abstract class Nat {
  def isZero: Boolean
  def predecessor: Nat
  def successor: Nat
  def + (that: Nat): Nat
  def -(that: Nat): Nat
}


object Zero extends Nat {
  def isZero: Boolean = true
  def predecessor: Nat = error("negative number")
  def successor: Nat = new Succ(Zero)
  def + (that: Nat): Nat = that
  def -(that: Nat): Nat = if (that.isZero) Zero
			  else error("negative number")
}

class Succ(x: Nat) extends Nat {
  def isZero: Boolean = false
  def predecessor: Nat = x
  def successor: Nat = new Succ(this)
  def +(that: Nat): Nat = x + that.successor
  def -(that: Nat): Nat = if (that.isZero) this
			  else x - that.predecessor
}