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Well-founded relation R x". As an example, consider the well-founded relation (N, "S"), where N is the set of all natural numbers, and "S" is the graph of the successor function "x" → "x" + 1. Then induction on "S" is the usual mathematical induction, and recursion on "S" gives primitive recursion. If we consider the order relation (N, <), we obtain complete induction, and course-of-values recursion. The statement that (N, <) is well-founded is also known as the well-ordering principle. There are other interesting special cases of well-founded induction. When the well-founded relation is the usual ordering on the class of