src/HOL/Bali/Trans.thy
author schirmer
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child 14981 e73f8140af78
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"Definite Assignment Analysis" included, with proof of correctness. Large adjustments of type safety proof and soundness proof of the axiomatic semantics were necessary. Completeness proof of the loop rule of the axiomatic semantic was altered. So the additional polymorphic variants of some rules could be removed.
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(*  Title:      HOL/Bali/Trans.thy
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    ID:         $Id$
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    Author:     David von Oheimb and Norbert Schirmer
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    License:    GPL (GNU GENERAL PUBLIC LICENSE)
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Operational transition (small-step) semantics of the 
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execution of Java expressions and statements
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PRELIMINARY!!!!!!!!
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*)
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theory Trans = Evaln:
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constdefs groundVar:: "var \<Rightarrow> bool"
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"groundVar v \<equiv> (case v of
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                   LVar ln \<Rightarrow> True
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                 | {accC,statDeclC,stat}e..fn \<Rightarrow> \<exists> a. e=Lit a
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                 | e1.[e2] \<Rightarrow> \<exists> a i. e1= Lit a \<and> e2 = Lit i
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                 | InsInitV c v \<Rightarrow> False)"
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lemma groundVar_cases [consumes 1, case_names LVar FVar AVar]:
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  assumes ground: "groundVar v" and
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          LVar: "\<And> ln. \<lbrakk>v=LVar ln\<rbrakk> \<Longrightarrow> P" and
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          FVar: "\<And> accC statDeclC stat a fn. 
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                    \<lbrakk>v={accC,statDeclC,stat}(Lit a)..fn\<rbrakk> \<Longrightarrow> P" and
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          AVar: "\<And> a i. \<lbrakk>v=(Lit a).[Lit i]\<rbrakk> \<Longrightarrow> P"
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  shows "P"
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proof -
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  from ground LVar FVar AVar
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  show ?thesis
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    apply (cases v)
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    apply (simp add: groundVar_def)
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    apply (simp add: groundVar_def,blast)
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    apply (simp add: groundVar_def,blast)
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    apply (simp add: groundVar_def)
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    done
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qed
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constdefs groundExprs:: "expr list \<Rightarrow> bool"
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"groundExprs es \<equiv> list_all (\<lambda> e. \<exists> v. e=Lit v) es"
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consts the_val:: "expr \<Rightarrow> val"
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primrec
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"the_val (Lit v) = v"
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consts the_var:: "prog \<Rightarrow> state \<Rightarrow> var \<Rightarrow> (vvar \<times> state)"
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primrec
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"the_var G s (LVar ln)                    =(lvar ln (store s),s)"
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the_var_FVar_def:
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"the_var G s ({accC,statDeclC,stat}a..fn) =fvar statDeclC stat fn (the_val a) s"
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the_var_AVar_def:
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"the_var G s(a.[i])                       =avar G (the_val i) (the_val a) s"
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lemma the_var_FVar_simp[simp]:
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"the_var G s ({accC,statDeclC,stat}(Lit a)..fn) = fvar statDeclC stat fn a s"
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by (simp)
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declare the_var_FVar_def [simp del]
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lemma the_var_AVar_simp:
