(* Title: HOL/MicroJava/BV/BVNoTypeError.thy
Author: Gerwin Klein
*)
section {* Welltyped Programs produce no Type Errors *}
theory BVNoTypeError
imports "../JVM/JVMDefensive" BVSpecTypeSafe
begin
text {*
Some simple lemmas about the type testing functions of the
defensive JVM:
*}
lemma typeof_NoneD [simp,dest]:
"typeof (\<lambda>v. None) v = Some x \<Longrightarrow> \<not>isAddr v"
by (cases v) auto
lemma isRef_def2:
"isRef v = (v = Null \<or> (\<exists>loc. v = Addr loc))"
by (cases v) (auto simp add: isRef_def)
lemma app'Store[simp]:
"app' (Store idx, G, pc, maxs, rT, (ST,LT)) = (\<exists>T ST'. ST = T#ST' \<and> idx < length LT)"
by (cases ST) auto
lemma app'GetField[simp]:
"app' (Getfield F C, G, pc, maxs, rT, (ST,LT)) =
(\<exists>oT vT ST'. ST = oT#ST' \<and> is_class G C \<and>
field (G,C) F = Some (C,vT) \<and> G \<turnstile> oT \<preceq> Class C)"
by (cases ST) auto
lemma app'PutField[simp]:
"app' (Putfield F C, G, pc, maxs, rT, (ST,LT)) =
(\<exists>vT vT' oT ST'. ST = vT#oT#ST' \<and> is_class G C \<and>
field (G,C) F = Some (C, vT') \<and>
G \<turnstile> oT \<preceq> Class C \<and> G \<turnstile> vT \<preceq> vT')"
apply rule
defer
apply clarsimp
apply (cases ST)
apply simp
apply (cases "tl ST")
apply auto
done
lemma app'Checkcast[simp]:
"app' (Checkcast C, G, pc, maxs, rT, (ST,LT)) =
(\<exists>rT ST'. ST = RefT rT#ST' \<and> is_class G C)"
apply rule
defer
apply clarsimp
apply (cases ST)
apply simp
apply (cases "hd ST")
defer
apply simp
apply simp
done
lemma app'Pop[simp]:
"app' (Pop, G, pc, maxs, rT, (ST,LT)) = (\<exists>T ST'. ST = T#ST')"
by (cases ST) auto
lemma app'Dup[simp]:
"app' (Dup, G, pc, maxs, rT, (ST,LT)) =
(\<exists>T ST'. ST = T#ST' \<and> length ST < maxs)"
by (cases ST) auto
lemma app'Dup_x1[simp]:
"app' (Dup_x1, G, pc, maxs, rT, (ST,LT)) =
(\<exists>T1 T2 ST'. ST = T1#T2#ST' \<and> length ST < maxs)"
by (cases ST, simp, cases "tl ST", auto)
lemma app'Dup_x2[simp]:
"app' (Dup_x2, G, pc, maxs, rT, (ST,LT)) =
(\<exists>T1 T2 T3 ST'. ST = T1#T2#T3#ST' \<and> length ST < maxs)"
by (cases ST, simp, cases "tl ST", simp, cases "tl (tl ST)", auto)
lemma app'Swap[simp]:
"app' (Swap, G, pc, maxs, rT, (ST,LT)) = (\<exists>T1 T2 ST'. ST = T1#T2#ST')"
by (cases ST, simp, cases "tl ST", auto)
lemma app'IAdd[simp]:
"app' (IAdd, G, pc, maxs, rT, (ST,LT)) =
(\<exists>ST'. ST = PrimT Integer#PrimT Integer#ST')"
apply (cases ST)
apply simp
apply simp
apply (case_tac a)
apply auto
apply (rename_tac prim_ty, case_tac prim_ty)
apply auto
apply (rename_tac prim_ty, case_tac prim_ty)
apply auto
apply (case_tac list)
apply auto
apply (case_tac a)
apply auto
apply (rename_tac prim_ty, case_tac prim_ty)
apply auto
done
lemma app'Ifcmpeq[simp]:
"app' (Ifcmpeq b, G, pc, maxs, rT, (ST,LT)) =
(\<exists>T1 T2 ST'. ST = T1#T2#ST' \<and> 0 \<le> b + int pc \<and>
((\<exists>p. T1 = PrimT p \<and> T1 = T2) \<or>
(\<exists>r r'. T1 = RefT r \<and> T2 = RefT r')))"
