ( A_1 = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),B_6) )
          & ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),B_6)) ) ) ).

tff(fact_638_mk__disjoint__insert,axiom,
    ! [A_2: fun_Ho882550911a_bool,A_1: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,A_2),A_1))
     => ? [B_6: fun_fu2026518849l_bool] :
          ( ( A_1 = hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,A_2),B_6) )
          & ~ hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,A_2),B_6)) ) ) ).

tff(fact_639_mk__disjoint__insert,axiom,
    ! [A_2: hoare_2076525193iple_a,A_1: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_2),A_1))
     => ? [B_6: fun_Ho882550911a_bool] :
          ( ( A_1 = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_2),B_6) )
          & ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,A_2),B_6)) ) ) ).

tff(fact_640_mk__disjoint__insert,axiom,
    ! [A_2: hoare_363350478_state,A_1: fun_Ho320143120e_bool] :
      ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_2),A_1))
     => ? [B_6: fun_Ho320143120e_bool] :
          ( ( A_1 = hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_2),B_6) )
          & ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,A_2),B_6)) ) ) ).

tff(fact_641_mk__disjoint__insert,axiom,
    ! [A_2: pname,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),A_1))
     => ? [B_6: fun_pname_bool] :
          ( ( A_1 = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),B_6) )
          & ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),B_6)) ) ) ).

tff(fact_642_evaln_OSemi,axiom,
    ! [C1_1: com,S2: state,C0_1: com,S0: state,N_2: nat,S1_2: state] :
      ( hBOOL(hAPP_state_bool(evaln(C0_1,S0,N_2),S1_2))
     => ( hBOOL(hAPP_state_bool(evaln(C1_1,S1_2,N_2),S2))
       => hBOOL(hAPP_state_bool(evaln(semi(C0_1,C1_1),S0,N_2),S2)) ) ) ).

tff(fact_643_evalc_OSemi,axiom,
    ! [C1_1: com,S2: state,C0_1: com,S0: state,S1_2: state] :
      ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C0_1),S0),S1_2))
     => ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C1_1),S1_2),S2))
       => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(semi(C0_1,C1_1)),S0),S2)) ) ) ).

tff(fact_644_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_645_com_Osimps_I48_J,axiom,
    ! [Com1_2: com,Com2_2: com,Pname_2: pname] : ( semi(Com1_2,Com2_2) != hAPP_pname_com(body,Pname_2) ) ).

tff(fact_646_com_Osimps_I49_J,axiom,
    ! [Pname_2: pname,Com1_2: com,Com2_2: com] : ( hAPP_pname_com(body,Pname_2) != semi(Com1_2,Com2_2) ) ).

tff(fact_647_com_Osimps_I47_J,axiom,
    ! [Fun: fun_state_bool,Com: com,Com1_1: com,Com2_1: com] : ( while(Fun,Com) != semi(Com1_1,Com2_1) ) ).

tff(fact_648_com_Osimps_I46_J,axiom,
    ! [Com1_1: com,Com2_1: com,Fun: fun_state_bool,Com: com] : ( semi(Com1_1,Com2_1) != while(Fun,Com) ) ).

tff(fact_649_com_Osimps_I13_J,axiom,
    ! [Com1_3: com,Com2_3: com] : ( semi(Com1_3,Com2_3) != skip ) ).

tff(fact_650_com_Osimps_I12_J,axiom,
    ! [Com1_3: com,Com2_3: com] : ( skip != semi(Com1_3,Com2_3) ) ).

tff(fact_651_evalc__elim__cases_I4_J,axiom,
    ! [C1_1: com,C2_1: com,S: state,T: state] :
      ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(semi(C1_1,C2_1)),S),T))
     => ~ ! [S1_1: state] :
            ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C1_1),S),S1_1))
           => ~ hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C2_1),S1_1),T)) ) ) ).

tff(fact_652_evaln__elim__cases_I4_J,axiom,
    ! [C1_1: com,C2_1: com,S: state,N_2: nat,T: state] :
      ( hBOOL(hAPP_state_bool(evaln(semi(C1_1,C2_1),S,N_2),T))
     => ~ ! [S1_1: state] :
            ( hBOOL(hAPP_state_bool(evaln(C1_1,S,N_2),S1_1))
           => ~ hBOOL(hAPP_state_bool(evaln(C2_1,S1_1,N_2),T)) ) ) ).

tff(fact_653_finite__imageI,axiom,
    ! [H: fun_pn1262351275_state,F_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
     => hBOOL(hAPP_f568432088l_bool(finite280062779_state,hAPP_f1763243027e_bool(image_1918575693_state(H),F_1))) ) ).

tff(fact_654_finite__imageI,axiom,
    ! [H: fun_na1240513468iple_a,F_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
     => hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,hAPP_f1985266946a_bool(image_344477316iple_a(H),F_1))) ) ).

tff(fact_655_finite__imageI,axiom,
    ! [H: fun_na1758826658a_bool,F_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
     => hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,hAPP_f2051260670l_bool(image_1007364804a_bool(H),F_1))) ) ).

tff(fact_656_finite__imageI,axiom,
    ! [H: fun_nat_pname,F_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
     => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f654413245e_bool(image_nat_pname(H),F_1))) ) ).

tff(fact_657_finite__imageI,axiom,
    ! [H: fun_nat_nat,F_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(image_nat_nat(H),F_1))) ) ).

tff(fact_658_finite__imageI,axiom,
    ! [H: fun_Ho864122878_a_nat,F_1: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,F_1))
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f306451970t_bool(image_1778392390_a_nat(H),F_1))) ) ).

tff(fact_659_finite__imageI,axiom,
    ! [H: fun_fu328701628ol_nat,F_1: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,F_1))
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f327789228t_bool(image_2009508184ol_nat(H),F_1))) ) ).

tff(fact_660_finite__imageI,axiom,
    ! [H: fun_pname_nat,F_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f1066163005t_bool(image_pname_nat(H),F_1))) ) ).

tff(fact_661_finite__imageI,axiom,
    ! [H: fun_pn706052866iple_a,F_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
     => hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,hAPP_f1451199292a_bool(image_217098954iple_a(H),F_1))) ) ).

tff(fact_662_finite_OinsertI,axiom,
    ! [A_2: fun_Ho882550911a_bool,A_1: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,A_1))
     => hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,A_2),A_1))) ) ).

tff(fact_663_finite_OinsertI,axiom,
    ! [A_2: pname,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
     => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),A_1))) ) ).

tff(fact_664_finite_OinsertI,axiom,
    ! [A_2: hoare_2076525193iple_a,A_1: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A_1))
     => hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_2),A_1))) ) ).

tff(fact_665_finite_OinsertI,axiom,
    ! [A_2: hoare_363350478_state,A_1: fun_Ho320143120e_bool] :
      ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_1))
     => hBOOL(hAPP_f568432088l_bool(finite280062779_state,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_2),A_1))) ) ).

tff(fact_666_finite_OinsertI,axiom,
    ! [A_2: nat,A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),A_1))) ) ).

tff(fact_667_finite_OemptyI,axiom,
    hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,bot_bo214064617a_bool)) ).

tff(fact_668_finite_OemptyI,axiom,
    hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,bot_bo1285271383l_bool)) ).

tff(fact_669_finite_OemptyI,axiom,
    hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,bot_bo844097828e_bool)) ).

tff(fact_670_finite_OemptyI,axiom,
    hBOOL(hAPP_f568432088l_bool(finite280062779_state,bot_bo39439654e_bool)) ).

tff(fact_671_finite_OemptyI,axiom,
    hBOOL(hAPP_f54304608l_bool(finite_finite_nat,bot_bot_fun_nat_bool)) ).

tff(fact_672_finite__Collect__conjI,axiom,
    ! [Qa: fun_pname_bool,Pa: fun_pname_bool] :
      ( ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(collect_pname,Pa)))
        | hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(collect_pname,Qa))) )
     => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,hAPP_f1402196763l_bool(cOMBB_675860798_pname(fconj),Pa)),Qa)))) ) ).

tff(fact_673_finite__Collect__conjI,axiom,
    ! [Qa: fun_Ho882550911a_bool,Pa: fun_Ho882550911a_bool] :
      ( ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,hAPP_f642208769a_bool(collec977865374iple_a,Pa)))
        | hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,hAPP_f642208769a_bool(collec977865374iple_a,Qa))) )
     => hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,hAPP_f642208769a_bool(collec977865374iple_a,hAPP_f642208769a_bool(hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,hAPP_f268025233l_bool(cOMBB_345279363iple_a(fconj),Pa)),Qa)))) ) ).

tff(fact_674_finite__Collect__conjI,axiom,
    ! [Qa: fun_fu2026518849l_bool,Pa: fun_fu2026518849l_bool] :
      ( ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,hAPP_f354682389l_bool(collec416005674a_bool,Pa)))
        | hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,hAPP_f354682389l_bool(collec416005674a_bool,Qa))) )
     => hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,hAPP_f354682389l_bool(collec416005674a_bool,hAPP_f354682389l_bool(hAPP_f1121033001l_bool(cOMBS_204124788l_bool,hAPP_f1684296317l_bool(cOMBB_120690437a_bool(fconj),Pa)),Qa)))) ) ).

tff(fact_675_finite__Collect__conjI,axiom,
    ! [Qa: fun_nat_bool,Pa: fun_nat_bool] :
      ( ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,Pa)))
        | hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,Qa))) )
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(cOMBB_1015721476ol_nat(fconj),Pa)),Qa)))) ) ).

tff(fact_676_finite__Un,axiom,
    ! [F_1: fun_fu2026518849l_bool,Ga: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,F_1),Ga)))
    <=> ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,F_1))
        & hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,Ga)) ) ) ).

tff(fact_677_finite__Un,axiom,
    ! [F_1: fun_pname_bool,Ga: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,F_1),Ga)))
    <=> ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
        & hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,Ga)) ) ) ).

tff(fact_678_finite__Un,axiom,
    ! [F_1: fun_Ho320143120e_bool,Ga: fun_Ho320143120e_bool] :
      ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,F_1),Ga)))
    <=> ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,F_1))
        & hBOOL(hAPP_f568432088l_bool(finite280062779_state,Ga)) ) ) ).

tff(fact_679_finite__Un,axiom,
    ! [F_1: fun_Ho882550911a_bool,Ga: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,F_1),Ga)))
    <=> ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,F_1))
        & hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,Ga)) ) ) ).

tff(fact_680_finite__Un,axiom,
    ! [F_1: fun_nat_bool,Ga: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,F_1),Ga)))
    <=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
        & hBOOL(hAPP_f54304608l_bool(finite_finite_nat,Ga)) ) ) ).

tff(fact_681_finite__UnI,axiom,
    ! [Ga: fun_fu2026518849l_bool,F_1: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,F_1))
     => ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,Ga))
       => hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,F_1),Ga))) ) ) ).

tff(fact_682_finite__UnI,axiom,
    ! [Ga: fun_pname_bool,F_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,Ga))
       => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,F_1),Ga))) ) ) ).

tff(fact_683_finite__UnI,axiom,
    ! [Ga: fun_Ho320143120e_bool,F_1: fun_Ho320143120e_bool] :
      ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,F_1))
     => ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,Ga))
       => hBOOL(hAPP_f568432088l_bool(finite280062779_state,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,F_1),Ga))) ) ) ).

tff(fact_684_finite__UnI,axiom,
    ! [Ga: fun_Ho882550911a_bool,F_1: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,F_1))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,Ga))
       => hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,F_1),Ga))) ) ) ).

tff(fact_685_finite__UnI,axiom,
    ! [Ga: fun_nat_bool,F_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,Ga))
       => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,F_1),Ga))) ) ) ).

tff(fact_686_finite__Collect__disjI,axiom,
    ! [Pa: fun_pname_bool,Qa: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,hAPP_f1402196763l_bool(cOMBB_675860798_pname(fdisj),Pa)),Qa))))
    <=> ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(collect_pname,Pa)))
        & hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(collect_pname,Qa))) ) ) ).

tff(fact_687_finite__Collect__disjI,axiom,
    ! [Pa: fun_Ho882550911a_bool,Qa: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,hAPP_f642208769a_bool(collec977865374iple_a,hAPP_f642208769a_bool(hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,hAPP_f268025233l_bool(cOMBB_345279363iple_a(fdisj),Pa)),Qa))))
    <=> ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,hAPP_f642208769a_bool(collec977865374iple_a,Pa)))
        & hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,hAPP_f642208769a_bool(collec977865374iple_a,Qa))) ) ) ).

tff(fact_688_finite__Collect__disjI,axiom,
    ! [Pa: fun_fu2026518849l_bool,Qa: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,hAPP_f354682389l_bool(collec416005674a_bool,hAPP_f354682389l_bool(hAPP_f1121033001l_bool(cOMBS_204124788l_bool,hAPP_f1684296317l_bool(cOMBB_120690437a_bool(fdisj),Pa)),Qa))))
    <=> ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,hAPP_f354682389l_bool(collec416005674a_bool,Pa)))
        & hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,hAPP_f354682389l_bool(collec416005674a_bool,Qa))) ) ) ).

tff(fact_689_finite__Collect__disjI,axiom,
    ! [Pa: fun_nat_bool,Qa: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(cOMBB_1015721476ol_nat(fdisj),Pa)),Qa))))
    <=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,Pa)))
        & hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,Qa))) ) ) ).

tff(fact_690_finite__insert,axiom,
    ! [A_2: fun_Ho882550911a_bool,A_1: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,A_2),A_1)))
    <=> hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,A_1)) ) ).

tff(fact_691_finite__insert,axiom,
    ! [A_2: pname,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),A_1)))
    <=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1)) ) ).

tff(fact_692_finite__insert,axiom,
    ! [A_2: hoare_2076525193iple_a,A_1: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_2),A_1)))
    <=> hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A_1)) ) ).

tff(fact_693_finite__insert,axiom,
    ! [A_2: hoare_363350478_state,A_1: fun_Ho320143120e_bool] :
      ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_2),A_1)))
    <=> hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_1)) ) ).

tff(fact_694_finite__insert,axiom,
    ! [A_2: nat,A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),A_1)))
    <=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1)) ) ).

tff(fact_695_finite__induct,axiom,
    ! [Pa: fun_fu1342882871l_bool,F_1: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,F_1))
     => ( hBOOL(hAPP_f1490793353l_bool(Pa,bot_bo1285271383l_bool))
       => ( ! [X: fun_Ho882550911a_bool,F_2: fun_fu2026518849l_bool] :
              ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,F_2))
             => ( ~ hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,X),F_2))
               => ( hBOOL(hAPP_f1490793353l_bool(Pa,F_2))
                 => hBOOL(hAPP_f1490793353l_bool(Pa,hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,X),F_2))) ) ) )
         => hBOOL(hAPP_f1490793353l_bool(Pa,F_1)) ) ) ) ).

tff(fact_696_finite__induct,axiom,
    ! [Pa: fun_fu2026518849l_bool,F_1: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,F_1))
     => ( hBOOL(hAPP_f500854239l_bool(Pa,bot_bo214064617a_bool))
       => ( ! [X: hoare_2076525193iple_a,F_2: fun_Ho882550911a_bool] :
              ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,F_2))
             => ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X),F_2))
               => ( hBOOL(hAPP_f500854239l_bool(Pa,F_2))
                 => hBOOL(hAPP_f500854239l_bool(Pa,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X),F_2))) ) ) )
         => hBOOL(hAPP_f500854239l_bool(Pa,F_1)) ) ) ) ).

tff(fact_697_finite__induct,axiom,
    ! [Pa: fun_fu1996308102l_bool,F_1: fun_Ho320143120e_bool] :
      ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,F_1))
     => ( hBOOL(hAPP_f568432088l_bool(Pa,bot_bo39439654e_bool))
       => ( ! [X: hoare_363350478_state,F_2: fun_Ho320143120e_bool] :
              ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,F_2))
             => ( ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X),F_2))
               => ( hBOOL(hAPP_f568432088l_bool(Pa,F_2))
                 => hBOOL(hAPP_f568432088l_bool(Pa,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X),F_2))) ) ) )
         => hBOOL(hAPP_f568432088l_bool(Pa,F_1)) ) ) ) ).

tff(fact_698_finite__induct,axiom,
    ! [Pa: fun_fu425979586l_bool,F_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(Pa,bot_bot_fun_nat_bool))
       => ( ! [X: nat,F_2: fun_nat_bool] :
              ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_2))
             => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),F_2))
               => ( hBOOL(hAPP_f54304608l_bool(Pa,F_2))
                 => hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X),F_2))) ) ) )
         => hBOOL(hAPP_f54304608l_bool(Pa,F_1)) ) ) ) ).

tff(fact_699_finite__induct,axiom,
    ! [Pa: fun_fu1430349052l_bool,F_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
     => ( hBOOL(hAPP_f1664156314l_bool(Pa,bot_bo844097828e_bool))
       => ( ! [X: pname,F_2: fun_pname_bool] :
              ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_2))
             => ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),F_2))
               => ( hBOOL(hAPP_f1664156314l_bool(Pa,F_2))
                 => hBOOL(hAPP_f1664156314l_bool(Pa,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X),F_2))) ) ) )
         => hBOOL(hAPP_f1664156314l_bool(Pa,F_1)) ) ) ) ).

tff(fact_700_finite_Osimps,axiom,
    ! [A_2: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,A_2))
    <=> ( ( A_2 = bot_bo1285271383l_bool )
        | ? [A_8: fun_fu2026518849l_bool,A_7: fun_Ho882550911a_bool] :
            ( ( A_2 = hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,A_7),A_8) )
            & hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,A_8)) ) ) ) ).

tff(fact_701_finite_Osimps,axiom,
    ! [A_2: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_2))
    <=> ( ( A_2 = bot_bo844097828e_bool )
        | ? [A_8: fun_pname_bool,A_7: pname] :
            ( ( A_2 = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_7),A_8) )
            & hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_8)) ) ) ) ).

tff(fact_702_finite_Osimps,axiom,
    ! [A_2: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A_2))
    <=> ( ( A_2 = bot_bo214064617a_bool )
        | ? [A_8: fun_Ho882550911a_bool,A_7: hoare_2076525193iple_a] :
            ( ( A_2 = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,A_7),A_8) )
            & hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A_8)) ) ) ) ).

tff(fact_703_finite_Osimps,axiom,
    ! [A_2: fun_Ho320143120e_bool] :
      ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_2))
    <=> ( ( A_2 = bot_bo39439654e_bool )
        | ? [A_8: fun_Ho320143120e_bool,A_7: hoare_363350478_state] :
            ( ( A_2 = hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,A_7),A_8) )
            & hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_8)) ) ) ) ).

tff(fact_704_finite_Osimps,axiom,
    ! [A_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_2))
    <=> ( ( A_2 = bot_bot_fun_nat_bool )
        | ? [A_8: fun_nat_bool,A_7: nat] :
            ( ( A_2 = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_7),A_8) )
            & hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_8)) ) ) ) ).

tff(fact_705_pigeonhole__infinite,axiom,
    ! [F: fun_fu328701628ol_nat,A_1: fun_fu2026518849l_bool] :
      ( ~ hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,A_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f327789228t_bool(image_2009508184ol_nat(F),A_1)))
       => ? [X: fun_Ho882550911a_bool] :
            ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,X),A_1))
            & ~ hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,hAPP_f354682389l_bool(collec416005674a_bool,hAPP_f354682389l_bool(hAPP_f1121033001l_bool(cOMBS_204124788l_bool,hAPP_f1684296317l_bool(cOMBB_120690437a_bool(fconj),hAPP_f354682389l_bool(hAPP_f638520823l_bool(cOMBC_1528738210l_bool,member186597612a_bool),A_1))),hAPP_n990327308l_bool(hAPP_f524575973l_bool(cOMBC_1205527147t_bool,hAPP_f1800675941t_bool(cOMBB_1649747131a_bool(fequal_nat),F)),hAPP_f190641160ol_nat(F,X)))))) ) ) ) ).

tff(fact_706_pigeonhole__infinite,axiom,
    ! [F: fun_Ho864122878_a_nat,A_1: fun_Ho882550911a_bool] :
      ( ~ hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f306451970t_bool(image_1778392390_a_nat(F),A_1)))
       => ? [X: hoare_2076525193iple_a] :
            ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X),A_1))
            & ~ hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,hAPP_f642208769a_bool(collec977865374iple_a,hAPP_f642208769a_bool(hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,hAPP_f268025233l_bool(cOMBB_345279363iple_a(fconj),hAPP_f642208769a_bool(hAPP_f325481667a_bool(cOMBC_1529670358l_bool,member276146780iple_a),A_1))),hAPP_n1335981428a_bool(hAPP_f1040149703a_bool(cOMBC_240829083t_bool,hAPP_f570970635t_bool(cOMBB_2017898829iple_a(fequal_nat),F)),hAPP_H174304918_a_nat(F,X)))))) ) ) ) ).

tff(fact_707_pigeonhole__infinite,axiom,
    ! [F: fun_pname_nat,A_1: fun_pname_bool] :
      ( ~ hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f1066163005t_bool(image_pname_nat(F),A_1)))
       => ? [X: pname] :
            ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),A_1))
            & ~ hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,hAPP_f1402196763l_bool(cOMBB_675860798_pname(fconj),hAPP_f759274231e_bool(hAPP_f661147897e_bool(cOMBC_1058051404l_bool,member_pname),A_1))),hAPP_n1025906991e_bool(hAPP_f1715247037e_bool(cOMBC_pname_nat_bool,hAPP_f1358769483t_bool(cOMBB_523834888_pname(fequal_nat),F)),hAPP_pname_nat(F,X)))))) ) ) ) ).

tff(fact_708_pigeonhole__infinite,axiom,
    ! [F: fun_nat_nat,A_1: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(image_nat_nat(F),A_1)))
       => ? [X: nat] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),A_1))
            & ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(cOMBB_1015721476ol_nat(fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),A_1))),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,hAPP_f618557131t_bool(cOMBB_800536526ol_nat(fequal_nat),F)),hAPP_nat_nat(F,X)))))) ) ) ) ).

tff(fact_709_pigeonhole__infinite,axiom,
    ! [F: fun_na1240513468iple_a,A_1: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,hAPP_f1985266946a_bool(image_344477316iple_a(F),A_1)))
       => ? [X: nat] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),A_1))
            & ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(cOMBB_1015721476ol_nat(fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),A_1))),hAPP_H904886772t_bool(hAPP_f1460432071t_bool(cOMBC_2097822937a_bool,hAPP_f1324734153a_bool(cOMBB_315605200ol_nat(fequal1589671816iple_a),F)),hAPP_n887873492iple_a(F,X)))))) ) ) ) ).

tff(fact_710_pigeonhole__infinite,axiom,
    ! [F: fun_na1758826658a_bool,A_1: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
     => ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,hAPP_f2051260670l_bool(image_1007364804a_bool(F),A_1)))
       => ? [X: nat] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),A_1))
            & ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(cOMBB_1015721476ol_nat(fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),A_1))),hAPP_f306451970t_bool(hAPP_f567756997t_bool(cOMBC_1759100375l_bool,hAPP_f947867151l_bool(cOMBB_262857860ol_nat(fequal291249344a_bool),F)),hAPP_n1335981428a_bool(F,X)))))) ) ) ) ).

tff(fact_711_pigeonhole__infinite,axiom,
    ! [F: fun_nat_pname,A_1: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f654413245e_bool(image_nat_pname(F),A_1)))
       => ? [X: nat] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),A_1))
            & ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(cOMBB_1015721476ol_nat(fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),A_1))),hAPP_p1499970991t_bool(hAPP_f998021053t_bool(cOMBC_nat_pname_bool,hAPP_f414474559e_bool(cOMBB_1212655066ol_nat(fequal_pname),F)),hAPP_nat_pname(F,X)))))) ) ) ) ).

tff(fact_712_pigeonhole__infinite,axiom,
    ! [F: fun_pn1262351275_state,A_1: fun_pname_bool] :
      ( ~ hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
     => ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,hAPP_f1763243027e_bool(image_1918575693_state(F),A_1)))
       => ? [X: pname] :
            ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),A_1))
            & ~ hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,hAPP_f1402196763l_bool(cOMBB_675860798_pname(fconj),hAPP_f759274231e_bool(hAPP_f661147897e_bool(cOMBC_1058051404l_bool,member_pname),A_1))),hAPP_H2033014795e_bool(hAPP_f443781369e_bool(cOMBC_1422551904e_bool,hAPP_f1284518107e_bool(cOMBB_1755629310_pname(fequal1026769423_state),F)),hAPP_p1337579261_state(F,X)))))) ) ) ) ).

tff(fact_713_pigeonhole__infinite,axiom,
    ! [F: fun_pn1811694044a_bool,A_1: fun_pname_bool] :
      ( ~ hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
     => ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,hAPP_f1243652676l_bool(image_1433143038a_bool(F),A_1)))
       => ? [X: pname] :
            ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),A_1))
            & ~ hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,hAPP_f1402196763l_bool(cOMBB_675860798_pname(fconj),hAPP_f759274231e_bool(hAPP_f661147897e_bool(cOMBC_1058051404l_bool,member_pname),A_1))),hAPP_f213795644e_bool(hAPP_f584257849e_bool(cOMBC_1126778257l_bool,hAPP_f940735259l_bool(cOMBB_1102432190_pname(fequal291249344a_bool),F)),hAPP_p1160247726a_bool(F,X)))))) ) ) ) ).

tff(fact_714_pigeonhole__infinite,axiom,
    ! [F: fun_pname_pname,A_1: fun_pname_bool] :
      ( ~ hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(image_pname_pname(F),A_1)))
       => ? [X: pname] :
            ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),A_1))
            & ~ hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,hAPP_f1402196763l_bool(cOMBB_675860798_pname(fconj),hAPP_f759274231e_bool(hAPP_f661147897e_bool(cOMBC_1058051404l_bool,member_pname),A_1))),hAPP_p61793385e_bool(hAPP_f22061361e_bool(cOMBC_1149511130e_bool,hAPP_f1336811455e_bool(cOMBB_542850580_pname(fequal_pname),F)),hAPP_pname_pname(F,X)))))) ) ) ) ).

tff(fact_715_pigeonhole__infinite,axiom,
    ! [F: fun_pn706052866iple_a,A_1: fun_pname_bool] :
      ( ~ hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,hAPP_f1451199292a_bool(image_217098954iple_a(F),A_1)))
       => ? [X: pname] :
            ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),A_1))
            & ~ hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,hAPP_f1402196763l_bool(cOMBB_675860798_pname(fconj),hAPP_f759274231e_bool(hAPP_f661147897e_bool(cOMBC_1058051404l_bool,member_pname),A_1))),hAPP_H1108528686e_bool(hAPP_f1658749755e_bool(cOMBC_726551967a_bool,hAPP_f830638537a_bool(cOMBB_1838759434_pname(fequal1589671816iple_a),F)),hAPP_p973116442iple_a(F,X)))))) ) ) ) ).

tff(fact_716_nonempty__iff,axiom,
    ! [A_1: fun_nat_bool] :
      ( ( A_1 != bot_bot_fun_nat_bool )
    <=> ? [X: nat,B_6: fun_nat_bool] :
          ( ( A_1 = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X),B_6) )
          & ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),B_6)) ) ) ).

tff(fact_717_nonempty__iff,axiom,
    ! [A_1: fun_fu2026518849l_bool] :
      ( ( A_1 != bot_bo1285271383l_bool )
    <=> ? [X: fun_Ho882550911a_bool,B_6: fun_fu2026518849l_bool] :
          ( ( A_1 = hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,X),B_6) )
          & ~ hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,X),B_6)) ) ) ).

tff(fact_718_nonempty__iff,axiom,
    ! [A_1: fun_Ho882550911a_bool] :
      ( ( A_1 != bot_bo214064617a_bool )
    <=> ? [X: hoare_2076525193iple_a,B_6: fun_Ho882550911a_bool] :
          ( ( A_1 = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X),B_6) )
          & ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X),B_6)) ) ) ).

