\documentclass{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{hyperref}
\usepackage[sets,operations,nn,colors]{cora-macs}

\title{The \texttt{cora-macs} Package}
\author{Tobias Ladner, Lukas Koller\\
        TUM - Cyber-Physical Systems Group\\
        \texttt{\small tobias.ladner@tum.de, lukas.koller@tum.de}}
\date{2024-10-25}

\begin{document}
\maketitle

\begin{abstract}
The \texttt{cora-macs} package provides tools for working with continuous sets, operations, neural networks, and color schemes tailored for use in the context of cyber-physical systems. This document serves as the official documentation for the package, detailing the available commands and options.
\end{abstract}

\tableofcontents

\section{Introduction}
The \texttt{cora-macs} package is designed to assist in the representation and manipulation of various mathematical objects such as sets, intervals, zonotopes, and operations often encountered in cyber-physical systems analysis. It also includes predefined color schemes based on the TUM corporate design and other custom colors.

\section{Installation}
To install the \texttt{cora-macs} package, follow these steps:
\begin{enumerate}
    \item Place the \texttt{cora.sty} file in your working directory or in a directory where LaTeX can find it (e.g., in the local \texttt{texmf} tree).
    \item Include the package in your LaTeX document with the command:
    \begin{verbatim}
    \usepackage{cora}
    \end{verbatim}
\end{enumerate}

\section{Package Options}
The \texttt{cora-macs} package provides several options that can be passed when loading the package. These options control the inclusion of different sets of commands.

\begin{itemize}
    \item \texttt{sets}: Includes commands for continuous set notation and operations.
    \item \texttt{operations}: Includes commands for various mathematical operations.
    \item \texttt{nn}: Includes commands for working with neural networks.
    \item \texttt{colors}: Defines a set of colors specific to the \texttt{cora-macs} package.
    \item \texttt{tumcolors}: Defines TUM-specific colors.
\end{itemize}

Example usage:
\begin{verbatim}
\usepackage[sets, operations]{cora}
\end{verbatim}

\section{Commands and Environments}

\subsection{Sets (\texttt{sets} option)}
When the \texttt{sets} option is enabled, the package provides several commands for defining and manipulating continuous sets.

\begin{itemize}
    \item \texttt{\textbackslash contSet\{name\}}: Defines a set in calligraphic font, e.g., $\contSet{A}$.
    \item \texttt{\textbackslash shortContSet[abbr]\{content\}\{set\}}: Defines a short-hand notation for continuous sets with optional abbreviation.
    \item \texttt{\textbackslash defContSet\{elements\}\{conditions\}}: Defines a set using the format $\{x \mid \text{conditions}\}$.
    \item \texttt{\textbackslash defContSetSplit\{elements\}\{conditions\}}: Defines a set with split conditions for better formatting in large equations.
    \item \texttt{\textbackslash shortI\{a\}\{b\}}: Defines a closed interval $[a, b]$.
    \item \texttt{\textbackslash defZ}: Defines a zonotope.
    \item \texttt{\textbackslash defPZ}: Defines a polynomial zonotope.
\end{itemize}

\subsection{Operations (\texttt{operations} option)}
Enabling the \texttt{operations} option introduces a variety of operations that are useful in mathematical and cyber-physical systems contexts.

\begin{itemize}
    \item \texttt{\textbackslash operator\{name\}\{args\}}: Defines a custom operator with the given name and arguments.
    \item \texttt{\textbackslash opEnclose\{set1\}\{set2\}}: Encapsulates a set within another.
    \item \texttt{\textbackslash opIntervalEnclosure\{interval\}}: Specifies an interval enclosure.
    \item \texttt{\textbackslash opProject\{dimension\}}: Projects a set onto a given dimension.
    \item \texttt{\textbackslash diag\{matrix\}}: Produces the diagonal matrix of the given input.
\end{itemize}

\subsection{Neural Networks (\texttt{nn} option)}
The \texttt{nn} option introduces commands related to neural networks, including notation for layers, inputs, outputs, and propagations.

\begin{itemize}
    \item \texttt{\textbackslash NN}: Represents the symbol for a neural network $\NN$.
    \item \texttt{\textbackslash nnLayer\{layer\}\{input\}}: Represents a neural network layer with a specified input.
    \item \texttt{\textbackslash nnInput}, \texttt{\textbackslash nnOutput}: Notation for input and output points in a neural network.
\end{itemize}

\subsection{Colors (\texttt{colors} and \texttt{tumcolors} options)}
The \texttt{colors} and \texttt{tumcolors} options define a variety of colors that can be used in your documents.

\begin{itemize}
    \item \texttt{\textbackslash definecolor\{CORAcolorReachSet\}}: A predefined color for reachability sets (blue).
    \item \texttt{\textbackslash definecolor\{TUMblue\}}: TUM corporate blue color.
\end{itemize}

\section{Examples}
Here are some examples demonstrating how to use the commands from the \texttt{cora-macs} package.

\subsection{Set Notation}
\begin{verbatim}
\contSet{X} = \shortContSet{a}{b}{X}
\end{verbatim}
This code will produce $\contSet{X} = \left<a, b\right>_X$.

\subsection{Operations}
\begin{verbatim}
\diag{A} = \operator{diag}{A}
\end{verbatim}
This code will produce $\diag{A} = \text{\normalfont\small diag}(A)$.

\subsection{Neural Networks}
\begin{verbatim}
\NN(\nnInput) = \nnLayer{1}{\nnInput}
\end{verbatim}
This code will produce $\NN(\nnInput) = L_1(\nnInput)$.

\section{Concluding Remarks}
The \texttt{cora-macs} package offers a comprehensive set of tools for researchers working in cyber-physical systems. By providing a consistent notation and color scheme, this package simplifies the process of documenting complex mathematical objects and operations.

For more information, visit the \href{https://www.ce.cit.tum.de/cps}{TUM CPS Group website}.
\end{document}