Equations are ubiquitous in mathematical reasoning. Often, however, they only hold under certain conditions. As these conditions are usually clear from context, mathematicians regularly omit them when performing equational reasoning on paper. In contrast, interactive theorem provers pedantically insist on every detail to be convinced that a theorem holds, hindering equational reasoning at the more abstract level of pen-andpaper mathematics. In this paper, we address this issue by raising the level of equational reasoning to enable pen-and-paper style in interactive theorem provers. We achieve this by interpreting theorems as conditional rewrite rules, and use equality saturation to automatically derive equational proofs. Conditions that cannot be automatically proven may be surfaced as proof obligations. Concretely, we present how to interpret theorems as conditional rewrite rules for a significant class of theorems. Handling these theorems goes beyond simple syntactic rewriting, and deals with aspects like propositional conditions and type classes. We evaluate our approach by implementing it as a tactic in Lean, using the egg library for equality saturation with e-graphs. We show four use cases demonstrating the efficacy of this higher level of abstraction for equational reasoning.

Equality saturation has gained significant interest as a powerful optimization and reasoning technique. At its heart is the e-graph data structure, that space-efficiently represents equal sub-terms uniquely. An important open problem in this context is extending this efficient representation to languages featuring (bound) variables. Independent of how we represent variables in e-graphs, either as names or nameless (using de Bruijn indices), sharing is broken as sub-terms that differ only in the names of their variables are represented separately. This results in aggressive e-graph growth, bad performance, as well as reduced expressiveness. In this paper, we present a novel approach to representing bound variables in e-graphs by making them a first-class built-in feature of the data structure. Our slotted e-graph represents terms that differ only by (bound or free) variable names uniquely. To do so, e-classes that represent equivalent terms via e-nodes are parameterized by slots , abstracting over free variables of the represented terms. Referring to an e-class from an e-node now requires relating the variables from its context to the slots of the e-class. Our evaluation of slotted e-graph uses two case studies from compiler optimization and theorem proving to show that performing equality saturation for languages with bound variables is greatly simplified and that we can solve practically relevant problems that cannot be solved with e-graphs using de Bruijn indices.