ΠΣ : A Core Language for Dependently Typed Programming

We introduce ΠΣ, a core language for dependently typed programming. Our intention is that ΠΣ should play the role extensions of System F are playing for conventional functional languages with polymorphism, like Haskell. The core language incorporates mutual dependent recursive definitions, Type : Type, Πand Σ-types, finite sets of labels and explicit constraints. We show that standard constructions in dependently typed programming can be easily encoded in our language. We address some important issues: having an equality checker which unfolds recursion only when needed, avoiding looping when typechecking sensible programs; the simplification of type checking for eliminators like case by using equational constraints, allowing the flexible use of case expressions within dependently typed programming and the representation of dependent datatypes using explicit constraints.