Language of CFG: $S \to aS | aSbS | \varepsilon$

Question

I'm trying to prove that the language L generated by the CFG $S \to aS | aSbS | \varepsilon$ is the language $L=\{ w \in \{a,b\}^*: \text{every prefix of $w$ has at least as many $a$'s as $b$'s} \}$.I proved the first inclusion $w \in L(G) \implies w \in L$. I can't prove the vice versa. By induction on the lenght of $w \in L$, I could say:

BASE: $|w|=0$, i.e. $w=\varepsilon$ and it's clear.

IND: suppose the thesis is true whenever $|w|<n+1$ and prove for a word of lenght $n+1$. Given $w \in L$, $w=a_1\dots a_{n+1}$ one has $a_1=a$ since $a_1$ is a prefix of $w$. Thus I can first apply the rule $S \to aS$ or $S \to aSbS$. But what now?

Any suggestion would be helpful. Thanks.

Answer 1

Let $w \in L$ be a word of length $n$. If $n = 0$ then clearly $w \in L(G)$, so suppose that $n > 0$. For $0 \leq i \leq n$, let $\delta_i$ be the difference between the number of $a$s and $b$s in the first $i$ characters of $w$; thus $\delta_i \geq 0$ for all $i$. We now consider two cases:

Case 1: $\delta_i > 0$ for all $i > 0$. In that case $w = ax$ for some $x \in L$. Induction tells us that $x \in L(G)$, and so $w \in L(G)$.

Case 2: $\delta_i = 0$ for some $i > 0$. In that case $w=axby$, where $|axb| = i$ and $x,y \in L$. Induction tells us that $x,y \in L(G)$, and so $w \in L(G)$.