Theory Detects

theory Detects
imports FP SubstAx
(*  Title:      HOL/UNITY/Detects.thy
    Author:     Tanja Vos, Cambridge University Computer Laboratory
    Copyright   2000  University of Cambridge

Detects definition (Section 3.8 of Chandy & Misra) using LeadsTo
*)

section‹The Detects Relation›

theory Detects imports FP SubstAx begin

definition Detects :: "['a set, 'a set] => 'a program set"  (infixl "Detects" 60)
  where "A Detects B = (Always (-A ∪ B)) ∩ (B LeadsTo A)"

definition Equality :: "['a set, 'a set] => 'a set"  (infixl "<==>" 60)
  where "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))"
supply [[simproc del: boolean_algebra_cancel_inf]] inf_compl_bot_right[simp del] 
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