TPTP Problem File: ITP025+2.p
View Solutions
- 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 v8.1.0, 0.97 v7.5.0
% Syntax : Number of formulae : 116 ( 29 unt; 0 def)
% Number of atoms : 536 ( 35 equ)
% Maximal formula atoms : 20 ( 4 avg)
% Number of connectives : 468 ( 48 ~; 32 |; 57 &)
% ( 72 <=>; 259 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 6 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 6 ( 3 usr; 2 prp; 0-2 aty)
% Number of functors : 52 ( 52 usr; 26 con; 0-5 aty)
% Number of variables : 214 ( 211 !; 3 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
include('Axioms/ITP001/ITP001+2.ax').
%------------------------------------------------------------------------------
fof(ne_ty_2Erealax_2Ereal,axiom,
ne(ty_2Erealax_2Ereal) ).
fof(ne_ty_2Epair_2Eprod,axiom,
! [A0] :
( ne(A0)
=> ! [A1] :
( ne(A1)
=> ne(ty_2Epair_2Eprod(A0,A1)) ) ) ).
fof(mem_c_2Emeasure_2Em__space,axiom,
! [A_27a] :
( ne(A_27a)
=> 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))) ) ).
fof(mem_c_2Ebool_2EBOUNDED,axiom,
mem(c_2Ebool_2EBOUNDED,arr(bool,bool)) ).
fof(mem_c_2Ecombin_2ES,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> 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)))) ) ) ) ).
fof(mem_c_2Ecombin_2EC,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> 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)))) ) ) ) ).
fof(mem_c_2Ecombin_2EI,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ecombin_2EI(A_27a),arr(A_27a,A_27a)) ) ).
fof(mem_c_2Ecombin_2Eo,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> 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)))) ) ) ) ).
fof(ne_ty_2Enum_2Enum,axiom,
ne(ty_2Enum_2Enum) ).
fof(ne_ty_2Eextreal_2Eextreal,axiom,
ne(ty_2Eextreal_2Eextreal) ).
fof(mem_c_2Eextreal_2EPosInf,axiom,
mem(c_2Eextreal_2EPosInf,ty_2Eextreal_2Eextreal) ).
fof(mem_c_2Eextreal_2ENegInf,axiom,
mem(c_2Eextreal_2ENegInf,ty_2Eextreal_2Eextreal) ).
fof(mem_c_2Eextreal_2Eextreal__inv,axiom,
mem(c_2Eextreal_2Eextreal__inv,arr(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal)) ).
fof(mem_c_2Eextreal_2Eextreal__div,axiom,
mem(c_2Eextreal_2Eextreal__div,arr(ty_2Eextreal_2Eextreal,arr(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal))) ).
fof(mem_c_2Eextreal_2Eextreal__le,axiom,
mem(c_2Eextreal_2Eextreal__le,arr(ty_2Eextreal_2Eextreal,arr(ty_2Eextreal_2Eextreal,bool))) ).
fof(mem_c_2Eextreal_2Eextreal__sup,axiom,
mem(c_2Eextreal_2Eextreal__sup,arr(arr(ty_2Eextreal_2Eextreal,bool),ty_2Eextreal_2Eextreal)) ).
fof(mem_c_2Eextreal_2ENormal,axiom,
mem(c_2Eextreal_2ENormal,arr(ty_2Erealax_2Ereal,ty_2Eextreal_2Eextreal)) ).
fof(mem_c_2Eextreal_2Eextreal__mul,axiom,
mem(c_2Eextreal_2Eextreal__mul,arr(ty_2Eextreal_2Eextreal,arr(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal))) ).
fof(mem_c_2Elebesgue_2Epsfis,axiom,
! [A_27a] :
( ne(A_27a)
=> 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)))) ) ).
fof(mem_c_2Eextreal_2Eextreal__of__num,axiom,
mem(c_2Eextreal_2Eextreal__of__num,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal)) ).
fof(mem_c_2Elebesgue_2Epos__fn__integral,axiom,
! [A_27a] :
( ne(A_27a)
=> 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))) ) ).
