Generating Bijections between HOAS and the Natural Numbers

TL;DR

This paper presents a method for automatically generating provably correct bijections between higher-order abstract syntax and natural numbers, facilitating encoding and manipulation of terms, sets, and maps in formal systems.

Contribution

It introduces a bijection generator with proofs of correctness for encoding HOAS in natural numbers, integrated into Twelf, simplifying the creation of reliable encodings.

Findings

Provides a library for sets and maps based on HOAS to natural numbers bijections

Automates the creation of bijections with correctness proofs

Enables encoding of

Abstract

A provably correct bijection between higher-order abstract syntax (HOAS) and the natural numbers enables one to define a "not equals" relationship between terms and also to have an adequate encoding of sets of terms, and maps from one term family to another. Sets and maps are useful in many situations and are preferably provided in a library of some sort. I have released a map and set library for use with Twelf which can be used with any type for which a bijection to the natural numbers exists. Since creating such bijections is tedious and error-prone, I have created a "bijection generator" that generates such bijections automatically together with proofs of correctness, all in the context of Twelf.