Using mathematical induction ( implicitly ) with the inductive hypothesis being that the statement is false for all natural numbers less than or equal to " m ", one may conclude that the statement cannot be true for any natural number " n ".
12.
Substitute 1 for " x ", and watch the one on the right become exactly [ 1 ] ^ m and go to 0 for each case due to our inductive hypothesis and the fact that 0 to any power greater than 0 is 0, and watch the left one become 0 ^ { n-( u + 1 ) } ( n ) _ { u + 1 }.
13.
At all lower levels, it is impossible for a process to wait forever, since either another process will enter the waiting room, setting and " liberating "; or this never happens, but then all processes that are also in the waiting rooms must be at higher levels and by the inductive hypothesis, they will eventually finish the loop and reset their levels, so that for all, and again exits the loop.
14.
The point you might be missing ( which is basically the hint restated ) is that you don't want to evaluate what Q is if P = x ^ k; it suffices to show you can deal with x ^ k ( in much the same way as the grounding case of the induction ) and then the left overs are of less degree so can be absorbed into the rest of P and dealt with by Inductive Hypothesis.
