TPTP Problem File: ITP025^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP025^2 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 set theory export of thm_2Elebesgue_2Epos__fn__integral__cmul.p, bushy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : thm_2Elebesgue_2Epos__fn__integral__cmul.p [Gau19]
% : HL412001^2.p [TPAP]
% Status : Theorem
% Rating : 1.00 v7.5.0
% Syntax : Number of formulae : 197 ( 29 unt; 67 typ; 0 def)
% Number of atoms : 985 ( 51 equ; 0 cnn)
% Maximal formula atoms : 36 ( 7 avg)
% Number of connectives : 1961 ( 48 ~; 32 |; 57 &;1588 @)
% ( 71 <=>; 165 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 8 avg)
% Number of types : 6 ( 4 usr)
% Number of type conns : 52 ( 52 >; 0 *; 0 +; 0 <<)
% Number of symbols : 72 ( 69 usr; 37 con; 0-3 aty)
% Number of variables : 245 ( 14 ^; 228 !; 3 ?; 245 :)
% SPC : TH0_THM_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
include('Axioms/ITP001/ITP001^2.ax').
%------------------------------------------------------------------------------
thf(tp_ty_2Erealax_2Ereal,type,
ty_2Erealax_2Ereal: del ).
thf(stp_ty_2Erealax_2Ereal,type,
tp__ty_2Erealax_2Ereal: $tType ).
thf(stp_inj_ty_2Erealax_2Ereal,type,
inj__ty_2Erealax_2Ereal: tp__ty_2Erealax_2Ereal > $i ).
thf(stp_surj_ty_2Erealax_2Ereal,type,
surj__ty_2Erealax_2Ereal: $i > tp__ty_2Erealax_2Ereal ).
thf(stp_inj_surj_ty_2Erealax_2Ereal,axiom,
! [X: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( inj__ty_2Erealax_2Ereal @ X ) )
= X ) ).
thf(stp_inj_mem_ty_2Erealax_2Ereal,axiom,
! [X: tp__ty_2Erealax_2Ereal] : ( mem @ ( inj__ty_2Erealax_2Ereal @ X ) @ ty_2Erealax_2Ereal ) ).
thf(stp_iso_mem_ty_2Erealax_2Ereal,axiom,
! [X: $i] :
( ( mem @ X @ ty_2Erealax_2Ereal )
=> ( X
= ( inj__ty_2Erealax_2Ereal @ ( surj__ty_2Erealax_2Ereal @ X ) ) ) ) ).
thf(tp_ty_2Epair_2Eprod,type,
ty_2Epair_2Eprod: del > del > del ).
thf(tp_c_2Emeasure_2Em__space,type,
c_2Emeasure_2Em__space: del > $i ).
thf(mem_c_2Emeasure_2Em__space,axiom,
! [A_27a: del] : ( mem @ ( c_2Emeasure_2Em__space @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( arr @ A_27a @ bool ) ) ) ).
thf(tp_c_2Ebool_2EBOUNDED,type,
c_2Ebool_2EBOUNDED: $i ).
thf(mem_c_2Ebool_2EBOUNDED,axiom,
mem @ c_2Ebool_2EBOUNDED @ ( arr @ bool @ bool ) ).
thf(tp_c_2Ecombin_2ES,type,
c_2Ecombin_2ES: del > del > del > $i ).
thf(mem_c_2Ecombin_2ES,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Ecombin_2ES @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27c ) ) ) ) ).
thf(tp_c_2Ecombin_2EC,type,
c_2Ecombin_2EC: del > del > del > $i ).
thf(mem_c_2Ecombin_2EC,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Ecombin_2EC @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) @ ( arr @ A_27b @ ( arr @ A_27a @ A_27c ) ) ) ) ).
thf(tp_c_2Ecombin_2EI,type,
c_2Ecombin_2EI: del > $i ).
thf(mem_c_2Ecombin_2EI,axiom,
! [A_27a: del] : ( mem @ ( c_2Ecombin_2EI @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) ).
thf(tp_c_2Ecombin_2Eo,type,
c_2Ecombin_2Eo: del > del > del > $i ).
thf(mem_c_2Ecombin_2Eo,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Ecombin_2Eo @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27c @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27c ) @ ( arr @ A_27a @ A_27b ) ) ) ) ).
thf(tp_ty_2Enum_2Enum,type,
ty_2Enum_2Enum: del ).
thf(stp_ty_2Enum_2Enum,type,
tp__ty_2Enum_2Enum: $tType ).
thf(stp_inj_ty_2Enum_2Enum,type,
inj__ty_2Enum_2Enum: tp__ty_2Enum_2Enum > $i ).
thf(stp_surj_ty_2Enum_2Enum,type,
surj__ty_2Enum_2Enum: $i > tp__ty_2Enum_2Enum ).
thf(stp_inj_surj_ty_2Enum_2Enum,axiom,
! [X: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( inj__ty_2Enum_2Enum @ X ) )
= X ) ).
thf(stp_inj_mem_ty_2Enum_2Enum,axiom,
! [X: tp__ty_2Enum_2Enum] : ( mem @ ( inj__ty_2Enum_2Enum @ X ) @ ty_2Enum_2Enum ) ).
thf(stp_iso_mem_ty_2Enum_2Enum,axiom,
! [X: $i] :
( ( mem @ X @ ty_2Enum_2Enum )
=> ( X
= ( inj__ty_2Enum_2Enum @ ( surj__ty_2Enum_2Enum @ X ) ) ) ) ).
