\begin{frame} \small \vspace{-1ex} \begin{example} \smallskip TRS $\RR$ \[ \begin{array}{lr@{~}c@{~}l@{\qquad}lr@{~}c@{~}l} \structure{\text{\ding{192}}} & \GREEN{x+\m{0}} & \GREEN{\to} & \GREEN{x} & \structure{\text{\ding{193}}} & \GREEN{x-\m{0}} & \GREEN{\to} & \GREEN{x} \\ \structure{\text{\ding{194}}} & \GREEN{x+\m{s}(y)} & \GREEN{\to} & \GREEN{\m{s}(x+y)} & \structure{\text{\ding{195}}} & \GREEN{x-\m{s}(y)} & \GREEN{\to} & \GREEN{\m{p}(x-y)} \\ \structure{\text{\ding{196}}} & \GREEN{\m{p}(\m{s}(x))} & \GREEN{\to} & \GREEN{x} & \structure{\text{\ding{197}}} & \GREEN{\m{s}(\m{p}(x))} & \GREEN{\to} & \GREEN{x} \\ \onslide<7->{\alert{\text{\ding{198}}}} & \onslide<7->{\alert{\m{s}(x+\m{p}(y))}} & \onslide<7->{\alert{\to}} & \onslide<7->{\alert{x+y}} & \onslide<8->{\alert{\text{\ding{199}}}} & \onslide<8->{\alert{\m{p}(x-\m{p}(y))}} & \onslide<8->{\alert{\to}} & \onslide<8->{\alert{x-y}} \end{array} \] \vspace{-3ex} \begin{itemize} \item<2-> \makebox[2cm][l]{SN \only<2>{?}} \onslide<3-> (e.g.) LPO with precedence $\GREEN{+} > \mG{s}$ and $\GREEN{-} > \mG{p}$ \item<4-> \makebox[2cm][l]{WCR ?} \onslide<5-> 4 critical pairs \vspace{-1ex} \begin{tabbing} \begin{tikzpicture}[on grid,baseline=(1).baseline,node distance=17mm] \node (1) {\MG{\ol{x+\ul{\m{s}(\m{p}(y))}}}}; \node (2) [below left=of 1] {\MGr{x+y}}; \node (3) [below right=of 1] {\MGl{\m{s}(x+\m{p}(y))}}; \draw[->] (1) -- (2) node[midway,left] {\ssd{197}}; \draw[->] (1) -- (3) node[midway,right] {\ssd{194}}; \onslide<7->{\alert{ \draw[->] (3) -- (2) node[midway,below] {\scriptsize \ding{198}}; }} \end{tikzpicture} \qquad\qquad\qquad \= \begin{tikzpicture}[on grid,baseline=(1).baseline,node distance=17mm] \node (1) {\MG{\ol{x-\ul{\m{s}(\m{p}(y))}}}}; \node (2) [below left=of 1] {\MGr{x-y}}; \node (3) [below right=of 1] {\MGl{\m{p}(x-\m{p}(y))}}; \draw[->] (1) -- (2) node[midway,left] {\ssd{197}}; \draw[->] (1) -- (3) node[midway,right] {\ssd{195}}; \onslide<8->{\alert{ \draw[->] (3) -- (2) node[midway,below] {\scriptsize \ding{199}}; }} \end{tikzpicture} \\[.5ex] \begin{tikzpicture}[on grid,baseline=(1).baseline,node distance=17mm] \node (1) {\MG{\ol{\m{p}(\ul{\m{s}(\m{p}(x))})}}}; \node (2) [below left=of 1] {\MGr{\m{p}(x)}}; \node (3) [below right=of 1] {\MGl{\m{p}(x)}}; \draw[->] (1) -- (2) node[midway,left] {\ssd{197}}; \draw[->] (1) -- (3) node[midway,right] {\ssd{196}}; \onslide<6->{ \path (2) -- (3) node[midway] {\strut $\alert<6>{=}$}; } \end{tikzpicture} \> \begin{tikzpicture}[on grid,baseline=(1).baseline,node distance=17mm] \node (1) {\MG{\ol{\m{s}(\ul{\m{p}(\m{s}(x))})}}}; \node (2) [below left=of 1] {\MGr{\m{s}(x)}}; \node (3) [below right=of 1] {\MGl{\m{s}(x)}}; \draw[->] (1) -- (2) node[midway,left] {\ssd{196}}; \draw[->] (1) -- (3) node[midway,right] {\ssd{197}}; \onslide<6->{ \path (2) -- (3) node[midway] {\strut $\alert<6>{=}$}; } \end{tikzpicture} \end{tabbing} \end{itemize} \end{example} \end{frame}