TPTP Problem File: SEU425+1.p
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%------------------------------------------------------------------------------
% File : SEU425+1 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Problem : First and Second Order Cutting of Binary Relations T21
% Version : [Urb08] axioms : Especial.
% English :
% Refs : [Ret05] Retel (2005), Properties of First and Second Order Cut
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : t21_relset_2 [Urb08]
% Status : Theorem
% Rating : 0.42 v8.1.0, 0.39 v7.5.0, 0.44 v7.4.0, 0.30 v7.3.0, 0.41 v7.1.0, 0.39 v7.0.0, 0.43 v6.4.0, 0.46 v6.2.0, 0.52 v6.1.0, 0.63 v6.0.0, 0.61 v5.5.0, 0.70 v5.4.0, 0.71 v5.3.0, 0.74 v5.2.0, 0.65 v5.1.0, 0.62 v5.0.0, 0.67 v4.1.0, 0.65 v4.0.0, 0.67 v3.7.0, 0.65 v3.5.0, 0.63 v3.4.0
% Syntax : Number of formulae : 49 ( 15 unt; 0 def)
% Number of atoms : 112 ( 12 equ)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 79 ( 16 ~; 2 |; 28 &)
% ( 5 <=>; 28 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 4 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 12 ( 10 usr; 1 prp; 0-3 aty)
% Number of functors : 9 ( 9 usr; 1 con; 0-3 aty)
% Number of variables : 82 ( 73 !; 9 ?)
% 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(t21_relset_2,conjecture,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> ( v1_funct_1(k5_relset_2(A,C))
& v1_funct_2(k5_relset_2(A,C),k1_zfmisc_1(A),k1_zfmisc_1(k2_relat_1(C)))
& m2_relset_1(k5_relset_2(A,C),k1_zfmisc_1(A),k1_zfmisc_1(k2_relat_1(C))) ) ) ).
fof(antisymmetry_r2_hidden,axiom,
! [A,B] :
( r2_hidden(A,B)
=> ~ r2_hidden(B,A) ) ).
fof(cc1_relat_1,axiom,
! [A] :
( v1_xboole_0(A)
=> v1_relat_1(A) ) ).
fof(cc1_relset_1,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> v1_relat_1(C) ) ).
fof(d1_funct_2,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
=> ( ( ( B = k1_xboole_0
=> A = k1_xboole_0 )
=> ( v1_funct_2(C,A,B)
<=> A = k4_relset_1(A,B,C) ) )
& ( B = k1_xboole_0
=> ( A = k1_xboole_0
| ( v1_funct_2(C,A,B)
<=> C = k1_xboole_0 ) ) ) ) ) ).
fof(d3_relset_2,axiom,
! [A,B] :
( v1_relat_1(B)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( C = k5_relset_2(A,B)
<=> ( k1_relat_1(C) = k1_zfmisc_1(A)
& ! [D] :
( r1_tarski(D,A)
=> k1_funct_1(C,D) = k9_relat_1(B,D) ) ) ) ) ) ).
fof(dt_k1_funct_1,axiom,
$true ).
fof(dt_k1_relat_1,axiom,
$true ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k2_relat_1,axiom,
$true ).
fof(dt_k2_zfmisc_1,axiom,
$true ).
fof(dt_k4_relset_1,axiom,
! [A,B,C] :
( m1_relset_1(C,A,B)
=> m1_subset_1(k4_relset_1(A,B,C),k1_zfmisc_1(A)) ) ).
fof(dt_k5_relset_2,axiom,
! [A,B] :
( v1_relat_1(B)
=> ( v1_relat_1(k5_relset_2(A,B))
& v1_funct_1(k5_relset_2(A,B)) ) ) ).
fof(dt_k9_relat_1,axiom,
$true ).
fof(dt_m1_relset_1,axiom,
$true ).
fof(dt_m1_subset_1,axiom,
$true ).
fof(dt_m2_relset_1,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
=> m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ).
fof(existence_m1_relset_1,axiom,
! [A,B] :
? [C] : m1_relset_1(C,A,B) ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : m1_subset_1(B,A) ).
fof(existence_m2_relset_1,axiom,
! [A,B] :
? [C] : m2_relset_1(C,A,B) ).
fof(fc12_relat_1,axiom,
( v1_xboole_0(k1_xboole_0)
& v1_relat_1(k1_xboole_0)
& v3_relat_1(k1_xboole_0) ) ).
fof(fc1_subset_1,axiom,
! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ).
fof(fc1_sysrel,axiom,
! [A,B] : v1_relat_1(k2_zfmisc_1(A,B)) ).
fof(fc4_relat_1,axiom,
( v1_xboole_0(k1_xboole_0)
& v1_relat_1(k1_xboole_0) ) ).
fof(fc4_subset_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B) )
=> ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ).
fof(fc5_relat_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_relat_1(A) )
=> ~ v1_xboole_0(k1_relat_1(A)) ) ).
fof(fc6_relat_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_relat_1(A) )
=> ~ v1_xboole_0(k2_relat_1(A)) ) ).
fof(fc7_relat_1,axiom,
! [A] :
( v1_xboole_0(A)
=> ( v1_xboole_0(k1_relat_1(A))
& v1_relat_1(k1_relat_1(A)) ) ) ).
fof(fc8_relat_1,axiom,
! [A] :
( v1_xboole_0(A)
=> ( v1_xboole_0(k2_relat_1(A))
& v1_relat_1(k2_relat_1(A)) ) ) ).
fof(rc1_relat_1,axiom,
? [A] :
( v1_xboole_0(A)
& v1_relat_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(rc2_partfun1,axiom,
! [A,B] :
? [C] :
( m1_relset_1(C,A,B)
& v1_relat_1(C)
& v1_funct_1(C) ) ).
fof(rc2_relat_1,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_relat_1(A) ) ).
fof(rc2_subset_1,axiom,
! [A] :
? [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
& v1_xboole_0(B) ) ).
fof(rc3_relat_1,axiom,
? [A] :
( v1_relat_1(A)
& v3_relat_1(A) ) ).
fof(redefinition_k4_relset_1,axiom,
! [A,B,C] :
( m1_relset_1(C,A,B)
=> k4_relset_1(A,B,C) = k1_relat_1(C) ) ).
fof(redefinition_m2_relset_1,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
<=> m1_relset_1(C,A,B) ) ).
fof(reflexivity_r1_tarski,axiom,
! [A,B] : r1_tarski(A,A) ).
fof(t11_relset_1,axiom,
! [A,B,C] :
( v1_relat_1(C)
=> ( ( r1_tarski(k1_relat_1(C),A)
& r1_tarski(k2_relat_1(C),B) )
=> m2_relset_1(C,A,B) ) ) ).
fof(t1_subset,axiom,
! [A,B] :
( r2_hidden(A,B)
=> m1_subset_1(A,B) ) ).
fof(t20_relset_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> r1_tarski(k2_relat_1(k5_relset_2(A,C)),k1_zfmisc_1(k2_relat_1(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) ) ).
%------------------------------------------------------------------------------