\begin{frame}{Exercise (1)}
  \begin{exampleblock}{}
    Use the pumping lemma to \emph{show that} 
    \begin{talign}
      L = \{\,a^nb^n\mid n\geq 0\,\}
    \end{talign}
    \emph{is not regular}.
    \pause
    Assume that $L$ was regular.
    \pause\medskip
    
    By the pumping lemma there exists $m>0$ such that
    \begin{talign}
      \mpause[1]{a^m b^m} = xyz
    \end{talign}
    with $|xy| \leq m$, $|y| \geq 1$, and $xy^i z \in L$ for every $i \geq 0$.
    \pause\pause\medskip
    
    Since $|xy| \leq m$ and $|y|\geq 1$, it follows that
    \begin{talign}
      x=a^{\,j} &&\text{and}&& y = a^k
    \end{talign}
    with $j\geq 0$ and $k\geq 1$.
    \pause\medskip
    
    However $xy^0z = xz = a^{m-k} b^m \not\in L$. \pause Contradiction!
    \pause\medskip
    
    Thus $L$ is not regular.\qed
  \end{exampleblock}  
\end{frame}