The following theorem is known as generalized Hensel lifting(see here). Can we prove this without using $P$-adic completion?
Theorem Let $A$ be a Dedekind domain. Let $P$ be a non-zero prime ideal of $A$. Let $f(x) \in A[x]$ be a polynomial. Let $a \in A$. Suppose $P^r||f'(a)$. Let $m = 2r + 1$. Let $k \geq m$ be an integer. Suppose $f(a) \equiv 0$ (mod $P^k$). Then there exists $b \in A$ such that $b \equiv a$ (mod $P^{k-r}$), $P^r||f'(b)$ and $f(b) \equiv 0$ (mod $P^{k + 1}$).
This can be an immediate corollary of the above theorem.
Motivation This theorem can be used in class field theory. I'm interested in proving CFT without $p$-adic numbers(see here).