TPTP Problem File: ITP024+2.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP024+2 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 set theory export of thm_2Ereal__topology_2ECLOSED__UNION__COMPACT__SUBSETS.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_2Ereal__topology_2ECLOSED__UNION__COMPACT__SUBSETS.p [Gau19]
% : HL411501+2.p [TPAP]
% Status : Theorem
% Rating : 1.00 v8.1.0, 0.97 v7.5.0
% Syntax : Number of formulae : 161 ( 47 unt; 0 def)
% Number of atoms : 803 ( 93 equ)
% Maximal formula atoms : 94 ( 4 avg)
% Number of connectives : 692 ( 50 ~; 35 |; 138 &)
% ( 102 <=>; 367 =>; 0 <=; 0 <~>)
% Maximal formula depth : 38 ( 6 avg)
% Maximal term depth : 9 ( 2 avg)
% Number of predicates : 6 ( 3 usr; 2 prp; 0-2 aty)
% Number of functors : 68 ( 68 usr; 44 con; 0-6 aty)
% Number of variables : 337 ( 321 !; 16 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
include('Axioms/ITP001/ITP001+2.ax').
%------------------------------------------------------------------------------
fof(ne_ty_2Enum_2Enum,axiom,
ne(ty_2Enum_2Enum) ).
fof(mem_c_2Earithmetic_2EEVEN,axiom,
mem(c_2Earithmetic_2EEVEN,arr(ty_2Enum_2Enum,bool)) ).
fof(mem_c_2Earithmetic_2EODD,axiom,
mem(c_2Earithmetic_2EODD,arr(ty_2Enum_2Enum,bool)) ).
fof(mem_c_2Earithmetic_2E_3E,axiom,
mem(c_2Earithmetic_2E_3E,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,bool))) ).
fof(mem_c_2Eprim__rec_2E_3C,axiom,
mem(c_2Eprim__rec_2E_3C,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,bool))) ).
fof(mem_c_2Eprim__rec_2EPRE,axiom,
mem(c_2Eprim__rec_2EPRE,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) ).
fof(mem_c_2Earithmetic_2EEXP,axiom,
mem(c_2Earithmetic_2EEXP,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))) ).
fof(mem_c_2Earithmetic_2E_2D,axiom,
mem(c_2Earithmetic_2E_2D,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))) ).
fof(mem_c_2Enumeral_2EiiSUC,axiom,
mem(c_2Enumeral_2EiiSUC,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) ).
fof(mem_c_2Enum_2ESUC,axiom,
mem(c_2Enum_2ESUC,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) ).
fof(mem_c_2Earithmetic_2EBIT2,axiom,
mem(c_2Earithmetic_2EBIT2,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) ).
fof(mem_c_2Enumeral_2EiDUB,axiom,
mem(c_2Enumeral_2EiDUB,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) ).
fof(mem_c_2Enumeral_2EiZ,axiom,
mem(c_2Enumeral_2EiZ,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) ).
fof(mem_c_2Ebool_2ET,axiom,
mem(c_2Ebool_2ET,bool) ).
fof(ax_true_p,axiom,
p(c_2Ebool_2ET) ).
fof(mem_c_2Epred__set_2EUNIV,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epred__set_2EUNIV(A_27a),arr(A_27a,bool)) ) ).
fof(mem_c_2Earithmetic_2EZERO,axiom,
mem(c_2Earithmetic_2EZERO,ty_2Enum_2Enum) ).
fof(mem_c_2Earithmetic_2EBIT1,axiom,
mem(c_2Earithmetic_2EBIT1,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) ).
fof(mem_c_2Earithmetic_2ENUMERAL,axiom,
mem(c_2Earithmetic_2ENUMERAL,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) ).
fof(ne_ty_2Erealax_2Ereal,axiom,
ne(ty_2Erealax_2Ereal) ).
fof(mem_c_2Erealax_2Ereal__neg,axiom,
mem(c_2Erealax_2Ereal__neg,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) ).
fof(mem_c_2Earithmetic_2E_2B,axiom,
mem(c_2Earithmetic_2E_2B,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))) ).
fof(mem_c_2Earithmetic_2E_3C_3D,axiom,
mem(c_2Earithmetic_2E_3C_3D,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,bool))) ).
fof(mem_c_2Earithmetic_2E_2A,axiom,
mem(c_2Earithmetic_2E_2A,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))) ).
fof(mem_c_2Erealax_2Ereal__mul,axiom,
mem(c_2Erealax_2Ereal__mul,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) ).
fof(mem_c_2Earithmetic_2E_3E_3D,axiom,
mem(c_2Earithmetic_2E_3E_3D,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,bool))) ).
fof(mem_c_2Ereal_2Ereal__ge,axiom,
mem(c_2Ereal_2Ereal__ge,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,bool))) ).
fof(mem_c_2Ereal_2Eabs,axiom,
mem(c_2Ereal_2Eabs,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) ).
fof(mem_c_2Erealax_2Ereal__lt,axiom,
mem(c_2Erealax_2Ereal__lt,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,bool))) ).
