Functor, Apply, Applicative And Monad

FUNCTORS, APPLY,
APPLICATIVE AND
MONADS

WHY SHOULD YOU
CARE?
How do Functors, Apply, Applicative and Monad help you
achieve early and continuous delivery of software?

IF A PIZZA COSTS $16 AND YOU BUY
TWO HOW MUCH DO YOU SPEND?

A MARIO KART COSTS $20 PER DAY
TO RENT. HOW MUCH DOES IT COST
TO RENT FOR TWO DAYS?

DON'T JUST THINK ABOUT THE
MATHS
Think about the process you are going through before you
get to the maths.
What information are you keeping and what information are
you dropping?

ABSTRACTION
Abstraction is an emphasis on the idea, qualities and
properties rather than the particulars
The importance of abstraction is derived from its ability to
hide irrelevant details
It doesn't matter if its pizza, mario carts or anything else we
take the cost and muliply it by some factor.

WHAT IS THE REALATION SHIP
BETWEEN A POLOGON:
A 3 sided triangle
A 4 sided quadrilateral

WHAT HAS AM × AN = AM+N GOT IN
COMMON WITH:
a2 × a3 = (a × a) × (a × a × a) = a5
a3 × a4 = (a × a × a) × (a × a × a × a) = a7

GENERALIZATION
Relationship that holds between all members of some set of
objects

IS A YEAR A LARGE AMOUNT OF
TIME?

IT ALL DEPENDS ON
CONTEXT
A mile is along way if you are an Ant but not if you are
flying in an aeroplane.
A year is a long time for a Mayfly that lives for only 5
minutes. But in geological time it's insignificant.

WHAT IS FUNCTIONAL
PROGRAMMING?
Construct our programs using only pure
functions.
Pure functions have no side effects.

WHY IS A FUNCTION
LIKE A PIPE?
Some thing goes into one end and something else comes
out the other end

Simple pipes simple can be joined together to form complex
systems?

WHAT'S SO GOOD ABOUT NO SIDE
EFFECTS?
It makes it easier to reason about what's going on

IT'S IMPORTANT THAT FUNCTIONS
LIKE PIPES DON'T LEAK

HOW TO DEFEAT A GOOMBA
Stomp it and it turns into a coin
case class Coin()
case class Goomba()
def stomp(g:Goomba) = Coin()
The function stomp is our pipe, that transforms from a
Goomba to a Coin.

LUIGIS VACUUM TO COLLECT
GOOMBAS
class Vacuum {
def collect(g:Goomba) = stomp(s)
}
val vacuum = new Vacuum()
val goomba = new Goomba()
vacuum.collect(goomba)
//Coin()

WHAT HAPPENS WHEN THE
GOOMBA ESCAPES THE SUCTION?
val vacuum = new Vacuum()
vacuum.collect(null)

CHECK FOR NULLS
class Vacuum {
def collect(s:Goomba) = if (s == null) null else stomp(s)
}
But then the calling class runs the risk of NullPointer
exceptions.

SOLUTION
Put the Goomba in a Cage

sealed trait Cage[T]
case class FullCage[T](value: T) extends Cage[T]
case class EmptyCage[T]() extends Cage[T]
object Cage {
def apply[T](x: T):Cage[T] =
if (x == null) EmptyCage[T]() else FullCage(x)
}
class Vacuum {
def collect(c:Cage[Goomba]):Cage[Coin] = c match {
case EmptyCage() => EmptyCage[Coin]()
case FullCage(s) => FullCage(stomp(s))
}
}
val vac = new Vacuum()
vac.collect(Cage(Goomba()))
//FullCage[Coin](Coin())
vac.collect(Cage(null))
//EmptyCage[Coin]()

CAN WE GENERALIZE
THE VACUUM CLASS?

WHY LIMIT OURSELF TO JUST
STOMPING GOOMBAS IN THE CAGE?
class Vacuum {
def collect[A,B](c:Cage[A], f:A => B):Cage[B] = c match {
case EmptyCage() => EmptyCage[B]()
case FullCage(s) => FullCage(f(s))
}
}

Turn it into a trait
trait Collector {
def collect[A,B](c:Cage[A], f:A => B):Cage[B]
}
class Vacuum extends Collector {
}
}

Parameterise the trait
trait Collector[F[_]] {
def collect[A,B](c:F[A], f:A => B): F[B]
}
object Vacuum extends Collector[Cage] {
}
}

THE FUNCTOR
A functor is basically for things that can be
mapped over.

