From Tagless-Final to Typed-Final: Program Transformations in the Final Style

From Tagless-Final
to Typed-Final
Program Transformations in the Final Style
Kenichi Suzuki
BizReach, Inc.
[email protected]

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໨࣍
• ࣗݾ঺հ
• tagless-ﬁnalͱ͸
• typed-ﬁnal styleͰϓϩάϥϜม׵

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ࣗݾ঺հ
• ླ໦ ݈Ұ (Kenichi Suzuki)
• [email protected]
• github.com/knih
• QUEΛ (http://logic.cs.tsukuba.ac.jp/~ken/quel/)
• ϓϩάϥϜݴޠ࿦ɺΫΤϦɺFunctional DDD
• BizReach
• yamory (https://yamory.io)

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The Death of Tagless Final

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Pure repository
trait UserRepository[F[_]] {
def findOneById(userId: UserId): F[Option[User]]
}
class UserService[F[_]](userRepository: UserRepository[F])
(implicit ME: MonadError[F, Error]) {
def getName(userId: String) = {
val userOptF: F[Option[User]] = userRepository.findOneById(userId)
Monad[F].map(userOptF) { userOpt =>
for {
user } yield user.name
}
}
}

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Pure repository
object UserRepositoryOnJdbc extends UserRepository[IO] {
override def findOneById(userId: UserId): IO[Option[User]] = ???
}
object UserController extends ApplicationController {
implicit val userRepository: UserRepository[IO] =
UserRepositoryOnJdbc
val service = new UserService[IO]
...
}

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What is Tagless Final?

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tagless-ﬁnalͱ͸
• ܕ҆શͳຒࠐΈDSLΛߏங͢Δख๏
• ܕ෇͚͕ϝλݴޠ (ϗετݴޠ) ͷͦΕʹؼண͢Δ
• ܕݕࠪΞϧΰϦζϜΛ࣮૷͠ͳͯ͘Α͍
• HOAS (ߴ֊ந৅ߏจ) ͕࢖͑Δ
• ϝλݴޠʹΑͬͯείʔϓ҆શੑ͕୲อ͞ΕΔ
• ݴޠίϯϙʔωϯτͷ߹੒Λαϙʔτ
• ύʔαΛ࡞Βͳͯ͘Α͍
• ϗετݴޠͷΤίγεςϜ͕࢖͑Δ
[J. Carette, O. Kiselyov and C.-c. Shan. (2009)] [1]

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ର৅ݴޠͱϝλݴޠ
• ର৅ݴޠ (object language)
• දݱ͍ͨ͠ݴޠɺDSL
• ϝλݴޠ (meta language)
• Scala, OCaml, Haskell౳ͷຒࠐΈઌϗετݴޠ

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Symantics
Symantics
ΠϯλϑΣʔε
Symantics
ΠϯλϓϦλ
ߏจ(Syntax)
ͱ ܕنଇ
Symantics
ΠϯλϓϦλ
Symantics
ΠϯλϓϦλ
ҙຯ࿦(Semantics)

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ର৅ݴޠΛઃܭ͢Δ
n is an integer
n : Z
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
e1 : Z e2 : Z
e1 + e2 : Z
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
e ::= n
| e + e
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
ߏจ (syntax) ܕ෇͚نଇ (typing rules)
දݱ͍ͨ͠ର৅ݴޠ
Symantics
trait AddSym {
type Repr[T]
def int(n: Int): Repr[Int]
def add(t1: Repr[Int], t2: Repr[Int]): Repr[Int]
}

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trait AddSym {
type Repr[T]
}
n is an integer
n : Z
e1 : Z e2 : Z
e1 + e2 : Z
e ::= n
| e + e
Symantics

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trait AddSym {
type Repr[T]
}
n is an integer
n : Z
e1 : Z e2 : Z
e1 + e2 : Z
e ::= n
| e + e
Symantics
Repr

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Symantics
• syntaxͱtyping rulesΛؔ਺ͷγάχνϟͰද͢
• දݱܕ (representation type) Ͱܕ෇͚نଇΛ
ද͢
• ߴ֊ΧΠϯυΛ࢖͏
• polymorphic lifting ͸࢖Θͳ͍
• semantics Λؔ਺ͷ࣮૷Ͱද͢

