Is nested induction necessary for all variables

Question

Let $S(n_1,n_2,n_3)$ be the statement to prove in natural numbers;

If I prove the following three

1. $S(1,1,1)$ is true

2. $\forall n_1n_2 S(n_1,n_2,n_3) \implies S(n_1,n_2,n_3+1)$

3. $\forall n_1n_3 S(n_1,n_2,n_3) \implies S(n_1,n_2+1,n_3)$

Now, Is it sufficient or we have to prove for $n_1$ also?

Answer 1

Yes, you need one for $n_1$ as well. As it stands, you don't even have any way of knowing whether $S(2,1,1)$ is true from those assumptions.

Answer 2

You have to prove it for $n_1$ as well. Consider the statement: $$ S(n_1,n_2,n_3) \leftrightarrow (n_1 < 2) \land (n_2 = n_2) \land (n_3 = n_3) $$ We have that $S(1,1,1)$ is true. For $n_1 > 1$, the statement is false, so the implications are all true; for $n_1=1$, the statement is true regardless of $n_2$ and $n_3$, so the implications are still true.

Note: I'm not sure if this is sufficient to prove $S(n_1,n_2,n_3)$ for every $n_1,n_2,n_3$. We'd have to prove it formally.