TPTP Problem File: ALG224+1.p
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%------------------------------------------------------------------------------
% File : ALG224+1 : TPTP v8.2.0. Released v3.4.0.
% Domain : General Algebra
% Problem : Algebraic Operation on Subsets of Many Sorted Sets T05
% Version : [Urb08] axioms : Especial.
% English :
% Refs : [Mar97] Marasik (1997), Algebraic Operation on Subsets of Many
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : t5_closure3 [Urb08]
% Status : Theorem
% Rating : 0.50 v8.2.0, 0.64 v8.1.0, 0.56 v7.5.0, 0.62 v7.4.0, 0.53 v7.3.0, 0.62 v7.2.0, 0.59 v7.1.0, 0.43 v7.0.0, 0.47 v6.4.0, 0.50 v6.3.0, 0.54 v6.2.0, 0.56 v6.1.0, 0.70 v6.0.0, 0.74 v5.5.0, 0.78 v5.4.0, 0.79 v5.3.0, 0.85 v5.2.0, 0.75 v5.1.0, 0.76 v5.0.0, 0.88 v4.1.0, 0.78 v4.0.0, 0.79 v3.7.0, 0.75 v3.5.0, 0.79 v3.4.0
% Syntax : Number of formulae : 42 ( 11 unt; 0 def)
% Number of atoms : 118 ( 2 equ)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 94 ( 18 ~; 1 |; 45 &)
% ( 2 <=>; 28 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 14 ( 12 usr; 1 prp; 0-4 aty)
% Number of functors : 3 ( 3 usr; 1 con; 0-2 aty)
% Number of variables : 75 ( 60 !; 15 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Normal version: includes the axioms (which may be theorems from
% other articles) and background that are possibly necessary.
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : The problem encoding is based on set theory.
%------------------------------------------------------------------------------
fof(t5_closure3,conjecture,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k1_closure2(A,B)))
=> ( ( r2_closure3(A,B,C,D)
& r2_closure3(A,B,D,E) )
=> r2_closure3(A,B,C,E) ) ) ) ) ) ).
fof(antisymmetry_r2_hidden,axiom,
! [A,B] :
( r2_hidden(A,B)
=> ~ r2_hidden(B,A) ) ).
fof(cc1_closure2,axiom,
! [A] :
( v1_xboole_0(A)
=> v1_fraenkel(A) ) ).
fof(cc1_finset_1,axiom,
! [A] :
( v1_xboole_0(A)
=> v1_finset_1(A) ) ).
fof(cc1_funct_1,axiom,
! [A] :
( v1_xboole_0(A)
=> v1_funct_1(A) ) ).
fof(cc2_finset_1,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> v1_finset_1(B) ) ) ).
fof(cc2_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_xboole_0(A)
& v1_funct_1(A) )
=> ( v1_relat_1(A)
& v1_funct_1(A)
& v2_funct_1(A) ) ) ).
fof(d2_closure3,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
=> ( r2_closure3(A,B,C,D)
<=> ! [E] :
~ ( r2_hidden(E,C)
& ! [F] :
~ ( r2_hidden(F,D)
& r1_tarski(F,E) ) ) ) ) ) ) ).
fof(dt_k1_closure2,axiom,
$true ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_m1_pboole,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ( v1_relat_1(B)
& v1_funct_1(B) ) ) ).
fof(dt_m1_subset_1,axiom,
$true ).
fof(existence_m1_pboole,axiom,
! [A] :
? [B] : m1_pboole(B,A) ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : m1_subset_1(B,A) ).
fof(fc1_subset_1,axiom,
! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ).
fof(fc1_xboole_0,axiom,
v1_xboole_0(k1_xboole_0) ).
fof(fc2_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ( ~ v1_xboole_0(k1_closure2(A,B))
& v1_fraenkel(k1_closure2(A,B))
& v1_pralg_2(k1_closure2(A,B)) ) ) ).
fof(rc1_closure2,axiom,
? [A] :
( v1_xboole_0(A)
& v1_relat_1(A)
& v1_funct_1(A)
& v2_funct_1(A)
& v1_finset_1(A)
& v1_fraenkel(A) ) ).
fof(rc1_finset_1,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_finset_1(A) ) ).
fof(rc1_funct_1,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A) ) ).
fof(rc1_subset_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
& ~ v1_xboole_0(B) ) ) ).
fof(rc1_xboole_0,axiom,
? [A] : v1_xboole_0(A) ).
fof(rc2_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
& ~ v1_xboole_0(C)
& v1_fraenkel(C)
& v1_pralg_2(C) ) ) ).
fof(rc2_funct_1,axiom,
? [A] :
( v1_relat_1(A)
& v1_xboole_0(A)
& v1_funct_1(A) ) ).
fof(rc2_subset_1,axiom,
! [A] :
? [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
& v1_xboole_0(B) ) ).
fof(rc2_xboole_0,axiom,
? [A] : ~ v1_xboole_0(A) ).
fof(rc3_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
& v1_xboole_0(C)
& v1_relat_1(C)
& v1_funct_1(C)
& v2_funct_1(C)
& v1_finset_1(C)
& v1_fraenkel(C) ) ) ).
fof(rc3_finset_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
& ~ v1_xboole_0(B)
& v1_finset_1(B) ) ) ).
fof(rc3_funct_1,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v2_funct_1(A) ) ).
fof(rc4_finset_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
& ~ v1_xboole_0(B)
& v1_finset_1(B) ) ) ).
fof(reflexivity_r1_tarski,axiom,
! [A,B] : r1_tarski(A,A) ).
fof(reflexivity_r2_closure3,axiom,
! [A,B,C,D] :
( ( m1_pboole(B,A)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
& m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B))) )
=> r2_closure3(A,B,C,C) ) ).
fof(t1_subset,axiom,
! [A,B] :
( r2_hidden(A,B)
=> m1_subset_1(A,B) ) ).
fof(t1_xboole_1,axiom,
! [A,B,C] :
( ( r1_tarski(A,B)
& r1_tarski(B,C) )
=> r1_tarski(A,C) ) ).
fof(t2_subset,axiom,
! [A,B] :
( m1_subset_1(A,B)
=> ( v1_xboole_0(B)
| r2_hidden(A,B) ) ) ).
fof(t3_subset,axiom,
! [A,B] :
( m1_subset_1(A,k1_zfmisc_1(B))
<=> r1_tarski(A,B) ) ).
fof(t4_subset,axiom,
! [A,B,C] :
( ( r2_hidden(A,B)
& m1_subset_1(B,k1_zfmisc_1(C)) )
=> m1_subset_1(A,C) ) ).
fof(t5_subset,axiom,
! [A,B,C] :
~ ( r2_hidden(A,B)
& m1_subset_1(B,k1_zfmisc_1(C))
& v1_xboole_0(C) ) ).
fof(t6_boole,axiom,
! [A] :
( v1_xboole_0(A)
=> A = k1_xboole_0 ) ).
fof(t7_boole,axiom,
! [A,B] :
~ ( r2_hidden(A,B)
& v1_xboole_0(B) ) ).
fof(t8_boole,axiom,
! [A,B] :
~ ( v1_xboole_0(A)
& A != B
& v1_xboole_0(B) ) ).
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