thf(tp_ty_2Eextreal_2Eextreal,type,
ty_2Eextreal_2Eextreal: del ).
thf(stp_ty_2Eextreal_2Eextreal,type,
tp__ty_2Eextreal_2Eextreal: $tType ).
thf(stp_inj_ty_2Eextreal_2Eextreal,type,
inj__ty_2Eextreal_2Eextreal: tp__ty_2Eextreal_2Eextreal > $i ).
thf(stp_surj_ty_2Eextreal_2Eextreal,type,
surj__ty_2Eextreal_2Eextreal: $i > tp__ty_2Eextreal_2Eextreal ).
thf(stp_inj_surj_ty_2Eextreal_2Eextreal,axiom,
! [X: tp__ty_2Eextreal_2Eextreal] :
( ( surj__ty_2Eextreal_2Eextreal @ ( inj__ty_2Eextreal_2Eextreal @ X ) )
= X ) ).
thf(stp_inj_mem_ty_2Eextreal_2Eextreal,axiom,
! [X: tp__ty_2Eextreal_2Eextreal] : ( mem @ ( inj__ty_2Eextreal_2Eextreal @ X ) @ ty_2Eextreal_2Eextreal ) ).
thf(stp_iso_mem_ty_2Eextreal_2Eextreal,axiom,
! [X: $i] :
( ( mem @ X @ ty_2Eextreal_2Eextreal )
=> ( X
= ( inj__ty_2Eextreal_2Eextreal @ ( surj__ty_2Eextreal_2Eextreal @ X ) ) ) ) ).
thf(tp_c_2Eextreal_2EPosInf,type,
c_2Eextreal_2EPosInf: $i ).
thf(mem_c_2Eextreal_2EPosInf,axiom,
mem @ c_2Eextreal_2EPosInf @ ty_2Eextreal_2Eextreal ).
thf(stp_fo_c_2Eextreal_2EPosInf,type,
fo__c_2Eextreal_2EPosInf: tp__ty_2Eextreal_2Eextreal ).
thf(stp_eq_fo_c_2Eextreal_2EPosInf,axiom,
( ( inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf )
= c_2Eextreal_2EPosInf ) ).
thf(tp_c_2Eextreal_2ENegInf,type,
c_2Eextreal_2ENegInf: $i ).
thf(mem_c_2Eextreal_2ENegInf,axiom,
mem @ c_2Eextreal_2ENegInf @ ty_2Eextreal_2Eextreal ).
thf(stp_fo_c_2Eextreal_2ENegInf,type,
fo__c_2Eextreal_2ENegInf: tp__ty_2Eextreal_2Eextreal ).
thf(stp_eq_fo_c_2Eextreal_2ENegInf,axiom,
( ( inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf )
= c_2Eextreal_2ENegInf ) ).
thf(tp_c_2Eextreal_2Eextreal__inv,type,
c_2Eextreal_2Eextreal__inv: $i ).
thf(mem_c_2Eextreal_2Eextreal__inv,axiom,
mem @ c_2Eextreal_2Eextreal__inv @ ( arr @ ty_2Eextreal_2Eextreal @ ty_2Eextreal_2Eextreal ) ).
thf(stp_fo_c_2Eextreal_2Eextreal__inv,type,
fo__c_2Eextreal_2Eextreal__inv: tp__ty_2Eextreal_2Eextreal > tp__ty_2Eextreal_2Eextreal ).
thf(stp_eq_fo_c_2Eextreal_2Eextreal__inv,axiom,
! [X0: tp__ty_2Eextreal_2Eextreal] :
( ( inj__ty_2Eextreal_2Eextreal @ ( fo__c_2Eextreal_2Eextreal__inv @ X0 ) )
= ( ap @ c_2Eextreal_2Eextreal__inv @ ( inj__ty_2Eextreal_2Eextreal @ X0 ) ) ) ).
thf(tp_c_2Eextreal_2Eextreal__div,type,
c_2Eextreal_2Eextreal__div: $i ).
thf(mem_c_2Eextreal_2Eextreal__div,axiom,
mem @ c_2Eextreal_2Eextreal__div @ ( arr @ ty_2Eextreal_2Eextreal @ ( arr @ ty_2Eextreal_2Eextreal @ ty_2Eextreal_2Eextreal ) ) ).
thf(stp_fo_c_2Eextreal_2Eextreal__div,type,
fo__c_2Eextreal_2Eextreal__div: tp__ty_2Eextreal_2Eextreal > tp__ty_2Eextreal_2Eextreal > tp__ty_2Eextreal_2Eextreal ).
thf(stp_eq_fo_c_2Eextreal_2Eextreal__div,axiom,
! [X0: tp__ty_2Eextreal_2Eextreal,X1: tp__ty_2Eextreal_2Eextreal] :
( ( inj__ty_2Eextreal_2Eextreal @ ( fo__c_2Eextreal_2Eextreal__div @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2Eextreal_2Eextreal__div @ ( inj__ty_2Eextreal_2Eextreal @ X0 ) ) @ ( inj__ty_2Eextreal_2Eextreal @ X1 ) ) ) ).
thf(tp_c_2Eextreal_2Eextreal__le,type,
c_2Eextreal_2Eextreal__le: $i ).
thf(mem_c_2Eextreal_2Eextreal__le,axiom,
mem @ c_2Eextreal_2Eextreal__le @ ( arr @ ty_2Eextreal_2Eextreal @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) ) ).
thf(tp_c_2Eextreal_2Eextreal__sup,type,
c_2Eextreal_2Eextreal__sup: $i ).