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(ne_ty_2Epair_2Eprod,axiom,
! [A0] :
( ne(A0)
=> ! [A1] :
( ne(A1)
=> ne(ty_2Epair_2Eprod(A0,A1)) ) ) ).
fof(mem_c_2Ereal__topology_2EDist,axiom,
mem(c_2Ereal__topology_2EDist,arr(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ty_2Erealax_2Ereal)) ).
fof(mem_c_2Erealax_2Ereal__add,axiom,
mem(c_2Erealax_2Ereal__add,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,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_2Ereal__topology_2Eball,axiom,
mem(c_2Ereal__topology_2Eball,arr(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,bool))) ).
fof(mem_c_2Epred__set_2ESUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epred__set_2ESUBSET(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) ) ).
fof(mem_c_2Ereal__topology_2Ebounded__def,axiom,
mem(c_2Ereal__topology_2Ebounded__def,arr(arr(ty_2Erealax_2Ereal,bool),bool)) ).
fof(mem_c_2Epred__set_2EINTER,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epred__set_2EINTER(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,bool),arr(A_27a,bool)))) ) ).
fof(mem_c_2Ereal__topology_2EClosed,axiom,
mem(c_2Ereal__topology_2EClosed,arr(arr(ty_2Erealax_2Ereal,bool),bool)) ).
fof(mem_c_2Ereal__topology_2Ecball,axiom,
mem(c_2Ereal__topology_2Ecball,arr(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,bool))) ).
fof(mem_c_2Ereal__topology_2Ecompact,axiom,
mem(c_2Ereal__topology_2Ecompact,arr(arr(ty_2Erealax_2Ereal,bool),bool)) ).
fof(mem_c_2Epair_2EUNCURRY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> mem(c_2Epair_2EUNCURRY(A_27a,A_27b,A_27c),arr(arr(A_27a,arr(A_27b,A_27c)),arr(ty_2Epair_2Eprod(A_27a,A_27b),A_27c))) ) ) ) ).
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_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_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_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_2Epred__set_2EBIGUNION,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epred__set_2EBIGUNION(A_27a),arr(arr(arr(A_27a,bool),bool),arr(A_27a,bool))) ) ).
fof(mem_c_2Ebool_2EF,axiom,
mem(c_2Ebool_2EF,bool) ).
fof(ax_false_p,axiom,
~ p(c_2Ebool_2EF) ).
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_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_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_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,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_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(conj_thm_2Earithmetic_2EZERO__LESS__EQ,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,c_2Enum_2E0),V0n)) ) ).
fof(conj_thm_2Earithmetic_2ENOT__LESS__EQUAL,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( ~ p(ap(ap(c_2Earithmetic_2E_3C_3D,V0m),V1n))
<=> p(ap(ap(c_2Eprim__rec_2E_3C,V1n),V0m)) ) ) ) ).
fof(conj_thm_2Earithmetic_2EADD__EQ__0,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( ap(ap(c_2Earithmetic_2E_2B,V0m),V1n) = c_2Enum_2E0
<=> ( V0m = c_2Enum_2E0
& V1n = c_2Enum_2E0 ) ) ) ) ).
fof(conj_thm_2Earithmetic_2ELE,axiom,
( ! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V0n),c_2Enum_2E0))
<=> V0n = c_2Enum_2E0 ) )
& ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1m),ap(c_2Enum_2ESUC,V2n)))
<=> ( V1m = ap(c_2Enum_2ESUC,V2n)
| p(ap(ap(c_2Earithmetic_2E_3C_3D,V1m),V2n)) ) ) ) ) ) ).
fof(ax_thm_2Ebool_2EIN__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> c_2Ebool_2EIN(A_27a) = f33(A_27a) ) ).
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_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_2EIMP__F,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ( p(V0t)
=> $false )
=> ~ p(V0t) ) ) ).
fof(conj_thm_2Ebool_2EF__IMP,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ~ p(V0t)
=> ( p(V0t)
=> $false ) ) ) ).
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_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_2ENOT__EXISTS__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ( ~ ? [V1x] :
( mem(V1x,A_27a)
& p(ap(V0P,V1x)) )
<=> ! [V2x] :
( mem(V2x,A_27a)
=> ~ p(ap(V0P,V2x)) ) ) ) ) ).
fof(conj_thm_2Ebool_2ELEFT__OR__EXISTS__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,bool)
=> ( ( ? [V2x] :
( mem(V2x,A_27a)
& p(ap(V0P,V2x)) )
| p(V1Q) )
<=> ? [V3x] :
( mem(V3x,A_27a)
& ( p(ap(V0P,V3x))
| p(V1Q) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ELEFT__EXISTS__AND__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,bool)
=> ( ? [V2x] :
( mem(V2x,A_27a)
& p(ap(V0P,V2x))
& p(V1Q) )
<=> ( ? [V3x] :
( mem(V3x,A_27a)
& p(ap(V0P,V3x)) )
& p(V1Q) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ERIGHT__EXISTS__AND__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,bool)
=> ! [V1Q] :
( mem(V1Q,arr(A_27a,bool))
=> ( ? [V2x] :
( mem(V2x,A_27a)
& p(V0P)
& p(ap(V1Q,V2x)) )
<=> ( p(V0P)
& ? [V3x] :
( mem(V3x,A_27a)
& p(ap(V1Q,V3x)) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EDE__MORGAN__THM,axiom,
! [V0A] :
( mem(V0A,bool)
=> ! [V1B] :
( mem(V1B,bool)
=> ( ( ~ ( p(V0A)
& p(V1B) )
<=> ( ~ p(V0A)
| ~ p(V1B) ) )
& ( ~ ( p(V0A)
| p(V1B) )
<=> ( ~ p(V0A)
& ~ p(V1B) ) ) ) ) ) ).
