TPTP Problem File: SEU335+1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SEU335+1 : TPTP v8.2.0. Released v3.3.0.
% Domain : Set theory
% Problem : MPTP bushy problem t13_tops_2
% 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-t13_tops_2 [Urb07]
% Status : Theorem
% Rating : 0.97 v8.2.0, 1.00 v7.4.0, 0.97 v7.1.0, 0.96 v7.0.0, 1.00 v6.0.0, 0.96 v5.4.0, 1.00 v3.3.0
% Syntax : Number of formulae : 53 ( 13 unt; 0 def)
% Number of atoms : 191 ( 29 equ)
% Maximal formula atoms : 21 ( 3 avg)
% Number of connectives : 151 ( 13 ~; 1 |; 61 &)
% ( 7 <=>; 69 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 5 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 20 ( 18 usr; 1 prp; 0-2 aty)
% Number of functors : 9 ( 9 usr; 1 con; 0-2 aty)
% Number of variables : 105 ( 97 !; 8 ?)
% 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(cc10_membered,axiom,
! [A] :
( v1_membered(A)
=> ! [B] :
( element(B,A)
=> v1_xcmplx_0(B) ) ) ).
fof(cc11_membered,axiom,
! [A] :
( v2_membered(A)
=> ! [B] :
( element(B,A)
=> ( v1_xcmplx_0(B)
& v1_xreal_0(B) ) ) ) ).
fof(cc12_membered,axiom,
! [A] :
( v3_membered(A)
=> ! [B] :
( element(B,A)
=> ( v1_xcmplx_0(B)
& v1_xreal_0(B)
& v1_rat_1(B) ) ) ) ).
fof(cc13_membered,axiom,
! [A] :
( v4_membered(A)
=> ! [B] :
( element(B,A)
=> ( v1_xcmplx_0(B)
& v1_xreal_0(B)
& v1_int_1(B)
& v1_rat_1(B) ) ) ) ).
fof(cc14_membered,axiom,
! [A] :
( v5_membered(A)
=> ! [B] :
( element(B,A)
=> ( v1_xcmplx_0(B)
& natural(B)
& v1_xreal_0(B)
& v1_int_1(B)
& v1_rat_1(B) ) ) ) ).
fof(cc15_membered,axiom,
! [A] :
( empty(A)
=> ( v1_membered(A)
& v2_membered(A)
& v3_membered(A)
& v4_membered(A)
& v5_membered(A) ) ) ).
fof(cc16_membered,axiom,
! [A] :
( v1_membered(A)
=> ! [B] :
( element(B,powerset(A))
=> v1_membered(B) ) ) ).
fof(cc17_membered,axiom,
! [A] :
( v2_membered(A)
=> ! [B] :
( element(B,powerset(A))
=> ( v1_membered(B)
& v2_membered(B) ) ) ) ).
fof(cc18_membered,axiom,
! [A] :
( v3_membered(A)
=> ! [B] :
( element(B,powerset(A))
=> ( v1_membered(B)
& v2_membered(B)
& v3_membered(B) ) ) ) ).
fof(cc19_membered,axiom,
! [A] :
( v4_membered(A)
=> ! [B] :
( element(B,powerset(A))
=> ( v1_membered(B)
& v2_membered(B)
& v3_membered(B)
& v4_membered(B) ) ) ) ).
fof(cc1_membered,axiom,
! [A] :
( v5_membered(A)
=> v4_membered(A) ) ).
fof(cc20_membered,axiom,
! [A] :
( v5_membered(A)
=> ! [B] :
( element(B,powerset(A))
=> ( v1_membered(B)
& v2_membered(B)
& v3_membered(B)
& v4_membered(B)
& v5_membered(B) ) ) ) ).
fof(cc2_membered,axiom,
! [A] :
( v4_membered(A)
=> v3_membered(A) ) ).
fof(cc3_membered,axiom,
! [A] :
( v3_membered(A)
=> v2_membered(A) ) ).
fof(cc4_membered,axiom,
! [A] :
( v2_membered(A)
=> v1_membered(A) ) ).
fof(d5_funct_1,axiom,
! [A] :
( ( relation(A)
& function(A) )
=> ! [B] :
( B = relation_rng(A)
<=> ! [C] :
( in(C,B)
<=> ? [D] :
( in(D,relation_dom(A))
& C = apply(A,D) ) ) ) ) ).
fof(d8_setfam_1,axiom,
! [A,B] :
( element(B,powerset(powerset(A)))
=> ! [C] :
( element(C,powerset(powerset(A)))
=> ( C = complements_of_subsets(A,B)
<=> ! [D] :
( element(D,powerset(A))
=> ( in(D,C)
<=> in(subset_complement(A,D),B) ) ) ) ) ) ).
fof(dt_k1_funct_1,axiom,
$true ).
fof(dt_k1_relat_1,axiom,
$true ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k2_relat_1,axiom,
$true ).
fof(dt_k3_subset_1,axiom,
! [A,B] :
( element(B,powerset(A))
=> element(subset_complement(A,B),powerset(A)) ) ).
