/core/src/main/scala/scalaz/Monoid.scala
http://github.com/scalaz/scalaz · Scala · 150 lines · 74 code · 24 blank · 52 comment · 4 complexity · 2bd2ac38bb80e2d44f0698dcef7a0939 MD5 · raw file
- package scalaz
- ////
- /**
- * Provides an identity element (`zero`) to the binary `append`
- * operation in [[scalaz.Semigroup]], subject to the monoid laws.
- *
- * Example instances:
- * - `Monoid[Int]`: `zero` and `append` are `0` and `Int#+` respectively
- * - `Monoid[List[A]]`: `zero` and `append` are `Nil` and `List#++` respectively
- *
- * References:
- * - [[http://mathworld.wolfram.com/Monoid.html]]
- *
- * @see [[scalaz.syntax.MonoidOps]]
- * @see [[scalaz.Monoid.MonoidLaw]]
- *
- */
- ////
- trait Monoid[F] extends Semigroup[F] { self =>
- ////
- /** The identity element for `append`. */
- def zero: F
- // derived functions
- /**
- * For `n = 0`, `zero`
- * For `n = 1`, `append(zero, value)`
- * For `n = 2`, `append(append(zero, value), value)`
- */
- def multiply(value: F, n: Int): F =
- if (n <= 0) zero else multiply1(value, n - 1)
- /** Whether `a` == `zero`. */
- def isMZero(a: F)(implicit eq: Equal[F]): Boolean =
- eq.equal(a, zero)
- final def ifEmpty[B](a: F)(t: => B)(f: => B)(implicit eq: Equal[F]): B =
- if (isMZero(a)) { t } else { f }
- final def onNotEmpty[B](a: F)(v: => B)(implicit eq: Equal[F], mb: Monoid[B]): B =
- ifEmpty(a)(mb.zero)(v)
- final def onEmpty[A,B](a: F)(v: => B)(implicit eq: Equal[F], mb: Monoid[B]): B =
- ifEmpty(a)(v)(mb.zero)
- def unfoldlSum[S](seed: S)(f: S => Maybe[(S, F)]): F =
- unfoldlSumOpt(seed)(f) getOrElse zero
- def unfoldrSum[S](seed: S)(f: S => Maybe[(F, S)]): F =
- unfoldrSumOpt(seed)(f) getOrElse zero
- /** Every `Monoid` gives rise to a [[scalaz.Category]], for which
- * the type parameters are phantoms.
- *
- * @note `category.monoid` = `this`
- */
- final def category: Category[λ[(α, β) => F]] =
- new Category[λ[(α, β) => F]] with SemigroupCompose {
- def id[A] = zero
- }
- /**
- * A monoidal applicative functor, that implements `point` and `ap`
- * with the operations `zero` and `append` respectively. Note that
- * the type parameter `α` in `Applicative[λ[α => F]]` is
- * discarded; it is a phantom type. As such, the functor cannot
- * support [[scalaz.Bind]].
- */
- final def applicative: Applicative[λ[α =>F]] =
- new Applicative[λ[α => F]] with SemigroupApply {
- def point[A](a: => A) = zero
- }
- /**
- * Monoid instances must satisfy [[scalaz.Semigroup.SemigroupLaw]] and 2 additional laws:
- *
- * - '''left identity''': `forall a. append(zero, a) == a`
- * - '''right identity''' : `forall a. append(a, zero) == a`
- */
- trait MonoidLaw extends SemigroupLaw {
- def leftIdentity(a: F)(implicit F: Equal[F]): Boolean = F.equal(a, append(zero, a))
- def rightIdentity(a: F)(implicit F: Equal[F]): Boolean = F.equal(a, append(a, zero))
- }
- def monoidLaw = new MonoidLaw {}
- ////
- val monoidSyntax: scalaz.syntax.MonoidSyntax[F] =
- new scalaz.syntax.MonoidSyntax[F] { def F = Monoid.this }
- }
- object Monoid {
- @inline def apply[F](implicit F: Monoid[F]): Monoid[F] = F
- import Isomorphism._
- def fromIso[F, G](D: F <=> G)(implicit M: Monoid[G]): Monoid[F] =
- new IsomorphismMonoid[F, G] {
- override def G: Monoid[G] = M
- override def iso: F <=> G = D
- }
- ////
- /** Make an append and zero into an instance. */
- def instance[A](f: (A, => A) => A, z: A): Monoid[A] =
- new Monoid[A] {
- def zero = z
- def append(f1: A, f2: => A): A = f(f1,f2)
- }
- private trait ApplicativeMonoid[F[_], M] extends Monoid[F[M]] with Semigroup.ApplySemigroup[F, M] {
- implicit def F: Applicative[F]
- implicit def M: Monoid[M]
- val zero = F.point(M.zero)
- }
- /**A monoid for sequencing Applicative effects. */
- def liftMonoid[F[_], M](implicit F0: Applicative[F], M0: Monoid[M]): Monoid[F[M]] =
- new ApplicativeMonoid[F, M] {
- implicit def F: Applicative[F] = F0
- implicit def M: Monoid[M] = M0
- }
- def liftPlusEmpty[A](implicit M0: Monoid[A]): PlusEmpty[λ[α => A]] =
- new PlusEmpty[λ[α => A]] {
- type A0[α] = A
- def empty[A]: A0[A] = M0.zero
- def plus[A](f1: A0[A], f2: => A0[A]): A0[A] = M0.append(f1, f2)
- }
- /** Monoid is an invariant functor. */
- implicit val monoidInvariantFunctor: InvariantFunctor[Monoid] =
- new InvariantFunctor[Monoid] {
- def xmap[A, B](ma: Monoid[A], f: A => B, g: B => A): Monoid[B] = new Monoid[B] {
- def zero: B = f(ma.zero)
- def append(x: B, y: => B): B = f(ma.append(g(x), g(y)))
- }
- }
- ////
- }
- trait IsomorphismMonoid[F, G] extends Monoid[F] with IsomorphismSemigroup[F, G]{
- implicit def G: Monoid[G]
- ////
- def zero: F = iso.from(G.zero)
- ////
- }