Translating Informal Proofs into Formal Proofs Using a Chain of States

TL;DR

This paper presents a novel two-stage framework that translates informal proofs into formal Lean4 proofs by extracting a chain of proof states, significantly improving proof success rates under computational constraints.

Contribution

The paper introduces the Chain of States approach, bridging informal and formal proofs, and develops datasets and benchmarks for training and evaluation.

Findings

Outperforms existing methods in proof success rate

Effective in aligning informal reasoning with formal tactics

Reduces complexity of tactic generation

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.