${\rm P{\small ROOF}W{\small ALA}}$: Multilingual Proof Data Synthesis and Theorem-Proving

Neural networks have shown substantial promise at automatic theorem-proving in interactive proof assistants (ITPs) like Lean and Coq. However, most neural theorem-proving models are restricted to specific ITPs, leaving out opportunities for cross-lingual $\textit{transfer}$ between ITPs. We address this weakness with a multilingual proof framework, ${\rm P{\small ROOF}W{\small ALA}}$, that allows a standardized form of interaction between neural theorem-provers and two established ITPs (Coq and Lean). It enables the collection of multilingual proof step data -- data recording the result of proof actions on ITP states -- for training neural provers. ${\rm P{\small ROOF}W{\small ALA}}$ allows the systematic evaluation of a model's performance across different ITPs and problem domains via efficient parallel proof search algorithms. We show that multilingual training enabled by ${\rm P{\small ROOF}W{\small ALA}}$ can lead to successful transfer across ITPs. Specifically, a model trained on a mix of ${\rm P{\small ROOF}W{\small ALA}}$-generated Coq and Lean data outperforms Lean-only and Coq-only models on the standard prove-at-$k$ metric. We open source all code including code for the $\href{https://github.com/trishullab/proof-wala}{ProofWala\; Framework}$, and the $\href{https://github.com/trishullab/itp-interface}{Multilingual\; ITP\; interaction\; framework}$.

PutnamBench: Evaluating Neural Theorem-Provers on the Putnam Mathematical Competition

We present PutnamBench, a new multilingual benchmark for evaluating the ability of neural theorem-provers to solve competition mathematics problems. PutnamBench consists of 1697 hand-constructed formalizations of 640 theorems sourced from the William Lowell Putnam Mathematical Competition, the premier undergraduate-level mathematics competition in North America. All the theorems have formalizations in Lean 4 and Isabelle; a substantial subset also has Coq formalizations. Proving the theorems requires significant problem-solving ability and proficiency in a broad range of topics taught in undergraduate mathematics courses. We use PutnamBench to evaluate several established neural and symbolic theorem-provers. These approaches can only solve a handful of the PutnamBench problems, establishing the benchmark as a difficult open challenge for research on neural theorem-proving. PutnamBench is available at https://github.com/trishullab/PutnamBench.