## TPTP Problem File: SEV191^5.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEV191^5 : TPTP v7.5.0. Released v4.0.0.
% Domain   : Set Theory (Sets of sets)
% Problem  : TPS problem S-JOINFN-MONOTONE
% Version  : Especial.
% English  : The function used to define JOIN as a LFP is monotone.

% Refs     : [Bro09] Brown (2009), Email to Geoff Sutcliffe
% Source   : [Bro09]
% Names    : tps_0503 [Bro09]
%          : S-JOINFN-MONOTONE [TPS]

% Status   : Theorem
% Rating   : 0.18 v7.5.0, 0.14 v7.4.0, 0.22 v7.2.0, 0.12 v7.1.0, 0.25 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.29 v5.5.0, 0.17 v5.4.0, 0.20 v5.1.0, 0.40 v5.0.0, 0.20 v4.1.0, 0.00 v4.0.0
% Syntax   : Number of formulae    :    4 (   0 unit;   3 type;   0 defn)
%            Number of atoms       :   82 (  14 equality;  54 variable)
%            Maximal formula depth :   24 (   8 average)
%            Number of connectives :   53 (   0   ~;   4   |;  15   &;  30   @)
%                                         (   0 <=>;   4  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   11 (  11   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    6 (   3   :;   0   =)
%            Number of variables   :   21 (   1 sgn;   9   !;  12   ?;   0   ^)
%                                         (  21   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% 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
%          : Polymorphic definitions expanded.
%------------------------------------------------------------------------------
thf(a_type,type,(
a: \$tType )).

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

thf(c0,type,(
c0: a )).

thf(cS_JOINFN_MONOTONE_pme,conjecture,
( ! [R: a > a > a > \$o] :
( \$true
=> \$true )
& ! [R: a > a > a > \$o,S: a > a > a > \$o] :
( ( \$true
& \$true
& ! [Xa: a,Xb: a,Xc: a] :
( ( R @ Xa @ Xb @ Xc )
=> ( S @ Xa @ Xb @ Xc ) ) )
=> ! [Xa: a,Xb: a,Xc: a] :
( ( ( ( Xa = c0 )
& ( Xb = Xc ) )
| ( ( Xb = c0 )
& ( Xa = Xc ) )
| ? [Xx1: a,Xx2: a,Xy1: a,Xy2: a,Xz1: a,Xz2: a] :
( ( Xa
= ( cP @ Xx1 @ Xx2 ) )
& ( Xb
= ( cP @ Xy1 @ Xy2 ) )
& ( Xc
= ( cP @ Xz1 @ Xz2 ) )
& ( R @ Xx1 @ Xy1 @ Xz1 )
& ( R @ Xx2 @ Xy2 @ Xz2 ) ) )
=> ( ( ( Xa = c0 )
& ( Xb = Xc ) )
| ( ( Xb = c0 )
& ( Xa = Xc ) )
| ? [Xx1: a,Xx2: a,Xy1: a,Xy2: a,Xz1: a,Xz2: a] :
( ( Xa
= ( cP @ Xx1 @ Xx2 ) )
& ( Xb
= ( cP @ Xy1 @ Xy2 ) )
& ( Xc
= ( cP @ Xz1 @ Xz2 ) )
& ( S @ Xx1 @ Xy1 @ Xz1 )
& ( S @ Xx2 @ Xy2 @ Xz2 ) ) ) ) ) )).

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