## TPTP Problem File: SEV192^5.p

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```%------------------------------------------------------------------------------
% File     : SEV192^5 : TPTP v8.0.0. Released v4.0.0.
% Domain   : Set Theory (Sets of sets)
% Problem  : TPS problem CS-DUC-RELNS
% Version  : Especial.
% English  : Given a pairing algebra B (with zero element 0(B) and pairing
%            operation P(BBB)), we can define a notion of join and a notion of
%            inclusion. A subset of the pairing algebra is a DUC-set (downward
%            union closed)  if it is downward closed with respect to
%            inclusion, and closed with respect to joins. A relation R between
%            any set C and the pairing algebra is DUC-valued if for any c in
%            C, {y | R(c,y)} is a DUC-set. The theorem states that the
%            DUC-valued relations form a closure system, i.e., an arbitrary
%            intersection of DUC-valued relations is a DUC-valued relation.

% Refs     : [Bro09] Brown (2009), Email to Geoff Sutcliffe
% Source   : [Bro09]
% Names    : tps_0573 [Bro09]
%          : CS-DUC-RELNS [TPS]

% Status   : Theorem
% Rating   : 0.82 v7.5.0, 0.71 v7.4.0, 0.44 v7.2.0, 0.38 v7.0.0, 0.29 v6.4.0, 0.50 v6.3.0, 0.60 v6.2.0, 0.71 v6.0.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.60 v4.1.0, 1.00 v4.0.0
% Syntax   : Number of formulae    :    5 (   0 unt;   4 typ;   0 def)
%            Number of atoms       :   32 (  28 equ;   0 cnn)
%            Maximal formula atoms :   32 (  32 avg)
%            Number of connectives :  169 (   0   ~;   8   |;  38   &; 103   @)
%                                         (   0 <=>;  20  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   36 (  36 avg)
%            Number of types       :    3 (   2 usr)
%            Number of type conns  :   31 (  31   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    4 (   2 usr;   2 con; 0-2 aty)
%            Number of variables   :   60 (   0   ^  36   !;  24   ?;  60   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This problem is from the TPS library. Copyright (c) 2009 The TPS
%            project in the Department of Mathematical Sciences at Carnegie
%            Mellon University. Distributed under the Creative Commons copyleft
%            license: http://creativecommons.org/licenses/by-sa/3.0/
%          : Polymorphic definitions expanded.
%------------------------------------------------------------------------------
thf(b_type,type,
b: \$tType ).

thf(c_type,type,
c: \$tType ).

thf(cP,type,
cP: b > b > b ).

thf(c0,type,
c0: b ).

