Theory Eval

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


header {* \isaheader{Operational Evaluation (big step) Semantics} *}

theory Eval imports State WellType begin


-- "Auxiliary notions"

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

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

definition lupd :: "vname => val => state => state" ("lupd'(_\<mapsto>_')"[10,10]1000) where
"lupd vn v ≡ λ (hp,loc). (hp, (loc(vn\<mapsto>v)))"

definition new_xcpt_var :: "vname => xstate => xstate" where
"new_xcpt_var vn ≡ λ(x,s). Norm (lupd(vn\<mapsto>the x) s)"


-- "Evaluation relations"

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

("_ \<turnstile> _ -_[\<succ>]_-> _" [51,82,60,51,82] 81)
and exec :: "[java_mb prog,xstate,stmt, xstate] => bool "
("_ \<turnstile> _ -_-> _" [51,82,60,82] 81)
for G :: "java_mb prog"
where

-- "evaluation of expressions"

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

-- "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 (¬ 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>]undefined-> (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; ¬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
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) ∧
(G\<turnstile>(x,s) -es[\<succ>]vs-> (x',s') --> x'=None --> x=None) ∧
(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\<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 exec_no_xcpt: "G \<turnstile> (x, s) -c-> (None, s')
==> 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\<turnstile>(x,s) -e \<succ> 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 ∧ s'=s) ∧
(G\<turnstile>(x,s) -c -> (x',s') --> x=Some xc --> x'=Some xc ∧ 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\<turnstile>(Some xc,s) -e\<succ>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\<turnstile>(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\<turnstile>Norm (h, l) -LAcc v\<succ>the (l v)-> Norm (h, l)"
using LAcc[of G "(h, l)" v] by simp

lemma eval_Call_code:
"[| 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,(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 => i => i => o => o => bool
and
evals: i => i => i => o => o => bool
and
exec: i => i => i => o => bool)
eval
proof -
case eval
from eval.prems show thesis
proof(cases (no_simp))
case LAcc with 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 Call_code show ?thesis
by(cases "s4")(simp, (erule meta_allE meta_impE|rule conjI refl| assumption)+)
qed(erule (3) that[OF refl]|assumption)+
next
case evals
from evals.prems show thesis
by(cases (no_simp))(erule (3) that[OF refl]|assumption)+
next
case exec
from exec.prems show thesis
proof(cases (no_simp))
case LoopT thus ?thesis by(rule LoopT_code[OF refl])
qed(erule (2) that[OF refl]|assumption)+
qed

end