Syntax for Free: Representing Syntax with Binding Using Parametricity

We show that, in a parametric model of polymorphism, the type *** *** . ((*** ****** ) ****** ) ***(*** ****** ****** ) ****** is isomorphic to closed de Bruijn terms. That is, the type of closed higher-order abstract syntax terms is isomorphic to a concrete representation. To demonstrate the proof we have constructed a model of parametric polymorphism inside the Coq proof assistant. The proof of the theorem requires parametricity over Kripke relations. We also investigate some variants of this representation.

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