(*<*)
theory Slides2
imports "~~/src/HOL/Library/LaTeXsugar"
begin

notation (latex output)
  set ("_") and
  Cons  ("_::/_" [66,65] 65) 

(*>*)


text_raw {*
  %\renewcommand{\slidecaption}{Cambridge, 9 November 2010}
  %\renewcommand{\slidecaption}{Nijmegen, 25 August 2011}
  \renewcommand{\slidecaption}{St Andrews, 19 November 2011}
  \newcommand{\bl}[1]{#1}                        
  \newcommand{\sout}[1]{\tikz[baseline=(X.base), inner sep=-0.1pt, outer sep=0pt]
  \node [cross out,red, ultra thick, draw] (X) {\textcolor{black}{#1}};}
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}
  \frametitle{%
  \begin{tabular}{@ {}c@ {}}
  \LARGE Formalising\\[-3mm] 
  \LARGE Regular Language Theory\\[-3mm] 
  \LARGE with Regular Expressions,\\[-3mm] 
  \LARGE \alert<2>{Only}\\[0mm] 
  \end{tabular}}
  
  \begin{center}
   Christian Urban\\
  \small King's College London
  \end{center}\bigskip
 
  \begin{center}
  \small joint work with Chunhan Wu and Xingyuan Zhang from the PLA
  University of Science and Technology in Nanjing
  \end{center}

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

text_raw {*
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[c]
  \frametitle{}

  \includegraphics[scale=0.5]{roy.jpg}\medskip

  Roy intertwined with my scientific life on many occasions, most 
  notably:\bigskip 

  \begin{itemize}
  \item he admitted me for M.Phil.~in St Andrews and\\ 
  made me like theory\smallskip
  \item sent me to Cambridge for Ph.D.\bigskip
  \item made me appreciate precision in proofs
  \end{itemize}

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

text_raw {*
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[c]
  \frametitle{}

  \begin{tabular}{c@ {\hspace{2mm}}c}
  \\[6mm]
  \begin{tabular}{c}
  \includegraphics[scale=0.11]{harper.jpg}\\[-2mm]
  {\footnotesize Bob Harper}\\[-2.5mm]
  {\footnotesize (CMU)}
  \end{tabular}
  \begin{tabular}{c}
  \includegraphics[scale=0.37]{pfenning.jpg}\\[-2mm]
  {\footnotesize Frank Pfenning}\\[-2.5mm]
  {\footnotesize (CMU)}
  \end{tabular} &

  \begin{tabular}{p{6cm}}
  \raggedright
  \color{gray}{published a proof in\\ {\bf ACM Transactions on Computational Logic}, 2005,
  $\sim$31pp}
  \end{tabular}\\

  \pause
  \\[0mm]
  
  \begin{tabular}{c}
  \includegraphics[scale=0.36]{appel.jpg}\\[-2mm] 
  {\footnotesize Andrew Appel}\\[-2.5mm]
  {\footnotesize (Princeton)}
  \end{tabular} &

  \begin{tabular}{p{6cm}}
  \raggedright
  \color{gray}{relied on their proof in a\\ {\bf security} critical application}
  \end{tabular}
  \end{tabular}\medskip\pause

  \small
  \begin{minipage}{1.0\textwidth}
  (I also found an {\bf error} in my Ph.D.-thesis about cut-elimination
  examined by Henk Barendregt and Andy Pitts.)
  \end{minipage}

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

text_raw {*
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[t]
  \frametitle{\normalsize Formal language theory\ldots\hfill\mbox{}}
  \mbox{}\\[-15mm]\mbox{}

  \begin{center}
  \huge\bf\textcolor{gray}{in Theorem Provers}\\
  \footnotesize\textcolor{gray}{e.g.~Isabelle, Coq, HOL4, \ldots}
  \end{center}

  \begin{itemize}
  \item automata @{text "\<Rightarrow>"} graphs, matrices, functions
  \item<2-> combining automata/graphs

  \onslide<2->{
  \begin{center}
  \begin{tabular}{ccc}
  \begin{tikzpicture}[scale=1]
  %\draw[step=2mm] (-1,-1) grid (1,1);
  
  \draw[rounded corners=1mm, very thick] (-1.0,-0.3) rectangle (-0.2,0.3);
  \draw[rounded corners=1mm, very thick] ( 0.2,-0.3) rectangle ( 1.0,0.3);

