I have the following language:
Let $A = \{a,e,i,o,u\} , S = \{l\}, \Sigma = A \cup S$. Then $L \in \Sigma^*$ is defined by the rules:
$\alpha \in A \Rightarrow \alpha \in L$
$X_1, \dots, X_n \in L \Rightarrow lX_1 \dots X_nl \in L$
A string in $\Sigma^*$ is in $L$ if and only if it can be obtained by finitely many applications of the previous rules
I'm asked to prove or disprove (I'm trying to prove it since I'm convinced it holds) that this language has a unique writing for words, that is, if a word in $L$ has length greater than $1$, then there is a unique construction chain (up to permutation) for it, In other words, if
$lX_1\dots X_nl = lY_1 \dots Y_kl$ then $n = k$ and for all $i$, $X_i = Y_i$
My strategy for proving this was:
Prove that no initial section of a word (except for the word itself) can be a word. Then using a property that says that since $X_1$ and $Y_1$ are initial segments of the same expression I must have (without loss of generality) $Y_1 = X_1E$ for some expression $E$. If $E$ is not empty, then $X_1$ can't be a word since it is a nontrivial initial segment of the word $Y_1$. Then inductively it follows that all words on both sides are equal
However, I'm struggling to prove this. Any help is appreciated