When does $p$ break the pumping lemma

Question

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?

Answer 1

Let's start with stating the pumping lemma, in its contrapositive form:

Suppose that $L$ is a language such that for all $p>0$, the following holds. There exists a word $w \in L$ of length at least $p$ such that for every decomposition $w=xyz$ in which $|xy| \leq p$ and $y \neq \epsilon$, there exists $t \geq 0$ such that $xy^tz \notin L$. Then $L$ is not regular.

Now let us prove that your language is not regular. Given $p > 0$, let $w = a^pb^p \in L$, which certainly has length at least $p$. Given a decomposition $w = xyz$ with $|xy| \leq p$ and $y \neq \epsilon$, we see that $y = a^i$ for some $i \neq 0$. Therefore $xy^0z = a^{p-i}b^p \notin L$. Hence the pumping lemma applies, and shows that $L$ is not regular.