\begin{definition}[Definition:Recursively Defined Mapping/Peano Structure]
Let $\struct {P, 0, s}$ be a Peano structure.
Let $T$ be a set.
Let $g: T \to T$ be a mapping.
Let  $f: P \to T$ be the mapping defined as:
:$\forall x \in P: \map f x = \begin{cases}
  a & : x = 0 \\
  \map g {\map f n} & : x = \map s n
\end{cases}$
where $a \in T$.
Then $f$ is said to be '''recursively defined''' on $P$.
\end{definition}