Theory Correct

theory Correct
imports BVSpec JVMExec
(*  Title:      HOL/MicroJava/BV/Correct.thy
Author: Cornelia Pusch, Gerwin Klein
Copyright 1999 Technische Universitaet Muenchen
*)


header {* \isaheader{BV Type Safety Invariant} *}

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\<turnstile>v::\<preceq>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 == λ(ST,LT) maxl ins (stk,loc,C,sig,pc).
approx_stk G hp stk ST ∧ approx_loc G hp loc LT ∧
pc < length ins ∧ 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
(∃ST LT rT maxs maxl ins et.
phi C sig ! pc = Some (ST,LT) ∧ is_class G C ∧
method (G,C) sig = Some(C,rT,(maxs,maxl,ins,et)) ∧
(∃C' mn pTs. ins!pc = (Invoke C' mn pTs) ∧
(mn,pTs) = sig0 ∧
(∃apTs D ST' LT'.
(phi C sig)!pc = Some ((rev apTs) @ (Class D) # ST', LT') ∧
length apTs = length pTs ∧
(∃D' rT' maxs' maxl' ins' et'.
method (G,D) sig0 = Some(D',rT',(maxs',maxl',ins',et')) ∧
G \<turnstile> rT0 \<preceq> rT') ∧
correct_frame G hp (ST, LT) maxl ins f ∧
correct_frames G hp phi rT sig frs))))"


definition correct_state :: "[jvm_prog,prog_type,jvm_state] => bool"
("_,_ |-JVM _ [ok]" [51,51] 50) where
"correct_state G phi == λ(xp,hp,frs).
case xp of
None => (case frs of
[] => True
| (f#fs) => G\<turnstile>h hp\<surd> ∧ preallocated hp ∧
(let (stk,loc,C,sig,pc) = f
in
∃rT maxs maxl ins et s.
is_class G C ∧
method (G,C) sig = Some(C,rT,(maxs,maxl,ins,et)) ∧
phi C sig ! pc = Some s ∧
correct_frame G hp s maxl ins f ∧
correct_frames G hp phi rT sig fs))
| Some x => frs = []"



notation (xsymbols)
correct_state ("_,_ \<turnstile>JVM _ \<surd>" [51,51] 50)


lemma sup_ty_opt_OK:
"(G \<turnstile> X <=o (OK T')) = (∃T. X = OK T ∧ G \<turnstile> T \<preceq> T')"
by (cases X) auto


section {* approx-val *}

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 \<turnstile> x ::\<preceq> 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\<turnstile> v::\<preceq>T; obj_ty obj = obj_ty obj'|]
==> 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 \<turnstile> T <=o T'; wf_prog wt G |]
==> approx_val G hp v T'"

by (cases T') (auto simp add: sup_ty_opt_OK intro: conf_widen)

section {* approx-loc *}

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 ∧ 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 |]
==> 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 \<turnstile> T \<preceq> T'; wf_prog wt G|]
==> 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 ∧ (∀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' |]
==> 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 \<turnstile> LT <=l LT'; wf_prog wt G |]
==> 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 \<turnstile> 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 |]
==> 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)
done

lemma approx_loc_append:
"length l1=length L1 ==>
approx_loc G hp (l1@l2) (L1@L2) =
(approx_loc G hp l1 L1 ∧ approx_loc G hp l2 L2)"

apply (unfold approx_loc_def list_all2_def)
apply (simp cong: conj_cong)
apply blast
done

section {* 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)
apply (simp add: rev_map [THEN sym])
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 \<turnstile> map OK ST <=l map OK ST'; wf_prog wt G |]
==> 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) ∧ 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') =
(∃s stk'. stk = s#stk' ∧ approx_val G hp s (OK S) ∧ 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') ==>
(∃s stk'. stk = s@stk' ∧ length s = length S ∧ length stk' = length S' ∧
approx_stk G hp s S ∧ 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 \<turnstile> x \<preceq> y; length ST = length ST'; wf_prog wt G |]
==> 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

section {* oconf *}

lemma oconf_field_update:
"[|map_of (fields (G, oT)) FD = Some T; G,hp\<turnstile>v::\<preceq>T; G,hp\<turnstile>(oT,fs)\<surd> |]
==> G,hp\<turnstile>(oT, fs(FD\<mapsto>v))\<surd>"

by (simp add: oconf_def lconf_def)

lemma oconf_newref:
"[|hp oref = None; G,hp \<turnstile> obj \<surd>; G,hp \<turnstile> obj' \<surd>|] ==> G,hp(oref\<mapsto>obj') \<turnstile> obj \<surd>"
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\<turnstile>obj\<surd> |]
==> G,hp(a\<mapsto>obj'')\<turnstile>obj\<surd>"

apply (unfold oconf_def lconf_def)
apply (fastforce intro: approx_val_heap_update)
done

section {* hconf *}

lemma hconf_newref:
"[| hp oref = None; G\<turnstile>h hp\<surd>; G,hp\<turnstile>obj\<surd> |] ==> G\<turnstile>h hp(oref\<mapsto>obj)\<surd>"
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> |]
==> 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

section {* preallocated *}

lemma preallocated_field_update:
"[| map_of (fields (G, oT)) X = Some T; hp a = Some(oT,fs);
G\<turnstile>h hp\<surd>; preallocated hp |]
==> 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\<mapsto>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

section {* correct-frames *}

lemmas [simp del] = fun_upd_apply

lemma correct_frames_field_update [rule_format]:
"∀rT C sig.
correct_frames G hp phi rT sig frs -->
hp a = Some (C,fs) -->
map_of (fields (G, C)) fl = Some fd -->
G,hp\<turnstile>v::\<preceq>fd
--> 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 -->
correct_frames G hp phi rT sig frs -->
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