[Documentation] Trying to rewrite typeclasses-new

docs/docs/reference/contextual/typeclasses-new.md

@@ -3,12 +3,13 @@ layout: doc-page
 title: "Implementing Typeclasses"
 ---
 
-Given instances, extension methods and context bounds
-allow a concise and natural expression of _typeclasses_. Typeclasses are just traits
-with canonical implementations defined by given instances. Here are some examples of standard typeclasses:
+In Scala 3, _typeclasses_ are just traits whose implementation are defined by given instances. 
+Here are some examples of standard typeclasses:
 
 ### Semigroups and monoids:
 
+Here's the `Monoid` typeclass definition:
+
 ```scala
 trait SemiGroup[T] {
  @infix def (x: T) combine (y: T): T
@@ -17,50 +18,204 @@ trait SemiGroup[T] {
 trait Monoid[T] extends SemiGroup[T] {
  def unit: T
 }
+```
 
-object Monoid {
- def apply[T] with (m: Monoid[T]) = m
-}
+An implementation of this `Monoid` typeclass for the type `String` can be the following: 
 
+```scala
 given as Monoid[String] {
  def (x: String) combine (y: String): String = x.concat(y)
  def unit: String = ""
 }
+```
 
+Whereas for the type `Int` one could write the following:
+```scala
 given as Monoid[Int] {
  def (x: Int) combine (y: Int): Int = x + y
  def unit: Int = 0
 }
+```
 
-def sum[T: Monoid](xs: List[T]): T =
+This monoid can now be used as _context bound_ in the following `combineAll` method:
+
+```scala
+def combineAll[T: Monoid](xs: List[T]): T =
+ xs.foldLeft(summon[Monoid[T]].unit)(_ combine _)
+```
+
+To get rid of the `summon[...]` we can define a `Monoid` object as follows:
+
+```scala
+object Monoid {
+ def apply[T] with (m: Monoid[T]) = m
+}
+```
+
+Which would allow to re-write the `combineAll` method this way: 
+
+```scala
+def combineAll[T: Monoid](xs: List[T]): T =
  xs.foldLeft(Monoid[T].unit)(_ combine _)
 ```
 
