1(* Title: HOL/UNITY/FP.thy 2 Author: Lawrence C Paulson, Cambridge University Computer Laboratory 3 Copyright 1998 University of Cambridge 4 5From Misra, "A Logic for Concurrent Programming", 1994 6*) 7 8section\<open>Fixed Point of a Program\<close> 9 10theory FP imports UNITY begin 11 12definition FP_Orig :: "'a program => 'a set" where 13 "FP_Orig F == \<Union>{A. \<forall>B. F \<in> stable (A \<inter> B)}" 14 15definition FP :: "'a program => 'a set" where 16 "FP F == {s. F \<in> stable {s}}" 17 18lemma stable_FP_Orig_Int: "F \<in> stable (FP_Orig F Int B)" 19apply (simp only: FP_Orig_def stable_def Int_Union2) 20apply (blast intro: constrains_UN) 21done 22 23lemma FP_Orig_weakest: 24 "(\<And>B. F \<in> stable (A \<inter> B)) \<Longrightarrow> A <= FP_Orig F" 25by (simp add: FP_Orig_def stable_def, blast) 26 27lemma stable_FP_Int: "F \<in> stable (FP F \<inter> B)" 28apply (subgoal_tac "FP F Int B = (UN x:B. FP F Int {x}) ") 29prefer 2 apply blast 30apply (simp (no_asm_simp) add: Int_insert_right) 31apply (simp add: FP_def stable_def) 32apply (rule constrains_UN) 33apply (simp (no_asm)) 34done 35 36lemma FP_equivalence: "FP F = FP_Orig F" 37apply (rule equalityI) 38 apply (rule stable_FP_Int [THEN FP_Orig_weakest]) 39apply (simp add: FP_Orig_def FP_def, clarify) 40apply (drule_tac x = "{x}" in spec) 41apply (simp add: Int_insert_right) 42done 43 44lemma FP_weakest: 45 "(\<And>B. F \<in> stable (A Int B)) \<Longrightarrow> A <= FP F" 46by (simp add: FP_equivalence FP_Orig_weakest) 47 48lemma Compl_FP: 49 "-(FP F) = (UN act: Acts F. -{s. act``{s} <= {s}})" 50by (simp add: FP_def stable_def constrains_def, blast) 51 52lemma Diff_FP: "A - (FP F) = (UN act: Acts F. A - {s. act``{s} <= {s}})" 53by (simp add: Diff_eq Compl_FP) 54 55lemma totalize_FP [simp]: "FP (totalize F) = FP F" 56by (simp add: FP_def) 57 58end 59