fof(mem_c_2Emeasure_2Emeasure__space,axiom,
! [A_27a] :
( ne(A_27a)
=> 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)) ) ).
fof(mem_c_2Ebool_2ET,axiom,
mem(c_2Ebool_2ET,bool) ).
fof(ax_true_p,axiom,
p(c_2Ebool_2ET) ).
fof(mem_c_2Epair_2E_2C,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epair_2E_2C(A_27a,A_27b),arr(A_27a,arr(A_27b,ty_2Epair_2Eprod(A_27a,A_27b)))) ) ) ).
fof(mem_c_2Ebool_2E_3F,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ebool_2E_3F(A_27a),arr(arr(A_27a,bool),bool)) ) ).
fof(ax_ex_p,axiom,
! [A] :
( ne(A)
=> ! [Q] :
( mem(Q,arr(A,bool))
=> ( p(ap(c_2Ebool_2E_3F(A),Q))
<=> ? [X] :
( mem(X,A)
& p(ap(Q,X)) ) ) ) ) ).
fof(mem_c_2Epred__set_2EGSPEC,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epred__set_2EGSPEC(A_27a,A_27b),arr(arr(A_27b,ty_2Epair_2Eprod(A_27a,bool)),arr(A_27a,bool))) ) ) ).
fof(mem_c_2Ebool_2EIN,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ebool_2EIN(A_27a),arr(A_27a,arr(arr(A_27a,bool),bool))) ) ).
fof(mem_c_2Erealax_2Ereal__lt,axiom,
mem(c_2Erealax_2Ereal__lt,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,bool))) ).
fof(mem_c_2Erealax_2Einv,axiom,
mem(c_2Erealax_2Einv,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) ).
fof(mem_c_2Enum_2E0,axiom,
mem(c_2Enum_2E0,ty_2Enum_2Enum) ).
fof(mem_c_2Ereal_2Ereal__of__num,axiom,
mem(c_2Ereal_2Ereal__of__num,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) ).
fof(mem_c_2Ereal_2Ereal__lte,axiom,
mem(c_2Ereal_2Ereal__lte,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,bool))) ).
fof(mem_c_2Ebool_2EF,axiom,
mem(c_2Ebool_2EF,bool) ).
fof(ax_false_p,axiom,
~ p(c_2Ebool_2EF) ).
fof(mem_c_2Ebool_2E_2F_5C,axiom,
mem(c_2Ebool_2E_2F_5C,arr(bool,arr(bool,bool))) ).
fof(ax_and_p,axiom,
! [Q] :
( mem(Q,bool)
=> ! [R] :
( mem(R,bool)
=> ( p(ap(ap(c_2Ebool_2E_2F_5C,Q),R))
<=> ( p(Q)
& p(R) ) ) ) ) ).
fof(mem_c_2Emin_2E_3D,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Emin_2E_3D(A_27a),arr(A_27a,arr(A_27a,bool))) ) ).
fof(ax_eq_p,axiom,
! [A] :
( ne(A)
=> ! [X] :
( mem(X,A)
=> ! [Y] :
( mem(Y,A)
=> ( p(ap(ap(c_2Emin_2E_3D(A),X),Y))
<=> X = Y ) ) ) ) ).
fof(mem_c_2Ebool_2E_5C_2F,axiom,
mem(c_2Ebool_2E_5C_2F,arr(bool,arr(bool,bool))) ).
fof(ax_or_p,axiom,
! [Q] :
( mem(Q,bool)
=> ! [R] :
( mem(R,bool)
=> ( p(ap(ap(c_2Ebool_2E_5C_2F,Q),R))
<=> ( p(Q)
| p(R) ) ) ) ) ).
fof(mem_c_2Ebool_2E_7E,axiom,
mem(c_2Ebool_2E_7E,arr(bool,bool)) ).
fof(ax_neg_p,axiom,
! [Q] :
( mem(Q,bool)
=> ( p(ap(c_2Ebool_2E_7E,Q))
<=> ~ p(Q) ) ) ).
