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When using the pumping lemma for a context free language, if I write any $w \in L$ as $uvxyz$, is my goal to show that a string will not pump for ANY arrangement of $uvxyz$ that I choose, or is my goal to show that THERE EXISTS some arrangement of $uvxyz$ over $w$ that will pump?

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The pumping lemma claims that if the language is context-free, then there exists a configuration such that you can pump the string. Therefore, to prove that the language isn't context-free, you must show that pumping fails for every possible arrangement.

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