(* Title: HOL/UNITY/Detects ID: $Id$ Author: Tanja Vos, Cambridge University Computer Laboratory Copyright 2000 University of Cambridge Detects definition (Section 3.8 of Chandy & Misra) using LeadsTo *) header{*The Detects Relation*} theory Detects = FP + SubstAx: consts op_Detects :: "['a set, 'a set] => 'a program set" (infixl "Detects" 60) op_Equality :: "['a set, 'a set] => 'a set" (infixl "<==>" 60) defs Detects_def: "A Detects B == (Always (-A Un B)) Int (B LeadsTo A)" Equality_def: "A <==> B == (-A Un B) Int (A Un -B)" (* Corollary from Sectiom 3.6.4 *) lemma Always_at_FP: "F: A LeadsTo B ==> F : Always (-((FP F) Int A Int -B))" apply (rule LeadsTo_empty) apply (subgoal_tac "F : (FP F Int A Int - B) LeadsTo (B Int (FP F Int -B))") apply (subgoal_tac [2] " (FP F Int A Int - B) = (A Int (FP F Int -B))") apply (subgoal_tac "(B Int (FP F Int -B)) = {}") apply auto apply (blast intro: PSP_Stable stable_imp_Stable stable_FP_Int) done lemma Detects_Trans: "[| F : A Detects B; F : B Detects C |] ==> F : A Detects C" apply (unfold Detects_def Int_def) apply (simp (no_asm)) apply safe apply (rule_tac [2] LeadsTo_Trans) apply auto apply (subgoal_tac "F : Always ((-A Un B) Int (-B Un C))") apply (blast intro: Always_weaken) apply (simp add: Always_Int_distrib) done lemma Detects_refl: "F : A Detects A" apply (unfold Detects_def) apply (simp (no_asm) add: Un_commute Compl_partition subset_imp_LeadsTo) done lemma Detects_eq_Un: "(A<==>B) = (A Int B) Un (-A Int -B)" apply (unfold Equality_def) apply blast done (*Not quite antisymmetry: sets A and B agree in all reachable states *) lemma Detects_antisym: "[| F : A Detects B; F : B Detects A|] ==> F : Always (A <==> B)" apply (unfold Detects_def Equality_def) apply (simp add: Always_Int_I Un_commute) done (* Theorem from Section 3.8 *) lemma Detects_Always: "F : A Detects B ==> F : Always ((-(FP F)) Un (A <==> B))" apply (unfold Detects_def Equality_def) apply (simp (no_asm) add: Un_Int_distrib Always_Int_distrib) apply (blast dest: Always_at_FP intro: Always_weaken) done (* Theorem from exercise 11.1 Section 11.3.1 *) lemma Detects_Imp_LeadstoEQ: "F : A Detects B ==> F : UNIV LeadsTo (A <==> B)" apply (unfold Detects_def Equality_def) apply (rule_tac B = "B" in LeadsTo_Diff) prefer 2 apply (blast intro: Always_LeadsTo_weaken) apply (blast intro: Always_LeadsToI subset_imp_LeadsTo) done end