more experiments in doing weird type parameterization stuff

typing.scala

class Mult[A, B](val l: A, val r: B) {
  def commute: Mult[B, A] = new Mult[B, A](r, l)
  def associate[C, D](implicit ev: B <:< Mult[C, D]): Mult[Mult[A, C], D] =
    new Mult[Mult[A, C], D](new Mult(l, r.l), r.r)

  def left[C](fn: A => C): Mult[C, B] = new Mult(fn(l), r)
  def right[C](fn: B => C): Mult[A, C] = new Mult(l, fn(r))
}

class Div[A, B](val l: A, val r: B) {
  def associate[C, D](implicit ev: B <:< Div[C, D]): Div[Mult[A, D], C] = {
    new Div[Mult[A, D], C](new Mult(l, r.r), r.l)
  }

  def left[C](fn: A => C): Div[C, B] = new Div(fn(l), r)
  def right[C](fn: B => C): Div[A, C] = new Div(l, fn(r))
}

object TryIt {
  class Cat
  class Dog
  class Mouse

  val x: Mult[Cat, Dog] = (new Mult(new Dog, new Cat)).commute
  val y: Mult[Mult[Cat, Dog], Mouse] = (new Mult(new Cat, new Mult(new Dog, new Mouse))).associate
  val z: Mult[Mouse, Mult[Cat, Dog]] = (new Mult(new Cat, new Mult(new Dog, new Mouse))).associate.commute
  val a: Div[Mult[Cat, Mouse], Dog] = new Div(new Cat, new Div(new Dog, new Mouse)).associate
  val b: Div[Mult[Mouse, Cat], Dog] = new Div(new Cat, new Div(new Dog, new Mouse)).associate.left(_.commute)
}