## 5.226. nequivalence

Origin
Constraint

$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi ½\pi ΄\pi \pi \pi Έ\pi  },\mathrm{\pi Ό},\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\right)$

Arguments
 $\mathrm{\pi ½\pi ΄\pi \pi \pi Έ\pi  }$ $\mathrm{\pi \pi \pi \pi }$ $\mathrm{\pi Ό}$ $\mathrm{\pi \pi \pi }$ $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$
Restrictions
 $\mathrm{\pi ½\pi ΄\pi \pi \pi Έ\pi  }\beta ₯\mathrm{\pi \pi \pi }\left(1,|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|\right)$ $\mathrm{\pi ½\pi ΄\pi \pi \pi Έ\pi  }\beta €\mathrm{\pi \pi \pi }\left(\mathrm{\pi Ό},|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|\right)$ $\mathrm{\pi Ό}>0$ $\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

$\mathrm{\pi ½\pi ΄\pi \pi \pi Έ\pi  }$ is the number of distinct rests obtained by dividing the variables of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ by $\mathrm{\pi Ό}$.

Example
$\left(\begin{array}{c}2,3,β©\begin{array}{c}\mathrm{\pi \pi \pi }-3,\hfill \\ \mathrm{\pi \pi \pi }-2,\hfill \\ \mathrm{\pi \pi \pi }-5,\hfill \\ \mathrm{\pi \pi \pi }-6,\hfill \\ \mathrm{\pi \pi \pi }-15,\hfill \\ \mathrm{\pi \pi \pi }-3,\hfill \\ \mathrm{\pi \pi \pi }-3\hfill \end{array}βͺ\hfill \end{array}\right)$

Since the expressions $3\mathrm{mod}3=0$, $2\mathrm{mod}3=2$, $5\mathrm{mod}3=2$, $6\mathrm{mod}3=0$, $15\mathrm{mod}3=0$, $3\mathrm{mod}3=0$, and $3\mathrm{mod}3=0$ involve two distinct values (values 0 and 2), the first argument $\mathrm{\pi ½\pi ΄\pi \pi \pi Έ\pi  }$ of the $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }$ constraint is set to value 2.

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

• An occurrence of a value $u$ of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }$ can be replaced by any other value $v$ such that $v$ is congruent to $u$ modulo $\mathrm{\pi Ό}$.

Algorithm

Since constraints $X=Y$ and $X\beta ‘Y\left(\mathrm{mod}M\right)$ are similar, one should also use a similar algorithm as the one [Beldiceanu01], [BeldiceanuCarlssonThiel02] provided for constraint $\mathrm{\pi \pi \pi \pi \pi \pi }$.

related: $\mathrm{\pi \pi \pi \pi \pi \pi }$Β ($\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }\mathrm{mod}\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$ replaced by $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }\beta \mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }$), $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }$Β ($\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }\mathrm{mod}\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$ replaced by $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }/\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$), $\mathrm{\pi \pi \pi \pi \pi }$Β ($\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }\mathrm{mod}\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$ replaced by $\mathrm{\pi \pi \pi \pi }$ of $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }$).

specialisation: $\mathrm{\pi \pi \pi \pi \pi \pi }$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)
$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi \pi \pi }\mathrm{mod}\mathrm{\pi Ό}=\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi \pi \pi }\mathrm{mod}\mathrm{\pi Ό}$
Graph property(ies)
$\mathrm{\pi \pi \pi \pi }$$=\mathrm{\pi ½\pi ΄\pi \pi \pi Έ\pi  }$

Graph model

PartsΒ (A) andΒ (B) of FigureΒ 5.226.1 respectively show the initial and final graph associated with the Example slot. Since we use the $\mathrm{\pi \pi \pi \pi }$ graph property we show the different strongly connected components of the final graph. Each strongly connected component corresponds to one equivalence class: We have two equivalence classes that respectively correspond to values $\left\{3,6,15\right\}$ and $\left\{2,5\right\}$.