GADTs, Functoriality, Parametricity: Pick Two

GADTs can be represented either as their Church encodings a la Atkey, or fixpoints Johann and Polonsky. While GADT its encoding need not support map function satisfying the functor laws, fixpoint representation of must such even to well-defined. The two representations thus same in general. This observation forces choice data types languages supporting GADTs. In this paper we show that choosing whether represent determines language have parametric models. has important consequences for how program with, reason about, these advanced types.