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+(*File: Reachability.thy
+  Author: L Beringer & M Hofmann, LMU Munich
+  Date: 05/12/2008
+  Purpose: Operational judgement that extends MultiStep by
+           entering subframes. Necessary for the interpretation
+           of invariants.            
+theory Reachability imports Language begin
+subsection{*Reachability relation*} 
+text{*The second auxiliary operational judgement is required for the
+interpretation of invariants and method invariants. Invariants are
+expected to be satisfied in all heap components of (future) states
+that occur either in the same frame as the current state or a subframe
+thereof. Likewise, method invariants are expected to be satisfied by
+all heap components of states observed during the execution of a
+method, including subframes. None of the previous three operational
+judgements allows us to express these interpretations, as @{text Step}
+injects the execution of an invoked method as a single step. Thus,
+states occurring in subframes cannot be related to states occuring in
+the parent frame using these judgements. This motivates the
+introduction of predicates relating states $s$ and $t$ whenever the
+latter can be reach from the former, i.e.~whenever $t$ occurs as a
+successor of $s$ in the same frame as $s$ or one of its
+subframes. Again, we first define a relation that includes an explicit
+derivation height index.*}
+  Reachable::"(Mbody \<times> Label \<times> State \<times> nat \<times> State) set" 
+Reachable_zero: "\<lbrakk>k=0; t=s\<rbrakk> \<Longrightarrow> (M,l,s,k,t):Reachable"
+  "\<lbrakk> (M,l,s,n,ll,r):Step; (M,ll,r,m,t):Reachable; k=Suc m+n\<rbrakk>
+  \<Longrightarrow> (M,l,s,k,t) : Reachable"
+  "\<lbrakk> mbody_is C m (par,code,l0); get_ins M l = Some (invokeS C m);
+      s = (ops,S,h); (ops,par,R,ops1):Frame;
+     ((par,code,l0), l0, ([],R,h), n, t):Reachable; k=Suc n\<rbrakk>
+  \<Longrightarrow> (M,l,s,k,t) : Reachable"
+text{*The following properties of are useful to notice.*}
+lemma ZeroHeightReachableElimAux[rule_format]:
+  "(M, l, s, k, r) \<in> Reachable \<Longrightarrow> 0=k \<longrightarrow> r=s"
+by (erule Reachable.induct, simp_all) 
+lemma ZeroHeightReachableElim: "(M,l,s,0,r) \<in> Reachable \<Longrightarrow> r=s"
+by (erule ZeroHeightReachableElimAux, simp)
+lemma ReachableSplit[rule_format]:
+  "(M, l,s, k, t) \<in> Reachable \<Longrightarrow> 
+    1 \<le> k \<longrightarrow> 
+   ((\<exists> n m r ll. (M,l,s,n,ll,r):Step \<and> 
+                 (M,ll,r,m,t):Reachable \<and> Suc m +n =k) \<or>
+    (\<exists> n ops S h c m par R ops1 code l0. 
+          s=(ops,S,h) \<and> get_ins M l = Some (invokeS c m) \<and> 
+          mbody_is c m (par,code,l0) \<and> (ops,par,R,ops1):Frame \<and> 
+          ((par,code,l0), l0, ([],R,h), n, t):Reachable \<and> Suc n = k))"
+apply (erule Reachable.induct, simp_all)
+apply clarsimp
+  apply (case_tac m, clarsimp) 
+    apply (drule ZeroHeightReachableElim[simplified], clarsimp) 
+      apply (rule, rule,rule, rule,rule, rule, rule, assumption) 
+      apply (rule,rule Reachable_zero) apply simp apply simp apply simp
+  apply clarsimp
+    apply (rule, rule,rule, rule,rule, rule,rule,assumption) apply (rule,assumption)
+    apply simp
+apply (rule disjI2)
+apply fast
+lemma Reachable_returnElim[rule_format]:
+"(M,l,s,k,t) \<in> Reachable \<Longrightarrow> get_ins M l = Some vreturn \<longrightarrow> t=s"
+apply (erule Reachable.induct)
+apply clarsimp
+apply clarsimp apply (drule RetElim1) apply simp apply clarsimp
+apply clarsimp
+text{*Similar to the operational semantics, we define a variation of
+the reachability relation that hides the index.*}
+constdefs Reach::"Mbody \<Rightarrow> Label \<Rightarrow> State \<Rightarrow> State \<Rightarrow> bool"
+"Reach M l s t \<equiv> \<exists> k . (M,l,s,k,t):Reachable"

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