TPTP Problem File: SEU385+1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SEU385+1 : TPTP v8.2.0. Released v3.3.0.
% Domain : Set theory
% Problem : MPTP bushy problem t16_waybel_9
% 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-t16_waybel_9 [Urb07]
% Status : Theorem
% Rating : 0.72 v8.2.0, 0.69 v8.1.0, 0.67 v7.5.0, 0.69 v7.4.0, 0.60 v7.3.0, 0.66 v7.1.0, 0.61 v7.0.0, 0.70 v6.4.0, 0.69 v6.3.0, 0.58 v6.2.0, 0.72 v6.1.0, 0.80 v6.0.0, 0.74 v5.5.0, 0.81 v5.4.0, 0.82 v5.3.0, 0.85 v5.2.0, 0.75 v5.1.0, 0.81 v5.0.0, 0.92 v4.1.0, 0.96 v4.0.1, 0.91 v4.0.0, 0.92 v3.7.0, 0.90 v3.5.0, 0.89 v3.3.0
% Syntax : Number of formulae : 79 ( 17 unt; 0 def)
% Number of atoms : 285 ( 20 equ)
% Maximal formula atoms : 14 ( 3 avg)
% Number of connectives : 253 ( 47 ~; 1 |; 131 &)
% ( 4 <=>; 70 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 24 ( 22 usr; 1 prp; 0-3 aty)
% Number of functors : 16 ( 16 usr; 1 con; 0-4 aty)
% Number of variables : 166 ( 147 !; 19 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Translated by MPTP 0.2 from the original problem in the Mizar
% library, www.mizar.org
%------------------------------------------------------------------------------
fof(abstractness_v6_waybel_0,axiom,
! [A,B] :
( ( one_sorted_str(A)
& net_str(B,A) )
=> ( strict_net_str(B,A)
=> B = net_str_of(A,the_carrier(B),the_InternalRel(B),the_mapping(A,B)) ) ) ).
fof(antisymmetry_r2_hidden,axiom,
! [A,B] :
( in(A,B)
=> ~ in(B,A) ) ).
fof(cc1_finset_1,axiom,
! [A] :
( empty(A)
=> finite(A) ) ).
fof(cc1_finsub_1,axiom,
! [A] :
( preboolean(A)
=> ( cup_closed(A)
& diff_closed(A) ) ) ).
fof(cc1_funct_1,axiom,
! [A] :
( empty(A)
=> function(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_relset_1,axiom,
! [A,B,C] :
( element(C,powerset(cartesian_product2(A,B)))
=> relation(C) ) ).
fof(cc2_finset_1,axiom,
! [A] :
( finite(A)
=> ! [B] :
( element(B,powerset(A))
=> finite(B) ) ) ).
fof(cc2_finsub_1,axiom,
! [A] :
( ( cup_closed(A)
& diff_closed(A) )
=> preboolean(A) ) ).
fof(cc2_funct_1,axiom,
! [A] :
( ( relation(A)
& empty(A)
& function(A) )
=> ( relation(A)
& function(A)
& one_to_one(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(d7_waybel_9,axiom,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ! [B] :
( ( ~ empty_carrier(B)
& net_str(B,A) )
=> ! [C] :
( element(C,the_carrier(B))
=> ! [D] :
( ( strict_net_str(D,A)
& net_str(D,A) )
=> ( D = netstr_restr_to_element(A,B,C)
<=> ( ! [E] :
( in(E,the_carrier(D))
<=> ? [F] :
( element(F,the_carrier(B))
& F = E
& related(B,C,F) ) )
& the_InternalRel(D) = relation_restriction_as_relation_of(the_InternalRel(B),the_carrier(D))
& the_mapping(A,D) = partfun_dom_restriction(the_carrier(B),the_carrier(A),the_mapping(A,B),the_carrier(D)) ) ) ) ) ) ) ).
fof(d8_waybel_0,axiom,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ! [B] :
( ( ~ empty_carrier(B)
& net_str(B,A) )
=> ! [C] :
( element(C,the_carrier(B))
=> apply_netmap(A,B,C) = apply_on_structs(B,A,the_mapping(A,B),C) ) ) ) ).
fof(dt_g1_waybel_0,axiom,
! [A,B,C,D] :
( ( one_sorted_str(A)
& relation_of2(C,B,B)
& function(D)
& quasi_total(D,B,the_carrier(A))
& relation_of2(D,B,the_carrier(A)) )
=> ( strict_net_str(net_str_of(A,B,C,D),A)
& net_str(net_str_of(A,B,C,D),A) ) ) ).