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"the_var G s ((Lit a).[Lit i]) = avar G i a s"
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by (simp)
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declare the_var_AVar_def [simp del]
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consts
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  step	:: "prog \<Rightarrow> ((term \<times> state) \<times> (term \<times> state)) set"
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syntax (symbols)
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  step :: "[prog,term \<times> state,term \<times> state] \<Rightarrow> bool" ("_\<turnstile>_ \<mapsto>1 _"[61,82,82] 81)
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  stepn:: "[prog, term \<times> state,nat,term \<times> state] \<Rightarrow> bool" 
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                                                  ("_\<turnstile>_ \<mapsto>_ _"[61,82,82] 81)
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"step*":: "[prog,term \<times> state,term \<times> state] \<Rightarrow> bool" ("_\<turnstile>_ \<mapsto>* _"[61,82,82] 81)
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  Ref  :: "loc \<Rightarrow> expr"
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  SKIP :: "expr"
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translations
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  "G\<turnstile>p \<mapsto>1 p' " == "(p,p') \<in> step G"
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  "G\<turnstile>p \<mapsto>n p' " == "(p,p') \<in> (step G)^n"
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  "G\<turnstile>p \<mapsto>* p' " == "(p,p') \<in> (step G)\<^sup>*"
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  "Ref a" == "Lit (Addr a)"
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  "SKIP"  == "Lit Unit"
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inductive "step G" intros 
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(* evaluation of expression *)
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  (* cf. 15.5 *)
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Abrupt:	         "\<lbrakk>\<forall>v. t \<noteq> \<langle>Lit v\<rangle>;
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                  \<forall> t. t \<noteq> \<langle>l\<bullet> Skip\<rangle>;
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                  \<forall> C vn c.  t \<noteq> \<langle>Try Skip Catch(C vn) c\<rangle>;
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                  \<forall> x c. t \<noteq> \<langle>Skip Finally c\<rangle> \<and> xc \<noteq> Xcpt x;
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                  \<forall> a c. t \<noteq> \<langle>FinA a c\<rangle>\<rbrakk> 
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                \<Longrightarrow> 
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                  G\<turnstile>(t,Some xc,s) \<mapsto>1 (\<langle>Lit arbitrary\<rangle>,Some xc,s)"
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InsInitE: "\<lbrakk>G\<turnstile>(\<langle>c\<rangle>,Norm s) \<mapsto>1 (\<langle>c'\<rangle>, s')\<rbrakk>
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           \<Longrightarrow> 
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           G\<turnstile>(\<langle>InsInitE c e\<rangle>,Norm s) \<mapsto>1 (\<langle>InsInitE c' e\<rangle>, s')"
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(* SeqE: "G\<turnstile>(\<langle>Seq Skip e\<rangle>,Norm s) \<mapsto>1 (\<langle>e\<rangle>, Norm s)" *)
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(* Specialised rules to evaluate: 
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   InsInitE Skip (NewC C), InisInitE Skip (NewA T[e]) *)
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  (* cf. 15.8.1 *)
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NewC: "G\<turnstile>(\<langle>NewC C\<rangle>,Norm s) \<mapsto>1 (\<langle>InsInitE (Init C) (NewC C)\<rangle>, Norm s)"
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NewCInited: "\<lbrakk>G\<turnstile> Norm s \<midarrow>halloc (CInst C)\<succ>a\<rightarrow> s'\<rbrakk> 
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             \<Longrightarrow> 
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             G\<turnstile>(\<langle>InsInitE Skip (NewC C)\<rangle>,Norm s) \<mapsto>1 (\<langle>Ref a\<rangle>, s')"
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(* Alternative when rule SeqE is present 
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NewCInited: "\<lbrakk>inited C (globs s); 
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              G\<turnstile> Norm s \<midarrow>halloc (CInst C)\<succ>a\<rightarrow> s'\<rbrakk> 
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             \<Longrightarrow> 
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              G\<turnstile>(\<langle>NewC C\<rangle>,Norm s) \<mapsto>1 (\<langle>Ref a\<rangle>, s')"
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NewC:
