Sum types

Sum types represent situations where we have a value which could be either one thing or another. Suppose A and B are types. Then:

• A + B is a sum type (also known as a disjoint union). It represents a disjoint union of the types A and B. That is, the values of A + B can be either a value of type A, or a value of type B.

• A value of type A + B can be written either left(a), where a is an arbitrary expression of type A, or right(b), where b is an arbitrary expression of type B. For example:

  Disco> left(3) : N + Bool
  left(3)
  Disco> right(false) : N + Bool
  right(false)
  

Note that the left or right ensures that A + B really does represent a disjoint union. For example, although the usual union operator is idempotent, that is, \(\mathbb{N} \cup \mathbb{N} = \mathbb{N}\), with a disjoint union of types N + N is not at all the same as N. Elements of N + N look like either left(3) or right(3), that is, N + N includes two copies of each natural number.