src/HOL/MicroJava/J/Eval.thy

Minor modifications

(*  Title:      HOL/MicroJava/J/Eval.thy
    ID:         $Id$
    Author:     David von Oheimb
    Copyright   1999 Technische Universitaet Muenchen
*)

header "Operational Evaluation (big-step) Semantics"

theory Eval = State + WellType:

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)
  eval :: "[java_mb prog,xstate,expr,val,xstate] => bool "
          ("_ \<turnstile> _ -_\<succ>_-> _" [51,82,60,82,82] 81)
  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 "
          ("_ \<turnstile> _ -_-> _" [51,82,60,82] 81)

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 *)

  (* cf. 15.5 *)
  XcptE:"G\<turnstile>(Some xc,s) -e\<succ>arbitrary-> (Some xc,s)"

  (* cf. 15.8.1 *)
  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);
            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"

  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"

  (* cf. 15.25.1 *)
  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); 
            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';
            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';
            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)"

  (* cf. 15.11.??? *)
  Nil:  "G\<turnstile>Norm s0 -[][\<succ>][]-> Norm s0"

  (* cf. 15.6.4 *)
  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)"

  (* cf. 14.5 *)
  Skip: "G\<turnstile>Norm s -Skip-> Norm s"

  (* cf. 14.7 *)
  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;
            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;
            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 |] ==>
         G\<turnstile>Norm s0 -While(e) c-> s1"
  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)]

lemma NewCI: "[|new_Addr (heap s) = (a,x);  
       s' = c_hupd (heap s(a\<mapsto>(C,init_vars (fields (G,C))))) (x,s)|] ==>  
       G\<turnstile>Norm s -NewC C\<succ>Addr a-> s'"
apply (simp (no_asm_simp))
apply (rule eval_evals_exec.NewC)
apply auto
done

lemma eval_evals_exec_no_xcpt: 
 "!!s s'. (G\<turnstile>(x,s) -e \<succ>  v -> (x',s') --> x'=None --> x=None) \<and>  
          (G\<turnstile>(x,s) -es[\<succ>]vs-> (x',s') --> x'=None --> x=None) \<and>  
          (G\<turnstile>(x,s) -c       -> (x',s') --> x'=None --> x=None)"
apply(simp (no_asm_simp) only: split_tupled_all)
apply(rule eval_evals_exec_induct)
apply(unfold c_hupd_def)
apply(simp_all)
done

lemma eval_no_xcpt: "G\<turnstile>(x,s) -e\<succ>v-> (None,s') ==> x=None"
apply (drule eval_evals_exec_no_xcpt [THEN conjunct1, THEN mp])
apply (fast)
done

lemma evals_no_xcpt: "G\<turnstile>(x,s) -e[\<succ>]v-> (None,s') ==> x=None"
apply (drule eval_evals_exec_no_xcpt [THEN conjunct2, THEN conjunct1, THEN mp])
apply (fast)
done

lemma eval_evals_exec_xcpt: 
"!!s s'. (G\<turnstile>(x,s) -e \<succ>  v -> (x',s') --> x=Some xc --> x'=Some xc \<and> s'=s) \<and>  
         (G\<turnstile>(x,s) -es[\<succ>]vs-> (x',s') --> x=Some xc --> x'=Some xc \<and> s'=s) \<and>  
         (G\<turnstile>(x,s) -c       -> (x',s') --> x=Some xc --> x'=Some xc \<and> s'=s)"
apply (simp (no_asm_simp) only: split_tupled_all)
apply (rule eval_evals_exec_induct)
apply (unfold c_hupd_def)
apply (simp_all)
done

lemma eval_xcpt: "G\<turnstile>(Some xc,s) -e\<succ>v-> (x',s') ==> x'=Some xc \<and>  s'=s"
apply (drule eval_evals_exec_xcpt [THEN conjunct1, THEN mp])
apply (fast)
done

lemma exec_xcpt: "G\<turnstile>(Some xc,s) -s0-> (x',s') ==> x'=Some xc \<and>  s'=s"
apply (drule eval_evals_exec_xcpt [THEN conjunct2, THEN conjunct2, THEN mp])
apply (fast)
done

end