(* Title: HOL/MicroJava/BV/Correct.thy
Author: Cornelia Pusch, Gerwin Klein
Copyright 1999 Technische Universitaet Muenchen
*)
section \<open>BV Type Safety Invariant\<close>
theory Correct
imports BVSpec "../JVM/JVMExec"
begin
definition approx_val :: "[jvm_prog,aheap,val,ty err] \ bool" where
"approx_val G h v any == case any of Err \ True | OK T \ G,h\v::\T"
definition approx_loc :: "[jvm_prog,aheap,val list,locvars_type] \ bool" where
"approx_loc G hp loc LT == list_all2 (approx_val G hp) loc LT"
definition approx_stk :: "[jvm_prog,aheap,opstack,opstack_type] \ bool" where
"approx_stk G hp stk ST == approx_loc G hp stk (map OK ST)"
definition correct_frame :: "[jvm_prog,aheap,state_type,nat,bytecode] \ frame \ bool" where
"correct_frame G hp == \(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"
primrec correct_frames :: "[jvm_prog,aheap,prog_type,ty,sig,frame list] \ bool" where
"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 maxs maxl ins et.
phi C sig ! pc = Some (ST,LT) \<and> is_class G C \<and>
method (G,C) sig = Some(C,rT,(maxs,maxl,ins,et)) \<and>
(\<exists>C' mn pTs. ins!pc = (Invoke C' mn pTs) \<and>
(mn,pTs) = sig0 \<and>
(\<exists>apTs D ST' LT'.
(phi C sig)!pc = Some ((rev apTs) @ (Class D) # ST', LT') \<and>
length apTs = length pTs \<and>
(\<exists>D' rT' maxs' maxl' ins' et'.
method (G,D) sig0 = Some(D',rT',(maxs',maxl',ins',et')) \<and>
G \<turnstile> rT0 \<preceq> rT') \<and>
correct_frame G hp (ST, LT) maxl ins f \<and>
correct_frames G hp phi rT sig frs))))"
definition correct_state :: "[jvm_prog,prog_type,jvm_state] \ bool"
("_,_ \JVM _ \" [51,51] 50) where
"correct_state G phi == \(xp,hp,frs).
case xp of
None \<Rightarrow> (case frs of
[] \<Rightarrow> True
| (f#fs) \<Rightarrow> G\<turnstile>h hp\<surd> \<and> preallocated hp \<and>
(let (stk,loc,C,sig,pc) = f
in
\<exists>rT maxs maxl ins et s.
is_class G C \<and>
method (G,C) sig = Some(C,rT,(maxs,maxl,ins,et)) \<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> frs = []"
lemma sup_ty_opt_OK:
"(G \ X <=o (OK T')) = (\T. X = OK T \ G \ T \ T')"
by (cases X) auto
subsection \<open>approx-val\<close>
lemma approx_val_Err [simp,intro!]:
"approx_val G hp x Err"
by (simp add: approx_val_def)
lemma approx_val_OK [iff]:
"approx_val G hp x (OK T) = (G,hp \ x ::\ T)"
by (simp add: approx_val_def)
lemma approx_val_Null [simp,intro!]:
"approx_val G hp Null (OK (RefT x))"
by (auto simp add: approx_val_def)
lemma approx_val_sup_heap:
"\ approx_val G hp v T; hp \| hp' \ \ approx_val G hp' v T"
by (cases T) (blast intro: conf_hext)+
lemma approx_val_heap_update:
"\ hp a = Some obj'; G,hp\ v::\T; obj_ty obj = obj_ty obj'\
\<Longrightarrow> G,hp(a\<mapsto>obj)\<turnstile> v::\<preceq>T"
by (cases v) (auto simp add: obj_ty_def conf_def)
lemma approx_val_widen:
"\ approx_val G hp v T; G \ T <=o T'; wf_prog wt G \
\<Longrightarrow> approx_val G hp v T'"
by (cases T') (auto simp add: sup_ty_opt_OK intro: conf_widen)
subsection \<open>approx-loc\<close>
lemma approx_loc_Nil [simp,intro!]:
"approx_loc G hp [] []"
by (simp add: approx_loc_def)
lemma approx_loc_Cons [iff]:
"approx_loc G hp (l#ls) (L#LT) =
(approx_val G hp l L \<and> approx_loc G hp ls LT)"
by (simp add: approx_loc_def)
lemma approx_loc_nth:
"\ approx_loc G hp loc LT; n < length LT \
