# Ralf's Ramblings: Coq

## Exponential blowup when using unbundled typeclasses to model algebraic hierarchies

When formalizing a proof in an interactive theorem prover like Coq, one reoccurring issue is the handling of algebraic hierarchies. Such hierarchies are everywhere: some operations are associative, while others 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 mechanisms 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. However as we learned the hard way in the Coq formalization of the original Iris paper, this approach quickly leads to terms that seem to be exponential in size. In this post, I will explain the cause of this exponential blowup.