\begin{frame}{Using the Pumping Lemma} \begin{alertblock}{Attention} A contradiction for specific $m$, $x$, $y$, or $z$ is \alert{not sufficient}! \end{alertblock} \pause \begin{goal}{} Pumping property as formula (\alert{note the quantifiers}): \begin{talign} &\exists m>0.\\[-.5ex] &\qquad \forall w\in L \text{ with } |w|\geq m.\\[-.5ex] &\qquad\qquad \exists x,y,z \text{ with } w=xyz,\; |xy| \leq m,\; |y|\geq 1.\\[-.5ex] &\qquad\qquad\qquad \forall i \geq 0.\, xy^iz\in L \end{talign} \end{goal} \pause \begin{goal}{} To \emph{contradict the pumping property}, we prove the negation: \begin{talign} &\alert{\forall} m>0.\\[-.5ex] &\qquad \alert{\exists} w\in L \text{ with } |w|\geq m.\\[-.5ex] &\qquad\qquad \alert{\forall} x,y,z \text{ with } w=xyz,\; |xy| \leq m,\; |y|\geq 1.\\[-.5ex] &\qquad\qquad\qquad \alert{\exists} i \geq 0.\, xy^iz \alert{\not\in} L \end{talign} \end{goal} \end{frame}