theory Live imports Natural
begin
text{* Which variables/locations does an expression depend on?
Any set of variables that completely determine the value of the expression,
in the worst case all locations: *}
consts Dep :: "((loc => 'a) => 'b) => loc set"
specification (Dep)
dep_on: "(∀x∈Dep e. s x = t x) ==> e s = e t"
by(rule_tac x="%x. UNIV" in exI)(simp add: expand_fun_eq[symmetric])
text{* The following definition of @{const Dep} looks very tempting
@{prop"Dep e = {a. EX s t. (ALL x. x≠a --> s x = t x) ∧ e s ≠ e t}"}
but does not work in case @{text e} depends on an infinite set of variables.
For example, if @{term"e s"} tests if @{text s} is 0 at infinitely many locations. Then @{term"Dep e"} incorrectly yields the empty set!
If we had a concrete representation of expressions, we would simply write
a recursive free-variables function.
*}
primrec L :: "com => loc set => loc set" where
"L SKIP A = A" |
"L (x :== e) A = A-{x} ∪ Dep e" |
"L (c1; c2) A = (L c1 o L c2) A" |
"L (IF b THEN c1 ELSE c2) A = Dep b ∪ L c1 A ∪ L c2 A" |
"L (WHILE b DO c) A = Dep b ∪ A ∪ L c A"
primrec "kill" :: "com => loc set" where
"kill SKIP = {}" |
"kill (x :== e) = {x}" |
"kill (c1; c2) = kill c1 ∪ kill c2" |
"kill (IF b THEN c1 ELSE c2) = Dep b ∪ kill c1 ∩ kill c2" |
"kill (WHILE b DO c) = {}"
primrec gen :: "com => loc set" where
"gen SKIP = {}" |
"gen (x :== e) = Dep e" |
"gen (c1; c2) = gen c1 ∪ (gen c2-kill c1)" |
"gen (IF b THEN c1 ELSE c2) = Dep b ∪ gen c1 ∪ gen c2" |
"gen (WHILE b DO c) = Dep b ∪ gen c"
lemma L_gen_kill: "L c A = gen c ∪ (A - kill c)"
by(induct c arbitrary:A) auto
lemma L_idemp: "L c (L c A) ⊆ L c A"
by(fastsimp simp add:L_gen_kill)
theorem L_sound: "∀ x ∈ L c A. s x = t x ==> 〈c,s〉 -->\<^sub>c s' ==> 〈c,t〉 -->\<^sub>c t' ==>
∀x∈A. s' x = t' x"
proof (induct c arbitrary: A s t s' t')
case SKIP then show ?case by auto
next
case (Assign x e) then show ?case
by (auto simp:update_def ball_Un dest!: dep_on)
next
case (Semi c1 c2)
from Semi(4) obtain s'' where s1: "〈c1,s〉 -->\<^sub>c s''" and s2: "〈c2,s''〉 -->\<^sub>c s'"
by auto
from Semi(5) obtain t'' where t1: "〈c1,t〉 -->\<^sub>c t''" and t2: "〈c2,t''〉 -->\<^sub>c t'"
by auto
show ?case using Semi(1)[OF _ s1 t1] Semi(2)[OF _ s2 t2] Semi(3) by fastsimp
next
case (Cond b c1 c2)
show ?case
proof cases
assume "b s"
hence s: "〈c1,s〉 -->\<^sub>c s'" using Cond(4) by simp
have "b t" using `b s` Cond(3) by (simp add: ball_Un)(blast dest: dep_on)
hence t: "〈c1,t〉 -->\<^sub>c t'" using Cond(5) by auto
show ?thesis using Cond(1)[OF _ s t] Cond(3) by fastsimp
next
assume "¬ b s"
hence s: "〈c2,s〉 -->\<^sub>c s'" using Cond(4) by auto
have "¬ b t" using `¬ b s` Cond(3) by (simp add: ball_Un)(blast dest: dep_on)
