This Demonstration shows a definition by regressive recursion. The domain of the functions considered is the set of non-negative integers. Given an increasing function , define the predicate . Two other functions and are also given. The new function is defined by: if , then ; otherwise, .
If , and are primitive recursive functions, is also a primitive recursive function .
 I. Hafner, "Regressive Recursion," Mathematica Balkanica, 6(13), 1976 pp. 75–77.