MathlibLemma: Folklore Lemma Generation and Benchmark for Formal Mathematics

2026-01-30

AI Summary

This work addresses a critical gap in the Mathlib library for Lean: the absence of widely used yet informal “folk lemmas” that hinder Lean’s practical utility in everyday mathematical research. To bridge this gap, the authors introduce the first large language model–driven multi-agent system that transforms LLMs from passive users into active contributors of formalized knowledge, autonomously discovering, generating, and verifying folk lemmas. The project establishes MathlibLemma, a benchmark comprising 4,028 type-checked Lean statements, and demonstrates real-world impact by successfully integrating a subset of these lemmas into the official Mathlib repository—marking the first instance of a self-evolving formal mathematical library.

Technology Category

• Machine Learning: Large Multimodal Models (LMMs)
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Multiagent Systems: Multiagent Learning

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Search and Retrieval-Augmented AI: Search Tool Learning with LLM: Teaching LLMs to invoke search and make use of retrieved information
• User Modeling, Personalization and Recommendation: Large Language Models (LLM) for user modeling and recommendation

Abstract

While the ecosystem of Lean and Mathlib has enjoyed celebrated success in formal mathematical reasoning with the help of large language models (LLMs), the absence of many folklore lemmas in Mathlib remains a persistent barrier that limits Lean's usability as an everyday tool for mathematicians like LaTeX or Maple. To address this, we introduce MathlibLemma, the first LLM-based multi-agent system to automate the discovery and formalization of mathematical folklore lemmas. This framework constitutes our primary contribution, proactively mining the missing connective tissue of mathematics. Its efficacy is demonstrated by the production of a verified library of folklore lemmas, a subset of which has already been formally merged into the latest build of Mathlib, thereby validating the system's real-world utility and alignment with expert standards. Leveraging this pipeline, we further construct the MathlibLemma benchmark, a suite of 4,028 type-checked Lean statements spanning a broad range of mathematical domains. By transforming the role of LLMs from passive consumers to active contributors, this work establishes a constructive methodology for the self-evolution of formal mathematical libraries.

Problem

• folklore lemmas
• formal mathematics
• Mathlib
• Lean
• mathematical formalization

Innovation

• folklore lemmas
• formal mathematics
• large language models
• multi-agent system
• Mathlib