thf(mem_c_2Eextreal_2Eextreal__sup,axiom,
mem @ c_2Eextreal_2Eextreal__sup @ ( arr @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) @ ty_2Eextreal_2Eextreal ) ).
thf(tp_c_2Eextreal_2ENormal,type,
c_2Eextreal_2ENormal: $i ).
thf(mem_c_2Eextreal_2ENormal,axiom,
mem @ c_2Eextreal_2ENormal @ ( arr @ ty_2Erealax_2Ereal @ ty_2Eextreal_2Eextreal ) ).
thf(stp_fo_c_2Eextreal_2ENormal,type,
fo__c_2Eextreal_2ENormal: tp__ty_2Erealax_2Ereal > tp__ty_2Eextreal_2Eextreal ).
thf(stp_eq_fo_c_2Eextreal_2ENormal,axiom,
! [X0: tp__ty_2Erealax_2Ereal] :
( ( inj__ty_2Eextreal_2Eextreal @ ( fo__c_2Eextreal_2ENormal @ X0 ) )
= ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ X0 ) ) ) ).
thf(tp_c_2Eextreal_2Eextreal__mul,type,
c_2Eextreal_2Eextreal__mul: $i ).
thf(mem_c_2Eextreal_2Eextreal__mul,axiom,
mem @ c_2Eextreal_2Eextreal__mul @ ( arr @ ty_2Eextreal_2Eextreal @ ( arr @ ty_2Eextreal_2Eextreal @ ty_2Eextreal_2Eextreal ) ) ).
thf(stp_fo_c_2Eextreal_2Eextreal__mul,type,
fo__c_2Eextreal_2Eextreal__mul: tp__ty_2Eextreal_2Eextreal > tp__ty_2Eextreal_2Eextreal > tp__ty_2Eextreal_2Eextreal ).
thf(stp_eq_fo_c_2Eextreal_2Eextreal__mul,axiom,
! [X0: tp__ty_2Eextreal_2Eextreal,X1: tp__ty_2Eextreal_2Eextreal] :
( ( inj__ty_2Eextreal_2Eextreal @ ( fo__c_2Eextreal_2Eextreal__mul @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( inj__ty_2Eextreal_2Eextreal @ X0 ) ) @ ( inj__ty_2Eextreal_2Eextreal @ X1 ) ) ) ).
thf(tp_c_2Elebesgue_2Epsfis,type,
c_2Elebesgue_2Epsfis: del > $i ).
thf(mem_c_2Elebesgue_2Epsfis,axiom,
! [A_27a: del] : ( mem @ ( c_2Elebesgue_2Epsfis @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( arr @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) ) ) ) ).
thf(tp_c_2Eextreal_2Eextreal__of__num,type,
c_2Eextreal_2Eextreal__of__num: $i ).
thf(mem_c_2Eextreal_2Eextreal__of__num,axiom,
mem @ c_2Eextreal_2Eextreal__of__num @ ( arr @ ty_2Enum_2Enum @ ty_2Eextreal_2Eextreal ) ).
thf(stp_fo_c_2Eextreal_2Eextreal__of__num,type,
fo__c_2Eextreal_2Eextreal__of__num: tp__ty_2Enum_2Enum > tp__ty_2Eextreal_2Eextreal ).
thf(stp_eq_fo_c_2Eextreal_2Eextreal__of__num,axiom,
! [X0: tp__ty_2Enum_2Enum] :
( ( inj__ty_2Eextreal_2Eextreal @ ( fo__c_2Eextreal_2Eextreal__of__num @ X0 ) )
= ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ X0 ) ) ) ).
thf(tp_c_2Elebesgue_2Epos__fn__integral,type,
c_2Elebesgue_2Epos__fn__integral: del > $i ).
thf(mem_c_2Elebesgue_2Epos__fn__integral,axiom,
! [A_27a: del] : ( mem @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ ( arr @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) @ ty_2Eextreal_2Eextreal ) ) ) ).
thf(tp_c_2Emeasure_2Emeasure__space,type,
c_2Emeasure_2Emeasure__space: del > $i ).
thf(mem_c_2Emeasure_2Emeasure__space,axiom,
! [A_27a: del] : ( mem @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) @ bool ) ) ).
thf(tp_c_2Ebool_2ET,type,
c_2Ebool_2ET: $i ).
thf(mem_c_2Ebool_2ET,axiom,
mem @ c_2Ebool_2ET @ bool ).
thf(ax_true_p,axiom,
p @ c_2Ebool_2ET ).
thf(tp_c_2Epair_2E_2C,type,
c_2Epair_2E_2C: del > del > $i ).
thf(mem_c_2Epair_2E_2C,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ ( arr @ A_27a @ ( arr @ A_27b @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) ) ) ).
thf(tp_c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F: del > $i ).
thf(mem_c_2Ebool_2E_3F,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebool_2E_3F @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ).
thf(ax_ex_p,axiom,
! [A: del,Q: $i] :
( ( mem @ Q @ ( arr @ A @ bool ) )
=> ( ( p @ ( ap @ ( c_2Ebool_2E_3F @ A ) @ Q ) )
<=> ? [X: $i] :
( ( mem @ X @ A )
& ( p @ ( ap @ Q @ X ) ) ) ) ) ).
thf(tp_c_2Epred__set_2EGSPEC,type,
c_2Epred__set_2EGSPEC: del > del > $i ).
thf(mem_c_2Epred__set_2EGSPEC,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ ( ty_2Epair_2Eprod @ A_27a @ bool ) ) @ ( arr @ A_27a @ bool ) ) ) ).
