 it returns false if it is not the case. After that it checks whether $L_0(s^\prime)$ can be evaluated true at any state $s^\prime$ such that the system can reach $s^\prime$ from $s_0$ through an environment transition. It returns true if it is the case. Otherwise, it chooses one candidate from the set of all possible input symbols from $s_0$, and feeds the sequence $\pi$ concatenated with the input symbol to $X$. If the output symbol of $X$ and the input symbol matches the annotation of an edge originating from the node, it moves forward to try the destination node of the edge with 