I want to prove that $\{A^iB^jC^k \mid i=j \text{ or } j=k\}$ is a not a regular language using the pumping lemma.
I've found that the only way to obtain a contradiction is when $x \in A^*$, $y \in B^*$, $z \in C^*$, as $y$ is unable to be pumped up and still fit in the language. However if I have any combination of $y \in B^*C^*$ or $y \in A^*B^*$ and either $x$ or $z$ is empty, it appears as if $y$ can be pumped and thus this would be a regular language. The only way I can think that this would violate the rules of pumping lemma is if $|xy| ≤ P$. Assuming that $P$ is $2$ (which may be completely wrong), then the above still wouldn't violate the pumping lemma and thus this would be a regular language, right?