This is a log of me going through Cats, a functional programming library for Scala
that is currently *experimental* and under active development.
I’ll try to follow the structure of learning Scalaz that I wrote in 2012 (!).

Cats’ website says the name is a playful shortening of the word “category.” Some people say coordinating programmers is like herding cats (as in attempting to move dozens of them in a direction like cattle, while cats have their own ideas), and it seems to be true at least for those of us using Scala. Keenly aware of this situation, the cats list approachability as their first motivation.

Cats also looks interesting from the technical perspective. With its approachability and Erik Asheim (@d6/@non)’s awesomeness, people are gathering around them with new ideas such as Michael Pilquist (@mpilquist)’s simulacrum and Miles Sabin (@milessabin)’s deriving typeclasses. Hopefully I’ll learn more in the coming days.

This is a web log of me learning Cats, based a log of me learning Scalaz. It’s sometimes called a tutorial, but you should treat it more like a travel log that was scribbled together. If you want to learn the stuff, I suggest you read books and go through the examples yourself.

Before we dive into the details, here is a prequel to ease you in.

I’m going to borrow material from Nick Partridge’s Scalaz talk given at Melbourne Scala Users Group on March 22, 2010:

Scalaz talk is up - http://bit.ly/c2eTVR Lots of code showing how/why the library exists

— Nick Partridge (@nkpart) March 28, 2010

Cats consists of two parts:

- New datatypes (
`Validated`

,`State`

, etc) - Implementation of many general functions you need (ad-hoc polymorphism, traits + implicits)

Nick says:

In this function

`head`

, it takes a list of`A`

’s, and returns an`A`

. And it doesn’t matter what the`A`

is: It could be`Int`

s,`String`

s,`Orange`

s,`Car`

s, whatever. Any`A`

would work, and the function is defined for every`A`

that there can be.

```
scala> def head[A](xs: List[A]): A = xs(0)
head: [A](xs: List[A])A
scala> head(1 :: 2 :: Nil)
res0: Int = 1
scala> case class Car(make: String)
defined class Car
scala> head(Car("Civic") :: Car("CR-V") :: Nil)
res1: Car = Car(Civic)
```

Haskell wiki says:

Parametric polymorphism refers to when the type of a value contains one or more (unconstrained)

type variables, so that the value may adopt any type that results from substituting those variables with concrete types.

Let’s think of a function `plus`

that can add two values of type `A`

:

```
scala> def plus[A](a1: A, a2: A): A = ???
plus: [A](a1: A, a2: A)A
```

Depending on the type `A`

, we need to provide different definition for what it means to add them.
One way to achieve this is through subtyping.

```
scala> trait PlusIntf[A] {
def plus(a2: A): A
}
defined trait PlusIntf
scala> def plusBySubtype[A <: PlusIntf[A]](a1: A, a2: A): A = a1.plus(a2)
plusBySubtype: [A <: PlusIntf[A]](a1: A, a2: A)A
```

We can at least provide different definitions of `plus`

for `A`

.
But, this is not flexible since trait `Plus`

needs to be mixed in at the time of defining the datatype.
So it can’t work for `Int`

and `String`

.

The third approach in Scala is to provide an implicit conversion or implicit parameters for the trait.

```
scala> trait CanPlus[A] {
def plus(a1: A, a2: A): A
}
defined trait CanPlus
scala> def plus[A: CanPlus](a1: A, a2: A): A = implicitly[CanPlus[A]].plus(a1, a2)
plus: [A](a1: A, a2: A)(implicit evidence$1: CanPlus[A])A
```

This is truely ad-hoc in the sense that

- we can provide separate function definitions for different types of
`A`

- we can provide function definitions to types (like
`Int`

) without access to its source code - the function definitions can be enabled or disabled in different scopes

The last point makes Scala’s ad-hoc polymorphism more powerful than that of Haskell. More on this topic can be found at Debasish Ghosh @debasishg’s Scala Implicits : Type Classes Here I Come.

Let’s look into `plus`

function in more detail.

Nick demonstrates an example of ad-hoc polymorphism by gradually making `sum`

function more general, starting from a simple function that adds up a list of `Int`

s:

```
scala> def sum(xs: List[Int]): Int = xs.foldLeft(0) { _ + _ }
sum: (xs: List[Int])Int
scala> sum(List(1, 2, 3, 4))
res2: Int = 10
```

If we try to generalize a little bit. I’m going to pull out a thing called

`Monoid`

. … It’s a type for which there exists a function`mappend`

, which produces another type in the same set; and also a function that produces a zero.

```
scala> object IntMonoid {
def mappend(a: Int, b: Int): Int = a + b
def mzero: Int = 0
}
defined object IntMonoid
```

If we pull that in, it sort of generalizes what’s going on here:

```
scala> def sum(xs: List[Int]): Int = xs.foldLeft(IntMonoid.mzero)(IntMonoid.mappend)
sum: (xs: List[Int])Int
scala> sum(List(1, 2, 3, 4))
res3: Int = 10
```

Now we’ll abstract on the type about

`Monoid`

, so we can define`Monoid`

for any type`A`

. So now`IntMonoid`

is a monoid on`Int`

:

```
scala> trait Monoid[A] {
def mappend(a1: A, a2: A): A
def mzero: A
}
defined trait Monoid
scala> object IntMonoid extends Monoid[Int] {
def mappend(a: Int, b: Int): Int = a + b
def mzero: Int = 0
}
defined object IntMonoid
```

What we can do is that `sum`

take a `List`

of `Int`

and a monoid on `Int`

to sum it:

```
scala> def sum(xs: List[Int], m: Monoid[Int]): Int = xs.foldLeft(m.mzero)(m.mappend)
sum: (xs: List[Int], m: Monoid[Int])Int
scala> sum(List(1, 2, 3, 4), IntMonoid)
res4: Int = 10
```

We are not using anything to do with

`Int`

here, so we can replace all`Int`

with a general type:

```
scala> def sum[A](xs: List[A], m: Monoid[A]): A = xs.foldLeft(m.mzero)(m.mappend)
sum: [A](xs: List[A], m: Monoid[A])A
scala> sum(List(1, 2, 3, 4), IntMonoid)
res5: Int = 10
```

The final change we have to take is to make the

`Monoid`

implicit so we don’t have to specify it each time.

```
scala> def sum[A](xs: List[A])(implicit m: Monoid[A]): A = xs.foldLeft(m.mzero)(m.mappend)
sum: [A](xs: List[A])(implicit m: Monoid[A])A
scala> implicit val intMonoid = IntMonoid
intMonoid: IntMonoid.type = IntMonoid$@56414743
scala> sum(List(1, 2, 3, 4))
res6: Int = 10
```

Nick didn’t do this, but the implicit parameter is often written as a context bound:

```
scala> def sum[A: Monoid](xs: List[A]): A = {
val m = implicitly[Monoid[A]]
xs.foldLeft(m.mzero)(m.mappend)
}
sum: [A](xs: List[A])(implicit evidence$1: Monoid[A])A
scala> sum(List(1, 2, 3, 4))
res7: Int = 10
```

Our

`sum`

function is pretty general now, appending any monoid in a list. We can test that by writing another`Monoid`

for`String`

. I’m also going to package these up in an object called`Monoid`

. The reason for that is Scala’s implicit resolution rules: When it needs an implicit parameter of some type, it’ll look for anything in scope. It’ll include the companion object of the type that you’re looking for.

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
trait Monoid[A] {
def mappend(a1: A, a2: A): A
def mzero: A
}
object Monoid {
implicit val IntMonoid: Monoid[Int] = new Monoid[Int] {
def mappend(a: Int, b: Int): Int = a + b
def mzero: Int = 0
}
implicit val StringMonoid: Monoid[String] = new Monoid[String] {
def mappend(a: String, b: String): String = a + b
def mzero: String = ""
}
}
def sum[A: Monoid](xs: List[A]): A = {
val m = implicitly[Monoid[A]]
xs.foldLeft(m.mzero)(m.mappend)
}
// Exiting paste mode, now interpreting.
defined trait Monoid
defined object Monoid
sum: [A](xs: List[A])(implicit evidence$1: Monoid[A])A
scala> sum(List("a", "b", "c"))
res8: String = abc
```

You can still provide different monoid directly to the function. We could provide an instance of monoid for

`Int`

using multiplications.

```
scala> val multiMonoid: Monoid[Int] = new Monoid[Int] {
def mappend(a: Int, b: Int): Int = a * b
def mzero: Int = 1
}
multiMonoid: Monoid[Int] = $anon$1@3fd2b713
scala> sum(List(1, 2, 3, 4))(multiMonoid)
res9: Int = 24
```

What we wanted was a function that generalized on

`List`

. … So we want to generalize on`foldLeft`

operation.

```
scala> object FoldLeftList {
def foldLeft[A, B](xs: List[A], b: B, f: (B, A) => B) = xs.foldLeft(b)(f)
}
defined object FoldLeftList
scala> def sum[A: Monoid](xs: List[A]): A = {
val m = implicitly[Monoid[A]]
FoldLeftList.foldLeft(xs, m.mzero, m.mappend)
}
sum: [A](xs: List[A])(implicit evidence$1: Monoid[A])A
scala> sum(List(1, 2, 3, 4))
res10: Int = 10
scala> sum(List("a", "b", "c"))
res11: String = abc
scala> sum(List(1, 2, 3, 4))(multiMonoid)
res12: Int = 24
```

Now we can apply the same abstraction to pull out

`FoldLeft`

typeclass.

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
trait FoldLeft[F[_]] {
def foldLeft[A, B](xs: F[A], b: B, f: (B, A) => B): B
}
object FoldLeft {
implicit val FoldLeftList: FoldLeft[List] = new FoldLeft[List] {
def foldLeft[A, B](xs: List[A], b: B, f: (B, A) => B) = xs.foldLeft(b)(f)
}
}
def sum[M[_]: FoldLeft, A: Monoid](xs: M[A]): A = {
val m = implicitly[Monoid[A]]
val fl = implicitly[FoldLeft[M]]
fl.foldLeft(xs, m.mzero, m.mappend)
}
// Exiting paste mode, now interpreting.
defined trait FoldLeft
defined object FoldLeft
sum: [M[_], A](xs: M[A])(implicit evidence$1: FoldLeft[M], implicit evidence$2: Monoid[A])A
scala> sum(List(1, 2, 3, 4))
res13: Int = 10
scala> sum(List("a", "b", "c"))
res14: String = abc
```

Both `Int`

and `List`

are now pulled out of `sum`

.

In the above example, the traits `Monoid`

and `FoldLeft`

correspond to Haskell’s typeclass.
Cats provides many typeclasses.

All this is broken down into just the pieces you need. So, it’s a bit like ultimate ducktyping because you define in your function definition that this is what you need and nothing more.

If we were to write a function that sums two types using the

`Monoid`

, we need to call it like this.

```
scala> def plus[A: Monoid](a: A, b: A): A = implicitly[Monoid[A]].mappend(a, b)
plus: [A](a: A, b: A)(implicit evidence$1: Monoid[A])A
scala> plus(3, 4)
res15: Int = 7
```

We would like to provide an operator. But we don’t want to enrich just one type,
but enrich all types that has an instance for `Monoid`

.
Let me do this in Cats style using Simulacrum.

```
scala> import simulacrum._
import simulacrum._
scala> :paste
// Entering paste mode (ctrl-D to finish)
@typeclass trait Monoid[A] {
@op("|+|") def mappend(a: A, b: A): A
def mzero: A
}
object Monoid {
// "ops" gets generated
val syntax = ops
implicit val IntMonoid: Monoid[Int] = new Monoid[Int] {
def mappend(a: Int, b: Int): Int = a + b
def mzero: Int = 0
}
implicit val StringMonoid: Monoid[String] = new Monoid[String] {
def mappend(a: String, b: String): String = a + b
def mzero: String = ""
}
}
// Exiting paste mode, now interpreting.
defined trait Monoid
defined object Monoid
scala> import Monoid.syntax._
import Monoid.syntax._
scala> 3 |+| 4
res0: Int = 7
scala> "a" |+| "b"
res1: String = ab
```

We were able to inject `|+|`

to both `Int`

and `String`

with just one definition.

Using the same technique, Cats *occasionally* provides method injections for standard library datatypes like `Option`

and `Vector`

:

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> 1.some
res2: Option[Int] = Some(1)
scala> 1.some.orEmpty
res3: Int = 1
```

But most operators in Cats are associated with typeclasses.

I hope you could get some feel on where Cats is coming from.

Learn You a Haskell for Great Good says:

A typeclass is a sort of interface that defines some behavior. If a type is a part of a typeclass, that means that it supports and implements the behavior the typeclass describes.

Cats says:

We are trying to make the library modular. It will have a tight core which will contain only the typeclasses and the bare minimum of data structures that are needed to support them. Support for using these typeclasses with the Scala standard library will be in the

`std`

project.

Let’s keep going with learning me a Haskell.

Cats is a new project under active development. Feedback and contributions are welcomed as we look to improve it. This project is evolving quickly and we are making no guarantees about stability until a 1.0 release is made (current est. around Q3 2016).

A released version of Cats is now available.

After that, you can test it using `build.sbt`

this:

```
val catsVersion = "0.9.0"
val catsAll = "org.typelevel" %% "cats" % catsVersion
val macroParadise = compilerPlugin("org.scalamacros" % "paradise" % "2.1.0" cross CrossVersion.full)
val kindProjector = compilerPlugin("org.spire-math" %% "kind-projector" % "0.6.3")
val resetAllAttrs = "org.scalamacros" %% "resetallattrs" % "1.0.0-M1"
val specs2Version = "3.6" // use the version used by discipline
val specs2Core = "org.specs2" %% "specs2-core" % specs2Version
val specs2Scalacheck = "org.specs2" %% "specs2-scalacheck" % specs2Version
val scalacheck = "org.scalacheck" %% "scalacheck" % "1.12.4"
lazy val root = (project in file(".")).
settings(
organization := "com.example",
name := "something",
scalaVersion := "2.11.6",
libraryDependencies ++= Seq(
catsAll,
specs2Core % Test, specs2Scalacheck % Test, scalacheck % Test,
macroParadise, kindProjector, resetAllAttrs
),
scalacOptions ++= Seq(
"-deprecation",
"-encoding", "UTF-8",
"-feature",
"-language:_"
)
)
```

You can then open the REPL using sbt 0.13.12:

```
$ sbt
> console
Welcome to Scala version 2.11.6 (Java HotSpot(TM) 64-Bit Server VM, Java 1.7.0_79).
Type in expressions to have them evaluated.
Type :help for more information.
scala>
```

There’s also API docs generated for Cats.

LYAHFGG:

`Eq`

is used for types that support equality testing. The functions its members implement are`==`

and`/=`

.

Cats’ equivalent for the `Eq`

typeclass is also called `Eq`

.
`Eq`

was moved from non/algebra into cats-kernel subproject, and became part of Cats:

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> 1 === 1
res4: Boolean = true
scala> 1 === "foo"
<console>:40: error: type mismatch;
found : String("foo")
required: Int
1 === "foo"
^
scala> 1 == "foo"
<console>:40: warning: comparing values of types Int and String using `==' will always yield false
1 == "foo"
^
res6: Boolean = false
scala> (Some(1): Option[Int]) =!= (Some(2): Option[Int])
res7: Boolean = true
```

Instead of the standard `==`

, `Eq`

enables `===`

and `=!=`

syntax by declaring `eqv`

method. The main difference is that `===`

would fail compilation if you tried to compare `Int`

and `String`

.

In algebra, `neqv`

is implemented based on `eqv`

.

```
/**
* A type class used to determine equality between 2 instances of the same
* type. Any 2 instances `x` and `y` are equal if `eqv(x, y)` is `true`.
* Moreover, `eqv` should form an equivalence relation.
*/
trait Eq[@sp A] extends Any with Serializable { self =>
/**
* Returns `true` if `x` and `y` are equivalent, `false` otherwise.
*/
def eqv(x: A, y: A): Boolean
/**
* Returns `false` if `x` and `y` are equivalent, `true` otherwise.
*/
def neqv(x: A, y: A): Boolean = !eqv(x, y)
....
}
```

This is an example of polymorphism. Whatever equality means for the type `A`

,
`neqv`

is the opposite of it. It does not matter if it’s `String`

, `Int`

, or whatever.
Another way of looking at it is that given `Eq[A]`

, `===`

is universally the opposite of `=!=`

.

I’m a bit concerned that `Eq`

seems to be using the terms “equal” and “equivalent”
interchangably. Equivalence relationship could include “having the same birthday”
whereas equality also requires substitution property.

LYAHFGG:

`Ord`

is for types that have an ordering.`Ord`

covers all the standard comparing functions such as`>`

,`<`

,`>=`

and`<=`

.

Cats’ equivalent for the `Ord`

typeclass is `Order`

.

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> 1 > 2.0
res0: Boolean = false
scala> 1 compare 2.0
<console>:21: error: type mismatch;
found : Double(2.0)
required: Int
1 compare 2.0
^
scala> 1.0 compare 2.0
res2: Int = -1
scala> 1.0 max 2.0
res3: Double = 2.0
```

`Order`

enables `compare`

syntax which returns `Int`

: negative, zero, or positive.
It also enables `min`

and `max`

operators.
Similar to `Eq`

, comparing `Int`

and `Double`

fails compilation.

In addition to `Order`

, Cats also defines `PartialOrder`

.

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> 1 tryCompare 2
res0: Option[Int] = Some(-1)
scala> 1.0 tryCompare Double.NaN
res1: Option[Int] = Some(-1)
```

`PartialOrder`

enables `tryCompare`

syntax which returns `Option[Int]`

.
According to algebra, it’ll return `None`

if operands are not comparable.
It’s returning `Some(-1)`

when comparing `1.0`

and `Double.NaN`

, so I’m not sure when things are incomparable.

```
scala> def lt[A: PartialOrder](a1: A, a2: A): Boolean = a1 <= a2
lt: [A](a1: A, a2: A)(implicit evidence$1: cats.PartialOrder[A])Boolean
scala> lt[Int](1, 2.0)
<console>:22: error: type mismatch;
found : Double(2.0)
required: Int
lt[Int](1, 2.0)
^
scala> lt(1, 2)
res3: Boolean = true
```

`PartialOrder`

also enables `>`

, `>=`

, `<`

, and `<=`

operators,
but these are tricky to use because if you’re not careful
you could end up using the built-in comparison operators.

LYAHFGG:

Members of

`Show`

can be presented as strings.

Cats’ equivalent for the `Show`

typeclass is `Show`

:

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> 3.show
res0: String = 3
scala> "hello".show
res1: String = hello
```

Here’s the typeclass contract:

```
@typeclass trait Show[T] {
def show(f: T): String
}
```

At first, it might seem silly to define `Show`

because Scala
already has `toString`

on `Any`

.
`Any`

also means anything would match the criteria, so you lose type safety.
The `toString`

could be junk supplied by some parent class:

```
scala> (new {}).toString
res2: String = $anon$1@61a19b19
scala> (new {}).show
<console>:21: error: value show is not a member of AnyRef
(new {}).show
^
```

`object Show`

provides two functions to create a `Show`

instance:

```
object Show {
/** creates an instance of [[Show]] using the provided function */
def show[A](f: A => String): Show[A] = new Show[A] {
def show(a: A): String = f(a)
}
/** creates an instance of [[Show]] using object toString */
def fromToString[A]: Show[A] = new Show[A] {
def show(a: A): String = a.toString
}
implicit val catsContravariantForShow: Contravariant[Show] = new Contravariant[Show] {
def contramap[A, B](fa: Show[A])(f: B => A): Show[B] =
show[B](fa.show _ compose f)
}
}
```

Let’s try using them:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
case class Person(name: String)
case class Car(model: String)
// Exiting paste mode, now interpreting.
defined class Person
defined class Car
scala> implicit val personShow = Show.show[Person](_.name)
personShow: cats.Show[Person] = cats.Show$$anon$2@6c4beffd
scala> Person("Alice").show
res4: String = Alice
scala> implicit val carShow = Show.fromToString[Car]
carShow: cats.Show[Car] = cats.Show$$anon$3@7ae69d9d
scala> Car("CR-V")
res5: Car = Car(CR-V)
```

LYAHFGG:

`Read`

is sort of the opposite typeclass of`Show`

. The`read`

function takes a string and returns a type which is a member of`Read`

.

I could not find Cats’ equivalent for this typeclass.

I find myself defining `Read`

and its variant `ReadJs`

time and time again.
Stringly typed programming is ugly.
At the same time, String is a robust data format to cross platform boundaries (e.g. JSON).
Also we humans know how to deal with them directly (e.g. command line options),
so it’s hard to get away from String parsing.
If we’re going to do it anyway, having `Read`

makes it easier.

LYAHFGG:

`Enum`

members are sequentially ordered types — they can be enumerated. The main advantage of the`Enum`

typeclass is that we can use its types in list ranges. They also have defined successors and predecessors, which you can get with the`succ`

and`pred`

functions.

I could not find Cats’ equivalent for this typeclass.

It’s not an `Enum`

or range, but non/spire has an interesting data structure called `Interval`

.
Check out Erik’s Intervals: Unifying Uncertainty, Ranges, and Loops talk from nescala 2015.

LYAHFGG:

`Num`

is a numeric typeclass. Its members have the property of being able to act like numbers.

I could not find Cats’ equivalent for this typeclass,
but spire defines `Numeric`

. Cats doesn’t define `Bounds`

either.

So far we’ve seen some of typeclasses that are *not* defined in Cats.
This is not necessarily a bad thing because having a tight core is part of the design goal of Cats.

I am now going to skip over to Chapter 8 Making Our Own Types and Typeclasses (Chapter 7 if you have the book) since the chapters in between are mostly about Haskell syntax.

```
data TrafficLight = Red | Yellow | Green
```

In Scala this would be:

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> :paste
// Entering paste mode (ctrl-D to finish)
sealed trait TrafficLight
object TrafficLight {
case object Red extends TrafficLight
case object Yellow extends TrafficLight
case object Green extends TrafficLight
}
// Exiting paste mode, now interpreting.
defined trait TrafficLight
defined object TrafficLight
```

Now let’s define an instance for `Eq`

.

```
scala> implicit val trafficLightEq: Eq[TrafficLight] =
new Eq[TrafficLight] {
def eqv(a1: TrafficLight, a2: TrafficLight): Boolean = a1 == a2
}
trafficLightEq: cats.Eq[TrafficLight] = $anon$1@28c75785
```

**Note**: The latest `algebra.Equal`

includes `Equal.instance`

and `Equal.fromUniversalEquals`

.

Can I use the `Eq`

?

```
scala> TrafficLight.Red === TrafficLight.Yellow
<console>:26: error: value === is not a member of object TrafficLight.Red
TrafficLight.Red === TrafficLight.Yellow
^
```

So apparently `Eq[TrafficLight]`

doesn’t get picked up because `Eq`

has nonvariant subtyping: `Eq[A]`

.
One way to workaround this issue is to define helper functions to cast them up to `TrafficLight`

:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
sealed trait TrafficLight
object TrafficLight {
def red: TrafficLight = Red
def yellow: TrafficLight = Yellow
def green: TrafficLight = Green
case object Red extends TrafficLight
case object Yellow extends TrafficLight
case object Green extends TrafficLight
}
// Exiting paste mode, now interpreting.
defined trait TrafficLight
defined object TrafficLight
scala> implicit val trafficLightEq: Eq[TrafficLight] =
new Eq[TrafficLight] {
def eqv(a1: TrafficLight, a2: TrafficLight): Boolean = a1 == a2
}
trafficLightEq: cats.Eq[TrafficLight] = $anon$1@6fb29781
scala> TrafficLight.red === TrafficLight.yellow
res1: Boolean = false
```

It is a bit of boilerplate, but it works.

Yesterday we reviewed a few basic typeclasses from Cats like `Eq`

by using Learn You a Haskell for Great Good as the guide.

LYAHFGG:

In JavaScript and some other weakly typed languages, you can put almost anything inside an if expression. …. Even though strictly using

`Bool`

for boolean semantics works better in Haskell, let’s try and implement that JavaScript-ish behavior anyway. For fun!

The conventional steps of defining a modular typeclass in Scala used to look like:

- Define typeclass contract trait
`Foo`

. - Define a companion object
`Foo`

with a helper method`apply`

that acts like`implcitly`

, and a way of defining`Foo`

instances typically from a function. - Define
`FooOps`

class that defines unibary or binary operators. - Define
`FooSyntax`

trait that implicitly provides`FooOps`

from a`Foo`

instance.

Frankly, these steps are mostly copy-paste boilerplate except for the first one.
Enter Michael Pilquist (@mpilquist)’s simulacrum.
simulacrum magically generates most of steps 2-4 just by putting `@typeclass`

annotation.
By chance, Stew O’Connor (@stewoconnor/@stew)’s #294 got merged,
which refactors Cats to use it.

In any case, let’s see if we can make our own truthy value typeclass.
Note the `@typeclass`

annotation:

```
scala> import simulacrum._
import simulacrum._
scala> :paste
// Entering paste mode (ctrl-D to finish)
@typeclass trait CanTruthy[A] { self =>
/** Return true, if `a` is truthy. */
def truthy(a: A): Boolean
}
object CanTruthy {
def fromTruthy[A](f: A => Boolean): CanTruthy[A] = new CanTruthy[A] {
def truthy(a: A): Boolean = f(a)
}
}
// Exiting paste mode, now interpreting.
defined trait CanTruthy
defined object CanTruthy
```

According to the README, the macro will generate all the operator enrichment stuff:

```
// This is the supposed generated code. You don't have to write it!
object CanTruthy {
def fromTruthy[A](f: A => Boolean): CanTruthy[A] = new CanTruthy[A] {
def truthy(a: A): Boolean = f(a)
}
def apply[A](implicit instance: CanTruthy[A]): CanTruthy[A] = instance
trait Ops[A] {
def typeClassInstance: CanTruthy[A]
def self: A
def truthy: A = typeClassInstance.truthy(self)
}
trait ToCanTruthyOps {
implicit def toCanTruthyOps[A](target: A)(implicit tc: CanTruthy[A]): Ops[A] = new Ops[A] {
val self = target
val typeClassInstance = tc
}
}
trait AllOps[A] extends Ops[A] {
def typeClassInstance: CanTruthy[A]
}
object ops {
implicit def toAllCanTruthyOps[A](target: A)(implicit tc: CanTruthy[A]): AllOps[A] = new AllOps[A] {
val self = target
val typeClassInstance = tc
}
}
}
```

To make sure it works, let’s define an instance for `Int`

and use it. The eventual goal is to get `1.truthy`

to return `true`

:

```
scala> implicit val intCanTruthy: CanTruthy[Int] = CanTruthy.fromTruthy({
case 0 => false
case _ => true
})
intCanTruthy: CanTruthy[Int] = CanTruthy$$anon$1@5734b94b
scala> import CanTruthy.ops._
import CanTruthy.ops._
scala> 10.truthy
res0: Boolean = true
```

It works. This is quite nifty.
One caveat is that this requires Macro Paradise plugin to compile. Once it’s compiled the user of `CanTruthy`

can use it without Macro Paradise.

For `CanTruthy`

the injected operator happened to be unary, and it matched the name of the function on the typeclass contract. simulacrum can also define operator with symbolic names using `@op`

annotation:

```
scala> @typeclass trait CanAppend[A] {
@op("|+|") def append(a1: A, a2: A): A
}
defined trait CanAppend
defined object CanAppend
scala> implicit val intCanAppend: CanAppend[Int] = new CanAppend[Int] {
def append(a1: Int, a2: Int): Int = a1 + a2
}
intCanAppend: CanAppend[Int] = $anon$1@459ce27e
scala> import CanAppend.ops._
import CanAppend.ops._
scala> 1 |+| 2
res1: Int = 3
```

LYAHFGG:

And now, we’re going to take a look at the

`Functor`

typeclass, which is basically for things that can be mapped over.

Like the book let’s look how it's implemented:

```
/**
* Functor.
*
* The name is short for "covariant functor".
*
* Must obey the laws defined in cats.laws.FunctorLaws.
*/
@typeclass trait Functor[F[_]] extends functor.Invariant[F] { self =>
def map[A, B](fa: F[A])(f: A => B): F[B]
....
}
```

Here’s how we can use this:

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> Functor[List].map(List(1, 2, 3)) { _ + 1 }
res0: List[Int] = List(2, 3, 4)
```

Let’s call the above usage the *function syntax*.

We now know that `@typeclass`

annotation will automatically turn a `map`

function into a `map`

operator.
The `fa`

part turns into the `this`

of the method, and the second parameter list will now be
the parameter list of `map`

operator:

```
// Supposed generated code
object Functor {
trait Ops[F[_], A] {
def typeClassInstance: Functor[F]
def self: F[A]
def map[B](f: A => B): F[B] = typeClassInstance.map(self)(f)
}
}
```

This looks almost like the `map`

method on Scala collection library,
except this `map`

doesn’t do the `CanBuildFrom`

auto conversion.

Cats defines a `Functor`

instance for `Either[A, B]`

.

```
scala> (Right(1): Either[String, Int]) map { _ + 1 }
res1: Either[String,Int] = Right(2)
scala> (Left("boom!"): Either[String, Int]) map { _ + 1 }
res2: Either[String,Int] = Left(boom!)
```

Note that the above demonstration only works because `Either[A, B]`

at the moment
does not implement its own `map`

.
If I used `List(1, 2, 3)`

it will call List’s implementation of `map`

instead of
the `Functor[List]`

’s `map`

. Therefore, even though the operator syntax looks familiar,
we should either avoid using it unless you’re sure that standard library doesn’t implement the `map`

or you’re using it from a polymorphic function.
One workaround is to opt for the function syntax.

Cats also defines a `Functor`

instance for `Function1`

.

```
scala> val h = ((x: Int) => x + 1) map {_ * 7}
h: Int => Int = <function1>
scala> h(3)
res3: Int = 28
```

This is interesting. Basically `map`

gives us a way to compose functions, except the order is in reverse from `f compose g`

. Another way of looking at `Function1`

is that it’s an infinite map from the domain to the range. Now let’s skip the input and output stuff and go to Functors, Applicative Functors and Monoids.

How are functions functors? …

What does the type

`fmap :: (a -> b) -> (r -> a) -> (r -> b)`

for this instance tell us? Well, we see that it takes a function from`a`

to`b`

and a function from`r`

to`a`

and returns a function from`r`

to`b`

. Does this remind you of anything? Yes! Function composition!

Oh man, LYAHFGG came to the same conclusion as I did about the function composition. But wait…

```
ghci> fmap (*3) (+100) 1
303
ghci> (*3) . (+100) $ 1
303
```

In Haskell, the `fmap`

seems to be working in the same order as `f compose g`

. Let’s check in Scala using the same numbers:

```
scala> (((_: Int) * 3) map {_ + 100}) (1)
res4: Int = 103
```

Something is not right. Let’s compare the declaration of `fmap`

and Cats’ `map`

function:

```
fmap :: (a -> b) -> f a -> f b
```

and here’s Cats:

```
def map[A, B](fa: F[A])(f: A => B): F[B]
```

So the order is flipped. Here’s Paolo Giarrusso (@blaisorblade)’s explanation:

That’s a common Haskell-vs-Scala difference.

In Haskell, to help with point-free programming, the “data” argument usually comes last. For instance, I can write

`map f . map g . map h`

and get a list transformer, because the argument order is`map f list`

. (Incidentally, map is an restriction of fmap to the List functor).In Scala instead, the “data” argument is usually the receiver. That’s often also important to help type inference, so defining map as a method on functions would not bring you very far: think the mess Scala type inference would make of

`(x => x + 1) map List(1, 2, 3)`

.

This seems to be the popular explanation.

LYAHFGG:

[We can think of

`fmap`

as] a function that takes a function and returns a new function that’s just like the old one, only it takes a functor as a parameter and returns a functor as the result. It takes an`a -> b`

function and returns a function`f a -> f b`

. This is calledliftinga function.

```
ghci> :t fmap (*2)
fmap (*2) :: (Num a, Functor f) => f a -> f a
ghci> :t fmap (replicate 3)
fmap (replicate 3) :: (Functor f) => f a -> f [a]
```

If the parameter order has been flipped, are we going to miss out on this lifting goodness?
Fortunately, Cats implements derived functions under the `Functor`

typeclass:

```
@typeclass trait Functor[F[_]] extends functor.Invariant[F] { self =>
def map[A, B](fa: F[A])(f: A => B): F[B]
....
// derived methods
/**
* Lift a function f to operate on Functors
*/
def lift[A, B](f: A => B): F[A] => F[B] = map(_)(f)
/**
* Empty the fa of the values, preserving the structure
*/
def void[A](fa: F[A]): F[Unit] = map(fa)(_ => ())
/**
* Tuple the values in fa with the result of applying a function
* with the value
*/
def fproduct[A, B](fa: F[A])(f: A => B): F[(A, B)] = map(fa)(a => a -> f(a))
/**
* Replaces the `A` value in `F[A]` with the supplied value.
*/
def as[A, B](fa: F[A], b: B): F[B] = map(fa)(_ => b)
}
```

As you see, we have `lift`

!

```
scala> val lifted = Functor[List].lift {(_: Int) * 2}
lifted: List[Int] => List[Int] = <function1>
scala> lifted(List(1, 2, 3))
res5: List[Int] = List(2, 4, 6)
```

We’ve just lifted the function `{(_: Int) * 2}`

to `List[Int] => List[Int]`

. Here the other derived functions using the operator syntax:

```
scala> List(1, 2, 3).void
res6: List[Unit] = List((), (), ())
scala> List(1, 2, 3) fproduct {(_: Int) * 2}
res7: List[(Int, Int)] = List((1,2), (2,4), (3,6))
scala> List(1, 2, 3) as "x"
res8: List[String] = List(x, x, x)
```

LYAHFGG:

In order for something to be a functor, it should satisfy some laws. All functors are expected to exhibit certain kinds of functor-like properties and behaviors. … The first functor law states that if we map the id function over a functor, the functor that we get back should be the same as the original functor.

We can check this for `Either[A, B]`

.

```
scala> val x: Either[String, Int] = Right(1)
x: Either[String,Int] = Right(1)
scala> assert { (x map identity) === x }
```

The second law says that 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.

In other words,

```
scala> val f = {(_: Int) * 3}
f: Int => Int = <function1>
scala> val g = {(_: Int) + 1}
g: Int => Int = <function1>
scala> assert { (x map (f map g)) === (x map f map g) }
```

These are laws the implementer of the functors must abide, and not something the compiler can check for you.

The compiler can’t check for the laws, but Cats ships with a `FunctorLaws`

trait that describes this in code:

```
/**
* Laws that must be obeyed by any [[Functor]].
*/
trait FunctorLaws[F[_]] extends InvariantLaws[F] {
implicit override def F: Functor[F]
def covariantIdentity[A](fa: F[A]): IsEq[F[A]] =
fa.map(identity) <-> fa
def covariantComposition[A, B, C](fa: F[A], f: A => B, g: B => C): IsEq[F[C]] =
fa.map(f).map(g) <-> fa.map(f andThen g)
}
```

This is based on a library called Discipline, which is a wrapper around ScalaCheck. We can run these tests from the REPL with ScalaCheck.

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> import cats.laws.discipline.FunctorTests
import cats.laws.discipline.FunctorTests
scala> val rs = FunctorTests[Either[Int, ?]].functor[Int, Int, Int]
rs: cats.laws.discipline.FunctorTests[[X_kp1]scala.util.Either[Int,X_kp1]]#RuleSet = cats.laws.discipline.FunctorTests$$anon$2@7993373d
scala> rs.all.check
+ functor.covariant composition: OK, passed 100 tests.
+ functor.covariant identity: OK, passed 100 tests.
+ functor.invariant composition: OK, passed 100 tests.
+ functor.invariant identity: OK, passed 100 tests.
```

`rs.all`

returns `org.scalacheck.Properties`

, which implements `check`

method.

You can also bake your own cake pattern into a test framework of choice. Here’s for specs2:

```
package example
import org.specs2.Specification
import org.typelevel.discipline.specs2.Discipline
import cats.instances.AllInstances
import cats.syntax.AllSyntax
trait CatsSpec extends Specification with Discipline with AllInstances with AllSyntax
```

Cats’ source include one for ScalaTest.

The spec to check the functor law for `Either[Int, Int]`

looks like this:

```
package example
import cats._
import cats.laws.discipline.FunctorTests
class EitherSpec extends CatsSpec { def is = s2"""
Either[Int, ?] forms a functor $e1
"""
def e1 = checkAll("Either[Int, Int]", FunctorTests[Either[Int, ?]].functor[Int, Int, Int])
}
```

The `Either[Int, ?]`

is using non/kind-projector.
Running the test from sbt displays the following output:

```
> test
[info] EitherSpec
[info]
[info]
[info] functor laws must hold for Either[Int, Int]
[info]
[info] + functor.covariant composition
[info] + functor.covariant identity
[info] + functor.invariant composition
[info] + functor.invariant identity
[info]
[info]
[info] Total for specification EitherSpec
[info] Finished in 14 ms
[info] 4 examples, 400 expectations, 0 failure, 0 error
[info] Passed: Total 4, Failed 0, Errors 0, Passed 4
```

LYAHFGG:

Let’s take a look at a pathological example of a type constructor being an instance of the Functor typeclass but not really being a functor, because it doesn’t satisfy the laws.

