\usepackage{stmaryrd}

% \usepackage{pala}
\definecolor{yellow}{rgb}{1,1,0.85}
\definecolor{orange}{rgb}{1.0,0.3759,0.0}
\definecolor{orange3}{rgb}{1.0,0.2538,0.1681}
\definecolor{green2}{rgb}{0,0.8125,0}
\definecolor{green3}{rgb}{0,0.55,0}
\definecolor{brown}{rgb}{0.7,0.1,0.1}
\definecolor{pink}{rgb}{0.8,0.0,0.8}
\definecolor{cyan1}{rgb}{0,.90,.90}
\definecolor{cyan3}{rgb}{0,.59,.59}
\definecolor{grey}{rgb}{.7,.7,.7}
% \definecolor{blau}{rgb}{0,.48627,0.78431}
\definecolor{blau}{rgb}{0,.39608,0.74118}
\definecolor{rot}{rgb}{0.79216,.12941,0.24706}
\newcommand\Color[2]{\textcolor{#1}{#2}}
\newcommand\Blue[1]{\Color{blue}{#1}}
\newcommand\Blau[1]{\Color{blau}{#1}}
\newcommand\Red[1]{\Color{red}{#1}}
\newcommand\Rot[1]{\Color{rot}{#1}}
\newcommand\Green[1]{\Color{green3}{#1}}
\newcommand\White[1]{\Color{white}{#1}}
\newcommand\Black[1]{\Color{black}{#1}}
\newcommand\Yellow[1]{\Color{yellow}{#1}}
\newcommand\Orange[1]{\Color{orange3}{#1}}
\newcommand\Brown[1]{\Color{brown}{#1}}
\newcommand\Pink[1]{\Color{pink}{#1}}
\newcommand\Cyan[1]{\Color{cyan3}{#1}}
\newcommand\Cyann[1]{\Color{cyan1}{#1}}
\newcommand\Grey[1]{\Color{grey}{#1}}
% \newcommand\Section[1]{\Color{red}{\section{ #1 }}}
% \newcommand\Subsection[1]{\Color{\red}{\subsection{ #1 }}}
\newcommand\Section[1]{\Red{{\Large\bf #1 }\\[2ex]}}
\newcommand\Subsection[1]{\Red{{\large\bf #1 }\\[2ex]}}
\newcommand\cinput[1]{\begin{center}\input{ #1 }\end{center}}
\newcommand\cpage[2]{\begin{center}\begin{minipage}{ #1 } #2 \end{minipage}\end{center}}
\newcommand\RR{=\!=\!\Rightarrow}
\newcommand\redRR{\Color{red}{$=\!=\!\Rightarrow$}}
\newcommand{\is}{\Blue{::\!{=}}}
\newcommand\Or{\quad\Blue{|}\quad}
\renewcommand\implies{\Rightarrow}
\newcommand{\T}[2]{\Blue{{\cal T}_\rho[\![} #1 \Blue{]\!]}\; #2 }
% \newcommand{\C}[2]{\Blue{{\cal C}_\rho[\![} #1 \Blue{]\!]}\; #2 }
\newcommand{\CC}[2]{\Blue{{\cal C}_\rho^\bullet[\![} #1 \Blue{]\!]}\; #2 }
\newcommand{\gT}{\Green{{\cal T}_{\rho^\sharp}^\sharp}}
\newcommand{\gC}{\Green{{\cal C}_{\rho^\sharp}^\sharp}}
\newcommand{\gCC}{\Green{({\cal C}_{\rho^\sharp}^\bullet)^\sharp}}
\newcommand{\oC}{\Orange{{\cal C}_{\rho^{\sharp\sharp}}^{\sharp\sharp}}}
\newcommand{\oCC}{\Orange{({\cal C}_{\rho^{\sharp\sharp}}^\bullet)^{\sharp\sharp}}}
\newcommand{\size}[1]{\Green{|} #1 \Green{|}}
\newcommand{\Min}{\Green{\sf min}}
\newcommand{\Max}{\Green{\sf max}}
\newcommand{\gplus}{\;\Green{+}\;}
\newcommand{\gdot}{\;\Green{\cdot}\;}
\newcommand{\sem}[1]{\left\llbracket #1 \right\rrbracket}
\newcommand{\asem}[1]{\sem{#1}^\sharp}
\newcommand{\eval}[1]{\sem{ #1 }_{\Red{r}}}
\newcommand{\E}{{\cal E}} 
% \newcommand{\U}{{\cal U}} 
\newcommand{\M}{{\mathbb M}_k(N)}
\newcommand{\D}{{\mathbb D}_{\Red{r}}}
\newcommand{\MM}{{\mathbb M}_3(N)}
% \newcommand{\Theorem}[1]{\Red{\large \bf Theorem}\\[1ex] #1 \\[2ex]}


