## 5.250. maximum_modulo

Origin
Constraint

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

Arguments
 $\mathrm{\pi Ό\pi °\pi }$ $\mathrm{\pi \pi \pi \pi }$ $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$ $\mathrm{\pi Ό}$ $\mathrm{\pi \pi \pi }$
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

$\mathrm{\pi Ό\pi °\pi }$ is a maximum 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
$\left(5,β©9,1,7,6,5βͺ,3\right)$

The $\mathrm{\pi \pi \pi ‘\pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi }$ constraint holds since its first argument $\mathrm{\pi Ό\pi °\pi }$ is set to value 5, where $5\mathrm{mod}3=2$ is greater than or equal to all the expressions $9\mathrm{mod}3=0$, $1\mathrm{mod}3=1$, $7\mathrm{mod}3=1$ and $6\mathrm{mod}3=0$.

Typical
 $\mathrm{\pi Ό}>1$ $\mathrm{\pi Ό}<$$\mathrm{\pi \pi \pi ‘\pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }\right)$ $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|>1$ $\mathrm{\pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }\right)>1$
Symmetry

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

Arg. properties

Functional dependency: $\mathrm{\pi Ό\pi °\pi }$ determined by $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ and $\mathrm{\pi Ό}$.

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 Έ\pi ½\pi Έ\pi ½\pi },\mathrm{\pi \pi \pi }\right)=\mathrm{\pi Ό\pi °\pi }$

Graph model

PartsΒ (A) andΒ (B) of FigureΒ 5.250.1 respectively show the initial and final graph associated with the Example slot. Since we use the $\mathrm{\pi \pi \pi \pi \pi }$ graph property, the vertex of rank 0 (without considering the loops) of the final graph is outlined with a thick circle.