Let’s try breaking the law.

```
package example
import cats._
sealed trait COption[+A]
case class CSome[A](counter: Int, a: A) extends COption[A]
case object CNone extends COption[Nothing]
object COption {
implicit def coptionEq[A]: Eq[COption[A]] = new Eq[COption[A]] {
def eqv(a1: COption[A], a2: COption[A]): Boolean = a1 == a2
}
implicit val coptionFunctor = new Functor[COption] {
def map[A, B](fa: COption[A])(f: A => B): COption[B] =
fa match {
case CNone => CNone
case CSome(c, a) => CSome(c + 1, f(a))
}
}
}
```

Here’s how we can use this:

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> import example._
import example._
scala> (CSome(0, "ho"): COption[String]) map {identity}
res0: example.COption[String] = CSome(1,ho)
```

This breaks the first law because the result of the `identity`

function is not equal to the input.
To catch this we need to supply an “arbitrary” `COption[A]`

implicitly:

```
package example
import cats._
import cats.laws.discipline.{ FunctorTests }
import org.scalacheck.{ Arbitrary, Gen }
class COptionSpec extends CatsSpec {
implicit def coptionArbiterary[A](implicit arbA: Arbitrary[A]): Arbitrary[COption[A]] =
Arbitrary {
val arbSome = for {
i <- implicitly[Arbitrary[Int]].arbitrary
a <- arbA.arbitrary
} yield (CSome(i, a): COption[A])
val arbNone = Gen.const(CNone: COption[Nothing])
Gen.oneOf(arbSome, arbNone)
}
def is = s2"""
COption[Int] forms a functor $e1
"""
def e1 = checkAll("COption[Int]", FunctorTests[COption].functor[Int, Int, Int])
}
```

Here’s the output:

```
[info] COptionSpec
[info]
[info]
[info] functor laws must hold for COption[Int]
[info]
[info] x functor.covariant composition
[error] A counter-example is [CSome(-1,-1), <function1>, <function1>] (after 0 try)
[error] (CSome(1,1358703086) ?== CSome(0,1358703086)) failed
[info]
[info] x functor.covariant identity
[error] A counter-example is 'CSome(1781926821,82888113)' (after 0 try)
[error] (CSome(1781926822,82888113) ?== CSome(1781926821,82888113)) failed
[info]
[info] x functor.invariant composition
[error] A counter-example is [CSome(-17878015,0), <function1>, <function1>, <function1>, <function1>] (after 1 try)
[error] (CSome(-17878013,-1351608161) ?== CSome(-17878014,-1351608161)) failed
[info]
[info] x functor.invariant identity
[error] A counter-example is 'CSome(-1699259031,1)' (after 0 try)
[error] (CSome(-1699259030,1) ?== CSome(-1699259031,1)) failed
[info]
[info]
[info]
[info] Total for specification COptionSpec
[info] Finished in 13 ms
[info] 4 examples, 4 failures, 0 error
```

The tests failed as expected.

Cats makes heavy use of implicits. Both as a user and an extender of the library, it will be useful to have general idea on where things are coming from. If you’re just starting out with Cats, you can use the following the imports and skip this page:

```
scala> import cats._, cats.data._, cats.implicits._
```

Let’s quickly review Scala’s imports and implicits! In Scala, imports are used for two purposes:

- To include names of values and types into the scope.
- To include implicits into the scope.

Implicits are for 4 purposes that I can think of:

- To provide typeclass instances.
- To inject methods and operators. (static monkey patching)
- To declare type constraints.
- To retrieve type information from compiler.

Implicits are selected in the following precedence:

- Values and converters accessible without prefix via local declaration, imports, outer scope, inheritance, and current package object. Inner scope can shadow values when they are named the same.
- Implicit scope. Values and converters declared in companion objects and package object of the type, its parts, or super types.

Now let’s see what gets imported with `import cats._`

.

First, the names. Typeclasses like `Show[A]`

and `Functor[F[_]]`

are implemented as trait, and are defined under the `cats`

package. So instead of writing `cats.Show[A]`

we can write `Show[A]`

.

Next, also the names, but type aliases. `cats`

’s package object declares type aliases like `Eq[A]`

and `~>[F[_], G[_]]`

. Again, these can also be accessed as `cats.Eq[A]`

if you want.

Finally, `catsInstancesForId`

is defined as typeclass instance of `Id[A]`

for `Traverse[F[_]]`

, `Monad[F[_]]`

etc, but it’s not relevant. By virtue of declaring an instance within its package object it will be available, so importing doesn’t add much. Let’s check this:

```
scala> cats.Functor[cats.Id]
res0: cats.Functor[cats.Id] = cats.package$$anon$1@3c201c09
```

No import needed, which is a good thing. So, the merit of `import cats._`

is for convenience, and it’s optional.

Next let’s see what gets imported with `import cats.data._`

.

First, more names. There are custom datatype defined under the `cats.data`

package such as `Validated[+E, +A]`

.

Next, the type aliases. The `cats.data`

package object defines type aliases such as `Reader[A, B]`

, which is treated as a specialization of `ReaderT`

transformer. We can still write this as `cats.data.Reader[A, B]`

.

What then is `import cats.implicits._`

doing? Here’s the definition of implicits object:

```
package cats
object implicits extends syntax.AllSyntax with instances.AllInstances
```

This is quite a nice way of organizing the imports. `implicits`

object itself doesn’t define anythig and it just mixes in the traits. We are going to look at each traits in detail, but they can also be imported a la carte, dim sum style. Back to the full course.

Thus far, I have been intentionally conflating the concept of typeclass instances and method injection (aka enrich my library). But the fact that `(Int, +)`

forms a `Monoid`

and that `Monoid`

introduces `|+|`

operator are two different things.

One of the interesting design of Cats is that it rigorously separates the two concepts into “instance” and “syntax.” Even if it makes logical sense to some users, the choice of symbolic operators can often be a point of contention with any libraries. Libraries and tools such as sbt, dispatch, and specs introduce its own DSL, and their effectiveness have been hotly debated.

`AllInstances`

is a trait that mixes in all the typeclass instances for built-in datatypes such as `Either[A, B]`

and `Option[A]`

.

```
package cats
package instances
trait AllInstances
extends FunctionInstances
with StringInstances
with EitherInstances
with ListInstances
with OptionInstances
with SetInstances
with StreamInstances
with VectorInstances
with AnyValInstances
with MapInstances
with BigIntInstances
with BigDecimalInstances
with FutureInstances
with TryInstances
with TupleInstances
with UUIDInstances
with SymbolInstances
```

`AllSyntax`

is a trait that mixes in all of the operators available in Cats.

```
package cats
package syntax
trait AllSyntax
extends ApplicativeSyntax
with ApplicativeErrorSyntax
with ApplySyntax
with BifunctorSyntax
with BifoldableSyntax
with BitraverseSyntax
with CartesianSyntax
with CoflatMapSyntax
with ComonadSyntax
with ComposeSyntax
with ContravariantSyntax
with CoproductSyntax
with EitherSyntax
with EqSyntax
....
```

Or, I’d like to call dim sum style, where they bring in a cart load of chinese dishes and you pick what you want.

If for whatever reason if you do not wish to import the entire `cats.implicits._`

, you can pick and choose.

Typeclass instances are broken down by the datatypes. Here’s how to get all typeclass instances for `Option`

:

```
scala> {
import cats.instances.option._
cats.Monad[Option].pure(0)
}
res0: Option[Int] = Some(0)
```

If you just want all instances, here’s how to load them all:

```
scala> {
import cats.instances.all._
cats.Monoid[Int].empty
}
res1: Int = 0
```

Because we have not injected any operators, you would have to work more with helper functions and functions under typeclass instances, which could be exactly what you want.

Typeclass syntax are broken down by the typeclass. Here’s how to get injected methods and operators for `Eq`

s:

```
scala> {
import cats.syntax.eq._
import cats.instances.all._
1 === 1
}
res2: Boolean = true
```

Cats datatype syntax like `Writer`

are also available under `cats.syntax`

package:

```
scala> {
import cats.syntax.writer._
import cats.instances.all._
1.tell
}
res3: cats.data.Writer[Int,Unit] = WriterT((1,()))
```

Standard datatype syntax are broken down by the datatypes. Here’s how to get injected methods and functions for `Option`

:

```
scala> {
import cats.syntax.option._
import cats.instances.all._
1.some
}
res4: Option[Int] = Some(1)
```

Here’s how to load all syntax and all instances.

```
scala> {
import cats.syntax.all._
import cats.instances.all._
1.some
}
res5: Option[Int] = Some(1)
```

This is the same as importing `cats.implicits._`

.
Again, if you are at all confused by this, just stick with the following first:

```
scala> import cats._, cats.data._, cats.implicits._
```

Yesterday we started with defining our own typeclasses using simulacrum, and ended with checking for functor laws using Discipline.

Learn You a Haskell For Great Good says:

Types are little labels that values carry so that we can reason about the values. But types have their own little labels, called kinds. A kind is more or less the type of a type. … What are kinds and what are they good for? Well, let’s examine the kind of a type by using the :k command in GHCI.

Scala 2.10 didn’t have a `:k`

command, so I wrote kind.scala.
Thanks to George Leontiev (@folone) and others, `:kind`

command is now part of Scala 2.11 (scala/scala#2340). Let’s try using it:

```
scala> :k Int
scala.Int's kind is A
scala> :k -v Int
scala.Int's kind is A
*
This is a proper type.
```

`Int`

and every other types that you can make a value out of are called a proper type and denoted with a `*`

symbol (read “type”). This is analogous to the value `1`

at value-level. Using Scala’s type variable notation this could be written as `A`

.

```
scala> :k -v Option
scala.Option's kind is F[+A]
* -(+)-> *
This is a type constructor: a 1st-order-kinded type.
scala> :k -v Either
scala.util.Either's kind is F[+A1,+A2]
* -(+)-> * -(+)-> *
This is a type constructor: a 1st-order-kinded type.
```

These are normally called *type constructors*. Another way of looking at it is that it’s one step removed from a proper type. So we can call it a first-order-kinded type. This is analogous to a first-order value `(_: Int) + 3`

, which we would normally call a function at the value level.

The curried notation uses arrows like `* -> *`

and `* -> * -> *`

. Note, `Option[Int]`

is `*`

; `Option`

is `* -> *`

. Using Scala’s type variable notation they could be written as `F[+A]`

and `F[+A1,+A2]`

.

```
scala> :k -v Eq
algebra.Eq's kind is F[A]
* -> *
This is a type constructor: a 1st-order-kinded type.
```

Scala encodes (or complects) the notion of typeclasses using type constructors.
When looking at this, think of it as `Eq`

is a typeclass for `A`

, a proper type.
This should make sense because you would pass `Int`

into `Eq`

.

```
scala> :k -v Functor
cats.Functor's kind is X[F[A]]
(* -> *) -> *
This is a type constructor that takes type constructor(s): a higher-kinded type.
```

Again, Scala encodes typeclasses using type constructors,
so when looking at this, think of it as `Functor`

is a typeclass for `F[A]`

, a type constructor.
This should also make sense because you would pass `List`

into `Functor`

.

In other words, this is a type constructor that accepts another type constructor.
This is analogous to a higher-order function, and thus called *higher-kinded type*.
These are denoted as `(* -> *) -> *`

. Using Scala’s type variable notation this could be written as `X[F[A]]`

.

The terminology around typeclasses tends to get jumbled up.
For example, the pair `(Int, +)`

forms a typeclass called monoid.
Colloquially, we say things like “is *X* a monoid?” to mean “can *X* form a monoid under some operation?”

An example of this is `Either[A, B]`

, which we implied “is-a” functor yesterday.
This is not completely accurate because, even though it might not be useful, we *could have* defined another left-biased functor.

Functors, Applicative Functors and Monoids:

So far, when we were mapping functions over functors, we usually mapped functions that take only one parameter. But what happens when we map a function like

`*`

, which takes two parameters, over a functor?

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> val hs = Functor[List].map(List(1, 2, 3, 4)) ({(_: Int) * (_:Int)}.curried)
hs: List[Int => Int] = List(<function1>, <function1>, <function1>, <function1>)
scala> Functor[List].map(hs) {_(9)}
res6: List[Int] = List(9, 18, 27, 36)
```

LYAHFGG:

But what if we have a functor value of

`Just (3 *)`

and a functor value of`Just 5`

, and we want to take out the function from`Just(3 *)`

and map it over`Just 5`

?Meet the

`Applicative`

typeclass. It lies in the`Control.Applicative`

module and it defines two methods,`pure`

and`<*>`

.

Cats splits this into `Cartesian`

, `Apply`

, and `Applicative`

. Here’s the contract for `Cartesian`

:

```
/**
* [[Cartesian]] captures the idea of composing independent effectful values.
* It is of particular interest when taken together with [[Functor]] - where [[Functor]]
* captures the idea of applying a unary pure function to an effectful value,
* calling `product` with `map` allows one to apply a function of arbitrary arity to multiple
* independent effectful values.
*
* That same idea is also manifested in the form of [[Apply]], and indeed [[Apply]] extends both
* [[Cartesian]] and [[Functor]] to illustrate this.
*/
@typeclass trait Cartesian[F[_]] {
def product[A, B](fa: F[A], fb: F[B]): F[(A, B)]
}
```

Cartesian defines `product`

function, which produces a pair of `(A, B)`

wrapped in effect `F[_]`

out of `F[A]`

and `F[B]`

. The symbolic alias for `product`

is `|@|`

also known as the applicative style.

Before we move on, let’s look at the syntax that Cats adds to create an `Option`

value.

```
scala> 9.some
res7: Option[Int] = Some(9)
scala> none[Int]
res8: Option[Int] = None
```

We can write `(Some(9): Option[Int])`

as `9.some`

.

LYAHFGG:

With the

`Applicative`

type class, we can chain the use of the`<*>`

function, thus enabling us to seamlessly operate on several applicative values instead of just one.

Here’s an example in Haskell:

```
ghci> pure (-) <*> Just 3 <*> Just 5
Just (-2)
```

Cats comes with the `CartesianBuilder`

syntax.

```
scala> (3.some |@| 5.some) map { _ - _ }
res9: Option[Int] = Some(-2)
scala> (none[Int] |@| 5.some) map { _ - _ }
res10: Option[Int] = None
scala> (3.some |@| none[Int]) map { _ - _ }
res11: Option[Int] = None
```

This shows that `Option`

forms `Cartesian`

.

LYAHFGG:

Lists (actually the list type constructor,

`[]`

) are applicative functors. What a surprise!

Let’s see if we can use the `CartesianBuilder`

sytax:

```
scala> (List("ha", "heh", "hmm") |@| List("?", "!", ".")) map {_ + _}
res12: List[String] = List(ha?, ha!, ha., heh?, heh!, heh., hmm?, hmm!, hmm.)
```

`Cartesian`

enables two operators, `<*`

and `*>`

, which are special cases of `Apply[F].product`

:

```
abstract class CartesianOps[F[_], A] extends Cartesian.Ops[F, A] {
def |@|[B](fb: F[B]): CartesianBuilder[F]#CartesianBuilder2[A, B] =
new CartesianBuilder[F] |@| self |@| fb
def *>[B](fb: F[B])(implicit F: Functor[F]): F[B] = F.map(typeClassInstance.product(self, fb)) { case (a, b) => b }
def <*[B](fb: F[B])(implicit F: Functor[F]): F[A] = F.map(typeClassInstance.product(self, fb)) { case (a, b) => a }
}
```

The definition looks simple enough, but the effect is cool:

```
scala> 1.some <* 2.some
res13: Option[Int] = Some(1)
scala> none[Int] <* 2.some
res14: Option[Int] = None
scala> 1.some *> 2.some
res15: Option[Int] = Some(2)
scala> none[Int] *> 2.some
res16: Option[Int] = None
```

If either side fails, we get `None`

.

`Cartesian`

has a single law called associativity:

```
trait CartesianLaws[F[_]] {
implicit def F: Cartesian[F]
def cartesianAssociativity[A, B, C](fa: F[A], fb: F[B], fc: F[C]): (F[(A, (B, C))], F[((A, B), C)]) =
(F.product(fa, F.product(fb, fc)), F.product(F.product(fa, fb), fc))
}
```

Functors, Applicative Functors and Monoids:

LYAHFGG:

But what if we have a functor value of

`Just (3 *)`

and a functor value of`Just 5`

, and we want to take out the function from`Just(3 *)`

and map it over`Just 5`

?Meet the

`Applicative`

typeclass. It lies in the`Control.Applicative`

module and it defines two methods,`pure`

and`<*>`

.

Cats splits `Applicative`

into `Cartesian`

, `Apply`

, and `Applicative`

. Here’s the contract for `Apply`

:

```
/**
* Weaker version of Applicative[F]; has apply but not pure.
*
* Must obey the laws defined in cats.laws.ApplyLaws.
*/
@typeclass(excludeParents = List("ApplyArityFunctions"))
trait Apply[F[_]] extends Functor[F] with Cartesian[F] with ApplyArityFunctions[F] { self =>
/**
* Given a value and a function in the Apply context, applies the
* function to the value.
*/
def ap[A, B](ff: F[A => B])(fa: F[A]): F[B]
....
}
```

Note that `Apply`

extends `Functor`

, `Cartesian`

, and `ApplyArityFunctions`

.
The `<*>`

function is called `ap`

in Cats’ `Apply`

. (This was originally called `apply`

, but was renamed to `ap`

. +1)

LYAHFGG:

You can think of

`<*>`

as a sort of a beefed-up`fmap`

. Whereas`fmap`

takes a function and a functor and applies the function inside the functor value,`<*>`

takes a functor that has a function in it and another functor and extracts that function from the first functor and then maps it over the second one.

Here’s how we can use it with `Apply[Option].ap`

:

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> Apply[Option].ap({{(_: Int) + 3}.some })(9.some)
res0: Option[Int] = Some(12)
scala> Apply[Option].ap({{(_: Int) + 3}.some })(10.some)
res1: Option[Int] = Some(13)
scala> Apply[Option].ap({{(_: String) + "hahah"}.some })(none[String])
res2: Option[String] = None
scala> Apply[Option].ap({ none[String => String] })("woot".some)
res3: Option[String] = None
```

If either side fails, we get `None`

.

If you remember Making our own typeclass with simulacrum from yesterday, simulacrum will automatically transpose the function defined on the typeclass contract into an operator, magically.

```
scala> ({(_: Int) + 3}.some) ap 9.some
res4: Option[Int] = Some(12)
scala> ({(_: Int) + 3}.some) ap 10.some
res5: Option[Int] = Some(13)
scala> ({(_: String) + "hahah"}.some) ap none[String]
res6: Option[String] = None
scala> (none[String => String]) ap "woot".some
res7: Option[String] = None
```

LYAHFGG:

`Control.Applicative`

defines a function that’s called`liftA2`

, which has a type of

```
liftA2 :: (Applicative f) => (a -> b -> c) -> f a -> f b -> f c .
```

Remember parameters are flipped around in Scala.
What we have is a function that takes `F[B]`

and `F[A]`

, then a function `(A, B) => C`

.
This is called `map2`

on `Apply`

.

```
@typeclass(excludeParents = List("ApplyArityFunctions"))
trait Apply[F[_]] extends Functor[F] with Cartesian[F] with ApplyArityFunctions[F] { self =>
/**
* Given a value and a function in the Apply context, applies the
* function to the value.
*/
def ap[A, B](ff: F[A => B])(fa: F[A]): F[B]
override def product[A, B](fa: F[A], fb: F[B]): F[(A, B)] =
ap(map(fa)(a => (b: B) => (a, b)))(fb)
/**
* ap2 is a binary version of ap, defined in terms of ap.
*/
def ap2[A, B, Z](ff: F[(A, B) => Z])(fa: F[A], fb: F[B]): F[Z] =
map(product(fa, product(fb, ff))) { case (a, (b, f)) => f(a, b) }
/**
* Applies the pure (binary) function f to the effectful values fa and fb.
*
* map2 can be seen as a binary version of [[cats.Functor]]#map.
*/
def map2[A, B, Z](fa: F[A], fb: F[B])(f: (A, B) => Z): F[Z] =
map(product(fa, fb)) { case (a, b) => f(a, b) }
....
}
```

For binary operators, `map2`

can be used to hide the applicative style.
Here we can write the same thing in two different ways:

```
scala> (3.some |@| List(4).some) map { _ :: _ }
res8: Option[List[Int]] = Some(List(3, 4))
scala> Apply[Option].map2(3.some, List(4).some) { _ :: _ }
res9: Option[List[Int]] = Some(List(3, 4))
```

The results match up.

The 2-parameter version of `Apply[F].ap`

is called `Apply[F].ap2`

:

```
scala> Apply[Option].ap2({{ (_: Int) :: (_: List[Int]) }.some })(3.some, List(4).some)
res10: Option[List[Int]] = Some(List(3, 4))
```

There’s a special case of `map2`

called `tuple2`

, which works like this:

```
scala> Apply[Option].tuple2(1.some, 2.some)
res11: Option[(Int, Int)] = Some((1,2))
scala> Apply[Option].tuple2(1.some, none[Int])
res12: Option[(Int, Int)] = None
```

If you are wondering what happens when you have a function with more than two
parameters, note that `Apply[F[_]]`

extends `ApplyArityFunctions[F]`

.
This is auto-generated code that defines `ap3`

, `map3`

, `tuple3`

, … up to
`ap22`

, `map22`

, `tuple22`

.

`Apply`

has a single law called composition:

```
trait ApplyLaws[F[_]] extends FunctorLaws[F] {
implicit override def F: Apply[F]
def applyComposition[A, B, C](fa: F[A], fab: F[A => B], fbc: F[B => C]): IsEq[F[C]] = {
val compose: (B => C) => (A => B) => (A => C) = _.compose
fa.ap(fab).ap(fbc) <-> fa.ap(fab.ap(fbc.map(compose)))
}
}
```

**Note**: If you jumped to this page because you’re interested in applicative functors,
you should definitely read Cartesian and Apply first.

Functors, Applicative Functors and Monoids:

Meet the

`Applicative`

typeclass. It lies in the`Control.Applicative`

module and it defines two methods,`pure`

and`<*>`

.

Let’s see Cats’ `Applicative`

:

```
@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]
....
}
```

It’s an extension of `Apply`

with `pure`

.

LYAHFGG:

`pure`

should take a value of any type and return an applicative value with that value inside it. … A better way of thinking about`pure`

would be to say that it takes a value and puts it in some sort of default (or pure) context—a minimal context that still yields that value.

It seems like it’s basically a constructor that takes value `A`

and returns `F[A]`

.

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> Applicative[List].pure(1)
res0: List[Int] = List(1)
scala> Applicative[Option].pure(1)
res1: Option[Int] = Some(1)
```

This actually comes in handy using `Apply[F].ap`

so we can avoid calling `{{...}.some}`

.

```
scala> val F = Applicative[Option]
F: cats.Applicative[Option] = cats.instances.OptionInstances$$anon$1@1caab648
scala> F.ap({ F.pure((_: Int) + 3) })(F.pure(9))
res2: Option[Int] = Some(12)
```

We’ve abstracted `Option`

away from the code.

LYAHFGG:

Let’s try implementing a function that takes a list of applicatives and returns an applicative that has a list as its result value. We’ll call it

`sequenceA`

.

```
sequenceA :: (Applicative f) => [f a] -> f [a]
sequenceA [] = pure []
sequenceA (x:xs) = (:) <$> x <*> sequenceA xs
```

Let’s try implementing this with Cats!

```
scala> import cats.syntax.cartesian._
import cats.syntax.cartesian._
scala> def sequenceA[F[_]: Applicative, A](list: List[F[A]]): F[List[A]] = list match {
case Nil => Applicative[F].pure(Nil: List[A])
case x :: xs => (x |@| sequenceA(xs)) map {_ :: _}
}
sequenceA: [F[_], A](list: List[F[A]])(implicit evidence$1: cats.Applicative[F])F[List[A]]
```

Let’s test it:

```
scala> sequenceA(List(1.some, 2.some))
res3: Option[List[Int]] = Some(List(1, 2))
scala> sequenceA(List(3.some, none[Int], 1.some))
res4: Option[List[Int]] = None
scala> sequenceA(List(List(1, 2, 3), List(4, 5, 6)))
res5: List[List[Int]] = List(List(1, 4), List(1, 5), List(1, 6), List(2, 4), List(2, 5), List(2, 6), List(3, 4), List(3, 5), List(3, 6))
```

We got the right answers. What’s interesting here is that we did end up needing
`Applicative`

after all, and `sequenceA`

is generic in a typeclassy way.

Using

`sequenceA`

is useful when we have a list of functions and we want to feed the same input to all of them and then view the list of results.

For `Function1`

with `Int`

fixed example, we need some type annotation:

```
scala> val f = sequenceA[Function1[Int, ?], Int](List((_: Int) + 3, (_: Int) + 2, (_: Int) + 1))
f: Int => List[Int] = <function1>
scala> f(3)
res6: List[Int] = List(6, 5, 4)
```

Here are the laws for `Applicative`

:

- identity:
`pure id <*> v = v`

- homomorphism:
`pure f <*> pure x = pure (f x)`

- interchange:
`u <*> pure y = pure ($ y) <*> u`

Cats defines another law

```
def applicativeMap[A, B](fa: F[A], f: A => B): IsEq[F[B]] =
fa.map(f) <-> fa.ap(F.pure(f))
```

This seem to say that if you combine `F.ap`

and `F.pure`

, you should get the same effect as `F.map`

.

It took us a while, but I am glad we got this far. We’ll pick it up from here later.

Yesterday we reviewed kinds and types, explored `Apply`

, applicative style, and ended with `sequenceA`

.

Let’s move on to `Semigroup`

and `Monoid`

today.

If you have the book *Learn You a Haskell for Great Good* you get to start a new chapter: “Monoids.” For the website, it’s still Functors, Applicative Functors and Monoids.

First, it seems like Cats is missing `newtype`

/tagged type facility.
We’ll implement our own later.

Haskell’s `Monoid`

is split into `Semigroup`

and `Monoid`

in Cats. They are also type aliases of `algebra.Semigroup`

and `algebra.Monoid`

. As with `Apply`

and `Applicative`

, `Semigroup`

is a weaker version of `Monoid`

. If you can solve the same problem, weaker is cooler because you’re making fewer assumptions.

LYAHFGG:

It doesn’t matter if we do

`(3 * 4) * 5`

or`3 * (4 * 5)`

. Either way, the result is`60`

. The same goes for`++`

. …We call this property

associativity.`*`

is associative, and so is`++`

, but`-`

, for example, is not.

Let’s check this:

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> assert { (3 * 2) * (8 * 5) === 3 * (2 * (8 * 5)) }
scala> assert { List("la") ++ (List("di") ++ List("da")) === (List("la") ++ List("di")) ++ List("da") }
```

No error means, they are equal.

Here’s the typeclass contract for `algebra.Semigroup`

.

```
/**
* A semigroup is any set `A` with an associative operation (`combine`).
*/
trait Semigroup[@sp(Int, Long, Float, Double) A] extends Any with Serializable {
/**
* Associative operation taking which combines two values.
*/
def combine(x: A, y: A): A
....
}
```

This enables `combine`

operator and its symbolic alias `|+|`

. Let’s try using this.

```
scala> List(1, 2, 3) |+| List(4, 5, 6)
res2: List[Int] = List(1, 2, 3, 4, 5, 6)
scala> "one" |+| "two"
res3: String = onetwo
```

Associativity is the only law for `Semigroup`

.

- associativity
`(x |+| y) |+| z = x |+| (y |+| z)`

Here’s how we can check the Semigroup laws from the REPL. Review Checking laws with discipline for the details:

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> import cats.kernel.laws.GroupLaws
import cats.kernel.laws.GroupLaws
scala> val rs1 = GroupLaws[Int].semigroup(Semigroup[Int])
rs1: cats.kernel.laws.GroupLaws[Int]#GroupProperties = cats.kernel.laws.GroupLaws$GroupProperties@5a077d1d
scala> rs1.all.check
+ semigroup.associativity: OK, passed 100 tests.
+ semigroup.combineN(a, 1) == a: OK, passed 100 tests.
+ semigroup.combineN(a, 2) == a |+| a: OK, passed 100 tests.
+ semigroup.serializable: OK, proved property.
```

```
scala> List(1, 2, 3) |+| List(4, 5, 6)
res4: List[Int] = List(1, 2, 3, 4, 5, 6)
```

For `Int`

a semigroup can be formed under both `+`

and `*`

.
Instead of tagged types, cats provides only the instance additive.

Trying to use operator syntax here is tricky.

```
scala> def doSomething[A: Semigroup](a1: A, a2: A): A =
a1 |+| a2
doSomething: [A](a1: A, a2: A)(implicit evidence$1: cats.Semigroup[A])A
scala> doSomething(3, 5)(Semigroup[Int])
res5: Int = 8
```

I might as well stick to function syntax:

```
scala> Semigroup[Int].combine(3, 5)
res6: Int = 8
```

LYAHFGG:

It seems that both

`*`

together with`1`

and`++`

along with`[]`

share some common properties:

- The function takes two parameters.
- The parameters and the returned value have the same type.
- There exists such a value that doesn’t change other values when used with the binary function.

Let’s check it out in Scala:

```
scala> 4 * 1
res0: Int = 4
scala> 1 * 9
res1: Int = 9
scala> List(1, 2, 3) ++ Nil
res2: List[Int] = List(1, 2, 3)
scala> Nil ++ List(0.5, 2.5)
res3: List[Double] = List(0.5, 2.5)
```

Looks right.

Here’s the typeclass contract of `algebra.Monoid`

:

```
/**
* A monoid is a semigroup with an identity. A monoid is a specialization of a
* semigroup, so its operation must be associative. Additionally,
* `combine(x, empty) == combine(empty, x) == x`. For example, if we have `Monoid[String]`,
* with `combine` as string concatenation, then `empty = ""`.
*/
trait Monoid[@sp(Int, Long, Float, Double) A] extends Any with Semigroup[A] {
/**
* Return the identity element for this monoid.
*/
def empty: A
...
}
```

In addition to the semigroup law, monoid must satify two more laws:

- associativity
`(x |+| y) |+| z = x |+| (y |+| z)`

- left identity
`Monoid[A].empty |+| x = x`

- right identity
`x |+| Monoid[A].empty = x`

Here’s how we can check monoid laws from the REPL:

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> import cats.kernel.laws.GroupLaws
import cats.kernel.laws.GroupLaws
scala> val rs1 = GroupLaws[Int].monoid(Monoid[Int])
rs1: cats.kernel.laws.GroupLaws[Int]#GroupProperties = cats.kernel.laws.GroupLaws$GroupProperties@17a695f0
scala> rs1.all.check
+ monoid.associativity: OK, passed 100 tests.
+ monoid.combineAll(Nil) == id: OK, passed 100 tests.
+ monoid.combineN(a, 0) == id: OK, passed 100 tests.
+ monoid.combineN(a, 1) == a: OK, passed 100 tests.
+ monoid.combineN(a, 2) == a |+| a: OK, passed 100 tests.
+ monoid.isEmpty: OK, passed 100 tests.
+ monoid.leftIdentity: OK, passed 100 tests.
+ monoid.rightIdentity: OK, passed 100 tests.
+ monoid.serializable: OK, proved property.
```

Here’s the spec2 specification of the above:

```
package example
import cats._
import algebra.laws.GroupLaws
class IntSpec extends CatsSpec { def is = s2"""
(Int, +) should
form a monoid $e1
"""
def e1 = checkAll("Int", GroupLaws[Int].monoid(Monoid[Int]))
}
```

LYAHFGG:

The

newtypekeyword in Haskell is made exactly for these cases when we want to just take one type and wrap it in something to present it as another type.

Cats does not ship with a tagged-type facility, but Scala now has value classes. This will remain unboxed under certain conditions, so it should work for simple examples.

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
class Wrapper(val unwrap: Int) extends AnyVal
// Exiting paste mode, now interpreting.
defined class Wrapper
```

LYAHFGG:

Another type which can act like a monoid in two distinct but equally valid ways is

`Bool`

. The first way is to have the or function`||`

act as the binary function along with`False`

as the identity value. … The other way for`Bool`

to be an instance of`Monoid`

is to kind of do the opposite: have`&&`

be the binary function and then make`True`

the identity value.

Cats does not provide this, but we can implement it ourselves.

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> :paste
// Entering paste mode (ctrl-D to finish)
class Disjunction(val unwrap: Boolean) extends AnyVal
object Disjunction {
@inline def apply(b: Boolean): Disjunction = new Disjunction(b)
implicit val disjunctionMonoid: Monoid[Disjunction] = new Monoid[Disjunction] {
def combine(a1: Disjunction, a2: Disjunction): Disjunction =
Disjunction(a1.unwrap || a2.unwrap)
def empty: Disjunction = Disjunction(false)
}
implicit val disjunctionEq: Eq[Disjunction] = new Eq[Disjunction] {
def eqv(a1: Disjunction, a2: Disjunction): Boolean =
a1.unwrap == a2.unwrap
}
}
// Exiting paste mode, now interpreting.
defined class Disjunction
defined object Disjunction
scala> val x1 = Disjunction(true) |+| Disjunction(false)
x1: Disjunction = Disjunction@4cf
scala> x1.unwrap
res0: Boolean = true
scala> val x2 = Monoid[Disjunction].empty |+| Disjunction(true)
x2: Disjunction = Disjunction@4cf
scala> x2.unwrap
res1: Boolean = true
```

Here’s conjunction:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
class Conjunction(val unwrap: Boolean) extends AnyVal
object Conjunction {
@inline def apply(b: Boolean): Conjunction = new Conjunction(b)
implicit val conjunctionMonoid: Monoid[Conjunction] = new Monoid[Conjunction] {
def combine(a1: Conjunction, a2: Conjunction): Conjunction =
Conjunction(a1.unwrap && a2.unwrap)
def empty: Conjunction = Conjunction(true)
}
implicit val conjunctionEq: Eq[Conjunction] = new Eq[Conjunction] {
def eqv(a1: Conjunction, a2: Conjunction): Boolean =
a1.unwrap == a2.unwrap
}
}
// Exiting paste mode, now interpreting.
defined class Conjunction
defined object Conjunction
scala> val x3 = Conjunction(true) |+| Conjunction(false)
x3: Conjunction = Conjunction@4d5
scala> x3.unwrap
res2: Boolean = false
scala> val x4 = Monoid[Conjunction].empty |+| Conjunction(true)
x4: Conjunction = Conjunction@4cf
scala> x4.unwrap
res3: Boolean = true
```

We should check if our custom new types satisfy the the monoid laws.

```
scala> import algebra.laws.GroupLaws
import algebra.laws.GroupLaws
scala> import org.scalacheck.{ Arbitrary, Gen }
import org.scalacheck.{Arbitrary, Gen}
scala> implicit def arbDisjunction(implicit ev: Arbitrary[Boolean]): Arbitrary[Disjunction] =
Arbitrary { ev.arbitrary map { Disjunction(_) } }
arbDisjunction: (implicit ev: org.scalacheck.Arbitrary[Boolean])org.scalacheck.Arbitrary[Disjunction]
scala> val rs1 = GroupLaws[Disjunction].monoid
rs1: algebra.laws.GroupLaws[Disjunction]#GroupProperties = algebra.laws.GroupLaws$GroupProperties@77663edf
scala> rs1.all.check
+ monoid.associativity: OK, passed 100 tests.
+ monoid.combineAll(Nil) == id: OK, passed 100 tests.
+ monoid.combineN(a, 0) == id: OK, passed 100 tests.
+ monoid.combineN(a, 1) == a: OK, passed 100 tests.
+ monoid.combineN(a, 2) == a |+| a: OK, passed 100 tests.
+ monoid.isEmpty: OK, passed 100 tests.
+ monoid.leftIdentity: OK, passed 100 tests.
+ monoid.rightIdentity: OK, passed 100 tests.
! monoid.serializable: Falsified after 0 passed tests.
```

The test failed because our monoid is not `Serializable`

.
I’m confused as to why the monoid law is checking for `Serializable`

.
non/algebra#13 says it’s convenient for Spark. I feel like this should be a separate thing.

Update: It turns out, the failure is due to the fact I’m using REPL to define the typeclass instances!

```
scala> implicit def arbConjunction(implicit ev: Arbitrary[Boolean]): Arbitrary[Conjunction] =
Arbitrary { ev.arbitrary map { Conjunction(_) } }
arbConjunction: (implicit ev: org.scalacheck.Arbitrary[Boolean])org.scalacheck.Arbitrary[Conjunction]
scala> val rs2 = GroupLaws[Conjunction].monoid
rs2: algebra.laws.GroupLaws[Conjunction]#GroupProperties = algebra.laws.GroupLaws$GroupProperties@15f279d
scala> rs2.all.check
+ monoid.associativity: OK, passed 100 tests.
+ monoid.combineAll(Nil) == id: OK, passed 100 tests.
+ monoid.combineN(a, 0) == id: OK, passed 100 tests.
+ monoid.combineN(a, 1) == a: OK, passed 100 tests.
+ monoid.combineN(a, 2) == a |+| a: OK, passed 100 tests.
+ monoid.isEmpty: OK, passed 100 tests.
+ monoid.leftIdentity: OK, passed 100 tests.
+ monoid.rightIdentity: OK, passed 100 tests.
! monoid.serializable: Falsified after 0 passed tests.
```

Apart from the serializable rule, `Conjunction`

looks ok.

LYAHFGG:

One way is to treat

`Maybe a`

as a monoid only if its type parameter a is a monoid as well and then implement mappend in such a way that it uses the mappend operation of the values that are wrapped with`Just`

.

Let’s see if this is how Cats does it.

```
implicit def optionMonoid[A](implicit ev: Semigroup[A]): Monoid[Option[A]] =
new Monoid[Option[A]] {
def empty: Option[A] = None
def combine(x: Option[A], y: Option[A]): Option[A] =
x match {
case None => y
case Some(xx) => y match {
case None => x
case Some(yy) => Some(ev.combine(xx,yy))
}
}
}
```

If we replace `mappend`

with the equivalent `combine`

, the rest is just pattern matching.
Let’s try using it.

```
scala> none[String] |+| "andy".some
res4: Option[String] = Some(andy)
scala> 1.some |+| none[Int]
res5: Option[Int] = Some(1)
```

It works.

LYAHFGG:

But if we don’t know if the contents are monoids, we can’t use

`mappend`

between them, so what are we to do? Well, one thing we can do is to just discard the second value and keep the first one. For this, the`First a`

type exists.

Haskell is using `newtype`

to implement `First`

type constructor. Since we can’t prevent allocation for generic value class, we can just make a normal case class.

