(* Title: HOL/MicroJava/BV/LBVComplete.thy
ID: $Id$
Author: Gerwin Klein
Copyright 2000 Technische Universitaet Muenchen
*)
header {* Completeness of the LBV *}
theory LBVComplete = BVSpec + LBVSpec + StepMono:
constdefs
is_approx :: "['a option list, 'a list] \\<Rightarrow> bool"
"is_approx a b \\<equiv> length a = length b \\<and> (\\<forall> n. n < length a \\<longrightarrow>
(a!n = None \\<or> a!n = Some (b!n)))"
contains_dead :: "[instr list, certificate, method_type, p_count] \\<Rightarrow> bool"
"contains_dead ins cert phi pc \\<equiv>
Suc pc \\<notin> succs (ins!pc) pc \\<and> Suc pc < length phi \\<longrightarrow>
cert ! (Suc pc) = Some (phi ! Suc pc)"
contains_targets :: "[instr list, certificate, method_type, p_count] \\<Rightarrow> bool"
"contains_targets ins cert phi pc \\<equiv> (
\\<forall> pc' \\<in> succs (ins!pc) pc. pc' \\<noteq> Suc pc \\<and> pc' < length phi \\<longrightarrow>
cert!pc' = Some (phi!pc'))"
fits :: "[instr list, certificate, method_type] \\<Rightarrow> bool"
"fits ins cert phi \\<equiv> is_approx cert phi \\<and> length ins < length phi \\<and>
(\\<forall> pc. pc < length ins \\<longrightarrow>
contains_dead ins cert phi pc \\<and>
contains_targets ins cert phi pc)"
is_target :: "[instr list, p_count] \\<Rightarrow> bool"
"is_target ins pc \\<equiv> \\<exists> pc'. pc' < length ins \\<and> pc \\<in> succs (ins!pc') pc'"
maybe_dead :: "[instr list, p_count] \\<Rightarrow> bool"
"maybe_dead ins pc \\<equiv> \\<exists> pc'. pc = pc'+1 \\<and> pc \\<notin> succs (ins!pc') pc'"
mdot :: "[instr list, p_count] \\<Rightarrow> bool"
"mdot ins pc \\<equiv> maybe_dead ins pc \\<or> is_target ins pc"
consts
option_filter_n :: "['a list, nat \\<Rightarrow> bool, nat] \\<Rightarrow> 'a option list"
primrec
"option_filter_n [] P n = []"
"option_filter_n (h#t) P n = (if (P n) then Some h # option_filter_n t P (n+1)
else None # option_filter_n t P (n+1))"
constdefs
option_filter :: "['a list, nat \\<Rightarrow> bool] \\<Rightarrow> 'a option list"
"option_filter l P \\<equiv> option_filter_n l P 0"
make_cert :: "[instr list, method_type] \\<Rightarrow> certificate"
"make_cert ins phi \\<equiv> option_filter phi (mdot ins)"
make_Cert :: "[jvm_prog, prog_type] \\<Rightarrow> prog_certificate"
"make_Cert G Phi \\<equiv> \\<lambda> C sig.