thf(tp_c_2Ebool_2EIN,type,
c_2Ebool_2EIN: del > $i ).
thf(mem_c_2Ebool_2EIN,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebool_2EIN @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Erealax_2Ereal__lt,type,
c_2Erealax_2Ereal__lt: $i ).
thf(mem_c_2Erealax_2Ereal__lt,axiom,
mem @ c_2Erealax_2Ereal__lt @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) ).
thf(tp_c_2Erealax_2Einv,type,
c_2Erealax_2Einv: $i ).
thf(mem_c_2Erealax_2Einv,axiom,
mem @ c_2Erealax_2Einv @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ).
thf(stp_fo_c_2Erealax_2Einv,type,
fo__c_2Erealax_2Einv: tp__ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal ).
thf(stp_eq_fo_c_2Erealax_2Einv,axiom,
! [X0: tp__ty_2Erealax_2Ereal] :
( ( inj__ty_2Erealax_2Ereal @ ( fo__c_2Erealax_2Einv @ X0 ) )
= ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ X0 ) ) ) ).
thf(tp_c_2Enum_2E0,type,
c_2Enum_2E0: $i ).
thf(mem_c_2Enum_2E0,axiom,
mem @ c_2Enum_2E0 @ ty_2Enum_2Enum ).
thf(stp_fo_c_2Enum_2E0,type,
fo__c_2Enum_2E0: tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Enum_2E0,axiom,
( ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 )
= c_2Enum_2E0 ) ).
thf(tp_c_2Ereal_2Ereal__of__num,type,
c_2Ereal_2Ereal__of__num: $i ).
thf(mem_c_2Ereal_2Ereal__of__num,axiom,
mem @ c_2Ereal_2Ereal__of__num @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ).
thf(stp_fo_c_2Ereal_2Ereal__of__num,type,
fo__c_2Ereal_2Ereal__of__num: tp__ty_2Enum_2Enum > tp__ty_2Erealax_2Ereal ).
thf(stp_eq_fo_c_2Ereal_2Ereal__of__num,axiom,
! [X0: tp__ty_2Enum_2Enum] :
( ( inj__ty_2Erealax_2Ereal @ ( fo__c_2Ereal_2Ereal__of__num @ X0 ) )
= ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ X0 ) ) ) ).
thf(tp_c_2Ereal_2Ereal__lte,type,
c_2Ereal_2Ereal__lte: $i ).
thf(mem_c_2Ereal_2Ereal__lte,axiom,
mem @ c_2Ereal_2Ereal__lte @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) ).
thf(tp_c_2Ebool_2EF,type,
c_2Ebool_2EF: $i ).
thf(mem_c_2Ebool_2EF,axiom,
mem @ c_2Ebool_2EF @ bool ).
thf(ax_false_p,axiom,
~ ( p @ c_2Ebool_2EF ) ).
thf(tp_c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $i ).
thf(mem_c_2Ebool_2E_2F_5C,axiom,
mem @ c_2Ebool_2E_2F_5C @ ( arr @ bool @ ( arr @ bool @ bool ) ) ).
thf(ax_and_p,axiom,
! [Q: $i] :
( ( mem @ Q @ bool )
=> ! [R: $i] :
( ( mem @ R @ bool )
=> ( ( p @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ Q ) @ R ) )
<=> ( ( p @ Q )
& ( p @ R ) ) ) ) ) ).
thf(tp_c_2Emin_2E_3D,type,
c_2Emin_2E_3D: del > $i ).
thf(mem_c_2Emin_2E_3D,axiom,
! [A_27a: del] : ( mem @ ( c_2Emin_2E_3D @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ).
thf(ax_eq_p,axiom,
! [A: del,X: $i] :
( ( mem @ X @ A )
=> ! [Y: $i] :
( ( mem @ Y @ A )
=> ( ( p @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A ) @ X ) @ Y ) )
<=> ( X = Y ) ) ) ) ).
thf(tp_c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $i ).
thf(mem_c_2Ebool_2E_5C_2F,axiom,
mem @ c_2Ebool_2E_5C_2F @ ( arr @ bool @ ( arr @ bool @ bool ) ) ).
thf(ax_or_p,axiom,
! [Q: $i] :
( ( mem @ Q @ bool )
=> ! [R: $i] :
( ( mem @ R @ bool )
=> ( ( p @ ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ Q ) @ R ) )
<=> ( ( p @ Q )
| ( p @ R ) ) ) ) ) ).
thf(tp_c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $i ).
thf(mem_c_2Ebool_2E_7E,axiom,
mem @ c_2Ebool_2E_7E @ ( arr @ bool @ bool ) ).
thf(ax_neg_p,axiom,
! [Q: $i] :
( ( mem @ Q @ bool )
=> ( ( p @ ( ap @ c_2Ebool_2E_7E @ Q ) )
<=> ~ ( p @ Q ) ) ) ).
thf(tp_c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $i ).
thf(mem_c_2Emin_2E_3D_3D_3E,axiom,
mem @ c_2Emin_2E_3D_3D_3E @ ( arr @ bool @ ( arr @ bool @ bool ) ) ).
thf(ax_imp_p,axiom,
! [Q: $i] :
( ( mem @ Q @ bool )
=> ! [R: $i] :
( ( mem @ R @ bool )
=> ( ( p @ ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ Q ) @ R ) )
<=> ( ( p @ Q )
=> ( p @ R ) ) ) ) ) ).
thf(tp_c_2Ebool_2E_21,type,
c_2Ebool_2E_21: del > $i ).
