\begin{frame}
  \frametitle{Transfinite Reductions}
  
  \begin{example}
  \vspace{-2ex}
  \begin{align*}
    f(x,x) &\to f(a,b)\\
    a &\to c(a)\\
    b &\to c(b)
  \end{align*}
  \pause
  A reduction of length $\omega\cdot 2 + 1$:
  \begin{align*}
    f(a,b) &\pause\to^\omega f(c^\omega,b) \pause\to^\omega f(c^\omega,c^\omega) \pause\to f(a,b)
  \end{align*}
  \pause
  A reduction of length $\omega + 1$:
  \begin{align*}
    f(a,b) &\pause\to^\omega f(c^\omega,c^\omega) \pause\to f(a,b)
  \end{align*}
  \pause
  by alternating $f(a,b) \to f(c(a),b) \to f(c(a),c(b)) \to \ldots$.
  \end{example}
\end{frame}