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\begin{frame}{Pumping Lemma for Context-Free Languages (1961)}
  \begin{block}{}
    Let $w \in L$ with $|w| \geq m$. 
    Consider a derivation tree for $w$.
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    There must be a path of length $> k$. Consider the longest path. 
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    As there are only $k$ variables, there must be 
    a variable $A$ that occurs twice among the last $k+1$ variable nodes of the path.
    \medskip
    
    \begin{minipage}{.39\textwidth}
    \begin{center}
      \input{pomp.pdf_t}
    \end{center}
    \end{minipage}  
    \begin{minipage}{.59\textwidth}
      \pause
      We have $w=uvxyz$ with
      \begin{itemize}
        \item \alert{$S \Rightarrow^* uAz$} 
        \item \alert{$A \Rightarrow^+ vAy$}
        \item \alert{$A \Rightarrow^+ x$}
      \end{itemize}
      \pause
      Hence
      \begin{itemize}
        \item \alert{$S \Rightarrow^+ uv^ixy^iz$} for every $i\geq 0$.
      \end{itemize}
    \end{minipage}
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    Then 
    \begin{itemize}
      \item $\alert{|vxy| \leq m}$ as the subtree generating $vxy$ has depth $\leq k+1$,
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      \item $\alert{|vy| \geq 1}$, since there are no $\lambda$ and unit productions.\hfill \qed
    \end{itemize}
  \end{block}  
\end{frame}