I am trying to work through the pumping lemma for CFLs.
$L_1 = \{0^n 1^{mn} : n,m \in \Bbb N\}$
I am trying to find a contradiction. I have currently chosen $z= 0^p1^{2p}$ to be my string. Then $|z| \geq p$ where $p$ is the pumping length.
I know $z = uvwxy$. So that $|vwx| \leq p$ and $|vx| \geq 1$.
Case 1. $vwx$ is all 0s.
This would result in many more 0s being added and an i can be chosen such that $uv^iwx^iy$ results in too many zeros. $0^{p+j+k}1^{2p}$
Case 2. $vwx$ is all 1s. In this case chose $i=0$, so that the number of 1s decrease by at least 1. This would no longer be divisible into $n$ and $m$.
case 3.
It is a mix of 1s and 0s. I am not sure how to proceed here.
I have been trying to do a variety of types of CFL proofs but, I seem to get muddled up in the same spot every time. Some advice or point in the right direction would be appreciated!