Translating Informal Proofs into Formal Proofs Using a Chain of States

2025-12-11

AI Summary

This work addresses the automatic translation of informal natural-language mathematical proofs into formal Lean 4 proofs—a task hindered by poor alignment between informal reasoning and structured verification languages, as well as weak controllability. We propose the novel “Chain of States” (CoS) intermediate representation, the first to explicitly decouple proof-structure modeling from tactic generation. CoS enables staged state extraction and structured tactic synthesis, and we further develop an interactive formalization framework alongside a domain-specific training dataset. Evaluated on multiple mathematical benchmarks, our approach significantly outperforms prior methods, achieving substantial gains in proof success rate. Results demonstrate that CoS effectively improves formalization accuracy and verifiability under limited computational budgets, validating its design principles and practical efficacy.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

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
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

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 like 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 generation and improves alignment with informal reasoning patterns. We build dedicated datasets and benchmarks for training and evaluation, and introduce an interactive framework to support tactic generation from formal states. Empirical results show that our method substantially outperforms existing baselines, achieving higher proof success rates.

Problem

• Translating informal proofs into formal Lean4 proofs
• Bridging the gap between informal and formal proof structures
• Generating formal tactics via an intermediate Chain of States representation

Innovation

• Extract Chain of States from informal proofs
• Generate tactics to transition between adjacent states
• Use interactive framework for tactic generation