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

2025-02-07

AI Summary

Existing neural theorem proving models rely heavily on a single interactive theorem prover (ITP), hindering cross-system transferability and limiting generalization and practical applicability. Method: We propose the first multi-ITP theorem proving framework supporting both Coq and Lean. It introduces a unified interaction protocol that models proof actions and state transitions as cross-language alignable, standardized representations; designs multi-ITP interfaces, parallelized search algorithms, and a cross-system proof-step synthesis mechanism; and enables joint multilingual training. Contribution/Results: Experiments demonstrate significant improvements over single-ITP baselines under the prove-at-k metric, providing the first empirical validation of effective knowledge transfer for neural theorem proving across heterogeneous ITPs. The complete codebase and framework are publicly released.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Natural Language Processing: Language Grounding & Multi-modal NLP
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

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 $ extit{transfer}$ between ITPs. We address this weakness with a multilingual proof framework, ${ m P{small ROOF}W{small ALA}}$, 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. ${ m P{small ROOF}W{small ALA}}$ 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 ${ m P{small ROOF}W{small ALA}}$ can lead to successful transfer across ITPs. Specifically, a model trained on a mix of ${ m P{small ROOF}W{small ALA}}$-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 $href{https://github.com/trishullab/itp-interface}{Multilingual; ITP; interaction; framework}$.

Problem

• Multilingual proof data synthesis
• Cross-lingual transfer in theorem-proving
• Neural theorem-proving performance evaluation

Innovation

• Multilingual proof framework for ITPs
• Standardized interaction between neural provers
• Efficient parallel proof search algorithms