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_2ESKOLEM__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,arr(A_27b,bool)))
=> ( ! [V1x] :
( mem(V1x,A_27a)
=> ? [V2y] :
( mem(V2y,A_27b)
& p(ap(ap(V0P,V1x),V2y)) ) )
<=> ? [V3f] :
( mem(V3f,arr(A_27a,A_27b))
& ! [V4x] :
( mem(V4x,A_27a)
=> p(ap(ap(V0P,V4x),ap(V3f,V4x))) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECONJ__EQ__IMP,axiom,
! [V0r] :
( mem(V0r,bool)
=> ! [V1p] :
( mem(V1p,bool)
=> ! [V2q] :
( mem(V2q,bool)
=> ( ( ( p(V1p)
& p(V2q) )
=> p(V0r) )
<=> ( p(V1p)
=> ( p(V2q)
=> p(V0r) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESIMP__REAL__ARCH,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ? [V1n] :
( mem(V1n,ty_2Enum_2Enum)
& p(ap(ap(c_2Ereal_2Ereal__lte,V0x),ap(c_2Ereal_2Ereal__of__num,V1n))) ) ) ).
fof(conj_thm_2Enumeral_2Enumeral__distrib,axiom,
( ! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ap(ap(c_2Earithmetic_2E_2B,c_2Enum_2E0),V0n) = V0n )
& ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ap(ap(c_2Earithmetic_2E_2B,V1n),c_2Enum_2E0) = V1n )
& ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ! [V3m] :
( mem(V3m,ty_2Enum_2Enum)
=> ap(ap(c_2Earithmetic_2E_2B,ap(c_2Earithmetic_2ENUMERAL,V2n)),ap(c_2Earithmetic_2ENUMERAL,V3m)) = ap(c_2Earithmetic_2ENUMERAL,ap(c_2Enumeral_2EiZ,ap(ap(c_2Earithmetic_2E_2B,V2n),V3m))) ) )
& ! [V4n] :
( mem(V4n,ty_2Enum_2Enum)
=> ap(ap(c_2Earithmetic_2E_2A,c_2Enum_2E0),V4n) = c_2Enum_2E0 )
& ! [V5n] :
( mem(V5n,ty_2Enum_2Enum)
=> ap(ap(c_2Earithmetic_2E_2A,V5n),c_2Enum_2E0) = c_2Enum_2E0 )
& ! [V6n] :
( mem(V6n,ty_2Enum_2Enum)
=> ! [V7m] :
( mem(V7m,ty_2Enum_2Enum)
=> ap(ap(c_2Earithmetic_2E_2A,ap(c_2Earithmetic_2ENUMERAL,V6n)),ap(c_2Earithmetic_2ENUMERAL,V7m)) = ap(c_2Earithmetic_2ENUMERAL,ap(ap(c_2Earithmetic_2E_2A,V6n),V7m)) ) )
& ! [V8n] :
( mem(V8n,ty_2Enum_2Enum)
=> ap(ap(c_2Earithmetic_2E_2D,c_2Enum_2E0),V8n) = c_2Enum_2E0 )
& ! [V9n] :
( mem(V9n,ty_2Enum_2Enum)
=> ap(ap(c_2Earithmetic_2E_2D,V9n),c_2Enum_2E0) = V9n )
& ! [V10n] :
( mem(V10n,ty_2Enum_2Enum)
=> ! [V11m] :
( mem(V11m,ty_2Enum_2Enum)
=> ap(ap(c_2Earithmetic_2E_2D,ap(c_2Earithmetic_2ENUMERAL,V10n)),ap(c_2Earithmetic_2ENUMERAL,V11m)) = ap(c_2Earithmetic_2ENUMERAL,ap(ap(c_2Earithmetic_2E_2D,V10n),V11m)) ) )
& ! [V12n] :
( mem(V12n,ty_2Enum_2Enum)
=> ap(ap(c_2Earithmetic_2EEXP,c_2Enum_2E0),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,V12n))) = c_2Enum_2E0 )
& ! [V13n] :
( mem(V13n,ty_2Enum_2Enum)
=> ap(ap(c_2Earithmetic_2EEXP,c_2Enum_2E0),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,V13n))) = c_2Enum_2E0 )
& ! [V14n] :
( mem(V14n,ty_2Enum_2Enum)
=> ap(ap(c_2Earithmetic_2EEXP,V14n),c_2Enum_2E0) = ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)) )
& ! [V15n] :
( mem(V15n,ty_2Enum_2Enum)
=> ! [V16m] :
( mem(V16m,ty_2Enum_2Enum)
=> ap(ap(c_2Earithmetic_2EEXP,ap(c_2Earithmetic_2ENUMERAL,V15n)),ap(c_2Earithmetic_2ENUMERAL,V16m)) = ap(c_2Earithmetic_2ENUMERAL,ap(ap(c_2Earithmetic_2EEXP,V15n),V16m)) ) )
& ap(c_2Enum_2ESUC,c_2Enum_2E0) = ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))
& ! [V17n] :
( mem(V17n,ty_2Enum_2Enum)
=> ap(c_2Enum_2ESUC,ap(c_2Earithmetic_2ENUMERAL,V17n)) = ap(c_2Earithmetic_2ENUMERAL,ap(c_2Enum_2ESUC,V17n)) )
& ap(c_2Eprim__rec_2EPRE,c_2Enum_2E0) = c_2Enum_2E0
& ! [V18n] :
( mem(V18n,ty_2Enum_2Enum)
=> ap(c_2Eprim__rec_2EPRE,ap(c_2Earithmetic_2ENUMERAL,V18n)) = ap(c_2Earithmetic_2ENUMERAL,ap(c_2Eprim__rec_2EPRE,V18n)) )
& ! [V19n] :
( mem(V19n,ty_2Enum_2Enum)
=> ( ap(c_2Earithmetic_2ENUMERAL,V19n) = c_2Enum_2E0