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
case class First[A: Eq](val unwrap: Option[A])
object First {
implicit def firstMonoid[A: Eq]: Monoid[First[A]] = new Monoid[First[A]] {
def combine(a1: First[A], a2: First[A]): First[A] =
First((a1.unwrap, a2.unwrap) match {
case (Some(x), _) => Some(x)
case (None, y) => y
})
def empty: First[A] = First(None: Option[A])
}
implicit def firstEq[A: Eq]: Eq[First[A]] = new Eq[First[A]] {
def eqv(a1: First[A], a2: First[A]): Boolean =
Eq[Option[A]].eqv(a1.unwrap, a2.unwrap)
}
}
// Exiting paste mode, now interpreting.
defined class First
defined object First
scala> First('a'.some) |+| First('b'.some)
res6: First[Char] = First(Some(a))
scala> First(none[Char]) |+| First('b'.some)
res7: First[Char] = First(Some(b))
```

Let’s check the laws:

```
scala> implicit def arbFirst[A: Eq](implicit ev: Arbitrary[Option[A]]): Arbitrary[First[A]] =
Arbitrary { ev.arbitrary map { First(_) } }
arbFirst: [A](implicit evidence$1: cats.Eq[A], implicit ev: org.scalacheck.Arbitrary[Option[A]])org.scalacheck.Arbitrary[First[A]]
scala> val rs3 = GroupLaws[First[Int]].monoid
rs3: algebra.laws.GroupLaws[First[Int]]#GroupProperties = algebra.laws.GroupLaws$GroupProperties@44736530
scala> rs3.all.check
+ monoid.associativity: OK, passed 100 tests.
+ monoid.combineAll(Nil) == id: OK, passed 100 tests.
+ monoid.combineN(a, 0) == id: OK, passed 100 tests.
+ monoid.combineN(a, 1) == a: OK, passed 100 tests.
+ monoid.combineN(a, 2) == a |+| a: OK, passed 100 tests.
+ monoid.isEmpty: OK, passed 100 tests.
+ monoid.leftIdentity: OK, passed 100 tests.
+ monoid.rightIdentity: OK, passed 100 tests.
! monoid.serializable: Falsified after 0 passed tests.
```

It thinks `First`

is not serializable either.

LYAHFGG:

If we want a monoid on

`Maybe a`

such that the second parameter is kept if both parameters of`mappend`

are`Just`

values,`Data.Monoid`

provides a the`Last a`

type.

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
case class Last[A: Eq](val unwrap: Option[A])
object Last {
implicit def lastMonoid[A: Eq]: Monoid[Last[A]] = new Monoid[Last[A]] {
def combine(a1: Last[A], a2: Last[A]): Last[A] =
Last((a1.unwrap, a2.unwrap) match {
case (_, Some(y)) => Some(y)
case (x, None) => x
})
def empty: Last[A] = Last(None: Option[A])
}
implicit def lastEq[A: Eq]: Eq[Last[A]] = new Eq[Last[A]] {
def eqv(a1: Last[A], a2: Last[A]): Boolean =
Eq[Option[A]].eqv(a1.unwrap, a2.unwrap)
}
}
// Exiting paste mode, now interpreting.
defined class Last
defined object Last
scala> Last('a'.some) |+| Last('b'.some)
res8: Last[Char] = Last(Some(b))
scala> Last('a'.some) |+| Last(none[Char])
res9: Last[Char] = Last(Some(a))
```

More law checking:

```
scala> implicit def arbLast[A: Eq](implicit ev: Arbitrary[Option[A]]): Arbitrary[Last[A]] =
Arbitrary { ev.arbitrary map { Last(_) } }
arbLast: [A](implicit evidence$1: cats.Eq[A], implicit ev: org.scalacheck.Arbitrary[Option[A]])org.scalacheck.Arbitrary[Last[A]]
scala> val rs4 = GroupLaws[Last[Int]].monoid
rs4: algebra.laws.GroupLaws[Last[Int]]#GroupProperties = algebra.laws.GroupLaws$GroupProperties@121fd6d9
scala> rs4.all.check
+ monoid.associativity: OK, passed 100 tests.
+ monoid.combineAll(Nil) == id: OK, passed 100 tests.
+ monoid.combineN(a, 0) == id: OK, passed 100 tests.
+ monoid.combineN(a, 1) == a: OK, passed 100 tests.
+ monoid.combineN(a, 2) == a |+| a: OK, passed 100 tests.
+ monoid.isEmpty: OK, passed 100 tests.
+ monoid.leftIdentity: OK, passed 100 tests.
+ monoid.rightIdentity: OK, passed 100 tests.
! monoid.serializable: Falsified after 0 passed tests.
```

I think we got a pretty good feel for monoids.

We covered lots of things around laws today. Why do we need laws anyway?

Laws are important because laws are important is a tautological statement, but there’s a grain of truth to it. Like traffic laws dictating that we drive on one side within a patch of land, some laws are convenient *if everyone follows them*.

What Cats/Haskell-style function programming allows us to do is
write code that abstracts out the data, container, execution model, etc.
The abstraction will make only the assumptions stated in the laws,
thus each `A: Monoid`

needs to satisfy the laws for the abstracted code to behave properly. We can call this the *utilitarian* view.

Even if we could accept the utility, why those laws? It’s because it’s on
HaskellWiki or one of SPJ’s papers. They might offer a starting point with an existing implementation, which we can mimic.
We can call this the *traditionalist* view. However, there is a danger of inheriting the design choices or even limitations that were made specifically for Haskell.
Functor in category theory, for instance, is a more general term than `Functor[F]`

. At least `fmap`

is a function that returns `F[A] => F[B]`

, which is related.
By the time we get to `map`

in Scala, we lose that because of type inference.

Eventually we should tie our understanding back to math. Monoid laws correspond with the mathematical definition of monoids, and from there we can reap the benefits of the known properties of monoids. This is relevant especially for monoid laws, because the three laws are the same as the axioms of the category, because a monoid is a special case of a category.

For the sake of learning, I think it’s ok to start out with cargo cult. We all learn language through imitation and pattern recognition.

LYAHFGG:

Because there are so many data structures that work nicely with folds, the

`Foldable`

type class was introduced. Much like`Functor`

is for things that can be mapped over, Foldable is for things that can be folded up!

The equivalent in Cats is also called `Foldable`

. Here’s the typeclass contract:

```
/**
* Data structures that can be folded to a summary value.
*
* In the case of a collection (such as `List` or `Set`), these
* methods will fold together (combine) the values contained in the
* collection to produce a single result. Most collection types have
* `foldLeft` methods, which will usually be used by the associationed
* `Fold[_]` instance.
*
* Foldable[F] is implemented in terms of two basic methods:
*
* - `foldLeft(fa, b)(f)` eagerly folds `fa` from left-to-right.
* - `foldLazy(fa, b)(f)` lazily folds `fa` from right-to-left.
*
* Beyond these it provides many other useful methods related to
* folding over F[A] values.
*
* See: [[https://www.cs.nott.ac.uk/~gmh/fold.pdf A tutorial on the universality and expressiveness of fold]]
*/
@typeclass trait Foldable[F[_]] extends Serializable { self =>
/**
* Left associative fold on 'F' using the function 'f'.
*/
def foldLeft[A, B](fa: F[A], b: B)(f: (B, A) => B): B
/**
* Right associative lazy fold on `F` using the folding function 'f'.
*
* This method evaluates `b` lazily (in some cases it will not be
* needed), and returns a lazy value. We are using `A => Fold[B]` to
* support laziness in a stack-safe way.
*
* For more detailed information about how this method works see the
* documentation for `Fold[_]`.
*/
def foldLazy[A, B](fa: F[A], lb: Lazy[B])(f: A => Fold[B]): Lazy[B] =
Lazy(partialFold[A, B](fa)(f).complete(lb))
/**
* Low-level method that powers `foldLazy`.
*/
def partialFold[A, B](fa: F[A])(f: A => Fold[B]): Fold[B]
....
}
```

We can use this as follows:

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> Foldable[List].foldLeft(List(1, 2, 3), 1) {_ * _}
res0: Int = 6
```

`Foldable`

comes with some useful functions/operators,
many of them taking advantage of the typeclasses.
Let’s try the `fold`

. `Monoid[A]`

gives us `empty`

and `combine`

, so that’s enough information to fold over things.

```
/**
* Fold implemented using the given Monoid[A] instance.
*/
def fold[A](fa: F[A])(implicit A: Monoid[A]): A =
foldLeft(fa, A.empty) { (acc, a) =>
A.combine(acc, a)
}
```

Let’s try this out.

```
scala> Foldable[List].fold(List(1, 2, 3))(Monoid[Int])
res1: Int = 6
```

There’s a variant called `foldMap`

that accepts a function.

```
/**
* Fold implemented by mapping `A` values into `B` and then
* combining them using the given `Monoid[B]` instance.
*/
def foldMap[A, B](fa: F[A])(f: A => B)(implicit B: Monoid[B]): B =
foldLeft(fa, B.empty) { (b, a) =>
B.combine(b, f(a))
}
```

Since the standard collection library doesn’t implement `foldMap`

,
we can now use this as an operator.

```
scala> List(1, 2, 3).foldMap(identity)(Monoid[Int])
res2: Int = 6
```

Another useful thing is that we can use this to convert the values into newtype.

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
class Conjunction(val unwrap: Boolean) extends AnyVal
object Conjunction {
@inline def apply(b: Boolean): Conjunction = new Conjunction(b)
implicit val conjunctionMonoid: Monoid[Conjunction] = new Monoid[Conjunction] {
def combine(a1: Conjunction, a2: Conjunction): Conjunction =
Conjunction(a1.unwrap && a2.unwrap)
def empty: Conjunction = Conjunction(true)
}
implicit val conjunctionEq: Eq[Conjunction] = new Eq[Conjunction] {
def eqv(a1: Conjunction, a2: Conjunction): Boolean =
a1.unwrap == a2.unwrap
}
}
// Exiting paste mode, now interpreting.
defined class Conjunction
defined object Conjunction
scala> val x = List(true, false, true) foldMap {Conjunction(_)}
x: Conjunction = Conjunction@4d5
scala> x.unwrap
res3: Boolean = false
```

This surely beats writing `Conjunction(true)`

for each of them and connecting them with `|+|`

.

We will pick it up from here later.

Derived from Bello Nock's Sky Walk by Chris Phutully

Yesterday we reviewed `Semigroup`

and `Monoid`

, implementing custom monoids along the way. We also looked at `Foldable`

that can `foldMap`

etc.

Starting with a few updates today. First, `Apply.apply`

that we looked at in day 3 has been renamed to `Apply.ap`

#308.

Second, do you remember that checking the monoid laws on a value class kept tripping on `Serializable`

?
That turned out not to be Cats’ fault. I went into Cats’ gitter chat and
Erik (@d6/@non) kindly pointed out that the reason my typeclass instances are not serializable is because they are defined on the REPL. Once I moved `First`

to `src/`

the laws passed fine:

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> import algebra.laws.GroupLaws
import algebra.laws.GroupLaws
scala> import org.scalacheck.{ Arbitrary, Gen }
import org.scalacheck.{Arbitrary, Gen}
scala> import example.First
import example.First
scala> implicit def arbFirst[A: Eq](implicit ev: Arbitrary[Option[A]]): Arbitrary[First[A]] =
| Arbitrary { ev.arbitrary map { First(_) } }
arbFirst: [A](implicit evidence$1: cats.Eq[A], implicit ev: org.scalacheck.Arbitrary[Option[A]])org.scalacheck.Arbitrary[example.First[A]]
scala> val rs = GroupLaws[First[Int]].monoid
rs: algebra.laws.GroupLaws[example.First[Int]]#GroupProperties = algebra.laws.GroupLaws$GroupProperties@77fac6ab
scala> rs.all.check
+ monoid.associativity: OK, passed 100 tests.
+ monoid.combineAll(Nil) == id: OK, passed 100 tests.
+ monoid.combineN(a, 0) == id: OK, passed 100 tests.
+ monoid.combineN(a, 1) == a: OK, passed 100 tests.
+ monoid.combineN(a, 2) == a |+| a: OK, passed 100 tests.
+ monoid.isEmpty: OK, passed 100 tests.
+ monoid.leftIdentity: OK, passed 100 tests.
+ monoid.rightIdentity: OK, passed 100 tests.
+ monoid.serializable: OK, proved property.
```

Jason Zaugg (@retronym) also pointed that, to support serialization beyond precisely the same version of Cats on both sides of the wire, we need to do more things like:

- Avoid anonymous classes (to avoid class name change)
- Tack
`@SerialVersionUID(0L)`

on everything

We get to start a new chapter today on Learn You a Haskell for Great Good.

Monads are a natural extension applicative functors, and they provide a solution to the following problem: If we have a value with context,

`m a`

, how do we apply it to a function that takes a normal`a`

and returns a value with a context.

Cats breaks down the Monad typeclass into two typeclasses: `FlatMap`

and `Monad`

.
Here’s the typeclass contract for FlatMap:

```
@typeclass trait FlatMap[F[_]] extends Apply[F] {
def flatMap[A, B](fa: F[A])(f: A => F[B]): F[B]
def tailRecM[A, B](a: A)(f: A => F[Either[A, B]]): F[B]
....
}
```

Note that `FlatMap`

extends `Apply`

, the weaker version of `Applicative`

. And here are the operators:

```
class FlatMapOps[F[_], A](fa: F[A])(implicit F: FlatMap[F]) {
def flatMap[B](f: A => F[B]): F[B] = F.flatMap(fa)(f)
def mproduct[B](f: A => F[B]): F[(A, B)] = F.mproduct(fa)(f)
def >>=[B](f: A => F[B]): F[B] = F.flatMap(fa)(f)
def >>[B](fb: F[B]): F[B] = F.flatMap(fa)(_ => fb)
}
```

It introduces the `flatMap`

operator and its symbolic alias `>>=`

. We’ll worry about the other operators later. We are used to `flapMap`

from the standard library:

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> (Right(3): Either[String, Int]) flatMap { x => Right(x + 1) }
res0: Either[String,Int] = Right(4)
```

Following the book, let’s explore `Option`

. In this section I’ll be less fussy about whether it’s using Cats’ typeclass or standard library’s implementation. Here’s `Option`

as a functor:

```
scala> "wisdom".some map { _ + "!" }
res1: Option[String] = Some(wisdom!)
scala> none[String] map { _ + "!" }
res2: Option[String] = None
```

Here’s `Option`

as an `Apply`

:

```
scala> ({(_: Int) + 3}.some) ap 3.some
res3: Option[Int] = Some(6)
scala> none[String => String] ap "greed".some
res4: Option[String] = None
scala> ({(_: String).toInt}.some) ap none[String]
res5: Option[Int] = None
```

Here’s `Option`

as a `FlatMap`

:

```
scala> 3.some flatMap { (x: Int) => (x + 1).some }
res6: Option[Int] = Some(4)
scala> "smile".some flatMap { (x: String) => (x + " :)").some }
res7: Option[String] = Some(smile :))
scala> none[Int] flatMap { (x: Int) => (x + 1).some }
res8: Option[Int] = None
scala> none[String] flatMap { (x: String) => (x + " :)").some }
res9: Option[String] = None
```

Just as expected, we get `None`

if the monadic value is `None`

.

FlatMap has a single law called associativity:

- associativity:
`(m flatMap f) flatMap g === m flatMap { x => f(x) flatMap {g} }`

Cats defines two more laws in `FlatMapLaws`

:

```
trait FlatMapLaws[F[_]] extends ApplyLaws[F] {
implicit override def F: FlatMap[F]
def flatMapAssociativity[A, B, C](fa: F[A], f: A => F[B], g: B => F[C]): IsEq[F[C]] =
fa.flatMap(f).flatMap(g) <-> fa.flatMap(a => f(a).flatMap(g))
def flatMapConsistentApply[A, B](fa: F[A], fab: F[A => B]): IsEq[F[B]] =
fab.ap(fa) <-> fab.flatMap(f => fa.map(f))
/**
* The composition of `cats.data.Kleisli` arrows is associative. This is
* analogous to [[flatMapAssociativity]].
*/
def kleisliAssociativity[A, B, C, D](f: A => F[B], g: B => F[C], h: C => F[D], a: A): IsEq[F[D]] = {
val (kf, kg, kh) = (Kleisli(f), Kleisli(g), Kleisli(h))
((kf andThen kg) andThen kh).run(a) <-> (kf andThen (kg andThen kh)).run(a)
}
}
```

Earlier I wrote that Cats breaks down the Monad typeclass into two typeclasses: `FlatMap`

and `Monad`

.
The `FlatMap`

-`Monad`

relationship forms a parallel with the `Apply`

-`Applicative`

relationship:

```
@typeclass trait Monad[F[_]] extends FlatMap[F] with Applicative[F] {
....
}
```

`Monad`

is a `FlatMap`

with `pure`

. Unlike Haskell, `Monad[F]`

extends `Applicative[F]`

so there’s no `return`

vs `pure`

discrepancies.

Derived from Bello Nock's Sky Walk by Chris Phutully

LYAHFGG:

Let’s say that [Pierre] keeps his balance if the number of birds on the left side of the pole and on the right side of the pole is within three. So if there’s one bird on the right side and four birds on the left side, he’s okay. But if a fifth bird lands on the left side, then he loses his balance and takes a dive.

Now let’s try implementing the `Pole`

example from the book.

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> type Birds = Int
defined type alias Birds
scala> case class Pole(left: Birds, right: Birds)
defined class Pole
```

I don’t think it’s common to alias `Int`

like this in Scala, but we’ll go with the flow. I am going to turn `Pole`

into a case class so I can implement `landLeft`

and `landRight`

as methods:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
case class Pole(left: Birds, right: Birds) {
def landLeft(n: Birds): Pole = copy(left = left + n)
def landRight(n: Birds): Pole = copy(right = right + n)
}
// Exiting paste mode, now interpreting.
defined class Pole
```

I think it looks better with some OO:

```
scala> Pole(0, 0).landLeft(2)
res0: Pole = Pole(2,0)
scala> Pole(1, 2).landRight(1)
res1: Pole = Pole(1,3)
scala> Pole(1, 2).landRight(-1)
res2: Pole = Pole(1,1)
```

We can chain these too:

```
scala> Pole(0, 0).landLeft(1).landRight(1).landLeft(2)
res3: Pole = Pole(3,1)
scala> Pole(0, 0).landLeft(1).landRight(4).landLeft(-1).landRight(-2)
res4: Pole = Pole(0,2)
```

As the book says, an intermediate value has failed but the calculation kept going. Now let’s introduce failures as `Option[Pole]`

:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
case class Pole(left: Birds, right: Birds) {
def landLeft(n: Birds): Option[Pole] =
if (math.abs((left + n) - right) < 4) copy(left = left + n).some
else none[Pole]
def landRight(n: Birds): Option[Pole] =
if (math.abs(left - (right + n)) < 4) copy(right = right + n).some
else none[Pole]
}
// Exiting paste mode, now interpreting.
defined class Pole
scala> Pole(0, 0).landLeft(2)
res5: Option[Pole] = Some(Pole(2,0))
scala> Pole(0, 3).landLeft(10)
res6: Option[Pole] = None
```

Now we can chain the `landLeft`

/`landRight`

using `flatMap`

or its symbolic alias `>>=`

.

```
scala> val rlr = Monad[Option].pure(Pole(0, 0)) >>= {_.landRight(2)} >>=
{_.landLeft(2)} >>= {_.landRight(2)}
rlr: Option[Pole] = Some(Pole(2,4))
```

Let’s see if monadic chaining simulates the pole balancing better:

```
scala> val lrlr = Monad[Option].pure(Pole(0, 0)) >>= {_.landLeft(1)} >>=
{_.landRight(4)} >>= {_.landLeft(-1)} >>= {_.landRight(-2)}
lrlr: Option[Pole] = None
```

It works. Take time to understand this example because this example highlights what a monad is.

- First,
`pure`

puts`Pole(0, 0)`

into a default context:`Pole(0, 0).some`

. - Then,
`Pole(0, 0).some >>= {_.landLeft(1)}`

happens. Since it’s a`Some`

value,`_.landLeft(1)`

gets applied to`Pole(0, 0)`

, resulting to`Pole(1, 0).some`

. - Next,
`Pole(1, 0).some >>= {_.landRight(4)}`

takes place. The result is`Pole(1, 4).some`

. Now we at at the max difference between left and right. `Pole(1, 4).some >>= {_.landLeft(-1)}`

happens, resulting to`none[Pole]`

. The difference is too great, and pole becomes off balance.`none[Pole] >>= {_.landRight(-2)}`

results automatically to`none[Pole]`

.

In this chain of monadic functions, the effect from one function is carried over to the next.

LYAHFGG:

We may also devise a function that ignores the current number of birds on the balancing pole and just makes Pierre slip and fall. We can call it

`banana`

.

Here’s the `banana`

that always fails:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
case class Pole(left: Birds, right: Birds) {
def landLeft(n: Birds): Option[Pole] =
if (math.abs((left + n) - right) < 4) copy(left = left + n).some
else none[Pole]
def landRight(n: Birds): Option[Pole] =
if (math.abs(left - (right + n)) < 4) copy(right = right + n).some
else none[Pole]
def banana: Option[Pole] = none[Pole]
}
// Exiting paste mode, now interpreting.
defined class Pole
scala> val lbl = Monad[Option].pure(Pole(0, 0)) >>= {_.landLeft(1)} >>=
{_.banana} >>= {_.landRight(1)}
lbl: Option[Pole] = None
```

LYAHFGG:

Instead of making functions that ignore their input and just return a predetermined monadic value, we can use the

`>>`

function.

Here’s how `>>`

behaves with `Option`

:

```
scala> none[Int] >> 3.some
res7: Option[Int] = None
scala> 3.some >> 4.some
res8: Option[Int] = Some(4)
scala> 3.some >> none[Int]
res9: Option[Int] = None
```

Let’s try replacing `banana`

with `>> none[Pole]`

:

```
scala> val lbl = Monad[Option].pure(Pole(0, 0)) >>= {_.landLeft(1)} >>
none[Pole] >>= {_.landRight(1)}
<console>:23: error: missing parameter type for expanded function ((x$1) => x$1.landLeft(1))
val lbl = Monad[Option].pure(Pole(0, 0)) >>= {_.landLeft(1)} >>
^
```

The type inference broke down all the sudden. The problem is likely the operator precedence. Programming in Scala says:

The one exception to the precedence rule, alluded to above, concerns assignment operators, which end in an equals character. If an operator ends in an equals character (

`=`

), and the operator is not one of the comparison operators`<=`

,`>=`

,`==`

, or`!=`

, then the precedence of the operator is the same as that of simple assignment (`=`

). That is, it is lower than the precedence of any other operator.

Note: The above description is incomplete. Another exception from the assignment operator rule is if it starts with (`=`

) like `===`

.

Because `>>=`

(bind) ends in the equals character, its precedence is the lowest, which forces `({_.landLeft(1)} >> (none: Option[Pole]))`

to evaluate first. There are a few unpalatable work arounds. First we can use dot-and-parens like normal method calls:

```
scala> Monad[Option].pure(Pole(0, 0)).>>=({_.landLeft(1)}).>>(none[Pole]).>>=({_.landRight(1)})
res10: Option[Pole] = None
```

Or we can recognize the precedence issue and place parens around just the right place:

```
scala> (Monad[Option].pure(Pole(0, 0)) >>= {_.landLeft(1)}) >> none[Pole] >>= {_.landRight(1)}
res11: Option[Pole] = None
```

Both yield the right result.

LYAHFGG:

Monads in Haskell are so useful that they got their own special syntax called

`do`

notation.

First, let’s write the nested lambda:

```
scala> 3.some >>= { x => "!".some >>= { y => (x.show + y).some } }
res12: Option[String] = Some(3!)
```

By using `>>=`

, any part of the calculation can fail:

```
scala> 3.some >>= { x => none[String] >>= { y => (x.show + y).some } }
res13: Option[String] = None
scala> (none: Option[Int]) >>= { x => "!".some >>= { y => (x.show + y).some } }
res14: Option[String] = None
scala> 3.some >>= { x => "!".some >>= { y => none[String] } }
res15: Option[String] = None
```

Instead of the `do`

notation in Haskell, Scala has the `for`

comprehension, which does similar things:

```
scala> for {
x <- 3.some
y <- "!".some
} yield (x.show + y)
res16: Option[String] = Some(3!)
```

LYAHFGG:

In a

`do`

expression, every line that isn’t a`let`

line is a monadic value.

That’s not quite accurate for `for`

, but we can come back to this later.

LYAHFGG:

Our tightwalker’s routine can also be expressed with

`do`

notation.

```
scala> def routine: Option[Pole] =
for {
start <- Monad[Option].pure(Pole(0, 0))
first <- start.landLeft(2)
second <- first.landRight(2)
third <- second.landLeft(1)
} yield third
routine: Option[Pole]
scala> routine
res17: Option[Pole] = Some(Pole(3,2))
```

We had to extract `third`

since `yield`

expects `Pole`

not `Option[Pole]`

.

LYAHFGG:

If we want to throw the Pierre a banana peel in

`do`

notation, we can do the following:

```
scala> def routine: Option[Pole] =
for {
start <- Monad[Option].pure(Pole(0, 0))
first <- start.landLeft(2)
_ <- none[Pole]
second <- first.landRight(2)
third <- second.landLeft(1)
} yield third
routine: Option[Pole]
scala> routine
res18: Option[Pole] = None
```

LYAHFGG:

In

`do`

notation, when we bind monadic values to names, we can utilize pattern matching, just like in let expressions and function parameters.

```
scala> def justH: Option[Char] =
for {
(x :: xs) <- "hello".toList.some
} yield x
justH: Option[Char]
scala> justH
res19: Option[Char] = Some(h)
```

When pattern matching fails in a do expression, the

`fail`

function is called. It’s part of the`Monad`

type class and it enables failed pattern matching to result in a failure in the context of the current monad instead of making our program crash.

```
scala> def wopwop: Option[Char] =
for {
(x :: xs) <- "".toList.some
} yield x
wopwop: Option[Char]
scala> wopwop
res20: Option[Char] = None
```

The failed pattern matching returns `None`

here. This is an interesting aspect of `for`

syntax that I haven’t thought about, but totally makes sense.

Monad had three laws:

- left identity:
`(Monad[F].pure(x) flatMap {f}) === f(x)`

- right identity:
`(m flatMap {Monad[F].pure(_)}) === m`

- associativity:
`(m flatMap f) flatMap g === m flatMap { x => f(x) flatMap {g} }`

LYAHFGG:

The first monad law states that if we take a value, put it in a default context with

`return`

and then feed it to a function by using`>>=`

, it’s the same as just taking the value and applying the function to it.

```
scala> assert { (Monad[Option].pure(3) >>= { x => (x + 100000).some }) ===
({ (x: Int) => (x + 100000).some })(3) }
```

LYAHFGG:

The second law states that if we have a monadic value and we use

`>>=`

to feed it to`return`

, the result is our original monadic value.

```
scala> assert { ("move on up".some >>= {Monad[Option].pure(_)}) === "move on up".some }
```

LYAHFGG:

The final monad law says that when we have a chain of monadic function applications with

`>>=`

, it shouldn’t matter how they’re nested.

```
scala> Monad[Option].pure(Pole(0, 0)) >>= {_.landRight(2)} >>= {_.landLeft(2)} >>= {_.landRight(2)}
res23: Option[Pole] = Some(Pole(2,4))
scala> Monad[Option].pure(Pole(0, 0)) >>= { x =>
x.landRight(2) >>= { y =>
y.landLeft(2) >>= { z =>
z.landRight(2)
}}}
res24: Option[Pole] = Some(Pole(2,4))
```

These laws look might look familiar if you remember monoid laws from day 4. That’s because monad is a special kind of a monoid.

You might be thinking, “But wait. Isn’t `Monoid`

for kind `A`

(or `*`

)?”
Yes, you’re right. And that’s the difference between monoid with lowercase *m* and `Monoid[A]`

.
Haskell-style functional programming allows you to abstract out the container and execution model.
In category theory, a notion like monoid can be generalized to `A`

, `F[A]`

, `F[A] => F[B]`

and all sorts of things.
Instead of thinking “omg so many laws,” know that there’s an underlying structure that connects many of them.

Here’s how to check Monad laws using Discipline:

```
scala> import cats._, cats.data._, cats.implicits._, cats.laws.discipline.MonadTests
import cats._
import cats.data._
import cats.implicits._
import cats.laws.discipline.MonadTests
scala> val rs = MonadTests[Option].monad[Int, Int, Int]
rs: cats.laws.discipline.MonadTests[Option]#RuleSet = cats.laws.discipline.MonadTests$$anon$2@35e8de37
scala> rs.all.check
+ monad.applicative homomorphism: OK, passed 100 tests.
+ monad.applicative identity: OK, passed 100 tests.
+ monad.applicative interchange: OK, passed 100 tests.
+ monad.applicative map: OK, passed 100 tests.
+ monad.apply composition: OK, passed 100 tests.
+ monad.covariant composition: OK, passed 100 tests.
+ monad.covariant identity: OK, passed 100 tests.
+ monad.flatMap associativity: OK, passed 100 tests.
+ monad.flatMap consistent apply: OK, passed 100 tests.
+ monad.invariant composition: OK, passed 100 tests.
+ monad.invariant identity: OK, passed 100 tests.
+ monad.monad left identity: OK, passed 100 tests.
+ monad.monad right identity: OK, passed 100 tests.
```

LYAHFGG:

On the other hand, a value like

`[3,8,9]`

contains several results, so we can view it as one value that is actually many values at the same time. Using lists as applicative functors showcases this non-determinism nicely.

Let’s look at using `List`

as Applicatives again:

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> (List(1, 2, 3) |@| List(10, 100, 100)) map { _ * _ }
res0: List[Int] = List(10, 100, 100, 20, 200, 200, 30, 300, 300)
```

let’s try feeding a non-deterministic value to a function:

```
scala> List(3, 4, 5) >>= { x => List(x, -x) }
res1: List[Int] = List(3, -3, 4, -4, 5, -5)
```

So in this monadic view, a `List`

context represents a mathematical value that could have multiple solutions. Other than that manipulating `List`

s using `for`

notation is just like plain Scala:

```
scala> for {
n <- List(1, 2)
ch <- List('a', 'b')
} yield (n, ch)
res2: List[(Int, Char)] = List((1,a), (1,b), (2,a), (2,b))
```

Scala’s `for`

comprehension allows filtering:

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> for {
x <- (1 to 50).toList if x.show contains '7'
} yield x
res0: List[Int] = List(7, 17, 27, 37, 47)
```

Here’s the typeclass contract for `FunctorFilter`:

```
@typeclass trait FunctorFilter[F[_]] extends Functor[F] {
/**
* A combined [[map]] and [[filter]]. Filtering is handled via `Option`
* instead of `Boolean` such that the output type `B` can be different than
* the input type `A`.
* ....
*
**/
def mapFilter[A, B](fa: F[A])(f: A => Option[B]): F[B]
}
```

We can use this like this:

```
scala> val english = Map(1 -> "one", 3 -> "three", 10 -> "ten")
english: scala.collection.immutable.Map[Int,String] = Map(1 -> one, 3 -> three, 10 -> ten)
scala> (1 to 50).toList mapFilter { english.get(_) }
res1: List[String] = List(one, three, ten)
```

This enables the derivative functions/operators `collect`

, `flattenOption`

, and `filter`

:

```
@typeclass trait FunctorFilter[F[_]] extends Functor[F] {
def mapFilter[A, B](fa: F[A])(f: A => Option[B]): F[B]
/**
* Similar to [[mapFilter]] but uses a partial function instead of a function
* that returns an `Option`.
*/
def collect[A, B](fa: F[A])(f: PartialFunction[A, B]): F[B] =
mapFilter(fa)(f.lift)
/**
* "Flatten" out a structure by collapsing `Option`s.
*/
def flattenOption[A](fa: F[Option[A]]): F[A] = mapFilter(fa)(identity)
/**
* Apply a filter to a structure such that the output structure contains all
* `A` elements in the input structure that satisfy the predicate `f` but none
* that don't.
*/
def filter[A](fa: F[A])(f: A => Boolean): F[A] =
mapFilter(fa)(a => if (f(a)) Some(a) else None)
}
```

We can use this like this:

```
scala> def collectEnglish[F[_]: FunctorFilter](f: F[Int]): F[String] =
f collect {
case 1 => "one"
case 3 => "three"
case 10 => "ten"
}
collectEnglish: [F[_]](f: F[Int])(implicit evidence$1: cats.FunctorFilter[F])F[String]
scala> collectEnglish((1 to 50).toList)
res2: List[String] = List(one, three, ten)
scala> def filterSeven[F[_]: FunctorFilter](f: F[Int]): F[Int] =
f filter { _.show contains '7' }
filterSeven: [F[_]](f: F[Int])(implicit evidence$1: cats.FunctorFilter[F])F[Int]
scala> filterSeven((1 to 50).toList)
res3: List[Int] = List(7, 17, 27, 37, 47)
```

Here’s the typeclass contract for `MonadFilter`:

```
@typeclass trait MonadFilter[F[_]] extends Monad[F] with FunctorFilter[F] {
def empty[A]: F[A]
...
}
```

We can use this to get an empty value of a datatype that supports it.

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> MonadFilter[List].empty[Int]
res0: List[Int] = List()
```

LYAHFGG:

Here’s a problem that really lends itself to being solved with non-determinism. Say you have a chess board and only one knight piece on it. We want to find out if the knight can reach a certain position in three moves.

Instead of type aliasing a pair, let’s make this into a case class again:

```
scala> case class KnightPos(c: Int, r: Int)
defined class KnightPos
```

Here’s the function to calculate all of the knight’s next positions:

```
scala> case class KnightPos(c: Int, r: Int) {
def move: List[KnightPos] =
for {
KnightPos(c2, r2) <- List(KnightPos(c + 2, r - 1), KnightPos(c + 2, r + 1),
KnightPos(c - 2, r - 1), KnightPos(c - 2, r + 1),
KnightPos(c + 1, r - 2), KnightPos(c + 1, r + 2),
KnightPos(c - 1, r - 2), KnightPos(c - 1, r + 2)) if (
((1 to 8).toList contains c2) && ((1 to 8).toList contains r2))
} yield KnightPos(c2, r2)
}
defined class KnightPos
scala> KnightPos(6, 2).move
res1: List[KnightPos] = List(KnightPos(8,1), KnightPos(8,3), KnightPos(4,1), KnightPos(4,3), KnightPos(7,4), KnightPos(5,4))
scala> KnightPos(8, 1).move
res2: List[KnightPos] = List(KnightPos(6,2), KnightPos(7,3))
```

The answers look good. Now we implement chaining this three times:

```
scala> case class KnightPos(c: Int, r: Int) {
def move: List[KnightPos] =
for {
KnightPos(c2, r2) <- List(KnightPos(c + 2, r - 1), KnightPos(c + 2, r + 1),
KnightPos(c - 2, r - 1), KnightPos(c - 2, r + 1),
KnightPos(c + 1, r - 2), KnightPos(c + 1, r + 2),
KnightPos(c - 1, r - 2), KnightPos(c - 1, r + 2)) if (
((1 to 8).toList contains c2) && ((1 to 8).toList contains r2))
} yield KnightPos(c2, r2)
def in3: List[KnightPos] =
for {
first <- move
second <- first.move
third <- second.move
} yield third
def canReachIn3(end: KnightPos): Boolean = in3 contains end
}
defined class KnightPos
scala> KnightPos(6, 2) canReachIn3 KnightPos(6, 1)
res3: Boolean = true
scala> KnightPos(6, 2) canReachIn3 KnightPos(7, 3)
res4: Boolean = false
```

As it turns out, from `(6, 2)`

you can reach `(6, 1)`

in three moves, but not `(7, 3)`

. As with Pierre’s bird example, one of key aspect of the monadic calculation is that the effect of one move can be passed on to the next.

We’ll pick up from here.

Yesterday we looked at `FlatMap`

and `Monad`

typeclass. We looked at how monadic chaining can add contexts to values. Because both `Option`

and `List`

already have `flatMap`

in the standard library, it was more about changing the way we see things rather than introducing new code. We also reviewed `for`

syntax as a way of chaining monadic operations.

Before I jump into the topic, I want to mention Pamflet, the Scala-based blog/book platform I’m using here. Nathan Hamblen (@n8han) started Pamflet and I’ve been contributing features to it too. Speaking of which, the source of these posts are also available on eed3si9n/herding-cats if you’re curious how this is built. A special thanks to Leif Wickland (@leifwickland) for proofreading all of the posts and sending me pull requests with fixes!

There are subtle differences in Haskell’s `do`

notation and Scala’s `for`

syntax. Here’s an example of `do`

notation:

```
foo = do
x <- Just 3
y <- Just "!"
Just (show x ++ y)
```

Typically one would write `return (show x ++ y)`

, but I wrote out `Just`

, so it’s clear that the last line is a monadic value. On the other hand, Scala would look as follows:

```
scala> def foo = for {
x <- Some(3)
y <- Some("!")
} yield x.toString + y
foo: Option[String]
```

Looks similar, but there are some differences.

- Scala doesn’t have a built-in
`Monad`

type. Instead, the compiler desugars`for`

comprehensions into`map`

,`withFilter`

,`flatMap`

, and`foreach`

calls mechanically. SLS 6.19 - For things like
`Option`

and`List`

that the standard library implements`map`

/`flatMap`

, the built-in implementations would be prioritized over the typeclasses provided by Cats. - The Scala collection library’s
`map`

etc accepts`CanBuildFrom`

, which may convert`F[A]`

into`G[B]`

. See The Architecture of Scala Collections `CanBuildFrom`

may convert some`G[A]`

into`F[B]`

.`yield`

with a pure value is required, otherwise`for`

turns into`Unit`

.

Here are some demonstration of these points:

```
scala> import collection.immutable.BitSet
import collection.immutable.BitSet
scala> val bits = BitSet(1, 2, 3)
bits: scala.collection.immutable.BitSet = BitSet(1, 2, 3)
scala> for {
x <- bits
} yield x.toFloat
res0: scala.collection.immutable.SortedSet[Float] = TreeSet(1.0, 2.0, 3.0)
scala> for {
i <- List(1, 2, 3)
j <- Some(1)
} yield i + j
res1: List[Int] = List(2, 3, 4)
scala> for {
i <- Map(1 -> 2)
j <- Some(3)
} yield j
res2: scala.collection.immutable.Iterable[Int] = List(3)
```

There are several DSLs around in Scala that transforms imperative-looking code into monadic or applicative function calls using macros:

Covering full array of Scala syntax in the macro is hard work,
but by copy-pasting code from Async and Effectful I put together
a toy macro that supports only simple expressions and `val`

s.
I’ll omit the details, but the key function is this:

```
def transform(group: BindGroup, isPure: Boolean): Tree =
group match {
case (binds, tree) =>
binds match {
case Nil =>
if (isPure) q"""$monadInstance.pure($tree)"""
else tree
case (name, unwrappedFrom) :: xs =>
val innerTree = transform((xs, tree), isPure)
val param = ValDef(Modifiers(Flag.PARAM), name, TypeTree(), EmptyTree)
q"""$monadInstance.flatMap($unwrappedFrom) { $param => $innerTree }"""
}
}
```

Here’s how we can use `actM`

:

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> import example.MonadSyntax._
import example.MonadSyntax._
scala> actM[Option, String] {
val x = 3.some.next
val y = "!".some.next
x.toString + y
}
res3: Option[String] = Some(3!)
```

`fa.next`

expands to a `Monad[F].flatMap(fa)()`

call.
So the above code expands into:

```
scala> Monad[Option].flatMap[String, String]({
val fa0: Option[Int] = 3.some
Monad[Option].flatMap[Int, String](fa0) { (arg0: Int) => {
val next0: Int = arg0
val x: Int = next0
val fa1: Option[String] = "!".some
Monad[Option].flatMap[String, String](fa1)((arg1: String) => {
val next1: String = arg1
val y: String = next1
Monad[Option].pure[String](x.toString + y)
})
}}
}) { (arg2: String) => Monad[Option].pure[String](arg2) }
res4: Option[String] = Some(3!)
```

Let’s see if this can prevent auto conversion from `Option`

to `List`

.

```
scala> actM[List, Int] {
val i = List(1, 2, 3).next
val j = 1.some.next
i + j
}
<console>:25: error: exception during macro expansion:
scala.reflect.macros.TypecheckException: type mismatch;
found : fa$macro$15.type (with underlying type Option[Int] @scala.reflect.internal.annotations.uncheckedBounds)
required: List[?]
at scala.reflect.macros.contexts.Typers$$anonfun$typecheck$2$$anonfun$apply$1.apply(Typers.scala:34)
at scala.reflect.macros.contexts.Typers$$anonfun$typecheck$2$$anonfun$apply$1.apply(Typers.scala:28)
at scala.reflect.macros.contexts.Typers$$anonfun$3.apply(Typers.scala:24)
at scala.reflect.macros.contexts.Typers$$anonfun$3.apply(Typers.scala:24)
at scala.reflect.macros.contexts.Typers$$anonfun$withContext$1$1.apply(Typers.scala:25)
at scala.reflect.macros.contexts.Typers$$anonfun$withContext$1$1.apply(Typers.scala:25)
at scala.reflect.macros.contexts.Typers$$anonfun$1.apply(Typers.scala:23)
at scala.reflect.macros.contexts.Typers$$anonfun$1.apply(Typers.scala:23)
at scala.reflect.macros.contexts.Typers$class.withContext$1(Typers.scala:25)
at scala.reflect.macros.contexts.Typers$$anonfun$typecheck$2.apply(Typers.scala:28)
at scala.reflect.macros.contexts.Typers$$anonfun$typecheck$2.apply(Typers.scala:28)
at scala.reflect.internal.Trees$class.wrappingIntoTerm(Trees.scala:1716)
at scala.reflect.internal.SymbolTable.wrappingIntoTerm(SymbolTable.scala:16)
at scala.reflect.macros.contexts.Typers$class.withWrapping$1(Typers.scala:26)
at scala.reflect.macros.contexts.Typers$class.typecheck(Typers.scala:28)
at scala.reflect.macros.contexts.Context.typecheck(Context.scala:6)
at scala.reflect.macros.contexts.Context.typecheck(Context.scala:6)
at example.internal.ActMTransform$class.actMTransform(ActMTransform.scala:27)
at example.internal.ActMMacro$$anon$1.actMTransform(ActMMacro.scala:8)
at example.internal.ActMBase.actMImpl(ActMBase.scala:13)
at example.internal.ActMImpl$.actMImpl(ActMImpl.scala:9)
actM[List, Int] {
^
```

The error message is a bit rough, but we were able to catch this at compile-time.
This will also work for any monads including `Future`

.