thf(mem_c_2Ebool_2E_21,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebool_2E_21 @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ).
thf(ax_all_p,axiom,
! [A: del,Q: $i] :
( ( mem @ Q @ ( arr @ A @ bool ) )
=> ( ( p @ ( ap @ ( c_2Ebool_2E_21 @ A ) @ Q ) )
<=> ! [X: $i] :
( ( mem @ X @ A )
=> ( p @ ( ap @ Q @ X ) ) ) ) ) ).
thf(ax_thm_2Ebool_2EETA__AX,axiom,
! [A_27a: del,A_27b: del,V0t: $i] :
( ( mem @ V0t @ ( arr @ A_27a @ A_27b ) )
=> ( ( lam @ A_27a
@ ^ [V1x: $i] : ( ap @ V0t @ V1x ) )
= V0t ) ) ).
thf(conj_thm_2Ebool_2ETRUTH,axiom,
$true ).
thf(conj_thm_2Ebool_2EIMP__ANTISYM__AX,axiom,
! [V0t1: $i] :
( ( mem @ V0t1 @ bool )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ bool )
=> ( ( ( p @ V0t1 )
=> ( p @ V1t2 ) )
=> ( ( ( p @ V1t2 )
=> ( p @ V0t1 ) )
=> ( ( p @ V0t1 )
<=> ( p @ V1t2 ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EFALSITY,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( $false
=> ( p @ V0t ) ) ) ).
thf(conj_thm_2Ebool_2EEXCLUDED__MIDDLE,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( p @ V0t )
| ~ ( p @ V0t ) ) ) ).
thf(conj_thm_2Ebool_2EFORALL__SIMP,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( p @ V0t ) )
<=> ( p @ V0t ) ) ) ).
thf(conj_thm_2Ebool_2EAND__CLAUSES,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( ( $true
& ( p @ V0t ) )
<=> ( p @ V0t ) )
& ( ( ( p @ V0t )
& $true )
<=> ( p @ V0t ) )
& ( ( $false
& ( p @ V0t ) )
<=> $false )
& ( ( ( p @ V0t )
& $false )
<=> $false )
& ( ( ( p @ V0t )
& ( p @ V0t ) )
<=> ( p @ V0t ) ) ) ) ).
thf(conj_thm_2Ebool_2EIMP__CLAUSES,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( ( $true
=> ( p @ V0t ) )
<=> ( p @ V0t ) )
& ( ( ( p @ V0t )
=> $true )
<=> $true )
& ( ( $false
=> ( p @ V0t ) )
<=> $true )
& ( ( ( p @ V0t )
=> ( p @ V0t ) )
<=> $true )
& ( ( ( p @ V0t )
=> $false )
<=> ~ ( p @ V0t ) ) ) ) ).
thf(conj_thm_2Ebool_2ENOT__CLAUSES,axiom,
( ! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ~ ~ ( p @ V0t )
<=> ( p @ V0t ) ) )
& ( ~ $true
<=> $false )
& ( ~ $false
<=> $true ) ) ).
thf(conj_thm_2Ebool_2EREFL__CLAUSE,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( V0x = V0x )
<=> $true ) ) ).
thf(conj_thm_2Ebool_2EEQ__SYM__EQ,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ( ( V0x = V1y )
<=> ( V1y = V0x ) ) ) ) ).
thf(conj_thm_2Ebool_2EFUN__EQ__THM,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ A_27b ) )
=> ( ( V0f = V1g )
<=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ap @ V0f @ V2x )
= ( ap @ V1g @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EEQ__CLAUSES,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( ( $true
<=> ( p @ V0t ) )
<=> ( p @ V0t ) )
& ( ( ( p @ V0t )
<=> $true )
<=> ( p @ V0t ) )
& ( ( $false
<=> ( p @ V0t ) )
<=> ~ ( p @ V0t ) )
& ( ( ( p @ V0t )
<=> $false )
<=> ~ ( p @ V0t ) ) ) ) ).
thf(conj_thm_2Ebool_2ERIGHT__AND__FORALL__THM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ bool )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ V0P )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( p @ ( ap @ V1Q @ V2x ) ) ) )
<=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ V0P )
& ( p @ ( ap @ V1Q @ V3x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ERIGHT__OR__OVER__AND,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ! [V1B: $i] :
( ( mem @ V1B @ bool )
=> ! [V2C: $i] :
( ( mem @ V2C @ bool )
=> ( ( ( ( p @ V1B )
& ( p @ V2C ) )
| ( p @ V0A ) )
<=> ( ( ( p @ V1B )
| ( p @ V0A ) )
& ( ( p @ V2C )
| ( p @ V0A ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EAND__IMP__INTRO,axiom,
! [V0t1: $i] :
( ( mem @ V0t1 @ bool )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ bool )
=> ! [V2t3: $i] :
( ( mem @ V2t3 @ bool )
=> ( ( ( p @ V0t1 )
=> ( ( p @ V1t2 )
=> ( p @ V2t3 ) ) )
<=> ( ( ( p @ V0t1 )
& ( p @ V1t2 ) )
=> ( p @ V2t3 ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EIMP__CONG,axiom,
! [V0x: $i] :
( ( mem @ V0x @ bool )
=> ! [V1x_27: $i] :
( ( mem @ V1x_27 @ bool )
=> ! [V2y: $i] :
( ( mem @ V2y @ bool )
=> ! [V3y_27: $i] :
( ( mem @ V3y_27 @ bool )
=> ( ( ( ( p @ V0x )
<=> ( p @ V1x_27 ) )
& ( ( p @ V1x_27 )
=> ( ( p @ V2y )
<=> ( p @ V3y_27 ) ) ) )
=> ( ( ( p @ V0x )
=> ( p @ V2y ) )
<=> ( ( p @ V1x_27 )
=> ( p @ V3y_27 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EUNWIND__THM2,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ( ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( V2x = V1a )
& ( p @ ( ap @ V0P @ V2x ) ) )
<=> ( p @ ( ap @ V0P @ V1a ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EBOUNDED__THM,axiom,
! [V0v: $i] :
( ( mem @ V0v @ bool )
=> ( ( p @ ( ap @ c_2Ebool_2EBOUNDED @ V0v ) )
<=> $true ) ) ).