let (C,x,y,mdecls) = \\<epsilon> (Cl,x,y,mdecls). (Cl,x,y,mdecls) \\<in> set G \\<and> Cl = C in
let (sig,rT,maxl,b) = \\<epsilon> (sg,rT,maxl,b). (sg,rT,maxl,b) \\<in> set mdecls \\<and> sg = sig in
make_cert b (Phi C sig)"
lemmas [simp del] = split_paired_Ex
lemma length_ofn [rulify]: "\\<forall>n. length (option_filter_n l P n) = length l"
by (induct l) auto
lemma is_approx_option_filter: "is_approx (option_filter l P) l"
proof -
{
fix a n
have "\\<forall>n. is_approx (option_filter_n a P n) a" (is "?P a")
proof (induct a)
show "?P []" by (auto simp add: is_approx_def)
fix l ls
assume Cons: "?P ls"
show "?P (l#ls)"
proof (unfold is_approx_def, intro allI conjI impI)
fix n
show "length (option_filter_n (l # ls) P n) = length (l # ls)"
by (simp only: length_ofn)
fix m
assume "m < length (option_filter_n (l # ls) P n)"
hence m: "m < Suc (length ls)" by (simp only: length_ofn) simp
show "option_filter_n (l # ls) P n ! m = None \\<or>
option_filter_n (l # ls) P n ! m = Some ((l # ls) ! m)"
proof (cases "m")
assume "m = 0"
with m show ?thesis by simp
next
fix m' assume Suc: "m = Suc m'"
from Cons
show ?thesis
proof (unfold is_approx_def, elim allE impE conjE)
from m Suc
show "m' < length (option_filter_n ls P (Suc n))" by (simp add: length_ofn)
assume "option_filter_n ls P (Suc n) ! m' = None \\<or>
option_filter_n ls P (Suc n) ! m' = Some (ls ! m')"
with m Suc
show ?thesis by auto
qed
qed
qed
qed
}
thus ?thesis
by (auto simp add: option_filter_def)
qed
lemma option_filter_Some:
"\\<lbrakk>n < length l; P n\\<rbrakk> \\<Longrightarrow> option_filter l P ! n = Some (l!n)"
proof -
assume 1: "n < length l" "P n"
have "\\<forall>n n'. n < length l \\<longrightarrow> P (n+n') \\<longrightarrow> option_filter_n l P n' ! n = Some (l!n)"
(is "?P l")
proof (induct l)
show "?P []" by simp
fix l ls assume Cons: "?P ls"
show "?P (l#ls)"
proof (intro)
fix n n'
assume l: "n < length (l # ls)"
assume P: "P (n + n')"
show "option_filter_n (l # ls) P n' ! n = Some ((l # ls) ! n)"
proof (cases "n")
assume "n=0"
with P show ?thesis by simp
next
fix m assume "n = Suc m"
with l P Cons
show ?thesis by simp
qed
qed
qed
with 1
show ?thesis by (auto simp add: option_filter_def)
qed
lemma option_filter_contains_dead:
"contains_dead ins (option_filter phi (mdot ins)) phi pc"
by (auto intro: option_filter_Some simp add: contains_dead_def mdot_def maybe_dead_def)
lemma option_filter_contains_targets:
"pc < length ins \\<Longrightarrow> contains_targets ins (option_filter phi (mdot ins)) phi pc"
proof (unfold contains_targets_def, clarsimp)
fix pc'
assume "pc < length ins"
"pc' \\<in> succs (ins ! pc) pc"
"pc' \\<noteq> Suc pc" and
pc': "pc' < length phi"
hence "is_target ins pc'" by (auto simp add: is_target_def)
with pc'
show "option_filter phi (mdot ins) ! pc' = Some (phi ! pc')"
by (auto intro: option_filter_Some simp add: mdot_def)
qed
lemma fits_make_cert:
"length ins < length phi \\<Longrightarrow> fits ins (make_cert ins phi) phi"
proof -
assume l: "length ins < length phi"
thus "fits ins (make_cert ins phi) phi"
proof (unfold fits_def make_cert_def, intro conjI allI impI)
show "is_approx (option_filter phi (mdot ins)) phi"
by (rule is_approx_option_filter)
show "length ins < length phi" .
