Given a language $L$ over the alphabet { 0
, 1
, [
, ,
, ]
} where if $x \in L$ then $x$ is of the form $[x_0, \dots, x_n]$, in which each of the $x_i$ represents unique binary strings (so $x_i = x_j$ only when $i=j$). Prove that this is not a regular language.
Here is what I have. I have used the pumping lemma.
Suppose $L$ is a regular language. Let $M$ be the DFA that accepts $L$. Let $p$ be the pumping length of $L$. Given $x = [x_0,\dots,x_p] \in L$ we see that $|x| > p$ so pumping lemma applies, so $x = UVW$ where $|UV| < p$, $|v| > 0$, and $UV^iW \in L$.
Now if we have $UVVW$ that means $V$ is repeating some of the characters in $x$ and therefore it must repeat some $x_i$. Therefore we have reached a contradiction, which shows that $L$ is not a regular language.
Is this a correct proof? If not, what can I do to improve it?