## 5.414. used_by_modulo

Origin
Constraint

$\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi ’}_\mathrm{\pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1},\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2},\mathrm{\pi Ό}\right)$

Arguments
 $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$ $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$ $\mathrm{\pi Ό}$ $\mathrm{\pi \pi \pi }$
Restrictions
 $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}|\beta ₯|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}|$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1},\mathrm{\pi \pi \pi }\right)$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2},\mathrm{\pi \pi \pi }\right)$ $\mathrm{\pi Ό}>0$
Purpose

For each integer $R$ in $\left[0,\mathrm{\pi Ό}-1\right]$, let $\mathrm{\pi }{\mathit{1}}_{R}$ (respectively $\mathrm{\pi }{\mathit{2}}_{R}$) denote the number of variables of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}$ (respectively $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}$) that have $R$ as a rest when divided by $\mathrm{\pi Ό}$. For all $R$ in $\left[0,\mathrm{\pi Ό}-1\right]$ we have $\mathrm{\pi }{\mathit{2}}_{R}>0\beta \mathrm{\pi }{\mathit{1}}_{R}\beta ₯\mathrm{\pi }{\mathit{2}}_{R}$.

Example
$\left(β©1,9,4,5,2,1βͺ,β©7,1,2,5βͺ,3\right)$

The values of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}=\beta ©7,1,2,5\beta ͺ$ are respectively associated with the equivalence classes $7\mathrm{mod}3=1$, $1\mathrm{mod}3=1$, $2\mathrm{mod}3=2$, $5\mathrm{mod}3=2$. Therefore the equivalence classes 1 and 2 are respectively used 2 and 2 times.

Similarly, the values of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}=\beta ©1,9,4,5,2,1\beta ͺ$ associated with the equivalence classes $1\mathrm{mod}3=1$, $9\mathrm{mod}3=0$, $4\mathrm{mod}3=1$, $5\mathrm{mod}3=2$, $2\mathrm{mod}3=2$, $1\mathrm{mod}3=1$. Therefore the equivalence classes 0, 1 and 2 are respectively used 1, 3 and 2 times.

Consequently, the $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi ’}_\mathrm{\pi \pi \pi \pi \pi \pi }$ constraint holds since, for each equivalence class associated with the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}=\beta ©7,1,2,5\beta ͺ$, its number of occurrences within $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}=\beta ©1,9,4,5,2,1\beta ͺ$ is greater than or equal to its number of occurrences within $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}$:

• The equivalence class 1 occurs 3 times within $\beta ©1,9,4,5,2,1\beta ͺ$ and 2 times within $\beta ©7,1,2,5\beta ͺ$.

• The equivalence class 2 occurs 2 times within $\beta ©1,9,4,5,2,1\beta ͺ$ and 2 times within $\beta ©7,1,2,5\beta ͺ$.

Typical
 $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}|>1$ $\mathrm{\pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}.\mathrm{\pi \pi \pi }\right)>1$ $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}|>1$ $\mathrm{\pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}.\mathrm{\pi \pi \pi }\right)>1$ $\mathrm{\pi Ό}>1$ $\mathrm{\pi Ό}<$$\mathrm{\pi \pi \pi ‘\pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}.\mathrm{\pi \pi \pi }\right)$ $\mathrm{\pi Ό}<$$\mathrm{\pi \pi \pi ‘\pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}.\mathrm{\pi \pi \pi }\right)$
Symmetries
• Items of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}$ are permutable.

• Items of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}$ are permutable.

• An occurrence of a value $u$ of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}.\mathrm{\pi \pi \pi }$ can be replaced by any other value $v$ such that $v$ is congruent to $u$ modulo $\mathrm{\pi Ό}$.

• An occurrence of a value $u$ of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}.\mathrm{\pi \pi \pi }$ can be replaced by any other value $v$ such that $v$ is congruent to $u$ modulo $\mathrm{\pi Ό}$.

Arg. properties
• Contractible wrt. $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}$.

• Extensible wrt. $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}$.

• Aggregate: $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}\left(\mathrm{\pi \pi \pi \pi \pi }\right)$, $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}\left(\mathrm{\pi \pi \pi \pi \pi }\right)$, $\mathrm{\pi Ό}\left(\mathrm{\pi \pi }\right)$.

Used in

specialisation: $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi ’}$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)

$\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}$ $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}$

Arc generator
$\mathrm{\pi \pi  \pi \pi ·\pi \pi Ά\pi }$$\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)
 $\beta ’\text{for}\text{all}\text{connected}\text{components:}$$\mathrm{\pi \pi \pi \pi \pi \pi \pi }$$\beta ₯$$\mathrm{\pi \pi \pi \pi \pi }$ $\beta ’$$\mathrm{\pi \pi \pi \pi \pi }$$=|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}|$

Graph model

PartsΒ (A) andΒ (B) of FigureΒ 5.414.1 respectively show the initial and final graph associated with the Example slot. Since we use the $\mathrm{\pi \pi \pi \pi \pi \pi \pi }$ and $\mathrm{\pi \pi \pi \pi \pi }$ graph properties, the source and sink vertices of the final graph are stressed with a double circle. Since there is a constraint on each connected component of the final graph we also show the different connected components. Each of them corresponds to an equivalence class according to the arc constraint. Note that the vertex corresponding to the variable that takes value 9 was removed from the final graph since there is no arc for which the associated equivalence constraint holds. The $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi ’}_\mathrm{\pi \pi \pi \pi \pi \pi }$ constraint holds since:

• For each connected component of the final graph the number of sources is greater than or equal to the number of sinks.

• The number of sinks of the final graph is equal to $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}|$.

Signature

Since the initial graph contains only sources and sinks, and since sources of the initial graph cannot become sinks of the final graph, we have that the maximum number of sinks of the final graph is equal to $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }2|$. Therefore we can rewrite $\mathrm{\pi \pi \pi \pi \pi }=|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }2|$ to $\mathrm{\pi \pi \pi \pi \pi }\beta ₯|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }2|$ and simplify $\underset{Μ²}{\stackrel{Β―}{\mathrm{\pi \pi \pi \pi \pi }}}$ to $\stackrel{Β―}{\mathrm{\pi \pi \pi \pi \pi }}$.