\begin{frame}{Exercises (1)}

  \begin{exampleblock}{}
    Show that $L = \{\, a^nb^n \mid n\geq 0 \,\}$ has the pumping property.
    \pause\medskip
    
    Let $m =2$. \pause Every word $w = a^n b^n$ with $|w| \ge m$ can be split
    \begin{talign}
      a^n b^n &= uvxyz & 
      u &= a^{n-1} &
      v &= a &
      x &= \lambda &
      y &= b &
      z &= b^{n-1}
    \end{talign}
    \pause
    We have $|vxy| \le m$, $|vy| \ge 1$\pause and 
    \begin{talign}
      uv^ixy^iz = a^{n-1+i} b^{n-1+i} \in L
    \end{talign}
    for every $i \ge 0$.
    \pause
    Thus the language has the pumping property.
  \end{exampleblock}
\end{frame}