### 2.2.2.1. Collection generators

The vertices of the initial graph are usually directly generated from collections of items that are arguments of the global constraint $\mathrm{\pi Ά}$ under consideration. However, it sometimes happens that we would like to derive a new collection from existing arguments of $\mathrm{\pi Ά}$ in order to produce the vertices of the initial graph.

EXAMPLE: This is for instance the case of the $\mathrm{\pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi Έ\pi ½\pi ³\pi ΄\pi },$ $\mathrm{\pi \pi °\pi ±\pi »\pi ΄},$ $\mathrm{\pi  \pi °\pi »\pi \pi ΄}\right)$ constraint, where $\mathrm{\pi Έ\pi ½\pi ³\pi ΄\pi }$ and $\mathrm{\pi  \pi °\pi »\pi \pi ΄}$ are domain variables that we would like to group as a single item $\mathrm{\beta }$ (with two attributes) of a new derived collection. This is in fact done in order to generate the following initial graph:

• The item $\mathrm{\beta }$ as well as all items of $\mathrm{\pi \pi °\pi ±\pi »\pi ΄}$ constitute the vertices,

• There is an arc from $\mathrm{\beta }$ to each item of the $\mathrm{\pi \pi °\pi ±\pi »\pi ΄}$ collection.

Β

We provide the following mechanism for deriving new collections:

• In a first phase we declare the name of the new collection as well as the names of its attributes and their respective types. This is achieved exactly in the same way as those collections that are used in the arguments of a global constraint (see page 2.1.2).

EXAMPLE: Consider again the example of the $\mathrm{\pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi Έ\pi ½\pi ³\pi ΄\pi },$ $\mathrm{\pi \pi °\pi ±\pi »\pi ΄},$ $\mathrm{\pi  \pi °\pi »\pi \pi ΄}\right)$ constraint. The declaration $\mathrm{\pi Έ\pi \pi ΄\pi Ό}-\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi ‘}-\mathrm{\pi \pi \pi \pi },\mathrm{\pi \pi \pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$ introduces a new collection called $\mathrm{\pi Έ\pi \pi ΄\pi Ό}$ where each item has an $\mathrm{\pi \pi \pi \pi \pi ‘}$ and a $\mathrm{\pi \pi \pi \pi \pi }$ attribute. Both attributes correspond to domain variables.

• In a second phase we give a list of patterns that are used for generating the items of the new collection. A pattern $o-\mathrm{\pi \pi \pi \pi }\left({\mathrm{\pi }}_{1}-{v}_{1},{\mathrm{\pi }}_{2}-{v}_{2},...,{\mathrm{\pi }}_{n}-{v}_{n}\right)$ or $\mathrm{\pi \pi \pi \pi }\left({\mathrm{\pi }}_{1}-{v}_{1},{\mathrm{\pi }}_{2}-{v}_{2},...,{\mathrm{\pi }}_{n}-{v}_{n}\right)$ specifies for each attribute ${\mathrm{\pi }}_{i}\left(1\beta €i\beta €n\right)$ of the new collection how to fill it.$o$ is one of the comparison operators . When omitted its default value is $=$. This is done by providing for each attribute ${\mathrm{\pi }}_{i}$ one of the following expression ${v}_{i}$:

• A constant.

• An argument of the global constraint $\mathrm{\pi Ά}$.

• An expression $\mathrm{\pi }.\mathrm{\pi }$, where $\mathrm{\pi }$ is an attribute of a collection $\mathrm{\pi }$, such that $\mathrm{\pi }$ is an argument of the global constraint $\mathrm{\pi Ά}$ or a derived collection that was previously declared. An expression of this form is called a direct reference to an attribute of a collection.

• An expression ${\mathrm{\pi }}_{1}.{\mathrm{\pi }}_{2}.\mathrm{\pi }$, where $\mathrm{\pi }$ is an attribute of a collection ${\mathrm{\pi }}_{2}$, and ${\mathrm{\pi }}_{2}$ is an attribute of a collection ${\mathrm{\pi }}_{1}$ such that ${\mathrm{\pi }}_{1}$ is an argument of the global constraint $\mathrm{\pi Ά}$ or a derived collection that was previously declared. An expression of this form is called an indirect reference to an attribute of a collection.

This expression ${v}_{i}$ must be compatible with the type declaration of the corresponding attribute of the new collection.

EXAMPLE: We continue the example of the $\mathrm{\pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi Έ\pi ½\pi ³\pi ΄\pi },\mathrm{\pi \pi °\pi ±\pi »\pi ΄},\mathrm{\pi  \pi °\pi »\pi \pi ΄}\right)$ constraint and the derived collection $\mathrm{\pi Έ\pi \pi ΄\pi Ό}-\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi ‘}-\mathrm{\pi \pi \pi \pi },\mathrm{\pi \pi \pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$. The pattern $\mathrm{\pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi ‘}-\mathrm{\pi Έ\pi ½\pi ³\pi ΄\pi },\mathrm{\pi \pi \pi \pi \pi }-\mathrm{\pi  \pi °\pi »\pi \pi ΄}\right)$ indicates that:

• The $\mathrm{\pi \pi \pi \pi \pi ‘}$ attribute of the $\mathrm{\pi Έ\pi \pi ΄\pi Ό}$ collection will be generated by using the $\mathrm{\pi Έ\pi ½\pi ³\pi ΄\pi }$ argument of the $\mathrm{\pi \pi \pi \pi \pi \pi \pi }$ constraint. Since $\mathrm{\pi Έ\pi ½\pi ³\pi ΄\pi }$ is a domain variable, it is compatible with the declaration $\mathrm{\pi Έ\pi \pi ΄\pi Ό}-\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi ‘}-\mathrm{\pi \pi \pi \pi },\mathrm{\pi \pi \pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$ of the new collection.

