## 5.10. all_incomparable

Origin

Inspired by incomparable rectangles.

Constraint

$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }\right)$

Synonym

$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }$.

Type
 $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$
Argument
 $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\right)$
Restrictions
 $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi },\mathrm{\pi \pi \pi }\right)$ $|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }|\beta ₯1$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi },\mathrm{\pi \pi \pi }\right)$ $|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }|\beta ₯1$ $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi £\pi }$$\left(\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi },\mathrm{\pi \pi \pi }\right)$
Purpose

Enforce for each pair of distinct vectors of the $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }$ collection the fact that they are incomparable. Two vectors $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{1}$ and $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{2}$ are incomparable if and only, when the components of both vectors are ordered, and respectively denoted by $\mathrm{\pi \pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{1}$ and $\mathrm{\pi \pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{2}$, we neither have $\mathrm{\pi \pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{1}\left[i\right].\mathrm{\pi \pi \pi }\beta €\mathrm{\pi \pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{2}\left[i\right].\mathrm{\pi \pi \pi }$ (for all $i\beta \left[1,|\mathrm{\pi \pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{1}|\right]$) nor have $\mathrm{\pi \pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{2}\left[i\right].\mathrm{\pi \pi \pi }\beta €\mathrm{\pi \pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{1}\left[i\right].\mathrm{\pi \pi \pi }$ (for all $i\beta \left[1,|\mathrm{\pi \pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{1}|\right]$).

Example
$\left(\begin{array}{c}β©\begin{array}{c}\mathrm{\pi \pi \pi }-β©1,18βͺ,\hfill \\ \mathrm{\pi \pi \pi }-β©2,16βͺ,\hfill \\ \mathrm{\pi \pi \pi }-β©3,13βͺ,\hfill \\ \mathrm{\pi \pi \pi }-β©4,11βͺ,\hfill \\ \mathrm{\pi \pi \pi }-β©5,10βͺ,\hfill \\ \mathrm{\pi \pi \pi }-β©6,9βͺ,\hfill \\ \mathrm{\pi \pi \pi }-β©7,7βͺ\hfill \end{array}βͺ\hfill \end{array}\right)$

The $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }$ constraint holds since all distinct pairs of vectors are incomparable as illustrated by FigureΒ 5.10.1.

All solutions

FigureΒ 5.10.2 gives all solutions to the following non ground instance of the $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }$ constraint: ${U}_{1}\beta \left[1,2\right]$, ${V}_{1}\beta \left[0,5\right]$, ${U}_{2}\beta \left[3,5\right]$, ${V}_{2}\beta \left[2,3\right]$, ${U}_{3}\beta \left[0,6\right]$, ${V}_{3}\beta \left[2,5\right]$, $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\beta ©\beta ©{U}_{1},{V}_{1}\beta ͺ,\beta ©{U}_{2},{V}_{2}\beta ͺ,\beta ©{U}_{3},{V}_{3}\beta ͺ\beta ͺ\right)$.

Typical
 $|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }|>1$ $|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }|>1$ $|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }|>|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }|$
Symmetry

Items of $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }$ are permutable.

Arg. properties

Contractible wrt. $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }$.

Keywords
Cond. implications

$\beta ’$ $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }\right)$

Β Β Β  withΒ  $|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }|=2$

Β Β implies $\mathrm{\pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi \pi ΄\pi \pi }:\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }\right)$.

$\beta ’$ $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }\right)$

Β Β Β  withΒ  $|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }|=2$

Β Β implies $\left(\mathrm{\pi Ώ\pi °\pi Έ\pi \pi }:\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }\right)$.

Arc input(s)

$\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }$

Arc generator
$\mathrm{\pi Ά\pi Ώ\pi Ό\pi \pi \pi Έ}$

Arc arity
Arc constraint(s)
$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi \pi \pi },\mathrm{\pi \pi \pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi \pi \pi }\right)$
Graph property(ies)
$\mathrm{\pi \pi \pi \pi }$$=|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }|*|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }|-|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi \pi }|$

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

Graph model

The Arc constraint(s) slot uses the $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }$ constraint defined in this catalogue.

PartsΒ (A) andΒ (B) of FigureΒ 5.10.3 respectively show the initial and final graph associated with the Example slot. Since we use the $\mathrm{\pi \pi \pi \pi }$ graph property, the arcs of the final graph are stressed in bold. The previous constraint holds since exactly $3Β·\left(3-1\right)=6$ arc constraints hold.