# Searching for a proof for a variant of the pumping lemma for context free languages

So I'm trying to understand the pumping lemma for CFL ( context free languages ).I've already used it to show that a language is not contextfree and I have considered the proof of this lemma (see the PDF below ) Now I've read that there is a variant of the pumping lemma for context free languages. You replace the condition " $$vy \neq \varepsilon$$ " with " $$v$$ and $$y$$ are not $$\varepsilon$$". Like I've said. Here is the proof of the"normal" pumping lemma for CFL.

• However, my first idea was this one: Take a larger constant that guarantees the existence of two non-interleaving loops in the derivation. If one of them produces pump factors on either side, take it for the proof of the lemma as before. If both produce non-terminals only on one side, take these two factors for the pumping. Here not just all $uv^iwx^iy$ are in the language, but all $uv^iwx^ky$; but the latter of course implies the former. So this kind of pumping is maybe not in the original spirit, but it still works. – Peter Leupold Nov 20 '18 at 10:56
• The constant might be significantly bigger and more complicated to calculate for a given grammar. More importantly, you loose the condition $|vwx|<c$ for the constant c, because the two loops might be far apart. Probably you can guarantee that one loop is below the other - but this makes the calculation of the constant even more complicated. – Peter Leupold Nov 20 '18 at 10:57