author | haftmann |
Tue, 24 Nov 2009 14:37:23 +0100 | |
changeset 33954 | 1bc3b688548c |
parent 32960 | 69916a850301 |
child 42150 | b0c0638c4aad |
permissions | -rwxr-xr-x |
13633 | 1 |
(* Title: HOL/MicroJava/BV/BVNoTypeErrors.thy |
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Author: Gerwin Klein |
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*) |
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header {* \isaheader{Welltyped Programs produce no Type Errors} *} |
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33954
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
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diff
changeset
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theory BVNoTypeError |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32960
diff
changeset
|
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imports "../JVM/JVMDefensive" BVSpecTypeSafe |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
32960
diff
changeset
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begin |
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text {* |
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Some simple lemmas about the type testing functions of the |
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defensive JVM: |
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*} |
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lemma typeof_NoneD [simp,dest]: |
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"typeof (\<lambda>v. None) v = Some x \<Longrightarrow> \<not>isAddr v" |
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by (cases v) auto |
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lemma isRef_def2: |
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"isRef v = (v = Null \<or> (\<exists>loc. v = Addr loc))" |
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by (cases v) (auto simp add: isRef_def) |
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lemma app'Store[simp]: |
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"app' (Store idx, G, pc, maxs, rT, (ST,LT)) = (\<exists>T ST'. ST = T#ST' \<and> idx < length LT)" |
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by (cases ST, auto) |
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lemma app'GetField[simp]: |
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"app' (Getfield F C, G, pc, maxs, rT, (ST,LT)) = |
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(\<exists>oT vT ST'. ST = oT#ST' \<and> is_class G C \<and> |
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field (G,C) F = Some (C,vT) \<and> G \<turnstile> oT \<preceq> Class C)" |
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by (cases ST, auto) |
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lemma app'PutField[simp]: |
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"app' (Putfield F C, G, pc, maxs, rT, (ST,LT)) = |
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(\<exists>vT vT' oT ST'. ST = vT#oT#ST' \<and> is_class G C \<and> |
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field (G,C) F = Some (C, vT') \<and> |
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G \<turnstile> oT \<preceq> Class C \<and> G \<turnstile> vT \<preceq> vT')" |
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apply rule |
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defer |
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apply clarsimp |
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apply (cases ST) |
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apply simp |
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apply (cases "tl ST") |
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apply auto |
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done |
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lemma app'Checkcast[simp]: |
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"app' (Checkcast C, G, pc, maxs, rT, (ST,LT)) = |
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(\<exists>rT ST'. ST = RefT rT#ST' \<and> is_class G C)" |
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apply rule |
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defer |
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apply clarsimp |
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apply (cases ST) |
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apply simp |
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apply (cases "hd ST") |
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defer |
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apply simp |
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apply simp |
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done |
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lemma app'Pop[simp]: |
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"app' (Pop, G, pc, maxs, rT, (ST,LT)) = (\<exists>T ST'. ST = T#ST')" |
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by (cases ST, auto) |
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lemma app'Dup[simp]: |
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"app' (Dup, G, pc, maxs, rT, (ST,LT)) = |
