# Theory Correct

```(*  Title:      HOL/MicroJava/BV/Correct.thy
Author:     Cornelia Pusch, Gerwin Klein
*)

section ‹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⊢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 == 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 ⊢ rT0 ≼ 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 _ √"  [51,51] 50) where
"correct_state G phi == λ(xp,hp,frs).
case xp of
None ⇒ (case frs of
[] ⇒ True
| (f#fs) ⇒ G⊢h hp√ ∧ 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 = []"

lemma sup_ty_opt_OK:
"(G ⊢ X <=o (OK T')) = (∃T. X = OK T ∧ G ⊢ T ≼ T')"
by (cases X) auto

subsection ‹approx-val›

lemma approx_val_Err [simp,intro!]:
"approx_val G hp x Err"

lemma approx_val_OK [iff]:
"approx_val G hp x (OK T) = (G,hp ⊢ x ::≼ T)"

lemma approx_val_Null [simp,intro!]:
"approx_val G hp Null (OK (RefT x))"

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'⟧
⟹ G,hp(a↦obj)⊢ v::≼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 ⟧
⟹ approx_val G hp v T'"
by (cases T') (auto simp add: sup_ty_opt_OK intro: conf_widen)

subsection ‹approx-loc›

lemma approx_loc_Nil [simp,intro!]:
"approx_loc G hp [] []"

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)"

lemma approx_loc_nth:
"⟦ approx_loc G hp loc LT; n < length LT ⟧
⟹ approx_val G hp (loc!n) (LT!n)"

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⟧
⟹ G,hp ⊢ (loc!n) ::≼ 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)))"

lemma approx_loc_sup_heap:
"⟦ approx_loc G hp loc LT; hp ≤| hp' ⟧
⟹ approx_loc G hp' loc LT"
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 ⟧
⟹ 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 ⟧
⟹ 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 ∧ approx_loc G hp l2 L2)"
apply (unfold approx_loc_def list_all2_iff)
apply (simp cong: conj_cong)
apply blast
done

subsection ‹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)
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 ⟧
⟹ 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 [] []"

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)"

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 ⊢ x ≼ 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

subsection ‹oconf›

lemma oconf_field_update:
"⟦map_of (fields (G, oT)) FD = Some T; G,hp⊢v::≼T; G,hp⊢(oT,fs)√ ⟧
⟹ G,hp⊢(oT, fs(FD↦v))√"

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√ ⟧
⟹ G,hp(a↦obj'')⊢obj√"
apply (unfold oconf_def lconf_def)
apply (fastforce intro: approx_val_heap_update)
done

subsection ‹hconf›

lemma hconf_newref:
"⟦ hp oref = None; G⊢h hp√; G,hp⊢obj√ ⟧ ⟹ G⊢h hp(oref↦obj)√"
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⊢v::≼T; G⊢h hp√ ⟧
⟹ G⊢h hp(a ↦ (oT, fs(X↦v)))√"
apply (fastforce intro: oconf_heap_update oconf_field_update
done

subsection ‹preallocated›

lemma preallocated_field_update:
"⟦ map_of (fields (G, oT)) X = Some T; hp a = Some(oT,fs);
G⊢h hp√; preallocated hp ⟧
⟹ preallocated (hp(a ↦ (oT, fs(X↦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 ‹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⊢v::≼fd
⟶ correct_frames G (hp(a ↦ (C, fs(fl↦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 ↦ 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
```