apply auto
apply (cases ST)
apply simp
apply (cases "tl ST")
apply (case_tac a)
apply auto
done
lemma app'Return[simp]:
"app' (Return, G, pc, maxs, rT, (ST,LT)) =
(\<exists>T ST'. ST = T#ST'\<and> G \<turnstile> T \<preceq> rT)"
by (cases ST) auto
lemma app'Throw[simp]:
"app' (Throw, G, pc, maxs, rT, (ST,LT)) =
(\<exists>ST' r. ST = RefT r#ST')"
apply (cases ST)
apply simp
apply (cases "hd ST")
apply auto
done
lemma app'Invoke[simp]:
"app' (Invoke C mn fpTs, G, pc, maxs, rT, ST, LT) =
(\<exists>apTs X ST' mD' rT' b'.
ST = (rev apTs) @ X # ST' \<and>
length apTs = length fpTs \<and> is_class G C \<and>
(\<forall>(aT,fT)\<in>set(zip apTs fpTs). G \<turnstile> aT \<preceq> fT) \<and>
method (G,C) (mn,fpTs) = Some (mD', rT', b') \<and> G \<turnstile> X \<preceq> Class C)"
(is "?app ST LT = ?P ST LT")
proof
assume "?P ST LT" thus "?app ST LT" by (auto simp add: list_all2_iff)
next
assume app: "?app ST LT"
hence l: "length fpTs < length ST" by simp
obtain xs ys where xs: "ST = xs @ ys" "length xs = length fpTs"
proof -
have "ST = take (length fpTs) ST @ drop (length fpTs) ST" by simp
moreover from l have "length (take (length fpTs) ST) = length fpTs"
by simp
ultimately show ?thesis ..
qed
obtain apTs where
"ST = (rev apTs) @ ys" and "length apTs = length fpTs"
proof -
from xs(1) have "ST = rev (rev xs) @ ys" by simp
then show thesis by (rule that) (simp add: xs(2))
qed
moreover
from l xs obtain X ST' where "ys = X#ST'" by (auto simp add: neq_Nil_conv)
ultimately
have "ST = (rev apTs) @ X # ST'" "length apTs = length fpTs" by auto
with app
show "?P ST LT"
apply (clarsimp simp add: list_all2_iff)
apply (intro exI conjI)
apply auto
done
qed
lemma approx_loc_len [simp]:
"approx_loc G hp loc LT \<Longrightarrow> length loc = length LT"
by (simp add: approx_loc_def list_all2_iff)
lemma approx_stk_len [simp]:
"approx_stk G hp stk ST \<Longrightarrow> length stk = length ST"
by (simp add: approx_stk_def)
lemma isRefI [intro, simp]: "G,hp \<turnstile> v ::\<preceq> RefT T \<Longrightarrow> isRef v"
apply (drule conf_RefTD)
apply (auto simp add: isRef_def)
done
lemma isIntgI [intro, simp]: "G,hp \<turnstile> v ::\<preceq> PrimT Integer \<Longrightarrow> isIntg v"
apply (unfold conf_def)
apply auto
apply (erule widen.cases)
apply auto
apply (cases v)
apply auto
done
lemma list_all2_approx:
"list_all2 (approx_val G hp) s (map OK S) = list_all2 (conf G hp) s S"
apply (induct S arbitrary: s)
apply (auto simp add: list_all2_Cons2 approx_val_def)
done
lemma list_all2_conf_widen:
"wf_prog mb G \<Longrightarrow>
list_all2 (conf G hp) a b \<Longrightarrow>
list_all2 (\<lambda>x y. G \<turnstile> x \<preceq> y) b c \<Longrightarrow>
list_all2 (conf G hp) a c"
apply (rule list_all2_trans)
defer
apply assumption
apply assumption
apply (drule conf_widen, assumption+)
done
text {*
The main theorem: welltyped programs do not produce type errors if they
are started in a conformant state.
