
diff -r 66ef2a2c64fb -r c3ff26204d2a Slides/Slides7.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Slides/Slides7.thy	Thu Apr 28 11:44:36 2011 +0800
@@ -0,0 +1,1086 @@
+(*<*)
+theory Slides7
+imports "~~/src/HOL/Library/LaTeXsugar" "Main"
+begin
+
+declare [[show_question_marks = false]]
+
+notation (latex output)
+  set ("_") and
+  Cons  ("_::/_" [66,65] 65) 
+
+(*>*)
+
+text_raw {*
+  \renewcommand{\slidecaption}{Hefei, 15.~April 2011}
+
+  \newcommand{\abst}[2]{#1.#2}% atom-abstraction
+  \newcommand{\pair}[2]{\langle #1,#2\rangle} % pairing
+  \newcommand{\susp}{{\boldsymbol{\cdot}}}% for suspensions
+  \newcommand{\unit}{\langle\rangle}% unit
+  \newcommand{\app}[2]{#1\,#2}% application
+  \newcommand{\eqprob}{\mathrel{{\approx}?}}
+  \newcommand{\freshprob}{\mathrel{\#?}}
+  \newcommand{\redu}[1]{\stackrel{#1}{\Longrightarrow}}% reduction
+  \newcommand{\id}{\varepsilon}% identity substitution
+  
+  \newcommand{\bl}[1]{\textcolor{blue}{#1}}
+  \newcommand{\gr}[1]{\textcolor{gray}{#1}}
+  \newcommand{\rd}[1]{\textcolor{red}{#1}}
+
+  \newcommand{\ok}{\includegraphics[scale=0.07]{ok.png}}
+  \newcommand{\notok}{\includegraphics[scale=0.07]{notok.png}}
+  \newcommand{\largenotok}{\includegraphics[scale=1]{notok.png}}
+
+  \renewcommand{\Huge}{\fontsize{61.92}{77}\selectfont}
+  \newcommand{\veryHuge}{\fontsize{74.3}{93}\selectfont}
+  \newcommand{\VeryHuge}{\fontsize{89.16}{112}\selectfont}
+  \newcommand{\VERYHuge}{\fontsize{107}{134}\selectfont}
+
+  \newcommand{\LL}{$\mathbb{L}\,$}
+
+
+  \pgfdeclareradialshading{smallbluesphere}{\pgfpoint{0.5mm}{0.5mm}}%
+  {rgb(0mm)=(0,0,0.9);
+  rgb(0.9mm)=(0,0,0.7);
+  rgb(1.3mm)=(0,0,0.5);
+  rgb(1.4mm)=(1,1,1)}
+
+  \def\myitemi{\begin{pgfpicture}{-1ex}{-0.55ex}{1ex}{1ex}
+    \usebeamercolor[fg]{subitem projected}
+    {\pgftransformscale{0.8}\pgftext{\normalsize\pgfuseshading{bigsphere}}}
+    \pgftext{%
+      \usebeamerfont*{subitem projected}}
+  \end{pgfpicture}}
+
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}<1>[t]
+  \frametitle{%
+  \begin{tabular}{@ {\hspace{-3mm}}c@ {}}
+  \\
+  \LARGE Verifying a Regular Expression\\[-1mm] 
+  \LARGE Matcher and Formal Language\\[-1mm]
+  \LARGE Theory\\[5mm]
+  \end{tabular}}
+  \begin{center}
+  Christian Urban\\
+  \small Technical University of Munich, Germany
+  \end{center}
+
+
+  \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{This Talk: 3 Points}
+  \large
+  \begin{itemize}
+  \item It is easy to make mistakes.\bigskip
+  \item Theorem provers can prevent mistakes, {\bf if} the problem
+  is formulated so that it is suitable for theorem provers.\bigskip
+  \item This re-formulation can be done, even in domains where
+  we do not expect it.
