# How can I prove given language is not regular?

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.

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 $$L_{reg}\cap L=L_{not}$$.