TPTP Problem File: SEU260+1.p
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%------------------------------------------------------------------------------
% File : SEU260+1 : TPTP v8.2.0. Released v3.3.0.
% Domain : Set theory
% Problem : MPTP bushy problem t53_wellord1
% 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-t53_wellord1 [Urb07]
% Status : CounterSatisfiable
% Rating : 1.00 v5.4.0, 0.86 v5.3.0, 1.00 v3.3.0
% Syntax : Number of formulae : 61 ( 24 unt; 0 def)
% Number of atoms : 181 ( 24 equ)
% Maximal formula atoms : 15 ( 2 avg)
% Number of connectives : 145 ( 25 ~; 1 |; 60 &)
% ( 14 <=>; 45 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 16 ( 14 usr; 1 prp; 0-3 aty)
% Number of functors : 14 ( 14 usr; 1 con; 0-2 aty)
% Number of variables : 102 ( 94 !; 8 ?)
% SPC : FOF_CSA_RFO_SEQ
% Comments : Translated by MPTP 0.2 from the original problem in the Mizar
% library, www.mizar.org
%------------------------------------------------------------------------------
fof(antisymmetry_r2_hidden,axiom,
! [A,B] :
( in(A,B)
=> ~ in(B,A) ) ).
fof(cc1_funct_1,axiom,
! [A] :
( empty(A)
=> function(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_k2_xboole_0,axiom,
! [A,B] : set_union2(A,B) = set_union2(B,A) ).
fof(d13_funct_1,axiom,
! [A] :
( ( relation(A)
& function(A) )
=> ! [B,C] :
( C = relation_inverse_image(A,B)
<=> ! [D] :
( in(D,C)
<=> ( in(D,relation_dom(A))
& in(apply(A,D),B) ) ) ) ) ).
fof(d1_wellord1,axiom,
! [A] :
( relation(A)
=> ! [B,C] :
( C = fiber(A,B)
<=> ! [D] :
( in(D,C)
<=> ( D != B
& in(ordered_pair(D,B),A) ) ) ) ) ).
fof(d2_wellord1,axiom,
! [A] :
( relation(A)
=> ( well_founded_relation(A)
<=> ! [B] :
~ ( subset(B,relation_field(A))
& B != empty_set
& ! [C] :
~ ( in(C,B)
& disjoint(fiber(A,C),B) ) ) ) ) ).
fof(d5_funct_1,axiom,
! [A] :
( ( relation(A)
& function(A) )
=> ! [B] :
( B = relation_rng(A)
<=> ! [C] :
( in(C,B)
<=> ? [D] :
( in(D,relation_dom(A))
& C = apply(A,D) ) ) ) ) ).
fof(d5_tarski,axiom,
! [A,B] : ordered_pair(A,B) = unordered_pair(unordered_pair(A,B),singleton(A)) ).
fof(d6_relat_1,axiom,
! [A] :
( relation(A)
=> relation_field(A) = set_union2(relation_dom(A),relation_rng(A)) ) ).
fof(d7_wellord1,axiom,
! [A] :
( relation(A)
=> ! [B] :
( relation(B)
=> ! [C] :
( ( relation(C)
& function(C) )
=> ( relation_isomorphism(A,B,C)
<=> ( relation_dom(C) = relation_field(A)
& relation_rng(C) = relation_field(B)
& one_to_one(C)
& ! [D,E] :
( in(ordered_pair(D,E),A)
<=> ( in(D,relation_field(A))
& in(E,relation_field(A))
& in(ordered_pair(apply(C,D),apply(C,E)),B) ) ) ) ) ) ) ) ).
fof(dt_k10_relat_1,axiom,
$true ).
fof(dt_k1_funct_1,axiom,
$true ).
fof(dt_k1_relat_1,axiom,
$true ).
fof(dt_k1_tarski,axiom,
$true ).
fof(dt_k1_wellord1,axiom,
$true ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k2_funct_1,axiom,
! [A] :
( ( relation(A)
& function(A) )
=> ( relation(function_inverse(A))
& function(function_inverse(A)) ) ) ).
fof(dt_k2_relat_1,axiom,
$true ).
fof(dt_k2_tarski,axiom,
$true ).
fof(dt_k2_xboole_0,axiom,
$true ).
fof(dt_k3_relat_1,axiom,
$true ).
fof(dt_k4_tarski,axiom,
$true ).
fof(dt_k5_relat_1,axiom,
! [A,B] :
( ( relation(A)
& relation(B) )
=> relation(relation_composition(A,B)) ) ).
fof(dt_m1_subset_1,axiom,
$true ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : element(B,A) ).
fof(fc1_funct_1,axiom,
! [A,B] :
( ( relation(A)
& function(A)
& relation(B)
& function(B) )
=> ( relation(relation_composition(A,B))
& function(relation_composition(A,B)) ) ) ).
fof(fc1_xboole_0,axiom,
empty(empty_set) ).
fof(fc1_zfmisc_1,axiom,
! [A,B] : ~ empty(ordered_pair(A,B)) ).
fof(fc2_xboole_0,axiom,
! [A,B] :
( ~ empty(A)
=> ~ empty(set_union2(A,B)) ) ).
