Large language models (LLMs) have been used to generate formal proofs of mathematical theorems in proofs assistants such as Lean. However, we often want to optimize a formal proof with respect to various criteria, depending on its downstream use. For example, we may want a proof to adhere to a certain style, or to be readable, concise, or modularly structured. Having suitably optimized proofs is also important for learning tasks, especially since human-written proofs may not optimal for that purpose. To this end, we study a new problem of automated proof optimization: rewriting a proof so that it is correct and optimizes for an arbitrary criterion, such as length or readability. As a first method for automated proof optimization, we present ImProver, a large-language-model agent that rewrites proofs to optimize arbitrary user-defined metrics in Lean. We find that naively applying LLMs to proof optimization falls short, and we incorporate various improvements into ImProver, such as the use of symbolic Lean context in a novel Chain-of-States technique, as well as error-correction and retrieval. We test ImProver on rewriting real-world undergraduate, competition, and research-level mathematics theorems, finding that ImProver is capable of rewriting proofs so that they are substantially shorter, more modular, and more readable.

本文针对现有形式证明在可读性、简洁性等方面的优化问题提出了新的解决方案。通过引入ImProver这一基于大型语言模型的智能体，本文实现了对形式证明的自动化优化，使得优化后的证明在长度、模块化程度及可读性等指标上显著提升。研究结果表明，该方法在多种数学定理的优化中表现出色，具有广泛的应用潜力。

ImProver：基于智能体的自动化证明优化