(* Title: HOL/MicroJava/BV/Correct.thy
ID: $Id$
Author: Cornelia Pusch
Copyright 1999 Technische Universitaet Muenchen
The invariant for the type safety proof.
*)
header "Type Safety Invariant"
theory Correct = BVSpec:
constdefs
approx_val :: "[jvm_prog,aheap,val,ty err] \<Rightarrow> bool"
"approx_val G h v any \<equiv> case any of Err \<Rightarrow> True | Ok T \<Rightarrow> G,h\<turnstile>v\<Colon>\<preceq>T"
approx_loc :: "[jvm_prog,aheap,val list,locvars_type] \<Rightarrow> bool"
"approx_loc G hp loc LT \<equiv> list_all2 (approx_val G hp) loc LT"
approx_stk :: "[jvm_prog,aheap,opstack,opstack_type] \<Rightarrow> bool"
"approx_stk G hp stk ST \<equiv> approx_loc G hp stk (map Ok ST)"
correct_frame :: "[jvm_prog,aheap,state_type,nat,bytecode] \<Rightarrow> frame \<Rightarrow> bool"
"correct_frame G hp \<equiv> \<lambda>(ST,LT) maxl ins (stk,loc,C,sig,pc).
approx_stk G hp stk ST \<and> approx_loc G hp loc LT \<and>
pc < length ins \<and> length loc=length(snd sig)+maxl+1"
correct_frame_opt :: "[jvm_prog,aheap,state_type option,nat,bytecode] \<Rightarrow> frame \<Rightarrow> bool"
"correct_frame_opt G hp s \<equiv> case s of None \<Rightarrow> \<lambda>maxl ins f. False | Some t \<Rightarrow> correct_frame G hp t"
consts
correct_frames :: "[jvm_prog,aheap,prog_type,ty,sig,frame list] \<Rightarrow> bool"
primrec
"correct_frames G hp phi rT0 sig0 [] = True"
"correct_frames G hp phi rT0 sig0 (f#frs) =
(let (stk,loc,C,sig,pc) = f in
(\<exists>ST LT rT maxl ins.
phi C sig ! pc = Some (ST,LT) \<and>
method (G,C) sig = Some(C,rT,(maxl,ins)) \<and>
(\<exists>C' mn pTs k. pc = k+1 \<and> ins!k = (Invoke C' mn pTs) \<and>
(mn,pTs) = sig0 \<and>
(\<exists>apTs D ST' LT'.
(phi C sig)!k = Some ((rev apTs) @ (Class D) # ST', LT') \<and>
length apTs = length pTs \<and>
(\<exists>D' rT' maxl' ins'.
method (G,D) sig0 = Some(D',rT',(maxl',ins')) \<and>
G \<turnstile> rT0 \<preceq> rT') \<and>
correct_frame G hp (tl ST, LT) maxl ins f \<and>
correct_frames G hp phi rT sig frs))))"
constdefs
correct_state :: "[jvm_prog,prog_type,jvm_state] \<Rightarrow> bool"
("_,_\<turnstile>JVM _\<surd>" [51,51] 50)
"correct_state G phi \<equiv> \<lambda>(xp,hp,frs).
case xp of
None \<Rightarrow> (case frs of
[] \<Rightarrow> True
| (f#fs) \<Rightarrow> G\<turnstile>h hp\<surd> \<and>
(let (stk,loc,C,sig,pc) = f
in
\<exists>rT maxl ins s.
method (G,C) sig = Some(C,rT,(maxl,ins)) \<and>
phi C sig ! pc = Some s \<and>
correct_frame G hp s maxl ins f \<and>
correct_frames G hp phi rT sig fs))
| Some x \<Rightarrow> True"
lemma sup_heap_newref:
"hp x = None \<Longrightarrow> hp \<le>| hp(newref hp \<mapsto> obj)"
apply (unfold hext_def)
apply clarsimp
apply (drule newref_None 1) back
apply simp
.
