--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/MicroJava/DFA/LBVSpec.thy Tue Nov 24 14:37:23 2009 +0100
@@ -0,0 +1,381 @@
+(* Title: HOL/MicroJava/BV/LBVSpec.thy
+ Author: Gerwin Klein
+ Copyright 1999 Technische Universitaet Muenchen
+*)
+
+header {* \isaheader{The Lightweight Bytecode Verifier} *}
+
+theory LBVSpec
+imports SemilatAlg Opt
+begin
+
+types
+ 's certificate = "'s list"
+
+consts
+merge :: "'s certificate \<Rightarrow> 's binop \<Rightarrow> 's ord \<Rightarrow> 's \<Rightarrow> nat \<Rightarrow> (nat \<times> 's) list \<Rightarrow> 's \<Rightarrow> 's"
+primrec
+"merge cert f r T pc [] x = x"
+"merge cert f r T pc (s#ss) x = merge cert f r T pc ss (let (pc',s') = s in
+ if pc'=pc+1 then s' +_f x
+ else if s' <=_r (cert!pc') then x
+ else T)"
+
+constdefs
+wtl_inst :: "'s certificate \<Rightarrow> 's binop \<Rightarrow> 's ord \<Rightarrow> 's \<Rightarrow>
+ 's step_type \<Rightarrow> nat \<Rightarrow> 's \<Rightarrow> 's"
+"wtl_inst cert f r T step pc s \<equiv> merge cert f r T pc (step pc s) (cert!(pc+1))"
+
+wtl_cert :: "'s certificate \<Rightarrow> 's binop \<Rightarrow> 's ord \<Rightarrow> 's \<Rightarrow> 's \<Rightarrow>
+ 's step_type \<Rightarrow> nat \<Rightarrow> 's \<Rightarrow> 's"
+"wtl_cert cert f r T B step pc s \<equiv>
+ if cert!pc = B then
+ wtl_inst cert f r T step pc s
+ else
+ if s <=_r (cert!pc) then wtl_inst cert f r T step pc (cert!pc) else T"
+
+consts
+wtl_inst_list :: "'a list \<Rightarrow> 's certificate \<Rightarrow> 's binop \<Rightarrow> 's ord \<Rightarrow> 's \<Rightarrow> 's \<Rightarrow>
+ 's step_type \<Rightarrow> nat \<Rightarrow> 's \<Rightarrow> 's"
+primrec
+"wtl_inst_list [] cert f r T B step pc s = s"
+"wtl_inst_list (i#is) cert f r T B step pc s =
+ (let s' = wtl_cert cert f r T B step pc s in
+ if s' = T \<or> s = T then T else wtl_inst_list is cert f r T B step (pc+1) s')"
+
+constdefs
+ cert_ok :: "'s certificate \<Rightarrow> nat \<Rightarrow> 's \<Rightarrow> 's \<Rightarrow> 's set \<Rightarrow> bool"
+ "cert_ok cert n T B A \<equiv> (\<forall>i < n. cert!i \<in> A \<and> cert!i \<noteq> T) \<and> (cert!n = B)"
+
+constdefs
+ bottom :: "'a ord \<Rightarrow> 'a \<Rightarrow> bool"
+ "bottom r B \<equiv> \<forall>x. B <=_r x"
+
+
+locale lbv = Semilat +
+ fixes T :: "'a" ("\<top>")
+ fixes B :: "'a" ("\<bottom>")
+ fixes step :: "'a step_type"
+ assumes top: "top r \<top>"
+ assumes T_A: "\<top> \<in> A"
+ assumes bot: "bottom r \<bottom>"
+ assumes B_A: "\<bottom> \<in> A"
+
+ fixes merge :: "'a certificate \<Rightarrow> nat \<Rightarrow> (nat \<times> 'a) list \<Rightarrow> 'a \<Rightarrow> 'a"
+ defines mrg_def: "merge cert \<equiv> LBVSpec.merge cert f r \<top>"
+
+ fixes wti :: "'a certificate \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"
+ defines wti_def: "wti cert \<equiv> wtl_inst cert f r \<top> step"
+
+ fixes wtc :: "'a certificate \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"
+ defines wtc_def: "wtc cert \<equiv> wtl_cert cert f r \<top> \<bottom> step"
+
+ fixes wtl :: "'b list \<Rightarrow> 'a certificate \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"
