Given is the the following language.
$L= \{ wa^{\vert w \vert} \mid w \in \{a,b\}^* \cup L(b^*a^*)\} $
Task: Prove that $L$ is not regular using pumping lemma.
I am not sure wether I did this correctly. Also I struggle to find the right often failing the first attempt. However I only have arround 15min to complete this prove. Any advice and feedback on how to be more efficient on a pumping lemma proves is greatly appreciated.
Here is my sollution:
Suppose that $w=a^nb^na^{2n}$. As you can see $w\in L$ and $\vert w \vert \ge n.$
Decompositions $w=xyz$
1: $x=a^c$ $y=a^d$ $z=a^eb^n a^{2n}$
Suppose $i=o$ with $w=a^c(a^d)^i a^eb^n a^{2n}$
Since $c+d+e=n$ and $d \ge1$ $ \Rightarrow \vert a^c a^eb^n \vert \lt 2n$.
Therefore $w \notin L$ which means L is not regular.
Why did I choose this word:
My thought process was to choose a word that is not in $L(b^*a^*)$ as this part is regular and I would not be able to produce a word that is not in L. But at the same time I needed a word that is easy to decompose. This is what I struggle with I often need to make a case distinction to have all the possible decompostions for my word. I started to make my words longer as this reduces the possible decompositions.
I apologize if this was hard to read my english is bad when it comes to math.
Is this prove sufficient?