# Some Wellfounded Trees in UniMath

## Abstract

UniMath, short for “Univalent Mathematics”, refers to both a language (a.k.a. a formal system) for mathematics, as well as to a computer-checked library of mathematics formalized in that system. The UniMath library, under active development, aims to coherently integrate machine-checked proofs of mathematical results from many different branches of mathematics.

The UniMath language is a dependent type theory, augmented by the univalence axiom. One goal is to keep the language as small as possible, to ease verification of the theory. In particular, general inductive types are not part of the language.

In this work, we partially remedy this lack by constructing some inductive (families of) sets. This involves a formalization of a standard category-theoretic result on the construction of initial algebras, as well as a mechanism to conveniently use the inductive sets thus obtained.

The present work constitutes one part of a construction of a framework for specifying, via a signature, programming languages with binders as nested datatypes. To this end, we are going to combine our work with previous work by Ahrens and Matthes (itself based on work by Matthes and Uustalu) on an axiomatisation of substitution for languages with variable binding. The languages specified via the framework will automatically be equipped with the structure of a monad, where the monadic operations and axioms correspond to a well-behaved substitution operation.

## Keywords

Proof assistant Univalent type theory Inductive datatypes UniMath Initial algebras## Notes

### Acknowledgments

We thank Dan Grayson, Ralph Matthes, Paige North and Vladimir Voevodsky for helpful discussions on the subject matter.

This material is based upon work supported by the National Science Foundation under agreement Nos. DMS-1128155 and CMU 1150129-338510. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

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