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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.Goal-directedapproaches are fast but limited in expressiveness, as they can only discover auxiliary lemmas which entail their goals.Theory explorationapproaches are expressive but inefficient, as they exhaustively enumerate candidate lemmas. We introducee-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 ofallgoal-oriented proofs. This allows us to explore only those auxiliary lemmasguaranteedto 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. 

Regular expressions are a popular target for programming by example (PBE) systems, which seek to learn regexes from user-provided examples. Synthesizing from only positive examples remains an unsolved challenge, as the unrestricted search space makes it difficult to avoid over- and under- generalizing. Prior work has approached this in two ways: search-based techniques which require extra input, such as user feedback and/or a natural language description, and neural techniques. The former puts an extra burden on the user, while the latter requires large representative training data sets which are almost nonexistent for this domain. To tackle this challenge we present Regex+, a search-based syn- thesizer that infers regexes from just a few positive examples. Regex+ avoids over/under-generalization by using minimum description length (MDL) learning, adapted to version space algebras in order to efficiently search for an optimal regex according to a compositional MDL ranking function. Our evaluation shows that Regex+ more than triples the accu- racy of existing neural and search-based regex synthesizers on benchmarks with only positive examples