Dipper Operation/Examples/n = 1

Examples of Dipper Operation
Let $m, n \in \Z$ be integers such that $m \ge 0, n > 0$.

Let $\N_{< \paren {m \mathop + n} }$ denote the initial segment of the natural numbers:
 * $\N_{< \paren {m \mathop + n} } := \set {0, 1, \ldots, m + n - 1}$

Let $+_{m, n}$ be the dipper operation on $\N_{< \paren {m \mathop + n} }$ defined as:
 * $\forall a, b \in \N_{< \paren {m \mathop + n} }: 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:
 * $m + k n \le a + b$

Let $n = 1$.

Then $+_{m, n}$ degenerates to the following operation on $\N_{< \paren {m \mathop + n} }$:


 * $\forall a, b \in \N_{< \paren {m \mathop + n} }: a +_{m, 1} b = \begin{cases}

a + b & : a + b < m \\ m & : a + b \ge m \end{cases}$