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| 3=''Projection function'' <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
{{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>
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