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ϢʔβϓϩάϥϜ
trait AddSym {
type Repr[T]
}
class Program[S import sym._
val ex1 = add(int(1), int(0))
}
ΠϯλϓϦλʢ࣮૷ʣʹ
ґଘ͠ͳ͍Ϣʔβϓϩά
ϥϜ͕هड़Ͱ͖Δ

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ΠϯλϓϦλ
final case class R[A](unR: A) // type Id[A] = A
object interp extends AddSym {
type Repr[T] = R[T]
def int(n: Int): R[Int] = R(n)
def add(t1: R[Int], t2: R[Int]): R[Int] =
R(t1.unR + t2.unR)
}
def eval[A](e: R[A]): A = e.unR
trait AddSym {
type Repr[T]
}
ܕ͕อଘ͞ΕΔ
ධՁؔ਺

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ಈ͔͢
object test {
val evaluator = Run.interp
val p1 = new Program(evaluator)
def main(args: Array[String]) = {
println("---------------------------------")
println(evaluator.eval(p1.ex1)) // => 1
}

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HOASͰλந৅Λѻ͏
• HOAS (Higher-Order Abstract Syntax; ߴ֊ந৅ߏจ)
• ม਺ଋറΛϗετݴޠͷλͰදݱ͢ΔςΫχοΫ
• ม਺Λਖ਼֬ʹѻ͏ͷ͸ׂͱ໘౗
• ม਺໊͕ඃΒͳ͍Α͏ʹ͢Δ
• de Bruijn ΠϯσοΫεΛ࢖͏৔߹Ͱ΋ɺ୅ೖૢ࡞࣌ͷΠ
ϯσοΫεͷγϑτܭࢉ͸ඞཁ
• capture-avoiding substitution
[Miller and Nadathur (1987); Pfenning and Elliott. (1988)][2,3]

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• ม਺ଋറΛϗετݴޠʹ·͔ͤΔ
trait LamSym { 
type Repr[T]
def lam[A, B](f: Repr[A] => Repr[B]): Repr[A => B]
def app[A, B](e1: Repr[A => B], e2: Repr[A]): Repr[B]
}
[x : ⌧1]
.
.
.
e : ⌧2
x.e : ⌧1
! ⌧2
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
e1 : ⌧1
! ⌧2 e2 : ⌧1
e1 e2 : ⌧2
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• ΠϯλϓϦλ͸ܕ߹ΘͤήʔϜ
def lam[A, B](f: R[A] => R[B]): R[A => B] =
R((x: A) => f(R(x)).unR)
def app[A, B](e1: R[A => B], e2: R[A]): R[B] =
R(e1.unR(e2.unR))

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ݴޠίϯϙʔωϯτΛ߹੒
trait AddSym {
type Repr[T]
def add(e1: Repr[Int], e2: Repr[Int]): Repr[Int]
}
trait SymLam {
type Repr[T]
}
trait Sym extends AddSym with SymLam

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ݴޠίϯϙʔωϯτΛ߹੒
trait AddSym[Repr[_]] {
def add(e1: Repr[Int], e2: Repr[Int]): Repr[Int]
}
trait SymLam[Repr[_]] {
}
trait Sym[Repr[_]] extends AddSym[Repr] with SymLam[Repr]

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tagless-ﬁnal
↓
typed-ﬁnal

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ਁಁ͍ͯ͠ͳ͍ͷͰ΋͏Ұ౓

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typed-ﬁnal

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ʢάάϥϏϦςΟΛߟ͍͑ͯͳ͔ͬͨʣ

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ϓϩάϥϜม׵
• ୯Ґݩ
• n + 0 = n ... ӈ୯Ґݩ
• 0 + m = m ... ࠨ୯Ґݩ
• tagless-ﬁnal͸ؔ਺ʹΑͬͯݴޠΛߏ੒͢Δ
• ؔ਺Ͱߏ੒͞ΕͨϓϩάϥϜΛͲͷΑ͏ʹม׵͢Δ͔
val program: R[Int] = add(int(1), int(0))