+  \end{itemize}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}<1->[t]
+  \frametitle{Regular Expressions}
+
+  \begin{textblock}{6}(2,4)
+  \begin{tabular}{@ {}rrl}
+  \bl{r} & \bl{$::=$}  & \bl{$\varnothing$}\\
+         & \bl{$\mid$} & \bl{[]}\\
+         & \bl{$\mid$} & \bl{c}\\
+         & \bl{$\mid$} & \bl{r$_1$ + r$_2$}\\
+         & \bl{$\mid$} & \bl{r$_1$ $\cdot$ r$_2$}\\
+         & \bl{$\mid$} & \bl{r$^*$}\\
+  \end{tabular}
+  \end{textblock}
+
+  \begin{textblock}{6}(8,3.5)
+  \includegraphics[scale=0.35]{Screen1.png}
+  \end{textblock}
+
+  \begin{textblock}{6}(10.2,2.8)
+  \footnotesize Isabelle:
+  \end{textblock}
+  
+  \only<2>{
+  \begin{textblock}{9}(3.6,11.8)
+  \bl{matches r s $\;\Longrightarrow\;$ true $\vee$ false}\\[3.5mm]
+
+  \hspace{10mm}\begin{tikzpicture}
+  \coordinate (m1) at (0.4,1);
+  \draw (0,0.3) node (m2) {\small\color{gray}rexp};
+  \path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (m2) edge (m1);
+  
+  \coordinate (s1) at (0.81,1);
+  \draw (1.3,0.3) node (s2) {\small\color{gray} string};
+  \path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (s2) edge (s1);
+  \end{tikzpicture}
+  \end{textblock}}
+
+
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}<1->[t]
+  \frametitle{Specification}
+
+  \small
+  \begin{textblock}{6}(0,3.5)
+  \begin{tabular}{r@ {\hspace{0.5mm}}r@ {\hspace{1.5mm}}c@ {\hspace{1.5mm}}l}
+  \multicolumn{4}{c}{rexp $\Rightarrow$ set of strings}\bigskip\\
+  &\bl{\LL ($\varnothing$)}   & \bl{$\dn$} & \bl{$\varnothing$}\\
+  &\bl{\LL ([])}              & \bl{$\dn$} & \bl{\{[]\}}\\
+  &\bl{\LL (c)}               & \bl{$\dn$} & \bl{\{c\}}\\
+  &\bl{\LL (r$_1$ + r$_2$)}   & \bl{$\dn$} & \bl{\LL (r$_1$) $\cup$ \LL (r$_2$)}\\
+  \rd{$\Rightarrow$} &\bl{\LL (r$_1$ $\cdot$ r$_2$)} & \bl{$\dn$} & \bl{\LL (r$_1$) ;; \LL (r$_2$)}\\
+  \rd{$\Rightarrow$} &\bl{\LL (r$^*$)}           & \bl{$\dn$} & \bl{(\LL (r))$^\star$}\\
+  \end{tabular}
+  \end{textblock}
+
+  \begin{textblock}{9}(7.3,3)
+  {\mbox{}\hspace{2cm}\footnotesize Isabelle:\smallskip}
+  \includegraphics[scale=0.325]{Screen3.png}
+  \end{textblock}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}<1->[t]
+  \frametitle{Version 1}
+  \small
+  \mbox{}\\[-8mm]\mbox{}
+
+  \begin{center}\def\arraystretch{1.05}
+  \begin{tabular}{@ {\hspace{-5mm}}l@ {\hspace{2.5mm}}c@ {\hspace{2.5mm}}l@ {}}
+  \bl{match [] []}                   & \bl{$=$} & \bl{true}\\
+  \bl{match [] (c::s)}               & \bl{$=$} & \bl{false}\\
+  \bl{match ($\varnothing$::rs) s}   & \bl{$=$} & \bl{false}\\
+  \bl{match ([]::rs) s}              & \bl{$=$} & \bl{match rs s}\\
+  \bl{match (c::rs) []}              & \bl{$=$} & \bl{false}\\
+  \bl{match (c::rs) (d::s)}          & \bl{$=$} & \bl{if c = d then match rs s else false}\\     
+  \bl{match (r$_1$ + r$_2$::rs) s} & \bl{$=$} & \bl{match (r$_1$::rs) s $\vee$ match (r$_2$::rs) s}\\ 
+  \bl{match (r$_1$ $\cdot$ r$_2$::rs) s} & \bl{$=$} & \bl{match (r$_1$::r$_2$::rs) s}\\
+  \bl{match (r$^*$::rs) s}          & \bl{$=$} & \bl{match rs s $\vee$ match (r::r$^*$::rs) s}\\