\newcommand{\N}{{\mathbb N}}
\newcommand{\nz}{\mathrm{I\!N}}
% \newcommand{\F}{\mathrm{I\!F}}
\newcommand{\nf}{{\colon\nz\Total\nz}}
\newcommand{\rz}{\mathrm{I\!R}}
\newcommand{\BB}{\mathbf{I\!B}}
\newcommand{\gz}{\mathrm{Z\!\!Z}}
%\newcommand{\qz}{{\rmfamily Q \hspace{-.6em}\prime\hspace{.4em}}}
%\newcommand{\cz}{{\rmfamily C \hspace{-.6em}\prime\hspace{.4em}}}
\newcommand{\qz}{\mathrm{Q \!\!\!\prime\;}}
\newcommand{\cz}{\mathrm{C \!\!\!\prime\;}}
\newcommand{\minus}{\stackrel{\bullet}{\raisebox{0pt}[.3ex]{$-$}}}
\newcommand{\head}{\mathit{head\/}}
\newcommand{\tail}{\mathit{tail\/}}
\newtheorem{DeFsAtZ}{\hspace{-.3em}}[section]
% \newenvironment{definition}{\begin{DeFsAtZ}
%            \Red{Definition}\quad\rmfamily}{\end{DeFsAtZ}}
% \newenvironment{theorem}{\begin{DeFsAtZ}
%            \Red{Theorem}\quad\rmfamily}{\end{DeFsAtZ}}
% \newenvironment{corollary}{\begin{DeFsAtZ}
%            \Pink{Corollary}\quad\rmfamily}{\end{DeFsAtZ}}
% \newenvironment{lemma}{\begin{DeFsAtZ}
%            \Pink{Lemma}\quad\rmfamily}{\end{DeFsAtZ}}
% \newenvironment{example}{\begin{DeFsAtZ}
%            \Green{Example}\rmfamily}{\end{DeFsAtZ}}
% \newcommand{\proof}{\noindent\brown{Proof:}}
\newcommand{\blitz}{\mbox{$\begin{array}{cc}\angle\\[-1ex]\swarrow\end{array}$}}
\newcommand{\existsone}{\exists!\hspace{.2em}}
\newcommand{\almev}{\mbox{\raisebox{-.4ex}{$\forall$}
           \hspace{-.95em}\raisebox{.3ex}{$^\infty$}}}
\newcommand{\QED}{\mbox{}\hfill$\Box$\vspace{1ex}\linebreak\mbox{}}
\newcommand{\GLA}{\begin{eqnarray*}}
\newcommand{\GLE}{\end{eqnarray*}}
\newcommand{\Def}{{\rmfamily Def}}
\newcommand{\Bild}{{\rmfamily Bild}}
\newcommand{\ABS}[1]{\left|#1\right|}
\newcommand{\OO}[1]{{\cal O}(\Orange{ #1 } )}
\newcommand\va{{\bf a}}
\newcommand\vb{{\bf b}}
\newcommand\vx{{\bf x}}
\newcommand\vy{{\bf y}}
\newcommand\vz{{\bf z}}
\newcommand\vX{{\bf X}}
\newcommand\vY{{\bf Y}}
\newcommand\Main{\sf Main}
\renewcommand{\binom}[2]{(\!\begin{array}{c}#1  \\[-.3ex]
                         #2 \end{array}\!)}
\newcommand{\unter}[2]{\begin{array}{c}\scriptstyle #1 \\[-.3ex]
                        \scriptstyle #2 \end{array}}
\newcommand{\stackrelsub}[2]{\raisebox{0pt}[0pt][0pt]{$
            \begin{array}{c}\scriptstyle\mbox{}\\[-1.8ex]#1\\[-1.8ex]
            \scriptstyle#2\end{array}$}}
\newcommand{\plusminus}{\begin{array}{c}+\\[-1.5ex]-\end{array}}
\newcommand{\stackrelall}[3]{\raisebox{0pt}[0pt][0pt]{$
            \begin{array}{@{}c@{}}\scriptstyle#1\\[-1.8ex]#2\\[-1.8ex]
            \scriptstyle#3\end{array}$}}
\newcommand{\Einschub}[1]{\mbox{}\\\mbox{}\begin{center}
               Einschub:\\\bfseries\large #1
             \end{center}\mbox{}\\[1ex]\mbox{}}
\newcommand{\angl}[1]{\langle #1\rangle}
\newcommand{\BNF}[1]{$\langle${\sl #1}\/$\rangle$}
\renewcommand{\sqsubset}{{\setlength{\unitlength}{1pt}\begin{picture}(10,10)
\put(2,0){\line(0,1){5}}
\put(2,0){\line(1,0){5}}
\put(2,5){\line(1,0){5}}
\end{picture}}}
\renewcommand{\sqsupset}{{\setlength{\unitlength}{1pt}\begin{picture}(9,9)
\put(7,0){\line(0,1){5}}
\put(2,0){\line(1,0){5}}
\put(2,5){\line(1,0){5}}
\end{picture}}}
\newcommand{\brl}{[\![}
\newcommand{\brr}{]\!]}

