By Richard Bird and Ross Paterson, in the Journal of Functional Programming, available from Ross Paterson’s website:
de Bruijn notation is a coding of lambda terms in which each occurrence of a bound variable x is replaced by a natural number, indicating the `distance’ from the occurrence to the abstraction that introduced x. One might suppose that in any datatype for representing de Bruijn terms, the distance restriction on numbers would have to maintained as an explicit datatype invariant. However, by using a nested (or non-regular) datatype, we can define a representation in which all terms are well-formed, so that the invariant is enforced automatically by the type system.
Programming with nested types is only a little more difficult than programming with regular types, provided we stick to well-established structuring techniques. These involve expressing inductively defined functions in terms of an appropriate fold function for the type, and using fusion laws to establish their properties. In particular, the definition of lambda abstraction and beta reduction is particularly simple, and the proof of their associated properties is entirely mechanical.
Associated Haskell files are here.