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ϓϩάϥϜม׵
• tagless-ﬁnal typed-ﬁnal ͸1ͭͷΠϯλϑΣʔ
ε(symantics)ʹରͯ͠ෳ਺ͷ࣮૷(ΠϯλϓϦ
λ)Λ࣋ͭ͜ͱ͕Ͱ͖Δ

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ϓϩάϥϜม׵
Symantics
ΠϯλϑΣʔε
Symantics
ΠϯλϓϦλ
Symantics
ΠϯλϓϦλ
Symantics
ΠϯλϓϦλ

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ϓϩάϥϜม׵
Symantics
ΠϯλϑΣʔε
Symantics
ΠϯλϓϦλ
Symantics
ΠϯλϓϦλ

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ϓϩάϥϜม׵
Symantics
ΠϯλϑΣʔε
ධՁΠϯλϓϦλ
ม׵ΠϯλϓϦλ

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ϓϩάϥϜม׵
• ධՁثΠϯλϓϦλ͸ͦͷ··࢖͍͍ͨ
• ม׵ϩδοΫͱධՁϩδοΫΛ෼཭͍ͨ͠
• ͞Βʹม׵ޙʹධՁ͍ͨ͠
• ܕ͸อଘ͍ͨ͠

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ϓϩάϥϜม׵
Symantics
ΠϯλϑΣʔε
ධՁΠϯλϓϦλ
ม׵ΠϯλϓϦλ

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ϓϩάϥϜม׵
ධՁΠϯλϓϦλ

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typed-ﬁnalϓϩάϥϜม׵
R(2)
R
ղऍ
mul(add(int(2),
int(0)),int(1))
߲ ධՁ݁Ռ
[Suzuki, Kiselyov and Yukiyoshi. (2016)][4]

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R(2)
R
ղऍ
mul(int(2),in
t(1))
Mul(Int(2), Int(1))
AddIdent(R)
AddIdent 
ϑΝϯΫλ
mul(add(int(2),
int(0)),int(1))
߲ ධՁ݁Ռ

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R(2)
R
ղऍ
mul(int(2),in
t(1))
Mul(Int(2), Int(1))
AddIdent(R)
reflect
reify
AddIdent 
ϑΝϯΫλ
mul(add(int(2),
int(0)),int(1))
Ճ๏୯Ґݩنଇ
("%5
߲ ධՁ݁Ռ

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R(2)
R
ղऍ
int(2)
Int(2)
AddIdent(R)
reflect
reify
AddIdent 
ϑΝϯΫλ
add(int(2),
int(0)
Ճ๏୯Ґݩنଇ
mul(add(int(2),
int(0)),int(1))
Add(Int(2),Int(0))
MulIdent(AddIdent(R))
...
...
...
MulIdent 
ϑΝϯΫλ
৐๏୯Ґݩنଇ
("%5
߲ ධՁ݁Ռ

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Normalization by Evaluation
• reiﬁcation
• ҙຯΦϒδΣΫτ͔Β߲ͷਖ਼نతදݱΛநग़͢Δ
ૢ࡞
• reﬂection
• ߲ͷਖ਼نతදݱ͔ΒҙຯΦϒδΣΫτ΁ͷม׵
• reifyͷٯؔ਺
• ਖ਼نԽ
•
[Per Martin-Löf. (1975); Danvy and Filinski. (1990); Dybjer and Filinski. (2000)][5,6,7] etc

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R(2)
R
ղऍ
int(2)
߲ ධՁ݁Ռ
Int(2)
AddIdent(R)
reflect
AddIdent 
ϑΝϯΫλ
add(int(2),
int(0)
Ճ๏୯Ґݩنଇ
mul(add(int(2),
int(0)),int(1))
Add(Int(2),Int(0))
...
...
...
MulIdent 
ϑΝϯΫλ
৐๏୯Ґݩنଇ
reify
reify

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R(2)
R
ղऍ
int(2)
߲ ධՁ݁Ռ
Int(2)
AddIdent(R)
reflect
AddIdent 
ϑΝϯΫλ
add(int(2),
int(0)
Ճ๏୯Ґݩنଇ
mul(add(int(2),
int(0)),int(1))
Add(Int(2),Int(0))
...
...
...
MulIdent 
ϑΝϯΫλ
৐๏୯Ґݩنଇ
reify
reify
؍ଌ
PCTFSWBUJPO

