TPTP Problem File: SEU443+1.p
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%------------------------------------------------------------------------------
% File : SEU443+1 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Problem : First and Second Order Cutting of Binary Relations T55
% 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 : t55_relset_2 [Urb08]
% Status : Theorem
% Rating : 1.00 v7.4.0, 0.97 v7.1.0, 0.96 v7.0.0, 0.97 v6.4.0, 1.00 v6.0.0, 0.96 v5.4.0, 1.00 v3.7.0, 0.95 v3.5.0, 1.00 v3.4.0
% Syntax : Number of formulae : 66 ( 21 unt; 0 def)
% Number of atoms : 146 ( 18 equ)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 94 ( 14 ~; 1 |; 26 &)
% ( 15 <=>; 38 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 4 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 10 ( 8 usr; 1 prp; 0-3 aty)
% Number of functors : 14 ( 14 usr; 1 con; 0-5 aty)
% Number of variables : 135 ( 124 !; 11 ?)
% 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(t55_relset_2,conjecture,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(B))
=> ! [D] :
( m2_relset_1(D,A,B)
=> ( r1_tarski(C,k2_funct_5(D))
<=> r1_tarski(C,k10_relset_1(B,B,k9_relset_2(B,A,B,k6_relset_1(A,B,D),D),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(commutativity_k2_tarski,axiom,
! [A,B] : k2_tarski(A,B) = k2_tarski(B,A) ).
fof(d13_relat_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( C = k9_relat_1(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ? [E] :
( r2_hidden(k4_tarski(E,D),A)
& r2_hidden(E,B) ) ) ) ) ).
fof(d1_tarski,axiom,
! [A,B] :
( B = k1_tarski(A)
<=> ! [C] :
( r2_hidden(C,B)
<=> C = A ) ) ).
fof(d2_funct_5,axiom,
! [A,B] :
( B = k2_funct_5(A)
<=> ! [C] :
( r2_hidden(C,B)
<=> ? [D] : r2_hidden(k4_tarski(D,C),A) ) ) ).
fof(d3_tarski,axiom,
! [A,B] :
( r1_tarski(A,B)
<=> ! [C] :
( r2_hidden(C,A)
=> r2_hidden(C,B) ) ) ).
fof(d5_tarski,axiom,
! [A,B] : k4_tarski(A,B) = k2_tarski(k2_tarski(A,B),k1_tarski(A)) ).
fof(d7_relat_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_relat_1(B)
=> ( B = k4_relat_1(A)
<=> ! [C,D] :
( r2_hidden(k4_tarski(C,D),B)
<=> r2_hidden(k4_tarski(D,C),A) ) ) ) ) ).
fof(d8_relat_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_relat_1(B)
=> ! [C] :
( v1_relat_1(C)
=> ( C = k5_relat_1(A,B)
<=> ! [D,E] :
( r2_hidden(k4_tarski(D,E),C)
<=> ? [F] :
( r2_hidden(k4_tarski(D,F),A)
& r2_hidden(k4_tarski(F,E),B) ) ) ) ) ) ) ).
fof(dt_k10_relset_1,axiom,
! [A,B,C,D] :
( m1_relset_1(C,A,B)
=> m1_subset_1(k10_relset_1(A,B,C,D),k1_zfmisc_1(B)) ) ).
fof(dt_k1_funct_5,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_funct_5,axiom,
$true ).
fof(dt_k2_tarski,axiom,
$true ).
fof(dt_k2_zfmisc_1,axiom,
$true ).
fof(dt_k4_relat_1,axiom,
! [A] :
( v1_relat_1(A)
=> v1_relat_1(k4_relat_1(A)) ) ).
fof(dt_k4_tarski,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_relset_1,axiom,
! [A,B,C] :
( m1_relset_1(C,A,B)
=> m2_relset_1(k6_relset_1(A,B,C),B,A) ) ).
fof(dt_k9_relat_1,axiom,
$true ).
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(fc11_relat_1,axiom,
! [A] :
( v1_xboole_0(A)
=> ( v1_xboole_0(k4_relat_1(A))
& v1_relat_1(k4_relat_1(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_subset_1,axiom,
! [A] : ~ v1_xboole_0(k1_tarski(A)) ).
fof(fc3_subset_1,axiom,
! [A,B] : ~ v1_xboole_0(k2_tarski(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(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(involutiveness_k4_relat_1,axiom,
! [A] :
( v1_relat_1(A)
=> k4_relat_1(k4_relat_1(A)) = A ) ).
fof(involutiveness_k6_relset_1,axiom,
! [A,B,C] :
( m1_relset_1(C,A,B)
=> k6_relset_1(A,B,k6_relset_1(A,B,C)) = C ) ).
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(redefinition_k10_relset_1,axiom,
! [A,B,C,D] :
( m1_relset_1(C,A,B)
=> k10_relset_1(A,B,C,D) = k9_relat_1(C,D) ) ).
fof(redefinition_k6_relset_1,axiom,
! [A,B,C] :
( m1_relset_1(C,A,B)
=> k6_relset_1(A,B,C) = k4_relat_1(C) ) ).
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(t156_relat_1,axiom,
! [A,B,C] :
( v1_relat_1(C)
=> ( r1_tarski(A,B)
=> r1_tarski(k9_relat_1(C,A),k9_relat_1(C,B)) ) ) ).
fof(t159_relat_1,axiom,
! [A,B] :
( v1_relat_1(B)
=> ! [C] :
( v1_relat_1(C)
=> k9_relat_1(k5_relat_1(B,C),A) = k9_relat_1(C,k9_relat_1(B,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(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(t51_relset_2,axiom,
! [A,B,C] :
( m2_relset_1(C,B,A)
=> ( k1_funct_5(C) = k10_relset_1(A,B,k6_relset_1(B,A,C),A)
& k2_funct_5(C) = k10_relset_1(B,A,C,B) ) ) ).
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) ) ).
fof(t9_relset_2,axiom,
! [A,B,C] :
( v1_relat_1(C)
=> ( r2_hidden(A,k9_relat_1(C,k1_tarski(B)))
<=> r2_hidden(k4_tarski(B,A),C) ) ) ).
%------------------------------------------------------------------------------