src/HOL/Bali/Evaln.thy

tuned header;

(*  Title:      HOL/Bali/Evaln.thy
    ID:         $Id$
    Author:     David von Oheimb
    Copyright   1999 Technische Universitaet Muenchen
*)
header {* Operational evaluation (big-step) semantics of Java expressions and 
          statements
*}

theory Evaln = Eval:

text {*
Variant of eval relation with counter for bounded recursive depth
Evaln could completely replace Eval.
*}

consts

  evaln	:: "prog \<Rightarrow> (state \<times> term \<times> nat \<times> vals \<times> state) set"

syntax

  evaln	:: "[prog, state, term,        nat, vals * state] => bool"
				("_|-_ -_>-_-> _"   [61,61,80,   61,61] 60)
  evarn	:: "[prog, state, var  , vvar        , nat, state] => bool"
				("_|-_ -_=>_-_-> _" [61,61,90,61,61,61] 60)
  eval_n:: "[prog, state, expr , val         , nat, state] => bool"
				("_|-_ -_->_-_-> _" [61,61,80,61,61,61] 60)
  evalsn:: "[prog, state, expr list, val list, nat, state] => bool"
				("_|-_ -_#>_-_-> _" [61,61,61,61,61,61] 60)
  execn	:: "[prog, state, stmt ,               nat, state] => bool"
				("_|-_ -_-_-> _"    [61,61,65,   61,61] 60)

syntax (xsymbols)

  evaln	:: "[prog, state, term,         nat, vals \<times> state] \<Rightarrow> bool"
				("_\<turnstile>_ \<midarrow>_\<succ>\<midarrow>_\<rightarrow> _"   [61,61,80,   61,61] 60)
  evarn	:: "[prog, state, var  , vvar         , nat, state] \<Rightarrow> bool"
				("_\<turnstile>_ \<midarrow>_=\<succ>_\<midarrow>_\<rightarrow> _" [61,61,90,61,61,61] 60)
  eval_n:: "[prog, state, expr , val ,          nat, state] \<Rightarrow> bool"
				("_\<turnstile>_ \<midarrow>_-\<succ>_\<midarrow>_\<rightarrow> _" [61,61,80,61,61,61] 60)
  evalsn:: "[prog, state, expr list, val  list, nat, state] \<Rightarrow> bool"
				("_\<turnstile>_ \<midarrow>_\<doteq>\<succ>_\<midarrow>_\<rightarrow> _" [61,61,61,61,61,61] 60)
  execn	:: "[prog, state, stmt ,                nat, state] \<Rightarrow> bool"
				("_\<turnstile>_ \<midarrow>_\<midarrow>_\<rightarrow> _"     [61,61,65,   61,61] 60)

translations

  "G\<turnstile>s \<midarrow>t    \<succ>\<midarrow>n\<rightarrow>  w___s' " == "(s,t,n,w___s') \<in> evaln G"
  "G\<turnstile>s \<midarrow>t    \<succ>\<midarrow>n\<rightarrow> (w,  s')" <= "(s,t,n,w,  s') \<in> evaln G"
  "G\<turnstile>s \<midarrow>t    \<succ>\<midarrow>n\<rightarrow> (w,x,s')" <= "(s,t,n,w,x,s') \<in> evaln G"
  "G\<turnstile>s \<midarrow>c     \<midarrow>n\<rightarrow> (x,s')" <= "G\<turnstile>s \<midarrow>In1r  c\<succ>\<midarrow>n\<rightarrow> (\<diamondsuit>    ,x,s')"
  "G\<turnstile>s \<midarrow>c     \<midarrow>n\<rightarrow>    s' " == "G\<turnstile>s \<midarrow>In1r  c\<succ>\<midarrow>n\<rightarrow> (\<diamondsuit>    ,  s')"
  "G\<turnstile>s \<midarrow>e-\<succ>v  \<midarrow>n\<rightarrow> (x,s')" <= "G\<turnstile>s \<midarrow>In1l e\<succ>\<midarrow>n\<rightarrow> (In1 v ,x,s')"
  "G\<turnstile>s \<midarrow>e-\<succ>v  \<midarrow>n\<rightarrow>    s' " == "G\<turnstile>s \<midarrow>In1l e\<succ>\<midarrow>n\<rightarrow> (In1 v ,  s')"
  "G\<turnstile>s \<midarrow>e=\<succ>vf \<midarrow>n\<rightarrow> (x,s')" <= "G\<turnstile>s \<midarrow>In2  e\<succ>\<midarrow>n\<rightarrow> (In2 vf,x,s')"
  "G\<turnstile>s \<midarrow>e=\<succ>vf \<midarrow>n\<rightarrow>    s' " == "G\<turnstile>s \<midarrow>In2  e\<succ>\<midarrow>n\<rightarrow> (In2 vf,  s')"
  "G\<turnstile>s \<midarrow>e\<doteq>\<succ>v  \<midarrow>n\<rightarrow> (x,s')" <= "G\<turnstile>s \<midarrow>In3  e\<succ>\<midarrow>n\<rightarrow> (In3 v ,x,s')"
  "G\<turnstile>s \<midarrow>e\<doteq>\<succ>v  \<midarrow>n\<rightarrow>    s' " == "G\<turnstile>s \<midarrow>In3  e\<succ>\<midarrow>n\<rightarrow> (In3 v ,  s')"