fix pc
show "contains_dead ins (option_filter phi (mdot ins)) phi pc"
by (rule option_filter_contains_dead)
assume "pc < length ins"
thus "contains_targets ins (option_filter phi (mdot ins)) phi pc"
by (rule option_filter_contains_targets)
qed
qed
lemma fitsD:
"\\<lbrakk>fits ins cert phi; pc' \\<in> succs (ins!pc) pc; pc' \\<noteq> Suc pc; pc < length ins; pc' < length ins\\<rbrakk>
\\<Longrightarrow> cert!pc' = Some (phi!pc')"
by (clarsimp simp add: fits_def contains_dead_def contains_targets_def)
lemma fitsD2:
"\\<lbrakk>fits ins cert phi; Suc pc \\<notin> succs (ins!pc) pc; pc < length ins\\<rbrakk>
\\<Longrightarrow> cert ! Suc pc = Some (phi ! Suc pc)"
by (clarsimp simp add: fits_def contains_dead_def contains_targets_def)
lemma wtl_inst_mono:
"\\<lbrakk>wtl_inst i G rT s1 s1' cert mpc pc; fits ins cert phi; pc < length ins;
G \\<turnstile> s2 <=s s1; i = ins!pc\\<rbrakk> \\<Longrightarrow>
\\<exists> s2'. wtl_inst (ins!pc) G rT s2 s2' cert mpc pc \\<and> (G \\<turnstile> s2' <=s s1')";
proof -
assume pc: "pc < length ins" "i = ins!pc"
assume wtl: "wtl_inst i G rT s1 s1' cert mpc pc"
assume fits: "fits ins cert phi"
assume G: "G \\<turnstile> s2 <=s s1"
let "?step s" = "step (i, G, s)"
from wtl G
have app: "app (i, G, rT, s2)" by (auto simp add: wtl_inst_def app_mono)
from wtl G
have step: "succs i pc \\<noteq> {} \\<Longrightarrow> G \\<turnstile> the (?step s2) <=s the (?step s1)"
by - (drule step_mono, auto simp add: wtl_inst_def)
with app
have some: "succs i pc \\<noteq> {} \\<Longrightarrow> ?step s2 \\<noteq> None" by (simp add: app_step_some)
{
fix pc'
assume 0: "pc' \\<in> succs i pc" "pc' \\<noteq> pc+1"
hence "succs i pc \\<noteq> {}" by auto
hence "G \\<turnstile> the (?step s2) <=s the (?step s1)" by (rule step)
also
from wtl 0
have "G \\<turnstile> the (?step s1) <=s the (cert!pc')"
by (auto simp add: wtl_inst_def)
finally
have "G\\<turnstile> the (?step s2) <=s the (cert!pc')" .
} note cert = this
have "\\<exists>s2'. wtl_inst i G rT s2 s2' cert mpc pc \\<and> G \\<turnstile> s2' <=s s1'"
proof (cases "pc+1 \\<in> succs i pc")
case True
with wtl
have s1': "s1' = the (?step s1)" by (simp add: wtl_inst_def)
have "wtl_inst i G rT s2 (the (?step s2)) cert mpc pc \\<and> G \\<turnstile> (the (?step s2)) <=s s1'"
(is "?wtl \\<and> ?G")
proof
from True s1'
show ?G by (auto intro: step)
from True app wtl
show ?wtl
by (clarsimp intro: cert simp add: wtl_inst_def)
qed
thus ?thesis by blast
next
case False
with wtl
have "s1' = the (cert ! Suc pc)" by (simp add: wtl_inst_def)
with False app wtl
have "wtl_inst i G rT s2 s1' cert mpc pc \\<and> G \\<turnstile> s1' <=s s1'"
by (clarsimp intro: cert simp add: wtl_inst_def)
thus ?thesis by blast
qed
with pc show ?thesis by simp
qed
lemma wtl_option_mono:
"\\<lbrakk>wtl_inst_option i G rT s1 s1' cert mpc pc; fits ins cert phi;
pc < length ins; G \\<turnstile> s2 <=s s1; i = ins!pc\\<rbrakk> \\<Longrightarrow>
\\<exists> s2'. wtl_inst_option (ins!pc) G rT s2 s2' cert mpc pc \\<and> (G \\<turnstile> s2' <=s s1')"
proof -
assume wtl: "wtl_inst_option i G rT s1 s1' cert mpc pc" and
fits: "fits ins cert phi" "pc < length ins"
"G \\<turnstile> s2 <=s s1" "i = ins!pc"
show ?thesis
proof (cases "cert!pc")
case None
