## 5.406. twin

Origin

Pairs of variables related by hiden $\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}$ constraints sharing the same table.

Constraint

$\mathrm{𝚝𝚠𝚒𝚗}\left(\mathrm{𝙿𝙰𝙸𝚁𝚂}\right)$

Argument
 $\mathrm{𝙿𝙰𝙸𝚁𝚂}$ $\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(𝚡-\mathrm{𝚍𝚟𝚊𝚛},𝚢-\mathrm{𝚍𝚟𝚊𝚛}\right)$
Restrictions
 $\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}$$\left(\mathrm{𝙿𝙰𝙸𝚁𝚂},𝚡\right)$ $\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}$$\left(\mathrm{𝙿𝙰𝙸𝚁𝚂},𝚢\right)$ $|\mathrm{𝙿𝙰𝙸𝚁𝚂}|>0$
Purpose

Enforce the condition $\mathrm{𝙿𝙰𝙸𝚁𝚂}\left[i\right].𝚡=u\wedge \mathrm{𝙿𝙰𝙸𝚁𝚂}\left[i\right].𝚢=v\left(i\in \left[1,|\mathrm{𝙿𝙰𝙸𝚁𝚂}|\right]\right)⇒\forall j\in \left[1,|\mathrm{𝙿𝙰𝙸𝚁𝚂}|\right]:\mathrm{𝙿𝙰𝙸𝚁𝚂}\left[j\right].𝚡=u⇔\mathrm{𝙿𝙰𝙸𝚁𝚂}\left[j\right].𝚢=v$.

Example
$\left(\begin{array}{c}〈\begin{array}{cc}𝚡-1\hfill & 𝚢-8,\hfill \\ 𝚡-9\hfill & 𝚢-6,\hfill \\ 𝚡-1\hfill & 𝚢-8,\hfill \\ 𝚡-5\hfill & 𝚢-0,\hfill \\ 𝚡-6\hfill & 𝚢-7,\hfill \\ 𝚡-9\hfill & 𝚢-6\hfill \end{array}〉\hfill \end{array}\right)$

The $\mathrm{𝚝𝚠𝚒𝚗}$ constraint holds since 1 is paired with 8, 9 is paired with 6, 5 is paired with 0, 6 is paired with 7.

Typical
 $|\mathrm{𝙿𝙰𝙸𝚁𝚂}|>1$ $|\mathrm{𝙿𝙰𝙸𝚁𝚂}|>$$\mathrm{𝚗𝚟𝚊𝚕}$$\left(\mathrm{𝙿𝙰𝙸𝚁𝚂}.𝚡\right)$ $|\mathrm{𝙿𝙰𝙸𝚁𝚂}|>$$\mathrm{𝚗𝚟𝚊𝚕}$$\left(\mathrm{𝙿𝙰𝙸𝚁𝚂}.𝚢\right)$ $\mathrm{𝚗𝚟𝚊𝚕}$$\left(\mathrm{𝙿𝙰𝙸𝚁𝚂}.𝚡\right)>1$ $\mathrm{𝚗𝚟𝚊𝚕}$$\left(\mathrm{𝙿𝙰𝙸𝚁𝚂}.𝚢\right)>1$ $\mathrm{𝚗𝚟𝚊𝚕}$$\left(\mathrm{𝙿𝙰𝙸𝚁𝚂}.𝚡\right)=$$\mathrm{𝚗𝚟𝚊𝚕}$$\left(\mathrm{𝙿𝙰𝙸𝚁𝚂}.𝚢\right)$ $\mathrm{𝚗𝚟𝚊𝚕}$$\left(\mathrm{𝙿𝙰𝙸𝚁𝚂}.𝚡\right)<|\mathrm{𝙿𝙰𝙸𝚁𝚂}|$ $\mathrm{𝚗𝚟𝚊𝚕}$$\left(\mathrm{𝙿𝙰𝙸𝚁𝚂}.𝚢\right)<|\mathrm{𝙿𝙰𝙸𝚁𝚂}|$
Arg. properties

Contractible wrt. $\mathrm{𝙿𝙰𝙸𝚁𝚂}$.

related: $\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}$ (pairs linked by an element with the same table).