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(\<exists>T ST'. ST = T#ST' \<and> length ST < maxs)" |
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by (cases ST, auto) |
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lemma app'Dup_x1[simp]: |
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"app' (Dup_x1, G, pc, maxs, rT, (ST,LT)) = |
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(\<exists>T1 T2 ST'. ST = T1#T2#ST' \<and> length ST < maxs)" |
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by (cases ST, simp, cases "tl ST", auto) |
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lemma app'Dup_x2[simp]: |
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"app' (Dup_x2, G, pc, maxs, rT, (ST,LT)) = |
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(\<exists>T1 T2 T3 ST'. ST = T1#T2#T3#ST' \<and> length ST < maxs)" |
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by (cases ST, simp, cases "tl ST", simp, cases "tl (tl ST)", auto) |
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lemma app'Swap[simp]: |
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"app' (Swap, G, pc, maxs, rT, (ST,LT)) = (\<exists>T1 T2 ST'. ST = T1#T2#ST')" |
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by (cases ST, simp, cases "tl ST", auto) |
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lemma app'IAdd[simp]: |
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"app' (IAdd, G, pc, maxs, rT, (ST,LT)) = |
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(\<exists>ST'. ST = PrimT Integer#PrimT Integer#ST')" |
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apply (cases ST) |
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apply simp |
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apply simp |
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apply (case_tac a) |
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apply auto |
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apply (case_tac prim_ty) |
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apply auto |
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apply (case_tac prim_ty) |
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apply auto |
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apply (case_tac list) |
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apply auto |
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apply (case_tac a) |
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apply auto |
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apply (case_tac prim_ty) |
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apply auto |
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done |
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lemma app'Ifcmpeq[simp]: |
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"app' (Ifcmpeq b, G, pc, maxs, rT, (ST,LT)) = |
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(\<exists>T1 T2 ST'. ST = T1#T2#ST' \<and> 0 \<le> b + int pc \<and> |
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((\<exists>p. T1 = PrimT p \<and> T1 = T2) \<or> |
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(\<exists>r r'. T1 = RefT r \<and> T2 = RefT r')))" |
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apply auto |
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apply (cases ST) |
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apply simp |
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apply (cases "tl ST") |
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apply (case_tac a) |
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apply auto |
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done |
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lemma app'Return[simp]: |
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"app' (Return, G, pc, maxs, rT, (ST,LT)) = |
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(\<exists>T ST'. ST = T#ST'\<and> G \<turnstile> T \<preceq> rT)" |
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by (cases ST, auto) |
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lemma app'Throw[simp]: |
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"app' (Throw, G, pc, maxs, rT, (ST,LT)) = |
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(\<exists>ST' r. ST = RefT r#ST')" |
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apply (cases ST, simp) |
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apply (cases "hd ST") |
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apply auto |
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done |
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lemma app'Invoke[simp]: |
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"app' (Invoke C mn fpTs, G, pc, maxs, rT, ST, LT) = |
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(\<exists>apTs X ST' mD' rT' b'. |
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ST = (rev apTs) @ X # ST' \<and> |
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length apTs = length fpTs \<and> is_class G C \<and> |