• The $\mathrm{\pi \pi \pi \pi \pi }$ attribute of the $\mathrm{\pi Έ\pi \pi ΄\pi Ό}$ collection will be generated by using the $\mathrm{\pi  \pi °\pi »\pi \pi ΄}$ argument of the $\mathrm{\pi \pi \pi \pi \pi \pi \pi }$ constraint. $\mathrm{\pi  \pi °\pi »\pi \pi ΄}$ is also compatible with the declaration statement of the new collection.

We now describe how we use the pattern for generating the items of a derived collection. We have the following two cases:

• If the pattern $o-\mathrm{\pi \pi \pi \pi }\left({\mathrm{\pi }}_{1}-{v}_{1},{\mathrm{\pi }}_{2}-{v}_{2},...,{\mathrm{\pi }}_{n}-{v}_{n}\right)$ does not contain any direct or indirect reference to an attribute of a collection then we generate one single item for such pattern.In this first case the value of $o$ is irrelevant. In this context the value ${v}_{i}$ of the attribute ${\mathrm{\pi }}_{i}$ $\left(1\beta €i\beta €n\right)$ corresponds to a constant, to an argument of the global constraint or to a new derived collection.

• If the pattern $o-\mathrm{\pi \pi \pi \pi }\left({\mathrm{\pi }}_{1}-{v}_{1},{\mathrm{\pi }}_{2}-{v}_{2},...,{\mathrm{\pi }}_{n}-{v}_{n}\right)$, where $o$ is one of the comparison operators , contains one or several direct or indirect references to an attribute of a collectionThis collection is an argument of the global constraint or corresponds to a newly derived collection. we denote by:

• $\mathrm{\pi }$ the set of indices of the positions corresponding to a direct reference to an attribute of a collection within $\mathrm{\pi \pi \pi \pi }\left({\mathrm{\pi }}_{1}-{v}_{1},{\mathrm{\pi }}_{2}-{v}_{2},...,{\mathrm{\pi }}_{n}-{v}_{n}\right)$. In this context, let ${\mathrm{\pi }}_{{\mathrm{\Xi ±}}_{1}},{\mathrm{\pi }}_{{\mathrm{\Xi ±}}_{2}},...,{\mathrm{\pi }}_{{\mathrm{\Xi ±}}_{m}}$ and ${\mathrm{\pi }}_{{\mathrm{\Xi ±}}_{1}},{\mathrm{\pi }}_{{\mathrm{\Xi ±}}_{2}},...,{\mathrm{\pi }}_{{\mathrm{\Xi ±}}_{m}}$ respectively denote the corresponding collections and attributes.

• $\mathrm{\beta }$ the set of indices of the positions corresponding to an indirect reference to an attribute of a collection within $\mathrm{\pi \pi \pi \pi }\left({\mathrm{\pi }}_{1}-{v}_{1},{\mathrm{\pi }}_{2}-{v}_{2},...,{\mathrm{\pi }}_{n}-{v}_{n}\right)$. In this context, let ${\mathrm{\pi }}_{{\mathrm{\Xi ²}}_{1}}^{1},{\mathrm{\pi }}_{{\mathrm{\Xi ²}}_{2}}^{1},...,{\mathrm{\pi }}_{{\mathrm{\Xi ²}}_{p}}^{1}$, ${\mathrm{\pi }}_{{\mathrm{\Xi ²}}_{1}}^{2},{\mathrm{\pi }}_{{\mathrm{\Xi ²}}_{2}}^{2},...,{\mathrm{\pi }}_{{\mathrm{\Xi ²}}_{p}}^{2}$ and ${\mathrm{\pi }}_{{\mathrm{\Xi ²}}_{1}},{\mathrm{\pi }}_{{\mathrm{\Xi ²}}_{2}},...,{\mathrm{\pi }}_{{\mathrm{\Xi ²}}_{p}}$ respectively denote the corresponding collections, attributes of type collection and attributes.

• Let ${\mathrm{\pi \pi \pi }}_{1},{\mathrm{\pi \pi \pi }}_{2},...,{\mathrm{\pi \pi \pi }}_{m}$, ${\mathrm{\pi \pi \pi }}_{1},{\mathrm{\pi \pi \pi }}_{2},...,{\mathrm{\pi \pi \pi }}_{p}$ and ${\mathrm{\pi \pi }}_{1},{\mathrm{\pi \pi }}_{2},...,{\mathrm{\pi \pi }}_{m+p}$ respectively denote the indices sorted in increasing order of $\mathrm{\pi }$, $\mathrm{\beta }$ and $\mathrm{\pi }\beta ͺ\mathrm{\beta }$.