lemma sup_heap_update_value:
"hp a = Some (C,od') \<Longrightarrow> hp \<le>| hp (a \<mapsto> (C,od))"
by (simp add: hext_def)
(** approx_val **)
lemma approx_val_Err:
"approx_val G hp x Err"
by (simp add: approx_val_def)
lemma approx_val_Null:
"approx_val G hp Null (Ok (RefT x))"
by (auto intro: null simp add: approx_val_def)
lemma approx_val_imp_approx_val_assConvertible [rule_format]:
"wf_prog wt G \<Longrightarrow> approx_val G hp v (Ok T) \<longrightarrow> G\<turnstile> T\<preceq>T' \<longrightarrow> approx_val G hp v (Ok T')"
by (cases T) (auto intro: conf_widen simp add: approx_val_def)
lemma approx_val_imp_approx_val_sup_heap [rule_format]:
"approx_val G hp v at \<longrightarrow> hp \<le>| hp' \<longrightarrow> approx_val G hp' v at"
apply (simp add: approx_val_def split: err.split)
apply (blast intro: conf_hext)
.
lemma approx_val_imp_approx_val_heap_update:
"\<lbrakk>hp a = Some obj'; G,hp\<turnstile> v\<Colon>\<preceq>T; obj_ty obj = obj_ty obj'\<rbrakk>
\<Longrightarrow> G,hp(a\<mapsto>obj)\<turnstile> v\<Colon>\<preceq>T"
by (cases v, auto simp add: obj_ty_def conf_def)
lemma approx_val_imp_approx_val_sup [rule_format]:
"wf_prog wt G \<Longrightarrow> (approx_val G h v us) \<longrightarrow> (G \<turnstile> us <=o us') \<longrightarrow> (approx_val G h v us')"
apply (simp add: sup_PTS_eq approx_val_def split: err.split)
apply (blast intro: conf_widen)
.
lemma approx_loc_imp_approx_val_sup:
"\<lbrakk>wf_prog wt G; approx_loc G hp loc LT; idx < length LT; v = loc!idx; G \<turnstile> LT!idx <=o at\<rbrakk>
\<Longrightarrow> approx_val G hp v at"
apply (unfold approx_loc_def)
apply (unfold list_all2_def)
apply (auto intro: approx_val_imp_approx_val_sup simp add: split_def all_set_conv_all_nth)
.
(** approx_loc **)
lemma approx_loc_Cons [iff]:
"approx_loc G hp (s#xs) (l#ls) = (approx_val G hp s l \<and> approx_loc G hp xs ls)"
by (simp add: approx_loc_def)
lemma assConv_approx_stk_imp_approx_loc [rule_format]:
"wf_prog wt G \<Longrightarrow> (\<forall>tt'\<in>set (zip tys_n ts). tt' \<in> widen G)
\<longrightarrow> length tys_n = length ts \<longrightarrow> approx_stk G hp s tys_n \<longrightarrow>
approx_loc G hp s (map Ok ts)"
apply (unfold approx_stk_def approx_loc_def list_all2_def)
apply (clarsimp simp add: all_set_conv_all_nth)
apply (rule approx_val_imp_approx_val_assConvertible)
apply auto
.
lemma approx_loc_imp_approx_loc_sup_heap [rule_format]:
"\<forall>lvars. approx_loc G hp lvars lt \<longrightarrow> hp \<le>| hp' \<longrightarrow> approx_loc G hp' lvars lt"
apply (unfold approx_loc_def list_all2_def)
apply (cases lt)
apply simp
apply clarsimp
apply (rule approx_val_imp_approx_val_sup_heap)
apply auto
.
lemma approx_loc_imp_approx_loc_sup [rule_format]:
"wf_prog wt G \<Longrightarrow> approx_loc G hp lvars lt \<longrightarrow> G \<turnstile> lt <=l lt' \<longrightarrow> approx_loc G hp lvars lt'"
apply (unfold sup_loc_def approx_loc_def list_all2_def)
apply (auto simp add: all_set_conv_all_nth)
apply (auto elim: approx_val_imp_approx_val_sup)
.
