(* Title: HOL/NanoJava/OpSem.thy
Author: David von Oheimb
Copyright 2001 Technische Universitaet Muenchen
*)
section "Operational Evaluation Semantics"
theory OpSem imports State begin
inductive
exec :: "[state,stmt, nat,state] => bool" ("_ -_-_\ _" [98,90, 65,98] 89)
and eval :: "[state,expr,val,nat,state] => bool" ("_ -_\_-_\ _"[98,95,99,65,98] 89)
where
Skip: " s -Skip-n\ s"
| Comp: "[| s0 -c1-n\ s1; s1 -c2-n\ s2 |] ==>
s0 -c1;; c2-n\<rightarrow> s2"
| Cond: "[| s0 -e\v-n\ s1; s1 -(if v\Null then c1 else c2)-n\ s2 |] ==>
s0 -If(e) c1 Else c2-n\<rightarrow> s2"
| LoopF:" s0 = Null ==>
s0 -While(x) c-n\<rightarrow> s0"
| LoopT:"[| s0 \ Null; s0 -c-n\ s1; s1 -While(x) c-n\ s2 |] ==>
s0 -While(x) c-n\<rightarrow> s2"
| LAcc: " s -LAcc x\s-n\ s"
| LAss: " s -e\v-n\ s' ==>
s -x:==e-n\<rightarrow> lupd(x\<mapsto>v) s'"
| FAcc: " s -e\Addr a-n\ s' ==>
s -e..f\<succ>get_field s' a f-n\<rightarrow> s'"
| FAss: "[| s0 -e1\Addr a-n\ s1; s1 -e2\v-n\ s2 |] ==>
s0 -e1..f:==e2-n\<rightarrow> upd_obj a f v s2"
| NewC: " new_Addr s = Addr a ==>
s -new C\<succ>Addr a-n\<rightarrow> new_obj a C s"
| Cast: "[| s -e\v-n\ s';
case v of Null => True | Addr a => obj_class s' a \C C |] ==>
s -Cast C e\<succ>v-n\<rightarrow> s'"
| Call: "[| s0 -e1\a-n\ s1; s1 -e2\p-n\ s2;
lupd(This\<mapsto>a)(lupd(Par\<mapsto>p)(del_locs s2)) -Meth (C,m)-n\<rightarrow> s3
|] ==> s0 -{C}e1..m(e2)\<succ>s3<Res>-n\<rightarrow> set_locs s2 s3"
| Meth: "[| s = Addr a; D = obj_class s a; D\C C;
init_locs D m s -Impl (D,m)-n\<rightarrow> s' |] ==>
s -Meth (C,m)-n\<rightarrow> s'"
| Impl: " s -body Cm- n\ s' ==>
s -Impl Cm-Suc n\<rightarrow> s'"
inductive_cases exec_elim_cases':
"s -Skip -n\ t"
"s -c1;; c2 -n\ t"
"s -If(e) c1 Else c2-n\ t"
"s -While(x) c -n\ t"
"s -x:==e -n\ t"
"s -e1..f:==e2 -n\ t"
inductive_cases Meth_elim_cases: "s -Meth Cm -n\ t"
inductive_cases Impl_elim_cases: "s -Impl Cm -n\ t"
lemmas exec_elim_cases = exec_elim_cases' Meth_elim_cases Impl_elim_cases
inductive_cases eval_elim_cases:
"s -new C \v-n\ t"
"s -Cast C e \v-n\ t"
"s -LAcc x \v-n\ t"
"s -e..f \v-n\ t"
"s -{C}e1..m(e2) \v-n\ t"
lemma exec_eval_mono [rule_format]:
"(s -c -n\ t \ (\m. n \ m \ s -c -m\ t)) \
(s -e\<succ>v-n\<rightarrow> t \<longrightarrow> (\<forall>m. n \<le> m \<longrightarrow> s -e\<succ>v-m\<rightarrow> t))"
apply (rule exec_eval.induct)
prefer 14 (* Impl *)
apply clarify
apply (rename_tac n)
apply (case_tac n)
apply (blast intro:exec_eval.intros)+
done
lemmas exec_mono = exec_eval_mono [THEN conjunct1, rule_format]
lemmas eval_mono = exec_eval_mono [THEN conjunct2, rule_format]
lemma exec_exec_max: "\s1 -c1- n1 \ t1 ; s2 -c2- n2\ t2\ \
s1 -c1-max n1 n2\<rightarrow> t1 \<and> s2 -c2-max n1 n2\<rightarrow> t2"
by (fast intro: exec_mono max.cobounded1 max.cobounded2)
lemma eval_exec_max: "\s1 -c- n1 \ t1 ; s2 -e\v- n2\ t2\ \
s1 -c-max n1 n2\<rightarrow> t1 \<and> s2 -e\<succ>v-max n1 n2\<rightarrow> t2"
by (fast intro: eval_mono exec_mono max.cobounded1 max.cobounded2)
lemma eval_eval_max: "\s1 -e1\v1- n1 \ t1 ; s2 -e2\v2- n2\ t2\ \
s1 -e1\<succ>v1-max n1 n2\<rightarrow> t1 \<and> s2 -e2\<succ>v2-max n1 n2\<rightarrow> t2"
by (fast intro: eval_mono max.cobounded1 max.cobounded2)
lemma eval_eval_exec_max:
"\s1 -e1\v1-n1\ t1; s2 -e2\v2-n2\ t2; s3 -c-n3\ t3\ \
s1 -e1\<succ>v1-max (max n1 n2) n3\<rightarrow> t1 \<and>
s2 -e2\<succ>v2-max (max n1 n2) n3\<rightarrow> t2 \<and>
s3 -c -max (max n1 n2) n3\<rightarrow> t3"
apply (drule (1) eval_eval_max, erule thin_rl)
by (fast intro: exec_mono eval_mono max.cobounded1 max.cobounded2)
lemma Impl_body_eq: "(\t. \n. Z -Impl M-n\ t) = (\t. \n. Z -body M-n\ t)"
apply (rule ext)
apply (fast elim: exec_elim_cases intro: exec_eval.Impl)
done
end
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