## 5.79. compare_and_count

Origin
Constraint

$\mathrm{\pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1},\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2},\mathrm{\pi ²\pi Ύ\pi Ό\pi Ώ\pi °\pi \pi ΄},\mathrm{\pi ²\pi Ύ\pi \pi ½\pi },\mathrm{\pi »\pi Έ\pi Ό\pi Έ\pi }\right)$

Arguments
 $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$ $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$ $\mathrm{\pi ²\pi Ύ\pi Ό\pi Ώ\pi °\pi \pi ΄}$ $\mathrm{\pi \pi \pi \pi }$ $\mathrm{\pi ²\pi Ύ\pi \pi ½\pi }$ $\mathrm{\pi \pi \pi \pi }$ $\mathrm{\pi »\pi Έ\pi Ό\pi Έ\pi }$ $\mathrm{\pi \pi \pi \pi }$
Restrictions
 $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}|=|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}|$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1},\mathrm{\pi \pi \pi }\right)$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2},\mathrm{\pi \pi \pi }\right)$ $\mathrm{\pi »\pi Έ\pi Ό\pi Έ\pi }\beta ₯0$
Purpose

Enforce the condition

$\left({\beta }_{i=1}^{|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}|}\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}\left[i\right].\mathrm{\pi \pi \pi }\mathrm{\pi ²\pi Ύ\pi Ό\pi Ώ\pi °\pi \pi ΄}\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}\left[i\right].\mathrm{\pi \pi \pi }\right)\mathrm{\pi ²\pi Ύ\pi \pi ½\pi }\mathrm{\pi »\pi Έ\pi Ό\pi Έ\pi }$.

Example
$\left(β©4,5,5,4,5βͺ,β©4,2,5,1,5βͺ,=,\beta €,3\right)$

The $\mathrm{\pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }$ constraint holds since no more than $\mathrm{\pi »\pi Έ\pi Ό\pi Έ\pi }=3$ pairs of variables are equal, i.e.,Β the first, third and fifth pairs.

Typical
 $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}|>1$ $\mathrm{\pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}.\mathrm{\pi \pi \pi }\right)>1$ $\mathrm{\pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}.\mathrm{\pi \pi \pi }\right)>1$ $\mathrm{\pi ²\pi Ύ\pi Ό\pi Ώ\pi °\pi \pi ΄}\beta \left[=\right]$ $\mathrm{\pi ²\pi Ύ\pi \pi ½\pi }\beta \left[=,<,\beta ₯,>,\beta €\right]$ $\mathrm{\pi »\pi Έ\pi Ό\pi Έ\pi }>0$ $\mathrm{\pi »\pi Έ\pi Ό\pi Έ\pi }<|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}|$
Arg. properties
• Contractible wrt. $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}$ and $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}$ (remove items from same position) when $\mathrm{\pi ²\pi Ύ\pi \pi ½\pi }\beta \left[<,\beta €\right]$.

• Extensible wrt. $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}$ and $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}$ (add items at same position) when $\mathrm{\pi ²\pi Ύ\pi \pi ½\pi }\beta \left[\beta ₯,>\right]$.