用AI辅助将数学证明转为计算机可验证形式，提升证明效率与可靠性。
Mathematical Formalized Problem Solving and Theorem Proving in Different Fields in Lean 4
- 用大模型将自然语言数学证明转化为Lean 4形式化步骤
- 对比传统与AI增强方法，显示AI能显著加速形式化过程
- 为AI辅助数学证明提供基础框架，适合数学与AI交叉研究者
使用像Lean 4这样的计算机化验证语言来形式化数学证明，有望深刻影响数学领域，具备推动数学推理的显著能力。然而，现有工作主要局限于对大型在线数学语料库中已有证明的形式化，难以跟上数学快速发展的步伐。为弥合传统与计算机化证明技术之间的差距，本文探索利用大语言模型（LLMs）生成形式化证明步骤并完成形式化证明。通过将自然语言（NL）数学证明转换为形式化版本，本工作介绍了Lean 4语言的基本结构与策略。目标是探究如何利用AI辅助数学形式化过程并提升其性能。文中提供了多个示例，展示传统与基于Lean 4的方法在解题中的应用。最终，本文阐述了Lean 4的基础，并对传统与AI增强的形式化过程进行了比较分析。结果表明，基于AI的工具在加速和提升数学证明形式化方面具有巨大潜力，为未来更高效、可靠的数学人工智能奠定基础。
原文摘要 · Abstract (English)
Formalizing mathematical proofs using computerized verification languages like Lean 4 has the potential to significantly impact the field of mathematics, it offers prominent capabilities for advancing mathematical reasoning. However, existing efforts are largely limited to creating formalized versions of proofs from extensive online mathematical corpora, struggling to keep pace with the rapidly evolving nature of mathematics. To bridge the gap between traditional and computerized proof techniques, this paper explores the use of Large Language Models (LLMs) to generate formal proof steps and complete formalized proofs. By converting natural language (NL) mathematical proofs into formalized versions, this work introduces the basic structure and tactics of the Lean 4 language. The goal is to determine how AI can be leveraged to assist the mathematical formalization process and improve its performance. Several examples are provided that demonstrate solving problems using both traditional and Lean 4-based approaches. Ultimately, this paper presents an explanation of the foundations of Lean 4 and comparative analyses of the mathematical formalization process using traditional and AI-augmented techniques. The findings indicate that AI- powered tools have significant potential to accelerate and enhance the formalization of mathematical proofs, paving the way for more efficient and reliable theorem-proving for AI for Math in the future.
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