\<Longrightarrow> approx_val G hp (loc!n) (LT!n)"
by (simp add: approx_loc_def list_all2_conv_all_nth)
lemma approx_loc_imp_approx_val_sup:
"\approx_loc G hp loc LT; n < length LT; LT ! n = OK T; G \ T \ T'; wf_prog wt G\
\<Longrightarrow> G,hp \<turnstile> (loc!n) ::\<preceq> T'"
apply (drule approx_loc_nth, assumption)
apply simp
apply (erule conf_widen, assumption+)
done
lemma approx_loc_conv_all_nth:
"approx_loc G hp loc LT =
(length loc = length LT \<and> (\<forall>n < length loc. approx_val G hp (loc!n) (LT!n)))"
by (simp add: approx_loc_def list_all2_conv_all_nth)
lemma approx_loc_sup_heap:
"\ approx_loc G hp loc LT; hp \| hp' \
\<Longrightarrow> approx_loc G hp' loc LT"
apply (clarsimp simp add: approx_loc_conv_all_nth)
apply (blast intro: approx_val_sup_heap)
done
lemma approx_loc_widen:
"\ approx_loc G hp loc LT; G \ LT <=l LT'; wf_prog wt G \
\<Longrightarrow> approx_loc G hp loc LT'"
apply (unfold Listn.le_def lesub_def sup_loc_def)
apply (simp (no_asm_use) only: list_all2_conv_all_nth approx_loc_conv_all_nth)
apply (simp (no_asm_simp))
apply clarify
apply (erule allE, erule impE)
apply simp
apply (erule approx_val_widen)
apply simp
apply assumption
done
lemma loc_widen_Err [dest]:
"\XT. G \ replicate n Err <=l XT \ XT = replicate n Err"
by (induct n) auto
lemma approx_loc_Err [iff]:
"approx_loc G hp (replicate n v) (replicate n Err)"
by (induct n) auto
lemma approx_loc_subst:
"\ approx_loc G hp loc LT; approx_val G hp x X \
\<Longrightarrow> approx_loc G hp (loc[idx:=x]) (LT[idx:=X])"
apply (unfold approx_loc_def list_all2_iff)
apply (auto dest: subsetD [OF set_update_subset_insert] simp add: zip_update)
done
lemma approx_loc_append:
"length l1=length L1 \
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_iff)
apply (simp cong: conj_cong)
apply blast
done
subsection \<open>approx-stk\<close>
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)
apply (simp add: rev_map [symmetric])
done
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_sup_heap:
"\ approx_stk G hp stk ST; hp \| hp' \ \ approx_stk G hp' stk ST"
by (auto intro: approx_loc_sup_heap simp add: approx_stk_def)
lemma approx_stk_widen:
"\ approx_stk G hp stk ST; G \ map OK ST <=l map OK ST'; wf_prog wt G \
\<Longrightarrow> approx_stk G hp stk ST'"
by (auto elim: approx_loc_widen simp add: approx_stk_def)
lemma approx_stk_Nil [iff]:
"approx_stk G hp [] []"
by (simp add: approx_stk_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)
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:
"approx_stk G hp stk (S@S') \
(\<exists>s stk'. stk = s@stk' \<and> length s = length S \<and> length stk' = length S' \<and>
approx_stk G hp s S \<and> approx_stk G hp stk' S')"
by (simp add: list_all2_append2 approx_stk_def approx_loc_def)
lemma approx_stk_all_widen:
"\ approx_stk G hp stk ST; \(x, y) \ set (zip ST ST'). G \ x \ y; length ST = length ST'; wf_prog wt G \
\<Longrightarrow> approx_stk G hp stk ST'"
apply (unfold approx_stk_def)
apply (clarsimp simp add: approx_loc_conv_all_nth all_set_conv_all_nth)
apply (erule allE, erule impE, assumption)
apply (erule allE, erule impE, assumption)
apply (erule conf_widen, assumption+)
done
subsection \<open>oconf\<close>
lemma oconf_field_update:
"\map_of (fields (G, oT)) FD = Some T; G,hp\v::\T; G,hp\(oT,fs)\ \