  \node (A) at (-1.0,0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
  \node (B) at ( 0.2,0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
  
  \node (C) at (-0.2, 0.13) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
  \node (D) at (-0.2,-0.13) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};

  \node (E) at (1.0, 0.2) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
  \node (F) at (1.0,-0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
  \node (G) at (1.0,-0.2) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};

  \draw (-0.6,0.0) node {\small$A_1$};
  \draw ( 0.6,0.0) node {\small$A_2$};
  \end{tikzpicture}}

  & 

  \onslide<3->{\raisebox{1.1mm}{\bf\Large$\;\Rightarrow\,$}}

  &

  \onslide<3->{\begin{tikzpicture}[scale=1]
  %\draw[step=2mm] (-1,-1) grid (1,1);
  
  \draw[rounded corners=1mm, very thick] (-1.0,-0.3) rectangle (-0.2,0.3);
  \draw[rounded corners=1mm, very thick] ( 0.2,-0.3) rectangle ( 1.0,0.3);

  \node (A) at (-1.0,0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
  \node (B) at ( 0.2,0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
  
  \node (C) at (-0.2, 0.13) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
  \node (D) at (-0.2,-0.13) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};

  \node (E) at (1.0, 0.2) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
  \node (F) at (1.0,-0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
  \node (G) at (1.0,-0.2) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
  
  \draw (C) to [red, very thick, bend left=45] (B);
  \draw (D) to [red, very thick, bend right=45] (B);

  \draw (-0.6,0.0) node {\small$A_1$};
  \draw ( 0.6,0.0) node {\small$A_2$};
  \end{tikzpicture}}

  \end{tabular}
  \end{center}\medskip

  \only<4-5>{
  \begin{tabular}{@ {\hspace{-5mm}}l@ {}}
  disjoint union:\\[2mm]
  \smath{A_1\uplus A_2 \dn \{(1, x)\,|\, x \in A_1\} \,\cup\, \{(2, y)\,|\, y \in A_2\}}
  \end{tabular}}
  \end{itemize}

  \only<5>{
  \begin{textblock}{13.9}(0.7,7.7)
  \begin{block}{}
  \medskip
  \begin{minipage}{14cm}\raggedright
  Problems with definition for regularity:\bigskip\\
  \smath{\;\text{is\_regular}(A) \dn \exists M.\;\text{is\_dfa}(M) \wedge {\cal L} (M) = A}\bigskip
  \end{minipage}
  \end{block}
  \end{textblock}}
  \medskip

  \only<6->{\underline{A solution}:\;\;use \smath{\text{nat}}s \;@{text "\<Rightarrow>"}\; state nodes\medskip}

  \only<7->{You have to \alert{rename} states!}

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

text_raw {*
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[t]
  \frametitle{\normalsize Formal language theory\ldots\hfill\mbox{}}
  \mbox{}\\[-15mm]\mbox{}

  \begin{center}
  \huge\bf\textcolor{gray}{in Theorem Provers}\\
  \footnotesize\textcolor{gray}{e.g.~Isabelle, Coq, HOL4, \ldots}
  \end{center}

  \begin{itemize}
  \item Kozen's ``paper'' proof of Myhill-Nerode:\\ 
  \hspace{2cm}requires absence of \alert{inaccessible states}
  \end{itemize}\bigskip\bigskip

  \begin{center}
  \smath{\;\text{is\_regular}(A) \dn \exists M.\;\text{is\_dfa}(M) \wedge {\cal L} (M) = A}
  \end{center}


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

text_raw {*
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[t]
  \frametitle{}
  \mbox{}\\[25mm]\mbox{}

  \begin{textblock}{13.9}(0.7,1.2)
  \begin{block}{}
  \begin{minipage}{13.4cm}\raggedright
  {\bf Definition:}\smallskip\\
  
  A language \smath{A} is \alert{regular}, provided there exists a\\ 
  \alert{regular expression} that matches all strings of \smath{A}.
  \end{minipage}
  \end{block}
  \end{textblock}\pause
  
  {\noindent\large\bf\alert{\ldots{}and forget about automata}}\bigskip\bigskip\pause

