TPTP Problem File: SEU451+1.p
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%------------------------------------------------------------------------------
% File : SEU451+1 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Problem : First and Second Order Cutting of Binary Relations T65
% 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 : t65_relset_2 [Urb08]
% Status : Theorem
% Rating : 0.33 v9.0.0, 0.39 v8.2.0, 0.36 v8.1.0, 0.39 v7.5.0, 0.44 v7.4.0, 0.27 v7.3.0, 0.38 v7.1.0, 0.26 v7.0.0, 0.27 v6.4.0, 0.31 v6.3.0, 0.33 v6.2.0, 0.48 v6.1.0, 0.53 v6.0.0, 0.43 v5.5.0, 0.56 v5.4.0, 0.57 v5.3.0, 0.63 v5.2.0, 0.50 v5.1.0, 0.57 v5.0.0, 0.58 v4.1.0, 0.57 v4.0.1, 0.52 v4.0.0, 0.58 v3.7.0, 0.55 v3.5.0, 0.58 v3.4.0
% Syntax : Number of formulae : 48 ( 13 unt; 0 def)
% Number of atoms : 113 ( 8 equ)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 77 ( 12 ~; 1 |; 42 &)
% ( 2 <=>; 20 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 4 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 16 ( 14 usr; 1 prp; 0-3 aty)
% Number of functors : 8 ( 8 usr; 1 con; 0-6 aty)
% Number of variables : 95 ( 85 !; 10 ?)
% 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(t65_relset_2,conjecture,
! [A,B,C] :
( m2_relset_1(C,A,B)
=> k9_relset_2(A,A,B,k6_partfun1(A),C) = k9_relset_2(A,B,B,C,k6_partfun1(B)) ) ).
fof(antisymmetry_r2_hidden,axiom,
! [A,B] :
( r2_hidden(A,B)
=> ~ r2_hidden(B,A) ) ).
fof(cc1_partfun1,axiom,
! [A] :
( ( v1_relat_1(A)
& v3_relat_2(A)
& v8_relat_2(A) )
=> ( v1_relat_1(A)
& v1_relat_2(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(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k2_zfmisc_1,axiom,
$true ).
fof(dt_k5_relat_1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_relat_1(B) )
=> v1_relat_1(k5_relat_1(A,B)) ) ).
fof(dt_k6_partfun1,axiom,
! [A] :
( v1_partfun1(k6_partfun1(A),A,A)
& m2_relset_1(k6_partfun1(A),A,A) ) ).
fof(dt_k6_relat_1,axiom,
! [A] : v1_relat_1(k6_relat_1(A)) ).
fof(dt_k7_relset_1,axiom,
! [A,B,C,D,E,F] :
( ( m1_relset_1(E,A,B)
& m1_relset_1(F,C,D) )
=> m2_relset_1(k7_relset_1(A,B,C,D,E,F),A,D) ) ).
fof(dt_k9_relset_2,axiom,
! [A,B,C,D,E] :
( ( m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(A,B)))
& m1_subset_1(E,k1_zfmisc_1(k2_zfmisc_1(B,C))) )
=> m2_relset_1(k9_relset_2(A,B,C,D,E),A,C) ) ).
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(fc10_relat_1,axiom,
! [A,B] :
( ( v1_xboole_0(A)
& v1_relat_1(B) )
=> ( v1_xboole_0(k5_relat_1(B,A))
& v1_relat_1(k5_relat_1(B,A)) ) ) ).
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(fc2_partfun1,axiom,
! [A] :
( v1_relat_1(k6_relat_1(A))
& v1_funct_1(k6_relat_1(A))
& v1_relat_2(k6_relat_1(A))
& v3_relat_2(k6_relat_1(A))
& v4_relat_2(k6_relat_1(A))
& v8_relat_2(k6_relat_1(A)) ) ).
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(fc9_relat_1,axiom,
! [A,B] :
( ( v1_xboole_0(A)
& v1_relat_1(B) )
=> ( v1_xboole_0(k5_relat_1(A,B))
& v1_relat_1(k5_relat_1(A,B)) ) ) ).
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_partfun1,axiom,
! [A] :
? [B] :
( m1_relset_1(B,A,A)
& v1_relat_1(B)
& v1_relat_2(B)
& v3_relat_2(B)
& v4_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A) ) ).
fof(rc3_relat_1,axiom,
? [A] :
( v1_relat_1(A)
& v3_relat_1(A) ) ).
fof(redefinition_k6_partfun1,axiom,
! [A] : k6_partfun1(A) = k6_relat_1(A) ).
fof(redefinition_k7_relset_1,axiom,
! [A,B,C,D,E,F] :
( ( m1_relset_1(E,A,B)
& m1_relset_1(F,C,D) )
=> k7_relset_1(A,B,C,D,E,F) = k5_relat_1(E,F) ) ).
fof(redefinition_k9_relset_2,axiom,
! [A,B,C,D,E] :
( ( m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(A,B)))
& m1_subset_1(E,k1_zfmisc_1(k2_zfmisc_1(B,C))) )
=> k9_relset_2(A,B,C,D,E) = k5_relat_1(D,E) ) ).
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(t1_subset,axiom,
! [A,B] :
( r2_hidden(A,B)
=> m1_subset_1(A,B) ) ).
fof(t23_funct_2,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
=> ( k7_relset_1(A,A,A,B,k6_partfun1(A),C) = C
& k7_relset_1(A,B,B,B,C,k6_partfun1(B)) = 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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