fof(mem_c_2Emin_2E_3D_3D_3E,axiom,
mem(c_2Emin_2E_3D_3D_3E,arr(bool,arr(bool,bool))) ).
fof(ax_imp_p,axiom,
! [Q] :
( mem(Q,bool)
=> ! [R] :
( mem(R,bool)
=> ( p(ap(ap(c_2Emin_2E_3D_3D_3E,Q),R))
<=> ( p(Q)
=> p(R) ) ) ) ) ).
fof(mem_c_2Ebool_2E_21,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ebool_2E_21(A_27a),arr(arr(A_27a,bool),bool)) ) ).
fof(ax_all_p,axiom,
! [A] :
( ne(A)
=> ! [Q] :
( mem(Q,arr(A,bool))
=> ( p(ap(c_2Ebool_2E_21(A),Q))
<=> ! [X] :
( mem(X,A)
=> p(ap(Q,X)) ) ) ) ) ).
fof(ax_thm_2Ebool_2EETA__AX,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0t] :
( mem(V0t,arr(A_27a,A_27b))
=> f31(A_27b,A_27a,V0t) = V0t ) ) ) ).
fof(conj_thm_2Ebool_2ETRUTH,axiom,
$true ).
fof(conj_thm_2Ebool_2EIMP__ANTISYM__AX,axiom,
! [V0t1] :
( mem(V0t1,bool)
=> ! [V1t2] :
( mem(V1t2,bool)
=> ( ( p(V0t1)
=> p(V1t2) )
=> ( ( p(V1t2)
=> p(V0t1) )
=> ( p(V0t1)
<=> p(V1t2) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EFALSITY,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( $false
=> p(V0t) ) ) ).
fof(conj_thm_2Ebool_2EEXCLUDED__MIDDLE,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( p(V0t)
| ~ p(V0t) ) ) ).
fof(conj_thm_2Ebool_2EFORALL__SIMP,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t] :
( mem(V0t,bool)
=> ( ! [V1x] :
( mem(V1x,A_27a)
=> p(V0t) )
<=> p(V0t) ) ) ) ).
fof(conj_thm_2Ebool_2EAND__CLAUSES,axiom,
! [V0t] :
( 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) ) ) ) ).
fof(conj_thm_2Ebool_2EIMP__CLAUSES,axiom,
! [V0t] :
( 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) ) ) ) ).
fof(conj_thm_2Ebool_2ENOT__CLAUSES,axiom,
( ! [V0t] :
( mem(V0t,bool)
=> ( ~ ~ p(V0t)
<=> p(V0t) ) )
& ( ~ $true
<=> $false )
& ( ~ $false
<=> $true ) ) ).
fof(conj_thm_2Ebool_2EREFL__CLAUSE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ( V0x = V0x
<=> $true ) ) ) ).
fof(conj_thm_2Ebool_2EEQ__SYM__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27a)
=> ( V0x = V1y
<=> V1y = V0x ) ) ) ) ).
fof(conj_thm_2Ebool_2EFUN__EQ__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g] :
( mem(V1g,arr(A_27a,A_27b))
=> ( V0f = V1g
<=> ! [V2x] :
( mem(V2x,A_27a)
=> ap(V0f,V2x) = ap(V1g,V2x) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EEQ__CLAUSES,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ( ( $true
<=> p(V0t) )
<=> p(V0t) )
& ( ( p(V0t)
<=> $true )
<=> p(V0t) )
& ( ( $false
<=> p(V0t) )
<=> ~ p(V0t) )
& ( ( p(V0t)
<=> $false )
<=> ~ p(V0t) ) ) ) ).