  Infrastructure for free. But do we lose anything?\medskip\pause

  \begin{minipage}{1.1\textwidth}
  \begin{itemize}
  \item pumping lemma\pause
  \item closure under complementation\pause
  \item \only<6>{regular expression matching}%
       \only<7->{\sout{regular expression matching}
  {\footnotesize(@{text "\<Rightarrow>"}Brozowski'64, Owens et al '09)}}
  \item<8-> most textbooks are about automata
  \end{itemize}
  \end{minipage}

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

*}


text_raw {*
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[c]
  \frametitle{\LARGE The Myhill-Nerode Theorem}

  \begin{itemize}
  \item provides necessary and suf\!ficient conditions\\ for a language 
  being regular\\ \textcolor{gray}{(pumping lemma only necessary)}\bigskip

  \item key is the equivalence relation:\medskip
  \begin{center}
  \smath{x \approx_{A} y \,\dn\, \forall z.\; x @ z \in A \Leftrightarrow y @ z \in A}
  \end{center}
  \end{itemize}

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

*}

text_raw {*
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[c]
  \frametitle{\LARGE The Myhill-Nerode Theorem}

  \begin{center}
  \only<1>{%
  \begin{tikzpicture}[scale=3]
  \draw[very thick] (0.5,0.5) circle (.6cm);
  \end{tikzpicture}}%
  \only<2->{%
  \begin{tikzpicture}[scale=3]
  \draw[very thick] (0.5,0.5) circle (.6cm);
  \clip[draw] (0.5,0.5) circle (.6cm);
  \draw[step=2mm, very thick] (-1.4,-1.4) grid (1.4,1.4);
  \end{tikzpicture}}
  \end{center}
  
  \begin{itemize}
  \item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_A) \;\Leftrightarrow\; A\; \text{is regular}}
  \end{itemize}

  \begin{textblock}{5}(2.1,5.3)
  \begin{tikzpicture}
  \node at (0,0) [single arrow, fill=red,text=white, minimum height=2cm]
  {$U\!N\!IV$};
  \draw (-0.3,-1.1) node {\begin{tabular}{l}set of all\\[-1mm] strings\end{tabular}};
  \end{tikzpicture}
  \end{textblock}

  \only<2->{%
  \begin{textblock}{5}(9.1,7.2)
  \begin{tikzpicture}
  \node at (0,0) [shape border rotate=180,single arrow, fill=red,text=white, minimum height=2cm]
  {@{text "\<lbrakk>x\<rbrakk>"}$_{\approx_{A}}$};
  \draw (0.9,-1.1) node {\begin{tabular}{l}an equivalence class\end{tabular}};
  \end{tikzpicture}
  \end{textblock}}

  \only<3->{
  \begin{textblock}{11.9}(1.7,3)
  \begin{block}{}
  \begin{minipage}{11.4cm}\raggedright
  Two directions:\medskip\\
  \begin{tabular}{@ {}ll}
  1.)\;finite $\Rightarrow$ regular\\
  \;\;\;\smath{\text{finite}\,(U\!N\!IV /\!/ \approx_A) \Rightarrow \exists r.\;A = {\cal L}(r)}\\[3mm]
  2.)\;regular $\Rightarrow$ finite\\
  \;\;\;\smath{\text{finite}\, (U\!N\!IV /\!/ \approx_{{\cal L}(r)})}
  \end{tabular}

  \end{minipage}
  \end{block}
  \end{textblock}}

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

*}


text_raw {*
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[c]
  \frametitle{\LARGE Initial and Final {\sout{\textcolor{gray}{States}}}}

  \begin{textblock}{8}(10, 2)
  \textcolor{black}{Equivalence Classes}
  \end{textblock}


  \begin{center}
  \begin{tikzpicture}[scale=3]
  \draw[very thick] (0.5,0.5) circle (.6cm);
  \clip[draw] (0.5,0.5) circle (.6cm);
  \draw[step=2mm, very thick] (-1.4,-1.4) grid (1.4,1.4);
  \only<2->{\draw[blue, fill] (0.0, 0.6) rectangle (0.2, 0.8);}
  \only<3->{\draw[red, fill] (0.2, 0.2) rectangle (0.4, 0.4);
  \draw[red, fill] (0.4, 0.8) rectangle (0.6, 1.0);
  \draw[red, fill] (0.6, 0.0) rectangle (0.8, 0.2);
  \draw[red, fill] (0.8, 0.4) rectangle (1.0, 0.6);}
  \end{tikzpicture}
  \end{center}

  \begin{itemize}
  \item \smath{\text{finals}\,A\,\dn \{[\!|x|\!]_{\approx_{A}}\;|\;x \in A\}}
  \smallskip
  \item we can prove: \smath{A = \bigcup \text{finals}\,A}
  \end{itemize}

