be non-trivial obstructions in $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$ and $\bigcup_{1\leq i\leq j\leq s-1}{\rm NTObs}(i,j)$, respectively. If there exist two words $w,w'\in\langle X\rangle$ such that $u_{j}=wv_{j}$ and $u'_{j}=v'_{j}w'$, then we have $${\rm o}_{j,s}(u_{j},u'_{j};w_{s},w'_{s})=-w{\rm o}_{i,j}(w_{i},w'_{i};v_{j},v'_{j})w'+
{\rm o}_{i,s}(ww_{i},w'_{i}w';w_{s},w'_{s})$$ where the inequalities ${\rm o}_{j,s}(u_{j},u'_{j};w_{s},w'_{s})
>_{\tau}{\rm o}_{i,j}(w_{i},w'_{i};v_{j},v'_{j})$ and ${\rm o}_{j,s}(u_{j},u'_{j};$ $w_{s},w'_{s})
>_{\tau}{\rm o}_{i,s}(ww_{i},w'_{i}w';w_{s},w'_{s})$ hold. Further, if the S-polynomials $S_{i,j}(w_{i},w'_{i};$ $v_{j},v'_{j})$ and $S_{i,s}(ww_{i},w'_{i}w';w_{s},w'_{s})$ have Gröbner representations in terms of $G$, then so does $S_{j,s}(u_{j},u'_{j};w_{s},w'_{s})$.

The claimed equality follows from Definition \[sec2def4\].a and from the conditions $u_{j}=wv_{j}$ and $u'_{j}=v'_{j}w'$. We have ${\rm o}_{j,s}(u_{j},u'_{j}; w_{s},w'_{s})>_{\tau}{\rm o}_{i,j}(w_{i},w'_{i};v_{j},v'_{j})$ for $w_{s}e_{s}w'_{s}>_{\tau}u_{j}e_{j}u'_{j}=
wv_{j}e_{j}v'_{j}w\geq_{\tau}v_{j}e_{j}v'_{j}$. From the inequality $u_{j}e_{j}u'_{j}=
wv_{j}e_{j}v'_{j}w>_{\tau}ww_{i}e_{i}w'_{i}w'$, it follows that ${\rm o}_{j,s}(u_{j},u'_{j};w_{s},w'_{s})
>_{\tau}{\rm o}_{i,s}(ww_{i},w'_{i}w';$ $w_{s},w'_{s})$. Again, we can prove the second part by following the same argument as in the proof of Lemma \[sec3lem3\].a.

Note that the obstruction ${\rm o}_{i,s}(ww_{i},w'_{i}w';w_{s},w'_{s})$ in Lemma \[sec3lem7\] can be either a multiple of a non-trivial obstruction or an obstruction without overlap. However, it suffices for us to consider only the latter case, since the former case has been considered in Proposition \[ncMCrit\], and, more precisely, in Lemma \[sec3lem3\].b.

\[ncTailRed\] Suppose that ${\rm o}_{j,s}(u_{j},u'_{j};w_{s},w'_{s})$ and ${\rm o}_{i,j}(w_{i},w'_{i};v_{j},v'_{j})$ are non-trivial obstructions in $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$ and $\bigcup_{1\leq i\leq j\leq s-1}{\rm NTObs}(i,j)$, respectively, such that there exist 