Universes for Generic Programs and Proofs in Dependent Type Theory

We show how to write generic programs and proofs in MartinLöf type theory. To this end we consider several extensions of MartinLöf’s logical framework for dependent types. Each extension has a universes of codes (signatures) for inductively defined sets with generic formation, introduction, elimination, and equality rules. These extensions are modeled on Dybjer and Setzer’s finitely axiomatized theories of inductive-recursive definitions, which also have a universe of codes for sets, and generic formation, introduction, elimination, and equality rules. However, here we consider several smaller universes of interest for generic programming and universal algebra. We thus formalize one-sorted and many-sorted term algebras, as well as iterated, parameterized, generalized, and indexed inductive definitions. We also show how to extend the techniques of generic programming to these universes. Most of the definitions in the paper have been implemented using the proof assistant Alfa for dependent type theory.