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\begin{frame} \frametitle{Why more than Cauchy-convergence?} We consider the TRS: \begin{align*} f(x,y) &\to f(y,x)\\ a &\to b \end{align*} \pause We start from $f(a,a)$ and trace the left occurrence of $a$: \pause \begin{align*} f(\overline{a},a) \to f(a,\overline{a}) \to f(\overline{a},a) \to f(a,\overline{a}) \to^\omega \alert{\ ?} \end{align*} \vspace{-2ex} The rewrite sequence without overlining is Cauchy-convergent. \medskip However, what are the residuals of the left $a$ after $\omega$-many steps? \pause\medskip Although it appears as if the term has a limit, this is only a syntactic accident.\\ The subterms get swapped all the time\ldots \end{frame}