FormalProofBench: Can Models Write Graduate Level Math Proofs That Are Formally Verified?

2026-03-27

AI Summary

This study presents the first systematic evaluation of state-of-the-art large language models’ ability to generate formally verifiable proofs in graduate-level mathematics, including analysis, algebra, probability, and logic. The authors construct a private benchmark that pairs natural language mathematical problems with their formalized statements in Lean 4 and employ an agent-based framework to automatically assess whether model-generated proofs are accepted by the Lean 4 proof checker. Experimental results show that the best base model achieves an accuracy of 33.5%, while other models exhibit substantially lower performance. The work further provides a detailed empirical analysis of tool usage, failure modes, reasoning costs, and latency, offering valuable insights into the current capabilities and limitations of large language models in formal mathematical reasoning.

Technology Category

• Natural Language Processing: (Large) Language Models
• Machine Learning: Large Multimodal Models (LMMs)
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving

Application Category

• Search and Retrieval-Augmented AI: Large language models for search
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

We present FormalProofBench, a private benchmark designed to evaluate whether AI models can produce formally verified mathematical proofs at the graduate level. Each task pairs a natural-language problem with a Lean~4 formal statement, and a model must output a Lean proof accepted by the Lean 4 checker. FormalProofBench targets advanced undergraduate and graduate mathematics, with problems drawn from qualifying exams and standard textbooks across topics including analysis, algebra, probability, and logic. We evaluate a range of frontier models with an agentic harness, and find that the best-performing foundation model achieves 33.5% accuracy, with performance dropping rapidly after that. In addition to the accuracy numbers, we also provide empirical analysis of tool-use, failure modes, cost and latency, thereby providing a thorough evaluation of the formal-theorem proving abilities of frontier models.

Problem

• formal verification
• mathematical proofs
• AI models
• graduate-level mathematics
• theorem proving

Innovation

• formal verification
• mathematical reasoning
• Lean 4
• AI benchmarking
• theorem proving