(* Author: Tobias Nipkow *)
header "Live Variable Analysis"
theory Live imports Vars Big_Step
begin
subsection "Liveness Analysis"
fun L :: "com \<Rightarrow> vname set \<Rightarrow> vname set" where
"L SKIP X = X" |
"L (x ::= a) X = X-{x} \<union> vars a" |
"L (c\<^isub>1; c\<^isub>2) X = (L c\<^isub>1 \<circ> L c\<^isub>2) X" |
"L (IF b THEN c\<^isub>1 ELSE c\<^isub>2) X = vars b \<union> L c\<^isub>1 X \<union> L c\<^isub>2 X" |
"L (WHILE b DO c) X = vars b \<union> X \<union> L c X"
value "show (L (''y'' ::= V ''z''; ''x'' ::= Plus (V ''y'') (V ''z'')) {''x''})"
value "show (L (WHILE Less (V ''x'') (V ''x'') DO ''y'' ::= V ''z'') {''x''})"
fun "kill" :: "com \<Rightarrow> vname set" where
"kill SKIP = {}" |
"kill (x ::= a) = {x}" |
"kill (c\<^isub>1; c\<^isub>2) = kill c\<^isub>1 \<union> kill c\<^isub>2" |
"kill (IF b THEN c\<^isub>1 ELSE c\<^isub>2) = kill c\<^isub>1 \<inter> kill c\<^isub>2" |
"kill (WHILE b DO c) = {}"
fun gen :: "com \<Rightarrow> vname set" where
"gen SKIP = {}" |
"gen (x ::= a) = vars a" |
"gen (c\<^isub>1; c\<^isub>2) = gen c\<^isub>1 \<union> (gen c\<^isub>2 - kill c\<^isub>1)" |
"gen (IF b THEN c\<^isub>1 ELSE c\<^isub>2) = vars b \<union> gen c\<^isub>1 \<union> gen c\<^isub>2" |
"gen (WHILE b DO c) = vars b \<union> gen c"
lemma L_gen_kill: "L c X = (X - kill c) \<union> gen c"
by(induct c arbitrary:X) auto
lemma L_While_subset: "L c (L (WHILE b DO c) X) \<subseteq> L (WHILE b DO c) X"
by(auto simp add:L_gen_kill)
subsection "Soundness"
theorem L_sound:
"(c,s) \<Rightarrow> s' \<Longrightarrow> s = t on L c X \<Longrightarrow>
\<exists> t'. (c,t) \<Rightarrow> t' & s' = t' on X"
proof (induction arbitrary: X t rule: big_step_induct)
case Skip then show ?case by auto
next
case Assign then show ?case
by (auto simp: ball_Un)
next
case (Semi c1 s1 s2 c2 s3 X t1)
from Semi.IH(1) Semi.prems obtain t2 where
t12: "(c1, t1) \<Rightarrow> t2" and s2t2: "s2 = t2 on L c2 X"
by simp blast
from Semi.IH(2)[OF s2t2] obtain t3 where
t23: "(c2, t2) \<Rightarrow> t3" and s3t3: "s3 = t3 on X"
by auto
show ?case using t12 t23 s3t3 by auto
next
case (IfTrue b s c1 s' c2)
hence "s = t on vars b" "s = t on L c1 X" by auto
from bval_eq_if_eq_on_vars[OF this(1)] IfTrue(1) have "bval b t" by simp
from IfTrue(3)[OF `s = t on L c1 X`] obtain t' where
"(c1, t) \<Rightarrow> t'" "s' = t' on X" by auto
thus ?case using `bval b t` by auto
next
case (IfFalse b s c2 s' c1)
hence "s = t on vars b" "s = t on L c2 X" by auto
from bval_eq_if_eq_on_vars[OF this(1)] IfFalse(1) have "~bval b t" by simp
from IfFalse(3)[OF `s = t on L c2 X`] obtain t' where
"(c2, t) \<Rightarrow> t'" "s' = t' on X" by auto
thus ?case using `~bval b t` by auto
next
case (WhileFalse b s c)
hence "~ bval b t" by (auto simp: ball_Un) (metis bval_eq_if_eq_on_vars)
thus ?case using WhileFalse(2) by auto
next
case (WhileTrue b s1 c s2 s3 X t1)
let ?w = "WHILE b DO c"
from `bval b s1` WhileTrue(6) have "bval b t1"
by (auto simp: ball_Un) (metis bval_eq_if_eq_on_vars)
have "s1 = t1 on L c (L ?w X)" using L_While_subset WhileTrue.prems
by (blast)
from WhileTrue.IH(1)[OF this] obtain t2 where
"(c, t1) \<Rightarrow> t2" "s2 = t2 on L ?w X" by auto
from WhileTrue.IH(2)[OF this(2)] obtain t3 where "(?w,t2) \<Rightarrow> t3" "s3 = t3 on X"
