321 lines
9.1 KiB
TeX
321 lines
9.1 KiB
TeX
\NeedsTeXFormat{LaTeX2e}
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\ProvidesPackage{mymath}[2021/03/31 - Math package by Maximilian Keßler (private use)]
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%%%%%%Provides the basic math packages used for my lecture-note write-ups
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%The language to pass to babel
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\newif\ifenglish\englishtrue
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\DeclareOption{german}{\englishfalse}
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\DeclareOption{english}{\englishtrue}
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% When working with beamer some things have to be done differently
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\newif\ifbeamer\beamerfalse
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\DeclareOption{nobeamer}{\beamerfalse}
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\DeclareOption{beamer}{\beamertrue}
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\newif\ifsubfig\subfigtrue
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\DeclareOption{nosubfig}{\subfigfalse}
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%Options for onehalf-spacing of lines
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\newif\ifonehalfspacing\onehalfspacingfalse %default is normalspace
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\DeclareOption{onehalfspacing}{\onehalfspacingtrue}
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\DeclareOption{normalspacing}{\onehalfspacingfalse}
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%Options for utf8 or utf8x - encoding
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\newif\ifutf\utftrue %by default, utf8 is not enabled
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\DeclareOption{utf8x}{\utffalse}
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\DeclareOption{utf8}{\utftrue}
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%Options for lukas
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\newif\iflukas\lukasfalse
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\DeclareOption{lukas}{\lukastrue}
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%Options for including theorem-environments or not.
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\newif\iffancythm\fancythmfalse
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\DeclareOption{fancythm}{\fancythmtrue}
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\DeclareOption*{\PackageWarning{mymath}{Unknown '\CurrentOption'}}
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\ProcessOptions\relax
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\RequirePackage[\ifenglish english \else ngerman\fi]{babel}
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\RequirePackage[T1]{fontenc}
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% When using utf8x, we have to load ucs and inputenc with utf8x option and must exclude bibtex
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% When not using utf8x (plan utf8), we must NOT load ucs, and then load inputenc with utf8-option as well as bibtex
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% also, when using plain utf8, we want to load the newunicodechar package
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\ifutf
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\RequirePackage{newunicodechar}
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\newunicodechar{α}{\ensuremath\alpha}
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\newunicodechar{β}{\ensuremath\beta}
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\newunicodechar{γ}{\ensuremath\gamma}
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\newunicodechar{δ}{\ensuremath\delta}
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\newunicodechar{ε}{\ensuremath\epsilon}
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\newunicodechar{ζ}{\ensuremath\zeta}
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\newunicodechar{η}{\ensuremath\eta}
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\newunicodechar{θ}{\ensuremath\theta}
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\newunicodechar{ι}{\ensuremath\iota}
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\newunicodechar{κ}{\ensuremath\kappa}
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\newunicodechar{λ}{\ensuremath\lambda}
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\newunicodechar{μ}{\ensuremath\mu}
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\newunicodechar{ν}{\ensuremath\nu}
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\newunicodechar{ξ}{\ensuremath\xi}
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\newunicodechar{π}{\ensuremath\pi}
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\newunicodechar{ρ}{\ensuremath\rho}
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\newunicodechar{σ}{\ensuremath\sigma}
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\newunicodechar{τ}{\ensuremath\tau}
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\newunicodechar{υ}{\ensuremath\upsilon}
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\newunicodechar{φ}{\ensuremath\phi}
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\newunicodechar{χ}{\ensuremath\chi}
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\newunicodechar{ψ}{\ensuremath\psi}
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\newunicodechar{ω}{\ensuremath\omega}
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\ifbeamer\else
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\RequirePackage[backend=biber,style=alphabetic]{biblatex}
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\fi
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\else
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\RequirePackage[mathletters]{ucs}
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\fi
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\RequirePackage[\ifutf utf8\else utf8x\fi]{inputenc}
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\RequirePackage{anyfontsize}
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\RequirePackage{lmodern}
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\RequirePackage{amsmath}
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\RequirePackage{mathtools}
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\RequirePackage{amsthm}
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\RequirePackage{amssymb}
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\ifbeamer\else
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\RequirePackage{bm} % bold math symbols
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\fi
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\RequirePackage{comment}
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\RequirePackage{soul}
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\ifsubfig
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\RequirePackage{subfig}\else\fi
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\RequirePackage{mathrsfs}
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\RequirePackage{bbm}
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\ifbeamer
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\else \RequirePackage[shortlabels]{enumitem}\fi
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\ifbeamer\else
