According to wikipedia the right quotient of a regular language with ANY other language is regular. I have not been able to find a proof of this fact. All the sources talk about quotient with another regular language. Can somebody point me to a proof?
P.S. Definition from Wikipedia: The right quotient (or simply quotient) of a formal language $L_1$ with a formal language $L_2$ is the language consisting of strings $w$ such that $wx$ is in $L_1$ for some string $x$ in $L_2$. In symbols, we write:
$$L_1 / L_2 = \{w \ | \ \exists x ((x \in L_2) \land (wx \in L_1)) \}$$
In other words, each string in L1 / L2 is the prefix of a string wx in L1, with the remainder of the word being a string in L2.
Best Answer
You can find a proof in this book: Introduction to Automata Theory, Languages and Computation by Hopcroft and Ullman, 1979 edition.
The short and non-constructive proof appears on page 63.
Here is a snapshot (which I got from google books by searching for quotient in that book):
Only the last line of the proof is incomplete in the above snapshot, which reads:
Here $R$ is a regular language and $L$ is an arbitrary language.