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(\<forall>(aT,fT)\<in>set(zip apTs fpTs). G \<turnstile> aT \<preceq> fT) \<and> |
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method (G,C) (mn,fpTs) = Some (mD', rT', b') \<and> G \<turnstile> X \<preceq> Class C)" |
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(is "?app ST LT = ?P ST LT") |
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proof |
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assume "?P ST LT" thus "?app ST LT" by (auto simp add: list_all2_def) |
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next |
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assume app: "?app ST LT" |
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hence l: "length fpTs < length ST" by simp |
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obtain xs ys where xs: "ST = xs @ ys" "length xs = length fpTs" |
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proof - |
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have "ST = take (length fpTs) ST @ drop (length fpTs) ST" by simp |
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moreover from l have "length (take (length fpTs) ST) = length fpTs" |
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by simp |
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ultimately show ?thesis .. |
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qed |
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obtain apTs where |
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"ST = (rev apTs) @ ys" and "length apTs = length fpTs" |
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proof - |
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from xs(1) have "ST = rev (rev xs) @ ys" by simp |
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then show thesis by (rule that) (simp add: xs(2)) |
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qed |
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moreover |
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from l xs obtain X ST' where "ys = X#ST'" by (auto simp add: neq_Nil_conv) |
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ultimately |
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have "ST = (rev apTs) @ X # ST'" "length apTs = length fpTs" by auto |
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with app |
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show "?P ST LT" |
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32443 | 172 |
apply (clarsimp simp add: list_all2_def) |
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apply ((rule exI)+, (rule conjI)?)+ |
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apply auto |
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done |
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qed |
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lemma approx_loc_len [simp]: |
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"approx_loc G hp loc LT \<Longrightarrow> length loc = length LT" |
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by (simp add: approx_loc_def list_all2_def) |
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lemma approx_stk_len [simp]: |
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"approx_stk G hp stk ST \<Longrightarrow> length stk = length ST" |
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by (simp add: approx_stk_def) |
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lemma isRefI [intro, simp]: "G,hp \<turnstile> v ::\<preceq> RefT T \<Longrightarrow> isRef v" |
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apply (drule conf_RefTD) |
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apply (auto simp add: isRef_def) |
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done |
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lemma isIntgI [intro, simp]: "G,hp \<turnstile> v ::\<preceq> PrimT Integer \<Longrightarrow> isIntg v" |
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apply (unfold conf_def) |
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apply auto |
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apply (erule widen.cases) |
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apply auto |
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apply (cases v) |
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apply auto |
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done |
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lemma list_all2_approx: |
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"\<And>s. list_all2 (approx_val G hp) s (map OK S) = |
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list_all2 (conf G hp) s S" |
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apply (induct S) |
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apply (auto simp add: list_all2_Cons2 approx_val_def) |
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done |
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lemma list_all2_conf_widen: |
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"wf_prog mb G \<Longrightarrow> |
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list_all2 (conf G hp) a b \<Longrightarrow> |
