“I have no data yet. It is a capital mistake to
theorise before one has data.”
Sir Arthur Conan Doyle
The Adventures of Sherlock Holmes
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 be 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