Programming proofs: a novel approach based on contextual types

We present an overview of Beluga, a dependently-typed programming and proof development environment. Beluga uses a two-level approach: it supports specifying formal systems within the logical framework LF and on top of LF, it provides a dependently-typed functional language that supports manipulating and analyzing LF objects via pattern matching. A distinct feature of Beluga is the explicit support for contexts and contextual objects, which concisely characterize objects depending on assumptions. The design of the dependently-typed functional language is generic and independent of the concrete specification language which in Beluga’s case is LF. Moreover, it is unique in its treatment of pattern matching in the presence of dependent types: it is centered around refinement substitutions which refine the types of patterns in case-expressions. This makes type checking decidable although constraint solving itself could be undecidable. In addition to a bi-directional decidable type system for Beluga we give an environment-based operational semantics and show that types are preserved. Beluga together with type reconstruction is implemented in OCaml and has been used on a wide variety of examples such as type uniqueness, proofs about compiler transformations, and preservation and progress for various ML-like languages.