## 5.221. lex_alldifferent_except_0

Origin

H. Simonis

Constraint

$\mathrm{𝚕𝚎𝚡}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚎𝚡𝚌𝚎𝚙𝚝}_\mathtt{0}\left(\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}\right)$

Synonyms

$\mathrm{𝚕𝚎𝚡}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏}_\mathrm{𝚎𝚡𝚌𝚎𝚙𝚝}_\mathtt{0}$, $\mathrm{𝚕𝚎𝚡}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}_\mathrm{𝚎𝚡𝚌𝚎𝚙𝚝}_\mathtt{0}$, $\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏}_\mathrm{𝚘𝚗}_\mathrm{𝚝𝚞𝚙𝚕𝚎𝚜}_\mathrm{𝚎𝚡𝚌𝚎𝚙𝚝}_\mathtt{0}$, $\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚘𝚗}_\mathrm{𝚝𝚞𝚙𝚕𝚎𝚜}_\mathrm{𝚎𝚡𝚌𝚎𝚙𝚝}_\mathtt{0}$, $\mathrm{𝚊𝚕𝚕𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}_\mathrm{𝚘𝚗}_\mathrm{𝚝𝚞𝚙𝚕𝚎𝚜}_\mathrm{𝚎𝚡𝚌𝚎𝚙𝚝}_\mathtt{0}$.

Type
 $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}$ $\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)$
Argument
 $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}$ $\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚎𝚌}-\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\right)$
Restrictions
 $|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}|\ge 1$ $\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}$$\left(\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁},\mathrm{𝚟𝚊𝚛}\right)$ $\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}$$\left(\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂},\mathrm{𝚟𝚎𝚌}\right)$ $\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚜𝚒𝚣𝚎}$$\left(\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂},\mathrm{𝚟𝚎𝚌}\right)$
Purpose

All the non null vectors of the collection $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}$ are distinct. A vector is null if all its components are equal to zero. Two non null vectors $\left({u}_{1},{u}_{2},\cdots ,{u}_{n}\right)$ and $\left({v}_{1},{v}_{2},\cdots ,{v}_{n}\right)$ are distinct if and only if there exists $i\in \left[1,n\right]$ such that ${u}_{i}\ne {v}_{i}$.

Example
$\left(\begin{array}{c}〈\begin{array}{c}\mathrm{𝚟𝚎𝚌}-〈0,0,0〉,\hfill \\ \mathrm{𝚟𝚎𝚌}-〈5,2,0〉,\hfill \\ \mathrm{𝚟𝚎𝚌}-〈5,8,0〉,\hfill \\ \mathrm{𝚟𝚎𝚌}-〈0,0,0〉\hfill \end{array}〉\hfill \end{array}\right)$

The $\mathrm{𝚕𝚎𝚡}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚎𝚡𝚌𝚎𝚙𝚝}_\mathtt{0}$ constraint holds since its two non null vectors, i.e. the second and third vectors are distinct (the vectors $〈5,2,0〉$ and $〈5,8,0〉$ differ in their second component.

Typical
 $|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}|>1$ $|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}|>1$
Arg. properties

Contractible wrt. $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}$.