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
val x = {
import scala.concurrent.{ExecutionContext, Future}
import ExecutionContext.Implicits.global
actM[Future, Int] {
val i = Future { 1 }.next
val j = Future { 2 }.next
i + j
}
}
// Exiting paste mode, now interpreting.
x: scala.concurrent.Future[Int] = List()
scala> x.value
res6: Option[scala.util.Try[Int]] = Some(Success(3))
```

This macro is incomplete toy code, but it demonstrates potential usefulness for having something like this.

Learn You a Haskell for Great Good says:

Whereas the

`Maybe`

monad is for values with an added context of failure, and the list monad is for nondeterministic values,`Writer`

monad is for values that have another value attached that acts as a sort of log value.

Let’s follow the book and implement `applyLog`

function:

```
scala> def isBigGang(x: Int): (Boolean, String) =
(x > 9, "Compared gang size to 9.")
isBigGang: (x: Int)(Boolean, String)
scala> implicit class PairOps[A](pair: (A, String)) {
def applyLog[B](f: A => (B, String)): (B, String) = {
val (x, log) = pair
val (y, newlog) = f(x)
(y, log ++ newlog)
}
}
defined class PairOps
scala> (3, "Smallish gang.") applyLog isBigGang
res0: (Boolean, String) = (false,Smallish gang.Compared gang size to 9.)
```

Since method injection is a common use case for implicits, Scala 2.10 adds a syntax sugar called implicit class to make the promotion from a class to an enriched class easier.
Here’s how we can generalize the log to a `Semigroup`

:

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> implicit class PairOps[A, B: Semigroup](pair: (A, B)) {
def applyLog[C](f: A => (C, B)): (C, B) = {
val (x, log) = pair
val (y, newlog) = f(x)
(y, log |+| newlog)
}
}
defined class PairOps
```

LYAHFGG:

To attach a monoid to a value, we just need to put them together in a tuple. The

`Writer w a`

type is just a`newtype`

wrapper for this.

In Cats, the equivalent is called `Writer`:

```
type Writer[L, V] = WriterT[Id, L, V]
object Writer {
def apply[L, V](l: L, v: V): WriterT[Id, L, V] = WriterT[Id, L, V]((l, v))
def value[L:Monoid, V](v: V): Writer[L, V] = WriterT.value(v)
def tell[L](l: L): Writer[L, Unit] = WriterT.tell(l)
}
```

`Writer[L, V]`

is a type alias for `WriterT[Id, L, V]`

Here’s the simplified version of `WriterT`:

```
final case class WriterT[F[_], L, V](run: F[(L, V)]) {
def tell(l: L)(implicit functorF: Functor[F], semigroupL: Semigroup[L]): WriterT[F, L, V] =
mapWritten(_ |+| l)
def written(implicit functorF: Functor[F]): F[L] =
functorF.map(run)(_._1)
def value(implicit functorF: Functor[F]): F[V] =
functorF.map(run)(_._2)
def mapBoth[M, U](f: (L, V) => (M, U))(implicit functorF: Functor[F]): WriterT[F, M, U] =
WriterT { functorF.map(run)(f.tupled) }
def mapWritten[M](f: L => M)(implicit functorF: Functor[F]): WriterT[F, M, V] =
mapBoth((l, v) => (f(l), v))
}
```

Here’s how we can create `Writer`

values:

```
scala> val w = Writer("Smallish gang.", 3)
w: cats.data.WriterT[cats.Id,String,Int] = WriterT((Smallish gang.,3))
scala> val v = Writer.value[String, Int](3)
v: cats.data.Writer[String,Int] = WriterT((,3))
scala> val l = Writer.tell[String]("Log something")
l: cats.data.Writer[String,Unit] = WriterT((Log something,()))
```

To run the `Writer`

datatype you can call its `run`

method:

```
scala> w.run
res1: cats.Id[(String, Int)] = (Smallish gang.,3)
```

LYAHFGG:

Now that we have a

`Monad`

instance, we’re free to use`do`

notation for`Writer`

values.

```
scala> def logNumber(x: Int): Writer[List[String], Int] =
Writer(List("Got number: " + x.show), 3)
logNumber: (x: Int)cats.data.Writer[List[String],Int]
scala> def multWithLog: Writer[List[String], Int] =
for {
a <- logNumber(3)
b <- logNumber(5)
} yield a * b
multWithLog: cats.data.Writer[List[String],Int]
scala> multWithLog.run
res2: cats.Id[(List[String], Int)] = (List(Got number: 3, Got number: 5),9)
```

Here’s the `gcd`

example:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
def gcd(a: Int, b: Int): Writer[List[String], Int] = {
if (b == 0) for {
_ <- Writer.tell(List("Finished with " + a.show))
} yield a
else
Writer.tell(List(s"${a.show} mod ${b.show} = ${(a % b).show}")) >>= { _ =>
gcd(b, a % b)
}
}
// Exiting paste mode, now interpreting.
gcd: (a: Int, b: Int)cats.data.Writer[List[String],Int]
scala> gcd(12, 16).run
res3: cats.Id[(List[String], Int)] = (List(12 mod 16 = 12, 16 mod 12 = 4, 12 mod 4 = 0, Finished with 4),4)
```

LYAHFGG:

When using the

`Writer`

monad, you have to be careful which monoid to use, because using lists can sometimes turn out to be very slow. That’s because lists use`++`

for`mappend`

and using`++`

to add something to the end of a list is slow if that list is really long.

Here’s the table of performance characteristics for major collections. What stands out for immutable collection is `Vector`

since it has effective constant for all operations. `Vector`

is a tree structure with the branching factor of 32, and it’s able to achieve fast updates by structure sharing.

Here’s the Vector version of `gcd`

:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
def gcd(a: Int, b: Int): Writer[Vector[String], Int] = {
if (b == 0) for {
_ <- Writer.tell(Vector("Finished with " + a.show))
} yield a
else
Writer.tell(Vector(s"${a.show} mod ${b.show} = ${(a % b).show}")) >>= { _ =>
gcd(b, a % b)
}
}
// Exiting paste mode, now interpreting.
gcd: (a: Int, b: Int)cats.data.Writer[Vector[String],Int]
scala> gcd(12, 16).run
res4: cats.Id[(Vector[String], Int)] = (Vector(12 mod 16 = 12, 16 mod 12 = 4, 12 mod 4 = 0, Finished with 4),4)
```

Like the book let’s write a microbenchmark to compare the performance:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
def vectorFinalCountDown(x: Int): Writer[Vector[String], Unit] = {
import annotation.tailrec
@tailrec def doFinalCountDown(x: Int, w: Writer[Vector[String], Unit]): Writer[Vector[String], Unit] = x match {
case 0 => w >>= { _ => Writer.tell(Vector("0")) }
case x => doFinalCountDown(x - 1, w >>= { _ =>
Writer.tell(Vector(x.show))
})
}
val t0 = System.currentTimeMillis
val r = doFinalCountDown(x, Writer.tell(Vector[String]()))
val t1 = System.currentTimeMillis
r >>= { _ => Writer.tell(Vector((t1 - t0).show + " msec")) }
}
def listFinalCountDown(x: Int): Writer[List[String], Unit] = {
import annotation.tailrec
@tailrec def doFinalCountDown(x: Int, w: Writer[List[String], Unit]): Writer[List[String], Unit] = x match {
case 0 => w >>= { _ => Writer.tell(List("0")) }
case x => doFinalCountDown(x - 1, w >>= { _ =>
Writer.tell(List(x.show))
})
}
val t0 = System.currentTimeMillis
val r = doFinalCountDown(x, Writer.tell(List[String]()))
val t1 = System.currentTimeMillis
r >>= { _ => Writer.tell(List((t1 - t0).show + " msec")) }
}
// Exiting paste mode, now interpreting.
vectorFinalCountDown: (x: Int)cats.data.Writer[Vector[String],Unit]
listFinalCountDown: (x: Int)cats.data.Writer[List[String],Unit]
```

Here’s the result I got on my machine:

```
scala> vectorFinalCountDown(10000).run._1.last
res17: String = 6 msec
scala> listFinalCountDown(10000).run._1.last
res18: String = 630 msec
```

As you can see, `List`

is 100 times slower.

Learn You a Haskell for Great Good says:

In the chapter about applicatives, we saw that the function type,

`(->) r`

is an instance of`Functor`

.

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> val f = (_: Int) * 2
f: Int => Int = <function1>
scala> val g = (_: Int) + 10
g: Int => Int = <function1>
scala> (g map f)(8)
res0: Int = 36
```

We’ve also seen that functions are applicative functors. They allow us to operate on the eventual results of functions as if we already had their results.

```
scala> val h = (f |@| g) map {_ + _}
h: Int => Int = <function1>
scala> h(3)
res1: Int = 19
```

Not only is the function type

`(->) r a`

functor and an applicative functor, but it’s also a monad. Just like other monadic values that we’ve met so far, a function can also be considered a value with a context. The context for functions is that that value is not present yet and that we have to apply that function to something in order to get its result value.

Let’s try implementing the example:

```
scala> val addStuff: Int => Int = for {
a <- (_: Int) * 2
b <- (_: Int) + 10
} yield a + b
addStuff: Int => Int = <function1>
scala> addStuff(3)
res2: Int = 19
```

Both

`(*2)`

and`(+10)`

get applied to the number`3`

in this case.`return (a+b)`

does as well, but it ignores it and always presents`a+b`

as the result. For this reason, the function monad is also called thereadermonad. All the functions read from a common source.

The `Reader`

monad lets us pretend the value is already there. I am guessing that this works only for functions that accepts one parameter.

At nescala 2012 on March 9th, Rúnar (@runarorama) gave a talk Dead-Simple Dependency Injection.
One of the ideas presented there was to use the `Reader`

monad for dependency injection. Later that year, he also gave a longer version of the talk Lambda: The Ultimate Dependency Injection Framework in YOW 2012.
In 2013, Jason Arhart wrote Scrap Your Cake Pattern Boilerplate: Dependency Injection Using the Reader Monad,
which I’m going to base my example on.

Imagine we have a case class for a user, and a trait that abstracts the data store to get them.

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
case class User(id: Long, parentId: Long, name: String, email: String)
trait UserRepo {
def get(id: Long): User
def find(name: String): User
}
// Exiting paste mode, now interpreting.
defined class User
defined trait UserRepo
```

Next we define a primitive reader for each operation defined in the `UserRepo`

trait:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
trait Users {
def getUser(id: Long): UserRepo => User = {
case repo => repo.get(id)
}
def findUser(name: String): UserRepo => User = {
case repo => repo.find(name)
}
}
// Exiting paste mode, now interpreting.
defined trait Users
```

That looks like boilerplate. (I thought we are scrapping it.) Moving on.

Based on the primitive readers, we can compose other readers, including the application.

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
object UserInfo extends Users {
def userInfo(name: String): UserRepo => Map[String, String] =
for {
user <- findUser(name)
boss <- getUser(user.parentId)
} yield Map(
"name" -> s"${user.name}",
"email" -> s"${user.email}",
"boss_name" -> s"${boss.name}"
)
}
trait Program {
def app: UserRepo => String =
for {
fredo <- UserInfo.userInfo("Fredo")
} yield fredo.toString
}
// Exiting paste mode, now interpreting.
defined object UserInfo
defined trait Program
```

To run this `app`

, we need something that provides an implementation for `UserRepo`

:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
val testUsers = List(User(0, 0, "Vito", "vito@example.com"),
User(1, 0, "Michael", "michael@example.com"),
User(2, 0, "Fredo", "fredo@example.com"))
object Main extends Program {
def run: String = app(mkUserRepo)
def mkUserRepo: UserRepo = new UserRepo {
def get(id: Long): User = (testUsers find { _.id === id }).get
def find(name: String): User = (testUsers find { _.name === name }).get
}
}
Main.run
// Exiting paste mode, now interpreting.
testUsers: List[User] = List(User(0,0,Vito,vito@example.com), User(1,0,Michael,michael@example.com), User(2,0,Fredo,fredo@example.com))
defined object Main
res3: String = Map(name -> Fredo, email -> fredo@example.com, boss_name -> Vito)
```

We got the boss man’s name.

We can try using `actM`

instead of a `for`

comprehension:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
object UserInfo extends Users {
import example.MonadSyntax._
def userInfo(name: String): UserRepo => Map[String, String] =
actM[UserRepo => ?, Map[String, String]] {
val user = findUser(name).next
val boss = getUser(user.parentId).next
Map(
"name" -> s"${user.name}",
"email" -> s"${user.email}",
"boss_name" -> s"${boss.name}"
)
}
}
trait Program {
import example.MonadSyntax._
def app: UserRepo => String =
actM[UserRepo => ?, String] {
val fredo = UserInfo.userInfo("Fredo").next
fredo.toString
}
}
object Main extends Program {
def run: String = app(mkUserRepo)
def mkUserRepo: UserRepo = new UserRepo {
def get(id: Long): User = (testUsers find { _.id === id }).get
def find(name: String): User = (testUsers find { _.name === name }).get
}
}
Main.run
// Exiting paste mode, now interpreting.
defined object UserInfo
defined trait Program
defined object Main
res4: String = Map(name -> Fredo, email -> fredo@example.com, boss_name -> Vito)
```

The code inside of the `actM`

block looks more natural than the `for`

version,
but the type annotations probably make it more difficult to use.

That’s all for today.

On day 6 we compared Scala’s `for`

comprehension with Haskell’s `do`

,
and implemented the `actM`

macro.
We also looked at the `Reader`

datatype, which is another way of thinking about `Function1[A, B]`

.

I had been a bit sloppy by describing `List`

and `Reader`

as monads,
but I want to correctly say `List`

datatype and `Reader`

datatype,
which form a monad under some operation.

When writing code using an immutable data structure,
one pattern that arises often is passing of a value that represents some state.
The example I like to use is Tetris. Imagine a functional implementation of Tetris
where `Tetrix.init`

creates the initial state, and then various
transition functions return a transformed state and some return value:

```
val (s0, _) = Tetrix.init()
val (s1, _) = Tetrix.nextBlock(s0)
val (s2, moved0) = Tetrix.moveBlock(s1, LEFT)
val (s3, moved1) =
if (moved0) Tetrix.moveBlock(s2, LEFT)
else (s2, moved0)
```

The passing of the state objects (`s0`

, `s1`

, `s2`

, …) becomes error-prone boilerplate.
The goal is to automate the explicit passing of the states.

To follow along the book, we’ll use the stack example from the book.
Here’s an implementation without using `State`

.

```
scala> type Stack = List[Int]
defined type alias Stack
scala> def pop(s0: Stack): (Stack, Int) =
s0 match {
case x :: xs => (xs, x)
case Nil => sys.error("stack is empty")
}
pop: (s0: Stack)(Stack, Int)
scala> def push(s0: Stack, a: Int): (Stack, Unit) = (a :: s0, ())
push: (s0: Stack, a: Int)(Stack, Unit)
scala> def stackManip(s0: Stack): (Stack, Int) = {
val (s1, _) = push(s0, 3)
val (s2, a) = pop(s1)
pop(s2)
}
stackManip: (s0: Stack)(Stack, Int)
scala> stackManip(List(5, 8, 2, 1))
res0: (Stack, Int) = (List(8, 2, 1),5)
```

Learn You a Haskell for Great Good says:

Haskell features the

`State`

monad, which makes dealing with stateful problems a breeze while still keeping everything nice and pure. ….We’ll say that a stateful computation is a function that takes some state and returns a value along with some new state. That function would have the following type:

```
s -> (a, s)
```

`State`

is a datatype that encapsulates a stateful computation: `S => (S, A)`

.
`State`

*forms* a monad which passes along the states represented by the type `S`

.
Haskell should’ve named this `Stater`

or `Program`

to avoid the confusion,
but now people already know this by `State`

, so it’s too late.

Cody Allen (@ceedubs) had a pull request open for `State`

/`StateT`

on
Cats #302, which was merged recently. (Thanks, Erik)
As it happens, `State`

is just a type alias:

```
package object data {
....
type State[S, A] = StateT[Eval, S, A]
object State extends StateFunctions
}
```

`StateT`

is a monad transformer, a type constructor for other datatypes.
`State`

partially applies `StateT`

with `Eval`

,
which emulates a call stack with heap memory to prevent overflow.
Here’s the definition of `StateT`

:

```
final class StateT[F[_], S, A](val runF: F[S => F[(S, A)]]) {
....
}
object StateT extends StateTInstances {
def apply[F[_], S, A](f: S => F[(S, A)])(implicit F: Applicative[F]): StateT[F, S, A] =
new StateT(F.pure(f))
def applyF[F[_], S, A](runF: F[S => F[(S, A)]]): StateT[F, S, A] =
new StateT(runF)
/**
* Run with the provided initial state value
*/
def run(initial: S)(implicit F: FlatMap[F]): F[(S, A)] =
F.flatMap(runF)(f => f(initial))
....
}
```

To construct a `State`

value, you pass the state transition function to `State.apply`

.

```
private[data] abstract class StateFunctions {
def apply[S, A](f: S => (S, A)): State[S, A] =
StateT.applyF(Now((s: S) => Now(f(s))))
....
}
```

As the `State`

implementation is fresh, a few bumps on the road are expected.
When I tried using `State`

on the REPL, I ran into an odd behavior where I can create
one state, but not the second. @retronym pointed me to
SI-7139: Type alias and object with the same name cause type mismatch in REPL, which I was able to workaround as #322.

Let’s consider how to implement stack with `State`

:

```
scala> type Stack = List[Int]
defined type alias Stack
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> val pop = State[Stack, Int] {
case x :: xs => (xs, x)
case Nil => sys.error("stack is empty")
}
pop: cats.data.State[Stack,Int] = cats.data.StateT@1647fded
scala> def push(a: Int) = State[Stack, Unit] {
case xs => (a :: xs, ())
}
push: (a: Int)cats.data.State[Stack,Unit]
```

These are the primitive programs. Now we can construct compound programs by composing the monad.

```
scala> def stackManip: State[Stack, Int] = for {
_ <- push(3)
a <- pop
b <- pop
} yield(b)
stackManip: cats.data.State[Stack,Int]
scala> stackManip.run(List(5, 8, 2, 1)).value
res0: (Stack, Int) = (List(8, 2, 1),5)
```

The first `run`

is for `StateT`

, and the second is to `run`

until the end `Eval`

.

Both `push`

and `pop`

are still purely functional, and we
were able to eliminate explicitly passing the state object (`s0`

, `s1`

, …).

LYAHFGG:

The

`Control.Monad.State`

module provides a type class that’s called`MonadState`

and it features two pretty useful functions, namely`get`

and`put`

.

The `State`

object defines a few helper functions:

```
private[data] abstract class StateFunctions {
def apply[S, A](f: S => (S, A)): State[S, A] =
StateT.applyF(Now((s: S) => Now(f(s))))
/**
* Return `a` and maintain the input state.
*/
def pure[S, A](a: A): State[S, A] = State(s => (s, a))
/**
* Modify the input state and return Unit.
*/
def modify[S](f: S => S): State[S, Unit] = State(s => (f(s), ()))
/**
* Inspect a value from the input state, without modifying the state.
*/
def inspect[S, T](f: S => T): State[S, T] = State(s => (s, f(s)))
/**
* Return the input state without modifying it.
*/
def get[S]: State[S, S] = inspect(identity)
/**
* Set the state to `s` and return Unit.
*/
def set[S](s: S): State[S, Unit] = State(_ => (s, ()))
}
```

These are confusing at first. But remember that the `State`

monad encapsulates
a pair of a state transition function and a return value.
So `State.get`

keeps the state as is, and returns it.

Similarly, `State.set(s)`

in this context means to overwrite the state with `s`

and return `()`

.

Let’s try using them with the `stackyStack`

example from the book:

```
scala> def stackyStack: State[Stack, Unit] = for {
stackNow <- State.get[Stack]
r <- if (stackNow === List(1, 2, 3)) State.set[Stack](List(8, 3, 1))
else State.set[Stack](List(9, 2, 1))
} yield r
stackyStack: cats.data.State[Stack,Unit]
scala> stackyStack.run(List(1, 2, 3)).value
res1: (Stack, Unit) = (List(8, 3, 1),())
```

We can also implement both `pop`

and `push`

in terms of `get`

and `set`

:

```
scala> val pop: State[Stack, Int] = for {
s <- State.get[Stack]
(x :: xs) = s
_ <- State.set[Stack](xs)
} yield x
pop: cats.data.State[Stack,Int] = cats.data.StateT@105f323
scala> def push(x: Int): State[Stack, Unit] = for {
xs <- State.get[Stack]
r <- State.set(x :: xs)
} yield r
push: (x: Int)cats.data.State[Stack,Unit]
```

As you can see, the `State`

monad on its own doesn’t do much
(encapsulate a state transition function and a return value),
but by chaining them we can remove some boilerplates.

`State.extract(f)`

and `State.modify(f)`

are slightly more
advanced variants of `State.get`

and `State.set(s)`

.

`State.extract(f)`

applies the function `f: S => T`

to `s`

,
and returns the result without modifying the state itself.

`State.modify`

applies the function `f: S => T`

to `s`

,
saves the result as the new state, and returns `()`

.

LYAHFGG:

The

`Either e a`

type on the other hand, allows us to incorporate a context of possible failure to our values while also being able to attach values to the failure, so that they can describe what went wrong or provide some other useful info regarding the failure.

We know `Either[A, B]`

from the standard library, and we’ve covered that
Cats implements a right-biased functor for it too.

There’s another datatype in Cats that we can use in place of `Either`

called Validated:

```
sealed abstract class Validated[+E, +A] extends Product with Serializable {
def fold[B](fe: E => B, fa: A => B): B =
this match {
case Invalid(e) => fe(e)
case Valid(a) => fa(a)
}
def isValid: Boolean = fold(_ => false, _ => true)
def isInvalid: Boolean = fold(_ => true, _ => false)
....
}
object Validated extends ValidatedInstances with ValidatedFunctions{
final case class Valid[+A](a: A) extends Validated[Nothing, A]
final case class Invalid[+E](e: E) extends Validated[E, Nothing]
}
```

Here’s how to create the values:

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> import Validated.{ valid, invalid }
import Validated.{valid, invalid}
scala> valid[String, String]("event 1 ok")
res0: cats.data.Validated[String,String] = Valid(event 1 ok)
scala> invalid[String, String]("event 1 failed!")
res1: cats.data.Validated[String,String] = Invalid(event 1 failed!)
```

What’s different about `Validation`

is that it is does not form a monad,
but forms an applicative functor.
Instead of chaining the result from first event to the next, `Validated`

validates all events:

```
scala> val result = (valid[String, String]("event 1 ok") |@|
invalid[String, String]("event 2 failed!") |@|
invalid[String, String]("event 3 failed!")) map {_ + _ + _}
result: cats.data.Validated[String,String] = Invalid(event 2 failed!event 3 failed!)
```

The final result is `Invalid(event 3 failed!event 2 failed!)`

.
Unlike the `Xor`

’s monad, which cuts the calculation short,
`Validated`

keeps going to report back all failures.
This would be useful for validating user input on an online bacon shop.

The problem, however, is that the error messages are mushed together into one string. Shouldn’t it be something like a list?

This is where `NonEmptyList`

datatype comes in handy.
For now, think of it as a list that’s guaranteed to have at least one element.

```
scala> import cats.data.{ NonEmptyList => NEL }
import cats.data.{NonEmptyList=>NEL}
scala> NEL.of(1)
res2: cats.data.NonEmptyList[Int] = NonEmptyList(1)
```

We can use `NEL[A]`

on the invalid side to accumulate the errors:

```
scala> val result =
(valid[NEL[String], String]("event 1 ok") |@|
invalid[NEL[String], String](NEL.of("event 2 failed!")) |@|
invalid[NEL[String], String](NEL.of("event 3 failed!"))) map {_ + _ + _}
result: cats.data.Validated[cats.data.NonEmptyList[String],String] = Invalid(NonEmptyList(event 2 failed!, event 3 failed!))
```

Inside `Invalid`

, we were able to accumulate all failed messages.

We can use the `fold`

method to extract the values:

```
scala> val errs: NEL[String] = result.fold(
{ l => l },
{ r => sys.error("invalid is expected") }
)
errs: cats.data.NonEmptyList[String] = NonEmptyList(event 2 failed!, event 3 failed!)
```

In Cats there is yet another datatype that represents an `A`

-`B`

pair called Ior.

```
/** Represents a right-biased disjunction that is either an `A`, or a `B`, or both an `A` and a `B`.
*
* An instance of `A [[Ior]] B` is one of:
* - `[[Ior.Left Left]][A]`
* - `[[Ior.Right Right]][B]`
* - `[[Ior.Both Both]][A, B]`
*
* `A [[Ior]] B` is similar to `A [[Xor]] B`, except that it can represent the simultaneous presence of
* an `A` and a `B`. It is right-biased like [[Xor]], so methods such as `map` and `flatMap` operate on the
* `B` value. Some methods, like `flatMap`, handle the presence of two [[Ior.Both Both]] values using a
* `[[Semigroup]][A]`, while other methods, like [[toXor]], ignore the `A` value in a [[Ior.Both Both]].
*
* `A [[Ior]] B` is isomorphic to `(A [[Xor]] B) [[Xor]] (A, B)`, but provides methods biased toward `B`
* values, regardless of whether the `B` values appear in a [[Ior.Right Right]] or a [[Ior.Both Both]].
* The isomorphic [[Xor]] form can be accessed via the [[unwrap]] method.
*/
sealed abstract class Ior[+A, +B] extends Product with Serializable {
final def fold[C](fa: A => C, fb: B => C, fab: (A, B) => C): C = this match {
case Ior.Left(a) => fa(a)
case Ior.Right(b) => fb(b)
case Ior.Both(a, b) => fab(a, b)
}
final def isLeft: Boolean = fold(_ => true, _ => false, (_, _) => false)
final def isRight: Boolean = fold(_ => false, _ => true, (_, _) => false)
final def isBoth: Boolean = fold(_ => false, _ => false, (_, _) => true)
....
}
object Ior extends IorInstances with IorFunctions {
final case class Left[+A](a: A) extends (A Ior Nothing)
final case class Right[+B](b: B) extends (Nothing Ior B)
final case class Both[+A, +B](a: A, b: B) extends (A Ior B)
}
```

These values are created using the `left`

, `right`

, and `both`

methods on `Ior`

:

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> import cats.data.{ NonEmptyList => NEL }
import cats.data.{NonEmptyList=>NEL}
scala> Ior.right[NEL[String], Int](1)
res0: cats.data.Ior[cats.data.NonEmptyList[String],Int] = Right(1)
scala> Ior.left[NEL[String], Int](NEL.of("error"))
res1: cats.data.Ior[cats.data.NonEmptyList[String],Int] = Left(NonEmptyList(error))
scala> Ior.both[NEL[String], Int](NEL.of("warning"), 1)
res2: cats.data.Ior[cats.data.NonEmptyList[String],Int] = Both(NonEmptyList(warning),1)
```

As noted in the scaladoc comment, `Ior`

’s `flatMap`

uses `Semigroup[A]`

to accumulate
failures when it sees an `Ior.both(...)`

value.
So we could probably use this as a hybrid of `Xor`

and `Validated`

.

Here’s how `flatMap`

behaves for all nine combinations:

```
scala> Ior.right[NEL[String], Int](1) >>=
{ x => Ior.right[NEL[String], Int](x + 1) }
res3: cats.data.Ior[cats.data.NonEmptyList[String],Int] = Right(2)
scala> Ior.left[NEL[String], Int](NEL.of("error 1")) >>=
{ x => Ior.right[NEL[String], Int](x + 1) }
res4: cats.data.Ior[cats.data.NonEmptyList[String],Int] = Left(NonEmptyList(error 1))
scala> Ior.both[NEL[String], Int](NEL.of("warning 1"), 1) >>=
{ x => Ior.right[NEL[String], Int](x + 1) }
res5: cats.data.Ior[cats.data.NonEmptyList[String],Int] = Both(NonEmptyList(warning 1),2)
scala> Ior.right[NEL[String], Int](1) >>=
{ x => Ior.left[NEL[String], Int](NEL.of("error 2")) }
res6: cats.data.Ior[cats.data.NonEmptyList[String],Int] = Left(NonEmptyList(error 2))
scala> Ior.left[NEL[String], Int](NEL.of("error 1")) >>=
{ x => Ior.left[NEL[String], Int](NEL.of("error 2")) }
res7: cats.data.Ior[cats.data.NonEmptyList[String],Int] = Left(NonEmptyList(error 1))
scala> Ior.both[NEL[String], Int](NEL.of("warning 1"), 1) >>=
{ x => Ior.left[NEL[String], Int](NEL.of("error 2")) }
res8: cats.data.Ior[cats.data.NonEmptyList[String],Int] = Left(NonEmptyList(warning 1, error 2))
scala> Ior.right[NEL[String], Int](1) >>=
{ x => Ior.both[NEL[String], Int](NEL.of("warning 2"), x + 1) }
res9: cats.data.Ior[cats.data.NonEmptyList[String],Int] = Both(NonEmptyList(warning 2),2)
scala> Ior.left[NEL[String], Int](NEL.of("error 1")) >>=
{ x => Ior.both[NEL[String], Int](NEL.of("warning 2"), x + 1) }
res10: cats.data.Ior[cats.data.NonEmptyList[String],Int] = Left(NonEmptyList(error 1))
scala> Ior.both[NEL[String], Int](NEL.of("warning 1"), 1) >>=
{ x => Ior.both[NEL[String], Int](NEL.of("warning 2"), x + 1) }
res11: cats.data.Ior[cats.data.NonEmptyList[String],Int] = Both(NonEmptyList(warning 1, warning 2),2)
```

Let’s try using it in `for`

comprehension:

```
scala> for {
e1 <- Ior.right[NEL[String], Int](1)
e2 <- Ior.both[NEL[String], Int](NEL.of("event 2 warning"), e1 + 1)
e3 <- Ior.both[NEL[String], Int](NEL.of("event 3 warning"), e2 + 1)
} yield (e1 |+| e2 |+| e3)
res12: cats.data.Ior[cats.data.NonEmptyList[String],Int] = Both(NonEmptyList(event 2 warning, event 3 warning),6)
```

So `Ior.left`

short curcuits like the failure values in `Xor[A, B]`

and `Either[A, B]`

,
but `Ior.both`

accumulates the failure values like `Validated[A, B]`

.

That’s it for today! We’ll pick it up from here later.

On day 7 we looked at the `State`

datatype, which encapsulates stateful computation.
We also looked at two alternatives for `Either[A, B]`

: the `Xor`

datatype, the `Validated`

datatype, and the `Ior`

datatype.

I’m going to diverge from Learn You a Haskell for Great Good a bit, and explore free objects.

Let’s first look into free monoids. Given a set of characters:

```
A = { 'a', 'b', 'c', ... }
```

We can form the *free monoid* on `A`

called `A*`

as follows:

```
A* = String
```

Here, the binary operation is `String`

concatenation `+`

operator.
We can show that this satisfies the monoid laws using the empty string `""`

as the identity.

Furthermore, a free monoid can be formed from any arbitrary set `A`

by concatenating them:

```
A* = List[A]
```

Here, the binary operation is `:::`

, and the identity is `Nil`

.
The definition of the free monoid *M(A)* is given as follows:

Universal Mapping Property of

M(A)

There is a functioni: A => |M(A)|, and given any monoidNand any functionf: A => |N|, there is auniquemonoid homomorphismf_hom = M(A) => Nsuch that|f_hom| ∘ i = f, all as indicated in the following diagram:

Instead of `A`

, I’ll use `X`

here. Also *|N|* means `Set[N]`

:

If we think in terms of Scala,

```
def i(x: X): Set[M[X]] = ???
def f(x: X): Set[N] = ???
// there exists a unique
def f_hom(mx: M[X]): N
// such that
def f_hom_set(smx: Set[M[X]]): Set[N] = smx map {f_hom}
f == f_hom_set compose i
```

Suppose `A`

is `Char`

and `N`

is `(Int, +)`

.
We can write a property test to see if `String`

is a free monoid.

```
scala> def i(x: Char): Set[String] = Set(x.toString)
i: (x: Char)Set[String]
scala> def f(x: Char): Set[Int] = Set(x.toInt) // example
f: (x: Char)Set[Int]
scala> val f_hom: PartialFunction[String, Int] =
{ case mx: String if mx.size == 1 => mx.charAt(0).toInt }
f_hom: PartialFunction[String,Int] = <function1>
scala> def f_hom_set(smx: Set[String]): Set[Int] = smx map {f_hom}
f_hom_set: (smx: Set[String])Set[Int]
scala> val g = (f_hom_set _) compose (i _)
g: Char => Set[Int] = <function1>
scala> import org.scalacheck.Prop.forAll
import org.scalacheck.Prop.forAll
scala> val propMAFree = forAll { c: Char => f(c) == g(c) }
propMAFree: org.scalacheck.Prop = Prop
scala> propMAFree.check
+ OK, passed 100 tests.
```

At least for this implemention of `f`

we were able to show that `String`

is free.

This intuitively shows that `Set[M[X]]`

needs to be lossless for `X`

to allow any *f*,
meaning no two values on `X`

can map into the same value in `M[X]`

.
In algebra, this is expressed as `i`

is *injective* for arrows from `Char`

.

Definitions: An arrowfsatisfying the property ‘for any pair of arrowsxand_{1}: T => Ax, if_{2}: T => Af ∘ xthen_{1}= f ∘ x_{2}x‘, it is said to be_{1}= x_{2}injective for arrows from T.

UMP also stipulates `f_hom`

to be unique, so that requires `Set[M[A]]`

to be
zero or more combinations of `A`

’s and nothing more.
Because `M[A]`

is unique for `A`

, conceptually there is one and only free monoid for a set `A`

.
We can however have the free monoid expressed in different ways like `String`

and `List[Char]`

,
so it ends up being more like a free monoid.

It turns out that the free monoid is an example of free objects,
which we can define using a functor `Set[A]: C[A] => Set[A]`

.

Comparing the diagram, we see that they are mostly similar.

We said that free monoids are examples of free objects. Similarly, free monads are examples of free objects.

I’m not going to get into the details, monad is a monoid in the category of endofunctors `F: C => C`

,
using `F × F => F`

as the binary operator.
Similar to the way we could derive `A*`

from `A`

,
we can derive the free monad `F*`

for a given endofunctor `F`

.

Here’s how Haskell does it:

```
data Free f a = Pure a | Free (f (Free f a))
instance Functor f => Monad (Free f) where
return = Pure
Pure a >>= f = f a
Free m >>= f = Free ((>>= f) <$> m)
```

Unlike

`List`

, which stores a list of values,`Free`

stores a list of functors, wrapped around an initial value. Accordingly, the`Functor`

and`Monad`

instances of`Free`

do nothing other than handing a given function down that list with`fmap`

.

Also note that `Free`

is a datatype, but
there are different free monads that gets formed for each `Functor`

.

In practice, we can view `Free`

as a clever way of forming `Monad`

out of `Functor`

.
This is particularly useful for interpreter pattern,
which is explained by
Gabriel Gonzalez (@gabrielg439)’s Why free monads matter:

Let’s try to come up with some sort of abstraction that represents the essence of a syntax tree. …

Our toy language will only have three commands:

```
output b -- prints a "b" to the console
bell -- rings the computer's bell
done -- end of execution
```

So we represent it as a syntax tree where subsequent commands are leaves of prior commands:

```
data Toy b next =
Output b next
| Bell next
| Done
```

Here’s `Toy`

translated into Scala as is:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
sealed trait Toy[+A, +Next]
object Toy {
case class Output[A, Next](a: A, next: Next) extends Toy[A, Next]
case class Bell[Next](next: Next) extends Toy[Nothing, Next]
case class Done() extends Toy[Nothing, Nothing]
}
// Exiting paste mode, now interpreting.
defined trait Toy
defined object Toy
scala> Toy.Output('A', Toy.Done())
res0: Toy.Output[Char,Toy.Done] = Output(A,Done())
scala> Toy.Bell(Toy.Output('A', Toy.Done()))
res1: Toy.Bell[Toy.Output[Char,Toy.Done]] = Bell(Output(A,Done()))
```

In WFMM’s original DSL, `Output b next`

has two type parameters `b`

and `next`

.
This allows `Output`

to be generic about the output type.
As demonstrated above as `Toy`

, Scala can do this too.
But doing so unnecessarily complicates the demonstration of `Free`

because of
Scala’s handling of partially applied types. So we’ll first hardcode the datatype to `Char`

as follows:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
sealed trait CharToy[+Next]
object CharToy {
case class CharOutput[Next](a: Char, next: Next) extends CharToy[Next]
case class CharBell[Next](next: Next) extends CharToy[Next]
case class CharDone() extends CharToy[Nothing]
def output[Next](a: Char, next: Next): CharToy[Next] = CharOutput(a, next)
def bell[Next](next: Next): CharToy[Next] = CharBell(next)
def done: CharToy[Nothing] = CharDone()
}
// Exiting paste mode, now interpreting.
defined trait CharToy
defined object CharToy
scala> { import CharToy._
output('A', done)
}
res2: CharToy[CharToy[Nothing]] = CharOutput(A,CharDone())
scala> { import CharToy._
bell(output('A', done))
}
res3: CharToy[CharToy[CharToy[Nothing]]] = CharBell(CharOutput(A,CharDone()))
```

I’ve added helper functions lowercase `output`

, `bell`

, and `done`

to unify the types to `CharToy`

.

WFMM:

but unfortunately this doesn’t work because every time I want to add a command, it changes the type.

Let’s define `Fix`

:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
case class Fix[F[_]](f: F[Fix[F]])
object Fix {
def fix(toy: CharToy[Fix[CharToy]]) = Fix[CharToy](toy)
}
// Exiting paste mode, now interpreting.
defined class Fix
defined object Fix
scala> { import Fix._, CharToy._
fix(output('A', fix(done)))
}
res4: Fix[CharToy] = Fix(CharOutput(A,Fix(CharDone())))
scala> { import Fix._, CharToy._
fix(bell(fix(output('A', fix(done)))))
}
res5: Fix[CharToy] = Fix(CharBell(Fix(CharOutput(A,Fix(CharDone())))))
```

Again, `fix`

is provided so that the type inference works.