fof(dt_k1_funct_1,axiom,
$true ).
fof(dt_k1_toler_1,axiom,
! [A,B] :
( relation(A)
=> relation_of2_as_subset(relation_restriction_as_relation_of(A,B),B,B) ) ).
fof(dt_k1_waybel_0,axiom,
! [A,B,C,D] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& ~ empty_carrier(B)
& one_sorted_str(B)
& function(C)
& quasi_total(C,the_carrier(A),the_carrier(B))
& relation_of2(C,the_carrier(A),the_carrier(B))
& element(D,the_carrier(A)) )
=> element(apply_on_structs(A,B,C,D),the_carrier(B)) ) ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k2_partfun1,axiom,
! [A,B,C,D] :
( ( function(C)
& relation_of2(C,A,B) )
=> ( function(partfun_dom_restriction(A,B,C,D))
& relation_of2_as_subset(partfun_dom_restriction(A,B,C,D),A,B) ) ) ).
fof(dt_k2_wellord1,axiom,
! [A,B] :
( relation(A)
=> relation(relation_restriction(A,B)) ) ).
fof(dt_k2_zfmisc_1,axiom,
$true ).
fof(dt_k3_waybel_0,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& ~ empty_carrier(B)
& net_str(B,A)
& element(C,the_carrier(B)) )
=> element(apply_netmap(A,B,C),the_carrier(A)) ) ).
fof(dt_k5_waybel_9,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& ~ empty_carrier(B)
& net_str(B,A)
& element(C,the_carrier(B)) )
=> ( strict_net_str(netstr_restr_to_element(A,B,C),A)
& net_str(netstr_restr_to_element(A,B,C),A) ) ) ).
fof(dt_k7_relat_1,axiom,
! [A,B] :
( relation(A)
=> relation(relation_dom_restriction(A,B)) ) ).
fof(dt_k7_yellow_2,axiom,
! [A,B,C,D] :
( ( ~ empty(A)
& ~ empty_carrier(B)
& one_sorted_str(B)
& function(C)
& quasi_total(C,A,the_carrier(B))
& relation_of2(C,A,the_carrier(B))
& element(D,A) )
=> element(apply_on_set_and_struct2(A,B,C,D),the_carrier(B)) ) ).
fof(dt_l1_orders_2,axiom,
! [A] :
( rel_str(A)
=> one_sorted_str(A) ) ).
fof(dt_l1_struct_0,axiom,
$true ).
fof(dt_l1_waybel_0,axiom,
! [A] :
( one_sorted_str(A)
=> ! [B] :
( net_str(B,A)
=> rel_str(B) ) ) ).
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_orders_2,axiom,
! [A] :
( rel_str(A)
=> relation_of2_as_subset(the_InternalRel(A),the_carrier(A),the_carrier(A)) ) ).
fof(dt_u1_struct_0,axiom,
$true ).
fof(dt_u1_waybel_0,axiom,
! [A,B] :
( ( one_sorted_str(A)
& net_str(B,A) )
=> ( function(the_mapping(A,B))
& quasi_total(the_mapping(A,B),the_carrier(B),the_carrier(A))
& relation_of2_as_subset(the_mapping(A,B),the_carrier(B),the_carrier(A)) ) ) ).
fof(existence_l1_orders_2,axiom,
? [A] : rel_str(A) ).
fof(existence_l1_struct_0,axiom,
? [A] : one_sorted_str(A) ).
fof(existence_l1_waybel_0,axiom,
! [A] :
( one_sorted_str(A)
=> ? [B] : net_str(B,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(fc14_finset_1,axiom,
! [A,B] :
( ( finite(A)
& finite(B) )
=> finite(cartesian_product2(A,B)) ) ).
fof(fc15_yellow_6,axiom,
! [A,B] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& ~ empty_carrier(B)
& net_str(B,A) )
=> ( ~ empty(the_mapping(A,B))
& relation(the_mapping(A,B))
& function(the_mapping(A,B))
& quasi_total(the_mapping(A,B),the_carrier(B),the_carrier(A)) ) ) ).
fof(fc1_finsub_1,axiom,
! [A] :
( ~ empty(powerset(A))
& cup_closed(powerset(A))
& diff_closed(powerset(A))
& preboolean(powerset(A)) ) ).
fof(fc1_struct_0,axiom,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ~ empty(the_carrier(A)) ) ).
fof(fc1_xboole_0,axiom,
empty(empty_set) ).
