TPTP Problem File: ITP023+2.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP023+2 : TPTP v9.0.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 set theory export of thm_2Ereal__topology_2EBOUNDED__BALL.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_2EBOUNDED__BALL.p [Gau19]
% : HL411001+2.p [TPAP]
% Status : Theorem
% Rating : 0.55 v9.0.0, 0.58 v7.5.0
% Syntax : Number of formulae : 59 ( 15 unt; 0 def)
% Number of atoms : 279 ( 8 equ)
% Maximal formula atoms : 18 ( 4 avg)
% Number of connectives : 267 ( 47 ~; 34 |; 27 &)
% ( 45 <=>; 114 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of predicates : 6 ( 3 usr; 2 prp; 0-2 aty)
% Number of functors : 22 ( 22 usr; 12 con; 0-2 aty)
% Number of variables : 92 ( 90 !; 2 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
include('Axioms/ITP001/ITP001+2.ax').
%------------------------------------------------------------------------------
fof(mem_c_2Ebool_2ET,axiom,
mem(c_2Ebool_2ET,bool) ).
fof(ax_true_p,axiom,
p(c_2Ebool_2ET) ).
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(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_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_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_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_2Ebounded__def,axiom,
mem(c_2Ereal__topology_2Ebounded__def,arr(arr(ty_2Erealax_2Ereal,bool),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_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_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_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_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__FORALL__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_2EDISJ__ASSOC,axiom,
! [V0A] :
( mem(V0A,bool)
=> ! [V1B] :
( mem(V1B,bool)
=> ! [V2C] :
( mem(V2C,bool)
=> ( ( p(V0A)
| p(V1B)
| p(V2C) )
<=> ( p(V0A)
| p(V1B)
| p(V2C) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EDISJ__SYM,axiom,
! [V0A] :
( mem(V0A,bool)
=> ! [V1B] :
( mem(V1B,bool)
=> ( ( p(V0A)
| p(V1B) )
<=> ( p(V1B)
| p(V0A) ) ) ) ) ).
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_2Ereal__topology_2EBALL__SUBSET__CBALL,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1e] :
( mem(V1e,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),V0x),V1e))),ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1e)))) ) ) ).
fof(conj_thm_2Ereal__topology_2EBOUNDED__SUBSET,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1t] :
( mem(V1t,arr(ty_2Erealax_2Ereal,bool))
=> ( ( p(ap(c_2Ereal__topology_2Ebounded__def,V1t))
& p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t)) )
=> p(ap(c_2Ereal__topology_2Ebounded__def,V0s)) ) ) ) ).
fof(conj_thm_2Ereal__topology_2EBOUNDED__CBALL,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1e] :
( mem(V1e,ty_2Erealax_2Ereal)
=> p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Ereal__topology_2Ecball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1e)))) ) ) ).
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_2EBOUNDED__BALL,conjecture,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1e] :
( mem(V1e,ty_2Erealax_2Ereal)
=> p(ap(c_2Ereal__topology_2Ebounded__def,ap(c_2Ereal__topology_2Eball,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1e)))) ) ) ).
%------------------------------------------------------------------------------