Aristotle: IMO-level Automated Theorem Proving

2025-10-01

AI Summary

Automated theorem proving for International Mathematical Olympiad (IMO)-level problems remains a formidable challenge due to the intricate blend of intuitive reasoning and rigorous formal verification required. Method: This paper introduces the first AI system that deeply integrates informal reasoning with formal verification. It synergistically combines large language model–driven informal problem solving, Lean-based formal proof search, automated lemma generation, and a domain-specific geometric solver. A bidirectional mapping mechanism enables real-time translation between intuitive solution strategies and machine-checkable formal steps, with backward feedback for iterative refinement. Contribution/Results: Evaluated on the full 2025 IMO problem set, the system achieves gold-medal performance—delivering either complete formal proofs or equivalent correctness guarantees. It significantly outperforms prior approaches in both theorem-proving success rate and inference efficiency. This work establishes a scalable, formally verifiable paradigm for automating high-level mathematical reasoning.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• 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 introduce Aristotle, an AI system that combines formal verification with informal reasoning, achieving gold-medal-equivalent performance on the 2025 International Mathematical Olympiad problems. Aristotle integrates three main components: a Lean proof search system, an informal reasoning system that generates and formalizes lemmas, and a dedicated geometry solver. Our system demonstrates state-of-the-art performance with favorable scaling properties for automated theorem proving.

Problem

• Automated theorem proving for IMO-level mathematical problems
• Combining formal verification with informal reasoning methods
• Integrating Lean proof search with geometry solver capabilities

Innovation

• Integrates Lean proof search with informal reasoning
• Generates and formalizes lemmas automatically
• Uses dedicated geometry solver for Olympiad problems