Primitive recursion

Given $g : \mathbb{N}^k \to \mathbb{N}$ and $h : \mathbb{N}^{k{+2}} \to \mathbb{N}$, the function $f : \mathbb{N}^{k{+1}} \to \mathbb{N}$ is defined by primitive recursion from $g$ and $h$ if the following equalities hold:

$$\begin{array}{lcl} f(0,n_1,n_2,\ldots,n_k) & = & g(n_1,n_2,... ...& = & h(n, f(n,n_1,n_2,\ldots,n_k), n_1,\ldots,n_k) \end{array}$$

Here are some examples of primitive recursive definitions.

• The function $\mathit{plus}(n,m) = n{+m}$ can be defined by primitive recursion from $g = \Pi ^1_1$ and $h = S \circ \Pi ^3_2$. In other words,

  $\begin{array}{lclclcl} \mathit{plus}(0,n) & = & g(n) & = & \Pi ^1_1(n) & = & ... ... \circ \Pi ^3_2(m, \mathit{plus}(m,n), n) &=& S(\mathit{plus}(m,n)) \end{array}$

• The function $\mathit{times}(n,m) = n \cdot m$ can be defined by primitive recursion from $g = Z$ and $h = \mathit{plus} \circ (\Pi ^3_3, \Pi ^3_2)$.