Refactoring Proofs: The Path to Readable AI-Generated Mathematics

Large Language Models (LLMs) have undeniably advanced the field of formal proof generation, yet their outputs frequently fall short readability and modularity. While these AI-generated proofs can impress with complexity, they often lack the finesse found in established formal mathematics libraries.

The Proof Gap

The primary issue isn't the ability to generate proofs, but rather the quality of these outputs. Most proof-generation systems focus on compiling proofs first, but this compile-first approach tends to produce proofs that are ad hoc and lack the polish expected from library-standard artifacts. The challenge lies in transforming these proofs into well-organized, reusable documents.

Existing methods aimed at enhancing proof quality generally rely on optimization techniques. However, these methods often prioritize measurable factors like proof length, overlooking other essential qualities such as readability and maintainability. Can we really consider a proof complete if it's unreadable?

Enter Proof-Refactor

Responding to this gap, a novel framework named Proof-Refactor has emerged. Inspired by human proof-refactoring workflows, it breaks down the process into four key phases: extracting candidate proof fragments, designing helper declarations, formally proving the components, and finally, repairing the original proof using verified components. This process aims to enhance the structure and clarity of proofs without reducing them to mere length metrics.

On tests using Lean proofs from PutnamBench and Putnam2025, Proof-Refactor has shown significant improvements. It outperformed the Claude Code refactoring baseline, particularly in enhancing signature quality and human readability. These findings highlight the potential of process-guided refactoring in elevating the quality of proofs.

Why This Matters

So, why should anyone pay attention to this development? The key finding is clear: improving AI-generated proof quality requires more than just shorter outputs. It's about nurturing a structure that humans can understand and build upon. This is vital for the integration of AI into formal mathematics, where collaboration between human and machine could redefine the field.

Could Proof-Refactor be the catalyst that transforms AI's role in mathematics from a tool to a partner? if these refined proofs will gain widespread acceptance. However, the framework sets a precedent for future innovations in AI-driven proof development, emphasizing the need for readability and modularity.

The paper's key contribution lies in its shift away from singular optimization metrics. By focusing on the process, Proof-Refactor offers a promising path for developing proofs that don't just work, but work well within human frameworks. The success of this approach may lead to more collaborative efforts between human mathematicians and AI, ultimately pushing the boundaries of what can be achieved in formal mathematics.