TPTP Problem File: SEU366+1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SEU366+1 : TPTP v8.2.0. Released v3.3.0.
% Domain : Set theory
% Problem : MPTP bushy problem t1_waybel_0
% 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-t1_waybel_0 [Urb07]
% Status : Theorem
% Rating : 1.00 v3.3.0
% Syntax : Number of formulae : 61 ( 23 unt; 0 def)
% Number of atoms : 177 ( 14 equ)
% Maximal formula atoms : 21 ( 2 avg)
% Number of connectives : 151 ( 35 ~; 3 |; 60 &)
% ( 11 <=>; 42 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 15 ( 13 usr; 1 prp; 0-3 aty)
% Number of functors : 7 ( 7 usr; 1 con; 0-3 aty)
% Number of variables : 116 ( 99 !; 17 ?)
% 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(cc2_finset_1,axiom,
! [A] :
( finite(A)
=> ! [B] :
( element(B,powerset(A))
=> finite(B) ) ) ).
fof(commutativity_k2_struct_0,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& element(B,the_carrier(A))
& element(C,the_carrier(A)) )
=> unordered_pair_as_carrier_subset(A,B,C) = unordered_pair_as_carrier_subset(A,C,B) ) ).
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(d1_tarski,axiom,
! [A,B] :
( B = singleton(A)
<=> ! [C] :
( in(C,B)
<=> C = A ) ) ).
fof(d1_waybel_0,axiom,
! [A] :
( rel_str(A)
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> ( directed_subset(B,A)
<=> ! [C] :
( element(C,the_carrier(A))
=> ! [D] :
( element(D,the_carrier(A))
=> ~ ( in(C,B)
& in(D,B)
& ! [E] :
( element(E,the_carrier(A))
=> ~ ( in(E,B)
& related(A,C,E)
& related(A,D,E) ) ) ) ) ) ) ) ) ).
fof(d2_tarski,axiom,
! [A,B,C] :
( C = unordered_pair(A,B)
<=> ! [D] :
( in(D,C)
<=> ( D = A
| D = B ) ) ) ).
fof(d2_xboole_0,axiom,
! [A,B,C] :
( C = set_union2(A,B)
<=> ! [D] :
( in(D,C)
<=> ( in(D,A)
| in(D,B) ) ) ) ).
fof(d9_lattice3,axiom,
! [A] :
( rel_str(A)
=> ! [B,C] :
( element(C,the_carrier(A))
=> ( relstr_set_smaller(A,B,C)
<=> ! [D] :
( element(D,the_carrier(A))
=> ( in(D,B)
=> related(A,D,C) ) ) ) ) ) ).
fof(dt_k1_tarski,axiom,
$true ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k2_struct_0,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& element(B,the_carrier(A))
& element(C,the_carrier(A)) )
=> element(unordered_pair_as_carrier_subset(A,B,C),powerset(the_carrier(A))) ) ).
fof(dt_k2_tarski,axiom,
$true ).
fof(dt_k2_xboole_0,axiom,
$true ).
fof(dt_l1_orders_2,axiom,
! [A] :
( rel_str(A)
=> one_sorted_str(A) ) ).
fof(dt_l1_struct_0,axiom,
$true ).
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_struct_0,axiom,
? [A] : one_sorted_str(A) ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : element(B,A) ).
fof(fc1_finset_1,axiom,
! [A] :
( ~ empty(singleton(A))
& finite(singleton(A)) ) ).