tff(fact_719_nonempty__iff,axiom,
    ! [A_1: fun_Ho320143120e_bool] :
      ( ( A_1 != bot_bo39439654e_bool )
    <=> ? [X: hoare_363350478_state,B_6: fun_Ho320143120e_bool] :
          ( ( A_1 = hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X),B_6) )
          & ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X),B_6)) ) ) ).

tff(fact_720_nonempty__iff,axiom,
    ! [A_1: fun_pname_bool] :
      ( ( A_1 != bot_bo844097828e_bool )
    <=> ? [X: pname,B_6: fun_pname_bool] :
          ( ( A_1 = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X),B_6) )
          & ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),B_6)) ) ) ).

tff(fact_721_folding__one__idem_Ounion__idem,axiom,
    ! [B_2: fun_fu2026518849l_bool,A_1: fun_fu2026518849l_bool,F: fun_fu2051904533a_bool,F_1: fun_fu1819779801a_bool] :
      ( hBOOL(finite344344197a_bool(F,F_1))
     => ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,A_1))
       => ( ( A_1 != bot_bo1285271383l_bool )
         => ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,B_2))
           => ( ( B_2 != bot_bo1285271383l_bool )
             => ( hAPP_f1758315691a_bool(F_1,hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_1),B_2)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(F,hAPP_f1758315691a_bool(F_1,A_1)),hAPP_f1758315691a_bool(F_1,B_2)) ) ) ) ) ) ) ).

tff(fact_722_folding__one__idem_Ounion__idem,axiom,
    ! [B_2: fun_pname_bool,A_1: fun_pname_bool,F: fun_pn1208287926_pname,F_1: fun_fu1499449723_pname] :
      ( hBOOL(finite89670078_pname(F,F_1))
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
       => ( ( A_1 != bot_bo844097828e_bool )
         => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B_2))
           => ( ( B_2 != bot_bo844097828e_bool )
             => ( hAPP_f1297739591_pname(F_1,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),B_2)) = hAPP_pname_pname(hAPP_p1170154830_pname(F,hAPP_f1297739591_pname(F_1,A_1)),hAPP_f1297739591_pname(F_1,B_2)) ) ) ) ) ) ) ).

tff(fact_723_folding__one__idem_Ounion__idem,axiom,
    ! [B_2: fun_Ho320143120e_bool,A_1: fun_Ho320143120e_bool,F: fun_Ho31865764_state,F_1: fun_fu700438391_state] :
      ( hBOOL(finite2031572_state(F,F_1))
     => ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_1))
       => ( ( A_1 != bot_bo39439654e_bool )
         => ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,B_2))
           => ( ( B_2 != bot_bo39439654e_bool )
             => ( hAPP_f1242401993_state(F_1,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_1),B_2)) = hAPP_H1982017439_state(hAPP_H230141040_state(F,hAPP_f1242401993_state(F_1,A_1)),hAPP_f1242401993_state(F_1,B_2)) ) ) ) ) ) ) ).

tff(fact_724_folding__one__idem_Ounion__idem,axiom,
    ! [B_2: fun_Ho882550911a_bool,A_1: fun_Ho882550911a_bool,F: fun_Ho483360497iple_a,F_1: fun_fu253747899iple_a] :
      ( hBOOL(finite1659845635iple_a(F,F_1))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A_1))
       => ( ( A_1 != bot_bo214064617a_bool )
         => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,B_2))
           => ( ( B_2 != bot_bo214064617a_bool )
             => ( hAPP_f2009966983iple_a(F_1,hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_1),B_2)) = hAPP_H1789220885iple_a(hAPP_H1417936777iple_a(F,hAPP_f2009966983iple_a(F_1,A_1)),hAPP_f2009966983iple_a(F_1,B_2)) ) ) ) ) ) ) ).

tff(fact_725_folding__one__idem_Ounion__idem,axiom,
    ! [B_2: fun_nat_bool,A_1: fun_nat_bool,F: fun_nat_fun_nat_nat,F_1: fun_fun_nat_bool_nat] :
      ( hBOOL(finite795500164em_nat(F,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
       => ( ( A_1 != bot_bot_fun_nat_bool )
         => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B_2))
           => ( ( B_2 != bot_bot_fun_nat_bool )
             => ( hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_1),B_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,hAPP_f22106695ol_nat(F_1,A_1)),hAPP_f22106695ol_nat(F_1,B_2)) ) ) ) ) ) ) ).

tff(fact_726_folding__one__idem_Oinsert__idem,axiom,
    ! [X_2: fun_Ho882550911a_bool,A_1: fun_fu2026518849l_bool,F: fun_fu2051904533a_bool,F_1: fun_fu1819779801a_bool] :
      ( hBOOL(finite344344197a_bool(F,F_1))
     => ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,A_1))
       => ( ( A_1 != bot_bo1285271383l_bool )
         => ( hAPP_f1758315691a_bool(F_1,hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,X_2),A_1)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(F,X_2),hAPP_f1758315691a_bool(F_1,A_1)) ) ) ) ) ).

tff(fact_727_folding__one__idem_Oinsert__idem,axiom,
    ! [X_2: pname,A_1: fun_pname_bool,F: fun_pn1208287926_pname,F_1: fun_fu1499449723_pname] :
      ( hBOOL(finite89670078_pname(F,F_1))
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
       => ( ( A_1 != bot_bo844097828e_bool )
         => ( hAPP_f1297739591_pname(F_1,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),A_1)) = hAPP_pname_pname(hAPP_p1170154830_pname(F,X_2),hAPP_f1297739591_pname(F_1,A_1)) ) ) ) ) ).

tff(fact_728_folding__one__idem_Oinsert__idem,axiom,
    ! [X_2: hoare_2076525193iple_a,A_1: fun_Ho882550911a_bool,F: fun_Ho483360497iple_a,F_1: fun_fu253747899iple_a] :
      ( hBOOL(finite1659845635iple_a(F,F_1))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A_1))
       => ( ( A_1 != bot_bo214064617a_bool )
         => ( hAPP_f2009966983iple_a(F_1,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_2),A_1)) = hAPP_H1789220885iple_a(hAPP_H1417936777iple_a(F,X_2),hAPP_f2009966983iple_a(F_1,A_1)) ) ) ) ) ).

tff(fact_729_folding__one__idem_Oinsert__idem,axiom,
    ! [X_2: hoare_363350478_state,A_1: fun_Ho320143120e_bool,F: fun_Ho31865764_state,F_1: fun_fu700438391_state] :
      ( hBOOL(finite2031572_state(F,F_1))
     => ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_1))
       => ( ( A_1 != bot_bo39439654e_bool )
         => ( hAPP_f1242401993_state(F_1,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_2),A_1)) = hAPP_H1982017439_state(hAPP_H230141040_state(F,X_2),hAPP_f1242401993_state(F_1,A_1)) ) ) ) ) ).

tff(fact_730_folding__one__idem_Oinsert__idem,axiom,
    ! [X_2: nat,A_1: fun_nat_bool,F: fun_nat_fun_nat_nat,F_1: fun_fun_nat_bool_nat] :
      ( hBOOL(finite795500164em_nat(F,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
       => ( ( A_1 != bot_bot_fun_nat_bool )
         => ( hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),A_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_2),hAPP_f22106695ol_nat(F_1,A_1)) ) ) ) ) ).

tff(fact_731_image__eq__fold__image,axiom,
    ! [F: fun_pn1262351275_state,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
     => ( hAPP_f1763243027e_bool(image_1918575693_state(F),A_1) = hAPP_f1763243027e_bool(hAPP_f981071554e_bool(finite517504007_pname(semila1579166468e_bool,hAPP_f265219918e_bool(hAPP_f1853922773e_bool(cOMBC_1001821703e_bool,hAPP_f43306888e_bool(cOMBB_1717464465_pname(insert1330351828_state),F)),bot_bo39439654e_bool)),bot_bo39439654e_bool),A_1) ) ) ).

tff(fact_732_image__eq__fold__image,axiom,
    ! [F: fun_Ho1518252029iple_a,A_1: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A_1))
     => ( hAPP_f642208769a_bool(image_1416816325iple_a(F),A_1) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(finite2067172233iple_a(semila531373451a_bool,hAPP_f219434403a_bool(hAPP_f1824299331a_bool(cOMBC_175592824a_bool,hAPP_f233660527a_bool(cOMBB_1751994227iple_a(insert1582918915iple_a),F)),bot_bo214064617a_bool)),bot_bo214064617a_bool),A_1) ) ) ).

tff(fact_733_image__eq__fold__image,axiom,
    ! [F: fun_fu253747899iple_a,A_1: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,A_1))
     => ( hAPP_f1758315691a_bool(image_1469988567iple_a(F),A_1) = hAPP_f1758315691a_bool(hAPP_f876057179a_bool(finite1197875071a_bool(semila531373451a_bool,hAPP_f1418638817a_bool(hAPP_f86610733a_bool(cOMBC_2005334440a_bool,hAPP_f15901907a_bool(cOMBB_1462920981a_bool(insert1582918915iple_a),F)),bot_bo214064617a_bool)),bot_bo214064617a_bool),A_1) ) ) ).

tff(fact_734_image__eq__fold__image,axiom,
    ! [F: fun_nat_nat,A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
     => ( hAPP_f800510211t_bool(image_nat_nat(F),A_1) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(finite2098682953ol_nat(semila465093516t_bool,hAPP_f1570313510t_bool(hAPP_f688831301t_bool(cOMBC_178881787t_bool,hAPP_f934069744t_bool(cOMBB_1141542387ol_nat(insert_nat),F)),bot_bot_fun_nat_bool)),bot_bot_fun_nat_bool),A_1) ) ) ).

tff(fact_735_image__eq__fold__image,axiom,
    ! [F: fun_na1758826658a_bool,A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
     => ( hAPP_f2051260670l_bool(image_1007364804a_bool(F),A_1) = hAPP_f2051260670l_bool(hAPP_f1523760350l_bool(finite794352476ol_nat(semila1787532597l_bool,hAPP_f2084083016l_bool(hAPP_f1288222677l_bool(cOMBC_1603421709l_bool,hAPP_f564435019l_bool(cOMBB_1921254728ol_nat(insert838260101a_bool),F)),bot_bo1285271383l_bool)),bot_bo1285271383l_bool),A_1) ) ) ).

tff(fact_736_image__eq__fold__image,axiom,
    ! [F: fun_nat_pname,A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
     => ( hAPP_f654413245e_bool(image_nat_pname(F),A_1) = hAPP_f654413245e_bool(hAPP_f1319825314e_bool(finite491497871ol_nat(semila278973382e_bool,hAPP_f1501416730e_bool(hAPP_f883303749e_bool(cOMBC_1771919087e_bool,hAPP_f446737578e_bool(cOMBB_1209446585ol_nat(insert_pname),F)),bot_bo844097828e_bool)),bot_bo844097828e_bool),A_1) ) ) ).

tff(fact_737_image__eq__fold__image,axiom,
    ! [F: fun_na1240513468iple_a,A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
     => ( hAPP_f1985266946a_bool(image_344477316iple_a(F),A_1) = hAPP_f1985266946a_bool(hAPP_f1404638946a_bool(finite898656778ol_nat(semila531373451a_bool,hAPP_f947130148a_bool(hAPP_f1795987397a_bool(cOMBC_58632185a_bool,hAPP_f917287855a_bool(cOMBB_1767297524ol_nat(insert1582918915iple_a),F)),bot_bo214064617a_bool)),bot_bo214064617a_bool),A_1) ) ) ).

tff(fact_738_image__eq__fold__image,axiom,
    ! [F: fun_na63265393_state,A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
     => ( hAPP_f1778645709e_bool(image_1316773907_state(F),A_1) = hAPP_f1778645709e_bool(hAPP_f559554044e_bool(finite1201106253ol_nat(semila1579166468e_bool,hAPP_f252359560e_bool(hAPP_f1331227093e_bool(cOMBC_1991019085e_bool,hAPP_f2008644348e_bool(cOMBB_1813251031ol_nat(insert1330351828_state),F)),bot_bo39439654e_bool)),bot_bo39439654e_bool),A_1) ) ) ).

tff(fact_739_image__eq__fold__image,axiom,
    ! [F: fun_pn706052866iple_a,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
     => ( hAPP_f1451199292a_bool(image_217098954iple_a(F),A_1) = hAPP_f1451199292a_bool(hAPP_f409981212a_bool(finite86305348_pname(semila531373451a_bool,hAPP_f406822750a_bool(hAPP_f1969289785a_bool(cOMBC_577917619a_bool,hAPP_f1984020655a_bool(cOMBB_1633811758_pname(insert1582918915iple_a),F)),bot_bo214064617a_bool)),bot_bo214064617a_bool),A_1) ) ) ).

tff(fact_740_finite__ne__induct,axiom,
    ! [Pa: fun_fu1342882871l_bool,F_1: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,F_1))
     => ( ( F_1 != bot_bo1285271383l_bool )
       => ( ! [X: fun_Ho882550911a_bool] : hBOOL(hAPP_f1490793353l_bool(Pa,hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,X),bot_bo1285271383l_bool)))
         => ( ! [X: fun_Ho882550911a_bool,F_2: fun_fu2026518849l_bool] :
                ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,F_2))
               => ( ( F_2 != bot_bo1285271383l_bool )
                 => ( ~ hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,X),F_2))
                   => ( hBOOL(hAPP_f1490793353l_bool(Pa,F_2))
                     => hBOOL(hAPP_f1490793353l_bool(Pa,hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,X),F_2))) ) ) ) )
           => hBOOL(hAPP_f1490793353l_bool(Pa,F_1)) ) ) ) ) ).

tff(fact_741_finite__ne__induct,axiom,
    ! [Pa: fun_fu2026518849l_bool,F_1: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,F_1))
     => ( ( F_1 != bot_bo214064617a_bool )
       => ( ! [X: hoare_2076525193iple_a] : hBOOL(hAPP_f500854239l_bool(Pa,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X),bot_bo214064617a_bool)))
         => ( ! [X: hoare_2076525193iple_a,F_2: fun_Ho882550911a_bool] :
                ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,F_2))
               => ( ( F_2 != bot_bo214064617a_bool )
                 => ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X),F_2))
                   => ( hBOOL(hAPP_f500854239l_bool(Pa,F_2))
                     => hBOOL(hAPP_f500854239l_bool(Pa,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X),F_2))) ) ) ) )
           => hBOOL(hAPP_f500854239l_bool(Pa,F_1)) ) ) ) ) ).

tff(fact_742_finite__ne__induct,axiom,
    ! [Pa: fun_fu1996308102l_bool,F_1: fun_Ho320143120e_bool] :
      ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,F_1))
     => ( ( F_1 != bot_bo39439654e_bool )
       => ( ! [X: hoare_363350478_state] : hBOOL(hAPP_f568432088l_bool(Pa,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X),bot_bo39439654e_bool)))
         => ( ! [X: hoare_363350478_state,F_2: fun_Ho320143120e_bool] :
                ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,F_2))
               => ( ( F_2 != bot_bo39439654e_bool )
                 => ( ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X),F_2))
                   => ( hBOOL(hAPP_f568432088l_bool(Pa,F_2))
                     => hBOOL(hAPP_f568432088l_bool(Pa,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X),F_2))) ) ) ) )
           => hBOOL(hAPP_f568432088l_bool(Pa,F_1)) ) ) ) ) ).

tff(fact_743_finite__ne__induct,axiom,
    ! [Pa: fun_fu425979586l_bool,F_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
     => ( ( F_1 != bot_bot_fun_nat_bool )
       => ( ! [X: nat] : hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X),bot_bot_fun_nat_bool)))
         => ( ! [X: nat,F_2: fun_nat_bool] :
                ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_2))
               => ( ( F_2 != bot_bot_fun_nat_bool )
                 => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),F_2))
                   => ( hBOOL(hAPP_f54304608l_bool(Pa,F_2))
                     => hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X),F_2))) ) ) ) )
           => hBOOL(hAPP_f54304608l_bool(Pa,F_1)) ) ) ) ) ).

tff(fact_744_finite__ne__induct,axiom,
    ! [Pa: fun_fu1430349052l_bool,F_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
     => ( ( F_1 != bot_bo844097828e_bool )
       => ( ! [X: pname] : hBOOL(hAPP_f1664156314l_bool(Pa,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X),bot_bo844097828e_bool)))
         => ( ! [X: pname,F_2: fun_pname_bool] :
                ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_2))
               => ( ( F_2 != bot_bo844097828e_bool )
                 => ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),F_2))
                   => ( hBOOL(hAPP_f1664156314l_bool(Pa,F_2))
                     => hBOOL(hAPP_f1664156314l_bool(Pa,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X),F_2))) ) ) ) )
           => hBOOL(hAPP_f1664156314l_bool(Pa,F_1)) ) ) ) ) ).

tff(fact_745_folding__one__idem_Oidem,axiom,
    ! [X_2: nat,F: fun_nat_fun_nat_nat,F_1: fun_fun_nat_bool_nat] :
      ( hBOOL(finite795500164em_nat(F,F_1))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_2),X_2) = X_2 ) ) ).

tff(fact_746_folding__one__idem_Oidem,axiom,
    ! [X_2: pname,F: fun_pn1208287926_pname,F_1: fun_fu1499449723_pname] :
      ( hBOOL(finite89670078_pname(F,F_1))
     => ( hAPP_pname_pname(hAPP_p1170154830_pname(F,X_2),X_2) = X_2 ) ) ).

tff(fact_747_folding__one__idem_Oidem,axiom,
    ! [X_2: hoare_2076525193iple_a,F: fun_Ho483360497iple_a,F_1: fun_fu253747899iple_a] :
      ( hBOOL(finite1659845635iple_a(F,F_1))
     => ( hAPP_H1789220885iple_a(hAPP_H1417936777iple_a(F,X_2),X_2) = X_2 ) ) ).

tff(fact_748_fold__image__empty,axiom,
    ! [F: fun_fu2051904533a_bool,G: fun_pn1811694044a_bool,Z_1: fun_Ho882550911a_bool] : ( hAPP_f1451199292a_bool(hAPP_f409981212a_bool(finite86305348_pname(F,G),Z_1),bot_bo844097828e_bool) = Z_1 ) ).

tff(fact_749_folding__one__idem_Oin__idem,axiom,
    ! [X_2: fun_Ho882550911a_bool,A_1: fun_fu2026518849l_bool,F: fun_fu2051904533a_bool,F_1: fun_fu1819779801a_bool] :
      ( hBOOL(finite344344197a_bool(F,F_1))
     => ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,A_1))
       => ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,X_2),A_1))
         => ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(F,X_2),hAPP_f1758315691a_bool(F_1,A_1)) = hAPP_f1758315691a_bool(F_1,A_1) ) ) ) ) ).

tff(fact_750_folding__one__idem_Oin__idem,axiom,
    ! [X_2: hoare_2076525193iple_a,A_1: fun_Ho882550911a_bool,F: fun_Ho483360497iple_a,F_1: fun_fu253747899iple_a] :
      ( hBOOL(finite1659845635iple_a(F,F_1))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A_1))
       => ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_2),A_1))
         => ( hAPP_H1789220885iple_a(hAPP_H1417936777iple_a(F,X_2),hAPP_f2009966983iple_a(F_1,A_1)) = hAPP_f2009966983iple_a(F_1,A_1) ) ) ) ) ).

tff(fact_751_folding__one__idem_Oin__idem,axiom,
    ! [X_2: nat,A_1: fun_nat_bool,F: fun_nat_fun_nat_nat,F_1: fun_fun_nat_bool_nat] :
      ( hBOOL(finite795500164em_nat(F,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
       => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A_1))
         => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_2),hAPP_f22106695ol_nat(F_1,A_1)) = hAPP_f22106695ol_nat(F_1,A_1) ) ) ) ) ).

tff(fact_752_folding__one__idem_Oin__idem,axiom,
    ! [X_2: pname,A_1: fun_pname_bool,F: fun_pn1208287926_pname,F_1: fun_fu1499449723_pname] :
      ( hBOOL(finite89670078_pname(F,F_1))
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
       => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A_1))
         => ( hAPP_pname_pname(hAPP_p1170154830_pname(F,X_2),hAPP_f1297739591_pname(F_1,A_1)) = hAPP_f1297739591_pname(F_1,A_1) ) ) ) ) ).

tff(fact_753_folding__one__idem_Ohom__commute,axiom,
    ! [N_3: fun_Ho882550911a_bool,H: fun_Ho1518252029iple_a,F: fun_Ho483360497iple_a,F_1: fun_fu253747899iple_a] :
      ( hBOOL(finite1659845635iple_a(F,F_1))
     => ( ! [X: hoare_2076525193iple_a,Y_8: hoare_2076525193iple_a] : ( hAPP_H1789220885iple_a(H,hAPP_H1789220885iple_a(hAPP_H1417936777iple_a(F,X),Y_8)) = hAPP_H1789220885iple_a(hAPP_H1417936777iple_a(F,hAPP_H1789220885iple_a(H,X)),hAPP_H1789220885iple_a(H,Y_8)) )
       => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,N_3))
         => ( ( N_3 != bot_bo214064617a_bool )
           => ( hAPP_H1789220885iple_a(H,hAPP_f2009966983iple_a(F_1,N_3)) = hAPP_f2009966983iple_a(F_1,hAPP_f642208769a_bool(image_1416816325iple_a(H),N_3)) ) ) ) ) ) ).

tff(fact_754_folding__one__idem_Ohom__commute,axiom,
    ! [N_3: fun_fu2026518849l_bool,H: fun_fu1663341283a_bool,F: fun_fu2051904533a_bool,F_1: fun_fu1819779801a_bool] :
      ( hBOOL(finite344344197a_bool(F,F_1))
     => ( ! [X: fun_Ho882550911a_bool,Y_8: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(H,hAPP_f642208769a_bool(hAPP_f1418638817a_bool(F,X),Y_8)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(F,hAPP_f642208769a_bool(H,X)),hAPP_f642208769a_bool(H,Y_8)) )
       => ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,N_3))
         => ( ( N_3 != bot_bo1285271383l_bool )
           => ( hAPP_f642208769a_bool(H,hAPP_f1758315691a_bool(F_1,N_3)) = hAPP_f1758315691a_bool(F_1,hAPP_f354682389l_bool(image_1317771953a_bool(H),N_3)) ) ) ) ) ) ).

tff(fact_755_folding__one__idem_Ohom__commute,axiom,
    ! [N_3: fun_pname_bool,H: fun_pname_pname,F: fun_pn1208287926_pname,F_1: fun_fu1499449723_pname] :
      ( hBOOL(finite89670078_pname(F,F_1))
     => ( ! [X: pname,Y_8: pname] : ( hAPP_pname_pname(H,hAPP_pname_pname(hAPP_p1170154830_pname(F,X),Y_8)) = hAPP_pname_pname(hAPP_p1170154830_pname(F,hAPP_pname_pname(H,X)),hAPP_pname_pname(H,Y_8)) )
       => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,N_3))
         => ( ( N_3 != bot_bo844097828e_bool )
           => ( hAPP_pname_pname(H,hAPP_f1297739591_pname(F_1,N_3)) = hAPP_f1297739591_pname(F_1,hAPP_f759274231e_bool(image_pname_pname(H),N_3)) ) ) ) ) ) ).

tff(fact_756_folding__one__idem_Ohom__commute,axiom,
    ! [N_3: fun_Ho320143120e_bool,H: fun_Ho901937921_state,F: fun_Ho31865764_state,F_1: fun_fu700438391_state] :
      ( hBOOL(finite2031572_state(F,F_1))
     => ( ! [X: hoare_363350478_state,Y_8: hoare_363350478_state] : ( hAPP_H1982017439_state(H,hAPP_H1982017439_state(hAPP_H230141040_state(F,X),Y_8)) = hAPP_H1982017439_state(hAPP_H230141040_state(F,hAPP_H1982017439_state(H,X)),hAPP_H1982017439_state(H,Y_8)) )
       => ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,N_3))
         => ( ( N_3 != bot_bo39439654e_bool )
           => ( hAPP_H1982017439_state(H,hAPP_f1242401993_state(F_1,N_3)) = hAPP_f1242401993_state(F_1,hAPP_f1749453397e_bool(image_2068641359_state(H),N_3)) ) ) ) ) ) ).

tff(fact_757_folding__one__idem_Ohom__commute,axiom,
    ! [N_3: fun_nat_bool,H: fun_nat_nat,F: fun_nat_fun_nat_nat,F_1: fun_fun_nat_bool_nat] :
      ( hBOOL(finite795500164em_nat(F,F_1))
     => ( ! [X: nat,Y_8: nat] : ( hAPP_nat_nat(H,hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X),Y_8)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,hAPP_nat_nat(H,X)),hAPP_nat_nat(H,Y_8)) )
       => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_3))
         => ( ( N_3 != bot_bot_fun_nat_bool )
           => ( hAPP_nat_nat(H,hAPP_f22106695ol_nat(F_1,N_3)) = hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(image_nat_nat(H),N_3)) ) ) ) ) ) ).

tff(fact_758_comm__monoid__big_OF__eq,axiom,
    ! [G: fun_pn1811694044a_bool,A_1: fun_pname_bool,F: fun_fu2051904533a_bool,Z_1: fun_Ho882550911a_bool,F_1: fun_fu1587922867a_bool] :
      ( hBOOL(big_co935026908_pname(F,Z_1,F_1))
     => ( ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
         => ( hAPP_f1451199292a_bool(hAPP_f205077375a_bool(F_1,G),A_1) = hAPP_f1451199292a_bool(hAPP_f409981212a_bool(finite86305348_pname(F,G),Z_1),A_1) ) )
        & ( ~ hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
         => ( hAPP_f1451199292a_bool(hAPP_f205077375a_bool(F_1,G),A_1) = Z_1 ) ) ) ) ).

tff(fact_759_folding__one_Oinsert,axiom,
    ! [X_2: fun_Ho882550911a_bool,A_1: fun_fu2026518849l_bool,F: fun_fu2051904533a_bool,F_1: fun_fu1819779801a_bool] :
      ( hBOOL(finite1520073154a_bool(F,F_1))
     => ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,A_1))
       => ( ~ hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,X_2),A_1))
         => ( ( A_1 != bot_bo1285271383l_bool )
           => ( hAPP_f1758315691a_bool(F_1,hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,X_2),A_1)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(F,X_2),hAPP_f1758315691a_bool(F_1,A_1)) ) ) ) ) ) ).

tff(fact_760_folding__one_Oinsert,axiom,
    ! [X_2: hoare_2076525193iple_a,A_1: fun_Ho882550911a_bool,F: fun_Ho483360497iple_a,F_1: fun_fu253747899iple_a] :
      ( hBOOL(finite232328454iple_a(F,F_1))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A_1))
       => ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_2),A_1))
         => ( ( A_1 != bot_bo214064617a_bool )
           => ( hAPP_f2009966983iple_a(F_1,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_2),A_1)) = hAPP_H1789220885iple_a(hAPP_H1417936777iple_a(F,X_2),hAPP_f2009966983iple_a(F_1,A_1)) ) ) ) ) ) ).

tff(fact_761_folding__one_Oinsert,axiom,
    ! [X_2: hoare_363350478_state,A_1: fun_Ho320143120e_bool,F: fun_Ho31865764_state,F_1: fun_fu700438391_state] :
      ( hBOOL(finite269920017_state(F,F_1))
     => ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_1))
       => ( ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_2),A_1))
         => ( ( A_1 != bot_bo39439654e_bool )
           => ( hAPP_f1242401993_state(F_1,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_2),A_1)) = hAPP_H1982017439_state(hAPP_H230141040_state(F,X_2),hAPP_f1242401993_state(F_1,A_1)) ) ) ) ) ) ).