THE FUNCTOR
DEFINITION IN SCALAZ
package scalaz
trait Functor[F[_]] extends InvariantFunctor[F] { self =>
...
/** Lift `f` into `F` and apply to `F[A]`. */
def map[A, B](fa: F[A])(f: A => B): F[B]
...
}

HOW DO YOU USE IT?
object CageFunctor extends Functor[Cage] {
def map[A,B](c:Cage[A])(f:A => B):Cage[B] = c match {
}
}
CageFunctor.map(Cage(Gummba()))(stomp)

WHY?
Extend existing classes
Without inheritance
Without altering original source
Keeps concerns seperate

HOW?
3 COMPONENTS
1. The type class
2. Instances for particular types
3. Interface methods for the api

THE TYPE CLASS
Provide a generic type of what we want to implement.
trait ProtoBuffWriter[A] {
def write(value: A): Array[Byte]
}

TYPE CLASS INSTANCES
Provide implementations for the types we care about.
Create concrete implementations of the type class and mark
them as implicit.
object DefaultProtoBuffWriters {
implicit val coinWriter = ProtoBuffWriter[Coin] { .... }
implicit val goombaWriter = ProtoBuffWriter[Goomba] { .... }
// etc ...
}

INTERFACES
What is exposed to clients.
Generic methods that accept instances of the type class as
implicit params.
TWO WAYS OF DOING IT

INTERFACE OBJECTS
All methods in a singleton.
object ProtoBuff {
def toProtoBuff[A](value: A)
(implicit writer: ProtoBuffWriter[A]): Array[Byte] {
writer.write(value)
}
}
import DefaultProtoBuffWriters._
val protoBuff: Array[Byte] = ProtoBuff.toProtoBuff(Coin())

INTERFACE SYNTAX
Pimp existing types with interface methods.
object ProtoBuffSyntax {
implicit class ProtoBuffWriter[A](value: A) {
def toProtoBuff(implicit writer: ProtoBuffWriter[A])
: Array[Byte] = {
writer.write(value)
}
}
}
import DefaultProtoBuffWriters._
import ProtBuffSyntax._
val protoBuff: Array[Byte] = Coin().toProtoBuff

WHAT ABOUT SCALAZ?
Pimp existing types
Uses the Type classes in Ops classes
Ops classes use the Type class and provide more methods
import scala.language.implicitConversions
sealed trait ToProtoBuffWriterOps {
implicit def ToProtoBuffWriterOps[A](v: A)
(implicit F: ProtoBuffWriter[A]) = new ProtoBuffWriterOps(v)
}
object protoBuffWriter extends ToProtoBuffWriterOps
import sun.misc.BASE64Encoder //for example only
class ProtoBuffWriterOps[A](val self: A)
(implicit val F: ProtoBuffWriter[A]) {
def write(value: A) = F.write(value)
def writeBase64(value: A) =
new BASE64Encoder().encodeBuffer(write(value))
}

!!WARNING!!
Implicits like warp pipes can be dangerous

SCALAZ FUNCTOR SYNTAX:
TOFUNCTOROPS
scalaz.syntax.FunctorSyntax.scala
trait ToFunctorOps extends ToFunctorOps0 with ToInvariantFunctorOps
{
implicit def ToFunctorOps[F[_],A](v: F[A])(implicit F0: Functor[F])
= new FunctorOps[F,A](v)
...
}
Given a F[A] and a implicit Functor[F] in scope add all the
FunctorOps to F[A]

SCALAZ FUNCTOR SYNTAX:
FUNCTOROPS
final class FunctorOps[F[_],A] private[syntax]
(val self: F[A])(implicit val F: Functor[F]) extends Ops[F[A]] {
...
final def map[B](f: A => B): F[B] = F.map(self)(f)
...
}
Given a F[A] and a implicit Functor[F] in scope delegate the
map method to the Functor[F] in scope

FINALLY
scalaz.syntax package object extends Syntaxes
trait Syntaxes {
object functor extends ToFunctorOps
}
import scalaz.syntax.functor

CAGE FUNCTOR AGAIN
import scalaz.Functor
implicit object CageFunctor extends Functor[Cage] {
}
}
Cage(Goomba()).map(stomp)

