follows: if $\pi
{\mathrm{lm}}(b) | {\mathrm{lm}}(f)$ for some $\pi \in G$ and $b \in B$, then subtract the multiple $({\mathrm{lt}}(f)/{\mathrm{lt}}(\pi b))\pi b$ of $\pi b$ from $f$, so as to lower the latter’s leading monomial. Do this until no such pair $(\pi,b)$ exists anymore. The resulting polynomial is called a [*$G$-remainder*]{} (or an [*equivariant remainder*]{}, if $G$ is fixed) of $f$ modulo $B$.

This procedure is non-deterministic, but necessarily finishes after a finite number of steps, since $\leq$ is a well-order. Any potential outcome is called an equivariant remainder of $f$ modulo $B$.

\[de:EqS\] Consider two polynomials $b_0,b_1$ with leading monomials $m_0,m_1$, respectively. Let $H$ be a set of pairs $(\sigma_0,\sigma_1) \in G \times G$ for which $Gb_0 \times Gb_1=\bigcup_{(\sigma_0,\sigma_1) \in H} \{(\pi
\sigma_0b_0,\pi \sigma_1b_1)
\mid \pi \in G\}$. For every element $(\sigma_0,\sigma_1) \in H$ we consider the ordinary S-polynomial $$S(\sigma_0b_0,\sigma_1b_1):=
{\mathrm{lc}}(b_1)\frac{{\operatorname{lcm}}(\sigma_0m_0,\sigma_1m_1)}{\sigma_0m_0}\sigma_0b_0 -
{\mathrm{lc}}(b_0)\frac{{\operatorname{lcm}}(\sigma_0m_0,\sigma_1m_1)}{\sigma_1m_1}\sigma_1b_1.$$ The set $\{S(\sigma_0b_0,\sigma_1b_1) \mid (\sigma_0,\sigma_1) \in H\}$ is called a [*complete set of equivariant S-polynomials*]{} for $b_0,b_1$. It depends on the choice of $H$. Under condition EGB4, $H$ can be chosen finite.

\[thm:Buchberger\] Under the assumptions EGB1, EGB2, and EGB3, let $B$ be a subset of $K[X]$ such that for all $b_0,b_1 \in B$ there exists 