We are also going to try to implement `FixE`

, which adds an exception to this. Since `throw`

and `catch`

are reserved, I am renaming them to `throwy`

and `catchy`

:

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> :paste
// Entering paste mode (ctrl-D to finish)
sealed trait FixE[F[_], E]
object FixE {
case class Fix[F[_], E](f: F[FixE[F, E]]) extends FixE[F, E]
case class Throwy[F[_], E](e: E) extends FixE[F, E]
def fix[E](toy: CharToy[FixE[CharToy, E]]): FixE[CharToy, E] =
Fix[CharToy, E](toy)
def throwy[F[_], E](e: E): FixE[F, E] = Throwy(e)
def catchy[F[_]: Functor, E1, E2](ex: => FixE[F, E1])
(f: E1 => FixE[F, E2]): FixE[F, E2] = ex match {
case Fix(x) => Fix[F, E2](Functor[F].map(x) {catchy(_)(f)})
case Throwy(e) => f(e)
}
}
// Exiting paste mode, now interpreting.
defined trait FixE
defined object FixE
```

We can only use this if Toy b is a functor, so we muddle around until we find something that type-checks (and satisfies the Functor laws).

Let’s define `Functor`

for `CharToy`

:

```
scala> implicit val charToyFunctor: Functor[CharToy] = new Functor[CharToy] {
def map[A, B](fa: CharToy[A])(f: A => B): CharToy[B] = fa match {
case o: CharToy.CharOutput[A] => CharToy.CharOutput(o.a, f(o.next))
case b: CharToy.CharBell[A] => CharToy.CharBell(f(b.next))
case CharToy.CharDone() => CharToy.CharDone()
}
}
charToyFunctor: cats.Functor[CharToy] = $anon$1@f0dc1c1
```

Here’s a sample usage:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
{
import FixE._, CharToy._
case class IncompleteException()
def subroutine = fix[IncompleteException](
output('A',
throwy[CharToy, IncompleteException](IncompleteException())))
def program = catchy[CharToy, IncompleteException, Nothing](subroutine) { _ =>
fix[Nothing](bell(fix[Nothing](done)))
}
}
// Exiting paste mode, now interpreting.
```

The fact that we need to supply type parameters everywhere is a bit unfortunate.

WFMM:

our

`FixE`

already exists, too, and it’s called the Free monad:

```
data Free f r = Free (f (Free f r)) | Pure r
```

As the name suggests, it is automatically a monad (if

`f`

is a functor):

```
instance (Functor f) => Monad (Free f) where
return = Pure
(Free x) >>= f = Free (fmap (>>= f) x)
(Pure r) >>= f = f r
```

The

`return`

was our`Throw`

, and`(>>=)`

was our`catch`

.

The datatype in Cats is called Free:

```
/**
* A free operational monad for some functor `S`. Binding is done
* using the heap instead of the stack, allowing tail-call
* elimination.
*/
sealed abstract class Free[S[_], A] extends Product with Serializable {
final def map[B](f: A => B): Free[S, B] =
flatMap(a => Pure(f(a)))
/**
* Bind the given continuation to the result of this computation.
* All left-associated binds are reassociated to the right.
*/
final def flatMap[B](f: A => Free[S, B]): Free[S, B] =
Gosub(this, f)
....
}
object Free {
/**
* Return from the computation with the given value.
*/
private final case class Pure[S[_], A](a: A) extends Free[S, A]
/** Suspend the computation with the given suspension. */
private final case class Suspend[S[_], A](a: S[A]) extends Free[S, A]
/** Call a subroutine and continue with the given function. */
private final case class Gosub[S[_], B, C](c: Free[S, C], f: C => Free[S, B]) extends Free[S, B]
/**
* Suspend a value within a functor lifting it to a Free.
*/
def liftF[F[_], A](value: F[A]): Free[F, A] = Suspend(value)
/** Suspend the Free with the Applicative */
def suspend[F[_], A](value: => Free[F, A])(implicit F: Applicative[F]): Free[F, A] =
liftF(F.pure(())).flatMap(_ => value)
/** Lift a pure value into Free */
def pure[S[_], A](a: A): Free[S, A] = Pure(a)
final class FreeInjectPartiallyApplied[F[_], G[_]] private[free] {
def apply[A](fa: F[A])(implicit I : Inject[F, G]): Free[G, A] =
Free.liftF(I.inj(fa))
}
def inject[F[_], G[_]]: FreeInjectPartiallyApplied[F, G] = new FreeInjectPartiallyApplied
....
}
```

To use these datatypes in Cats, use `Free.liftF`

:

```
scala> import cats.free.Free
import cats.free.Free
scala> :paste
// Entering paste mode (ctrl-D to finish)
sealed trait CharToy[+Next]
object CharToy {
case class CharOutput[Next](a: Char, next: Next) extends CharToy[Next]
case class CharBell[Next](next: Next) extends CharToy[Next]
case class CharDone() extends CharToy[Nothing]
implicit val charToyFunctor: Functor[CharToy] = new Functor[CharToy] {
def map[A, B](fa: CharToy[A])(f: A => B): CharToy[B] = fa match {
case o: CharOutput[A] => CharOutput(o.a, f(o.next))
case b: CharBell[A] => CharBell(f(b.next))
case CharDone() => CharDone()
}
}
def output(a: Char): Free[CharToy, Unit] =
Free.liftF[CharToy, Unit](CharOutput(a, ()))
def bell: Free[CharToy, Unit] = Free.liftF[CharToy, Unit](CharBell(()))
def done: Free[CharToy, Unit] = Free.liftF[CharToy, Unit](CharDone())
def pure[A](a: A): Free[CharToy, A] = Free.pure[CharToy, A](a)
}
// Exiting paste mode, now interpreting.
defined trait CharToy
defined object CharToy
```

Here’s the command sequence:

```
scala> import CharToy._
import CharToy._
scala> val subroutine = output('A')
subroutine: cats.free.Free[CharToy,Unit] = Free(...)
scala> val program = for {
_ <- subroutine
_ <- bell
_ <- done
} yield ()
program: cats.free.Free[CharToy,Unit] = Free(...)
```

This is where things get magical. We now have

`do`

notation for something that hasn’t even been interpreted yet: it’s pure data.

Next we’d like to define `showProgram`

to prove that what we have is just data:

```
scala> def showProgram[R: Show](p: Free[CharToy, R]): String =
p.fold({ r: R => "return " + Show[R].show(r) + "\n" },
{
case CharOutput(a, next) =>
"output " + Show[Char].show(a) + "\n" + showProgram(next)
case CharBell(next) =>
"bell " + "\n" + showProgram(next)
case CharDone() =>
"done\n"
})
showProgram: [R](p: cats.free.Free[CharToy,R])(implicit evidence$1: cats.Show[R])String
scala> showProgram(program)
res7: String =
"output A
bell
done
"
```

We can manually check that the monad generated using `Free`

satisfies the monad laws:

```
scala> showProgram(output('A'))
res8: String =
"output A
return ()
"
scala> showProgram(pure('A') flatMap output)
res9: String =
"output A
return ()
"
scala> showProgram(output('A') flatMap pure)
res10: String =
"output A
return ()
"
scala> showProgram((output('A') flatMap { _ => done }) flatMap { _ => output('C') })
res11: String =
"output A
done
"
scala> showProgram(output('A') flatMap { _ => (done flatMap { _ => output('C') }) })
res12: String =
"output A
done
"
```

Looking good. Notice the abort-like semantics of `done`

.
Also, due to type inference limitation, I was not able to use `>>=`

and `>>`

here.

WFMM:

The free monad is the interpreter’s best friend. Free monads “free the interpreter” as much as possible while still maintaining the bare minimum necessary to form a monad.

Another way of looking at it is that the `Free`

datatype provides a way of building a syntax tree given a container.

One of the reasons the `Free`

datatype is gaining popularity I think is that people
are running into the limitation of combining different monads.
It’s not impossible with monad transformer, but the type signature gets hairy quickly, and the stacking leaks into
various places in code. On the other hand, `Free`

essentially gives up on encoding meaning into the monad.
You gain flexibility because you can do whatever in the interpreter function, for instance run sequentially
during testing, but run in parallel for production.

The notion of free monad goes beyond just the interpreter pattern. I think people are still discovering new ways of harnessing its power.

Rúnar (@runarorama) has been a proponent of using `Free`

in Scala.
His talk that we covered on day 6, Dead-Simple Dependency Injection, uses `Free`

to implement
a mini language to implement a key-value store.
The same year, Rúnar also gave a talk at Scala Days 2012 called
Stackless Scala With Free Monads.
I recommend watching the talk before reading the paper, but it’s easier to quote the paper version,
Stackless Scala With Free Monads.

Rúnar starts out with code that uses an implementation of the `State`

monad to zip a list with its indices.
It blows the stack when the list is larger than the stack limit.
Then he introduces trampoline, which is a single loop that drives the entire program.

```
sealed trait Trampoline [+ A] {
final def runT : A =
this match {
case More (k) => k().runT
case Done (v) => v
}
}
case class More[+A](k: () => Trampoline[A])
extends Trampoline[A]
case class Done [+A](result: A)
extends Trampoline [A]
```

In the above code, `Function0`

`k`

is used as a thunk for the next step.

To extend its usage for State monad, he then reifies `flatMap`

into a data structure called `FlatMap`

:

```
case class FlatMap [A,+B](
sub: Trampoline [A],
k: A => Trampoline[B]) extends Trampoline[B]
```

Next, it is revealed that `Trampoline`

is a free monad of `Function0`

. Here’s how it’s defined in Cats:

```
type Trampoline[A] = Free[Function0, A]
```

Using Trampoline any program can be transformed into a stackless one.
`Trampoline`

object defines a few useful functions for trampolining:

```
object Trampoline {
def done[A](a: A): Trampoline[A] =
Free.Pure[Function0,A](a)
def suspend[A](a: => Trampoline[A]): Trampoline[A] =
Free.Suspend[Function0, A](() => a)
def delay[A](a: => A): Trampoline[A] =
suspend(done(a))
}
```

Let’s try implementing `even`

and `odd`

from the talk:

```
scala> import cats._, cats.data._, cats.implicits._, cats.free.{ Free, Trampoline }
import cats._
import cats.data._
import cats.implicits._
import cats.free.{Free, Trampoline}
scala> import Trampoline._
import Trampoline._
scala> :paste
// Entering paste mode (ctrl-D to finish)
def even[A](ns: List[A]): Trampoline[Boolean] =
ns match {
case Nil => done(true)
case x :: xs => suspend(odd(xs))
}
def odd[A](ns: List[A]): Trampoline[Boolean] =
ns match {
case Nil => done(false)
case x :: xs => suspend(even(xs))
}
// Exiting paste mode, now interpreting.
even: [A](ns: List[A])cats.free.Trampoline[Boolean]
odd: [A](ns: List[A])cats.free.Trampoline[Boolean]
scala> even(List(1, 2, 3)).run
res0: Boolean = false
scala> even((0 to 3000).toList).run
res1: Boolean = false
```

While implementing the above I ran into SI-7139 again, so I had to tweak the Cats’ code. #322

In addition, Rúnar introduces several datatypes that can be derived using `Free`

:

```
type Pair[+A] = (A, A)
type BinTree[+A] = Free[Pair, A]
type Tree[+A] = Free[List, A]
type FreeMonoid[+A] = Free[({type λ[+α] = (A,α)})#λ, Unit]
type Trivial[+A] = Unit
type Option[+A] = Free[Trivial, A]
```

There’s also an iteratees implementation based on free monads. Finally, he summarizes free monads in nice bullet points:

- A model for any recursive data type with data at the leaves.
- A free monad is an expression tree with variables at the leaves and flatMap is variable substitution.

Let’s try defining “List” using `Free`

.

```
scala> type FreeMonoid[A] = Free[(A, +?), Unit]
defined type alias FreeMonoid
scala> def cons[A](a: A): FreeMonoid[A] =
Free.liftF[(A, +?), Unit]((a, ()))
cons: [A](a: A)FreeMonoid[A]
scala> val x = cons(1)
x: FreeMonoid[Int] = Free(...)
scala> val xs = cons(1) flatMap {_ => cons(2)}
xs: cats.free.Free[[+β](Int, β),Unit] = Free(...)
```

As a way of interpreting the result, let’s try converting this to a standard `List`

:

```
scala> implicit def tuple2Functor[A]: Functor[(A, ?)] =
new Functor[(A, ?)] {
def map[B, C](fa: (A, B))(f: B => C) =
(fa._1, f(fa._2))
}
tuple2Functor: [A]=> cats.Functor[[β](A, β)]
scala> def toList[A](list: FreeMonoid[A]): List[A] =
list.fold(
{ _ => Nil },
{ case (x: A @unchecked, xs: FreeMonoid[A]) => x :: toList(xs) })
toList: [A](list: FreeMonoid[A])List[A]
scala> toList(xs)
res2: List[Int] = List(1, 2)
```

In 2015 Phil Freeman (@paf31) wrote Stack Safety for Free working with PureScript hosted on JavaScript, a strict language like Java:

I've written up some work on stack safe free monad transformers. Feedback would be very much appreciated http://t.co/1rH7OwaWpy

— Phil Freeman (@paf31) August 8, 2015

The paper gives a hat tip to Rúnar (@runarorama)’s Stackless Scala With Free Monads, but presents a more drastic solution to the stack safety problem.

As a background, in Scala the compiler is able to optimize on self-recursive tail calls. For example, here’s an example of a self-recursive tail calls.

```
scala> import scala.annotation.tailrec
import scala.annotation.tailrec
scala> :paste
// Entering paste mode (ctrl-D to finish)
def pow(n: Long, exp: Long): Long =
{
@tailrec def go(acc: Long, p: Long): Long =
(acc, p) match {
case (acc, 0) => acc
case (acc, p) => go(acc * n, p - 1)
}
go(1, exp)
}
// Exiting paste mode, now interpreting.
pow: (n: Long, exp: Long)Long
scala> pow(2, 3)
res3: Long = 8
```

Here’s an example that’s not self-recursive. It’s blowing up the stack.

```
scala> :paste
object OddEven0 {
def odd(n: Int): String = even(n - 1)
def even(n: Int): String = if (n <= 0) "done" else odd(n - 1)
}
// Exiting paste mode, now interpreting.
defined object OddEven0
scala> OddEven0.even(200000)
java.lang.StackOverflowError
at OddEven0$.even(<console>:15)
at OddEven0$.odd(<console>:14)
at OddEven0$.even(<console>:15)
at OddEven0$.odd(<console>:14)
at OddEven0$.even(<console>:15)
....
```

Next, we’d try to add Writer datatype to do the `pow`

calculation using `LongProduct`

monoid.

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> :paste
// Entering paste mode (ctrl-D to finish)
case class LongProduct(value: Long)
implicit val longProdMonoid: Monoid[LongProduct] = new Monoid[LongProduct] {
def empty: LongProduct = LongProduct(1)
def combine(x: LongProduct, y: LongProduct): LongProduct = LongProduct(x.value * y.value)
}
def powWriter(x: Long, exp: Long): Writer[LongProduct, Unit] =
exp match {
case 0 => Writer(LongProduct(1L), ())
case m =>
Writer(LongProduct(x), ()) >>= { _ => powWriter(x, exp - 1) }
}
// Exiting paste mode, now interpreting.
defined class LongProduct
longProdMonoid: cats.Monoid[LongProduct] = $anon$1@7e4ff5d6
powWriter: (x: Long, exp: Long)cats.data.Writer[LongProduct,Unit]
scala> powWriter(2, 3).run
res4: cats.Id[(LongProduct, Unit)] = (LongProduct(8),())
```

This is no longer self-recursive, so it will blow the stack with large `exp`

.

```
scala> powWriter(1, 10000).run
java.lang.StackOverflowError
at $anonfun$powWriter$1.apply(<console>:35)
at $anonfun$powWriter$1.apply(<console>:35)
at cats.data.WriterT$$anonfun$flatMap$1.apply(WriterT.scala:37)
at cats.data.WriterT$$anonfun$flatMap$1.apply(WriterT.scala:36)
at cats.package$$anon$1.flatMap(package.scala:34)
at cats.data.WriterT.flatMap(WriterT.scala:36)
at cats.data.WriterTFlatMap1$class.flatMap(WriterT.scala:249)
at cats.data.WriterTInstances2$$anon$4.flatMap(WriterT.scala:130)
at cats.data.WriterTInstances2$$anon$4.flatMap(WriterT.scala:130)
at cats.FlatMap$class.$greater$greater$eq(FlatMap.scala:26)
at cats.data.WriterTInstances2$$anon$4.$greater$greater$eq(WriterT.scala:130)
at cats.FlatMap$Ops$class.$greater$greater$eq(FlatMap.scala:20)
at cats.syntax.FlatMapSyntax1$$anon$1.$greater$greater$eq(flatMap.scala:6)
at .powWriter1(<console>:35)
at $anonfun$powWriter$1.apply(<console>:35)
```

This characteristic of Scala limits the usefulness of monadic composition where `flatMap`

can call
monadic function `f`

, which then can call `flatMap`

etc..

Our solution is to reduce the candidates for the target monad

`m`

from an arbitrary monad, to the class of so-called tail-recursive monads.

```
class (Monad m) <= MonadRec m where
tailRecM :: forall a b. (a -> m (Either a b)) -> a -> m b
```

Here’s the same function in Scala:

```
/**
* Keeps calling `f` until a `scala.util.Right[B]` is returned.
*/
def tailRecM[A, B](a: A)(f: A => F[Either[A, B]]): F[B]
```

As it turns out, Oscar Boykin (@posco) brought `tailRecM`

into `FlatMap`

in #1280
(Remove FlatMapRec make all FlatMap implement tailRecM), and it’s now part of Cats 0.7.0.
In other words, all FlatMap/Monads in Cats 0.7.0 are tail-recursive.

We can for example, obtain the `tailRecM`

for `Writer`

:

```
scala> def tailRecM[A, B] = FlatMap[Writer[Vector[String], ?]].tailRecM[A, B] _
tailRecM: [A, B]=> A => ((A => cats.data.WriterT[[A]A,scala.collection.immutable.Vector[String],Either[A,B]]) => cats.data.WriterT[[A]A,scala.collection.immutable.Vector[String],B])
```

Here’s how we can make a stack-safe `powWriter`

:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
def powWriter2(x: Long, exp: Long): Writer[LongProduct, Unit] =
FlatMap[Writer[LongProduct, ?]].tailRecM(exp) {
case 0L => Writer.value[LongProduct, Either[Long, Unit]](Right(()))
case m: Long => Writer.tell(LongProduct(x)) >>= { _ => Writer.value(Left(m - 1)) }
}
// Exiting paste mode, now interpreting.
powWriter2: (x: Long, exp: Long)cats.data.Writer[LongProduct,Unit]
scala> powWriter2(2, 3).run
res5: cats.Id[(LongProduct, Unit)] = (LongProduct(8),())
scala> powWriter2(1, 10000).run
res6: cats.Id[(LongProduct, Unit)] = (LongProduct(1),())
```

This guarantees greater safety on the user of `FlatMap`

typeclass, but it would mean that
each the implementers of the instances would need to provide a safe `tailRecM`

.

Here’s the one for `Option`

for example:

```
@tailrec
def tailRecM[A, B](a: A)(f: A => Option[Either[A, B]]): Option[B] =
f(a) match {
case None => None
case Some(Left(a1)) => tailRecM(a1)(f)
case Some(Right(b)) => Some(b)
}
```

That’s it for today!

On day 8 we reviewed the `Ior`

datatype, free monoids,
free monads, and the `Trampoline`

datatype.
Let’s continue on.

Learn You a Haskell for Great Good says:

In this section, we’re going to explore a few functions that either operate on monadic values or return monadic values as their results (or both!). Such functions are usually referred to as

monadic functions.

Unlike Haskell’s standard `Monad`

, Cats’ `Monad`

is more granularly designed with the hindsight of
weaker typeclasses.

`Monad`

- extends
`FlatMap`

and`Applicative`

- extends
`Apply`

- extends
`Functor`

Here there’s no question that all monads are applicative functors
as well as functors. This means we can use `ap`

or `map`

operator
on the datatypes that form a monad.

LYAHFGG:

It turns out that any nested monadic value can be flattened and that this is actually a property unique to monads. For this, the

`join`

function exists.

In Cats, the equivalent function called `flatten`

on `FlatMap`

.
Thanks to simulacrum, `flatten`

can also be injected as a method.

```
@typeclass trait FlatMap[F[_]] extends Apply[F] {
def flatMap[A, B](fa: F[A])(f: A => F[B]): F[B]
/**
* also commonly called join
*/
def flatten[A](ffa: F[F[A]]): F[A] =
flatMap(ffa)(fa => fa)
....
}
```

Since `Option[A]`

already implements `flatten`

we need to
make an abtract function to turn it into an abtract type.

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> def join[F[_]: FlatMap, A](fa: F[F[A]]): F[A] =
fa.flatten
join: [F[_], A](fa: F[F[A]])(implicit evidence$1: cats.FlatMap[F])F[A]
scala> join(1.some.some)
res0: Option[Int] = Some(1)
```

If I’m going to make it into a function, I could’ve used the function syntax:

```
scala> FlatMap[Option].flatten(1.some.some)
res1: Option[Int] = Some(1)
```

I tried using `flatten`

method for an `Xor`

of an `Xor`

value,
but it didn’t seem to work:

```
scala> val xorOfXor = Xor.right[String, Xor[String, Int]](Xor.right[String, Int](1))
<console>:20: error: not found: value Xor
val xorOfXor = Xor.right[String, Xor[String, Int]](Xor.right[String, Int](1))
^
<console>:20: error: not found: type Xor
val xorOfXor = Xor.right[String, Xor[String, Int]](Xor.right[String, Int](1))
^
<console>:20: error: not found: value Xor
val xorOfXor = Xor.right[String, Xor[String, Int]](Xor.right[String, Int](1))
^
scala> xorOfXor.flatten
<console>:21: error: not found: value xorOfXor
xorOfXor.flatten
^
```

LYAHFGG:

The

`filterM`

function from`Control.Monad`

does just what we want! … The predicate returns a monadic value whose result is a`Bool`

.

Cats does not have `filterM`

, but there’s `filterA`

on TraverseFilter.

LYAHFGG:

The monadic counterpart to

`foldl`

is`foldM`

.

I did not find `foldM`

in Cats, so implemented it myself, but it wasn’t stack-safe. Tomas Mikula added a better implementation and that got merged as #925.

:

```
/**
* Left associative monadic folding on `F`.
*/
def foldM[G[_], A, B](fa: F[A], z: B)(f: (B, A) => G[B])(implicit G: Monad[G]): G[B] =
foldLeft(fa, G.pure(z))((gb, a) => G.flatMap(gb)(f(_, a)))
```

Let’s try using this.

```
scala> def binSmalls(acc: Int, x: Int): Option[Int] =
if (x > 9) none[Int] else (acc + x).some
binSmalls: (acc: Int, x: Int)Option[Int]
scala> (Foldable[List].foldM(List(2, 8, 3, 1), 0) {binSmalls})
res3: Option[Int] = Some(14)
scala> (Foldable[List].foldM(List(2, 11, 3, 1), 0) {binSmalls})
res4: Option[Int] = None
```

In the above, `binSmalls`

returns `None`

when it finds a number larger than 9.

LYAHFGG:

When we were solving the problem of implementing a RPN calculator, we noted that it worked fine as long as the input that it got made sense.

We have not covered the chapter on RPN calculator, so let’s translate it into Scala.

```
scala> def foldingFunction(list: List[Double], next: String): List[Double] =
(list, next) match {
case (x :: y :: ys, "*") => (y * x) :: ys
case (x :: y :: ys, "+") => (y + x) :: ys
case (x :: y :: ys, "-") => (y - x) :: ys
case (xs, numString) => numString.toInt :: xs
}
foldingFunction: (list: List[Double], next: String)List[Double]
scala> def solveRPN(s: String): Double =
(s.split(' ').toList.
foldLeft(Nil: List[Double]) {foldingFunction}).head
solveRPN: (s: String)Double
scala> solveRPN("10 4 3 + 2 * -")
res0: Double = -4.0
```

Looks like it’s working.

The next step is to change the folding function to handle errors gracefully. We can implement `parseInt`

as follows:

```
scala> import scala.util.Try
import scala.util.Try
scala> def parseInt(x: String): Option[Int] =
(scala.util.Try(x.toInt) map { Some(_) }
recover { case _: NumberFormatException => None }).get
parseInt: (x: String)Option[Int]
scala> parseInt("1")
res1: Option[Int] = Some(1)
scala> parseInt("foo")
res2: Option[Int] = None
```

Here’s the updated folding function:

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> def foldingFunction(list: List[Double], next: String): Option[List[Double]] =
(list, next) match {
case (x :: y :: ys, "*") => ((y * x) :: ys).some
case (x :: y :: ys, "+") => ((y + x) :: ys).some
case (x :: y :: ys, "-") => ((y - x) :: ys).some
case (xs, numString) => parseInt(numString) map {_ :: xs}
}
foldingFunction: (list: List[Double], next: String)Option[List[Double]]
scala> foldingFunction(List(3, 2), "*")
res3: Option[List[Double]] = Some(List(6.0))
scala> foldingFunction(Nil, "*")
res4: Option[List[Double]] = None
scala> foldingFunction(Nil, "wawa")
res5: Option[List[Double]] = None
```

Finally, here’s the updated `solveRPN`

using `foldM`

:

```
scala> def solveRPN(s: String): Option[Double] =
for {
List(x) <- (Foldable[List].foldM(s.split(' ').toList, Nil: List[Double]) {foldingFunction})
} yield x
solveRPN: (s: String)Option[Double]
scala> solveRPN("1 2 * 4 +")
res6: Option[Double] = Some(6.0)
scala> solveRPN("1 2 * 4")
res7: Option[Double] = None
scala> solveRPN("1 8 garbage")
res8: Option[Double] = None
```

LYAHFGG:

When we were learning about the monad laws, we said that the

`<=<`

function is just like composition, only instead of working for normal functions like`a -> b`

, it works for monadic functions like`a -> m b`

.

In Cats there’s a special wrapper for a function of type `A => F[B]`

called Kleisli:

```
/**
* Represents a function `A => F[B]`.
*/
final case class Kleisli[F[_], A, B](run: A => F[B]) { self =>
....
}
object Kleisli extends KleisliInstances with KleisliFunctions
private[data] sealed trait KleisliFunctions {
def pure[F[_], A, B](x: B)(implicit F: Applicative[F]): Kleisli[F, A, B] =
Kleisli(_ => F.pure(x))
def ask[F[_], A](implicit F: Applicative[F]): Kleisli[F, A, A] =
Kleisli(F.pure)
def local[M[_], A, R](f: R => R)(fa: Kleisli[M, R, A]): Kleisli[M, R, A] =
Kleisli(f andThen fa.run)
}
```

We can use the `Kleisli()`

constructor to construct a `Kliesli`

value:

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> val f = Kleisli { (x: Int) => (x + 1).some }
f: cats.data.Kleisli[Option,Int,Int] = Kleisli(<function1>)
scala> val g = Kleisli { (x: Int) => (x * 100).some }
g: cats.data.Kleisli[Option,Int,Int] = Kleisli(<function1>)
```

We can then compose the functions using `compose`

, which runs the right-hand side first:

```
scala> 4.some >>= (f compose g).run
res0: Option[Int] = Some(401)
```

There’s also `andThen`

, which runs the left-hand side first:

```
scala> 4.some >>= (f andThen g).run
res1: Option[Int] = Some(500)
```

Both `compose`

and `andThen`

work like function composition
but note that they retain the monadic context.

Kleisli also has some interesting methods like `lift`

,
which allows you to lift a monadic function into another applicative functor.
When I tried using it, I realized it’s broken, so here’s the fixed version #354:

```
def lift[G[_]](implicit G: Applicative[G]): Kleisli[λ[α => G[F[α]]], A, B] =
Kleisli[λ[α => G[F[α]]], A, B](a => Applicative[G].pure(run(a)))
```

Here’s how we can use it:

```
scala> val l = f.lift[List]
l: cats.data.Kleisli[[α]List[Option[α]],Int,Int] = Kleisli(<function1>)
scala> List(1, 2, 3) >>= l.run
res2: List[Option[Int]] = List(Some(2), Some(3), Some(4))
```

LYAHFGG:

In this section, we’re going to look at an example of how a type gets made, identified as a monad and then given the appropriate

`Monad`

instance. … What if we wanted to model a non-deterministic value like`[3,5,9]`

, but we wanted to express that`3`

has a 50% chance of happening and`5`

and`9`

both have a 25% chance of happening?

Since Scala doesn’t have a built-in rational, let’s just use `Double`

. Here’s the case class:

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> :paste
// Entering paste mode (ctrl-D to finish)
case class Prob[A](list: List[(A, Double)])
trait ProbInstances {
implicit def probShow[A]: Show[Prob[A]] = Show.fromToString
}
case object Prob extends ProbInstances
// Exiting paste mode, now interpreting.
defined class Prob
defined trait ProbInstances
defined object Prob
```

Is this a functor? Well, the list is a functor, so this should probably be a functor as well, because we just added some stuff to the list.

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
case class Prob[A](list: List[(A, Double)])
trait ProbInstances {
implicit val probInstance: Functor[Prob] = new Functor[Prob] {
def map[A, B](fa: Prob[A])(f: A => B): Prob[B] =
Prob(fa.list map { case (x, p) => (f(x), p) })
}
implicit def probShow[A]: Show[Prob[A]] = Show.fromToString
}
case object Prob extends ProbInstances
// Exiting paste mode, now interpreting.
defined class Prob
defined trait ProbInstances
defined object Prob
scala> Prob((3, 0.5) :: (5, 0.25) :: (9, 0.25) :: Nil) map {-_}
res0: Prob[Int] = Prob(List((-3,0.5), (-5,0.25), (-9,0.25)))
```

Just like the book, we are going to implement `flatten`

first.

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
case class Prob[A](list: List[(A, Double)])
trait ProbInstances {
def flatten[B](xs: Prob[Prob[B]]): Prob[B] = {
def multall(innerxs: Prob[B], p: Double) =
innerxs.list map { case (x, r) => (x, p * r) }
Prob((xs.list map { case (innerxs, p) => multall(innerxs, p) }).flatten)
}
implicit val probInstance: Functor[Prob] = new Functor[Prob] {
def map[A, B](fa: Prob[A])(f: A => B): Prob[B] =
Prob(fa.list map { case (x, p) => (f(x), p) })
}
implicit def probShow[A]: Show[Prob[A]] = Show.fromToString
}
case object Prob extends ProbInstances
// Exiting paste mode, now interpreting.
defined class Prob
defined trait ProbInstances
defined object Prob
```

This should be enough prep work for monad:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
import scala.annotation.tailrec
case class Prob[A](list: List[(A, Double)])
trait ProbInstances { self =>
def flatten[B](xs: Prob[Prob[B]]): Prob[B] = {
def multall(innerxs: Prob[B], p: Double) =
innerxs.list map { case (x, r) => (x, p * r) }
Prob((xs.list map { case (innerxs, p) => multall(innerxs, p) }).flatten)
}
implicit val probInstance: Monad[Prob] = new Monad[Prob] {
def pure[A](a: A): Prob[A] = Prob((a, 1.0) :: Nil)
def flatMap[A, B](fa: Prob[A])(f: A => Prob[B]): Prob[B] = self.flatten(map(fa)(f))
override def map[A, B](fa: Prob[A])(f: A => B): Prob[B] =
Prob(fa.list map { case (x, p) => (f(x), p) })
def tailRecM[A, B](a: A)(f: A => Prob[Either[A, B]]): Prob[B] = {
val buf = List.newBuilder[(B, Double)]
@tailrec def go(lists: List[List[(Either[A, B], Double)]]): Unit =
lists match {
case (ab :: abs) :: tail => ab match {
case (Right(b), p) =>
buf += ((b, p))
go(abs :: tail)
case (Left(a), p) =>
go(f(a).list :: abs :: tail)
}
case Nil :: tail => go(tail)
case Nil => ()
}
go(f(a).list :: Nil)
Prob(buf.result)
}
}
implicit def probShow[A]: Show[Prob[A]] = Show.fromToString
}
case object Prob extends ProbInstances
// Exiting paste mode, now interpreting.
import scala.annotation.tailrec
defined class Prob
defined trait ProbInstances
defined object Prob
```

The book says it satisfies the monad laws. Let’s implement the `Coin`

example:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
sealed trait Coin
object Coin {
case object Heads extends Coin
case object Tails extends Coin
implicit val coinEq: Eq[Coin] = new Eq[Coin] {
def eqv(a1: Coin, a2: Coin): Boolean = a1 == a2
}
def heads: Coin = Heads
def tails: Coin = Tails
}
import Coin.{heads, tails}
def coin: Prob[Coin] = Prob(heads -> 0.5 :: tails -> 0.5 :: Nil)
def loadedCoin: Prob[Coin] = Prob(heads -> 0.1 :: tails -> 0.9 :: Nil)
// Exiting paste mode, now interpreting.
defined trait Coin
defined object Coin
import Coin.{heads, tails}
coin: Prob[Coin]
loadedCoin: Prob[Coin]
```

Here’s how we can implement `flipThree`

:

```
scala> def flipThree: Prob[Boolean] = for {
a <- coin
b <- coin
c <- loadedCoin
} yield { List(a, b, c) forall {_ === tails} }
flipThree: Prob[Boolean]
scala> flipThree
res1: Prob[Boolean] = Prob(List((false,0.025), (false,0.225), (false,0.025), (false,0.225), (false,0.025), (false,0.225), (false,0.025), (true,0.225)))
```

So the probability of having all three coins on `Tails`

even with a loaded coin is pretty low.

The intuition for `FlatMap`

and `Monad`

we built on day 5
via the tightrope walking example is that a monadic chaining `>>=`

can carry context from one operation to the next.
A single `None`

in an intermediate value banishes the entire chain.

The context monad instances carry along is different.
The `State`

datatype we saw on day 7 for example
automates the explicit passing of the state object with `>>=`

.

This is a useful intuition of monads to have in comparison to `Functor`

,
`Apply`

, and `Applicative`

, but it doesn’t tell the whole story.

Another intuition about monads (technically `FlatMap`

) is
that they are fractals, like the above Sierpinski triangle.
Each part of a fractal is self-similar to the whole shape.

Take `List`

for example. A `List`

of `List`

s can be treated a flat `List`

.

```
scala> val xss = List(List(1), List(2, 3), List(4))
xss: List[List[Int]] = List(List(1), List(2, 3), List(4))
scala> xss.flatten
res0: List[Int] = List(1, 2, 3, 4)
```

The `flatten`

function embodies the crunching of the `List`

data structure.
If we think in terms of the type signature, it would be `F[F[A]] => F[A]`

.

We can get a better idea of the flattening by reimplementing it using `foldLeft`

:

```
scala> xss.foldLeft(List(): List[Int]) { _ ++ _ }
res1: List[Int] = List(1, 2, 3, 4)
```

We can say that `List`

forms a monad under `++`

.

Now let try to figure out under what operation does `Option`

form a monad:

```
scala> val o1 = Some(None: Option[Int]): Option[Option[Int]]
o1: Option[Option[Int]] = Some(None)
scala> val o2 = Some(Some(1): Option[Int]): Option[Option[Int]]
o2: Option[Option[Int]] = Some(Some(1))
scala> val o3 = None: Option[Option[Int]]
o3: Option[Option[Int]] = None
```

And here’s the `foldLeft`

:

```
scala> o1.foldLeft(None: Option[Int]) { (_, _)._2 }
res2: Option[Int] = None
scala> o2.foldLeft(None: Option[Int]) { (_, _)._2 }
res3: Option[Int] = Some(1)
scala> o3.foldLeft(None: Option[Int]) { (_, _)._2 }
res4: Option[Int] = None
```

It seems like `Option`

forms a monad under `(_, _)._2`

.

If we come back to the `State`

datatype from the point of view of fractals,
it becomes clear that a `State`

of `State`

is also a `State`

.
This property allows us to create mini-programs like `pop`

and `push`

,
and compose them into a larger `State`

using `for`

comprehension:

```
def stackManip: State[Stack, Int] = for {
_ <- push(3)
a <- pop
b <- pop
} yield(b)
```

We also saw similar composition with the `Free`

monad.

In short, *monadic values* compose within the same monad instance.

When you want to find your own monad, keep a lookout for the fractal structure.
From there, see if you can implement the `flatten`

function `F[F[A]] => F[A]`

.

On day 9 we reviewed monadic functions `flatten`

, `filterM`

, and the implementation of `foldM`

. Next we made a safe RPN calculator using `foldM`

, looked at `Kleisli`

composition, implemented our own monad `Prob`

, and thought about fractals.

