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| 2=''Successor function'': The 1-ary successor function ''S'', which returns the successor of its argument (see [[Peano postulates]]), that is, <math>S(x) \ \stackrel{\mathrm{def}}{=}\ x + 1</math>, is primitive recursive.
| 3=''Projection functions'' <math>P_i^k</math>: For all natural numbers <math>i, k</math> such that <math>1\le i\le k</math>, the ''k''-ary function defined by <math>P_i^k(x_1,\ldots,x_k) \ \stackrel{\mathrm{def}}{=}\ x_i</math> is primitive recursive.
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More complex primitive recursive functions can be obtained by applying the [[operation (mathematics)|operation]]s given by these axioms:
{{ordered list|start=4
| 4=''Composition operator'' <math>\circ\,</math> (also called the ''substitution operator''): Given an ''m''-ary function <math>h(x_1,\ldots,x_m)\,</math> and ''m'' ''k''-ary functions <math>g_1(x_1,\ldots,x_k),\ldots,g_m(x_1,\ldots, x_k)</math>: <math display="block">h \circ (g_1, \ldots, g_m) \ \stackrel{\mathrm{def}}{=}\ f, \quad\text{where}\quad f(x_1,\ldots,x_k) = h(g_1(x_1,\ldots,x_k),\ldots,g_m(x_1,\ldots,x_k)).</math> For <math>m=1</math>, the ordinary [[function composition]] <math>h \circ g_1</math> is obtained.
| 5=''Primitive recursion operator'' <math>\rho</math>: Given the ''k''-ary function <math>g(x_1,\ldots,x_k)\,</math> and the (''k'' + 2)-ary function <math>h(y,z,x_1,\ldots,x_k)\,</math>:<math display="block">\begin{align}
\rho(g, h) &\ \stackrel{\mathrm{def}}{=}\ f, \quad\text{where the }(k+1)\text{-ary function } f \text{ is defined by}\\
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