Formalizing Gr\"obner Basis Theory in Lean

Abstract

We present a formalization of Gr\"obner basis theory in Lean 4, built on top of Mathlib's infrastructure for multivariate polynomials and monomial orders. Our development covers the core foundations of Gr\"obner basis theory, including polynomial division with remainder, Buchberger's criterion, and the existence and uniqueness of reduced Gr\"obner bases. We develop the theory uniformly for polynomial rings indexed by arbitrary types, enabling the treatment of Gr\"obner bases in rings with infinitely many variables. Furthermore, we connect the finite and infinite settings by showing that infinite-variable reduced Gr\"obner bases can be characterized via reduced Gr\"obner bases on finite-variable subrings through monomial-order embeddings and filter-based limit constructions.