## 5.40. between_min_max

Origin
Constraint

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

$\mathrm{\pi  \pi °\pi }$ is greater than or equal to at least one variable of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ and less than or equal to at least one variable of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$.

Example
$\left(3,β©1,1,4,8βͺ\right)$

The constraint holds since its first argument 3 is greater than or equal to the minimum value of the values of the collection $\beta ©1,1,4,8\beta ͺ$ and less than or equal to the maximum value of $\beta ©1,1,4,8\beta ͺ$.

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

• $\mathrm{\pi  \pi °\pi }$ can be set to any value of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }$.

Reformulation

By introducing two extra variables $\mathrm{\pi Ό\pi Έ\pi ½}$ and $\mathrm{\pi Ό\pi °\pi }$, the $\left(\mathrm{\pi  \pi °\pi },\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\right)$ constraint can be expressed in term of the following conjunction of constraints:

Β Β Β $\mathrm{\pi \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 }\right)$,

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

Β Β Β $\mathrm{\pi  \pi °\pi }\beta ₯\mathrm{\pi Ό\pi Έ\pi ½}$,

Β Β Β $\mathrm{\pi  \pi °\pi }\beta €\mathrm{\pi Ό\pi °\pi }$.

Used in
Keywords
Derived Collection
$\mathrm{\pi \pi \pi }\left(\mathrm{\pi Έ\pi \pi ΄\pi Ό}-\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right),\left[\mathrm{\pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi  \pi °\pi }\right)\right]\right)$
Arc input(s)

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

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 },\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\right)$

Arc arity
Arc constraint(s)
$\mathrm{\pi \pi \pi \pi }.\mathrm{\pi \pi \pi }\beta ₯\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }.\mathrm{\pi \pi \pi }$
Graph property(ies)
$\mathrm{\pi \pi \pi \pi }$$\beta ₯1$

Graph class
 $\beta ’$$\mathrm{\pi °\pi ²\pi \pi ²\pi »\pi Έ\pi ²}$ $\beta ’$$\mathrm{\pi ±\pi Έ\pi Ώ\pi °\pi \pi \pi Έ\pi \pi ΄}$ $\beta ’$$\mathrm{\pi ½\pi Ύ}_\mathrm{\pi »\pi Ύ\pi Ύ\pi Ώ}$

Arc input(s)

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

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 },\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\right)$

Arc arity
Arc constraint(s)
$\mathrm{\pi \pi \pi \pi }.\mathrm{\pi \pi \pi }\beta €\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }.\mathrm{\pi \pi \pi }$
Graph property(ies)
$\mathrm{\pi \pi \pi \pi }$$\beta ₯1$

Graph class
 $\beta ’$$\mathrm{\pi °\pi ²\pi \pi ²\pi »\pi Έ\pi ²}$ $\beta ’$$\mathrm{\pi ±\pi Έ\pi Ώ\pi °\pi \pi \pi Έ\pi \pi ΄}$ $\beta ’$$\mathrm{\pi ½\pi Ύ}_\mathrm{\pi »\pi Ύ\pi Ύ\pi Ώ}$

Graph model

PartsΒ (A) andΒ (B) of FigureΒ 5.40.1 respectively show the initial and final graph associated with the second graph constraint of the Example slot. Since we use the $\mathrm{\pi \pi \pi \pi }$ graph property, the two arcs of the final graph are stressed in bold. The constraint holds since 3 is greater than 1 and since 3 is less than 8.

Automaton

FigureΒ 5.40.2 depicts the automaton associated with the constraint. To each pair $\left(\mathrm{\pi  \pi °\pi },{\mathrm{\pi  \pi °\pi }}_{i}\right)$, where ${\mathrm{\pi  \pi °\pi }}_{i}$ is a variable of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ corresponds a signature variable ${\mathrm{\pi }}_{i}$. The following signature constraint links $\mathrm{\pi  \pi °\pi }$, ${\mathrm{\pi  \pi °\pi }}_{i}$ and ${\mathrm{\pi }}_{i}$: $\left(\mathrm{\pi  \pi °\pi }<{\mathrm{\pi  \pi °\pi }}_{i}\beta {\mathrm{\pi }}_{i}=0\right)\beta §\left(\mathrm{\pi  \pi °\pi }={\mathrm{\pi  \pi °\pi }}_{i}\beta {\mathrm{\pi }}_{i}=1\right)\beta §\left(\mathrm{\pi  \pi °\pi }>{\mathrm{\pi  \pi °\pi }}_{i}\beta {\mathrm{\pi }}_{i}=2\right)$.