## 5.184. incomparable

Origin

Inspired by incomparable rectangles.

Constraint

$\mathrm{𝚒𝚗𝚌𝚘𝚖𝚙𝚊𝚛𝚊𝚋𝚕𝚎}\left(\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1},\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}\right)$

Synonym

$\mathrm{𝚒𝚗𝚌𝚘𝚖𝚙𝚊𝚛𝚊𝚋𝚕𝚎𝚜}$.

Arguments
 $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}$ $\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)$ $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}$ $\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)$
Restrictions
 $\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}$$\left(\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1},\mathrm{𝚟𝚊𝚛}\right)$ $\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}$$\left(\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2},\mathrm{𝚟𝚊𝚛}\right)$ $|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}|\ge 1$ $|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}|\ge 1$ $|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}|=|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}|$
Purpose

Enforce that when the components of $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}$ and $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}$ are ordered, and respectively denoted by $\mathrm{𝚂𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}$ and $\mathrm{𝚂𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}$, we neither have $\mathrm{𝚂𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}\left[i\right].\mathrm{𝚟𝚊𝚛}\le \mathrm{𝚂𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}\left[i\right].\mathrm{𝚟𝚊𝚛}$ (for all $i\in \left[1,|\mathrm{𝚂𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}|\right]$) nor have $\mathrm{𝚂𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}\left[i\right].\mathrm{𝚟𝚊𝚛}\le \mathrm{𝚂𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}\left[i\right].\mathrm{𝚟𝚊𝚛}$ (for all $i\in \left[1,|\mathrm{𝚂𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}|\right]$).

Example
$\left(〈16,2〉,〈4,11〉\right)$

The $\mathrm{𝚒𝚗𝚌𝚘𝚖𝚙𝚊𝚛𝚊𝚋𝚕𝚎}$ constraint holds since $16>4$ and $2<11$.

Typical
$|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}|>1$
Symmetries
• Items of $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}$ are permutable.

• Items of $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}$ are permutable.

• Arguments are permutable w.r.t. permutation $\left(\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1},\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}\right)$.

Used in
Keywords
Cond. implications

$•$ $\mathrm{𝚒𝚗𝚌𝚘𝚖𝚙𝚊𝚛𝚊𝚋𝚕𝚎}\left(\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1},\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}\right)$

with  $|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}|=2$

implies $\mathrm{𝚍𝚒𝚜𝚓𝚘𝚒𝚗𝚝}$$\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}:\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1},\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}:\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}\right)$.

$•$ $\mathrm{𝚒𝚗𝚌𝚘𝚖𝚙𝚊𝚛𝚊𝚋𝚕𝚎}\left(\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1},\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}\right)$

with  $|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}|=2$

implies $\mathrm{𝚒𝚗𝚝}_\mathrm{𝚟𝚊𝚕𝚞𝚎}_\mathrm{𝚙𝚛𝚎𝚌𝚎𝚍𝚎}_\mathrm{𝚌𝚑𝚊𝚒𝚗}$$\left(\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}:\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1},\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}:\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}\right)$.