For each combination of items ${\mathrm{\pi }}_{{\mathrm{\Xi ±}}_{1}}\left[{i}_{1}\right],{\mathrm{\pi }}_{{\mathrm{\Xi ±}}_{2}}\left[{i}_{2}\right],...,{\mathrm{\pi }}_{{\mathrm{\Xi ±}}_{m}}\left[{i}_{m}\right]$, ${\mathrm{\pi }}_{{\mathrm{\Xi ²}}_{1}}^{1}\left[{j}_{1}\right].{\mathrm{\pi }}_{{\mathrm{\Xi ²}}_{1}}^{2}\left[{k}_{1}\right],{\mathrm{\pi }}_{{\mathrm{\Xi ²}}_{2}}^{1}\left[{j}_{2}\right].{\mathrm{\pi }}_{{\mathrm{\Xi ²}}_{2}}^{2}\left[{k}_{2}\right],...,{\mathrm{\pi }}_{{\mathrm{\Xi ²}}_{p}}^{1}\left[{j}_{p}\right].{\mathrm{\pi }}_{{\mathrm{\Xi ²}}_{p}}^{2}\left[{k}_{p}\right]$ such that:

$\left\{\begin{array}{cccc}{i}_{1}\beta \left[1,|{\mathrm{\pi }}_{{\mathrm{\Xi ±}}_{1}}|\right],\hfill & {i}_{2}\beta \left[1,|{\mathrm{\pi }}_{{\mathrm{\Xi ±}}_{2}}|\right],\hfill & ...,\hfill & {i}_{m}\beta \left[1,|{\mathrm{\pi }}_{{\mathrm{\Xi ±}}_{m}}|\right]\hfill \\ {j}_{1}\beta \left[1,|{\mathrm{\pi }}_{{\mathrm{\Xi ²}}_{1}}^{1}|\right],\hfill & {j}_{2}\beta \left[1,|{\mathrm{\pi }}_{{\mathrm{\Xi ²}}_{2}}^{1}|\right],\hfill & ...,\hfill & {j}_{p}\beta \left[1,|{\mathrm{\pi }}_{{\mathrm{\Xi ²}}_{p}}^{1}|\right]\hfill \\ {k}_{1}\beta \left[1,|{\mathrm{\pi }}_{{\mathrm{\Xi ²}}_{1}}^{1}\left[{j}_{1}\right].{\mathrm{\pi }}_{{\mathrm{\Xi ²}}_{1}}^{2}|\right],\hfill & {k}_{2}\beta \left[1,|{\mathrm{\pi }}_{{\mathrm{\Xi ²}}_{2}}^{1}\left[{j}_{2}\right].{\mathrm{\pi }}_{{\mathrm{\Xi ²}}_{2}}^{2}|\right],\hfill & ...,\hfill & {k}_{p}\beta \left[1,|{\mathrm{\pi }}_{{\mathrm{\Xi ²}}_{p}}^{1}\left[{j}_{p}\right].{\mathrm{\pi }}_{{\mathrm{\Xi ²}}_{p}}^{2}|\right]\hfill \\ {\mathrm{\pi \pi }}_{1}o{\mathrm{\pi \pi }}_{2}o...o{\mathrm{\pi \pi }}_{m+p}\hfill \end{array}\right\$

we generate an item of the new derived collection $\beta ©{\mathrm{\pi }}_{1}-{w}_{1}{\mathrm{\pi }}_{2}-{w}_{2}...{\mathrm{\pi }}_{n}-{w}_{n}\beta ͺ$ defined by:

${w}_{j}\left(1\beta €j\beta €n\right)=\left\{\begin{array}{cc}\hfill {\mathrm{\pi }}_{{\mathrm{\Xi ±}}_{r}}\left[{i}_{r}\right].{\mathrm{\pi }}_{{\mathrm{\Xi ±}}_{r}}& \mathrm{if}j\beta \mathrm{\pi },j={\mathrm{\pi \pi \pi }}_{r}\hfill \\ \hfill {\mathrm{\pi }}_{{\mathrm{\Xi ²}}_{r}}^{1}\left[{j}_{r}\right].{\mathrm{\pi }}_{{\mathrm{\Xi ²}}_{r}}^{2}\left[{k}_{r}\right].{\mathrm{\pi }}_{{\mathrm{\Xi ²}}_{r}}& \mathrm{if}j\beta \mathrm{\beta },j={\mathrm{\pi \pi \pi }}_{r}\hfill \\ \hfill {v}_{j}& \mathrm{if}j\beta \mathrm{\pi }\beta ͺ\mathrm{\beta }\hfill \end{array}\right\.$

We illustrate this generation process on a set of examples. Each example is described by providing:

• The global constraint and its arguments,

• The declaration of the new derived collection,

• The pattern used for creating an item of the new collection,

• The items generated by applying this pattern to the global constraint,

• A comment about the generation process.

We first start with four examples that do not mention any references to an attribute of a collection. A box surrounds an argument of a global constraint that is mentioned in a generated item.

EXAMPLE

$\mathrm{\pi ²\pi Ύ\pi ½\pi \pi \pi \pi °\pi Έ\pi ½\pi }$:

$\mathrm{\pi \pi \pi \pi \pi \pi \pi }$$\left(\begin{array}{|c|}\hline \mathrm{\pi Έ\pi ½\pi ³\pi ΄\pi }\\ \hline\end{array},\mathrm{\pi \pi °\pi ±\pi »\pi ΄},\begin{array}{|c|}\hline \mathrm{\pi  \pi °\pi »\pi \pi ΄}\\ \hline\end{array}\right)$

$\mathrm{\pi ³\pi ΄\pi \pi Έ\pi  \pi ΄\pi ³}\mathrm{\pi ²\pi Ύ\pi »\pi »\pi ΄\pi ²\pi \pi Έ\pi Ύ\pi ½}$:

$\mathrm{\pi Έ\pi \pi ΄\pi Ό}-\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi ‘}-\mathrm{\pi \pi \pi \pi },\mathrm{\pi \pi \pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$

$\mathrm{\pi Ώ\pi °\pi \pi \pi ΄\pi \pi ½}\left(\mathrm{\pi }\right)$Β Β Β Β Β Β Β Β Β :

$\mathrm{\pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi ‘}-\mathrm{\pi Έ\pi ½\pi ³\pi ΄\pi },\mathrm{\pi \pi \pi \pi \pi }-\mathrm{\pi  \pi °\pi »\pi \pi ΄}\right)$

