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## Part VI Infinite Data TypesChapter 19 Inductive and Co-Inductive Types

### Abstract

The inductive and the coinductive types are two important classes of recursive types. 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 an iterator, 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. The motivating example of an inductive type is the type of natural numbers. It is the least type containing the introductory forms z and s(e)

Year: 2011
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