+  \end{tabular}
+  \end{center}
+
+  \begin{textblock}{9}(0.2,1.6)
+  \hspace{10mm}\begin{tikzpicture}
+  \coordinate (m1) at (0.44,-0.5);
+  \draw (0,0.3) node (m2) {\small\color{gray}\mbox{}\hspace{-9mm}list of rexps};
+  \path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (m2) edge (m1);
+  
+  \coordinate (s1) at (0.86,-0.5);
+  \draw (1.5,0.3) node (s2) {\small\color{gray} string};
+  \path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (s2) edge (s1);
+  \end{tikzpicture}
+  \end{textblock}
+
+  \begin{textblock}{9}(2.8,11.8)
+  \bl{matches$_1$ r s $\;=\;$ match [r] s}
+  \end{textblock}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}<1->[c]
+  \frametitle{Testing}
+  
+  \small
+  Every good programmer should do thourough tests: 
+
+  \begin{center}
+  \begin{tabular}{@ {\hspace{-20mm}}lcl}
+  \bl{matches$_1$ (a$\cdot$b)$^*\;$ []}     & \bl{$\mapsto$} & \bl{true}\\
+  \bl{matches$_1$ (a$\cdot$b)$^*\;$ ab}   & \bl{$\mapsto$} & \bl{true}\\ 
+  \bl{matches$_1$ (a$\cdot$b)$^*\;$ aba}  & \bl{$\mapsto$} & \bl{false}\\
+  \bl{matches$_1$ (a$\cdot$b)$^*\;$ abab} & \bl{$\mapsto$} & \bl{true}\\ 
+  \bl{matches$_1$ (a$\cdot$b)$^*\;$ abaa} & \bl{$\mapsto$} & \bl{false}\medskip\\
+  \onslide<2->{\bl{matches$_1$ x$\cdot$(0$|$1)$^*\;$ x}   & \bl{$\mapsto$} & \bl{true}}\\
+  \onslide<2->{\bl{matches$_1$ x$\cdot$(0$|$1)$^*\;$ x0}  & \bl{$\mapsto$} & \bl{true}}\\
+  \onslide<2->{\bl{matches$_1$ x$\cdot$(0$|$1)$^*\;$ x3}  & \bl{$\mapsto$} & \bl{false}}
+  \end{tabular}
+  \end{center}
+ 
+  \onslide<3->
+  {looks OK \ldots let's ship it to customers\hspace{5mm} 
+   \raisebox{-5mm}{\includegraphics[scale=0.05]{sun.png}}}
+  
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}<1->[c]
+  \frametitle{Version 1}
+
+  \only<1->{Several hours later\ldots}\pause
+
+
+  \begin{center}
+  \begin{tabular}{@ {\hspace{0mm}}lcl}
+  \bl{matches$_1$ []$^*$ s}     & \bl{$\mapsto$} & loops\\
+  \onslide<4->{\bl{matches$_1$ ([] + \ldots)$^*$ s}   & \bl{$\mapsto$} & loops\\} 
+  \end{tabular}
+  \end{center}
+
+  \small
+  \onslide<3->{
+  \begin{center}
+  \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}}
+  \ldots\\
+  \bl{match ([]::rs) s}           & \bl{$=$} & \bl{match rs s}\\
+  \ldots\\
+  \bl{match (r$^*$::rs) s}        & \bl{$=$} & \bl{match rs s $\vee$ match (r::r$^*$::rs) s}\\
+  \end{tabular}
+  \end{center}}
+  
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}<1->[t]
+  \frametitle{Testing}
+
+  \begin{itemize}
+  \item While testing is an important part in the process of programming development\pause\ldots
+
+  \item we can only test a {\bf finite} amount of examples.\bigskip\pause
+
+  \begin{center}
+  \colorbox{cream}
+  {\gr{\begin{minipage}{10cm}
+  ``Testing can only show the presence of errors, never their
+  absence.'' (Edsger W.~Dijkstra)
+  \end{minipage}}}
+  \end{center}\bigskip\pause
+
+  \item In a theorem prover we can establish properties that apply to 
+  {\bf all} input and {\bf all} output. 
