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\begin{frame}
  \small
  
  \begin{definition}
  \smallskip
  A property of TRSs is \alert{modular} if it is preserved under union.
  \smallskip
  \end{definition}
  
  \medskip
  
  \begin{remark}<2->
  \smallskip
  Without further restrictions `no' property of TRSs is modular:
  \onslide<3->
  \[
  \begin{tabular}{l@{\qquad\qquad}ll}
  termination &
  \GREEN{$\begin{array}{r@{~}c@{~}l} \m{a} & \to & \m{b} \end{array}$} &
  \BlUE{$\begin{array}{r@{~}c@{~}l} \m{b} & \to & \m{a} \end{array}$}
  \onslide<4->
  \\[.5ex]
  confluence &
  \GREEN{$\begin{array}{r@{~}c@{~}l} \m{a} & \to & \m{b} \end{array}$} &
  \BlUE{$\begin{array}{r@{~}c@{~}l} \m{a} & \to & \m{c} \end{array}$}
  \end{tabular}
  \]
  \end{remark}
  
  \medskip
  
  \begin{definition}<5->
  \smallskip
  A property $P$ is \alert{preserved under signature extension} if:
  \[
  (\FF,\RR) \vDash P \quad\Longrightarrow\quad
  (\FF \cup \GG,\RR) \vDash P
  \]
  for all TRSs $(\FF,\RR)$ and signatures $\GG$ with
  $\FF \cap \GG \neq \varnothing$.
  \smallskip
  \end{definition}
\end{frame}