So I am trying to see how the Kendall $\tau$ distance is considered a metric; i.e. that it satisfies the triangle inequality.
The Kendall $\tau$ distance is defined as follows:
$K(\tau_1,\tau_2) = |(i,j): i < j, ( \tau_1(i) < \tau_1(j) \land \tau_2(i) > \tau_2(j) ) \lor ( \tau_1(i) > \tau_1(j) \land \tau_2(i) < \tau_2(j) )|$
Thank you in advance.