slides/slides02.tex
author Christian Urban <christian.urban@kcl.ac.uk>
Thu, 02 Nov 2023 13:53:37 +0000
changeset 474 b528d1d3d3c3
parent 446 99dcfdf5aed8
child 481 e03a0100ec46
permissions -rw-r--r--
updated

% !TEX program = xelatex
\documentclass[dvipsnames,14pt,t,xelatex,aspectratio=169,xcolor={table}]{beamer}
%\usepackage{chessboard}
%\usepackage[LSBC4,T1]{fontenc}
\usepackage{../styles/slides}
\usepackage{../styles/mygraphs}
\usepackage{../styles/langs}
\usetikzlibrary{shapes}
% \usepackage{../data}

\hfuzz=220pt 

%\setmonofont[Scale=.88]{Consolas}
%\newfontfamily{\consolas}{Consolas}

\lstset{language=Scala,
        style=mystyle,
        numbersep=0pt,
        numbers=none,
        xleftmargin=0mm}

\newcommand{\bl}[1]{\textcolor{blue}{#1}}     

% beamer stuff 
\renewcommand{\slidecaption}{PEP (Scala) 02, King's College London}

\newcommand{\UParrow}[3]{%
\begin{textblock}{0}(#2,#3)%
\onslide<#1>{%
\begin{tikzpicture}%
\node at (0,0) [single arrow, shape border rotate=90, fill=red,text=red]{a};%
\end{tikzpicture}}%
\end{textblock}}

\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t]
\frametitle{%
  \begin{tabular}{@ {}c@ {}}
  \\[5mm]
  \huge PEP Scala (2) 
  \end{tabular}}


  \normalsize
  \begin{center}
  \begin{tabular}{ll}
    Email:  & christian.urban at kcl.ac.uk\\
    %Office: & N\liningnums{7.07} (North Wing, Bush House)\bigskip\\
    Slides \& Code: & KEATS\bigskip\\

    Office Hour: &  Fridays 11:00 -- 12:00\\
    Location: & N7.07 (North Wing, Bush House)\bigskip\\

    Pollev: & \texttt{\alert{https://pollev.com/cfltutoratki576}}\\  \\
  \end{tabular}
  \end{center}

\end{frame}


% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
% \begin{frame}[t,fragile]
% \frametitle{For-Comprehensions}

% %\small
% \begin{lstlisting}[language=Scala,numbers=none]
% for (n <- List(1, 2, 3, 4, 5)) yield n * n
% \end{lstlisting}

% \begin{textblock}{5}(2,6)
% \includegraphics[scale=0.3]{../pics/fun.png}
% \end{textblock}  

% \begin{textblock}{5}(9,6)
% \includegraphics[scale=0.3]{../pics/fun.png}
% \end{textblock}  


% \end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
\begin{frame}[t]
\frametitle{For-Comprehensions}

\begin{center}
  \begin{tikzpicture}[scale=1,
                      node/.style={
                      rectangle,rounded corners=3mm,
                      very thick,draw=black!50,
                      minimum height=18mm, minimum width=20mm,
                      top color=white,bottom color=black!20}]

  \node (A0) at (0.1,0) {\texttt{\textcolor{purple}{\textbf{for}} (\alert<2->{n} <- List(}};
  \node (A1) at (2.3,0) {\texttt{\phantom{,}1,}};
  \node (A2) at (3.2,0) {\texttt{\phantom{,}2,}};
  \node (A3) at (4.1,0) {\texttt{\phantom{,}3,}};
  \node (A4) at (5.0,0) {\texttt{\phantom{,}4,}};
  \node (A5) at (5.9,0) {\texttt{\phantom{))}5))}};
  \node (A6) at (8,0) {\texttt{\textcolor{purple}{\textbf{yield}} \alert<2->{n\,*\,n}}};

  \onslide<2->{
  \node (B0) at (1.4,-3) {\texttt{List(}};
  \node (B1) at (2.3,-3) {\texttt{\phantom{,}1,}};
  \node (B2) at (3.6,-3) {\texttt{\phantom{,}4,}};
  \node (B3) at (4.9,-3) {\texttt{\phantom{,}9,}};
  \node (B4) at (6.2,-3) {\texttt{\phantom{,}16,}};
  \node (B5) at (7.5,-3) {\texttt{\phantom{,}25)}};}

  \onslide<2->{
  \draw [->,line width=1mm] (A1.south) -- (B1.north);
  \draw [->,line width=1mm] (A2.south) -- (B2.north);
  \draw [->,line width=1mm] (A3.south) -- (B3.north);
  \draw [->,line width=1mm] (A4.south) -- (B4.north);
  \draw [->,line width=1mm] (A5.south) -- (B5.north);}