\<Longrightarrow> G,hp\<turnstile>(oT, fs(FD\<mapsto>v))\<surd>"
by (simp add: oconf_def lconf_def)
lemma oconf_newref:
"\hp oref = None; G,hp \ obj \; G,hp \ obj' \\ \ G,hp(oref\obj') \ obj \"
apply (unfold oconf_def lconf_def)
apply simp
apply (blast intro: conf_hext hext_new)
done
lemma oconf_heap_update:
"\ hp a = Some obj'; obj_ty obj' = obj_ty obj''; G,hp\obj\ \
\<Longrightarrow> G,hp(a\<mapsto>obj'')\<turnstile>obj\<surd>"
apply (unfold oconf_def lconf_def)
apply (fastforce intro: approx_val_heap_update)
done
subsection \<open>hconf\<close>
lemma hconf_newref:
"\ hp oref = None; G\h hp\; G,hp\obj\ \ \ G\h hp(oref\obj)\"
apply (simp add: hconf_def)
apply (fast intro: oconf_newref)
done
lemma hconf_field_update:
"\ map_of (fields (G, oT)) X = Some T; hp a = Some(oT,fs);
G,hp\<turnstile>v::\<preceq>T; G\<turnstile>h hp\<surd> \<rbrakk>
\<Longrightarrow> G\<turnstile>h hp(a \<mapsto> (oT, fs(X\<mapsto>v)))\<surd>"
apply (simp add: hconf_def)
apply (fastforce intro: oconf_heap_update oconf_field_update
simp add: obj_ty_def)
done
subsection \<open>preallocated\<close>
lemma preallocated_field_update:
"\ map_of (fields (G, oT)) X = Some T; hp a = Some(oT,fs);
G\<turnstile>h hp\<surd>; preallocated hp \<rbrakk>
\<Longrightarrow> preallocated (hp(a \<mapsto> (oT, fs(X\<mapsto>v))))"
apply (unfold preallocated_def)
apply (rule allI)
apply (erule_tac x=x in allE)
apply simp
apply (rule ccontr)
apply (unfold hconf_def)
apply (erule allE, erule allE, erule impE, assumption)
apply (unfold oconf_def lconf_def)
apply (simp del: split_paired_All)
done
lemma
assumes none: "hp oref = None" and alloc: "preallocated hp"
shows preallocated_newref: "preallocated (hp(oref\obj))"
proof (cases oref)
case (XcptRef x)
with none alloc have False by (auto elim: preallocatedE [of _ x])
thus ?thesis ..
next
case (Loc l)
with alloc show ?thesis by (simp add: preallocated_def)
qed
subsection \<open>correct-frames\<close>
lemmas [simp del] = fun_upd_apply
lemma correct_frames_field_update [rule_format]:
"\rT C sig.
correct_frames G hp phi rT sig frs \<longrightarrow>
hp a = Some (C,fs) \<longrightarrow>
map_of (fields (G, C)) fl = Some fd \<longrightarrow>
G,hp\<turnstile>v::\<preceq>fd
\<longrightarrow> correct_frames G (hp(a \<mapsto> (C, fs(fl\<mapsto>v)))) phi rT sig frs"
apply (induct frs)
apply simp
apply clarify
apply (simp (no_asm_use))
apply clarify
apply (unfold correct_frame_def)
apply (simp (no_asm_use))
apply clarify
apply (intro exI conjI)
apply assumption+
apply (erule approx_stk_sup_heap)
apply (erule hext_upd_obj)
apply (erule approx_loc_sup_heap)
apply (erule hext_upd_obj)
apply assumption+
apply blast
done
lemma correct_frames_newref [rule_format]:
"\rT C sig.
hp x = None \<longrightarrow>
correct_frames G hp phi rT sig frs \<longrightarrow>
correct_frames G (hp(x \<mapsto> obj)) phi rT sig frs"
apply (induct frs)
apply simp
apply clarify
apply (simp (no_asm_use))
apply clarify
apply (unfold correct_frame_def)
apply (simp (no_asm_use))
apply clarify
apply (intro exI conjI)
apply assumption+
apply (erule approx_stk_sup_heap)
apply (erule hext_new)
apply (erule approx_loc_sup_heap)
apply (erule hext_new)
apply assumption+
apply blast
done
end
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