I have the language $L = \{ 0^{2l} 1^m | l,m >= 0 \} \ where \ \Sigma= \{0,1\} $ which I am trying to find the class of language for, e.g. not context-free, context-free, regular.
By this notion I assume $L$ is context-free and has some pumping length $p$, and I will use the pumping lemma.
$w = 0^{2p}1^p$ and can be split into $w = uv^ixy^iz, i >= 0.$
$|w| > p$
I know that $vxy$ can only be in 3 places as it is of length at most $p$, these scenarios are:
-all 0's
-A mix of 0's and 1's.
-All 1's.
My thinking is that scenario 1 is invalid as it must always be an even length, and pumping up may break this. Scenarios 2 I am unsure although I think it follows the same logic as the first scenario, and scenario 3 could be pumped and still be in the language.
Does this mean $L$ is context-free? Can I also infer if $L$ is regular?