--- a/src/HOL/NanoJava/OpSem.thy Tue Aug 07 22:42:22 2001 +0200
+++ b/src/HOL/NanoJava/OpSem.thy Wed Aug 08 12:36:48 2001 +0200
@@ -9,81 +9,116 @@
theory OpSem = State:
consts
- exec :: "(state \<times> stmt \<times> nat \<times> state) set"
+ exec :: "(state \<times> stmt \<times> nat \<times> state) set"
+ eval :: "(state \<times> expr \<times> val \<times> nat \<times> state) set"
syntax (xsymbols)
- exec :: "[state, stmt, nat, state] => bool" ("_ -_-_\<rightarrow> _" [98,90,99,98] 89)
+ exec :: "[state,stmt, nat,state] => bool" ("_ -_-_\<rightarrow> _" [98,90, 99,98] 89)
+ eval :: "[state,expr,val,nat,state] => bool" ("_ -_\<succ>_-_\<rightarrow> _"[98,95,99,99,98] 89)
syntax
- exec :: "[state, stmt, nat, state] => bool" ("_ -_-_-> _" [98,90,99,98] 89)
+ exec :: "[state,stmt, nat,state] => bool" ("_ -_-_-> _" [98,90, 99,98]89)
+ eval :: "[state,expr,val,nat,state] => bool" ("_ -_>_-_-> _"[98,95,99,99,98]89)
translations
- "s -c-n-> s'" == "(s, c, n, s') \<in> exec"
+ "s -c -n-> s'" == "(s, c, n, s') \<in> exec"
+ "s -e>v-n-> s'" == "(s, e, v, n, s') \<in> eval"
-inductive exec intros
+inductive exec eval intros
Skip: " s -Skip-n-> s"
Comp: "[| s0 -c1-n-> s1; s1 -c2-n-> s2 |] ==>
s0 -c1;; c2-n-> s2"
- Cond: "[| s -(if s<e> \<noteq> Null then c1 else c2)-n-> s' |] ==>
- s -If(e) c1 Else c2-n-> s'"
+ Cond: "[| s0 -e>v-n-> s1; s1 -(if v\<noteq>Null then c1 else c2)-n-> s2 |] ==>
+ s0 -If(e) c1 Else c2-n-> s2"
+
+ LoopF:" s0<x> = Null ==>
+ s0 -While(x) c-n-> s0"
+ LoopT:"[| s0<x> \<noteq> Null; s0 -c-n-> s1; s1 -While(x) c-n-> s2 |] ==>
+ s0 -While(x) c-n-> s2"
- LoopF:" s0<e> = Null ==>
- s0 -While(e) c-n-> s0"
- LoopT:"[| s0<e> \<noteq> Null; s0 -c-n-> s1; s1 -While(e) c-n-> s2 |] ==>
- s0 -While(e) c-n-> s2"
+ LAcc: " s -LAcc x>s<x>-n-> s"
+
+ LAss: " s -e>v-n-> s' ==>
+ s -x:==e-n-> lupd(x\<mapsto>v) s'"
+
+ FAcc: " s -e>Addr a-n-> s' ==>
+ s -e..f>get_field s' a f-n-> s'"
+
+ FAss: "[| s0 -e1>Addr a-n-> s1; s1 -e2>v-n-> s2 |] ==>
+ s0 -e1..f:==e2-n-> upd_obj a f v s2"
NewC: " new_Addr s = Addr a ==>
- s -x:=new C-n-> lupd(x\<mapsto>Addr a)(new_obj a C s)"
-
- Cast: " (case s<y> of Null => True | Addr a => obj_class s a \<preceq>C C) ==>
- s -x:=(C)y-n-> lupd(x\<mapsto>s<y>) s"
+ s -new C>Addr a-n-> new_obj a C s"
- FAcc: " s<y> = Addr a ==>
- s -x:=y..f-n-> lupd(x\<mapsto>get_field s a f) s"
+ Cast: "[| s -e>v-n-> s';
+ case v of Null => True | Addr a => obj_class s' a \<preceq>C C |] ==>
+ s -Cast C e>v-n-> s'"
- FAss: " s<y> = Addr a ==>
- s -y..f:=x-n-> upd_obj a f (s<x>) s"
-
- Call: "lupd(This\<mapsto>s<y>)(lupd(Param\<mapsto>s<p>)(init_locs C m s))-Meth C m -n-> s'==>
- s -x:={C}y..m(p)-n-> lupd(x\<mapsto>s'<Res>)(set_locs s s')"
+ Call: "[| s0 -e1>a-n-> s1; s1 -e2>p-n-> s2;
+ lupd(This\<mapsto>a)(lupd(Param\<mapsto>p)(init_locs C m s2)) -Meth C m-n-> s3
+ |] ==> s0 -{C}e1..m(e2)>s3<Res>-n-> set_locs s2 s3"
Meth: "[| s<This> = Addr a; obj_class s a\<preceq>C C;
s -Impl (obj_class s a) m-n-> s' |] ==>
s -Meth C m-n-> s'"
