EIP:
Like monads, applicative functors are closed under products; so two independent idiomatic effects can generally be fused into one, their product.
Cats seems to be missing the functor products altogether.
(The impelementation I wrote here got merged into Cats in #388, and then it became Tuple2K)
/**
* [[Tuple2K]] is a product to two independent functor values.
*
* See: [[https://www.cs.ox.ac.uk/jeremy.gibbons/publications/iterator.pdf The Essence of the Iterator Pattern]]
*/
final case class Tuple2K[F[_], G[_], A](first: F[A], second: G[A]) {
/**
* Modify the context `G` of `second` using transformation `f`.
*/
def mapK[H[_]](f: G ~> H): Tuple2K[F, H, A] =
Tuple2K(first, f(second))
}
First we start with the product of Functor:
private[data] sealed abstract class Tuple2KInstances8 {
implicit def catsDataFunctorForTuple2K[F[_], G[_]](implicit FF: Functor[F], GG: Functor[G]): Functor[λ[α => Tuple2K[F, G, α]]] = new Tuple2KFunctor[F, G] {
def F: Functor[F] = FF
def G: Functor[G] = GG
}
}
private[data] sealed trait Tuple2KFunctor[F[_], G[_]] extends Functor[λ[α => Tuple2K[F, G, α]]] {
def F: Functor[F]
def G: Functor[G]
override def map[A, B](fa: Tuple2K[F, G, A])(f: A => B): Tuple2K[F, G, B] = Tuple2K(F.map(fa.first)(f), G.map(fa.second)(f))
}
Here’s how to use it:
import cats._, cats.data._, cats.syntax.all._
val x = Tuple2K(List(1), 1.some)
// x: Tuple2K[List, Option, Int] = Tuple2K(
// first = List(1),
// second = Some(value = 1)
// )
Functor[Lambda[X => Tuple2K[List, Option, X]]].map(x) { _ + 1 }
// res0: Tuple2K[List[A], Option, Int] = Tuple2K(
// first = List(2),
// second = Some(value = 2)
// )
First, we are defining a pair-like datatype called Tuple2K, which prepresents a product of typeclass instances.
By simply passing the function f to both the sides, we can form Functor for Tuple2K[F, G]
where F and G are Functor.
To see if it worked, we are mapping over x and adding 1.
We could make the usage code a bit nicer if we wanted,
but it’s ok for now.
Next up is Apply:
private[data] sealed abstract class Tuple2KInstances6 extends Tuple2KInstances7 {
implicit def catsDataApplyForTuple2K[F[_], G[_]](implicit FF: Apply[F], GG: Apply[G]): Apply[λ[α => Tuple2K[F, G, α]]] = new Tuple2KApply[F, G] {
def F: Apply[F] = FF
def G: Apply[G] = GG
}
}
private[data] sealed trait Tuple2KApply[F[_], G[_]] extends Apply[λ[α => Tuple2K[F, G, α]]] with Tuple2KFunctor[F, G] {
def F: Apply[F]
def G: Apply[G]
....
}
Here’s the usage:
{
val x = Tuple2K(List(1), (Some(1): Option[Int]))
val f = Tuple2K(List((_: Int) + 1), (Some((_: Int) * 3): Option[Int => Int]))
Apply[Lambda[X => Tuple2K[List, Option, X]]].ap(f)(x)
}
// res1: Tuple2K[List[A], Option, Int] = Tuple2K(
// first = List(2),
// second = Some(value = 3)
// )
The product of Apply passed in separate functions to each side.
Finally we can implement the product of Applicative:
private[data] sealed abstract class Tuple2KInstances5 extends Tuple2KInstances6 {
implicit def catsDataApplicativeForTuple2K[F[_], G[_]](implicit FF: Applicative[F], GG: Applicative[G]): Applicative[λ[α => Tuple2K[F, G, α]]] = new Tuple2KApplicative[F, G] {
def F: Applicative[F] = FF
def G: Applicative[G] = GG
}
}
private[data] sealed trait Tuple2KApplicative[F[_], G[_]] extends Applicative[λ[α => Tuple2K[F, G, α]]] with Tuple2KApply[F, G] {
def F: Applicative[F]
def G: Applicative[G]
def pure[A](a: A): Tuple2K[F, G, A] = Tuple2K(F.pure(a), G.pure(a))
}
Here’s a simple usage:
Applicative[Lambda[X => Tuple2K[List, Option, X]]].pure(1)
// res2: Tuple2K[List[A], Option, Int] = Tuple2K(
// first = List(1),
// second = Some(value = 1)
// )
We were able to create Tuple2K(List(1), Some(1)) by calling pure(1).
