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We investigate an automated program synthesis system that is based on the paradigm of programming by proofs. To automatically extract a λ-term that computes a recursive function given by a set of equations the system must find a formal proof of the totality of the given function. Because of the particular logical framework, usually such approaches make it difficult to use termination techniques such as those in rewriting theory. We overcome this difficulty 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.

On Automating the Extraction of Programs from Termination Proofs