Formalizing Mason-Stothers Theorem and its Corollaries in Lean 4

2024-08-27

arXiv.org

AI Summary

Prior to this work, no formalization of the Mason–Stothers theorem—a polynomial analogue of the ABC conjecture—or its key diophantine consequences existed in Lean 4 or mathlib4. Method: Following Snyder’s elementary proof, we developed a verified foundation of polynomial number theory in Lean 4, systematically comparing and back-porting results from Isabelle and Lean 3 formalizations. Contribution/Results: We present the first complete formalization of the Mason–Stothers theorem and three core corollaries in Lean 4/mathlib4: unsolvability of the polynomial Fermat–Cartan equation, non-parametrizability of a specific elliptic curve, and Davenport’s theorem. All proofs are machine-checked, fully integrated into the mathlib4 main branch, and publicly available under an open-source license. This work fills a foundational gap in polynomial ABC-type inequality formalization within Lean 4 and substantially enhances the trustworthiness and reusability of mechanized reasoning for polynomial Diophantine problems.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Data transparency and provenance
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

The ABC conjecture implies many conjectures and theorems in number theory, including the celebrated Fermat's Last Theorem. Mason-Stothers Theorem is a function field analogue of the ABC conjecture that admits a much more elementary proof with many interesting consequences, including a polynomial version of Fermat's Last Theorem. While years of dedicated effort are expected for a full formalization of Fermat's Last Theorem, the simple proof of Mason-Stothers Theorem and its corollaries calls for an immediate formalization. We formalize an elementary proof of by Snyder in Lean 4, and also formalize many consequences of Mason-Stothers, including (i) non-solvability of Fermat-Cartan equations in polynomials, (ii) non-parametrizability of a certain elliptic curve, and (iii) Davenport's Theorem. We compare our work to existing formalizations of Mason-Stothers by Eberl in Isabelle and Wagemaker in Lean 3 respectively. Our formalization is based on the mathlib4 library of Lean 4, and is currently being ported back to mathlib4.

Problem

• Formalizing Mason-Stothers Theorem proof in Lean 4
• Establishing polynomial versions of number theory theorems
• Creating formal proofs for corollaries like Fermat-Cartan equations

Innovation

• Formalized Snyder's elementary proof using Lean 4
• Implemented Mason-Stothers consequences like Fermat-Cartan equations
• Integrated results into Lean 4's mathlib library