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I was thinking about whether $\{ 0^{a}10^{a}10^{a}|a\in\mathbb{N} \}$ is a context-free language, and I found this post. I am not sure if my understanding is correct or not, but I guess $R = \{ (a,1,a,1,a)|a\in\mathbb{N} \}$ is not a stratified set, so this language is not context-free?

Before looking this up, I have been wondering if $L =\{ 0^{a}10^{b}10^{c} |(a,b,c)\in R\}$ is context free, and my intuition is that for $(a_{0},b_{0},c_{0}) \in R$, if we modify $a_{0}$ to $a_{1} \neq a_{0}$, we must modify both $b_{0}$ and $c_{0}$ to $b_{1}\neq b_{0}$, and $c_{1}\neq c_{0}$, so that $(a_{1},b_{1},c_{1}) \in R$, then it is not context free. In other words, if we change one of a,b,c we must change the other two (not only one), for the changed 3-tuple to be in $R$, then it must not be context-free. Is my intuition correct?

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