lemma approx_loc_imp_approx_loc_subst [rule_format]:
"\<forall>loc idx x X. (approx_loc G hp loc LT) \<longrightarrow> (approx_val G hp x X)
\<longrightarrow> (approx_loc G hp (loc[idx:=x]) (LT[idx:=X]))"
apply (unfold approx_loc_def list_all2_def)
apply (auto dest: subsetD [OF set_update_subset_insert] simp add: zip_update)
.
lemmas [cong] = conj_cong
lemma approx_loc_append [rule_format]:
"\<forall>L1 l2 L2. length l1=length L1 \<longrightarrow>
approx_loc G hp (l1@l2) (L1@L2) = (approx_loc G hp l1 L1 \<and> approx_loc G hp l2 L2)"
apply (unfold approx_loc_def list_all2_def)
apply simp
apply blast
.
lemmas [cong del] = conj_cong
(** approx_stk **)
lemma approx_stk_rev_lem:
"approx_stk G hp (rev s) (rev t) = approx_stk G hp s t"
apply (unfold approx_stk_def approx_loc_def list_all2_def)
apply (auto simp add: zip_rev sym [OF rev_map])
.
lemma approx_stk_rev:
"approx_stk G hp (rev s) t = approx_stk G hp s (rev t)"
by (auto intro: subst [OF approx_stk_rev_lem])
lemma approx_stk_imp_approx_stk_sup_heap [rule_format]:
"\<forall>lvars. approx_stk G hp lvars lt \<longrightarrow> hp \<le>| hp' \<longrightarrow> approx_stk G hp' lvars lt"
by (auto intro: approx_loc_imp_approx_loc_sup_heap simp add: approx_stk_def)
lemma approx_stk_imp_approx_stk_sup [rule_format]:
"wf_prog wt G \<Longrightarrow> approx_stk G hp lvars st \<longrightarrow> (G \<turnstile> map Ok st <=l (map Ok st'))
\<longrightarrow> approx_stk G hp lvars st'"
by (auto intro: approx_loc_imp_approx_loc_sup simp add: approx_stk_def)
lemma approx_stk_Nil [iff]:
"approx_stk G hp [] []"
by (simp add: approx_stk_def approx_loc_def)
lemma approx_stk_Cons [iff]:
"approx_stk G hp (x # stk) (S#ST) =
(approx_val G hp x (Ok S) \<and> approx_stk G hp stk ST)"
by (simp add: approx_stk_def approx_loc_def)
lemma approx_stk_Cons_lemma [iff]:
"approx_stk G hp stk (S#ST') =
(\<exists>s stk'. stk = s#stk' \<and> approx_val G hp s (Ok S) \<and> approx_stk G hp stk' ST')"
by (simp add: list_all2_Cons2 approx_stk_def approx_loc_def)
lemma approx_stk_append_lemma:
"approx_stk G hp stk (S@ST') \<Longrightarrow>
(\<exists>s stk'. stk = s@stk' \<and> length s = length S \<and> length stk' = length ST' \<and>
approx_stk G hp s S \<and> approx_stk G hp stk' ST')"
by (simp add: list_all2_append2 approx_stk_def approx_loc_def)
(** oconf **)
lemma correct_init_obj:
"\<lbrakk>is_class G C; wf_prog wt G\<rbrakk> \<Longrightarrow>
G,h \<turnstile> (C, map_of (map (\<lambda>(f,fT).(f,default_val fT)) (fields(G,C)))) \<surd>"
apply (unfold oconf_def lconf_def)
apply (simp add: map_of_map)
apply (force intro: defval_conf dest: map_of_SomeD is_type_fields)
.
