Formalizing Wu-Ritt Method in Lean 4

2026-04-16

AI Summary

This work presents the first complete formalization of the Wu–Ritt characteristic set method in Lean 4, providing a rigorously verified foundation for triangular decomposition of polynomial systems. By formally modeling core algebraic notions—such as initials, admissible orderings, and pseudo-division—the authors define basic sets and characteristic sets, and implement a zero decomposition algorithm whose termination and correctness are formally proved. The primary contribution is a machine-verified statement and proof of the zero decomposition theorem: the zero set of the original polynomial system equals the union of the zero sets of finitely many triangular sets, each excluding the zeros of its corresponding initials. This result establishes a formal basis for trustworthy symbolic computation and automated geometric theorem proving.

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

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• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We formalize the Wu-Ritt characteristic set method for the triangular decomposition of polynomial systems in the Lean 4 theorem prover. Our development includes the core algebraic notions of the method, such as polynomial initials, orders, pseudo-division, pseudo-remainders with respect to a polynomial or a triangular set, and standard and weak ascending sets. On this basis, we formalize algorithms for computing basic sets, characteristic sets, and zero decompositions, and prove their termination and correctness. In particular, we formalize the well-ordering principle relating a polynomial system to its characteristic set and verify that zero decomposition expresses the zero set of the original system as a union of zero sets of triangular sets away from the zeros of the corresponding initials. This work provides a machine-checked verification of Wu-Ritt's method in Lean 4 and establishes a foundation for certified polynomial system solving and geometric theorem proving.

Problem

• Wu-Ritt method
• polynomial systems
• triangular decomposition
• formal verification
• characteristic set

Innovation

• Wu-Ritt method
• characteristic set
• triangular decomposition
• formal verification
• Lean 4