The proof of Lemma $2.41$ says that $(q, x)$ is a looping situation for a $DPDA \:P$ if when $P$ is started in state $q$ with $x \in \Gamma$ on the top of the stack, it never pops anything below $x$ and it never reads an input symbol. Show that $F$ is decidable, where $F = \{ \langle P, q, x \rangle \mid (q, x)\: \text{is a looping situation for P}\}$.