(* Title : Filter.thy
ID : $Id$
Author : Jacques D. Fleuriot
Copyright : 1998 University of Cambridge
Description : Filters and Ultrafilters
*)
Filter = Zorn +
constdefs
is_Filter :: ['a set set,'a set] => bool
"is_Filter F S == (F <= Pow(S) & S : F & {} ~: F &
(ALL u: F. ALL v: F. u Int v : F) &
(ALL u v. u: F & u <= v & v <= S --> v: F))"
Filter :: 'a set => 'a set set set
"Filter S == {X. is_Filter X S}"
(* free filter does not contain any finite set *)
Freefilter :: 'a set => 'a set set set
"Freefilter S == {X. X: Filter S & (ALL x: X. ~ finite x)}"
Ultrafilter :: 'a set => 'a set set set
"Ultrafilter S == {X. X: Filter S & (ALL A: Pow(S). A: X | S - A : X)}"
FreeUltrafilter :: 'a set => 'a set set set
"FreeUltrafilter S == {X. X: Ultrafilter S & (ALL x: X. ~ finite x)}"
(* A locale makes proof of Ultrafilter Theorem more modular *)
locale UFT =
fixes frechet :: "'a set => 'a set set"
superfrechet :: "'a set => 'a set set set"
assumes not_finite_UNIV "~finite (UNIV :: 'a set)"
defines frechet_def "frechet S == {A. finite (S - A)}"
superfrechet_def "superfrechet S ==
{G. G: Filter S & frechet S <= G}"
end