Infinite regular languages

Question

There is a formal proof for the following sentence?

For every 2 languages $A,B$, we write A@B if A subset of B and B\A infinite. Prove that if $A,B$ regular languages and A@b, than exists regular language $C$ such that A@C@B.

thanks.

Answer 1

If $A$ and $B$ are regular languages, so is $B\setminus A$, so the problem really boils down to showing that if $L$ is an infinite regular language, there is an infinite regular language $L'@L$.

HINT: Let $p$ be the pumping length for $L$, and let $w\in L$ be any word of length at least $p$. Decompose $w$ as $w=xyz$, where $|xy|\le p$, $|y|\ge 1$, and $xy^kz\in L$ for each $k\ge 0$. Take $L'$ to be a suitable subset of $\{xy^kz:k\ge 0\}$.