 The key result in dynamic programming is the Bellman equation, which writes the value of the optimization problem at an earlier time (or earlier step) in terms of its value at a later time (or later step).In set theory, this is a theorem guaranteeing that recursively defined functions exist.Given a set "X", an element "a" of "X" and a function formula_7, the theorem states that there is a unique function formula_8 (where formula_4 denotes the set of natural numbers including zero) such that
for any natural number "n".