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\RequirePackage[colorlinks=true]{hyperref} % Hyperlinks
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\fi
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% TIKZ
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\RequirePackage{tikz}
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\usetikzlibrary{calc, intersections, through, quotes, angles, babel, positioning}
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\RequirePackage{tkz-euclide}
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\RequirePackage[export]{adjustbox}
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%Commutative diagrams
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\RequirePackage{tikz-cd}
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%%for small diagrams, similar to tikz-cd
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\usepackage{xy} % for small diagrams, e.g. arrows
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\xyoption{all}
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%%% AMSTHM
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\iffancythm
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\RequirePackage{mkessler-fancythm}
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\else
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\theoremstyle{plain}
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\ifbeamer\else
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\newtheorem{theorem}{\ifenglish Theorem\else Satz\fi}
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\newtheorem{corollary}{\ifenglish Corollary\else Korollar\fi}
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\newtheorem*{lemma}{Lemma}
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\newtheorem{proposition}{Proposition}
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\fi
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\theoremstyle{definition}
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\ifbeamer\else
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\newtheorem{definition}{Definition}
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\newtheorem{example}{\ifenglish Example\else Beispiel\fi}
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\newtheorem*{remark}{\ifenglish Remark\else Bemerkung\fi}
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\newtheorem*{notation}{Notation}
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\fi
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%%Theorems needed in any case (fancy or not fancy theorems)
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\theoremstyle{plain}
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\newtheorem{claim}{\ifenglish Claim\else Behauptung\fi}
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\newtheorem{variant}{\ifenglish Variant\else Variante\fi}
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\newtheorem{assumption}{\ifenglish Assumption\else Annahme\fi}
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\theoremstyle{definition}
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\ifbeamer\else
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\newtheorem*{fact}{\ifenglish Fact\else Fakt\fi}
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\newtheorem*{note}{\ifenglish Note\else Anmerkung\fi}
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\newtheorem*{problem}{Problem}
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\fi
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\newtheorem*{warning}{\color{red}\ifenglish Warning \else Warnung\fi}
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\newtheorem*{goal}{\ifenglish Goal \else Ziel\fi}
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\newtheorem*{question}{\ifenglish Question \else Frage \fi}
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\newtheorem*{info}{Information}
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\newtheorem*{moral}{Moral}
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\newtheorem*{answer}{\ifenglish Answer\else Antwort\fi}
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\newtheorem*{observe}{\ifenglish Observe\else Beobachte\fi}
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\newtheorem*{property}{\ifenglish Property\else Eigenschaft\fi}
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\newtheorem*{intuition}{Intuition}
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\newtheorem*{recall}{\ifenglish Recall\else Erinnerung\fi}
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\newtheorem*{idea}{\ifenglish Idea\else Idee\fi}
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\newtheorem{exercise}{\ifenglish Exercise\else Aufgabe\fi}
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\ifbeamer\else\newenvironment{solution}[1][]{\begin{proof}[\ifenglish{}Solution\else{}Lösung\fi{}#1]}{\end{proof}}\fi
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\fi %end of theorem definitions
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% Für Formeln
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\RequirePackage{mathabx}
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\RequirePackage{faktor}
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\RequirePackage[mathscr]{eucal}
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\RequirePackage{blindtext}
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\RequirePackage{tabto}
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\ifbeamer\else
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\RequirePackage{lastpage} %\pageref{LastPage} for reference on the last page
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\fi
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% Für Tabulatoren
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\RequirePackage{pdfpages} % PDF-Dateien einbinden
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% Für Zeilenumbruch
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\RequirePackage{microtype}
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\RequirePackage{changepage}
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%Für Zeilenabstand 1,5
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\ifonehalfspacing
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\RequirePackage[onehalfspacing]{setspace} \else
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\fi
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%For setting counters of itemns in \enemerate
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\makeatletter
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\newcommand\setItemnumber[1]{\setcounter{enum\romannumeral\@enumdepth}{\numexpr#1-1\relax}}
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\makeatother
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%Make circled symbols
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\newcommand*\circled[1]{\tikz[baseline=(char.base)]{
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\node[shape=circle,draw,inner sep=2pt] (char) {#1};}}
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%Print a warning sign
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\newcommand\Warning{%
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\makebox[1.4em][c]{%
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\makebox[0pt][c]{\raisebox{.1em}{\small!}}%
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\makebox[0pt][c]{\color{red}\Large$\bigtriangleup$}}}%
