# Reference request: proof that if $L \in DCFL$, then $L \Sigma^* \in DCFL$

So, it's fairly easy to prove that if $L \in DCFL$, then $L \Sigma^* \in DCFL$. Basically, you take the DPDA accepting $L$. You remove all transitions on final states, and then for each $a \in \Sigma$ and each final state $q$, you add a transition looping from $q$ to $q$ on $a$.

I'm using this in a paper, and I'd love to not have to actually prove this construction is valid. It's easy, but it's about a half-page long. Since DPDAs have been studied almost exhaustively, I was wondering, does anybody know of a paper that proves this property?

• For what's worth, this result holds for any $LR$ where $R$ is regular (see here). – sdcvvc Jul 10 '13 at 1:38
• That's awesome. You wouldn't know of a paper with that in it that I could cite, do you? – jmite Jul 10 '13 at 5:11