tff(fact_762_folding__one_Oinsert,axiom,
    ! [X_2: nat,A_1: fun_nat_bool,F: fun_nat_fun_nat_nat,F_1: fun_fun_nat_bool_nat] :
      ( hBOOL(finite988810631ne_nat(F,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
       => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A_1))
         => ( ( A_1 != bot_bot_fun_nat_bool )
           => ( hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),A_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_2),hAPP_f22106695ol_nat(F_1,A_1)) ) ) ) ) ) ).

tff(fact_763_folding__one_Oinsert,axiom,
    ! [X_2: pname,A_1: fun_pname_bool,F: fun_pn1208287926_pname,F_1: fun_fu1499449723_pname] :
      ( hBOOL(finite1282449217_pname(F,F_1))
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
       => ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A_1))
         => ( ( A_1 != bot_bo844097828e_bool )
           => ( hAPP_f1297739591_pname(F_1,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),A_1)) = hAPP_pname_pname(hAPP_p1170154830_pname(F,X_2),hAPP_f1297739591_pname(F_1,A_1)) ) ) ) ) ) ).

tff(fact_764_folding__one_Osingleton,axiom,
    ! [X_2: nat,F: fun_nat_fun_nat_nat,F_1: fun_fun_nat_bool_nat] :
      ( hBOOL(finite988810631ne_nat(F,F_1))
     => ( hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),bot_bot_fun_nat_bool)) = X_2 ) ) ).

tff(fact_765_folding__one_Osingleton,axiom,
    ! [X_2: fun_Ho882550911a_bool,F: fun_fu2051904533a_bool,F_1: fun_fu1819779801a_bool] :
      ( hBOOL(finite1520073154a_bool(F,F_1))
     => ( hAPP_f1758315691a_bool(F_1,hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,X_2),bot_bo1285271383l_bool)) = X_2 ) ) ).

tff(fact_766_folding__one_Osingleton,axiom,
    ! [X_2: pname,F: fun_pn1208287926_pname,F_1: fun_fu1499449723_pname] :
      ( hBOOL(finite1282449217_pname(F,F_1))
     => ( hAPP_f1297739591_pname(F_1,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),bot_bo844097828e_bool)) = X_2 ) ) ).

tff(fact_767_folding__one_Osingleton,axiom,
    ! [X_2: hoare_2076525193iple_a,F: fun_Ho483360497iple_a,F_1: fun_fu253747899iple_a] :
      ( hBOOL(finite232328454iple_a(F,F_1))
     => ( hAPP_f2009966983iple_a(F_1,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_2),bot_bo214064617a_bool)) = X_2 ) ) ).

tff(fact_768_folding__one_Osingleton,axiom,
    ! [X_2: hoare_363350478_state,F: fun_Ho31865764_state,F_1: fun_fu700438391_state] :
      ( hBOOL(finite269920017_state(F,F_1))
     => ( hAPP_f1242401993_state(F_1,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_2),bot_bo39439654e_bool)) = X_2 ) ) ).

tff(fact_769_folding__one_Oclosed,axiom,
    ! [A_1: fun_fu2026518849l_bool,F: fun_fu2051904533a_bool,F_1: fun_fu1819779801a_bool] :
      ( hBOOL(finite1520073154a_bool(F,F_1))
     => ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,A_1))
       => ( ( A_1 != bot_bo1285271383l_bool )
         => ( ! [X: fun_Ho882550911a_bool,Y_8: fun_Ho882550911a_bool] : hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,hAPP_f642208769a_bool(hAPP_f1418638817a_bool(F,X),Y_8)),hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,X),hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,Y_8),bot_bo1285271383l_bool))))
           => hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,hAPP_f1758315691a_bool(F_1,A_1)),A_1)) ) ) ) ) ).

tff(fact_770_folding__one_Oclosed,axiom,
    ! [A_1: fun_Ho882550911a_bool,F: fun_Ho483360497iple_a,F_1: fun_fu253747899iple_a] :
      ( hBOOL(finite232328454iple_a(F,F_1))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A_1))
       => ( ( A_1 != bot_bo214064617a_bool )
         => ( ! [X: hoare_2076525193iple_a,Y_8: hoare_2076525193iple_a] : hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,hAPP_H1789220885iple_a(hAPP_H1417936777iple_a(F,X),Y_8)),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,Y_8),bot_bo214064617a_bool))))
           => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,hAPP_f2009966983iple_a(F_1,A_1)),A_1)) ) ) ) ) ).

tff(fact_771_folding__one_Oclosed,axiom,
    ! [A_1: fun_Ho320143120e_bool,F: fun_Ho31865764_state,F_1: fun_fu700438391_state] :
      ( hBOOL(finite269920017_state(F,F_1))
     => ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_1))
       => ( ( A_1 != bot_bo39439654e_bool )
         => ( ! [X: hoare_363350478_state,Y_8: hoare_363350478_state] : hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,hAPP_H1982017439_state(hAPP_H230141040_state(F,X),Y_8)),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,Y_8),bot_bo39439654e_bool))))
           => hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,hAPP_f1242401993_state(F_1,A_1)),A_1)) ) ) ) ) ).

tff(fact_772_folding__one_Oclosed,axiom,
    ! [A_1: fun_nat_bool,F: fun_nat_fun_nat_nat,F_1: fun_fun_nat_bool_nat] :
      ( hBOOL(finite988810631ne_nat(F,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
       => ( ( A_1 != bot_bot_fun_nat_bool )
         => ( ! [X: nat,Y_8: nat] : hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X),Y_8)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Y_8),bot_bot_fun_nat_bool))))
           => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_f22106695ol_nat(F_1,A_1)),A_1)) ) ) ) ) ).

tff(fact_773_folding__one_Oclosed,axiom,
    ! [A_1: fun_pname_bool,F: fun_pn1208287926_pname,F_1: fun_fu1499449723_pname] :
      ( hBOOL(finite1282449217_pname(F,F_1))
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
       => ( ( A_1 != bot_bo844097828e_bool )
         => ( ! [X: pname,Y_8: pname] : hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,hAPP_pname_pname(hAPP_p1170154830_pname(F,X),Y_8)),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,Y_8),bot_bo844097828e_bool))))
           => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,hAPP_f1297739591_pname(F_1,A_1)),A_1)) ) ) ) ) ).

tff(fact_774_Set_Oset__insert,axiom,
    ! [X_2: nat,A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A_1))
     => ~ ! [B_6: fun_nat_bool] :
            ( ( A_1 = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),B_6) )
           => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),B_6)) ) ) ).

tff(fact_775_Set_Oset__insert,axiom,
    ! [X_2: fun_Ho882550911a_bool,A_1: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,X_2),A_1))
     => ~ ! [B_6: fun_fu2026518849l_bool] :
            ( ( A_1 = hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,X_2),B_6) )
           => hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,X_2),B_6)) ) ) ).

tff(fact_776_Set_Oset__insert,axiom,
    ! [X_2: hoare_2076525193iple_a,A_1: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_2),A_1))
     => ~ ! [B_6: fun_Ho882550911a_bool] :
            ( ( A_1 = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_2),B_6) )
           => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_2),B_6)) ) ) ).

tff(fact_777_Set_Oset__insert,axiom,
    ! [X_2: hoare_363350478_state,A_1: fun_Ho320143120e_bool] :
      ( hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_2),A_1))
     => ~ ! [B_6: fun_Ho320143120e_bool] :
            ( ( A_1 = hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_2),B_6) )
           => hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,X_2),B_6)) ) ) ).

tff(fact_778_Set_Oset__insert,axiom,
    ! [X_2: pname,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A_1))
     => ~ ! [B_6: fun_pname_bool] :
            ( ( A_1 = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),B_6) )
           => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),B_6)) ) ) ).

tff(fact_779_equals0I,axiom,
    ! [A_1: fun_nat_bool] :
      ( ! [Y_8: nat] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Y_8),A_1))
     => ( A_1 = bot_bot_fun_nat_bool ) ) ).

tff(fact_780_equals0I,axiom,
    ! [A_1: fun_fu2026518849l_bool] :
      ( ! [Y_8: fun_Ho882550911a_bool] : ~ hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,Y_8),A_1))
     => ( A_1 = bot_bo1285271383l_bool ) ) ).

tff(fact_781_equals0I,axiom,
    ! [A_1: fun_Ho882550911a_bool] :
      ( ! [Y_8: hoare_2076525193iple_a] : ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,Y_8),A_1))
     => ( A_1 = bot_bo214064617a_bool ) ) ).

tff(fact_782_equals0I,axiom,
    ! [A_1: fun_Ho320143120e_bool] :
      ( ! [Y_8: hoare_363350478_state] : ~ hBOOL(hAPP_f568432088l_bool(hAPP_H40373975l_bool(member1253905979_state,Y_8),A_1))
     => ( A_1 = bot_bo39439654e_bool ) ) ).

tff(fact_783_equals0I,axiom,
    ! [A_1: fun_pname_bool] :
      ( ! [Y_8: pname] : ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,Y_8),A_1))
     => ( A_1 = bot_bo844097828e_bool ) ) ).

tff(fact_784_Sup__fin_Ounion__idem,axiom,
    ! [B_2: fun_fu425979586l_bool,A_1: fun_fu425979586l_bool] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A_1))
     => ( ( A_1 != bot_bo1701429464l_bool )
       => ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B_2))
         => ( ( B_2 != bot_bo1701429464l_bool )
           => ( hAPP_f582319405t_bool(big_la1704749377t_bool,hAPP_f1246832597l_bool(hAPP_f540020688l_bool(semila653644470l_bool,A_1),B_2)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,hAPP_f582319405t_bool(big_la1704749377t_bool,A_1)),hAPP_f582319405t_bool(big_la1704749377t_bool,B_2)) ) ) ) ) ) ).

tff(fact_785_Sup__fin_Ounion__idem,axiom,
    ! [B_2: fun_fu1342882871l_bool,A_1: fun_fu1342882871l_bool] :
      ( hBOOL(hAPP_f249932695l_bool(finite629390228l_bool,A_1))
     => ( ( A_1 != bot_bo1683866273l_bool )
       => ( hBOOL(hAPP_f249932695l_bool(finite629390228l_bool,B_2))
         => ( ( B_2 != bot_bo1683866273l_bool )
           => ( hAPP_f1453855623l_bool(big_la1867421504l_bool,hAPP_f1122542193l_bool(hAPP_f392660249l_bool(semila1183405891l_bool,A_1),B_2)) = hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,hAPP_f1453855623l_bool(big_la1867421504l_bool,A_1)),hAPP_f1453855623l_bool(big_la1867421504l_bool,B_2)) ) ) ) ) ) ).

tff(fact_786_Sup__fin_Ounion__idem,axiom,
    ! [B_2: fun_fu1996308102l_bool,A_1: fun_fu1996308102l_bool] :
      ( hBOOL(hAPP_f1287348838l_bool(finite2030652771e_bool,A_1))
     => ( ( A_1 != bot_bo133096688l_bool )
       => ( hBOOL(hAPP_f1287348838l_bool(finite2030652771e_bool,B_2))
         => ( ( B_2 != bot_bo133096688l_bool )
           => ( hAPP_f1180969799e_bool(big_la624482575e_bool,hAPP_f85652879l_bool(hAPP_f1175211816l_bool(semila862847378l_bool,A_1),B_2)) = hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,hAPP_f1180969799e_bool(big_la624482575e_bool,A_1)),hAPP_f1180969799e_bool(big_la624482575e_bool,B_2)) ) ) ) ) ) ).

tff(fact_787_Sup__fin_Ounion__idem,axiom,
    ! [B_2: fun_fu1430349052l_bool,A_1: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A_1))
     => ( ( A_1 != bot_bo1649642514l_bool )
       => ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B_2))
         => ( ( B_2 != bot_bo1649642514l_bool )
           => ( hAPP_f674760225e_bool(big_la841148155e_bool,hAPP_f559147733l_bool(hAPP_f230953622l_bool(semila1782091504l_bool,A_1),B_2)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f674760225e_bool(big_la841148155e_bool,A_1)),hAPP_f674760225e_bool(big_la841148155e_bool,B_2)) ) ) ) ) ) ).

tff(fact_788_Sup__fin_Ounion__idem,axiom,
    ! [B_2: fun_bool_bool,A_1: fun_bool_bool] :
      ( hBOOL(hAPP_f387058535l_bool(finite_finite_bool,A_1))
     => ( ( A_1 != bot_bo806936373l_bool )
       => ( hBOOL(hAPP_f387058535l_bool(finite_finite_bool,B_2))
         => ( ( B_2 != bot_bo806936373l_bool )
           => ( hBOOL(hAPP_f387058535l_bool(big_la1480321694n_bool,hAPP_f961197973l_bool(hAPP_f1172769267l_bool(semila1866150931l_bool,A_1),B_2)))
            <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,hAPP_f387058535l_bool(big_la1480321694n_bool,A_1)),hAPP_f387058535l_bool(big_la1480321694n_bool,B_2))) ) ) ) ) ) ).

tff(fact_789_Sup__fin_Ounion__idem,axiom,
    ! [B_2: fun_fu2026518849l_bool,A_1: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,A_1))
     => ( ( A_1 != bot_bo1285271383l_bool )
       => ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,B_2))
         => ( ( B_2 != bot_bo1285271383l_bool )
           => ( hAPP_f1758315691a_bool(big_la1025929536a_bool,hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_1),B_2)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,hAPP_f1758315691a_bool(big_la1025929536a_bool,A_1)),hAPP_f1758315691a_bool(big_la1025929536a_bool,B_2)) ) ) ) ) ) ).

tff(fact_790_Sup__fin_Ounion__idem,axiom,
    ! [B_2: fun_nat_bool,A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
     => ( ( A_1 != bot_bot_fun_nat_bool )
       => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B_2))
         => ( ( B_2 != bot_bot_fun_nat_bool )
           => ( hAPP_f22106695ol_nat(big_la43341705in_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_1),B_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,hAPP_f22106695ol_nat(big_la43341705in_nat,A_1)),hAPP_f22106695ol_nat(big_la43341705in_nat,B_2)) ) ) ) ) ) ).

tff(fact_791_Sup__fin_Oinsert,axiom,
    ! [X_2: fun_nat_bool,A_1: fun_fu425979586l_bool] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A_1))
     => ( ~ hBOOL(hAPP_f1637334154l_bool(hAPP_f1951378235l_bool(member_fun_nat_bool,X_2),A_1))
       => ( ( A_1 != bot_bo1701429464l_bool )
         => ( hAPP_f582319405t_bool(big_la1704749377t_bool,hAPP_f1246832597l_bool(hAPP_f633452666l_bool(insert_fun_nat_bool,X_2),A_1)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,X_2),hAPP_f582319405t_bool(big_la1704749377t_bool,A_1)) ) ) ) ) ).

tff(fact_792_Sup__fin_Oinsert,axiom,
    ! [X_2: fun_fu2026518849l_bool,A_1: fun_fu1342882871l_bool] :
      ( hBOOL(hAPP_f249932695l_bool(finite629390228l_bool,A_1))
     => ( ~ hBOOL(hAPP_f249932695l_bool(hAPP_f1351571149l_bool(member991318164l_bool,X_2),A_1))
       => ( ( A_1 != bot_bo1683866273l_bool )
         => ( hAPP_f1453855623l_bool(big_la1867421504l_bool,hAPP_f1122542193l_bool(hAPP_f674363047l_bool(insert851727547l_bool,X_2),A_1)) = hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,X_2),hAPP_f1453855623l_bool(big_la1867421504l_bool,A_1)) ) ) ) ) ).

tff(fact_793_Sup__fin_Oinsert,axiom,
    ! [X_2: fun_Ho320143120e_bool,A_1: fun_fu1996308102l_bool] :
      ( hBOOL(hAPP_f1287348838l_bool(finite2030652771e_bool,A_1))
     => ( ~ hBOOL(hAPP_f1287348838l_bool(hAPP_f1610329741l_bool(member1534059107e_bool,X_2),A_1))
       => ( ( A_1 != bot_bo133096688l_bool )
         => ( hAPP_f1180969799e_bool(big_la624482575e_bool,hAPP_f85652879l_bool(hAPP_f1941218614l_bool(insert975831690e_bool,X_2),A_1)) = hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,X_2),hAPP_f1180969799e_bool(big_la624482575e_bool,A_1)) ) ) ) ) ).

tff(fact_794_Sup__fin_Oinsert,axiom,
    ! [X_2: fun_pname_bool,A_1: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A_1))
     => ( ~ hBOOL(hAPP_f1935102916l_bool(hAPP_f556039215l_bool(member799430823e_bool,X_2),A_1))
       => ( ( A_1 != bot_bo1649642514l_bool )
         => ( hAPP_f674760225e_bool(big_la841148155e_bool,hAPP_f559147733l_bool(hAPP_f1320879424l_bool(insert1325755072e_bool,X_2),A_1)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),hAPP_f674760225e_bool(big_la841148155e_bool,A_1)) ) ) ) ) ).

tff(fact_795_Sup__fin_Oinsert,axiom,
    ! [X_2: bool,A_1: fun_bool_bool] :
      ( hBOOL(hAPP_f387058535l_bool(finite_finite_bool,A_1))
     => ( ~ hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,X_2),A_1))
       => ( ( A_1 != bot_bo806936373l_bool )
         => ( hBOOL(hAPP_f387058535l_bool(big_la1480321694n_bool,hAPP_f961197973l_bool(hAPP_b496459037l_bool(insert_bool,X_2),A_1)))
          <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),hAPP_f387058535l_bool(big_la1480321694n_bool,A_1))) ) ) ) ) ).

tff(fact_796_Sup__fin_Oinsert,axiom,
    ! [X_2: fun_Ho882550911a_bool,A_1: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,A_1))
     => ( ~ hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,X_2),A_1))
       => ( ( A_1 != bot_bo1285271383l_bool )
         => ( hAPP_f1758315691a_bool(big_la1025929536a_bool,hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,X_2),A_1)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,X_2),hAPP_f1758315691a_bool(big_la1025929536a_bool,A_1)) ) ) ) ) ).

tff(fact_797_Sup__fin_Oinsert,axiom,
    ! [X_2: nat,A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
     => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A_1))
       => ( ( A_1 != bot_bot_fun_nat_bool )
         => ( hAPP_f22106695ol_nat(big_la43341705in_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),A_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_2),hAPP_f22106695ol_nat(big_la43341705in_nat,A_1)) ) ) ) ) ).

tff(fact_798_Sup__fin_Oinsert__idem,axiom,
    ! [X_2: fun_nat_bool,A_1: fun_fu425979586l_bool] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A_1))
     => ( ( A_1 != bot_bo1701429464l_bool )
       => ( hAPP_f582319405t_bool(big_la1704749377t_bool,hAPP_f1246832597l_bool(hAPP_f633452666l_bool(insert_fun_nat_bool,X_2),A_1)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,X_2),hAPP_f582319405t_bool(big_la1704749377t_bool,A_1)) ) ) ) ).

tff(fact_799_Sup__fin_Oinsert__idem,axiom,
    ! [X_2: fun_fu2026518849l_bool,A_1: fun_fu1342882871l_bool] :
      ( hBOOL(hAPP_f249932695l_bool(finite629390228l_bool,A_1))
     => ( ( A_1 != bot_bo1683866273l_bool )
       => ( hAPP_f1453855623l_bool(big_la1867421504l_bool,hAPP_f1122542193l_bool(hAPP_f674363047l_bool(insert851727547l_bool,X_2),A_1)) = hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,X_2),hAPP_f1453855623l_bool(big_la1867421504l_bool,A_1)) ) ) ) ).

tff(fact_800_Sup__fin_Oinsert__idem,axiom,
    ! [X_2: fun_Ho320143120e_bool,A_1: fun_fu1996308102l_bool] :
      ( hBOOL(hAPP_f1287348838l_bool(finite2030652771e_bool,A_1))
     => ( ( A_1 != bot_bo133096688l_bool )
       => ( hAPP_f1180969799e_bool(big_la624482575e_bool,hAPP_f85652879l_bool(hAPP_f1941218614l_bool(insert975831690e_bool,X_2),A_1)) = hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,X_2),hAPP_f1180969799e_bool(big_la624482575e_bool,A_1)) ) ) ) ).

tff(fact_801_Sup__fin_Oinsert__idem,axiom,
    ! [X_2: fun_pname_bool,A_1: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A_1))
     => ( ( A_1 != bot_bo1649642514l_bool )
       => ( hAPP_f674760225e_bool(big_la841148155e_bool,hAPP_f559147733l_bool(hAPP_f1320879424l_bool(insert1325755072e_bool,X_2),A_1)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,X_2),hAPP_f674760225e_bool(big_la841148155e_bool,A_1)) ) ) ) ).

tff(fact_802_Sup__fin_Oinsert__idem,axiom,
    ! [X_2: bool,A_1: fun_bool_bool] :
      ( hBOOL(hAPP_f387058535l_bool(finite_finite_bool,A_1))
     => ( ( A_1 != bot_bo806936373l_bool )
       => ( hBOOL(hAPP_f387058535l_bool(big_la1480321694n_bool,hAPP_f961197973l_bool(hAPP_b496459037l_bool(insert_bool,X_2),A_1)))
        <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila1168014441p_bool,X_2),hAPP_f387058535l_bool(big_la1480321694n_bool,A_1))) ) ) ) ).

tff(fact_803_Sup__fin_Oinsert__idem,axiom,
    ! [X_2: fun_Ho882550911a_bool,A_1: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,A_1))
     => ( ( A_1 != bot_bo1285271383l_bool )
       => ( hAPP_f1758315691a_bool(big_la1025929536a_bool,hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,X_2),A_1)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,X_2),hAPP_f1758315691a_bool(big_la1025929536a_bool,A_1)) ) ) ) ).

tff(fact_804_Sup__fin_Oinsert__idem,axiom,
    ! [X_2: nat,A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
     => ( ( A_1 != bot_bot_fun_nat_bool )
       => ( hAPP_f22106695ol_nat(big_la43341705in_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),A_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila972727038up_nat,X_2),hAPP_f22106695ol_nat(big_la43341705in_nat,A_1)) ) ) ) ).

tff(fact_805_folding__one_Ounion__disjoint,axiom,
    ! [B_2: fun_fu2026518849l_bool,A_1: fun_fu2026518849l_bool,F: fun_fu2051904533a_bool,F_1: fun_fu1819779801a_bool] :
      ( hBOOL(finite1520073154a_bool(F,F_1))
     => ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,A_1))
       => ( ( A_1 != bot_bo1285271383l_bool )
         => ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,B_2))
           => ( ( B_2 != bot_bo1285271383l_bool )
             => ( ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1706849871l_bool,A_1),B_2) = bot_bo1285271383l_bool )
               => ( hAPP_f1758315691a_bool(F_1,hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_1),B_2)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(F,hAPP_f1758315691a_bool(F_1,A_1)),hAPP_f1758315691a_bool(F_1,B_2)) ) ) ) ) ) ) ) ).

tff(fact_806_folding__one_Ounion__disjoint,axiom,
    ! [B_2: fun_pname_bool,A_1: fun_pname_bool,F: fun_pn1208287926_pname,F_1: fun_fu1499449723_pname] :
      ( hBOOL(finite1282449217_pname(F,F_1))
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
       => ( ( A_1 != bot_bo844097828e_bool )
         => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B_2))
           => ( ( B_2 != bot_bo844097828e_bool )
             => ( ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,A_1),B_2) = bot_bo844097828e_bool )
               => ( hAPP_f1297739591_pname(F_1,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),B_2)) = hAPP_pname_pname(hAPP_p1170154830_pname(F,hAPP_f1297739591_pname(F_1,A_1)),hAPP_f1297739591_pname(F_1,B_2)) ) ) ) ) ) ) ) ).

tff(fact_807_folding__one_Ounion__disjoint,axiom,
    ! [B_2: fun_Ho320143120e_bool,A_1: fun_Ho320143120e_bool,F: fun_Ho31865764_state,F_1: fun_fu700438391_state] :
      ( hBOOL(finite269920017_state(F,F_1))
     => ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_1))
       => ( ( A_1 != bot_bo39439654e_bool )
         => ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,B_2))
           => ( ( B_2 != bot_bo39439654e_bool )
             => ( ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1129477150e_bool,A_1),B_2) = bot_bo39439654e_bool )
               => ( hAPP_f1242401993_state(F_1,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_1),B_2)) = hAPP_H1982017439_state(hAPP_H230141040_state(F,hAPP_f1242401993_state(F_1,A_1)),hAPP_f1242401993_state(F_1,B_2)) ) ) ) ) ) ) ) ).

tff(fact_808_folding__one_Ounion__disjoint,axiom,
    ! [B_2: fun_Ho882550911a_bool,A_1: fun_Ho882550911a_bool,F: fun_Ho483360497iple_a,F_1: fun_fu253747899iple_a] :
      ( hBOOL(finite232328454iple_a(F,F_1))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A_1))
       => ( ( A_1 != bot_bo214064617a_bool )
         => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,B_2))
           => ( ( B_2 != bot_bo214064617a_bool )
             => ( ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,A_1),B_2) = bot_bo214064617a_bool )
               => ( hAPP_f2009966983iple_a(F_1,hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_1),B_2)) = hAPP_H1789220885iple_a(hAPP_H1417936777iple_a(F,hAPP_f2009966983iple_a(F_1,A_1)),hAPP_f2009966983iple_a(F_1,B_2)) ) ) ) ) ) ) ) ).

tff(fact_809_folding__one_Ounion__disjoint,axiom,
    ! [B_2: fun_nat_bool,A_1: fun_nat_bool,F: fun_nat_fun_nat_nat,F_1: fun_fun_nat_bool_nat] :
      ( hBOOL(finite988810631ne_nat(F,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
       => ( ( A_1 != bot_bot_fun_nat_bool )
         => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B_2))
           => ( ( B_2 != bot_bot_fun_nat_bool )
             => ( ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A_1),B_2) = bot_bot_fun_nat_bool )
               => ( hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_1),B_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,hAPP_f22106695ol_nat(F_1,A_1)),hAPP_f22106695ol_nat(F_1,B_2)) ) ) ) ) ) ) ) ).

tff(fact_810_folding__one_Ounion__inter,axiom,
    ! [B_2: fun_fu2026518849l_bool,A_1: fun_fu2026518849l_bool,F: fun_fu2051904533a_bool,F_1: fun_fu1819779801a_bool] :
      ( hBOOL(finite1520073154a_bool(F,F_1))
     => ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,A_1))
       => ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,B_2))
         => ( ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1706849871l_bool,A_1),B_2) != bot_bo1285271383l_bool )
           => ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(F,hAPP_f1758315691a_bool(F_1,hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,A_1),B_2))),hAPP_f1758315691a_bool(F_1,hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1706849871l_bool,A_1),B_2))) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(F,hAPP_f1758315691a_bool(F_1,A_1)),hAPP_f1758315691a_bool(F_1,B_2)) ) ) ) ) ) ).

tff(fact_811_folding__one_Ounion__inter,axiom,
    ! [B_2: fun_pname_bool,A_1: fun_pname_bool,F: fun_pn1208287926_pname,F_1: fun_fu1499449723_pname] :
      ( hBOOL(finite1282449217_pname(F,F_1))
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
       => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B_2))
         => ( ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,A_1),B_2) != bot_bo844097828e_bool )
           => ( hAPP_pname_pname(hAPP_p1170154830_pname(F,hAPP_f1297739591_pname(F_1,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,A_1),B_2))),hAPP_f1297739591_pname(F_1,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,A_1),B_2))) = hAPP_pname_pname(hAPP_p1170154830_pname(F,hAPP_f1297739591_pname(F_1,A_1)),hAPP_f1297739591_pname(F_1,B_2)) ) ) ) ) ) ).