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     "\<lbrakk>\<not> inited C (globs s)\<rbrakk> 
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     \<Longrightarrow> 
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      G\<turnstile>(\<langle>NewC C\<rangle>,Norm s) \<mapsto>1 (\<langle>Seq (Init C) (NewC C)\<rangle>, Norm s)"
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*)
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  (* cf. 15.9.1 *)
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NewA: 
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   "G\<turnstile>(\<langle>New T[e]\<rangle>,Norm s) \<mapsto>1 (\<langle>InsInitE (init_comp_ty T) (New T[e])\<rangle>,Norm s)"
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InsInitNewAIdx: 
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   "\<lbrakk>G\<turnstile>(\<langle>e\<rangle>,Norm s) \<mapsto>1 (\<langle>e'\<rangle>, s')\<rbrakk>
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    \<Longrightarrow>  
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    G\<turnstile>(\<langle>InsInitE Skip (New T[e])\<rangle>,Norm s) \<mapsto>1 (\<langle>InsInitE Skip (New T[e'])\<rangle>,s')"
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InsInitNewA: 
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   "\<lbrakk>G\<turnstile>abupd (check_neg i) (Norm s) \<midarrow>halloc (Arr T (the_Intg i))\<succ>a\<rightarrow> s' \<rbrakk>
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    \<Longrightarrow>
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    G\<turnstile>(\<langle>InsInitE Skip (New T[Lit i])\<rangle>,Norm s) \<mapsto>1 (\<langle>Ref a\<rangle>,s')"
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  (* cf. 15.15 *)
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CastE:	
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   "\<lbrakk>G\<turnstile>(\<langle>e\<rangle>,Norm s) \<mapsto>1 (\<langle>e'\<rangle>,s')\<rbrakk> 
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    \<Longrightarrow>
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    G\<turnstile>(\<langle>Cast T e\<rangle>,None,s) \<mapsto>1 (\<langle>Cast T e'\<rangle>,s')" 
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Cast:	
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   "\<lbrakk>s' = abupd (raise_if (\<not>G,s\<turnstile>v fits T)  ClassCast) (Norm s)\<rbrakk> 
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    \<Longrightarrow> 
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    G\<turnstile>(\<langle>Cast T (Lit v)\<rangle>,Norm s) \<mapsto>1 (\<langle>Lit v\<rangle>,s')"
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  (* can be written without abupd, since we know Norm s *)
12854
00d4a435777f Isabelle/Bali sources;
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00d4a435777f Isabelle/Bali sources;
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InstE: "\<lbrakk>G\<turnstile>(\<langle>e\<rangle>,Norm s) \<mapsto>1 (\<langle>e'::expr\<rangle>,s')\<rbrakk> 
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        \<Longrightarrow> 
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        G\<turnstile>(\<langle>e InstOf T\<rangle>,Norm s) \<mapsto>1 (\<langle>e'\<rangle>,s')" 
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Inst:  "\<lbrakk>b = (v\<noteq>Null \<and> G,s\<turnstile>v fits RefT T)\<rbrakk> 
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        \<Longrightarrow> 
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        G\<turnstile>(\<langle>(Lit v) InstOf T\<rangle>,Norm s) \<mapsto>1 (\<langle>Lit (Bool b)\<rangle>,s')"
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  (* cf. 15.7.1 *)
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(*Lit				"G\<turnstile>(Lit v,None,s) \<mapsto>1 (Lit v,None,s)"*)
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UnOpE:  "\<lbrakk>G\<turnstile>(\<langle>e\<rangle>,Norm s) \<mapsto>1 (\<langle>e'\<rangle>,s') \<rbrakk>
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         \<Longrightarrow> 
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         G\<turnstile>(\<langle>UnOp unop e\<rangle>,Norm s) \<mapsto>1 (\<langle>UnOp unop e'\<rangle>,s')"
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UnOp:   "G\<turnstile>(\<langle>UnOp unop (Lit v)\<rangle>,Norm s) \<mapsto>1 (\<langle>Lit (eval_unop unop v)\<rangle>,Norm s)"
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BinOpE1:  "\<lbrakk>G\<turnstile>(\<langle>e1\<rangle>,Norm s) \<mapsto>1 (\<langle>e1'\<rangle>,s') \<rbrakk>
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           \<Longrightarrow> 
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           G\<turnstile>(\<langle>BinOp binop e1 e2\<rangle>,Norm s) \<mapsto>1 (\<langle>BinOp binop e1' e2\<rangle>,s')"
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BinOpE2:  "\<lbrakk>need_second_arg binop v1; G\<turnstile>(\<langle>e2\<rangle>,Norm s) \<mapsto>1 (\<langle>e2'\<rangle>,s') \<rbrakk>
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           \<Longrightarrow> 
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           G\<turnstile>(\<langle>BinOp binop (Lit v1) e2\<rangle>,Norm s) 
