src/HOL/MicroJava/J/Eval.thy

  "new_xcpt_var vn \<equiv>  \<lambda>(x,s). Norm (lupd(vn\<mapsto>the x) s)"
 
 
   -- "Evaluation relations"
 
-consts
-  eval  :: "java_mb prog => (xstate \<times> expr      \<times> val      \<times> xstate) set"
-  evals :: "java_mb prog => (xstate \<times> expr list \<times> val list \<times> xstate) set"
-  exec  :: "java_mb prog => (xstate \<times> stmt                 \<times> xstate) set"
-
-syntax (xsymbols)
+inductive2
   eval :: "[java_mb prog,xstate,expr,val,xstate] => bool "
           ("_ \<turnstile> _ -_\<succ>_-> _" [51,82,60,82,82] 81)
-  evals:: "[java_mb prog,xstate,expr list,
+  and evals :: "[java_mb prog,xstate,expr list,
                         val list,xstate] => bool "
           ("_ \<turnstile> _ -_[\<succ>]_-> _" [51,82,60,51,82] 81)
-  exec :: "[java_mb prog,xstate,stmt,    xstate] => bool "
+  and exec :: "[java_mb prog,xstate,stmt,    xstate] => bool "
           ("_ \<turnstile> _ -_-> _" [51,82,60,82] 81)
+  for G :: "java_mb prog"
+where
 
-syntax
-  eval :: "[java_mb prog,xstate,expr,val,xstate] => bool "
-          ("_ |- _ -_>_-> _" [51,82,60,82,82] 81)
-  evals:: "[java_mb prog,xstate,expr list,
-                        val list,xstate] => bool "
-          ("_ |- _ -_[>]_-> _" [51,82,60,51,82] 81)
-  exec :: "[java_mb prog,xstate,stmt,    xstate] => bool "
-          ("_ |- _ -_-> _" [51,82,60,82] 81)
-
-
-translations
-  "G\<turnstile>s -e \<succ> v-> (x,s')" <= "(s, e, v, x, s') \<in> eval  G"
-  "G\<turnstile>s -e \<succ> v->    s' " == "(s, e, v,    s') \<in> eval  G"
-  "G\<turnstile>s -e[\<succ>]v-> (x,s')" <= "(s, e, v, x, s') \<in> evals G"
-  "G\<turnstile>s -e[\<succ>]v->    s' " == "(s, e, v,    s') \<in> evals G"
-  "G\<turnstile>s -c    -> (x,s')" <= "(s, c, x, s') \<in> exec G"
-  "G\<turnstile>s -c    ->    s' " == "(s, c,    s') \<in> exec G"
-
-
-inductive "eval G" "evals G" "exec G" intros
-
-  (* evaluation of expressions *)
+  -- "evaluation of expressions"
 
   XcptE:"G\<turnstile>(Some xc,s) -e\<succ>arbitrary-> (Some xc,s)"  -- "cf. 15.5"
 
   -- "cf. 15.8.1"
-  NewC: "[| h = heap s; (a,x) = new_Addr h;
+| NewC: "[| h = heap s; (a,x) = new_Addr h;
             h'= h(a\<mapsto>(C,init_vars (fields (G,C)))) |] ==>
          G\<turnstile>Norm s -NewC C\<succ>Addr a-> c_hupd h' (x,s)"
 
   -- "cf. 15.15"
-  Cast: "[| G\<turnstile>Norm s0 -e\<succ>v-> (x1,s1);
+| Cast: "[| G\<turnstile>Norm s0 -e\<succ>v-> (x1,s1);
             x2 = raise_if (\<not> cast_ok G C (heap s1) v) ClassCast x1 |] ==>
          G\<turnstile>Norm s0 -Cast C e\<succ>v-> (x2,s1)"
 
   -- "cf. 15.7.1"
-  Lit:  "G\<turnstile>Norm s -Lit v\<succ>v-> Norm s"
+| Lit:  "G\<turnstile>Norm s -Lit v\<succ>v-> Norm s"
 
-  BinOp:"[| G\<turnstile>Norm s -e1\<succ>v1-> s1;
+| BinOp:"[| G\<turnstile>Norm s -e1\<succ>v1-> s1;
             G\<turnstile>s1     -e2\<succ>v2-> s2;
             v = (case bop of Eq  => Bool (v1 = v2)
                            | Add => Intg (the_Intg v1 + the_Intg v2)) |] ==>
          G\<turnstile>Norm s -BinOp bop e1 e2\<succ>v-> s2"
 
   -- "cf. 15.13.1, 15.2"
-  LAcc: "G\<turnstile>Norm s -LAcc v\<succ>the (locals s v)-> Norm s"
+| LAcc: "G\<turnstile>Norm s -LAcc v\<succ>the (locals s v)-> Norm s"
 
   -- "cf. 15.25.1"
-  LAss: "[| G\<turnstile>Norm s -e\<succ>v-> (x,(h,l));
+| LAss: "[| G\<turnstile>Norm s -e\<succ>v-> (x,(h,l));
             l' = (if x = None then l(va\<mapsto>v) else l) |] ==>
          G\<turnstile>Norm s -va::=e\<succ>v-> (x,(h,l'))"
 