  \onslide<2->{
  \node (Q1) at (-0.45,-0.1) {};
  \node (Q2) at (-0.45,-2.8) {};
  \node (Q3) at (-0.45,-2.95) {\alert<2->{\texttt{n\,*\,n:}}};
  \draw [->,red,line width=1mm] (Q1.south) -- (Q2.north);}
 \end{tikzpicture}
\end{center}

\onslide<3>{This is for when the for-comprehension\\ \textbf{yields / produces} a result.}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
% \begin{frame}[t]
% \frametitle{For-Comprehensions Again}

% \begin{center}
%   \begin{tikzpicture}[scale=1,
%                       node/.style={
%                       rectangle,rounded corners=3mm,
%                       very thick,draw=black!50,
%                       minimum height=18mm, minimum width=20mm,
%                       top color=white,bottom color=black!20}]

%   \node (A0) at (0,0)
%     {\texttt{\textcolor{purple}{\textbf{for}} (n <- List(1, 2, 3, 4, 5))
%              \textcolor{purple}{\textbf{yield}} n\,*\,n}};

%   \node (A1) at (0,-1.5) {\LARGE\textbf{vs}};       
         
%   \node (A2) at (0,-3)
%     {\texttt{\textcolor{purple}{\textbf{for}} (n <- List(1, 2, 3, 4, 5)) println(n)}};
%  \end{tikzpicture}
% \end{center}\bigskip


% The second version is in case the for \textbf{does not}
% produce any result.

% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
\begin{frame}[c,fragile]
  %\frametitle{Option Type}
  
  Find something below 4 in a list. What do you think Scala answers?\bigskip\bigskip
  
  \begin{onlyenv}<1>
  \begin{lstlisting}[language=Scala, numbers=none, xleftmargin=-1mm]
  List(7,2,3,4,5,6).find(_ < 4)
       
  List(5,6,7,8,9).find(_ < 4)
  \end{lstlisting}
  \end{onlyenv}
  \begin{onlyenv}<2>
  \begin{lstlisting}[language=Scala, numbers=none, xleftmargin=-1mm]
  List(7,2,3,4,5,6).find(_ < 4)
  res: Option[Int] = Some(2)
   
  List(5,6,7,8,9).find(_ < 4)
  res: Option[Int] = None
  \end{lstlisting}
  \end{onlyenv}
  
  \end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
\begin{frame}[c]
\frametitle{Option Type}
    
\begin{itemize}
\item if the value is present, you use\bigskip
\begin{center}\pcode{Some(value)}\end{center}\bigskip\bigskip

\item if no value is present, you use\bigskip
\begin{center}\pcode{None}\end{center}\bigskip\bigskip
\end{itemize}

\small e.g.~\code{Option[Int]}, then \code{Some(42)} and \code{None}\\
good for error handling      
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
\begin{frame}[c,fragile]
\frametitle{Option Type}

\small   
\begin{onlyenv}<1>
\begin{lstlisting}[language=Scala, numbers=none, xleftmargin=-1mm]
Integer.parseInt("1234")

// vs.

def get_me_an_int(s: String) : Option[Int] = 
 Try(Some(Integer.parseInt(s))).getOrElse(None)
\end{lstlisting}
\end{onlyenv}\bigskip\bigskip\bigskip
 
in the Scala code it is clear from the type I that have to deal 
with the \pcode{None}-case; no JavaDoc needed
  
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
\begin{frame}[c,fragile]
\frametitle{Higher-Order Functions}
  
In Scala, functions can take other functions as arguments and can return 
a function as a result.\bigskip\bigskip  

\begin{lstlisting}[language=Scala, numbers=none, xleftmargin=1mm]
List(7,2,3,4,5,6).find(_ < 4)
\end{lstlisting}
    
\UParrow{1}{8}{11}    
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
\begin{frame}[c,fragile]
\frametitle{Higher-Order Functions (2)}
  

\begin{lstlisting}[language=Scala, numbers=none, xleftmargin=1mm]
def even(x: Int) : Boolean = x % 2 == 0

List(1, 2, 3, 4, 5).filter(even)
  res : List[Int] = List(2, 4)

List(1, 2, 3, 4, 5).count(even)
  res : Int = 2

List(1, 2, 3, 4, 5).find(even)
  res: Option[Int] = Some(2)
\end{lstlisting}
    
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
\begin{frame}[c,fragile]
\frametitle{Anonymous Functions}
  

\begin{lstlisting}[language=Scala, numbers=none, xleftmargin=1mm]
def less4(x: Int) = x < 4
\end{lstlisting}

\begin{center}
vs
\end{center}  

\begin{lstlisting}[language=Scala, numbers=none, xleftmargin=1mm]
      (x: Int) => x < 4
\end{lstlisting}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
\begin{frame}[c,fragile]
\frametitle{map (lower case)}

 applies a function to each element of a list (and more)

\begin{center}
\begin{tikzpicture}[scale=0.9]
                      
