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.