This article The purpose of this paper is to introduce b, a simply typed -calculus that supports type-based recursive definitions. Although heavily inspired from previous work by Giménez (Giménez 1998) and closely related to recent work by Amadio and Coupet (Amadio and Coupet-Grimal 1998), the technical machinery behind our system puts a slightly different emphasis on the interpretation of types. More precisely, we formalize the notion of type-based termination using a restricted form of type dependency (a.k.a. indexed types), as popularized by (Xi and Pfenning 1998; Xi and Pfenning 1999). This leads to a simple and intuitive system which is robust under several extensions, such as mutually inductive datatypes and mutually recursive function definitions; however, such extensions are not treated in the paper

In proof checkers and theorem provers (e.g. Coq [4] and ProPre [13]) recursive de nitions of functions are shown to terminate automatically.

We investigate an automated program synthesis system based on the paradigm of programming by proofs. To automatically extract a -term that computes a recursive function given by a set of equations the system must nd a formal proof of the totality of the given function. Because of the particular logical framework, usually such approaches make it dicult to use techniques such as those in rewriting theory. We overcome this diculty for the automated system that we consider by exploiting product types. As a consequence, this would enable the incorporation of termination techniques used in other areas while still extracting programs.