Natural metrics and their properties. P. 1. Submetrics and overmetrics

Criteria for integer-valued function $\mu\colon X\times X\to\{0,1,\dots\}$ to be a metric (where $X$ is a finite set) are given. Notions of submetric, overmetric, natural and canonical metrics are introduced. Classification of metrics admitting no more than 5 values is constructed, some of their submetrics and overmetrics are described.