# Theory Detects

Up to index of Isabelle/HOL/UNITY

theory Detects
imports FP SubstAx
begin

    (*  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)