Since Cats does not provide Zippers, we are now done with Learn You a Haskell for Great Good, and we need to find our own topic.

One topic we have been dancing around, but haven’t gotten into is the notion of the monad transformer. Luckily there’s another good Haskell book that I’ve read that’s also available online.

Real World Haskell says:

It would be ideal if we could somehow take the standard

`State`

monad and add failure handling to it, without resorting to the wholesale construction of custom monads by hand. The standard monads in the`mtl`

library don’t allow us to combine them. Instead, the library provides a set ofmonad transformersto achieve the same result.A monad transformer is similar to a regular monad, but it’s not a standalone entity: instead, it modifies the behaviour of an underlying monad.

Let’s look into the idea of using `Reader`

datatype (`Function1`

)
for dependency injection, which we saw on day 6.

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
case class User(id: Long, parentId: Long, name: String, email: String)
trait UserRepo {
def get(id: Long): User
def find(name: String): User
}
// Exiting paste mode, now interpreting.
defined class User
defined trait UserRepo
```

Jason Arhart’s Scrap Your Cake Pattern Boilerplate: Dependency Injection Using the Reader Monad generalizes the notion of `Reader`

datatype for supporting multiple services by creating a `Config`

object:

```
scala> import java.net.URI
import java.net.URI
scala> :paste
// Entering paste mode (ctrl-D to finish)
trait HttpService {
def get(uri: URI): String
}
trait Config {
def userRepo: UserRepo
def httpService: HttpService
}
// Exiting paste mode, now interpreting.
defined trait HttpService
defined trait Config
```

To use this, we would construct mini-programs of type `Config => A`

, and compose them.

Suppose we want to also encode the notion of failure using `Option`

.

We can use the `Kleisli`

datatype we saw yesterday as `ReaderT`

, a monad transformer version of the `Reader`

datatype, and stack it on top of `Option`

like this:

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> :paste
// Entering paste mode (ctrl-D to finish)
type ReaderTOption[A, B] = Kleisli[Option, A, B]
object ReaderTOption {
def ro[A, B](f: A => Option[B]): ReaderTOption[A, B] = Kleisli(f)
}
// Exiting paste mode, now interpreting.
defined type alias ReaderTOption
defined object ReaderTOption
```

We can modify the `Config`

to make `httpService`

optional:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
trait UserRepo {
def get(id: Long): Option[User]
def find(name: String): Option[User]
}
trait Config {
def userRepo: UserRepo
def httpService: Option[HttpService]
}
// Exiting paste mode, now interpreting.
defined trait UserRepo
defined trait Config
```

Next we can rewrite the “primitive” readers to return `ReaderTOption[Config, A]`

:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
trait Users {
def getUser(id: Long): ReaderTOption[Config, User] =
ReaderTOption.ro {
case config => config.userRepo.get(id)
}
def findUser(name: String): ReaderTOption[Config, User] =
ReaderTOption.ro {
case config => config.userRepo.find(name)
}
}
trait Https {
def getHttp(uri: URI): ReaderTOption[Config, String] =
ReaderTOption.ro {
case config => config.httpService map {_.get(uri)}
}
}
// Exiting paste mode, now interpreting.
defined trait Users
defined trait Https
```

We can compose these mini-programs into compound programs:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
trait Program extends Users with Https {
def userSearch(id: Long): ReaderTOption[Config, String] =
for {
u <- getUser(id)
r <- getHttp(new URI(s"http://www.google.com/?q=${u.name}"))
} yield r
}
object Main extends Program {
def run(config: Config): Option[String] =
userSearch(2).run(config)
}
val dummyConfig: Config = new Config {
val testUsers = List(User(0, 0, "Vito", "vito@example.com"),
User(1, 0, "Michael", "michael@example.com"),
User(2, 0, "Fredo", "fredo@example.com"))
def userRepo: UserRepo = new UserRepo {
def get(id: Long): Option[User] =
testUsers find { _.id === id }
def find(name: String): Option[User] =
testUsers find { _.name === name }
}
def httpService: Option[HttpService] = None
}
// Exiting paste mode, now interpreting.
defined trait Program
defined object Main
dummyConfig: Config = $anon$1@7ff6c34d
```

The above `ReaderTOption`

datatype combines `Reader`

’s ability to read from some configuration once, and the `Option`

’s ability to express failure.

RWH:

When we stack a monad transformer on a normal monad, the result is another monad. This suggests the possibility that we can again stack a monad transformer on top of our combined monad, to give a new monad, and in fact this is a common thing to do.

We can stack `StateT`

on top of `ReaderTOption`

to represent state transfer.

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
type StateTReaderTOption[C, S, A] = StateT[({type l[X] = ReaderTOption[C, X]})#l, S, A]
object StateTReaderTOption {
def state[C, S, A](f: S => (S, A)): StateTReaderTOption[C, S, A] =
StateT[({type l[X] = ReaderTOption[C, X]})#l, S, A] {
s: S => Monad[({type l[X] = ReaderTOption[C, X]})#l].pure(f(s))
}
def get[C, S]: StateTReaderTOption[C, S, S] =
state { s => (s, s) }
def put[C, S](s: S): StateTReaderTOption[C, S, Unit] =
state { _ => (s, ()) }
def ro[C, S, A](f: C => Option[A]): StateTReaderTOption[C, S, A] =
StateT[({type l[X] = ReaderTOption[C, X]})#l, S, A] {
s: S =>
ReaderTOption.ro[C, (S, A)]{
c: C => f(c) map {(s, _)}
}
}
}
// Exiting paste mode, now interpreting.
defined type alias StateTReaderTOption
defined object StateTReaderTOption
```

This is a bit confusing, so let’s break it down. Ultimately the point of `State`

datatype is to wrap `S => (S, A)`

, so I kept those parameter names for `state`

. Next, I needed to modify the kind of `ReaderTOption`

to `* -> *`

(a type constructor that takes exactly one type as its parameter).

Similarly, we need a way of using this datatype as a `ReaderTOption`

, which is expressed as `C => Option[A]`

in `ro`

.

We also can implement a `Stack`

again. This time let’s use `String`

instead.

```
scala> type Stack = List[String]
defined type alias Stack
scala> val pop = StateTReaderTOption.state[Config, Stack, String] {
case x :: xs => (xs, x)
case _ => ???
}
pop: StateTReaderTOption[Config,Stack,String] = cats.data.StateT@2c70ef50
```

Here’s a version of `pop`

and `push`

using `get`

and `put`

primitive:

```
scala> import StateTReaderTOption.{get, put}
import StateTReaderTOption.{get, put}
scala> val pop: StateTReaderTOption[Config, Stack, String] =
for {
s <- get[Config, Stack]
(x :: xs) = s
_ <- put(xs)
} yield x
pop: StateTReaderTOption[Config,Stack,String] = cats.data.StateT@706e633
scala> def push(x: String): StateTReaderTOption[Config, Stack, Unit] =
for {
xs <- get[Config, Stack]
r <- put(x :: xs)
} yield r
push: (x: String)StateTReaderTOption[Config,Stack,Unit]
```

We can also port `stackManip`

:

```
scala> def stackManip: StateTReaderTOption[Config, Stack, String] =
for {
_ <- push("Fredo")
a <- pop
b <- pop
} yield(b)
stackManip: StateTReaderTOption[Config,Stack,String]
```

Here’s how we can use this:

```
scala> stackManip.run(List("Hyman Roth")).run(dummyConfig)
res0: Option[(Stack, String)] = Some((List(),Hyman Roth))
```

So far we have the same feature as the `State`

version.
We can modify `Users`

to use `StateTReaderTOption.ro`

:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
trait Users {
def getUser[S](id: Long): StateTReaderTOption[Config, S, User] =
StateTReaderTOption.ro[Config, S, User] {
case config => config.userRepo.get(id)
}
def findUser[S](name: String): StateTReaderTOption[Config, S, User] =
StateTReaderTOption.ro[Config, S, User] {
case config => config.userRepo.find(name)
}
}
// Exiting paste mode, now interpreting.
defined trait Users
```

Using this we can now manipulate the stack using the read-only configuration:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
trait Program extends Users {
def stackManip: StateTReaderTOption[Config, Stack, Unit] =
for {
u <- getUser(2)
a <- push(u.name)
} yield(a)
}
object Main extends Program {
def run(s: Stack, config: Config): Option[(Stack, Unit)] =
stackManip.run(s).run(config)
}
// Exiting paste mode, now interpreting.
defined trait Program
defined object Main
```

We can run this program like this:

```
scala> Main.run(List("Hyman Roth"), dummyConfig)
res1: Option[(Stack, Unit)] = Some((List(Fredo, Hyman Roth),()))
```

Now we have `StateT`

, `ReaderT`

and `Option`

working all at the same time.
Maybe I am not doing it right, but setting up the monad constructor functions like `state`

and `ro`

to set up `StateTReaderTOption`

is a rather mind-bending excercise.

Once the primitive monadic values are constructed, the usage code (like `stackManip`

) looks relatively clean.
It sure does avoid the cake pattern, but the stacked monad type `StateTReaderTOption`

is sprinkled all over the code base.

If all we wanted was being able to use `getUser(id: Long)`

and `push`

etc.,
an alternative is to construct a DSL with those commands using Free monads we saw on day 8.

One use of monad transformers that seem to come up often is stacking `Future`

datatype with `Either`

. There’s a blog post by Yoshida-san (@xuwei_k) in Japanese called How to combine Future and Either nicely in Scala.

A little known fact about Yoshida-san outside of Tokyo, is that he majored in Chinese calligraphy. Apparently he spent his collge days writing ancient seal scripts and carving seals:

「大学では、はんこを刻ったり、篆書を書いてました」
「えっ？なぜプログラマに？？？」 pic.twitter.com/DEhqy4ELpF

— Kenji Yoshida (@xuwei_k) October 21, 2013

His namesake Xu Wei was a Ming-era painter, poet, writer and dramatist famed for his free-flowing style. Here’s Yoshida-san writing functional programming language.

In any case, why would one want to stack `Future`

and `Either`

together?
The blog post explains like this:

`Future[A]`

comes up a lot in Scala.- You don’t want to block future, so you end up with
`Future`

everywhere. - Since
`Future`

is asynchronous, any errors need to be captured in there. `Future`

is designed to handle`Throwable`

, but that’s all you get.- When the program becomes complex enough, you want to type your own error states.
- How can we make
`Future`

of`Either`

?

Here’s the prepration step:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
case class User(id: Long, name: String)
// In actual code, probably more than 2 errors
sealed trait Error
object Error {
final case class UserNotFound(userId: Long) extends Error
final case class ConnectionError(message: String) extends Error
}
object UserRepo {
def followers(userId: Long): Either[Error, List[User]] = ???
}
import UserRepo.followers
// Exiting paste mode, now interpreting.
defined class User
defined trait Error
defined object Error
defined object UserRepo
import UserRepo.followers
```

Suppose our application allows the users to follow each other like Twitter.
The `followers`

function returns the list of followers.

Now let’s try writing a function that checks if two users follow each other.

```
scala> def isFriends0(user1: Long, user2: Long): Either[Error, Boolean] =
for {
a <- followers(user1).right
b <- followers(user2).right
} yield a.exists(_.id == user2) && b.exists(_.id == user1)
isFriends0: (user1: Long, user2: Long)Either[Error,Boolean]
```

Now suppose we want to make the database access async
so we changed the `followers`

to return a `Future`

like this:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
import scala.concurrent.{ Future, ExecutionContext }
object UserRepo {
def followers(userId: Long): Future[Either[Error, List[User]]] = ???
}
import UserRepo.followers
// Exiting paste mode, now interpreting.
import scala.concurrent.{Future, ExecutionContext}
defined object UserRepo
import UserRepo.followers
```

Now, how would `isFriends0`

look like? Here’s one way of writing this:

```
scala> def isFriends1(user1: Long, user2: Long)
(implicit ec: ExecutionContext): Future[Either[Error, Boolean]] =
for {
a <- followers(user1)
b <- followers(user2)
} yield for {
x <- a.right
y <- b.right
} yield x.exists(_.id == user2) && y.exists(_.id == user1)
isFriends1: (user1: Long, user2: Long)(implicit ec: scala.concurrent.ExecutionContext)scala.concurrent.Future[Either[Error,Boolean]]
```

And here’s another version:

```
scala> def isFriends2(user1: Long, user2: Long)
(implicit ec: ExecutionContext): Future[Either[Error, Boolean]] =
followers(user1) flatMap {
case Right(a) =>
followers(user2) map {
case Right(b) =>
Right(a.exists(_.id == user2) && b.exists(_.id == user1))
case Left(e) =>
Left(e)
}
case Left(e) =>
Future.successful(Left(e))
}
isFriends2: (user1: Long, user2: Long)(implicit ec: scala.concurrent.ExecutionContext)scala.concurrent.Future[Either[Error,Boolean]]
```

What is the difference between the two versions?
They’ll both behave the same if `followers`

return normally,
but suppose `followers(user1)`

returns an `Error`

state.
`isFriend1`

would still call `followers(user2)`

, whereas `isFriend2`

would short-circuit and return the error.

Regardless, both functions became convoluted compared to the original.
And it’s mostly a boilerplate to satisfy the types.
I don’t want to imagine doing this for *every* functions that uses `Future[Either[Error, A]]`

.

Cats comes with `EitherT`

datatype, which is a monad transformer for `Either`

.

```
/**
* Transformer for `Either`, allowing the effect of an arbitrary type constructor `F` to be combined with the
* fail-fast effect of `Either`.
*
* `EitherT[F, A, B]` wraps a value of type `F[Either[A, B]]`. An `F[C]` can be lifted in to `EitherT[F, A, C]` via `EitherT.right`,
* and lifted in to a `EitherT[F, C, B]` via `EitherT.left`.
*/
case class EitherT[F[_], A, B](value: F[Either[A, B]]) {
....
}
```

Here’s `UserRepo.followers`

with a dummy implementation:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
import cats._, cats.data._, cats.implicits._
object UserRepo {
def followers(userId: Long)
(implicit ec: ExecutionContext): EitherT[Future, Error, List[User]] =
userId match {
case 0L =>
EitherT.right(Future { List(User(1, "Michael")) })
case 1L =>
EitherT.right(Future { List(User(0, "Vito")) })
case x =>
println("not found")
EitherT.left(Future.successful { Error.UserNotFound(x) })
}
}
import UserRepo.followers
// Exiting paste mode, now interpreting.
import cats._
import cats.data._
import cats.implicits._
defined object UserRepo
import UserRepo.followers
```

Now let’s try writing `isFriends0`

function again.

```
scala> def isFriends3(user1: Long, user2: Long)
(implicit ec: ExecutionContext): EitherT[Future, Error, Boolean] =
for{
a <- followers(user1)
b <- followers(user2)
} yield a.exists(_.id == user2) && b.exists(_.id == user1)
isFriends3: (user1: Long, user2: Long)(implicit ec: scala.concurrent.ExecutionContext)cats.data.EitherT[scala.concurrent.Future,Error,Boolean]
```

Isn’t this great? Except for the type signature and the `ExecutionContext`

,
`isFriends3`

is identical to `isFriends0`

.

Now let’s try using this.

```
scala> implicit val ec = scala.concurrent.ExecutionContext.global
ec: scala.concurrent.ExecutionContextExecutor = scala.concurrent.impl.ExecutionContextImpl@321e273f
scala> import scala.concurrent.Await
import scala.concurrent.Await
scala> import scala.concurrent.duration._
import scala.concurrent.duration._
scala> Await.result(isFriends3(0, 1).value, 1 second)
res0: Either[Error,Boolean] = Right(true)
```

When the first user is not found, `EitherT`

will short circuit.

```
scala> Await.result(isFriends3(2, 3).value, 1 second)
not found
res34: cats.data.Xor[Error,Boolean] = Left(UserNotFound(2))
```

Note that `"not found"`

is printed only once.

Unlike the `StateTReaderTOption`

example, `EitherT`

seems usable in many situations.

That’s it for today!

On day 10 we looked at the idea of monad transformers, first using `Kliesli`

as `ReaderT`

,
and also stacking `Future`

and `Either`

(`XorT`

).

In the grand scheme of things, functional programming is about abstracting things out. Skimming over Jeremy Gibbons’s 2006 book Datatype-Generic Programming, I found a nice overview.

Generic programmingis about making programming languages more flexible without compromising safety.

One of the first and most fundamental techniques that any programmer learns is how to parametrize computations by values

```
scala> def triangle4: Unit = {
println("*")
println("**")
println("***")
println("****")
}
triangle4: Unit
```

We can abstract out 4 into a parameter:

```
scala> def triangle(side: Int): Unit = {
(1 to side) foreach { row =>
(1 to row) foreach { col =>
println("*")
}
}
}
triangle: (side: Int)Unit
```

`List[A]`

is a *polymorphic datatype* parametrized by another type,
the type of list elements. This enables *parametric polymorphism*.

```
scala> def head[A](xs: List[A]): A = xs(0)
head: [A](xs: List[A])A
```

The above function would work for all proper types.

Higher-orderprograms are programs parametrized by other programs.

For example `foldLeft`

can be used to `append`

two lists:

```
scala> def append[A](list: List[A], ys: List[A]): List[A] =
list.foldLeft(ys) { (acc, x) => x :: acc }
append: [A](list: List[A], ys: List[A])List[A]
scala> append(List(1, 2, 3), List(4, 5, 6))
res0: List[Int] = List(3, 2, 1, 4, 5, 6)
```

Or it can also be used to add numbers:

```
scala> def sum(list: List[Int]): Int =
list.foldLeft(0) { _ + _ }
sum: (list: List[Int])Int
```

“Generic programming” embodied in the sense of a collection library,
like Scala Collections.
In the case of C++’s Standard Template Library,
the parametric datatypes are called *containers*, and various abstractions are provided via *iterators*,
such as input iterators and forward iterators.

The notion of the typeclass fits in here too.

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
trait Read[A] {
def reads(s: String): Option[A]
}
object Read extends ReadInstances {
def read[A](f: String => Option[A]): Read[A] = new Read[A] {
def reads(s: String): Option[A] = f(s)
}
def apply[A: Read]: Read[A] = implicitly[Read[A]]
}
trait ReadInstances {
implicit val stringRead: Read[String] =
Read.read[String] { Some(_) }
implicit val intRead: Read[Int] =
Read.read[Int] { s =>
try {
Some(s.toInt)
} catch {
case e: NumberFormatException => None
}
}
}
// Exiting paste mode, now interpreting.
defined trait Read
defined object Read
defined trait ReadInstances
scala> Read[Int].reads("1")
res1: Option[Int] = Some(1)
```

The typeclass captures the requirements required of types, called typeclass contract.
It also lets us list the types providing these requirements by defining
typeclass instances.
This enables *ad-hoc polymorphism* because `A`

in `Read[A]`

is not universal.

In Scala Collection library, some of the concepts promised are more elaborate than the list of operations covered by the type.

as well as signatures of operations, the concept might specify the laws these operations satisfy, and non-functional characteristics such as the asymptotic complexities of the operations in terms of time and space.

Typeclasses with laws fit in here too. For example `Monoid[A]`

comes with the monoid laws.
The laws need to be validated for each instance using property-based testing tools.

Various flavors of *metaprogramming* can be though of as the development
or program that construct or manipulate other programs.
This could include code generation and macros.

Let’s say there’s a polymorphic datatype of binary trees:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
sealed trait Btree[A]
object Btree {
case class Tip[A](a: A) extends Btree[A]
case class Bin[A](left: Btree[A], right: Btree[A]) extends Btree[A]
}
// Exiting paste mode, now interpreting.
defined trait Btree
defined object Btree
```

Let’s write `foldB`

as a way of abstracting similar programs.

```
scala> def foldB[A, B](tree: Btree[A], b: (B, B) => B)(t: A => B): B =
tree match {
case Btree.Tip(a) => t(a)
case Btree.Bin(xs, ys) => b(foldB(xs, b)(t), foldB(ys, b)(t))
}
foldB: [A, B](tree: Btree[A], b: (B, B) => B)(t: A => B)B
```

The next goal is to abstract `foldB`

and `foldLeft`

.

In fact, what differs between the two fold operators is the

shapeof the data on which they operate, and hence the shape of the programs themselves. The kind of parametrization required is by this shape; that is, by the datatype or type constructor (such as`List`

or`Tree`

) concerned. We call thisdatatype genericity.

For example, `fold`

apparently could be expressed as

```
scala> import cats._, cats.data._, cats.implicits._, cats.functor.Bifunctor
import cats._
import cats.data._
import cats.implicits._
import cats.functor.Bifunctor
scala> :paste
// Entering paste mode (ctrl-D to finish)
trait Fix[F[_,_], A]
def cata[S[_,_]: Bifunctor, A, B](t: Fix[S, A])(f: S[A, B] => B): B = ???
// Exiting paste mode, now interpreting.
defined trait Fix
cata: [S[_, _], A, B](t: Fix[S,A])(f: S[A,B] => B)(implicit evidence$1: cats.functor.Bifunctor[S])B
```

In the above, `S`

represents the shape of the datatype.
By abstracting out the shapes, we can construct *parametrically datatype-generic* programs.
We’ll come back to this later.

Alternatively, such programs might be

ad-hoc datatype-generic, when the behaviour exploits that shape in some essential manner. Typical examples of the latter are pretty printers and marshallers.

The example that fits in this category might be Scala Pickling. Pickling defines picklers for common types, and it derives pickler instances for different shapes using macro.

This approach to datatype genericity has been variously called

polytypism,structural polymorphismortypecase, and is the meaning given to ‘generic programming’ by the Generic Haskell team. Whatever the name, functions are defined inductively by case analysis on the structure of datatypes ….We consider parametric datatype genericity to be the ‘gold standard’, and in the remainder of these lecture notes, we concentrate on parametric datatype-generic definitions where possible.

In Scala, shapeless is focused on abstracting out the shape.

Moving on the section 3.6 in Datatype-Generic Programming called “Datatype genericity.” Gibbons tries to call this Origami programming, but I don’t think the name stuck, so we’ll go with datatype-generic programming.

As we have already argued, data structure determines program structure. It therefore makes sense to abstract from the determining shape, leaving only what programs of different shape have in common. What datatypes such as

`List`

and`Tree`

have in common is the fact that they are recursive — which is to say, a datatype`Fix`

```
data Fix s a = In {out :: s a (Fix s a)}
```

Here are three instances of Fix using different shapes: lists and internally labelled binary trees as seen before, and also a datatype of externally labelled binary trees.

```
data ListF a b = NilF | ConsF a b
type List a = Fix ListF a
data TreeF a b = EmptyF | NodeF a b b
type Tree a = Fix TreeF a
data BtreeF a b = TipF a | BinF b b
type Btree a = Fix BtreeF a
```

From Why free monads matter we saw on day 8
we actually know this is similar as `Free`

datatype, but
the semantics around `Functor`

etc is going to different,
so let’s implement it from scratch:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
sealed abstract class Fix[S[_], A] extends Serializable {
def out: S[Fix[S, A]]
}
object Fix {
case class In[S[_], A](out: S[Fix[S, A]]) extends Fix[S, A]
}
// Exiting paste mode, now interpreting.
defined class Fix
defined object Fix
```

Following `Free`

, I am putting `S[_]`

on the left, and `A`

on the right.

Let’s try implementing the `List`

first.

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
sealed trait ListF[+Next, +A]
object ListF {
case class NilF() extends ListF[Nothing, Nothing]
case class ConsF[A, Next](a: A, n: Next) extends ListF[Next, A]
}
type GenericList[A] = Fix[ListF[+?, A], A]
object GenericList {
def nil[A]: GenericList[A] = Fix.In[ListF[+?, A], A](ListF.NilF())
def cons[A](a: A, xs: GenericList[A]): GenericList[A] =
Fix.In[ListF[+?, A], A](ListF.ConsF(a, xs))
}
// Exiting paste mode, now interpreting.
defined trait ListF
defined object ListF
defined type alias GenericList
defined object GenericList
scala> import GenericList.{ cons, nil }
import GenericList.{cons, nil}
```

Here’s how we can use it:

```
scala> cons(1, nil)
res0: GenericList[Int] = In(ConsF(1,In(NilF())))
```

So far this is similar to what we saw with the free monad.

Not all valid binary type constructors s are suitable for

Fixing; for example, function types with the parameter appearing incontravariant(source) positions cause problems. It turns out that we should restrict attention to (covariant)bifunctors, which support abimapoperation ‘locating’ all the elements.

Cats ships with `Bifunctor`

:

```
/**
* A typeclass of types which give rise to two independent, covariant
* functors.
*/
trait Bifunctor[F[_, _]] extends Serializable { self =>
/**
* The quintessential method of the Bifunctor trait, it applies a
* function to each "side" of the bifunctor.
*/
def bimap[A, B, C, D](fab: F[A, B])(f: A => C, g: B => D): F[C, D]
....
}
```

Here is the `Bifunctor`

instance for `GenericList`

.

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> import cats.functor.Bifunctor
import cats.functor.Bifunctor
scala> :paste
// Entering paste mode (ctrl-D to finish)
implicit val listFBifunctor: Bifunctor[ListF] = new Bifunctor[ListF] {
def bimap[S1, A1, S2, A2](fab: ListF[S1, A1])(f: S1 => S2, g: A1 => A2): ListF[S2, A2] =
fab match {
case ListF.NilF() => ListF.NilF()
case ListF.ConsF(a, next) => ListF.ConsF(g(a), f(next))
}
}
// Exiting paste mode, now interpreting.
listFBifunctor: cats.functor.Bifunctor[ListF] = $anon$1@7a453673
```

It turns out that the class

`Bifunctor`

provides sufficient flexibility to capture a wide variety of recursion patterns as datatype-generic programs.

First, we can implement `map`

in terms of `bimap`

.

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
object DGP {
def map[F[_, _]: Bifunctor, A1, A2](fa: Fix[F[?, A1], A1])(f: A1 => A2): Fix[F[?, A2], A2] =
Fix.In[F[?, A2], A2](Bifunctor[F].bimap(fa.out)(map(_)(f), f))
}
// Exiting paste mode, now interpreting.
defined object DGP
scala> DGP.map(cons(1, nil)) { _ + 1 }
res1: Fix[[α]ListF[α,Int],Int] = In(ConsF(2,In(NilF())))
```

The above definition of `map`

is independent from `GenericList`

, abstracted by `Bifunctor`

and `Fix`

.
Another way of looking at it is that we can get `Functor`

for free out of `Bifunctor`

and `Fix`

.

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
trait FixInstances {
implicit def fixFunctor[F[_, _]: Bifunctor]: Functor[Lambda[L => Fix[F[?, L], L]]] =
new Functor[Lambda[L => Fix[F[?, L], L]]] {
def map[A1, A2](fa: Fix[F[?, A1], A1])(f: A1 => A2): Fix[F[?, A2], A2] =
Fix.In[F[?, A2], A2](Bifunctor[F].bimap(fa.out)(map(_)(f), f))
}
}
// Exiting paste mode, now interpreting.
defined trait FixInstances
scala> {
val instances = new FixInstances {}
import instances._
import cats.syntax.functor._
cons(1, nil) map { _ + 1 }
}
res2: GenericList[Int] = In(ConsF(2,In(NilF())))
```

Intense amount of type lambdas, but I think it’s clear that I translated `DB.map`

into a `Functor`

instance.

We can also implement `fold`

, also known as `cata`

from catamorphism:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
object DGP {
// catamorphism
def fold[F[_, _]: Bifunctor, A1, A2](fa: Fix[F[?, A1], A1])(f: F[A2, A1] => A2): A2 =
{
val g = (fa1: F[Fix[F[?, A1], A1], A1]) =>
Bifunctor[F].leftMap(fa1) { (fold(_)(f)) }
f(g(fa.out))
}
}
// Exiting paste mode, now interpreting.
defined object DGP
scala> DGP.fold[ListF, Int, Int](cons(2, cons(1, nil))) {
case ListF.NilF() => 0
case ListF.ConsF(x, n) => x + n
}
res3: Int = 3
```

The

`unfold`

operator for a datatype grows a data structure from a value. In a precise technical sense, it is the dual of the`fold`

operator.

The `unfold`

is also called `ana`

from anamorphism:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
object DGP {
// catamorphism
def fold[F[_, _]: Bifunctor, A1, A2](fa: Fix[F[?, A1], A1])(f: F[A2, A1] => A2): A2 =
{
val g = (fa1: F[Fix[F[?, A1], A1], A1]) =>
Bifunctor[F].leftMap(fa1) { (fold(_)(f)) }
f(g(fa.out))
}
// anamorphism
def unfold[F[_, _]: Bifunctor, A1, A2](x: A2)(f: A2 => F[A2, A1]): Fix[F[?, A1], A1] =
Fix.In[F[?, A1], A1](Bifunctor[F].leftMap(f(x))(unfold[F, A1, A2](_)(f)))
}
// Exiting paste mode, now interpreting.
defined object DGP
```

Here’s how we can construct list of numbers counting down:

```
scala> def pred(n: Int): GenericList[Int] =
DGP.unfold[ListF, Int, Int](n) {
case 0 => ListF.NilF()
case n => ListF.ConsF(n, n - 1)
}
pred: (n: Int)GenericList[Int]
scala> pred(4)
res4: GenericList[Int] = In(ConsF(4,In(ConsF(3,In(ConsF(2,In(ConsF(1,In(NilF())))))))))
```

There are several more we can derive, too.

The point of the datatype-generic programming is to abstract out the shape.
Let’s introduce some other datatype, like a binary `Tree`

:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
sealed trait TreeF[+Next, +A]
object TreeF {
case class EmptyF() extends TreeF[Nothing, Nothing]
case class NodeF[Next, A](a: A, left: Next, right: Next) extends TreeF[Next, A]
}
type Tree[A] = Fix[TreeF[?, A], A]
object Tree {
def empty[A]: Tree[A] =
Fix.In[TreeF[+?, A], A](TreeF.EmptyF())
def node[A, Next](a: A, left: Tree[A], right: Tree[A]): Tree[A] =
Fix.In[TreeF[+?, A], A](TreeF.NodeF(a, left, right))
}
// Exiting paste mode, now interpreting.
defined trait TreeF
defined object TreeF
defined type alias Tree
defined object Tree
```

Here’s how to create this tree:

```
scala> import Tree.{empty,node}
import Tree.{empty, node}
scala> node(2, node(1, empty, empty), empty)
res5: Tree[Int] = In(NodeF(2,In(NodeF(1,In(EmptyF()),In(EmptyF()))),In(EmptyF())))
```

Now, all we have to do should be to define a `Bifunctor`

instance:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
implicit val treeFBifunctor: Bifunctor[TreeF] = new Bifunctor[TreeF] {
def bimap[A, B, C, D](fab: TreeF[A, B])(f: A => C, g: B => D): TreeF[C, D] =
fab match {
case TreeF.EmptyF() => TreeF.EmptyF()
case TreeF.NodeF(a, left, right) =>
TreeF.NodeF(g(a), f(left), f(right))
}
}
// Exiting paste mode, now interpreting.
treeFBifunctor: cats.functor.Bifunctor[TreeF] = $anon$1@60e4a628
```

First, let’s try `Functor`

:

```
scala> {
val instances = new FixInstances {}
import instances._
import cats.syntax.functor._
node(2, node(1, empty, empty), empty) map { _ + 1 }
}
res6: Tree[Int] = In(NodeF(3,In(NodeF(2,In(EmptyF()),In(EmptyF()))),In(EmptyF())))
```

Looking good. Next, let’s try folding.

```
scala> def sum(tree: Tree[Int]): Int =
DGP.fold[TreeF, Int, Int](tree) {
case TreeF.EmptyF() => 0
case TreeF.NodeF(a, l, r) => a + l + r
}
sum: (tree: Tree[Int])Int
scala> sum(node(2, node(1, empty, empty), empty))
res7: Int = 3
```

We got the `fold`

.

Here’s a function named `grow`

that generates a binary search tree from a list.

```
scala> def grow[A: PartialOrder](xs: List[A]): Tree[A] =
DGP.unfold[TreeF, A, List[A]](xs) {
case Nil => TreeF.EmptyF()
case x :: xs =>
import cats.syntax.partialOrder._
TreeF.NodeF(x, xs filter {_ <= x}, xs filter {_ > x})
}
grow: [A](xs: List[A])(implicit evidence$1: cats.PartialOrder[A])Tree[A]
scala> grow(List(3, 1, 4, 2))
res8: Tree[Int] = In(NodeF(3,In(NodeF(1,In(EmptyF()),In(NodeF(2,In(EmptyF()),In(EmptyF()))))),In(NodeF(4,In(EmptyF()),In(EmptyF())))))
```

Looks like `unfold`

is working too.

For more details on DGP in Scala, Oliveira and Gibbons wrote a paper translating the ideas and many more called Scala for Generic Programmers (2008), and its updated version Scala for Generic Programmers (2010).

Datatype-generic programming described here has a limitation of datatype being recursive, so I don’t think it’s very usable as is, but there are some interesting concepts that can be exploited.

Next, Gibbons claims that the design patterns are evidence of a “evidence of a lack of expressivity in those mainstream programming languages.” He then sets out to replace the patterns using higher-order datatype-generic programming.

Chapter 5 of Datatype-Generic Programming, the last one before Conclusions, is called “The Essence of the Iterator pattern,” the same name of the paper Gibbons and Oliveira wrote in 2006. The one available online as The Essence of the Iterator Pattern is from 2009. Reading this paper as a continuation of DGP gives it a better context.

Here’s the example given at the beginning of this paper, translated to Java.

```
public static <E> int loop(Collection<E> coll) {
int n = 0;
for (E elem: coll) {
n = n + 1;
doSomething(elem);
}
return n;
}
```

EIP:

We emphasize that we want to capture both aspects of the method

loopand iterations like it:mappingover the elements, and simultaneouslyaccumulatingsome measure of those elements.

The first half of the paper reviews functional iterations and applicative style. For applicative functors, it brings up the fact that there are three kinds of applicatives:

- Monadic applicative functors
- Naperian applicative functors
- Monoidal applicative functors

We’ve brought up the fact that all monads are applicatives many times. Naperian applicative functor zips together data structure that are fixed in shape.

Appliactive functors were originally named *idom* by McBride and Paterson,
so Gibbons uses the term *idiomatic* interchangably with *applicative* througout this paper,
even though McBride and Paterson renamed it to applicative functors.

A second family of applicative functors, this time non-monadic, arises from constant functors with monoidal targets.

We can derive an applicative functor from any `Monoid`

,
by using `empty`

for `pure`

, and `|+|`

for `ap`

.
The `Const`

datatype is also called `Const`

in Cats:

```
/**
* [[Const]] is a phantom type, it does not contain a value of its second type parameter `B`
* [[Const]] can be seen as a type level version of `Function.const[A, B]: A => B => A`
*/
final case class Const[A, B](getConst: A) {
/**
* changes the type of the second type parameter
*/
def retag[C]: Const[A, C] =
this.asInstanceOf[Const[A, C]]
....
}
```

In the above, the type parameter `A`

represents the value,
but `B`

is a phantom type used to make `Functor`

happy.

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> Const(1) map { (_: String) + "!" }
res0: cats.data.Const[Int,String] = Const(1)
```

When `A`

forms a `Semigroup`

, an `Apply`

is derived,
and when `A`

form a `Monoid`

, an `Applicative`

is derived automatically.

Computations within this applicative functor accumulate some measure: for the monoid of integers with addition, they count or sum…

```
scala> Const(2).retag[String => String] ap Const(1).retag[String]
res1: cats.data.Const[Int,String] = Const(3)
```

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)

```
/**
* [[Prod]] 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]]
*/
sealed trait Prod[F[_], G[_], A] {
def first: F[A]
def second: G[A]
}
object Prod extends ProdInstances {
def apply[F[_], G[_], A](first0: => F[A], second0: => G[A]): Prod[F, G, A] = new Prod[F, G, A] {
val firstThunk: Eval[F[A]] = Later(first0)
val secondThunk: Eval[G[A]] = Later(second0)
def first: F[A] = firstThunk.value
def second: G[A] = secondThunk.value
}
def unapply[F[_], G[_], A](x: Prod[F, G, A]): Option[(F[A], G[A])] =
Some((x.first, x.second))
}
```

First we start with the product of `Functor`

:

```
private[data] sealed abstract class ProdInstances4 {
implicit def prodFunctor[F[_], G[_]](implicit FF: Functor[F], GG: Functor[G]): Functor[Lambda[X => Prod[F, G, X]]] = new ProdFunctor[F, G] {
def F: Functor[F] = FF
def G: Functor[G] = GG
}
}
sealed trait ProdFunctor[F[_], G[_]] extends Functor[Lambda[X => Prod[F, G, X]]] {
def F: Functor[F]
def G: Functor[G]
def map[A, B](fa: Prod[F, G, A])(f: A => B): Prod[F, G, B] = Prod(F.map(fa.first)(f), G.map(fa.second)(f))
}
```

Here’s how to use it:

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> val x = Prod(List(1), (Some(1): Option[Int]))
x: cats.data.Prod[List,Option,Int] = Prod(List(1),Some(1))
scala> Functor[Lambda[X => Prod[List, Option, X]]].map(x) { _ + 1 }
res0: cats.data.Prod[[+A]List[A],Option,Int] = Prod(List(2),Some(2))
```

First, we are defining a pair-like datatype called `Prod`

, which prepresents a product of typeclass instances.
By simply passing the function `f`

to both the sides, we can form `Functor`

for `Prod[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 ProdInstances3 extends ProdInstances4 {
implicit def prodApply[F[_], G[_]](implicit FF: Apply[F], GG: Apply[G]): Apply[Lambda[X => Prod[F, G, X]]] = new ProdApply[F, G] {
def F: Apply[F] = FF
def G: Apply[G] = GG
}
}
sealed trait ProdApply[F[_], G[_]] extends Apply[Lambda[X => Prod[F, G, X]]] with ProdFunctor[F, G] {
def F: Apply[F]
def G: Apply[G]
def ap[A, B](fa: Prod[F, G, A])(f: Prod[F, G, A => B]): Prod[F, G, B] =
Prod(F.ap(f.first)(fa.first), G.ap(f.second)(fa.second))
def product[A, B](fa: Prod[F, G, A], fb: Prod[F, G, B]): Prod[F, G, (A, B)] =
Prod(F.product(fa.first, fb.first), G.product(fa.second, fb.second))
}
```

Here’s the usage:

```
scala> val x = Prod(List(1), (Some(1): Option[Int]))
x: cats.data.Prod[List,Option,Int] = Prod(List(1),Some(1))
scala> val f = Prod(List((_: Int) + 1), (Some((_: Int) * 3): Option[Int => Int]))
f: cats.data.Prod[List,Option,Int => Int] = Prod(List(<function1>),Some(<function1>))
scala> Apply[Lambda[X => Prod[List, Option, X]]].ap(f)(x)
res1: cats.data.Prod[[+A]List[A],Option,Int] = Prod(List(2),Some(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 ProdInstances2 extends ProdInstances3 {
implicit def prodApplicative[F[_], G[_]](implicit FF: Applicative[F], GG: Applicative[G]): Applicative[Lambda[X => Prod[F, G, X]]] = new ProdApplicative[F, G] {
def F: Applicative[F] = FF
def G: Applicative[G] = GG
}
}
sealed trait ProdApplicative[F[_], G[_]] extends Applicative[Lambda[X => Prod[F, G, X]]] with ProdApply[F, G] {
def F: Applicative[F]
def G: Applicative[G]
def pure[A](a: A): Prod[F, G, A] = Prod(F.pure(a), G.pure(a))
}
```

Here’s a simple usage:

```
scala> Applicative[Lambda[X => Prod[List, Option, X]]].pure(1)
res2: cats.data.Prod[[+A]List[A],Option,Int] = Prod(List(1),Some(1))
```

We were able to create `Prod(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.