\newcommand{\Vw}{$\uparrow$}
\newcommand{\id}{\hspace*{1em}} 
\newcommand{\idx}[1]{#1} 
\newcommand{\idxem}[1]{{\em #1}} 
\renewcommand{\H}[1]{${\cal H}${\it #1\/}}

\newlength\breite
\newenvironment{xlist}{%
  \settowidth\breite{--}\begin{list}{--}{%
      \setlength\itemsep{0pt}\setlength\topsep{.7ex}\setlength\parsep{.7ex}%
      \setlength\itemindent{0mm}\setlength\listparindent{0mm}\setlength\labelsep{1ex}%
      \setlength\labelwidth{\breite}%
      \addtolength\breite{1ex}\addtolength\breite{.7ex}\setlength\leftmargin{\breite}%
      }}{\end{list}}
\newenvironment{ylist}[1]{%
  \settowidth\breite{#1}\begin{list}{#1}{%
      \setlength\itemsep{0pt}\setlength\topsep{.7ex}\setlength\parsep{.7ex}%
      \setlength\itemindent{0mm}\setlength\listparindent{0mm}\setlength\labelsep{1ex}%
      \setlength\labelwidth{\breite}%
      \addtolength\breite{1ex}\addtolength\breite{.7ex}\setlength\leftmargin{\breite}%
      }}{\end{list}}
\setcounter{section}{-1}

\newcommand{\MBlau}[1]{\Blau{\ {#1}\ }}
\newcommand{\MRot}[1]{\Rot{\ {#1}\ }}

\newcommand{\neginfty}{{-\infty}}

\newcommand{\I}{\mathcal{I}}

\renewcommand{\L}{\mathbf{L}}
\newcommand{\var}[1]{\Rot{\left<#1\right>}}


\newcommand\BackgroundPicture[1]{%
\setbeamertemplate{background}{%
\parbox[c][\paperheight]{\paperwidth}{%
\vfill \hfill
\includegraphics[width=1\paperwidth,height=1\paperheight]{#1}
\hfill \vfill
}}}