THE FUNCTOR LAWS
Mapping preserves identity
If we map the id function over a functor, the
functor that we get back should be the
same as the original functor
Mapping respects composition
Composing two functions and then
mapping the resulting function over a
functor should be the same as first
mapping one function over the functor and
then mapping the other one

DOES YOUR FUNCTOR BREAK LAWS?
import org.scalacheck.Arbitrary
import org.specs2.scalaz.Spec
import scalaz.Equal
import scalaz.scalacheck.ScalazProperties
class CageFunctorSpec extends Spec {
implicit val abrCage = Arbitrary[Cage[Int]] {
for {
ns <- Arbitrary.arbInt.arbitrary
} yield Cage(ns)
}
implicit val cageEqual = Equal.equal[Cage[Int]]((a, b) => a == b)
checkAll(ScalazProperties.functor.laws[Cage])
}
val scalazVersion = "7.1.3"
libraryDependencies ++= Seq(
"org.scalaz" %% "scalaz-core" % scalazVersion,
"org.specs2" %% "specs2-core" % "2.4" % "test",
"org.typelevel" %% "scalaz-specs2" % "0.3.0" % "test"
)

REMEMBER THE THREE POINTS?
Abstraction: Functor is a abstract concept
Generalization: There is a set of objects that can be
mapped over
What about the context?

CONTEXT
Context is the environment the function is applied in.

SCALA.OPTION
sealed abstract class Option[+A] extends Product with Serializable {
...
final def map[B](f: A => B): Option[B] =
if (isEmpty) None else Some(f(this.get))
...
}
Scalaz provides implicits to convert Option to a Functor trait
import scalaz.std.option._
import scalaz.std.anyVal._
checkAll(ScalazProperties.functor.laws[Option])
Option is context for a computation that might fail

LIST ARE ALSO FUNCTORS
sealed abstract class List[+A] { ....
final def map[B](f: (A) ⇒ B): List[B]
...
}
Scalaz provides implicits to convert List to a Functor trait
import scalaz.std.list._
import scalaz.std.anyVal._
class ListFunctorSpec extends Spec {
checkAll(ScalazProperties.functor.laws[List])
}
If 6 is deterministic and having one value.
The List context such as List(1,10,3,4) can be thought of as
having multipule values at once.
Or no values if empty

DISJUNCTIONS ARE FUNCTORS
class ListFunctorSpec extends Spec {
implicit val abrStringIntEither = Arbitrary[/[String, Int]] {
for {
} yield /-(ns)
}
implicit val disjuncEqual =
Equal.equal[/[String, Int]]((a, b) => { (a,b) match {
case(-/(l1), -/(l2)) => l1 == l2
case(/-(r1), /-(r2)) => r1 == r2
case _ => false
}
})
//left type param is fixed
checkAll(ScalazProperties.functor.laws[({type λ[α] = /[String, α]})#λ])
}

ORDINARY FUNCTIONS ARE SIMPLER
TO:
read
write
use
reason about

FUNCTIONS IN A CONTEXT HAVE
USEFUL PROPERTIES
Functors let us write ordinary functions
Then promote those functions into every context that might
need that code
As new contexts arise we just define new functors to
promote our ordinary code to work in those contexts.

///Ordinary function
def stomp(g:Goomba) = g.stomp()
///The context
sealed trait Cage[T]
case class FullCage[T](value: T) extends Cage[T]
case class EmptyCage[T]() extends Cage[T]
object Cage {
def apply[T](x: T):Cage[T] =
if (x == null) EmptyCage[T]() else FullCage(x)
}
///Promote into context
import scalaz.Functor
implicit object CageFunctor extends Functor[Cage] {
}
}
///use
Cage(Goomba()).map(stomp)

PIRANHA PLANT
case class Coin()
case class Fireball()
case class PiranhaPlant()
def shoot(plant:PiranhaPlant, fireball:Fireball): Coin = Coin()

THE PLANT IS GENERATED FROM A
UNRELIABLE SOURCE
Wrap the plant in a Cage and map over it?
Cage(PiranhaPlant()).map(shoot _)
<console>:31: error: type mismatch;
found : (PiranhaPlant, Fireball) => Coin
required: PiranhaPlant => ?
Cage(PiranhaPlant()).map(shoot _)
</console>

CURRING IS PARTIAL APPLICATION
Translating the evaluation of a function that takes multiple
arguments into evaluating a sequence of functions, each
with a single argument
(shoot _).curried
//res41: PiranhaPlant => (Fireball => Coin) = <function1>
</function1>

MAP THE CURRIED SHOOT FUNCTION
Cage(PiranhaPlant()) map {shoot _}.curried
//Cage[Fireball => Coin] = ...