$\mathrm{\pi Ά\pi ΄\pi ½\pi ΄\pi \pi °\pi \pi ΄\pi ³}\mathrm{\pi Έ\pi \pi ΄\pi Ό}\left(\mathrm{\pi }\right)$Β Β :

$\beta ©\mathrm{\pi \pi \pi \pi \pi ‘}-\begin{array}{|c|}\hline \mathrm{\pi Έ\pi ½\pi ³\pi ΄\pi }\\ \hline\end{array}\mathrm{\pi \pi \pi \pi \pi }-\begin{array}{|c|}\hline \mathrm{\pi  \pi °\pi »\pi \pi ΄}\\ \hline\end{array}\beta ͺ$

We generate one single item where the two attributes $\mathrm{\pi \pi \pi \pi \pi ‘}$ and $\mathrm{\pi \pi \pi \pi \pi }$ respectively take the first argument $\mathrm{\pi Έ\pi ½\pi ³\pi ΄\pi }$ and the third argument $\mathrm{\pi  \pi °\pi »\pi \pi ΄}$ of the $\mathrm{\pi \pi \pi \pi \pi \pi \pi }$ constraint.

EXAMPLE

$\mathrm{\pi ²\pi Ύ\pi ½\pi \pi \pi \pi °\pi Έ\pi ½\pi }$:

$\mathrm{\pi \pi \pi ‘}_\mathrm{\pi \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)$

$\mathrm{\pi ³\pi ΄\pi \pi Έ\pi  \pi ΄\pi ³}\mathrm{\pi ²\pi Ύ\pi »\pi »\pi ΄\pi ²\pi \pi Έ\pi Ύ\pi ½}$:

$\mathrm{\pi ³\pi ΄\pi \pi \pi Έ\pi ½\pi °\pi \pi Έ\pi Ύ\pi ½}-\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi ‘}-\mathrm{\pi \pi \pi },\mathrm{\pi ‘}-\mathrm{\pi \pi \pi },\mathrm{\pi ’}-\mathrm{\pi \pi \pi }\right)$

$\mathrm{\pi Ώ\pi °\pi \pi \pi ΄\pi \pi ½}\left(\mathrm{\pi }\right)$Β Β Β Β Β Β Β Β Β Β :

$\mathrm{\pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi ‘}-\mathtt{0},\mathrm{\pi ‘}-\mathtt{0},\mathrm{\pi ’}-\mathtt{0}\right)$

$\mathrm{\pi Ά\pi ΄\pi ½\pi ΄\pi \pi °\pi \pi ΄\pi ³}\mathrm{\pi Έ\pi \pi ΄\pi Ό}\left(\mathrm{\pi }\right)$Β Β :

$\beta ©\mathrm{\pi \pi \pi \pi \pi ‘}-\mathtt{0}\mathrm{\pi ‘}-\mathtt{0}\mathrm{\pi ’}-\mathtt{0}\beta ͺ$

We generate one single item where the three attributes $\mathrm{\pi \pi \pi \pi \pi ‘}$, $\mathrm{\pi ‘}$ and $\mathrm{\pi ’}$ take value 0.

EXAMPLE

$\mathrm{\pi ²\pi Ύ\pi ½\pi \pi \pi \pi °\pi Έ\pi ½\pi }$:

$\mathrm{\pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\begin{array}{|c|}\hline \mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\\ \hline\end{array},\mathrm{\pi \pi \pi Ώ\pi »\pi ΄\pi }_\mathrm{\pi Ύ\pi ΅}_\mathrm{\pi  \pi °\pi »\pi }\right)$

$\mathrm{\pi ³\pi ΄\pi \pi Έ\pi  \pi ΄\pi ³}\mathrm{\pi ²\pi Ύ\pi »\pi »\pi ΄\pi ²\pi \pi Έ\pi Ύ\pi ½}$:

$\mathrm{\pi \pi \pi Ώ\pi »\pi ΄\pi }_\mathrm{\pi Ύ\pi ΅}_\mathrm{\pi  \pi °\pi \pi }-\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi Ώ\pi »\pi ΄}_\mathrm{\pi Ύ\pi ΅}_\mathrm{\pi  \pi °\pi \pi }\right)$

$\mathrm{\pi Ώ\pi °\pi \pi \pi ΄\pi \pi ½}\left(\mathrm{\pi }\right)$Β Β Β Β Β Β Β Β Β :

$\mathrm{\pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\right)$

$\mathrm{\pi Ά\pi ΄\pi ½\pi ΄\pi \pi °\pi \pi ΄\pi ³}\mathrm{\pi Έ\pi \pi ΄\pi Ό}\left(\mathrm{\pi }\right)$Β Β :

$\beta ©\mathrm{\pi \pi \pi }-\begin{array}{|c|}\hline \mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\\ \hline\end{array}\beta ͺ$

We generate one single item where the unique attribute $\mathrm{\pi \pi \pi }$ takes the first argument of the $\mathrm{\pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$ constraint as its value.

