We define the language $L = \{cw_1cw_2c| w_1,w_2 \in \{a,b\}^*, \text{num of a's in }w_1 = 2|w_2|\}$
We are asked to show this language is irregular using pumping lemma.
I know we gotta look at the word $z = ca^{2n}cb^nc$ or something to that effect and see that the three conditions of the pumping lemma don't hold, but honestly its very new material for me and im just trying to help my girlfriend - i'm more on the mathematics side of things, not really theoretical computer science.
Any help would be appreciated.
First, language $L' = ca^*ca^*c$ is obviously regular, so if $L$ was regular, so would be their intersection $K = L' \cap L = \{c w_1 c w_2 c: w_1, w_2 \in {a}^*, |w_1| = 2|w_2| \}$. It's enough now to prove that $K$ is irregular, and it's simpler because we don't have to worry about those pesky $b$s.
Then, the pumping lemma says that words long enough can be decomposed as $uvw$ so that $uv^nw$ also belongs to $K$ for any natural $n$. Since all words of $K$ have exactly 3 $c$'s, in our decomposition $v$ cannot have any $c$, otherwise after pumping we'd have more $c$ than available. It means then that $v$ must fall into either first segment of $a$s or into the second, but then pumping it will make it too long, as the length of one segment is fully determined by the length of the other. Thus $K$ cannot be pumped, and so it's not regular.