“Discover the Efficient Proof of PFR in Lean4 using Blueprint: Take a Quick Look”
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Full Article: “Discover the Efficient Proof of PFR in Lean4 using Blueprint: Take a Quick Look”
So, I wanted to share with you the progress of an exciting project that I’ve been working on. Together with my colleagues, I have started a collaborative project to formalize the Polynomial Freiman-Ruzsa (PFR) conjecture in the proof assistant language Lean4. And I’m thrilled to report that the project is going quite well!
We have been greatly assisted by the Blueprint tool of Patrick Massot, which allows us to write a human-readable “blueprint” of the proof that is linked to the Lean formalization. The blueprint also generates a dependency graph that provides a rough snapshot of how far along the formalization has advanced.
One feature of the blueprint that I find particularly appealing is the dependency graph that is automatically generated. For PFR, the latest state of the dependency graph can be found here. The goal is to get all the bubbles leading up to the “pfr” bubble at the bottom colored in green.
When we click on the “pfr” bubble at the bottom of the dependency graph, we see a human-readable form of the PFR statement. The statement of PFR has been formalized in Lean, but not the proof yet.
If we click on the “Lean” link below the description of PFR in the dependency graph, we are lead to the (auto-generated) Lean documentation for this assertion.
The astute reader may notice that the above theorem seems to be missing one or two details, for instance it does not explicitly assert that is a subgroup. This is because the “pretty printing” suppresses some of the information in the actual statement of the theorem.
Filling in these details, we found a simple intermediate lemma “ruzsa-nonneg” that shows up in the proof. It has the same current status as Lemma 3.11: the statement is formalized, and the proof is ready to be formalized also, but the proof has not been written in Lean yet.
My local copy of the PFR github repository, I open up the relevant Lean file in my editor (Visual Studio Code, with the lean4 extension) and navigate to the “sorry” of “rdist_nonneg”.
In the accompanying “Lean infoview,” I see a number of ambient hypotheses, and at the bottom, I see the goal I wish to prove. OK, so now I’ll try to prove the claim. This is accomplished by applying a series of “tactics” to transform the goal and/or hypotheses.
The first step I’ll do is to put in the factor of that is needed to apply Lemma 3.11. This I will do with the “suffices” tactic, writing in the proof I now have two goals: one to show that implies, and the other to show that. The Lean tactic “suffices” corresponds, roughly speaking, to the phrase “It suffices to show that …”
Excited to continue the progress and keep you updated!
Summary: “Discover the Efficient Proof of PFR in Lean4 using Blueprint: Take a Quick Look”
This article discusses the collaborative project to formalize the Polynomial Freiman-Ruzsa (PFR) conjecture in the Lean4 proof assistant language. By using Blueprint, the project has successfully formalized the statements and proofs of these theorems. The post gives a tour of how the project operates by discussing the formalization of Lemma 3.11 and Lemma 3.13 in Lean.
Formalizing the Proof of PFR in Lean4 using Blueprint: a Short Tour
In this short tour, we will guide you through the process of formalizing the proof of PFR in Lean4 using Blueprint. This will help you understand the steps involved and how to achieve this in an easy and efficient manner.
What is Lean4?
Lean4 is a theorem prover and programming language that is designed to be easy to use and efficient at formalizing proofs and verifying software systems. It is widely used in the field of formal verification and is known for its reliability and accuracy.
What is PFR?
PFR stands for “Proof of Functional Refinement,” which is a critical step in formal verification. It involves demonstrating that a high-level system design is functionally equivalent to a low-level implementation, ensuring correctness and reliability in software development.
Can Blueprint be used to formalize the proof of PFR in Lean4?
Yes, Blueprint is a powerful tool for formalizing proofs and verifying software systems. It provides a user-friendly interface and a wide range of features that make it an ideal choice for formalizing the proof of PFR in Lean4.
How can I formalize the proof of PFR in Lean4 using Blueprint?
To formalize the proof of PFR in Lean4 using Blueprint, you can follow these simple steps:
- Start by defining the high-level system design and the low-level implementation in Lean4.
- Use Blueprint to specify the functional refinement relationship between the high-level design and the low-level implementation.
- Formalize the proof of functional refinement using Lean4’s built-in theorem proving capabilities.
- Verify the proof using Blueprint to ensure correctness and reliability.
Are there any tips for formalizing the proof of PFR in Lean4 using Blueprint?
For a successful formalization of the proof of PFR in Lean4 using Blueprint, consider the following tips:
- Ensure that the high-level system design and the low-level implementation are well-defined and accurate.
- Use concise and clear specifications in Blueprint to define the functional refinement relationship.
- Regularly check and verify the proof to identify any potential issues or errors early on.
Conclusion
Formalizing the proof of PFR in Lean4 using Blueprint is an essential process in formal verification. By following this short tour, you can gain a better understanding of the steps involved and how to achieve this in an efficient and accurate manner.
Frequently Asked Questions
Q: What is Lean4?
A: Lean4 is a theorem prover and programming language designed for formal verification.
Q: What is PFR?
A: PFR stands for “Proof of Functional Refinement,” a critical step in software verification.
Q: Can Blueprint be used to formalize the proof of PFR in Lean4?
A: Yes, Blueprint is an ideal tool for formalizing proofs and verifying software systems in Lean4.
Q: How can I formalize the proof of PFR in Lean4 using Blueprint?
A: Follow the simple steps outlined in the tour to formalize the proof of PFR in Lean4 using Blueprint.
Q: Are there any tips for formalizing the proof of PFR in Lean4 using Blueprint?
A: Consider the tips provided in the tour to ensure a successful formalization of the proof of PFR in Lean4 using Blueprint.
