Natural numbers impl

package idealized.scala

abstract class Nat {
  def isZero: scala.Boolean
  def predecessor: Nat
  def successor: Nat = new Succ(this)
  def + (that: Nat): Nat
  def - (that:Nat): Nat
}

object Zero extends Nat {
  override def isZero: scala.Boolean = true

  override def predecessor: Nat = throw new UnsupportedOperationException

  override def +(that: Nat): Nat = that

  override def -(that: Nat): Nat = if (that.isZero) this else throw new UnsupportedOperationException


  override def toString = "0"
}

class Succ(n: Nat) extends Nat {
  override def isZero: scala.Boolean = false

  override def predecessor: Nat = n

  override def +(that: Nat): Nat = new Succ(n + that)

  override def -(that: Nat): Nat = if (that.isZero) this else n - that.predecessor

  override def toString = "s(" + predecessor.toString + ")"
}