by auto
with `bval b t1` `(c, t1) \<Rightarrow> t2` show ?case by auto
qed
subsection "Program Optimization"
text{* Burying assignments to dead variables: *}
fun bury :: "com \<Rightarrow> vname set \<Rightarrow> com" where
"bury SKIP X = SKIP" |
"bury (x ::= a) X = (if x:X then x::= a else SKIP)" |
"bury (c\<^isub>1; c\<^isub>2) X = (bury c\<^isub>1 (L c\<^isub>2 X); bury c\<^isub>2 X)" |
"bury (IF b THEN c\<^isub>1 ELSE c\<^isub>2) X = IF b THEN bury c\<^isub>1 X ELSE bury c\<^isub>2 X" |
"bury (WHILE b DO c) X = WHILE b DO bury c (vars b \<union> X \<union> L c X)"
text{* We could prove the analogous lemma to @{thm[source]L_sound}, and the
proof would be very similar. However, we phrase it as a semantics
preservation property: *}
theorem bury_sound:
"(c,s) \<Rightarrow> s' \<Longrightarrow> s = t on L c X \<Longrightarrow>
\<exists> t'. (bury c X,t) \<Rightarrow> t' & s' = t' on X"
proof (induction arbitrary: X t rule: big_step_induct)
case Skip then show ?case by auto
next
case Assign then show ?case
by (auto simp: ball_Un)
next
case (Semi c1 s1 s2 c2 s3 X t1)
from Semi.IH(1) Semi.prems obtain t2 where
t12: "(bury c1 (L c2 X), t1) \<Rightarrow> t2" and s2t2: "s2 = t2 on L c2 X"
by simp blast
from Semi.IH(2)[OF s2t2] obtain t3 where
t23: "(bury c2 X, t2) \<Rightarrow> t3" and s3t3: "s3 = t3 on X"
by auto
show ?case using t12 t23 s3t3 by auto
next
case (IfTrue b s c1 s' c2)
hence "s = t on vars b" "s = t on L c1 X" by auto
from bval_eq_if_eq_on_vars[OF this(1)] IfTrue(1) have "bval b t" by simp
from IfTrue(3)[OF `s = t on L c1 X`] obtain t' where
"(bury c1 X, t) \<Rightarrow> t'" "s' =t' on X" by auto
thus ?case using `bval b t` by auto
next
case (IfFalse b s c2 s' c1)
hence "s = t on vars b" "s = t on L c2 X" by auto
from bval_eq_if_eq_on_vars[OF this(1)] IfFalse(1) have "~bval b t" by simp
from IfFalse(3)[OF `s = t on L c2 X`] obtain t' where
"(bury c2 X, t) \<Rightarrow> t'" "s' = t' on X" by auto
thus ?case using `~bval b t` by auto
next
case (WhileFalse b s c)
hence "~ bval b t" by (auto simp: ball_Un) (metis bval_eq_if_eq_on_vars)
thus ?case using WhileFalse(2) by auto
next
case (WhileTrue b s1 c s2 s3 X t1)
let ?w = "WHILE b DO c"
from `bval b s1` WhileTrue(6) have "bval b t1"
by (auto simp: ball_Un) (metis bval_eq_if_eq_on_vars)
have "s1 = t1 on L c (L ?w X)"
using L_While_subset WhileTrue.prems by blast
from WhileTrue.IH(1)[OF this] obtain t2 where
"(bury c (L ?w X), t1) \<Rightarrow> t2" "s2 = t2 on L ?w X" by auto
from WhileTrue.IH(2)[OF this(2)] obtain t3
where "(bury ?w X,t2) \<Rightarrow> t3" "s3 = t3 on X"
by auto
with `bval b t1` `(bury c (L ?w X), t1) \<Rightarrow> t2` show ?case by auto
qed
corollary final_bury_sound: "(c,s) \<Rightarrow> s' \<Longrightarrow> (bury c UNIV,s) \<Rightarrow> s'"
using bury_sound[of c s s' UNIV]
by (auto simp: fun_eq_iff[symmetric])
text{* Now the opposite direction. *}
lemma SKIP_bury[simp]:
"SKIP = bury c X \<longleftrightarrow> c = SKIP | (EX x a. c = x::=a & x \<notin> X)"
by (cases c) auto
lemma Assign_bury[simp]: "x::=a = bury c X \<longleftrightarrow> c = x::=a & x : X"
by (cases c) auto
lemma Semi_bury[simp]: "bc\<^isub>1;bc\<^isub>2 = bury c X \<longleftrightarrow>
