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Assume $ \Sigma=\{0,1\}$, is $L$ a regular language? If it is not, how should we prove it with pumping lemma?

$$L = \{1^{a_1} 0^{a_2}\ldots 01^{a_k} \mid k \in \mathbb N , a_i \geq 0 , \text{ the $a_i$ are all different}\} $$

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    $\begingroup$ Should the last $0$ in $L = \{1^{a_1} 0^{a_2}...01^{a_k}|\ ... \}$ be $ 0^{a_k-1}$? $\endgroup$ – Thumbnail Apr 29 at 15:10
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The definition of your language is not completely clear, but whatever it is, if you consider $$ L \cap 1^* 0^* = \{ 1^a 0^b \mid a \neq b \} $$ then you immediately see that $L$ is not regular.

If you have to prove it using the pumping lemma, then simply take a proof that works for the above language; it will work for $L$ as well.

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