### 3.7.35. Border

A constraint that can be related to the notion of border, which we define now. Given a sequence $s=urv$, $r$ is a prefix of $s$ when $u$ is empty, $r$ is a suffix of $s$ when $v$ is empty, $r$ is a proper factor of $s$ when $r\ne s$. A border of a non -empty sequence $s$ is a proper factor of $s$, which is both a prefix and a suffix of $s$. We have that the smallest period of a sequence $s$ is equal to the size of $s$ minus the length of the longest border of $s$.