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list_all2 (\<lambda>x y. G \<turnstile> x \<preceq> y) b c \<Longrightarrow> |
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list_all2 (conf G hp) a c" |
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apply (rule list_all2_trans) |
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defer |
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apply assumption |
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apply assumption |
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apply (drule conf_widen, assumption+) |
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done |
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text {* |
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The main theorem: welltyped programs do not produce type errors if they |
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are started in a conformant state. |
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*} |
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theorem no_type_error: |
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assumes welltyped: "wt_jvm_prog G Phi" and conforms: "G,Phi \<turnstile>JVM s \<surd>" |
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shows "exec_d G (Normal s) \<noteq> TypeError" |
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proof - |
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from welltyped obtain mb where wf: "wf_prog mb G" by (fast dest: wt_jvm_progD) |
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obtain xcp hp frs where s [simp]: "s = (xcp, hp, frs)" by (cases s) |
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from conforms have "xcp \<noteq> None \<or> frs = [] \<Longrightarrow> check G s" |
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by (unfold correct_state_def check_def) auto |
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moreover { |
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assume "\<not>(xcp \<noteq> None \<or> frs = [])" |
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then obtain stk loc C sig pc frs' where |
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xcp [simp]: "xcp = None" and |
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frs [simp]: "frs = (stk,loc,C,sig,pc)#frs'" |
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by (clarsimp simp add: neq_Nil_conv) fast |
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from conforms obtain ST LT rT maxs maxl ins et where |
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hconf: "G \<turnstile>h hp \<surd>" and |
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"class": "is_class G C" and |
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meth: "method (G, C) sig = Some (C, rT, maxs, maxl, ins, et)" and |
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phi: "Phi C sig ! pc = Some (ST,LT)" and |
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frame: "correct_frame G hp (ST,LT) maxl ins (stk,loc,C,sig,pc)" and |
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frames: "correct_frames G hp Phi rT sig frs'" |
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23467 | 248 |
by (auto simp add: correct_state_def) (rule that) |
13633 | 249 |
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from frame obtain |
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stk: "approx_stk G hp stk ST" and |
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loc: "approx_loc G hp loc LT" and |
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pc: "pc < length ins" and |
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len: "length loc = length (snd sig) + maxl + 1" |
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by (auto simp add: correct_frame_def) |
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note approx_val_def [simp] |
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from welltyped meth conforms |
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have "wt_instr (ins!pc) G rT (Phi C sig) maxs (length ins) et pc" |
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by simp (rule wt_jvm_prog_impl_wt_instr_cor) |
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then obtain |
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app': "app (ins!pc) G maxs rT pc et (Phi C sig!pc) " and |
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eff: "\<forall>(pc', s')\<in>set (eff (ins ! pc) G pc et (Phi C sig ! pc)). pc' < length ins" |
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by (simp add: wt_instr_def phi) blast |
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from eff |
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have pc': "\<forall>pc' \<in> set (succs (ins!pc) pc). pc' < length ins" |
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by (simp add: eff_def) blast |
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from app' phi |
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have app: |
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"xcpt_app (ins!pc) G pc et \<and> app' (ins!pc, G, pc, maxs, rT, (ST,LT))" |
