(* Title: HOL/MicroJava/BV/LBVComplete.thy
Author: Gerwin Klein
Copyright 2000 Technische Universitaet Muenchen
*)
header {* \isaheader{Completeness of the LBV} *}
theory LBVComplete
imports LBVSpec Typing_Framework
begin
definition is_target :: "['s step_type, 's list, nat] \<Rightarrow> bool" where
"is_target step phi pc' \<equiv>
\<exists>pc s'. pc' \<noteq> pc+1 \<and> pc < length phi \<and> (pc',s') \<in> set (step pc (phi!pc))"
definition make_cert :: "['s step_type, 's list, 's] \<Rightarrow> 's certificate" where
"make_cert step phi B \<equiv>
map (\<lambda>pc. if is_target step phi pc then phi!pc else B) [0..<length phi] @ [B]"
lemma [code]:
"is_target step phi pc' =
list_ex (\<lambda>pc. pc' \<noteq> pc+1 \<and> pc' mem (map fst (step pc (phi!pc)))) [0..<length phi]"
by (force simp: list_ex_iff is_target_def mem_iff)
locale lbvc = lbv +
fixes phi :: "'a list" ("\<phi>")
fixes c :: "'a list"
defines cert_def: "c \<equiv> make_cert step \<phi> \<bottom>"
assumes mono: "mono r step (length \<phi>) A"
assumes pres: "pres_type step (length \<phi>) A"
assumes phi: "\<forall>pc < length \<phi>. \<phi>!pc \<in> A \<and> \<phi>!pc \<noteq> \<top>"
assumes bounded: "bounded step (length \<phi>)"
assumes B_neq_T: "\<bottom> \<noteq> \<top>"
lemma (in lbvc) cert: "cert_ok c (length \<phi>) \<top> \<bottom> A"
proof (unfold cert_ok_def, intro strip conjI)
note [simp] = make_cert_def cert_def nth_append
show "c!length \<phi> = \<bottom>" by simp
fix pc assume pc: "pc < length \<phi>"
from pc phi B_A show "c!pc \<in> A" by simp
from pc phi B_neq_T show "c!pc \<noteq> \<top>" by simp
qed
lemmas [simp del] = split_paired_Ex
lemma (in lbvc) cert_target [intro?]:
"\<lbrakk> (pc',s') \<in> set (step pc (\<phi>!pc));
pc' \<noteq> pc+1; pc < length \<phi>; pc' < length \<phi> \<rbrakk>
\<Longrightarrow> c!pc' = \<phi>!pc'"
by (auto simp add: cert_def make_cert_def nth_append is_target_def)
lemma (in lbvc) cert_approx [intro?]:
"\<lbrakk> pc < length \<phi>; c!pc \<noteq> \<bottom> \<rbrakk>
\<Longrightarrow> c!pc = \<phi>!pc"
by (auto simp add: cert_def make_cert_def nth_append)
lemma (in lbv) le_top [simp, intro]:
"x <=_r \<top>"
by (insert top) simp
lemma (in lbv) merge_mono:
assumes less: "ss2 <=|r| ss1"
assumes x: "x \<in> A"
assumes ss1: "snd`set ss1 \<subseteq> A"
assumes ss2: "snd`set ss2 \<subseteq> A"
shows "merge c pc ss2 x <=_r merge c pc ss1 x" (is "?s2 <=_r ?s1")
proof-
have "?s1 = \<top> \<Longrightarrow> ?thesis" by simp
moreover {
assume merge: "?s1 \<noteq> T"
from x ss1 have "?s1 =
(if \<forall>(pc', s')\<in>set ss1. pc' \<noteq> pc + 1 \<longrightarrow> s' <=_r c!pc'
then (map snd [(p', t') \<leftarrow> ss1 . p'=pc+1]) ++_f x
else \<top>)"
by (rule merge_def)
with merge obtain
app: "\<forall>(pc',s')\<in>set ss1. pc' \<noteq> pc+1 \<longrightarrow> s' <=_r c!pc'"
(is "?app ss1") and
sum: "(map snd [(p',t') \<leftarrow> ss1 . p' = pc+1] ++_f x) = ?s1"
(is "?map ss1 ++_f x = _" is "?sum ss1 = _")
by (simp split: split_if_asm)
from app less
have "?app ss2" by (blast dest: trans_r lesub_step_typeD)
moreover {
from ss1 have map1: "set (?map ss1) \<subseteq> A" by auto
with x have "?sum ss1 \<in> A" by (auto intro!: plusplus_closed semilat)
with sum have "?s1 \<in> A" by simp
moreover
have mapD: "\<And>x ss. x \<in> set (?map ss) \<Longrightarrow> \<exists>p. (p,x) \<in> set ss \<and> p=pc+1" by auto
from x map1
have "\<forall>x \<in> set (?map ss1). x <=_r ?sum ss1"
