(* Title: HOL/UNITY/FP.thy Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1998 University of Cambridge From Misra, "A Logic for Concurrent Programming", 1994 \<open>Fixed Point of a Program\<close> theory FP imports UNITY begin definition FP_Orig :: "'a program => 'a set" where "FP_Orig F == \{A. \B. F \ stable (A \ B)}" definition FP :: "'a program => 'a set" where "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 \ B)) \ A <= FP_Orig F" by (simp add: FP_Orig_def stable_def, blast)lemma stable_FP_Int: "F \ stable (FP F \ B)" proof -have "F \ stable (\x\B. FP F \ {x})" apply (simp only: Int_insert_right FP_def stable_def)apply (rule constrains_UN)apply simpdone also have "(\x\B. FP F \ {x}) = FP F \ B" by simpfinally show ?thesis .qed 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 quality 100%
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