<=> V19n = c_2Earithmetic_2EZERO ) )
& ! [V20n] :
( mem(V20n,ty_2Enum_2Enum)
=> ( c_2Enum_2E0 = ap(c_2Earithmetic_2ENUMERAL,V20n)
<=> V20n = c_2Earithmetic_2EZERO ) )
& ! [V21n] :
( mem(V21n,ty_2Enum_2Enum)
=> ! [V22m] :
( mem(V22m,ty_2Enum_2Enum)
=> ( ap(c_2Earithmetic_2ENUMERAL,V21n) = ap(c_2Earithmetic_2ENUMERAL,V22m)
<=> V21n = V22m ) ) )
& ! [V23n] :
( mem(V23n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Eprim__rec_2E_3C,V23n),c_2Enum_2E0))
<=> $false ) )
& ! [V24n] :
( mem(V24n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),ap(c_2Earithmetic_2ENUMERAL,V24n)))
<=> p(ap(ap(c_2Eprim__rec_2E_3C,c_2Earithmetic_2EZERO),V24n)) ) )
& ! [V25n] :
( mem(V25n,ty_2Enum_2Enum)
=> ! [V26m] :
( mem(V26m,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Eprim__rec_2E_3C,ap(c_2Earithmetic_2ENUMERAL,V25n)),ap(c_2Earithmetic_2ENUMERAL,V26m)))
<=> p(ap(ap(c_2Eprim__rec_2E_3C,V25n),V26m)) ) ) )
& ! [V27n] :
( mem(V27n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3E,c_2Enum_2E0),V27n))
<=> $false ) )
& ! [V28n] :
( mem(V28n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3E,ap(c_2Earithmetic_2ENUMERAL,V28n)),c_2Enum_2E0))
<=> p(ap(ap(c_2Eprim__rec_2E_3C,c_2Earithmetic_2EZERO),V28n)) ) )
& ! [V29n] :
( mem(V29n,ty_2Enum_2Enum)
=> ! [V30m] :
( mem(V30m,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3E,ap(c_2Earithmetic_2ENUMERAL,V29n)),ap(c_2Earithmetic_2ENUMERAL,V30m)))
<=> p(ap(ap(c_2Eprim__rec_2E_3C,V30m),V29n)) ) ) )
& ! [V31n] :
( mem(V31n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,c_2Enum_2E0),V31n))
<=> $true ) )
& ! [V32n] :
( mem(V32n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Earithmetic_2ENUMERAL,V32n)),c_2Enum_2E0))
<=> p(ap(ap(c_2Earithmetic_2E_3C_3D,V32n),c_2Earithmetic_2EZERO)) ) )
& ! [V33n] :
( mem(V33n,ty_2Enum_2Enum)
=> ! [V34m] :
( mem(V34m,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Earithmetic_2ENUMERAL,V33n)),ap(c_2Earithmetic_2ENUMERAL,V34m)))
<=> p(ap(ap(c_2Earithmetic_2E_3C_3D,V33n),V34m)) ) ) )
& ! [V35n] :
( mem(V35n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3E_3D,V35n),c_2Enum_2E0))
<=> $true ) )
& ! [V36n] :
( mem(V36n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3E_3D,c_2Enum_2E0),V36n))
<=> V36n = c_2Enum_2E0 ) )
& ! [V37n] :
( mem(V37n,ty_2Enum_2Enum)
=> ! [V38m] :
( mem(V38m,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3E_3D,ap(c_2Earithmetic_2ENUMERAL,V37n)),ap(c_2Earithmetic_2ENUMERAL,V38m)))
<=> p(ap(ap(c_2Earithmetic_2E_3C_3D,V38m),V37n)) ) ) )
& ! [V39n] :
( mem(V39n,ty_2Enum_2Enum)
=> ( p(ap(c_2Earithmetic_2EODD,ap(c_2Earithmetic_2ENUMERAL,V39n)))
<=> p(ap(c_2Earithmetic_2EODD,V39n)) ) )
& ! [V40n] :
( mem(V40n,ty_2Enum_2Enum)
=> ( p(ap(c_2Earithmetic_2EEVEN,ap(c_2Earithmetic_2ENUMERAL,V40n)))
<=> p(ap(c_2Earithmetic_2EEVEN,V40n)) ) )
& ~ p(ap(c_2Earithmetic_2EODD,c_2Enum_2E0))
& p(ap(c_2Earithmetic_2EEVEN,c_2Enum_2E0)) ) ).
fof(conj_thm_2Enumeral_2Enumeral__add,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ( ap(c_2Enumeral_2EiZ,ap(ap(c_2Earithmetic_2E_2B,c_2Earithmetic_2EZERO),V0n)) = V0n
& ap(c_2Enumeral_2EiZ,ap(ap(c_2Earithmetic_2E_2B,V0n),c_2Earithmetic_2EZERO)) = V0n
& ap(c_2Enumeral_2EiZ,ap(ap(c_2Earithmetic_2E_2B,ap(c_2Earithmetic_2EBIT1,V0n)),ap(c_2Earithmetic_2EBIT1,V1m))) = ap(c_2Earithmetic_2EBIT2,ap(c_2Enumeral_2EiZ,ap(ap(c_2Earithmetic_2E_2B,V0n),V1m)))
& ap(c_2Enumeral_2EiZ,ap(ap(c_2Earithmetic_2E_2B,ap(c_2Earithmetic_2EBIT1,V0n)),ap(c_2Earithmetic_2EBIT2,V1m))) = ap(c_2Earithmetic_2EBIT1,ap(c_2Enum_2ESUC,ap(ap(c_2Earithmetic_2E_2B,V0n),V1m)))
