TPTP Problem File: ALG228+1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ALG228+1 : TPTP v8.2.0. Released v3.4.0.
% Domain : General Algebra
% Problem : Algebraic Operation on Subsets of Many Sorted Sets T09
% 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 : t9_closure3 [Urb08]
% Status : Theorem
% Rating : 1.00 v3.7.0, 0.95 v3.5.0, 1.00 v3.4.0
% Syntax : Number of formulae : 109 ( 27 unt; 0 def)
% Number of atoms : 353 ( 31 equ)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 288 ( 44 ~; 1 |; 128 &)
% ( 17 <=>; 98 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 26 ( 24 usr; 1 prp; 0-4 aty)
% Number of functors : 16 ( 16 usr; 1 con; 0-3 aty)
% Number of variables : 216 ( 189 !; 27 ?)
% 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(t9_closure3,conjecture,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_closure2(C,A,B,k6_closure2(A,B))
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
~ ( r2_hidden(E,k1_funct_1(C,D))
& ! [F] :
( m1_closure2(F,A,B,k6_closure2(A,B))
=> ~ ( r2_hidden(E,k1_funct_1(F,D))
& v1_pre_circ(F,A)
& v1_finset_1(k1_closure3(A,F))
& r2_pboole(A,F,C) ) ) ) ) ) ) ) ).
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(cc1_mssubfam,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ( v3_relat_1(B)
=> v1_pre_circ(B,A) ) ) ).
fof(cc1_pboole,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,A)
=> ( v2_relat_1(B)
=> ~ v3_relat_1(B) ) ) ) ).
fof(cc2_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ( v2_closure2(C,A,B)
=> ( v1_fraenkel(C)
& v1_closure2(C,A,B) ) ) ) ) ).
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(cc2_mssubfam,axiom,
! [A,B] :
( ( v1_pre_circ(B,A)
& m1_pboole(B,A) )
=> ! [C] :
( m4_pboole(C,A,B)
=> v1_pre_circ(C,A) ) ) ).
fof(cc2_pboole,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,A)
=> ( v3_relat_1(B)
=> ~ v2_relat_1(B) ) ) ) ).
fof(cc3_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ( v4_closure2(C,A,B)
=> ( v1_fraenkel(C)
& v3_closure2(C,A,B) ) ) ) ) ).
fof(cc4_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ( v4_closure2(C,A,B)
=> ( v1_fraenkel(C)
& v5_closure2(C,A,B) ) ) ) ) ).
fof(cc5_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ( v5_closure2(C,A,B)
=> ( ~ v1_xboole_0(C)
& v1_fraenkel(C) ) ) ) ) ).
fof(cc6_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ( v2_closure2(C,A,B)
=> ( v1_fraenkel(C)
& v6_closure2(C,A,B) ) ) ) ) ).
fof(cc7_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ( v6_closure2(C,A,B)
=> ( ~ v1_xboole_0(C)
& v1_fraenkel(C) ) ) ) ) ).
fof(commutativity_k2_xboole_0,axiom,
! [A,B] : k2_xboole_0(A,B) = k2_xboole_0(B,A) ).
fof(d10_xboole_0,axiom,
! [A,B] :
( A = B
<=> ( r1_tarski(A,B)
& r1_tarski(B,A) ) ) ).
fof(d1_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( C = k1_closure2(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> m4_pboole(D,A,B) ) ) ) ).
fof(d1_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( C = k1_funct_4(A,B)
<=> ( k1_relat_1(C) = k2_xboole_0(k1_relat_1(A),k1_relat_1(B))
& ! [D] :
( r2_hidden(D,k2_xboole_0(k1_relat_1(A),k1_relat_1(B)))
=> ( ( r2_hidden(D,k1_relat_1(B))
=> k1_funct_1(C,D) = k1_funct_1(B,D) )
& ( ~ r2_hidden(D,k1_relat_1(B))
=> k1_funct_1(C,D) = k1_funct_1(A,D) ) ) ) ) ) ) ) ) ).
fof(d1_tarski,axiom,
! [A,B] :
( B = k1_tarski(A)
<=> ! [C] :
( r2_hidden(C,B)
<=> C = A ) ) ).
fof(d23_pboole,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ( m4_pboole(C,A,B)
<=> r2_pboole(A,C,B) ) ) ) ).
fof(d3_closure3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,A)
=> ! [C] :
( C = k1_closure3(A,B)
<=> C = a_2_0_closure3(A,B) ) ) ) ).
fof(d3_pboole,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( m1_pboole(B,A)
<=> k1_relat_1(B) = A ) ) ).
fof(d3_pre_circ,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ( v1_pre_circ(B,A)
<=> ! [C] :
( r2_hidden(C,A)
=> v1_finset_1(k1_funct_1(B,C)) ) ) ) ).