  \only<2->{%
  \begin{textblock}{5}(2.1,4.6)
  \begin{tikzpicture}
  \node at (0,0) [single arrow, fill=blue,text=white, minimum height=2cm]
  {$[] \in X$};
  \end{tikzpicture}
  \end{textblock}}

  \only<3->{%
  \begin{textblock}{5}(10,7.4)
  \begin{tikzpicture}
  \node at (0,0) [shape border rotate=180,single arrow, fill=red,text=white, minimum height=2cm]
  {a final};
  \end{tikzpicture}
  \end{textblock}}

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


text_raw {*
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}<-1>[c]
  \frametitle{\begin{tabular}{@ {}l}\LARGE% 
  Transitions between Eq-Classes\end{tabular}}

  \begin{center}
  \begin{tikzpicture}[scale=3]
  \draw[very thick] (0.5,0.5) circle (.6cm);
  \clip[draw] (0.5,0.5) circle (.6cm);
  \draw[step=2mm, very thick] (-1.4,-1.4) grid (1.4,1.4);
  \draw[blue, fill] (0.0, 0.6) rectangle (0.2, 0.8);
  \draw[blue, fill] (0.8, 0.4) rectangle (1.0, 0.6);
  \draw[white] (0.1,0.7) node (X) {$X$};
  \draw[white] (0.9,0.5) node (Y) {$Y$};
  \draw[blue, ->, line width = 2mm, bend left=45] (X) -- (Y);
  \node [inner sep=1pt,label=above:\textcolor{blue}{$c$}] at ($ (X)!.5!(Y) $) {};
  \end{tikzpicture}
  \end{center}

  \begin{center}
  \smath{X \stackrel{c}{\longrightarrow} Y \;\dn\; X ; c \subseteq Y}
  \end{center}

  \onslide<8>{
  \begin{tabular}{c}
  \begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick]
  \tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
  \node[state,initial] (q_0) {$R_1$};
  \end{tikzpicture}
  \end{tabular}}

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


text_raw {*
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[c]
  \frametitle{\LARGE Systems of Equations}

  Inspired by a method of Brzozowski\;'64:\bigskip\bigskip

  \begin{center}
  \begin{tabular}{@ {\hspace{-20mm}}c}
  \\[-13mm]
  \begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick]
  \tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]

  %\draw[help lines] (0,0) grid (3,2);

  \node[state,initial]   (p_0)                  {$X_1$};
  \node[state,accepting] (p_1) [right of=q_0]   {$X_2$};

  \path[->] (p_0) edge [bend left]   node        {a} (p_1)
                  edge [loop above]   node       {b} ()
            (p_1) edge [loop above]   node       {a} ()
                  edge [bend left]   node        {b} (p_0);
  \end{tikzpicture}\\
  \\[-13mm]
  \end{tabular}
  \end{center}

  \begin{center}
  \begin{tabular}{@ {\hspace{-6mm}}ll@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
  & \smath{X_1} & \smath{=} & \smath{X_1;b + X_2;b \onslide<2->{\alert<2>{+ \lambda;[]}}}\\
  & \smath{X_2} & \smath{=} & \smath{X_1;a + X_2;a}\medskip\\
  \end{tabular}
  \end{center}

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


text_raw {*
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}<1-2,4->[t]
  \small

  \begin{center}
  \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
  \onslide<1->{\smath{X_1}} & \onslide<1->{\smath{=}} 
      & \onslide<1->{\smath{X_1; b + X_2; b + \lambda;[]}}\\
  \onslide<1->{\smath{X_2}} & \onslide<1->{\smath{=}}    
      & \onslide<1->{\smath{X_1; a + X_2; a}}\\

