Translating Informal Proofs into Formal Proofs Using a Chain of States

📝 Original Info

• Title: Translating Informal Proofs into Formal Proofs Using a Chain of States
• ArXiv ID: 2512.10317
• Date: 2025-12-11
• Authors: Ziyu Wang, Bowen Yang, Chenyi Li, Yuan Zhang, Shihao Zhou, Bin Dong, Zaiwen Wen

📝 Abstract

We address the problem of translating informal mathematical proofs expressed in natural language into formal proofs in Lean4 under a constrained computational budget. Our approach is grounded in two key insights. First, informal proofs tend to proceed via a sequence of logical transitions-often implications or equivalences-without explicitly specifying intermediate results or auxiliary lemmas. In contrast, formal systems such as Lean require an explicit representation of each proof state and the tactics that connect them. Second, each informal reasoning step can be viewed as an abstract transformation between proof states, but identifying the corresponding formal tactics often requires nontrivial domain knowledge and precise control over proof context. To bridge this gap, we propose a two-stage framework. Rather than generating formal tactics directly, we first extract a Chain of States (CoS), a sequence of intermediate formal proof states aligned with the logical structure of the informal argument. We then generate tactics to transition between adjacent states in the CoS, thereby constructing the full formal proof. This intermediate representation significantly reduces the complexity of tactic generati...

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