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            \<mapsto>1 (\<langle>BinOp binop (Lit v1) e2'\<rangle>,s')"
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BinOpTerm:  "\<lbrakk>\<not> need_second_arg binop v1\<rbrakk>
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             \<Longrightarrow> 
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             G\<turnstile>(\<langle>BinOp binop (Lit v1) e2\<rangle>,Norm s) 
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              \<mapsto>1 (\<langle>Lit v1\<rangle>,Norm s)"
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BinOp:    "G\<turnstile>(\<langle>BinOp binop (Lit v1) (Lit v2)\<rangle>,Norm s) 
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            \<mapsto>1 (\<langle>Lit (eval_binop binop v1 v2)\<rangle>,Norm s)"
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(* Maybe its more convenient to add: need_second_arg as precondition to BinOp 
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   to make the choice between BinOpTerm and BinOp deterministic *)
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Super: "G\<turnstile>(\<langle>Super\<rangle>,Norm s) \<mapsto>1 (\<langle>Lit (val_this s)\<rangle>,Norm s)"
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AccVA: "\<lbrakk>G\<turnstile>(\<langle>va\<rangle>,Norm s) \<mapsto>1 (\<langle>va'\<rangle>,s') \<rbrakk>
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        \<Longrightarrow> 
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        G\<turnstile>(\<langle>Acc va\<rangle>,Norm s) \<mapsto>1 (\<langle>Acc va'\<rangle>,s')"
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Acc:  "\<lbrakk>groundVar va; ((v,vf),s') = the_var G (Norm s) va\<rbrakk>
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       \<Longrightarrow>  
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       G\<turnstile>(\<langle>Acc va\<rangle>,Norm s) \<mapsto>1 (\<langle>Lit v\<rangle>,s')"
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(*
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AccLVar: "G\<turnstile>(\<langle>Acc (LVar vn)\<rangle>,Norm s) \<mapsto>1 (\<langle>Lit (fst (lvar vn s))\<rangle>,Norm s)"
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AccFVar: "\<lbrakk>((v,vf),s') = fvar statDeclC stat fn a (Norm s)\<rbrakk>
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          \<Longrightarrow>  
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          G\<turnstile>(\<langle>Acc ({accC,statDeclC,stat}(Lit a)..fn)\<rangle>,Norm s) 
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           \<mapsto>1 (\<langle>Lit v\<rangle>,s')"
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AccAVar: "\<lbrakk>((v,vf),s') = avar G i a (Norm s)\<rbrakk>
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          \<Longrightarrow>  
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          G\<turnstile>(\<langle>Acc ((Lit a).[Lit i])\<rangle>,Norm s) \<mapsto>1 (\<langle>Lit v\<rangle>,s')"
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*) 
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AssVA:  "\<lbrakk>G\<turnstile>(\<langle>va\<rangle>,Norm s) \<mapsto>1 (\<langle>va'\<rangle>,s')\<rbrakk> 
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         \<Longrightarrow> 
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         G\<turnstile>(\<langle>va:=e\<rangle>,Norm s) \<mapsto>1 (\<langle>va':=e\<rangle>,s')"
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AssE:   "\<lbrakk>groundVar va; G\<turnstile>(\<langle>e\<rangle>,Norm s) \<mapsto>1 (\<langle>e'\<rangle>,s')\<rbrakk> 
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         \<Longrightarrow> 
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         G\<turnstile>(\<langle>va:=e\<rangle>,Norm s) \<mapsto>1 (\<langle>va:=e'\<rangle>,s')"
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Ass:    "\<lbrakk>groundVar va; ((w,f),s') = the_var G (Norm s) va\<rbrakk> 
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         \<Longrightarrow> 
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         G\<turnstile>(\<langle>va:=(Lit v)\<rangle>,Norm s) \<mapsto>1 (\<langle>Lit v\<rangle>,assign f v s')"
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CondC: "\<lbrakk>G\<turnstile>(\<langle>e0\<rangle>,Norm s) \<mapsto>1 (\<langle>e0'\<rangle>,s')\<rbrakk> 
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        \<Longrightarrow> 
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        G\<turnstile>(\<langle>e0? e1:e2\<rangle>,Norm s) \<mapsto>1 (\<langle>e0'? e1:e2\<rangle>,s')"
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Cond:  "G\<turnstile>(\<langle>Lit b? e1:e2\<rangle>,Norm s) \<mapsto>1 (\<langle>if the_Bool b then e1 else e2\<rangle>,Norm s)"
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CallTarget: "\<lbrakk>G\<turnstile>(\<langle>e\<rangle>,Norm s) \<mapsto>1 (\<langle>e'\<rangle>,s')\<rbrakk> 
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             \<Longrightarrow>
