Pragmatic isomorphism proofs between Coq representations: application to lambda-term families

TL;DR

This paper explores different formal representations of lambda-term families in Coq, demonstrating their isomorphisms, and provides tools for conversion and random generation, enhancing practical handling of these structures.

Contribution

It introduces multiple isomorphic representations of lambda-term families in Coq and provides conversion tools and random generators, improving practical formalization.

Findings

Established isomorphisms between different lambda-term representations

Implemented conversion tools in Coq for these representations

Developed random generators for each representation using QuickChick

Abstract

There are several ways to formally represent families of data, such as lambda terms, in a type theory such as the dependent type theory of Coq. Mathematical representations are very compact ones and usually rely on the use of dependent types, but they tend to be difficult to handle in practice. On the contrary, implementations based on a larger (and simpler) data structure combined with a restriction property are much easier to deal with. In this work, we study several families related to lambda terms, among which Motzkin trees, seen as lambda term skeletons, closable Motzkin trees, corresponding to closed lambda terms, and a parameterized family of open lambda terms. For each of these families, we define two different representations, show that they are isomorphic and provide tools to switch from one representation to another. All these datatypes and their associated transformations are implemented in the Coq proof assistant. Furthermore we implement random generators for each representation, using the QuickChick plugin.