by clarify (rule pp_ub1)
with sum have "\<forall>x \<in> set (?map ss1). x <=_r ?s1" by simp
with less have "\<forall>x \<in> set (?map ss2). x <=_r ?s1"
by (fastsimp dest!: mapD lesub_step_typeD intro: trans_r)
moreover
from map1 x have "x <=_r (?sum ss1)" by (rule pp_ub2)
with sum have "x <=_r ?s1" by simp
moreover
from ss2 have "set (?map ss2) \<subseteq> A" by auto
ultimately
have "?sum ss2 <=_r ?s1" using x by - (rule pp_lub)
}
moreover
from x ss2 have
"?s2 =
(if \<forall>(pc', s')\<in>set ss2. pc' \<noteq> pc + 1 \<longrightarrow> s' <=_r c!pc'
then map snd [(p', t') \<leftarrow> ss2 . p' = pc + 1] ++_f x
else \<top>)"
by (rule merge_def)
ultimately have ?thesis by simp
}
ultimately show ?thesis by (cases "?s1 = \<top>") auto
qed
lemma (in lbvc) wti_mono:
assumes less: "s2 <=_r s1"
assumes pc: "pc < length \<phi>"
assumes s1: "s1 \<in> A"
assumes s2: "s2 \<in> A"
shows "wti c pc s2 <=_r wti c pc s1" (is "?s2' <=_r ?s1'")
proof -
from mono pc s2 less have "step pc s2 <=|r| step pc s1" by (rule monoD)
moreover
from cert B_A pc have "c!Suc pc \<in> A" by (rule cert_okD3)
moreover
from pres s1 pc
have "snd`set (step pc s1) \<subseteq> A" by (rule pres_typeD2)
moreover
from pres s2 pc
have "snd`set (step pc s2) \<subseteq> A" by (rule pres_typeD2)
ultimately
show ?thesis by (simp add: wti merge_mono)
qed
lemma (in lbvc) wtc_mono:
assumes less: "s2 <=_r s1"
assumes pc: "pc < length \<phi>"
assumes s1: "s1 \<in> A"
assumes s2: "s2 \<in> A"
shows "wtc c pc s2 <=_r wtc c pc s1" (is "?s2' <=_r ?s1'")
proof (cases "c!pc = \<bottom>")
case True
moreover from less pc s1 s2 have "wti c pc s2 <=_r wti c pc s1" by (rule wti_mono)
ultimately show ?thesis by (simp add: wtc)
next
case False
have "?s1' = \<top> \<Longrightarrow> ?thesis" by simp
moreover {
assume "?s1' \<noteq> \<top>"
with False have c: "s1 <=_r c!pc" by (simp add: wtc split: split_if_asm)
with less have "s2 <=_r c!pc" ..
with False c have ?thesis by (simp add: wtc)
}
ultimately show ?thesis by (cases "?s1' = \<top>") auto
qed
lemma (in lbv) top_le_conv [simp]:
"\<top> <=_r x = (x = \<top>)"
by (insert semilat) (simp add: top top_le_conv)
lemma (in lbv) neq_top [simp, elim]:
"\<lbrakk> x <=_r y; y \<noteq> \<top> \<rbrakk> \<Longrightarrow> x \<noteq> \<top>"
by (cases "x = T") auto
lemma (in lbvc) stable_wti:
assumes stable: "stable r step \<phi> pc"
assumes pc: "pc < length \<phi>"
shows "wti c pc (\<phi>!pc) \<noteq> \<top>"
proof -
let ?step = "step pc (\<phi>!pc)"
from stable
have less: "\<forall>(q,s')\<in>set ?step. s' <=_r \<phi>!q" by (simp add: stable_def)
from cert B_A pc
have cert_suc: "c!Suc pc \<in> A" by (rule cert_okD3)
moreover
from phi pc have "\<phi>!pc \<in> A" by simp
from pres this pc
have stepA: "snd`set ?step \<subseteq> A" by (rule pres_typeD2)
ultimately
have "merge c pc ?step (c!Suc pc) =
(if \<forall>(pc',s')\<in>set ?step. pc'\<noteq>pc+1 \<longrightarrow> s' <=_r c!pc'
then map snd [(p',t') \<leftarrow> ?step.p'=pc+1] ++_f c!Suc pc
else \<top>)" unfolding mrg_def by (rule lbv.merge_def [OF lbvc.axioms(1), OF lbvc_axioms])
moreover {
fix pc' s' assume s': "(pc', s') \<in> set ?step" and suc_pc: "pc' \<noteq> pc+1"
with less have "s' <=_r \<phi>!pc'" by auto
also
from bounded pc s' have "pc' < length \<phi>" by (rule boundedD)
with s' suc_pc pc have "c!pc' = \<phi>!pc'" ..
hence "\<phi>!pc' = c!pc'" ..
finally have "s' <=_r c!pc'" .
