# Pumping lemma for two words that “completely different”

Let $"x"$ and $"y"$ be a words, we will say that two words are "completely different" if for all $1\leq i\leq |x|$ the $i$ letter in $x$ diffrent from the $i$ letter in $y$.

Prove that the language $\mathscr{L}=\{xy \mid x,y\in \{0,1\}^*,|x|=|y|,x,y \text{ completely different} \}$ is not a free-context-language

I need some hints please about choosing the word for applaying the pumping lemma, I tried $0^n1^n$ but it did not work well

If $p$ is the proposed pumping length, try the word $0^p1^p0^p1^p0^p1^p$. Note that if it is decomposed as $uvwxy$ as in the Pumping Lemma, then as $| vwx | \leq p$ this substring is either part of a $0^p$ or part of a $1^p$ or straddles a $0^p1^p$ or straddles a $1^p0^p$. Argue based on positioning that pumping zero times results in a string not in the language.