  \node (A0) at (1.2,0) {\texttt{List(\,}};
  \node (A1) at (2.0,0) {\texttt{1\makebox[0mm]{ ,}}};
  \node (A2) at (2.9,0) {\texttt{2\makebox[0mm]{ ,}}};
  \node (A3) at (3.8,0) {\texttt{3\makebox[0mm]{ ,}}};
  \node (A4) at (4.7,0) {\texttt{4\makebox[0mm]{ ,}}};
  \node (A5) at (5.6,0) {\texttt{5\makebox[0mm]{ ,}}};
  \node (A6) at (6.5,0) {\texttt{6\makebox[0mm]{ ,}}};
  \node (A7) at (7.4,0) {\texttt{7\makebox[0mm]{ ,}}};
  \node (A8) at (8.3,0) {\texttt{8)}};

  \node (B0) at (1.2,-3) {\texttt{List(\,}};
  \node (B1) at (2.0,-3) {\texttt{1\makebox[0mm]{ ,}}};
  \node (B2) at (3.0,-3) {\texttt{4\makebox[0mm]{ ,}}};
  \node (B3) at (4.1,-3) {\texttt{9\makebox[0mm]{ ,}}};
  \node (B4) at (5.2,-3) {\texttt{16\makebox[0mm]{ ,}}};
  \node (B5) at (6.3,-3) {\texttt{25\makebox[0mm]{ ,}}};
  \node (B6) at (7.4,-3) {\texttt{36\makebox[0mm]{ ,}}};
  \node (B7) at (8.4,-3) {\texttt{49\makebox[0mm]{ ,}}};
  \node (B8) at (9.4,-3) {\texttt{64\makebox[0mm]{ )}}};

  \draw [->,line width=1mm] (A1.south) -- (B1.north);
  \draw [->,line width=1mm] (A2.south) -- (B2.north);
  \draw [->,line width=1mm] (A3.south) -- (B3.north);
  \draw [->,line width=1mm] (A4.south) -- (B4.north);
  \draw [->,line width=1mm] (A5.south) -- (B5.north);
  \draw [->,line width=1mm] (A6.south) -- (B6.north);
  \draw [->,line width=1mm] (A7.south) -- (B7.north);
  \draw [->,line width=1mm] (A8.south) -- (B8.north);

  \node [red] (Q0) at (-0.5,-0.3) {\large\texttt{n}}; 
  \node (Q1) at (-0.5,-0.4) {};
  \node (Q2) at (-0.5,-2.5) {};
  \node [red] (Q3) at (-0.5,-2.65) {\large\texttt{n\,*\,n}};
  \draw [->,red,line width=1mm] (Q1.south) -- (Q2.north);

  \node [red] at (-1.5,-1.5) {\Large{}\it\textbf{map}};
 \end{tikzpicture}
\end{center}\bigskip

\begin{lstlisting}[language=Scala, numbers=none, xleftmargin=1mm]
List(1,2,3,4,5,6,7,8).map(n => n * n)
\end{lstlisting}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
\begin{frame}[c,fragile]
\frametitle{For-Comprehensions are maps}
  
\begin{lstlisting}[language=Scala, numbers=none, xleftmargin=1mm]
for (n <- List(1,2,3,4,5,6,7,8)) 
  yield n * n


// is just syntactic sugar for


List(1,2,3,4,5,6,7,8).map(n => n * n)
\end{lstlisting}
    
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
\begin{frame}[c,fragile]
\frametitle{Map (upper case)}

a type, representing a key-value association datastructure\bigskip\bigskip

\begin{lstlisting}[language=Scala, numbers=none, xleftmargin=-2mm]
val ascii = 
     ('a' to 'z').map(c => (c, c.toInt))

val ascii_Map = ascii.toMap

ascii_Map.get('a')   // -> 97
\end{lstlisting}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
\begin{frame}[c,fragile]
\frametitle{Pattern Matching}

\ldots on pairs:\bigskip

\begin{lstlisting}[language=Scala, numbers=none, xleftmargin=-2mm]
def fizz_buzz(n: Int) : String = 
 (n % 3, n % 5) match {
   case (0, 0) => "fizz buzz"
   case (0, _) => "fizz"
   case (_, 0) => "buzz"
   case _ => n.toString  
 }
\end{lstlisting}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
\begin{frame}[c,fragile]
\frametitle{Recursion}

\begin{lstlisting}[language=Scala, numbers=none, xleftmargin=-2mm]
def fib(n: Int) : Int = { 
  if (n == 0 || n == 1) 1
   else fib(n - 1) + fib(n - 2)
}
\end{lstlisting}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
\begin{frame}[c,fragile]
\frametitle{Recursion}