tff(fact_812_folding__one_Ounion__inter,axiom,
    ! [B_2: fun_Ho320143120e_bool,A_1: fun_Ho320143120e_bool,F: fun_Ho31865764_state,F_1: fun_fu700438391_state] :
      ( hBOOL(finite269920017_state(F,F_1))
     => ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_1))
       => ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,B_2))
         => ( ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1129477150e_bool,A_1),B_2) != bot_bo39439654e_bool )
           => ( hAPP_H1982017439_state(hAPP_H230141040_state(F,hAPP_f1242401993_state(F_1,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,A_1),B_2))),hAPP_f1242401993_state(F_1,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1129477150e_bool,A_1),B_2))) = hAPP_H1982017439_state(hAPP_H230141040_state(F,hAPP_f1242401993_state(F_1,A_1)),hAPP_f1242401993_state(F_1,B_2)) ) ) ) ) ) ).

tff(fact_813_folding__one_Ounion__inter,axiom,
    ! [B_2: fun_Ho882550911a_bool,A_1: fun_Ho882550911a_bool,F: fun_Ho483360497iple_a,F_1: fun_fu253747899iple_a] :
      ( hBOOL(finite232328454iple_a(F,F_1))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A_1))
       => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,B_2))
         => ( ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,A_1),B_2) != bot_bo214064617a_bool )
           => ( hAPP_H1789220885iple_a(hAPP_H1417936777iple_a(F,hAPP_f2009966983iple_a(F_1,hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,A_1),B_2))),hAPP_f2009966983iple_a(F_1,hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,A_1),B_2))) = hAPP_H1789220885iple_a(hAPP_H1417936777iple_a(F,hAPP_f2009966983iple_a(F_1,A_1)),hAPP_f2009966983iple_a(F_1,B_2)) ) ) ) ) ) ).

tff(fact_814_folding__one_Ounion__inter,axiom,
    ! [B_2: fun_nat_bool,A_1: fun_nat_bool,F: fun_nat_fun_nat_nat,F_1: fun_fun_nat_bool_nat] :
      ( hBOOL(finite988810631ne_nat(F,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
       => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B_2))
         => ( ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A_1),B_2) != bot_bot_fun_nat_bool )
           => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,A_1),B_2))),hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A_1),B_2))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,hAPP_f22106695ol_nat(F_1,A_1)),hAPP_f22106695ol_nat(F_1,B_2)) ) ) ) ) ) ).

tff(fact_815_folding__one_Oinsert__remove,axiom,
    ! [X_2: fun_Ho882550911a_bool,A_1: fun_fu2026518849l_bool,F: fun_fu2051904533a_bool,F_1: fun_fu1819779801a_bool] :
      ( hBOOL(finite1520073154a_bool(F,F_1))
     => ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,A_1))
       => ( ( ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(minus_1051402706l_bool,A_1),hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,X_2),bot_bo1285271383l_bool)) = bot_bo1285271383l_bool )
           => ( hAPP_f1758315691a_bool(F_1,hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,X_2),A_1)) = X_2 ) )
          & ( ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(minus_1051402706l_bool,A_1),hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,X_2),bot_bo1285271383l_bool)) != bot_bo1285271383l_bool )
           => ( hAPP_f1758315691a_bool(F_1,hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,X_2),A_1)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(F,X_2),hAPP_f1758315691a_bool(F_1,hAPP_f354682389l_bool(hAPP_f1976119377l_bool(minus_1051402706l_bool,A_1),hAPP_f354682389l_bool(hAPP_f1803426171l_bool(insert838260101a_bool,X_2),bot_bo1285271383l_bool)))) ) ) ) ) ) ).

tff(fact_816_folding__one_Oinsert__remove,axiom,
    ! [X_2: pname,A_1: fun_pname_bool,F: fun_pn1208287926_pname,F_1: fun_fu1499449723_pname] :
      ( hBOOL(finite1282449217_pname(F,F_1))
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
       => ( ( ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A_1),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),bot_bo844097828e_bool)) = bot_bo844097828e_bool )
           => ( hAPP_f1297739591_pname(F_1,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),A_1)) = X_2 ) )
          & ( ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A_1),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),bot_bo844097828e_bool)) != bot_bo844097828e_bool )
           => ( hAPP_f1297739591_pname(F_1,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),A_1)) = hAPP_pname_pname(hAPP_p1170154830_pname(F,X_2),hAPP_f1297739591_pname(F_1,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A_1),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),bot_bo844097828e_bool)))) ) ) ) ) ) ).

tff(fact_817_folding__one_Oinsert__remove,axiom,
    ! [X_2: hoare_2076525193iple_a,A_1: fun_Ho882550911a_bool,F: fun_Ho483360497iple_a,F_1: fun_fu253747899iple_a] :
      ( hBOOL(finite232328454iple_a(F,F_1))
     => ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A_1))
       => ( ( ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A_1),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_2),bot_bo214064617a_bool)) = bot_bo214064617a_bool )
           => ( hAPP_f2009966983iple_a(F_1,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_2),A_1)) = X_2 ) )
          & ( ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A_1),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_2),bot_bo214064617a_bool)) != bot_bo214064617a_bool )
           => ( hAPP_f2009966983iple_a(F_1,hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_2),A_1)) = hAPP_H1789220885iple_a(hAPP_H1417936777iple_a(F,X_2),hAPP_f2009966983iple_a(F_1,hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A_1),hAPP_f642208769a_bool(hAPP_H1940471919a_bool(insert1582918915iple_a,X_2),bot_bo214064617a_bool)))) ) ) ) ) ) ).

tff(fact_818_folding__one_Oinsert__remove,axiom,
    ! [X_2: hoare_363350478_state,A_1: fun_Ho320143120e_bool,F: fun_Ho31865764_state,F_1: fun_fu700438391_state] :
      ( hBOOL(finite269920017_state(F,F_1))
     => ( hBOOL(hAPP_f568432088l_bool(finite280062779_state,A_1))
       => ( ( ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_1),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_2),bot_bo39439654e_bool)) = bot_bo39439654e_bool )
           => ( hAPP_f1242401993_state(F_1,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_2),A_1)) = X_2 ) )
          & ( ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_1),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_2),bot_bo39439654e_bool)) != bot_bo39439654e_bool )
           => ( hAPP_f1242401993_state(F_1,hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_2),A_1)) = hAPP_H1982017439_state(hAPP_H230141040_state(F,X_2),hAPP_f1242401993_state(F_1,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_1),hAPP_f1749453397e_bool(hAPP_H122293036e_bool(insert1330351828_state,X_2),bot_bo39439654e_bool)))) ) ) ) ) ) ).

tff(fact_819_folding__one_Oinsert__remove,axiom,
    ! [X_2: nat,A_1: fun_nat_bool,F: fun_nat_fun_nat_nat,F_1: fun_fun_nat_bool_nat] :
      ( hBOOL(finite988810631ne_nat(F,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
       => ( ( ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),bot_bot_fun_nat_bool)) = bot_bot_fun_nat_bool )
           => ( hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),A_1)) = X_2 ) )
          & ( ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),bot_bot_fun_nat_bool)) != bot_bot_fun_nat_bool )
           => ( hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),A_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_2),hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_1),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),bot_bot_fun_nat_bool)))) ) ) ) ) ) ).

tff(fact_820_inf1I,axiom,
    ! [B_2: fun_nat_bool,A_1: fun_nat_bool,X_2: nat] :
      ( hBOOL(hAPP_nat_bool(A_1,X_2))
     => ( hBOOL(hAPP_nat_bool(B_2,X_2))
       => hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A_1),B_2),X_2)) ) ) ).

tff(fact_821_inf1I,axiom,
    ! [B_2: fun_Ho882550911a_bool,A_1: fun_Ho882550911a_bool,X_2: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_H454055633a_bool(A_1,X_2))
     => ( hBOOL(hAPP_H454055633a_bool(B_2,X_2))
       => hBOOL(hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,A_1),B_2),X_2)) ) ) ).

tff(fact_822_inf1I,axiom,
    ! [B_2: fun_pname_bool,A_1: fun_pname_bool,X_2: pname] :
      ( hBOOL(hAPP_pname_bool(A_1,X_2))
     => ( hBOOL(hAPP_pname_bool(B_2,X_2))
       => hBOOL(hAPP_pname_bool(hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,A_1),B_2),X_2)) ) ) ).

tff(fact_823_IntI,axiom,
    ! [B_2: fun_nat_bool,C_2: nat,A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),A_1))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),B_2))
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A_1),B_2))) ) ) ).

tff(fact_824_IntI,axiom,
    ! [B_2: fun_fu2026518849l_bool,C_2: fun_Ho882550911a_bool,A_1: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,C_2),A_1))
     => ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,C_2),B_2))
       => hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,C_2),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1706849871l_bool,A_1),B_2))) ) ) ).

tff(fact_825_IntI,axiom,
    ! [B_2: fun_Ho882550911a_bool,C_2: hoare_2076525193iple_a,A_1: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,C_2),A_1))
     => ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,C_2),B_2))
       => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,C_2),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,A_1),B_2))) ) ) ).

tff(fact_826_IntI,axiom,
    ! [B_2: fun_pname_bool,C_2: pname,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_2),A_1))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_2),B_2))
       => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,A_1),B_2))) ) ) ).

tff(fact_827_IntE,axiom,
    ! [C_2: nat,A_1: fun_nat_bool,B_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A_1),B_2)))
     => ~ ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),A_1))
         => ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),B_2)) ) ) ).

tff(fact_828_IntE,axiom,
    ! [C_2: fun_Ho882550911a_bool,A_1: fun_fu2026518849l_bool,B_2: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,C_2),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1706849871l_bool,A_1),B_2)))
     => ~ ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,C_2),A_1))
         => ~ hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,C_2),B_2)) ) ) ).

tff(fact_829_IntE,axiom,
    ! [C_2: hoare_2076525193iple_a,A_1: fun_Ho882550911a_bool,B_2: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,C_2),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,A_1),B_2)))
     => ~ ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,C_2),A_1))
         => ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,C_2),B_2)) ) ) ).

tff(fact_830_IntE,axiom,
    ! [C_2: pname,A_1: fun_pname_bool,B_2: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,A_1),B_2)))
     => ~ ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_2),A_1))
         => ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_2),B_2)) ) ) ).

tff(fact_831_inf1E,axiom,
    ! [A_1: fun_nat_bool,B_2: fun_nat_bool,X_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A_1),B_2),X_2))
     => ~ ( hBOOL(hAPP_nat_bool(A_1,X_2))
         => ~ hBOOL(hAPP_nat_bool(B_2,X_2)) ) ) ).

tff(fact_832_inf1E,axiom,
    ! [A_1: fun_Ho882550911a_bool,B_2: fun_Ho882550911a_bool,X_2: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,A_1),B_2),X_2))
     => ~ ( hBOOL(hAPP_H454055633a_bool(A_1,X_2))
         => ~ hBOOL(hAPP_H454055633a_bool(B_2,X_2)) ) ) ).

tff(fact_833_inf1E,axiom,
    ! [A_1: fun_pname_bool,B_2: fun_pname_bool,X_2: pname] :
      ( hBOOL(hAPP_pname_bool(hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,A_1),B_2),X_2))
     => ~ ( hBOOL(hAPP_pname_bool(A_1,X_2))
         => ~ hBOOL(hAPP_pname_bool(B_2,X_2)) ) ) ).

tff(fact_834_DiffI,axiom,
    ! [B_2: fun_nat_bool,C_2: nat,A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),A_1))
     => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),B_2))
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_1),B_2))) ) ) ).

tff(fact_835_DiffI,axiom,
    ! [B_2: fun_fu2026518849l_bool,C_2: fun_Ho882550911a_bool,A_1: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,C_2),A_1))
     => ( ~ hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,C_2),B_2))
       => hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,C_2),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(minus_1051402706l_bool,A_1),B_2))) ) ) ).

tff(fact_836_DiffI,axiom,
    ! [B_2: fun_Ho882550911a_bool,C_2: hoare_2076525193iple_a,A_1: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,C_2),A_1))
     => ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,C_2),B_2))
       => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,C_2),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A_1),B_2))) ) ) ).

tff(fact_837_DiffI,axiom,
    ! [B_2: fun_pname_bool,C_2: pname,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_2),A_1))
     => ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_2),B_2))
       => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A_1),B_2))) ) ) ).

tff(fact_838_DiffE,axiom,
    ! [C_2: nat,A_1: fun_nat_bool,B_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_1),B_2)))
     => ~ ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),A_1))
         => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),B_2)) ) ) ).

tff(fact_839_DiffE,axiom,
    ! [C_2: fun_Ho882550911a_bool,A_1: fun_fu2026518849l_bool,B_2: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,C_2),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(minus_1051402706l_bool,A_1),B_2)))
     => ~ ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,C_2),A_1))
         => hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,C_2),B_2)) ) ) ).

tff(fact_840_DiffE,axiom,
    ! [C_2: hoare_2076525193iple_a,A_1: fun_Ho882550911a_bool,B_2: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,C_2),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A_1),B_2)))
     => ~ ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,C_2),A_1))
         => hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,C_2),B_2)) ) ) ).

tff(fact_841_DiffE,axiom,
    ! [C_2: pname,A_1: fun_pname_bool,B_2: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A_1),B_2)))
     => ~ ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_2),A_1))
         => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_2),B_2)) ) ) ).

tff(fact_842_finite__Int,axiom,
    ! [Ga: fun_Ho882550911a_bool,F_1: fun_Ho882550911a_bool] :
      ( ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,F_1))
        | hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,Ga)) )
     => hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,F_1),Ga))) ) ).

tff(fact_843_finite__Int,axiom,
    ! [Ga: fun_fu2026518849l_bool,F_1: fun_fu2026518849l_bool] :
      ( ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,F_1))
        | hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,Ga)) )
     => hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1706849871l_bool,F_1),Ga))) ) ).

tff(fact_844_finite__Int,axiom,
    ! [Ga: fun_pname_bool,F_1: fun_pname_bool] :
      ( ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
        | hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,Ga)) )
     => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,F_1),Ga))) ) ).

tff(fact_845_finite__Int,axiom,
    ! [Ga: fun_nat_bool,F_1: fun_nat_bool] :
      ( ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
        | hBOOL(hAPP_f54304608l_bool(finite_finite_nat,Ga)) )
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,F_1),Ga))) ) ).

tff(fact_846_finite__Diff,axiom,
    ! [B_2: fun_Ho882550911a_bool,A_1: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,A_1))
     => hBOOL(hAPP_f500854239l_bool(finite217552220iple_a,hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A_1),B_2))) ) ).

tff(fact_847_finite__Diff,axiom,
    ! [B_2: fun_fu2026518849l_bool,A_1: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,A_1))
     => hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,hAPP_f354682389l_bool(hAPP_f1976119377l_bool(minus_1051402706l_bool,A_1),B_2))) ) ).

tff(fact_848_finite__Diff,axiom,
    ! [B_2: fun_pname_bool,A_1: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
     => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A_1),B_2))) ) ).

tff(fact_849_finite__Diff,axiom,
    ! [B_2: fun_nat_bool,A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_1),B_2))) ) ).

tff(fact_850_inf__Sup__absorb,axiom,
    ! [A_2: fun_Ho882550911a_bool,A_1: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(finite1832883692a_bool,A_1))
     => ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,A_2),A_1))
       => ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,A_2),hAPP_f1758315691a_bool(big_la1025929536a_bool,A_1)) = A_2 ) ) ) ).

tff(fact_851_inf__Sup__absorb,axiom,
    ! [A_2: fun_nat_bool,A_1: fun_fu425979586l_bool] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A_1))
     => ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1951378235l_bool(member_fun_nat_bool,A_2),A_1))
       => ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A_2),hAPP_f582319405t_bool(big_la1704749377t_bool,A_1)) = A_2 ) ) ) ).

tff(fact_852_inf__Sup__absorb,axiom,
    ! [A_2: fun_pname_bool,A_1: fun_fu1430349052l_bool] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A_1))
     => ( hBOOL(hAPP_f1935102916l_bool(hAPP_f556039215l_bool(member799430823e_bool,A_2),A_1))
       => ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,A_2),hAPP_f674760225e_bool(big_la841148155e_bool,A_1)) = A_2 ) ) ) ).

tff(fact_853_inf__Sup__absorb,axiom,
    ! [A_2: nat,A_1: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),A_1))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,A_2),hAPP_f22106695ol_nat(big_la43341705in_nat,A_1)) = A_2 ) ) ) ).

tff(fact_854_Diff__Int,axiom,
    ! [A_1: fun_nat_bool,B_2: fun_nat_bool,C_1: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,B_2),C_1)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_1),B_2)),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_1),C_1)) ) ).

tff(fact_855_Diff__Int,axiom,
    ! [A_1: fun_pname_bool,B_2: fun_pname_bool,C_1: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A_1),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,B_2),C_1)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A_1),B_2)),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A_1),C_1)) ) ).

tff(fact_856_Diff__Int,axiom,
    ! [A_1: fun_fu2026518849l_bool,B_2: fun_fu2026518849l_bool,C_1: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(minus_1051402706l_bool,A_1),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1706849871l_bool,B_2),C_1)) = hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,hAPP_f354682389l_bool(hAPP_f1976119377l_bool(minus_1051402706l_bool,A_1),B_2)),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(minus_1051402706l_bool,A_1),C_1)) ) ).

tff(fact_857_Diff__Int,axiom,
    ! [A_1: fun_Ho320143120e_bool,B_2: fun_Ho320143120e_bool,C_1: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_1),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1129477150e_bool,B_2),C_1)) = hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_1),B_2)),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_1),C_1)) ) ).

tff(fact_858_Diff__Int,axiom,
    ! [A_1: fun_Ho882550911a_bool,B_2: fun_Ho882550911a_bool,C_1: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A_1),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,B_2),C_1)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A_1),B_2)),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A_1),C_1)) ) ).

tff(fact_859_Diff__Un,axiom,
    ! [A_1: fun_nat_bool,B_2: fun_nat_bool,C_1: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,B_2),C_1)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_1),B_2)),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_1),C_1)) ) ).

tff(fact_860_Diff__Un,axiom,
    ! [A_1: fun_pname_bool,B_2: fun_pname_bool,C_1: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A_1),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,B_2),C_1)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A_1),B_2)),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A_1),C_1)) ) ).

tff(fact_861_Diff__Un,axiom,
    ! [A_1: fun_fu2026518849l_bool,B_2: fun_fu2026518849l_bool,C_1: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(minus_1051402706l_bool,A_1),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,B_2),C_1)) = hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1706849871l_bool,hAPP_f354682389l_bool(hAPP_f1976119377l_bool(minus_1051402706l_bool,A_1),B_2)),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(minus_1051402706l_bool,A_1),C_1)) ) ).

tff(fact_862_Diff__Un,axiom,
    ! [A_1: fun_Ho320143120e_bool,B_2: fun_Ho320143120e_bool,C_1: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_1),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,B_2),C_1)) = hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1129477150e_bool,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_1),B_2)),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_1),C_1)) ) ).

tff(fact_863_Diff__Un,axiom,
    ! [A_1: fun_Ho882550911a_bool,B_2: fun_Ho882550911a_bool,C_1: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A_1),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,B_2),C_1)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A_1),B_2)),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A_1),C_1)) ) ).

tff(fact_864_Un__Diff__Int,axiom,
    ! [A_1: fun_nat_bool,B_2: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila465093516t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_1),B_2)),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A_1),B_2)) = A_1 ) ).

tff(fact_865_Un__Diff__Int,axiom,
    ! [A_1: fun_pname_bool,B_2: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila278973382e_bool,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A_1),B_2)),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,A_1),B_2)) = A_1 ) ).

tff(fact_866_Un__Diff__Int,axiom,
    ! [A_1: fun_fu2026518849l_bool,B_2: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1787532597l_bool,hAPP_f354682389l_bool(hAPP_f1976119377l_bool(minus_1051402706l_bool,A_1),B_2)),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1706849871l_bool,A_1),B_2)) = A_1 ) ).

tff(fact_867_Un__Diff__Int,axiom,
    ! [A_1: fun_Ho320143120e_bool,B_2: fun_Ho320143120e_bool] : ( hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1579166468e_bool,hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(minus_1060678561e_bool,A_1),B_2)),hAPP_f1749453397e_bool(hAPP_f1043393922e_bool(semila1129477150e_bool,A_1),B_2)) = A_1 ) ).

tff(fact_868_Un__Diff__Int,axiom,
    ! [A_1: fun_Ho882550911a_bool,B_2: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A_1),B_2)),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,A_1),B_2)) = A_1 ) ).

tff(fact_869_Collect__conj__eq,axiom,
    ! [Pa: fun_pname_bool,Qa: fun_pname_bool] : ( hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,hAPP_f1402196763l_bool(cOMBB_675860798_pname(fconj),Pa)),Qa)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,hAPP_f759274231e_bool(collect_pname,Pa)),hAPP_f759274231e_bool(collect_pname,Qa)) ) ).

tff(fact_870_Collect__conj__eq,axiom,
    ! [Pa: fun_Ho882550911a_bool,Qa: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(collec977865374iple_a,hAPP_f642208769a_bool(hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,hAPP_f268025233l_bool(cOMBB_345279363iple_a(fconj),Pa)),Qa)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,hAPP_f642208769a_bool(collec977865374iple_a,Pa)),hAPP_f642208769a_bool(collec977865374iple_a,Qa)) ) ).

tff(fact_871_Collect__conj__eq,axiom,
    ! [Pa: fun_nat_bool,Qa: fun_nat_bool] : ( hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(cOMBB_1015721476ol_nat(fconj),Pa)),Qa)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,hAPP_f800510211t_bool(collect_nat,Pa)),hAPP_f800510211t_bool(collect_nat,Qa)) ) ).

tff(fact_872_Int__Collect,axiom,
    ! [X_2: fun_Ho882550911a_bool,A_1: fun_fu2026518849l_bool,Pa: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,X_2),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1706849871l_bool,A_1),hAPP_f354682389l_bool(collec416005674a_bool,Pa))))
    <=> ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,X_2),A_1))
        & hBOOL(hAPP_f500854239l_bool(Pa,X_2)) ) ) ).

tff(fact_873_Int__Collect,axiom,
    ! [X_2: nat,A_1: fun_nat_bool,Pa: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A_1),hAPP_f800510211t_bool(collect_nat,Pa))))
    <=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A_1))
        & hBOOL(hAPP_nat_bool(Pa,X_2)) ) ) ).

tff(fact_874_Int__Collect,axiom,
    ! [X_2: hoare_2076525193iple_a,A_1: fun_Ho882550911a_bool,Pa: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_2),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,A_1),hAPP_f642208769a_bool(collec977865374iple_a,Pa))))
    <=> ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X_2),A_1))
        & hBOOL(hAPP_H454055633a_bool(Pa,X_2)) ) ) ).

tff(fact_875_Int__Collect,axiom,
    ! [X_2: pname,A_1: fun_pname_bool,Pa: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,A_1),hAPP_f759274231e_bool(collect_pname,Pa))))
    <=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A_1))
        & hBOOL(hAPP_pname_bool(Pa,X_2)) ) ) ).

tff(fact_876_inf__Int__eq,axiom,
    ! [R_1: fun_fu2026518849l_bool,S_2: fun_fu2026518849l_bool,X: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1706849871l_bool,hAPP_f354682389l_bool(hAPP_f638520823l_bool(cOMBC_1528738210l_bool,member186597612a_bool),R_1)),hAPP_f354682389l_bool(hAPP_f638520823l_bool(cOMBC_1528738210l_bool,member186597612a_bool),S_2)),X))
    <=> hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,X),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1706849871l_bool,R_1),S_2))) ) ).

tff(fact_877_inf__Int__eq,axiom,
    ! [R_1: fun_nat_bool,S_2: fun_nat_bool,X: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),R_1)),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),S_2)),X))
    <=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,R_1),S_2))) ) ).

tff(fact_878_inf__Int__eq,axiom,
    ! [R_1: fun_Ho882550911a_bool,S_2: fun_Ho882550911a_bool,X: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,hAPP_f642208769a_bool(hAPP_f325481667a_bool(cOMBC_1529670358l_bool,member276146780iple_a),R_1)),hAPP_f642208769a_bool(hAPP_f325481667a_bool(cOMBC_1529670358l_bool,member276146780iple_a),S_2)),X))
    <=> hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,R_1),S_2))) ) ).

tff(fact_879_inf__Int__eq,axiom,
    ! [R_1: fun_pname_bool,S_2: fun_pname_bool,X: pname] :
      ( hBOOL(hAPP_pname_bool(hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,hAPP_f759274231e_bool(hAPP_f661147897e_bool(cOMBC_1058051404l_bool,member_pname),R_1)),hAPP_f759274231e_bool(hAPP_f661147897e_bool(cOMBC_1058051404l_bool,member_pname),S_2)),X))
    <=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,R_1),S_2))) ) ).

tff(fact_880_Int__absorb,axiom,
    ! [A_1: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A_1),A_1) = A_1 ) ).

tff(fact_881_Int__absorb,axiom,
    ! [A_1: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,A_1),A_1) = A_1 ) ).

tff(fact_882_Int__absorb,axiom,
    ! [A_1: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,A_1),A_1) = A_1 ) ).

tff(fact_883_inf_Oidem,axiom,
    ! [A_2: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A_2),A_2) = A_2 ) ).

tff(fact_884_inf_Oidem,axiom,
    ! [A_2: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,A_2),A_2) = A_2 ) ).

tff(fact_885_inf_Oidem,axiom,
    ! [A_6: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,A_6),A_6) = A_6 ) ).

tff(fact_886_inf_Oidem,axiom,
    ! [A_2: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,A_2),A_2) = A_2 ) ).

tff(fact_887_inf__idem,axiom,
    ! [X_2: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,X_2),X_2) = X_2 ) ).

tff(fact_888_inf__idem,axiom,
    ! [X_2: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,X_2),X_2) = X_2 ) ).

tff(fact_889_inf__idem,axiom,
    ! [X_9: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,X_9),X_9) = X_9 ) ).

tff(fact_890_inf__idem,axiom,
    ! [X_2: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,X_2),X_2) = X_2 ) ).

tff(fact_891_fun__diff__def,axiom,
    ! [A_1: fun_Ho882550911a_bool,B_2: fun_Ho882550911a_bool,X: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A_1),B_2),X))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(minus_minus_bool,hAPP_H454055633a_bool(A_1,X)),hAPP_H454055633a_bool(B_2,X))) ) ).

tff(fact_892_fun__diff__def,axiom,
    ! [A_1: fun_pname_bool,B_2: fun_pname_bool,X: pname] :
      ( hBOOL(hAPP_pname_bool(hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A_1),B_2),X))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(minus_minus_bool,hAPP_pname_bool(A_1,X)),hAPP_pname_bool(B_2,X))) ) ).

tff(fact_893_fun__diff__def,axiom,
    ! [A_1: fun_nat_bool,B_2: fun_nat_bool,X: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_1),B_2),X))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(minus_minus_bool,hAPP_nat_bool(A_1,X)),hAPP_nat_bool(B_2,X))) ) ).

tff(fact_894_inf__fun__def,axiom,
    ! [F: fun_nat_bool,G: fun_nat_bool,X: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,F),G),X))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila310582991f_bool,hAPP_nat_bool(F,X)),hAPP_nat_bool(G,X))) ) ).

tff(fact_895_inf__fun__def,axiom,
    ! [F: fun_Ho882550911a_bool,G: fun_Ho882550911a_bool,X: hoare_2076525193iple_a] :
      ( hBOOL(hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,F),G),X))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila310582991f_bool,hAPP_H454055633a_bool(F,X)),hAPP_H454055633a_bool(G,X))) ) ).

