Theory Fold

theory Fold
imports Sem_Equiv Vars
header "Constant Folding"

theory Fold imports Sem_Equiv Vars begin

subsection "Simple folding of arithmetic expressions"

type_synonym
tab = "vname => val option"

fun afold :: "aexp => tab => aexp" where
"afold (N n) _ = N n" |
"afold (V x) t = (case t x of None => V x | Some k => N k)" |
"afold (Plus e1 e2) t = (case (afold e1 t, afold e2 t) of
(N n1, N n2) => N(n1+n2) | (e1',e2') => Plus e1' e2')"


definition "approx t s <-> (ALL x k. t x = Some k --> s x = k)"

theorem aval_afold[simp]:
assumes "approx t s"
shows "aval (afold a t) s = aval a s"
using assms
by (induct a) (auto simp: approx_def split: aexp.split option.split)

theorem aval_afold_N:
assumes "approx t s"
shows "afold a t = N n ==> aval a s = n"
by (metis assms aval.simps(1) aval_afold)

definition
"merge t1 t2 = (λm. if t1 m = t2 m then t1 m else None)"

primrec "defs" :: "com => tab => tab" where
"defs SKIP t = t" |
"defs (x ::= a) t =
(case afold a t of N k => t(x \<mapsto> k) | _ => t(x:=None))"
|
"defs (c1;;c2) t = (defs c2 o defs c1) t" |
"defs (IF b THEN c1 ELSE c2) t = merge (defs c1 t) (defs c2 t)" |
"defs (WHILE b DO c) t = t |` (-lvars c)"

primrec fold where
"fold SKIP _ = SKIP" |
"fold (x ::= a) t = (x ::= (afold a t))" |
"fold (c1;;c2) t = (fold c1 t;; fold c2 (defs c1 t))" |
"fold (IF b THEN c1 ELSE c2) t = IF b THEN fold c1 t ELSE fold c2 t" |
"fold (WHILE b DO c) t = WHILE b DO fold c (t |` (-lvars c))"

lemma approx_merge:
"approx t1 s ∨ approx t2 s ==> approx (merge t1 t2) s"
by (fastforce simp: merge_def approx_def)

lemma approx_map_le:
"approx t2 s ==> t1 ⊆m t2 ==> approx t1 s"
by (clarsimp simp: approx_def map_le_def dom_def)

lemma restrict_map_le [intro!, simp]: "t |` S ⊆m t"
by (clarsimp simp: restrict_map_def map_le_def)

lemma merge_restrict:
assumes "t1 |` S = t |` S"
assumes "t2 |` S = t |` S"
shows "merge t1 t2 |` S = t |` S"
proof -
from assms
have "∀x. (t1 |` S) x = (t |` S) x"
and "∀x. (t2 |` S) x = (t |` S) x" by auto
thus ?thesis
by (auto simp: merge_def restrict_map_def
split: if_splits)
qed


lemma defs_restrict:
"defs c t |` (- lvars c) = t |` (- lvars c)"
proof (induction c arbitrary: t)
case (Seq c1 c2)
hence "defs c1 t |` (- lvars c1) = t |` (- lvars c1)"
by simp
hence "defs c1 t |` (- lvars c1) |` (-lvars c2) =
t |` (- lvars c1) |` (-lvars c2)"
by simp
moreover
from Seq
have "defs c2 (defs c1 t) |` (- lvars c2) =
defs c1 t |` (- lvars c2)"

by simp
hence "defs c2 (defs c1 t) |` (- lvars c2) |` (- lvars c1) =
defs c1 t |` (- lvars c2) |` (- lvars c1)"

