TPTP Problem File: SEU353+1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SEU353+1 : TPTP v8.2.0. Released v3.3.0.
% Domain : Set theory
% Problem : MPTP bushy problem t91_tmap_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-t91_tmap_1 [Urb07]
% Status : Theorem
% Rating : 0.33 v8.2.0, 0.31 v7.5.0, 0.34 v7.4.0, 0.17 v7.3.0, 0.31 v7.2.0, 0.28 v7.1.0, 0.26 v7.0.0, 0.23 v6.4.0, 0.27 v6.3.0, 0.29 v6.2.0, 0.32 v6.1.0, 0.37 v6.0.0, 0.26 v5.5.0, 0.41 v5.4.0, 0.46 v5.3.0, 0.52 v5.2.0, 0.40 v5.1.0, 0.38 v5.0.0, 0.46 v4.1.0, 0.43 v4.0.0, 0.46 v3.7.0, 0.40 v3.5.0, 0.42 v3.4.0, 0.37 v3.3.0
% Syntax : Number of formulae : 57 ( 19 unt; 0 def)
% Number of atoms : 173 ( 7 equ)
% Maximal formula atoms : 10 ( 3 avg)
% Number of connectives : 140 ( 24 ~; 1 |; 80 &)
% ( 2 <=>; 33 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 21 ( 19 usr; 1 prp; 0-3 aty)
% Number of functors : 9 ( 9 usr; 1 con; 0-4 aty)
% Number of variables : 99 ( 84 !; 15 ?)
% SPC : FOF_THM_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_2,axiom,
! [A,B,C] :
( relation_of2(C,A,B)
=> ( ( function(C)
& v1_partfun1(C,A,B) )
=> ( function(C)
& quasi_total(C,A,B) ) ) ) ).
fof(cc1_partfun1,axiom,
! [A] :
( ( relation(A)
& symmetric(A)
& transitive(A) )
=> ( relation(A)
& reflexive(A) ) ) ).
fof(cc1_relset_1,axiom,
! [A,B,C] :
( element(C,powerset(cartesian_product2(A,B)))
=> relation(C) ) ).
fof(cc2_funct_2,axiom,
! [A,B,C] :
( relation_of2(C,A,B)
=> ( ( function(C)
& quasi_total(C,A,B)
& bijective(C,A,B) )
=> ( function(C)
& one_to_one(C)
& quasi_total(C,A,B)
& onto(C,A,B) ) ) ) ).
fof(cc3_funct_2,axiom,
! [A,B,C] :
( relation_of2(C,A,B)
=> ( ( function(C)
& one_to_one(C)
& quasi_total(C,A,B)
& onto(C,A,B) )
=> ( function(C)
& quasi_total(C,A,B)
& bijective(C,A,B) ) ) ) ).
fof(cc4_funct_2,axiom,
! [A,B] :
( relation_of2(B,A,A)
=> ( ( function(B)
& v1_partfun1(B,A,A)
& reflexive(B)
& quasi_total(B,A,A) )
=> ( function(B)
& one_to_one(B)
& quasi_total(B,A,A)
& onto(B,A,A)
& bijective(B,A,A) ) ) ) ).
fof(cc5_funct_2,axiom,
! [A,B] :
( ~ empty(B)
=> ! [C] :
( relation_of2(C,A,B)
=> ( ( function(C)
& quasi_total(C,A,B) )
=> ( function(C)
& v1_partfun1(C,A,B)
& quasi_total(C,A,B) ) ) ) ) ).
fof(cc6_funct_2,axiom,
! [A,B] :
( ( ~ empty(A)
& ~ empty(B) )
=> ! [C] :
( relation_of2(C,A,B)
=> ( ( function(C)
& quasi_total(C,A,B) )
=> ( function(C)
& ~ empty(C)
& v1_partfun1(C,A,B)
& quasi_total(C,A,B) ) ) ) ) ).
fof(d11_grcat_1,axiom,
! [A] :
( one_sorted_str(A)
=> identity_on_carrier(A) = identity_as_relation_of(the_carrier(A)) ) ).
fof(dt_k1_funct_1,axiom,
$true ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k2_zfmisc_1,axiom,
$true ).
fof(dt_k6_partfun1,axiom,
! [A] :
( v1_partfun1(identity_as_relation_of(A),A,A)
& relation_of2_as_subset(identity_as_relation_of(A),A,A) ) ).
fof(dt_k6_relat_1,axiom,
! [A] : relation(identity_relation(A)) ).
fof(dt_k7_grcat_1,axiom,
! [A] :
( one_sorted_str(A)
=> ( function(identity_on_carrier(A))
& quasi_total(identity_on_carrier(A),the_carrier(A),the_carrier(A))
& relation_of2_as_subset(identity_on_carrier(A),the_carrier(A),the_carrier(A)) ) ) ).
