what does it mean?

what-does-it-mean.scala

// List[Either[A, B]] <=> (List[A], List[B])
// Option[These[A, B]] <=> (Option[A], Option[B])
// what does it mean that this compiles? what can be derived as a relationship? what can be induced?

trait Iso[A, B] {
  def from(a: A): B
  def to(b: B): A
}

sealed trait These[A, B]
case class This[A, B](run: A) extends These[A, B]
case class That[A, B](run: B) extends These[A, B]
case class Both[A, B](run: (A, B)) extends These[A, B]

object Alg {
  def CopProd[A, B]: Iso[List[Either[A, B]], (List[A], List[B])] =
    new Iso[List[Either[A, B]], (List[A], List[B])] {
      def from(a: List[Either[A, B]]): (List[A], List[B]) =
        a.foldLeft((List.empty[A], List.empty[B]))((acc, el) =>
          el match {
            case Left(l) => (acc._1 :+ l, acc._2)
            case Right(r) => (acc._1, acc._2 :+ r)
          })
      def to(b: (List[A], List[B])): List[Either[A, B]] =
        b._1.map(Left(_)) ++ b._2.map(Right(_))
    }

  def OpsThese[A, B]: Iso[Option[These[A, B]], (Option[A], Option[B])] =
    new Iso[Option[These[A, B]], (Option[A], Option[B])] {
      def from(a: Option[These[A, B]]): (Option[A], Option[B]) =
        a match {
          case None => (None, None)
          case Some(x) => x match {
            case This(a) => (Some(a), None)
            case That(b) => (None, Some(b))
            case Both((a, b)) => (Some(a), Some(b))
          }
        }
      def to(b: (Option[A], Option[B])): Option[These[A, B]] =
        b match {
          case (None, None) => None
          case (Some(x), None) => Some(This(x))
          case (None, Some(y)) => Some(That(y))
          case (Some(x), Some(y)) => Some(Both((x, y)))
        }
    }
}

F[G[A, B]] <=> H[F[A], F[B]] for what (F, G, H)

List, Either, (List, List)
Option, These, (Option, Option)

only true if List is changed to SortedSet