Names For Free: Polymorphic Views of Names and Binders

We propose a novel technique to represent names and binders in
Haskell. The dynamic (run-time) representation is based on de Bruijn
indices, but it features an interface to write and manipulate
variables conviently, using Haskell-level lambdas and variables. The
key idea is to use rich types: a subterm with an additional free variable
is viewed either as $\forall v. v \rightarrow \mathsf{Term} (a + v)$
or $\exists v. v \times \mathsf{Term} (a + v)$ depending on whether it
is constructed or analysed. We demonstrate on a number of examples how
this approach permits to express term construction and manipulation in
a natural way, while retaining the good properties of representations
based on de Bruijn indices.