*}
theorem no_type_error:
assumes welltyped: "wt_jvm_prog G Phi" and conforms: "G,Phi \<turnstile>JVM s \<surd>"
shows "exec_d G (Normal s) \<noteq> TypeError"
proof -
from welltyped obtain mb where wf: "wf_prog mb G" by (fast dest: wt_jvm_progD)
obtain xcp hp frs where s [simp]: "s = (xcp, hp, frs)" by (cases s)
from conforms have "xcp \<noteq> None \<or> frs = [] \<Longrightarrow> check G s"
by (unfold correct_state_def check_def) auto
moreover {
assume "\<not>(xcp \<noteq> None \<or> frs = [])"
then obtain stk loc C sig pc frs' where
xcp [simp]: "xcp = None" and
frs [simp]: "frs = (stk,loc,C,sig,pc)#frs'"
by (clarsimp simp add: neq_Nil_conv)
from conforms obtain ST LT rT maxs maxl ins et where
hconf: "G \<turnstile>h hp \<surd>" and
"class": "is_class G C" and
meth: "method (G, C) sig = Some (C, rT, maxs, maxl, ins, et)" and
phi: "Phi C sig ! pc = Some (ST,LT)" and
frame: "correct_frame G hp (ST,LT) maxl ins (stk,loc,C,sig,pc)" and
frames: "correct_frames G hp Phi rT sig frs'"
by (auto simp add: correct_state_def)
from frame obtain
stk: "approx_stk G hp stk ST" and
loc: "approx_loc G hp loc LT" and
pc: "pc < length ins" and
len: "length loc = length (snd sig) + maxl + 1"
by (auto simp add: correct_frame_def)
note approx_val_def [simp]
from welltyped meth conforms
have "wt_instr (ins!pc) G rT (Phi C sig) maxs (length ins) et pc"
by simp (rule wt_jvm_prog_impl_wt_instr_cor)
then obtain
app': "app (ins!pc) G maxs rT pc et (Phi C sig!pc) " and
eff: "\<forall>(pc', s')\<in>set (eff (ins ! pc) G pc et (Phi C sig ! pc)). pc' < length ins"
by (simp add: wt_instr_def phi) blast
from eff
have pc': "\<forall>pc' \<in> set (succs (ins!pc) pc). pc' < length ins"
by (simp add: eff_def) blast
from app' phi
have app:
"xcpt_app (ins!pc) G pc et \<and> app' (ins!pc, G, pc, maxs, rT, (ST,LT))"
by (clarsimp simp add: app_def)
with eff stk loc pc'
have "check_instr (ins!pc) G hp stk loc C sig pc maxs frs'"
proof (cases "ins!pc")
case (Getfield F C)
with app stk loc phi obtain v vT stk' where
"class": "is_class G C" and
field: "field (G, C) F = Some (C, vT)" and
stk: "stk = v # stk'" and
conf: "G,hp \<turnstile> v ::\<preceq> Class C"
apply clarsimp
apply (blast dest: conf_widen [OF wf])
done
from conf have isRef: "isRef v" ..