& ap(c_2Enumeral_2EiZ,ap(ap(c_2Earithmetic_2E_2B,ap(c_2Earithmetic_2EBIT2,V0n)),ap(c_2Earithmetic_2EBIT1,V1m))) = ap(c_2Earithmetic_2EBIT1,ap(c_2Enum_2ESUC,ap(ap(c_2Earithmetic_2E_2B,V0n),V1m)))
& ap(c_2Enumeral_2EiZ,ap(ap(c_2Earithmetic_2E_2B,ap(c_2Earithmetic_2EBIT2,V0n)),ap(c_2Earithmetic_2EBIT2,V1m))) = ap(c_2Earithmetic_2EBIT2,ap(c_2Enum_2ESUC,ap(ap(c_2Earithmetic_2E_2B,V0n),V1m)))
& ap(c_2Enum_2ESUC,ap(ap(c_2Earithmetic_2E_2B,c_2Earithmetic_2EZERO),V0n)) = ap(c_2Enum_2ESUC,V0n)
& ap(c_2Enum_2ESUC,ap(ap(c_2Earithmetic_2E_2B,V0n),c_2Earithmetic_2EZERO)) = ap(c_2Enum_2ESUC,V0n)
& ap(c_2Enum_2ESUC,ap(ap(c_2Earithmetic_2E_2B,ap(c_2Earithmetic_2EBIT1,V0n)),ap(c_2Earithmetic_2EBIT1,V1m))) = ap(c_2Earithmetic_2EBIT1,ap(c_2Enum_2ESUC,ap(ap(c_2Earithmetic_2E_2B,V0n),V1m)))
& ap(c_2Enum_2ESUC,ap(ap(c_2Earithmetic_2E_2B,ap(c_2Earithmetic_2EBIT1,V0n)),ap(c_2Earithmetic_2EBIT2,V1m))) = ap(c_2Earithmetic_2EBIT2,ap(c_2Enum_2ESUC,ap(ap(c_2Earithmetic_2E_2B,V0n),V1m)))
& ap(c_2Enum_2ESUC,ap(ap(c_2Earithmetic_2E_2B,ap(c_2Earithmetic_2EBIT2,V0n)),ap(c_2Earithmetic_2EBIT1,V1m))) = ap(c_2Earithmetic_2EBIT2,ap(c_2Enum_2ESUC,ap(ap(c_2Earithmetic_2E_2B,V0n),V1m)))
& ap(c_2Enum_2ESUC,ap(ap(c_2Earithmetic_2E_2B,ap(c_2Earithmetic_2EBIT2,V0n)),ap(c_2Earithmetic_2EBIT2,V1m))) = ap(c_2Earithmetic_2EBIT1,ap(c_2Enumeral_2EiiSUC,ap(ap(c_2Earithmetic_2E_2B,V0n),V1m)))
& ap(c_2Enumeral_2EiiSUC,ap(ap(c_2Earithmetic_2E_2B,c_2Earithmetic_2EZERO),V0n)) = ap(c_2Enumeral_2EiiSUC,V0n)
& ap(c_2Enumeral_2EiiSUC,ap(ap(c_2Earithmetic_2E_2B,V0n),c_2Earithmetic_2EZERO)) = ap(c_2Enumeral_2EiiSUC,V0n)
& ap(c_2Enumeral_2EiiSUC,ap(ap(c_2Earithmetic_2E_2B,ap(c_2Earithmetic_2EBIT1,V0n)),ap(c_2Earithmetic_2EBIT1,V1m))) = ap(c_2Earithmetic_2EBIT2,ap(c_2Enum_2ESUC,ap(ap(c_2Earithmetic_2E_2B,V0n),V1m)))
& ap(c_2Enumeral_2EiiSUC,ap(ap(c_2Earithmetic_2E_2B,ap(c_2Earithmetic_2EBIT1,V0n)),ap(c_2Earithmetic_2EBIT2,V1m))) = ap(c_2Earithmetic_2EBIT1,ap(c_2Enumeral_2EiiSUC,ap(ap(c_2Earithmetic_2E_2B,V0n),V1m)))
& ap(c_2Enumeral_2EiiSUC,ap(ap(c_2Earithmetic_2E_2B,ap(c_2Earithmetic_2EBIT2,V0n)),ap(c_2Earithmetic_2EBIT1,V1m))) = ap(c_2Earithmetic_2EBIT1,ap(c_2Enumeral_2EiiSUC,ap(ap(c_2Earithmetic_2E_2B,V0n),V1m)))
& ap(c_2Enumeral_2EiiSUC,ap(ap(c_2Earithmetic_2E_2B,ap(c_2Earithmetic_2EBIT2,V0n)),ap(c_2Earithmetic_2EBIT2,V1m))) = ap(c_2Earithmetic_2EBIT2,ap(c_2Enumeral_2EiiSUC,ap(ap(c_2Earithmetic_2E_2B,V0n),V1m))) ) ) ) ).
fof(conj_thm_2Enumeral_2EiDUB__removal,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ( ap(c_2Enumeral_2EiDUB,ap(c_2Earithmetic_2EBIT1,V0n)) = ap(c_2Earithmetic_2EBIT2,ap(c_2Enumeral_2EiDUB,V0n))
& ap(c_2Enumeral_2EiDUB,ap(c_2Earithmetic_2EBIT2,V0n)) = ap(c_2Earithmetic_2EBIT2,ap(c_2Earithmetic_2EBIT1,V0n))
& ap(c_2Enumeral_2EiDUB,c_2Earithmetic_2EZERO) = c_2Earithmetic_2EZERO ) ) ).
fof(conj_thm_2Enumeral_2Enumeral__mult,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ( ap(ap(c_2Earithmetic_2E_2A,c_2Earithmetic_2EZERO),V0n) = c_2Earithmetic_2EZERO
& ap(ap(c_2Earithmetic_2E_2A,V0n),c_2Earithmetic_2EZERO) = c_2Earithmetic_2EZERO
& ap(ap(c_2Earithmetic_2E_2A,ap(c_2Earithmetic_2EBIT1,V0n)),V1m) = ap(c_2Enumeral_2EiZ,ap(ap(c_2Earithmetic_2E_2B,ap(c_2Enumeral_2EiDUB,ap(ap(c_2Earithmetic_2E_2A,V0n),V1m))),V1m))
& ap(ap(c_2Earithmetic_2E_2A,ap(c_2Earithmetic_2EBIT2,V0n)),V1m) = ap(c_2Enumeral_2EiDUB,ap(c_2Enumeral_2EiZ,ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Earithmetic_2E_2A,V0n),V1m)),V1m))) ) ) ) ).