+ defines wtl_def: "wtl ins cert \<equiv> wtl_inst_list ins cert f r \<top> \<bottom> step"
+
+
+lemma (in lbv) wti:
+ "wti c pc s \<equiv> merge c pc (step pc s) (c!(pc+1))"
+ by (simp add: wti_def mrg_def wtl_inst_def)
+
+lemma (in lbv) wtc:
+ "wtc c pc s \<equiv> if c!pc = \<bottom> then wti c pc s else if s <=_r c!pc then wti c pc (c!pc) else \<top>"
+ by (unfold wtc_def wti_def wtl_cert_def)
+
+
+lemma cert_okD1 [intro?]:
+ "cert_ok c n T B A \<Longrightarrow> pc < n \<Longrightarrow> c!pc \<in> A"
+ by (unfold cert_ok_def) fast
+
+lemma cert_okD2 [intro?]:
+ "cert_ok c n T B A \<Longrightarrow> c!n = B"
+ by (simp add: cert_ok_def)
+
+lemma cert_okD3 [intro?]:
+ "cert_ok c n T B A \<Longrightarrow> B \<in> A \<Longrightarrow> pc < n \<Longrightarrow> c!Suc pc \<in> A"
+ by (drule Suc_leI) (auto simp add: le_eq_less_or_eq dest: cert_okD1 cert_okD2)
+
+lemma cert_okD4 [intro?]:
+ "cert_ok c n T B A \<Longrightarrow> pc < n \<Longrightarrow> c!pc \<noteq> T"
+ by (simp add: cert_ok_def)
+
+declare Let_def [simp]
+
+section "more semilattice lemmas"
+
+
+lemma (in lbv) sup_top [simp, elim]:
+ assumes x: "x \<in> A"
+ shows "x +_f \<top> = \<top>"
+proof -
+ from top have "x +_f \<top> <=_r \<top>" ..
+ moreover from x T_A have "\<top> <=_r x +_f \<top>" ..
+ ultimately show ?thesis ..
+qed
+
+lemma (in lbv) plusplussup_top [simp, elim]:
+ "set xs \<subseteq> A \<Longrightarrow> xs ++_f \<top> = \<top>"
+ by (induct xs) auto
+
+
+
+lemma (in Semilat) pp_ub1':
+ assumes S: "snd`set S \<subseteq> A"
+ assumes y: "y \<in> A" and ab: "(a, b) \<in> set S"
+ shows "b <=_r map snd [(p', t') \<leftarrow> S . p' = a] ++_f y"
+proof -
+ from S have "\<forall>(x,y) \<in> set S. y \<in> A" by auto
+ with semilat y ab show ?thesis by - (rule ub1')
+qed
+
+lemma (in lbv) bottom_le [simp, intro]:
+ "\<bottom> <=_r x"
+ by (insert bot) (simp add: bottom_def)
+
+lemma (in lbv) le_bottom [simp]:
+ "x <=_r \<bottom> = (x = \<bottom>)"
+ by (blast intro: antisym_r)
+
+
+
+section "merge"
+
+lemma (in lbv) merge_Nil [simp]:
+ "merge c pc [] x = x" by (simp add: mrg_def)
+
+lemma (in lbv) merge_Cons [simp]:
+ "merge c pc (l#ls) x = merge c pc ls (if fst l=pc+1 then snd l +_f x
+ else if snd l <=_r (c!fst l) then x
+ else \<top>)"
+ by (simp add: mrg_def split_beta)
+
+lemma (in lbv) merge_Err [simp]:
+ "snd`set ss \<subseteq> A \<Longrightarrow> merge c pc ss \<top> = \<top>"
+ by (induct ss) auto
+
+lemma (in lbv) merge_not_top:
+ "\<And>x. snd`set ss \<subseteq> A \<Longrightarrow> merge c pc ss x \<noteq> \<top> \<Longrightarrow>
+ \<forall>(pc',s') \<in> set ss. (pc' \<noteq> pc+1 \<longrightarrow> s' <=_r (c!pc'))"
+ (is "\<And>x. ?set ss \<Longrightarrow> ?merge ss x \<Longrightarrow> ?P ss")
+proof (induct ss)
+ show "?P []" by simp
+next
+ fix x ls l
+ assume "?set (l#ls)" then obtain set: "snd`set ls \<subseteq> A" by simp
+ assume merge: "?merge (l#ls) x"
+ moreover
+ obtain pc' s' where l: "l = (pc',s')" by (cases l)
+ ultimately
+ obtain x' where merge': "?merge ls x'" by simp
+ assume "\<And>x. ?set ls \<Longrightarrow> ?merge ls x \<Longrightarrow> ?P ls" hence "?P ls" using set merge' .