moreover {
assume "v \<noteq> Null"
with conf field isRef wf
have "\<exists>D vs. hp (the_Addr v) = Some (D,vs) \<and> G \<turnstile> D \<preceq>C C"
by (auto dest!: non_np_objD)
}
ultimately show ?thesis using Getfield field "class" stk hconf wf
apply clarsimp
apply (fastforce dest!: hconfD widen_cfs_fields oconf_objD)
done
next
case (Putfield F C)
with app stk loc phi obtain v ref vT stk' where
"class": "is_class G C" and
field: "field (G, C) F = Some (C, vT)" and
stk: "stk = v # ref # stk'" and
confv: "G,hp \<turnstile> v ::\<preceq> vT" and
confr: "G,hp \<turnstile> ref ::\<preceq> Class C"
apply clarsimp
apply (blast dest: conf_widen [OF wf])
done
from confr have isRef: "isRef ref" ..
moreover {
assume "ref \<noteq> Null"
with confr field isRef wf
have "\<exists>D vs. hp (the_Addr ref) = Some (D,vs) \<and> G \<turnstile> D \<preceq>C C"
by (auto dest: non_np_objD)
}
ultimately show ?thesis using Putfield field "class" stk confv
by clarsimp
next
case (Invoke C mn ps)
with app
obtain apTs X ST' where
ST: "ST = rev apTs @ X # ST'" and
ps: "length apTs = length ps" and
w: "\<forall>(x, y)\<in>set (zip apTs ps). G \<turnstile> x \<preceq> y" and
C: "G \<turnstile> X \<preceq> Class C" and
mth: "method (G, C) (mn, ps) \<noteq> None"
by (simp del: app'.simps) blast
from ST stk
obtain aps x stk' where
stk': "stk = aps @ x # stk'" and
aps: "approx_stk G hp aps (rev apTs)" and
x: "G,hp \<turnstile> x ::\<preceq> X" and
l: "length aps = length apTs"
by (auto dest!: approx_stk_append)
from stk' l ps
have [simp]: "stk!length ps = x" by (simp add: nth_append)
from stk' l ps
have [simp]: "take (length ps) stk = aps" by simp
from w ps
have widen: "list_all2 (\<lambda>x y. G \<turnstile> x \<preceq> y) apTs ps"
by (simp add: list_all2_iff)
from stk' l ps have "length ps < length stk" by simp
moreover
from wf x C
have x: "G,hp \<turnstile> x ::\<preceq> Class C" by (rule conf_widen)
hence "isRef x" by simp
moreover
{ assume "x \<noteq> Null"
with x
obtain D fs where
hp: "hp (the_Addr x) = Some (D,fs)" and
D: "G \<turnstile> D \<preceq>C C"
by - (drule non_npD, assumption, clarsimp)
hence "hp (the_Addr x) \<noteq> None" (is ?P1) by simp
moreover
from wf mth hp D
have "method (G, cname_of hp x) (mn, ps) \<noteq> None" (is ?P2)
by (auto dest: subcls_widen_methd)
moreover
from aps have "list_all2 (conf G hp) aps (rev apTs)"
by (simp add: list_all2_approx approx_stk_def approx_loc_def)
hence "list_all2 (conf G hp) (rev aps) (rev (rev apTs))"
by (simp only: list_all2_rev)
hence "list_all2 (conf G hp) (rev aps) apTs" by simp
with wf widen
have "list_all2 (conf G hp) (rev aps) ps" (is ?P3)
by - (rule list_all2_conf_widen)
ultimately
have "?P1 \<and> ?P2 \<and> ?P3" by blast
}
moreover
note Invoke
ultimately
show ?thesis by simp
next
case Return with stk app phi meth frames
show ?thesis
apply clarsimp
apply (drule conf_widen [OF wf], assumption)
apply (clarsimp simp add: neq_Nil_conv isRef_def2)
done
qed auto
hence "check G s" by (simp add: check_def meth pc)
} ultimately
have "check G s" by blast
thus "exec_d G (Normal s) \<noteq> TypeError" ..
qed
text {*
The theorem above tells us that, in welltyped programs, the
defensive machine reaches the same result as the aggressive
one (after arbitrarily many steps).