inductive "evaln G" intros

(* propagation of abrupt completion *)

  Abrupt:   "G\<turnstile>(Some xc,s) \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (arbitrary3 t,(Some xc,s))"


(* evaluation of variables *)

  LVar:	"G\<turnstile>Norm s \<midarrow>LVar vn=\<succ>lvar vn s\<midarrow>n\<rightarrow> Norm s"

  FVar:	"\<lbrakk>G\<turnstile>Norm s0 \<midarrow>Init C\<midarrow>n\<rightarrow> s1; G\<turnstile>s1 \<midarrow>e-\<succ>a'\<midarrow>n\<rightarrow> s2;
	  (v,s2') = fvar C stat fn a' s2\<rbrakk> \<Longrightarrow>
	  G\<turnstile>Norm s0 \<midarrow>{C,stat}e..fn=\<succ>v\<midarrow>n\<rightarrow> s2'"

  AVar:	"\<lbrakk>G\<turnstile> Norm s0 \<midarrow>e1-\<succ>a\<midarrow>n\<rightarrow> s1 ; G\<turnstile>s1 \<midarrow>e2-\<succ>i\<midarrow>n\<rightarrow> s2; 
	  (v,s2') = avar G i a s2\<rbrakk> \<Longrightarrow>
	              G\<turnstile>Norm s0 \<midarrow>e1.[e2]=\<succ>v\<midarrow>n\<rightarrow> s2'"




(* evaluation of expressions *)

  NewC:	"\<lbrakk>G\<turnstile>Norm s0 \<midarrow>Init C\<midarrow>n\<rightarrow> s1;
	  G\<turnstile>     s1 \<midarrow>halloc (CInst C)\<succ>a\<rightarrow> s2\<rbrakk> \<Longrightarrow>
	                          G\<turnstile>Norm s0 \<midarrow>NewC C-\<succ>Addr a\<midarrow>n\<rightarrow> s2"

  NewA:	"\<lbrakk>G\<turnstile>Norm s0 \<midarrow>init_comp_ty T\<midarrow>n\<rightarrow> s1; G\<turnstile>s1 \<midarrow>e-\<succ>i'\<midarrow>n\<rightarrow> s2; 
	  G\<turnstile>abupd (check_neg i') s2 \<midarrow>halloc (Arr T (the_Intg i'))\<succ>a\<rightarrow> s3\<rbrakk> \<Longrightarrow>
	                        G\<turnstile>Norm s0 \<midarrow>New T[e]-\<succ>Addr a\<midarrow>n\<rightarrow> s3"

  Cast:	"\<lbrakk>G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<midarrow>n\<rightarrow> s1;
	  s2 = abupd (raise_if (\<not>G,snd s1\<turnstile>v fits T) ClassCast) s1\<rbrakk> \<Longrightarrow>
			        G\<turnstile>Norm s0 \<midarrow>Cast T e-\<succ>v\<midarrow>n\<rightarrow> s2"

