I need to prove a statement involving two variables over non-negative integers. That is $P(a, b)$ for all $a\in \mathbb{Z_{\ge 0}}$ and $b \in \mathbb{Z_{\ge 0}}$
I did the following steps
1.$ P(a, 0)$ for all $a\in \mathbb{Z_{\ge 0}}$
2.$ P(a, 1)$ for all $a\in \mathbb{Z_{\ge 0}}$
3.$ P(a, b-1) \land P(a, b) \implies P(a+1, b)$
Is my proof complete? Any statement to prove more than the statements provided?
No, this is not complete. How, for example, do you get that $P(0,2)$ from any of this?
You either need to show that $P(0,b)$ for any $b$, and then induct over the $a$, or show that $P(a, b-1) \land P(a,b) \Rightarrow P(a,b+1)$