Making Written Theorems Explorable by Grounding Them in Formal Representations

2026-04-02

AI Summary

This work addresses the limited interactivity of static textual presentations of theorems and proofs, which hinders readers’ ability to deeply understand and verify mathematical reasoning. The authors propose a novel approach that leverages large language models (LLMs) to automatically translate natural-language proofs into formal Lean code, establishing fine-grained alignment between the two representations. This integration enables step-level exploration, customizable example testing, and logical dependency tracing. For the first time, the system combines LLM-generated mathematical explanations with executable formal proofs, offering an interactive comprehension experience that transcends static text. A user study (n=16) demonstrates that participants using the system achieved significantly deeper understanding of proofs, producing responses that were more accurate, detailed, and logically coherent.

Abstract

LLM-generated explanations can make technical content more accessible, but there is a ceiling on what they can support interactively. Because LLM outputs are static text, they cannot be executed or stepped through. We argue that grounding explanations in a formalized representation enables interactive affordances beyond what static text supports. We instantiate this idea for mathematical proof comprehension with explorable theorems, a system that uses LLMs to translate a theorem and its written proof into Lean, a programming language for machine-checked proofs, and links the written proof with the Lean code. Readers can work through the proof at a step-level granularity, test custom examples or counterexamples, and trace the logical dependencies bridging each step. Each worked-out step is produced by executing the Lean proof on that example and extracting its intermediate state. A user study ($n = 16$) shows potential advantages of this approach: in a proof-reading task, participants who had access to the provided explorability features gave better, more correct, and more detailed answers to comprehension questions, demonstrating a stronger overall understanding of the underlying mathematics.

Problem

• interactive theorem comprehension
• formal representation
• mathematical proof
• explorable explanations
• Lean

Innovation

• explorable theorems
• formal verification
• Lean
• interactive proof comprehension
• LLM-grounded formalization