  Inst:	"\<lbrakk>G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<midarrow>n\<rightarrow> s1;
	  b = (v\<noteq>Null \<and> G,store s1\<turnstile>v fits RefT T)\<rbrakk> \<Longrightarrow>
			      G\<turnstile>Norm s0 \<midarrow>e InstOf T-\<succ>Bool b\<midarrow>n\<rightarrow> s1"

  Lit:			   "G\<turnstile>Norm s \<midarrow>Lit v-\<succ>v\<midarrow>n\<rightarrow> Norm s"

  Super:		   "G\<turnstile>Norm s \<midarrow>Super-\<succ>val_this s\<midarrow>n\<rightarrow> Norm s"

  Acc:	"\<lbrakk>G\<turnstile>Norm s0 \<midarrow>va=\<succ>(v,f)\<midarrow>n\<rightarrow> s1\<rbrakk> \<Longrightarrow>
	                          G\<turnstile>Norm s0 \<midarrow>Acc va-\<succ>v\<midarrow>n\<rightarrow> s1"

  Ass:	"\<lbrakk>G\<turnstile>Norm s0 \<midarrow>va=\<succ>(w,f)\<midarrow>n\<rightarrow> s1;
          G\<turnstile>     s1 \<midarrow>e-\<succ>v     \<midarrow>n\<rightarrow> s2\<rbrakk> \<Longrightarrow>
				   G\<turnstile>Norm s0 \<midarrow>va:=e-\<succ>v\<midarrow>n\<rightarrow> assign f v s2"

  Cond:	"\<lbrakk>G\<turnstile>Norm s0 \<midarrow>e0-\<succ>b\<midarrow>n\<rightarrow> s1;
          G\<turnstile>     s1 \<midarrow>(if the_Bool b then e1 else e2)-\<succ>v\<midarrow>n\<rightarrow> s2\<rbrakk> \<Longrightarrow>
			    G\<turnstile>Norm s0 \<midarrow>e0 ? e1 : e2-\<succ>v\<midarrow>n\<rightarrow> s2"

  Call:	
  "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>e-\<succ>a'\<midarrow>n\<rightarrow> s1; G\<turnstile>s1 \<midarrow>args\<doteq>\<succ>vs\<midarrow>n\<rightarrow> s2;
    D = invocation_declclass G mode (store s2) a' statT \<lparr>name=mn,parTs=pTs\<rparr>; 
    G\<turnstile>init_lvars G D \<lparr>name=mn,parTs=pTs\<rparr> mode a' vs s2
            \<midarrow>Methd D \<lparr>name=mn,parTs=pTs\<rparr>-\<succ>v\<midarrow>n\<rightarrow> s3\<rbrakk>
   \<Longrightarrow> G\<turnstile>Norm s0 \<midarrow>{statT,mode}e\<cdot>mn({pTs}args)-\<succ>v\<midarrow>n\<rightarrow> (restore_lvars s2 s3)"

  Methd:"\<lbrakk>G\<turnstile>Norm s0 \<midarrow>body G D sig-\<succ>v\<midarrow>n\<rightarrow> s1\<rbrakk> \<Longrightarrow>
				G\<turnstile>Norm s0 \<midarrow>Methd D sig-\<succ>v\<midarrow>Suc n\<rightarrow> s1"

  Body:	"\<lbrakk>G\<turnstile>Norm s0\<midarrow>Init D\<midarrow>n\<rightarrow> s1; G\<turnstile>s1 \<midarrow>c\<midarrow>n\<rightarrow> s2\<rbrakk>\<Longrightarrow>
 G\<turnstile>Norm s0 \<midarrow>Body D c-\<succ>the (locals (store s2) Result)\<midarrow>n\<rightarrow>abupd (absorb Ret) s2"

(* evaluation of expression lists *)

  Nil:
				"G\<turnstile>Norm s0 \<midarrow>[]\<doteq>\<succ>[]\<midarrow>n\<rightarrow> Norm s0"