by simp
ultimately
show ?case by (clarsimp simp: Int_commute)
next
case (If b c1 c2)
hence "defs c1 t |` (- lvars c1) = t |` (- lvars c1)" by simp
hence "defs c1 t |` (- lvars c1) |` (-lvars c2) =
t |` (- lvars c1) |` (-lvars c2)"
by simp
moreover
from If
have "defs c2 t |` (- lvars c2) = t |` (- lvars c2)" by simp
hence "defs c2 t |` (- lvars c2) |` (-lvars c1) =
t |` (- lvars c2) |` (-lvars c1)"
by simp
ultimately
show ?case by (auto simp: Int_commute intro: merge_restrict)
qed (auto split: aexp.split)


lemma big_step_pres_approx:
"(c,s) => s' ==> approx t s ==> approx (defs c t) s'"
proof (induction arbitrary: t rule: big_step_induct)
case Skip thus ?case by simp
next
case Assign
thus ?case
by (clarsimp simp: aval_afold_N approx_def split: aexp.split)
next
case (Seq c1 s1 s2 c2 s3)
have "approx (defs c1 t) s2" by (rule Seq.IH(1)[OF Seq.prems])
hence "approx (defs c2 (defs c1 t)) s3" by (rule Seq.IH(2))
thus ?case by simp
next
case (IfTrue b s c1 s')
hence "approx (defs c1 t) s'" by simp
thus ?case by (simp add: approx_merge)
next
case (IfFalse b s c2 s')
hence "approx (defs c2 t) s'" by simp
thus ?case by (simp add: approx_merge)
next
case WhileFalse
thus ?case by (simp add: approx_def restrict_map_def)
next
case (WhileTrue b s1 c s2 s3)
hence "approx (defs c t) s2" by simp
with WhileTrue
have "approx (defs c t |` (-lvars c)) s3" by simp
thus ?case by (simp add: defs_restrict)
qed


lemma big_step_pres_approx_restrict:
"(c,s) => s' ==> approx (t |` (-lvars c)) s ==> approx (t |` (-lvars c)) s'"
proof (induction arbitrary: t rule: big_step_induct)
case Assign
thus ?case by (clarsimp simp: approx_def)
next
case (Seq c1 s1 s2 c2 s3)
hence "approx (t |` (-lvars c2) |` (-lvars c1)) s1"
by (simp add: Int_commute)
hence "approx (t |` (-lvars c2) |` (-lvars c1)) s2"
by (rule Seq)
hence "approx (t |` (-lvars c1) |` (-lvars c2)) s2"
by (simp add: Int_commute)
hence "approx (t |` (-lvars c1) |` (-lvars c2)) s3"
by (rule Seq)
thus ?case by simp
next
case (IfTrue b s c1 s' c2)
hence "approx (t |` (-lvars c2) |` (-lvars c1)) s"
by (simp add: Int_commute)
hence "approx (t |` (-lvars c2) |` (-lvars c1)) s'"
by (rule IfTrue)
thus ?case by (simp add: Int_commute)
next
case (IfFalse b s c2 s' c1)
hence "approx (t |` (-lvars c1) |` (-lvars c2)) s"
by simp
hence "approx (t |` (-lvars c1) |` (-lvars c2)) s'"
by (rule IfFalse)
thus ?case by simp
qed auto


declare assign_simp [simp]

lemma approx_eq:
"approx t \<Turnstile> c ∼ fold c t"
proof (induction c arbitrary: t)
case SKIP show ?case by simp
next
case Assign
show ?case by (simp add: equiv_up_to_def)
next
case Seq
thus ?case by (auto intro!: equiv_up_to_seq big_step_pres_approx)
next
case If
thus ?case by (auto intro!: equiv_up_to_if_weak)
next
case (While b c)
hence "approx (t |` (- lvars c)) \<Turnstile>
WHILE b DO c ∼ WHILE b DO fold c (t |` (- lvars c))"

by (auto intro: equiv_up_to_while_weak big_step_pres_approx_restrict)
thus ?case
by (auto intro: equiv_up_to_weaken approx_map_le)
qed


lemma approx_empty [simp]:
"approx empty = (λ_. True)"
by (auto simp: approx_def)


