# Prove that $L=\{a^n b^l : n \leq l\}$ is not regular by pumping lemma

I'm currently trying to prove that $$L=\{a^n b^l : n \leq l\}$$ is not regular by pumping lemma

My proof:

If we choose $$w$$ such that $$w=a^P b^P$$, then since $$|xy| \leq p$$, $$y$$ must be $$a^P$$, meaning it can be pumped any number of times $$i$$, such that $$a^{iP} b^P \in L$$. Since $$iP \leq P$$ is not true for any $$i > 1$$, it cannot exist in $$L$$, therefore $$L$$ cannot be regular.

Is this a valid proof? I don't fully comprehend the pumping lemma, so please let me know if/where I may be going wrong.

Thank you

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– D.W.
Apr 11 at 2:32
• Crosspost with this question on math.stackexchange. Apr 12 at 5:59

• what is $$P$$? If it is the pumping length, then say it so;
• what is $$p$$? Is it a typo and should be a $$P$$?
• what are $$x$$ and $$y$$? Don't hesitate to remind the hypotheses of the pumping lemma;
• why would $$y$$ be $$a^P$$? The only conclusion you can make is that $$y = a^k$$ for $$1\leqslant k \leqslant P$$.