Applicative.scala

import scala.language.higherKinds

trait Functor[F[_]] {
  def fmap[A, B](f: A => B): F[A] => F[B]
}

trait Apply[F[_]] {
  val functor: Functor[F]

  def ap[A, B](f: F[A => B]): F[A] => F[B]

  def fmap[A, B](f: A => B): F[A] => F[B] =
    functor.fmap(f)
}

trait Applicative[F[_]] {
  val apply: Apply[F]

  def ap[A, B](f: F[A => B]): F[A] => F[B] =
    apply.ap(f)

  def pure[A](a: A): F[A]

  def functor: Functor[F] = 
    apply.functor

  def fmap[A, B](f: A => B): F[A] => F[B] =
    functor.fmap(f)

  object Derived {
    def functor: Functor[F] = 
      new Functor[F] {
        def fmap[A, B](f: A => B): F[A] => F[B] =
          ap(pure(f))  
      }

    def fmap[A, B](f: A => B): F[A] => F[B] =
      functor.fmap(f)
    }
}