theorem constant_folding_equiv:
"fold c empty ∼ c"
using approx_eq [of empty c]
by (simp add: equiv_up_to_True sim_sym)



subsection {* More ambitious folding including boolean expressions *}


fun bfold :: "bexp => tab => bexp" where
"bfold (Less a1 a2) t = less (afold a1 t) (afold a2 t)" |
"bfold (And b1 b2) t = and (bfold b1 t) (bfold b2 t)" |
"bfold (Not b) t = not(bfold b t)" |
"bfold (Bc bc) _ = Bc bc"

theorem bval_bfold [simp]:
assumes "approx t s"
shows "bval (bfold b t) s = bval b s"
using assms by (induct b) auto

lemma not_Bc [simp]: "not (Bc v) = Bc (¬v)"
by (cases v) auto

lemma not_Bc_eq [simp]: "(not b = Bc v) = (b = Bc (¬v))"
by (cases b) auto

lemma and_Bc_eq:
"(and a b = Bc v) =
(a = Bc False ∧ ¬v ∨ b = Bc False ∧ ¬v ∨
(∃v1 v2. a = Bc v1 ∧ b = Bc v2 ∧ v = v1 ∧ v2))"

by (rule and.induct) auto

lemma less_Bc_eq [simp]:
"(less a b = Bc v) = (∃n1 n2. a = N n1 ∧ b = N n2 ∧ v = (n1 < n2))"
by (rule less.induct) auto

theorem bval_bfold_Bc:
assumes "approx t s"
shows "bfold b t = Bc v ==> bval b s = v"
using assms
by (induct b arbitrary: v)
(auto simp: approx_def and_Bc_eq aval_afold_N
split: bexp.splits bool.splits)


primrec "bdefs" :: "com => tab => tab" where
"bdefs SKIP t = t" |
"bdefs (x ::= a) t =
(case afold a t of N k => t(x \<mapsto> k) | _ => t(x:=None))"
|
"bdefs (c1;;c2) t = (bdefs c2 o bdefs c1) t" |
"bdefs (IF b THEN c1 ELSE c2) t = (case bfold b t of
Bc True => bdefs c1 t
| Bc False => bdefs c2 t
| _ => merge (bdefs c1 t) (bdefs c2 t))"
|
"bdefs (WHILE b DO c) t = t |` (-lvars c)"


primrec fold' where
"fold' SKIP _ = SKIP" |
"fold' (x ::= a) t = (x ::= (afold a t))" |
"fold' (c1;;c2) t = (fold' c1 t;; fold' c2 (bdefs c1 t))" |
"fold' (IF b THEN c1 ELSE c2) t = (case bfold b t of
Bc True => fold' c1 t
| Bc False => fold' c2 t
| _ => IF bfold b t THEN fold' c1 t ELSE fold' c2 t)"
|
"fold' (WHILE b DO c) t = (case bfold b t of
Bc False => SKIP
| _ => WHILE bfold b (t |` (-lvars c)) DO fold' c (t |` (-lvars c)))"



lemma bdefs_restrict:
"bdefs c t |` (- lvars c) = t |` (- lvars c)"
proof (induction c arbitrary: t)
case (Seq c1 c2)
hence "bdefs c1 t |` (- lvars c1) = t |` (- lvars c1)"
by simp
hence "bdefs c1 t |` (- lvars c1) |` (-lvars c2) =
t |` (- lvars c1) |` (-lvars c2)"
by simp
moreover
from Seq
have "bdefs c2 (bdefs c1 t) |` (- lvars c2) =
bdefs c1 t |` (- lvars c2)"

by simp
hence "bdefs c2 (bdefs c1 t) |` (- lvars c2) |` (- lvars c1) =
bdefs c1 t |` (- lvars c2) |` (- lvars c1)"