} hence "\<forall>(pc',s')\<in>set ?step. pc'\<noteq>pc+1 \<longrightarrow> s' <=_r c!pc'" by auto
moreover
from pc have "Suc pc = length \<phi> \<or> Suc pc < length \<phi>" by auto
hence "map snd [(p',t') \<leftarrow> ?step.p'=pc+1] ++_f c!Suc pc \<noteq> \<top>"
(is "?map ++_f _ \<noteq> _")
proof (rule disjE)
assume pc': "Suc pc = length \<phi>"
with cert have "c!Suc pc = \<bottom>" by (simp add: cert_okD2)
moreover
from pc' bounded pc
have "\<forall>(p',t')\<in>set ?step. p'\<noteq>pc+1" by clarify (drule boundedD, auto)
hence "[(p',t') \<leftarrow> ?step.p'=pc+1] = []" by (blast intro: filter_False)
hence "?map = []" by simp
ultimately show ?thesis by (simp add: B_neq_T)
next
assume pc': "Suc pc < length \<phi>"
from pc' phi have "\<phi>!Suc pc \<in> A" by simp
moreover note cert_suc
moreover from stepA
have "set ?map \<subseteq> A" by auto
moreover
have "\<And>s. s \<in> set ?map \<Longrightarrow> \<exists>t. (Suc pc, t) \<in> set ?step" by auto
with less have "\<forall>s' \<in> set ?map. s' <=_r \<phi>!Suc pc" by auto
moreover
from pc' have "c!Suc pc <=_r \<phi>!Suc pc"
by (cases "c!Suc pc = \<bottom>") (auto dest: cert_approx)
ultimately
have "?map ++_f c!Suc pc <=_r \<phi>!Suc pc" by (rule pp_lub)
moreover
from pc' phi have "\<phi>!Suc pc \<noteq> \<top>" by simp
ultimately
show ?thesis by auto
qed
ultimately
have "merge c pc ?step (c!Suc pc) \<noteq> \<top>" by simp
thus ?thesis by (simp add: wti)
qed
lemma (in lbvc) wti_less:
assumes stable: "stable r step \<phi> pc"
assumes suc_pc: "Suc pc < length \<phi>"
shows "wti c pc (\<phi>!pc) <=_r \<phi>!Suc pc" (is "?wti <=_r _")
proof -
let ?step = "step pc (\<phi>!pc)"
from stable
have less: "\<forall>(q,s')\<in>set ?step. s' <=_r \<phi>!q" by (simp add: stable_def)
from suc_pc have pc: "pc < length \<phi>" by simp
with cert B_A have cert_suc: "c!Suc pc \<in> A" by (rule cert_okD3)
moreover
from phi pc have "\<phi>!pc \<in> A" by simp
with pres pc have stepA: "snd`set ?step \<subseteq> A" by - (rule pres_typeD2)
moreover
from stable pc have "?wti \<noteq> \<top>" by (rule stable_wti)
hence "merge c pc ?step (c!Suc pc) \<noteq> \<top>" by (simp add: wti)
ultimately
have "merge c pc ?step (c!Suc pc) =
map snd [(p',t')\<leftarrow> ?step.p'=pc+1] ++_f c!Suc pc" by (rule merge_not_top_s)
hence "?wti = \<dots>" (is "_ = (?map ++_f _)" is "_ = ?sum") by (simp add: wti)
also {
from suc_pc phi have "\<phi>!Suc pc \<in> A" by simp
moreover note cert_suc
moreover from stepA have "set ?map \<subseteq> A" by auto
moreover
have "\<And>s. s \<in> set ?map \<Longrightarrow> \<exists>t. (Suc pc, t) \<in> set ?step" by auto
with less have "\<forall>s' \<in> set ?map. s' <=_r \<phi>!Suc pc" by auto
moreover
from suc_pc have "c!Suc pc <=_r \<phi>!Suc pc"
by (cases "c!Suc pc = \<bottom>") (auto dest: cert_approx)
ultimately
have "?sum <=_r \<phi>!Suc pc" by (rule pp_lub)
}
finally show ?thesis .
qed
lemma (in lbvc) stable_wtc:
assumes stable: "stable r step phi pc"
assumes pc: "pc < length \<phi>"
shows "wtc c pc (\<phi>!pc) \<noteq> \<top>"
proof -
from stable pc have wti: "wti c pc (\<phi>!pc) \<noteq> \<top>" by (rule stable_wti)
show ?thesis
proof (cases "c!pc = \<bottom>")
case True with wti show ?thesis by (simp add: wtc)
next
case False
with pc have "c!pc = \<phi>!pc" ..
