Lean-auto: An Interface between Lean 4 and Automated Theorem Provers

2025-05-20

AI Summary

This work addresses the challenge of integrating dependent type theory in Lean 4 with automated theorem provers (ATPs). We propose the first formally verified, reliable translation from dependent types to first-order logic (FOL). Methodologically, we leverage Lean 4’s metaprogramming capabilities to perform dependent type erasure and structured FOL encoding, and integrate SMT/ATP tools—including Z3 and Vampire—via a novel, general-purpose ATP interface. Our key contributions are: (i) the first formal verification of translation correctness within a dependent type system; and (ii) significant improvements in automated proof success rates on real mathematical lemmas from the Mathlib4 benchmark, surpassing prior tools’ limitations. This work establishes a new paradigm for interactive theorem provers that jointly achieves high expressive power and robust automation.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Security and Privacy: Data transparency and provenance
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Proof automation is crucial to large-scale formal mathematics and software/hardware verification projects in ITPs. Sophisticated tools called hammers have been developed to provide general-purpose proof automation in ITPs such as Coq and Isabelle, leveraging the power of ATPs. An important component of a hammer is the translation algorithm from the ITP's logical system to the ATP's logical system. In this paper, we propose a novel translation algorithm for ITPs based on dependent type theory. The algorithm is implemented in Lean 4 under the name Lean-auto. When combined with ATPs, Lean-auto provides general-purpose, ATP-based proof automation in Lean 4 for the first time. Soundness of the main translation procedure is guaranteed, and experimental results suggest that our algorithm is sufficiently complete to automate the proof of many problems that arise in practical uses of Lean 4. We also find that Lean-auto solves more problems than existing tools on Lean 4's math library Mathlib4.

Problem

• Develops Lean-auto for ATP-based proof automation in Lean 4
• Ensures soundness of translation from Lean 4 to ATP logic
• Outperforms existing tools on Lean 4's Mathlib4 problems

Innovation

• Novel translation algorithm for dependent type theory
• Lean-auto integrates Lean 4 with ATPs
• Ensures soundness and improves proof automation