  Cons:	"\<lbrakk>G\<turnstile>Norm s0 \<midarrow>e -\<succ> v \<midarrow>n\<rightarrow> s1;
          G\<turnstile>     s1 \<midarrow>es\<doteq>\<succ>vs\<midarrow>n\<rightarrow> s2\<rbrakk> \<Longrightarrow>
			     G\<turnstile>Norm s0 \<midarrow>e#es\<doteq>\<succ>v#vs\<midarrow>n\<rightarrow> s2"


(* execution of statements *)

  Skip:	 			    "G\<turnstile>Norm s \<midarrow>Skip\<midarrow>n\<rightarrow> Norm s"

  Expr:	"\<lbrakk>G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<midarrow>n\<rightarrow> s1\<rbrakk> \<Longrightarrow>
				  G\<turnstile>Norm s0 \<midarrow>Expr e\<midarrow>n\<rightarrow> s1"

  Lab:  "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>c \<midarrow>n\<rightarrow> s1\<rbrakk> \<Longrightarrow>
                             G\<turnstile>Norm s0 \<midarrow>l\<bullet> c\<midarrow>n\<rightarrow> abupd (absorb (Break l)) s1"

  Comp:	"\<lbrakk>G\<turnstile>Norm s0 \<midarrow>c1 \<midarrow>n\<rightarrow> s1;
	  G\<turnstile>     s1 \<midarrow>c2 \<midarrow>n\<rightarrow> s2\<rbrakk> \<Longrightarrow>
				 G\<turnstile>Norm s0 \<midarrow>c1;; c2\<midarrow>n\<rightarrow> s2"

  If:	"\<lbrakk>G\<turnstile>Norm s0 \<midarrow>e-\<succ>b\<midarrow>n\<rightarrow> s1;
	  G\<turnstile>     s1\<midarrow>(if the_Bool b then c1 else c2)\<midarrow>n\<rightarrow> s2\<rbrakk> \<Longrightarrow>
		       G\<turnstile>Norm s0 \<midarrow>If(e) c1 Else c2 \<midarrow>n\<rightarrow> s2"

  Loop:	"\<lbrakk>G\<turnstile>Norm s0 \<midarrow>e-\<succ>b\<midarrow>n\<rightarrow> s1;
	  if normal s1 \<and> the_Bool b 
             then (G\<turnstile>s1 \<midarrow>c\<midarrow>n\<rightarrow> s2 \<and> 
                   G\<turnstile>(abupd (absorb (Cont l)) s2) \<midarrow>l\<bullet> While(e) c\<midarrow>n\<rightarrow> s3)
	     else s3 = s1\<rbrakk> \<Longrightarrow>
			      G\<turnstile>Norm s0 \<midarrow>l\<bullet> While(e) c\<midarrow>n\<rightarrow> s3"
  
  Do: "G\<turnstile>Norm s \<midarrow>Do j\<midarrow>n\<rightarrow> (Some (Jump j), s)"
  
  Throw:"\<lbrakk>G\<turnstile>Norm s0 \<midarrow>e-\<succ>a'\<midarrow>n\<rightarrow> s1\<rbrakk> \<Longrightarrow>
				 G\<turnstile>Norm s0 \<midarrow>Throw e\<midarrow>n\<rightarrow> abupd (throw a') s1"

  Try:	"\<lbrakk>G\<turnstile>Norm s0 \<midarrow>c1\<midarrow>n\<rightarrow> s1; G\<turnstile>s1 \<midarrow>sxalloc\<rightarrow> s2;
	  if G,s2\<turnstile>catch tn then G\<turnstile>new_xcpt_var vn s2 \<midarrow>c2\<midarrow>n\<rightarrow> s3 else s3 = s2\<rbrakk>
          \<Longrightarrow>
		  G\<turnstile>Norm s0 \<midarrow>Try c1 Catch(tn vn) c2\<midarrow>n\<rightarrow> s3"

  Fin:	"\<lbrakk>G\<turnstile>Norm s0 \<midarrow>c1\<midarrow>n\<rightarrow> (x1,s1);
	  G\<turnstile>Norm s1 \<midarrow>c2\<midarrow>n\<rightarrow> s2\<rbrakk> \<Longrightarrow>
              G\<turnstile>Norm s0 \<midarrow>c1 Finally c2\<midarrow>n\<rightarrow> abupd (abrupt_if (x1\<noteq>None) x1) s2"
  
