---
title: "Exponential blowup when using typeclasses with nested indices"
categories: research coq
---
When formalizing a proof in an interactive theorem prover like [Coq](https://coq.inria.fr/), one reoccuring issue is handling algebraic hierarchies.
Such hierarchies are everywhere: Some operations are associative, while others also commute; some types have an equivalence relation, while others also have a (pre-)order or maybe even a well-ordering; and so on.
So the question arises: What is the best way to actually encode these hierarchies in Coq?
Coq offers two mechanism that are suited to solve this task: typeclasses and canonical structures.
Both can be instrumented in different ways to obtain a (more or less) convenient-to-use algebraic hierarchy.
A common approach using typeclasses is the ["unbundled" approach by Bas Spitters and Eelis van der Weegen](https://arxiv.org/abs/1102.1323).
However, as [has been observed before](https://pastel.archives-ouvertes.fr/pastel-00649586), and as learned the hard way in the Coq formalization of the [original Iris paper](https://iris-project.org/pdfs/2015-popl-iris1-final.pdf), this approach quickly leads to terms that seem to be exponential in size.