+
+  \end{itemize}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}<1->[t]
+  \frametitle{Version 2}
+  \mbox{}\\[-14mm]\mbox{}
+
+  \small
+  \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}ll@ {}}
+  \bl{nullable ($\varnothing$)}   & \bl{$=$} & \bl{false} &\\
+  \bl{nullable ([])}              & \bl{$=$} & \bl{true}  &\\
+  \bl{nullable (c)}               & \bl{$=$} & \bl{false} &\\
+  \bl{nullable (r$_1$ + r$_2$)}   & \bl{$=$} & \bl{nullable r$_1$ $\vee$ nullable r$_2$} & \\ 
+  \bl{nullable (r$_1$ $\cdot$ r$_2$)} & \bl{$=$} & \bl{nullable r$_1$ $\wedge$ nullable r$_2$} & \\
+  \bl{nullable (r$^*$)}           & \bl{$=$} & \bl{true} & \\
+  \end{tabular}\medskip
+
+  \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}}
+  \bl{der c ($\varnothing$)}       & \bl{$=$} & \bl{$\varnothing$} & \\
+  \bl{der c ([])}                  & \bl{$=$} & \bl{$\varnothing$} & \\
+  \bl{der c (d)}                   & \bl{$=$} & \bl{if c = d then [] else $\varnothing$} & \\
+  \bl{der c (r$_1$ + r$_2$)}       & \bl{$=$} & \bl{(der c r$_1$) + (der c r$_2$)} & \\
+  \bl{der c (r$_1$ $\cdot$ r$_2$)} & \bl{$=$} & \bl{((der c r$_1$) $\cdot$ r$_2$)} & \\
+       &          & \bl{\;\;+ (if nullable r$_1$ then der c r$_2$ else $\varnothing$)}\\
+  \bl{der c (r$^*$)}          & \bl{$=$} & \bl{(der c r) $\cdot$ r$^*$} &\smallskip\\
+
+  \bl{derivative r []}     & \bl{$=$} & \bl{r} & \\
+  \bl{derivative r (c::s)} & \bl{$=$} & \bl{derivative (der c r) s} & \\
+  \end{tabular}\medskip
+
+  \bl{matches$_2$ r s $=$ nullable (derivative r s)}
+
+  \begin{textblock}{6}(9.5,0.9)
+  \begin{flushright}
+  \color{gray}``if r matches []'' 
+  \end{flushright}
+  \end{textblock}
+
+  \begin{textblock}{6}(9.5,6.18)
+  \begin{flushright}
+  \color{gray}``derivative w.r.t.~a char'' 
+  \end{flushright}
+  \end{textblock}
+
+  \begin{textblock}{6}(9.5,12.1)
+  \begin{flushright}
+  \color{gray}``deriv.~w.r.t.~a string'' 
+  \end{flushright}
+  \end{textblock}
+
+  \begin{textblock}{6}(9.5,13.98)
+  \begin{flushright}
+  \color{gray}``main'' 
+  \end{flushright}
+  \end{textblock}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}<1->[t]
+  \frametitle{Is the Matcher Error-Free?}
+
+  We expect that
+
+  \begin{center}
+  \begin{tabular}{lcl}
+  \bl{matches$_2$ r s = true}  & \only<1>{\rd{$\Longrightarrow\,\,$}}\only<2>{\rd{$\Longleftarrow\,\,$}}% 
+  \only<3->{\rd{$\Longleftrightarrow$}} & \bl{s $\in$ \LL(r)}\\
+  \bl{matches$_2$ r s = false} & \only<1>{\rd{$\Longrightarrow\,\,$}}\only<2>{\rd{$\Longleftarrow\,\,$}}%
+  \only<3->{\rd{$\Longleftrightarrow$}} & \bl{s $\notin$ \LL(r)}\\
+  \end{tabular}
+  \end{center}
+  \pause\pause\bigskip
+  ??? By \alert<4->{induction}, we can {\bf prove} these properties.\bigskip
+
+  \begin{tabular}{lrcl}
+  Lemmas:  & \bl{nullable (r)}          & \bl{$\Longleftrightarrow$} & \bl{[] $\in$ \LL (r)}\\
+           & \bl{s $\in$ \LL (der c r)} & \bl{$\Longleftrightarrow$} & \bl{(c::s) $\in$ \LL (r)}\\
+  \end{tabular}
+  
+  \only<4->{
+  \begin{textblock}{3}(0.9,4.5)
+  \rd{\huge$\forall$\large{}r s.}
+  \end{textblock}}
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}<1>[c]
+  \frametitle{
+  \begin{tabular}{c}
+  \mbox{}\\[23mm]
+  \LARGE Demo
+  \end{tabular}}
+  
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}<1->[t]
+
+  \mbox{}\\[-2mm]
+
+  \small
+  \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}ll@ {}}
+  \bl{nullable (NULL)}            & \bl{$=$} & \bl{false} &\\
+  \bl{nullable (EMPTY)}           & \bl{$=$} & \bl{true}  &\\
+  \bl{nullable (CHR c)}           & \bl{$=$} & \bl{false} &\\
+  \bl{nullable (ALT r$_1$ r$_2$)} & \bl{$=$} & \bl{(nullable r$_1$) orelse (nullable r$_2$)} & \\ 
+  \bl{nullable (SEQ r$_1$ r$_2$)} & \bl{$=$} & \bl{(nullable r$_1$) andalso (nullable r$_2$)} & \\
