I have a question related to the Skorokhod distance.

Let $\Omega:=D([0,1],R)$ be the space of cadlag functions $x$ defined on $[0,1]$. Let $\Lambda$ be the collection of non-increasing continuous onto functions $\lambda: [0,1]\to [0,1]$. Then define

$$\rho(x, x'):=\inf_{\lambda\in\Lambda}\Big\{\max\Big(||x\circ \lambda-x'||, ||\lambda-I||\Big)\Big\},~ \forall x, x'\in\Omega$$

Here $\circ$ denotes the composition of functions. Compared to the Skorokhod distance, $\rho$ is not a distance since it is no longer symetric, i.e.

$$\rho(x,x')\neq \rho(x',x)$$

Now it is easy to show the triangle inequality. I would like to know whether we have

$$\rho(x,x')=0\Longrightarrow x=x'$$

Thx a lot for the reply!