\begin{frame}
  \begin{definition}
  We write $s \ired t$ if the rewrite sequence is strongly convergent and with limit $t$.
  \end{definition}
  \pause

  \begin{example}
  $R = \{\;a \to c(a)\;\}$. Then $a \ired c^\omega$.
  \end{example}
  \pause\bigskip
  
  \begin{lemma}
  Every proper prefix of a (even divergent) rewrite sequence is strongly convergent.
  \end{lemma}
  \pause
 
  \begin{example}
    $R = \{\; f(x,x) \to f(a,b),\; a \to c(a),\; b \to c(b \; \}$
    $$f(a,b) \to^{\omega\cdot 2 + 1} f(a,b) \to^{\omega\cdot 2 + 1} f(a,b) \to^{\omega\cdot 2 + 1} \ldots$$
    \ldots is a divergent rewrite sequence of length $\omega^2$.
    \pause
    %
    But every prefix is convergent!
  \end{example}
\end{frame}