fof(d3_tarski,axiom,
! [A,B] :
( r1_tarski(A,B)
<=> ! [C] :
( r2_hidden(C,A)
=> r2_hidden(C,B) ) ) ).
fof(d5_pboole,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ( r2_pboole(A,B,C)
<=> ! [D] :
( r2_hidden(D,A)
=> r1_tarski(k1_funct_1(B,D),k1_funct_1(C,D)) ) ) ) ) ).
fof(d6_pboole,axiom,
! [A] : k1_pboole(A) = k2_funcop_1(A,k1_xboole_0) ).
fof(dt_k1_closure2,axiom,
$true ).
fof(dt_k1_closure3,axiom,
$true ).
fof(dt_k1_funct_1,axiom,
$true ).
fof(dt_k1_funct_4,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_relat_1(B)
& v1_funct_1(B) )
=> ( v1_relat_1(k1_funct_4(A,B))
& v1_funct_1(k1_funct_4(A,B)) ) ) ).
fof(dt_k1_pboole,axiom,
! [A] : m1_pboole(k1_pboole(A),A) ).
fof(dt_k1_relat_1,axiom,
$true ).
fof(dt_k1_tarski,axiom,
$true ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k2_funcop_1,axiom,
$true ).
fof(dt_k2_pre_circ,axiom,
! [A,B] : m1_pboole(k2_pre_circ(A,B),A) ).
fof(dt_k2_relat_1,axiom,
$true ).
fof(dt_k2_xboole_0,axiom,
$true ).
fof(dt_k6_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ( v1_closure2(k6_closure2(A,B),A,B)
& v2_closure2(k6_closure2(A,B),A,B)
& v3_closure2(k6_closure2(A,B),A,B)
& v4_closure2(k6_closure2(A,B),A,B)
& v5_closure2(k6_closure2(A,B),A,B)
& v6_closure2(k6_closure2(A,B),A,B)
& m1_subset_1(k6_closure2(A,B),k1_zfmisc_1(k1_closure2(A,B))) ) ) ).
fof(dt_m1_closure2,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
=> ! [D] :
( m1_closure2(D,A,B,C)
=> m4_pboole(D,A,B) ) ) ).
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(dt_m4_pboole,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m4_pboole(C,A,B)
=> m1_pboole(C,A) ) ) ).
fof(existence_m1_closure2,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
=> ? [D] : m1_closure2(D,A,B,C) ) ).
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(existence_m4_pboole,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ? [C] : m4_pboole(C,A,B) ) ).
fof(fc1_finset_1,axiom,
! [A] :
( ~ v1_xboole_0(k1_tarski(A))
& v1_finset_1(k1_tarski(A)) ) ).
fof(fc1_mssubfam,axiom,
! [A] :
( v1_relat_1(k1_pboole(A))
& v3_relat_1(k1_pboole(A))
& v1_funct_1(k1_pboole(A))
& v1_pre_circ(k1_pboole(A),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(fc2_pboole,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v2_relat_1(B)
& m1_pboole(B,A)
& m1_subset_1(C,A) )
=> ~ v1_xboole_0(k1_funct_1(B,C)) ) ).
fof(fc2_subset_1,axiom,
! [A] : ~ v1_xboole_0(k1_tarski(A)) ).
fof(fc2_xboole_0,axiom,
! [A,B] :
( ~ v1_xboole_0(A)
=> ~ v1_xboole_0(k2_xboole_0(A,B)) ) ).
fof(fc3_pboole,axiom,
! [A] :
( v1_relat_1(k1_pboole(A))
& v3_relat_1(k1_pboole(A))
& v1_funct_1(k1_pboole(A)) ) ).
fof(fc3_xboole_0,axiom,
! [A,B] :
( ~ v1_xboole_0(A)
=> ~ v1_xboole_0(k2_xboole_0(B,A)) ) ).
fof(fc6_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v2_relat_1(A)
& v1_funct_1(A) )
=> v1_setfam_1(k2_relat_1(A)) ) ).
fof(fc9_finset_1,axiom,
! [A,B] :
( ( v1_finset_1(A)
& v1_finset_1(B) )
=> v1_finset_1(k2_xboole_0(A,B)) ) ).
fof(fraenkel_a_2_0_closure3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& m1_pboole(C,B) )
=> ( r2_hidden(A,a_2_0_closure3(B,C))
<=> ? [D] :
( m1_subset_1(D,B)
& A = D
& k1_funct_1(C,D) != k1_xboole_0 ) ) ) ).
fof(fraenkel_a_3_1_closure3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& m1_pboole(C,B)
& m1_closure2(D,B,C,k6_closure2(B,C)) )
=> ( r2_hidden(A,a_3_1_closure3(B,C,D))
<=> ? [E] :
( m1_subset_1(E,B)
& A = E
& k1_funct_1(D,E) != k1_xboole_0 ) ) ) ).
fof(idempotence_k1_funct_4,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_relat_1(B)
& v1_funct_1(B) )
=> k1_funct_4(A,A) = A ) ).
fof(idempotence_k2_xboole_0,axiom,
! [A,B] : k2_xboole_0(A,A) = A ).
