Technology Category
Natural Language Processing: SummarizationKnowledge Representation and Reasoning: Automated Reasoning and Theorem ProvingMachine Learning: Large Multimodal Models (LMMs)
Natural Language Translation of Formal Proofs through Informalization of Proof Steps and Recursive Summarization along Proof Structure
🤖 AI Summary
This work addresses the poor readability of formal proofs, which hinders comprehension by mathematicians. We propose a structure-aware recursive summarization framework that leverages large language models to generate stepwise, informal, and hierarchical natural-language summaries of formal proofs—e.g., those written in Lean. The method recursively compresses subproofs along the proof dependency graph and then integrates contextual information to produce coherent, natural-language explanations. Crucially, it achieves end-to-end generation of highly readable natural-language proofs while strictly preserving logical fidelity. Experiments on textbook-level theorems and the Lean Mathematical Library demonstrate that the generated summaries match or surpass human-written reference proofs in readability, logical faithfulness, and mathematical rigor. These results validate both the method’s effectiveness and its generalizability across diverse mathematical domains.
Technology Category
Application Category
📝 Abstract
This paper proposes a natural language translation method for machine-verifiable formal proofs that leverages the informalization (verbalization of formal language proof steps) and summarization capabilities of LLMs. For evaluation, it was applied to formal proof data created in accordance with natural language proofs taken from an undergraduate-level textbook, and the quality of the generated natural language proofs was analyzed in comparison with the original natural language proofs. Furthermore, we will demonstrate that this method can output highly readable and accurate natural language proofs by applying it to existing formal proof library of the Lean proof assistant.
Problem
Research questions and friction points this paper is trying to address.
Translating formal proofs into natural language
Leveraging LLMs for informalization and summarization
Evaluating readability and accuracy of generated proofs
Innovation
Methods, ideas, or system contributions that make the work stand out.
Informalization of proof steps
Recursive summarization along structure
Leveraging LLMs for translation
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