Codifying guarded definitions with recursive schemes

Abstract

We formalize an extension of the Calculus of Constructions with inductive and coinductive types which allows a more direct description of recursive definitions. The approach we follow is close to the one proposed for Martin-Löf's type theory in [5]. Recursive objects can be defined by fixed-point definitions as in functional programming languages, and a syntactical checking of these definitions avoids the introduction of non-normalizable terms. We show that the conditions for accepting a recursive definition proposed in [5] are not sufficient for the Calculus of Constructions, and we modify them. As a way of justifying our conditions, we develop a general method to codify a fix point definition satisfying them using well-known recursive schemes, like primitive recursion and co-recursion. We also propose different reduction rules from the ones used in [5] in order to obtain a decidable conversion relation for the system.

This research was partially supported by ESPRIT Basic Research Action “Types for Proofs and Programs” and by Programme de Recherches Coordonnées and CNRS Groupement de Recherche “Programmation”.