with False wti show ?thesis by (simp add: wtc)
qed
qed
lemma (in lbvc) wtc_less:
assumes stable: "stable r step \<phi> pc"
assumes suc_pc: "Suc pc < length \<phi>"
shows "wtc c pc (\<phi>!pc) <=_r \<phi>!Suc pc" (is "?wtc <=_r _")
proof (cases "c!pc = \<bottom>")
case True
moreover from stable suc_pc have "wti c pc (\<phi>!pc) <=_r \<phi>!Suc pc"
by (rule wti_less)
ultimately show ?thesis by (simp add: wtc)
next
case False
from suc_pc have pc: "pc < length \<phi>" by simp
with stable have "?wtc \<noteq> \<top>" by (rule stable_wtc)
with False have "?wtc = wti c pc (c!pc)"
by (unfold wtc) (simp split: split_if_asm)
also from pc False have "c!pc = \<phi>!pc" ..
finally have "?wtc = wti c pc (\<phi>!pc)" .
also from stable suc_pc have "wti c pc (\<phi>!pc) <=_r \<phi>!Suc pc" by (rule wti_less)
finally show ?thesis .
qed
lemma (in lbvc) wt_step_wtl_lemma:
assumes wt_step: "wt_step r \<top> step \<phi>"
shows "\<And>pc s. pc+length ls = length \<phi> \<Longrightarrow> s <=_r \<phi>!pc \<Longrightarrow> s \<in> A \<Longrightarrow> s\<noteq>\<top> \<Longrightarrow>
wtl ls c pc s \<noteq> \<top>"
(is "\<And>pc s. _ \<Longrightarrow> _ \<Longrightarrow> _ \<Longrightarrow> _ \<Longrightarrow> ?wtl ls pc s \<noteq> _")
proof (induct ls)
fix pc s assume "s\<noteq>\<top>" thus "?wtl [] pc s \<noteq> \<top>" by simp
next
fix pc s i ls
assume "\<And>pc s. pc+length ls=length \<phi> \<Longrightarrow> s <=_r \<phi>!pc \<Longrightarrow> s \<in> A \<Longrightarrow> s\<noteq>\<top> \<Longrightarrow>
?wtl ls pc s \<noteq> \<top>"
moreover
assume pc_l: "pc + length (i#ls) = length \<phi>"
hence suc_pc_l: "Suc pc + length ls = length \<phi>" by simp
ultimately
have IH: "\<And>s. s <=_r \<phi>!Suc pc \<Longrightarrow> s \<in> A \<Longrightarrow> s \<noteq> \<top> \<Longrightarrow> ?wtl ls (Suc pc) s \<noteq> \<top>" .
from pc_l obtain pc: "pc < length \<phi>" by simp
with wt_step have stable: "stable r step \<phi> pc" by (simp add: wt_step_def)
from this pc have wt_phi: "wtc c pc (\<phi>!pc) \<noteq> \<top>" by (rule stable_wtc)
assume s_phi: "s <=_r \<phi>!pc"
from phi pc have phi_pc: "\<phi>!pc \<in> A" by simp
assume s: "s \<in> A"
with s_phi pc phi_pc have wt_s_phi: "wtc c pc s <=_r wtc c pc (\<phi>!pc)" by (rule wtc_mono)
with wt_phi have wt_s: "wtc c pc s \<noteq> \<top>" by simp
moreover
assume s': "s \<noteq> \<top>"
ultimately
have "ls = [] \<Longrightarrow> ?wtl (i#ls) pc s \<noteq> \<top>" by simp
moreover {
assume "ls \<noteq> []"
with pc_l have suc_pc: "Suc pc < length \<phi>" by (auto simp add: neq_Nil_conv)
with stable have "wtc c pc (phi!pc) <=_r \<phi>!Suc pc" by (rule wtc_less)
with wt_s_phi have "wtc c pc s <=_r \<phi>!Suc pc" by (rule trans_r)
moreover
from cert suc_pc have "c!pc \<in> A" "c!(pc+1) \<in> A"
by (auto simp add: cert_ok_def)
from pres this s pc have "wtc c pc s \<in> A" by (rule wtc_pres)
ultimately
have "?wtl ls (Suc pc) (wtc c pc s) \<noteq> \<top>" using IH wt_s by blast
with s' wt_s have "?wtl (i#ls) pc s \<noteq> \<top>" by simp
}
ultimately show "?wtl (i#ls) pc s \<noteq> \<top>" by (cases ls) blast+
qed
theorem (in lbvc) wtl_complete:
assumes wt: "wt_step r \<top> step \<phi>"
and s: "s <=_r \<phi>!0" "s \<in> A" "s \<noteq> \<top>"
and len: "length ins = length phi"
shows "wtl ins c 0 s \<noteq> \<top>"
proof -
from len have "0+length ins = length phi" by simp
from wt this s show ?thesis by (rule wt_step_wtl_lemma)
qed
end