## TPTP Problem File: SEV319^5.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEV319^5 : TPTP v7.5.0. Released v4.0.0.
% Domain   : Set Theory
% Problem  : TPS problem THM145L
% Version  : Especial.
% English  : Tarski's (actually Knaster's) Fixed Point Theorem for lattices:
%            In a complete lattice, every monotone function has a fixed point.

% Refs     : [Bro09] Brown (2009), Email to Geoff Sutcliffe
% Source   : [Bro09]
% Names    : tps_0570 [Bro09]
%          : THM145 [TPS]
%          : THM145L [TPS]

% Status   : Theorem
% Rating   : 0.92 v7.5.0, 1.00 v4.0.0
% Syntax   : Number of formulae    :    2 (   0 unit;   1 type;   0 defn)
%            Number of atoms       :   40 (   0 equality;  40 variable)
%            Maximal formula depth :   13 (   8 average)
%            Number of connectives :   39 (   0   ~;   0   |;   4   &;  28   @)
%                                         (   0 <=>;   7  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :    6 (   6   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    3 (   1   :;   0   =)
%            Number of variables   :   13 (   0 sgn;  12   !;   1   ?;   0   ^)
%                                         (  13   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH0_THM_NEQ_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(cTHM145L_pme,conjecture,(
! [L: a > a > \$o,U: ( a > \$o ) > a] :
( ( ! [Xx: a,Xy: a,Xz: a] :
( ( ( L @ Xx @ Xy )
& ( L @ Xy @ Xz ) )
=> ( L @ Xx @ Xz ) )
& ! [Xs: a > \$o] :
( ! [Xz: a] :
( ( Xs @ Xz )
=> ( L @ Xz @ ( U @ Xs ) ) )
& ! [Xj: a] :
( ! [Xk: a] :
( ( Xs @ Xk )
=> ( L @ Xk @ Xj ) )
=> ( L @ ( U @ Xs ) @ Xj ) ) ) )
=> ! [Xf: a > a] :
( ! [Xx: a,Xy: a] :
( ( L @ Xx @ Xy )
=> ( L @ ( Xf @ Xx ) @ ( Xf @ Xy ) ) )
=> ? [Xw: a] :
( ( L @ Xw @ ( Xf @ Xw ) )
& ( L @ ( Xf @ Xw ) @ Xw ) ) ) ) )).

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