tff(fact_896_inf__fun__def,axiom,
    ! [F: fun_pname_bool,G: fun_pname_bool,X: pname] :
      ( hBOOL(hAPP_pname_bool(hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,F),G),X))
    <=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(semila310582991f_bool,hAPP_pname_bool(F,X)),hAPP_pname_bool(G,X))) ) ).

tff(fact_897_set__diff__eq,axiom,
    ! [A_1: fun_fu2026518849l_bool,B_2: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(minus_1051402706l_bool,A_1),B_2) = hAPP_f354682389l_bool(collec416005674a_bool,hAPP_f354682389l_bool(hAPP_f1121033001l_bool(cOMBS_204124788l_bool,hAPP_f1684296317l_bool(cOMBB_120690437a_bool(fconj),hAPP_f354682389l_bool(hAPP_f638520823l_bool(cOMBC_1528738210l_bool,member186597612a_bool),A_1))),hAPP_f354682389l_bool(cOMBB_1757941347a_bool(fNot),hAPP_f354682389l_bool(hAPP_f638520823l_bool(cOMBC_1528738210l_bool,member186597612a_bool),B_2)))) ) ).

tff(fact_898_set__diff__eq,axiom,
    ! [A_1: fun_nat_bool,B_2: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_1),B_2) = hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(cOMBB_1015721476ol_nat(fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),A_1))),hAPP_f800510211t_bool(cOMBB_bool_bool_nat(fNot),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),B_2)))) ) ).

tff(fact_899_set__diff__eq,axiom,
    ! [A_1: fun_Ho882550911a_bool,B_2: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A_1),B_2) = hAPP_f642208769a_bool(collec977865374iple_a,hAPP_f642208769a_bool(hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,hAPP_f268025233l_bool(cOMBB_345279363iple_a(fconj),hAPP_f642208769a_bool(hAPP_f325481667a_bool(cOMBC_1529670358l_bool,member276146780iple_a),A_1))),hAPP_f642208769a_bool(cOMBB_361863589iple_a(fNot),hAPP_f642208769a_bool(hAPP_f325481667a_bool(cOMBC_1529670358l_bool,member276146780iple_a),B_2)))) ) ).

tff(fact_900_set__diff__eq,axiom,
    ! [A_1: fun_pname_bool,B_2: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A_1),B_2) = hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,hAPP_f1402196763l_bool(cOMBB_675860798_pname(fconj),hAPP_f759274231e_bool(hAPP_f661147897e_bool(cOMBC_1058051404l_bool,member_pname),A_1))),hAPP_f759274231e_bool(cOMBB_647938656_pname(fNot),hAPP_f759274231e_bool(hAPP_f661147897e_bool(cOMBC_1058051404l_bool,member_pname),B_2)))) ) ).

tff(fact_901_Int__def,axiom,
    ! [A_1: fun_fu2026518849l_bool,B_2: fun_fu2026518849l_bool] : ( hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1706849871l_bool,A_1),B_2) = hAPP_f354682389l_bool(collec416005674a_bool,hAPP_f354682389l_bool(hAPP_f1121033001l_bool(cOMBS_204124788l_bool,hAPP_f1684296317l_bool(cOMBB_120690437a_bool(fconj),hAPP_f354682389l_bool(hAPP_f638520823l_bool(cOMBC_1528738210l_bool,member186597612a_bool),A_1))),hAPP_f354682389l_bool(hAPP_f638520823l_bool(cOMBC_1528738210l_bool,member186597612a_bool),B_2))) ) ).

tff(fact_902_Int__def,axiom,
    ! [A_1: fun_nat_bool,B_2: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A_1),B_2) = hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(cOMBB_1015721476ol_nat(fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),A_1))),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),B_2))) ) ).

tff(fact_903_Int__def,axiom,
    ! [A_1: fun_Ho882550911a_bool,B_2: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,A_1),B_2) = hAPP_f642208769a_bool(collec977865374iple_a,hAPP_f642208769a_bool(hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,hAPP_f268025233l_bool(cOMBB_345279363iple_a(fconj),hAPP_f642208769a_bool(hAPP_f325481667a_bool(cOMBC_1529670358l_bool,member276146780iple_a),A_1))),hAPP_f642208769a_bool(hAPP_f325481667a_bool(cOMBC_1529670358l_bool,member276146780iple_a),B_2))) ) ).

tff(fact_904_Int__def,axiom,
    ! [A_1: fun_pname_bool,B_2: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,A_1),B_2) = hAPP_f759274231e_bool(collect_pname,hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,hAPP_f1402196763l_bool(cOMBB_675860798_pname(fconj),hAPP_f759274231e_bool(hAPP_f661147897e_bool(cOMBC_1058051404l_bool,member_pname),A_1))),hAPP_f759274231e_bool(hAPP_f661147897e_bool(cOMBC_1058051404l_bool,member_pname),B_2))) ) ).

tff(fact_905_Int__commute,axiom,
    ! [A_1: fun_nat_bool,B_2: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A_1),B_2) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,B_2),A_1) ) ).

tff(fact_906_Int__commute,axiom,
    ! [A_1: fun_Ho882550911a_bool,B_2: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,A_1),B_2) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,B_2),A_1) ) ).

tff(fact_907_Int__commute,axiom,
    ! [A_1: fun_pname_bool,B_2: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,A_1),B_2) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,B_2),A_1) ) ).

tff(fact_908_inf_Ocommute,axiom,
    ! [A_2: fun_nat_bool,B_1: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A_2),B_1) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,B_1),A_2) ) ).

tff(fact_909_inf_Ocommute,axiom,
    ! [A_2: fun_Ho882550911a_bool,B_1: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,A_2),B_1) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,B_1),A_2) ) ).

tff(fact_910_inf_Ocommute,axiom,
    ! [A_5: nat,B_5: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,A_5),B_5) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,B_5),A_5) ) ).

tff(fact_911_inf_Ocommute,axiom,
    ! [A_2: fun_pname_bool,B_1: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,A_2),B_1) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,B_1),A_2) ) ).

tff(fact_912_inf__sup__aci_I1_J,axiom,
    ! [X_2: fun_nat_bool,Y_1: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,X_2),Y_1) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,Y_1),X_2) ) ).

tff(fact_913_inf__sup__aci_I1_J,axiom,
    ! [X_2: fun_Ho882550911a_bool,Y_1: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,X_2),Y_1) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,Y_1),X_2) ) ).

tff(fact_914_inf__sup__aci_I1_J,axiom,
    ! [X_8: nat,Y_7: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,X_8),Y_7) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,Y_7),X_8) ) ).

tff(fact_915_inf__sup__aci_I1_J,axiom,
    ! [X_2: fun_pname_bool,Y_1: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,X_2),Y_1) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,Y_1),X_2) ) ).

tff(fact_916_inf__commute,axiom,
    ! [X_2: fun_nat_bool,Y_1: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,X_2),Y_1) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,Y_1),X_2) ) ).

tff(fact_917_inf__commute,axiom,
    ! [X_2: fun_Ho882550911a_bool,Y_1: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,X_2),Y_1) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,Y_1),X_2) ) ).

tff(fact_918_inf__commute,axiom,
    ! [X_7: nat,Y_6: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,X_7),Y_6) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,Y_6),X_7) ) ).

tff(fact_919_inf__commute,axiom,
    ! [X_2: fun_pname_bool,Y_1: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,X_2),Y_1) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,Y_1),X_2) ) ).

tff(fact_920_Int__left__absorb,axiom,
    ! [A_1: fun_nat_bool,B_2: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A_1),B_2)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A_1),B_2) ) ).

tff(fact_921_Int__left__absorb,axiom,
    ! [A_1: fun_Ho882550911a_bool,B_2: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,A_1),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,A_1),B_2)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,A_1),B_2) ) ).

tff(fact_922_Int__left__absorb,axiom,
    ! [A_1: fun_pname_bool,B_2: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,A_1),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,A_1),B_2)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,A_1),B_2) ) ).

tff(fact_923_inf_Oleft__idem,axiom,
    ! [A_2: fun_nat_bool,B_1: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A_2),B_1)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A_2),B_1) ) ).

tff(fact_924_inf_Oleft__idem,axiom,
    ! [A_2: fun_Ho882550911a_bool,B_1: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,A_2),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,A_2),B_1)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,A_2),B_1) ) ).

tff(fact_925_inf_Oleft__idem,axiom,
    ! [A_4: nat,B_4: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,A_4),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,A_4),B_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,A_4),B_4) ) ).

tff(fact_926_inf_Oleft__idem,axiom,
    ! [A_2: fun_pname_bool,B_1: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,A_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,A_2),B_1)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,A_2),B_1) ) ).

tff(fact_927_inf__sup__aci_I4_J,axiom,
    ! [X_2: fun_nat_bool,Y_1: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,X_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,X_2),Y_1)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,X_2),Y_1) ) ).

tff(fact_928_inf__sup__aci_I4_J,axiom,
    ! [X_2: fun_Ho882550911a_bool,Y_1: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,X_2),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,X_2),Y_1)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,X_2),Y_1) ) ).

tff(fact_929_inf__sup__aci_I4_J,axiom,
    ! [X_6: nat,Y_5: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,X_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,X_6),Y_5)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,X_6),Y_5) ) ).

tff(fact_930_inf__sup__aci_I4_J,axiom,
    ! [X_2: fun_pname_bool,Y_1: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,X_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,X_2),Y_1)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,X_2),Y_1) ) ).

tff(fact_931_inf__left__idem,axiom,
    ! [X_2: fun_nat_bool,Y_1: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,X_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,X_2),Y_1)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,X_2),Y_1) ) ).

tff(fact_932_inf__left__idem,axiom,
    ! [X_2: fun_Ho882550911a_bool,Y_1: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,X_2),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,X_2),Y_1)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,X_2),Y_1) ) ).

tff(fact_933_inf__left__idem,axiom,
    ! [X_5: nat,Y_4: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,X_5),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,X_5),Y_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,X_5),Y_4) ) ).

tff(fact_934_inf__left__idem,axiom,
    ! [X_2: fun_pname_bool,Y_1: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,X_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,X_2),Y_1)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,X_2),Y_1) ) ).

tff(fact_935_Int__left__commute,axiom,
    ! [A_1: fun_nat_bool,B_2: fun_nat_bool,C_1: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,B_2),C_1)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,B_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A_1),C_1)) ) ).

tff(fact_936_Int__left__commute,axiom,
    ! [A_1: fun_Ho882550911a_bool,B_2: fun_Ho882550911a_bool,C_1: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,A_1),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,B_2),C_1)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,B_2),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,A_1),C_1)) ) ).

tff(fact_937_Int__left__commute,axiom,
    ! [A_1: fun_pname_bool,B_2: fun_pname_bool,C_1: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,A_1),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,B_2),C_1)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,B_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,A_1),C_1)) ) ).

tff(fact_938_inf_Oleft__commute,axiom,
    ! [B_1: fun_nat_bool,A_2: fun_nat_bool,C_2: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,B_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A_2),C_2)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,B_1),C_2)) ) ).

tff(fact_939_inf_Oleft__commute,axiom,
    ! [B_1: fun_Ho882550911a_bool,A_2: fun_Ho882550911a_bool,C_2: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,B_1),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,A_2),C_2)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,A_2),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,B_1),C_2)) ) ).

tff(fact_940_inf_Oleft__commute,axiom,
    ! [B_3: nat,A_3: nat,C_3: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,B_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,A_3),C_3)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,B_3),C_3)) ) ).

tff(fact_941_inf_Oleft__commute,axiom,
    ! [B_1: fun_pname_bool,A_2: fun_pname_bool,C_2: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,B_1),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,A_2),C_2)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,A_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,B_1),C_2)) ) ).

tff(fact_942_inf__sup__aci_I3_J,axiom,
    ! [X_2: fun_nat_bool,Y_1: fun_nat_bool,Z_1: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,X_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,Y_1),Z_1)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,Y_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,X_2),Z_1)) ) ).

tff(fact_943_inf__sup__aci_I3_J,axiom,
    ! [X_2: fun_Ho882550911a_bool,Y_1: fun_Ho882550911a_bool,Z_1: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,X_2),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,Y_1),Z_1)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,Y_1),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,X_2),Z_1)) ) ).

tff(fact_944_inf__sup__aci_I3_J,axiom,
    ! [X_4: nat,Y_3: nat,Z_3: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,X_4),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,Y_3),Z_3)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,Y_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,X_4),Z_3)) ) ).

tff(fact_945_inf__sup__aci_I3_J,axiom,
    ! [X_2: fun_pname_bool,Y_1: fun_pname_bool,Z_1: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,X_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,Y_1),Z_1)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,Y_1),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,X_2),Z_1)) ) ).

tff(fact_946_inf__left__commute,axiom,
    ! [X_2: fun_nat_bool,Y_1: fun_nat_bool,Z_1: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,X_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,Y_1),Z_1)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,Y_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,X_2),Z_1)) ) ).

tff(fact_947_inf__left__commute,axiom,
    ! [X_2: fun_Ho882550911a_bool,Y_1: fun_Ho882550911a_bool,Z_1: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,X_2),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,Y_1),Z_1)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,Y_1),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,X_2),Z_1)) ) ).

tff(fact_948_inf__left__commute,axiom,
    ! [X_3: nat,Y_2: nat,Z_2: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,X_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,Y_2),Z_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,Y_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(semila80283416nf_nat,X_3),Z_2)) ) ).

tff(fact_949_inf__left__commute,axiom,
    ! [X_2: fun_pname_bool,Y_1: fun_pname_bool,Z_1: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,X_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,Y_1),Z_1)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,Y_1),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,X_2),Z_1)) ) ).

tff(fact_950_Diff__iff,axiom,
    ! [C_2: fun_Ho882550911a_bool,A_1: fun_fu2026518849l_bool,B_2: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,C_2),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(minus_1051402706l_bool,A_1),B_2)))
    <=> ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,C_2),A_1))
        & ~ hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,C_2),B_2)) ) ) ).

tff(fact_951_Diff__iff,axiom,
    ! [C_2: nat,A_1: fun_nat_bool,B_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_1),B_2)))
    <=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),A_1))
        & ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),B_2)) ) ) ).

tff(fact_952_Diff__iff,axiom,
    ! [C_2: hoare_2076525193iple_a,A_1: fun_Ho882550911a_bool,B_2: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,C_2),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A_1),B_2)))
    <=> ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,C_2),A_1))
        & ~ hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,C_2),B_2)) ) ) ).

tff(fact_953_Diff__iff,axiom,
    ! [C_2: pname,A_1: fun_pname_bool,B_2: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A_1),B_2)))
    <=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_2),A_1))
        & ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_2),B_2)) ) ) ).

tff(fact_954_Int__iff,axiom,
    ! [C_2: fun_Ho882550911a_bool,A_1: fun_fu2026518849l_bool,B_2: fun_fu2026518849l_bool] :
      ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,C_2),hAPP_f354682389l_bool(hAPP_f1976119377l_bool(semila1706849871l_bool,A_1),B_2)))
    <=> ( hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,C_2),A_1))
        & hBOOL(hAPP_f1490793353l_bool(hAPP_f844821689l_bool(member186597612a_bool,C_2),B_2)) ) ) ).

tff(fact_955_Int__iff,axiom,
    ! [C_2: nat,A_1: fun_nat_bool,B_2: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,A_1),B_2)))
    <=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),A_1))
        & hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),B_2)) ) ) ).

tff(fact_956_Int__iff,axiom,
    ! [C_2: hoare_2076525193iple_a,A_1: fun_Ho882550911a_bool,B_2: fun_Ho882550911a_bool] :
      ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,C_2),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,A_1),B_2)))
    <=> ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,C_2),A_1))
        & hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,C_2),B_2)) ) ) ).

tff(fact_957_Int__iff,axiom,
    ! [C_2: pname,A_1: fun_pname_bool,B_2: fun_pname_bool] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,A_1),B_2)))
    <=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_2),A_1))
        & hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_2),B_2)) ) ) ).

tff(fact_958_Diff__Int__distrib,axiom,
    ! [C_1: fun_Ho882550911a_bool,A_1: fun_Ho882550911a_bool,B_2: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,C_1),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,A_1),B_2)) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(minus_1649626414a_bool,hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,C_1),A_1)),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila1851231473a_bool,C_1),B_2)) ) ).

tff(fact_959_Diff__Int__distrib,axiom,
    ! [C_1: fun_pname_bool,A_1: fun_pname_bool,B_2: fun_pname_bool] : ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,C_1),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A_1),B_2)) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,C_1),A_1)),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(semila1635148844e_bool,C_1),B_2)) ) ).

tff(fact_960_Diff__Int__distrib,axiom,
    ! [C_1: fun_nat_bool,A_1: fun_nat_bool,B_2: fun_nat_bool] : ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,C_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_1),B_2)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,C_1),A_1)),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(semila840543986t_bool,C_1),B_2)) ) ).

tff(fact_961_diff__0__eq__0,axiom,
    ! [N_2: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),N_2) = zero_zero_nat ) ).

tff(fact_962_minus__nat_Odiff__0,axiom,
    ! [M_2: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_2),zero_zero_nat) = M_2 ) ).

tff(fact_963_diff__self__eq__0,axiom,
    ! [M_2: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_2),M_2) = zero_zero_nat ) ).

tff(fact_964_diffs0__imp__equal,axiom,
    ! [M_2: nat,N_2: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_2),N_2) = zero_zero_nat )
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_2),M_2) = zero_zero_nat )
       => ( M_2 = N_2 ) ) ) ).

tff(fact_965_diff__Suc__Suc,axiom,
    ! [M_2: nat,N_2: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M_2)),hAPP_nat_nat(suc,N_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_2),N_2) ) ).

tff(fact_966_Suc__diff__diff,axiom,
    ! [M_2: nat,N_2: nat,K_2: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M_2)),N_2)),hAPP_nat_nat(suc,K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_2),N_2)),K_2) ) ).

tff(fact_967_diff__commute,axiom,
    ! [I: nat,J_1: nat,K_2: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),J_1)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),K_2)),J_1) ) ).

tff(fact_968_zero__induct__lemma,axiom,
    ! [I_1: nat,Pa: fun_nat_bool,K_1: nat] :
      ( hBOOL(hAPP_nat_bool(Pa,K_1))
     => ( ! [N: nat] :
            ( hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(suc,N)))
           => hBOOL(hAPP_nat_bool(Pa,N)) )
       => hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,K_1),I_1))) ) ) ).

tff(fact_969_diff__Suc,axiom,
    ! [M: nat,N_1: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(nat_case_nat(zero_zero_nat,cOMBI_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1)) ) ).

tff(fact_970_One__nat__def,axiom,
    one_one_nat = hAPP_nat_nat(suc,zero_zero_nat) ).

tff(fact_971_diff__Suc__1,axiom,
    ! [N_2: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N_2)),one_one_nat) = N_2 ) ).

tff(fact_972_diff__Suc__eq__diff__pred,axiom,
    ! [M_2: nat,N_2: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_2),hAPP_nat_nat(suc,N_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_2),one_one_nat)),N_2) ) ).

tff(fact_973_plus__nat_Oadd__0,axiom,
    ! [N_2: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),N_2) = N_2 ) ).

tff(fact_974_Nat_Oadd__0__right,axiom,
    ! [M_2: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_2),zero_zero_nat) = M_2 ) ).

tff(fact_975_add__is__0,axiom,
    ! [M: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        & ( N_1 = zero_zero_nat ) ) ) ).

tff(fact_976_add__eq__self__zero,axiom,
    ! [M_2: nat,N_2: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_2),N_2) = M_2 )
     => ( N_2 = zero_zero_nat ) ) ).

tff(fact_977_add__Suc__right,axiom,
    ! [M_2: nat,N_2: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_2),hAPP_nat_nat(suc,N_2)) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_2),N_2)) ) ).

tff(fact_978_add__Suc,axiom,
    ! [M_2: nat,N_2: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(suc,M_2)),N_2) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_2),N_2)) ) ).

tff(fact_979_add__Suc__shift,axiom,
    ! [M_2: nat,N_2: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(suc,M_2)),N_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_2),hAPP_nat_nat(suc,N_2)) ) ).

tff(fact_980_nat__add__commute,axiom,
    ! [M_2: nat,N_2: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_2),N_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_2),M_2) ) ).

tff(fact_981_nat__add__left__commute,axiom,
    ! [X_1: nat,Y: nat,Z: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_1),Z)) ) ).

tff(fact_982_nat__add__assoc,axiom,
    ! [M_2: nat,N_2: nat,K_2: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_2),N_2)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_2),K_2)) ) ).

tff(fact_983_nat__add__left__cancel,axiom,
    ! [K_1: nat,M: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),N_1) )
    <=> ( M = N_1 ) ) ).

tff(fact_984_nat__add__right__cancel,axiom,
    ! [M: nat,K_1: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),K_1) )
    <=> ( M = N_1 ) ) ).

tff(fact_985_diff__add__inverse2,axiom,
    ! [M_2: nat,N_2: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_2),N_2)),N_2) = M_2 ) ).

tff(fact_986_diff__add__inverse,axiom,
    ! [N_2: nat,M_2: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_2),M_2)),N_2) = M_2 ) ).

tff(fact_987_diff__diff__left,axiom,
    ! [I: nat,J_1: nat,K_2: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),J_1)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_1),K_2)) ) ).

tff(fact_988_diff__cancel,axiom,
    ! [K_2: nat,M_2: nat,N_2: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),M_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),N_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_2),N_2) ) ).

tff(fact_989_diff__cancel2,axiom,
    ! [M_2: nat,K_2: nat,N_2: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_2),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_2),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_2),N_2) ) ).

tff(fact_990_add__is__1,axiom,
    ! [M: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1) = hAPP_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( M = hAPP_nat_nat(suc,zero_zero_nat) )
          & ( N_1 = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N_1 = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

tff(fact_991_one__is__add,axiom,
    ! [M: nat,N_1: nat] :
      ( ( hAPP_nat_nat(suc,zero_zero_nat) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1) )
    <=> ( ( ( M = hAPP_nat_nat(suc,zero_zero_nat) )
          & ( N_1 = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N_1 = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

tff(fact_992_diff__add__0,axiom,
    ! [N_2: nat,M_2: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_2),M_2)) = zero_zero_nat ) ).

tff(fact_993_Suc__eq__plus1,axiom,
    ! [N_2: nat] : ( hAPP_nat_nat(suc,N_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_2),one_one_nat) ) ).

tff(fact_994_Suc__eq__plus1__left,axiom,
    ! [N_2: nat] : ( hAPP_nat_nat(suc,N_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N_2) ) ).

tff(fact_995_add__eq__if,axiom,
    ! [N_2: nat,M_2: nat] :
      ( ( ( M_2 = zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_2),N_2) = N_2 ) )
      & ( ( M_2 != zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_2),N_2) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_2),one_one_nat)),N_2)) ) ) ) ).

tff(fact_996_com_Osize_I4_J,axiom,
    ! [Com1_2: com,Com2_2: com] : ( com_size(semi(Com1_2,Com2_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,com_size(Com1_2)),com_size(Com2_2))),hAPP_nat_nat(suc,zero_zero_nat)) ) ).

tff(fact_997_com_Osize_I6_J,axiom,
    ! [Fun_1: fun_state_bool,Com_1: com] : ( com_size(while(Fun_1,Com_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,com_size(Com_1)),hAPP_nat_nat(suc,zero_zero_nat)) ) ).

tff(fact_998_com_Osize_I7_J,axiom,
    ! [Pname_1: pname] : ( com_size(hAPP_pname_com(body,Pname_1)) = zero_zero_nat ) ).

tff(fact_999_com_Osize_I1_J,axiom,
    com_size(skip) = zero_zero_nat ).

tff(fact_1000_com_Osize_I12_J,axiom,
    ! [Com1_2: com,Com2_2: com] : ( hAPP_com_nat(size_size_com,semi(Com1_2,Com2_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_com_nat(size_size_com,Com1_2)),hAPP_com_nat(size_size_com,Com2_2))),hAPP_nat_nat(suc,zero_zero_nat)) ) ).

tff(fact_1001_com_Osize_I14_J,axiom,
    ! [Fun_1: fun_state_bool,Com_1: com] : ( hAPP_com_nat(size_size_com,while(Fun_1,Com_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_com_nat(size_size_com,Com_1)),hAPP_nat_nat(suc,zero_zero_nat)) ) ).

tff(fact_1002_add__mult__distrib2,axiom,
    ! [K_2: nat,M_2: nat,N_2: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_2),N_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_2)) ) ).

tff(fact_1003_add__mult__distrib,axiom,
    ! [M_2: nat,N_2: nat,K_2: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_2),N_2)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_2),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_2),K_2)) ) ).

tff(fact_1004_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1) = one_one_nat )
    <=> ( ( M = one_one_nat )
        & ( N_1 = one_one_nat ) ) ) ).

tff(fact_1005_nat__mult__1__right,axiom,
    ! [N_2: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_2),one_one_nat) = N_2 ) ).

tff(fact_1006_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N_1: nat] :
      ( ( one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1) )
    <=> ( ( M = one_one_nat )
        & ( N_1 = one_one_nat ) ) ) ).

tff(fact_1007_nat__mult__1,axiom,
    ! [N_2: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),N_2) = N_2 ) ).

tff(fact_1008_mult__0,axiom,
    ! [N_2: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,zero_zero_nat),N_2) = zero_zero_nat ) ).

tff(fact_1009_mult__0__right,axiom,
    ! [M_2: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_2),zero_zero_nat) = zero_zero_nat ) ).

tff(fact_1010_mult__is__0,axiom,
    ! [M: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        | ( N_1 = zero_zero_nat ) ) ) ).

tff(fact_1011_mult__cancel1,axiom,
    ! [K_1: nat,M: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N_1) )
    <=> ( ( M = N_1 )
        | ( K_1 = zero_zero_nat ) ) ) ).

tff(fact_1012_mult__cancel2,axiom,
    ! [M: nat,K_1: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),K_1) )
    <=> ( ( M = N_1 )
        | ( K_1 = zero_zero_nat ) ) ) ).

tff(fact_1013_Suc__mult__cancel1,axiom,
    ! [K_1: nat,M: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),N_1) )
    <=> ( M = N_1 ) ) ).

tff(fact_1014_diff__mult__distrib,axiom,
    ! [M_2: nat,N_2: nat,K_2: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_2),N_2)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_2),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_2),K_2)) ) ).

tff(fact_1015_diff__mult__distrib2,axiom,
    ! [K_2: nat,M_2: nat,N_2: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_2),N_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N_2)) ) ).

tff(fact_1016_mult__eq__1__iff,axiom,
    ! [M: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1) = hAPP_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = hAPP_nat_nat(suc,zero_zero_nat) )
        & ( N_1 = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ).

tff(fact_1017_mult__Suc,axiom,
    ! [M_2: nat,N_2: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,M_2)),N_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_2),N_2)) ) ).

tff(fact_1018_mult__Suc__right,axiom,
    ! [M_2: nat,N_2: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_2),hAPP_nat_nat(suc,N_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_2),N_2)) ) ).

tff(fact_1019_mult__eq__self__implies__10,axiom,
    ! [M_2: nat,N_2: nat] :
      ( ( M_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_2),N_2) )
     => ( ( N_2 = one_one_nat )
        | ( M_2 = zero_zero_nat ) ) ) ).

tff(fact_1020_com_Osize_I15_J,axiom,
    ! [Pname_1: pname] : ( hAPP_com_nat(size_size_com,hAPP_pname_com(body,Pname_1)) = zero_zero_nat ) ).

tff(fact_1021_com_Osize_I9_J,axiom,
    hAPP_com_nat(size_size_com,skip) = zero_zero_nat ).