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_2EEXTENSION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ( V0s = V1t
<=> ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0s))
<=> p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1t)) ) ) ) ) ) ) ).
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_2Epred__set_2EIN__UNIV,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),c_2Epred__set_2EUNIV(A_27a))) ) ) ).
fof(ax_thm_2Epred__set_2ESUBSET__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V1t))
<=> ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0s))
=> p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1t)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUBSET__TRANS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ! [V2u] :
( mem(V2u,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V1t))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1t),V2u)) )
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V2u)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EIN__INTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),ap(ap(c_2Epred__set_2EINTER(A_27a),V0s),V1t)))
<=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0s))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1t)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINTER__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(ap(c_2Epred__set_2EINTER(A_27a),V0s),V1t)),V0s)) ) )
& ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t] :
( mem(V3t,arr(A_27a,bool))
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(ap(c_2Epred__set_2EINTER(A_27a),V3t),V2s)),V2s)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUBSET__INTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ! [V2u] :
( mem(V2u,arr(A_27a,bool))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),ap(ap(c_2Epred__set_2EINTER(A_27a),V1t),V2u)))
<=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V1t))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V2u)) ) ) ) ) ) ) ).
fof(conj_thm_2Ereal_2EREAL__ADD__SYM,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ap(ap(c_2Erealax_2Ereal__add,V0x),V1y) = ap(ap(c_2Erealax_2Ereal__add,V1y),V0x) ) ) ).
fof(conj_thm_2Ereal_2EREAL__ADD__ASSOC,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ! [V2z] :
( mem(V2z,ty_2Erealax_2Ereal)
=> ap(ap(c_2Erealax_2Ereal__add,V0x),ap(ap(c_2Erealax_2Ereal__add,V1y),V2z)) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Erealax_2Ereal__add,V0x),V1y)),V2z) ) ) ) ).
fof(conj_thm_2Ereal_2EREAL__ADD__LID,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ap(ap(c_2Erealax_2Ereal__add,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0x) = V0x ) ).
fof(conj_thm_2Ereal_2EREAL__ADD__LINV,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ap(ap(c_2Erealax_2Ereal__add,ap(c_2Erealax_2Ereal__neg,V0x)),V0x) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ).
fof(conj_thm_2Ereal_2EREAL__LT__TRANS,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ! [V2z] :
( mem(V2z,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,V0x),V1y))
& p(ap(ap(c_2Erealax_2Ereal__lt,V1y),V2z)) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,V0x),V2z)) ) ) ) ) ).
fof(conj_thm_2Ereal_2EREAL__MUL__LID,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V0x) = V0x ) ).
fof(ax_thm_2Ereal_2Ereal__ge,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ereal_2Ereal__ge,V0x),V1y))
<=> p(ap(ap(c_2Ereal_2Ereal__lte,V1y),V0x)) ) ) ) ).
fof(conj_thm_2Ereal_2EREAL__ADD__RID,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ap(ap(c_2Erealax_2Ereal__add,V0x),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)) = V0x ) ).
fof(conj_thm_2Ereal_2EREAL__ADD__RINV,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ap(ap(c_2Erealax_2Ereal__add,V0x),ap(c_2Erealax_2Ereal__neg,V0x)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ).
fof(conj_thm_2Ereal_2EREAL__MUL__RID,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ap(ap(c_2Erealax_2Ereal__mul,V0x),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))) = V0x ) ).
fof(conj_thm_2Ereal_2EREAL__NEG__ADD,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ap(c_2Erealax_2Ereal__neg,ap(ap(c_2Erealax_2Ereal__add,V0x),V1y)) = ap(ap(c_2Erealax_2Ereal__add,ap(c_2Erealax_2Ereal__neg,V0x)),ap(c_2Erealax_2Ereal__neg,V1y)) ) ) ).
fof(conj_thm_2Ereal_2EREAL__MUL__LZERO,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0x) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ).
fof(conj_thm_2Ereal_2EREAL__LT__LADD,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ! [V2z] :
( mem(V2z,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Erealax_2Ereal__add,V0x),V1y)),ap(ap(c_2Erealax_2Ereal__add,V0x),V2z)))
<=> p(ap(ap(c_2Erealax_2Ereal__lt,V1y),V2z)) ) ) ) ) ).
fof(conj_thm_2Ereal_2EREAL__LTE__TRANS,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ! [V2z] :
( mem(V2z,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,V0x),V1y))
& p(ap(ap(c_2Ereal_2Ereal__lte,V1y),V2z)) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,V0x),V2z)) ) ) ) ) ).
fof(conj_thm_2Ereal_2EREAL__LET__TRANS,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ! [V2z] :
( mem(V2z,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V0x),V1y))
& p(ap(ap(c_2Erealax_2Ereal__lt,V1y),V2z)) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,V0x),V2z)) ) ) ) ) ).
fof(conj_thm_2Ereal_2EREAL__LE__TRANS,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ! [V2z] :
( mem(V2z,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V0x),V1y))
& p(ap(ap(c_2Ereal_2Ereal__lte,V1y),V2z)) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,V0x),V2z)) ) ) ) ) ).
fof(conj_thm_2Ereal_2EREAL__LE__ANTISYM,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V0x),V1y))
& p(ap(ap(c_2Ereal_2Ereal__lte,V1y),V0x)) )
<=> V0x = V1y ) ) ) ).
fof(conj_thm_2Ereal_2EREAL__ADD,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ap(ap(c_2Erealax_2Ereal__add,ap(c_2Ereal_2Ereal__of__num,V0m)),ap(c_2Ereal_2Ereal__of__num,V1n)) = ap(c_2Ereal_2Ereal__of__num,ap(ap(c_2Earithmetic_2E_2B,V0m),V1n)) ) ) ).
