TPTP Problem File: SEU225+1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SEU225+1 : TPTP v8.2.0. Released v3.3.0.
% Domain : Set theory
% Problem : MPTP bushy problem t72_funct_1
% Version : [Urb07] axioms : Especial.
% English :
% Refs : [Ban01] Bancerek et al. (2001), On the Characterizations of Co
% : [Urb07] Urban (2006), Email to G. Sutcliffe
% Source : [Urb07]
% Names : bushy-t72_funct_1 [Urb07]
% Status : Theorem
% Rating : 0.33 v8.2.0, 0.28 v8.1.0, 0.31 v7.5.0, 0.34 v7.4.0, 0.20 v7.3.0, 0.24 v7.2.0, 0.21 v7.1.0, 0.30 v7.0.0, 0.27 v6.4.0, 0.31 v6.3.0, 0.25 v6.2.0, 0.28 v6.1.0, 0.43 v5.5.0, 0.52 v5.4.0, 0.50 v5.3.0, 0.56 v5.2.0, 0.40 v5.1.0, 0.38 v4.1.0, 0.39 v4.0.0, 0.42 v3.7.0, 0.40 v3.5.0, 0.42 v3.4.0, 0.47 v3.3.0
% Syntax : Number of formulae : 47 ( 20 unt; 0 def)
% Number of atoms : 107 ( 14 equ)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 72 ( 12 ~; 1 |; 33 &)
% ( 4 <=>; 22 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 4 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 9 ( 7 usr; 1 prp; 0-2 aty)
% Number of functors : 8 ( 8 usr; 1 con; 0-2 aty)
% Number of variables : 61 ( 52 !; 9 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Translated by MPTP 0.2 from the original problem in the Mizar
% library, www.mizar.org
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fof(antisymmetry_r2_hidden,axiom,
! [A,B] :
( in(A,B)
=> ~ in(B,A) ) ).
fof(cc1_funct_1,axiom,
! [A] :
( empty(A)
=> function(A) ) ).
fof(cc1_relat_1,axiom,
! [A] :
( empty(A)
=> relation(A) ) ).
fof(cc2_funct_1,axiom,
! [A] :
( ( relation(A)
& empty(A)
& function(A) )
=> ( relation(A)
& function(A)
& one_to_one(A) ) ) ).
fof(commutativity_k2_tarski,axiom,
! [A,B] : unordered_pair(A,B) = unordered_pair(B,A) ).
fof(commutativity_k3_xboole_0,axiom,
! [A,B] : set_intersection2(A,B) = set_intersection2(B,A) ).
fof(d4_funct_1,axiom,
! [A] :
( ( relation(A)
& function(A) )
=> ! [B,C] :
( ( in(B,relation_dom(A))
=> ( C = apply(A,B)
<=> in(ordered_pair(B,C),A) ) )
& ( ~ in(B,relation_dom(A))
=> ( C = apply(A,B)
<=> C = empty_set ) ) ) ) ).
fof(d5_tarski,axiom,
! [A,B] : ordered_pair(A,B) = unordered_pair(unordered_pair(A,B),singleton(A)) ).
fof(dt_k1_funct_1,axiom,
$true ).
fof(dt_k1_relat_1,axiom,
$true ).
fof(dt_k1_tarski,axiom,
$true ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k2_tarski,axiom,
$true ).
fof(dt_k3_xboole_0,axiom,
$true ).
fof(dt_k4_tarski,axiom,
$true ).
fof(dt_k7_relat_1,axiom,
! [A,B] :
( relation(A)
=> relation(relation_dom_restriction(A,B)) ) ).
fof(dt_m1_subset_1,axiom,
$true ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : element(B,A) ).
fof(fc12_relat_1,axiom,
( empty(empty_set)
& relation(empty_set)
& relation_empty_yielding(empty_set) ) ).
