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Datei: meson.ML Sprache: Isabelle

Original von: Isabelle^©

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

section \<open>Operational Evaluation (big step) Semantics\<close>

theory Eval imports State WellType begin

  \<comment> \<open>Auxiliary notions\<close>

definition fits :: "java_mb prog \ state \ val \ ty \ bool" ("_,_\_ fits _"[61,61,61,61]60) where
"G,s\a' fits T \ case T of PrimT T' \ False | RefT T' \ a'=Null \ G\obj_ty(lookup_obj s a')\T"

definition catch :: "java_mb prog \ xstate \ cname \ bool" ("_,_\catch _"[61,61,61]60) where
"G,s\catch C\ case abrupt s of None \ False | Some a \ G,store s\ a fits Class C"

definition lupd :: "vname \ val \ state \ state" ("lupd'(_\_')"[10,10]1000) where
"lupd vn v \ \ (hp,loc). (hp, (loc(vn\v)))"

definition new_xcpt_var :: "vname \ xstate \ xstate" where
"new_xcpt_var vn \ \(x,s). Norm (lupd(vn\the x) s)"

  \<comment> \<open>Evaluation relations\<close>

inductive
  eval :: "[java_mb prog,xstate,expr,val,xstate] => bool "
          ("_ \ _ -_\_-> _" [51,82,60,82,82] 81)
  and evals :: "[java_mb prog,xstate,expr list,
                        val list,xstate] => bool "
          ("_ \ _ -_[\]_-> _" [51,82,60,51,82] 81)
  and exec :: "[java_mb prog,xstate,stmt, xstate] => bool "
          ("_ \ _ -_-> _" [51,82,60,82] 81)
  for G :: "java_mb prog"
where

  \<comment> \<open>evaluation of expressions\<close>

  XcptE:"G\(Some xc,s) -e\undefined-> (Some xc,s)" \ \cf. 15.5\

  \<comment> \<open>cf. 15.8.1\<close>
| NewC: "[| h = heap s; (a,x) = new_Addr h;
            h'= h(a\(C,init_vars (fields (G,C)))) |] ==>
         G\<turnstile>Norm s -NewC C\<succ>Addr a-> c_hupd h' (x,s)"

  \<comment> \<open>cf. 15.15\<close>
| Cast: "[| G\Norm s0 -e\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)"

  \<comment> \<open>cf. 15.7.1\<close>
| Lit:  "G\Norm s -Lit v\v-> Norm s"

| BinOp:"[| G\Norm s -e1\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"

  \<comment> \<open>cf. 15.13.1, 15.2\<close>
| LAcc: "G\Norm s -LAcc v\the (locals s v)-> Norm s"

  \<comment> \<open>cf. 15.25.1\<close>
| LAss: "[| G\Norm s -e\v-> (x,(h,l));
            l' = (if x = None then l(va\v) else l) |] ==>
         G\<turnstile>Norm s -va::=e\<succ>v-> (x,(h,l'))"

  \<comment> \<open>cf. 15.10.1, 15.2\<close>
| FAcc: "[| G\Norm s0 -e\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)"

  \<comment> \<open>cf. 15.25.1\<close>
| FAss: "[| G\ Norm s0 -e1\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\(c,(fs((fn,T)\v)))) |] ==>
         G\<turnstile>Norm s0 -{T}e1..fn:=e2\<succ>v-> c_hupd h' (x2,s2)"

  \<comment> \<open>cf. 15.11.4.1, 15.11.4.2, 15.11.4.4, 15.11.4.5, 14.15\<close>
| Call: "[| G\Norm s0 -e\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))"

  \<comment> \<open>evaluation of expression lists\<close>

  \<comment> \<open>cf. 15.5\<close>
| XcptEs:"G\(Some xc,s) -e[\]undefined-> (Some xc,s)"

  \<comment> \<open>cf. 15.11.???\<close>
| Nil:  "G\Norm s0 -[][\][]-> Norm s0"

  \<comment> \<open>cf. 15.6.4\<close>
| Cons: "[| G\Norm s0 -e \ v -> s1;
            G\<turnstile>     s1 -es[\<succ>]vs-> s2 |] ==>
         G\<turnstile>Norm s0 -e#es[\<succ>]v#vs-> s2"

  \<comment> \<open>execution of statements\<close>

  \<comment> \<open>cf. 14.1\<close>
| XcptS:"G\(Some xc,s) -c-> (Some xc,s)"

  \<comment> \<open>cf. 14.5\<close>
| Skip: "G\Norm s -Skip-> Norm s"