  Init:	"\<lbrakk>the (class G C) = c;
	  if inited C (globs s0) then s3 = Norm s0
	  else (G\<turnstile>Norm (init_class_obj G C s0)
	          \<midarrow>(if C = Object then Skip else Init (super c))\<midarrow>n\<rightarrow> s1 \<and>
	        G\<turnstile>set_lvars empty s1 \<midarrow>init c\<midarrow>n\<rightarrow> s2 \<and> 
                s3 = restore_lvars s1 s2)\<rbrakk>
          \<Longrightarrow>
		 G\<turnstile>Norm s0 \<midarrow>Init C\<midarrow>n\<rightarrow> s3"
monos
  if_def2

lemma evaln_eval: "\<And>ws. G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n\<rightarrow> ws \<Longrightarrow> G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> ws"
apply (simp (no_asm_simp) only: split_tupled_all)
apply (erule evaln.induct)
apply (rule eval.intros, (assumption+)?,(force split del: split_if)?)+
done


lemma Suc_le_D_lemma: "\<lbrakk>Suc n <= m'; (\<And>m. n <= m \<Longrightarrow> P (Suc m)) \<rbrakk> \<Longrightarrow> P m'"
apply (frule Suc_le_D)
apply fast
done

lemma evaln_nonstrict [rule_format (no_asm), elim]: 
  "\<And>ws. G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n\<rightarrow> ws \<Longrightarrow> \<forall>m. n\<le>m \<longrightarrow> G\<turnstile>s \<midarrow>t\<succ>\<midarrow>m\<rightarrow> ws"
apply (simp (no_asm_simp) only: split_tupled_all)
apply (erule evaln.induct)
apply (tactic {* ALLGOALS (EVERY'[strip_tac, TRY o etac (thm "Suc_le_D_lemma"),
  REPEAT o smp_tac 1, 
  resolve_tac (thms "evaln.intros") THEN_ALL_NEW TRY o atac]) *})
(* 3 subgoals *)
apply (auto split del: split_if)
done

lemmas evaln_nonstrict_Suc = evaln_nonstrict [OF _ le_refl [THEN le_SucI]]

lemma evaln_max2: "\<lbrakk>G\<turnstile>s1 \<midarrow>t1\<succ>\<midarrow>n1\<rightarrow> ws1; G\<turnstile>s2 \<midarrow>t2\<succ>\<midarrow>n2\<rightarrow> ws2\<rbrakk> \<Longrightarrow> 
             G\<turnstile>s1 \<midarrow>t1\<succ>\<midarrow>max n1 n2\<rightarrow> ws1 \<and> G\<turnstile>s2 \<midarrow>t2\<succ>\<midarrow>max n1 n2\<rightarrow> ws2"
apply (fast intro: le_maxI1 le_maxI2)
done

lemma evaln_max3: 
"\<lbrakk>G\<turnstile>s1 \<midarrow>t1\<succ>\<midarrow>n1\<rightarrow> ws1; G\<turnstile>s2 \<midarrow>t2\<succ>\<midarrow>n2\<rightarrow> ws2; G\<turnstile>s3 \<midarrow>t3\<succ>\<midarrow>n3\<rightarrow> ws3\<rbrakk> \<Longrightarrow>
 G\<turnstile>s1 \<midarrow>t1\<succ>\<midarrow>max (max n1 n2) n3\<rightarrow> ws1 \<and>
 G\<turnstile>s2 \<midarrow>t2\<succ>\<midarrow>max (max n1 n2) n3\<rightarrow> ws2 \<and> 
 G\<turnstile>s3 \<midarrow>t3\<succ>\<midarrow>max (max n1 n2) n3\<rightarrow> ws3"
apply (drule (1) evaln_max2, erule thin_rl)
apply (fast intro!: le_maxI1 le_maxI2)
done