EXAMPLE

$\mathrm{\pi ²\pi Ύ\pi ½\pi \pi \pi \pi °\pi Έ\pi ½\pi }$:

$\mathrm{\pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\begin{array}{|c|}\hline \mathrm{\pi  \pi °\pi }\\ \hline\end{array},\mathrm{\pi  \pi °\pi »\pi \pi ΄\pi }\right)$

$\mathrm{\pi ³\pi ΄\pi \pi Έ\pi  \pi ΄\pi ³}\mathrm{\pi ²\pi Ύ\pi »\pi »\pi ΄\pi ²\pi \pi Έ\pi Ύ\pi ½}$:

$\mathrm{\pi  \pi °\pi »\pi \pi ΄}-\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }\mathtt{01}-\mathrm{\pi \pi \pi },\mathrm{\pi \pi \pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$

$\mathrm{\pi Ώ\pi °\pi \pi \pi ΄\pi \pi ½}\left(\mathrm{\pi }\right)$Β Β Β Β Β Β Β Β Β :

$\mathrm{\pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }\mathtt{01}-\mathtt{1},\mathrm{\pi \pi \pi \pi \pi }-\mathrm{\pi  \pi °\pi }\right)$

$\mathrm{\pi Ά\pi ΄\pi ½\pi ΄\pi \pi °\pi \pi ΄\pi ³}\mathrm{\pi Έ\pi \pi ΄\pi Ό}\left(\mathrm{\pi }\right)$Β Β :

$\beta ©\mathrm{\pi \pi \pi }\mathtt{01}-\mathtt{1}\mathrm{\pi \pi \pi \pi \pi }-\begin{array}{|c|}\hline \mathrm{\pi  \pi °\pi }\\ \hline\end{array}\beta ͺ$

We generate one single item where the two attributes $\mathrm{\pi \pi \pi }\mathtt{01}$ and $\mathrm{\pi \pi \pi \pi \pi }$ respectively take value 1 and the first argument of the $\mathrm{\pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }$ constraint.

We continue with three examples that mention one or several direct references to an attribute of some collections. We now need to explicitly give the items of these collections in order to generate the items of the derived collection.

EXAMPLE

$\mathrm{\pi ²\pi Ύ\pi ½\pi \pi \pi \pi °\pi Έ\pi ½\pi }$:

$\mathrm{\pi \pi \pi ‘}_\mathrm{\pi \pi \pi \pi \pi \pi }$$\left(\begin{array}{|c|}\hline \mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{1}\\ \hline\end{array},\begin{array}{|c|}\hline \mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{2}\\ \hline\end{array}\right)$

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

$\beta ©\mathrm{\pi \pi \pi }-\mathtt{5},\mathrm{\pi \pi \pi }-\mathtt{2},\mathrm{\pi \pi \pi }-\mathtt{3},\mathrm{\pi \pi \pi }-\mathtt{1}\beta ͺ$

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

$\beta ©\mathrm{\pi \pi \pi }-\mathtt{5},\mathrm{\pi \pi \pi }-\mathtt{2},\mathrm{\pi \pi \pi }-\mathtt{6},\mathrm{\pi \pi \pi }-\mathtt{2}\beta ͺ$

$\mathrm{\pi ³\pi ΄\pi \pi Έ\pi  \pi ΄\pi ³}\mathrm{\pi ²\pi Ύ\pi »\pi »\pi ΄\pi ²\pi \pi Έ\pi Ύ\pi ½}$:

$\mathrm{\pi ²\pi Ύ\pi Ό\pi Ώ\pi Ύ\pi ½\pi ΄\pi ½\pi \pi }-\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi ‘}-\mathrm{\pi \pi \pi },$

$\mathrm{\pi ‘}-\mathrm{\pi \pi \pi \pi },\mathrm{\pi ’}-\mathrm{\pi \pi \pi \pi }\right)$

$\mathrm{\pi Ώ\pi °\pi \pi \pi ΄\pi \pi ½}\left(\mathrm{\pi }\right)$Β Β Β Β Β Β Β Β Β Β :

$\mathrm{\pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi ‘}-\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{1}.\mathrm{\pi \pi \pi ’}$As defined in SectionΒ 2.1.2 on page 2.1.2, $\mathrm{\pi \pi \pi ’}$ is an implicit attribute corresponding to the position of an item within a collection.$,$

$\mathrm{\pi ‘}-\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{1}.\mathrm{\pi \pi \pi },\mathrm{\pi ’}-\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{2}.\mathrm{\pi \pi \pi }\right)$

$\mathrm{\pi Ά\pi ΄\pi ½\pi ΄\pi \pi °\pi \pi ΄\pi ³}\mathrm{\pi Έ\pi \pi ΄\pi Ό}\left(\mathrm{\pi }\right)$Β Β :

$\beta ©\mathrm{\pi \pi \pi \pi \pi ‘}-\mathtt{1}\mathrm{\pi ‘}-\mathtt{5}\mathrm{\pi ’}-\mathtt{5},\mathrm{\pi \pi \pi \pi \pi ‘}-\mathtt{2}\mathrm{\pi ‘}-\mathtt{2}\mathrm{\pi ’}-\mathtt{2},$

$\mathrm{\pi \pi \pi \pi \pi ‘}-\mathtt{3}\mathrm{\pi ‘}-\mathtt{3}\mathrm{\pi ’}-\mathtt{6},\mathrm{\pi \pi \pi \pi \pi ‘}-\mathtt{4}\mathrm{\pi ‘}-\mathtt{1}\mathrm{\pi ’}-\mathtt{2}\beta ͺ$

The pattern mentions three references $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{1}.\mathrm{\pi \pi \pi ’}$, $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{1}.\mathrm{\pi \pi \pi }$ and $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{2}.\mathrm{\pi \pi \pi }$ to the collections $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{1}$ and $\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{2}$ used in the arguments of the $\mathrm{\pi \pi \pi ‘}_\mathrm{\pi \pi \pi \pi \pi \pi }$ constraint. $\beta {i}_{1}\beta \left[1,|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{1}|\right],\beta {i}_{2}\beta \left[1,|\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{2}|\right]$ such that ${i}_{1}={i}_{2}$We use an equality since this is the default value of the comparison operator $o$ when we do not use a pattern of the form $o-\mathrm{\pi \pi \pi \pi }\left(...\right)$. we generate an item $\mathrm{\pi \pi \pi \pi \pi ‘}-{v}_{1}\mathrm{\pi ‘}-{v}_{2}\mathrm{\pi ’}-{v}_{3}$ where:

