# How to prove that a language is not context-free using pumping lemma

I'm trying to prove that that language isn't a context free:

$$L = \{ w11w \mid w\in \Sigma^* = \{0,1\}\}$$

I succeed to prove that $$L = ww$$ isn't context free, but not the language above. What am I doing wrong?

## 1 Answer

Here is the simplest way. Suppose that your language were context-free, and consider its intersection with $$0^*(10)^*110^*(10)^*$$. Applying further simple manipulations, you reach the language $$\{a^nb^mc^nb^m : n,m \geq 0\}$$, which is known not to be context-free.

If you insist on using the pumping lemma, perhaps you can get inspiration from this argument.

• Not sure how this inspires arguments using the pumping lemma, but at least it proves the statement. – D. Ben Knoble May 18 '19 at 18:07
• It could suggest which word to pump. – Yuval Filmus May 18 '19 at 18:08