lemma eval_evaln: "\<And>ws. G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> ws \<Longrightarrow> (\<exists>n. G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n\<rightarrow> ws)"
apply (simp (no_asm_simp) only: split_tupled_all)
apply (erule eval.induct)
apply (tactic {* ALLGOALS 
         (asm_full_simp_tac (HOL_basic_ss addsplits [split_if_asm])) *})
apply (tactic {* ALLGOALS (EVERY'[
   REPEAT o eresolve_tac [exE, conjE], rtac exI,
                     TRY o datac (thm "evaln_max3") 2, REPEAT o etac conjE,
  resolve_tac (thms "evaln.intros") THEN_ALL_NEW 
  force_tac (HOL_cs, HOL_ss)]) *})
done

declare split_if     [split del] split_if_asm     [split del]
        option.split [split del] option.split_asm [split del]
inductive_cases evaln_cases: "G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n\<rightarrow> vs'"

inductive_cases evaln_elim_cases:
	"G\<turnstile>(Some xc, s) \<midarrow>t                        \<succ>\<midarrow>n\<rightarrow> vs'"
	"G\<turnstile>Norm s \<midarrow>In1r Skip                      \<succ>\<midarrow>n\<rightarrow> xs'"
        "G\<turnstile>Norm s \<midarrow>In1r (Do j)                    \<succ>\<midarrow>n\<rightarrow> xs'"
        "G\<turnstile>Norm s \<midarrow>In1r (l\<bullet> c)                    \<succ>\<midarrow>n\<rightarrow> xs'"
	"G\<turnstile>Norm s \<midarrow>In3  ([])                      \<succ>\<midarrow>n\<rightarrow> vs'"
	"G\<turnstile>Norm s \<midarrow>In3  (e#es)                    \<succ>\<midarrow>n\<rightarrow> vs'"
	"G\<turnstile>Norm s \<midarrow>In1l (Lit w)                   \<succ>\<midarrow>n\<rightarrow> vs'"
	"G\<turnstile>Norm s \<midarrow>In2  (LVar vn)                 \<succ>\<midarrow>n\<rightarrow> vs'"
	"G\<turnstile>Norm s \<midarrow>In1l (Cast T e)                \<succ>\<midarrow>n\<rightarrow> vs'"
	"G\<turnstile>Norm s \<midarrow>In1l (e InstOf T)              \<succ>\<midarrow>n\<rightarrow> vs'"
	"G\<turnstile>Norm s \<midarrow>In1l (Super)                   \<succ>\<midarrow>n\<rightarrow> vs'"
	"G\<turnstile>Norm s \<midarrow>In1l (Acc va)                  \<succ>\<midarrow>n\<rightarrow> vs'"
	"G\<turnstile>Norm s \<midarrow>In1r (Expr e)                  \<succ>\<midarrow>n\<rightarrow> xs'"
	"G\<turnstile>Norm s \<midarrow>In1r (c1;; c2)                 \<succ>\<midarrow>n\<rightarrow> xs'"
	"G\<turnstile>Norm s \<midarrow>In1l (Methd C sig)             \<succ>\<midarrow>n\<rightarrow> xs'"
	"G\<turnstile>Norm s \<midarrow>In1l (Body D c)                \<succ>\<midarrow>n\<rightarrow> xs'"
	"G\<turnstile>Norm s \<midarrow>In1l (e0 ? e1 : e2)            \<succ>\<midarrow>n\<rightarrow> vs'"
	"G\<turnstile>Norm s \<midarrow>In1r (If(e) c1 Else c2)        \<succ>\<midarrow>n\<rightarrow> xs'"
	"G\<turnstile>Norm s \<midarrow>In1r (l\<bullet> While(e) c)           \<succ>\<midarrow>n\<rightarrow> xs'"
	"G\<turnstile>Norm s \<midarrow>In1r (c1 Finally c2)           \<succ>\<midarrow>n\<rightarrow> xs'"
	"G\<turnstile>Norm s \<midarrow>In1r (Throw e)                 \<succ>\<midarrow>n\<rightarrow> xs'"
	"G\<turnstile>Norm s \<midarrow>In1l (NewC C)                  \<succ>\<midarrow>n\<rightarrow> vs'"
	"G\<turnstile>Norm s \<midarrow>In1l (New T[e])                \<succ>\<midarrow>n\<rightarrow> vs'"
	"G\<turnstile>Norm s \<midarrow>In1l (Ass va e)                \<succ>\<midarrow>n\<rightarrow> vs'"
	"G\<turnstile>Norm s \<midarrow>In1r (Try c1 Catch(tn vn) c2)  \<succ>\<midarrow>n\<rightarrow> xs'"
	"G\<turnstile>Norm s \<midarrow>In2  ({C,stat}e..fn)           \<succ>\<midarrow>n\<rightarrow> vs'"
	"G\<turnstile>Norm s \<midarrow>In2  (e1.[e2])                 \<succ>\<midarrow>n\<rightarrow> vs'"
	"G\<turnstile>Norm s \<midarrow>In1l ({statT,mode}e\<cdot>mn({pT}p)) \<succ>\<midarrow>n\<rightarrow> vs'"
	"G\<turnstile>Norm s \<midarrow>In1r (Init C)                  \<succ>\<midarrow>n\<rightarrow> xs'"
declare split_if     [split] split_if_asm     [split] 
        option.split [split] option.split_asm [split]