${v}_{1}={i}_{1},{v}_{2}=\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{1}\left[{i}_{1}\right].\mathrm{\pi \pi \pi },{v}_{3}=\mathrm{\pi  \pi ΄\pi ²\pi \pi Ύ\pi }\mathtt{2}\left[{i}_{1}\right].\mathrm{\pi \pi \pi }.$

This leads to the four items listed in the $\mathrm{\pi Ά\pi ΄\pi ½\pi ΄\pi \pi °\pi \pi ΄\pi ³}\mathrm{\pi Έ\pi \pi ΄\pi Ό}\left(\mathrm{\pi }\right)$ field.

EXAMPLE

$\mathrm{\pi ²\pi Ύ\pi ½\pi \pi \pi \pi °\pi Έ\pi ½\pi }$:

$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\begin{array}{|c|}\hline \mathrm{\pi \pi °\pi \pi Ί\pi }\\ \hline\end{array},\mathrm{\pi Ό\pi °\pi ²\pi ·\pi Έ\pi ½\pi ΄\pi },\mathrm{\pi ²\pi \pi }\right)$

$\mathrm{\pi \pi °\pi \pi Ί\pi }$:

$\beta ©\mathrm{\pi \pi \pi \pi \pi \pi \pi }-\mathtt{1}\mathrm{\pi \pi \pi \pi \pi \pi }-\mathtt{1}\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }-\mathtt{4}\mathrm{\pi \pi \pi }-\mathtt{5}\mathrm{\pi \pi \pi \pi \pi \pi }-\mathtt{1},$

$\mathrm{\pi \pi \pi \pi \pi \pi \pi }-\mathtt{1}\mathrm{\pi \pi \pi \pi \pi \pi }-\mathtt{4}\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }-\mathtt{2}\mathrm{\pi \pi \pi }-\mathtt{6}\mathrm{\pi \pi \pi \pi \pi \pi }-\mathtt{3},$

$\mathrm{\pi \pi \pi \pi \pi \pi \pi }-\mathtt{1}\mathrm{\pi \pi \pi \pi \pi \pi }-\mathtt{2}\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }-\mathtt{3}\mathrm{\pi \pi \pi }-\mathtt{5}\mathrm{\pi \pi \pi \pi \pi \pi }-\mathtt{2},$

$\mathrm{\pi \pi \pi \pi \pi \pi \pi }-\mathtt{2}\mathrm{\pi \pi \pi \pi \pi \pi }-\mathtt{5}\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }-\mathtt{2}\mathrm{\pi \pi \pi }-\mathtt{7}\mathrm{\pi \pi \pi \pi \pi \pi }-\mathtt{2}\beta ͺ$

$\mathrm{\pi ³\pi ΄\pi \pi Έ\pi  \pi ΄\pi ³}\mathrm{\pi ²\pi Ύ\pi »\pi »\pi ΄\pi ²\pi \pi Έ\pi Ύ\pi ½}$:

$\mathrm{\pi \pi Έ\pi Ό\pi ΄}_\mathrm{\pi Ώ\pi Ύ\pi Έ\pi ½\pi \pi }-\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi },$

$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }-\mathrm{\pi \pi \pi \pi },\mathrm{\pi \pi \pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$

$\mathrm{\pi Ώ\pi °\pi \pi \pi ΄\pi \pi ½}\left(\mathrm{\pi }\right)$Β Β Β Β Β Β Β Β Β :

$\mathrm{\pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi °\pi \pi Ί\pi }.\mathrm{\pi \pi \pi \pi \pi \pi \pi },$

$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }-\mathrm{\pi \pi °\pi \pi Ί\pi }.\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi },\mathrm{\pi \pi \pi \pi \pi }-\mathrm{\pi \pi °\pi \pi Ί\pi }.\mathrm{\pi \pi \pi \pi \pi \pi }\right)$

$\mathrm{\pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi °\pi \pi Ί\pi }.\mathrm{\pi \pi \pi \pi \pi \pi \pi },$

$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }-\mathrm{\pi \pi °\pi \pi Ί\pi }.\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi },\mathrm{\pi \pi \pi \pi \pi }-\mathrm{\pi \pi °\pi \pi Ί\pi }.\mathrm{\pi \pi \pi }\right)$

$\mathrm{\pi Ά\pi ΄\pi ½\pi ΄\pi \pi °\pi \pi ΄\pi ³}\mathrm{\pi Έ\pi \pi ΄\pi Ό}\left(\mathrm{\pi }\right)$Β Β :

$\beta ©\mathrm{\pi \pi \pi }-\mathtt{1}\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }-\mathtt{4}\mathrm{\pi \pi \pi \pi \pi }-\mathtt{1},$

$\mathrm{\pi \pi \pi }-\mathtt{1}\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }-\mathtt{2}\mathrm{\pi \pi \pi \pi \pi }-\mathtt{4},$

$\mathrm{\pi \pi \pi }-\mathtt{1}\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }-\mathtt{3}\mathrm{\pi \pi \pi \pi \pi }-\mathtt{2},$

$\mathrm{\pi \pi \pi }-\mathtt{2}\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }-\mathtt{2}\mathrm{\pi \pi \pi \pi \pi }-\mathtt{5},$