with wtl fits;
show ?thesis;
by - (rule wtl_inst_mono [elimify], (simp add: wtl_inst_option_def)+);
next
case Some
with wtl fits;
have G: "G \\<turnstile> s2 <=s a"
by - (rule sup_state_trans, (simp add: wtl_inst_option_def)+)
from Some wtl
have "wtl_inst i G rT a s1' cert mpc pc"; by (simp add: wtl_inst_option_def)
with G fits
have "\\<exists> s2'. wtl_inst (ins!pc) G rT a s2' cert mpc pc \\<and> (G \\<turnstile> s2' <=s s1')"
by - (rule wtl_inst_mono, (simp add: wtl_inst_option_def)+);
with Some G;
show ?thesis; by (simp add: wtl_inst_option_def);
qed
qed
lemma wt_instr_imp_wtl_inst:
"\\<lbrakk>wt_instr (ins!pc) G rT phi max_pc pc; fits ins cert phi;
pc < length ins; length ins = max_pc\\<rbrakk> \\<Longrightarrow>
\\<exists> s. wtl_inst (ins!pc) G rT (phi!pc) s cert max_pc pc \\<and> G \\<turnstile> s <=s phi ! Suc pc";
proof -
assume wt: "wt_instr (ins!pc) G rT phi max_pc pc"
assume fits: "fits ins cert phi"
assume pc: "pc < length ins" "length ins = max_pc"
from wt
have app: "app (ins!pc, G, rT, phi!pc)" by (simp add: wt_instr_def);
from wt pc
have pc': "!!pc'. pc' \\<in> succs (ins!pc) pc \\<Longrightarrow> pc' < length ins"
by (simp add: wt_instr_def)
let ?s' = "the (step (ins!pc,G, phi!pc))"
from wt
have G: "!!pc'. pc' \\<in> succs (ins!pc) pc \\<Longrightarrow> G \\<turnstile> ?s' <=s phi ! pc'"
by (simp add: wt_instr_def)
from wt fits pc
have cert: "!!pc'. \\<lbrakk>pc' \\<in> succs (ins!pc) pc; pc' < max_pc; pc' \\<noteq> pc+1\\<rbrakk>
\\<Longrightarrow> cert!pc' \\<noteq> None \\<and> G \\<turnstile> ?s' <=s the (cert!pc')"
by (auto dest: fitsD simp add: wt_instr_def)
show ?thesis
proof (cases "pc+1 \\<in> succs (ins!pc) pc")
case True
have "wtl_inst (ins!pc) G rT (phi!pc) ?s' cert max_pc pc \\<and> G \\<turnstile> ?s' <=s phi ! Suc pc" (is "?wtl \\<and> ?G")
proof
from True
show "G \\<turnstile> ?s' <=s phi ! Suc pc" by (simp add: G)
from True fits app pc cert pc'
show "?wtl"
by (auto simp add: wtl_inst_def)
qed
thus ?thesis by blast
next
let ?s'' = "the (cert ! Suc pc)"
case False
with fits app pc cert pc'
have "wtl_inst (ins ! pc) G rT (phi ! pc) ?s'' cert max_pc pc \\<and> G \\<turnstile> ?s'' <=s phi ! Suc pc"
by (auto dest: fitsD2 simp add: wtl_inst_def)
thus ?thesis by blast
qed
qed
lemma wt_instr_imp_wtl_option:
"\\<lbrakk>fits ins cert phi; pc < length ins; wt_instr (ins!pc) G rT phi max_pc pc; max_pc = length ins\\<rbrakk> \\<Longrightarrow>
\\<exists> s. wtl_inst_option (ins!pc) G rT (phi!pc) s cert max_pc pc \\<and> G \\<turnstile> s <=s phi ! Suc pc";
proof -
assume fits : "fits ins cert phi" "pc < length ins"
and "wt_instr (ins!pc) G rT phi max_pc pc"
"max_pc = length ins";
hence * : "\\<exists> s. wtl_inst (ins!pc) G rT (phi!pc) s cert max_pc pc \\<and> G \\<turnstile> s <=s phi ! Suc pc";
by - (rule wt_instr_imp_wtl_inst, simp+);
show ?thesis;
proof (cases "cert ! pc");
case None;
with *;
show ?thesis; by (simp add: wtl_inst_option_def);
next;
case Some;
from fits;
have "pc < length phi"; by (clarsimp simp add: fits_def);
with fits Some;
have "cert ! pc = Some (phi!pc)"; by (auto simp add: fits_def is_approx_def);
with *;
show ?thesis; by (simp add: wtl_inst_option_def);
qed
qed
text {*
\medskip
Main induction over the instruction list.
