lemma helps us to implement the first step, that is, to detect unnecessary obstructions in the set $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$ of newly constructed obstructions via other obstructions in this set.

\[sec3lem3\] Let ${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})$ and ${\rm o}_{j,s}(w_{j},w'_{j};v_{s},v'_{s})$ be two distinct non-trivial obstructions in $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$ with two words $w,w'\in\langle X\rangle$ satisfying $u_{s}=$ $wv_{s}$ and $u'_{s}=v'_{s}w'$.

-   If $i<j$ and $ww'\neq1$, then we have $${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})=w{\rm o}_{j,s}(w_{j},w'_{j};v_{s},v'_{s})w'+{\rm o}_{i,j}(w_{i},w'_{i};ww_{j},w'_{j}w')$$ with ${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})>_{\tau}{\rm o}_{j,s}(w_{j},w'_{j};v_{s},v'_{s})$ and ${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})>_{\tau}{\rm o}_{i,j}(w_{i},w'_{i};ww_{j},w'_{j}w')$. Further, if the S-polynomials $S_{j,s}(w_{j},w'_{j};v_{s},v'_{s})$ and $S_{i,j}(w_{i},w'_{i};ww_{j},w'_{j}w')$ have Gröbner representations in terms of $G$, then so does $S_{i,s}(w_{i},w'_{i};u_{s},u'_{s})$.

-   If $i>j$, then we have $${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})=w{\rm o}_{j,s}(w_{j},w'_{j};v_{s},v'_{s})w'-{\rm o}_{j,i}(ww_{j},w'_{j}w';w_{i},w'_{i})$$ with ${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})>_{\tau}{\rm o}_{j,s}(w_{j},w'_{j};v_{s},v'_{s})$ and ${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})>_{\tau}{\rm o}_{j,i}(ww_{j},w'_{j}w';w_{i},w'_{i})$. Further, if the S-polynomials $S_{j,s}(w_{j},w'_{j};v_{s},v'_{s})$ and $S_{j,i}(ww_{j},w'_{j}w';w_{i},w'_{i})$ have Gröbner representations in terms of $G$, then so does $S_{i,s}(w_{i},w'_{i};u_{s},u'_{s})$.

-   If $i=j$ and $ww'\neq 1$ or if $i=j$ and $ww'=1$ and $w_i>_{\sigma}w_j$, then we have $${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})=w{\rm o}_{j,s}(w_{j},w'_{j};v_{s},v'_{s})w'+{\rm o}_{i,j}(w_{i},w'_{i};ww_{j},w'_{j}w')$$ with ${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})>_{\tau}{\rm o}_{j,s}(w_{j},w'_{j};v_{s},v'_{s})$ and ${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})>_{\tau}{\rm o}_{i,j}(w_{i},w'_{i};ww_{j},w'_{j}w')$. Further, if the S-polynomials $S_{j,s}(w_{j},w'_{j};v_{s},v'_{s})$ and $S_{i,j}(w_{i},w'_{i};ww_{j},w'_{j}w')$ have Gröbner representations in terms of $G$, then so does $_{i,s}(w_{i},w'_{i};u_{s},u'_{s})$.

We prove case (a). Cases (b) and (c) can be proved similarly. The equation in case (a) follows from Definition \[sec2def4\].a and 