We present a generic and executable formalization of signature-based algorithms (such as Faugère's $F_5$) for computing Gröbner bases, as well as their mathematical background, in the Isabelle/HOL proof assistant. Said algorithms are currently the best known algorithms for computing Gröbner bases in terms of computational efficiency. The formal development attempts to be as generic as possible, generalizing most known variants of signature-based algorithms, but at the same time the implemented functions are effectively executable on concrete input for efficiently computing mechanically verified Gröbner bases. Besides correctness the formalization also proves that under certain conditions the algorithms a-priori detect and avoid all useless reductions to zero, and return minimal signature Gröbner bases.

To the best of our knowledge, the formalization presented here is the only formalization of signature-based Gröbner basis algorithms in existence so far.

我们介绍了基于签名的算法（例如Faugère's $F_5$），以便在Isabelle / HOL证明助手中计算Gröbner基础及其数学背景。就计算效率而言，所述算法是当前最著名的用于计算Gröbner基的算法。正式的开发尝试尽可能通用，以概括基于签名的算法的大多数已知变体，但同时，所实现的功能可以在具体输入上有效执行，以有效地计算经过机械验证的Gröbner基。除了正确性以外，形式化还证明了在某些条件下，先验算法可以检测并避免所有无用的归零，并返回最小的特征Gröbner基。

据我们所知，此处介绍的形式化是迄今为止存在的基于签名的Gröbner基算法的唯一形式化。