tff(fact_1022_mult__eq__if,axiom,
    ! [N_2: nat,M_2: nat] :
      ( ( ( M_2 = zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_2),N_2) = zero_zero_nat ) )
      & ( ( M_2 != zero_zero_nat )
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_2),N_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_2),one_one_nat)),N_2)) ) ) ) ).

tff(fact_1023_nat__mult__commute,axiom,
    ! [M_2: nat,N_2: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_2),N_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_2),M_2) ) ).

tff(fact_1024_nat__mult__assoc,axiom,
    ! [M_2: nat,N_2: nat,K_2: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_2),N_2)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_2),K_2)) ) ).

tff(fact_1025_left__add__mult__distrib,axiom,
    ! [I: nat,U_1: nat,J_1: nat,K_2: nat] : ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),U_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_1),U_1)),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J_1)),U_1)),K_2) ) ).

tff(fact_1026_nat__mult__eq__cancel__disj,axiom,
    ! [K_1: nat,M: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N_1) )
    <=> ( ( K_1 = zero_zero_nat )
        | ( M = N_1 ) ) ) ).

tff(fact_1027_com_Osize_I13_J,axiom,
    ! [Fun_1: fun_state_bool,Com1_1: com,Com2_1: com] : ( hAPP_com_nat(size_size_com,cond(Fun_1,Com1_1,Com2_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_com_nat(size_size_com,Com1_1)),hAPP_com_nat(size_size_com,Com2_1))),hAPP_nat_nat(suc,zero_zero_nat)) ) ).

tff(fact_1028_com_Osize_I5_J,axiom,
    ! [Fun_1: fun_state_bool,Com1_1: com,Com2_1: com] : ( com_size(cond(Fun_1,Com1_1,Com2_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,com_size(Com1_1)),com_size(Com2_1))),hAPP_nat_nat(suc,zero_zero_nat)) ) ).

tff(fact_1029_finite__Collect__le__nat,axiom,
    ! [K_1: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_eq_nat),K_1)))) ).

tff(fact_1030_le0,axiom,
    ! [N_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N_2)) ).

tff(fact_1031_evaln__elim__cases_I5_J,axiom,
    ! [B_1: fun_state_bool,C1: com,C2: com,S_1: state,N_1: nat,T_1: state] :
      ( hBOOL(hAPP_state_bool(evaln(cond(B_1,C1,C2),S_1,N_1),T_1))
     => ( ( hBOOL(hAPP_state_bool(B_1,S_1))
         => ~ hBOOL(hAPP_state_bool(evaln(C1,S_1,N_1),T_1)) )
       => ~ ( ~ hBOOL(hAPP_state_bool(B_1,S_1))
           => ~ hBOOL(hAPP_state_bool(evaln(C2,S_1,N_1),T_1)) ) ) ) ).

tff(fact_1032_evaln_OIfTrue,axiom,
    ! [C1: com,C0: com,N_1: nat,S1: state,B_1: fun_state_bool,S_1: state] :
      ( hBOOL(hAPP_state_bool(B_1,S_1))
     => ( hBOOL(hAPP_state_bool(evaln(C0,S_1,N_1),S1))
       => hBOOL(hAPP_state_bool(evaln(cond(B_1,C0,C1),S_1,N_1),S1)) ) ) ).

tff(fact_1033_evaln_OIfFalse,axiom,
    ! [C0: com,C1: com,N_1: nat,S1: state,B_1: fun_state_bool,S_1: state] :
      ( ~ hBOOL(hAPP_state_bool(B_1,S_1))
     => ( hBOOL(hAPP_state_bool(evaln(C1,S_1,N_1),S1))
       => hBOOL(hAPP_state_bool(evaln(cond(B_1,C0,C1),S_1,N_1),S1)) ) ) ).

tff(fact_1034_evalc_OIfFalse,axiom,
    ! [C0: com,C1: com,S1: state,B_1: fun_state_bool,S_1: state] :
      ( ~ hBOOL(hAPP_state_bool(B_1,S_1))
     => ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C1),S_1),S1))
       => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(cond(B_1,C0,C1)),S_1),S1)) ) ) ).

tff(fact_1035_evalc_OIfTrue,axiom,
    ! [C1: com,C0: com,S1: state,B_1: fun_state_bool,S_1: state] :
      ( hBOOL(hAPP_state_bool(B_1,S_1))
     => ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C0),S_1),S1))
       => hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(cond(B_1,C0,C1)),S_1),S1)) ) ) ).

tff(fact_1036_evalc__elim__cases_I5_J,axiom,
    ! [B_1: fun_state_bool,C1: com,C2: com,S_1: state,T_1: state] :
      ( hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(cond(B_1,C1,C2)),S_1),T_1))
     => ( ( hBOOL(hAPP_state_bool(B_1,S_1))
         => ~ hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C1),S_1),T_1)) )
       => ~ ( ~ hBOOL(hAPP_state_bool(B_1,S_1))
           => ~ hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(evalc(C2),S_1),T_1)) ) ) ) ).

tff(fact_1037_com_Osimps_I55_J,axiom,
    ! [Pname: pname,Fun_1: fun_state_bool,Com1_1: com,Com2_1: com] : ( hAPP_pname_com(body,Pname) != cond(Fun_1,Com1_1,Com2_1) ) ).

tff(fact_1038_com_Osimps_I54_J,axiom,
    ! [Fun_1: fun_state_bool,Com1_1: com,Com2_1: com,Pname: pname] : ( cond(Fun_1,Com1_1,Com2_1) != hAPP_pname_com(body,Pname) ) ).

tff(fact_1039_com_Osimps_I4_J,axiom,
    ! [Fun_1: fun_state_bool,Com1_1: com,Com2_1: com,Fun: fun_state_bool,Com1: com,Com2: com] :
      ( ( cond(Fun_1,Com1_1,Com2_1) = cond(Fun,Com1,Com2) )
    <=> ( ( Fun_1 = Fun )
        & ( Com1_1 = Com1 )
        & ( Com2_1 = Com2 ) ) ) ).

tff(fact_1040_le__antisym,axiom,
    ! [M_2: nat,N_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),M_2))
       => ( M_2 = N_2 ) ) ) ).

tff(fact_1041_le__trans,axiom,
    ! [K_2: nat,I: nat,J_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_1),K_2))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),K_2)) ) ) ).

tff(fact_1042_eq__imp__le,axiom,
    ! [M_2: nat,N_2: nat] :
      ( ( M_2 = N_2 )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_2)) ) ).

tff(fact_1043_nat__le__linear,axiom,
    ! [M_2: nat,N_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_2))
      | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),M_2)) ) ).

tff(fact_1044_le__refl,axiom,
    ! [N_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),N_2)) ).

tff(fact_1045_Suc__leD,axiom,
    ! [M_2: nat,N_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M_2)),N_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_2)) ) ).

tff(fact_1046_le__SucE,axiom,
    ! [M_2: nat,N_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),hAPP_nat_nat(suc,N_2)))
     => ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_2))
       => ( M_2 = hAPP_nat_nat(suc,N_2) ) ) ) ).

tff(fact_1047_le__SucI,axiom,
    ! [M_2: nat,N_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),hAPP_nat_nat(suc,N_2))) ) ).

tff(fact_1048_Suc__le__mono,axiom,
    ! [N_1: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_1)),hAPP_nat_nat(suc,M)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M)) ) ).

tff(fact_1049_le__Suc__eq,axiom,
    ! [M: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(suc,N_1)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
        | ( M = hAPP_nat_nat(suc,N_1) ) ) ) ).

tff(fact_1050_not__less__eq__eq,axiom,
    ! [M: nat,N_1: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_1)),M)) ) ).

tff(fact_1051_Suc__n__not__le__n,axiom,
    ! [N_2: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_2)),N_2)) ).

tff(fact_1052_le__0__eq,axiom,
    ! [N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),zero_zero_nat))
    <=> ( N_1 = zero_zero_nat ) ) ).

tff(fact_1053_less__eq__nat_Osimps_I1_J,axiom,
    ! [N_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N_2)) ).

tff(fact_1054_evaln__nonstrict,axiom,
    ! [M_2: nat,C: com,S: state,N_2: nat,T: state] :
      ( hBOOL(hAPP_state_bool(evaln(C,S,N_2),T))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),M_2))
       => hBOOL(hAPP_state_bool(evaln(C,S,M_2),T)) ) ) ).

tff(fact_1055_diff__le__self,axiom,
    ! [M_2: nat,N_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_2),N_2)),M_2)) ).

tff(fact_1056_diff__le__mono2,axiom,
    ! [L: nat,M_2: nat,N_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L),N_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L),M_2))) ) ).

tff(fact_1057_diff__le__mono,axiom,
    ! [L: nat,M_2: nat,N_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_2),L)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_2),L))) ) ).

tff(fact_1058_diff__diff__cancel,axiom,
    ! [I: nat,N_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),N_2))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_2),I)) = I ) ) ).

tff(fact_1059_eq__diff__iff,axiom,
    ! [N_1: nat,K_1: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),M))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_1))
       => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),K_1) )
        <=> ( M = N_1 ) ) ) ) ).

tff(fact_1060_Nat_Odiff__diff__eq,axiom,
    ! [N_2: nat,K_2: nat,M_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),M_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N_2))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_2),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_2),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_2),N_2) ) ) ) ).

tff(fact_1061_le__diff__iff,axiom,
    ! [N_1: nat,K_1: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),M))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_1))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),K_1)))
        <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) ) ) ) ).

tff(fact_1062_add__leE,axiom,
    ! [M_2: nat,K_2: nat,N_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_2),K_2)),N_2))
     => ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_2))
         => ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N_2)) ) ) ).

tff(fact_1063_add__leD1,axiom,
    ! [M_2: nat,K_2: nat,N_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_2),K_2)),N_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_2)) ) ).

tff(fact_1064_add__leD2,axiom,
    ! [M_2: nat,K_2: nat,N_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_2),K_2)),N_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N_2)) ) ).

tff(fact_1065_add__le__mono,axiom,
    ! [K_2: nat,L: nat,I: nat,J_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),L))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_1),L))) ) ) ).

tff(fact_1066_add__le__mono1,axiom,
    ! [K_2: nat,I: nat,J_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_1),K_2))) ) ).

tff(fact_1067_trans__le__add2,axiom,
    ! [M_2: nat,I: nat,J_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_2),J_1))) ) ).

tff(fact_1068_trans__le__add1,axiom,
    ! [M_2: nat,I: nat,J_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_1),M_2))) ) ).

tff(fact_1069_nat__add__left__cancel__le,axiom,
    ! [K_1: nat,M: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) ) ).

tff(fact_1070_le__iff__add,axiom,
    ! [M: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
    <=> ? [K: nat] : ( N_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K) ) ) ).

tff(fact_1071_le__add1,axiom,
    ! [N_2: nat,M_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_2),M_2))) ).

tff(fact_1072_le__add2,axiom,
    ! [N_2: nat,M_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_2),N_2))) ).

tff(fact_1073_card__Collect__le__nat,axiom,
    ! [N_1: nat] : ( hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(collect_nat,hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_eq_nat),N_1))) = hAPP_nat_nat(suc,N_1) ) ).

tff(fact_1074_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M)),N_1))
    <=> hBOOL(hAPP_nat_bool(nat_case_bool(fFalse,hAPP_n1699378549t_bool(ord_less_eq_nat,M)),N_1)) ) ).

tff(fact_1075_mult__le__mono,axiom,
    ! [K_2: nat,L: nat,I: nat,J_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),L))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_1),L))) ) ) ).

tff(fact_1076_mult__le__mono2,axiom,
    ! [K_2: nat,I: nat,J_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),I)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),J_1))) ) ).

tff(fact_1077_mult__le__mono1,axiom,
    ! [K_2: nat,I: nat,J_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_1),K_2))) ) ).

tff(fact_1078_le__cube,axiom,
    ! [M_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_2),M_2)))) ).

tff(fact_1079_le__square,axiom,
    ! [M_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_2),M_2))) ).

tff(fact_1080_com_Osimps_I52_J,axiom,
    ! [Fun_1: fun_state_bool,Com1_1: com,Com2_1: com,Fun: fun_state_bool,Com: com] : ( cond(Fun_1,Com1_1,Com2_1) != while(Fun,Com) ) ).

tff(fact_1081_com_Osimps_I53_J,axiom,
    ! [Fun: fun_state_bool,Com: com,Fun_1: fun_state_bool,Com1_1: com,Com2_1: com] : ( while(Fun,Com) != cond(Fun_1,Com1_1,Com2_1) ) ).

tff(fact_1082_com_Osimps_I45_J,axiom,
    ! [Fun: fun_state_bool,Com1: com,Com2: com,Com1_1: com,Com2_1: com] : ( cond(Fun,Com1,Com2) != semi(Com1_1,Com2_1) ) ).

tff(fact_1083_com_Osimps_I44_J,axiom,
    ! [Com1_1: com,Com2_1: com,Fun: fun_state_bool,Com1: com,Com2: com] : ( semi(Com1_1,Com2_1) != cond(Fun,Com1,Com2) ) ).

tff(fact_1084_com_Osimps_I15_J,axiom,
    ! [Fun: fun_state_bool,Com1: com,Com2: com] : ( cond(Fun,Com1,Com2) != skip ) ).

tff(fact_1085_com_Osimps_I14_J,axiom,
    ! [Fun: fun_state_bool,Com1: com,Com2: com] : ( skip != cond(Fun,Com1,Com2) ) ).

tff(fact_1086_diff__is__0__eq_H,axiom,
    ! [M_2: nat,N_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_2))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_2),N_2) = zero_zero_nat ) ) ).

tff(fact_1087_diff__is__0__eq,axiom,
    ! [M: nat,N_1: nat] :
      ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_1) = zero_zero_nat )
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) ) ).

tff(fact_1088_Suc__diff__le,axiom,
    ! [N_2: nat,M_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),M_2))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M_2)),N_2) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_2),N_2)) ) ) ).

tff(fact_1089_Suc__mult__le__cancel1,axiom,
    ! [K_1: nat,M: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) ) ).

tff(fact_1090_diff__diff__right,axiom,
    ! [I: nat,K_2: nat,J_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_1))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K_2)),J_1) ) ) ).

tff(fact_1091_le__diff__conv,axiom,
    ! [J_2: nat,K_1: nat,I_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_1)),I_1))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1))) ) ).

tff(fact_1092_le__add__diff,axiom,
    ! [M_2: nat,K_2: nat,N_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_2),M_2)),K_2))) ) ).

tff(fact_1093_le__add__diff__inverse,axiom,
    ! [N_2: nat,M_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),M_2))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_2),N_2)) = M_2 ) ) ).

tff(fact_1094_add__diff__assoc,axiom,
    ! [I: nat,K_2: nat,J_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_1))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J_1)),K_2) ) ) ).

tff(fact_1095_le__diff__conv2,axiom,
    ! [I_1: nat,K_1: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_1)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),J_2)) ) ) ).

tff(fact_1096_le__add__diff__inverse2,axiom,
    ! [N_2: nat,M_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),M_2))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_2),N_2)),N_2) = M_2 ) ) ).

tff(fact_1097_le__imp__diff__is__add,axiom,
    ! [K_1: nat,I_1: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_2))
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),I_1) = K_1 )
      <=> ( J_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),I_1) ) ) ) ).

tff(fact_1098_diff__add__assoc,axiom,
    ! [I: nat,K_2: nat,J_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_1))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J_1)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),K_2)) ) ) ).

tff(fact_1099_add__diff__assoc2,axiom,
    ! [I: nat,K_2: nat,J_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_1))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),K_2)),I) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_1),I)),K_2) ) ) ).

tff(fact_1100_diff__add__assoc2,axiom,
    ! [I: nat,K_2: nat,J_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_1))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_1),I)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),K_2)),I) ) ) ).

tff(fact_1101_one__le__mult__iff,axiom,
    ! [M: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),M))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),N_1)) ) ) ).

tff(fact_1102_diff__Suc__diff__eq2,axiom,
    ! [M_2: nat,K_2: nat,J_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_1))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),K_2))),M_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,J_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),M_2)) ) ) ).

tff(fact_1103_diff__Suc__diff__eq1,axiom,
    ! [M_2: nat,K_2: nat,J_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_1))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_2),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),K_2))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_2),K_2)),hAPP_nat_nat(suc,J_1)) ) ) ).

tff(fact_1104_nat__le__add__iff1,axiom,
    ! [U: nat,M: nat,N_1: nat,J_2: nat,I_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),I_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_2),U)),N_1)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),J_2)),U)),M)),N_1)) ) ) ).

tff(fact_1105_nat__diff__add__eq1,axiom,
    ! [U_1: nat,M_2: nat,N_2: nat,J_1: nat,I: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_1),I))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),U_1)),M_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_1),U_1)),N_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),J_1)),U_1)),M_2)),N_2) ) ) ).

tff(fact_1106_nat__eq__add__iff1,axiom,
    ! [U: nat,M: nat,N_1: nat,J_2: nat,I_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),I_1))
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_2),U)),N_1) )
      <=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),J_2)),U)),M) = N_1 ) ) ) ).

tff(fact_1107_nat__le__add__iff2,axiom,
    ! [U: nat,M: nat,N_1: nat,I_1: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_2),U)),N_1)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),I_1)),U)),N_1))) ) ) ).

tff(fact_1108_nat__diff__add__eq2,axiom,
    ! [U_1: nat,M_2: nat,N_2: nat,I: nat,J_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_1))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),U_1)),M_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_1),U_1)),N_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),I)),U_1)),N_2)) ) ) ).

tff(fact_1109_nat__eq__add__iff2,axiom,
    ! [U: nat,M: nat,N_1: nat,I_1: nat,J_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_2))
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_2),U)),N_1) )
      <=> ( M = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),I_1)),U)),N_1) ) ) ) ).

tff(fact_1110_Suc__le__D,axiom,
    ! [N_2: nat,M_4: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_2)),M_4))
     => ? [M_1: nat] : ( M_4 = hAPP_nat_nat(suc,M_1) ) ) ).

tff(fact_1111_Suc__le__D__lemma,axiom,
    ! [Pa: fun_nat_bool,N_1: nat,M_3: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_1)),M_3))
     => ( ! [M_1: nat] :
            ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_1))
           => hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(suc,M_1))) )
       => hBOOL(hAPP_nat_bool(Pa,M_3)) ) ) ).

tff(fact_1112_finite__nat__set__iff__bounded__le,axiom,
    ! [N_3: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_3))
    <=> ? [M_1: nat] :
        ! [X: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),N_3))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),M_1)) ) ) ).

tff(fact_1113_finite__less__ub,axiom,
    ! [U: nat,F: fun_nat_nat] :
      ( ! [N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),hAPP_nat_nat(F,N)))
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,hAPP_f618557131t_bool(cOMBB_800536526ol_nat(ord_less_eq_nat),F)),U)))) ) ).

tff(fact_1114_termination__basic__simps_I3_J,axiom,
    ! [Z: nat,X_1: nat,Y: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z))) ) ).

tff(fact_1115_termination__basic__simps_I4_J,axiom,
    ! [Y: nat,X_1: nat,Z: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Z))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z))) ) ).

tff(fact_1116_less__zeroE,axiom,
    ! [N_2: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_2),zero_zero_nat)) ).

tff(fact_1117_lessI,axiom,
    ! [N_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_2),hAPP_nat_nat(suc,N_2))) ).

tff(fact_1118_Suc__mono,axiom,
    ! [M_2: nat,N_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M_2)),hAPP_nat_nat(suc,N_2))) ) ).

tff(fact_1119_finite__Collect__less__nat,axiom,
    ! [K_1: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),K_1)))) ).

tff(fact_1120_zero__less__Suc,axiom,
    ! [N_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(suc,N_2))) ).

tff(fact_1121_finite__M__bounded__by__nat,axiom,
    ! [Pa: fun_nat_bool,I_1: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(collect_nat,hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(cOMBB_1015721476ol_nat(fconj),Pa)),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),I_1))))) ).

tff(fact_1122_finite__nat__set__iff__bounded,axiom,
    ! [N_3: fun_nat_bool] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_3))
    <=> ? [M_1: nat] :
        ! [X: nat] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X),N_3))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),M_1)) ) ) ).

tff(fact_1123_less__or__eq__imp__le,axiom,
    ! [M_2: nat,N_2: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_2))
        | ( M_2 = N_2 ) )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_2)) ) ).

tff(fact_1124_le__neq__implies__less,axiom,
    ! [M_2: nat,N_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_2))
     => ( ( M_2 != N_2 )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_2)) ) ) ).

tff(fact_1125_less__imp__le__nat,axiom,
    ! [M_2: nat,N_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_2)) ) ).

tff(fact_1126_le__eq__less__or__eq,axiom,
    ! [M: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
        | ( M = N_1 ) ) ) ).

tff(fact_1127_nat__less__le,axiom,
    ! [M: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
        & ( M != N_1 ) ) ) ).

tff(fact_1128_Suc__le__lessD,axiom,
    ! [M_2: nat,N_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M_2)),N_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_2)) ) ).

tff(fact_1129_le__less__Suc__eq,axiom,
    ! [M: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(suc,M)))
      <=> ( N_1 = M ) ) ) ).

tff(fact_1130_Suc__leI,axiom,
    ! [M_2: nat,N_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M_2)),N_2)) ) ).

tff(fact_1131_le__imp__less__Suc,axiom,
    ! [M_2: nat,N_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),hAPP_nat_nat(suc,N_2))) ) ).

tff(fact_1132_Suc__le__eq,axiom,
    ! [M: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M)),N_1))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) ) ).

tff(fact_1133_less__Suc__eq__le,axiom,
    ! [M: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) ) ).

tff(fact_1134_less__eq__Suc__le,axiom,
    ! [N_1: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),M))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_1)),M)) ) ).

tff(fact_1135_less__diff__iff,axiom,
    ! [N_1: nat,K_1: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),M))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_1))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),K_1)))
        <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) ) ) ) ).

tff(fact_1136_diff__less__mono,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C),A))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,A),C)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,B),C))) ) ) ).

tff(fact_1137_not__add__less1,axiom,
    ! [I: nat,J_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J_1)),I)) ).

tff(fact_1138_not__add__less2,axiom,
    ! [J_1: nat,I: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_1),I)),I)) ).

tff(fact_1139_nat__add__left__cancel__less,axiom,
    ! [K_1: nat,M: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) ) ).

tff(fact_1140_trans__less__add1,axiom,
    ! [M_2: nat,I: nat,J_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_1),M_2))) ) ).

tff(fact_1141_trans__less__add2,axiom,
    ! [M_2: nat,I: nat,J_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_2),J_1))) ) ).

tff(fact_1142_add__less__mono1,axiom,
    ! [K_2: nat,I: nat,J_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_1),K_2))) ) ).

tff(fact_1143_add__less__mono,axiom,
    ! [K_2: nat,L: nat,I: nat,J_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_2),L))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_1),L))) ) ) ).

tff(fact_1144_less__add__eq__less,axiom,
    ! [M_2: nat,N_2: nat,K_2: nat,L: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_2),L))
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_2),L) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),N_2) )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_2)) ) ) ).

tff(fact_1145_add__lessD1,axiom,
    ! [I: nat,J_1: nat,K_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J_1)),K_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),K_2)) ) ).

tff(fact_1146_diff__less__mono2,axiom,
    ! [L: nat,M_2: nat,N_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),L))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L),N_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L),M_2))) ) ) ).

tff(fact_1147_less__imp__diff__less,axiom,
    ! [N_2: nat,J_1: nat,K_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),K_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),N_2)),K_2)) ) ).

tff(fact_1148_not__less0,axiom,
    ! [N_2: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_2),zero_zero_nat)) ).

tff(fact_1149_neq0__conv,axiom,
    ! [N_1: nat] :
      ( ( N_1 != zero_zero_nat )
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) ) ).

tff(fact_1150_less__nat__zero__code,axiom,
    ! [N_2: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_2),zero_zero_nat)) ).

tff(fact_1151_gr__implies__not0,axiom,
    ! [M_2: nat,N_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_2))
     => ( N_2 != zero_zero_nat ) ) ).

tff(fact_1152_gr0I,axiom,
    ! [N_2: nat] :
      ( ( N_2 != zero_zero_nat )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2)) ) ).

tff(fact_1153_not__less__eq,axiom,
    ! [M: nat,N_1: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(suc,M))) ) ).

tff(fact_1154_less__Suc__eq,axiom,
    ! [M: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N_1)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
        | ( M = N_1 ) ) ) ).

tff(fact_1155_Suc__less__eq,axiom,
    ! [M: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) ) ).

tff(fact_1156_not__less__less__Suc__eq,axiom,
    ! [N_1: nat,M: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),M))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(suc,M)))
      <=> ( N_1 = M ) ) ) ).

tff(fact_1157_less__antisym,axiom,
    ! [N_2: nat,M_2: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_2),M_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_2),hAPP_nat_nat(suc,M_2)))
       => ( M_2 = N_2 ) ) ) ).

tff(fact_1158_less__SucI,axiom,
    ! [M_2: nat,N_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),hAPP_nat_nat(suc,N_2))) ) ).

tff(fact_1159_Suc__lessI,axiom,
    ! [M_2: nat,N_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_2))
     => ( ( hAPP_nat_nat(suc,M_2) != N_2 )
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M_2)),N_2)) ) ) ).

tff(fact_1160_less__trans__Suc,axiom,
    ! [K_2: nat,I: nat,J_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),K_2))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,I)),K_2)) ) ) ).

tff(fact_1161_less__SucE,axiom,
    ! [M_2: nat,N_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),hAPP_nat_nat(suc,N_2)))
     => ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_2))
       => ( M_2 = N_2 ) ) ) ).

tff(fact_1162_Suc__lessD,axiom,
    ! [M_2: nat,N_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M_2)),N_2))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_2)) ) ).

tff(fact_1163_Suc__less__SucD,axiom,
    ! [M_2: nat,N_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M_2)),hAPP_nat_nat(suc,N_2)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_2)) ) ).

tff(fact_1164_less__not__refl,axiom,
    ! [N_2: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_2),N_2)) ).

tff(fact_1165_nat__neq__iff,axiom,
    ! [M: nat,N_1: nat] :
      ( ( M != N_1 )
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
        | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),M)) ) ) ).

tff(fact_1166_linorder__neqE__nat,axiom,
    ! [X_1: nat,Y: nat] :
      ( ( X_1 != Y )
     => ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y),X_1)) ) ) ).

tff(fact_1167_less__irrefl__nat,axiom,
    ! [N_2: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_2),N_2)) ).

tff(fact_1168_less__not__refl2,axiom,
    ! [N_2: nat,M_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_2),M_2))
     => ( M_2 != N_2 ) ) ).

tff(fact_1169_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,S),T))
     => ( S != T ) ) ).

tff(fact_1170_nat__less__cases,axiom,
    ! [Pa: fun_nat_fun_nat_bool,M: nat,N_1: nat] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(Pa,N_1),M)) )
     => ( ( ( M = N_1 )
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(Pa,N_1),M)) )
       => ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),M))
           => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(Pa,N_1),M)) )
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(Pa,N_1),M)) ) ) ) ).

tff(fact_1171_less__diff__conv,axiom,
    ! [I_1: nat,J_2: nat,K_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),J_2)) ) ).

tff(fact_1172_add__diff__inverse,axiom,
    ! [M_2: nat,N_2: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_2))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_2),N_2)) = M_2 ) ) ).

tff(fact_1173_Suc__mult__less__cancel1,axiom,
    ! [K_1: nat,M: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),N_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) ) ).

tff(fact_1174_diff__less__Suc,axiom,
    ! [M_2: nat,N_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_2),N_2)),hAPP_nat_nat(suc,M_2))) ).

tff(fact_1175_nat__0__less__mult__iff,axiom,
    ! [M: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N_1)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) ) ) ).

tff(fact_1176_mult__less__cancel1,axiom,
    ! [K_1: nat,M: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N_1)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) ) ) ).

tff(fact_1177_mult__less__cancel2,axiom,
    ! [M: nat,K_1: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),K_1)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) ) ) ).

tff(fact_1178_mult__less__mono1,axiom,
    ! [K_2: nat,I: nat,J_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_1),K_2))) ) ) ).

tff(fact_1179_mult__less__mono2,axiom,
    ! [K_2: nat,I: nat,J_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),I)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),J_1))) ) ) ).

tff(fact_1180_nat__mult__eq__cancel1,axiom,
    ! [M: nat,N_1: nat,K_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
     => ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N_1) )
      <=> ( M = N_1 ) ) ) ).

tff(fact_1181_nat__mult__less__cancel1,axiom,
    ! [M: nat,N_1: nat,K_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N_1)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) ) ) ).

tff(fact_1182_diff__less,axiom,
    ! [M_2: nat,N_2: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M_2))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_2),N_2)),M_2)) ) ) ).

tff(fact_1183_zero__less__diff,axiom,
    ! [N_1: nat,M: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),M)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) ) ).

tff(fact_1184_less__add__Suc1,axiom,
    ! [I: nat,M_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),M_2)))) ).

tff(fact_1185_less__add__Suc2,axiom,
    ! [I: nat,M_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_2),I)))) ).

tff(fact_1186_less__iff__Suc__add,axiom,
    ! [M: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))
    <=> ? [K: nat] : ( N_1 = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K)) ) ) ).

tff(fact_1187_add__gr__0,axiom,
    ! [M: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_1)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
        | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) ) ) ).

tff(fact_1188_gr0__conv__Suc,axiom,
    ! [N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
    <=> ? [M_1: nat] : ( N_1 = hAPP_nat_nat(suc,M_1) ) ) ).

tff(fact_1189_less__Suc0,axiom,
    ! [N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(suc,zero_zero_nat)))
    <=> ( N_1 = zero_zero_nat ) ) ).

tff(fact_1190_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N_1: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N_1)))
    <=> ( ( M = zero_zero_nat )
        | ? [J: nat] :
            ( ( M = hAPP_nat_nat(suc,J) )
            & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J),N_1)) ) ) ) ).

tff(fact_1191_card__Collect__less__nat,axiom,
    ! [N_1: nat] : ( hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(collect_nat,hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),N_1))) = N_1 ) ).

tff(fact_1192_less__eq__Suc__le__raw,axiom,
    ! [X: nat] : ( hAPP_n1699378549t_bool(ord_less_nat,X) = hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,X)) ) ).

tff(fact_1193_termination__basic__simps_I1_J,axiom,
    ! [Z: nat,X_1: nat,Y: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z))) ) ).

tff(fact_1194_termination__basic__simps_I2_J,axiom,
    ! [Y: nat,X_1: nat,Z: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Z))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z))) ) ).

tff(fact_1195_termination__basic__simps_I5_J,axiom,
    ! [X_1: nat,Y: nat] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y)) ) ).

