TPTP Problem File: SEU329+1.p
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%------------------------------------------------------------------------------
% File : SEU329+1 : TPTP v8.2.0. Released v3.3.0.
% Domain : Set theory
% Problem : MPTP bushy problem s1_xboole_0__e4_5_1__funct_1
% 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-s1_xboole_0__e4_5_1__funct_1 [Urb07]
% Status : Theorem
% Rating : 0.61 v7.5.0, 0.72 v7.4.0, 0.53 v7.3.0, 0.59 v7.1.0, 0.57 v7.0.0, 0.63 v6.4.0, 0.62 v6.3.0, 0.58 v6.2.0, 0.60 v6.1.0, 0.73 v6.0.0, 0.65 v5.5.0, 0.70 v5.4.0, 0.71 v5.3.0, 0.78 v5.2.0, 0.70 v5.1.0, 0.71 v5.0.0, 0.75 v4.1.0, 0.70 v4.0.1, 0.74 v4.0.0, 0.75 v3.5.0, 0.74 v3.3.0
% Syntax : Number of formulae : 34 ( 7 unt; 0 def)
% Number of atoms : 129 ( 18 equ)
% Maximal formula atoms : 23 ( 3 avg)
% Number of connectives : 103 ( 8 ~; 0 |; 48 &)
% ( 2 <=>; 45 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 5 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 17 ( 15 usr; 1 prp; 0-2 aty)
% Number of functors : 6 ( 6 usr; 0 con; 1-2 aty)
% Number of variables : 72 ( 58 !; 14 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Translated by MPTP 0.2 from the original problem in the Mizar
% library, www.mizar.org
%------------------------------------------------------------------------------
fof(s1_xboole_0__e4_7_1__tops_2__1,conjecture,
! [A,B] :
( ( one_sorted_str(A)
& element(B,powerset(powerset(the_carrier(A)))) )
=> ! [C] :
? [D] :
! [E] :
( in(E,D)
<=> ( in(E,cartesian_product2(complements_of_subsets(the_carrier(A),B),C))
& ? [F,G] :
( ordered_pair(F,G) = E
& in(F,complements_of_subsets(the_carrier(A),B))
& ! [H] :
( element(H,powerset(the_carrier(A)))
=> ( H = F
=> G = subset_complement(the_carrier(A),H) ) ) ) ) ) ) ).
fof(cc1_membered,axiom,
! [A] :
( v5_membered(A)
=> v4_membered(A) ) ).
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(rc1_membered,axiom,
? [A] :
( ~ empty(A)
& v1_membered(A)
& v2_membered(A)
& v3_membered(A)
& v4_membered(A)
& v5_membered(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(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(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(cc15_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(fc4_subset_1,axiom,
! [A,B] :
( ( ~ empty(A)
& ~ empty(B) )
=> ~ empty(cartesian_product2(A,B)) ) ).
fof(rc2_subset_1,axiom,
! [A] :
? [B] :
( element(B,powerset(A))
& empty(B) ) ).
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(antisymmetry_r2_hidden,axiom,
! [A,B] :
( in(A,B)
=> ~ in(B,A) ) ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k2_zfmisc_1,axiom,
$true ).
fof(dt_k3_subset_1,axiom,
! [A,B] :
( element(B,powerset(A))
=> element(subset_complement(A,B),powerset(A)) ) ).
fof(dt_k4_tarski,axiom,
$true ).
fof(dt_k7_setfam_1,axiom,
! [A,B] :
( element(B,powerset(powerset(A)))
=> element(complements_of_subsets(A,B),powerset(powerset(A))) ) ).
fof(dt_l1_struct_0,axiom,
$true ).
fof(dt_m1_subset_1,axiom,
$true ).
fof(dt_u1_struct_0,axiom,
$true ).
fof(cc1_relset_1,axiom,
! [A,B,C] :
( element(C,powerset(cartesian_product2(A,B)))
=> relation(C) ) ).
fof(fc1_subset_1,axiom,
! [A] : ~ empty(powerset(A)) ).
fof(s1_tarski__e4_7_1__tops_2__2,axiom,
! [A,B] :
( ( one_sorted_str(A)
& element(B,powerset(powerset(the_carrier(A)))) )
=> ! [C] :
( ! [D,E,F] :
( ( D = E
& ? [G,H] :
( ordered_pair(G,H) = E
& in(G,complements_of_subsets(the_carrier(A),B))
& ! [I] :
( element(I,powerset(the_carrier(A)))
=> ( I = G
=> H = subset_complement(the_carrier(A),I) ) ) )
& D = F
& ? [J,K] :
( ordered_pair(J,K) = F
& in(J,complements_of_subsets(the_carrier(A),B))
& ! [L] :
( element(L,powerset(the_carrier(A)))
=> ( L = J
=> K = subset_complement(the_carrier(A),L) ) ) ) )
=> E = F )
=> ? [D] :
! [E] :
( in(E,D)
<=> ? [F] :
( in(F,cartesian_product2(complements_of_subsets(the_carrier(A),B),C))
& F = E
& ? [M,N] :
( ordered_pair(M,N) = E
& in(M,complements_of_subsets(the_carrier(A),B))
& ! [O] :
( element(O,powerset(the_carrier(A)))
=> ( O = M
=> N = subset_complement(the_carrier(A),O) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------