by simp
ultimately
show ?case by (clarsimp simp: Int_commute)
next
case (If b c1 c2)
hence "bdefs c1 t |` (- lvars c1) = t |` (- lvars c1)" by simp
hence "bdefs c1 t |` (- lvars c1) |` (-lvars c2) =
t |` (- lvars c1) |` (-lvars c2)"
by simp
moreover
from If
have "bdefs c2 t |` (- lvars c2) = t |` (- lvars c2)" by simp
hence "bdefs c2 t |` (- lvars c2) |` (-lvars c1) =
t |` (- lvars c2) |` (-lvars c1)"
by simp
ultimately
show ?case
by (auto simp: Int_commute intro: merge_restrict
split: bexp.splits bool.splits)
qed (auto split: aexp.split bexp.split bool.split)


lemma big_step_pres_approx_b:
"(c,s) => s' ==> approx t s ==> approx (bdefs c t) s'"
proof (induction arbitrary: t rule: big_step_induct)
case Skip thus ?case by simp
next
case Assign
thus ?case
by (clarsimp simp: aval_afold_N approx_def split: aexp.split)
next
case (Seq c1 s1 s2 c2 s3)
have "approx (bdefs c1 t) s2" by (rule Seq.IH(1)[OF Seq.prems])
hence "approx (bdefs c2 (bdefs c1 t)) s3" by (rule Seq.IH(2))
thus ?case by simp
next
case (IfTrue b s c1 s')
hence "approx (bdefs c1 t) s'" by simp
thus ?case using `bval b s` `approx t s`
by (clarsimp simp: approx_merge bval_bfold_Bc
split: bexp.splits bool.splits)
next
case (IfFalse b s c2 s')
hence "approx (bdefs c2 t) s'" by simp
thus ?case using `¬bval b s` `approx t s`
by (clarsimp simp: approx_merge bval_bfold_Bc
split: bexp.splits bool.splits)
next
case WhileFalse
thus ?case
by (clarsimp simp: bval_bfold_Bc approx_def restrict_map_def
split: bexp.splits bool.splits)
next
case (WhileTrue b s1 c s2 s3)
hence "approx (bdefs c t) s2" by simp
with WhileTrue
have "approx (bdefs c t |` (- lvars c)) s3"
by simp
thus ?case
by (simp add: bdefs_restrict)
qed

lemma fold'_equiv:
"approx t \<Turnstile> c ∼ fold' c t"
proof (induction c arbitrary: t)
case SKIP show ?case by simp
next
case Assign
thus ?case by (simp add: equiv_up_to_def)
next
case Seq
thus ?case by (auto intro!: equiv_up_to_seq big_step_pres_approx_b)
next
case (If b c1 c2)
hence "approx t \<Turnstile> IF b THEN c1 ELSE c2 ∼
IF bfold b t THEN fold' c1 t ELSE fold' c2 t"

by (auto intro: equiv_up_to_if_weak simp: bequiv_up_to_def)
thus ?case using If
by (fastforce simp: bval_bfold_Bc split: bexp.splits bool.splits
intro: equiv_up_to_trans)
next
case (While b c)
hence "approx (t |` (- lvars c)) \<Turnstile>
WHILE b DO c ∼
WHILE bfold b (t |` (- lvars c))
DO fold' c (t |` (- lvars c))"
(is "_ \<Turnstile> ?W ∼ ?W'")
by (auto intro: equiv_up_to_while_weak big_step_pres_approx_restrict
simp: bequiv_up_to_def)
hence "approx t \<Turnstile> ?W ∼ ?W'"
by (auto intro: equiv_up_to_weaken approx_map_le)
thus ?case
by (auto split: bexp.splits bool.splits
intro: equiv_up_to_while_False
simp: bval_bfold_Bc)
qed


theorem constant_folding_equiv':
"fold' c empty ∼ c"
using fold'_equiv [of empty c]
by (simp add: equiv_up_to_True sim_sym)


end