lemma evaln_Inj_elim: "G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (w,s') \<Longrightarrow> case t of In1 ec \<Rightarrow>  
  (case ec of Inl e \<Rightarrow> (\<exists>v. w = In1 v) | Inr c \<Rightarrow> w = \<diamondsuit>)  
  | In2 e \<Rightarrow> (\<exists>v. w = In2 v) | In3 e \<Rightarrow> (\<exists>v. w = In3 v)"
apply (erule evaln_cases , auto)
apply (induct_tac "t")
apply   (induct_tac "a")
apply auto
done

ML_setup {*
fun enf nam inj rhs =
let
  val name = "evaln_" ^ nam ^ "_eq"
  val lhs = "G\<turnstile>s \<midarrow>" ^ inj ^ " t\<succ>\<midarrow>n\<rightarrow> (w, s')"
  val () = qed_goal name (the_context()) (lhs ^ " = (" ^ rhs ^ ")") 
	(K [Auto_tac, ALLGOALS (ftac (thm "evaln_Inj_elim")) THEN Auto_tac])
  fun is_Inj (Const (inj,_) $ _) = true
    | is_Inj _                   = false
  fun pred (_ $ (Const ("Pair",_) $ _ $ (Const ("Pair", _) $ _ $ 
    (Const ("Pair", _) $ _ $ (Const ("Pair", _) $ x $ _ )))) $ _ ) = is_Inj x
in
  make_simproc name lhs pred (thm name)
end;

val evaln_expr_proc = enf "expr" "In1l" "\<exists>v.  w=In1 v  \<and> G\<turnstile>s \<midarrow>t-\<succ>v \<midarrow>n\<rightarrow> s'";
val evaln_var_proc  = enf "var"  "In2"  "\<exists>vf. w=In2 vf \<and> G\<turnstile>s \<midarrow>t=\<succ>vf\<midarrow>n\<rightarrow> s'";
val evaln_exprs_proc= enf "exprs""In3"  "\<exists>vs. w=In3 vs \<and> G\<turnstile>s \<midarrow>t\<doteq>\<succ>vs\<midarrow>n\<rightarrow> s'";
val evaln_stmt_proc = enf "stmt" "In1r" "     w=\<diamondsuit>      \<and> G\<turnstile>s \<midarrow>t     \<midarrow>n\<rightarrow> s'";
Addsimprocs [evaln_expr_proc,evaln_var_proc,evaln_exprs_proc,evaln_stmt_proc];

bind_thms ("evaln_AbruptIs", sum3_instantiate (thm "evaln.Abrupt"))
*}
declare evaln_AbruptIs [intro!]