fof(fc13_relat_1,axiom,
! [A,B] :
( ( relation(A)
& relation_empty_yielding(A) )
=> ( relation(relation_dom_restriction(A,B))
& relation_empty_yielding(relation_dom_restriction(A,B)) ) ) ).
fof(fc1_relat_1,axiom,
! [A,B] :
( ( relation(A)
& relation(B) )
=> relation(set_intersection2(A,B)) ) ).
fof(fc1_xboole_0,axiom,
empty(empty_set) ).
fof(fc1_zfmisc_1,axiom,
! [A,B] : ~ empty(ordered_pair(A,B)) ).
fof(fc2_subset_1,axiom,
! [A] : ~ empty(singleton(A)) ).
fof(fc3_subset_1,axiom,
! [A,B] : ~ empty(unordered_pair(A,B)) ).
fof(fc4_funct_1,axiom,
! [A,B] :
( ( relation(A)
& function(A) )
=> ( relation(relation_dom_restriction(A,B))
& function(relation_dom_restriction(A,B)) ) ) ).
fof(fc4_relat_1,axiom,
( empty(empty_set)
& relation(empty_set) ) ).
fof(fc5_relat_1,axiom,
! [A] :
( ( ~ empty(A)
& relation(A) )
=> ~ empty(relation_dom(A)) ) ).
fof(fc7_relat_1,axiom,
! [A] :
( empty(A)
=> ( empty(relation_dom(A))
& relation(relation_dom(A)) ) ) ).
fof(idempotence_k3_xboole_0,axiom,
! [A,B] : set_intersection2(A,A) = A ).
fof(l82_funct_1,axiom,
! [A,B,C] :
( ( relation(C)
& function(C) )
=> ( in(B,relation_dom(relation_dom_restriction(C,A)))
<=> ( in(B,relation_dom(C))
& in(B,A) ) ) ) ).
fof(rc1_funct_1,axiom,
? [A] :
( relation(A)
& function(A) ) ).
fof(rc1_relat_1,axiom,
? [A] :
( empty(A)
& relation(A) ) ).
fof(rc1_xboole_0,axiom,
? [A] : empty(A) ).
fof(rc2_funct_1,axiom,
? [A] :
( relation(A)
& empty(A)
& function(A) ) ).
fof(rc2_relat_1,axiom,
? [A] :
( ~ empty(A)
& relation(A) ) ).
fof(rc2_xboole_0,axiom,
? [A] : ~ empty(A) ).
fof(rc3_funct_1,axiom,
? [A] :
( relation(A)
& function(A)
& one_to_one(A) ) ).
fof(rc3_relat_1,axiom,
? [A] :
( relation(A)
& relation_empty_yielding(A) ) ).
fof(t1_subset,axiom,
! [A,B] :
( in(A,B)
=> element(A,B) ) ).
fof(t2_boole,axiom,
! [A] : set_intersection2(A,empty_set) = empty_set ).
fof(t2_subset,axiom,
! [A,B] :
( element(A,B)
=> ( empty(B)
| in(A,B) ) ) ).
fof(t68_funct_1,axiom,
! [A,B] :
( ( relation(B)
& function(B) )
=> ! [C] :
( ( relation(C)
& function(C) )
=> ( B = relation_dom_restriction(C,A)
<=> ( relation_dom(B) = set_intersection2(relation_dom(C),A)
& ! [D] :
( in(D,relation_dom(B))
=> apply(B,D) = apply(C,D) ) ) ) ) ) ).
fof(t6_boole,axiom,
! [A] :
( empty(A)
=> A = empty_set ) ).
fof(t72_funct_1,conjecture,
! [A,B,C] :
( ( relation(C)
& function(C) )
=> ( in(B,A)
=> apply(relation_dom_restriction(C,A),B) = apply(C,B) ) ) ).
fof(t7_boole,axiom,
! [A,B] :
~ ( in(A,B)
& empty(B) ) ).
fof(t8_boole,axiom,
! [A,B] :
~ ( empty(A)
& A != B
& empty(B) ) ).
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