thf(cCS_DUC_RELNS_pme,conjecture,
! [S: ( c > b > \$o ) > \$o] :
( ! [Xx: c > b > \$o] :
( ( S @ Xx )
=> ! [Xc: c] :
( ( Xx @ Xc @ c0 )
& ! [Xx0: b,Xy: b] :
( ( ( Xx @ Xc @ Xy )
& ! [R: b > b > b > \$o] :
( ( \$true
& ! [Xa: b,Xb: b,Xc0: b] :
( ( ( ( Xa = c0 )
& ( Xb = Xc0 ) )
| ( ( Xb = c0 )
& ( Xa = Xc0 ) )
| ? [Xx1: b,Xx2: b,Xy1: b,Xy2: b,Xz1: b,Xz2: b] :
( ( Xa
= ( cP @ Xx1 @ Xx2 ) )
& ( Xb
= ( cP @ Xy1 @ Xy2 ) )
& ( Xc0
= ( cP @ Xz1 @ Xz2 ) )
& ( R @ Xx1 @ Xy1 @ Xz1 )
& ( R @ Xx2 @ Xy2 @ Xz2 ) ) )
=> ( R @ Xa @ Xb @ Xc0 ) ) )
=> ( R @ Xx0 @ Xy @ Xy ) ) )
=> ( Xx @ Xc @ Xx0 ) )
& ! [Xx0: b,Xy: b,Xz: b] :
( ( ( Xx @ Xc @ Xx0 )
& ( Xx @ Xc @ Xy )
& ! [R: b > b > b > \$o] :
( ( \$true
& ! [Xa: b,Xb: b,Xc0: b] :
( ( ( ( Xa = c0 )
& ( Xb = Xc0 ) )
| ( ( Xb = c0 )
& ( Xa = Xc0 ) )
| ? [Xx1: b,Xx2: b,Xy1: b,Xy2: b,Xz1: b,Xz2: b] :
( ( Xa
= ( cP @ Xx1 @ Xx2 ) )
& ( Xb
= ( cP @ Xy1 @ Xy2 ) )
& ( Xc0
= ( cP @ Xz1 @ Xz2 ) )
& ( R @ Xx1 @ Xy1 @ Xz1 )
& ( R @ Xx2 @ Xy2 @ Xz2 ) ) )
=> ( R @ Xa @ Xb @ Xc0 ) ) )
=> ( R @ Xx0 @ Xy @ Xz ) ) )
=> ( Xx @ Xc @ Xz ) ) ) )
=> ! [Xc: c] :
( ! [R: c > b > \$o] :
( ( S @ R )
=> ( R @ Xc @ c0 ) )
& ! [Xx: b,Xy: b] :
( ( ! [R: c > b > \$o] :
( ( S @ R )
=> ( R @ Xc @ Xy ) )
& ! [R: b > b > b > \$o] :
( ( \$true
& ! [Xa: b,Xb: b,Xc0: b] :
( ( ( ( Xa = c0 )
& ( Xb = Xc0 ) )
| ( ( Xb = c0 )
& ( Xa = Xc0 ) )
| ? [Xx1: b,Xx2: b,Xy1: b,Xy2: b,Xz1: b,Xz2: b] :
( ( Xa
= ( cP @ Xx1 @ Xx2 ) )
& ( Xb
= ( cP @ Xy1 @ Xy2 ) )
& ( Xc0
= ( cP @ Xz1 @ Xz2 ) )
& ( R @ Xx1 @ Xy1 @ Xz1 )
& ( R @ Xx2 @ Xy2 @ Xz2 ) ) )
=> ( R @ Xa @ Xb @ Xc0 ) ) )
=> ( R @ Xx @ Xy @ Xy ) ) )
=> ! [R: c > b > \$o] :
( ( S @ R )
=> ( R @ Xc @ Xx ) ) )
& ! [Xx: b,Xy: b,Xz: b] :
( ( ! [R: c > b > \$o] :
( ( S @ R )
=> ( R @ Xc @ Xx ) )
& ! [R: c > b > \$o] :
( ( S @ R )
=> ( R @ Xc @ Xy ) )
& ! [R: b > b > b > \$o] :
( ( \$true
& ! [Xa: b,Xb: b,Xc0: b] :
( ( ( ( Xa = c0 )
& ( Xb = Xc0 ) )
| ( ( Xb = c0 )
& ( Xa = Xc0 ) )
| ? [Xx1: b,Xx2: b,Xy1: b,Xy2: b,Xz1: b,Xz2: b] :
( ( Xa
= ( cP @ Xx1 @ Xx2 ) )
& ( Xb
= ( cP @ Xy1 @ Xy2 ) )
& ( Xc0
= ( cP @ Xz1 @ Xz2 ) )
& ( R @ Xx1 @ Xy1 @ Xz1 )
& ( R @ Xx2 @ Xy2 @ Xz2 ) ) )
=> ( R @ Xa @ Xb @ Xc0 ) ) )
=> ( R @ Xx @ Xy @ Xz ) ) )
=> ! [R: c > b > \$o] :
( ( S @ R )
=> ( R @ Xc @ Xz ) ) ) ) ) ).

%------------------------------------------------------------------------------
```