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GADT (Generalized Algebraic Data Types)
• දݱܕΛGADTͰߏ੒ͨ͠ܕʹ͢Δ
sealed trait Term[T]
final case class Unknown[A](x: sym.Repr[A]) extends Term[A]
final case class IntLit(e: Int) extends Term[Int]
final case class Add(e1: Term[Int], e2: Term[Int]) extends Term[Int]
type Repr[T] = Term[T]
trait AddSym {
type Repr[T]
}

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දݱܕͷม׵
trait RR {
import cats.~>
type From[_]
type To[_] // Term
def fwd: From ~> To // reflection
def bwd: To ~> From // reification
def map[A, B](f: From[A] => From[B]): To[A] => To[B] =
(t: To[A]) => fwd(f(bwd(t)))
def map2[A, B, C](f: (From[A], From[B]) => From[C]): (To[A],
To[B]) => To[C] =
(t1: To[A], t2: To[B]) => fwd(f(bwd(t1), bwd(t2)))
}
R.Repr
AddIdent(R).Repr
.Repr
From To
From To

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Reiﬁcation and Reﬂection
// AdditiveIdentity
class AddIdent[FS sealed trait Term[T]
object AddRR extends RR {
override type From[T] = sym.Repr[T]
override type To[T] = Term[T]
def fwd: From ~> To = new (From ~> To) {
def apply[A](x: sym.Repr[A]): Term[A] = Unknown(x)
}
def bwd: To ~> From = new (To ~> From) {
def apply[A](term: Term[A]): sym.Repr[A] = term match {
case Unknown(x) => x
case IntLit(n) => sym.int(n)
case Add(e1, e2) => sym.add(bwd(e1), bwd(e2))
}
}
}
... 

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Reiﬁcation and Reﬂection
// AdditiveIdentity
class AddIdent[FS sealed trait Term[T]
object AddRR extends RR {
override type From[T] = sym.Repr[T]
override type To[T] = Term[T]
def fwd: From ~> To = new (From ~> To) {
def apply[A](x: sym.Repr[A]): Term[A] = Unknown(x)
}
def bwd: To ~> From = new (To ~> From) {
def apply[A](term: Term[A]): sym.Repr[A] = term match {
case Unknown(x) => x
case IntLit(n) => sym.int(n)
case Add(e1, e2) => sym.add(bwd(e1), bwd(e2))
}
}
}
... 
R.Repr
AddIdent(R).Repr
.Repr
From To
From To

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GADTͱNbEͰܕอଘͭͭ͠ม׵
• GADTʹΑͬͯSymanticsͷܕʹै͏
class AddIdent[FS ...
val addIdentInterp = new GenericTrans(AddRR)(sym) {
def add(t1: Term[Int], t2: Term[Int]): Term[Int] = (t1, t2) match {
case (IntLit(0), _) => t2
case (_, IntLit(0)) => t1
case (_, _) => Add(t1, t2)
}
}
}
ࠨ୯Ґݩ
ӈ୯Ґݩ

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R(2)
R
ղऍ
int(2)
߲ ධՁ݁Ռ
Int(2)
AddIdent(R)
reflect
AddIdent 
ϑΝϯΫλ
add(int(2),
int(0)
Ճ๏୯Ґݩنଇ
mul(add(int(2),
int(0)),int(1))
Add(Int(2),Int(0))
...
...
...
MulIdent 
ϑΝϯΫλ
৐๏୯Ґݩنଇ
reify
reify
؍ଌ
PCTFSWBUJPO

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R(2)
R
ղऍ
int(2)
߲ ධՁ݁Ռ
Int(2)
AddIdent(R)
reflect
AddIdent 
ϑΝϯΫλ
add(int(2),
int(0)
Ճ๏୯Ґݩنଇ
mul(add(int(2),
int(0)),int(1))
Add(Int(2),Int(0))
...
...
...
MulIdent 
ϑΝϯΫλ
৐๏୯Ґݩنଇ
reify
reify
؍ଌ
PCTFSWBUJPO

