2.1.2.43 Exercise. Prove that Ar. [log2] E PR. Remove the undefinedness at x = 0 in some convenient manner. For example, arrange that [log2 0] = 0 2.1.2.6 Exercise. (The Grzegorczyk (1953) Substitution Operations) PR is closed under the following operations: (i) Substitution of a function invocation for a variable: From Azyz. f(x, y, z) and Aw.g(w) obtain Aruz. f(x, g(w), z). (ii) Substitution of a constant for a variable: From Axyz.f(, y, z) obtain A. f(x, k, ž). (iii) Interchange of two variables: From Azyzw. f(x, y, z, w, u) obtain Ayzw. f(, w, z, y, u). (iv) Identification of two variables: From Azyzwu.f(z, y, z, w, u) obtain Azyzu. f(x, y, z, y, u). (v) Introduction of "don't care" variables: From A. f() obtain A. f(x). By 2.1.2.6 composition can simulate the Grzegorczyk operations if the initial functions I are present. Of course, (i) alone can in turn simulate composition. With these comments out of the way, we see that the "rigidity" of Definition 2.1.1.13 is gone.