fof(dt_k7_setfam_1,axiom,
! [A,B] :
( element(B,powerset(powerset(A)))
=> element(complements_of_subsets(A,B),powerset(powerset(A))) ) ).
fof(dt_k9_relat_1,axiom,
$true ).
fof(dt_l1_struct_0,axiom,
$true ).
fof(dt_m1_subset_1,axiom,
$true ).
fof(dt_u1_struct_0,axiom,
$true ).
fof(existence_l1_struct_0,axiom,
? [A] : one_sorted_str(A) ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : element(B,A) ).
fof(fc1_subset_1,axiom,
! [A] : ~ empty(powerset(A)) ).
fof(fc6_membered,axiom,
( empty(empty_set)
& v1_membered(empty_set)
& v2_membered(empty_set)
& v3_membered(empty_set)
& v4_membered(empty_set)
& v5_membered(empty_set) ) ).
fof(involutiveness_k3_subset_1,axiom,
! [A,B] :
( element(B,powerset(A))
=> subset_complement(A,subset_complement(A,B)) = B ) ).
fof(involutiveness_k7_setfam_1,axiom,
! [A,B] :
( element(B,powerset(powerset(A)))
=> complements_of_subsets(A,complements_of_subsets(A,B)) = B ) ).
fof(rc1_membered,axiom,
? [A] :
( ~ empty(A)
& v1_membered(A)
& v2_membered(A)
& v3_membered(A)
& v4_membered(A)
& v5_membered(A) ) ).
fof(rc1_subset_1,axiom,
! [A] :
( ~ empty(A)
=> ? [B] :
( element(B,powerset(A))
& ~ empty(B) ) ) ).
fof(rc2_subset_1,axiom,
! [A] :
? [B] :
( element(B,powerset(A))
& empty(B) ) ).
fof(reflexivity_r1_tarski,axiom,
! [A,B] : subset(A,A) ).
fof(s2_funct_1__e4_7_1__tops_2,axiom,
! [A,B] :
( ( one_sorted_str(A)
& element(B,powerset(powerset(the_carrier(A)))) )
=> ( ( ! [C,D,E] :
( ( in(C,complements_of_subsets(the_carrier(A),B))
& ! [F] :
( element(F,powerset(the_carrier(A)))
=> ( F = C
=> D = subset_complement(the_carrier(A),F) ) )
& ! [G] :
( element(G,powerset(the_carrier(A)))
=> ( G = C
=> E = subset_complement(the_carrier(A),G) ) ) )
=> D = E )
& ! [C] :
~ ( in(C,complements_of_subsets(the_carrier(A),B))
& ! [D] :
~ ! [H] :
( element(H,powerset(the_carrier(A)))
=> ( H = C
=> D = subset_complement(the_carrier(A),H) ) ) ) )
=> ? [C] :
( relation(C)
& function(C)
& relation_dom(C) = complements_of_subsets(the_carrier(A),B)
& ! [D] :
( in(D,complements_of_subsets(the_carrier(A),B))
=> ! [I] :
( element(I,powerset(the_carrier(A)))
=> ( I = D
=> apply(C,D) = subset_complement(the_carrier(A),I) ) ) ) ) ) ) ).
fof(s2_funct_1__e4_7_2__tops_2,axiom,
! [A,B] :
( ( one_sorted_str(A)
& element(B,powerset(powerset(the_carrier(A)))) )
=> ( ( ! [C,D,E] :
( ( in(C,B)
& ! [F] :
( element(F,powerset(the_carrier(A)))
=> ( F = C
=> D = subset_complement(the_carrier(A),F) ) )
& ! [G] :
( element(G,powerset(the_carrier(A)))
=> ( G = C
=> E = subset_complement(the_carrier(A),G) ) ) )
=> D = E )
& ! [C] :
~ ( in(C,B)
& ! [D] :
~ ! [H] :
( element(H,powerset(the_carrier(A)))
=> ( H = C
=> D = subset_complement(the_carrier(A),H) ) ) ) )
=> ? [C] :
( relation(C)
& function(C)
& relation_dom(C) = B
& ! [D] :
( in(D,B)
=> ! [I] :
( element(I,powerset(the_carrier(A)))
=> ( I = D
=> apply(C,D) = subset_complement(the_carrier(A),I) ) ) ) ) ) ) ).
fof(t13_tops_2,conjecture,
! [A] :
( one_sorted_str(A)
=> ! [B] :
( element(B,powerset(powerset(the_carrier(A))))
=> ( finite(complements_of_subsets(the_carrier(A),B))
<=> finite(B) ) ) ) ).
fof(t146_relat_1,axiom,
! [A] :
( relation(A)
=> relation_image(A,relation_dom(A)) = relation_rng(A) ) ).
fof(t17_finset_1,axiom,
! [A,B] :
( ( relation(B)
& function(B) )
=> ( finite(A)
=> finite(relation_image(B,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(t2_tarski,axiom,
! [A,B] :
( ! [C] :
( in(C,A)
<=> in(C,B) )
=> 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) ) ).
%------------------------------------------------------------------------------