fof(conj_thm_2Ereal_2EREAL__MUL__RNEG,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ap(ap(c_2Erealax_2Ereal__mul,V0x),ap(c_2Erealax_2Ereal__neg,V1y)) = ap(c_2Erealax_2Ereal__neg,ap(ap(c_2Erealax_2Ereal__mul,V0x),V1y)) ) ) ).
fof(conj_thm_2Ereal_2EREAL__MUL__LNEG,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Erealax_2Ereal__neg,V0x)),V1y) = ap(c_2Erealax_2Ereal__neg,ap(ap(c_2Erealax_2Ereal__mul,V0x),V1y)) ) ) ).
fof(conj_thm_2Ereal_2Ereal__lt,axiom,
! [V0y] :
( mem(V0y,ty_2Erealax_2Ereal)
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,V1x),V0y))
<=> ~ p(ap(ap(c_2Ereal_2Ereal__lte,V0y),V1x)) ) ) ) ).
fof(conj_thm_2Ereal_2EREAL__LE__LADD__IMP,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ! [V2z] :
( mem(V2z,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,V1y),V2z))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Erealax_2Ereal__add,V0x),V1y)),ap(ap(c_2Erealax_2Ereal__add,V0x),V2z))) ) ) ) ) ).
fof(conj_thm_2Ereal_2EREAL__LE__LNEG,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Erealax_2Ereal__neg,V0x)),V1y))
<=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Erealax_2Ereal__add,V0x),V1y))) ) ) ) ).
fof(conj_thm_2Ereal_2EREAL__LE__NEG2,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Erealax_2Ereal__neg,V0x)),ap(c_2Erealax_2Ereal__neg,V1y)))
<=> p(ap(ap(c_2Ereal_2Ereal__lte,V1y),V0x)) ) ) ) ).
fof(conj_thm_2Ereal_2EREAL__NEG__NEG,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ap(c_2Erealax_2Ereal__neg,ap(c_2Erealax_2Ereal__neg,V0x)) = V0x ) ).
fof(conj_thm_2Ereal_2EREAL__LE__RNEG,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,V0x),ap(c_2Erealax_2Ereal__neg,V1y)))
<=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Erealax_2Ereal__add,V0x),V1y)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ) ).
fof(conj_thm_2Ereal_2EREAL__ADD__RDISTRIB,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ! [V2z] :
( mem(V2z,ty_2Erealax_2Ereal)
=> ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Erealax_2Ereal__add,V0x),V1y)),V2z) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Erealax_2Ereal__mul,V0x),V2z)),ap(ap(c_2Erealax_2Ereal__mul,V1y),V2z)) ) ) ) ).
fof(conj_thm_2Ereal_2EREAL__OF__NUM__ADD,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ap(ap(c_2Erealax_2Ereal__add,ap(c_2Ereal_2Ereal__of__num,V0m)),ap(c_2Ereal_2Ereal__of__num,V1n)) = ap(c_2Ereal_2Ereal__of__num,ap(ap(c_2Earithmetic_2E_2B,V0m),V1n)) ) ) ).
fof(conj_thm_2Ereal_2EREAL__OF__NUM__LE,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,V0m)),ap(c_2Ereal_2Ereal__of__num,V1n)))
<=> p(ap(ap(c_2Earithmetic_2E_3C_3D,V0m),V1n)) ) ) ) ).
fof(conj_thm_2Ereal_2EREAL__OF__NUM__MUL,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,V0m)),ap(c_2Ereal_2Ereal__of__num,V1n)) = ap(c_2Ereal_2Ereal__of__num,ap(ap(c_2Earithmetic_2E_2A,V0m),V1n)) ) ) ).
fof(conj_thm_2Ereal__topology_2EREAL__OF__NUM__GE,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ereal_2Ereal__ge,ap(c_2Ereal_2Ereal__of__num,V0m)),ap(c_2Ereal_2Ereal__of__num,V1n)))
<=> p(ap(ap(c_2Earithmetic_2E_3E_3D,V0m),V1n)) ) ) ) ).
fof(conj_thm_2Ereal__topology_2EDIST__REFL,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V0x)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ).
fof(conj_thm_2Ereal__topology_2EIN__CBALL__0,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1e] :
( mem(V1e,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0x),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1e))))
<=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,V0x)),V1e)) ) ) ) ).
fof(conj_thm_2Ereal__topology_2EBOUNDED__SUBSET__CBALL,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ( p(ap(c_2Ereal__topology_2Ebounded__def,V0s))
=> ? [V2r] :
( mem(V2r,ty_2Erealax_2Ereal)
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2r))
& p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1x),V2r)))) ) ) ) ) ).
fof(conj_thm_2Ereal__topology_2ESUBSET__BALLS,axiom,
( ! [V0a] :
( mem(V0a,ty_2Erealax_2Ereal)
=> ! [V1a_27] :
( mem(V1a_27,ty_2Erealax_2Ereal)
=> ! [V2r] :
( mem(V2r,ty_2Erealax_2Ereal)
=> ! [V3r_27] :
( mem(V3r_27,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V2r))),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a_27),V3r_27))))
<=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Erealax_2Ereal__add,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0a),V1a_27))),V2r)),V3r_27))
| p(ap(ap(c_2Ereal_2Ereal__lte,V2r),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ) ) ) )
& ! [V4a] :
( mem(V4a,ty_2Erealax_2Ereal)
=> ! [V5a_27] :
( mem(V5a_27,ty_2Erealax_2Ereal)
=> ! [V6r] :
( mem(V6r,ty_2Erealax_2Ereal)
=> ! [V7r_27] :
( mem(V7r_27,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V4a),V6r))),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V5a_27),V7r_27))))
<=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Erealax_2Ereal__add,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V4a),V5a_27))),V6r)),V7r_27))
| p(ap(ap(c_2Ereal_2Ereal__lte,V6r),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ) ) ) )
& ! [V8a] :
( mem(V8a,ty_2Erealax_2Ereal)
=> ! [V9a_27] :
( mem(V9a_27,ty_2Erealax_2Ereal)
=> ! [V10r] :
( mem(V10r,ty_2Erealax_2Ereal)
=> ! [V11r_27] :
( mem(V11r_27,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V8a),V10r))),ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V9a_27),V11r_27))))
<=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Erealax_2Ereal__add,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V8a),V9a_27))),V10r)),V11r_27))
| p(ap(ap(c_2Erealax_2Ereal__lt,V10r),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ) ) ) )
& ! [V12a] :
( mem(V12a,ty_2Erealax_2Ereal)
=> ! [V13a_27] :
( mem(V13a_27,ty_2Erealax_2Ereal)
=> ! [V14r] :
( mem(V14r,ty_2Erealax_2Ereal)
=> ! [V15r_27] :
( mem(V15r_27,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V12a),V14r))),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V13a_27),V15r_27))))
<=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Erealax_2Ereal__add,ap(c_2Ereal__topology_2EDist,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V12a),V13a_27))),V14r)),V15r_27))
| p(ap(ap(c_2Erealax_2Ereal__lt,V14r),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ereal__topology_2ECOMPACT__IMP__BOUNDED,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ( p(ap(c_2Ereal__topology_2Ecompact,V0s))
=> p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) ) ) ).