fof(conj_thm_2Ebool_2ERIGHT__AND__FORALL__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,bool)
=> ! [V1Q] :
( mem(V1Q,arr(A_27a,bool))
=> ( ( p(V0P)
& ! [V2x] :
( mem(V2x,A_27a)
=> p(ap(V1Q,V2x)) ) )
<=> ! [V3x] :
( mem(V3x,A_27a)
=> ( p(V0P)
& p(ap(V1Q,V3x)) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ERIGHT__OR__OVER__AND,axiom,
! [V0A] :
( mem(V0A,bool)
=> ! [V1B] :
( mem(V1B,bool)
=> ! [V2C] :
( mem(V2C,bool)
=> ( ( ( p(V1B)
& p(V2C) )
| p(V0A) )
<=> ( ( p(V1B)
| p(V0A) )
& ( p(V2C)
| p(V0A) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EAND__IMP__INTRO,axiom,
! [V0t1] :
( mem(V0t1,bool)
=> ! [V1t2] :
( mem(V1t2,bool)
=> ! [V2t3] :
( mem(V2t3,bool)
=> ( ( p(V0t1)
=> ( p(V1t2)
=> p(V2t3) ) )
<=> ( ( p(V0t1)
& p(V1t2) )
=> p(V2t3) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EIMP__CONG,axiom,
! [V0x] :
( mem(V0x,bool)
=> ! [V1x_27] :
( mem(V1x_27,bool)
=> ! [V2y] :
( mem(V2y,bool)
=> ! [V3y_27] :
( 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) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EUNWIND__THM2,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1a] :
( mem(V1a,A_27a)
=> ( ? [V2x] :
( mem(V2x,A_27a)
& V2x = V1a
& p(ap(V0P,V2x)) )
<=> p(ap(V0P,V1a)) ) ) ) ) ).
fof(conj_thm_2Ebool_2EBOUNDED__THM,axiom,
! [V0v] :
( mem(V0v,bool)
=> ( p(ap(c_2Ebool_2EBOUNDED,V0v))
<=> $true ) ) ).
fof(ax_thm_2Ecombin_2ES__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> c_2Ecombin_2ES(A_27a,A_27b,A_27c) = f71(A_27c,A_27a,A_27b) ) ) ) ).
fof(ax_thm_2Ecombin_2EC__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> c_2Ecombin_2EC(A_27a,A_27b,A_27c) = f74(A_27a,A_27c,A_27b) ) ) ) ).
fof(ax_thm_2Ecombin_2Eo__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0f] :
( mem(V0f,arr(A_27c,A_27b))
=> ! [V1g] :
( mem(V1g,arr(A_27a,A_27c))
=> ap(ap(c_2Ecombin_2Eo(A_27a,A_27b,A_27c),V0f),V1g) = f77(A_27c,A_27b,A_27a,V1g,V0f) ) ) ) ) ) ).
fof(conj_thm_2Ecombin_2EI__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ap(c_2Ecombin_2EI(A_27a),V0x) = V0x ) ) ).
fof(conj_thm_2Ecombin_2EI__o__ID,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( 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 ) ) ) ) ).
fof(ax_thm_2Eextreal_2Eextreal__of__num__def,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ap(c_2Eextreal_2Eextreal__of__num,V0n) = ap(c_2Eextreal_2ENormal,ap(c_2Ereal_2Ereal__of__num,V0n)) ) ).
fof(conj_thm_2Eextreal_2Eextreal__le__def,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ! [V2a] :
( mem(V2a,ty_2Eextreal_2Eextreal)
=> ! [V3v2] :
( mem(V3v2,ty_2Erealax_2Ereal)
=> ! [V4v3] :
( mem(V4v3,ty_2Erealax_2Ereal)
=> ! [V5v5] :
( mem(V5v5,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2ENormal,V0x)),ap(c_2Eextreal_2ENormal,V1y)))
<=> p(ap(ap(c_2Ereal_2Ereal__lte,V0x),V1y)) )
& ( p(ap(ap(c_2Eextreal_2Eextreal__le,c_2Eextreal_2ENegInf),V2a))
<=> $true )
& ( p(ap(ap(c_2Eextreal_2Eextreal__le,c_2Eextreal_2EPosInf),c_2Eextreal_2EPosInf))
<=> $true )
& ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2ENormal,V3v2)),c_2Eextreal_2EPosInf))
<=> $true )
& ( p(ap(ap(c_2Eextreal_2Eextreal__le,c_2Eextreal_2EPosInf),c_2Eextreal_2ENegInf))
<=> $false )
& ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2ENormal,V4v3)),c_2Eextreal_2ENegInf))
<=> $false )
& ( p(ap(ap(c_2Eextreal_2Eextreal__le,c_2Eextreal_2EPosInf),ap(c_2Eextreal_2ENormal,V5v5)))
<=> $false ) ) ) ) ) ) ) ) ).