WHAT IF THE FIREBALL PARAMETER
GENERATION IS IN A CONTEXT?
Functor only support mapping functions over functor
We need to map function in a functor over a value in a
functor

APPLY
package scalaz
trait Apply[F[_]] extends Functor[F] { self =>
////
def ap[A,B](fa: => F[A])(f: => F[A => B]): F[B]
...
}
package scalaz
package syntax
final class ApplyOps[F[_],A] private[syntax](val self: F[A])
(implicit val F: Apply[F]) extends Ops[F[A]] {
final def <*>[B](f: F[A => B]): F[B] = F.ap(self)(f)
...
}
trait ToApplyOps extends ToApplyOps0 with ToFunctorOps {
implicit def ToApplyOps[F[_],A](v: F[A])(implicit F0: Apply[F]) =
new ApplyOps[F,A](v)
...
}

WHAT WOULD OUR VACUUM LOOK LIKE?
implicit object CageApply extends Apply[Cage]{
override def ap[A, B](fa: => Cage[A])
(fab: => Cage[(A) => B]): Cage[B] = fab match {
case FullCage(f) => fa match {
case FullCage(x) => FullCage(f(x))
}
}
override def map[A, B](fa: Cage[A])
(f: (A) => B): Cage[B] = CageFunctor.map(fa)(f)
}

HOW WOULD YOU USE IT?
val partialShoot = Cage(PiranhaPlant()) <*> Cage((shoot _).curried)
val optCoin = Cage(Fireball()) <*> partialShoot
//optCoin: Cage[Coin] = FullCage(Coin())
val optCoin = EmptyCage[Fireball]() <*> partialShoot
//optCoin: Cage[Coin] = EmptyCage[Coin]()

WHAT HAVE WE DONE?
Taken a function that takes two values.
Turned it into a function that takes two values in a context.

TESTING THE LAWS
import org.scalacheck.Arbitrary
import scalaz.Equal
class CageApplySpec extends Spec {
implicit val abrCage = Arbitrary[Cage[Int]] {
for {
} yield Cage(ns)
}
implicit val arbCageIntToInt = Arbitrary[Cage[Int => Int]] {
for{
multi <- Arbitrary.arbInt.arbitrary
} yield Cage((x:Int) => x * multi)
}
implicit val cageEqual = Equal.equal[Cage[Int]]((a, b) => a == b)
checkAll(ScalazProperties.apply.laws[Cage])
}

OPTION APPLY: OPTIONINSTANCES
package scalaz
package std
override def ap[A, B](fa: => Option[A])
(f: => Option[A => B]) = f match {
case Some(f) => fa match {
case Some(x) => Some(f(x))
case None => None
}
case None => None
}

SHOOT THAT PIRANHA PLANT
val partialShoot =
Option(PiranhaPlant()) <*> Option((shoot _).curried)
val optCoin = Option(Fireball()) <*> partialShoot
//optCoin: Option[Coin] = Some(Coin())

SOME NICER SYNTAX
import scalaz.syntax.apply._
^(Option(PiranhaPlant()), Option(Fireball()))(shoot)
//res69: Option[Coin] = Some(Coin())
import scalaz.Apply
Apply[Option]
.lift2(shoot)(Option(PiranhaPlant()), Option(Fireball()))
//res70: Option[Coin] = Some(Coin())

LIST AS AN APPLY CONTEXT
val partial = List(PiranhaPlant(), PiranhaPlant(),
PiranhaPlant()) <*> List((shoot _).curried)
List(Fireball()) <*> partial
//res23: List[Coin] = List(Coin(), Coin(), Coin())

DISJUNCTION AS AN APPLY
/.right[String, Goomba](Goomba()) <*>
/.right[String, Goomba => Coin]((_:Goomba) => Coin())
//res: scalaz./[String,Coin] = /-(Coin())

