A proof of the triangle inequality for the Tanimoto distance

By A. H. Lipkus. 1999

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Generalized Intersection Over Union: A Metric and a Loss for Bounding Box Regression (2019, background): “LIoU = 1− IoU , is a metric has been provided in different articles [1, 3, 4, 5].”
A Normalized Levenshtein Distance Metric (2007, background): “Although dN GLD is not always shown to be a metric over for an arbitrary weight function, Theorem 2, inspired by the work reported in [19], can be shown to hold.”
Analysis of Similarity Measures in Times Series Clustering for the Discovery of Building Energy Patterns (2013, background): “…options are the cosine measure [22], which is good for patterns with different or variable size or length; or Jaccard and Tanimoto similarity measures, that can also be intuitively understood as a combination of Euclidean distances and correlations assessed by means of the inner product [23].”
A note on the triangle inequality for the Jaccard distance (2016, background): “, [15, 11, 4]), minwise independent permutations [1], or sometimes cumbersome arithmetics [10, 3].”
A Probabilistic U-Net for Segmentation of Ambiguous Images (2018, background): “In our case we choose d(x, y) = 1− IoU(x, y), which as proved in [30, 31], is a metric.”