Can someone help with this Problem 7.12 from Computability and Logic 5th ed. (Boolos, Burgess & Jeffrey, 2007)? I can't find the solution anywhere online. Much appreciated.
If f and g are n- and (n+2)-place primitive recursive functions obtainable from initial functions (zero, successor, identity) by composition, without use of recursion, we have shown in Proposition 7.24 [of the aforementioned book] that there are numbers a and b such that for $x_1, \ldots, x_n, y$, and z, we have:
$f(x_1, \ldots, x_n) = x + a$, where x is the largest $x_1, \ldots, x_n$
$g(x_1, \ldots, x_n, y, z) = x + b$, where x is the largest $x_1, \ldots, x_n, y$ and z.
Show now that if $h = Pr[f,g]$, then there is a number c such that for all $x_1, \ldots, x_n$ and y we have $h(x_1, \ldots, x_n, y) < cx + c$, where x is the largest $x_1, \ldots, x_n$ and y.
(N.b. Proposition 7.24 demonstrated that it is impossible to obtain the sum or addition function from the basic functions (zero, successor, and identity) by composition, without using recursion.)