Formalizing Wu-Ritt Method in Lean 4

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.

• The paper presents a formalization of the Wu-Ritt method in Lean 4, verifying the correctness and termination of algorithms for triangular decomposition and polynomial system solving.
• It rigorously defines algebraic structures, including triangular sets and pseudo-division, to establish well-ordering and minimality properties.
• The work underpins certified symbolic computation and automated geometric theorem proving within Lean 4, advancing formal verification in algebraic geometry.

Formalization of the Wu-Ritt Characteristic Set Method in Lean 4

Introduction

The paper "Formalizing Wu-Ritt Method in Lean 4" (2604.14912) presents a rigorous formalization of the Wu-Ritt characteristic set method within the Lean 4 theorem prover. The Wu-Ritt method constitutes a cornerstone for symbolic computation in multivariate polynomial system solving and automated geometric theorem proving. This formalization integrates the algebraic theory of triangular decomposition with verified algorithmic procedures, leveraging Lean 4's extensible and expressive environment to establish correctness and termination of core computational primitives.

Algebraic Structures and Foundational Definitions

The formalization begins by specifying the algebraic context: multivariate polynomial rings $R[X_i]_{i \in \sigma}$ over a commutative ring $R$ equipped with a linear order on $\sigma$. Principal notions such as main variable, initial, main degree, and polynomial order are formalized using Lean's type classes and ordering mechanisms. The lexicographically inspired order introduced distinguishes this formalization from Gröbner basis theory, providing an order grounded in variable indices rather than monomial orders.

Triangular sets are defined as sequences of nonzero polynomials with strictly ascending main variables. The ordering and equivalence relations on these sets are precisely specified, and their well-ordering property is established in the context of finite index sets, which is vital for the termination guarantees in subsequent algorithms.

Ascending Sets, Basic Sets, and Characteristic Sets

Ascending sets are triangular sets in which each polynomial is reduced with respect to its predecessors. The reduction relation is implemented with careful dependency on polynomial degrees in main variables. This structure is extended to basic sets, defined as minimal ascending sets for a given polynomial set. The formalization includes algorithmic procedures to compute these sets, proofs of minimality, and strict order reduction, ensuring algorithmic progress.

Characteristic sets are triangular sets satisfying the property that pseudo-remainders of the input system vanish and the corresponding zero set of the characteristic set contains the zero set of the original system. The well-ordering principle, central to Wu-Ritt theory, is formalized: it relates polynomial systems and their characteristic sets, and establishes the set-theoretic inclusion and decomposition properties for solutions.

Pseudo-Division and Zero Decomposition Algorithms

The pseudo-division algorithm is essential for the Wu-Ritt approach. The paper formalizes this both for single polynomials and recursively with respect to triangular sets. The implementation abstracts the algebraic process and rigorously proves its correctness. Crucially, the degree reduction ensures algorithmic termination, a property also exploited in computing basic and characteristic sets.

The zero decomposition theorem is formalized and proved: the zero set of a polynomial system is expressed as a union of zero sets of triangular sets, each avoiding the vanishing of their initials. The recursive decomposition algorithm's correctness and completeness are verified, showing that each triangular set in the output corresponds to a characteristic set for some subsystem arising in the recursive construction.

Numerical and Structural Results

The paper does not provide explicit numerical benchmarks but contains strong claims on algorithmic termination and correctness. The well-ordering principle and zero decomposition theorem are proved in Lean 4, establishing that the algorithms compute finite decompositions and terminate on input polynomial systems with finite variables. The correctness guarantees encompass the mechanical verification of vanishing sets and pseudo-remainders, certifying the formal approach.

Practical Implications and Theoretical Impact

The formalization in Lean 4 establishes a foundation for certified symbolic computation in algebraic geometry and automated theorem proving. Its integration with the Mathlib algebraic infrastructure positions Lean 4 as a viable environment for developing executable, formally verified algorithms for polynomial system solving. The work surpasses prior efforts in Coq by encompassing the entire characteristic set formalism, thus opening the path for broader applications in certified computer algebra and geometry.

From a theoretical standpoint, the paper provides machine-checked assurance of deep algebraic principles related to triangular decomposition, pseudo-division, and solution set decompositions. The abstraction and implementation in Lean 4 also prepare the ground for formalizing more advanced methods, such as differential characteristic sets or the integration with Gröbner basis theory (Guo et al., 13 Feb 2026).

Speculation on Future Extensions

Future development will likely include extraction of executable code from Lean 4 formalizations, enabling practical certified symbolic computations. Such extensions could encompass algorithms for differential polynomials, mechanical formalization of geometric decision procedures, and integration with larger formal mathematical libraries (e.g., Mathlib). The formal foundation laid by this work sets a precedent for collaborations between algebraic computation and formal verification communities, potentially impacting certifiable AI-driven theorem proving and algebraic reasoning systems.

Conclusion

The formalization of the Wu-Ritt characteristic set method in Lean 4 rigorously verifies the algebraic primitives, data structures, and algorithms essential for triangular decomposition of polynomial systems. By proving termination and correctness of the core algorithms and establishing foundational decomposition theorems, the paper provides a reusable, certifiable substrate for symbolic computation and automated geometric theorem proving within the Lean ecosystem. The results bear significant practical and theoretical implications for certified algebraic computation, and the approach is poised to influence further developments in verified mathematical software and formalized mathematics.