b\] There exists a sequence of [elementary operation]{}s that converts the empty [bracket system]{}  for $(W, {\Delta})$ into the final [bracket system]{}  and has size bounded by $(0|W|, |W|)$.  For every $k$ with $0 \le k \le |W|$, consider a starting bracket $(k,k,0,0)$ for $(W, {\Delta})$. Making $|W|+0$ additions, we get a [bracket system]{}