fof(conj_thm_2Ereal__topology_2ECLOSED__INTER__COMPACT,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1t] :
( mem(V1t,arr(ty_2Erealax_2Ereal,bool))
=> ( ( p(ap(c_2Ereal__topology_2EClosed,V0s))
& p(ap(c_2Ereal__topology_2Ecompact,V1t)) )
=> p(ap(c_2Ereal__topology_2Ecompact,ap(ap(c_2Epred__set_2EINTER(ty_2Erealax_2Ereal),V0s),V1t))) ) ) ) ).
fof(conj_thm_2Ereal__topology_2ECOMPACT__CBALL,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1e] :
( mem(V1e,ty_2Erealax_2Ereal)
=> p(ap(c_2Ereal__topology_2Ecompact,ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1e)))) ) ) ).
fof(conj_thm_2Ereal__topology_2EBIGUNION__GSPEC,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [A_27e] :
( ne(A_27e)
=> ! [A_27f] :
( ne(A_27f)
=> ! [A_27g] :
( ne(A_27g)
=> ! [A_27h] :
( ne(A_27h)
=> ! [A_27i] :
( ne(A_27i)
=> ( ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,arr(A_27b,bool)))
=> ap(c_2Epred__set_2EBIGUNION(A_27b),ap(c_2Epred__set_2EGSPEC(arr(A_27b,bool),A_27a),f3575(A_27b,A_27a,V1f,V0P))) = ap(c_2Epred__set_2EGSPEC(A_27b,A_27b),f3768(A_27a,A_27b,V1f,V0P)) ) )
& ! [V5P] :
( mem(V5P,arr(A_27c,arr(A_27d,bool)))
=> ! [V6f] :
( mem(V6f,arr(A_27c,arr(A_27d,arr(A_27e,bool))))
=> ap(c_2Epred__set_2EBIGUNION(A_27e),ap(c_2Epred__set_2EGSPEC(arr(A_27e,bool),ty_2Epair_2Eprod(A_27c,A_27d)),ap(c_2Epair_2EUNCURRY(A_27c,A_27d,ty_2Epair_2Eprod(arr(A_27e,bool),bool)),f3579(A_27d,A_27e,A_27c,V6f,V5P)))) = ap(c_2Epred__set_2EGSPEC(A_27e,A_27e),f3771(A_27d,A_27c,A_27e,V5P,V6f)) ) )
& ! [V12P] :
( mem(V12P,arr(A_27f,arr(A_27g,arr(A_27h,bool))))
=> ! [V13f] :
( mem(V13f,arr(A_27f,arr(A_27g,arr(A_27h,arr(A_27i,bool)))))
=> ap(c_2Epred__set_2EBIGUNION(A_27i),ap(c_2Epred__set_2EGSPEC(arr(A_27i,bool),ty_2Epair_2Eprod(A_27f,ty_2Epair_2Eprod(A_27g,A_27h))),ap(c_2Epair_2EUNCURRY(A_27f,ty_2Epair_2Eprod(A_27g,A_27h),ty_2Epair_2Eprod(arr(A_27i,bool),bool)),f3585(A_27g,A_27i,A_27h,A_27f,V12P,V13f)))) = ap(c_2Epred__set_2EGSPEC(A_27i,A_27i),f3775(A_27h,A_27g,A_27f,A_27i,V13f,V12P)) ) ) ) ) ) ) ) ) ) ) ) ) ).
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_2Ereal__topology_2ECLOSED__UNION__COMPACT__SUBSETS,conjecture,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ( p(ap(c_2Ereal__topology_2EClosed,V0s))
=> ? [V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(ty_2Erealax_2Ereal,bool)))
& ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> p(ap(c_2Ereal__topology_2Ecompact,ap(V1f,V2n))) )
& ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(V1f,V3n)),V0s)) )
& ! [V4n] :
( mem(V4n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),ap(V1f,V4n)),ap(V1f,ap(ap(c_2Earithmetic_2E_2B,V4n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))) )
& ap(c_2Epred__set_2EBIGUNION(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(arr(ty_2Erealax_2Ereal,bool),ty_2Enum_2Enum),f3855(V1f))) = V0s
& ! [V6k] :
( mem(V6k,arr(ty_2Erealax_2Ereal,bool))
=> ( ( p(ap(c_2Ereal__topology_2Ecompact,V6k))
& p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V6k),V0s)) )
=> ? [V7N] :
( mem(V7N,ty_2Enum_2Enum)
& ! [V8n] :
( mem(V8n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3E_3D,V8n),V7N))
=> p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V6k),ap(V1f,V8n))) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------