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	     G\<turnstile>(\<langle>{accC,statT,mode}e\<cdot>mn({pTs}args)\<rangle>,Norm s) 
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              \<mapsto>1 (\<langle>{accC,statT,mode}e'\<cdot>mn({pTs}args)\<rangle>,s')"
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CallArgs:   "\<lbrakk>G\<turnstile>(\<langle>args\<rangle>,Norm s) \<mapsto>1 (\<langle>args'\<rangle>,s')\<rbrakk> 
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             \<Longrightarrow>
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	     G\<turnstile>(\<langle>{accC,statT,mode}Lit a\<cdot>mn({pTs}args)\<rangle>,Norm s) 
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              \<mapsto>1 (\<langle>{accC,statT,mode}Lit a\<cdot>mn({pTs}args')\<rangle>,s')"
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Call:       "\<lbrakk>groundExprs args; vs = map the_val args;
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              D = invocation_declclass G mode s a statT \<lparr>name=mn,parTs=pTs\<rparr>;
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              s'=init_lvars G D \<lparr>name=mn,parTs=pTs\<rparr> mode a' vs (Norm s)\<rbrakk> 
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             \<Longrightarrow> 
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             G\<turnstile>(\<langle>{accC,statT,mode}Lit a\<cdot>mn({pTs}args)\<rangle>,Norm s) 
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              \<mapsto>1 (\<langle>Callee (locals s) (Methd D \<lparr>name=mn,parTs=pTs\<rparr>)\<rangle>,s')"
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Callee:     "\<lbrakk>G\<turnstile>(\<langle>e\<rangle>,Norm s) \<mapsto>1 (\<langle>e'::expr\<rangle>,s')\<rbrakk>
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             \<Longrightarrow> 
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             G\<turnstile>(\<langle>Callee lcls_caller e\<rangle>,Norm s) \<mapsto>1 (\<langle>e'\<rangle>,s')"
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CalleeRet:   "G\<turnstile>(\<langle>Callee lcls_caller (Lit v)\<rangle>,Norm s) 
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               \<mapsto>1 (\<langle>Lit v\<rangle>,(set_lvars lcls_caller (Norm s)))"
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Methd: "G\<turnstile>(\<langle>Methd D sig\<rangle>,Norm s) \<mapsto>1 (\<langle>body G D sig\<rangle>,Norm s)"
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Body: "G\<turnstile>(\<langle>Body D c\<rangle>,Norm s) \<mapsto>1 (\<langle>InsInitE (Init D) (Body D c)\<rangle>,Norm s)"
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InsInitBody: 
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    "\<lbrakk>G\<turnstile>(\<langle>c\<rangle>,Norm s) \<mapsto>1 (\<langle>c'\<rangle>,s')\<rbrakk>
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     \<Longrightarrow> 
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     G\<turnstile>(\<langle>InsInitE Skip (Body D c)\<rangle>,Norm s) \<mapsto>1(\<langle>InsInitE Skip (Body D c')\<rangle>,s')"
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InsInitBodyRet: 
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     "G\<turnstile>(\<langle>InsInitE Skip (Body D Skip)\<rangle>,Norm s)
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       \<mapsto>1 (\<langle>Lit (the ((locals s) Result))\<rangle>,abupd (absorb Ret) (Norm s))"
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(*   LVar: "G\<turnstile>(LVar vn,Norm s)" is already evaluated *)
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FVar: "\<lbrakk>\<not> inited statDeclC (globs s)\<rbrakk>
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       \<Longrightarrow> 
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       G\<turnstile>(\<langle>{accC,statDeclC,stat}e..fn\<rangle>,Norm s) 
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        \<mapsto>1 (\<langle>InsInitV (Init statDeclC) ({accC,statDeclC,stat}e..fn)\<rangle>,Norm s)"
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InsInitFVarE:
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      "\<lbrakk>G\<turnstile>(\<langle>e\<rangle>,Norm s) \<mapsto>1 (\<langle>e'\<rangle>,s')\<rbrakk>
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       \<Longrightarrow>
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       G\<turnstile>(\<langle>InsInitV Skip ({accC,statDeclC,stat}e..fn)\<rangle>,Norm s) 
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        \<mapsto>1 (\<langle>InsInitV Skip ({accC,statDeclC,stat}e'..fn)\<rangle>,s')"
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InsInitFVar:
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      "G\<turnstile>(\<langle>InsInitV Skip ({accC,statDeclC,stat}Lit a..fn)\<rangle>,Norm s) 
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        \<mapsto>1 (\<langle>{accC,statDeclC,stat}Lit a..fn\<rangle>,Norm s)"
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--  {* Notice, that we do not have literal values for @{text vars}. 