fof(ax_thm_2Eextreal_2Eextreal__inv__def,axiom,
( ap(c_2Eextreal_2Eextreal__inv,c_2Eextreal_2ENegInf) = ap(c_2Eextreal_2ENormal,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))
& ap(c_2Eextreal_2Eextreal__inv,c_2Eextreal_2EPosInf) = ap(c_2Eextreal_2ENormal,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))
& ! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ap(c_2Eextreal_2Eextreal__inv,ap(c_2Eextreal_2ENormal,V0x)) = ap(c_2Eextreal_2ENormal,ap(c_2Erealax_2Einv,V0x)) ) ) ).
fof(ax_thm_2Eextreal_2Eextreal__div__def,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ap(ap(c_2Eextreal_2Eextreal__div,V0x),V1y) = ap(ap(c_2Eextreal_2Eextreal__mul,V0x),ap(c_2Eextreal_2Eextreal__inv,V1y)) ) ) ).
fof(conj_thm_2Eextreal_2Emul__lzero,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V0x) = ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0) ) ).
fof(conj_thm_2Eextreal_2Ele__lmul__imp,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0)),V2z))
& p(ap(ap(c_2Eextreal_2Eextreal__le,V0x),V1y)) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__mul,V2z),V0x)),ap(ap(c_2Eextreal_2Eextreal__mul,V2z),V1y))) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Emul__comm,axiom,
! [V0x] :
( mem(V0x,ty_2Eextreal_2Eextreal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ap(ap(c_2Eextreal_2Eextreal__mul,V0x),V1y) = ap(ap(c_2Eextreal_2Eextreal__mul,V1y),V0x) ) ) ).
fof(conj_thm_2Eextreal_2Ele__rdiv,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0x))
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__mul,V1y),ap(c_2Eextreal_2ENormal,V0x))),V2z))
<=> p(ap(ap(c_2Eextreal_2Eextreal__le,V1y),ap(ap(c_2Eextreal_2Eextreal__div,V2z),ap(c_2Eextreal_2ENormal,V0x)))) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Ele__ldiv,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Eextreal_2Eextreal)
=> ! [V2z] :
( mem(V2z,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0x))
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,V1y),ap(ap(c_2Eextreal_2Eextreal__mul,V2z),ap(c_2Eextreal_2ENormal,V0x))))
<=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__div,V1y),ap(c_2Eextreal_2ENormal,V0x))),V2z)) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Esup__le,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1x] :
( mem(V1x,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__sup,V0p)),V1x))
<=> ! [V2y] :
( mem(V2y,ty_2Eextreal_2Eextreal)
=> ( p(ap(V0p,V2y))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V2y),V1x)) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Ele__sup,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1x] :
( mem(V1x,ty_2Eextreal_2Eextreal)
=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,V1x),ap(c_2Eextreal_2Eextreal__sup,V0p)))
<=> ! [V2y] :
( mem(V2y,ty_2Eextreal_2Eextreal)
=> ( ! [V3z] :
( mem(V3z,ty_2Eextreal_2Eextreal)
=> ( p(ap(V0p,V3z))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V3z),V2y)) ) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V1x),V2y)) ) ) ) ) ) ).
fof(conj_thm_2Eextreal_2Esup__eq,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1x] :
( mem(V1x,ty_2Eextreal_2Eextreal)
=> ( ap(c_2Eextreal_2Eextreal__sup,V0p) = V1x
<=> ( ! [V2y] :
( mem(V2y,ty_2Eextreal_2Eextreal)
=> ( p(ap(V0p,V2y))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V2y),V1x)) ) )
& ! [V3y] :
( mem(V3y,ty_2Eextreal_2Eextreal)
=> ( ! [V4z] :
( mem(V4z,ty_2Eextreal_2Eextreal)
=> ( p(ap(V0p,V4z))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V4z),V3y)) ) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,V1x),V3y)) ) ) ) ) ) ) ).
