\begin{frame} \small \begin{definition} A \alert{one-step} strategy maps every reducible term to a set of one-step reductions. \end{definition} \begin{example} There exists no fair one-step strategy for $\atrs = \{I(x) \to I(x)\}$. \medskip\pause For the term: \begin{align*} t = I(I(x)) \end{align*} there are only 3 possible mappings: \begin{itemize} \pause \item $\SS(t) = \{\alert{I}(I(x)) \to_{\varepsilon} I(I(x))\}$, \pause \item $\SS(t) = \{I(\alert{I}(x)) \to_{1} I(I(x))\}$, or \pause \item $\SS(t) = \{\alert{I}(I(x)) \to_{\varepsilon} I(I(x)), \; I(\alert{I}(x)) \to_1 I(I(x))\}$. \end{itemize} \pause None of these is fair as we can always continue to reduce the same occurrence of $I$. \end{example} \end{frame}