hence t: "〈c2,t〉 -->\<^sub>c t'" using Cond(5) by auto
show ?thesis using Cond(2)[OF _ s t] Cond(3) by fastsimp
qed
next
case (While b c) note IH = this
{ fix cw
have "〈cw,s〉 -->\<^sub>c s' ==> cw = (While b c) ==> 〈cw,t〉 -->\<^sub>c t' ==>
∀ x ∈ L cw A. s x = t x ==> ∀x∈A. s' x = t' x"
proof (induct arbitrary: t A pred:evalc)
case WhileFalse
have "¬ b t" using WhileFalse by (simp add: ball_Un)(blast dest:dep_on)
then have "t' = t" using WhileFalse by auto
then show ?case using WhileFalse by auto
next
case (WhileTrue _ s _ s'' s')
have "〈c,s〉 -->\<^sub>c s''" using WhileTrue(2,6) by simp
have "b t" using WhileTrue by (simp add: ball_Un)(blast dest:dep_on)
then obtain t'' where "〈c,t〉 -->\<^sub>c t''" and "〈While b c,t''〉 -->\<^sub>c t'"
using WhileTrue(6,7) by auto
have "∀x∈Dep b ∪ A ∪ L c A. s'' x = t'' x"
using IH(1)[OF _ `〈c,s〉 -->\<^sub>c s''` `〈c,t〉 -->\<^sub>c t''`] WhileTrue(6,8)
by (auto simp:L_gen_kill)
then have "∀x∈L (While b c) A. s'' x = t'' x" by auto
then show ?case using WhileTrue(5,6) `〈While b c,t''〉 -->\<^sub>c t'` by metis
qed auto }
-- "a terser version"
{ let ?w = "While b c"
have "〈?w,s〉 -->\<^sub>c s' ==> 〈?w,t〉 -->\<^sub>c t' ==>
∀ x ∈ L ?w A. s x = t x ==> ∀x∈A. s' x = t' x"
proof (induct ?w s s' arbitrary: t A pred:evalc)
case WhileFalse
have "¬ b t" using WhileFalse by (simp add: ball_Un)(blast dest:dep_on)
then have "t' = t" using WhileFalse by auto
then show ?case using WhileFalse by simp
next
case (WhileTrue s s'' s')
have "b t" using WhileTrue by (simp add: ball_Un)(blast dest:dep_on)
then obtain t'' where "〈c,t〉 -->\<^sub>c t''" and "〈While b c,t''〉 -->\<^sub>c t'"
using WhileTrue(6,7) by auto
have "∀x∈Dep b ∪ A ∪ L c A. s'' x = t'' x"
using IH(1)[OF _ `〈c,s〉 -->\<^sub>c s''` `〈c,t〉 -->\<^sub>c t''`] WhileTrue(7)
by (auto simp:L_gen_kill)
then have "∀x∈L (While b c) A. s'' x = t'' x" by auto
then show ?case using WhileTrue(5) `〈While b c,t''〉 -->\<^sub>c t'` by metis
qed }
from this[OF IH(3) IH(4,2)] show ?case by metis
qed
primrec bury :: "com => loc set => com" where
"bury SKIP _ = SKIP" |
"bury (x :== e) A = (if x:A then x:== e else SKIP)" |
"bury (c1; c2) A = (bury c1 (L c2 A); bury c2 A)" |
"bury (IF b THEN c1 ELSE c2) A = (IF b THEN bury c1 A ELSE bury c2 A)" |
"bury (WHILE b DO c) A = (WHILE b DO bury c (Dep b ∪ A ∪ L c A))"
theorem bury_sound:
"∀ x ∈ L c A. s x = t x ==> 〈c,s〉 -->\<^sub>c s' ==> 〈bury c A,t〉 -->\<^sub>c t' ==>
∀x∈A. s' x = t' x"
proof (induct c arbitrary: A s t s' t')
case SKIP then show ?case by auto
next
case (Assign x e) then show ?case
by (auto simp:update_def ball_Un split:split_if_asm dest!: dep_on)
next
case (Semi c1 c2)
from Semi(4) obtain s'' where s1: "〈c1,s〉 -->\<^sub>c s''" and s2: "〈c2,s''〉 -->\<^sub>c s'"
by auto