+  \bl{nullable (STAR r)}          & \bl{$=$} & \bl{true} & \\
+  \end{tabular}\medskip
+
+  \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}}
+  \bl{der c (NULL)}            & \bl{$=$} & \bl{NULL} & \\
+  \bl{der c (EMPTY)}           & \bl{$=$} & \bl{NULL} & \\
+  \bl{der c (CHR d)}           & \bl{$=$} & \bl{if c=d then EMPTY else NULL} & \\
+  \bl{der c (ALT r$_1$ r$_2$)} & \bl{$=$} & \bl{ALT (der c r$_1$) (der c r$_2$)} & \\
+  \bl{der c (SEQ r$_1$ r$_2$)} & \bl{$=$} & \bl{ALT (SEQ (der c r$_1$) r$_2$)} & \\
+       &          & \bl{\phantom{ALT} (if nullable r$_1$ then der c r$_2$ else NULL)}\\
+  \bl{der c (STAR r)}          & \bl{$=$} & \bl{SEQ (der c r) (STAR r)} &\smallskip\\
+
+  \bl{derivative r []}     & \bl{$=$} & \bl{r} & \\
+  \bl{derivative r (c::s)} & \bl{$=$} & \bl{derivative (der c r) s} & \\
+  \end{tabular}\medskip
+
+  \bl{matches r s $=$ nullable (derivative r s)}
+  
+  \only<2>{
+  \begin{textblock}{8}(1.5,4)
+  \includegraphics[scale=0.3]{approved.png}
+  \end{textblock}}
+  
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{No Automata?}
+
+  You might be wondering why I did not use any automata?
+
+  \begin{itemize}
+  \item {\bf Def.:} A \alert{regular language} is one where there is a DFA that 
+  recognises it.\bigskip\pause
+  \end{itemize}
+
+
+  There are many reasons why this is a good definition:\medskip
+  \begin{itemize}
+  \item pumping lemma
+  \item closure properties of regular languages\\ (e.g.~closure under complement)
+  \end{itemize}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[t]
+  \frametitle{Really Bad News!}
+
+  DFAs are bad news for formalisations in theorem provers. They might
+  be represented as:
+
+  \begin{itemize}
+  \item graphs
+  \item matrices
+  \item partial functions
+  \end{itemize}
+
+  All constructions are messy to reason about.\bigskip\bigskip 
+  \pause
+
+  \small
+  \only<2>{
+  Constable et al needed (on and off) 18 months for a 3-person team 
+  to formalise automata theory in Nuprl including Myhill-Nerode. There is 
+  only very little other formalised work on regular languages I know of
+  in Coq, Isabelle and HOL.}
+  \only<3>{Typical textbook reasoning goes like: ``\ldots if \smath{M} and \smath{N} are any two
+  automata with no inaccessible states \ldots''
+  }
+  
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{}
+  \large
+  \begin{center}
+  \begin{tabular}{p{9cm}}
+  My point:\bigskip\\
+
+  The theory about regular languages can be reformulated 
+  to be more suitable for theorem proving.
+  \end{tabular}
+  \end{center}
+  \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 (pumping lemma only necessary)\medskip
+
+  \item will help with closure properties of regular languages\bigskip\pause
+
+  \item key is the equivalence relation:\smallskip
+  \begin{center}
+  \smath{x \approx_{L} y \,\dn\, \forall z.\; x @ z \in L \Leftrightarrow y @ z \in L}
+  \end{center}
+  \end{itemize}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{\LARGE The Myhill-Nerode Theorem}
+
+  \mbox{}\\[5cm]
+
+  \begin{itemize}
+  \item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_L) \;\Leftrightarrow\; L\; \text{is regular}}
+  \end{itemize}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{\LARGE Equivalence Classes}
+
+  \begin{itemize}
+  \item \smath{L = []}
+  \begin{center}
+  \smath{\Big\{\{[]\},\; U\!N\!IV - \{[]\}\Big\}}
+  \end{center}\bigskip\bigskip
+
+  \item \smath{L = [c]}
+  \begin{center}
+  \smath{\Big\{\{[]\},\; \{[c]\},\; U\!N\!IV - \{[], [c]\}\Big\}}
+  \end{center}\bigskip\bigskip
+
+  \item \smath{L = \varnothing}
+  \begin{center}
+  \smath{\Big\{U\!N\!IV\Big\}}
+  \end{center}
+
+  \end{itemize}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{\LARGE Regular Languages}
