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Interactive proofs of theorems often require auxiliary helper lemmas to prove the desired theorem. Existing approaches for automatically synthesizing helper lemmas fall into two broad categories. Some approaches are goal-directed, producing lemmas specifically to help a user make progress from a given proof state, but they have limited expressiveness in terms of the lemmas that can be produced. Other approaches are highly expressive, able to generate arbitrary lemmas from a given grammar, but they are completely undirected and hence not amenable to interactive usage. In this paper, we develop an approach to lemma synthesis that is both goal-directed and expressive. The key novelty is a technique for reducing lemma synthesis to a data-driven program synthesis problem, whereby examples for synthesis are generated from the current proof state. We also describe a technique to systematically introduce new variables for lemma synthesis, as well as techniques for filtering and ranking candidate lemmas for presentation to the user. We implement these ideas in a tool called lfind, which can be run as a Coq tactic. In an evaluation on four benchmark suites, lfind produces useful lemmas in 68% of the cases where a human prover used a lemma to make progress. In these cases lfind synthesizes a lemma that either enables a fully automated proof of the original goal or that matches the human-provided lemma. 

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