|<math>S \circ S</math> is a 1-ary function that adds 2 to its input, <math>(S \circ S)(x) = x+2</math>.
|<math>S \circ C_0^1</math> is a 1-ary function which returns 1 for every input: <math>(S \circ C_0^1)(x) = S(C_0^1(x)) = S(0) = 1</math>. That is, <math>S \circ C_0^1</math> and <math>C_1^1</math> are the same function: <math>S \circ C_0^1 = C_1^1</math>. In a similar way, every <math>C_n^1k</math> can be expressed as a composition of appropriately many <math>S</math> and <math>C_0^k</math>. Moreover, <math>C_0^k</math> equals <math>C_0^1 \circ P_1^k</math>, since <math>C_0^k(x_1,\ldots,x_k) = 0 = C_0^1(x_1) = C_0^1(P_1^k(x_1,\ldots,x_k)) = (C_0^1 \circ P_1^k)(x_1,\ldots,x_k)</math>. For thisthese reasonreasons, some authors<ref>E.g.: {{cite book | author=Henk Barendregt | author-link=Henk Barendregt | contribution=Functional Programming and Lambda Calculus | pages=321–364 | isbn=0-444-88074-7 | editor=Jan van Leeuwen | editor-link=Jan van Leeuwen | title=Formal Models and Semantics | publisher=Elsevier | series=Handbook of Theoretical Computer Science | volume=B | year=1990}} Here: 2.2.6 ''initial functions'', Def.2.2.7 ''primitive recursion'', p.331-332.</ref> omit the definition ofdefine <math>C_n^k</math> only for <math>n >= 0</math> and <math>k = 1</math>.