ͦΕͧΕͷϞδϡʔϧ͸
4ZNBOUJDTΛ९कͯ͠࿈࠯͢Δ

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R(2)
R
ղऍ
int(2)
߲ ධՁ݁Ռ
Int(2)
AddIdent(R)
reflect
AddIdent 
ϑΝϯΫλ
add(int(2),
int(0)
Ճ๏୯Ґݩنଇ
mul(add(int(2),
int(0)),int(1))
Add(Int(2),Int(0))
...
...
...
MulIdent 
ϑΝϯΫλ
৐๏୯Ґݩنଇ
reify
reify
؍ଌ
PCTFSWBUJPO

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R(2)
R
ղऍ
int(2)
߲ ධՁ݁Ռ
Int(2)
AddIdent(R)
reflect
AddIdent 
ϑΝϯΫλ
add(int(2),
int(0)
Ճ๏୯Ґݩنଇ
mul(add(int(2),
int(0)),int(1))
Add(Int(2),Int(0))
...
...
...
MulIdent 
ϑΝϯΫλ
৐๏୯Ґݩنଇ
reify
reify
؍ଌ
PCTFSWBUJPO

ίʔυੜ੒ثΛ༩͑Δ
͜ͱ΋Ͱ͖Δ

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Finally, Tagless
ҙ༁: ࠷ޙʹλά͕ͳ͚Ε͹͍͍ΑͶ

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typed-ﬁnal

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·ͱΊ

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·ͱΊ
• typed-final͸DSLΛߏங͢ΔΞϓϩʔν
• දݱܕreprͰର৅֎ͷߏ੒ࢠ͕ೖΓࠐ·ͳ͍
Α͏ʹ͢Δ
• GADTͱNbEతΞϓϩʔνΛซ༻͢Δ͜ͱͰɺ
ܕΛอଘ͠ͳ͕Βfinal styleͰϓϩάϥϜม׵
• tagless-final͸typed-finalʹͳͬͨ

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ࢀߟจݙ
• [1] Jacques Carette, Oleg Kiselyov, and Chung-chieh Shan. 2009. Finally tagless, partially evaluated: Tagless
staged interpreters for simpler typed languages. J. Funct. Program. 19, 5 (September 2009), 509-543.
• [2] Miller, Dale & Nadathur, Gopalan (1987) A logic programming approach to manipulating formulas and
programs. In IEEE Symposium on Logic Programming, Seif Haridi (ed). Washington, DC: IEEE Comp. Society
Press, pp. 379–388.
• [3] Pfenning, Frank & Elliott, Conal (1988) Higher-order abstract syntax. In PLDI 88: Proceedings of the ACM
Conference on Programming Language Design and Implementation. ACM SIGPLAN Notices, vol. 23(7).
New York: ACM, pp. 199–208.
• [4] Kenichi Suzuki, Oleg Kiselyov, and Yukiyoshi Kameyama. 2016. Finally, safely-extensible and efﬁcient
language-integrated query. PEPM '16. ACM, New York, NY, USA, 37-48.
• [5] Per Martin-Löf. About models for intuitionistic type theories and the notion of deﬁnitional equality. In
Proceedings of the Third Scandinavian Logic Symposium (1972), volume 82 of Studies in Logic and the
Foundation of Mathematics, pages 81–109. North-Holland, 1975.
• [6] Olivier Danvy and Andrzej Filinski. Abstracting control. In Mitchell Wand, editor, Proceedings of the 1990
ACM Conference on Lisp and Functional Programming, pages 151–160, Nice, France, June 1990. ACM Press
• [7] Peter Dybjer and Andrzej Filinski. Normalization and partial evaluation. In Gilles Barthe, Peter Dybjer,
Luís Pinto, and João Saraiva, editors, Applied Semantics – Advanced Lectures, number 2395 in Lecture
Notes in Computer Science, pages 137–192, Caminha, Portugal, September 2000. Springer-Verlag.

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From Tagless-Final to Typed-Final: Program Transformations in the Final Style

From Tagless-Final to Typed-Final: Program Transformations in the Final Style

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