fof(ax_thm_2Elebesgue_2Epos__fn__integral__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( 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] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),V1f) = ap(c_2Eextreal_2Eextreal__sup,ap(c_2Epred__set_2EGSPEC(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f4000(A_27a,V0m,V1f))) ) ) ) ).
fof(conj_thm_2Elebesgue_2Epsfis__cmul,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( 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] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2a] :
( mem(V2a,ty_2Eextreal_2Eextreal)
=> ! [V3z] :
( mem(V3z,ty_2Erealax_2Ereal)
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& p(ap(ap(c_2Ebool_2EIN(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,c_2Enum_2E0)),V3z)) )
=> p(ap(ap(c_2Ebool_2EIN(ty_2Eextreal_2Eextreal),ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V3z)),V2a)),ap(ap(c_2Elebesgue_2Epsfis(A_27a),V0m),f4026(A_27a,V1f,V3z)))) ) ) ) ) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__fn__integral__zero,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( 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))
=> ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),k(A_27a,ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0))) = ap(c_2Eextreal_2Eextreal__of__num,c_2Enum_2E0) ) ) ) ).
fof(conj_thm_2Epair_2ECLOSED__PAIR__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27b)
=> ! [V2a] :
( mem(V2a,A_27a)
=> ! [V3b] :
( 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 ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESPECIFICATION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),V0P))
<=> p(ap(V0P,V1x)) ) ) ) ) ).
fof(ax_thm_2Epred__set_2EGSPECIFICATION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27b,ty_2Epair_2Eprod(A_27a,bool)))
=> ! [V1v] :
( mem(V1v,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V1v),ap(c_2Epred__set_2EGSPEC(A_27a,A_27b),V0f)))
<=> ? [V2x] :
( mem(V2x,A_27b)
& ap(ap(c_2Epair_2E_2C(A_27a,bool),V1v),c_2Ebool_2ET) = ap(V0f,V2x) ) ) ) ) ) ) ).
fof(conj_thm_2Ereal_2EREAL__LT__LE,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,V0x),V1y))
<=> ( p(ap(ap(c_2Ereal_2Ereal__lte,V0x),V1y))
& V0x != V1y ) ) ) ) ).
fof(conj_thm_2Ereal_2EREAL__LE__INV,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0x))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Erealax_2Einv,V0x))) ) ) ).
fof(conj_thm_2Esat_2ENOT__NOT,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ~ ~ p(V0t)
<=> p(V0t) ) ) ).
fof(conj_thm_2Esat_2EAND__INV__IMP,axiom,
! [V0A] :
( mem(V0A,bool)
=> ( p(V0A)
=> ( ~ p(V0A)
=> $false ) ) ) ).
fof(conj_thm_2Esat_2EOR__DUAL2,axiom,
! [V0A] :
( mem(V0A,bool)
=> ! [V1B] :
( mem(V1B,bool)
=> ( ( ~ ( p(V0A)
| p(V1B) )
=> $false )
<=> ( ( p(V0A)
=> $false )
=> ( ~ p(V1B)
=> $false ) ) ) ) ) ).
fof(conj_thm_2Esat_2EOR__DUAL3,axiom,
! [V0A] :
( mem(V0A,bool)
=> ! [V1B] :
( mem(V1B,bool)
=> ( ( ~ ( ~ p(V0A)
| p(V1B) )
=> $false )
<=> ( p(V0A)
=> ( ~ p(V1B)
=> $false ) ) ) ) ) ).
fof(conj_thm_2Esat_2EAND__INV2,axiom,
! [V0A] :
( mem(V0A,bool)
=> ( ( ~ p(V0A)
=> $false )
=> ( ( p(V0A)
=> $false )
=> $false ) ) ) ).
fof(conj_thm_2Esat_2Edc__eq,axiom,
! [V0p] :
( mem(V0p,bool)
=> ! [V1q] :
( mem(V1q,bool)
=> ! [V2r] :
( 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) ) ) ) ) ) ) ).
