Theory Fix2

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theory Fix2
imports HOLCF
begin

(*  Title:      HOLCF/ex/Fix2.thy
    ID:         $Id: Fix2.thy,v 1.9 2007/10/21 14:27:43 wenzelm Exp $
    Author:     Franz Regensburger

Show that fix is the unique least fixed-point operator.
From axioms gix1_def,gix2_def it follows that fix = gix
*)

theory Fix2
imports HOLCF
begin

axiomatization
  gix :: "('a->'a)->'a" where
  gix1_def: "F$(gix$F) = gix$F" and
  gix2_def: "F$y=y ==> gix$F << y"


lemma lemma1: "fix = gix"
apply (rule ext_cfun)
apply (rule antisym_less)
apply (rule fix_least)
apply (rule gix1_def)
apply (rule gix2_def)
apply (rule fix_eq [symmetric])
done

lemma lemma2: "gix$F=lub(range(%i. iterate i$F$UU))"
apply (rule lemma1 [THEN subst])
apply (rule fix_def2)
done

end

lemma lemma1:

  fix = gix

lemma lemma2:

  gix·F = (LUB i. iterate i·F·UU)