lemma oconf_imp_oconf_field_update [rule_format]:
"\<lbrakk>map_of (fields (G, oT)) FD = Some T; G,hp\<turnstile>v\<Colon>\<preceq>T; G,hp\<turnstile>(oT,fs)\<surd> \<rbrakk>
\<Longrightarrow> G,hp\<turnstile>(oT, fs(FD\<mapsto>v))\<surd>"
by (simp add: oconf_def lconf_def)
lemma oconf_imp_oconf_heap_newref [rule_format]:
"hp x = None \<longrightarrow> G,hp\<turnstile>obj\<surd> \<longrightarrow> G,hp\<turnstile>obj'\<surd> \<longrightarrow> G,(hp(newref hp\<mapsto>obj'))\<turnstile>obj\<surd>"
apply (unfold oconf_def lconf_def)
apply simp
apply (fast intro: conf_hext sup_heap_newref)
.
lemma oconf_imp_oconf_heap_update [rule_format]:
"hp a = Some obj' \<longrightarrow> obj_ty obj' = obj_ty obj'' \<longrightarrow> G,hp\<turnstile>obj\<surd>
\<longrightarrow> G,hp(a\<mapsto>obj'')\<turnstile>obj\<surd>"
apply (unfold oconf_def lconf_def)
apply simp
apply (force intro: approx_val_imp_approx_val_heap_update)
.
(** hconf **)
lemma hconf_imp_hconf_newref [rule_format]:
"hp x = None \<longrightarrow> G\<turnstile>h hp\<surd> \<longrightarrow> G,hp\<turnstile>obj\<surd> \<longrightarrow> G\<turnstile>h hp(newref hp\<mapsto>obj)\<surd>"
apply (simp add: hconf_def)
apply (fast intro: oconf_imp_oconf_heap_newref)
.
lemma hconf_imp_hconf_field_update [rule_format]:
"map_of (fields (G, oT)) (F, D) = Some T \<and> hp oloc = Some(oT,fs) \<and>
G,hp\<turnstile>v\<Colon>\<preceq>T \<and> G\<turnstile>h hp\<surd> \<longrightarrow> G\<turnstile>h hp(oloc \<mapsto> (oT, fs((F,D)\<mapsto>v)))\<surd>"
apply (simp add: hconf_def)
apply (force intro: oconf_imp_oconf_heap_update oconf_imp_oconf_field_update
simp add: obj_ty_def)
.
(** correct_frames **)
lemmas [simp del] = fun_upd_apply
lemma correct_frames_imp_correct_frames_field_update [rule_format]:
"\<forall>rT C sig. correct_frames G hp phi rT sig frs \<longrightarrow>
hp a = Some (C,od) \<longrightarrow> map_of (fields (G, C)) fl = Some fd \<longrightarrow>
G,hp\<turnstile>v\<Colon>\<preceq>fd
\<longrightarrow> correct_frames G (hp(a \<mapsto> (C, od(fl\<mapsto>v)))) phi rT sig frs";
apply (induct frs)
apply simp
apply (clarsimp simp add: correct_frame_def) (*takes long*)
apply (intro exI conjI)
apply simp
apply simp
apply force
apply simp
apply (rule approx_stk_imp_approx_stk_sup_heap)
apply simp
apply (rule sup_heap_update_value)
apply simp
apply (rule approx_loc_imp_approx_loc_sup_heap)
apply simp
apply (rule sup_heap_update_value)
apply simp
.
lemma correct_frames_imp_correct_frames_newref [rule_format]:
"\<forall>rT C sig. hp x = None \<longrightarrow> correct_frames G hp phi rT sig frs \<and> oconf G hp obj
\<longrightarrow> correct_frames G (hp(newref hp \<mapsto> obj)) phi rT sig frs"
apply (induct frs)
apply simp
apply (clarsimp simp add: correct_frame_def)
apply (intro exI conjI)
apply simp
apply simp
apply force
apply simp
apply (rule approx_stk_imp_approx_stk_sup_heap)
apply simp
apply (rule sup_heap_newref)
apply simp
apply (rule approx_loc_imp_approx_loc_sup_heap)
apply simp
apply (rule sup_heap_newref)
apply simp
.
end