Pumping Lemma - non regular

Question

Can everyone help me to show that:

the language

$$L = \{a,b\}^* \setminus \{a^m b^{2m} a^n\mid m,n \ge 0\}$$

is not regular.

I don't know what is the meaning for the proof.

Answer 1

HINT: If $L$ were regular, its complement would also be regular. (If you’ve not yet proved this, you should do so; it can be done quite easily in terms of DFAs.) Its complement is

$$\left\{a^mb^{2m}a^n:m,n\ge 0\right\}\;,$$

and it’s very straightforward to use the pumping lemma to prove that this is not regular. In fact, the argument is very similar to the example used in the Wikipedia article.