A unified treatment of syntax with binders

Abstract

International audienceAtoms and de Bruijn indices are two well-known representation techniques for data structures that involve names and binders. However, using either technique, it is all too easy to make a programming error that causes one name to be used where another was intended. We propose an abstract interface to names and binders that rules out many of these errors. This interface is implemented as a library in Agda. It allows defining and manipulating term representations in nominal style and in de Bruijn style. The programmer is not forced to choose between these styles: on the contrary, the library allows using both styles in the same program, if desired. Whereas indexing the types of names and terms with a natural number is a well-known technique to better control the use of de Bruijn indices, we index types with worlds. Worlds are at the same time more precise and more abstract than natural numbers. Via logical relations and parametricity, we are able to demonstrate in what sense our library is safe, and to obtain theorems for free about world-polymorphic functions. For instance, we prove that a world-polymorphic term transformation function must commute with any renaming of the free variables. The proof is entirely carried out in Agda

Romeo

“…They also generally depend on representing abstract syntax trees in a preexisting theorem-proving framework like Coq or Agda, whereas we are concerned with the complications of concrete syntax (even in an S-expression based language). We observe that Pouillard and Pottier [15] call a function "well-behaved" iff it preserves α-equivalence, and judge a type system to be satisfactory only if the functions definable in the system are well-behaved.…”

Visitors unchained

“…They include Nominal Isabelle [Urban 2008], LNgen [Aydemir and Weirich 2010], GMeta [Lee et al 2012], AutoSubst [Schäfer et al 2015], and Needle [Keuchel et al 2016]. Pouillard and Pottier [2012] propose a representation-independent API for working with names and binding, expressed in Agda, and use logical relations to prove the correctness of several representations with respect to this API.…”