from the conditions $u_{s}=wv_{s}, u'_{s}=v'_{s}w'$ and $i<j$. Because of $ww'>1$, we have $u_{s}\lw(g_{s})u'_{s}=wv_{s}\lw(g_{s})v'_{s}w'>_{\sigma}v_{s}\lw(g_{s})v'_{s}$. Consequently, we get $u_{s}e_{s}u'_{s}>_{\tau}v_{s}e_{s}v'_{s}$ and ${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})>_{\tau}{\rm o}_{j,s}(w_{j},w'_{j};v_{s},v'_{s})$. From $u_{s}\lw(g_{s})u'_{s}$ $=w_{i}\lw(g_{i})w'_{i}=ww_{j}\lw(g_{j})w'_{j}w'$ and $s>j$, we get the inequalities $u_{s}e_{s}u'_{s}>_{\tau}ww_{j}e_{j}w'_{j}w'$ and ${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})>_{\tau}{\rm o}_{i,j}(w_{i},w'_{i};ww_{j},w'_{j}w')$.

Next we show that, if $S_{j,s}(w_{j},w'_{j};v_{s},v'_{s})$ and $S_{i,j}(w_{i},w'_{i};ww_{j},w'_{j}w')$ have Gröbner representations in terms of $G$, then so does $S_{i,s}(w_{i},w'_{i};u_{s},u'_{s})$. Clearly we have $$S_{i,s}(w_{i},w'_{i};u_{s},u'_{s})=wS_{j,s}(w_{j},w'_{j};v_{s},v'_{s})w'+S_{i,j}(w_{i},w'_{i};ww_{j},w'_{j}w').$$ Without loss of generality, we assume that $S_{i,s}(w_{i},w'_{i};u_{s},u'_{s}), S_{j,s}(w_{j},w'_{j};v_{s},v'_{s})$ and $S_{i,j}(w_{i},w'_{i};ww_{j},w'_{j}w')$ are non-zero. Since there is a Gröbner representation for $S_{j,s}(w_{j},w'_{j};v_{s},v'_{s})$, we have $$S_{j,s}(w_{j},w'_{j};v_{s},v'_{s})=\sum^\mu_{k=1}a_kw_kg_{i_k}w'_k$$ with $a_k\in K\setminus\{0\},\ w_k,w'_k\in\langle X\rangle,\ g_{i_k}\in G$ for all $k\in\{1,\dots,\mu\}$, such that $\lw_{\sigma}(v_{s}g_{s}v'_{s})>_{\sigma}\lw_{\sigma}(a_kw_kg_{i_k}w'_k)$. Similarly, for $S_{i,j}(w_{i},w'_{i};ww_{j},w'_{j}w')$ we have $$S_{i,j}(w_{i},w'_{i};ww_{j},w'_{j}w')=\sum^\nu_{l=1}b_lw_lg_{i_l}w'_l$$ with $b_l\in K\setminus\{0\},\ w_l,w'_l\in\langle X\rangle,\ g_{i_l}\in G$ for all $l\in\{1,\dots,\nu\}$, such that $\lw_{\sigma}(ww_{j}g_{j}w'_{j}w')>_{\sigma}\lw_{\sigma}(b_lw_lg_{i_l}w'_l)$. Therefore we have $$\begin{aligned}
S_{i,s}(w_{i},w'_{i};u_{s},u'_{s})&=&w(\sum^\mu_{k=1}a_kw_kg_{i_k}w'_k)w'+\sum^\nu_{l=1}b_lw_lg_{i_l}w'_l\\
&=&\sum^\mu_{k=1}a_kww_kg_{i_k}w'_kw'+\sum^\nu_{l=1}b_lw_lg_{i_l}w'_l.\end{aligned}$$ As $u_{s}\lw_{\sigma}(g_{s})u'_{s}=wv_{s}\lw_{\sigma}(g_{s})v'_{s}w'$, we have $\lw_{\sigma}(u_{s}g_{s}u'_{s})=\lw_{\sigma}(wv_{s}g_{s}v'_{s}w')$ $>_{\sigma}\lw_{\sigma}(ww_kg_{i_k}w'_kw')$ for all $k\in\{1,\dots,\mu\}$. By Definition \[sec2def4\], we have $\lw_{\sigma}(u_{s}g_{s}u'_{s})=\lw_{\sigma}(w_{i}g_iw'_{i})=\lw_{\sigma}(ww_{j}g_jw'_{j}w')>_{\sigma}\lw_{\sigma}(b_lw_lg_{i_l}w'_l)$ for all $l\in\{1,\dots,\nu\}$. Therefore $$S_{i,s}(w_{i},w'_{i};u_{s},u'_{s})=\sum^\mu_{k=1}a_kww_kg_{i_k}w'_kw'+\sum^\nu_{l=1}b_lw_lg_{i_l}w'_l$$ is a Gröbner representation of $S_{i,s}(w_{i},w'_{i};u_{s},u'_{s})$.

The following example shows that the obstruction ${\rm o}_{i,j}(w_{i},w'_{i};ww_{j},w'_{j}w')$ in case (a) of Lemma \[sec3lem3\] can be a non-trivial obstruction, i.e. a multiple of a non-trivial obstruction or an obstruction without overlap. Similar phenomena occur in cases (b) and (c) of Lemma \[sec3lem3\], as well as in Lemmas \[sec3lem7\] and \[sec3lem10\].

\[sec3exa4\] Consider polynomials 