## 5.374. stage_element

Origin
Constraint

$\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi Έ\pi \pi ΄\pi Ό},\mathrm{\pi \pi °\pi ±\pi »\pi ΄}\right)$

Usual name

$\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }$

Synonym

$\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$.

Arguments
 $\mathrm{\pi Έ\pi \pi ΄\pi Ό}$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi ‘}-\mathrm{\pi \pi \pi \pi },\mathrm{\pi \pi \pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$ $\mathrm{\pi \pi °\pi ±\pi »\pi ΄}$
Restrictions
 $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi Έ\pi \pi ΄\pi Ό},\left[\mathrm{\pi \pi \pi \pi \pi ‘},\mathrm{\pi \pi \pi \pi \pi }\right]\right)$ $|\mathrm{\pi Έ\pi \pi ΄\pi Ό}|=1$ $|\mathrm{\pi \pi °\pi ±\pi »\pi ΄}|>0$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }$
Purpose

Let , ${\mathrm{\pi \pi }}_{i}$ and ${\mathrm{\pi \pi \pi \pi \pi }}_{i}$ respectively denote the values of the , $\mathrm{\pi \pi }$ and $\mathrm{\pi \pi \pi \pi \pi }$ attributes of the ${i}^{th}$ item of the $\mathrm{\pi \pi °\pi ±\pi »\pi ΄}$ collection. First we have that: and .

Second, the $\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi }$ constraint forces the following equivalence:

.

Example

FigureΒ 5.374.1 depicts the function associated with the items of the $\mathrm{\pi \pi °\pi ±\pi »\pi ΄}$ collection. The $\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi }$ constraint holds since:

• The value of $\mathrm{\pi Έ\pi \pi ΄\pi Ό}\left[1\right].\mathrm{\pi \pi \pi \pi \pi ‘}$ is located between the values of the and $\mathrm{\pi \pi }$ attributes of the first item of the $\mathrm{\pi \pi °\pi ±\pi »\pi ΄}$ collection (i.e.,Β $5\beta \left[3,7\right]$).

• The value of $\mathrm{\pi Έ\pi \pi ΄\pi Ό}\left[1\right].\mathrm{\pi \pi \pi \pi \pi }$ corresponds to the $\mathrm{\pi \pi \pi \pi \pi }$ attribute of the first item of the $\mathrm{\pi \pi °\pi ±\pi »\pi ΄}$ collection (i.e.,Β 6).

Typical
 $|\mathrm{\pi \pi °\pi ±\pi »\pi ΄}|>1$ $\mathrm{\pi \pi \pi \pi \pi }$$\left(\mathrm{\pi \pi °\pi ±\pi »\pi ΄}.\mathrm{\pi \pi \pi \pi \pi }\right)>1$
Symmetry

All occurrences of two distinct values in $\mathrm{\pi Έ\pi \pi ΄\pi Ό}.\mathrm{\pi \pi \pi \pi \pi }$ or $\mathrm{\pi \pi °\pi ±\pi »\pi ΄}.\mathrm{\pi \pi \pi \pi \pi }$ can be swapped; all occurrences of a value in $\mathrm{\pi Έ\pi \pi ΄\pi Ό}.\mathrm{\pi \pi \pi \pi \pi }$ or $\mathrm{\pi \pi °\pi ±\pi »\pi ΄}.\mathrm{\pi \pi \pi \pi \pi }$ can be renamed to any unused value.

Arg. properties
• Functional dependency: $\mathrm{\pi Έ\pi \pi ΄\pi Ό}.\mathrm{\pi \pi \pi \pi \pi }$ determined by $\mathrm{\pi Έ\pi \pi ΄\pi Ό}.\mathrm{\pi \pi \pi \pi \pi ‘}$ and $\mathrm{\pi \pi °\pi ±\pi »\pi ΄}$.

• Suffix-extensible wrt. $\mathrm{\pi \pi °\pi ±\pi »\pi ΄}$.

Keywords
Arc input(s)

$\mathrm{\pi \pi °\pi ±\pi »\pi ΄}$

Arc generator
$\mathrm{\pi \pi ΄\pi \pi »}$$\beta ¦\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi }\mathtt{1},\mathrm{\pi \pi \pi \pi \pi }\mathtt{2}\right)$

Arc arity
Arc constraint(s)
Graph property(ies)
$\mathrm{\pi \pi \pi \pi }$$=|\mathrm{\pi \pi °\pi ±\pi »\pi ΄}|-1$

Arc input(s)

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

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 }\right)$

Arc arity
Arc constraint(s)
 $\beta ’\mathrm{\pi \pi \pi \pi }.\mathrm{\pi \pi \pi \pi \pi ‘}\beta €\mathrm{\pi \pi \pi \pi \pi }.\mathrm{\pi \pi }$ $\beta ’\mathrm{\pi \pi \pi \pi }.\mathrm{\pi \pi \pi \pi \pi }=\mathrm{\pi \pi \pi \pi \pi }.\mathrm{\pi \pi \pi \pi \pi }$
Graph property(ies)
$\mathrm{\pi \pi \pi \pi }$$=1$

Graph model

The first graph constraint models the restrictions on the and $\mathrm{\pi \pi }$ attributes of the $\mathrm{\pi \pi °\pi ±\pi »\pi ΄}$ collection, while the second graph constraint is similar to the one used for defining the $\mathrm{\pi \pi \pi \pi \pi \pi \pi }$ constraint.

PartsΒ (A) andΒ (B) of FigureΒ 5.374.2 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 unique arc of the final graph is stressed in bold.

Automaton

FigureΒ 5.374.3 depicts the automaton associated with the $\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi }$ constraint. Let $\mathrm{\pi Έ\pi ½\pi ³\pi ΄\pi }$ and $\mathrm{\pi  \pi °\pi »\pi \pi ΄}$ respectively be the $\mathrm{\pi \pi \pi \pi \pi ‘}$ and the $\mathrm{\pi \pi \pi \pi \pi }$ attributes of the unique item of the $\mathrm{\pi Έ\pi \pi ΄\pi Ό}$ collection. Let ${\mathrm{\pi »\pi Ύ\pi }}_{i}$, ${\mathrm{\pi \pi Ώ}}_{i}$ and ${\mathrm{\pi  \pi °\pi »\pi \pi ΄}}_{i}$ respectively be the , the $\mathrm{\pi \pi }$ and the $\mathrm{\pi \pi \pi \pi \pi }$ attributes of the ${i}^{th}$ item of the $\mathrm{\pi \pi °\pi ±\pi »\pi ΄}$ collection. To each quintuple $\left(\mathrm{\pi Έ\pi ½\pi ³\pi ΄\pi },\mathrm{\pi  \pi °\pi »\pi \pi ΄},{\mathrm{\pi »\pi Ύ\pi }}_{i},{\mathrm{\pi \pi Ώ}}_{i},{\mathrm{\pi  \pi °\pi »\pi \pi ΄}}_{i}\right)$ corresponds a 0-1 signature variable ${S}_{i}$ as well as the following signature constraint: $\left(\left({\mathrm{\pi »\pi Ύ\pi }}_{i}\beta €\mathrm{\pi Έ\pi ½\pi ³\pi ΄\pi }\right)\beta §\left(\mathrm{\pi Έ\pi ½\pi ³\pi ΄\pi }\beta €{\mathrm{\pi \pi Ώ}}_{i}\right)\beta §\left(\mathrm{\pi  \pi °\pi »\pi \pi ΄}={\mathrm{\pi  \pi °\pi »\pi \pi ΄}}_{i}\right)\right)\beta {S}_{i}$.