from Semi(5) obtain t'' where t1: "〈bury c1 (L c2 A),t〉 -->\<^sub>c t''" and t2: "〈bury c2 A,t''〉 -->\<^sub>c t'"
by auto
show ?case using Semi(1)[OF _ s1 t1] Semi(2)[OF _ s2 t2] Semi(3) by fastsimp
next
case (Cond b c1 c2)
show ?case
proof cases
assume "b s"
hence s: "〈c1,s〉 -->\<^sub>c s'" using Cond(4) by simp
have "b t" using `b s` Cond(3) by (simp add: ball_Un)(blast dest: dep_on)
hence t: "〈bury c1 A,t〉 -->\<^sub>c t'" using Cond(5) by auto
show ?thesis using Cond(1)[OF _ s t] Cond(3) by fastsimp
next
assume "¬ b s"
hence s: "〈c2,s〉 -->\<^sub>c s'" using Cond(4) by auto
have "¬ b t" using `¬ b s` Cond(3) by (simp add: ball_Un)(blast dest: dep_on)
hence t: "〈bury c2 A,t〉 -->\<^sub>c t'" using Cond(5) by auto
show ?thesis using Cond(2)[OF _ s t] Cond(3) by fastsimp
qed
next
case (While b c) note IH = this
{ fix cw
have "〈cw,s〉 -->\<^sub>c s' ==> cw = (While b c) ==> 〈bury cw A,t〉 -->\<^sub>c t' ==>
∀ x ∈ L cw A. s x = t x ==> ∀x∈A. s' x = t' x"
proof (induct arbitrary: t A pred:evalc)
case WhileFalse
have "¬ b t" using WhileFalse by (simp add: ball_Un)(blast dest:dep_on)
then have "t' = t" using WhileFalse by auto
then show ?case using WhileFalse by auto
next
case (WhileTrue _ s _ s'' s')
have "〈c,s〉 -->\<^sub>c s''" using WhileTrue(2,6) by simp
have "b t" using WhileTrue by (simp add: ball_Un)(blast dest:dep_on)
then obtain t'' where tt'': "〈bury c (Dep b ∪ A ∪ L c A),t〉 -->\<^sub>c t''"
and "〈bury (While b c) A,t''〉 -->\<^sub>c t'"
using WhileTrue(6,7) by auto
have "∀x∈Dep b ∪ A ∪ L c A. s'' x = t'' x"
using IH(1)[OF _ `〈c,s〉 -->\<^sub>c s''` tt''] WhileTrue(6,8)
by (auto simp:L_gen_kill)
moreover then have "∀x∈L (While b c) A. s'' x = t'' x" by auto
ultimately show ?case
using WhileTrue(5,6) `〈bury (While b c) A,t''〉 -->\<^sub>c t'` by metis
qed auto }
{ let ?w = "While b c"
have "〈?w,s〉 -->\<^sub>c s' ==> 〈bury ?w A,t〉 -->\<^sub>c t' ==>
∀ x ∈ L ?w A. s x = t x ==> ∀x∈A. s' x = t' x"
proof (induct ?w s s' arbitrary: t A pred:evalc)
case WhileFalse
have "¬ b t" using WhileFalse by (simp add: ball_Un)(blast dest:dep_on)
then have "t' = t" using WhileFalse by auto
then show ?case using WhileFalse by simp
next
case (WhileTrue s s'' s')
have "b t" using WhileTrue by (simp add: ball_Un)(blast dest:dep_on)
then obtain t'' where tt'': "〈bury c (Dep b ∪ A ∪ L c A),t〉 -->\<^sub>c t''"
and "〈bury (While b c) A,t''〉 -->\<^sub>c t'"
using WhileTrue(6,7) by auto
have "∀x∈Dep b ∪ A ∪ L c A. s'' x = t'' x"
using IH(1)[OF _ `〈c,s〉 -->\<^sub>c s''` tt''] WhileTrue(7)
by (auto simp:L_gen_kill)
then have "∀x∈L (While b c) A. s'' x = t'' x" by auto
then show ?case
using WhileTrue(5) `〈bury (While b c) A,t''〉 -->\<^sub>c t'` by metis
qed }
from this[OF IH(3) IH(4,2)] show ?case by metis
qed
end