$\mathrm{\pi \pi \pi }-\mathtt{1}\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }-\mathtt{4}\mathrm{\pi \pi \pi \pi \pi }-\mathtt{5},$

$\mathrm{\pi \pi \pi }-\mathtt{1}\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }-\mathtt{2}\mathrm{\pi \pi \pi \pi \pi }-\mathtt{6},$

$\mathrm{\pi \pi \pi }-\mathtt{1}\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }-\mathtt{3}\mathrm{\pi \pi \pi \pi \pi }-\mathtt{5},$

$\mathrm{\pi \pi \pi }-\mathtt{2}\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }-\mathtt{2}\mathrm{\pi \pi \pi \pi \pi }-\mathtt{7}\beta ͺ$

The two patterns mention the references $\mathrm{\pi \pi °\pi \pi Ί\pi }.\mathrm{\pi \pi \pi \pi \pi \pi \pi }$, $\mathrm{\pi \pi °\pi \pi Ί\pi }.\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$, $\mathrm{\pi \pi °\pi \pi Ί\pi }.\mathrm{\pi \pi \pi \pi \pi \pi }$ and $\mathrm{\pi \pi °\pi \pi Ί\pi }.\mathrm{\pi \pi \pi }$ of the $\mathrm{\pi \pi °\pi \pi Ί\pi }$ collection used in the arguments of the $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }$ constraint. $\beta i\beta \left[1,|\mathrm{\pi \pi °\pi \pi Ί\pi }|\right]$, we generate two items $\mathrm{\pi \pi \pi }-{u}_{1}\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }-{u}_{2}\mathrm{\pi \pi \pi \pi \pi }-{u}_{3}$, $\mathrm{\pi \pi \pi }-{v}_{1}\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }-{v}_{2}\mathrm{\pi \pi \pi \pi \pi }-{v}_{3}$ where:

${u}_{1}=\mathrm{\pi \pi °\pi \pi Ί\pi }\left[i\right].\mathrm{\pi \pi \pi \pi \pi \pi \pi },{u}_{2}=\mathrm{\pi \pi °\pi \pi Ί\pi }\left[i\right].\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi },{u}_{3}=\mathrm{\pi \pi °\pi \pi Ί\pi }\left[i\right].\mathrm{\pi \pi \pi \pi \pi \pi },$

${v}_{1}=\mathrm{\pi \pi °\pi \pi Ί\pi }\left[i\right].\mathrm{\pi \pi \pi \pi \pi \pi \pi },{v}_{2}=\mathrm{\pi \pi °\pi \pi Ί\pi }\left[i\right].\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi },{v}_{3}=\mathrm{\pi \pi °\pi \pi Ί\pi }\left[i\right].\mathrm{\pi \pi \pi }$.

This leads to the eight items listed in the $\mathrm{\pi Ά\pi ΄\pi ½\pi ΄\pi \pi °\pi \pi ΄\pi ³}\mathrm{\pi Έ\pi \pi ΄\pi Ό}\left(\mathrm{\pi }\right)$ field.

EXAMPLE

$\mathrm{\pi ²\pi Ύ\pi ½\pi \pi \pi \pi °\pi Έ\pi ½\pi }$:

$\mathrm{\pi \pi \pi \pi \pi \pi }$$\left(\begin{array}{|c|}\hline \mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\\ \hline\end{array}\right)$

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

$\beta ©\mathrm{\pi \pi \pi }-\mathtt{0},\mathrm{\pi \pi \pi }-\mathtt{1},\mathrm{\pi \pi \pi }-\mathtt{4},\mathrm{\pi \pi \pi }-\mathtt{6}\beta ͺ$

$\mathrm{\pi ³\pi ΄\pi \pi Έ\pi  \pi ΄\pi ³}\mathrm{\pi ²\pi Ύ\pi »\pi »\pi ΄\pi ²\pi \pi Έ\pi Ύ\pi ½}$:

$\mathrm{\pi Ώ\pi °\pi Έ\pi \pi }-\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi ‘}-\mathrm{\pi \pi \pi \pi },\mathrm{\pi ’}-\mathrm{\pi \pi \pi \pi }\right)$

$\mathrm{\pi Ώ\pi °\pi \pi \pi ΄\pi \pi ½}\left(\mathrm{\pi }\right)$Β Β Β Β Β Β Β Β Β :

$>-\mathrm{\pi \pi \pi \pi }\left(\mathrm{\pi ‘}-\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi },\mathrm{\pi ’}-\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }\right)$

$\mathrm{\pi Ά\pi ΄\pi ½\pi ΄\pi \pi °\pi \pi ΄\pi ³}\mathrm{\pi Έ\pi \pi ΄\pi Ό}\left(\mathrm{\pi }\right)$Β Β :

$\beta ©\mathrm{\pi ‘}-\mathtt{1}\mathrm{\pi ’}-\mathtt{0},$

$\mathrm{\pi ‘}-\mathtt{4}\mathrm{\pi ’}-\mathtt{0},\mathrm{\pi ‘}-\mathtt{4}\mathrm{\pi ’}-\mathtt{1},$

$\mathrm{\pi ‘}-\mathtt{6}\mathrm{\pi ’}-\mathtt{0},\mathrm{\pi ‘}-\mathtt{6}\mathrm{\pi ’}-\mathtt{1},\mathrm{\pi ‘}-\mathtt{6}\mathrm{\pi ’}-\mathtt{4}\beta ͺ$

The pattern mentions two references $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }$ and $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }$ to the $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ collection used in the arguments of the $\mathrm{\pi \pi \pi \pi \pi \pi }$ constraint. $\beta {i}_{1}\beta \left[1,|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|\right]$, $\beta {i}_{2}\beta \left[1,|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|\right]$ such that ${i}_{1}>{i}_{2}$We use the comparison operator $>$ since we have a pattern of the form $>-\mathrm{\pi \pi \pi \pi }\left(...\right)$. we generate the item $\mathrm{\pi ‘}-{u}_{1}\mathrm{\pi ’}-{u}_{2}$ where:

${u}_{1}=\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\left[{i}_{1}\right].\mathrm{\pi \pi \pi },{u}_{2}=\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\left[{i}_{2}\right].\mathrm{\pi \pi \pi }$.

