I am having trouble understanding the definition of primitive recursion. I would like to have clarification of the definition with simple applications of the definition with examples.
The definition of Primitive Recursion is: $h(x,0)=f(x)$, $h(x,s(y))=g(x,y,h(x,y))$.
The short hand is $h=\Pr[f,g]$.
What exactly does: $h(x,s(y))=g(x,y,h(x,y))$ mean?
The example the book gives is Pr for addition: $s(x,0)=x$, $\text{sum}(x,s(y))=s(\text{sum}(x,y))$. Where s means "successor". I do not understand the example, nor how the example correlates to the definition.
Specifically how does $h(x,s(y))=g(x,y,h(x,y))$ correlate to $\text{sum}(x,s(y))=s(\text{sum}(x,y))$ and what is $\text{sum}(x,s(y))=s(\text{sum}(x,y))$ doing? I'd like to see an example so I can see how the definition works.