Definition:Recursively Defined Mapping/Natural Numbers

Definition
Let $p \in \N$ be a natural number.

Let $S = \left\{{x \in \N: p \le x}\right\}$.

Let $T$ be a set.

Let $g: T \to T$ be a mapping.

Let $f: S \to T$ be the mapping defined as:


 * $\forall x \in S: f \left({x}\right) = \begin{cases}

a & : x = p \\ g \left({f \left({n}\right)}\right) & : x = n + 1 \end{cases}$

for $a \in T$.

Then $f$ is said to be recursively defined on $S$.

Also see

 * Principle of Recursive Definition/Natural Numbers