+ moreover
+ from merge set
+ have "pc' \<noteq> pc+1 \<longrightarrow> s' <=_r (c!pc')" by (simp add: l split: split_if_asm)
+ ultimately
+ show "?P (l#ls)" by (simp add: l)
+qed
+
+
+lemma (in lbv) merge_def:
+ shows
+ "\<And>x. x \<in> A \<Longrightarrow> snd`set ss \<subseteq> A \<Longrightarrow>
+ merge c pc ss x =
+ (if \<forall>(pc',s') \<in> set ss. pc'\<noteq>pc+1 \<longrightarrow> s' <=_r c!pc' then
+ map snd [(p',t') \<leftarrow> ss. p'=pc+1] ++_f x
+ else \<top>)"
+ (is "\<And>x. _ \<Longrightarrow> _ \<Longrightarrow> ?merge ss x = ?if ss x" is "\<And>x. _ \<Longrightarrow> _ \<Longrightarrow> ?P ss x")
+proof (induct ss)
+ fix x show "?P [] x" by simp
+next
+ fix x assume x: "x \<in> A"
+ fix l::"nat \<times> 'a" and ls
+ assume "snd`set (l#ls) \<subseteq> A"
+ then obtain l: "snd l \<in> A" and ls: "snd`set ls \<subseteq> A" by auto
+ assume "\<And>x. x \<in> A \<Longrightarrow> snd`set ls \<subseteq> A \<Longrightarrow> ?P ls x"
+ hence IH: "\<And>x. x \<in> A \<Longrightarrow> ?P ls x" using ls by iprover
+ obtain pc' s' where [simp]: "l = (pc',s')" by (cases l)
+ hence "?merge (l#ls) x = ?merge ls
+ (if pc'=pc+1 then s' +_f x else if s' <=_r c!pc' then x else \<top>)"
+ (is "?merge (l#ls) x = ?merge ls ?if'")
+ by simp
+ also have "\<dots> = ?if ls ?if'"
+ proof -
+ from l have "s' \<in> A" by simp
+ with x have "s' +_f x \<in> A" by simp
+ with x T_A have "?if' \<in> A" by auto
+ hence "?P ls ?if'" by (rule IH) thus ?thesis by simp
+ qed
+ also have "\<dots> = ?if (l#ls) x"
+ proof (cases "\<forall>(pc', s')\<in>set (l#ls). pc'\<noteq>pc+1 \<longrightarrow> s' <=_r c!pc'")
+ case True
+ hence "\<forall>(pc', s')\<in>set ls. pc'\<noteq>pc+1 \<longrightarrow> s' <=_r c!pc'" by auto
+ moreover
+ from True have
+ "map snd [(p',t')\<leftarrow>ls . p'=pc+1] ++_f ?if' =
+ (map snd [(p',t')\<leftarrow>l#ls . p'=pc+1] ++_f x)"
+ by simp
+ ultimately
+ show ?thesis using True by simp
+ next
+ case False
+ moreover
+ from ls have "set (map snd [(p', t')\<leftarrow>ls . p' = Suc pc]) \<subseteq> A" by auto
+ ultimately show ?thesis by auto
+ qed
+ finally show "?P (l#ls) x" .
