TPTP Problem File: ALG226+1.p
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%------------------------------------------------------------------------------
% File : ALG226+1 : TPTP v8.2.0. Released v3.4.0.
% Domain : General Algebra
% Problem : Algebraic Operation on Subsets of Many Sorted Sets T07
% 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 : t7_closure3 [Urb08]
% Status : Theorem
% Rating : 0.61 v8.2.0, 0.72 v8.1.0, 0.67 v7.5.0, 0.72 v7.4.0, 0.57 v7.3.0, 0.59 v7.1.0, 0.57 v7.0.0, 0.60 v6.4.0, 0.65 v6.3.0, 0.62 v6.2.0, 0.80 v6.1.0, 0.87 v6.0.0, 0.83 v5.5.0, 0.85 v5.4.0, 0.86 v5.3.0, 0.89 v5.2.0, 0.85 v5.1.0, 0.90 v5.0.0, 0.83 v4.1.0, 0.87 v4.0.1, 0.91 v4.0.0, 0.96 v3.7.0, 0.90 v3.5.0, 0.95 v3.4.0
% Syntax : Number of formulae : 57 ( 14 unt; 0 def)
% Number of atoms : 167 ( 22 equ)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 129 ( 19 ~; 1 |; 56 &)
% ( 9 <=>; 44 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 15 ( 13 usr; 1 prp; 0-3 aty)
% Number of functors : 8 ( 8 usr; 1 con; 0-2 aty)
% Number of variables : 100 ( 84 !; 16 ?)
% 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(t7_closure3,conjecture,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v2_relat_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,A) )
=> ( ( k1_closure3(A,B) = k1_closure3(A,C)
& k7_relat_1(B,k1_closure3(A,B)) = k7_relat_1(C,k1_closure3(A,C)) )
=> r6_pboole(A,B,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_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_pboole,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,A)
=> ( v3_relat_1(B)
=> ~ v2_relat_1(B) ) ) ) ).
fof(commutativity_k3_xboole_0,axiom,
! [A,B] : k3_xboole_0(A,B) = k3_xboole_0(B,A) ).
fof(d16_pboole,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ( v2_relat_1(B)
<=> ! [C] :
~ ( r2_hidden(C,A)
& v1_xboole_0(k1_funct_1(B,C)) ) ) ) ).
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_xboole_0,axiom,
! [A,B,C] :
( C = k3_xboole_0(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ( r2_hidden(D,A)
& r2_hidden(D,B) ) ) ) ).
fof(dt_k1_closure3,axiom,
$true ).
fof(dt_k1_funct_1,axiom,
$true ).
fof(dt_k1_relat_1,axiom,
$true ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k3_xboole_0,axiom,
$true ).
fof(dt_k7_relat_1,axiom,
! [A,B] :
( v1_relat_1(A)
=> v1_relat_1(k7_relat_1(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(existence_m1_pboole,axiom,
! [A] :
? [B] : m1_pboole(B,A) ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : m1_subset_1(B,A) ).
fof(fc10_finset_1,axiom,
! [A,B] :
( v1_finset_1(B)
=> v1_finset_1(k3_xboole_0(A,B)) ) ).
fof(fc11_finset_1,axiom,
! [A,B] :
( v1_finset_1(A)
=> v1_finset_1(k3_xboole_0(A,B)) ) ).
fof(fc1_xboole_0,axiom,
v1_xboole_0(k1_xboole_0) ).
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(fc4_funct_1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v1_relat_1(k7_relat_1(A,B))
& v1_funct_1(k7_relat_1(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_2_1_closure3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& v2_relat_1(C)
& m1_pboole(C,B) )
=> ( r2_hidden(A,a_2_1_closure3(B,C))
<=> ? [D] :
( m1_subset_1(D,B)
& A = D
& k1_funct_1(C,D) != k1_xboole_0 ) ) ) ).
fof(idempotence_k3_xboole_0,axiom,
! [A,B] : k3_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_pboole,axiom,
? [A] :
( v1_relat_1(A)
& v3_relat_1(A)
& v1_funct_1(A) ) ).
fof(rc1_xboole_0,axiom,
? [A] : v1_xboole_0(A) ).
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_xboole_0,axiom,
? [A] : ~ v1_xboole_0(A) ).
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_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(redefinition_r6_pboole,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& m1_pboole(C,A) )
=> ( r6_pboole(A,B,C)
<=> B = C ) ) ).
fof(reflexivity_r6_pboole,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& m1_pboole(C,A) )
=> r6_pboole(A,B,B) ) ).
fof(symmetry_r6_pboole,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& m1_pboole(C,A) )
=> ( r6_pboole(A,B,C)
=> r6_pboole(A,C,B) ) ) ).
fof(t1_subset,axiom,
! [A,B] :
( r2_hidden(A,B)
=> m1_subset_1(A,B) ) ).
fof(t2_boole,axiom,
! [A] : k3_xboole_0(A,k1_xboole_0) = k1_xboole_0 ).
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(t6_boole,axiom,
! [A] :
( v1_xboole_0(A)
=> A = k1_xboole_0 ) ).
fof(t70_funct_1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r2_hidden(B,k1_relat_1(k7_relat_1(C,A)))
=> k1_funct_1(k7_relat_1(C,A),B) = k1_funct_1(C,B) ) ) ).
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) ) ).
fof(t90_relat_1,axiom,
! [A,B] :
( v1_relat_1(B)
=> k1_relat_1(k7_relat_1(B,A)) = k3_xboole_0(k1_relat_1(B),A) ) ).
fof(t9_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( ( k1_relat_1(A) = k1_relat_1(B)
& ! [C] :
( r2_hidden(C,k1_relat_1(A))
=> k1_funct_1(A,C) = k1_funct_1(B,C) ) )
=> A = B ) ) ) ).
%------------------------------------------------------------------------------