(EX c\<^isub>1 c\<^isub>2. c = c\<^isub>1;c\<^isub>2 & bc\<^isub>2 = bury c\<^isub>2 X & bc\<^isub>1 = bury c\<^isub>1 (L c\<^isub>2 X))"
by (cases c) auto
lemma If_bury[simp]: "IF b THEN bc1 ELSE bc2 = bury c X \<longleftrightarrow>
(EX c1 c2. c = IF b THEN c1 ELSE c2 &
bc1 = bury c1 X & bc2 = bury c2 X)"
by (cases c) auto
lemma While_bury[simp]: "WHILE b DO bc' = bury c X \<longleftrightarrow>
(EX c'. c = WHILE b DO c' & bc' = bury c' (vars b \<union> X \<union> L c X))"
by (cases c) auto
theorem bury_sound2:
"(bury c X,s) \<Rightarrow> s' \<Longrightarrow> s = t on L c X \<Longrightarrow>
\<exists> t'. (c,t) \<Rightarrow> t' & s' = t' on X"
proof (induction "bury c X" s s' arbitrary: c X t rule: big_step_induct)
case Skip then show ?case by auto
next
case Assign then show ?case
by (auto simp: ball_Un)
next
case (Semi bc1 s1 s2 bc2 s3 c X t1)
then obtain c1 c2 where c: "c = c1;c2"
and bc2: "bc2 = bury c2 X" and bc1: "bc1 = bury c1 (L c2 X)" by auto
note IH = Semi.hyps(2,4)
from IH(1)[OF bc1, of t1] Semi.prems c obtain t2 where
t12: "(c1, t1) \<Rightarrow> t2" and s2t2: "s2 = t2 on L c2 X" by auto
from IH(2)[OF bc2 s2t2] obtain t3 where
t23: "(c2, t2) \<Rightarrow> t3" and s3t3: "s3 = t3 on X"
by auto
show ?case using c t12 t23 s3t3 by auto
next
case (IfTrue b s bc1 s' bc2)
then obtain c1 c2 where c: "c = IF b THEN c1 ELSE c2"
and bc1: "bc1 = bury c1 X" and bc2: "bc2 = bury c2 X" by auto
have "s = t on vars b" "s = t on L c1 X" using IfTrue.prems c by auto
from bval_eq_if_eq_on_vars[OF this(1)] IfTrue(1) have "bval b t" by simp
note IH = IfTrue.hyps(3)
from IH[OF bc1 `s = t on L c1 X`] obtain t' where
"(c1, t) \<Rightarrow> t'" "s' =t' on X" by auto
thus ?case using c `bval b t` by auto
next
case (IfFalse b s bc2 s' bc1)
then obtain c1 c2 where c: "c = IF b THEN c1 ELSE c2"
and bc1: "bc1 = bury c1 X" and bc2: "bc2 = bury c2 X" by auto
have "s = t on vars b" "s = t on L c2 X" using IfFalse.prems c by auto
from bval_eq_if_eq_on_vars[OF this(1)] IfFalse(1) have "~bval b t" by simp
note IH = IfFalse.hyps(3)
from IH[OF bc2 `s = t on L c2 X`] obtain t' where
"(c2, t) \<Rightarrow> t'" "s' =t' on X" by auto
thus ?case using c `~bval b t` by auto
next
case (WhileFalse b s c)
hence "~ bval b t" by (auto simp: ball_Un dest: bval_eq_if_eq_on_vars)
thus ?case using WhileFalse by auto
next
case (WhileTrue b s1 bc' s2 s3 c X t1)
then obtain c' where c: "c = WHILE b DO c'"
and bc': "bc' = bury c' (vars b \<union> X \<union> L c' X)" by auto
let ?w = "WHILE b DO c'"
from `bval b s1` WhileTrue.prems c have "bval b t1"
by (auto simp: ball_Un) (metis bval_eq_if_eq_on_vars)
note IH = WhileTrue.hyps(3,5)
have "s1 = t1 on L c' (L ?w X)"
using L_While_subset WhileTrue.prems c by blast
with IH(1)[OF bc', of t1] obtain t2 where
"(c', t1) \<Rightarrow> t2" "s2 = t2 on L ?w X" by auto
from IH(2)[OF WhileTrue.hyps(6), of t2] c this(2) obtain t3
where "(?w,t2) \<Rightarrow> t3" "s3 = t3 on X"
by auto
with `bval b t1` `(c', t1) \<Rightarrow> t2` c show ?case by auto
qed
corollary final_bury_sound2: "(bury c UNIV,s) \<Rightarrow> s' \<Longrightarrow> (c,s) \<Rightarrow> s'"
using bury_sound2[of c UNIV]
by (auto simp: fun_eq_iff[symmetric])
corollary bury_iff: "(bury c UNIV,s) \<Rightarrow> s' \<longleftrightarrow> (c,s) \<Rightarrow> s'"
by(metis final_bury_sound final_bury_sound2)
end