fof(fc22_waybel_9,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& ~ empty_carrier(B)
& directed_relstr(B)
& net_str(B,A)
& element(C,the_carrier(B)) )
=> ( ~ empty_carrier(netstr_restr_to_element(A,B,C))
& strict_net_str(netstr_restr_to_element(A,B,C),A) ) ) ).
fof(fc4_funct_1,axiom,
! [A,B] :
( ( relation(A)
& function(A) )
=> ( relation(relation_dom_restriction(A,B))
& function(relation_dom_restriction(A,B)) ) ) ).
fof(fc6_waybel_0,axiom,
! [A,B,C,D] :
( ( one_sorted_str(A)
& ~ empty(B)
& relation_of2(C,B,B)
& function(D)
& quasi_total(D,B,the_carrier(A))
& relation_of2(D,B,the_carrier(A)) )
=> ( ~ empty_carrier(net_str_of(A,B,C,D))
& strict_net_str(net_str_of(A,B,C,D),A) ) ) ).
fof(free_g1_waybel_0,axiom,
! [A,B,C,D] :
( ( one_sorted_str(A)
& relation_of2(C,B,B)
& function(D)
& quasi_total(D,B,the_carrier(A))
& relation_of2(D,B,the_carrier(A)) )
=> ! [E,F,G,H] :
( net_str_of(A,B,C,D) = net_str_of(E,F,G,H)
=> ( A = E
& B = F
& C = G
& D = H ) ) ) ).
fof(rc1_finset_1,axiom,
? [A] :
( ~ empty(A)
& finite(A) ) ).
fof(rc1_funct_1,axiom,
? [A] :
( relation(A)
& function(A) ) ).
fof(rc1_funct_2,axiom,
! [A,B] :
? [C] :
( relation_of2(C,A,B)
& relation(C)
& function(C)
& quasi_total(C,A,B) ) ).
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_finset_1,axiom,
! [A] :
( ~ empty(A)
=> ? [B] :
( element(B,powerset(A))
& ~ empty(B)
& finite(B) ) ) ).
fof(rc3_funct_1,axiom,
? [A] :
( relation(A)
& function(A)
& one_to_one(A) ) ).
fof(rc3_struct_0,axiom,
? [A] :
( one_sorted_str(A)
& ~ empty_carrier(A) ) ).
fof(rc4_finset_1,axiom,
! [A] :
( ~ empty(A)
=> ? [B] :
( element(B,powerset(A))
& ~ empty(B)
& finite(B) ) ) ).
fof(rc4_waybel_0,axiom,
! [A] :
( one_sorted_str(A)
=> ? [B] :
( net_str(B,A)
& strict_net_str(B,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_k1_toler_1,axiom,
! [A,B] :
( relation(A)
=> relation_restriction_as_relation_of(A,B) = relation_restriction(A,B) ) ).
fof(redefinition_k1_waybel_0,axiom,
! [A,B,C,D] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& ~ empty_carrier(B)
& one_sorted_str(B)
& function(C)
& quasi_total(C,the_carrier(A),the_carrier(B))
& relation_of2(C,the_carrier(A),the_carrier(B))
& element(D,the_carrier(A)) )
=> apply_on_structs(A,B,C,D) = apply(C,D) ) ).
fof(redefinition_k2_partfun1,axiom,
! [A,B,C,D] :
( ( function(C)
& relation_of2(C,A,B) )
=> partfun_dom_restriction(A,B,C,D) = relation_dom_restriction(C,D) ) ).
fof(redefinition_k7_yellow_2,axiom,
! [A,B,C,D] :
( ( ~ empty(A)
& ~ empty_carrier(B)
& one_sorted_str(B)
& function(C)
& quasi_total(C,A,the_carrier(B))
& relation_of2(C,A,the_carrier(B))
& element(D,A) )
=> apply_on_set_and_struct2(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(t16_waybel_9,conjecture,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ! [B] :
( ( ~ empty_carrier(B)
& directed_relstr(B)
& net_str(B,A) )
=> ! [C] :
( element(C,the_carrier(B))
=> ! [D] :
( element(D,the_carrier(B))
=> ! [E] :
( element(E,the_carrier(netstr_restr_to_element(A,B,C)))
=> ( D = E
=> apply_netmap(A,B,D) = apply_netmap(A,netstr_restr_to_element(A,B,C),E) ) ) ) ) ) ) ).
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(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(t72_funct_1,axiom,
! [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) ) ).
%------------------------------------------------------------------------------