I'm trying to answer the following question:
Show that $f(x,y)=xy$ is primitive recursive.
Basically, multiply function.
Here's my try:
To start with, we try to define it in terms of itself:
$$ f(x,0)=0, \ f(x,1)=f(x,0)+x, \ f(x,2)=f(x,1)+x, \ ... , \ f(x,n+1)=f(x,n)+x $$
Now I'll try to show that it is indeed primitive recursive.
By zero rule: $g(x)=0$ is primitive recursive.
Then we devise $h$ such that $h(x,y,z)=S^x (\pi_3 (x,y,z))=z+x$.
Here, $S$ is primitive recursive by successor rule, $\pi_3$ is primitive recursive by projection rule, and $S^x (\pi_3 (x,y,z))$ is primitive recursive by composition rule and composing $S^x$ and $\pi_3$.
Then, we try to show that $f$ is primitive recursive by recursion rule.
$f(x,0)=g(x)=0$
$f(x,n+1)=h(x,n,f(x,n))=S^x (\pi_3 (x,n,f(x,n)))=S^x (f(x,n))=f(x,n)+x$
$\square$
Are the steps correct and reasonable? Thanks a bunch!