+
+  \begin{itemize}
+  \item \smath{L} is regular \smath{\dn} if there is an automaton \smath{M} 
+  such that \smath{\mathbb{L}(M) = L}\\[1.5cm]
+
+  \item Myhill-Nerode:
+
+  \begin{center}
+  \begin{tabular}{l}
+  finite $\Rightarrow$ regular\\
+  \;\;\;\smath{\text{finite}\,(U\!N\!IV /\!/ \approx_L) \Rightarrow \exists r. L = \mathbb{L}(r)}\\[3mm]
+  regular $\Rightarrow$ finite\\
+  \;\;\;\smath{\text{finite}\, (U\!N\!IV /\!/ \approx_{\mathbb{L}(r)})}
+  \end{tabular}
+  \end{center}
+
+  \end{itemize}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{\LARGE Final States}
+
+  \mbox{}\\[3cm]
+
+  \begin{itemize}
+  \item ??? \smath{\text{final}_L\,X \dn \{[|s|]_\approx\;|\; s \in X\}}\\
+  \medskip
+
+  \item we can prove: \smath{L = \bigcup \{X\;|\;\text{final}_L\,X\}}
+
+  \end{itemize}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{\LARGE Transitions between\\[-3mm] Equivalence Classes}
+
+  \smath{L = \{[c]\}}
+
+  \begin{tabular}{@ {\hspace{-7mm}}cc}
+  \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]
+
+  %\draw[help lines] (0,0) grid (3,2);
+
+  \node[state,initial]   (q_0)                        {$R_1$};
+  \node[state,accepting] (q_1) [above right of=q_0]   {$R_2$};
+  \node[state]           (q_2) [below right of=q_0]   {$R_3$};
+
+  \path[->] (q_0) edge                node        {c} (q_1)
+                  edge                node [swap] {$\Sigma-{c}$} (q_2)
+            (q_2) edge [loop below]   node        {$\Sigma$} ()
+            (q_1) edge                node        {$\Sigma$} (q_2);
+  \end{tikzpicture}
+  \end{tabular}
+  &
+  \begin{tabular}[t]{ll}
+  \\[-20mm]
+  \multicolumn{2}{l}{\smath{U\!N\!IV /\!/\approx_L} produces}\\[4mm]
+
+  \smath{R_1}: & \smath{\{[]\}}\\
+  \smath{R_2}: & \smath{\{[c]\}}\\
+  \smath{R_3}: & \smath{U\!N\!IV - \{[], [c]\}}\\[6mm]
+  \multicolumn{2}{l}{\onslide<2->{\smath{X \stackrel{c}{\longrightarrow} Y \dn X ;; [c] \subseteq Y}}}
+  \end{tabular}
+
+  \end{tabular}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{\LARGE Systems of Equations}
+
+  Inspired by a method of Brzozowski\;'64, we can build an equational system
+  characterising the equivalence classes:
+
+  \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)                  {$R_1$};
+  \node[state,accepting] (p_1) [right of=q_0]   {$R_2$};
+
+  \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{R_1} & \smath{\equiv} & \smath{R_1;b + R_2;b \onslide<2->{\alert<2>{+ \lambda;[]}}}\\
+  & \smath{R_2} & \smath{\equiv} & \smath{R_1;a + R_2;a}\medskip\\
+  \onslide<3->{we can prove} 
+  & \onslide<3->{\smath{R_1}} & \onslide<3->{\smath{=}} 
+      & \onslide<3->{\smath{R_1; \mathbb{L}(b) \,\cup\, R_2;\mathbb{L}(b) \,\cup\, \{[]\};\{[]\}}}\\
+  & \onslide<3->{\smath{R_2}} & \onslide<3->{\smath{=}}    
+      & \onslide<3->{\smath{R_1; \mathbb{L}(a) \,\cup\, R_2;\mathbb{L}(a)}}\\
+  \end{tabular}
+  \end{center}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}<1>[t]
+  \small
+
+  \begin{center}
+  \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
+  \onslide<1->{\smath{R_1}} & \onslide<1->{\smath{=}} 
+      & \onslide<1->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
+  \onslide<1->{\smath{R_2}} & \onslide<1->{\smath{=}}    
+      & \onslide<1->{\smath{R_1; a + R_2; a}}\\
+
+  & & & \onslide<2->{by Arden}\\
+
+  \onslide<2->{\smath{R_1}} & \onslide<2->{\smath{=}} 
+      & \onslide<2->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
+  \onslide<2->{\smath{R_2}} & \onslide<2->{\smath{=}}    
+      & \only<2>{\smath{R_1; a + R_2; a}}%
+        \only<3->{\smath{R_1; a\cdot a^\star}}\\
+
+  & & & \onslide<4->{by Arden}\\
+
+  \onslide<4->{\smath{R_1}} & \onslide<4->{\smath{=}} 
+      & \onslide<4->{\smath{R_2; b \cdot b^\star+ \lambda;b^\star}}\\
