\begin{frame}
  \frametitle{Why do we need the conditions?}
  
  \begin{example}[$>$ not well-founded]
  \smallskip
  Let $R = \{\; {\rm f}(x) \to {\rm f}({\rm f}(x)) \;\}$ with the $\Sigma$-algebra $(\mathbb{Z},\interpret{\cdot})$ and
  \begin{align*}
    \interpret{{\rm f}}(x) &= x - 1
  \end{align*}
  
  and $>$ as usual on $\mathbb{Z}$.\\[.5em]

  \pause  
  Then $R$ is not terminating
  $${\rm f}(x) \to {\rm f}({\rm f}(x)) \to {\rm f}({\rm f}({\rm f}(x))) \to \ldots$$
  \pause  
  but 
  \begin{itemize}
    \item $\interpret{f}$ is monotone, and
    \item $\interpret{{\rm f}(x),\alpha} = \alpha(x)-1 > \alpha(x)-2 = \interpret{{\rm f}({\rm f}(x)),\alpha}$.
  \end{itemize}
  \smallskip
  \pause
  Hence $>$ needs to be well-founded!
  \smallskip
  \end{example}
\end{frame}