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% Add \contra symbol to denote contradiction
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\usepackage{stmaryrd} % for \lightning
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\newcommand\contra{\scalebox{1.5}{$\lightning$}}
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%Emphasize \vocabulary
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\newcommand{\vocab}[1]{\textbf{\color{blue} #1}}
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%%%%math operators
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\usepackage{operators}
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%Short commands for \mathbb{}
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\newcommand{\C}{\ensuremath{\mathbb{C}}}
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\newcommand{\F}{\ensuremath{\mathbb{F}}}
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\newcommand{\K}{\ensuremath{\mathbb{K}}}
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\newcommand{\N}{\ensuremath{\mathbb{N}}}
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\newcommand{\Q}{\ensuremath{\mathbb{Q}}}
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\newcommand{\R}{\ensuremath{\mathbb{R}}}
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\newcommand{\Z}{\ensuremath{\mathbb{Z}}}
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%Paired Delimiters
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\DeclarePairedDelimiter\ceil{\lceil}{\rceil}
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\DeclarePairedDelimiter\floor{\lfloor}{\rfloor}
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%Norm and absolute value
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%Make them scaling by default and have \abs*{} as the non-scaling version of the command
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\DeclarePairedDelimiter\abs{\lvert}{\rvert}
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\makeatletter
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\let\oldabs\abs
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\def\abs{\@ifstar{\oldabs}{\oldabs*}}
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\DeclarePairedDelimiter\norm{\lVert}{\rVert}
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\let\oldnorm\norm
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\def\norm{\@ifstar{\oldnorm}{\oldnorm*}}
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\makeatother
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%Fixes of common misbehaviour
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\renewcommand{\degree}{^\circ}
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%Used to start an exercise and resetting the equation number
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\newcommand{\nr}[1]{\setcounter{equation}{0}\section*{\ifenglish Exercise \else Aufgabe \fi #1}~}
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\ifbeamer\else\RequirePackage{esint}\fi
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\RequirePackage{IEEEtrantools}
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\RequirePackage[ngerman,ruled,vlined]{algorithm2e}
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\iflukas
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\newcommand{\RrClA}{\begin{IEEEeqnarray*}{RrCl}}
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\newcommand{\RrClZ}{\end{IEEEeqnarray*}}
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\newcommand{\RrCla}{\begin{IEEEeqnarray}{RrCl}}
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\newcommand{\RrClz}{\end{IEEEeqnarray}}
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\newcommand{\ud}{\,\mathrm{d}}
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\newcommand{\dLx}[1]{\;\ud\mathcal{L}^{#1}(x)}
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\setenumerate[1]{label=(\alph*)}
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%%%%%%%%%%%%% INDUKTION %%%%%%%%%%%%%%%%%
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\newcommand{\its}{Das gilt offentsichtlich.}
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\newcommand\induktion[6][\its]{
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\def\n{#2}
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\def\kn{#2}
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Es sei $P(\n)$ die Aussage, dass:
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\RrClA
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P(\n) : \equiv \qquad \left(#5 \right)
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\RrClZ
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Diese soll nun per Induktion über $#3$ gezeigt werden.
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\begin{enumerate}
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\item [\underline{IA:}]Für den Induktionsanfang muss $P(#4)$ gezeigt werden, d.h.:
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\def\n{#4}
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\def\kn{#4}
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\RrClA
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P(#4) \equiv \qquad \left(#5\right)
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\RrClZ
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#1\ Also ist der Induktionsanfang gezeigt.
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\def\n{#2}
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\def\kn{#2}
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\item [\underline{IV:}] Angenommen $P(\n)$ gilt für ein $\n \in #3$, d.h. es gilt:
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\RrClA
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P(\n) \equiv \qquad \left(#5\right)
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\RrClZ
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\item [\underline{IS:}]
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Im Induktionsschritt von $\n$ auf $\n+1$ muss gezeigt werden, dass unter Annahme der Induktionsvoraussetzung auch die Induktionsbehauptung gilt:
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\def\n{#2+1}
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\def\kn{(#2+1)}
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\RrClA
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P(\n) \equiv \qquad \left(#5\right)
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\RrClZ
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#6
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\def\n{#2}
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\def\kn{#2}
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Damit ist der Induktionsschritt erbracht und es gilt für alle $#2$ in $#3$:
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\RrClA
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\forall \n \in #3 : #5
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\RrClZ
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\end{enumerate}
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}
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%[Begründung IA]{$Variablenname$}{$Ursprungsmenge$}{$Startwert$}{$Aussage (variable ist \n, \kn)$}{Begründung IS}
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%\n ist n oder n+1, \kn ist n oder (n+1)
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\else
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\fi
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