In you need to give either a context$-$free grammar or a PDA. Make sure to include a sentence explaining why the language is context$-$free $($i$.$e$."$L is context$-$free because $...").$

In $19$c you do NOT need to write the whole proof $-$ you can simply $($i$)$ specify the resulting language and $($ii$)$ in one sentence refer to the same$/$similar proof that was done during lecture or in a previous exercise in this homework.

$19.$ a$)$ Prove that the language $L_1=\{a^i{b}^i{c}^j{d}^j|i,j0\}$ is context$-$free.

b$)$ Prove that the language $L_2=\{a^j{b}^i{c}^i{d}^k|i,j,k0\}$ is context$-$free.

c$)$ Prove that $L_1{L}_2$ is not context$-$free.