Bridging Syntax and Semantics of Lean Expressions in E-Graphs

Abstract

Interactive theorem provers, like Isabelle/HOL, Coq and Lean, have expressive languages that allow the formalization of general mathematical objects and proofs. In this context, an important goal is to reduce the time and effort needed to prove theorems. A significant means of achieving this is by improving proof automation. We have implemented an early prototype of proof automation for equational reasoning in Lean by using equality saturation. To achieve this, we need to bridge the gap between Lean's expression semantics and the syntactically driven e-graphs in equality saturation. This involves handling bound variables, implicit typing, as well as Lean's definitional equality, which is more general than syntactic equality and involves notions like $\alpha$-equivalence, $\beta$-reduction, and $\eta$-reduction. In this extended abstract, we highlight how we attempt to bridge this gap, and which challenges remain to be solved. Notably, while our techniques are partially unsound, the resulting proof automation remains sound by virtue of Lean's proof checking.