(* Title: HOL/UNITY/Detects ID: $Id: Detects.thy,v 1.6 2005/06/17 14:13:10 haftmann Exp $ 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 imports FP SubstAx begin 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 ∪ B)) ∩ (B LeadsTo A)" Equality_def: "A <==> B == (-A ∪ B) ∩ (A ∪ -B)" (* Corollary from Sectiom 3.6.4 *) lemma Always_at_FP: "[|F ∈ A LeadsTo B; all_total F|] ==> F ∈ Always (-((FP F) ∩ A ∩ -B))" apply (rule LeadsTo_empty) apply (subgoal_tac "F ∈ (FP F ∩ A ∩ - B) LeadsTo (B ∩ (FP F ∩ -B))") apply (subgoal_tac [2] " (FP F ∩ A ∩ - B) = (A ∩ (FP F ∩ -B))") apply (subgoal_tac "(B ∩ (FP F ∩ -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, auto) apply (subgoal_tac "F ∈ Always ((-A ∪ B) ∩ (-B ∪ 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 ∩ B) ∪ (-A ∩ -B)" by (unfold Equality_def, blast) (*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; all_total F|] ==> F ∈ Always (-(FP F) ∪ (A <==> B))" apply (unfold Detects_def Equality_def) apply (simp 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) apply (blast intro: Always_LeadsToI subset_imp_LeadsTo) apply (blast intro: Always_LeadsTo_weaken) done end
lemma Always_at_FP:
[| F ∈ A LeadsTo B; all_total F |] ==> F ∈ Always (- (FP F ∩ A ∩ - B))
lemma Detects_Trans:
[| F ∈ A Detects B; F ∈ B Detects C |] ==> F ∈ A Detects C
lemma Detects_refl:
F ∈ A Detects A
lemma Detects_eq_Un:
A <==> B = A ∩ B ∪ - A ∩ - B
lemma Detects_antisym:
[| F ∈ A Detects B; F ∈ B Detects A |] ==> F ∈ Always (A <==> B)
lemma Detects_Always:
[| F ∈ A Detects B; all_total F |] ==> F ∈ Always (- FP F ∪ (A <==> B))
lemma Detects_Imp_LeadstoEQ:
F ∈ A Detects B ==> F ∈ UNIV LeadsTo (A <==> B)