*}
theorem welltyped_aggressive_imp_defensive:
"wt_jvm_prog G Phi \<Longrightarrow> G,Phi \<turnstile>JVM s \<surd> \<Longrightarrow> G \<turnstile> s \<midarrow>jvm\<rightarrow> t
\<Longrightarrow> G \<turnstile> (Normal s) \<midarrow>jvmd\<rightarrow> (Normal t)"
apply (unfold exec_all_def)
apply (erule rtrancl_induct)
apply (simp add: exec_all_d_def)
apply simp
apply (fold exec_all_def)
apply (frule BV_correct, assumption+)
apply (drule no_type_error, assumption, drule no_type_error_commutes, simp)
apply (simp add: exec_all_d_def)
apply (rule rtrancl_trans, assumption)
apply blast
done
lemma neq_TypeError_eq [simp]: "s \<noteq> TypeError = (\<exists>s'. s = Normal s')"
by (cases s, auto)
theorem no_type_errors:
"wt_jvm_prog G Phi \<Longrightarrow> G,Phi \<turnstile>JVM s \<surd>
\<Longrightarrow> G \<turnstile> (Normal s) \<midarrow>jvmd\<rightarrow> t \<Longrightarrow> t \<noteq> TypeError"
apply (unfold exec_all_d_def)
apply (erule rtrancl_induct)
apply simp
apply (fold exec_all_d_def)
apply (auto dest: defensive_imp_aggressive BV_correct no_type_error)
done
corollary no_type_errors_initial:
fixes G ("\<Gamma>") and Phi ("\<Phi>")
assumes wt: "wt_jvm_prog \<Gamma> \<Phi>"
assumes is_class: "is_class \<Gamma> C"
and "method": "method (\<Gamma>,C) (m,[]) = Some (C, b)"
and m: "m \<noteq> init"
defines start: "s \<equiv> start_state \<Gamma> C m"
assumes s: "\<Gamma> \<turnstile> (Normal s) \<midarrow>jvmd\<rightarrow> t"
shows "t \<noteq> TypeError"
proof -
from wt is_class "method" have "\<Gamma>,\<Phi> \<turnstile>JVM s \<surd>"
unfolding start by (rule BV_correct_initial)
from wt this s show ?thesis by (rule no_type_errors)
qed
text {*
As corollary we get that the aggressive and the defensive machine
are equivalent for welltyped programs (if started in a conformant
state or in the canonical start state)
*}
corollary welltyped_commutes:
fixes G ("\<Gamma>") and Phi ("\<Phi>")
assumes wt: "wt_jvm_prog \<Gamma> \<Phi>" and *: "\<Gamma>,\<Phi> \<turnstile>JVM s \<surd>"
shows "\<Gamma> \<turnstile> (Normal s) \<midarrow>jvmd\<rightarrow> (Normal t) = \<Gamma> \<turnstile> s \<midarrow>jvm\<rightarrow> t"
apply rule
apply (erule defensive_imp_aggressive, rule welltyped_aggressive_imp_defensive)
apply (rule wt)
apply (rule *)
apply assumption
done
corollary welltyped_initial_commutes:
fixes G ("\<Gamma>") and Phi ("\<Phi>")
assumes wt: "wt_jvm_prog \<Gamma> \<Phi>"
assumes is_class: "is_class \<Gamma> C"
and "method": "method (\<Gamma>,C) (m,[]) = Some (C, b)"
and m: "m \<noteq> init"
defines start: "s \<equiv> start_state \<Gamma> C m"
shows "\<Gamma> \<turnstile> (Normal s) \<midarrow>jvmd\<rightarrow> (Normal t) = \<Gamma> \<turnstile> s \<midarrow>jvm\<rightarrow> t"
proof -
from wt is_class "method" have "\<Gamma>,\<Phi> \<turnstile>JVM s \<surd>"
unfolding start by (rule BV_correct_initial)
with wt show ?thesis by (rule welltyped_commutes)
qed
end