\newcommand\NoBackgroundPicture{\setbeamertemplate{background}{}}

\newcommand{\pos}[2]{\textblockcolor{}\begin{textblock}{30}(#1)#2\end{textblock}}

\newcommand{\speechbubble}[4]{%
  {%\textblockcolor{orange}  
   %\TPMargin{3pt} 
    \begin{textblock}{10}(#1)
      \begin{tikzpicture}[fill=orange]
        \pgfpathmoveto{\pgfpointorigin} 
        \pgfpathlineto{\pgfpoint{0cm}{#3cm-0.2cm}} 
        \pgfpathlineto{\pgfpoint{-0.4cm}{#3cm+0.2cm}} 
        \pgfpathlineto{\pgfpoint{0.2cm}{#3cm}} 
        \pgfpathlineto{\pgfpoint{#2cm}{#3cm}} 
        \pgfpathlineto{\pgfpoint{#2cm}{0cm}} 
        \pgfusepath{fill} 
        {\color{black}
          \pgftext[at=\pgfpoint{#2cm / 2}{#3cm / 2 - 0.15cm},center,base]{#4}
        }
      \end{tikzpicture}    
    \end{textblock}
  }%
}

\renewcommand{\emph}[1]{\Rot{\bf #1}}

\newcommand{\Cite}[1]{\Rot{\cite{#1}}}

\newcommand{\bvec}[1]{\mathbf{#1}}

\newcommand{\R}{{\mathbb R}}
\newcommand{\CR}{\overline{\R}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\CQ}{\overline{\Q}}

\newcommand{\omu}{\Orange{\mu}}
\newcommand{\onu}{\Orange{\nu}}

\newcommand{\handr}[1]{%
  \pos{#1}{\includegraphics[width=10mm]{logos/hand_rechts.pdf}}%
}

\newcommand{\marker}[4]{\pos{#3}{%
  \begin{tikzpicture}[fill=#1,opacity=#2]%
    \pgfsetstrokeopacity{#2}
    \pgfsetstrokecolor{#1}
    \pgfsetlinewidth{0pt}
    \filldraw (0,0) rectangle (#4);
  \end{tikzpicture}    
}}

\newcounter{cnt}
%
\newcommand\schachbrett{{%
  \forloop[2]{cnt}{0}{\value{cnt} < 13}{\marker{orange}{0.2}{\value{cnt},0}{1,1}}
  \forloop[2]{cnt}{1}{\value{cnt} < 12}{\marker{orange}{0.2}{\value{cnt},1}{1,1}}
  \forloop[2]{cnt}{0}{\value{cnt} < 13}{\marker{orange}{0.2}{\value{cnt},2}{1,1}}
  \forloop[2]{cnt}{1}{\value{cnt} < 12}{\marker{orange}{0.2}{\value{cnt},3}{1,1}}
  \forloop[2]{cnt}{0}{\value{cnt} < 13}{\marker{orange}{0.2}{\value{cnt},4}{1,1}}
  \forloop[2]{cnt}{1}{\value{cnt} < 12}{\marker{orange}{0.2}{\value{cnt},5}{1,1}}
  \forloop[2]{cnt}{0}{\value{cnt} < 13}{\marker{orange}{0.2}{\value{cnt},6}{1,1}}
  \forloop[2]{cnt}{1}{\value{cnt} < 12}{\marker{orange}{0.2}{\value{cnt},7}{1,1}}
  \forloop[2]{cnt}{0}{\value{cnt} < 13}{\marker{orange}{0.2}{\value{cnt},8}{1,1}}
  \forloop[2]{cnt}{1}{\value{cnt} < 12}{\marker{orange}{0.2}{\value{cnt},9}{1,1}}
}}

\newcommand\Z{\mathbb{Z}}
\newcommand\CZ{\overline\Z}