on coq: Why is the proof complete after proving only for one induction when we have more than one variable?
You do not have to prove things by induction. For example, you can prove $\forall n : \mathbb{N} \,.\, n = n$ without induction by applying reflexivity. In your proof, we use induction on $a$, but then we do not need to use induction on $b$ and $c$ because we can finish the proof simply using other methods. We could have used induction on $b$ and $c$, and you should try to do so, to see how unecessary applications of induction just make your proof longer and less clear.

P.S. This has absolutely nothing to do with constructivity.