Unifiers as equivalences: Proof-relevant unification of dependently typed data

Abstract

Dependently typed languages such as Agda, Coq and Idris use a syntactic first-order unification algorithm to check definitions by dependent pattern matching. However, these algorithms don't adequately consider the types of the terms being unified, leading to various unintended results. As a consequence, they require ad hoc restrictions to preserve soundness, but this makes them very hard to prove correct, modify, or extend. This paper proposes a framework for reasoning formally about unification in a dependently typed setting. In this framework, unification rules compute not just a unifier but also a corresponding correctness proof in the form of an equivalence between two sets of equations. By rephrasing the standard unification rules in a proof-relevant manner, they are guaranteed to preserve soundness of the theory. In addition, it enables us to safely add new rules that can exploit the dependencies between the types of equations. Using our framework, we reimplemented the unification algorithm used by Agda. As a result, we were able to replace previous ad hoc restrictions with formally verified unification rules, fixing a number of bugs in the process. We are convinced this will also enable the addition of new and interesting unification rules in the future, without compromising soundness along the way.