Let $G_r=$ be a right$-$linear regular grammar. We want to convert it to an equivalent left$-$linear regular grammar $G_l.$ Naturally, we need $L(G_l)=L(G_r).$

$3+5+2+4=14$

$($a$)$ Note that just flipping the terminal and non$-$terminal of a production $AaB$ to $ABa$ may not work. For example, for $G_r-=\{AaB,BaB|bB|lon\}$ show that $Ba|Bb|lon$ is not the corresponding left$-$linear grammar $($as is misguidedly claimed at: How to convert right linear grammar to left linear grammar?$).$ That is$,L(G_l)L(G_r).$

$($b$)$ Design a process to convert a $G_r$ to an equivalent $G_l.$

$($c$)$ Illustrate your process with the grammar $G_r$ in $Q(2$a$).$

$($d$)$ Prove the correctness of your suggested process in Q $(2$b$).$