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The rules for accessing variables (@{text Acc}) and assigning to variables 
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(@{text Ass}), test this with the predicate @{text groundVar}.  After 
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initialisation is done and the @{text FVar} is evaluated, we can't just 
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throw away the @{text InsInitFVar} term and return a literal value, as in the 
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cases of @{text New}  or @{text NewC}. Instead we just return the evaluated 
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@{text FVar} and test for initialisation in the rule @{text FVar}. 
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*}
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AVarE1: "\<lbrakk>G\<turnstile>(\<langle>e1\<rangle>,Norm s) \<mapsto>1 (\<langle>e1'\<rangle>,s')\<rbrakk> 
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         \<Longrightarrow> 
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         G\<turnstile>(\<langle>e1.[e2]\<rangle>,Norm s) \<mapsto>1 (\<langle>e1'.[e2]\<rangle>,s')"
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AVarE2: "G\<turnstile>(\<langle>e2\<rangle>,Norm s) \<mapsto>1 (\<langle>e2'\<rangle>,s') 
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         \<Longrightarrow> 
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         G\<turnstile>(\<langle>Lit a.[e2]\<rangle>,Norm s) \<mapsto>1 (\<langle>Lit a.[e2']\<rangle>,s')"
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(* AVar: \<langle>(Lit a).[Lit i]\<rangle> is fully evaluated *)
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(* evaluation of expression lists *)
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  -- {* @{text Nil}  is fully evaluated *}
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ConsHd: "\<lbrakk>G\<turnstile>(\<langle>e::expr\<rangle>,Norm s) \<mapsto>1 (\<langle>e'::expr\<rangle>,s')\<rbrakk> 
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         \<Longrightarrow>
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         G\<turnstile>(\<langle>e#es\<rangle>,Norm s) \<mapsto>1 (\<langle>e'#es\<rangle>,s')"
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ConsTl: "\<lbrakk>G\<turnstile>(\<langle>es\<rangle>,Norm s) \<mapsto>1 (\<langle>es'\<rangle>,s')\<rbrakk> 
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         \<Longrightarrow>
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         G\<turnstile>(\<langle>(Lit v)#es\<rangle>,Norm s) \<mapsto>1 (\<langle>(Lit v)#es'\<rangle>,s')"
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(* execution of statements *)
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  (* cf. 14.5 *)
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Skip: "G\<turnstile>(\<langle>Skip\<rangle>,Norm s) \<mapsto>1 (\<langle>SKIP\<rangle>,Norm s)"
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ExprE: "\<lbrakk>G\<turnstile>(\<langle>e\<rangle>,Norm s) \<mapsto>1 (\<langle>e'\<rangle>,s')\<rbrakk> 
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        \<Longrightarrow> 
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        G\<turnstile>(\<langle>Expr e\<rangle>,Norm s) \<mapsto>1 (\<langle>Expr e'\<rangle>,s')"
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Expr:  "G\<turnstile>(\<langle>Expr (Lit v)\<rangle>,Norm s) \<mapsto>1 (\<langle>Skip\<rangle>,Norm s)"
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LabC: "\<lbrakk>G\<turnstile>(\<langle>c\<rangle>,Norm s) \<mapsto>1 (\<langle>c'\<rangle>,s')\<rbrakk> 
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       \<Longrightarrow>  
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       G\<turnstile>(\<langle>l\<bullet> c\<rangle>,Norm s) \<mapsto>1 (\<langle>l\<bullet> c'\<rangle>,s')"
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Lab:  "G\<turnstile>(\<langle>l\<bullet> Skip\<rangle>,s) \<mapsto>1 (\<langle>Skip\<rangle>, abupd (absorb l) s)"
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  (* cf. 14.2 *)
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CompC1:	"\<lbrakk>G\<turnstile>(\<langle>c1\<rangle>,Norm s) \<mapsto>1 (\<langle>c1'\<rangle>,s')\<rbrakk> 
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         \<Longrightarrow> 
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         G\<turnstile>(\<langle>c1;; c2\<rangle>,Norm s) \<mapsto>1 (\<langle>c1';; c2\<rangle>,s')"
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Comp:   "G\<turnstile>(\<langle>Skip;; c2\<rangle>,Norm s) \<mapsto>1 (\<langle>c2\<rangle>,Norm s)"
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  (* cf. 14.8.2 *)
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IfE: "\<lbrakk>G\<turnstile>(\<langle>e\<rangle> ,Norm s) \<mapsto>1 (\<langle>e'\<rangle>,s')\<rbrakk> 
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      \<Longrightarrow>
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      G\<turnstile>(\<langle>If(e) s1 Else s2\<rangle>,Norm s) \<mapsto>1 (\<langle>If(e') s1 Else s2\<rangle>,s')"
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If:  "G\<turnstile>(\<langle>If(Lit v) s1 Else s2\<rangle>,Norm s) 
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       \<mapsto>1 (\<langle>if the_Bool v then s1 else s2\<rangle>,Norm s)"