```
scala> Applicative[List].compose[Option].pure(1)
res3: List[Option[Int]] = List(Some(1))
```

So much nicer.

For some reason people seem to overlook is that Gibbons also introduces
applicative function composition operators.
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
}
/** applicative function using [[Unapply]]. */
def appFuncU[A, R](f: A => R)(implicit RR: Unapply[Applicative, R]): AppFunc[RR.M, A, RR.A] =
appFunc({ a: A => RR.subst(f(a)) })(RR.TC)
}
....
/**
* 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:

```
scala> val f = Func.appFunc { x: Int => List(x.toString + "!") }
f: cats.data.AppFunc[List,Int,String] = cats.data.Func$$anon$6@13b41ee2
scala> val g = Func.appFunc { x: Int => (Some(x.toString + "?"): Option[String]) }
g: cats.data.AppFunc[Option,Int,String] = cats.data.Func$$anon$6@47d50669
scala> val h = f product g
h: cats.data.AppFunc[[α]cats.data.Prod[List,Option,α],Int,String] = cats.data.Func$$anon$6@7cde3c58
scala> h.run(1)
res4: cats.data.Prod[List,Option,String] = Prod(List(1!),Some(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)
```

```
scala> val f = Func.appFunc { x: Int => List(x.toString + "!") }
f: cats.data.AppFunc[List,Int,String] = cats.data.Func$$anon$6@101fb548
scala> val g = Func.appFunc { x: String => (Some(x + "?"): Option[String]) }
g: cats.data.AppFunc[Option,String,String] = cats.data.Func$$anon$6@3ffc8b9c
scala> val h = f andThen g
h: cats.data.AppFunc[[γ$3$]cats.data.Nested[List,Option,γ$3$],Int,String] = cats.data.Func$$anon$6@60f96668
scala> h.run(1)
res5: cats.data.Nested[List,Option,String] = Nested(List(Some(1!?)))
```

EIP:

The two operators [snip] allow us to combine idiomatic computations in two different ways; we call them

parallelandsequentialcomposition, respectively.

Even though we had to introduce a new datatype `Prod`

,
the combining applicative computation is an abstract concept for all `Applicative`

.

We’ll continue from here.

On day 11 we started reading Jeremy Gibbons’s
Datatype-Generic Programming.
We saw the creative use of `Fix`

and `Bifunctor`

,

When we moved on to The Essence of the Iterator Pattern,
we found that Cats uses `Const`

to represent
monoidal applicatives like `Int`

, but it’s currently missing
a way to compose applicative functions.

We started with a custon datatype called `Prod`

that represents a pair of `F[A]`

and `G[A]`

,
then we also added `AppFunc`

that represents applicative functions.
#388

The Essence of the Iterator Pattern:

Two of the three motivating examples McBride and Paterson provide for idiomatic computations — sequencing a list of monadic effects and transposing a matrix — are instances of a general scheme they call

traversal. This involves iterating over the elements of a data structure, in the style of a`map`

, but interpreting certain function applications idiomatically. … We capture this via a type class ofTraversabledata structures.

In Cats, this typeclass is called Traverse:

```
@typeclass trait Traverse[F[_]] extends Functor[F] with Foldable[F] { self =>
/**
* given a function which returns a G effect, thread this effect
* through the running of this function on all the values in F,
* returning an F[A] in a G context
*/
def traverse[G[_]: Applicative, A, B](fa: F[A])(f: A => G[B]): G[F[B]]
/**
* thread all the G effects through the F structure to invert the
* structure from F[G[_]] to G[F[_]]
*/
def sequence[G[_]: Applicative, A](fga: F[G[A]]): G[F[A]] =
traverse(fga)(ga => ga)
....
}
```

Note that the `f`

takes the shape of `A => G[B]`

.

When

mis specialised to the identity applicative functor, traversal reduces precisely (modulo the wrapper) to the functorial map over lists.

Cats’ identity applicative functor is defined as follows:

```
type Id[A] = A
implicit val Id: Bimonad[Id] =
new Bimonad[Id] {
def pure[A](a: A): A = a
def extract[A](a: A): A = a
def flatMap[A, B](a: A)(f: A => B): B = f(a)
def coflatMap[A, B](a: A)(f: A => B): B = f(a)
override def map[A, B](fa: A)(f: A => B): B = f(fa)
override def ap[A, B](fa: A)(ff: A => B): B = ff(fa)
override def flatten[A](ffa: A): A = ffa
override def map2[A, B, Z](fa: A, fb: B)(f: (A, B) => Z): Z = f(fa, fb)
override def lift[A, B](f: A => B): A => B = f
override def imap[A, B](fa: A)(f: A => B)(fi: B => A): B = f(fa)
}
```

Here’s how we can traverse over `List(1, 2, 3)`

using `Id`

.

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> List(1, 2, 3) traverse[Id, Int] { (x: Int) => x + 1 }
res0: cats.Id[List[Int]] = List(2, 3, 4)
```

In the case of a monadic applicative functor, traversal specialises to monadic map, and has the same uses. In fact, traversal is really just a slight generalisation of monadic map.

Let’s try using this for `List`

:

```
scala> List(1, 2, 3) traverse { (x: Int) => (Some(x + 1): Option[Int]) }
res1: Option[List[Int]] = Some(List(2, 3, 4))
scala> List(1, 2, 3) traverse { (x: Int) => None }
res2: Option[List[Nothing]] = None
```

For a Naperian applicative functor, traversal transposes results.

We’re going to skip this one.

For a monoidal applicative functor, traversal accumulates values. The function

`reduce`

performs that accumulation, given an argument that assigns a value to each element

```
scala> def reduce[A, B, F[_]](fa: F[A])(f: A => B)
(implicit FF: Traverse[F], BB: Monoid[B]): B =
{
val g: A => Const[B, Unit] = { (a: A) => Const((f(a))) }
val x = FF.traverse[Const[B, ?], A, Unit](fa)(g)
x.getConst
}
reduce: [A, B, F[_]](fa: F[A])(f: A => B)(implicit FF: cats.Traverse[F], implicit BB: cats.Monoid[B])B
```

Here’s how we can use this:

```
scala> reduce(List('a', 'b', 'c')) { c: Char => c.toInt }
res3: Int = 294
```

It sort of works, but it would be nicer if we can reduce the type annotation a bit.
The problem is that as it stands, Scala 2.11 compiler cannot infer `Const[B, ?]`

.
There’s a actually a trick to nudge the compiler into looking into several shapes called `Unapply`

,
and we can use `traverseU`

instead of `traverse`

to take advantage of it:

```
scala> def reduce[A, B, F[_]](fa: F[A])(f: A => B)
(implicit FF: Traverse[F], BB: Monoid[B]): B =
{
val x = fa traverseU { (a: A) => Const((f(a))) }
x.getConst
}
reduce: [A, B, F[_]](fa: F[A])(f: A => B)(implicit FF: cats.Traverse[F], implicit BB: cats.Monoid[B])B
```

We’ll find out what this is about later.

`Applicative`

and `Traverse`

are mentioned together by
McBride and Paterson in Applicative programming with effects.

As a background, until a few months ago (March 2015), the sequence function
in `Control.Monad`

package used to look like this:

```
-- | Evaluate each action in the sequence from left to right,
-- and collect the results.
sequence :: Monad m => [m a] -> m [a]
```

If I translate this into Scala, it would look like:

```
def sequence[G[_]: Monad, A](gas: List[G[A]]): G[List[A]]
```

It takes a list of monadic values, and returns a monadic value of a list.
This already looks pretty useful on its own,
but whenever you find a hardcoded `List`

like this, we should suspect if it should
be replaced with a better typeclass.

McBride and Paterson first generalized the `sequence`

function to
`dist`

, by replacing `Monad`

with `Applicative`

:

```
def dist[G[_]: Applicative, A](gas: List[G[A]]): G[List[A]]
```

Next, they realized that `dist`

is often called with `map`

so they
added another parameter for applicative function, and called it `traverse`

:

```
def traverse[G[_]: Applicative, A, B](as: List[A])(f: A => G[B]): G[List[B]]
```

Finally they generalized the above signature into a typeclass named `Traversible`

:

```
@typeclass trait Traverse[F[_]] extends Functor[F] with Foldable[F] { self =>
/**
* given a function which returns a G effect, thread this effect
* through the running of this function on all the values in F,
* returning an F[A] in a G context
*/
def traverse[G[_]: Applicative, A, B](fa: F[A])(f: A => G[B]): G[F[B]]
/**
* thread all the G effects through the F structure to invert the
* structure from F[G[_]] to G[F[_]]
*/
def sequence[G[_]: Applicative, A](fga: F[G[A]]): G[F[A]] =
traverse(fga)(ga => ga)
....
}
```

Thus as the matter of course, `Traverse`

implements a datatype-generic,
`sequence`

function, which is just `traverse`

with `identity`

, conceptually easier to remember,
because it simply flips `F[G[A]]`

into `G[F[A]]`

.
You might have seen this function for `Future`

in the standard library.

```
scala> import scala.concurrent.{ Future, ExecutionContext, Await }
import scala.concurrent.{Future, ExecutionContext, Await}
scala> import scala.concurrent.duration._
import scala.concurrent.duration._
scala> val x = {
implicit val ec = scala.concurrent.ExecutionContext.global
List(Future { 1 }, Future { 2 }).sequence
}
x: scala.concurrent.Future[List[Int]] = List()
scala> Await.result(x, 1 second)
res4: List[Int] = List(1, 2)
```

Another useseful thing might be to turn a `List`

of `Either`

into an `Either`

.

```
scala> List(Right(1): Either[String, Int]).sequenceU
res5: scala.util.Either[String,List[Int]] = Right(List(1))
scala> List(Right(1): Either[String, Int], Left("boom"): Either[String, Int]).sequenceU
res6: scala.util.Either[String,List[Int]] = Left(boom)
```

Note we just used `sequenceU`

, which is the `Unapply`

variant of `sequence`

.
Let’s look into that next.

Cats 0.7.0 adds a new typeclass called TraverseFilter.

```
/**
* `TraverseFilter`, also known as `Witherable`, represents list-like structures
* that can essentially have a [[traverse]] and a [[filter]] applied as a single
* combined operation ([[traverseFilter]]).
*
* Must obey the laws defined in cats.laws.TraverseFilterLaws.
*
* Based on Haskell's [[https://hackage.haskell.org/package/witherable-0.1.3.3/docs/Data-Witherable.html Data.Witherable]]
*/
@typeclass trait TraverseFilter[F[_]] extends Traverse[F] with FunctorFilter[F] { self =>
/**
* A combined [[traverse]] and [[filter]]. Filtering is handled via `Option`
* instead of `Boolean` such that the output type `B` can be different than
* the input type `A`.
*/
def traverseFilter[G[_]: Applicative, A, B](fa: F[A])(f: A => G[Option[B]]): G[F[B]]
/**
*
* Filter values inside a `G` context.
*
* This is a generalized version of Haskell's [[http://hackage.haskell.org/package/base-4.9.0.0/docs/Control-Monad.html#v:filterM filterM]].
* [[http://stackoverflow.com/questions/28872396/haskells-filterm-with-filterm-x-true-false-1-2-3 This StackOverflow question]] about `filterM` may be helpful in understanding how it behaves.
*/
def filterA[G[_], A](fa: F[A])(f: A => G[Boolean])(implicit G: Applicative[G]): G[F[A]] =
traverseFilter(fa)(a => G.map(f(a))(if (_) Some(a) else None))
}
```

As noted in the comment, `filterA`

is a more generalized (or weaker) version of `filterM`

. Instead of requiring a `Monad[G]`

it needs `Applicative[G]`

.

Here’s how we can use this:

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> List(1, 2, 3) filterA { x => List(true, false) }
res7: List[List[Int]] = List(List(1, 2, 3), List(1, 2), List(1, 3), List(1), List(2, 3), List(2), List(3), List())
scala> Vector(1, 2, 3) filterA { x => Vector(true, false) }
res8: scala.collection.immutable.Vector[scala.collection.immutable.Vector[Int]] = Vector(Vector(1, 2, 3), Vector(1, 2), Vector(1, 3), Vector(1), Vector(2, 3), Vector(2), Vector(3), Vector())
```

EIP:

We will have a number of new datatypes with coercion functions like

`Id`

,`unId`

,`Const`

and`unConst`

. To reduce clutter, we introduce a common notation for such coercions.

In Scala, implicits and type inference take us pretty far. But by dealing with typeclass a lot, we also end up seeing some of the weaknesses of Scala’s type inference. One of the issues we come across frequently is its inability to infer partially applied parameterized type, also known as SI-2712. If you’re reading this, please jump to the page, upvote the bug, or help solving the problem.

To work around this issue, Cats uses a typeclass called `Unapply`

:

```
/**
* A typeclass that is used to help guide scala's type inference to
* find typeclass instances for types which have shapes which differ
* from what their typeclasses are looking for.
*
* For example, [[Functor]] is defined for types in the shape
* F[_]. Scala has no problem finding instance of Functor which match
* this shape, such as Functor[Option], Functor[List], etc. There is
* also a functor defined for some types which have the Shape F[_,_]
* when one of the two 'holes' is fixed. For example. there is a
* Functor for Map[A,?] for any A, and for Either[A,?] for any A,
* however the scala compiler will not find them without some coercing.
*/
trait Unapply[TC[_[_]], MA] {
// a type constructor which is properly kinded for the typeclass
type M[_]
// the type applied to the type constructor to make an MA
type A
// the actual typeclass instance found
def TC: TC[M]
// a function which will coerce the MA value into one of type M[A]
// this will end up being the identity function, but we can't supply
// it until we have proven that MA and M[A] are the same type
def subst: MA => M[A]
}
```

I think it’s easier to demonstrate this using an example.

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> def foo[F[_]: Applicative](fa: F[Int]): F[Int] = fa
foo: [F[_]](fa: F[Int])(implicit evidence$1: cats.Applicative[F])F[Int]
```

In the above, `foo`

is a simple function that returns the passed in value `fa: F[Int]`

where `F`

forms an `Applicative`

.
Since `Either[String, Int]`

is an applicative, it should qualify.

```
scala> foo(Right(1): Either[String, Int])
<console>:22: error: no type parameters for method foo: (fa: F[Int])(implicit evidence$1: cats.Applicative[F])F[Int] exist so that it can be applied to arguments (Either[String,Int])
--- because ---
argument expression's type is not compatible with formal parameter type;
found : Either[String,Int]
required: ?F[Int]
foo(Right(1): Either[String, Int])
^
<console>:22: error: type mismatch;
found : Either[String,Int]
required: F[Int]
foo(Right(1): Either[String, Int])
^
<console>:22: error: could not find implicit value for evidence parameter of type cats.Applicative[F]
foo(Right(1): Either[String, Int])
^
```

We got the error “argument expression’s type is not compatible with formal parameter type.”
We can make an `Unapply`

version of `foo`

as follows:

```
scala> def fooU[FA](fa: FA)(implicit U: Unapply[Applicative, FA]): U.M[U.A] =
U.subst(fa)
fooU: [FA](fa: FA)(implicit U: cats.Unapply[cats.Applicative,FA])U.M[U.A]
```

Now let’s try passing in exactly same parameter as we tried:

```
scala> fooU(Right(1): Either[String, Int])
res1: scala.util.Either[String,Int] = Right(1)
```

It works. Let’s look into how this is implemented.
For `Either`

, a monad is formed for `Either[AA, ?]`

, which means
during `map`

the right side of the parameter might change like
`List[Int]`

changing to `List[String]`

, but the left side stays put.

```
sealed abstract class Unapply2Instances extends Unapply3Instances {
type Aux2Right[TC[_[_]], MA, F[_,_], AA, B] = Unapply[TC, MA] {
type M[X] = F[AA,X]
type A = B
}
implicit def unapply2right[TC[_[_]], F[_,_], AA, B](implicit tc: TC[F[AA,?]]): Aux2Right[TC,F[AA,B], F, AA, B] = new Unapply[TC, F[AA,B]] {
type M[X] = F[AA, X]
type A = B
def TC: TC[F[AA, ?]] = tc
def subst: F[AA, B] => M[A] = identity
}
....
}
```

First Cats is defining `Aux2Right`

as a type alias that defines the abstract types `M[_]`

and `A`

.
Next, it defines an implicit converter from an arbitray typeclass instance `TC[F[AA,?]]`

to `Aux2Right[TC,F[AA,B], F, AA, B]`

.

One area where `Unapply`

is used in Cats is where infix operators, also known as “syntax”,
is injected. These implicit converters are called “bedazzlers” in cecc3a.

```
package cats
package syntax
trait FunctorSyntax1 {
implicit def functorSyntaxU[FA](fa: FA)(implicit U: Unapply[Functor,FA]): Functor.Ops[U.M, U.A] =
new Functor.Ops[U.M, U.A]{
val self = U.subst(fa)
val typeClassInstance = U.TC
}
}
trait FunctorSyntax extends Functor.ToFunctorOps with FunctorSyntax1
```

Let’s try using `*>`

operator from `Apply`

:

```
scala> (Right(1): Either[String, Int]) *> Right(2)
res2: scala.util.Either[String,Int] = Right(2)
```

This works, likely thanks to the `Unapply`

.

AppFunc we looked at on day 11 lets us create ever more complex
composition of applicative functor instances.
It would be half as useful without the help of `Unapply`

, since part of the benefit
is that it can derive the instances automatically.
At the same time, `Unapply`

cannot possibly be the ultimate solution because
it requires all shapes to be spelled out up front, and the most complex type it handles currently is `F[AA, B, ?]`

.
I could go beyond this by making a product of compositions of monad instances with two parameters.

I can only guess that this type inference problem is a problem that needs to be solved by the Scala compiler itself by treating sequential composition and parallel composition (product) as a first-class constuct in the type system.

EIP:

In addition to being parametrically polymorphic in the collection elements, the generic

`traverse`

operation is parametrised along two further dimensions: the datatype being traversed, and the applicative functor in which the traversal is interpreted. Specialising the latter to lists as a monoid yields a generic`contents`

operation:

Here’s how we can implement this with Cats:

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> def contents[F[_], A](fa: F[A])(implicit FF: Traverse[F]): Const[List[A], F[Unit]] =
{
val contentsBody: A => Const[List[A], Unit] = { (a: A) => Const(List(a)) }
FF.traverseU(fa)(contentsBody)
}
contents: [F[_], A](fa: F[A])(implicit FF: cats.Traverse[F])cats.data.Const[List[A],F[Unit]]
```

Now we can take any datatype that supports `Traverse`

and turn it into a `List`

.

```
scala> contents(Vector(1, 2, 3)).getConst
res0: List[Int] = List(1, 2, 3)
```

I am actually not sure if the result suppose to come out in the reverse order here.

The other half of the decomposition is obtained simply by a map, which is to say, a traversal interpreted in the identity idiom.

When Gibbons say identity idiom, he means the identity applicative functor, `Id[_]`

.

```
scala> def shape[F[_], A](fa: F[A])(implicit FF: Traverse[F]): Id[F[Unit]] =
{
val shapeBody: A => Id[Unit] = { (a: A) => () }
FF.traverseU(fa)(shapeBody)
}
shape: [F[_], A](fa: F[A])(implicit FF: cats.Traverse[F])cats.Id[F[Unit]]
```

Here’s the shape for `Vector(1, 2, 3)`

:

```
scala> shape(Vector(1, 2, 3))
res1: cats.Id[scala.collection.immutable.Vector[Unit]] = Vector((), (), ())
```

EIP:

This pair of traversals nicely illustrates the two aspects of iterations that we are focussing on, namely mapping and accumulation.

Next EIP demonstrates applicative composition by first combining `shape`

and `contents`

together like this:

```
scala> def decompose[F[_], A](fa: F[A])(implicit FF: Traverse[F]) =
Prod[Const[List[A], ?], Id, F[Unit]](contents(fa), shape(fa))
decompose: [F[_], A](fa: F[A])(implicit FF: cats.Traverse[F])cats.data.Prod[[β]cats.data.Const[List[A],β],cats.Id,F[Unit]]
scala> val d = decompose(Vector(1, 2, 3))
d: cats.data.Prod[[β]cats.data.Const[List[Int],β],cats.Id,scala.collection.immutable.Vector[Unit]] = Prod(Const(List(1, 2, 3)),Vector((), (), ()))
scala> d.first
res2: cats.data.Const[List[Int],scala.collection.immutable.Vector[Unit]] = Const(List(1, 2, 3))
scala> d.second
res3: cats.Id[scala.collection.immutable.Vector[Unit]] = Vector((), (), ())
```

The problem here is that we are running `traverse`

twice.

Is it possible to fuse the two traversals into one? The product of applicative functors allows exactly this.

Let’s try rewriting this using `AppFunc`

.

```
scala> import cats.data.Func.appFunc
import cats.data.Func.appFunc
scala> def contentsBody[A]: AppFunc[Const[List[A], ?], A, Unit] =
appFunc[Const[List[A], ?], A, Unit] { (a: A) => Const(List(a)) }
contentsBody: [A]=> cats.data.AppFunc[[β]cats.data.Const[List[A],β],A,Unit]
scala> def shapeBody[A]: AppFunc[Id, A, Unit] =
appFunc { (a: A) => ((): Id[Unit]) }
shapeBody: [A]=> cats.data.AppFunc[cats.Id,A,Unit]
scala> def decompose[F[_], A](fa: F[A])(implicit FF: Traverse[F]) =
(contentsBody[A] product shapeBody[A]).traverse(fa)
decompose: [F[_], A](fa: F[A])(implicit FF: cats.Traverse[F])cats.data.Prod[[β]cats.data.Const[List[A],β],cats.Id,F[Unit]]
scala> val d = decompose(Vector(1, 2, 3))
d: cats.data.Prod[[β]cats.data.Const[List[Int],β],cats.Id,scala.collection.immutable.Vector[Unit]] = Prod(Const(List(1, 2, 3)),Vector((), (), ()))
scala> d.first
res4: cats.data.Const[List[Int],scala.collection.immutable.Vector[Unit]] = Const(List(1, 2, 3))
scala> d.second
res5: cats.Id[scala.collection.immutable.Vector[Unit]] = Vector((), (), ())
```

The return type of the `decompose`

is a bit messy, but it’s infered by `AppFunc`

:
`Prod[[X_kp1]Const[List[A], X_kp1], Id, F[Unit]]`

.

Skipping over to section 6 of EIP “Modular programming with applicative functors.”

EIP:

There is an additional benefit of applicative functors over monads, which concerns the modular development of complex iterations from simpler aspects. ….

As an illustration, we consider the Unix word-counting utility

`wc`

, which computes the numbers of characters, words and lines in a text file.

We can translate the full example using applicative function composition, which is only available on my personal branch. (PR #388 is pending)

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> import Func.{ appFunc, appFuncU }
import Func.{appFunc, appFuncU}
```

The character-counting slice of the

`wc`

program accumulates a result in the integers-as-monoid applicative functor:

Here’s a type alias to treat `Int`

as a monoidal applicative:

```
scala> type Count[A] = Const[Int, A]
defined type alias Count
```

In the above, `A`

is a phantom type we don’t need, so let’s just hardcode it to `Unit`

:

```
scala> def liftInt(i: Int): Count[Unit] = Const(i)
liftInt: (i: Int)Count[Unit]
scala> def count[A](a: A): Count[Unit] = liftInt(1)
count: [A](a: A)Count[Unit]
```

The body of the iteration simply yields 1 for every element:

```
scala> val countChar: AppFunc[Count, Char, Unit] = appFunc(count)
countChar: cats.data.AppFunc[Count,Char,Unit] = cats.data.Func$$anon$6@4f7da1db
```

To use this `AppFunc`

, we would call `traverse`

with `List[Char]`

.
Here’s a quote I found from Hamlet.

```
scala> val text = ("Faith, I must leave thee, love, and shortly too.\n" +
"My operant powers their functions leave to do.\n").toList
text: List[Char] =
List(F, a, i, t, h, ,, , I, , m, u, s, t, , l, e, a, v, e, , t, h, e, e, ,, , l, o, v, e, ,, , a, n, d, , s, h, o, r, t, l, y, , t, o, o, .,
, M, y, , o, p, e, r, a, n, t, , p, o, w, e, r, s, , t, h, e, i, r, , f, u, n, c, t, i, o, n, s, , l, e, a, v, e, , t, o, , d, o, .,
)
scala> countChar traverse text
res0: Count[List[Unit]] = Const(96)
```

This looks ok.

Counting the lines (in fact, the newline characters, thereby ignoring a final ‘line’ that is not terminated with a newline character) is similar: the difference is simply what number to use for each element, namely 1 for a newline and 0 for anything else.

```
scala> def testIf(b: Boolean): Int = if (b) 1 else 0
testIf: (b: Boolean)Int
scala> val countLine: AppFunc[Count, Char, Unit] =
appFunc { (c: Char) => liftInt(testIf(c === '\n')) }
countLine: cats.data.AppFunc[Count,Char,Unit] = cats.data.Func$$anon$6@318107a7
```

Again, to use this we’ll just call `traverse`

:

```
scala> countLine traverse text
res1: Count[List[Unit]] = Const(2)
```

Counting the words is trickier, because it necessarily involves state. Here, we use the

`State`

monad with a boolean state, indicating whether we are currently within a word, and compose this with the applicative functor for counting.

```
scala> def isSpace(c: Char): Boolean = (c === ' ' || c === '\n' || c === '\t')
isSpace: (c: Char)Boolean
scala> val countWord =
appFuncU { (c: Char) =>
import cats.data.State.{ get, set }
for {
x <- get[Boolean]
y = !isSpace(c)
_ <- set(y)
} yield testIf(y && !x)
} andThen appFunc(liftInt)
countWord: cats.data.AppFunc[[γ$3$]cats.data.Nested[[X]cats.data.StateT[cats.Eval,Boolean,X],Count,γ$3$],Char,Unit] = cats.data.Func$$anon$6@19a4ff28
```

Traversing this `AppFunc`

returns a `State`

datatype:

```
scala> val x = countWord traverse text
x: cats.data.Nested[[X]cats.data.StateT[cats.Eval,Boolean,X],Count,List[Unit]] = Nested(cats.data.StateT@2be3e2b5)
```

We then need to run this state machine with an initial value `false`

to get the result:

```
scala> x.value.runA(false).value
res2: Count[List[Unit]] = Const(17)
```

17 words.

Like we did with `shape`

and `content`

, we can *fuse* the traversal into one shot
by combining the applicative functions.

```
scala> val countAll = countWord product countLine product countChar
countAll: cats.data.AppFunc[[α]cats.data.Prod[[α]cats.data.Prod[[γ$3$]cats.data.Nested[[X]cats.data.StateT[cats.Eval,Boolean,X],Count,γ$3$],Count,α],Count,α],Char,Unit] = cats.data.Func$$anon$6@6176bbe2
scala> val allResults = countAll traverse text
allResults: cats.data.Prod[[α]cats.data.Prod[[γ$3$]cats.data.Nested[[X]cats.data.StateT[cats.Eval,Boolean,X],Count,γ$3$],Count,α],Count,List[Unit]] = Prod(Prod(Nested(cats.data.StateT@30755fd0),Const(2)),Const(96))
scala> val charCount = allResults.second
charCount: Count[List[Unit]] = Const(96)
scala> val lineCount = allResults.first.second
lineCount: Count[List[Unit]] = Const(2)
scala> val wordCountState = allResults.first.first
wordCountState: cats.data.Nested[[X]cats.data.StateT[cats.Eval,Boolean,X],Count,List[Unit]] = Nested(cats.data.StateT@30755fd0)
scala> val wordCount = wordCountState.value.runA(false).value
wordCount: Count[List[Unit]] = Const(17)
```

EIP:

Applicative functors have a richer algebra of composition operators, which can often replace the use of monad transformers; there is the added advantage of being able to compose applicative but non-monadic computations.

That’s it for today.

On day 12 we continued exploring The Essence of the Iterator Pattern,
and looked at `Traverse`

, `Unapply`

, shape and contents, and the applicative wordcount.

We saw `Id`

in passing while reading EIP, but it’s an interesting tool, so we should revisit it.
This is also called Identity, Identity functor, or Identity monad, depending on the context.
The definition of the datatype is quite simple:

```
type Id[A] = A
```

Here’s with the documentation and the typeclass instances:

```
/**
* Identity, encoded as `type Id[A] = A`, a convenient alias to make
* identity instances well-kinded.
*
* The identity monad can be seen as the ambient monad that encodes
* the effect of having no effect. It is ambient in the sense that
* plain pure values are values of `Id`.
*
* For instance, the [[cats.Functor]] instance for `[[cats.Id]]`
* allows us to apply a function `A => B` to an `Id[A]` and get an
* `Id[B]`. However, an `Id[A]` is the same as `A`, so all we're doing
* is applying a pure function of type `A => B` to a pure value of
* type `A` to get a pure value of type `B`. That is, the instance
* encodes pure unary function application.
*/
type Id[A] = A
implicit val Id: Bimonad[Id] with Traverse[Id] =
new Bimonad[Id] with Traverse[Id] {
def pure[A](a: A): A = a
def extract[A](a: A): A = a
def flatMap[A, B](a: A)(f: A => B): B = f(a)
def coflatMap[A, B](a: A)(f: A => B): B = f(a)
override def map[A, B](fa: A)(f: A => B): B = f(fa)
override def ap[A, B](fa: A)(ff: A => B): B = ff(fa)
override def flatten[A](ffa: A): A = ffa
override def map2[A, B, Z](fa: A, fb: B)(f: (A, B) => Z): Z = f(fa, fb)
override def lift[A, B](f: A => B): A => B = f
override def imap[A, B](fa: A)(f: A => B)(fi: B => A): B = f(fa)
def foldLeft[A, B](a: A, b: B)(f: (B, A) => B) = f(b, a)
def foldRight[A, B](a: A, lb: Eval[B])(f: (A, Eval[B]) => Eval[B]): Eval[B] =
f(a, lb)
def traverse[G[_], A, B](a: A)(f: A => G[B])(implicit G: Applicative[G]): G[B] =
f(a)
}
```

Here’s how to create a value of `Id`

:

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> val one: Id[Int] = 1
one: cats.Id[Int] = 1
```

The functor instance for `Id`

is same as function application:

```
scala> Functor[Id].map(one) { _ + 1 }
res0: cats.Id[Int] = 2
```

The apply’s `ap`

method, which takes `Id[A => B]`

, but in reality just `A => B`

is also implemented as function application:

```
scala> Apply[Id].ap({ _ + 1 }: Id[Int => Int])(one)
res1: cats.Id[Int] = 2
```

The FlatMap’s `flatMap`

method, which takes `A => Id[B]`

is the same story. It’s implemented function application:

```
scala> FlatMap[Id].flatMap(one) { _ + 1 }
res2: cats.Id[Int] = 2
```

At a glance, `Id`

datatype is not very useful. The hint is in the Scaladoc of the definition: a convenient alias to make identity instances well-kinded. In other words, there are situations where we need to lift some type `A`

into `F[A]`

, and `Id`

can be used to just that without introducing any effects. We’ll see an example of that later.

Cats also comes with `Eval`

datatype that controls evaluation.

```
sealed abstract class Eval[+A] extends Serializable { self =>
/**
* Evaluate the computation and return an A value.
*
* For lazy instances (Later, Always), any necessary computation
* will be performed at this point. For eager instances (Now), a
* value will be immediately returned.
*/
def value: A
/**
* Ensure that the result of the computation (if any) will be
* memoized.
*
* Practically, this means that when called on an Always[A] a
* Later[A] with an equivalent computation will be returned.
*/
def memoize: Eval[A]
}
```

There are several ways to create an `Eval`

value:

```
object Eval extends EvalInstances {
/**
* Construct an eager Eval[A] value (i.e. Now[A]).
*/
def now[A](a: A): Eval[A] = Now(a)
/**
* Construct a lazy Eval[A] value with caching (i.e. Later[A]).
*/
def later[A](a: => A): Eval[A] = new Later(a _)
/**
* Construct a lazy Eval[A] value without caching (i.e. Always[A]).
*/
def always[A](a: => A): Eval[A] = new Always(a _)
/**
* Defer a computation which produces an Eval[A] value.
*
* This is useful when you want to delay execution of an expression
* which produces an Eval[A] value. Like .flatMap, it is stack-safe.
*/
def defer[A](a: => Eval[A]): Eval[A] =
new Eval.Call[A](a _) {}
/**
* Static Eval instances for some common values.
*
* These can be useful in cases where the same values may be needed
* many times.
*/
val Unit: Eval[Unit] = Now(())
val True: Eval[Boolean] = Now(true)
val False: Eval[Boolean] = Now(false)
val Zero: Eval[Int] = Now(0)
val One: Eval[Int] = Now(1)
....
}
```

The most useful one is `Eval.later`

, which captures a by-name parameter in a `lazy val`

.

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> var g: Int = 0
g: Int = 0
scala> val x = Eval.later {
g = g + 1
g
}
x: cats.Eval[Int] = cats.Later@216b0ebb
scala> g = 2
g: Int = 2
scala> x.value
res0: Int = 3
scala> x.value
res1: Int = 3
```

The `value`

is cached, so the second evaluation doesn’t happen.

`Eval.now`

evaluates eagerly, and then captures the result in a field, so the second evaluation doesn’t happen.

```
scala> val y = Eval.now {
g = g + 1
g
}
y: cats.Eval[Int] = Now(4)
scala> y.value
res2: Int = 4
scala> y.value
res3: Int = 4
```

`Eval.always`

doesn’t cache.

```
scala> val z = Eval.always {
g = g + 1
g
}
z: cats.Eval[Int] = cats.Always@6a5c80bc
scala> z.value
res4: Int = 5
scala> z.value
res5: Int = 6
```

One useful feature of `Eval`

is that it supports stack-safe lazy computation via `map`

and `flatMap`

methods,
which use an internal trampoline to avoid stack overflow.

You can also defer a computation which produces `Eval[A]`

value using `Eval.defer`

. Here’s how `foldRight`

is implemented for `List`

for example:

```
def foldRight[A, B](fa: List[A], lb: Eval[B])(f: (A, Eval[B]) => Eval[B]): Eval[B] = {
def loop(as: List[A]): Eval[B] =
as match {
case Nil => lb
case h :: t => f(h, Eval.defer(loop(t)))
}
Eval.defer(loop(fa))
}
```

Let’s try blowing up the stack on purpose:

```
scala> :paste
object OddEven0 {
def odd(n: Int): String = even(n - 1)
def even(n: Int): String = if (n <= 0) "done" else odd(n - 1)
}
// Exiting paste mode, now interpreting.
defined object OddEven0
scala> OddEven0.even(200000)
java.lang.StackOverflowError
at OddEven0$.even(<console>:15)
at OddEven0$.odd(<console>:14)
at OddEven0$.even(<console>:15)
at OddEven0$.odd(<console>:14)
at OddEven0$.even(<console>:15)
....
```

Here’s my attempt at making a safer version:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
object OddEven1 {
def odd(n: Int): Eval[String] = Eval.defer {even(n - 1)}
def even(n: Int): Eval[String] =
Eval.now { n <= 0 } flatMap {
case true => Eval.now {"done"}
case _ => Eval.defer { odd(n - 1) }
}
}
// Exiting paste mode, now interpreting.
defined object OddEven1
scala> OddEven1.even(200000).value
res6: String = done
```

In the earlier versions of Cats the above caused stack overflow, but as BryanM let me know in the comment, David Gregory fixed it in #769, so it works now.

One blog post that I occasionally see being mentioned as a poweful application of monad, especially in the context of building large application is The Abstract Future. It was originally posted to the precog.com engineering blog on November 27, 2012 by Kris Nuttycombe (@nuttycom).

At Precog, we use Futures extensively, both in a direct fashion and to allow us a composable way to interact with subsystems that are implemented atop Akka’s actor framework. Futures are arguably one of the best tools we have for reining in the complexity of asynchronous programming, and so our many of our early versions of APIs in our codebase exposed Futures directly. ….

What this means is that from the perspective of the consumer of the DatasetModule interface, the only aspect of Future that we’re relying upon is the ability to order operations in a statically checked fashion; the sequencing, rather than any particular semantics related to Future’s asynchrony, is the relevant piece of information provided by the type. So, the following generalization becomes natural.

Here I’ll use similar example as the Yoshida-san’s.

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> :paste
// Entering paste mode (ctrl-D to finish)
case class User(id: Long, name: String)
// In actual code, probably more than 2 errors
sealed trait Error
object Error {
final case class UserNotFound(userId: Long) extends Error
final case class ConnectionError(message: String) extends Error
}
trait UserRepos[F[_]] {
implicit def F: Monad[F]
def userRepo: UserRepo
trait UserRepo {
def followers(userId: Long): F[List[User]]
}
}
// Exiting paste mode, now interpreting.
defined class User
defined trait Error
defined object Error
defined trait UserRepos
```

Let’s start implementing the `UserRepos`

module using `Future`

.

