LeanSide: A Formally Verified Co-Reasoning System for Natural-language Proofs

2026-09-30

AI Summary

This study addresses the challenges of reasoning hallucinations in large language models (LLMs) and the high entry barriers associated with formal proof tools by proposing LeanSide, a system that integrates LLMs with the Lean proof assistant. LeanSide enables the automated translation of natural language into formal verification and provides informalized feedback on verification outcomes, thereby significantly lowering the usability threshold and facilitating human-AI collaborative reasoning in mathematics education. The effectiveness of the system is validated through real-world classroom deployments. Furthermore, this work distills key design insights for human-AI collaborative reasoning, offering a novel paradigm for AI-assisted mathematics education.

Abstract

Large language models are increasingly used as collaborators on deductive-reasoning tasks, but their outputs can hallucinate or pull users away from intended reasoning. Formal proof assistants provide machine-checked verification, but have a steep learning curve and require more granular reasoning than human written proofs. We explore an interface that combines these strengths, allowing users to write and revise free-form natural-language proofs while a verified backend checks their reasoning and returns feedback at the user's granularity. We study this interface in the context of undergraduate mathematics education by developing LeanSide, a formally verified co-reasoning system, which auto-formalizes student reasoning into Lean and informalizes verifier output into understandable feedback. We conducted user studies through classroom deployment and analyzed which system properties helped students make progress and which caused them to get stuck. We use these findings to derive design implications for using a formally verified backend in human-AI co-reasoning systems.

Problem

• Large language models
• Formal verification
• Deductive reasoning
• Human-AI co-reasoning
• Mathematics education

Innovation

• Formal Verification
• Co-Reasoning
• Auto-formalization
• Natural Language Proofs
• Lean