# How are we sure choose is going to halt? question regarding a section from Elaine Rich's Automata, Computability and Complexity book?

I have a problem regarding the choose algorithm, I provide the algorithm's definition and it's use in the book,I attached a picture for the Illustration and for how the algorithm is used.
choose (x from S: P(x)) is defined as follows:

• Return some element x of S such that P(x) halts with a value other than False, if there is one.
• If there is no such element, then choose will:
• Halt and return False if it can be determined that, for all elements x of S, P(x) is not satisfied. This will happen if S is finite and there is a procedure for checking P that always halts. It may also happen, even if S is infinite, if there is some way, short of checking all the elements, to determine that no elements that satisfy P exist.
• Fail to halt if there is no mechanism for determining that no elements of S that satisfy P exist. This may happen either because S is infinite or because there is no algorithm, guaranteed to halt on all inputs, that checks for P and returns False when necessary.

How does choose "picks one with the property that, once it has been appended to position-list, solve-15 can continue and find a solution"?, do we code that in, or do we accept any legal position ? if the latter is true, what's to guarantee that it will ever arrive to a solution, what if it gets stuck in loop where it does a legal action and then undoes it, thus getting stuck.