Small Induction Recursion, Indexed Containers and Dependent Polynomials are equivalent∗

There are several different approaches to the theory of data types. At the simplest level, polynomials and containers give a theory of data types as free standing entities. At a second level of complexity, dependent polynomials and indexed containers handle more sophisticated data types in which the data have an associated indices which can be used to store important computational information. The crucial and salient feature of dependent polynomials and indexed containers is that the index types are defined in advance of the data. At the most sophisticated level, induction-recursion allows us to define the data and the indices simultaneously. The aim of this work is to investigate the relationship between the theory of small inductive recursive definitions and the theory of dependent polynomials and indexed containers. Our central result is that the expressiveness of small inductive recursive definitions is exactly the same as that of dependent polynomials and indexed containers. Formally, this result applies not just to the data types definable in these theories, but also to the morphisms between such data types. Indeed, we introduce the category of small inductive-recursive definitions and prove the equivalence of this category with the category of dependent polynomials/indexed containers. 1998 ACM Subject Classification F.3.3 Studies of Program Constructs