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My first post here, so glad I found this great place. Hoping I could improve and learn a lot from you and contribute in the future if I can.

I have a problem with the following scenario:

Given $\Sigma= \{a,b,c,d\}$, prove that $L$ is not regular, where $$L = \{a^ib^jcd^k \mid i \geq 0; k > j > 0\}$$ using: 1) pumping lemma 2) closure properties.

Regarding 1)

Assuming that $L$ is regular, using the pumping lemma there should exist some integer $n$ (pumping length). Choosing a word $w \in L$ (not sure if I chose the right word). If $w=a^ib^jcd^k$ ($|w|>n$), but I cannot find a way to obtain $xy^i z \notin L$, would very appreciate an explanation or example of how you found the term so it can be concluded that $L$ is not regular using the pumping lemma.

Regarding 2)

Assuming that $L$ is regular, it means that $a^ib^jcd^k$ is also regular. So it should be closed under intersection, however, the form of $\{a^ib^jcd^k \mid i\geq 0;k>j>0\}$ is not regular, and it can be achieved using completion, so contradiction and because of that $L$ is not regular.

Hoping to learn from my mistakes and improve.

Thank you very much for your aid.

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For (1): You assume the language satisfies the pumping lemma for some $n$. You want to choose a word that would have at least one of its letters related to $n$. A good start would be taking one of your letters $l\in w$ to be $l^n$, and using that to show that for some $i$, $xy^iz\notin L$

For (2): You want to show that if $L$ is regular, then some other language $L_{not}$ which you know is not regular, is the intersection of another regular language $L_{reg}\cap L=L_{not}$.

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