thf(ax_thm_2Ecombin_2ES__DEF,axiom,
! [A_27a: del,A_27b: del,A_27c: del] :
( ( c_2Ecombin_2ES @ A_27a @ A_27b @ A_27c )
= ( lam @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) )
@ ^ [V0f: $i] :
( lam @ ( arr @ A_27a @ A_27b )
@ ^ [V1g: $i] :
( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ V0f @ V2x ) @ ( ap @ V1g @ V2x ) ) ) ) ) ) ).
thf(ax_thm_2Ecombin_2EC__DEF,axiom,
! [A_27a: del,A_27b: del,A_27c: del] :
( ( c_2Ecombin_2EC @ A_27a @ A_27b @ A_27c )
= ( lam @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) )
@ ^ [V0f: $i] :
( lam @ A_27b
@ ^ [V1x: $i] :
( lam @ A_27a
@ ^ [V2y: $i] : ( ap @ ( ap @ V0f @ V2y ) @ V1x ) ) ) ) ) ).
thf(ax_thm_2Ecombin_2Eo__DEF,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27c @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ A_27c ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1g )
= ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ V0f @ ( ap @ V1g @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Ecombin_2EI__THM,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( ap @ ( c_2Ecombin_2EI @ A_27a ) @ V0x )
= V0x ) ) ).
thf(conj_thm_2Ecombin_2EI__o__ID,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27b @ A_27b ) @ ( c_2Ecombin_2EI @ A_27b ) ) @ V0f )
= V0f )
& ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27b @ A_27a ) @ V0f ) @ ( c_2Ecombin_2EI @ A_27a ) )
= V0f ) ) ) ).
thf(ax_thm_2Eextreal_2Eextreal__of__num__def,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ c_2Eextreal_2ENormal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ).
thf(conj_thm_2Eextreal_2Eextreal__le__def,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2a: tp__ty_2Eextreal_2Eextreal,V3v2: tp__ty_2Erealax_2Ereal,V4v3: tp__ty_2Erealax_2Ereal,V5v5: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
& ( ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V2a ) ) )
<=> $true )
& ( ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf ) ) @ ( inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf ) ) )
<=> $true )
& ( ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V3v2 ) ) ) @ ( inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf ) ) )
<=> $true )
& ( ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf ) ) @ ( inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf ) ) )
<=> $false )
& ( ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V4v3 ) ) ) @ ( inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf ) ) )
<=> $false )
& ( ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf ) ) @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V5v5 ) ) ) )
<=> $false ) ) ).
thf(ax_thm_2Eextreal_2Eextreal__inv__def,axiom,
( ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ c_2Eextreal_2Eextreal__inv @ ( inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ c_2Eextreal_2ENormal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
& ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ c_2Eextreal_2Eextreal__inv @ ( inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ c_2Eextreal_2ENormal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
& ! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ c_2Eextreal_2Eextreal__inv @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ c_2Eextreal_2ENormal @ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ) ).
thf(ax_thm_2Eextreal_2Eextreal__div__def,axiom,
! [V0x: tp__ty_2Eextreal_2Eextreal,V1y: tp__ty_2Eextreal_2Eextreal] :
( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__div @ ( inj__ty_2Eextreal_2Eextreal @ V0x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V1y ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( inj__ty_2Eextreal_2Eextreal @ V0x ) ) @ ( ap @ c_2Eextreal_2Eextreal__inv @ ( inj__ty_2Eextreal_2Eextreal @ V1y ) ) ) ) ) ).
thf(conj_thm_2Eextreal_2Emul__lzero,axiom,
! [V0x: tp__ty_2Eextreal_2Eextreal] :
( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V0x ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Eextreal_2Ele__lmul__imp,axiom,
! [V0x: tp__ty_2Eextreal_2Eextreal,V1y: tp__ty_2Eextreal_2Eextreal,V2z: tp__ty_2Eextreal_2Eextreal] :
( ( ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V2z ) ) )
& ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( inj__ty_2Eextreal_2Eextreal @ V0x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V1y ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( inj__ty_2Eextreal_2Eextreal @ V2z ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V0x ) ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( inj__ty_2Eextreal_2Eextreal @ V2z ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V1y ) ) ) ) ) ).
thf(conj_thm_2Eextreal_2Emul__comm,axiom,
! [V0x: tp__ty_2Eextreal_2Eextreal,V1y: tp__ty_2Eextreal_2Eextreal] :
( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( inj__ty_2Eextreal_2Eextreal @ V0x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V1y ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( inj__ty_2Eextreal_2Eextreal @ V1y ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V0x ) ) ) ) ).
thf(conj_thm_2Eextreal_2Ele__rdiv,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Eextreal_2Eextreal,V2z: tp__ty_2Eextreal_2Eextreal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( inj__ty_2Eextreal_2Eextreal @ V1y ) ) @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V2z ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( inj__ty_2Eextreal_2Eextreal @ V1y ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__div @ ( inj__ty_2Eextreal_2Eextreal @ V2z ) ) @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ) ) ).