\small
\begin{lstlisting}[language=Scala, numbers=none, xleftmargin=-4mm]
def my_flatten(xs: List[Option[Int]]): List[Int] = 
 xs match {
   case Nil => Nil 
   case None :: rest => my_flatten(rest)
   case Some(v) :: rest => v :: my_flatten(rest)
 }
\end{lstlisting}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{\begin{tabular}{c}\\[0cm]\alert{Questions?}\end{tabular}}

\begin{center}
  \begin{tabular}[t]{@{}l@{}l@{}}
    \includegraphics[scale=0.1]{../pics/mand4.png} & \hspace{4mm}
    \raisebox{0mm}{\includegraphics[scale=0.1]{../pics/mand3.png}}      
  \end{tabular}     
\end{center}

\begin{center}
  My Office Hours: Thursdays 12 -- 14\\
  And specifically for Scala: Tuesdays 10:45 -- 11:45
\end{center}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}[t]

%   \begin{center}  
%   \includegraphics[scale=0.3]{../pics/blow.png}
%   \end{center}
  
%   \begin{textblock}{14}(2,11.4)
%   \large\bf{}Mind-Blowing Programming Languages:\\ 
%   Overloading in any language is great but it makes a difference\; \code{10/3}
%   \;or\; \code{10.0/3}
%   \end{textblock}
%   \end{frame}
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}[t]

%   \begin{center}  
%   \includegraphics[scale=0.3]{../pics/blow.png}
%   \end{center}
  
%   \begin{textblock}{14}(2,11.4)
%   \large\bf{}Mind-Blowing Programming Languages:\\ 
%   \centering PHP \textcolor{gray}{(7.0)}
%   \end{textblock}
%   \end{frame}
%   %%%
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     

\begin{frame}[c]
\frametitle{Jumping Towers}

\begin{center}
\begin{tikzpicture}[scale=1.2]
  \draw[line width=1mm,cap=round] (0,0) -- (5,0);
  \draw[line width=1mm,cap=round] (0,1) -- (5,1);

  \draw[line width=1mm,cap=round] (0,0) -- (0,1);
  \node at (0.5,0.5) {\textbf{\Large 3}};

  \draw[line width=1mm,cap=round] (1,0) -- (1,1);
  \node at (1.5,0.5) {\textbf{\Large 4}};

  \draw[line width=1mm,cap=round] (2,0) -- (2,1);
  \node at (2.5,0.5) {\textbf{\Large 2}};

  \draw[line width=1mm,cap=round] (3,0) -- (3,1);
  \node at (3.5,0.5) {\textbf{\Large 0}};
  
  \draw[line width=1mm,cap=round] (4,0) -- (4,1);

  \node at (4.5,0.5) {\textbf{\Large 1}};
  
  \draw[line width=1mm,cap=round] (5,0) -- (5,1);

  \draw[->,line width=0.5mm,cap=round,out=90,in=90,relative] (0.5,1) to (1.5,1);
  \draw[->,line width=0.5mm,cap=round,out=90,in=90,relative] (0.5,1) to (2.5,1);
  \draw[->,line width=0.5mm,cap=round,out=90,in=90,relative] (0.5,1) to (3.5,1);

  \draw[->,line width=0.5mm,cap=round,out=-90,in=-90,relative] (2.5,0) to (3.5,0);
  \draw[->,line width=0.5mm,cap=round,out=-90,in=-90,relative] (2.5,0) to (4.5,0);

  \draw[->,line width=0.5mm,cap=round,out=90,in=90,relative] (4.5,1) to (5.7,1);
  \node at (5.7, 0.8) {End};
\end{tikzpicture}
\end{center}\bigskip


shortest: 3 $\rightarrow$ 4 $\rightarrow$ End

\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     

\begin{frame}[c]
\frametitle{``Children'' / moves}

\begin{center}
  \begin{tikzpicture}
    [grow=right,level distance=30mm,child anchor=north,line width=0.5mm]
  \node {$[3,4,2,0,1]$}
     child {node {$[0,1]$}}
     child {node {$[2,0,1]$}
        child {node {$[1]$} child [level distance=13mm] {node {End}}}
        child {node {$[0,1]$}}
     }
     child {node {$[4,2,0,1]$\ldots}};
\end{tikzpicture}
\end{center}
\end{frame}

\begin{frame}<1-10>[t]
  
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
\begin{frame}[t,fragile]
\frametitle{\mbox{}\hspace{40mm}\textbf{???}}

\begin{textblock}{5}(2,6)
\includegraphics[scale=0.35]{../pics/commits.png}
\end{textblock}

\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}

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