Apply lets you take a function that takes values and turn it
into a function that takes values in a context.
Write the code once and reuse it in the context you need

REMEMBER THE THREE POINTS?
Abstraction: Apply is a abstract concept
Generalization: There is a set of objects that implement
the Apply trait
Context: How the funciton is used depends on the Apply
Specialization we are using

HAS TO BE:
1. Koopa Paratroopa shot to a Koopa Troopa
2. Koopa Troopa is shot to a Shell
3. Shell is shot to a Coin

THE CODE
case class KoopaParatroopa()
case class KoopaTroopa()
case class Shell()
case class Coin()
case class Fireball()
def shootKP(fb: Fireball, kt:KoopaParatroopa) = KoopaTroopa()
def shootKT(fb: Fireball, kt:KoopaTroopa) = Shell()
def shootS(fb: Fireball, kt:Shell) = Coin()
val cagedKoopa = ^(Cage(Fireball()),
Cage(KoopaParatroopa()))(shootKP)
val cagedShell = ^(Cage(Fireball()), cagedKoopa)(shootKT)
val cagedCoin = ^(Cage(Fireball()), cagedShell)(shootS)
//cagedCoin: Cage[Coin] = FullCage(Coin())

APPLICATIVE
trait Applicative[F[_]] extends Apply[F] { self =>
////
def point[A](a: => A): F[A]
...
}
implicit object CageApplicative extends Applicative[Cage] {
override def ap[A, B](fa: => Cage[A])
(fab: => Cage[(A) => B]): Cage[B] = fab match {
case FullCage(f) => fa match {
case FullCage(x) => FullCage(f(x))
}
}
override def point[A](a: => A): Cage[A] = Cage(a)
}
We no longer need to define map
override def map[A, B](fa: F[A])(f: A => B): F[B] =
ap(fa)(point(f))

USING APPLICATIVE
import scalaz.syntax.applicative._
val cagedKoopa = ^(Fireball().point[Cage],
KoopaParatroopa().point[Cage])(shootKP)
val cagedShell = ^(Fireball().point[Cage], cagedKoopa)(shootKT)
val cagedCoin = ^(Fireball().point[Cage], cagedShell)(shootS)
//cagedCoin: Cage[Coin] = FullCage(Coin())

SAME CODE DIFFERENT CONTEXT
val cagedKoopa = ^(Fireball().point[List],
KoopaParatroopa().point[List])(shootKP)
val cagedShell = ^(Fireball().point[List], cagedKoopa)(shootKT)
val cagedCoin = ^(Fireball().point[List], cagedShell)(shootS)
//cagedCoin: List[Coin] = List(Coin())

REMEMBER
1. Abstraction: The Applicative
2. Generalisation: The Applicative trait
3. The context: Different behaviours for the same code

THE RULES
Mario can hit Bowser
Bowser can hit Mario
Mario dies if at any point hits on Mario > hits on Bowser + 2

FIRST TRY
case class Hits(mario:Int, bowser:Int)
def hitBowser(hits: Hits) = hits.copy(bowser = hits.bowser + 1)
def hitMario(hits: Hits) = hits.copy(mario = hits.mario + 1)
def marioWins = hitMario _ andThen hitBowser
andThen hitBowser andThen hitBowser
marioWins(Hits(0,0))
//Hits = Hits(1,3)

HOW ABOUT THIS?
def marioWins = hitMario _ andThen hitMario andThen
hitMario andThen hitBowser andThen
hitBowser andThen hitBowser
//hits = Hits(3,3)
//marioWins(Hits(6,3))
Mario should have died

FAILING THE COMPUTATION?
Hits => Cage[Hits]
def hitMario2(hits: Hits):Cage[Hits] = hits match {
case ko:Hits if ko.mario + 1 - ko.bowser > 2 => EmptyCage[Hits]()
case Hits(mario, bowser) => Cage(Hits(mario + 1, bowser))
}
def hitBowser2(hits: Hits):Cage[Hits] = hits match {
case ko:Hits if ko.mario + 1- ko.bowser > 2 => EmptyCage[Hits]()
case Hits(mario, bowser) => Cage(Hits(mario, bowser + 1))
}
What's the problem?

THE HITS ARE TRAPPED IN THE CAGE!!

