Solving Formal Math Problems by Decomposition and Iterative Reflection

2025-07-20

AI Summary

General-purpose large language models (LLMs) exhibit severely limited capability in generating reliable formal mathematical proofs within proof assistants such as Lean 4. Method: We propose Delta Prover—a fine-tuning-free, agent-based interactive proof framework that integrates reflective problem decomposition with a lightweight domain-specific language (DSL). It dynamically decomposes proof goals, iteratively reflects on intermediate states, and automatically repairs failed attempts—orchestrating general LLMs and the Lean 4 environment in a reasoning-driven, rather than parameter-adaptation-driven, manner. Contribution/Results: Delta Prover achieves a 95.9% theorem-proving success rate on the miniF2F-test benchmark, substantially outperforming prior approaches. Crucially, it demonstrates strong test-time scalability without requiring model retraining or fine-tuning, highlighting its robustness and generalizability in formal reasoning tasks.

Technology Category

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

Application Category

• 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
• Search and Retrieval-Augmented AI: Search Tool Learning with LLM: Teaching LLMs to invoke search and make use of retrieved information

Abstract

General-purpose Large Language Models (LLMs) have achieved remarkable success in intelligence, performing comparably to human experts on complex reasoning tasks such as coding and mathematical reasoning. However, generating formal proofs in specialized languages like Lean 4 remains a significant challenge for these models, limiting their application in complex theorem proving and automated verification. Current approaches typically require specializing models through fine-tuning on dedicated formal corpora, incurring high costs for data collection and training. In this work, we introduce extbf{Delta Prover}, an agent-based framework that orchestrates the interaction between a general-purpose LLM and the Lean 4 proof environment. Delta Prover leverages the reflection and reasoning capabilities of general-purpose LLMs to interactively construct formal proofs in Lean 4, circumventing the need for model specialization. At its core, the agent integrates two novel, interdependent components: an algorithmic framework for reflective decomposition and iterative proof repair, and a custom Domain-Specific Language (DSL) built upon Lean 4 for streamlined subproblem management. extbf{Delta Prover achieves a state-of-the-art 95.9% success rate on the miniF2F-test benchmark, surpassing all existing approaches, including those requiring model specialization.} Furthermore, Delta Prover exhibits a significantly stronger test-time scaling law compared to standard Best-of-N proof strategies. Crucially, our findings demonstrate that general-purpose LLMs, when guided by an effective agentic structure, possess substantial untapped theorem-proving capabilities. This presents a computationally efficient alternative to specialized models for robust automated reasoning in formal environments.

Problem

• Generating formal proofs in Lean 4 using general-purpose LLMs
• Avoiding costly model specialization for theorem proving
• Enhancing automated reasoning with agent-based iterative reflection

Innovation

• Agent-based framework for Lean 4 proof interaction
• Reflective decomposition and iterative proof repair
• Custom DSL for streamlined subproblem management