*}
theorem wt_imp_wtl_inst_list:
"\\<forall> pc. (\\<forall>pc'. pc' < length all_ins \\<longrightarrow> wt_instr (all_ins ! pc') G rT phi (length all_ins) pc') \\<longrightarrow>
fits all_ins cert phi \\<longrightarrow>
(\\<exists>l. pc = length l \\<and> all_ins = l@ins) \\<longrightarrow>
pc < length all_ins \\<longrightarrow>
(\\<forall> s. (G \\<turnstile> s <=s (phi!pc)) \\<longrightarrow>
(\\<exists>s'. wtl_inst_list ins G rT s s' cert (length all_ins) pc))"
(is "\\<forall>pc. ?wt \\<longrightarrow> ?fits \\<longrightarrow> ?l pc ins \\<longrightarrow> ?len pc \\<longrightarrow> ?wtl pc ins" is "\\<forall>pc. ?C pc ins" is "?P ins")
proof (induct "?P" "ins")
case Nil
show "?P []" by simp
next
fix i ins'
assume Cons: "?P ins'"
show "?P (i#ins')"
proof (intro allI impI, elim exE conjE)
fix pc s l
assume wt : "\\<forall>pc'. pc' < length all_ins \\<longrightarrow>
wt_instr (all_ins ! pc') G rT phi (length all_ins) pc'"
assume fits: "fits all_ins cert phi"
assume G : "G \\<turnstile> s <=s phi ! pc"
assume l : "pc < length all_ins"
assume pc : "all_ins = l@i#ins'" "pc = length l"
hence i : "all_ins ! pc = i"
by (simp add: nth_append)
from l wt
have "wt_instr (all_ins!pc) G rT phi (length all_ins) pc" by blast
with fits l
obtain s1 where
"wtl_inst_option (all_ins!pc) G rT (phi!pc) s1 cert (length all_ins) pc" and
s1: "G \\<turnstile> s1 <=s phi ! (Suc pc)"
by - (drule wt_instr_imp_wtl_option, assumption+, simp, elim exE conjE, simp)
with fits l
obtain s2 where
o: "wtl_inst_option (all_ins!pc) G rT s s2 cert (length all_ins) pc"
and "G \\<turnstile> s2 <=s s1"
by - (drule wtl_option_mono, assumption+, simp, elim exE conjE, rule that)
with s1
have G': "G \\<turnstile> s2 <=s phi ! (Suc pc)"
by - (rule sup_state_trans, auto)
from Cons
have "?C (Suc pc) ins'" by blast
with wt fits pc
have IH: "?len (Suc pc) \\<longrightarrow> ?wtl (Suc pc) ins'" by auto
show "\\<exists>s'. wtl_inst_list (i#ins') G rT s s' cert (length all_ins) pc"
proof (cases "?len (Suc pc)")
case False
with pc
have "ins' = []" by simp
with i o
show ?thesis by auto
next
case True
with IH
have "?wtl (Suc pc) ins'" by blast
with G'
obtain s' where
"wtl_inst_list ins' G rT s2 s' cert (length all_ins) (Suc pc)"
by - (elim allE impE, auto)
with i o
show ?thesis by auto
qed
qed
qed
lemma fits_imp_wtl_method_complete:
"\\<lbrakk>wt_method G C pTs rT mxl ins phi; fits ins cert phi; wf_prog wf_mb G\\<rbrakk> \\<Longrightarrow> wtl_method G C pTs rT mxl ins cert"
by (simp add: wt_method_def wtl_method_def del: split_paired_Ex)