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_mssubfam,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ? [C] :
( m4_pboole(C,A,B)
& v1_relat_1(C)
& v3_relat_1(C)
& v1_funct_1(C)
& v1_pre_circ(C,A) ) ) ).
fof(rc1_pboole,axiom,
? [A] :
( v1_relat_1(A)
& v3_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_pboole,axiom,
! [A] :
? [B] :
( m1_pboole(B,A)
& v1_relat_1(B)
& v3_relat_1(B)
& v1_funct_1(B) ) ).
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(rc3_pboole,axiom,
! [A] :
? [B] :
( m1_pboole(B,A)
& v1_relat_1(B)
& v2_relat_1(B)
& v1_funct_1(B) ) ).
fof(rc4_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)
& v1_closure2(C,A,B)
& v2_closure2(C,A,B)
& v3_closure2(C,A,B)
& v4_closure2(C,A,B)
& v5_closure2(C,A,B)
& v6_closure2(C,A,B) ) ) ).
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(rc4_funct_1,axiom,
? [A] :
( v1_relat_1(A)
& v3_relat_1(A)
& v1_funct_1(A) ) ).
fof(rc5_funct_1,axiom,
? [A] :
( v1_relat_1(A)
& v2_relat_1(A)
& v1_funct_1(A) ) ).
fof(rc5_pboole,axiom,
! [A,B] :
( ( v2_relat_1(B)
& m1_pboole(B,A) )
=> ? [C] :
( m4_pboole(C,A,B)
& v1_relat_1(C)
& v2_relat_1(C)
& v1_funct_1(C) ) ) ).
fof(redefinition_k2_pre_circ,axiom,
! [A,B] : k2_pre_circ(A,B) = k2_funcop_1(A,B) ).
fof(redefinition_k6_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> k6_closure2(A,B) = k1_closure2(A,B) ) ).
fof(redefinition_m1_closure2,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
=> ! [D] :
( m1_closure2(D,A,B,C)
<=> m1_subset_1(D,C) ) ) ).
fof(reflexivity_r1_tarski,axiom,
! [A,B] : r1_tarski(A,A) ).
fof(reflexivity_r2_pboole,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& m1_pboole(C,A) )
=> r2_pboole(A,B,B) ) ).
fof(t13_funcop_1,axiom,
! [A,B,C] :
( r2_hidden(B,A)
=> k1_funct_1(k2_funcop_1(A,C),B) = C ) ).
fof(t14_funcop_1,axiom,
! [A,B] :
( A != k1_xboole_0
=> k2_relat_1(k2_funcop_1(A,B)) = k1_tarski(B) ) ).
fof(t14_funct_4,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r2_hidden(A,k1_relat_1(B))
=> k1_funct_1(k1_funct_4(C,B),A) = k1_funct_1(B,A) ) ) ) ).
fof(t19_funcop_1,axiom,
! [A,B] :
( k1_relat_1(k2_funcop_1(A,B)) = A
& r1_tarski(k2_relat_1(k2_funcop_1(A,B)),k1_tarski(B)) ) ).
fof(t1_boole,axiom,
! [A] : k2_xboole_0(A,k1_xboole_0) = A ).
fof(t1_subset,axiom,
! [A,B] :
( r2_hidden(A,B)
=> m1_subset_1(A,B) ) ).
fof(t2_subset,axiom,
! [A,B] :
( m1_subset_1(A,B)
=> ( v1_xboole_0(B)
| r2_hidden(A,B) ) ) ).
fof(t2_tarski,axiom,
! [A,B] :
( ! [C] :
( r2_hidden(C,A)
<=> r2_hidden(C,B) )
=> A = B ) ).
fof(t2_xboole_1,axiom,
! [A] : r1_tarski(k1_xboole_0,A) ).
fof(t3_subset,axiom,
! [A,B] :
( m1_subset_1(A,k1_zfmisc_1(B))
<=> r1_tarski(A,B) ) ).
fof(t46_zfmisc_1,axiom,
! [A,B] :
( r2_hidden(A,B)
=> k2_xboole_0(k1_tarski(A),B) = 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) ) ).
%------------------------------------------------------------------------------