$2.$ Construct CFGs to prove that the languages below are context$-$free:

$($a$)$ L$1=\{0^$i$1^$j $|$ j $<=$ i $<=2$j$\}$

$($b$)$ L$2=\{0$i$1$j $|$ i $<$ j or i $>2$j$\}$

$($c$)$ L$3=\{$a$^$w b$^$xc$^$y d$^$z e$^$w $|$ w$,$ x$,$ y$,$ z $>=0$ x $+$ z $=$ y$\}$

$($d$)$ L$4=$ L$1,$ where L$1=\{0^$i$1^$j $|$ j $<=$ i $<=2$j$\}$

Hint: a regular expression for $01$ is $01+0(0+1)$