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  (* cf. 14.10, 14.10.1 *)
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Loop: "G\<turnstile>(\<langle>l\<bullet> While(e) c\<rangle>,Norm s) 
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        \<mapsto>1 (\<langle>If(e) (Cont l\<bullet>c;; l\<bullet> While(e) c) Else Skip\<rangle>,Norm s)"
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13688
a0b16d42d489 "Definite Assignment Analysis" included, with proof of correctness. Large adjustments of type safety proof and soundness proof of the axiomatic semantics were necessary. Completeness proof of the loop rule of the axiomatic semantic was altered. So the additional polymorphic variants of some rules could be removed.
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Jmp: "G\<turnstile>(\<langle>Jmp j\<rangle>,Norm s) \<mapsto>1 (\<langle>Skip\<rangle>,(Some (Jump j), s))"
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ThrowE: "\<lbrakk>G\<turnstile>(\<langle>e\<rangle>,Norm s) \<mapsto>1 (\<langle>e'\<rangle>,s')\<rbrakk> 
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         \<Longrightarrow>
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         G\<turnstile>(\<langle>Throw e\<rangle>,Norm s) \<mapsto>1 (\<langle>Throw e'\<rangle>,s')"
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Throw:  "G\<turnstile>(\<langle>Throw (Lit a)\<rangle>,Norm s) \<mapsto>1 (\<langle>Skip\<rangle>,abupd (throw a) (Norm s))"
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TryC1: "\<lbrakk>G\<turnstile>(\<langle>c1\<rangle>,Norm s) \<mapsto>1 (\<langle>c1'\<rangle>,s')\<rbrakk> 
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        \<Longrightarrow>
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diff changeset
   335
        G\<turnstile>(\<langle>Try c1 Catch(C vn) c2\<rangle>, Norm s) \<mapsto>1 (\<langle>Try c1' Catch(C vn) c2\<rangle>,s')"
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
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diff changeset
   336
Try:   "\<lbrakk>G\<turnstile>s \<midarrow>sxalloc\<rightarrow> s'\<rbrakk>
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
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diff changeset
   337
        \<Longrightarrow>
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
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diff changeset
   338
        G\<turnstile>(\<langle>Try Skip Catch(C vn) c2\<rangle>, s) 
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
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diff changeset
   339
         \<mapsto>1 (if G,s'\<turnstile>catch C then (\<langle>c2\<rangle>,new_xcpt_var vn s')
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
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   340
                              else (\<langle>Skip\<rangle>,s'))"
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
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diff changeset
   341
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
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diff changeset
   342
FinC1: "\<lbrakk>G\<turnstile>(\<langle>c1\<rangle>,Norm s) \<mapsto>1 (\<langle>c1'\<rangle>,s')\<rbrakk> 
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
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diff changeset
   343
        \<Longrightarrow>
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
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diff changeset
   344
        G\<turnstile>(\<langle>c1 Finally c2\<rangle>,Norm s) \<mapsto>1 (\<langle>c1' Finally c2\<rangle>,s')"
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
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parents: 12925
diff changeset
   345
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
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diff changeset
   346
Fin:    "G\<turnstile>(\<langle>Skip Finally c2\<rangle>,(a,s)) \<mapsto>1 (\<langle>FinA a c2\<rangle>,Norm s)"
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
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parents: 12925
diff changeset
   347
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
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diff changeset
   348
FinAC: "\<lbrakk>G\<turnstile>(\<langle>c\<rangle>,s) \<mapsto>1 (\<langle>c'\<rangle>,s')\<rbrakk>
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
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diff changeset
   349
        \<Longrightarrow>
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
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diff changeset
   350
        G\<turnstile>(\<langle>FinA a c\<rangle>,s) \<mapsto>1 (\<langle>FinA a c'\<rangle>,s')"
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
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diff changeset
   351
FinA: "G\<turnstile>(\<langle>FinA a Skip\<rangle>,s) \<mapsto>1 (\<langle>Skip\<rangle>,abupd (abrupt_if (a\<noteq>None) a) s)"
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   352
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   353
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
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diff changeset
   354
Init1: "\<lbrakk>inited C (globs s)\<rbrakk> 
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
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diff changeset
   355
        \<Longrightarrow> 
