Groebner basis theory is fundamental to computational algebra, but lacked a comprehensive formalization in Lean 4 that covers the core foundations uniformly for polynomial rings indexed by arbitrary types.

The authors formalize Groebner basis theory in Lean 4 on top of Mathlib's multivariate polynomial infrastructure. The development covers polynomial division with remainder, Buchberger's criterion, and the existence and uniqueness of reduced Groebner bases. The theory is developed uniformly for polynomial rings indexed by arbitrary types, enabling treatment of infinitely many variables. They connect finite and infinite settings by showing infinite-variable Groebner bases reduce to finite cases.

The formalization provides a complete treatment of core Groebner basis theory in Lean 4, including the reduction from infinite to finite variable cases. It integrates with Mathlib's existing monomial order infrastructure and establishes the key algorithmic foundations (division, Buchberger's criterion, reduced bases).