Lower Type Bounds

While upper type bounds limit a type to a subtype of another type, lower type bounds declare a type to be a supertype of another type. The term T >: A expresses that the type parameter T or the abstract type T refer to a supertype of type A.

Here is an example where this is useful:

case class ListNode[T](h: T, t: ListNode[T]) {
  def head: T = h
  def tail: ListNode[T] = t
  def prepend(elem: T): ListNode[T] =
    ListNode(elem, this)
}

The program above implements a linked list with a prepend operation. Unfortunately, this type is invariant in the type parameter of class ListNode; i.e. type ListNode[String] is not a subtype of type List[Object]. With the help of variance annotations we can express such a subtype semantics:

case class ListNode[+T](h: T, t: ListNode[T]) { ... }

Unfortunately, this program does not compile, because a covariance annotation is only possible if the type variable is used only in covariant positions. Since type variable T appears as a parameter type of method prepend, this rule is broken. With the help of a lower type bound, though, we can implement a prepend method where T only appears in covariant positions.

Here is the corresponding code:

case class ListNode[+T](h: T, t: ListNode[T]) {
  def head: T = h
  def tail: ListNode[T] = t
  def prepend[U >: T](elem: U): ListNode[U] =
    ListNode(elem, this)
}

Note: the new prepend method has a slightly less restrictive type. It allows, for instance, to prepend an object of a supertype to an existing list. The resulting list will be a list of this supertype.

Here is some code which illustrates this:

object LowerBoundTest extends App {
  val empty: ListNode[Null] = ListNode(null, null)
  val strList: ListNode[String] = empty.prepend("hello")
                                       .prepend("world")
  val anyList: ListNode[Any] = strList.prepend(12345)
}

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