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by (clarsimp simp add: app_def) |
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with eff stk loc pc' |
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13822 | 277 |
have "check_instr (ins!pc) G hp stk loc C sig pc maxs frs'" |
13633 | 278 |
proof (cases "ins!pc") |
279 |
case (Getfield F C) |
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with app stk loc phi obtain v vT stk' where |
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18551 | 281 |
"class": "is_class G C" and |
13633 | 282 |
field: "field (G, C) F = Some (C, vT)" and |
283 |
stk: "stk = v # stk'" and |
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conf: "G,hp \<turnstile> v ::\<preceq> Class C" |
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apply clarsimp |
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apply (blast dest: conf_widen [OF wf]) |
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done |
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from conf have isRef: "isRef v" .. |
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moreover { |
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assume "v \<noteq> Null" |
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with conf field isRef wf |
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have "\<exists>D vs. hp (the_Addr v) = Some (D,vs) \<and> G \<turnstile> D \<preceq>C C" |
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by (auto dest!: non_np_objD) |
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} |
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18551 | 295 |
ultimately show ?thesis using Getfield field "class" stk hconf wf |
13633 | 296 |
apply clarsimp |
14045 | 297 |
apply (fastsimp intro: wf_prog_ws_prog |
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dest!: hconfD widen_cfs_fields oconf_objD) |
13633 | 299 |
done |
300 |
next |
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301 |
case (Putfield F C) |
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with app stk loc phi obtain v ref vT stk' where |
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18551 | 303 |
"class": "is_class G C" and |
13633 | 304 |
field: "field (G, C) F = Some (C, vT)" and |
305 |
stk: "stk = v # ref # stk'" and |
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confv: "G,hp \<turnstile> v ::\<preceq> vT" and |
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confr: "G,hp \<turnstile> ref ::\<preceq> Class C" |
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apply clarsimp |
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309 |
apply (blast dest: conf_widen [OF wf]) |
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310 |
done |
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311 |
from confr have isRef: "isRef ref" .. |
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312 |
moreover { |
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assume "ref \<noteq> Null" |
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with confr field isRef wf |
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315 |
have "\<exists>D vs. hp (the_Addr ref) = Some (D,vs) \<and> G \<turnstile> D \<preceq>C C" |
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316 |
by (auto dest: non_np_objD) |
|
317 |
} |
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18551 | 318 |
ultimately show ?thesis using Putfield field "class" stk confv |
13633 | 319 |
by clarsimp |
320 |
next |
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321 |
case (Invoke C mn ps) |
|
322 |
with app |
|
323 |
obtain apTs X ST' where |
|
324 |
ST: "ST = rev apTs @ X # ST'" and |
|
325 |
ps: "length apTs = length ps" and |
|
22271 | 326 |
w: "\<forall>(x, y)\<in>set (zip apTs ps). G \<turnstile> x \<preceq> y" and |
13633 | 327 |
C: "G \<turnstile> X \<preceq> Class C" and |
328 |
mth: "method (G, C) (mn, ps) \<noteq> None" |
|
329 |
by (simp del: app'.simps) blast |
|
330 |
||
331 |
from ST stk |
|
332 |
obtain aps x stk' where |
|
333 |
stk': "stk = aps @ x # stk'" and |
|
334 |
aps: "approx_stk G hp aps (rev apTs)" and |
|
335 |
x: "G,hp \<turnstile> x ::\<preceq> X" and |
|
336 |
l: "length aps = length apTs" |
|
337 |
by (auto dest!: approx_stk_append) |
|
338 |
||
339 |
from stk' l ps |
|
340 |
have [simp]: "stk!length ps = x" by (simp add: nth_append) |
|
341 |
from stk' l ps |
|
342 |
have [simp]: "take (length ps) stk = aps" by simp |
|
343 |
from w ps |
|
344 |
have widen: "list_all2 (\<lambda>x y. G \<turnstile> x \<preceq> y) apTs ps" |
|
345 |
by (simp add: list_all2_def) |
|
346 |
||
347 |
from stk' l ps have "length ps < length stk" by simp |
|
348 |
moreover |
|
349 |
from wf x C |
|
350 |
have x: "G,hp \<turnstile> x ::\<preceq> Class C" by (rule conf_widen) |
|
351 |
hence "isRef x" by simp |
|
352 |
moreover |
|
353 |
{ assume "x \<noteq> Null" |
|
354 |
with x |
|
355 |
obtain D fs where |
|
356 |
hp: "hp (the_Addr x) = Some (D,fs)" and |
|
357 |
D: "G \<turnstile> D \<preceq>C C" |
|
358 |
by - (drule non_npD, assumption, clarsimp, blast) |
|
359 |
hence "hp (the_Addr x) \<noteq> None" (is ?P1) by simp |