thf(conj_thm_2Eextreal_2Ele__ldiv,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Eextreal_2Eextreal,V2z: tp__ty_2Eextreal_2Eextreal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( inj__ty_2Eextreal_2Eextreal @ V1y ) ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( inj__ty_2Eextreal_2Eextreal @ V2z ) ) @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__div @ ( inj__ty_2Eextreal_2Eextreal @ V1y ) ) @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V2z ) ) ) ) ) ).
thf(conj_thm_2Eextreal_2Esup__le,axiom,
! [V0p: $i] :
( ( mem @ V0p @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) )
=> ! [V1x: tp__ty_2Eextreal_2Eextreal] :
( ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__sup @ V0p ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V1x ) ) )
<=> ! [V2y: tp__ty_2Eextreal_2Eextreal] :
( ( p @ ( ap @ V0p @ ( inj__ty_2Eextreal_2Eextreal @ V2y ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( inj__ty_2Eextreal_2Eextreal @ V2y ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V1x ) ) ) ) ) ) ).
thf(conj_thm_2Eextreal_2Ele__sup,axiom,
! [V0p: $i] :
( ( mem @ V0p @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) )
=> ! [V1x: tp__ty_2Eextreal_2Eextreal] :
( ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( inj__ty_2Eextreal_2Eextreal @ V1x ) ) @ ( ap @ c_2Eextreal_2Eextreal__sup @ V0p ) ) )
<=> ! [V2y: tp__ty_2Eextreal_2Eextreal] :
( ! [V3z: tp__ty_2Eextreal_2Eextreal] :
( ( p @ ( ap @ V0p @ ( inj__ty_2Eextreal_2Eextreal @ V3z ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( inj__ty_2Eextreal_2Eextreal @ V3z ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V2y ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( inj__ty_2Eextreal_2Eextreal @ V1x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V2y ) ) ) ) ) ) ).
thf(conj_thm_2Eextreal_2Esup__eq,axiom,
! [V0p: $i] :
( ( mem @ V0p @ ( arr @ ty_2Eextreal_2Eextreal @ bool ) )
=> ! [V1x: tp__ty_2Eextreal_2Eextreal] :
( ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ c_2Eextreal_2Eextreal__sup @ V0p ) )
= V1x )
<=> ( ! [V2y: tp__ty_2Eextreal_2Eextreal] :
( ( p @ ( ap @ V0p @ ( inj__ty_2Eextreal_2Eextreal @ V2y ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( inj__ty_2Eextreal_2Eextreal @ V2y ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V1x ) ) ) )
& ! [V3y: tp__ty_2Eextreal_2Eextreal] :
( ! [V4z: tp__ty_2Eextreal_2Eextreal] :
( ( p @ ( ap @ V0p @ ( inj__ty_2Eextreal_2Eextreal @ V4z ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( inj__ty_2Eextreal_2Eextreal @ V4z ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V3y ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( inj__ty_2Eextreal_2Eextreal @ V1x ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V3y ) ) ) ) ) ) ) ).
thf(ax_thm_2Elebesgue_2Epos__fn__integral__def,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ( ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ V1f ) )
= ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ c_2Eextreal_2Eextreal__sup
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Eextreal_2Eextreal @ ty_2Eextreal_2Eextreal )
@ ( lam @ ty_2Eextreal_2Eextreal
@ ^ [V2r: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Eextreal_2Eextreal @ bool ) @ V2r )
@ ( ap @ ( c_2Ebool_2E_3F @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
@ ( lam @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal )
@ ^ [V3g: $i] :
( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ V2r ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Epsfis @ A_27a ) @ V0m ) @ V3g ) ) )
@ ( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ V3g @ V4x ) ) @ ( ap @ V1f @ V4x ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epsfis__cmul,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2a: tp__ty_2Eextreal_2Eextreal,V3z: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ ( inj__ty_2Eextreal_2Eextreal @ V2a ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Epsfis @ A_27a ) @ V0m ) @ V1f ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3z ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Eextreal_2Eextreal ) @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V3z ) ) ) @ ( inj__ty_2Eextreal_2Eextreal @ V2a ) ) )
@ ( ap @ ( ap @ ( c_2Elebesgue_2Epsfis @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V3z ) ) ) @ ( ap @ V1f @ V4x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__fn__integral__zero,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
=> ( ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V1x: $i] : ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2ECLOSED__PAIR__EQ,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27b )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ! [V3b: $i] :
( ( mem @ V3b @ A_27b )
=> ( ( ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V0x ) @ V1y )
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2a ) @ V3b ) )
<=> ( ( V0x = V2a )
& ( V1y = V3b ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESPECIFICATION,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V0P ) )
<=> ( p @ ( ap @ V0P @ V1x ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2EGSPECIFICATION,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ ( ty_2Epair_2Eprod @ A_27a @ bool ) ) )
=> ! [V1v: $i] :
( ( mem @ V1v @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1v ) @ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27b ) @ V0f ) ) )
<=> ? [V2x: $i] :
( ( mem @ V2x @ A_27b )
& ( ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V1v ) @ c_2Ebool_2ET )
= ( ap @ V0f @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Ereal_2EREAL__LT__LE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) )
& ( V0x != V1y ) ) ) ).
thf(conj_thm_2Ereal_2EREAL__LE__INV,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Esat_2ENOT__NOT,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ~ ~ ( p @ V0t )
<=> ( p @ V0t ) ) ) ).
