Add Alternative docs

docs/src/main/resources/microsite/data/menu.yml

@@ -79,6 +79,9 @@ options:
        url: typeclasses/invariantmonoidal.html
        menu_section: variance
 
+  - title: Alternative
+    url: typeclasses/alternative.html
+    menu_type: typeclasses
 
   - title: Eq
     url: typeclasses/eq.html

docs/src/main/tut/typeclasses/alternative.md

@@ -0,0 +1,145 @@
+---
+layout: docs
+title: "Alternative"
+section: "typeclasses"
+source: "core/src/main/scala/cats/Alternative.scala"
+scaladoc: "#cats.Alternative"
+---
+# Alternative
+Alternative extends [`Applicative`](`applicative.html`) with a [`MonoidK`](`monoidk.html`).
+Let's stub out all the operations just to remind ourselves what that gets us.
+
+```tut:book:silent
+import cats.{Applicative, MonoidK}
+
+trait Alternative[F[_]] extends Applicative[F] with MonoidK[F] {
+  def ap[A, B](ff: F[A => B])(fa: F[A]): F[B]
+
+  def pure[A](a: A): F[A]
+
+  def empty[A]: F[A]
+
+  def combineK[A](x: F[A], y: F[A]): F[A]
+}
+```
+
+As you might recall, `pure` wraps values in the context and `ap` allows us to do calculations in the context, while `combineK` allows us to combine, for any given type `A`, any two contextual values `F[A]` and `empty` provides the identity element for the combine operation.
+
+Of course, like other type classes, `Alternative` instances must obey some laws, in addition to those otherwise applying to `MonoidK` and `Applicative instances`:
+
+* Right Absorption: Applying a contextual function `F[A => B]` to `empty [A]` should be `empty [B]`.
+  * `ff ap F.empty[A] = F.empty[B]`.
+* Left Distributivity:  Mapping over a combined element must be the combinations of the mapped elements.
+  * `(fa <+> fa2) map f = ((fa map f) <+> (fa2 map f))` where `fa: F[A]` and `fb: F[B]` and `f: A => B`.
+* Right Distributivity: Applying the combination of two functions must be the combination of their applications.
+  * `(ff <+> fg) ap fa = (ff ap fa) <+> (fg ap fa)` where `ff: F[A => B]`, `fg: F[A => B]`, and `fa: F[A]`.
+
+Morally, these laws guarantee the compatibility of the otherwise possibly independent `Applicative` and `MonoidK` structures.
+
+## Vector as an Alternative
+
+Let's examine an instance to get a feel for things. A useful instance of `Alternative` is that for `Vector`.
+
+The relevant imports:
+
+```tut:book:reset:silent
+import cats.Alternative
+import cats.implicits._
+```
+
+And what we can do with them:
+
+```tut:book
+val empty = Alternative[Vector].empty[Int]
+val pureOfFive = 5.pure[Vector]
+val concatenated = 7.pure[Vector] <+> 8.pure[Vector]
+val double: Int => Int = _ * 2
+val addFive: Int => Int = _ + 5
+val apForVectors = (double.pure[Vector] <+> addFive.pure[Vector]) ap concatenated
+```
+
+## Making choices with `Alternative`
+
+### Alternative Parsing
+
+Suppose we have a simple parser library with an interface something like:
+
+```tut:book:silent
+trait Decoder[A] {
+  def decode(in: String): Either[Throwable, A]
+}
+object Decoder {
+  def from[A](f: String => Either[Throwable, A]): Decoder[A] =
+    new Decoder[A] {
+      def decode(in: String) = f(in)
+    }
+}
+```
+
+Then, we can implement an `Alternative` instance for this type like so:
+
+```tut:book
+implicit val decoderAlternative = new Alternative[Decoder] {
+  def pure[A](a: A) = Decoder.from(Function.const(Right(a)))
+
+  def empty[A] = Decoder.from(Function.const(Left(new Error("No dice."))))
+
+  def combineK[A](l: Decoder[A], r: Decoder[A]): Decoder[A] =
+    new Decoder[A] {
+      def decode(in: String) = l.decode(in).orElse(r.decode(in))
+    }
+
+  def ap[A, B](ff: Decoder[A => B])(fa: Decoder[A]): Decoder[B] =
+    new Decoder[B] {
+      def decode(in: String) = fa.decode(in) ap ff.decode(in)
+    }
+}
+```
+
+The addition of the `Alternative` methods allows us to prioritize multiple strategies, compensating for inconsistencies in the source data.
+
+```tut:book
+def parseInt(s: String) = Either.catchNonFatal(s.toInt)
+def parseIntFirstChar(s: String) = Either.catchNonFatal(2 * Character.digit(s.charAt(0), 10))
+
+// Try first parsing the whole, then just the first character.
+val decoder = Decoder.from(parseInt _) <+> Decoder.from(parseIntFirstChar _)
+
+decoder.decode("555")
+decoder.decode("5a")
+```
+
+### Partitioning Results
+
+`Alternative` gives us a notion of partitioning, as long as we also have a `Monad` available. This is what `separate` does.
+
+The trick here is that the inner type constructor (essentially the replacement for `Boolean` as the target of our "predicate") must have a `Bifoldable` available. A great example of a `Bifoldable` is `Either`, and another is `Tuple2`.
+
+Let's imagine that we're trying to make a bunch of independent possibly failure prone calls with a bunch of different `Int` inputs (say it's the id of a resource), each returning `Either[Int, Int]` where a left of an integer is the code modeling the failure we're given (say it's the HTTP code returned by a remote API we're calling), while right of an integer is the output of the calculation.
+
+We can use `separate` to pull apart the failures and successes zipped with the input, letting us log failures and proceed with successes intelligently.
+
+```tut:book
+// Resource holder returns (Request, Status)
+def requestResource(a: Int): Either[(Int, Int), (Int, Int)] = {
+  if (a % 4 == 0) Left((a, 400))
+  else if (a % 3 == 0) Left((a, 500))
+  else Right((a, 200))
+}
+
+val partitionedResults = 
+  ((requestResource _).pure[Vector] ap Vector(5, 6, 7, 99, 1200, 8, 22)).separate
+```
+
+Alternatively (no pun intended), in a totally different version of the same idea, we can lean on the instance of `Bitraverse` for `Tuple2` to try and partition cleanly the outcomes of two different calculations that each depend pairwise on the same input (maybe we have a primary key and we want to select both the remote id of the resources with that key, and those that are foreign key linked to the input key).
+
+```tut:book
+// Surprising regularity in this politico-geographical data model!
+def getRegionAndDistrict(pkey: Int): (Int, Vector[Int]) = (5 * pkey, (double.pure[Vector] <+> addFive.pure[Vector]) ap pkey.pure[Vector])
+
+val regionsWithDistricts = (getRegionAndDistrict _).pure[Vector] ap Vector(5, 6, 7, 97, 1200, 8, 25)
+val regionIds = regionsWithDistricts.separate._1
+val districtIds = regionsWithDistricts.separate._2.flatten
+```
+
+