Prove a function is primitive recursive

Question

Help me please $f(x)=x+a$, where $a$ is a constant.

Answer 1

First of all I think that your statement is wrong when $a\notin \mathbb{Z}$, and the function you define when $a\notin \mathbb{N}$ should have the domain restricted, because you cannot use the subtraction but only the limited subtraction.

So let us suppose $a\in \mathbb{N}$ for simplicity. If $a=0$ then $f(x)=x$ is the identity function, and this is known to be primitive recursive. Indeed $f(x) = P_1^1(x)$.

Now let us proceed by induction and suppose that $f_n(x)= x+n$ is primitive recursive. By $S$ we denote the successor function $S(k)=k+1$ which is axiomatically primitive recursive. Then $S(f_n(x))= (x+n)+1=f_{n+1}(x)$.

Since the composition of primitive recursive functions is primitive recursive we deduce that $f_{n+1}(x)=x+(n+1)$ is primitive recursive as well. By the induction hypothesis we are done.

Note: This has been brutally copied by Wikipedia: Primitive recursive function.