The problem of automatically proving the equality of terms over recursive functions and inductive data types is challenging, as such proofs often require auxiliary lemmas which must themselves be proven. Previous attempts at lemma discovery compromise on either efficiency or efficacy. \emph{Goal-directed} approaches are fast but limited in expressiveness, as they can only discover auxiliary lemmas which entail their goals. \emph{Theory exploration} approaches are expressive but inefficient, as they exhaustively enumerate candidate lemmas.

We introduce \emph{e-graph guided lemma discovery}, a new approach to finding equational proofs that makes theory exploration goal-directed. We accomplish this by using e-graphs and equality saturation to efficiently construct and compactly represent the space of \emph{all} goal-oriented proofs. This allows us to explore only those auxiliary lemmas \emph{guaranteed} to help make progress on some of these proofs. We implemented our method in a new prover called CCLemma and compared it with three state-of-the-art provers across a variety of benchmarks. CCLemma performs consistently well on two standard benchmarks and additionally solves 50% more problems than the next best tool on a new challenging set.