TPTP Problem File: SEV319^5.p
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% File : SEV319^5 : TPTP v9.0.0. 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 : 0.75 v9.0.0, 0.92 v8.2.0, 0.91 v8.1.0, 0.92 v7.5.0, 1.00 v4.0.0
% Syntax : Number of formulae : 2 ( 0 unt; 1 typ; 0 def)
% Number of atoms : 0 ( 0 equ; 0 cnn)
% Maximal formula atoms : 0 ( 0 avg)
% Number of connectives : 39 ( 0 ~; 0 |; 4 &; 28 @)
% ( 0 <=>; 7 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 13 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 6 ( 6 >; 0 *; 0 +; 0 <<)
% Number of symbols : 0 ( 0 usr; 0 con; --- aty)
% Number of variables : 13 ( 0 ^; 12 !; 1 ?; 13 :)
% 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
% Mellon University. Distributed under the Creative Commons copyleft
% license: http://creativecommons.org/licenses/by-sa/3.0/
% : Polymorphic definitions expanded.
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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 ) ) ) ) ).
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