Two notions of equivalence for valuations (reviewed)

Let $v_1$ and $v_2$ are valuations of a ring $R$, $p \in \mathbb{R}_{>1}$, and we set $\lvert x \rvert_1 = p^{-v_1(x)}$ and $\lvert x \rvert_2 = p^{-v_2(x)}$, the following are equivalent:

• there exists a positive real number $c$ such that $|x|_1 = |x|_2^c$ for all $x\in R$,
• for all elements $r, s \in R$, $v_1(r) \le v_1(s)$ if and only if $v_2(r) \le v_2(s)$.

The second form is the definition used in mathlib.