two words $w,w'\in\langle X\rangle$ satisfying $u_{j}=wv_{j}$ and $u'_{j}=v'_{j}w'$. If the word $ww_{i}$ is a multiple of $w_{s}\lw_{\sigma}(g_{s})$, or if the word $w'_{i}w'$ is a multiple of $\lw_{\sigma}(g_{s})w'_{s}$, then ${\rm o}_{j,s}(u_{j},u'_{j};w_{s},w'_{s})$ can be removed from $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$ in the execution of the Buchberger Procedure.

By Lemma \[sec3lem7\], the obstruction ${\rm o}_{j,s}(u_{j},u'_{j};w_{s},w'_{s})$ can be represented as $${\rm o}_{j,s}(u_{j},u'_{j};w_{s},w'_{s})=
-w{\rm o}_{i,j}(w_{i},w'_{i};v_{j},v'_{j})w'+{\rm o}_{i,s}(ww_{i},w'_{i}w';w_{s},w'_{s})$$ where the inequalitites ${\rm o}_{j,s}(u_{j},u'_{j};w_{s},w'_{s})>_\tau{\rm o}_{i,j}(w_{i},w'_{i};v_{j},v'_{j})$ 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 $S_{j,s}(u_{j},u'_{j};w_{s},w'_{s})$. Theorem \[sec2the11\] ensures that $S_{i,j}(w_{i},w'_{i};v_{j},v'_{j})$ has a Gröbner representation in terms of $G$. Note that $ww_{i}$ is a multiple of $w_{s}\lw_{\sigma}(g_{s})$ or $w'_{i}w'$ is a multiple of $\lw_{\sigma}(g_{s})w'_{s}$. This implies that ${\rm o}_{i,s}(ww_{i},$ $w'_{i}w';w_{s},w'_{s})$ has no overlap. By Lemma \[sec2lem8\], $S_{i,s}(ww_{i},w'_{i}w';w_{s},w'_{s})$ has a Gröbner representation in terms of $G$. Now the conclusion follows from Proposition \[sec2pro10\] and Theorem \[sec2the11\].

\[sec3rem9\] Our experiments in the final section show that, after applying the previous two criteria, the Non-Commutative Tail Reduction is unlikely to apply in the Buchberger Procedure. This may be due to the fact that frequently the Non-Commutative Multiply Criterion and the Non-Commutative Leading Word Criterion have already detected 