Definition:Dipper Operation

Definition
Let $m, n \in \Z$ be integers such that $m \ge 0, n > 0$.

Let $+_{m, n}$ be the binary operation on $\Z_{>0}$ defined as:
 * $\forall a, b \in \Z_{>0}: a +_{m, n} b = \begin{cases}

a + b & : a + b < m \\ a + b - k n & : a + b \ge m \end{cases}$ where $k$ is the largest integer satisfying:
 * $m + k n \le a + b$

The operation $+_{m, n}$ is known as the Big Dipper.