TPTP Problem File: SEU386+1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SEU386+1 : TPTP v9.0.0. Released v3.3.0.
% Domain : Set theory
% Problem : MPTP bushy problem t29_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-t29_waybel_9 [Urb07]
% Status : Theorem
% Rating : 0.24 v9.0.0, 0.22 v8.2.0, 0.19 v8.1.0, 0.17 v7.5.0, 0.19 v7.4.0, 0.20 v7.3.0, 0.21 v7.2.0, 0.17 v7.0.0, 0.20 v6.4.0, 0.27 v6.3.0, 0.17 v6.2.0, 0.24 v6.1.0, 0.33 v6.0.0, 0.22 v5.5.0, 0.26 v5.4.0, 0.36 v5.3.0, 0.44 v5.2.0, 0.30 v5.1.0, 0.29 v5.0.0, 0.42 v4.1.0, 0.39 v4.0.0, 0.42 v3.7.0, 0.35 v3.5.0, 0.37 v3.3.0
% Syntax : Number of formulae : 45 ( 13 unt; 0 def)
% Number of atoms : 139 ( 3 equ)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 123 ( 29 ~; 1 |; 49 &)
% ( 4 <=>; 40 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 23 ( 21 usr; 1 prp; 0-3 aty)
% Number of functors : 4 ( 4 usr; 1 con; 0-2 aty)
% Number of variables : 73 ( 60 !; 13 ?)
% 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_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(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(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(d9_waybel_9,axiom,
! [A] :
( ( ~ empty_carrier(A)
& topological_space(A)
& top_str(A) )
=> ! [B] :
( ( ~ empty_carrier(B)
& net_str(B,A) )
=> ! [C] :
( element(C,the_carrier(A))
=> ( is_a_cluster_point_of_netstr(A,B,C)
<=> ! [D] :
( point_neighbourhood(D,A,C)
=> is_often_in(A,B,D) ) ) ) ) ) ).
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_xboole_0,axiom,
$true ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
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_subset_1,axiom,
$true ).
fof(dt_u1_struct_0,axiom,
$true ).
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_subset_1,axiom,
! [A] :
? [B] : element(B,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(rc1_finset_1,axiom,
? [A] :
( ~ empty(A)
& finite(A) ) ).
fof(rc1_xboole_0,axiom,
? [A] : empty(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_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(rc5_struct_0,axiom,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ? [B] :
( element(B,powerset(the_carrier(A)))
& ~ empty(B) ) ) ).
fof(reflexivity_r1_tarski,axiom,
! [A,B] : subset(A,A) ).
fof(t1_subset,axiom,
! [A,B] :
( in(A,B)
=> element(A,B) ) ).
fof(t28_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] :
( is_eventually_in(A,B,C)
=> is_often_in(A,B,C) ) ) ) ).
fof(t29_waybel_9,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] :
( element(C,the_carrier(A))
=> ( in(C,lim_points_of_net(A,B))
=> is_a_cluster_point_of_netstr(A,B,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(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) ) ).
%------------------------------------------------------------------------------