+  \onslide<4->{\smath{R_2}} & \onslide<4->{\smath{=}}    
+      & \onslide<4->{\smath{R_1; a\cdot a^\star}}\\
+
+  & & & \onslide<5->{by substitution}\\
+
+  \onslide<5->{\smath{R_1}} & \onslide<5->{\smath{=}} 
+      & \onslide<5->{\smath{R_1; a\cdot a^\star \cdot b \cdot b^\star+ \lambda;b^\star}}\\
+  \onslide<5->{\smath{R_2}} & \onslide<5->{\smath{=}}    
+      & \onslide<5->{\smath{R_1; a\cdot a^\star}}\\
+
+  & & & \onslide<6->{by Arden}\\
+
+  \onslide<6->{\smath{R_1}} & \onslide<6->{\smath{=}} 
+      & \onslide<6->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
+  \onslide<6->{\smath{R_2}} & \onslide<6->{\smath{=}}    
+      & \onslide<6->{\smath{R_1; a\cdot a^\star}}\\
+
+  & & & \onslide<7->{by substitution}\\
+
+  \onslide<7->{\smath{R_1}} & \onslide<7->{\smath{=}} 
+      & \onslide<7->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
+  \onslide<7->{\smath{R_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}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{\LARGE A Variant of Arden's Lemma}
+
+  {\bf Arden's Lemma:}\smallskip 
+
+  If \smath{[] \not\in A} then
+  \begin{center}
+  \smath{X = X; A + \text{something}}
+  \end{center}
+  has the (unique) solution
+  \begin{center}
+  \smath{X = \text{something} ; A^\star}
+  \end{center}
+
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}<1->[t]
+  \small
+
+  \begin{center}
+  \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
+  \onslide<1->{\smath{R_1}} & \onslide<1->{\smath{=}} 
+      & \onslide<1->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
+  \onslide<1->{\smath{R_2}} & \onslide<1->{\smath{=}}    
+      & \onslide<1->{\smath{R_1; a + R_2; a}}\\
+
+  & & & \onslide<2->{by Arden}\\
+
+  \onslide<2->{\smath{R_1}} & \onslide<2->{\smath{=}} 
+      & \onslide<2->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
+  \onslide<2->{\smath{R_2}} & \onslide<2->{\smath{=}}    
+      & \only<2>{\smath{R_1; a + R_2; a}}%
+        \only<3->{\smath{R_1; a\cdot a^\star}}\\
+
+  & & & \onslide<4->{by Arden}\\
+
+  \onslide<4->{\smath{R_1}} & \onslide<4->{\smath{=}} 
+      & \onslide<4->{\smath{R_2; b \cdot b^\star+ \lambda;b^\star}}\\
+  \onslide<4->{\smath{R_2}} & \onslide<4->{\smath{=}}    
+      & \onslide<4->{\smath{R_1; a\cdot a^\star}}\\
+
+  & & & \onslide<5->{by substitution}\\
+
+  \onslide<5->{\smath{R_1}} & \onslide<5->{\smath{=}} 
+      & \onslide<5->{\smath{R_1; a\cdot a^\star \cdot b \cdot b^\star+ \lambda;b^\star}}\\
+  \onslide<5->{\smath{R_2}} & \onslide<5->{\smath{=}}    
+      & \onslide<5->{\smath{R_1; a\cdot a^\star}}\\
+
+  & & & \onslide<6->{by Arden}\\
+
+  \onslide<6->{\smath{R_1}} & \onslide<6->{\smath{=}} 
+      & \onslide<6->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
+  \onslide<6->{\smath{R_2}} & \onslide<6->{\smath{=}}    
+      & \onslide<6->{\smath{R_1; a\cdot a^\star}}\\
+
+  & & & \onslide<7->{by substitution}\\
+
+  \onslide<7->{\smath{R_1}} & \onslide<7->{\smath{=}} 
+      & \onslide<7->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
+  \onslide<7->{\smath{R_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)                  {$R_1$};
+  \node[state,accepting] (p_1) [right of=q_0]   {$R_2$};
+
+  \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}}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{\LARGE The Equ's Solving Algorithm}
+
+  \begin{itemize}
+  \item The algorithm must terminate: Arden makes one equation smaller; 
+  substitution deletes one variable from the right-hand sides.\bigskip
+
+  \item We need to maintain the invariant that Arden is applicable
+  (if \smath{[] \not\in A} then \ldots):\medskip
+
+  \begin{center}\small
+  \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
+  \smath{R_1} & \smath{=} & \smath{R_1; b + R_2; b + \lambda;[]}\\