  \<comment> \<open>cf. 14.7\<close>
| Expr: "[| G\Norm s0 -e\v-> s1 |] ==>
         G\<turnstile>Norm s0 -Expr e-> s1"

  \<comment> \<open>cf. 14.2\<close>
| Comp: "[| G\Norm s0 -c1-> s1;
            G\<turnstile>     s1 -c2-> s2|] ==>
         G\<turnstile>Norm s0 -c1;; c2-> s2"

  \<comment> \<open>cf. 14.8.2\<close>
| Cond: "[| G\Norm s0 -e\v-> s1;
            G\<turnstile> s1 -(if the_Bool v then c1 else c2)-> s2|] ==>
         G\<turnstile>Norm s0 -If(e) c1 Else c2-> s2"

  \<comment> \<open>cf. 14.10, 14.10.1\<close>
| LoopF:"[| G\Norm s0 -e\v-> s1; \the_Bool v |] ==>
         G\<turnstile>Norm s0 -While(e) c-> s1"
| LoopT:"[| G\Norm s0 -e\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\(C,init_vars (fields (G,C))))) (x,s)|] ==>
       G\<turnstile>Norm s -NewC C\<succ>Addr a-> s'"
apply simp
apply (rule eval_evals_exec.NewC)
apply auto
done

lemma eval_evals_exec_no_xcpt:
"!!s s'. (G\(x,s) -e \ v -> (x',s') --> x'=None --> x=None) \
          (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 only: split_tupled_all)
apply(rule eval_evals_exec_induct)
apply(unfold c_hupd_def)
apply(simp_all)
done

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

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

lemma exec_no_xcpt: "G \ (x, s) -c-> (None, s')
\<Longrightarrow> x = None"
apply (drule eval_evals_exec_no_xcpt [THEN conjunct2 [THEN conjunct2], rule_format])
apply simp+
done

lemma eval_evals_exec_xcpt:
"!!s s'. (G\(x,s) -e \ v -> (x',s') --> x=Some xc --> x'=Some xc \ s'=s) \
         (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 only: split_tupled_all)
apply (rule eval_evals_exec_induct)
apply (unfold c_hupd_def)
apply (simp_all)
done

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

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

lemma eval_LAcc_code: "G\Norm (h, l) -LAcc v\the (l v)-> Norm (h, l)"
using LAcc[of G "(h, l)" v] by simp

lemma eval_Call_code:
  "[| G\Norm s0 -e\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,(h4, l4)) |] ==>
   G\<turnstile>Norm s0 -{C}e..mn({pTs}ps)\<succ>v-> (x4,(h4,l))"
using Call[of G s0 e a' s1 a ps pvs x h l dynT md rT pns lvars blk res mn pTs s3 v x4 "(h4, l4)" C]
by simp

lemmas [code_pred_intro] = XcptE NewC Cast Lit BinOp
lemmas [code_pred_intro LAcc_code] = eval_LAcc_code
lemmas [code_pred_intro] = LAss FAcc FAss
lemmas [code_pred_intro Call_code] = eval_Call_code
lemmas [code_pred_intro] = XcptEs Nil Cons XcptS Skip Expr Comp Cond LoopF
lemmas [code_pred_intro LoopT_code] = LoopT

code_pred
  (modes:
    eval: i \<Rightarrow> i \<Rightarrow> i \<Rightarrow> o \<Rightarrow> o \<Rightarrow> bool
  and
    evals: i \<Rightarrow> i \<Rightarrow> i \<Rightarrow> o \<Rightarrow> o \<Rightarrow> bool
  and
    exec: i \<Rightarrow> i \<Rightarrow> i \<Rightarrow> o \<Rightarrow> bool)
  eval
proof -
  case eval
  from eval.prems show thesis
  proof(cases (no_simp))
    case LAcc with eval.LAcc_code show ?thesis by auto
  next
    case (Call a b c d e f g h i j k l m n o p q r s t u v s4)
    with eval.Call_code show ?thesis
      by(cases "s4") auto
  qed(erule (3) eval.that[OF refl]|assumption)+
next
  case evals
  from evals.prems show thesis
    by(cases (no_simp))(erule (3) evals.that[OF refl]|assumption)+
next
  case exec
  from exec.prems show thesis
  proof(cases (no_simp))
    case LoopT thus ?thesis by(rule exec.LoopT_code[OF refl])
  qed(erule (2) exec.that[OF refl]|assumption)+
qed

end

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