Use pumping lemma to show that the following language is not

Use pumping lemma to show that the following language is not regular

L = {a²ⁿbⁿ | n > 0}

Solution

Assuming regularity of L and using the Pumping Lemma, we have a2p=xyza2p=xyz where:

a) y>0y>0

b) |xy|p|xy|p

c) xyizL i0xyizL i0

(also |xyz| =2pp|xyz| =2pp)

(notice both x and z can be empty)

I choose i=2i=2 to get xy2zxy2z so (since y>0y>0) |xy2z|>2p|xy2z|>2p

I understand that the next step is trying to prove that |xy2z|<2p+1|xy2z|<2p+1 so that the final result is 2p<|xy2z|<2p+12p<|xy2z|<2p+1 and so xy2zxy2z cannot be an element of L.

However if |y|=p|y|=p and so |x|=0, |z|=0|x|=0, |z|=0 then this is not possible as taking y2y2 is just doubling the length of the word which makes another word that fits the language.