Book of Abstracts - IRIT

A very generic implementation of data-types
with binders in Coq
Benjamin Werner
Ecole Polytechnique
Abstract
Formalizing structures with binders, like first-order logic, lambda-calculus or programming
languages in a system like Coq is a very common problem. The recent years have
seen a lot of work devoted to it, which now gives us a quite clear picture of the respective
advantages and drawbacks of various approaches: De Bruijn indices, nominal approach,
the locally nameless approach which compromises between the former two, and many other
variants. In any case, when starting a formalization, the user has to make an early choice
about what encoding will be used. Furthermore, every approach still comes with a boilerplate
part of work (typically defining the lifting functions for de Bruijn indices or stating
and proving the co-finite induction principles for the locally nameless approach).
We propose a generic data-type which allows to encode all languages with binders,
provided all operators have a fixed arity. This data-type comes with a full set of functions
and induction principles, thus allowing the user to avoid boilerplate work, and also to
switch between diverse approaches in the same development.
1 Motivation and setting
When defining the syntax of a specific language in type theory, the obvious way is to use an
inductive type: each operator of the language is mapped to a corresponding constructor of the
inductive type; both having the same fixed arity. This approach generalized to many-sorted
languages by using mutual inductive types.
Things are more complicated when the formalized language yields binders. It is notorious
one has to make an almost religious choice between various solutions :
• named variables, which involves dealing with explicit α-conversion and various tricky
points about freshness of identifiers,
• “full” De Bruijn indices, which involves defining various lifting operators, which then
appear in further definitions,
• locally nameless approach, where De Bruijn indices are restricted to variables bound in
the term,
• trying to mimic at least the spirit of the Higher-Order Abstract Syntax approach, in which
the binder of the meta formalism is used, as it can be done in weaker formalisms like LF.
In all cases where De Bruijn indices are involved, one has to build a specific induction principle.
This is particularly the case in the locally nameless approach, where Aydemir et. al have
designed a clever co-finite quantification induction scheme.
In all cases, there is thus a non-negligible amount of boilerplate work involved. The aim
of this work is to provide a package which allows the user to handle, in a smooth way, all the
approaches but the one using only named variables. We do this by proposing a generic type
which :
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