fof(conj_thm_2Esat_2Edc__conj,axiom,
! [V0p] :
( mem(V0p,bool)
=> ! [V1q] :
( mem(V1q,bool)
=> ! [V2r] :
( mem(V2r,bool)
=> ( ( p(V0p)
<=> ( p(V1q)
& p(V2r) ) )
<=> ( ( p(V0p)
| ~ p(V1q)
| ~ p(V2r) )
& ( p(V1q)
| ~ p(V0p) )
& ( p(V2r)
| ~ p(V0p) ) ) ) ) ) ) ).
fof(conj_thm_2Esat_2Edc__disj,axiom,
! [V0p] :
( mem(V0p,bool)
=> ! [V1q] :
( mem(V1q,bool)
=> ! [V2r] :
( mem(V2r,bool)
=> ( ( p(V0p)
<=> ( p(V1q)
| p(V2r) ) )
<=> ( ( p(V0p)
| ~ p(V1q) )
& ( p(V0p)
| ~ p(V2r) )
& ( p(V1q)
| p(V2r)
| ~ p(V0p) ) ) ) ) ) ) ).
fof(conj_thm_2Esat_2Edc__imp,axiom,
! [V0p] :
( mem(V0p,bool)
=> ! [V1q] :
( mem(V1q,bool)
=> ! [V2r] :
( mem(V2r,bool)
=> ( ( p(V0p)
<=> ( p(V1q)
=> p(V2r) ) )
<=> ( ( p(V0p)
| p(V1q) )
& ( p(V0p)
| ~ p(V2r) )
& ( ~ p(V1q)
| p(V2r)
| ~ p(V0p) ) ) ) ) ) ) ).
fof(conj_thm_2Esat_2Edc__neg,axiom,
! [V0p] :
( mem(V0p,bool)
=> ! [V1q] :
( mem(V1q,bool)
=> ( ( p(V0p)
<=> ~ p(V1q) )
<=> ( ( p(V0p)
| p(V1q) )
& ( ~ p(V1q)
| ~ p(V0p) ) ) ) ) ) ).
fof(conj_thm_2Esat_2Epth__ni1,axiom,
! [V0p] :
( mem(V0p,bool)
=> ! [V1q] :
( mem(V1q,bool)
=> ( ~ ( p(V0p)
=> p(V1q) )
=> p(V0p) ) ) ) ).
fof(conj_thm_2Esat_2Epth__ni2,axiom,
! [V0p] :
( mem(V0p,bool)
=> ! [V1q] :
( mem(V1q,bool)
=> ( ~ ( p(V0p)
=> p(V1q) )
=> ~ p(V1q) ) ) ) ).
fof(conj_thm_2Esat_2Epth__no1,axiom,
! [V0p] :
( mem(V0p,bool)
=> ! [V1q] :
( mem(V1q,bool)
=> ( ~ ( p(V0p)
| p(V1q) )
=> ~ p(V0p) ) ) ) ).
fof(conj_thm_2Esat_2Epth__no2,axiom,
! [V0p] :
( mem(V0p,bool)
=> ! [V1q] :
( mem(V1q,bool)
=> ( ~ ( p(V0p)
| p(V1q) )
=> ~ p(V1q) ) ) ) ).
fof(conj_thm_2Esat_2Epth__nn,axiom,
! [V0p] :
( mem(V0p,bool)
=> ( ~ ~ p(V0p)
=> p(V0p) ) ) ).
fof(conj_thm_2Elebesgue_2Epos__fn__integral__cmul,conjecture,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( 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] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2c] :
( mem(V2c,ty_2Erealax_2Ereal)
=> ( ( p(ap(c_2Emeasure_2Emeasure__space(A_27a),V0m))
& ! [V3x] :
( 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,c_2Enum_2E0)),ap(V1f,V3x))) ) )
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2c)) )
=> ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),f3354(A_27a,V1f,V2c)) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,V2c)),ap(ap(c_2Elebesgue_2Epos__fn__integral(A_27a),V0m),V1f)) ) ) ) ) ) ).
%------------------------------------------------------------------------------