(*  Title:      HOL/UNITY/FP

    ID:         $Id$

    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

    Copyright   1998  University of Cambridge

From Misra, "A Logic for Concurrent Programming", 1994

*)

header{*Fixed Point of a Program*}

theory FP imports UNITY begin

constdefs

  FP_Orig :: "'a program => 'a set"

    "FP_Orig F == Union{A. ALL B. F : stable (A Int B)}"

  FP :: "'a program => 'a set"

    "FP F == {s. F : stable {s}}"

lemma stable_FP_Orig_Int: "F : stable (FP_Orig F Int B)"

apply (simp only: FP_Orig_def stable_def Int_Union2)

apply (blast intro: constrains_UN)

done

lemma FP_Orig_weakest:

    "(!!B. F : stable (A Int B)) ==> A <= FP_Orig F"

by (simp add: FP_Orig_def stable_def, blast)

lemma stable_FP_Int: "F : stable (FP F Int B)"

apply (subgoal_tac "FP F Int B = (UN x:B. FP F Int {x}) ")

prefer 2 apply blast

apply (simp (no_asm_simp) add: Int_insert_right)

apply (simp add: FP_def stable_def)

apply (rule constrains_UN)

apply (simp (no_asm))

done

lemma FP_equivalence: "FP F = FP_Orig F"

apply (rule equalityI) 

 apply (rule stable_FP_Int [THEN FP_Orig_weakest])

apply (simp add: FP_Orig_def FP_def, clarify)

apply (drule_tac x = "{x}" in spec)

apply (simp add: Int_insert_right)

done

lemma FP_weakest:

    "(!!B. F : stable (A Int B)) ==> A <= FP F"

by (simp add: FP_equivalence FP_Orig_weakest)

lemma Compl_FP: 

    "-(FP F) = (UN act: Acts F. -{s. act``{s} <= {s}})"

by (simp add: FP_def stable_def constrains_def, blast)

lemma Diff_FP: "A - (FP F) = (UN act: Acts F. A - {s. act``{s} <= {s}})"

by (simp add: Diff_eq Compl_FP)

lemma totalize_FP [simp]: "FP (totalize F) = FP F"

by (simp add: FP_def)

end