Dipper Operation is Commutative

Theorem
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:
 * $\N_{< \paren {m \mathop + n} } := \set {0, 1, \ldots, m + n - 1}$

The dipper operation on $\N_{< \paren {m \mathop + n} }$ is commutative.

Proof
Recall the definition of the dipper operation on $\N_{< \paren {m \mathop + n} }$ 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:
 * $m + k n \le a + b$

Let $a + b < m$.

Then:

Otherwise $a + b \ge m$.

Then:

So in both cases:
 * $a +_{m, n} b = b +_{m, n} a$

and the result follows by definition of commutative operation.

Also see

 * Dipper Operation is Associative