Let $D$ be a $\mathbb{Q}$-divisor in a smooth variety $X$. In Lazarsfeld book "Positivity in Algebraic Geometry 2" I found Proposition 9.5.13 saying that if for any $x\in D$ we have $mult_xD < 1$ then the pair $(X,D)$ is klt.

I am wondering if it possible to check that a pair is klt by compunting the discrepancies of a partial resolution and apply the fact above. I will be more precise.

Assume we have a birational divisor contraction $f:Y\rightarrow X$ such that $f^{-1}D\cup Exc(f)$ is simple normal crossing and the discrepancies of $(X,D)$ with respect any exceptional divisor are gretaer than $-1$. Let $Z = f(Exc(f))\subset X$. If for any $x\in D\setminus (D\cap Z)$ we have $mult_xD< 1$ is it true that $(X,D)$ is klt?