  & & & \onslide<2->{by Arden}\\

  \onslide<2->{\smath{X_1}} & \onslide<2->{\smath{=}} 
      & \onslide<2->{\smath{X_1; b + X_2; b + \lambda;[]}}\\
  \onslide<2->{\smath{X_2}} & \onslide<2->{\smath{=}}    
      & \only<2->{\smath{X_1; a\cdot a^\star}}\\

  & & & \onslide<4->{by Arden}\\

  \onslide<4->{\smath{X_1}} & \onslide<4->{\smath{=}} 
      & \onslide<4->{\smath{X_2; b \cdot b^\star+ \lambda;b^\star}}\\
  \onslide<4->{\smath{X_2}} & \onslide<4->{\smath{=}}    
      & \onslide<4->{\smath{X_1; a\cdot a^\star}}\\

  & & & \onslide<5->{by substitution}\\

  \onslide<5->{\smath{X_1}} & \onslide<5->{\smath{=}} 
      & \onslide<5->{\smath{X_1; a\cdot a^\star \cdot b \cdot b^\star+ \lambda;b^\star}}\\
  \onslide<5->{\smath{X_2}} & \onslide<5->{\smath{=}}    
      & \onslide<5->{\smath{X_1; a\cdot a^\star}}\\

  & & & \onslide<6->{by Arden}\\

  \onslide<6->{\smath{X_1}} & \onslide<6->{\smath{=}} 
      & \onslide<6->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
  \onslide<6->{\smath{X_2}} & \onslide<6->{\smath{=}}    
      & \onslide<6->{\smath{X_1; a\cdot a^\star}}\\

  & & & \onslide<7->{by substitution}\\

  \onslide<7->{\smath{X_1}} & \onslide<7->{\smath{=}} 
      & \onslide<7->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
  \onslide<7->{\smath{X_2}} & \onslide<7->{\smath{=}}    
      & \onslide<7->{\smath{\lambda; b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star 
          \cdot a\cdot a^\star}}\\
  \end{tabular}
  \end{center}

  \only<8->{
  \begin{textblock}{6}(2.5,4)
  \begin{block}{}
  \begin{minipage}{8cm}\raggedright
  
  \begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick, inner sep=1mm]
  \tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]

  %\draw[help lines] (0,0) grid (3,2);

  \node[state,initial]   (p_0)                  {$X_1$};
  \node[state,accepting] (p_1) [right of=q_0]   {$X_2$};

  \path[->] (p_0) edge [bend left]   node        {a} (p_1)
                  edge [loop above]   node       {b} ()
            (p_1) edge [loop above]   node       {a} ()
                  edge [bend left]   node        {b} (p_0);
  \end{tikzpicture}

  \end{minipage}
  \end{block}
  \end{textblock}}

  \only<1,2>{%
  \begin{textblock}{3}(0.6,1.2)
  \begin{tikzpicture}
  \node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm]
  {\textcolor{red}{a}};
  \end{tikzpicture}
  \end{textblock}}
  \only<2>{%
  \begin{textblock}{3}(0.6,3.6)
  \begin{tikzpicture}
  \node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm]
  {\textcolor{red}{a}};
  \end{tikzpicture}
  \end{textblock}}
  \only<4>{%
  \begin{textblock}{3}(0.6,2.9)
  \begin{tikzpicture}
  \node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm]
  {\textcolor{red}{a}};
  \end{tikzpicture}
  \end{textblock}}
  \only<4>{%
  \begin{textblock}{3}(0.6,5.3)
  \begin{tikzpicture}
  \node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm]
  {\textcolor{red}{a}};
  \end{tikzpicture}
  \end{textblock}}
  \only<5>{%
  \begin{textblock}{3}(1.0,5.6)
  \begin{tikzpicture}
  \node at (0,0) (A) {};
  \node at (0,1) (B) {};
  \draw[<-, line width=2mm, red] (B) to  (A);
  \end{tikzpicture}
  \end{textblock}}
  \only<5,6>{%
  \begin{textblock}{3}(0.6,7.7)
  \begin{tikzpicture}
  \node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm]
  {\textcolor{red}{a}};
  \end{tikzpicture}
  \end{textblock}}
  \only<6>{%
  \begin{textblock}{3}(0.6,10.1)
  \begin{tikzpicture}
  \node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm]
  {\textcolor{red}{a}};
  \end{tikzpicture}
  \end{textblock}}
  \only<7>{%
  \begin{textblock}{3}(1.0,10.3)
  \begin{tikzpicture}
  \node at (0,0) (A) {};
  \node at (0,1) (B) {};
  \draw[->, line width=2mm, red] (B) to  (A);
  \end{tikzpicture}
  \end{textblock}}

