## 5.196. lex_equal

Origin

Initially introduced for defining $\mathrm{\pi \pi \pi \pi \pi \pi \pi }$

Constraint

$\mathrm{\pi \pi \pi ‘}_\mathrm{\pi \pi \pi \pi \pi }\left(\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{1},\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{2}\right)$

Synonyms

$\mathrm{\pi \pi \pi \pi \pi }$, $\mathrm{\pi \pi }$.

Arguments
 $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{1}$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$ $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{2}$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$
Restrictions
 $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{1},\mathrm{\pi \pi \pi }\right)$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{2},\mathrm{\pi \pi \pi }\right)$ $|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{1}|=|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{2}|$
Purpose

$\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{1}$ is equal to $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{2}$. Given two vectors, $\stackrel{\beta }{X}$ and $\stackrel{\beta }{Y}$ of $n$ components, $\beta ©{X}_{0},...,{X}_{n-1}\beta ͺ$ and $\beta ©{Y}_{0},...,{Y}_{n-1}\beta ͺ$, $\stackrel{\beta }{X}$ is equal to $\stackrel{\beta }{Y}$ if and only if $n=0$ or ${X}_{0}={Y}_{0}\beta §{X}_{1}={Y}_{1}\beta §...\beta §{X}_{n-1}={Y}_{n-1}$.

Example
$\left(\begin{array}{c}β©1,9,1,5βͺ,\hfill \\ β©1,9,1,5βͺ\hfill \end{array}\right)$

The $\mathrm{\pi \pi \pi ‘}_\mathrm{\pi \pi \pi \pi \pi }$ constraint holds since (1)Β the first component of the first vector is equal to the first component of the second vector, (2)Β the second component of the first vector is equal to the second component of the second vector, (3)Β the third component of the first vector is equal to the third component of the second vector and (4)Β the fourth component of the first vector is equal to the fourth component of the second vector.

Symmetries
• Arguments are permutable w.r.t. permutation $\left(\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{1},\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{2}\right)$.

• Items of $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{1}$ and $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{2}$ are permutable (same permutation used).

Used in

specialisation: $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi }_\mathrm{\pi \pi \pi \pi \pi }$Β ($\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$ replaced by $\mathrm{\pi \pi \pi \pi \pi \pi \pi }$ in second argument).

Keywords
Arc input(s)

$\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{1}$ $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{2}$

Arc generator
$\mathrm{\pi \pi  \pi \pi ·\pi \pi Ά\pi }$$\left(=\right)\beta ¦\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi \pi }\mathtt{1},\mathrm{\pi \pi \pi \pi \pi \pi }\mathtt{2}\right)$

Arc arity
Arc constraint(s)
$\mathrm{\pi \pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi \pi \pi }=\mathrm{\pi \pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi \pi \pi }$
Graph property(ies)
$\mathrm{\pi \pi \pi \pi }$$=|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{1}|$

Graph class
 $\beta ’$$\mathrm{\pi °\pi ²\pi \pi ²\pi »\pi Έ\pi ²}$ $\beta ’$$\mathrm{\pi ±\pi Έ\pi Ώ\pi °\pi \pi \pi Έ\pi \pi ΄}$ $\beta ’$$\mathrm{\pi ½\pi Ύ}_\mathrm{\pi »\pi Ύ\pi Ύ\pi Ώ}$

Graph model

PartsΒ (A) andΒ (B) of FigureΒ 5.196.1 respectively show the initial and final graphs associated with the Example slot. Since we use the $\mathrm{\pi \pi \pi \pi }$ graph property, the arcs of the final graph are stressed in bold.

Automaton

FigureΒ 5.196.2 depicts the automaton associated with the $\mathrm{\pi \pi \pi ‘}_\mathrm{\pi \pi \pi \pi \pi }$ constraint. Let $\mathrm{\pi  \pi °\pi }{\mathtt{1}}_{i}$ and $\mathrm{\pi  \pi °\pi }{\mathtt{2}}_{i}$ respectively be the $\mathrm{\pi \pi \pi }$ attributes of the ${i}^{th}$ items of the $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{1}$ and the $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{2}$ collections. To each pair $\left(\mathrm{\pi  \pi °\pi }{\mathtt{1}}_{i},\mathrm{\pi  \pi °\pi }{\mathtt{2}}_{i}\right)$ corresponds a signature variable ${\mathrm{\pi }}_{i}$ as well as the following signature constraint: .