MONAD
trait Bind[F[_]] extends Apply[F] { self =>
def bind[A, B](fa: F[A])(f: A => F[B]): F[B]
override def ap[A, B](fa: => F[A])(f: => F[A => B]): F[B] = {
lazy val fa0 = fa
bind(f)(map(fa0))
}
...
}
trait Monad[F[_]] extends Applicative[F] with Bind[F] { self =>
...
override def map[A,B](fa: F[A])(f: A => B) = bind(fa)(a => point(f(a)))
...
}
Define point and bind and we get map and ap for free

CAGE MONAD
implicit object CageMonad extends Monad[Cage]{
override def bind[A, B](fa: Cage[A])(f: (A) => Cage[B]):
Cage[B] = fa match {
case FullCage(a) => f(a)
}
override def point[A](a: => A): Cage[A] = Cage(a)
}

BINDOPS
final class BindOps[F[_],A] private[syntax](val self: F[A])
(implicit val F: Bind[F]) extends Ops[F[A]] {
...
def flatMap[B](f: A => F[B]) = F.bind(self)(f)
def >>=[B](f: A => F[B]) = F.bind(self)(f)
...
}
trait ToBindOps extends ToBindOps0 with ToApplyOps {
implicit def ToBindOps[F[_],A](v: F[A])(implicit F0: Bind[F]) =
new BindOps[F,A](v)
}

NOW
import scalaz.syntax.monad._
Cage(Hits(0,0)) >>= hitMario2 >>= hitMario2 >>=
hitMario2 >>= hitBowser2 >>= hitBowser2 >>= hitBowser2
//Cage[Hits] = EmptyCage()
Cage(Hits(0,2)) >>= hitMario2 >>= hitMario2 >>=
hitMario2 >>= hitBowser2 >>= hitBowser2 >>= hitBowser2
//Cage[Hits] = FullCage(Hits(3,5))

FOR COMPREHENSION FLAT MAP
val x = for {
r1 <- Cage(Hits(0,0))
r2 <- hitMario2(r1)
r3 <- hitMario2(r2)
r4 <- hitMario2(r3)
r5 <- hitBowser2(r4)
r6 <- hitBowser2(r5)
result <- hitBowser2(r6)
} yield result

OTHER MONADS
def addTwo(x:Int) = Some(x + 2)
1.some >>= addTwo
//addTwo: (x: Int)List[Int]
def addTwo(x:Int) = List(x + 2)
List(1,2,3) >>= addTwo
//List[Int] = List(3, 4, 5)
Reader Monad
Writer Monad
State Monad

MONAD LAWS
Inherit the laws from Bind and Applicative
Right Identity
Left Identity

HANG ON I CAN'T REUSE THESE
FUNCTIONS
def hitMario3[F[_]](hits: Hits)(implicit F0: Monad[F])
:F[Hits] = hits match {
case ko:Hits if ko.mario + 1 - ko.bowser > 2 =>
F0.point(null: Hits)
case Hits(mario, bowser) =>
F0.point(Hits(mario + 1, bowser))
}
def hitBowser3[F[_]](hits: Hits)(implicit F0: Monad[F])
:F[Hits] = hits match {
case ko:Hits if ko.mario + 1- ko.bowser > 2 =>
F0.point(null: Hits)
case Hits(mario, bowser) =>
F0.point(Hits(mario, bowser + 1))
}
Cage(Hits(1,2)) >>= hitMario3[Cage]
Cage(Hits(1,2)) >>= hitBowser3[Cage]

MONAD
Lets us call a function that takes a value and returns a value
in a context with a value in a context.

REMEMBER
1. Abstraction: The Monad
2. Generalisation: The Monad trait
3. The context: Different behaviours for the same code

SUMMARY
Functor
Apply
Applicative
Monad
def ap[A,B](fa: => F[A])
(f: => F[A => B]): F[B]
def point[A](a: => A): F[A]
def bind[A, B](fa: F[A]) (f: A => F[B]): F[B]

REFERENCES
Learn you a Haskell for Greater Good
Leaning Scalaz
Functional Programming in Scala
Bartosz Blog
http://learnyouahaskell.com/
http://eed3si9n.com/learning-scalaz/
https://www.manning.com/books/functional-
programming-in-scala
http://bartoszmilewski.com/2014/10/28/category-theory-
for-programmers-the-preface/

Functor, Apply, Applicative And Monad

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