TPTP Problem File: SEU416+1.p
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% File : SEU416+1 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Problem : First and Second Order Cutting of Binary Relations T05
% 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 : t5_relset_2 [Urb08]
% Status : Theorem
% Rating : 0.94 v8.1.0, 0.92 v7.5.0, 0.97 v7.4.0, 1.00 v7.3.0, 0.97 v7.1.0, 0.96 v7.0.0, 0.97 v6.4.0, 0.92 v6.3.0, 0.96 v6.2.0, 1.00 v4.1.0, 0.96 v4.0.0, 0.92 v3.7.0, 0.85 v3.5.0, 0.95 v3.4.0
% Syntax : Number of formulae : 38 ( 14 unt; 0 def)
% Number of atoms : 74 ( 9 equ)
% Maximal formula atoms : 4 ( 1 avg)
% Number of connectives : 47 ( 11 ~; 1 |; 18 &)
% ( 6 <=>; 11 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 4 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 9 ( 7 usr; 1 prp; 0-2 aty)
% Number of functors : 5 ( 5 usr; 1 con; 0-2 aty)
% Number of variables : 58 ( 50 !; 8 ?)
% 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.
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fof(t5_relset_2,conjecture,
! [A,B] : k3_pua2mss1(k4_xboole_0(A,B)) = k4_xboole_0(k3_pua2mss1(A),k3_pua2mss1(B)) ).
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(d10_xboole_0,axiom,
! [A,B] :
( A = B
<=> ( r1_tarski(A,B)
& r1_tarski(B,A) ) ) ).
fof(d3_tarski,axiom,
! [A,B] :
( r1_tarski(A,B)
<=> ! [C] :
( r2_hidden(C,A)
=> r2_hidden(C,B) ) ) ).
fof(d4_xboole_0,axiom,
! [A,B,C] :
( C = k4_xboole_0(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ( r2_hidden(D,A)
& ~ r2_hidden(D,B) ) ) ) ).
fof(dt_k1_tarski,axiom,
$true ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k3_pua2mss1,axiom,
! [A] : m1_eqrel_1(k3_pua2mss1(A),A) ).
fof(dt_k4_xboole_0,axiom,
$true ).
fof(dt_m1_eqrel_1,axiom,
! [A,B] :
( m1_eqrel_1(B,A)
=> m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) ).
fof(dt_m1_subset_1,axiom,
$true ).
fof(existence_m1_eqrel_1,axiom,
! [A] :
? [B] : m1_eqrel_1(B,A) ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : m1_subset_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(fc2_subset_1,axiom,
! [A] : ~ v1_xboole_0(k1_tarski(A)) ).
fof(fc3_relat_1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_relat_1(B) )
=> v1_relat_1(k4_xboole_0(A,B)) ) ).
fof(fc4_relat_1,axiom,
( v1_xboole_0(k1_xboole_0)
& v1_relat_1(k1_xboole_0) ) ).
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_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(reflexivity_r1_tarski,axiom,
! [A,B] : r1_tarski(A,A) ).
fof(t1_relset_2,axiom,
! [A,B] :
( r2_hidden(A,k3_pua2mss1(B))
<=> ? [C] :
( A = k1_tarski(C)
& r2_hidden(C,B) ) ) ).
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(t3_boole,axiom,
! [A] : k4_xboole_0(A,k1_xboole_0) = A ).
fof(t3_subset,axiom,
! [A,B] :
( m1_subset_1(A,k1_zfmisc_1(B))
<=> r1_tarski(A,B) ) ).
fof(t4_boole,axiom,
! [A] : k4_xboole_0(k1_xboole_0,A) = k1_xboole_0 ).
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(t6_zfmisc_1,axiom,
! [A,B] :
( r1_tarski(k1_tarski(A),k1_tarski(B))
=> A = 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) ) ).
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