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
import scala.concurrent.{ Future, ExecutionContext, Await }
import scala.concurrent.duration.Duration
class UserRepos0(implicit ec: ExecutionContext) extends UserRepos[Future] {
override val F = implicitly[Monad[Future]]
override val userRepo: UserRepo = new UserRepo0 {}
trait UserRepo0 extends UserRepo {
def followers(userId: Long): Future[List[User]] = Future.successful { Nil }
}
}
// Exiting paste mode, now interpreting.
import scala.concurrent.{Future, ExecutionContext, Await}
import scala.concurrent.duration.Duration
defined class UserRepos0
```

Here’s how to use it:

```
scala> val service = new UserRepos0()(ExecutionContext.global)
service: UserRepos0 = UserRepos0@64cfd52a
scala> val xs = service.userRepo.followers(1L)
xs: scala.concurrent.Future[List[User]] = scala.concurrent.impl.Promise$KeptPromise@4d806892
```

Now we have an asynchronous result. Let’s say during testing we would like it to be synchronous.

In a test, we probably don’t want to worry about the fact that the computation is being performed asynchronously; all that we care about is that we obtain a correct result. ….

For most cases, we’ll use the identity monad for testing. Suppose that we’re testing the piece of functionality described earlier, which has computed a result from the combination of a load, a sort, take and reduce. The test framework need never consider the monad that it’s operating in.

This is where Id datatype can be used.

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
class TestUserRepos extends UserRepos[Id] {
override val F = implicitly[Monad[Id]]
override val userRepo: UserRepo = new UserRepo0 {}
trait UserRepo0 extends UserRepo {
def followers(userId: Long): List[User] =
userId match {
case 0L => List(User(1, "Michael"))
case 1L => List(User(0, "Vito"))
case x => sys.error("not found")
}
}
}
// Exiting paste mode, now interpreting.
defined class TestUserRepos
```

Here’s how to use it:

```
scala> val testRepo = new TestUserRepos {}
testRepo: TestUserRepos = $anon$1@6743fb21
scala> val ys = testRepo.userRepo.followers(1L)
ys: cats.Id[List[User]] = List(User(0,Vito))
```

Now that we were able to abtract the type constructor of the followers, let’s try implementing `isFriends`

from day 10 that checks if two users follow each other.

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
trait UserServices[F[_]] { this: UserRepos[F] =>
def userService: UserService = new UserService
class UserService {
def isFriends(user1: Long, user2: Long): F[Boolean] =
F.flatMap(userRepo.followers(user1)) { a =>
F.map(userRepo.followers(user2)) { b =>
a.exists(_.id == user2) && b.exists(_.id == user1)
}
}
}
}
// Exiting paste mode, now interpreting.
defined trait UserServices
```

Here’s how to use it:

```
scala> val testService = new TestUserRepos with UserServices[Id] {}
testService: TestUserRepos with UserServices[cats.Id] = $anon$1@49d37b72
scala> testService.userService.isFriends(0L, 1L)
res0: cats.Id[Boolean] = true
```

The above demonstrates that `isFriends`

can be written without knowing anything about `F[]`

apart from the fact that it forms a `Monad`

. It would be nice if I could use infix `flatMap`

and `map`

method while keeping `F`

abstract. I tried creating `FlatMapOps(fa)`

manually, but that resulted in abstract method error during runtime. The `actM`

macro that we implemented on day 6 seems to work ok:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
trait UserServices[F[_]] { this: UserRepos[F] =>
def userService: UserService = new UserService
class UserService {
import example.MonadSyntax._
def isFriends(user1: Long, user2: Long): F[Boolean] =
actM[F, Boolean] {
val a = userRepo.followers(user1).next
val b = userRepo.followers(user2).next
a.exists(_.id == user2) && b.exists(_.id == user1)
}
}
}
// Exiting paste mode, now interpreting.
defined trait UserServices
scala> val testService = new TestUserRepos with UserServices[Id] {}
testService: TestUserRepos with UserServices[cats.Id] = $anon$1@3a8408d1
scala> testService.userService.isFriends(0L, 1L)
res1: cats.Id[Boolean] = true
```

We can also use this with the `EitherT`

with `Future`

to carry a custom error type.

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
class UserRepos1(implicit ec: ExecutionContext) extends UserRepos[EitherT[Future, Error, ?]] {
override val F = implicitly[Monad[EitherT[Future, Error, ?]]]
override val userRepo: UserRepo = new UserRepo1 {}
trait UserRepo1 extends UserRepo {
def followers(userId: Long): EitherT[Future, Error, List[User]] =
userId match {
case 0L => EitherT.right(Future { List(User(1, "Michael")) })
case 1L => EitherT.right(Future { List(User(0, "Vito")) })
case x =>
EitherT.left(Future.successful { Error.UserNotFound(x) })
}
}
}
// Exiting paste mode, now interpreting.
defined class UserRepos1
```

Here’s how to use it:

```
scala> val service1 = {
import ExecutionContext.Implicits._
new UserRepos1 with UserServices[EitherT[Future, Error, ?]] {}
}
service1: UserRepos1 with UserServices[[γ]cats.data.EitherT[scala.concurrent.Future,Error,γ]] = $anon$1@41e51baf
scala> {
import scala.concurrent.duration._
Await.result(service1.userService.isFriends(0L, 1L).value, 1 second)
}
res2: Either[Error,Boolean] = Right(true)
scala> {
import scala.concurrent.duration._
Await.result(service1.userService.isFriends(0L, 2L).value, 1 second)
}
res3: Either[Error,Boolean] = Left(UserNotFound(2))
```

Note that for all three versions of services, I was able to reuse the `UserServices`

trait without any changes.

That’s it for today.

On day 13 we looked at Id datatype, Eval datatype, and The Abstract Future.

Semigroup we saw on day 4 is a bread and butter of functional programming that shows up in many places.

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> List(1, 2, 3) |+| List(4, 5, 6)
res0: List[Int] = List(1, 2, 3, 4, 5, 6)
scala> "one" |+| "two"
res1: String = onetwo
```

There’s a similar typeclass called `SemigroupK`

for type constructors `F[_]`

.

```
@typeclass trait SemigroupK[F[_]] { self =>
/**
* Combine two F[A] values.
*/
@simulacrum.op("<+>", alias = true)
def combineK[A](x: F[A], y: F[A]): F[A]
/**
* Given a type A, create a concrete Semigroup[F[A]].
*/
def algebra[A]: Semigroup[F[A]] =
new Semigroup[F[A]] {
def combine(x: F[A], y: F[A]): F[A] = self.combineK(x, y)
}
}
```

This enables `combineK`

operator and its symbolic alias `<+>`

. Let’s try using this.

```
scala> List(1, 2, 3) <+> List(4, 5, 6)
res2: List[Int] = List(1, 2, 3, 4, 5, 6)
```

Unlike `Semigroup`

, `SemigroupK`

works with any type parameter of `F[_]`

.

`Option[A]`

can form a `Semigroup`

only when the type parameter `A`

forms a `Semigroup`

.

```
scala> case class Foo(x: String)
defined class Foo
```

So this won’t work:

```
scala> Foo("x").some |+| Foo("y").some
<console>:33: error: value |+| is not a member of Option[Foo]
Foo("x").some |+| Foo("y").some
^
```

But this works fine:

```
scala> Foo("x").some <+> Foo("y").some
res3: Option[Foo] = Some(Foo(x))
```

There’s also a subtle difference in the behaviors of two typeclasses.

```
scala> 1.some |+| 2.some
res4: Option[Int] = Some(3)
scala> 1.some <+> 2.some
res5: Option[Int] = Some(1)
```

The `Semigroup`

will combine the inner value of the `Option`

whereas `SemigroupK`

will just pick the first one.

```
trait SemigroupKLaws[F[_]] {
implicit def F: SemigroupK[F]
def semigroupKAssociative[A](a: F[A], b: F[A], c: F[A]): IsEq[F[A]] =
F.combineK(F.combineK(a, b), c) <-> F.combineK(a, F.combineK(b, c))
}
```

There’s also `MonoidK`

.

```
@typeclass trait MonoidK[F[_]] extends SemigroupK[F] { self =>
/**
* Given a type A, create an "empty" F[A] value.
*/
def empty[A]: F[A]
/**
* Given a type A, create a concrete Monoid[F[A]].
*/
override def algebra[A]: Monoid[F[A]] =
new Monoid[F[A]] {
def empty: F[A] = self.empty
def combine(x: F[A], y: F[A]): F[A] = self.combineK(x, y)
}
....
}
```

This adds `empty[A]`

function to the contract.
The notion of emptiness here is defined in terms of the left and right identity laws with regards to `combineK`

.
Given that `combine`

and `combineK`

behave differently, `Monoid[F[A]].empty`

and `MonoidK[F].empty[A]`

could also be different.

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> Monoid[Option[Int]].empty
res0: Option[Int] = None
scala> MonoidK[Option].empty[Int]
res1: Option[Int] = None
```

In case of `Option[Int]`

they happened to be both `None`

.

```
trait MonoidKLaws[F[_]] extends SemigroupKLaws[F] {
override implicit def F: MonoidK[F]
def monoidKLeftIdentity[A](a: F[A]): IsEq[F[A]] =
F.combineK(F.empty, a) <-> a
def monoidKRightIdentity[A](a: F[A]): IsEq[F[A]] =
F.combineK(a, F.empty) <-> a
}
```

There’s a typeclass that combines `Applicative`

and `MonoidK`

called `Alternative`

:

```
@typeclass trait Alternative[F[_]] extends Applicative[F] with MonoidK[F] { self =>
....
}
```

`Alternative`

on its own does not introduce any new methods or operators.

It’s more of a weaker (thus better) `Applicative`

version of `MonadPlus`

that adds `filter`

on top of `Monad`

.
See day 3 to review the applicative style of coding.

```
trait AlternativeLaws[F[_]] extends ApplicativeLaws[F] with MonoidKLaws[F] {
implicit override def F: Alternative[F]
implicit def algebra[A]: Monoid[F[A]] = F.algebra[A]
def alternativeRightAbsorption[A, B](ff: F[A => B]): IsEq[F[B]] =
(ff ap F.empty[A]) <-> F.empty[B]
def alternativeLeftDistributivity[A, B](fa: F[A], fa2: F[A], f: A => B): IsEq[F[B]] =
((fa |+| fa2) map f) <-> ((fa map f) |+| (fa2 map f))
def alternativeRightDistributivity[A, B](fa: F[A], ff: F[A => B], fg: F[A => B]): IsEq[F[B]] =
((ff |+| fg) ap fa) <-> ((ff ap fa) |+| (fg ap fa))
}
```

There’s an open question by Yoshida-san on whether the last law is necessary or not.

Here’s Justin Le (@mstk)’s 2013 'Wolf, Goat, Cabbage: The List MonadPlus & Logic Problems.'.

Wolf, Goat, Cabbage: Solving simple logic problems in #haskell using the List MonadPlus :) http://t.co/YkKi6EQdDy

— Justin Le (@mstk) December 26, 2013

We can try implementing this using `Alternative`

.

A farmer has a wolf, a goat, and a cabbage that he wishes to transport across a river. Unfortunately, his boat can carry only one thing at a time with him. He can’t leave the wolf alone with the goat, or the wolf will eat the goat. He can’t leave the goat alone with the cabbage, or the goat will eat the cabbage. How can he properly transport his belongings to the other side one at a time, without any disasters?

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> :paste
// Entering paste mode (ctrl-D to finish)
sealed trait Character
case object Farmer extends Character
case object Wolf extends Character
case object Goat extends Character
case object Cabbage extends Character
case class Move(x: Character)
case class Plan(moves: List[Move])
sealed trait Position
case object West extends Position
case object East extends Position
implicit val moveShow = Show.show[Move](_ match {
case Move(Farmer) => "F"
case Move(Wolf) => "W"
case Move(Goat) => "G"
case Move(Cabbage) => "C"
})
// Exiting paste mode, now interpreting.
defined trait Character
defined object Farmer
defined object Wolf
defined object Goat
defined object Cabbage
defined class Move
defined class Plan
defined trait Position
defined object West
defined object East
moveShow: cats.Show[Move] = cats.Show$$anon$2@5956e1c0
```

Here’s making `n`

moves

```
scala> val possibleMoves = List(Farmer, Wolf, Goat, Cabbage) map {Move(_)}
possibleMoves: List[Move] = List(Move(Farmer), Move(Wolf), Move(Goat), Move(Cabbage))
scala> :paste
// Entering paste mode (ctrl-D to finish)
def makeMove(ps: List[List[Move]]): List[List[Move]] =
(ps |@| possibleMoves) map { (p, m) => List(m) <+> p }
def makeNMoves(n: Int): List[List[Move]] =
n match {
case 0 => Nil
case 1 => makeMove(List(Nil))
case n => makeMove(makeNMoves(n - 1))
}
// Exiting paste mode, now interpreting.
makeMove: (ps: List[List[Move]])List[List[Move]]
makeNMoves: (n: Int)List[List[Move]]
```

We can test this as follows:

```
scala> makeNMoves(1)
res0: List[List[Move]] = List(List(Move(Farmer)), List(Move(Wolf)), List(Move(Goat)), List(Move(Cabbage)))
scala> makeNMoves(2)
res1: List[List[Move]] = List(List(Move(Farmer), Move(Farmer)), List(Move(Wolf), Move(Farmer)), List(Move(Goat), Move(Farmer)), List(Move(Cabbage), Move(Farmer)), List(Move(Farmer), Move(Wolf)), List(Move(Wolf), Move(Wolf)), List(Move(Goat), Move(Wolf)), List(Move(Cabbage), Move(Wolf)), List(Move(Farmer), Move(Goat)), List(Move(Wolf), Move(Goat)), List(Move(Goat), Move(Goat)), List(Move(Cabbage), Move(Goat)), List(Move(Farmer), Move(Cabbage)), List(Move(Wolf), Move(Cabbage)), List(Move(Goat), Move(Cabbage)), List(Move(Cabbage), Move(Cabbage)))
```

Let’s define our helper function

`isSolution :: Plan -> Bool`

. Basically, we want to check if the positions of all of the characters are`East`

.

We can define `filter`

using just what’s available in `Alternative`

:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
def filterA[F[_]: Alternative, A](fa: F[A])(cond: A => Boolean): F[A] =
{
var acc = Alternative[F].empty[A]
Alternative[F].map(fa) { x =>
if (cond(x)) acc = Alternative[F].combineK(acc, Alternative[F].pure(x))
else ()
}
acc
}
def positionOf(p: List[Move], c: Character): Position =
{
def positionFromCount(n: Int): Position = {
if (n % 2 == 0) West
else East
}
c match {
case Farmer => positionFromCount(p.size)
case x => positionFromCount(filterA(p)(_ == Move(c)).size)
}
}
// Exiting paste mode, now interpreting.
filterA: [F[_], A](fa: F[A])(cond: A => Boolean)(implicit evidence$1: cats.Alternative[F])F[A]
positionOf: (p: List[Move], c: Character)Position
scala> val p = List(Move(Goat), Move(Farmer), Move(Wolf), Move(Goat))
p: List[Move] = List(Move(Goat), Move(Farmer), Move(Wolf), Move(Goat))
scala> positionOf(p, Farmer)
res2: Position = West
scala> positionOf(p, Wolf)
res3: Position = East
```

Here’s how we can check all positions are `East`

:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
def isSolution(p: List[Move]) =
{
val pos = (List(p) |@| possibleMoves) map { (p, m) => positionOf(p, m.x) }
(filterA(pos)(_ == West)).isEmpty
}
// Exiting paste mode, now interpreting.
isSolution: (p: List[Move])Boolean
```

What makes a move legal? Well, the farmer has to be on the same side as whatever is being moved.

```
scala> def moveLegal(p: List[Move], m: Move): Boolean =
positionOf(p, Farmer) == positionOf(p, m.x)
moveLegal: (p: List[Move], m: Move)Boolean
scala> moveLegal(p, Move(Wolf))
res4: Boolean = false
```

The plan is safe if nothing can eat anything else. That means if the wolf and goat or goat and cabbage sit on the same bank, so too must the farmer.

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
def safePlan(p: List[Move]): Boolean =
{
val posGoat = positionOf(p, Goat)
val posFarmer = positionOf(p, Farmer)
val safeGoat = posGoat != positionOf(p, Wolf)
val safeCabbage = positionOf(p, Cabbage) != posGoat
(posFarmer == posGoat) || (safeGoat && safeCabbage)
}
// Exiting paste mode, now interpreting.
safePlan: (p: List[Move])Boolean
```

Using these functions we can now re-implement `makeMove`

:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
def makeMove(ps: List[List[Move]]): List[List[Move]] =
(ps |@| possibleMoves) map { (p, m) =>
if (!moveLegal(p, m)) Nil
else if (!safePlan(List(m) <+> p)) Nil
else List(m) <+> p
}
def makeNMoves(n: Int): List[List[Move]] =
n match {
case 0 => Nil
case 1 => makeMove(List(Nil))
case n => makeMove(makeNMoves(n - 1))
}
def findSolution(n: Int): Unit =
filterA(makeNMoves(n))(isSolution) map { p =>
println(p map {_.show})
}
// Exiting paste mode, now interpreting.
makeMove: (ps: List[List[Move]])List[List[Move]]
makeNMoves: (n: Int)List[List[Move]]
findSolution: (n: Int)Unit
```

Let’s try solving the puzzle:

```
scala> findSolution(6)
scala> findSolution(7)
List(G, F, C, G, W, F, G)
List(G, F, W, G, C, F, G)
scala> findSolution(8)
List(G, F, C, G, W, F, G)
List(G, F, W, G, C, F, G)
List(G, F, C, G, W, F, G)
List(G, F, W, G, C, F, G)
List(G, F, C, G, W, F, G)
List(G, F, W, G, C, F, G)
```

It worked. That’s all for today.

It’s no secret that some of the fundamentals of Cats like Monoid and Functor comes from category theory. Let’s try studying category theory and see if we can use the knowledge to further our understanding of Cats.

The most accessible category theory book I’ve come across is Lawvere and Schanuel’s Conceptual Mathematics: A First Introduction to Categories 2nd ed. The book mixes Articles, which is written like a normal textbook; and Sessions, which is kind of written like a recitation class.

Even in the Article section, Lawvere uses many pages to go over the basic concept compared to other books, which is good for self learners.

Lawvere:

Before giving a precise definition of ‘category’, we should become familiar with one example, the

category of finite sets and maps. Anobjectin this category is a finite set or collection. … You are probably familiar with some notations for finite sets:

```
{ John, Mary, Sam }
```

There are two ways that I can think of to express this in Scala. One is by using a value `a: Set[Person]`

:

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
sealed trait Person {}
case object John extends Person {}
case object Mary extends Person {}
case object Sam extends Person {}
val a: Set[Person] = Set[Person](John, Mary, Sam)
// Exiting paste mode, now interpreting.
defined trait Person
defined object John
defined object Mary
defined object Sam
a: Set[Person] = Set(John, Mary, Sam)
```

Another way of looking at it, is that `Person`

as the type is a finite set already without `Set`

. **Note**: In Lawvere uses the term “map”, but I’m going to change to *arrow* like Mac Lane and others.

A

arrowfin this cateogry consists of three things:

- a set A, called the
domainof the arrow,- a set B, called the
codomainof the arrow,- a rule assigning to each element
ain the domain, an elementbin the codomain. Thisbis denoted byf ∘ a(or sometimes ’f(a)‘), read ’fofa‘.(Other words for arrow are ‘function’, ‘transformation’, ‘operator’, ‘map’, and ‘morphism’.)

Let’s try implementing the favorite breakfast arrow.

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
sealed trait Breakfast {}
case object Eggs extends Breakfast {}
case object Oatmeal extends Breakfast {}
case object Toast extends Breakfast {}
case object Coffee extends Breakfast {}
val favoriteBreakfast: Person => Breakfast = {
case John => Eggs
case Mary => Coffee
case Sam => Coffee
}
// Exiting paste mode, now interpreting.
defined trait Breakfast
defined object Eggs
defined object Oatmeal
defined object Toast
defined object Coffee
favoriteBreakfast: Person => Breakfast = <function1>
```

Note here that an “object” in this category is `Set[Person]`

or `Person`

, but the “arrow” `favoriteBreakfast`

accepts a value whose type is `Person`

. Here’s the *internal diagram* of the arrow.

The important thing is: For each dot in the domain, we have exactly one arrow leaving, and the arrow arrives at some dot in the codomain.

I get that a map can be more general than `Function1[A, B]`

but it’s ok for this category. Here’s the implementation of `favoritePerson`

:

```
scala> val favoritePerson: Person => Person = {
case John => Mary
case Mary => John
case Sam => Mary
}
favoritePerson: Person => Person = <function1>
```

An arrow in which the domain and codomain are the same object is called an

endomorphism.

An arrow, in which the domain and codomain are the same set

A, and for each ofainA,f(a)=a, is called anidentity arrow.

The “identity arrow on A” is denoted as 1_{A}.

Again, identity is an arrow, so it works on an element in the set, not the set itself. So in this case we can just use `scala.Predef.identity`

.

```
scala> identity(John)
res0: John.type = John
```

Here are the *external diagrams* corresponding to the three internal diagrams from the above.

This reiterates the point that *in the category of finite sets*, the “objects” translate to types like `Person`

and `Breakfast`

, and arrows translate to functions like `Person => Person`

. The external diagram looks a lot like the type-level signatures like `Person => Person`

.

The final basic ingredient, which is what lends all the dynamics to the notion of category is

composition of arrows, by which two arrows are combined to obtain a third arrow.

We can do this in scala using `scala.Function1`

’s `andThen`

or `compose`

.

```
scala> val favoritePersonsBreakfast = favoriteBreakfast compose favoritePerson
favoritePersonsBreakfast: Person => Breakfast = <function1>
```

Here’s the internal diagram:

and the external diagram:

After composition the external diagram becomes as follows:

’

f ∘ g’ is read ’f following g‘, or sometimes ’f of g‘.

*Data* for a category consists of the four ingredients:

- objects:
*A*,*B*,*C*, … - arrows:
*f: A => B* - identity arrows:
*1*_{A}: A => A - composition of arrows

These data must satisfy the following rules:

The identity laws:

- If
*1*, then_{A}: A => A, g: A => B*g ∘ 1*_{A}= g - If
*f: A => B, 1*, then_{B}: B => B*1*_{A}∘ f = f

The associative law:

- If
*f: A => B, g: B => C, h: C => D*, then*h ∘ (g ∘ f) = (h ∘ g) ∘ f*

Lawvere:

One very useful sort of set is a ‘singleton’ set, a set with exactly one element. Fix one of these, say

`{me}`

, and call this set ’1‘.

Definition: Apointof a set X is an arrows1 => X. … (IfAis some familiar set, an arrow fromAtoXis called an ’A-element’ ofX; thus ’1-elements’ are points.) Since a point is an arrow, we can compose it with another arrow, and get a point again.

If I understand what’s going on, it seems like Lawvere is redefining the concept of the element as a special case of arrow. Another name for singleton is unit set, and in Scala it is `(): Unit`

. So it’s analogous to saying that values are sugar for `Unit => X`

.

```
scala> val johnPoint: Unit => Person = { case () => John }
johnPoint: Unit => Person = <function1>
scala> val johnFav = favoriteBreakfast compose johnPoint
johnFav: Unit => Breakfast = <function1>
scala> johnFav(())
res1: Breakfast = Eggs
```

Session 2 and 3 contain nice review of Article I, so you should read them if you own the book.

As we saw, an *arrow* (or *morphism*) is a mapping between a *domain* and a *codomain*.
Another way of thinking about it, is that its an abstract notion for things that behave like functions.

In Cats, an Arrow instance is provided for `Function1[A, B]`

, `Kleisli[F[_], A, B]`

, and `Cokleisli[F[_], A, B]`

.

Here’s the typeclass contract for `Arrow`

:

```
package cats
package arrow
import cats.functor.Strong
import simulacrum.typeclass
@typeclass trait Arrow[F[_, _]] extends Split[F] with Strong[F] with Category[F] { self =>
/**
* Lift a function into the context of an Arrow
*/
def lift[A, B](f: A => B): F[A, B]
....
}
```

Here’s the typeclass contract for `Category`

:

```
package cats
package arrow
import simulacrum.typeclass
/**
* Must obey the laws defined in cats.laws.CategoryLaws.
*/
@typeclass trait Category[F[_, _]] extends Compose[F] { self =>
def id[A]: F[A, A]
....
}
```

Here’s the typeclass contract for `Compose`

:

```
package cats
package arrow
import simulacrum.typeclass
/**
* Must obey the laws defined in cats.laws.ComposeLaws.
*/
@typeclass trait Compose[F[_, _]] { self =>
@simulacrum.op("<<<", alias = true)
def compose[A, B, C](f: F[B, C], g: F[A, B]): F[A, C]
@simulacrum.op(">>>", alias = true)
def andThen[A, B, C](f: F[A, B], g: F[B, C]): F[A, C] =
compose(g, f)
....
}
```

This enables two operators `<<<`

and `>>>`

.

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> val f = (_:Int) + 1
f: Int => Int = <function1>
scala> val g = (_:Int) * 100
g: Int => Int = <function1>
scala> (f >>> g)(2)
res0: Int = 300
scala> (f <<< g)(2)
res1: Int = 201
```

Let’s read Haskell’s Arrow tutorial:

First and second make a new arrow out of an existing arrow. They perform a transformation (given by their argument) on either the first or the second item of a pair. These definitions are arrow-specific.

Here’s Cat’s `Strong`

:

```
package cats
package functor
import simulacrum.typeclass
/**
* Must obey the laws defined in cats.laws.StrongLaws.
*/
@typeclass trait Strong[F[_, _]] extends Profunctor[F] {
/**
* Create a new `F` that takes two inputs, but only modifies the first input
*/
def first[A, B, C](fa: F[A, B]): F[(A, C), (B, C)]
/**
* Create a new `F` that takes two inputs, but only modifies the second input
*/
def second[A, B, C](fa: F[A, B]): F[(C, A), (C, B)]
}
```

This enables two methods `first[C]`

and `second[C]`

.

```
scala> val f_first = f.first[Int]
f_first: ((Int, Int)) => (Int, Int) = <function1>
scala> f_first((1, 1))
res2: (Int, Int) = (2,1)
scala> val f_second = f.second[Int]
f_second: ((Int, Int)) => (Int, Int) = <function1>
scala> f_second((1, 1))
res3: (Int, Int) = (1,2)
```

Given that `f`

here is a function to add one, I think it’s clear what `f_first`

and `f_second`

are doing.

`(***)`

combines two arrows into a new arrow by running the two arrows on a pair of values (one arrow on the first item of the pair and one arrow on the second item of the pair).

This is called `split`

in Cats.

```
package cats
package arrow
import simulacrum.typeclass
@typeclass trait Split[F[_, _]] extends Compose[F] { self =>
/**
* Create a new `F` that splits its input between `f` and `g`
* and combines the output of each.
*/
def split[A, B, C, D](f: F[A, B], g: F[C, D]): F[(A, C), (B, D)]
}
```

We can use it as `split`

operator:

```
scala> (f split g)((1, 1))
res4: (Int, Int) = (2,100)
```

Lawvere:

Definitions: An arrowf: A => Bis called anisomorphism, orinvertible arrow, if there is a map g: B => A, for whichg ∘ f = 1and_{A}f ∘ g = 1. An arrow_{B}grelated tofby satisfying these equations is called aninverse for f. Two objectsAandBare said to beisomorphicif there is at least one isomorphismf: A => B.

Unfortunately, Cats doesn’t seem to have a datatype to represent isomorphisms, so we have to define one.

```
scala> import cats._, cats.data._, cats.implicits._, cats.arrow.Arrow
import cats._
import cats.data._
import cats.implicits._
import cats.arrow.Arrow
scala> :paste
// Entering paste mode (ctrl-D to finish)
object Isomorphisms {
trait Isomorphism[Arrow[_, _], A, B] { self =>
def to: Arrow[A, B]
def from: Arrow[B, A]
}
type IsoSet[A, B] = Isomorphism[Function1, A, B]
type <=>[A, B] = IsoSet[A, B]
}
import Isomorphisms._
// Exiting paste mode, now interpreting.
defined object Isomorphisms
import Isomorphisms._
```

This is now we can define an isomorphism from `Family`

to `Relic`

.

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
sealed trait Family {}
case object Mother extends Family {}
case object Father extends Family {}
case object Child extends Family {}
sealed trait Relic {}
case object Feather extends Relic {}
case object Stone extends Relic {}
case object Flower extends Relic {}
val isoFamilyRelic = new (Family <=> Relic) {
val to: Family => Relic = {
case Mother => Feather
case Father => Stone
case Child => Flower
}
val from: Relic => Family = {
case Feather => Mother
case Stone => Father
case Flower => Child
}
}
// Exiting paste mode, now interpreting.
defined trait Family
defined object Mother
defined object Father
defined object Child
defined trait Relic
defined object Feather
defined object Stone
defined object Flower
isoFamilyRelic: Isomorphisms.<=>[Family,Relic]{val to: Family => Relic; val from: Relic => Family} = $anon$1@e7d65c7
```

To test this, we could first implement a test for comparing two functions. Two arrows are equal when they have the same three ingredients:

- domain
*A* - codomain
*B* - a rule that assigns
*f ∘ a*

We can express this using ScalaCheck as follows:

```
scala> import org.scalacheck.{Prop, Arbitrary, Gen}
import org.scalacheck.{Prop, Arbitrary, Gen}
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> def func1EqualsProp[A, B](f: A => B, g: A => B)
(implicit ev1: Eq[B], ev2: Arbitrary[A]): Prop =
Prop.forAll { a: A =>
f(a) === g(a)
}
func1EqualsProp: [A, B](f: A => B, g: A => B)(implicit ev1: cats.Eq[B], implicit ev2: org.scalacheck.Arbitrary[A])org.scalacheck.Prop
scala> val p1 = func1EqualsProp((_: Int) + 2, 1 + (_: Int))
p1: org.scalacheck.Prop = Prop
scala> p1.check
! Falsified after 0 passed tests.
> ARG_0: 0
scala> val p2 = func1EqualsProp((_: Int) + 2, 2 + (_: Int))
p2: org.scalacheck.Prop = Prop
scala> p2.check
+ OK, passed 100 tests.
```

```
scala> :paste
implicit val familyEqual = Eq.fromUniversalEquals[Family]
implicit val relicEqual = Eq.fromUniversalEquals[Relic]
implicit val arbFamily: Arbitrary[Family] = Arbitrary {
Gen.oneOf(Mother, Father, Child)
}
implicit val arbRelic: Arbitrary[Relic] = Arbitrary {
Gen.oneOf(Feather, Stone, Flower)
}
// Exiting paste mode, now interpreting.
familyEqual: cats.kernel.Eq[Family] = cats.kernel.Eq$$anon$116@99f2e3d
relicEqual: cats.kernel.Eq[Relic] = cats.kernel.Eq$$anon$116@159bd786
arbFamily: org.scalacheck.Arbitrary[Family] = org.scalacheck.ArbitraryLowPriority$$anon$1@799b3915
arbRelic: org.scalacheck.Arbitrary[Relic] = org.scalacheck.ArbitraryLowPriority$$anon$1@36c230c0
scala> func1EqualsProp(isoFamilyRelic.from compose isoFamilyRelic.to, identity[Family] _).check
+ OK, passed 100 tests.
scala> func1EqualsProp(isoFamilyRelic.to compose isoFamilyRelic.from, identity[Relic] _).check
+ OK, passed 100 tests.
```

This shows that the test was successful. That’s it for today.

On day 15 we started looking at basic concepts in category theory using Lawvere and Schanuel’s Conceptual Mathematics. The book is a good introduction book into the notion of category because it spends a lot of pages explaining the basic concepts using concrete examples. The very aspect gets a bit tedious when you want to move on to more advanced concept, since it’s goes winding around.

Today I’m switching to Steve Awodey’s Category Theory. This is also a book written for non-mathematicians, but goes at faster pace, and more emphasis is placed on thinking in abstract terms.

A definition or a theorem is called *abstract*, when it relies only on category theoric notions, rather than some additional information about the objects and arrows. The advantage of an abstract notion is that it applies in any category immediately.

Definition 1.3In any categoryC, an arrowf: A => Bis called anisomorphism, if there is an arrow g: B => A inCsuch that:

g ∘ f = 1and_{A}f ∘ g = 1._{B}

Awodey names the above definition to be an abstract notion as it does make use only of category theoric notion.

Another good book to read together is Bartosz Milewski (@bartoszmilewski)’s Category Theory for Programmers that he’s been writing online.

Before we go abtract, we’re going to look at some concrete categories. This is actually a good thing, since we only saw one category yesterday.

The category of sets and total functions are denoted by **Sets** written in bold.
In Scala, this can be encoded roughly as types and functions between them, such as `Int => String`

. There’s apprently a philosophical debate on whether it is correct because programming languages admit bottom type (`Nothing`

), exceptions, and non-terminating code. For the sake of convenience, I’m happy to ignore it for now and pretend we can encode **Sets**.

The category of all finite sets and total functions between them are called **Sets _{fin}**.

Awodey:

Another kind of example one often sees in mathematics is categories of

structured sets, that is, sets with some further “structure” and functions that “preserve it,” where these notions are determined in some independent way.

A partially ordered set or

posetis a setAequipped with a binary relationa ≤such that the following conditions hold for all_{A}ba, b, c ∈ A:

- reflexivity: a ≤
_{A}a- transitivity: if a ≤
_{A}b and b ≤_{A}c, then a ≤_{A}c- antisymmetry: if a ≤
_{A}b and b ≤_{A}a, then a = bAn arrow from a poset

Ato a posetBis a function m: A => B that ismonotone, in the sense that, for all a, a’ ∈ A,

- a ≤
_{A}a’ implies m(a) ≤_{A}m(a’).

As long as the functions are *monotone*, the objects will continue to be in the category, so the “structure” is preserved. The category of posets and monotone functions is denoted as **Pos**. Awodey likes posets, so it’s important we understand it.

An example of poset is the `Int`

type where the binary operation of `≤`

is integer `<=`

just as it is defined with `PartialOrder`

typeclass.
Another example could be `case class LString(value: String)`

where `≤`

is defined by comparing the length of `value`

.

```
scala> :paste
// Entering paste mode (ctrl-D to finish)
case class LString(value: String)
val f: Int => LString = (x: Int) => LString(if (x < 0) "" else x.toString)
// Exiting paste mode, now interpreting.
defined class LString
f: Int => LString = <function1>
scala> f(0)
res0: LString = LString(0)
scala> f(10)
res1: LString = LString(10)
```

`f`

in the above is monotone since `f(0) ≤ f(10)`

, and any other pairs of `Int`

s that are `a <= a'`

.

Awodey:

Of course, the objects of a category do not have to be sets, either. Here are some very simple examples:

- The category
1looks like this:

It has one object and its identity arrow, which we do not draw.- The category
2looks like this:

It has two objects, their required identity arrows, and exactly one arrow between the objects.- The category
3looks like this:

It has three objects, their required identity arrows, exactly one arrow from the first to the second object, exactly one arrow from the second to the third object, and exactly one arrow from the first to the third object (which is therefore the composite of the other two).- The category
0looks like this:

It has no objects or arrows.

We don’t have a direct application for these categories, but it’s good to keep them in mind to expand our notion of categories, and also so we can use them to form other concepts.

Awodey:

Definition 1.2.A functor

F:C=>D

between categoriesCandDis a mapping of objects to objects and arrows to arrows, in such a way that.

- F(f: A => B) = F(f): F(A) => F(B)
- F(1
_{A}) = 1_{F(A)}- F(g ∘ f) = F(g) ∘ F(f)
That is,

F, preserves domains and codomains, identity arrows, and composition.

Now we are talking. Functor is an arrow between two categories. Here’s the external diagram:

The fact that the positions of *F(A)*, *F(B)*, and *F(C)* are distorted is intentional. That’s what *F* is doing, slightly distorting the picture, but still preserving the composition.

This category of categories and functors is denoted as **Cat**.

Let me remind you of the typographical conventions.
The italic uppercase *A*, *B*, and *C* represent objects (which in case of **Sets** corresponds to types like `Int`

and `String`

).
On the other hand, the bold uppercase **C** and **D** represent categories. Categories can be all kinds of things, including the datatypes we’ve seen ealier like `List[A]`

. So a functor *F*: **C** => **D** is not some function, it’s **an arrow between two categories**.

In that sense, the way programmers use the term `Functor`

is an extremely limited variety of the functor where **C** is hardcoded to **Sets**.

Awodey:

A

monoid(sometimes called a semigroup with unit) is a set M equipped with a binary operation·: M × M => Mand a distinguished “unit” element u ∈ M such that for all x, y, z ∈ M,

- x · (y · z) = (x · y) · z
- u · x = x = x · u
Equivalently, a monoid is a category with just one object. The arrows of the category are the elements of the monoid. In particular, the identity arrow is the unit element

u. Composition of arrows is the binary operation m · n for the monoid.

See Monoid from day 4 for how Monoid is encoded in Cats.

```
trait Monoid[@sp(Int, Long, Float, Double) A] extends Any with Semigroup[A] {
def empty: A
....
}
trait Semigroup[@sp(Int, Long, Float, Double) A] extends Any with Serializable {
def combine(x: A, y: A): A
....
}
```

Here is addition of `Int`

and `0`

:

```
scala> 10 |+| Monoid[Int].empty
res26: Int = 10
```

The idea that these monoids are categories with one object and that elements are arrows used to sound so alien to me, but now it’s a bit more understandable since we were exposed to singleton.

Note in the above monoid (Int, +), arrows are literally 0, 1, 2, etc, and that they are *not functions*.

There’s another category related to monoids.
The category of monoids and functions that preserve the monoid structure is denoted by **Mon**. These arrows that preserve structure are called *homomorphism*.

In detail, a homomorphism from a monoid M to a monoid N is a function h: M => N such that for all m, n ∈ M,

- h(m ·
_{M}n) = h(m) ·_{N}h(n)- h(u
_{M}) = u_{N}

Since a monoid is a category, a monoid homomorphism is a special case of functors.

Awodey:

Definition 1.4AgroupG is a monoid with an inverse g^{-1}for every element g. Thus, G is a category with one object, in which every arrow is an isomorphism.

Here’s the typeclass contract of `cats.kernel.Monoid`

:

```
/**
* A group is a monoid where each element has an inverse.
*/
trait Group[@sp(Int, Long, Float, Double) A] extends Any with Monoid[A] {
/**
* Find the inverse of `a`.
*
* `combine(a, inverse(a))` = `combine(inverse(a), a)` = `empty`.
*/
def inverse(a: A): A
}
```

This enables `inverse`

method if the syntax is imported.

```
scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> 1.inverse
res0: Int = -1
scala> assert((1 |+| 1.inverse) === Monoid[Int].empty)
```

The category of groups and group homomorphism (functions that preserve the monoid structure) is denoted by **Grp**.

We’ve seen the term homomorphism a few times, but it’s possible to think of a function that doesn’t preserve the structure.
Because every group *G* is also a monoid we can think of a function *f: G => M* where *f* loses the inverse ability from *G* and returns underlying monoid as *M*. Since both groups and monoids are categories, *f* is a functor.

We can extend this to the entire **Grp**, and think of a functor *F: Grp => Mon*. These kinds of functors that strips the structure is called *forgetful functors*. If we try to express this using Scala, you would start with `A: Group`

, and somehow downgrade to `A: Monoid`

as the ruturn value.

That’s it for today.

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