fof(dt_k8_funct_2,axiom,
! [A,B,C,D] :
( ( ~ empty(A)
& function(C)
& quasi_total(C,A,B)
& relation_of2(C,A,B)
& element(D,A) )
=> element(apply_as_element(A,B,C,D),B) ) ).
fof(dt_l1_struct_0,axiom,
$true ).
fof(dt_m1_relset_1,axiom,
$true ).
fof(dt_m1_subset_1,axiom,
$true ).
fof(dt_m2_relset_1,axiom,
! [A,B,C] :
( relation_of2_as_subset(C,A,B)
=> element(C,powerset(cartesian_product2(A,B))) ) ).
fof(dt_u1_struct_0,axiom,
$true ).
fof(existence_l1_struct_0,axiom,
? [A] : one_sorted_str(A) ).
fof(existence_m1_relset_1,axiom,
! [A,B] :
? [C] : relation_of2(C,A,B) ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : element(B,A) ).
fof(existence_m2_relset_1,axiom,
! [A,B] :
? [C] : relation_of2_as_subset(C,A,B) ).
fof(fc1_struct_0,axiom,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ~ empty(the_carrier(A)) ) ).
fof(fc1_subset_1,axiom,
! [A] : ~ empty(powerset(A)) ).
fof(fc1_xboole_0,axiom,
empty(empty_set) ).
fof(fc2_partfun1,axiom,
! [A] :
( relation(identity_relation(A))
& function(identity_relation(A))
& reflexive(identity_relation(A))
& symmetric(identity_relation(A))
& antisymmetric(identity_relation(A))
& transitive(identity_relation(A)) ) ).
fof(fc4_subset_1,axiom,
! [A,B] :
( ( ~ empty(A)
& ~ empty(B) )
=> ~ empty(cartesian_product2(A,B)) ) ).
fof(rc1_funct_2,axiom,
! [A,B] :
? [C] :
( relation_of2(C,A,B)
& relation(C)
& function(C)
& quasi_total(C,A,B) ) ).
fof(rc1_partfun1,axiom,
? [A] :
( relation(A)
& function(A)
& one_to_one(A)
& empty(A) ) ).
fof(rc1_subset_1,axiom,
! [A] :
( ~ empty(A)
=> ? [B] :
( element(B,powerset(A))
& ~ empty(B) ) ) ).
fof(rc1_xboole_0,axiom,
? [A] : empty(A) ).
fof(rc2_funct_2,axiom,
! [A] :
? [B] :
( relation_of2(B,A,A)
& relation(B)
& function(B)
& one_to_one(B)
& quasi_total(B,A,A)
& onto(B,A,A)
& bijective(B,A,A) ) ).
fof(rc2_partfun1,axiom,
! [A,B] :
? [C] :
( relation_of2(C,A,B)
& relation(C)
& function(C) ) ).
fof(rc2_subset_1,axiom,
! [A] :
? [B] :
( element(B,powerset(A))
& empty(B) ) ).
fof(rc2_xboole_0,axiom,
? [A] : ~ empty(A) ).
fof(rc3_partfun1,axiom,
! [A] :
? [B] :
( relation_of2(B,A,A)
& relation(B)
& reflexive(B)
& symmetric(B)
& antisymmetric(B)
& transitive(B)
& v1_partfun1(B,A,A) ) ).
fof(rc3_struct_0,axiom,
? [A] :
( one_sorted_str(A)
& ~ empty_carrier(A) ) ).
fof(rc5_struct_0,axiom,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ? [B] :
( element(B,powerset(the_carrier(A)))
& ~ empty(B) ) ) ).
fof(redefinition_k6_partfun1,axiom,
! [A] : identity_as_relation_of(A) = identity_relation(A) ).
fof(redefinition_k8_funct_2,axiom,
! [A,B,C,D] :
( ( ~ empty(A)
& function(C)
& quasi_total(C,A,B)
& relation_of2(C,A,B)
& element(D,A) )
=> apply_as_element(A,B,C,D) = apply(C,D) ) ).
fof(redefinition_m2_relset_1,axiom,
! [A,B,C] :
( relation_of2_as_subset(C,A,B)
<=> relation_of2(C,A,B) ) ).
fof(reflexivity_r1_tarski,axiom,
! [A,B] : subset(A,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(t35_funct_1,axiom,
! [A,B] :
( in(B,A)
=> apply(identity_relation(A),B) = B ) ).
fof(t3_subset,axiom,
! [A,B] :
( element(A,powerset(B))
<=> subset(A,B) ) ).
fof(t4_subset,axiom,
! [A,B,C] :
( ( in(A,B)
& element(B,powerset(C)) )
=> element(A,C) ) ).
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) ) ).
fof(t91_tmap_1,conjecture,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ! [B] :
( element(B,the_carrier(A))
=> apply_as_element(the_carrier(A),the_carrier(A),identity_on_carrier(A),B) = B ) ) ).
%------------------------------------------------------------------------------