-### Functors and monads:
+We can also benefit from [extension methods](extension-methods-new.html) to make this `combineAll` function accessible as a method on the `List` type:
+
+
+```scala
+def [T: Monoid](xs: List[T]).combineAll: T =
+ xs.foldLeft(Monoid[T].unit)(_ combine _) 
+```
+
+Which allows one to write: 
+
+```scala
+assert("ab" == List("a", "b").combineAll)
+```
+or:
+```scala
+assert(3 == List(1, 2).combineAll)
+```
+
+### Functors:
+
+A `Functor` represents the ability for a type containing zero or more elements to be "mapped over", i.e. apply a function to every of its elements.
+Let's name our "type containing zero or more elements" `F`. It's a [higher-kinded type](http://guillaume.martres.me/publications/dotty-hk.pdf), since it contains elements of an other type.
+Therefore we'll write it `F[_]` since we don't really care about the type of the elements it contains.
+The definition of the `Functor` ability would thus be written as:
+
+```scala
+trait Functor[F[_]] {
+ def map[A, B](original: F[A], mapper: A => B): F[B]
+}
+```
+
+Which could read as follows: "The `Functor` ability for a wrapper type `F` represents the ability to transform `F[A]` to `F[B]` through the application of the `mapper` function whose type is `A => B`".
+This way, we could define an instance of `Functor` for the `List` type: 
+
+```scala
+given as Functor[List] {
+ def map[A, B](original: List[A], mapper: A => B): List[B] =
+ original.map(mapper) // List already has a `map` method
+}
+```
+
+With this `given` instance in scope, everywhere a `Functor` is expected, the compiler will accept a `List` to be used.
+
+For instance, we may write such a testing method:
+```scala
+def assertTransformation[F[_]: Functor, A, B](expected: F[B], original: F[A], mapping: A => B): Unit =
+ assert(expected == summon[Functor[F]].map(original, mapping))
+```
+
+And use it this way, for example:
+
+```scala
+assertTransformation(List("a1", "b1"), List("a", "b"), elt => s"${elt}1")
+```
+
+That's a first step, but in practice we probably would like the `map` function to be a method directly accessible on the type `F`. So that we can call `map` directly on instances of `F`, and get rid of the `summon[Functor[F]]` part.
+As in the previous example of Monoids, [`extension` methods](extension-methods-new.html) help achieving that. Let's re-define the `Functor` _typeclass_ with extension methods.
 
 ```scala
 trait Functor[F[_]] {
- def [A, B](x: F[A]).map(f: A => B): F[B]
+ def [A, B](original: F[A]).map(mapper: A => B): F[B]
+}
+```
+
+The instance of `Functor` for `List` now becomes:
+
+```scala
+given as Functor[List] {
+ def [A, B](original: List[A]).map(mapper: A => B): List[B] =
+ original.map(mapper) // List already has a `map` method
 }
+```
+
+It simplifies the `assertTransformation` method:
+
+```scala
+def assertTransformation[F[_]: Functor, A, B](expected: F[B], original: F[A], mapping: A => B): Unit =
+ assert(expected == original.map(mapping))
+```
+
+The `map` method is now directly used on `original` since it is of type `F[A]` (where `F` is a `Functor`).
+
+
+### Monads
+
+Now we have a `Functor` for `List`.
+
+Applying the `List.map` ability with the following mapping function as parameter: `mapping: A => B` would result in a `List[B]`.
+
+Now, applying the `List.map` ability with the following mapping function as parameter: `mapping: A => List[B]` would result in a `List[List[B]]`. 
 
-trait Monad[F[_]] extends Functor[F] {
- def [A, B](x: F[A]).flatMap(f: A => F[B]): F[B]
- def [A, B](x: F[A]).map(f: A => B) = x.flatMap(f `andThen` pure)
+To avoid avoid managing lists of lists, we may want to "flatten" the values in a single list.
 
- def pure[A](x: A): F[A]
+That's where `Monad` enter the party. A `Monad` for type `F[_]` is a `Functor[F]` with 2 more abilities: 
+* the flatten ability we just described: turning `F[A]` to `F[B]` when given a `mapping: A => F[B]` function
+* the ability to create `F[A]` from a single value `A`
+
+Here is the translation of this definition in Scala 3:
+
+```scala
+trait Monad[F[_]] extends Functor[F] { // "A `Monad` for type `F[_]` is a `Functor[F]`" => thus has the `map` ability
+ def pure[A](x: A): F[A] // `pure` can construct F[A] from a single value A
+ def [A, B](x: F[A]).flatMap(f: A => F[B]): F[B] // the flattening ability is named `flatMap`, using extension methods as previous examples
+ def [A, B](x: F[A]).map(f: A => B) = x.flatMap(f `andThen` pure) // the `map(f)` ability is simply a combination of applying `f` then turning the result into an `F[A]` then applying `flatMap` to it
 }
+```
+
+#### List
 
+Let us declare the `Monad` ability for type `List` 
+```scala
 given listMonad as Monad[List] {
- def [A, B](xs: List[A]).flatMap(f: A => List[B]): List[B] =
- xs.flatMap(f)
  def pure[A](x: A): List[A] =
  List(x)
+ def [A, B](xs: List[A]).flatMap(f: A => List[B]): List[B] =
+ xs.flatMap(f) // let's rely on the existing `flatMap` method of `List`
 }
+```
+
+`map` implementation is no longer needed.
+
+#### Option
+
+`Option` is an other type having the same kind of behaviour:
+* the `map` ability turning `Option[A]` into `Option[B]` if passed a function `f: A => B`
+* the `flatMap` ability turning `Option[A]` into `Option[B]` if passed a function `f: A => Option[B]`
+* the `pure` ability turning `A` into `Option[A]`
 
+```scala
+given optionMonad as Monad[Option] {
+ def pure[A](x: A): Option[A] =
+ Option(x)
+ def [A, B](xs: Option[A]).flatMap(f: A => Option[B]): Option[B] =
+ xs.flatMap(f) // let's rely on the existing `flatMap` method of `Option`
+}
+```
+
+#### The Reader Monad
+
+Another example of a `Monad` is the Reader Monad. It no longer acts on a type like `List` or `Option`, but on a function.
+It can be used for example for combining functions that all need the same type of parameter. For instance multiple functions needing access to some configuration, context, environment variables, etc.
+
+The Reader monad allows to abstract over such a configuration dependency (or context, environment, ...), named `Ctx` in the following examples. It is therefore _parameterized_ by `Ctx`:
+
+```scala
 given readerMonad[Ctx] as Monad[[X] =>> Ctx => X] {
  def [A, B](r: Ctx => A).flatMap(f: A => Ctx => B): Ctx => B =
  ctx => f(r(ctx))(ctx)
  def pure[A](x: A): Ctx => A =
  ctx => x
 }
 ```
+
+### Summary
+
+The definition of a _typeclass_ is expressed in Scala 3 via a `trait`.
+The main difference with other traits resides in how these traits are implemented. 
+In the case of a _typeclass_ the trait's implementations are expressed through `given ... as` type definitions, and not through classes that `extends` the trait linearly.
+
+In addition to these given instances, other constructs like extension methods, context bounds and type lambdas allow a concise and natural expression of _typeclasses_.