## 5.220. minimum_modulo

Origin
Constraint

$\mathrm{\pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi Ό\pi Έ\pi ½},\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi },\mathrm{\pi Ό}\right)$

Arguments
 $\mathrm{\pi Ό\pi Έ\pi ½}$ $\mathrm{\pi \pi \pi \pi }$ $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$ $\mathrm{\pi Ό}$ $\mathrm{\pi \pi \pi }$
Restrictions
 $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|>0$ $\mathrm{\pi Ό}>0$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi },\mathrm{\pi \pi \pi }\right)$
Purpose

$\mathrm{\pi Ό\pi Έ\pi ½}$ is a minimum value of the collection of domain variables $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ according to the following partial ordering: $\left(X\mathrm{mod}\mathrm{\pi Ό}\right)<\left(Y\mathrm{mod}\mathrm{\pi Ό}\right)$.

Example
 $\left(6,β©9,1,7,6,5βͺ,3\right)$ $\left(9,β©9,1,7,6,5βͺ,3\right)$

The $\mathrm{\pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi }$ constraints hold since $\mathrm{\pi Ό\pi Έ\pi ½}$ is respectively set to values 6 and 9, where $6\mathrm{mod}3=0$ and $9\mathrm{mod}3=0$ are both less than or equal to all the expressions $9\mathrm{mod}3=0$, $1\mathrm{mod}3=1$, $7\mathrm{mod}3=1$, $6\mathrm{mod}3=0$, and $5\mathrm{mod}3=2$.

Symmetry

Items of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ are permutable.

specialisation: $\mathrm{\pi \pi \pi \pi \pi \pi \pi }$Β ($\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }\mathrm{mod}\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$ replaced by $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$).

Keywords
Arc input(s)

$\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$

Arc generator
$\mathrm{\pi Ά\pi Ώ\pi Ό\pi \pi \pi Έ}$$\beta ¦\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{1},\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{2}\right)$

Arc arity
Arc constraint(s)
$\beta \left(\begin{array}{c}\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi \pi \pi ’}=\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi \pi \pi ’},\hfill \\ \mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi \pi \pi }\mathrm{mod}\mathrm{\pi Ό}<\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi \pi \pi }\mathrm{mod}\mathrm{\pi Ό}\hfill \end{array}\right)$
Graph property(ies)
$\mathrm{\pi \pi \pi \pi \pi }$$\left(0,\mathrm{\pi Ό\pi °\pi \pi Έ\pi ½\pi },\mathrm{\pi \pi \pi }\right)=\mathrm{\pi Ό\pi Έ\pi ½}$

Graph model

We use a similar definition that the one that was utilised for the $\mathrm{\pi \pi \pi \pi \pi \pi \pi }$ constraint. Within the arc constraint we replace the condition $X by the condition $\left(X\mathrm{mod}\mathrm{\pi Ό}\right)<\left(Y\mathrm{mod}\mathrm{\pi Ό}\right)$.

PartsΒ (A) andΒ (B) of FigureΒ 5.220.1 respectively show the initial and final graph associated with the second example of the Example slot. Since we use the $\mathrm{\pi \pi \pi \pi \pi }$ graph property, the vertex of rank 0 (without considering the loops) associated with value 9 is outlined with a thick circle.