\begin{frame} \small \begin{block}{Confluence via Critical Pairs} \smallskip \alert<1>{Control interference} of rewrite rules: \begin{itemize} \item<2-> Critical Pair Lemma (lecture 5): \alert<2>{$\text{WCR} \quad\Longleftrightarrow\quad {\cp} \:\subseteq\: {\join}$} \quad (all critical pairs are convergent) \item<3-> Combine with Newman's Lemma: \alert<3>{$\text{SN} \quad\&\quad {\cp} \:\subseteq\: {\join} \quad\Longrightarrow\quad \text{CR}$} % \item<4-> % Observe from preceding examples: % % \alert<4>{ % ${\cp} \:=\: \varnothing \quad\not\Longrightarrow\quad \text{CR}$} \smallskip \end{itemize} \end{block} \medskip \onslide<4-> Observe from preceding examples: \begin{block}<4->{Confluence via Orthogonality} \smallskip \alert<4>{Forbid interference} of rewrite rules \begin{itemize} \item<5-> \alert<5>{no critical pairs} \item<6-> \alert<6>{no equality checks} (duplicated variables in the left-hand sides) \smallskip \end{itemize} \end{block} \end{frame}