Proof-Refactor: Refactoring Generated Formal Proofs into Modular Artifacts
The paper introduces Proof-Refactor, an agentic framework that improves the readability, modularity, and maintainability of LLM-generated formal proofs by employing a process-guided, four-phase refactoring workflow instead of relying on single-metric optimization like proof length.
Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer
The Problem: The "Fast Food" vs. "Home-Cooked Meal" of Math
Imagine you ask a very smart robot (a Large Language Model) to write a formal mathematical proof. The robot is great at its job: it follows the rules, gets the answer right, and the computer says, "Yes, this is correct."
However, the proof it writes is often like a fast-food burger. It gets the job done, but it's messy. It's all squished together, uses weird ingredients specific to that one meal, and if you tried to use a piece of it for a different meal later, it wouldn't fit. It's hard to read, hard to fix, and hard to share with others.
In the world of formal math (using tools like Lean), these proofs are often "monolithic"—one giant block of code that works but is a nightmare to maintain. The current way of making these proofs better is to try to make them shorter. But making a proof shorter is like "golfing" (trying to hit a ball in the fewest strokes possible); it often leads to clever but unreadable tricks, rather than a clean, logical structure.
The Solution: The "Renovation Crew" (Proof-Refactor)
The authors of this paper propose a new approach called Proof-Refactor. Instead of trying to make the proof shorter, they treat it like a house renovation.
They argue that the best way to fix a messy proof isn't to just compress it, but to refactor it. This means taking the messy parts, breaking them down, and rebuilding them into clean, reusable rooms that fit into a standard neighborhood (the library of math).
To do this, they built a team of AI agents that work in four distinct phases, much like a construction crew:
The Demolition Crew (Extraction):
First, they look at the messy proof and identify small chunks of logic that are doing a specific job. They "cut" these chunks out of the main proof and turn them into standalone, temporary blueprints called scaffolds. Think of this as taking a weird, custom-built shelf out of a wall and setting it on a table to examine it.The Architect (Helper Design):
This is the most important step. A human architect (or in this case, an external AI assistant) looks at those temporary blueprints. They ask: "Is this just a weird shelf for this one house, or is this a standard bookshelf that could be used in any house?"
They redesign the shelf to be a standard, reusable component. They give it a clean name and a clear description so it fits into the neighborhood's building codes.The Builders (Proving):
Now, the team goes back and actually builds those new, standard components. They prove that these new, clean blueprints actually work. This is easier than building the whole house at once because they are only building one small, perfect room at a time.The Finishers (Repair):
Finally, they go back to the original messy house. They tear down the old, weird wall and replace it with the new, standard bookshelf they just built. The house still stands, but now it's cleaner, easier to understand, and the new shelf can be used in other houses too.
Why This Works Better
The paper tested this method on difficult math problems (from the Putnam competition). They compared their "Renovation Crew" against a standard robot that just tries to make proofs shorter.
- The Result: The Proof-Refactor team created proofs that were much more readable, modular (easy to break into parts), and reusable.
- The Trade-off: Sometimes, the new proofs weren't actually shorter. In fact, they were sometimes longer! But that's okay. Just like a well-organized kitchen might take up more space than a cluttered one, a well-structured proof is better for humans to read and for computers to verify in the long run.
The Secret Sauce: "Two Brains"
A key part of their success was separation of labor.
- One AI (the "Builder") is great at talking to the computer code, checking for errors, and typing commands.
- Another AI (the "Architect") is great at high-level thinking and math concepts.
The paper found that if you ask the "Builder" to do the "Architect's" job (designing the new structure) while it's also trying to fix code errors, it gets overwhelmed and makes bad designs. By letting the "Architect" think about the big picture separately, the final result is much higher quality.
In Summary
Proof-Refactor doesn't just try to make math proofs shorter. It treats them like software code that needs cleaning up. It breaks messy proofs apart, redesigns the pieces to be standard and reusable, and then stitches them back together. The result is math that is not just "correct," but also beautiful, understandable, and useful for future mathematicians.
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