TPTP Problem File: SEU380+1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SEU380+1 : TPTP v8.2.0. Released v3.3.0.
% Domain : Set theory
% Problem : MPTP bushy problem t41_yellow_6
% 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-t41_yellow_6 [Urb07]
% Status : Theorem
% Rating : 1.00 v3.3.0
% Syntax : Number of formulae : 78 ( 16 unt; 0 def)
% Number of atoms : 304 ( 10 equ)
% Maximal formula atoms : 20 ( 3 avg)
% Number of connectives : 280 ( 54 ~; 2 |; 149 &)
% ( 7 <=>; 68 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 6 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 26 ( 24 usr; 1 prp; 0-3 aty)
% Number of functors : 12 ( 12 usr; 1 con; 0-5 aty)
% Number of variables : 157 ( 133 !; 24 ?)
% 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_1,axiom,
! [A] :
( empty(A)
=> function(A) ) ).
fof(cc1_relat_1,axiom,
! [A] :
( empty(A)
=> relation(A) ) ).
fof(cc1_relset_1,axiom,
! [A,B,C] :
( element(C,powerset(cartesian_product2(A,B)))
=> relation(C) ) ).
fof(cc1_yellow_3,axiom,
! [A] :
( rel_str(A)
=> ( empty_carrier(A)
=> v1_yellow_3(A) ) ) ).
fof(cc2_funct_1,axiom,
! [A] :
( ( relation(A)
& empty(A)
& function(A) )
=> ( relation(A)
& function(A)
& one_to_one(A) ) ) ).
fof(cc2_yellow_3,axiom,
! [A] :
( rel_str(A)
=> ( ~ v1_yellow_3(A)
=> ~ empty_carrier(A) ) ) ).
fof(d11_waybel_0,axiom,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ! [B] :
( ( ~ empty_carrier(B)
& net_str(B,A) )
=> ! [C] :
( is_eventually_in(A,B,C)
<=> ? [D] :
( element(D,the_carrier(B))
& ! [E] :
( element(E,the_carrier(B))
=> ( related(B,D,E)
=> in(apply_netmap(A,B,E),C) ) ) ) ) ) ) ).
fof(d12_yellow_6,axiom,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ! [B] :
( ( ~ empty_carrier(B)
& transitive_relstr(B)
& directed_relstr(B)
& net_str(B,A) )
=> ! [C] :
( ( ~ empty_carrier(C)
& transitive_relstr(C)
& directed_relstr(C)
& net_str(C,A) )
=> ( subnet(C,A,B)
<=> ? [D] :
( function(D)
& quasi_total(D,the_carrier(C),the_carrier(B))
& relation_of2_as_subset(D,the_carrier(C),the_carrier(B))
& the_mapping(A,C) = function_of_composition(the_carrier(C),the_carrier(B),the_carrier(A),D,the_mapping(A,B))
& ! [E] :
( element(E,the_carrier(B))
=> ? [F] :
( element(F,the_carrier(C))
& ! [G] :
( element(G,the_carrier(C))
=> ( related(C,F,G)
=> related(B,E,apply_on_set_and_struct(the_carrier(C),B,D,G)) ) ) ) ) ) ) ) ) ) ).
fof(d18_yellow_6,axiom,
! [A] :
( ( ~ empty_carrier(A)
& topological_space(A)
& top_str(A) )
=> ! [B] :
( ( ~ empty_carrier(B)
& transitive_relstr(B)
& directed_relstr(B)
& net_str(B,A) )
=> ! [C] :
( element(C,powerset(the_carrier(A)))
=> ( C = lim_points_of_net(A,B)
<=> ! [D] :
( element(D,the_carrier(A))
=> ( in(D,C)
<=> ! [E] :
( point_neighbourhood(E,A,D)
=> is_eventually_in(A,B,E) ) ) ) ) ) ) ) ).
fof(d3_tarski,axiom,
! [A,B] :
( subset(A,B)
<=> ! [C] :
( in(C,A)
=> in(C,B) ) ) ).
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_k11_yellow_6,axiom,
! [A,B] :
( ( ~ empty_carrier(A)
& topological_space(A)
& top_str(A)
& ~ empty_carrier(B)
& transitive_relstr(B)
& directed_relstr(B)
& net_str(B,A) )
=> element(lim_points_of_net(A,B),powerset(the_carrier(A))) ) ).
fof(dt_k1_funct_1,axiom,
$true ).
