From signatures to monads in UniMath

TL;DR

This paper develops a framework within UniMath to construct datatypes and their induction principles from basic type constructors, enabling formalization of languages with variable binding and associated substitution operations.

Contribution

It introduces a method to derive inductive datatypes and monadic substitution in UniMath without native inductive types, formalizing language syntax and semantics.

Findings

Constructed datatypes with induction principles from basic type constructors.

Formalized monadic substitution for variable binding languages.

Applied framework to lambda calculus and Martin-Löf type theory syntax.

Abstract

The term UniMath refers both to a formal system for mathematics, as well as a computer-checked library of mathematics formalized in that system. The UniMath system is a core dependent type theory, augmented by the univalence axiom. The system is kept as small as possible in order to ease verification of it - in particular, general inductive types are not part of the system. In this work, we partially remedy the lack of inductive types by constructing some datatypes and their associated induction principles from other type constructors. This involves a formalization of a category-theoretic result on the construction of initial algebras, as well as a mechanism to conveniently use the datatypes obtained. We also connect this construction to a previous formalization of substitution for languages with variable binding. Altogether, we construct a framework that allows us to concisely specify, via a simple notion of binding signature, a language with variable binding. From such a specification we obtain the datatype of terms of that language, equipped with a certified monadic substitution operation and a suitable recursion scheme. Using this we formalize the untyped lambda calculus and the raw syntax of Martin-L\"of type theory.