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_finset_1,axiom,
! [A,B] :
( ~ empty(unordered_pair(A,B))
& finite(unordered_pair(A,B)) ) ).
fof(fc2_subset_1,axiom,
! [A] : ~ empty(singleton(A)) ).
fof(fc2_xboole_0,axiom,
! [A,B] :
( ~ empty(A)
=> ~ empty(set_union2(A,B)) ) ).
fof(fc3_subset_1,axiom,
! [A,B] : ~ empty(unordered_pair(A,B)) ).
fof(fc3_xboole_0,axiom,
! [A,B] :
( ~ empty(A)
=> ~ empty(set_union2(B,A)) ) ).
fof(fc9_finset_1,axiom,
! [A,B] :
( ( finite(A)
& finite(B) )
=> finite(set_union2(A,B)) ) ).
fof(idempotence_k2_xboole_0,axiom,
! [A,B] : set_union2(A,A) = A ).
fof(rc1_finset_1,axiom,
? [A] :
( ~ empty(A)
& finite(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_subset_1,axiom,
! [A] :
? [B] :
( element(B,powerset(A))
& empty(B) ) ).
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(redefinition_k2_struct_0,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& element(B,the_carrier(A))
& element(C,the_carrier(A)) )
=> unordered_pair_as_carrier_subset(A,B,C) = unordered_pair(B,C) ) ).
fof(reflexivity_r1_tarski,axiom,
! [A,B] : subset(A,A) ).
fof(s2_finset_1__e11_2_1__waybel_0,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& transitive_relstr(A)
& rel_str(A)
& element(B,powerset(the_carrier(A)))
& finite(C)
& element(C,powerset(B)) )
=> ( ( finite(C)
& ? [D] :
( element(D,the_carrier(A))
& in(D,B)
& relstr_set_smaller(A,empty_set,D) )
& ! [E,F] :
( ( in(E,C)
& subset(F,C)
& ? [G] :
( element(G,the_carrier(A))
& in(G,B)
& relstr_set_smaller(A,F,G) ) )
=> ? [H] :
( element(H,the_carrier(A))
& in(H,B)
& relstr_set_smaller(A,set_union2(F,singleton(E)),H) ) ) )
=> ? [I] :
( element(I,the_carrier(A))
& in(I,B)
& relstr_set_smaller(A,C,I) ) ) ) ).
fof(t1_boole,axiom,
! [A] : set_union2(A,empty_set) = A ).
fof(t1_subset,axiom,
! [A,B] :
( in(A,B)
=> element(A,B) ) ).
fof(t1_waybel_0,conjecture,
! [A] :
( ( ~ empty_carrier(A)
& transitive_relstr(A)
& rel_str(A) )
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> ( ( ~ empty(B)
& directed_subset(B,A) )
<=> ! [C] :
( ( finite(C)
& element(C,powerset(B)) )
=> ? [D] :
( element(D,the_carrier(A))
& in(D,B)
& relstr_set_smaller(A,C,D) ) ) ) ) ) ).
fof(t26_orders_2,axiom,
! [A] :
( ( transitive_relstr(A)
& rel_str(A) )
=> ! [B] :
( element(B,the_carrier(A))
=> ! [C] :
( element(C,the_carrier(A))
=> ! [D] :
( element(D,the_carrier(A))
=> ( ( related(A,B,C)
& related(A,C,D) )
=> related(A,B,D) ) ) ) ) ) ).
fof(t2_subset,axiom,
! [A,B] :
( element(A,B)
=> ( empty(B)
| in(A,B) ) ) ).
fof(t2_xboole_1,axiom,
! [A] : subset(empty_set,A) ).
fof(t38_zfmisc_1,axiom,
! [A,B,C] :
( subset(unordered_pair(A,B),C)
<=> ( in(A,C)
& in(B,C) ) ) ).
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(t6_yellow_0,axiom,
! [A] :
( rel_str(A)
=> ! [B] :
( element(B,the_carrier(A))
=> ( relstr_set_smaller(A,empty_set,B)
& relstr_element_smaller(A,empty_set,B) ) ) ) ).
fof(t7_boole,axiom,
! [A,B] :
~ ( in(A,B)
& empty(B) ) ).
fof(t8_boole,axiom,
! [A,B] :
~ ( empty(A)
& A != B
& empty(B) ) ).
%------------------------------------------------------------------------------