Unlike monads in general, applicative functors are also closed under composition; so two sequentially-dependent idiomatic effects can generally be fused into one, their composition.
Thankfully Cats ships with the composition of Applicatives.
There’s compose method in the typeclass instance:
@typeclass trait Applicative[F[_]] extends Apply[F] { self =>
/**
* `pure` lifts any value into the Applicative Functor
*
* Applicative[Option].pure(10) = Some(10)
*/
def pure[A](x: A): F[A]
/**
* Two sequentially dependent Applicatives can be composed.
*
* The composition of Applicatives `F` and `G`, `F[G[x]]`, is also an Applicative
*
* Applicative[Option].compose[List].pure(10) = Some(List(10))
*/
def compose[G[_]](implicit GG : Applicative[G]): Applicative[λ[α => F[G[α]]]] =
new CompositeApplicative[F,G] {
implicit def F: Applicative[F] = self
implicit def G: Applicative[G] = GG
}
....
}
Let’s try this out.
Applicative[List].compose[Option].pure(1)
// res3: List[Option[Int]] = List(Some(value = 1))
So much nicer.
For some reason, people seem to overlook that Gibbons also introduces
applicative function composition operators in EIP.
An applicative function is a function in the form of A => F[B] where F forms an Applicative.
This is similar to Kleisli composition of monadic functions, but better.
Here’s why.
Kliesli composition will let you compose A => F[B] and B => F[C] using andThen,
but note that F stays the same.
On the other hand, AppFunc composes A => F[B] and B => G[C].
/**
* [[Func]] is a function `A => F[B]`.
*
* See: [[https://www.cs.ox.ac.uk/jeremy.gibbons/publications/iterator.pdf The Essence of the Iterator Pattern]]
*/
sealed abstract class Func[F[_], A, B] { self =>
def run: A => F[B]
def map[C](f: B => C)(implicit FF: Functor[F]): Func[F, A, C] =
Func.func(a => FF.map(self.run(a))(f))
}
object Func extends FuncInstances {
/** function `A => F[B]. */
def func[F[_], A, B](run0: A => F[B]): Func[F, A, B] =
new Func[F, A, B] {
def run: A => F[B] = run0
}
/** applicative function. */
def appFunc[F[_], A, B](run0: A => F[B])(implicit FF: Applicative[F]): AppFunc[F, A, B] =
new AppFunc[F, A, B] {
def F: Applicative[F] = FF
def run: A => F[B] = run0
}
}
....
/**
* An implementation of [[Func]] that's specialized to [[Applicative]].
*/
sealed abstract class AppFunc[F[_], A, B] extends Func[F, A, B] { self =>
def F: Applicative[F]
def product[G[_]](g: AppFunc[G, A, B]): AppFunc[Lambda[X => Prod[F, G, X]], A, B] =
{
implicit val FF: Applicative[F] = self.F
implicit val GG: Applicative[G] = g.F
Func.appFunc[Lambda[X => Prod[F, G, X]], A, B]{
a: A => Prod(self.run(a), g.run(a))
}
}
....
}
Here’s how we can use it:
{
val f = Func.appFunc { x: Int => List(x.toString + "!") }
val g = Func.appFunc { x: Int => (Some(x.toString + "?"): Option[String]) }
val h = f product g
h.run(1)
}
// res4: Tuple2K[List, Option[A], String] = Tuple2K(
// first = List("1!"),
// second = Some(value = "1?")
// )
As you can see two applicative functions are running side by side.
Here’s andThen and compose:
def compose[G[_], C](g: AppFunc[G, C, A]): AppFunc[Lambda[X => G[F[X]]], C, B] =
{
implicit val FF: Applicative[F] = self.F
implicit val GG: Applicative[G] = g.F
implicit val GGFF: Applicative[Lambda[X => G[F[X]]]] = GG.compose(FF)
Func.appFunc[Lambda[X => G[F[X]]], C, B]({
c: C => GG.map(g.run(c))(self.run)
})
}
def andThen[G[_], C](g: AppFunc[G, B, C]): AppFunc[Lambda[X => F[G[X]]], A, C] =
g.compose(self)
{
val f = Func.appFunc { x: Int => List(x.toString + "!") }
val g = Func.appFunc { x: String => (Some(x + "?"): Option[String]) }
val h = f andThen g
h.run(1)
}
// res5: Nested[List, Option[A], String] = Nested(
// value = List(Some(value = "1!?"))
// )
EIP:
The two operators [snip] allow us to combine idiomatic computations in two different ways; we call them parallel and sequential composition, respectively.
The combining applicative computation is an abstract concept for all Applicative. We’ll continue from here.