\begin{frame} \begin{remark} \begin{itemize} \item \makebox[4cm][l]{$\SNi \not\Rightarrow \SN \onslide<2->{ \vee \WN}$} \GREEN{$a \to c(a)$}\\ \makebox[4cm][l]{} \GREEN{Here, $a \ired c^\omega$ which is a normal form.} \pause\pause \item \makebox[4cm][l]{$\SN \mathrel{\onslide<4->{\not\Rightarrow}} \SNi \onslide<5->{\vee \WNi}$} \pause \GREEN{$I(x) \to x$}\\ \makebox[4cm][l]{} \GREEN{Here, $I(I(I(\ldots)))$ rewrites only to itself.} \pause\pause \item \makebox[4cm][l]{$\CRi \mathrel{\onslide<7->{\not\Rightarrow}} \CR$} \pause \GREEN{$a\to b,\; a\to c,\; b \to d(b),\; c \to d(c)$} \makebox[4cm][l]{} \GREEN{Here, $\neg (b \join c)$, but $b \ired d^\omega \iredi c$.} \pause \item \makebox[4cm][l]{$\CR \mathrel{\onslide<9->{\not\Rightarrow}} \CRi$} \pause \GREEN{$A(x) \to x,\; B(x) \to x$}\\ \makebox[4cm][l]{} \GREEN{Here, $A^\omega \iredi (AB)^\omega \ired B^\omega$.} \end{itemize} \end{remark} \pause \begin{remark} The example $A(x) \to x,\; B(x) \to x$ shows: not every orthogonal TRSs is $\CRi$. \pause\medskip Even one collapsing rule is sufficient to violate $\CRi$.\\ \pause Take $R = \{\;f(x,y) \to y\;\}$. Then \begin{align*} f(x,f(x,f(x,\ldots))) \iredi f(x,f(y,f(x,f(y,\ldots)))) \ired f(y,f(y,f(y,\ldots))) \end{align*} \end{remark} \end{frame}