   -- "cf. 15.10.1, 15.2"
-  FAcc: "[| G\<turnstile>Norm s0 -e\<succ>a'-> (x1,s1); 
+| FAcc: "[| G\<turnstile>Norm s0 -e\<succ>a'-> (x1,s1); 
             v = the (snd (the (heap s1 (the_Addr a'))) (fn,T)) |] ==>
          G\<turnstile>Norm s0 -{T}e..fn\<succ>v-> (np a' x1,s1)"
 
   -- "cf. 15.25.1"
-  FAss: "[| G\<turnstile>     Norm s0  -e1\<succ>a'-> (x1,s1); a = the_Addr a';
+| FAss: "[| G\<turnstile>     Norm s0  -e1\<succ>a'-> (x1,s1); a = the_Addr a';
             G\<turnstile>(np a' x1,s1) -e2\<succ>v -> (x2,s2);
             h  = heap s2; (c,fs) = the (h a);
             h' = h(a\<mapsto>(c,(fs((fn,T)\<mapsto>v)))) |] ==>
          G\<turnstile>Norm s0 -{T}e1..fn:=e2\<succ>v-> c_hupd h' (x2,s2)"
 
   -- "cf. 15.11.4.1, 15.11.4.2, 15.11.4.4, 15.11.4.5, 14.15"
-  Call: "[| G\<turnstile>Norm s0 -e\<succ>a'-> s1; a = the_Addr a';
+| Call: "[| G\<turnstile>Norm s0 -e\<succ>a'-> s1; a = the_Addr a';
             G\<turnstile>s1 -ps[\<succ>]pvs-> (x,(h,l)); dynT = fst (the (h a));
             (md,rT,pns,lvars,blk,res) = the (method (G,dynT) (mn,pTs));
             G\<turnstile>(np a' x,(h,(init_vars lvars)(pns[\<mapsto>]pvs)(This\<mapsto>a'))) -blk-> s3;
             G\<turnstile> s3 -res\<succ>v -> (x4,s4) |] ==>
          G\<turnstile>Norm s0 -{C}e..mn({pTs}ps)\<succ>v-> (x4,(heap s4,l))"
 
 
   -- "evaluation of expression lists"
 
   -- "cf. 15.5"
-  XcptEs:"G\<turnstile>(Some xc,s) -e[\<succ>]arbitrary-> (Some xc,s)"
+| XcptEs:"G\<turnstile>(Some xc,s) -e[\<succ>]arbitrary-> (Some xc,s)"
 
   -- "cf. 15.11.???"
-  Nil:  "G\<turnstile>Norm s0 -[][\<succ>][]-> Norm s0"
+| Nil:  "G\<turnstile>Norm s0 -[][\<succ>][]-> Norm s0"
 
   -- "cf. 15.6.4"
-  Cons: "[| G\<turnstile>Norm s0 -e  \<succ> v -> s1;
+| Cons: "[| G\<turnstile>Norm s0 -e  \<succ> v -> s1;
             G\<turnstile>     s1 -es[\<succ>]vs-> s2 |] ==>
          G\<turnstile>Norm s0 -e#es[\<succ>]v#vs-> s2"
 
 
   -- "execution of statements"
 
   -- "cf. 14.1"
-  XcptS:"G\<turnstile>(Some xc,s) -c-> (Some xc,s)"
+| XcptS:"G\<turnstile>(Some xc,s) -c-> (Some xc,s)"
 
   -- "cf. 14.5"
-  Skip: "G\<turnstile>Norm s -Skip-> Norm s"
+| Skip: "G\<turnstile>Norm s -Skip-> Norm s"
 
   -- "cf. 14.7"
-  Expr: "[| G\<turnstile>Norm s0 -e\<succ>v-> s1 |] ==>
+| Expr: "[| G\<turnstile>Norm s0 -e\<succ>v-> s1 |] ==>
          G\<turnstile>Norm s0 -Expr e-> s1"
 
   -- "cf. 14.2"
-  Comp: "[| G\<turnstile>Norm s0 -c1-> s1;
+| Comp: "[| G\<turnstile>Norm s0 -c1-> s1;
             G\<turnstile>     s1 -c2-> s2|] ==>
          G\<turnstile>Norm s0 -c1;; c2-> s2"
 
   -- "cf. 14.8.2"
-  Cond: "[| G\<turnstile>Norm s0  -e\<succ>v-> s1;
+| Cond: "[| G\<turnstile>Norm s0  -e\<succ>v-> s1;
             G\<turnstile> s1 -(if the_Bool v then c1 else c2)-> s2|] ==>
          G\<turnstile>Norm s0 -If(e) c1 Else c2-> s2"
 
   -- "cf. 14.10, 14.10.1"
-  LoopF:"[| G\<turnstile>Norm s0 -e\<succ>v-> s1; \<not>the_Bool v |] ==>
+| LoopF:"[| G\<turnstile>Norm s0 -e\<succ>v-> s1; \<not>the_Bool v |] ==>
          G\<turnstile>Norm s0 -While(e) c-> s1"
-  LoopT:"[| G\<turnstile>Norm s0 -e\<succ>v-> s1;  the_Bool v;
+| LoopT:"[| G\<turnstile>Norm s0 -e\<succ>v-> s1;  the_Bool v;
       G\<turnstile>s1 -c-> s2; G\<turnstile>s2 -While(e) c-> s3 |] ==>
          G\<turnstile>Norm s0 -While(e) c-> s3"
 
 
 lemmas eval_evals_exec_induct = eval_evals_exec.induct [split_format (complete)]