## 5.97. deepest_valley

Origin
Constraint

$\mathrm{\pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi ’}\left(\mathrm{\pi ³\pi ΄\pi Ώ\pi \pi ·},\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\right)$

Arguments
 $\mathrm{\pi ³\pi ΄\pi Ώ\pi \pi ·}$ $\mathrm{\pi \pi \pi \pi }$ $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$
Restrictions
 $\mathrm{\pi ³\pi ΄\pi Ώ\pi \pi ·}\beta ₯0$ $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }\beta ₯0$ $\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

A variable ${V}_{k}$ $\left(1 of the sequence of variables $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }={V}_{1},...,{V}_{m}$ is a valley if and only if there exists an $i$ $\left(1 such that ${V}_{i-1}>{V}_{i}$ and ${V}_{i}={V}_{i+1}=...={V}_{k}$ and ${V}_{k}<{V}_{k+1}$. $\mathrm{\pi ³\pi ΄\pi Ώ\pi \pi ·}$ is the minimum value of the valley variables. If no such variable exists $\mathrm{\pi ³\pi ΄\pi Ώ\pi \pi ·}$ is equal to the default value $\mathrm{\pi Ό\pi °\pi \pi Έ\pi ½\pi }$.

Example
$\left(\begin{array}{c}2,β©\begin{array}{c}\mathrm{\pi \pi \pi }-5,\hfill \\ \mathrm{\pi \pi \pi }-3,\hfill \\ \mathrm{\pi \pi \pi }-4,\hfill \\ \mathrm{\pi \pi \pi }-8,\hfill \\ \mathrm{\pi \pi \pi }-8,\hfill \\ \mathrm{\pi \pi \pi }-2,\hfill \\ \mathrm{\pi \pi \pi }-7,\hfill \\ \mathrm{\pi \pi \pi }-1\hfill \end{array}βͺ\hfill \end{array}\right)$

The $\mathrm{\pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi ’}$ constraint holds since 2 is the deepest valley of the sequence $53488271$.

Typical
 $\mathrm{\pi ³\pi ΄\pi Ώ\pi \pi ·}\beta €$$\mathrm{\pi \pi \pi ‘\pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }\right)$ $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|>1$ $\mathrm{\pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }\right)>1$
Symmetry

Items of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ can be reversed.

FigureΒ 5.97.2 depicts the automaton associated with the $\mathrm{\pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi ’}$ constraint. To each pair of consecutive variables $\left({\mathrm{\pi  \pi °\pi }}_{i},{\mathrm{\pi  \pi °\pi }}_{i+1}\right)$ of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ corresponds a signature variable ${\mathrm{\pi }}_{i}$. The following signature constraint links ${\mathrm{\pi  \pi °\pi }}_{i}$, ${\mathrm{\pi  \pi °\pi }}_{i+1}$ and ${\mathrm{\pi }}_{i}$:
${\mathrm{\pi  \pi °\pi }}_{i}<{\mathrm{\pi  \pi °\pi }}_{i+1}\beta {\mathrm{\pi }}_{i}=0\beta §{\mathrm{\pi  \pi °\pi }}_{i}={\mathrm{\pi  \pi °\pi }}_{i+1}\beta {\mathrm{\pi }}_{i}=1\beta §{\mathrm{\pi  \pi °\pi }}_{i}>{\mathrm{\pi  \pi °\pi }}_{i+1}\beta {\mathrm{\pi }}_{i}=2$.