Determine whether this d is a metric? Suppose we have a function $d:\mathbb{Z}^{2}\times\mathbb{Z}^{2}\rightarrow\mathbb{R}$ where   $\forall(x_{0},y_{0}),(x_{1},y_{1})\in \mathbb{Z}^{2}$, we have..
$d\left( \left( x_{0},y_{0} \right), \left( x_{1},y_{1} \right) \right)\;\;=\;\;\left| \left| x_{1}-x_{0} \right| - \left| y_{1}-y_{0} \right| \right|\;\;+\;\;\sqrt{2}\cdot \min \left \{ \left | x_{1}-x_{0} \right | , \left | y_{1}-y_{0} \right |\right \}$
I am trying to determine whether $d$ is a metric in $\mathbb{Z}^{2}$ and I am a bit stuck on the triangle inequality.
Graphically, I think of this d as somewhere in-between a Euclidean and Manhattan distance;
 
I've created a model of this function in geogebra:

The bold text will switch to saying "Counter-example" once the inequality fails when we move the 3 points around.
The geogebra file is uploaded here (8kb). So far, no counter-example have been found.  

The algebraic problem aside, I will be using this $d$ as a heuristic in an A* search algorithm, so what would be the significance, if any, of $d$ being a metric (if it is) (as opposed to an arbitrary assignment of numbers for a heuristic, say)?

Hint: let $d_1$ and $d_2$ be two metrics on the same metric space $X$, with $\lambda>0$.
Then
$$\lambda d_1, \lambda d_2~~\text{are both metrics }. $$
  $$ d_1+d_2~~\text{is a metric }. $$
In particular, $d:=d_1+\lambda d_2$ is a metric.
