## 5.33. arith_sliding

Origin

Used in the definition of some automaton

Constraint

$\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi },\mathrm{\pi \pi ΄\pi »\pi Ύ\pi Ώ},\mathrm{\pi  \pi °\pi »\pi \pi ΄}\right)$

Arguments
 $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$ $\mathrm{\pi \pi ΄\pi »\pi Ύ\pi Ώ}$ $\mathrm{\pi \pi \pi \pi }$ $\mathrm{\pi  \pi °\pi »\pi \pi ΄}$ $\mathrm{\pi \pi \pi }$
Restrictions
 $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi },\mathrm{\pi \pi \pi }\right)$
Purpose

Enforce for all sequences of variables ${\mathrm{\pi \pi \pi }}_{1},{\mathrm{\pi \pi \pi }}_{2},\beta ―,{\mathrm{\pi \pi \pi }}_{i}$ $\left(1\beta €i\beta €|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|\right)$ of the $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ collection to have $\left({\mathrm{\pi \pi \pi }}_{1}+{\mathrm{\pi \pi \pi }}_{2}+\beta ―+{\mathrm{\pi \pi \pi }}_{i}\right)\mathrm{\pi \pi ΄\pi »\pi Ύ\pi Ώ}\mathrm{\pi  \pi °\pi »\pi \pi ΄}$.

Example
$\left(β©0,0,1,2,0,0,-3βͺ,<,4\right)$

The $\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi }$ constraint holds since all the following seven inequalities hold:

• $0<4$,

• $0+0<4$,

• $0+0+1<4$,

• $0+0+1+2<4$,

• $0+0+1+2+0<4$,

• $0+0+1+2+0+0<4$,

• $0+0+1+2+0+0-3<4$.

Typical
 $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|>1$ $\mathrm{\pi \pi ΄\pi »\pi Ύ\pi Ώ}\beta \left[<,\beta ₯,>,\beta €\right]$
Arg. properties
• Contractible wrt. $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ when $\mathrm{\pi \pi ΄\pi »\pi Ύ\pi Ώ}\beta \left[<,\beta €\right]$ and $\mathrm{\pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }\right)\beta ₯0$.

• Suffix-contractible wrt. $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$.

Keywords
Arc input(s)

$\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$

Arc generator
$\mathrm{\pi \pi ΄\pi \pi »}_\mathit{1}$$\beta ¦\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }$

Arc arity
$*$
Arc constraint(s)
$\mathrm{\pi \pi \pi \pi \pi }$$\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi },\mathrm{\pi \pi ΄\pi »\pi Ύ\pi Ώ},\mathrm{\pi  \pi °\pi »\pi \pi ΄}\right)$
Graph property(ies)
$\mathrm{\pi \pi \pi \pi }$$=|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|$

Automaton

FigureΒ 5.33.1 depicts the automaton associated with the $\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi }$ constraint. To each item of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ corresponds a signature variable ${S}_{i}$ that is equal to 0.