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_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_k3_yellow_6,axiom,
! [A,B,C,D] :
( ( ~ empty(A)
& ~ empty_carrier(B)
& rel_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_struct(A,B,C,D),the_carrier(B)) ) ).
fof(dt_k5_relat_1,axiom,
! [A,B] :
( ( relation(A)
& relation(B) )
=> relation(relation_composition(A,B)) ) ).
fof(dt_k7_funct_2,axiom,
! [A,B,C,D,E] :
( ( ~ empty(B)
& function(D)
& quasi_total(D,A,B)
& relation_of2(D,A,B)
& function(E)
& quasi_total(E,B,C)
& relation_of2(E,B,C) )
=> ( function(function_of_composition(A,B,C,D,E))
& quasi_total(function_of_composition(A,B,C,D,E),A,C)
& relation_of2_as_subset(function_of_composition(A,B,C,D,E),A,C) ) ) ).
fof(dt_l1_orders_2,axiom,
! [A] :
( rel_str(A)
=> one_sorted_str(A) ) ).
fof(dt_l1_pre_topc,axiom,
! [A] :
( top_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_connsp_2,axiom,
! [A,B] :
( ( ~ empty_carrier(A)
& topological_space(A)
& top_str(A)
& element(B,the_carrier(A)) )
=> ! [C] :
( point_neighbourhood(C,A,B)
=> element(C,powerset(the_carrier(A))) ) ) ).
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_m2_yellow_6,axiom,
! [A,B] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& ~ empty_carrier(B)
& transitive_relstr(B)
& directed_relstr(B)
& net_str(B,A) )
=> ! [C] :
( subnet(C,A,B)
=> ( ~ empty_carrier(C)
& transitive_relstr(C)
& directed_relstr(C)
& net_str(C,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_pre_topc,axiom,
? [A] : top_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_connsp_2,axiom,
! [A,B] :
( ( ~ empty_carrier(A)
& topological_space(A)
& top_str(A)
& element(B,the_carrier(A)) )
=> ? [C] : point_neighbourhood(C,A,B) ) ).
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(existence_m2_yellow_6,axiom,
! [A,B] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& ~ empty_carrier(B)
& transitive_relstr(B)
& directed_relstr(B)
& net_str(B,A) )
=> ? [C] : subnet(C,A,B) ) ).
fof(fc10_relat_1,axiom,
! [A,B] :
( ( empty(A)
& relation(B) )
=> ( empty(relation_composition(B,A))
& relation(relation_composition(B,A)) ) ) ).
fof(fc12_relat_1,axiom,
( empty(empty_set)
& relation(empty_set)
& relation_empty_yielding(empty_set) ) ).
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_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_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(fc4_relat_1,axiom,
( empty(empty_set)
& relation(empty_set) ) ).
fof(fc4_subset_1,axiom,
! [A,B] :
( ( ~ empty(A)
& ~ empty(B) )
=> ~ empty(cartesian_product2(A,B)) ) ).
fof(fc9_relat_1,axiom,
! [A,B] :
( ( empty(A)
& relation(B) )
=> ( empty(relation_composition(A,B))
& relation(relation_composition(A,B)) ) ) ).
fof(rc1_funct_1,axiom,
? [A] :
( relation(A)
& function(A) ) ).
fof(rc1_pboole,axiom,
? [A] :
( relation(A)
& relation_empty_yielding(A)
& function(A) ) ).
fof(rc1_relat_1,axiom,
? [A] :
( empty(A)
& relation(A) ) ).
fof(rc1_subset_1,axiom,
! [A] :
( ~ empty(A)
=> ? [B] :
( element(B,powerset(A))
& ~ empty(B) ) ) ).
fof(rc2_funct_1,axiom,
? [A] :
( relation(A)
& empty(A)
& function(A) ) ).
fof(rc2_relat_1,axiom,
? [A] :
( ~ empty(A)
& relation(A) ) ).
fof(rc2_subset_1,axiom,
! [A] :
? [B] :
( element(B,powerset(A))
& empty(B) ) ).
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(rc3_struct_0,axiom,
? [A] :
( one_sorted_str(A)
& ~ empty_carrier(A) ) ).
fof(rc4_funct_1,axiom,
? [A] :
( relation(A)
& relation_empty_yielding(A)
& function(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_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_k3_yellow_6,axiom,
! [A,B,C,D] :
( ( ~ empty(A)
& ~ empty_carrier(B)
& rel_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_struct(A,B,C,D) = apply(C,D) ) ).
fof(redefinition_k7_funct_2,axiom,
! [A,B,C,D,E] :
( ( ~ empty(B)
& function(D)
& quasi_total(D,A,B)
& relation_of2(D,A,B)
& function(E)
& quasi_total(E,B,C)
& relation_of2(E,B,C) )
=> function_of_composition(A,B,C,D,E) = relation_composition(D,E) ) ).
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(t21_funct_2,axiom,
! [A,B,C,D] :
( ( function(D)
& quasi_total(D,A,B)
& relation_of2_as_subset(D,A,B) )
=> ! [E] :
( ( relation(E)
& function(E) )
=> ( in(C,A)
=> ( B = empty_set
| apply(relation_composition(D,E),C) = apply(E,apply(D,C)) ) ) ) ) ).
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(t41_yellow_6,conjecture,
! [A] :
( ( ~ empty_carrier(A)
& topological_space(A)
& top_str(A) )
=> ! [B] :
( ( ~ empty_carrier(B)
& transitive_relstr(B)
& directed_relstr(B)
& net_str(B,A) )
=> ! [C] :
( subnet(C,A,B)
=> subset(lim_points_of_net(A,B),lim_points_of_net(A,C)) ) ) ) ).
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) ) ).
%------------------------------------------------------------------------------