- Impl: " s -body C m- n-> s' ==>
+ Impl: " s -body C m-n-> s' ==>
s -Impl C m-Suc n-> 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(e) c -n-> t"
- "s -x:=new C -n-> t"
- "s -x:=(C)y -n-> t"
- "s -x:=y..f -n-> t"
- "s -y..f:=x -n-> t"
- "s -x:={C}y..m(p) -n-> t"
-inductive_cases Meth_elim_cases: "s -Meth C m -n-> t"
-inductive_cases Impl_elim_cases: "s -Impl C m -n-> t"
+ "s -Skip -n\<rightarrow> t"
+ "s -c1;; c2 -n\<rightarrow> t"
+ "s -If(e) c1 Else c2-n\<rightarrow> t"
+ "s -While(x) c -n\<rightarrow> t"
+ "s -x:==e -n\<rightarrow> t"
+ "s -e1..f:==e2 -n\<rightarrow> t"
+inductive_cases Meth_elim_cases: "s -Meth C m-n\<rightarrow> t"
+inductive_cases Impl_elim_cases: "s -Impl C m-n\<rightarrow> t"
lemmas exec_elim_cases = exec_elim_cases' Meth_elim_cases Impl_elim_cases
+inductive_cases eval_elim_cases:
+ "s -new C \<succ>v-n\<rightarrow> t"
+ "s -Cast C e \<succ>v-n\<rightarrow> t"
+ "s -LAcc x \<succ>v-n\<rightarrow> t"
+ "s -e..f \<succ>v-n\<rightarrow> t"
+ "s -{C}e1..m(e2) \<succ>v-n\<rightarrow> t"
-lemma exec_mono [rule_format]: "s -c-n\<rightarrow> t \<Longrightarrow> \<forall>m. n \<le> m \<longrightarrow> s -c-m\<rightarrow> t"
-apply (erule exec.induct)
-prefer 12 (* Impl *)
+lemma exec_eval_mono [rule_format]:
+ "(s -c -n\<rightarrow> t \<longrightarrow> (\<forall>m. n \<le> m \<longrightarrow> s -c -m\<rightarrow> t)) \<and>
+ (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.intros)+
+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: "\<lbrakk>s1 -c1- n1 \<rightarrow> t1 ; s2 -c2- n2\<rightarrow> t2\<rbrakk> \<Longrightarrow>
+ s1 -c1-max n1 n2\<rightarrow> t1 \<and> s2 -c2-max n1 n2\<rightarrow> t2"
+by (fast intro: exec_mono le_maxI1 le_maxI2)
-lemma exec_max2: "\<lbrakk>s1 -c1- n1 \<rightarrow> t1 ; s2 -c2- n2\<rightarrow> t2\<rbrakk> \<Longrightarrow>
- s1 -c1-max n1 n2\<rightarrow> t1 \<and> s2 -c2-max n1 n2\<rightarrow> t2"
-by (fast intro: exec_mono le_maxI1 le_maxI2)
+lemma eval_exec_max: "\<lbrakk>s1 -c- n1 \<rightarrow> t1 ; s2 -e\<succ>v- n2\<rightarrow> t2\<rbrakk> \<Longrightarrow>
+ s1 -c-max n1 n2\<rightarrow> t1 \<and> s2 -e\<succ>v-max n1 n2\<rightarrow> t2"
+by (fast intro: eval_mono exec_mono le_maxI1 le_maxI2)
+
+lemma eval_eval_max: "\<lbrakk>s1 -e1\<succ>v1- n1 \<rightarrow> t1 ; s2 -e2\<succ>v2- n2\<rightarrow> t2\<rbrakk> \<Longrightarrow>
+ s1 -e1\<succ>v1-max n1 n2\<rightarrow> t1 \<and> s2 -e2\<succ>v2-max n1 n2\<rightarrow> t2"
+by (fast intro: eval_mono le_maxI1 le_maxI2)
+
+lemma eval_eval_exec_max:
+ "\<lbrakk>s1 -e1\<succ>v1-n1\<rightarrow> t1; s2 -e2\<succ>v2-n2\<rightarrow> t2; s3 -c-n3\<rightarrow> t3\<rbrakk> \<Longrightarrow>
+ 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 le_maxI1 le_maxI2)
lemma Impl_body_eq: "(\<lambda>t. \<exists>n. z -Impl C m-n\<rightarrow> t) = (\<lambda>t. \<exists>n. z -body C m-n\<rightarrow> t)"
apply (rule ext)
-apply (fast elim: exec_elim_cases intro: exec.Impl)
+apply (fast elim: exec_elim_cases intro: exec_eval.Impl)
done
end
\ No newline at end of file