This leads to the six items listed in the $\mathrm{\pi Ά\pi ΄\pi ½\pi ΄\pi \pi °\pi \pi ΄\pi ³}\mathrm{\pi Έ\pi \pi ΄\pi Ό}\left(\mathrm{\pi }\right)$ field.

Β

We finish with an example that mentions an indirect reference to an attribute of a collection.

EXAMPLE

$\mathrm{\pi ²\pi Ύ\pi ½\pi \pi \pi \pi °\pi Έ\pi ½\pi }$:

$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi ‘}$$\left(\begin{array}{|c|}\hline \mathrm{\pi \pi °\pi \pi Ί\pi }\\ \hline\end{array},\mathrm{\pi »\pi Έ\pi Ό\pi Έ\pi }\right)$

$\mathrm{\pi \pi °\pi \pi Ί\pi }$:

$\beta ©\mathrm{\pi \pi \pi \pi \pi \pi }-\beta ©\mathrm{\pi \pi \pi }-\mathtt{2},\mathrm{\pi \pi \pi }-\mathtt{1},\mathrm{\pi \pi \pi }-\mathtt{5}\beta ͺ\mathrm{\pi \pi \pi \pi \pi \pi }-\mathtt{1},$

$\mathrm{\pi \pi \pi \pi \pi \pi }-\beta ©\mathrm{\pi \pi \pi }-\mathtt{4},\mathrm{\pi \pi \pi }-\mathtt{5},\mathrm{\pi \pi \pi }-\mathtt{7}\beta ͺ\mathrm{\pi \pi \pi \pi \pi \pi }-\mathtt{2},$

$\mathrm{\pi \pi \pi \pi \pi \pi }-\beta ©\mathrm{\pi \pi \pi }-\mathtt{14},\mathrm{\pi \pi \pi }-\mathtt{15}\beta ͺ\mathrm{\pi \pi \pi \pi \pi \pi }-\mathtt{2}\beta ͺ$

$\mathrm{\pi ³\pi ΄\pi \pi Έ\pi  \pi ΄\pi ³}\mathrm{\pi ²\pi Ύ\pi »\pi »\pi ΄\pi ²\pi \pi Έ\pi Ύ\pi ½}$:

$\mathrm{\pi Έ\pi ½\pi \pi \pi °\pi ½\pi \pi }-\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi }-\mathrm{\pi \pi \pi }\right)$

$\mathrm{\pi Ώ\pi °\pi \pi \pi ΄\pi \pi ½}\left(\mathrm{\pi }\right)$Β Β Β Β Β Β Β Β Β Β :

$\mathrm{\pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi }-\mathrm{\pi \pi °\pi \pi Ί\pi }.\mathrm{\pi \pi \pi \pi \pi \pi }.\mathrm{\pi \pi \pi }\right)$

$\mathrm{\pi Ά\pi ΄\pi ½\pi ΄\pi \pi °\pi \pi ΄\pi ³}\mathrm{\pi Έ\pi \pi ΄\pi Ό}\left(\mathrm{\pi }\right)$Β Β :

$\beta ©\mathrm{\pi \pi \pi \pi \pi \pi \pi }-\mathtt{2},\mathrm{\pi \pi \pi \pi \pi \pi \pi }-\mathtt{1},\mathrm{\pi \pi \pi \pi \pi \pi \pi }-\mathtt{5},\mathrm{\pi \pi \pi \pi \pi \pi \pi }-\mathtt{4},$

$\mathrm{\pi \pi \pi \pi \pi \pi \pi }-\mathtt{5},\mathrm{\pi \pi \pi \pi \pi \pi \pi }-\mathtt{7},\mathrm{\pi \pi \pi \pi \pi \pi \pi }-\mathtt{14},\mathrm{\pi \pi \pi \pi \pi \pi \pi }-\mathtt{15}\beta ͺ$

The pattern mentions the indirect reference $\mathrm{\pi \pi °\pi \pi Ί\pi }.\mathrm{\pi \pi \pi \pi \pi \pi }.\mathrm{\pi \pi \pi }$ of the $\mathrm{\pi \pi °\pi \pi Ί\pi }$ collection used in the arguments of the $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi ‘}$ constraint. $\beta i\beta \left[1,|\mathrm{\pi \pi °\pi \pi Ί\pi }|\right]$, $\beta j\beta \left[1,|\mathrm{\pi \pi °\pi \pi Ί\pi }\left[i\right].\mathrm{\pi \pi \pi \pi \pi \pi }|\right]$ we generate the item $\mathrm{\pi \pi \pi \pi \pi \pi \pi }-{u}_{ij}$ where:

${u}_{ij}=\mathrm{\pi \pi °\pi \pi Ί\pi }\left[i\right].\mathrm{\pi \pi \pi \pi \pi \pi }\left[j\right]$.

This leads to the eight items listed in the $\mathrm{\pi Ά\pi ΄\pi ½\pi ΄\pi \pi °\pi \pi ΄\pi ³}\mathrm{\pi Έ\pi \pi ΄\pi Ό}\left(\mathrm{\pi }\right)$ field.