set of provable propositions is the smallest set of propositions satisfying these conditions. Finite subdivision rules Finite subdivision rules are a geometric form of recursion, which can be used to create fractal-like images. A subdivision rule starts with a collection of polygons labelled by finitely many labels, and then each polygon is subdivided into smaller labelled polygons in a way that depends only on the labels of the original polygon. This process can be iterated. The standard `middle thirds' technique for creating the Cantor set is a subdivision rule, as is barycentric subdivision. Functional recursion A function may be recursively defined in terms of itself. A familiar example is the Fibonacci number sequence: F(n) = F(n − 1) + F(n − 2). For such a definition to be useful, it must be reducible to non-recursively defined values: in this case F(0) = 0 and F(1) = 1. A famous recursive function is the Ackermann function, which, unlike the Fibonacci sequence, cannot be expressed without recursion. Proofs involving recursive definitions Applying the standard technique of proof by cases to recursively defined sets or functions, as in the preceding sections, yields structural induction — a powerful generalization of mathematical induction widely used to derive proofs in mathematical logic and computer science. Recursive optimization Dynamic programming is an approach to optimization that restates a multiperiod or multistep optimization problem in recursive form. 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). The recursion theorem In set theory, this is a theorem guaranteeing that recursively defined functions exist. Given a set , an element of and a function , the theorem states that there is a unique function (where denotes the set of natural numbers including zero) such that for any natural number . Proof of uniqueness Take two functions and such that: where is an element of . It can be proved by mathematical induction that for all natural numbers : Base Case: so the equality holds for . Inductive Step: Suppose for some Then . Hence implies . By induction, for all . In computer science A common method of simplification is to divide a problem into subproblems of the same type. As a computer programming technique, this is called divide and conquer and