  Jim\'enez, Becerra, and Gelbukh (2013) defined a family of "symmetric Tversky
ratio models" $S_{\alpha,\beta}$, $0\le\alpha\le 1$, $\beta>0$. Each function
$D_{\alpha,\beta}=1-S_{\alpha,\beta}$ is a semimetric on the powerset of a
given finite set.
  We show that $D_{\alpha,\beta}$ is a metric if and only if $0\le\alpha \le
\frac12$ and $\beta\ge 1/(1-\alpha)$. This result is formally verified in the
Lean proof assistant.
  The extreme points of this parametrized space of metrics are $\mathcal
V_1=D_{1/2,2}$, the Jaccard distance, and $\mathcal V_{\infty}=D_{0,1}$, an
analogue of the normalized information distance of M. Li, Chen, X. Li, Ma, and
Vit\'anyi (2004).
  As a second interpolation, in general we also show that $\mathcal V_p$ is a
metric, $1\le p\le\infty$, where $$\Delta_p(A,B)=(|B\setminus A|^p+|A\setminus
B|^p)^{1/p},$$ $$\mathcal V_p(A,B)=\frac{\Delta_p(A,B)}{|A\cap B| +
\Delta_p(A,B)}.$$
