\begin{frame} \small \begin{block}{} \begin{center} rewrite step $A\colon s \xrightarrow[\ell \to r]{p} t$ at position $p \in \Pos(s)$ \quad\quad set of positions $Q \subseteq \Pos(s)$ \end{center} \end{block} \bigskip \begin{definition}[Descendants after Rewrite Step] \smallskip \begin{itemize} \item The \alert{descendants} of $q$ after $A$ in $t$ \[ \alert{q \project A} = \begin{cases} \{ q \} & \text{if $q < p$ or $q \parallel p$} \\ \{ p p_3 p_2 \mid r|_{p_3} = \ell|_{p_1} \} & \text{if $q = p \ct p_1 \ct p_2$ with $p_1 \in \Pos_\VV(\ell)$} \\ \varnothing & \text{otherwise} \end{cases} \] \item<2-> The descendants of $Q$ after $A$ in $t$ are \[ \alert{Q \project A} = \bigcup_{q \in Q} q \project A \] \end{itemize} \end{definition} \bigskip %AM something to ask during the lecture, to practice during the exercises, % but (perhaps) not to put on slide % \begin{block}{Question} % What are the possibilites for `otherwise'? % \end{block} %AM 21.06 TODO add (abstract) picture \end{frame}