Prove that a language is not context free

I was solving some hard exercises on context free grammer.

Consider the language L={w∈{a,b}^{*} :the length of the longest substring of all b’s in w is longer than any of the length of substring of just a’s in w}.

Prove that L is not context free.

I have tried this problem using pumping lemma and closure properties. How to go about it? Any help would be appreciated.

• Got it. Used pumping lemma on a^mb^{m+1}c^m. – user3523469 Oct 21 '17 at 2:19

Let $p$ be the pumping length. Then $b^pa^{p+1}\in L$ and also $b^{r}a^{p+1}$ for many $r>p$, contradiction.
Let $$p$$ be the pumping length. Then $$а^{p}b^{p+1}a^{p}\in L$$. $$w = xyzuv. |yu|>=1 |yzu|<= p$$.
If yzu is only of 1 letter it's easy to see that $$xy^{0}zu^{0}v$$ it's not in L.
If yzu consist of 2 letters , then $$xy^{0}zu^{0}v = а^{p-r}b^{p+1-t}a^{p}$$ or $$а^{p}b^{p+1-t}a^{p-r}$$ and since $$p>=p+1-t$$ the string obviously are not in L