thf(conj_thm_2Esat_2EAND__INV__IMP,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ( ( p @ V0A )
=> ( ~ ( p @ V0A )
=> $false ) ) ) ).
thf(conj_thm_2Esat_2EOR__DUAL2,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ! [V1B: $i] :
( ( mem @ V1B @ bool )
=> ( ( ~ ( ( p @ V0A )
| ( p @ V1B ) )
=> $false )
<=> ( ( ( p @ V0A )
=> $false )
=> ( ~ ( p @ V1B )
=> $false ) ) ) ) ) ).
thf(conj_thm_2Esat_2EOR__DUAL3,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ! [V1B: $i] :
( ( mem @ V1B @ bool )
=> ( ( ~ ( ~ ( p @ V0A )
| ( p @ V1B ) )
=> $false )
<=> ( ( p @ V0A )
=> ( ~ ( p @ V1B )
=> $false ) ) ) ) ) ).
thf(conj_thm_2Esat_2EAND__INV2,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ( ( ~ ( p @ V0A )
=> $false )
=> ( ( ( p @ V0A )
=> $false )
=> $false ) ) ) ).
thf(conj_thm_2Esat_2Edc__eq,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ! [V2r: $i] :
( ( mem @ V2r @ bool )
=> ( ( ( p @ V0p )
<=> ( ( p @ V1q )
<=> ( p @ V2r ) ) )
<=> ( ( ( p @ V0p )
| ( p @ V1q )
| ( p @ V2r ) )
& ( ( p @ V0p )
| ~ ( p @ V2r )
| ~ ( p @ V1q ) )
& ( ( p @ V1q )
| ~ ( p @ V2r )
| ~ ( p @ V0p ) )
& ( ( p @ V2r )
| ~ ( p @ V1q )
| ~ ( p @ V0p ) ) ) ) ) ) ) ).
thf(conj_thm_2Esat_2Edc__conj,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ! [V2r: $i] :
( ( mem @ V2r @ bool )
=> ( ( ( p @ V0p )
<=> ( ( p @ V1q )
& ( p @ V2r ) ) )
<=> ( ( ( p @ V0p )
| ~ ( p @ V1q )
| ~ ( p @ V2r ) )
& ( ( p @ V1q )
| ~ ( p @ V0p ) )
& ( ( p @ V2r )
| ~ ( p @ V0p ) ) ) ) ) ) ) ).
thf(conj_thm_2Esat_2Edc__disj,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ! [V2r: $i] :
( ( mem @ V2r @ bool )
=> ( ( ( p @ V0p )
<=> ( ( p @ V1q )
| ( p @ V2r ) ) )
<=> ( ( ( p @ V0p )
| ~ ( p @ V1q ) )
& ( ( p @ V0p )
| ~ ( p @ V2r ) )
& ( ( p @ V1q )
| ( p @ V2r )
| ~ ( p @ V0p ) ) ) ) ) ) ) ).
thf(conj_thm_2Esat_2Edc__imp,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ! [V2r: $i] :
( ( mem @ V2r @ bool )
=> ( ( ( p @ V0p )
<=> ( ( p @ V1q )
=> ( p @ V2r ) ) )
<=> ( ( ( p @ V0p )
| ( p @ V1q ) )
& ( ( p @ V0p )
| ~ ( p @ V2r ) )
& ( ~ ( p @ V1q )
| ( p @ V2r )
| ~ ( p @ V0p ) ) ) ) ) ) ) ).
thf(conj_thm_2Esat_2Edc__neg,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ( ( ( p @ V0p )
<=> ~ ( p @ V1q ) )
<=> ( ( ( p @ V0p )
| ( p @ V1q ) )
& ( ~ ( p @ V1q )
| ~ ( p @ V0p ) ) ) ) ) ) ).
thf(conj_thm_2Esat_2Epth__ni1,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ( ~ ( ( p @ V0p )
=> ( p @ V1q ) )
=> ( p @ V0p ) ) ) ) ).
thf(conj_thm_2Esat_2Epth__ni2,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ( ~ ( ( p @ V0p )
=> ( p @ V1q ) )
=> ~ ( p @ V1q ) ) ) ) ).
thf(conj_thm_2Esat_2Epth__no1,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ( ~ ( ( p @ V0p )
| ( p @ V1q ) )
=> ~ ( p @ V0p ) ) ) ) ).
thf(conj_thm_2Esat_2Epth__no2,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ( ~ ( ( p @ V0p )
| ( p @ V1q ) )
=> ~ ( p @ V1q ) ) ) ) ).
thf(conj_thm_2Esat_2Epth__nn,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ( ~ ~ ( p @ V0p )
=> ( p @ V0p ) ) ) ).
thf(conj_thm_2Elebesgue_2Epos__fn__integral__cmul,conjecture,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Eextreal_2Eextreal ) )
=> ! [V2c: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Emeasure_2Emeasure__space @ A_27a ) @ V0m ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( c_2Emeasure_2Em__space @ A_27a ) @ V0m ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__le @ ( ap @ c_2Eextreal_2Eextreal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V3x ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) )
=> ( ( surj__ty_2Eextreal_2Eextreal
@ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) @ ( ap @ V1f @ V4x ) ) ) ) )
= ( surj__ty_2Eextreal_2Eextreal @ ( ap @ ( ap @ c_2Eextreal_2Eextreal__mul @ ( ap @ c_2Eextreal_2ENormal @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) @ ( ap @ ( ap @ ( c_2Elebesgue_2Epos__fn__integral @ A_27a ) @ V0m ) @ V1f ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------