Towards Pen-and-Paper-Style Equational Reasoning in Interactive Theorem Provers by Equality Saturation

Abstract

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-and-paper 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.