Applicative.scala

import scala.language.higherKinds // lol

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

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

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] = 
    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)
}

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

trait Monad[F[_]] {
  val bind: Bind[F]

  def bnd[A, B](f: A => F[B]): F[A] => F[B] =
    bind.bnd(f)

  val applicative: Applicative[F]

  def pure[A](a: A): F[A] =
    applicative.pure(a)

  def apply: Apply[F] =
    applicative.apply

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

  def functor: Functor[F] = 
    applicative.functor

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