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   356
        G\<turnstile>(\<langle>Init C\<rangle>,Norm s) \<mapsto>1 (\<langle>Skip\<rangle>,Norm s)"
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
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diff changeset
   357
Init: "\<lbrakk>the (class G C)=c; \<not> inited C (globs s)\<rbrakk>  
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
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diff changeset
   358
       \<Longrightarrow> 
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   359
       G\<turnstile>(\<langle>Init C\<rangle>,Norm s) 
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
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parents: 12925
diff changeset
   360
        \<mapsto>1 (\<langle>(if C = Object then Skip else (Init (super c)));;
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   361
              Expr (Callee (locals s) (InsInitE (init c) SKIP))\<rangle>
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
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parents: 12925
diff changeset
   362
             ,Norm (init_class_obj G C s))"
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   363
-- {* @{text InsInitE} is just used as trick to embed the statement 
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   364
@{text "init c"} into an expression*} 
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   365
InsInitESKIP:
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   366
  "G\<turnstile>(\<langle>InsInitE Skip SKIP\<rangle>,Norm s) \<mapsto>1 (\<langle>SKIP\<rangle>,Norm s)"
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   367
         
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   368
(* Equivalenzen:
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   369
  Bigstep zu Smallstep komplett.
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   370
  Smallstep zu Bigstep, nur wenn nicht die Ausdrücke Callee, FinA ,\<dots>
12925
99131847fb93 Added check for field/method access to operational semantics and proved the acesses valid.
schirmer
parents: 12858
diff changeset
   371
*)
12854
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   372
13337
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
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diff changeset
   373
lemma rtrancl_imp_rel_pow: "p \<in> R^* \<Longrightarrow> \<exists>n. p \<in> R^n"
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
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diff changeset
   374
proof -
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   375
  assume "p \<in> R\<^sup>*"
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   376
  moreover obtain x y where p: "p = (x,y)" by (cases p)
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   377
  ultimately have "(x,y) \<in> R\<^sup>*" by hypsubst
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   378
  hence "\<exists>n. (x,y) \<in> R^n"
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   379
  proof induct
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   380
    fix a have "(a,a) \<in> R^0" by simp
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   381
    thus "\<exists>n. (a,a) \<in> R ^ n" ..
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   382
  next
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   383
    fix a b c assume "\<exists>n. (a,b) \<in> R ^ n"
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   384
    then obtain n where "(a,b) \<in> R^n" ..
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   385
    moreover assume "(b,c) \<in> R"
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   386
    ultimately have "(a,c) \<in> R^(Suc n)" by auto
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   387
    thus "\<exists>n. (a,c) \<in> R^n" ..
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   388
  qed
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   389
  with p show ?thesis by hypsubst
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   390
qed  
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   391
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   392
(*
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   393
lemma imp_eval_trans:
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   394
  assumes eval: "G\<turnstile>s0 \<midarrow>t\<succ>\<rightarrow> (v,s1)" 
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   395
    shows trans: "G\<turnstile>(t,s0) \<mapsto>* (\<langle>Lit v\<rangle>,s1)"
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   396
*)
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   397
(* Jetzt muss man bei trans natürlich wieder unterscheiden: Stmt, Expr, Var!
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   398
   Sowas blödes:
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   399
   Am besten den Terminus ground auf Var,Stmt,Expr hochziehen und dann
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   400
   the_vals definieren\<dots>
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   401
  G\<turnstile>(t,s0) \<mapsto>* (t',s1) \<and> the_vals t' = v
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   402
*)
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   403
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   404
f75dfc606ac7 Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
schirmer
parents: 12925
diff changeset
   405
end