Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In computer science, a loop variant is a mathematical function defined on the state space of a computer program whose value is monotonically decreased with respect to a well-founded relation by the iteration of a while loop under some invariant conditions, thereby ensuring its termination. A loop variant whose range is restricted to the non-negative integers is also known as a bound function, because in this case it provides a trivial upper bound on the number of iterations of a loop before it terminates. In general, however, the smallest set sufficient to represent the range of any loop variant that might be useful in a termination proof is an ordinal number at least ¿1CK, but no more than ¿1. The existence of a variant proves the termination of a while loop in a computer program by well-founded descent. A basic property of a well-founded relation is that it has no infinite descending chains. Therefore a loop possessing a variant will terminate after a finite number of iterations, as long as its body terminates each time.