Quantities and quantified sets

Quantity.scala

package jspha.comms

import scala.language.higherKinds

sealed trait Quantity {
  type T[_]
}

object Singular extends Quantity { self =>
  case class T[A](v: A) extends Quantity.Set[A] {
    type T = self.type
  }
}

object Optional extends Quantity { self =>
  case class T[A](v: Option[A]) extends Quantity.Set[A] {
    type T = self.type
  }
}

object Multiple extends Quantity { self =>
  case class T[A](v: List[A]) extends Quantity.Set[A] {
    type T = self.type
  }
}

object Quantity {

  type Singular = Singular.type
  type Optional = Optional.type
  type Multiple = Multiple.type

  /**
    * A "quantified set" of values A consists of either one value A, the option
    * of having a value A, or many values of A. In this sense, it's easy to
    * subsume a quantified set into a simple list, but quantified sets have a
    * special relationship with Cardinalities.
    */
  sealed trait Set[A] {
    type T <: Quantity
  }

  trait Narrowing[Q <: Quantity] {
    def pf[A]: PartialFunction[Set[A], Q#T[A]]
    def apply[A](set: Set[A]): Option[Q#T[A]] = pf.lift(set)
  }

  object Narrowing {

    implicit val SingularNarrowing: Narrowing[Singular] =
      new Narrowing[Singular] {
        def pf[A] = { case Singular.T(v) => Singular.T(v) }
      }

    implicit val OptionalNarrowing: Narrowing[Optional] =
      new Narrowing[Optional] {
        def pf[A] = { case Optional.T(v) => Optional.T(v) }
      }

    implicit val MultipleNarrowing: Narrowing[Multiple] =
      new Narrowing[Multiple] {
        def pf[A] = { case Multiple.T(v) => Multiple.T(v) }
      }

  }

}