# Prove the following language is not regular using the Pumping Lemma for Regular Languages

I am trying to use the Pumping Lemma to prove the language $$L=\{a^nb^mc^md^n\}$$ is not regular. However, I am having trouble when selecting the values of x, y, and z to show that xyz is contained within the language (before proceeding to show that it is not contained in the language when selecting a value of i for v, as uvw = y, where |v| > 0.)

Any ideas?

Thank you.

• What values can $n$ and $m$ take? – angryavian May 7 at 18:31

Suppose for sake of contradiction that $$L$$ is regular, and let $$p$$ be the pumping length.
Consider the word $$a^p bc d^p$$. If we write this word in the form $$xyz$$ such that $$|xy| \le p$$ and $$|y| > 0$$, then $$y=a^k$$ for some $$0 < k \le p$$. But then $$xy^2z = a^{p+k} bc d^p \notin L$$.