The inductive and the coinductive types are two important forms of recursive type. Inductive types correspond to least, or initial, solutions of certain type isomorphism equations, and coinductive types correspond to their greatest, or final, solutions. Intuitively, the elements of an inductive type are those that may be obtained by a finite composition of its introductory forms. Consequently, if we specify the behavior of a function on each of the introductory forms of an inductive type, then its behavior is determined for all values of that type. Such a function is called a recursor, or catamorphism. Dually, the elements of a coinductive type are those that behave properly in response to a finite composition of its elimination forms. Consequently, if we specify the behavior of an element on each elimination form, then we have fully specified that element as a value of that type. Such an element is called an generator, or anamorphism.

Motivating Examples

The most important example of an inductive type is the type of natural numbers as formalized in Chapter 9. The type nat is defined to be the least type containing z and closed under s(−). The minimality condition is witnessed by the existence of the recursor, iter e {z ⇒ e0 ∣ s(x) ⇒ e1}, which transforms a natural number into a value of type τ, given its value for zero, and a transformation from its value on a number to its value on the successor of that number.