(rule wt_imp_wtl_inst_list [rulify, elimify], auto simp add: wt_start_def simp del: split_paired_Ex);
lemma wtl_method_complete:
"\\<lbrakk>wt_method G C pTs rT mxl ins phi; wf_prog wf_mb G\\<rbrakk> \\<Longrightarrow> wtl_method G C pTs rT mxl ins (make_cert ins phi)";
proof -;
assume * : "wt_method G C pTs rT mxl ins phi" "wf_prog wf_mb G";
hence "fits ins (make_cert ins phi) phi";
by - (rule fits_make_cert, simp add: wt_method_def);
with *;
show ?thesis; by - (rule fits_imp_wtl_method_complete);
qed;
lemma unique_set:
"(a,b,c,d)\\<in>set l \\<longrightarrow> unique l \\<longrightarrow> (a',b',c',d') \\<in> set l \\<longrightarrow> a = a' \\<longrightarrow> b=b' \\<and> c=c' \\<and> d=d'";
by (induct "l") auto;
lemma unique_epsilon:
"(a,b,c,d)\\<in>set l \\<longrightarrow> unique l \\<longrightarrow> (\\<epsilon> (a',b',c',d'). (a',b',c',d') \\<in> set l \\<and> a'=a) = (a,b,c,d)";
by (auto simp add: unique_set);
theorem wtl_complete: "\\<lbrakk>wt_jvm_prog G Phi\\<rbrakk> \\<Longrightarrow> wtl_jvm_prog G (make_Cert G Phi)";
proof (simp only: wt_jvm_prog_def);
assume wfprog: "wf_prog (\\<lambda>G C (sig,rT,maxl,b). wt_method G C (snd sig) rT maxl b (Phi C sig)) G";
thus ?thesis;
proof (simp add: wtl_jvm_prog_def wf_prog_def wf_cdecl_def wf_mdecl_def, auto);
fix a aa ab b ac ba ad ae bb;
assume 1 : "\\<forall>(C,sc,fs,ms)\\<in>set G.
Ball (set fs) (wf_fdecl G) \\<and>
unique fs \\<and>
(\\<forall>(sig,rT,mb)\\<in>set ms. wf_mhead G sig rT \\<and> (\\<lambda>(maxl,b). wt_method G C (snd sig) rT maxl b (Phi C sig)) mb) \\<and>
unique ms \\<and>
(case sc of None \\<Rightarrow> C = Object
| Some D \\<Rightarrow>
is_class G D \\<and>
(D, C) \\<notin> (subcls1 G)^* \\<and>
(\\<forall>(sig,rT,b)\\<in>set ms. \\<forall>D' rT' b'. method (G, D) sig = Some (D', rT', b') \\<longrightarrow> G\\<turnstile>rT\\<preceq>rT'))"
"(a, aa, ab, b) \\<in> set G";
assume uG : "unique G";
assume b : "((ac, ba), ad, ae, bb) \\<in> set b";
from 1;
show "wtl_method G a ba ad ae bb (make_Cert G Phi a (ac, ba))";
proof (rule bspec [elimify], clarsimp);
assume ub : "unique b";
assume m: "\\<forall>(sig,rT,mb)\\<in>set b. wf_mhead G sig rT \\<and> (\\<lambda>(maxl,b). wt_method G a (snd sig) rT maxl b (Phi a sig)) mb";
from m b;
show ?thesis;
proof (rule bspec [elimify], clarsimp);
assume "wt_method G a ba ad ae bb (Phi a (ac, ba))";
with wfprog uG ub b 1;
show ?thesis;
by - (rule wtl_method_complete [elimify], assumption+, simp add: make_Cert_def unique_epsilon);
qed;
qed;
qed;
qed
lemmas [simp] = split_paired_Ex
end