object Proofs extends App {
  trait Implication[A, B] {
    type Out = B
  }

  sealed trait Base
  sealed trait Left
    extends Base
  sealed trait Right
    extends Base
  case class LeftOne(x: Int, q: Double)
    extends Left
  case class LeftTwo(y: String)
    extends Left
  case class RightOne(z: Int)
    extends Right
  case class RightTwo(q: String)
    extends Right

  object Chirality {
    private def evidence[A, B]: Implication[A, B] = new Implication[A, B] {}

    implicit val left_implies_right_one: Implication[LeftOne, RightOne] = evidence
    implicit val left_implies_right_two: Implication[LeftTwo, RightTwo] = evidence
  }

  def map[A, B](a: A)(f: A => B)(implicit ev: Implication[A, B]): ev.Out = f(a)

  import Chirality._

  val x = map(LeftTwo("hi mom"))(_ => RightTwo("hi pop"))
  println(x.q)
}