lemma evaln_abrupt_lemma: "G\<turnstile>s \<midarrow>e\<succ>\<midarrow>n\<rightarrow> (v,s') \<Longrightarrow> 
 fst s = Some xc \<longrightarrow> s' = s \<and> v = arbitrary3 e"
apply (erule evaln_cases , auto)
done

lemma evaln_abrupt: 
 "\<And>s'. G\<turnstile>(Some xc,s) \<midarrow>e\<succ>\<midarrow>n\<rightarrow> (w,s') = (s' = (Some xc,s) \<and>  
  w=arbitrary3 e \<and> G\<turnstile>(Some xc,s) \<midarrow>e\<succ>\<midarrow>n\<rightarrow> (arbitrary3 e,(Some xc,s)))"
apply auto
apply (frule evaln_abrupt_lemma, auto)+
done

ML {*
local
  fun is_Some (Const ("Pair",_) $ (Const ("Option.option.Some",_) $ _)$ _) =true
    | is_Some _ = false
  fun pred (_ $ (Const ("Pair",_) $
     _ $ (Const ("Pair", _) $ _ $ (Const ("Pair", _) $ _ $
       (Const ("Pair", _) $ _ $ x)))) $ _ ) = is_Some x
in
  val evaln_abrupt_proc = 
 make_simproc "evaln_abrupt" "G\<turnstile>(Some xc,s) \<midarrow>e\<succ>\<midarrow>n\<rightarrow> (w,s')" pred (thm "evaln_abrupt")
end;
Addsimprocs [evaln_abrupt_proc]
*}

lemma evaln_LitI: "G\<turnstile>s \<midarrow>Lit v-\<succ>(if normal s then v else arbitrary)\<midarrow>n\<rightarrow> s"
apply (case_tac "s", case_tac "a = None")
by (auto intro!: evaln.Lit)

lemma CondI: 
 "\<And>s1. \<lbrakk>G\<turnstile>s \<midarrow>e-\<succ>b\<midarrow>n\<rightarrow> s1; G\<turnstile>s1 \<midarrow>(if the_Bool b then e1 else e2)-\<succ>v\<midarrow>n\<rightarrow> s2\<rbrakk> \<Longrightarrow> 
  G\<turnstile>s \<midarrow>e ? e1 : e2-\<succ>(if normal s1 then v else arbitrary)\<midarrow>n\<rightarrow> s2"
apply (case_tac "s", case_tac "a = None")
by (auto intro!: evaln.Cond)

lemma evaln_SkipI [intro!]: "G\<turnstile>s \<midarrow>Skip\<midarrow>n\<rightarrow> s"
apply (case_tac "s", case_tac "a = None")
by (auto intro!: evaln.Skip)

lemma evaln_ExprI: "G\<turnstile>s \<midarrow>e-\<succ>v\<midarrow>n\<rightarrow> s' \<Longrightarrow> G\<turnstile>s \<midarrow>Expr e\<midarrow>n\<rightarrow> s'"
apply (case_tac "s", case_tac "a = None")
by (auto intro!: evaln.Expr)

lemma evaln_CompI: "\<lbrakk>G\<turnstile>s \<midarrow>c1\<midarrow>n\<rightarrow> s1; G\<turnstile>s1 \<midarrow>c2\<midarrow>n\<rightarrow> s2\<rbrakk> \<Longrightarrow> G\<turnstile>s \<midarrow>c1;; c2\<midarrow>n\<rightarrow> s2"
apply (case_tac "s", case_tac "a = None")
by (auto intro!: evaln.Comp)

lemma evaln_IfI: 
 "\<lbrakk>G\<turnstile>s \<midarrow>e-\<succ>v\<midarrow>n\<rightarrow> s1; G\<turnstile>s1 \<midarrow>(if the_Bool v then c1 else c2)\<midarrow>n\<rightarrow> s2\<rbrakk> \<Longrightarrow> 
  G\<turnstile>s \<midarrow>If(e) c1 Else c2\<midarrow>n\<rightarrow> s2"
apply (case_tac "s", case_tac "a = None")
by (auto intro!: evaln.If)

lemma evaln_SkipD [dest!]: "G\<turnstile>s \<midarrow>Skip\<midarrow>n\<rightarrow> s' \<Longrightarrow> s' = s" 
by (erule evaln_cases, auto)

lemma evaln_Skip_eq [simp]: "G\<turnstile>s \<midarrow>Skip\<midarrow>n\<rightarrow> s' = (s = s')"
apply auto
done

end