+qed
+
+lemma (in lbv) merge_not_top_s:
+ assumes x: "x \<in> A" and ss: "snd`set ss \<subseteq> A"
+ assumes m: "merge c pc ss x \<noteq> \<top>"
+ shows "merge c pc ss x = (map snd [(p',t') \<leftarrow> ss. p'=pc+1] ++_f x)"
+proof -
+ from ss m have "\<forall>(pc',s') \<in> set ss. (pc' \<noteq> pc+1 \<longrightarrow> s' <=_r c!pc')"
+ by (rule merge_not_top)
+ with x ss m show ?thesis by - (drule merge_def, auto split: split_if_asm)
+qed
+
+section "wtl-inst-list"
+
+lemmas [iff] = not_Err_eq
+
+lemma (in lbv) wtl_Nil [simp]: "wtl [] c pc s = s"
+ by (simp add: wtl_def)
+
+lemma (in lbv) wtl_Cons [simp]:
+ "wtl (i#is) c pc s =
+ (let s' = wtc c pc s in if s' = \<top> \<or> s = \<top> then \<top> else wtl is c (pc+1) s')"
+ by (simp add: wtl_def wtc_def)
+
+lemma (in lbv) wtl_Cons_not_top:
+ "wtl (i#is) c pc s \<noteq> \<top> =
+ (wtc c pc s \<noteq> \<top> \<and> s \<noteq> T \<and> wtl is c (pc+1) (wtc c pc s) \<noteq> \<top>)"
+ by (auto simp del: split_paired_Ex)
+
+lemma (in lbv) wtl_top [simp]: "wtl ls c pc \<top> = \<top>"
+ by (cases ls) auto
+
+lemma (in lbv) wtl_not_top:
+ "wtl ls c pc s \<noteq> \<top> \<Longrightarrow> s \<noteq> \<top>"
+ by (cases "s=\<top>") auto
+
+lemma (in lbv) wtl_append [simp]:
+ "\<And>pc s. wtl (a@b) c pc s = wtl b c (pc+length a) (wtl a c pc s)"
+ by (induct a) auto
+
+lemma (in lbv) wtl_take:
+ "wtl is c pc s \<noteq> \<top> \<Longrightarrow> wtl (take pc' is) c pc s \<noteq> \<top>"
+ (is "?wtl is \<noteq> _ \<Longrightarrow> _")
+proof -
+ assume "?wtl is \<noteq> \<top>"
+ hence "?wtl (take pc' is @ drop pc' is) \<noteq> \<top>" by simp
+ thus ?thesis by (auto dest!: wtl_not_top simp del: append_take_drop_id)
+qed
+
+lemma take_Suc:
+ "\<forall>n. n < length l \<longrightarrow> take (Suc n) l = (take n l)@[l!n]" (is "?P l")
+proof (induct l)
+ show "?P []" by simp
+next
+ fix x xs assume IH: "?P xs"
+ show "?P (x#xs)"
+ proof (intro strip)
+ fix n assume "n < length (x#xs)"
+ with IH show "take (Suc n) (x # xs) = take n (x # xs) @ [(x # xs) ! n]"
+ by (cases n, auto)
+ qed
+qed
+
+lemma (in lbv) wtl_Suc:
+ assumes suc: "pc+1 < length is"
+ assumes wtl: "wtl (take pc is) c 0 s \<noteq> \<top>"
+ shows "wtl (take (pc+1) is) c 0 s = wtc c pc (wtl (take pc is) c 0 s)"
+proof -
+ from suc have "take (pc+1) is=(take pc is)@[is!pc]" by (simp add: take_Suc)
+ with suc wtl show ?thesis by (simp add: min_max.inf_absorb2)
+qed
+
+lemma (in lbv) wtl_all:
+ assumes all: "wtl is c 0 s \<noteq> \<top>" (is "?wtl is \<noteq> _")
+ assumes pc: "pc < length is"
+ shows "wtc c pc (wtl (take pc is) c 0 s) \<noteq> \<top>"
+proof -
+ from pc have "0 < length (drop pc is)" by simp
+ then obtain i r where Cons: "drop pc is = i#r"
+ by (auto simp add: neq_Nil_conv simp del: length_drop drop_eq_Nil)
+ hence "i#r = drop pc is" ..