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


text_raw {*
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[c]
  \frametitle{\LARGE The Other Direction}

  One has to prove

  \begin{center}
  \smath{\text{finite} (U\!N\!IV /\!/ \approx_{{\cal L}(r)})}
  \end{center}

  by induction on \smath{r}. Not trivial, but after a bit 
  of thinking, one can find a \alert{refined} relation:\bigskip

  
  \begin{center}
  \mbox{\begin{tabular}{c@ {\hspace{7mm}}c@ {\hspace{7mm}}c}
  \begin{tikzpicture}[scale=1.1]
  %Circle
  \draw[thick] (0,0) circle (1.1);    
  \end{tikzpicture}
  &
  \begin{tikzpicture}[scale=1.1]
  %Circle
  \draw[thick] (0,0) circle (1.1);    
  %Main rays
  \foreach \a in {0, 90,...,359}
    \draw[very thick] (0, 0) -- (\a:1.1);
  \foreach \a / \l in {45/1, 135/2, 225/3, 315/4}
      \draw (\a: 0.65) node {\small$a_\l$};
  \end{tikzpicture}
  &
  \begin{tikzpicture}[scale=1.1]
  %Circle
  \draw[red, thick] (0,0) circle (1.1);    
  %Main rays
  \foreach \a in {0, 45,...,359}
     \draw[red, very thick] (0, 0) -- (\a:1.1);
  \foreach \a / \l in {22.5/1.1, 67.5/1.2, 112.5/2.1, 157.5/2.2, 202.4/3.1, 247.5/3.2, 292.5/4.1, 337.5/4.2}
      \draw (\a: 0.77) node {\textcolor{red}{\footnotesize$a_{\l}$}};
  \end{tikzpicture}\\
  \small\smath{U\!N\!IV} & 
  \small\smath{U\!N\!IV /\!/ \approx_{{\cal L}(r)}} &
  \small\smath{U\!N\!IV /\!/ \alert{R}}
  \end{tabular}}
  \end{center}

  \begin{textblock}{5}(9.8,2.6)
  \begin{tikzpicture}
  \node at (0,0) [shape border rotate=270,single arrow, fill=red,text=white, minimum height=0cm]{\textcolor{red}{a}};
  \end{tikzpicture}
  \end{textblock}
  

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

text_raw {*
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[t]
  \frametitle{\LARGE\begin{tabular}{c}Derivatives of RExps\end{tabular}}

  \begin{itemize}
  \item introduced by Brozowski~'64
  \item a regular expressions after a character has been parsed\\[-18mm]\mbox{}
  \end{itemize}

  \only<1>{%
  \textcolor{blue}{%
  \begin{center}
  \begin{tabular}{@ {}lc@ {\hspace{3mm}}l@ {}}
  der c $\varnothing$     & $\dn$ & $\varnothing$\\
  der c []                & $\dn$ & $\varnothing$\\
  der c d                 & $\dn$ & if c $=$ d then [] else $\varnothing$\\
  der c ($r_1 + r_2$)     & $\dn$ & (der c $r_1$) $+$ (der c $r_2$)\\
  der c ($r^\star$)       & $\dn$ & (der c $r$) $\cdot$ $r^\star$\\
  der c ($r_1 \cdot r_2$) & $\dn$ & if nullable $r_1$\\
                          &       & then (der c $r_1$) $\cdot$ $r_2$ $+$ (der c $r_2$)\\
                          &       & else (der c $r_1$) $\cdot$ $r_2$\\
  \end{tabular}
  \end{center}}}
  \only<2>{%
  \textcolor{blue}{%
  \begin{center}
  \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}}
  pder c $\varnothing$     & $\dn$ & \alert{$\{\}$}\\
  pder c []                & $\dn$ & \alert{$\{\}$}\\
  pder c d                 & $\dn$ & if c $=$ d then $\{$[]$\}$ else $\{\}$\\
  pder c ($r_1 + r_2$)     & $\dn$ & (pder c $r_1$) \alert{$\cup$} (der c $r_2$)\\
  pder c ($r^\star$)       & $\dn$ & (pder c $r$) $\cdot$ $r^\star$\\
  pder c ($r_1 \cdot r_2$) & $\dn$ & if nullable $r_1$\\
                          &       & then (pder c $r_1$) $\cdot$ $r_2$ \alert{$\cup$} (pder c $r_2$)\\
                          &       & else (pder c $r_1$) $\cdot$ $r_2$\\
  \end{tabular}
  \end{center}}}

