Constrained Type Families (extended version)

We present an approach to support partiality in type-level computation without compromising expressiveness or type safety. Existing frameworks for type-level computation either require totality or implicitly assume it. For example, type families in Haskell provide a powerful, modular means of de€ning type-level computation. However, their current design implicitly assumes that type families are total, introducing nonsensical types and signi€cantly complicating the metatheory of type families and their extensions. We propose an alternative design, using quali€ed types to pair type-level computations with predicates that capture their domains. Our approach naturally captures the intuitive partiality of type families, simplifying their metatheory. As evidence, we present the €rst complete proof of consistency for a language with closed type families.