+ with all have take: "?wtl (take pc is@i#r) \<noteq> \<top>" by simp
+ from pc have "is!pc = drop pc is ! 0" by simp
+ with Cons have "is!pc = i" by simp
+ with take pc show ?thesis by (auto simp add: min_max.inf_absorb2)
+qed
+
+section "preserves-type"
+
+lemma (in lbv) merge_pres:
+ assumes s0: "snd`set ss \<subseteq> A" and x: "x \<in> A"
+ shows "merge c pc ss x \<in> A"
+proof -
+ from s0 have "set (map snd [(p', t')\<leftarrow>ss . p'=pc+1]) \<subseteq> A" by auto
+ with x have "(map snd [(p', t')\<leftarrow>ss . p'=pc+1] ++_f x) \<in> A"
+ by (auto intro!: plusplus_closed semilat)
+ with s0 x show ?thesis by (simp add: merge_def T_A)
+qed
+
+
+lemma pres_typeD2:
+ "pres_type step n A \<Longrightarrow> s \<in> A \<Longrightarrow> p < n \<Longrightarrow> snd`set (step p s) \<subseteq> A"
+ by auto (drule pres_typeD)
+
+
+lemma (in lbv) wti_pres [intro?]:
+ assumes pres: "pres_type step n A"
+ assumes cert: "c!(pc+1) \<in> A"
+ assumes s_pc: "s \<in> A" "pc < n"
+ shows "wti c pc s \<in> A"
+proof -
+ from pres s_pc have "snd`set (step pc s) \<subseteq> A" by (rule pres_typeD2)
+ with cert show ?thesis by (simp add: wti merge_pres)
+qed
+
+
+lemma (in lbv) wtc_pres:
+ assumes pres: "pres_type step n A"
+ assumes cert: "c!pc \<in> A" and cert': "c!(pc+1) \<in> A"
+ assumes s: "s \<in> A" and pc: "pc < n"
+ shows "wtc c pc s \<in> A"
+proof -
+ have "wti c pc s \<in> A" using pres cert' s pc ..
+ moreover have "wti c pc (c!pc) \<in> A" using pres cert' cert pc ..
+ ultimately show ?thesis using T_A by (simp add: wtc)
+qed
+
+
+lemma (in lbv) wtl_pres:
+ assumes pres: "pres_type step (length is) A"
+ assumes cert: "cert_ok c (length is) \<top> \<bottom> A"
+ assumes s: "s \<in> A"
+ assumes all: "wtl is c 0 s \<noteq> \<top>"
+ shows "pc < length is \<Longrightarrow> wtl (take pc is) c 0 s \<in> A"
+ (is "?len pc \<Longrightarrow> ?wtl pc \<in> A")
+proof (induct pc)
+ from s show "?wtl 0 \<in> A" by simp
+next
+ fix n assume IH: "Suc n < length is"
+ then have n: "n < length is" by simp
+ from IH have n1: "n+1 < length is" by simp
+ assume prem: "n < length is \<Longrightarrow> ?wtl n \<in> A"
+ have "wtc c n (?wtl n) \<in> A"
+ using pres _ _ _ n
+ proof (rule wtc_pres)
+ from prem n show "?wtl n \<in> A" .
+ from cert n show "c!n \<in> A" by (rule cert_okD1)
+ from cert n1 show "c!(n+1) \<in> A" by (rule cert_okD1)
+ qed
+ also
+ from all n have "?wtl n \<noteq> \<top>" by - (rule wtl_take)
+ with n1 have "wtc c n (?wtl n) = ?wtl (n+1)" by (rule wtl_Suc [symmetric])
+ finally show "?wtl (Suc n) \<in> A" by simp
+qed
+
+end