  \only<2>{
  \begin{textblock}{6}(8.5,4.7)
  \begin{block}{}
  \begin{quote}
  \begin{minipage}{6cm}\raggedright
  \begin{itemize}
  \item partial derivatives
  \item by Antimirov~'95
  \end{itemize}
  \end{minipage}
  \end{quote}
  \end{block}
  \end{textblock}}

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


text_raw {*
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[t]
  \frametitle{\LARGE Partial Derivatives}

  \mbox{}\\[0mm]\mbox{}

  \begin{itemize}

  \item \alt<1>{\smath{\text{pders $x$ $r$ \mbox{$=$} pders $y$ $r$}}}
            {\smath{\underbrace{\text{pders $x$ $r$ \mbox{$=$} pders $y$ $r$}}_{R}}} 
        refines \textcolor{blue}{$x$ $\approx_{{\cal L}(r)}$ $y$}\\[16mm]\pause
  \item \smath{\text{finite} (U\!N\!IV /\!/ R)} \bigskip\pause
  \item Therefore \smath{\text{finite} (U\!N\!IV /\!/ \approx_{{\cal L}(r)})}. Qed.
  \end{itemize}
  
  \only<2->{%
  \begin{textblock}{5}(3.9,7.2)
  \begin{tikzpicture}
  \node at (0,0) [shape border rotate=270,single arrow, fill=red,text=white, minimum height=0cm]{\textcolor{red}{a}};
  \draw (2.2,0) node {Antimirov '95};
  \end{tikzpicture}
  \end{textblock}}

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



text_raw {*
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[t]
  \frametitle{\LARGE What Have We Achieved?}

  \begin{itemize}
  \item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_A) \;\Leftrightarrow\; A\; \text{is regular}}
  \medskip\pause
  \item regular languages are closed under complementation; this is now easy
  \begin{center}
  \smath{U\!N\!IV /\!/ \approx_A \;\;=\;\; U\!N\!IV /\!/ \approx_{\overline{A}}}
  \end{center}\pause\medskip
  
  \item non-regularity (\smath{a^nb^n})\medskip\pause\pause

  \item take \alert{\bf any} language; build the language of substrings\\
  \pause

  then this language \alert{\bf is} regular\;\; (\smath{a^nb^n} $\Rightarrow$ \smath{a^\star{}b^\star})
  
  \end{itemize}

\only<2>{
\begin{textblock}{10}(4,14)
\small
\smath{x \approx_{A} y \,\dn\, \forall z.\; x @ z \in A \Leftrightarrow y @ z \in A}
\end{textblock}} 

\only<4>{
\begin{textblock}{5}(2,8.6)
\begin{minipage}{8.8cm}
\begin{block}{}
\begin{minipage}{8.6cm}
If there exists a sufficiently large set \smath{B} (for example infinitely large), 
such that

\begin{center}
\smath{\forall x,y \in B.\; x \not= y \;\Rightarrow\; x \not\approx_{A} y}. 
\end{center}  

then \smath{A} is not regular.\hspace{1.3cm}\small(\smath{B \dn \bigcup_n a^n})
\end{minipage}
\end{block}
\end{minipage}
\end{textblock}
}

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


text_raw {*
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[c]
  \frametitle{\LARGE Conclusion}

  \begin{itemize}
  \item We have never seen a proof of Myhill-Nerode based on
  regular expressions.\smallskip\pause

  \item great source of examples (inductions)\smallskip\pause

  \item no need to fight the theorem prover:\\ 
  \begin{itemize}
  \item first direction (790 loc)\\
  \item second direction (400 / 390 loc)
  \end{itemize}
  \end{itemize}

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

text_raw {*
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[b]
  \frametitle{\mbox{}\\[2cm]\textcolor{red}{Thank you!\\[5mm]Questions?}}

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

(*<*)
end
(*>*)