After thinking for a bit, I am not able to prove a double inclusion proof for the following problem. It seems very interesting to me.
Consider the regular expression $r= ((1(00)^∗1 + 0)1)^∗$ and the right-linear grammar $G= ({S,A},{0,1},S,P)$, where $P$ consists of the following rules:
$Srightarrow 1A|01S|lambda$
$Arightarrow 00A|11S$
Prove that $L(G)subseteq L(r)$ and vice versa.
In general, how exactly do I prove that a regular grammar describes the same language as a regular expression?