+  \smath{R_2} & \smath{=} & \smath{R_1; a + R_2; a}\\
+
+  & & & by Arden\\
+
+  \smath{R_1} & \smath{=} & \smath{R_1; b + R_2; b + \lambda;[]}\\
+  \smath{R_2} & \smath{=} & \smath{R_1; a\cdot a^\star}\\
+  \end{tabular}
+  \end{center}
+
+  \end{itemize}
+
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{\LARGE Other Direction}
+
+  One has to prove
+
+  \begin{center}
+  \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r)})}
+  \end{center}
+
+  by induction on \smath{r}. Not trivial, but after a bit 
+  of thinking, one can prove that if
+
+  \begin{center}
+  \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_1)})}\hspace{5mm}
+  \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_2)})}
+  \end{center}
+
+  then
+
+  \begin{center}
+  \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_1) \,\cup\, \mathbb{L}(r_2)})}
+  \end{center}
+  
+  
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{\LARGE What Have We Achieved?}
+
+  \begin{itemize}
+  \item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_L) \;\Leftrightarrow\; L\; \text{is regular}}
+  \bigskip\pause
+  \item regular languages are closed under complementation; this is now easy\medskip
+  \begin{center}
+  \smath{U\!N\!IV /\!/ \approx_L \;\;=\;\; U\!N\!IV /\!/ \approx_{-L}}
+  \end{center}
+  \end{itemize}
+
+  
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{\LARGE Examples}
+
+  \begin{itemize}
+  \item \smath{L \equiv \Sigma^\star 0 \Sigma} is regular
+  \begin{quote}\small
+  \begin{tabular}{lcl}
+  \smath{A_1} & \smath{=} & \smath{\Sigma^\star 00}\\
+  \smath{A_2} & \smath{=} & \smath{\Sigma^\star 01}\\
+  \smath{A_3} & \smath{=} & \smath{\Sigma^\star 10 \cup \{0\}}\\
+  \smath{A_4} & \smath{=} & \smath{\Sigma^\star 11 \cup \{1\} \cup \{[]\}}\\
+  \end{tabular}
+  \end{quote}
+
+  \item \smath{L \equiv \{ 0^n 1^n \,|\, n \ge 0\}} is not regular
+  \begin{quote}\small
+  \begin{tabular}{lcl}
+  \smath{B_0} & \smath{=} & \smath{\{0^n 1^n \,|\,     n \ge 0\}}\\
+  \smath{B_1} & \smath{=} & \smath{\{0^n 1^{(n-1)} \,|\, n \ge 1\}}\\
+  \smath{B_2} & \smath{=} & \smath{\{0^n 1^{(n-2)} \,|\, n \ge 2\}}\\
+  \smath{B_3} & \smath{=} & \smath{\{0^n 1^{(n-3)} \,|\, n \ge 3\}}\\
+              & \smath{\vdots} &\\
+  \end{tabular}
+  \end{quote}
+  \end{itemize}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{\LARGE What We Have Not Achieved}
+
+  \begin{itemize}
+  \item regular expressions are not good if you look for a minimal
+  one for a language (DFAs have this notion)\pause\bigskip
+
+  \item Is there anything to be said about context free languages:\medskip
+  
+  \begin{quote}
+  A context free language is where every string can be recognised by
+  a pushdown automaton.\bigskip
+  \end{quote}
+  \end{itemize}
+
+  \textcolor{gray}{\footnotesize Yes. Derivatives also work for c-f grammars. Ongoing work.}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{\LARGE Conclusion}
+
+  \begin{itemize}
+  \item We formalised the Myhill-Nerode theorem based on 
+  regular expressions only (DFAs are difficult to deal with in a theorem prover).\smallskip
+
+  \item Seems to be a common theme: algorithms need to be reformulated
+  to better suit formal treatment.\smallskip
+
+  \item The most interesting aspect is that we are able to
+  implement the matcher directly inside the theorem prover
+  (ongoing work).\smallskip
+
+  \item Parsing is a vast field which seem to offer new results. 
+  \end{itemize}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}<1>[b]
+  \frametitle{
+  \begin{tabular}{c}
+  \mbox{}\\[13mm]
+  \alert{\LARGE Thank you very much!}\\
+  \alert{\Large Questions?}
+  \end{tabular}}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+
+(*<*)
+end
+(*>*)
\ No newline at end of file




