TPTP Problem File: SEV192^5.p
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%------------------------------------------------------------------------------
% File : SEV192^5 : TPTP v8.2.0. 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 : 0.70 v8.2.0, 0.77 v8.1.0, 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 ) ) ) ) ) ).
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