Similar to regular languages, CFLs are closed under certain properties.

For regular languages, we developed a pumping lemma that can help us prove that a language is not regular. While we can't use the same pumping lemma for CFLs, we will see that there is a similar argument to be made that will lead to a CFL pumping lemma that we can make use of to prove certain languages are not CFL.

Prove that $L = \{a^nb^nc^n: n\ge 1\}$ is not a CFL.

Prove that $L = \{a^nb^nc^p : p > n > 0\}$ is not a CFL.

Prove that $L = \{a^jb^k: k = j^2\}$ is not a CFL.

Prove that the following language is not a CFL:
$L = \{w{\bar w}w : w\in \Sigma^*\},\ \Sigma = \{a, b\}$, where $\bar w$ is the string $w$ with each occurence of $a$ replaced by $b$ and each occurence of $b$ replaced by $a$.

Prove that $L = \{ a^{2n}b^{2m}c^nd^m : n,m \ge 0 \}$ is not a CFL