%----Helper facts (160)
tff(help_fNot_1_1_U,axiom,
    ! [P: bool] :
      ( ~ hBOOL(hAPP_bool_bool(fNot,P))
      | ~ hBOOL(P) ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(help_fequal_1_1_fequal_000tc__Nat__Onat_T,axiom,
    ! [X_1: nat,Y: nat] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X_1),Y))
      | ( X_1 = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__Nat__Onat_T,axiom,
    ! [X_1: nat,Y: nat] :
      ( ( X_1 != Y )
      | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X_1),Y)) ) ).

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

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

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

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

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

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

tff(help_COMBK_1_1_COMBK_000tc__Nat__Onat_000tc__Com__Opname_U,axiom,
    ! [P: nat,Q: pname] : ( hAPP_pname_nat(hAPP_n362732366me_nat(cOMBK_nat_pname,P),Q) = P ) ).

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

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

tff(help_fequal_1_1_fequal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_T,axiom,
    ! [X_1: fun_nat_bool,Y: fun_nat_bool] :
      ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(fequal_fun_nat_bool,X_1),Y))
      | ( X_1 = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_T,axiom,
    ! [X_1: fun_nat_bool,Y: fun_nat_bool] :
      ( ( X_1 != Y )
      | hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(fequal_fun_nat_bool,X_1),Y)) ) ).

tff(help_fequal_1_1_fequal_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_T,axiom,
    ! [X_1: fun_pname_bool,Y: fun_pname_bool] :
      ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(fequal533582459e_bool,X_1),Y))
      | ( X_1 = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_T,axiom,
    ! [X_1: fun_pname_bool,Y: fun_pname_bool] :
      ( ( X_1 != Y )
      | hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(fequal533582459e_bool,X_1),Y)) ) ).

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

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

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

tff(help_COMBC_1_1_COMBC_000tc__Com__Opname_000tc__Nat__Onat_000tc__HOL__Obool_U,axiom,
    ! [P: fun_pn406123357t_bool,Q: nat,R: pname] : ( hAPP_pname_bool(hAPP_n1025906991e_bool(hAPP_f1715247037e_bool(cOMBC_pname_nat_bool,P),Q),R) = hAPP_nat_bool(hAPP_p1499970991t_bool(P,R),Q) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Com__Opname_000tc__HOL__Obool_U,axiom,
    ! [P: fun_na936072029e_bool,Q: pname,R: nat] : ( hAPP_nat_bool(hAPP_p1499970991t_bool(hAPP_f998021053t_bool(cOMBC_nat_pname_bool,P),Q),R) = hAPP_pname_bool(hAPP_n1025906991e_bool(P,R),Q) ) ).

tff(help_COMBK_1_1_COMBK_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000t__a_U,axiom,
    ! [P: fun_state_bool,Q: x_a] : ( hAPP_a2036067514e_bool(hAPP_f762886889e_bool(cOMBK_1458035955bool_a,P),Q) = P ) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Com__Opname_U,axiom,
    ! [P: fun_bool_bool,Q: fun_pname_bool,R: pname] : ( hAPP_pname_bool(hAPP_f759274231e_bool(cOMBB_647938656_pname(P),Q),R) = hAPP_bool_bool(P,hAPP_pname_bool(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Com__Ostate_U,axiom,
    ! [P: fun_bool_bool,Q: fun_state_bool,R: state] : ( hAPP_state_bool(hAPP_f1759915619e_bool(cOMBB_160679318_state(P),Q),R) = hAPP_bool_bool(P,hAPP_state_bool(Q,R)) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Com__Ostate_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
    ! [P: fun_st2063251938l_bool,Q: bool,R: state] : ( hAPP_state_bool(hAPP_b2019457360e_bool(hAPP_f167292325e_bool(cOMBC_231445413l_bool,P),Q),R) = hAPP_bool_bool(hAPP_s58564346l_bool(P,R),Q) ) ).

tff(help_COMBS_1_1_COMBS_000tc__Com__Opname_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
    ! [P: fun_pn250273176l_bool,Q: fun_pname_bool,R: pname] : ( hAPP_pname_bool(hAPP_f759274231e_bool(hAPP_f649174806e_bool(cOMBS_568398431l_bool,P),Q),R) = hAPP_bool_bool(hAPP_p393069232l_bool(P,R),hAPP_pname_bool(Q,R)) ) ).

tff(help_COMBS_1_1_COMBS_000tc__Com__Ostate_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
    ! [P: fun_st2063251938l_bool,Q: fun_state_bool,R: state] : ( hAPP_state_bool(hAPP_f1759915619e_bool(hAPP_f644196280e_bool(cOMBS_1378840469l_bool,P),Q),R) = hAPP_bool_bool(hAPP_s58564346l_bool(P,R),hAPP_state_bool(Q,R)) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Com__Opname_000tc__Com__Opname_000tc__HOL__Obool_U,axiom,
    ! [P: fun_pn800050071e_bool,Q: pname,R: pname] : ( hAPP_pname_bool(hAPP_p61793385e_bool(hAPP_f22061361e_bool(cOMBC_1149511130e_bool,P),Q),R) = hAPP_pname_bool(hAPP_p61793385e_bool(P,R),Q) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Com__Ostate_000tc__Com__Ostate_000tc__HOL__Obool_U,axiom,
    ! [P: fun_st1506752259e_bool,Q: state,R: state] : ( hAPP_state_bool(hAPP_s1806633685e_bool(hAPP_f817621513e_bool(cOMBC_2027030106e_bool,P),Q),R) = hAPP_state_bool(hAPP_s1806633685e_bool(P,R),Q) ) ).

tff(help_fequal_1_1_fequal_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_,axiom,
    ! [X_1: hoare_2076525193iple_a,Y: hoare_2076525193iple_a] :
      ( ~ hBOOL(hAPP_H454055633a_bool(hAPP_H2077279219a_bool(fequal1589671816iple_a,X_1),Y))
      | ( X_1 = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_,axiom,
    ! [X_1: hoare_2076525193iple_a,Y: hoare_2076525193iple_a] :
      ( ( X_1 != Y )
      | hBOOL(hAPP_H454055633a_bool(hAPP_H2077279219a_bool(fequal1589671816iple_a,X_1),Y)) ) ).

tff(help_COMBK_1_1_COMBK_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc__Com__O,axiom,
    ! [P: fun_state_bool,Q: state] : ( hAPP_s1806633685e_bool(hAPP_f1863945078e_bool(cOMBK_631994958_state,P),Q) = P ) ).

tff(help_fequal_1_1_fequal_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com,axiom,
    ! [X_1: hoare_363350478_state,Y: hoare_363350478_state] :
      ( ~ hBOOL(hAPP_H1645464494e_bool(hAPP_H705653759e_bool(fequal1026769423_state,X_1),Y))
      | ( X_1 = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com,axiom,
    ! [X_1: hoare_363350478_state,Y: hoare_363350478_state] :
      ( ( X_1 != Y )
      | hBOOL(hAPP_H1645464494e_bool(hAPP_H705653759e_bool(fequal1026769423_state,X_1),Y)) ) ).

tff(help_COMBC_1_1_COMBC_000t__a_000tc__HOL__Obool_000tc__fun_Itc__Com__Ostate_Mtc__,axiom,
    ! [P: fun_a_998512028e_bool,Q: bool,R: x_a] : ( hAPP_a2036067514e_bool(hAPP_b540892988e_bool(hAPP_f1824947087e_bool(cOMBC_41962815e_bool,P),Q),R) = hAPP_b2019457360e_bool(hAPP_a723219176e_bool(P,R),Q) ) ).

tff(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Hoare____Mirabelle____wdcyoqcnvy__,axiom,
    ! [P: bool,Q: hoare_2076525193iple_a] : ( hAPP_H454055633a_bool(hAPP_b1299643615a_bool(cOMBK_861658160iple_a,P),Q) = P ) ).

tff(help_COMBB_1_1_COMBB_000tc__Option__Ooption_Itc__Com__Ocom_J_000tc__Com__Ocom_00,axiom,
    ! [P: fun_option_com_com,Q: fun_pname_option_com,R: pname] : ( hAPP_pname_com(hAPP_f96342628me_com(cOMBB_923936821_pname(P),Q),R) = hAPP_option_com_com(P,hAPP_p799580910on_com(Q,R)) ) ).

tff(help_COMBK_1_1_COMBK_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_00,axiom,
    ! [P: hoare_2076525193iple_a,Q: pname] : ( hAPP_p973116442iple_a(hAPP_H1272595598iple_a(cOMBK_159560841_pname,P),Q) = P ) ).

tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_,axiom,
    ! [P: fun_nat_fun_nat_bool,Q: fun_nat_nat,R: nat] : ( hAPP_n1699378549t_bool(hAPP_f618557131t_bool(cOMBB_800536526ol_nat(P),Q),R) = hAPP_n1699378549t_bool(P,hAPP_nat_nat(Q,R)) ) ).

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

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

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

tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool__470,axiom,
    ! [P: fun_nat_fun_nat_bool,Q: fun_pname_nat,R: pname] : ( hAPP_p1499970991t_bool(hAPP_f1358769483t_bool(cOMBB_523834888_pname(P),Q),R) = hAPP_n1699378549t_bool(P,hAPP_pname_nat(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__Com__Opname_000tc__fun_Itc__Com__Opname_Mtc__HOL__Ob,axiom,
    ! [P: fun_pn800050071e_bool,Q: fun_nat_pname,R: nat] : ( hAPP_n1025906991e_bool(hAPP_f414474559e_bool(cOMBB_1212655066ol_nat(P),Q),R) = hAPP_p61793385e_bool(P,hAPP_nat_pname(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_471,axiom,
    ! [P: fun_bo1549164019l_bool,Q: fun_pname_bool,R: pname] : ( hAPP_p393069232l_bool(hAPP_f1402196763l_bool(cOMBB_675860798_pname(P),Q),R) = hAPP_b589554111l_bool(P,hAPP_pname_bool(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_472,axiom,
    ! [P: fun_bo1549164019l_bool,Q: fun_state_bool,R: state] : ( hAPP_s58564346l_bool(hAPP_f1259673775l_bool(cOMBB_188601460_state(P),Q),R) = hAPP_b589554111l_bool(P,hAPP_state_bool(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__HOL__O,axiom,
    ! [P: fun_fu1430349052l_bool,Q: fun_pn800050071e_bool,R: pname] : ( hAPP_pname_bool(hAPP_f42430548e_bool(cOMBB_530759491_pname(P),Q),R) = hAPP_f1664156314l_bool(P,hAPP_p61793385e_bool(Q,R)) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Com__Opname_000tc__fun_Itc__Com__Opname_Mtc__HOL__Ob,axiom,
    ! [P: fun_pn422929397l_bool,Q: fun_pname_bool,R: pname] : ( hAPP_pname_bool(hAPP_f759274231e_bool(hAPP_f661147897e_bool(cOMBC_1058051404l_bool,P),Q),R) = hAPP_f1664156314l_bool(hAPP_p338031245l_bool(P,R),Q) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Com__Ostate_000tc__HOL__Obool_000tc__fun_Itc__Com__O,axiom,
    ! [P: fun_st1483075957e_bool,Q: bool,R: state] : ( hAPP_s1806633685e_bool(hAPP_b1095269219e_bool(hAPP_f701449317e_bool(cOMBC_471052088e_bool,P),Q),R) = hAPP_b2019457360e_bool(hAPP_s1874344717e_bool(P,R),Q) ) ).

tff(help_COMBB_1_1_COMBB_000tc__Com__Opname_000tc__fun_Itc__Com__Opname_Mtc__HOL__Ob_473,axiom,
    ! [P: fun_pn800050071e_bool,Q: fun_pname_pname,R: pname] : ( hAPP_p61793385e_bool(hAPP_f1336811455e_bool(cOMBB_542850580_pname(P),Q),R) = hAPP_p61793385e_bool(P,hAPP_pname_pname(Q,R)) ) ).

tff(help_fequal_1_1_fequal_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple,axiom,
    ! [X_1: fun_Ho882550911a_bool,Y: fun_Ho882550911a_bool] :
      ( ~ hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(fequal291249344a_bool,X_1),Y))
      | ( X_1 = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple,axiom,
    ! [X_1: fun_Ho882550911a_bool,Y: fun_Ho882550911a_bool] :
      ( ( X_1 != Y )
      | hBOOL(hAPP_f500854239l_bool(hAPP_f256727871l_bool(fequal291249344a_bool,X_1),Y)) ) ).

tff(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Hoare____Mirabelle____wdcyoqcnvy___474,axiom,
    ! [P: bool,Q: hoare_363350478_state] : ( hAPP_H1645464494e_bool(hAPP_b230077104e_bool(cOMBK_574827367_state,P),Q) = P ) ).

tff(help_COMBK_1_1_COMBK_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com__,axiom,
    ! [P: hoare_363350478_state,Q: pname] : ( hAPP_p1337579261_state(hAPP_H1607320188_state(cOMBK_92056634_pname,P),Q) = P ) ).

tff(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_00,axiom,
    ! [P: fun_Ho1349715618t_bool,Q: nat,R: hoare_2076525193iple_a] : ( hAPP_H454055633a_bool(hAPP_n1335981428a_bool(hAPP_f1040149703a_bool(cOMBC_240829083t_bool,P),Q),R) = hAPP_nat_bool(hAPP_H904886772t_bool(P,R),Q) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Hoare____Mirabelle____wdcyoqcnvy__O,axiom,
    ! [P: fun_na1758826658a_bool,Q: hoare_2076525193iple_a,R: nat] : ( hAPP_nat_bool(hAPP_H904886772t_bool(hAPP_f1460432071t_bool(cOMBC_2097822937a_bool,P),Q),R) = hAPP_H454055633a_bool(hAPP_n1335981428a_bool(P,R),Q) ) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Hoare____Mirabel,axiom,
    ! [P: fun_bool_bool,Q: fun_Ho882550911a_bool,R: hoare_2076525193iple_a] : ( hAPP_H454055633a_bool(hAPP_f642208769a_bool(cOMBB_361863589iple_a(P),Q),R) = hAPP_bool_bool(P,hAPP_H454055633a_bool(Q,R)) ) ).

tff(help_COMBS_1_1_COMBS_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_00,axiom,
    ! [P: fun_Ho2037816211l_bool,Q: fun_Ho882550911a_bool,R: hoare_2076525193iple_a] : ( hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_f1004093413a_bool(cOMBS_1108095524l_bool,P),Q),R) = hAPP_bool_bool(hAPP_H421140395l_bool(P,R),hAPP_H454055633a_bool(Q,R)) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Com__Opname_000tc__Hoare____Mirabelle____wdcyoqcnvy_,axiom,
    ! [P: fun_pn1811694044a_bool,Q: hoare_2076525193iple_a,R: pname] : ( hAPP_pname_bool(hAPP_H1108528686e_bool(hAPP_f1658749755e_bool(cOMBC_726551967a_bool,P),Q),R) = hAPP_H454055633a_bool(hAPP_p1160247726a_bool(P,R),Q) ) ).

tff(help_fequal_1_1_fequal_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_475,axiom,
    ! [X_1: fun_Ho320143120e_bool,Y: fun_Ho320143120e_bool] :
      ( ~ hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(fequal731041167e_bool,X_1),Y))
      | ( X_1 = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_476,axiom,
    ! [X_1: fun_Ho320143120e_bool,Y: fun_Ho320143120e_bool] :
      ( ( X_1 != Y )
      | hBOOL(hAPP_f568432088l_bool(hAPP_f726209793l_bool(fequal731041167e_bool,X_1),Y)) ) ).

tff(help_COMBC_1_1_COMBC_000t__a_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc,axiom,
    ! [P: fun_a_2117018159e_bool,Q: fun_state_bool,R: x_a] : ( hAPP_a2036067514e_bool(hAPP_f762886889e_bool(hAPP_f1261923407e_bool(cOMBC_892787026e_bool,P),Q),R) = hAPP_f1759915619e_bool(hAPP_a1200519163e_bool(P,R),Q) ) ).

tff(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__fun_Itc__Hoare____Mirabelle____wdc,axiom,
    ! [P: bool,Q: fun_Ho882550911a_bool] : ( hAPP_f500854239l_bool(hAPP_b1036872929l_bool(cOMBK_1968875032a_bool,P),Q) = P ) ).

tff(help_COMBK_1_1_COMBK_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_I,axiom,
    ! [P: fun_Ho882550911a_bool,Q: pname] : ( hAPP_p1160247726a_bool(hAPP_f406822750a_bool(cOMBK_579861065_pname,P),Q) = P ) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Hoare____Mirabel_477,axiom,
    ! [P: fun_bool_bool,Q: fun_Ho320143120e_bool,R: hoare_363350478_state] : ( hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(cOMBB_2059354034_state(P),Q),R) = hAPP_bool_bool(P,hAPP_H1645464494e_bool(Q,R)) ) ).

tff(help_COMBS_1_1_COMBS_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com__,axiom,
    ! [P: fun_Ho544647106l_bool,Q: fun_Ho320143120e_bool,R: hoare_363350478_state] : ( hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(hAPP_f1612626488e_bool(cOMBS_56025283l_bool,P),Q),R) = hAPP_bool_bool(hAPP_H1173243534l_bool(P,R),hAPP_H1645464494e_bool(Q,R)) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Com__Opname_000tc__Hoare____Mirabelle____wdcyoqcnvy__478,axiom,
    ! [P: fun_pn1521100553e_bool,Q: hoare_363350478_state,R: pname] : ( hAPP_pname_bool(hAPP_H2033014795e_bool(hAPP_f443781369e_bool(cOMBC_1422551904e_bool,P),Q),R) = hAPP_H1645464494e_bool(hAPP_p2009489185e_bool(P,R),Q) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool__479,axiom,
    ! [P: fun_na35521688t_bool,Q: fun_nat_bool,R: nat] : ( hAPP_n1699378549t_bool(hAPP_f1570313510t_bool(hAPP_f688831301t_bool(cOMBC_178881787t_bool,P),Q),R) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(P,R),Q) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Com__Opname_Mtc__HOL__Oboo,axiom,
    ! [P: fun_na1889826572e_bool,Q: fun_pname_bool,R: nat] : ( hAPP_n1025906991e_bool(hAPP_f1501416730e_bool(hAPP_f883303749e_bool(cOMBC_1771919087e_bool,P),Q),R) = hAPP_f759274231e_bool(hAPP_n60670500e_bool(P,R),Q) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Com__Opname_000tc__fun_Itc__Com__Opname_Mtc__HOL__Ob_480,axiom,
    ! [P: fun_pn479984722e_bool,Q: fun_pname_bool,R: pname] : ( hAPP_p61793385e_bool(hAPP_f180827860e_bool(hAPP_f698292281e_bool(cOMBC_1123258281e_bool,P),Q),R) = hAPP_f759274231e_bool(hAPP_p905327722e_bool(P,R),Q) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Com__Ostate_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Ob,axiom,
    ! [P: fun_st277393160e_bool,Q: fun_state_bool,R: state] : ( hAPP_s1806633685e_bool(hAPP_f1863945078e_bool(hAPP_f915354021e_bool(cOMBC_952831051e_bool,P),Q),R) = hAPP_f1759915619e_bool(hAPP_s1226857760e_bool(P,R),Q) ) ).

tff(help_fequal_1_1_fequal_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy,axiom,
    ! [X_1: fun_fu2026518849l_bool,Y: fun_fu2026518849l_bool] :
      ( ~ hBOOL(hAPP_f1490793353l_bool(hAPP_f1238945379l_bool(fequal261524416l_bool,X_1),Y))
      | ( X_1 = Y ) ) ).

tff(help_fequal_2_1_fequal_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy,axiom,
    ! [X_1: fun_fu2026518849l_bool,Y: fun_fu2026518849l_bool] :
      ( ( X_1 != Y )
      | hBOOL(hAPP_f1490793353l_bool(hAPP_f1238945379l_bool(fequal261524416l_bool,X_1),Y)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool__481,axiom,
    ! [P: fun_nat_fun_nat_bool,Q: fun_Ho864122878_a_nat,R: hoare_2076525193iple_a] : ( hAPP_H904886772t_bool(hAPP_f570970635t_bool(cOMBB_2017898829iple_a(P),Q),R) = hAPP_n1699378549t_bool(P,hAPP_H174304918_a_nat(Q,R)) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Hoare____Mirabelle____wdcy,axiom,
    ! [P: fun_na2131225204l_bool,Q: fun_Ho882550911a_bool,R: nat] : ( hAPP_nat_bool(hAPP_f306451970t_bool(hAPP_f567756997t_bool(cOMBC_1759100375l_bool,P),Q),R) = hAPP_f500854239l_bool(hAPP_n990327308l_bool(P,R),Q) ) ).

tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_I,axiom,
    ! [P: fun_fu2145798756t_bool,Q: nat,R: fun_Ho882550911a_bool] : ( hAPP_f500854239l_bool(hAPP_n990327308l_bool(hAPP_f524575973l_bool(cOMBC_1205527147t_bool,P),Q),R) = hAPP_nat_bool(hAPP_f306451970t_bool(P,R),Q) ) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__fun_Itc__Hoare__,axiom,
    ! [P: fun_bool_bool,Q: fun_fu2026518849l_bool,R: fun_Ho882550911a_bool] : ( hAPP_f500854239l_bool(hAPP_f354682389l_bool(cOMBB_1757941347a_bool(P),Q),R) = hAPP_bool_bool(P,hAPP_f500854239l_bool(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_482,axiom,
    ! [P: fun_bo1549164019l_bool,Q: fun_Ho882550911a_bool,R: hoare_2076525193iple_a] : ( hAPP_H421140395l_bool(hAPP_f268025233l_bool(cOMBB_345279363iple_a(P),Q),R) = hAPP_b589554111l_bool(P,hAPP_H454055633a_bool(Q,R)) ) ).

tff(help_COMBS_1_1_COMBS_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_I,axiom,
    ! [P: fun_fu1788233809l_bool,Q: fun_fu2026518849l_bool,R: fun_Ho882550911a_bool] : ( hAPP_f500854239l_bool(hAPP_f354682389l_bool(hAPP_f1121033001l_bool(cOMBS_204124788l_bool,P),Q),R) = hAPP_bool_bool(hAPP_f1845133085l_bool(P,R),hAPP_f500854239l_bool(Q,R)) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Com__Opname_000tc__fun_Itc__Hoare____Mirabelle____wd,axiom,
    ! [P: fun_pn1644896442l_bool,Q: fun_Ho882550911a_bool,R: pname] : ( hAPP_pname_bool(hAPP_f213795644e_bool(hAPP_f584257849e_bool(cOMBC_1126778257l_bool,P),Q),R) = hAPP_f500854239l_bool(hAPP_p715838802l_bool(P,R),Q) ) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc__fun_It,axiom,
    ! [P: fun_fu1658206819l_bool,Q: fun_a_fun_state_bool,R: x_a] : ( hAPP_a849909144l_bool(hAPP_f1509969235l_bool(cOMBB_1355796797bool_a(P),Q),R) = hAPP_f1259673775l_bool(P,hAPP_a2036067514e_bool(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_483,axiom,
    ! [P: fun_bo1549164019l_bool,Q: fun_Ho320143120e_bool,R: hoare_363350478_state] : ( hAPP_H1173243534l_bool(hAPP_f1422032445l_bool(cOMBB_877416532_state(P),Q),R) = hAPP_b589554111l_bool(P,hAPP_H1645464494e_bool(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HO,axiom,
    ! [P: fun_na35521688t_bool,Q: fun_nat_nat,R: nat] : ( hAPP_n1512601776t_bool(hAPP_f934069744t_bool(cOMBB_1141542387ol_nat(P),Q),R) = hAPP_n1512601776t_bool(P,hAPP_nat_nat(Q,R)) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_00_484,axiom,
    ! [P: fun_Ho943022881a_bool,Q: hoare_2076525193iple_a,R: hoare_2076525193iple_a] : ( hAPP_H454055633a_bool(hAPP_H2077279219a_bool(hAPP_f877684549a_bool(cOMBC_1047656090a_bool,P),Q),R) = hAPP_H454055633a_bool(hAPP_H2077279219a_bool(P,R),Q) ) ).

tff(help_COMBB_1_1_COMBB_000tc__Com__Opname_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc,axiom,
    ! [P: fun_pn479984722e_bool,Q: fun_nat_pname,R: nat] : ( hAPP_n60670500e_bool(hAPP_f446737578e_bool(cOMBB_1209446585ol_nat(P),Q),R) = hAPP_p905327722e_bool(P,hAPP_nat_pname(Q,R)) ) ).