|
360 |
moreover |
|
361 |
from wf mth hp D |
|
362 |
have "method (G, cname_of hp x) (mn, ps) \<noteq> None" (is ?P2) |
|
363 |
by (auto dest: subcls_widen_methd) |
|
364 |
moreover |
|
365 |
from aps have "list_all2 (conf G hp) aps (rev apTs)" |
|
366 |
by (simp add: list_all2_approx approx_stk_def approx_loc_def) |
|
367 |
hence "list_all2 (conf G hp) (rev aps) (rev (rev apTs))" |
|
368 |
by (simp only: list_all2_rev) |
|
369 |
hence "list_all2 (conf G hp) (rev aps) apTs" by simp |
|
370 |
with wf widen |
|
371 |
have "list_all2 (conf G hp) (rev aps) ps" (is ?P3) |
|
372 |
by - (rule list_all2_conf_widen) |
|
373 |
ultimately |
|
374 |
have "?P1 \<and> ?P2 \<and> ?P3" by blast |
|
375 |
} |
|
376 |
moreover |
|
377 |
note Invoke |
|
378 |
ultimately |
|
379 |
show ?thesis by simp |
|
380 |
next |
|
381 |
case Return with stk app phi meth frames |
|
382 |
show ?thesis |
|
383 |
apply clarsimp |
|
384 |
apply (drule conf_widen [OF wf], assumption) |
|
385 |
apply (clarsimp simp add: neq_Nil_conv isRef_def2) |
|
386 |
done |
|
387 |
qed auto |
|
13822 | 388 |
hence "check G s" by (simp add: check_def meth pc) |
13633 | 389 |
} ultimately |
390 |
have "check G s" by blast |
|
391 |
thus "exec_d G (Normal s) \<noteq> TypeError" .. |
|
392 |
qed |
|
393 |
||
394 |
||
395 |
text {* |
|
396 |
The theorem above tells us that, in welltyped programs, the |
|
397 |
defensive machine reaches the same result as the aggressive |
|
398 |
one (after arbitrarily many steps). |
|
399 |
*} |
|
400 |
theorem welltyped_aggressive_imp_defensive: |
|
401 |
"wt_jvm_prog G Phi \<Longrightarrow> G,Phi \<turnstile>JVM s \<surd> \<Longrightarrow> G \<turnstile> s -jvm\<rightarrow> t |
|
402 |
\<Longrightarrow> G \<turnstile> (Normal s) -jvmd\<rightarrow> (Normal t)" |
|
403 |
apply (unfold exec_all_def) |
|
404 |
apply (erule rtrancl_induct) |
|
405 |
apply (simp add: exec_all_d_def) |
|
406 |
apply simp |
|
407 |
apply (fold exec_all_def) |
|
408 |
apply (frule BV_correct, assumption+) |
|
409 |
apply (drule no_type_error, assumption, drule no_type_error_commutes, simp) |
|
410 |
apply (simp add: exec_all_d_def) |
|
411 |
apply (rule rtrancl_trans, assumption) |
|
412 |
apply blast |
|
413 |
done |
|
414 |
||
415 |
||
13822 | 416 |
lemma neq_TypeError_eq [simp]: "s \<noteq> TypeError = (\<exists>s'. s = Normal s')" |
417 |
by (cases s, auto) |
|
418 |
||
419 |
theorem no_type_errors: |
|
420 |
"wt_jvm_prog G Phi \<Longrightarrow> G,Phi \<turnstile>JVM s \<surd> |
|
421 |
\<Longrightarrow> G \<turnstile> (Normal s) -jvmd\<rightarrow> t \<Longrightarrow> t \<noteq> TypeError" |
|
422 |
apply (unfold exec_all_d_def) |
|
423 |
apply (erule rtrancl_induct) |
|
424 |
apply simp |
|
425 |
apply (fold exec_all_d_def) |
|
426 |
apply (auto dest: defensive_imp_aggressive BV_correct no_type_error) |
|
427 |
done |
|
428 |
||
429 |
corollary no_type_errors_initial: |
|
430 |
fixes G ("\<Gamma>") and Phi ("\<Phi>") |
|
23467 | 431 |
assumes wt: "wt_jvm_prog \<Gamma> \<Phi>" |
432 |
assumes is_class: "is_class \<Gamma> C" |
|
433 |
and method: "method (\<Gamma>,C) (m,[]) = Some (C, b)" |
|
434 |
and m: "m \<noteq> init" |
|
13822 | 435 |
defines start: "s \<equiv> start_state \<Gamma> C m" |
436 |
||
23467 | 437 |
assumes s: "\<Gamma> \<turnstile> (Normal s) -jvmd\<rightarrow> t" |
13822 | 438 |
shows "t \<noteq> TypeError" |
439 |
proof - |
|
23467 | 440 |
from wt is_class method have "\<Gamma>,\<Phi> \<turnstile>JVM s \<surd>" |
441 |
unfolding start by (rule BV_correct_initial) |
|
442 |
from wt this s show ?thesis by (rule no_type_errors) |
|
13822 | 443 |
qed |
444 |
||
13633 | 445 |
text {* |
446 |
As corollary we get that the aggressive and the defensive machine |
|
447 |
are equivalent for welltyped programs (if started in a conformant |
|
448 |
state or in the canonical start state) |
|
449 |
*} |
|
450 |
corollary welltyped_commutes: |
|
451 |
fixes G ("\<Gamma>") and Phi ("\<Phi>") |
|
23467 | 452 |
assumes wt: "wt_jvm_prog \<Gamma> \<Phi>" and *: "\<Gamma>,\<Phi> \<turnstile>JVM s \<surd>" |
13633 | 453 |
shows "\<Gamma> \<turnstile> (Normal s) -jvmd\<rightarrow> (Normal t) = \<Gamma> \<turnstile> s -jvm\<rightarrow> t" |
23467 | 454 |
apply rule |
455 |
apply (erule defensive_imp_aggressive, rule welltyped_aggressive_imp_defensive) |
|
456 |
apply (rule wt) |
|
457 |
apply (rule *) |
|
458 |
apply assumption |
|
459 |
done |
|
13633 | 460 |
|
461 |
corollary welltyped_initial_commutes: |
|
462 |
fixes G ("\<Gamma>") and Phi ("\<Phi>") |
|
23467 | 463 |
assumes wt: "wt_jvm_prog \<Gamma> \<Phi>" |
464 |
assumes is_class: "is_class \<Gamma> C" |
|
465 |
and method: "method (\<Gamma>,C) (m,[]) = Some (C, b)" |
|
466 |
and m: "m \<noteq> init" |
|
13633 | 467 |
defines start: "s \<equiv> start_state \<Gamma> C m" |
468 |
shows "\<Gamma> \<turnstile> (Normal s) -jvmd\<rightarrow> (Normal t) = \<Gamma> \<turnstile> s -jvm\<rightarrow> t" |
|
469 |
proof - |
|
23467 | 470 |
from wt is_class method have "\<Gamma>,\<Phi> \<turnstile>JVM s \<surd>" |
471 |
unfolding start by (rule BV_correct_initial) |
|
472 |
with wt show ?thesis by (rule welltyped_commutes) |
|
13633 | 473 |
qed |
474 |
||
475 |
end |