fof(fc3_xboole_0,axiom,
! [A,B] :
( ~ empty(A)
=> ~ empty(set_union2(B,A)) ) ).
fof(idempotence_k2_xboole_0,axiom,
! [A,B] : set_union2(A,A) = A ).
fof(l1_wellord1,axiom,
! [A] :
( relation(A)
=> ( reflexive(A)
<=> ! [B] :
( in(B,relation_field(A))
=> in(ordered_pair(B,B),A) ) ) ) ).
fof(l2_wellord1,axiom,
! [A] :
( relation(A)
=> ( transitive(A)
<=> ! [B,C,D] :
( ( in(ordered_pair(B,C),A)
& in(ordered_pair(C,D),A) )
=> in(ordered_pair(B,D),A) ) ) ) ).
fof(l3_wellord1,axiom,
! [A] :
( relation(A)
=> ( antisymmetric(A)
<=> ! [B,C] :
( ( in(ordered_pair(B,C),A)
& in(ordered_pair(C,B),A) )
=> B = C ) ) ) ).
fof(l4_wellord1,axiom,
! [A] :
( relation(A)
=> ( connected(A)
<=> ! [B,C] :
~ ( in(B,relation_field(A))
& in(C,relation_field(A))
& B != C
& ~ in(ordered_pair(B,C),A)
& ~ in(ordered_pair(C,B),A) ) ) ) ).
fof(rc1_funct_1,axiom,
? [A] :
( relation(A)
& function(A) ) ).
fof(rc1_xboole_0,axiom,
? [A] : empty(A) ).
fof(rc2_funct_1,axiom,
? [A] :
( relation(A)
& empty(A)
& function(A) ) ).
fof(rc2_xboole_0,axiom,
? [A] : ~ empty(A) ).
fof(rc3_funct_1,axiom,
? [A] :
( relation(A)
& function(A)
& one_to_one(A) ) ).
fof(reflexivity_r1_tarski,axiom,
! [A,B] : subset(A,A) ).
fof(symmetry_r1_xboole_0,axiom,
! [A,B] :
( disjoint(A,B)
=> disjoint(B,A) ) ).
fof(t167_relat_1,axiom,
! [A,B] :
( relation(B)
=> subset(relation_inverse_image(B,A),relation_dom(B)) ) ).
fof(t174_relat_1,axiom,
! [A,B] :
( relation(B)
=> ~ ( A != empty_set
& subset(A,relation_rng(B))
& relation_inverse_image(B,A) = empty_set ) ) ).
fof(t1_boole,axiom,
! [A] : set_union2(A,empty_set) = A ).
fof(t1_subset,axiom,
! [A,B] :
( in(A,B)
=> element(A,B) ) ).
fof(t2_subset,axiom,
! [A,B] :
( element(A,B)
=> ( empty(B)
| in(A,B) ) ) ).
fof(t30_relat_1,axiom,
! [A,B,C] :
( relation(C)
=> ( in(ordered_pair(A,B),C)
=> ( in(A,relation_field(C))
& in(B,relation_field(C)) ) ) ) ).
fof(t3_subset,axiom,
! [A,B] :
( element(A,powerset(B))
<=> subset(A,B) ) ).
fof(t3_xboole_0,axiom,
! [A,B] :
( ~ ( ~ disjoint(A,B)
& ! [C] :
~ ( in(C,A)
& in(C,B) ) )
& ~ ( ? [C] :
( in(C,A)
& in(C,B) )
& disjoint(A,B) ) ) ).
fof(t4_subset,axiom,
! [A,B,C] :
( ( in(A,B)
& element(B,powerset(C)) )
=> element(A,C) ) ).
fof(t53_wellord1,conjecture,
! [A] :
( relation(A)
=> ! [B] :
( relation(B)
=> ! [C] :
( ( relation(C)
& function(C) )
=> ( relation_isomorphism(A,B,C)
=> ( ( reflexive(A)
=> reflexive(B) )
& ( transitive(A)
=> transitive(B) )
& ( connected(A)
=> connected(B) )
& ( antisymmetric(A)
=> antisymmetric(B) )
& ( well_founded_relation(A)
=> well_founded_relation(B) ) ) ) ) ) ) ).
fof(t55_funct_1,axiom,
! [A] :
( ( relation(A)
& function(A) )
=> ( one_to_one(A)
=> ( relation_rng(A) = relation_dom(function_inverse(A))
& relation_dom(A) = relation_rng(function_inverse(A)) ) ) ) ).
fof(t57_funct_1,axiom,
! [A,B] :
( ( relation(B)
& function(B) )
=> ( ( one_to_one(B)
& in(A,relation_rng(B)) )
=> ( A = apply(B,apply(function_inverse(B),A))
& A = apply(relation_composition(function_inverse(B),B),A) ) ) ) ).
fof(t5_subset,axiom,
! [A,B,C] :
~ ( in(A,B)
& element(B,powerset(C))
& empty(C) ) ).
fof(t6_boole,axiom,
! [A] :
( empty(A)
=> A = empty_set ) ).
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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