GADTs were introduced in Haskell’s eco-system more than a decade ago, but their interaction with several mainstream features such as type classes and functional dependencies has a lot of room for improvement. More specifically, functions with qualified types cannot perform induction over some GADTs, significantly undermining their use.

In this paper we identify the source of this shortcoming and address it by introducing a conservative extension to Haskell’s type classes: Bidirectional Type Class Instances. In essence, under our interpretation class instances correspond to logical bi-implications, in contrast to their traditional unidirectional interpretation.

We present a fully-fledged design of bidirectional instances, covering the specification of typing and elaboration into System FC, as well as an algorithm for type inference and elaboration. We provide a proof-of-concept implementation of our algorithm, and revisit the meta-theory of type classes in the presence of our extension.