tff(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__fun_It__a_Mtc__fun_Itc__Com__Ostate,axiom,
    ! [P: fun_na602334557iple_a,Q: fun_na186072911e_bool,R: nat] : ( hAPP_n887873492iple_a(hAPP_f1101337346iple_a(hAPP_f165988002iple_a(cOMBS_2043478954iple_a,P),Q),R) = hAPP_f860089282iple_a(hAPP_n368284661iple_a(P,R),hAPP_n1063102567e_bool(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc__fun_It_485,axiom,
    ! [P: fun_fu1658206819l_bool,Q: fun_st1506752259e_bool,R: state] : ( hAPP_s2001034685l_bool(hAPP_f1283379615l_bool(cOMBB_1759179140_state(P),Q),R) = hAPP_f1259673775l_bool(P,hAPP_s1806633685e_bool(Q,R)) ) ).

tff(help_COMBS_1_1_COMBS_000tc__Com__Opname_000tc__fun_It__a_Mtc__fun_Itc__Com__Osta,axiom,
    ! [P: fun_pn84361379iple_a,Q: fun_pn1683930517e_bool,R: pname] : ( hAPP_p973116442iple_a(hAPP_f1246077570iple_a(hAPP_f345351132iple_a(cOMBS_970288740iple_a,P),Q),R) = hAPP_f860089282iple_a(hAPP_p1511644219iple_a(P,R),hAPP_p635540397e_bool(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_00,axiom,
    ! [P: fun_Ho1658235120_state,Q: fun_pn706052866iple_a,R: pname] : ( hAPP_p1337579261_state(hAPP_f1641181080_state(cOMBB_911045627_pname(P),Q),R) = hAPP_H2095461698_state(P,hAPP_p973116442iple_a(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com__,axiom,
    ! [P: fun_Ho1876430636iple_a,Q: fun_pn1262351275_state,R: pname] : ( hAPP_p973116442iple_a(hAPP_f1024620500iple_a(cOMBB_1765583813_pname(P),Q),R) = hAPP_H1835089400iple_a(P,hAPP_p1337579261_state(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool__486,axiom,
    ! [P: fun_nat_fun_nat_bool,Q: fun_fu328701628ol_nat,R: fun_Ho882550911a_bool] : ( hAPP_f306451970t_bool(hAPP_f1800675941t_bool(cOMBB_1649747131a_bool(P),Q),R) = hAPP_n1699378549t_bool(P,hAPP_f190641160ol_nat(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_487,axiom,
    ! [P: fun_bo1549164019l_bool,Q: fun_fu2026518849l_bool,R: fun_Ho882550911a_bool] : ( hAPP_f1845133085l_bool(hAPP_f1684296317l_bool(cOMBB_120690437a_bool(P),Q),R) = hAPP_b589554111l_bool(P,hAPP_f500854239l_bool(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__fun_Itc__HOL__Obool_Mtc__H,axiom,
    ! [P: fun_fu1219323149e_bool,Q: fun_a_1632297036l_bool,R: x_a] : ( hAPP_a723219176e_bool(hAPP_f340725611e_bool(cOMBB_1348041619bool_a(P),Q),R) = hAPP_f167292325e_bool(P,hAPP_a849909144l_bool(Q,R)) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com__,axiom,
    ! [P: fun_Ho424838195e_bool,Q: hoare_363350478_state,R: hoare_363350478_state] : ( hAPP_H1645464494e_bool(hAPP_H705653759e_bool(hAPP_f114072425e_bool(cOMBC_13083556e_bool,P),Q),R) = hAPP_H1645464494e_bool(hAPP_H705653759e_bool(P,R),Q) ) ).

tff(help_COMBB_1_1_COMBB_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_00_488,axiom,
    ! [P: fun_Ho943022881a_bool,Q: fun_na1240513468iple_a,R: nat] : ( hAPP_n1335981428a_bool(hAPP_f1324734153a_bool(cOMBB_315605200ol_nat(P),Q),R) = hAPP_H2077279219a_bool(P,hAPP_n887873492iple_a(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_I,axiom,
    ! [P: fun_fu2026518849l_bool,Q: fun_Ho943022881a_bool,R: hoare_2076525193iple_a] : ( hAPP_H454055633a_bool(hAPP_f338738851a_bool(cOMBB_1676209667iple_a(P),Q),R) = hAPP_f500854239l_bool(P,hAPP_H2077279219a_bool(Q,R)) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_00_489,axiom,
    ! [P: fun_Ho1910759733l_bool,Q: fun_Ho882550911a_bool,R: hoare_2076525193iple_a] : ( hAPP_H454055633a_bool(hAPP_f642208769a_bool(hAPP_f325481667a_bool(cOMBC_1529670358l_bool,P),Q),R) = hAPP_f500854239l_bool(hAPP_H1566092493l_bool(P,R),Q) ) ).

tff(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__fun_Itc__Com__Ostate_Mtc__fun_Itc__,axiom,
    ! [P: fun_na648618217_state,Q: fun_na1057458470e_bool,R: nat] : ( hAPP_n622160579_state(hAPP_f1886686490_state(hAPP_f77533864_state(cOMBS_976781756_state,P),Q),R) = hAPP_f83715540_state(hAPP_n2018653441_state(P,R),hAPP_n1252169848e_bool(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_00_490,axiom,
    ! [P: fun_Ho943022881a_bool,Q: fun_pn706052866iple_a,R: pname] : ( hAPP_p1160247726a_bool(hAPP_f830638537a_bool(cOMBB_1838759434_pname(P),Q),R) = hAPP_H2077279219a_bool(P,hAPP_p973116442iple_a(Q,R)) ) ).

tff(help_COMBS_1_1_COMBS_000tc__Com__Opname_000tc__fun_Itc__Com__Ostate_Mtc__fun_Itc,axiom,
    ! [P: fun_pn640817967_state,Q: fun_pn12888928e_bool,R: pname] : ( hAPP_p1337579261_state(hAPP_f19336718_state(hAPP_f720869462_state(cOMBS_1110920566_state,P),Q),R) = hAPP_f83715540_state(hAPP_p435788871_state(P,R),hAPP_p1637813682e_bool(Q,R)) ) ).

tff(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__Com__Ocom_000tc__fun_Itc__fun_It__a,axiom,
    ! [P: fun_na1964826320iple_a,Q: fun_nat_com,R: nat] : ( hAPP_n368284661iple_a(hAPP_f198456565iple_a(hAPP_f603309282iple_a(cOMBS_803657245iple_a,P),Q),R) = hAPP_c426811957iple_a(hAPP_n94278504iple_a(P,R),hAPP_nat_com(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__fun_Itc__HOL__Obool_Mtc__H_491,axiom,
    ! [P: fun_fu1219323149e_bool,Q: fun_st2116860965l_bool,R: state] : ( hAPP_s1874344717e_bool(hAPP_f873506917e_bool(cOMBB_1757942702_state(P),Q),R) = hAPP_f167292325e_bool(P,hAPP_s2001034685l_bool(Q,R)) ) ).

tff(help_COMBS_1_1_COMBS_000tc__Com__Opname_000tc__Com__Ocom_000tc__fun_Itc__fun_It_,axiom,
    ! [P: fun_pn1410192150iple_a,Q: fun_pname_com,R: pname] : ( hAPP_p1511644219iple_a(hAPP_f804419189iple_a(hAPP_f900936988iple_a(cOMBS_1123526615iple_a,P),Q),R) = hAPP_c426811957iple_a(hAPP_p1294005422iple_a(P,R),hAPP_pname_com(Q,R)) ) ).

tff(help_COMBS_1_1_COMBS_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_00_492,axiom,
    ! [P: fun_Ho1253050270iple_a,Q: fun_Ho1146995216e_bool,R: hoare_2076525193iple_a] : ( hAPP_H1789220885iple_a(hAPP_f1255182146iple_a(hAPP_f2049934177iple_a(cOMBS_159040809iple_a,P),Q),R) = hAPP_f860089282iple_a(hAPP_H1369455158iple_a(P,R),hAPP_H1347006248e_bool(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_I_493,axiom,
    ! [P: fun_fu1996308102l_bool,Q: fun_Ho424838195e_bool,R: hoare_363350478_state] : ( hAPP_H1645464494e_bool(hAPP_f265222234e_bool(cOMBB_1356223341_state(P),Q),R) = hAPP_f568432088l_bool(P,hAPP_H705653759e_bool(Q,R)) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com___494,axiom,
    ! [P: fun_Ho177530809l_bool,Q: fun_Ho320143120e_bool,R: hoare_363350478_state] : ( hAPP_H1645464494e_bool(hAPP_f1749453397e_bool(hAPP_f1072682137e_bool(cOMBC_1415895650l_bool,P),Q),R) = hAPP_f568432088l_bool(hAPP_H40373975l_bool(P,R),Q) ) ).

tff(help_COMBB_1_1_COMBB_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com___495,axiom,
    ! [P: fun_Ho424838195e_bool,Q: fun_pn1262351275_state,R: pname] : ( hAPP_p2009489185e_bool(hAPP_f1284518107e_bool(cOMBB_1755629310_pname(P),Q),R) = hAPP_H705653759e_bool(P,hAPP_p1337579261_state(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__fun_Itc__HOL__Obool_Mtc__H_496,axiom,
    ! [P: fun_fu1047394976e_bool,Q: fun_a_1632297036l_bool,R: x_a] : ( hAPP_a1200519163e_bool(hAPP_f963367678e_bool(cOMBB_145932198bool_a(P),Q),R) = hAPP_f644196280e_bool(P,hAPP_a849909144l_bool(Q,R)) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Hoare____Mirabelle____wdcy_497,axiom,
    ! [P: fun_na555126550a_bool,Q: fun_Ho882550911a_bool,R: nat] : ( hAPP_n1335981428a_bool(hAPP_f947130148a_bool(hAPP_f1795987397a_bool(cOMBC_58632185a_bool,P),Q),R) = hAPP_f642208769a_bool(hAPP_n156313134a_bool(P,R),Q) ) ).

tff(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__Com__Ocom_000tc__fun_Itc__fun_Itc__,axiom,
    ! [P: fun_na522192988_state,Q: fun_nat_com,R: nat] : ( hAPP_n2018653441_state(hAPP_f1844653185_state(hAPP_f1469189858_state(cOMBS_1435179817_state,P),Q),R) = hAPP_c1279058753_state(hAPP_n549479028_state(P,R),hAPP_nat_com(Q,R)) ) ).

tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_I_498,axiom,
    ! [P: fun_fu1789210355l_bool,Q: fun_Ho882550911a_bool,R: fun_Ho882550911a_bool] : ( hAPP_f500854239l_bool(hAPP_f256727871l_bool(hAPP_f246259945l_bool(cOMBC_1515890438l_bool,P),Q),R) = hAPP_f500854239l_bool(hAPP_f256727871l_bool(P,R),Q) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Com__Opname_000tc__fun_Itc__Hoare____Mirabelle____wd_499,axiom,
    ! [P: fun_pn236364380a_bool,Q: fun_Ho882550911a_bool,R: pname] : ( hAPP_p1160247726a_bool(hAPP_f406822750a_bool(hAPP_f1969289785a_bool(cOMBC_577917619a_bool,P),Q),R) = hAPP_f642208769a_bool(hAPP_p649799284a_bool(P,R),Q) ) ).

tff(help_COMBS_1_1_COMBS_000tc__Com__Opname_000tc__Com__Ocom_000tc__fun_Itc__fun_Itc,axiom,
    ! [P: fun_pn203430818_state,Q: fun_pname_com,R: pname] : ( hAPP_p435788871_state(hAPP_f1146170881_state(hAPP_f1311198492_state(cOMBS_124505571_state,P),Q),R) = hAPP_c1279058753_state(hAPP_p1042965178_state(P,R),hAPP_pname_com(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__fun_Itc__HOL__Obool_Mtc__H_500,axiom,
    ! [P: fun_fu1047394976e_bool,Q: fun_st2116860965l_bool,R: state] : ( hAPP_s1226857760e_bool(hAPP_f1138284024e_bool(cOMBB_20296667_state(P),Q),R) = hAPP_f644196280e_bool(P,hAPP_s2001034685l_bool(Q,R)) ) ).

tff(help_COMBS_1_1_COMBS_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_00_501,axiom,
    ! [P: fun_Ho312102314_state,Q: fun_Ho1217407269e_bool,R: hoare_2076525193iple_a] : ( hAPP_H2095461698_state(hAPP_f1907710744_state(hAPP_f1734768933_state(cOMBS_8231099_state,P),Q),R) = hAPP_f83715540_state(hAPP_H1066460610_state(P,R),hAPP_H166769783e_bool(Q,R)) ) ).

tff(help_COMBS_1_1_COMBS_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_00_502,axiom,
    ! [P: fun_Ho630565521iple_a,Q: fun_Ho1421867454_a_com,R: hoare_2076525193iple_a] : ( hAPP_H1369455158iple_a(hAPP_f1031856565iple_a(hAPP_f1391221537iple_a(cOMBS_164138524iple_a,P),Q),R) = hAPP_c426811957iple_a(hAPP_H90235689iple_a(P,R),hAPP_H45551702_a_com(Q,R)) ) ).

tff(help_COMBS_1_1_COMBS_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_I_503,axiom,
    ! [P: fun_fu102784604iple_a,Q: fun_fu1307637710e_bool,R: fun_Ho882550911a_bool] : ( hAPP_f2009966983iple_a(hAPP_f375308812iple_a(hAPP_f2030579535iple_a(cOMBS_736671097iple_a,P),Q),R) = hAPP_f860089282iple_a(hAPP_f1109526440iple_a(P,R),hAPP_f369538458e_bool(Q,R)) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Hoare____Mirabelle____wdcy_504,axiom,
    ! [P: fun_na653256560e_bool,Q: fun_Ho320143120e_bool,R: nat] : ( hAPP_n753050587e_bool(hAPP_f252359560e_bool(hAPP_f1331227093e_bool(cOMBC_1991019085e_bool,P),Q),R) = hAPP_f1749453397e_bool(hAPP_n1098771208e_bool(P,R),Q) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Com__Opname_000tc__fun_Itc__Hoare____Mirabelle____wd_505,axiom,
    ! [P: fun_pn73701814e_bool,Q: fun_Ho320143120e_bool,R: pname] : ( hAPP_p2009489185e_bool(hAPP_f265219918e_bool(hAPP_f1853922773e_bool(cOMBC_1001821703e_bool,P),Q),R) = hAPP_f1749453397e_bool(hAPP_p678758990e_bool(P,R),Q) ) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_I_506,axiom,
    ! [P: fun_fu1789210355l_bool,Q: fun_na1758826658a_bool,R: nat] : ( hAPP_n990327308l_bool(hAPP_f947867151l_bool(cOMBB_262857860ol_nat(P),Q),R) = hAPP_f256727871l_bool(P,hAPP_n1335981428a_bool(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__,axiom,
    ! [P: fun_fu1342882871l_bool,Q: fun_fu1789210355l_bool,R: fun_Ho882550911a_bool] : ( hAPP_f500854239l_bool(hAPP_f1605242827l_bool(cOMBB_4714191a_bool(P),Q),R) = hAPP_f1490793353l_bool(P,hAPP_f256727871l_bool(Q,R)) ) ).

tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_I_507,axiom,
    ! [P: fun_fu330842395l_bool,Q: fun_fu2026518849l_bool,R: fun_Ho882550911a_bool] : ( hAPP_f500854239l_bool(hAPP_f354682389l_bool(hAPP_f638520823l_bool(cOMBC_1528738210l_bool,P),Q),R) = hAPP_f1490793353l_bool(hAPP_f844821689l_bool(P,R),Q) ) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_I_508,axiom,
    ! [P: fun_fu1789210355l_bool,Q: fun_pn1811694044a_bool,R: pname] : ( hAPP_p715838802l_bool(hAPP_f940735259l_bool(cOMBB_1102432190_pname(P),Q),R) = hAPP_f256727871l_bool(P,hAPP_p1160247726a_bool(Q,R)) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_00_509,axiom,
    ! [P: fun_Ho112928087a_bool,Q: fun_Ho882550911a_bool,R: hoare_2076525193iple_a] : ( hAPP_H2077279219a_bool(hAPP_f219434403a_bool(hAPP_f1824299331a_bool(cOMBC_175592824a_bool,P),Q),R) = hAPP_f642208769a_bool(hAPP_H1940471919a_bool(P,R),Q) ) ).

tff(help_COMBS_1_1_COMBS_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_00_510,axiom,
    ! [P: fun_Ho79994525_state,Q: fun_Ho1421867454_a_com,R: hoare_2076525193iple_a] : ( hAPP_H1066460610_state(hAPP_f595441985_state(hAPP_f646848545_state(cOMBS_672164008_state,P),Q),R) = hAPP_c1279058753_state(hAPP_H186154165_state(P,R),hAPP_H45551702_a_com(Q,R)) ) ).

tff(help_COMBS_1_1_COMBS_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_I_511,axiom,
    ! [P: fun_fu1800449896_state,Q: fun_fu857054695e_bool,R: fun_Ho882550911a_bool] : ( hAPP_f762576208_state(hAPP_f1319084128_state(hAPP_f1676710173_state(cOMBS_1608831243_state,P),Q),R) = hAPP_f83715540_state(hAPP_f576364596_state(P,R),hAPP_f1470859141e_bool(Q,R)) ) ).

tff(help_COMBS_1_1_COMBS_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_I_512,axiom,
    ! [P: fun_fu28937807iple_a,Q: fun_fu886446204ol_com,R: fun_Ho882550911a_bool] : ( hAPP_f1109526440iple_a(hAPP_f1229970779iple_a(hAPP_f1723480683iple_a(cOMBS_282424940iple_a,P),Q),R) = hAPP_c426811957iple_a(hAPP_f2071128987iple_a(P,R),hAPP_f61887944ol_com(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_It__a_Mtc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_,axiom,
    ! [P: fun_fu1192048830iple_a,Q: fun_na186072911e_bool,R: nat] : ( hAPP_n94278504iple_a(hAPP_f391471894iple_a(cOMBB_1836265421_a_nat(P),Q),R) = hAPP_f1838364758iple_a(P,hAPP_n1063102567e_bool(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_It__a_Mtc__fun_Itc__Com__Ostate_Mtc__HOL__Obool__513,axiom,
    ! [P: fun_fu1192048830iple_a,Q: fun_pn1683930517e_bool,R: pname] : ( hAPP_p1294005422iple_a(hAPP_f1471296150iple_a(cOMBB_1308294791_pname(P),Q),R) = hAPP_f1838364758iple_a(P,hAPP_p635540397e_bool(Q,R)) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__fun_Itc__Hoare____Mirabell,axiom,
    ! [P: fun_na15769264l_bool,Q: fun_fu2026518849l_bool,R: nat] : ( hAPP_n990327308l_bool(hAPP_f2084083016l_bool(hAPP_f1288222677l_bool(cOMBC_1603421709l_bool,P),Q),R) = hAPP_f354682389l_bool(hAPP_n221063752l_bool(P,R),Q) ) ).

tff(help_COMBB_1_1_COMBB_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_00_514,axiom,
    ! [P: fun_Ho112928087a_bool,Q: fun_na1240513468iple_a,R: nat] : ( hAPP_n156313134a_bool(hAPP_f917287855a_bool(cOMBB_1767297524ol_nat(P),Q),R) = hAPP_H1940471919a_bool(P,hAPP_n887873492iple_a(Q,R)) ) ).

tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_I_515,axiom,
    ! [P: fun_fu2051904533a_bool,Q: fun_Ho882550911a_bool,R: fun_Ho882550911a_bool] : ( hAPP_f642208769a_bool(hAPP_f1418638817a_bool(hAPP_f86610733a_bool(cOMBC_2005334440a_bool,P),Q),R) = hAPP_f642208769a_bool(hAPP_f1418638817a_bool(P,R),Q) ) ).

tff(help_COMBS_1_1_COMBS_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_I_516,axiom,
    ! [P: fun_fu2018970971_state,Q: fun_fu886446204ol_com,R: fun_Ho882550911a_bool] : ( hAPP_f576364596_state(hAPP_f1517110247_state(hAPP_f995418987_state(cOMBS_1312987384_state,P),Q),R) = hAPP_c1279058753_state(hAPP_f1811804711_state(P,R),hAPP_f61887944ol_com(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_00_517,axiom,
    ! [P: fun_Ho112928087a_bool,Q: fun_pn706052866iple_a,R: pname] : ( hAPP_p649799284a_bool(hAPP_f1984020655a_bool(cOMBB_1633811758_pname(P),Q),R) = hAPP_H1940471919a_bool(P,hAPP_p973116442iple_a(Q,R)) ) ).

tff(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com___518,axiom,
    ! [P: fun_Ho1500656352e_bool,Q: fun_Ho320143120e_bool,R: hoare_363350478_state] : ( hAPP_H705653759e_bool(hAPP_f596150574e_bool(hAPP_f961492031e_bool(cOMBC_1774708299e_bool,P),Q),R) = hAPP_f1749453397e_bool(hAPP_H122293036e_bool(P,R),Q) ) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__fun_Itc__Com__Ostate_Mtc__,axiom,
    ! [P: fun_fu515363031_state,Q: fun_na1057458470e_bool,R: nat] : ( hAPP_n549479028_state(hAPP_f15168243_state(cOMBB_186380832te_nat(P),Q),R) = hAPP_f960042659_state(P,hAPP_n1252169848e_bool(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__fun_Itc__Com__Ostate_Mtc___519,axiom,
    ! [P: fun_fu515363031_state,Q: fun_pn12888928e_bool,R: pname] : ( hAPP_p1042965178_state(hAPP_f387017471_state(cOMBB_1212270426_pname(P),Q),R) = hAPP_f960042659_state(P,hAPP_p1637813682e_bool(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_It__a_Mtc__fun_Itc__Com__Ostate_Mtc__HOL__Obool__520,axiom,
    ! [P: fun_fu1192048830iple_a,Q: fun_Ho1146995216e_bool,R: hoare_2076525193iple_a] : ( hAPP_H90235689iple_a(hAPP_f1151385430iple_a(cOMBB_782889804iple_a(P),Q),R) = hAPP_f1838364758iple_a(P,hAPP_H1347006248e_bool(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com___521,axiom,
    ! [P: fun_Ho1500656352e_bool,Q: fun_na63265393_state,R: nat] : ( hAPP_n1098771208e_bool(hAPP_f2008644348e_bool(cOMBB_1813251031ol_nat(P),Q),R) = hAPP_H122293036e_bool(P,hAPP_n622160579_state(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_Itc__Com___522,axiom,
    ! [P: fun_Ho1500656352e_bool,Q: fun_pn1262351275_state,R: pname] : ( hAPP_p678758990e_bool(hAPP_f43306888e_bool(cOMBB_1717464465_pname(P),Q),R) = hAPP_H122293036e_bool(P,hAPP_p1337579261_state(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_00_523,axiom,
    ! [P: fun_Ho112928087a_bool,Q: fun_Ho1518252029iple_a,R: hoare_2076525193iple_a] : ( hAPP_H1940471919a_bool(hAPP_f233660527a_bool(cOMBB_1751994227iple_a(P),Q),R) = hAPP_H1940471919a_bool(P,hAPP_H1789220885iple_a(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_It__a_Mtc__fun_Itc__Com__Ostate_Mtc__HOL__Obool__524,axiom,
    ! [P: fun_fu1192048830iple_a,Q: fun_fu1307637710e_bool,R: fun_Ho882550911a_bool] : ( hAPP_f2071128987iple_a(hAPP_f42286368iple_a(cOMBB_1339444220a_bool(P),Q),R) = hAPP_f1838364758iple_a(P,hAPP_f369538458e_bool(Q,R)) ) ).

tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_I_525,axiom,
    ! [P: fun_fu25215279l_bool,Q: fun_fu2026518849l_bool,R: fun_Ho882550911a_bool] : ( hAPP_f256727871l_bool(hAPP_f1662919455l_bool(hAPP_f2041002367l_bool(cOMBC_1141206332l_bool,P),Q),R) = hAPP_f354682389l_bool(hAPP_f1803426171l_bool(P,R),Q) ) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__fun_Itc__Com__Ostate_Mtc___526,axiom,
    ! [P: fun_fu515363031_state,Q: fun_Ho1217407269e_bool,R: hoare_2076525193iple_a] : ( hAPP_H186154165_state(hAPP_f1793699445_state(cOMBB_1475458975iple_a(P),Q),R) = hAPP_f960042659_state(P,hAPP_H166769783e_bool(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_It__a_J_00_527,axiom,
    ! [P: fun_Ho112928087a_bool,Q: fun_fu253747899iple_a,R: fun_Ho882550911a_bool] : ( hAPP_f1418638817a_bool(hAPP_f15901907a_bool(cOMBB_1462920981a_bool(P),Q),R) = hAPP_H1940471919a_bool(P,hAPP_f2009966983iple_a(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____wdcyoqcnvy__Otriple_I_528,axiom,
    ! [P: fun_fu25215279l_bool,Q: fun_na1758826658a_bool,R: nat] : ( hAPP_n221063752l_bool(hAPP_f564435019l_bool(cOMBB_1921254728ol_nat(P),Q),R) = hAPP_f1803426171l_bool(P,hAPP_n1335981428a_bool(Q,R)) ) ).

tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__fun_Itc__Com__Ostate_Mtc___529,axiom,
    ! [P: fun_fu515363031_state,Q: fun_fu857054695e_bool,R: fun_Ho882550911a_bool] : ( hAPP_f1811804711_state(hAPP_f1181766445_state(cOMBB_498853865a_bool(P),Q),R) = hAPP_f960042659_state(P,hAPP_f1470859141e_bool(Q,R)) ) ).

%----Conjectures (2)
tff(conj_0,hypothesis,
    ! [N: nat] :
      ( ! [X: hoare_2076525193iple_a] :
          ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X),hAPP_f642208769a_bool(hAPP_f1418638817a_bool(semila531373451a_bool,g),hAPP_f1451199292a_bool(image_217098954iple_a(hAPP_f1246077570iple_a(hAPP_f345351132iple_a(cOMBS_970288740iple_a,hAPP_f804419189iple_a(hAPP_f900936988iple_a(cOMBS_1123526615iple_a,hAPP_f1471296150iple_a(cOMBB_1308294791_pname(hoare_1876769875iple_a),p)),body)),q)),procs))))
         => hBOOL(hAPP_H454055633a_bool(hAPP_n1335981428a_bool(hoare_526497939alid_a,N),X)) )
     => ! [X: hoare_2076525193iple_a] :
          ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X),hAPP_f1451199292a_bool(image_217098954iple_a(hAPP_f1246077570iple_a(hAPP_f345351132iple_a(cOMBS_970288740iple_a,hAPP_f804419189iple_a(hAPP_f900936988iple_a(cOMBS_1123526615iple_a,hAPP_f1471296150iple_a(cOMBB_1308294791_pname(hoare_1876769875iple_a),p)),hAPP_f96342628me_com(cOMBB_923936821_pname(the_com),body_1))),q)),procs)))
         => hBOOL(hAPP_H454055633a_bool(hAPP_n1335981428a_bool(hoare_526497939alid_a,N),X)) ) ) ).

% tff(conj_1,conjecture,
%     ( ! [X: hoare_2076525193iple_a] :
%         ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X),g))
%        => hBOOL(hAPP_H454055633a_bool(hAPP_n1335981428a_bool(hoare_526497939alid_a,n),X)) )
%    => ! [X: hoare_2076525193iple_a] :
%         ( hBOOL(hAPP_f500854239l_bool(hAPP_H1566092493l_bool(member276146780iple_a,X),hAPP_f1451199292a_bool(image_217098954iple_a(hAPP_f1246077570iple_a(hAPP_f345351132iple_a(cOMBS_970288740iple_a,hAPP_f804419189iple_a(hAPP_f900936988iple_a(cOMBS_1123526615iple_a,hAPP_f1471296150iple_a(cOMBB_1308294791_pname(hoare_1876769875iple_a),p)),body)),q)),procs)))
%        => hBOOL(hAPP_H454055633a_bool(hAPP_n1335981428a_bool(hoare_526497939alid_a,n),X)) ) ) ).

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