0|W|, k_0)$.  Assume that the sequence $\Omega$ has an addition operation which introduces a starting bracket $c = (k,k,0,0)$ with $0 \le k \le |W|$. Since the final [bracket system]{} contains no starting bracket, $c$ must disappear and an [elementary operation]{} is applied to $c$. If a merger is applied to $c$ and another bracket $b$ and the merger yields ${\widehat}c$, then ${\widehat}c = b$. This means that the addition of $c$ and the merger could be skipped without affecting the sequence otherwise. Note that the size of the new sequence $\Omega'$ is bounded by $(k_0-0, k_0)$. Therefore, we may assume that no merger is applied to a starting bracket in $\Omega$. 