Small Induction Recursion

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 data and indices simultaneously. This work investigates 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. A second contribution of this paper is the definition of morphisms of small inductive recursive definitions. This allows us to extend our main result to an equivalence between the category of small inductive recursive definitions and the category of dependent polynomials/indexed containers. We comment on both the theoretical and practical ramifications of this result.