
diff -r fb201e383f1b -r da575186d492 Slides/Slides8.thy
--- a/Slides/Slides8.thy	Tue Feb 19 05:38:46 2013 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,1419 +0,0 @@
-(*<*)
-theory Slides8
-imports "~~/src/HOL/Library/LaTeXsugar" "Nominal"
-begin
-
-declare [[show_question_marks = false]]
-
-notation (latex output)
-  set ("_") and
-  Cons  ("_::/_" [66,65] 65) 
-
-(*>*)
-
-text_raw {*
-  \renewcommand{\slidecaption}{Copenhagen, 23rd~May 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: 4 Points}
-  \large
-  \begin{itemize}
-  \item It is easy to make mistakes.\medskip
-  \item Theorem provers can prevent mistakes, {\bf if} the problem
-  is formulated so that it is suitable for theorem provers.\medskip
-  \item This re-formulation can be done, even in domains where
-  we least expect it.\medskip 
-  \item Where theorem provers are superior to the {\color{gray}{(best)}} human reasoners. ;o)
-  \end{itemize}
-
-  \end{frame}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
-*}
-
-
-text_raw {*
-
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-  \mode<presentation>{
-  \begin{frame}[c]
-  \frametitle{}
-
-  \begin{tabular}{c@ {\hspace{2mm}}c}
-  \\[6mm]
-  \begin{tabular}{c}
-  \includegraphics[scale=0.12]{harper.jpg}\\[-2mm]
-  {\footnotesize Bob Harper}\\[-2.5mm]
-  {\footnotesize (CMU)}
-  \end{tabular}
-  \begin{tabular}{c}
-  \includegraphics[scale=0.36]{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}
-
-
-  
-
-
-  \end{frame}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
-
-*}
-
-text_raw {*
-
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-  \mode<presentation>{
-  \begin{frame}
-  \frametitle{Proof-Carrying Code}
-
-  \begin{textblock}{10}(2.5,2.2)
-  \begin{block}{Idea:}
-  \begin{center}
-  \begin{tikzpicture}
-  \draw[help lines,cream] (0,0.2) grid (8,4);
-  
-  \draw[line width=1mm, red] (5.5,0.6) rectangle (7.5,4);
-  \node[anchor=base] at (6.5,2.8) 
-     {\small\begin{tabular}{@ {}p{1.9cm}@ {}}\centering  user: untrusted code\end{tabular}};
-
-  \draw[line width=1mm, red] (0.5,0.6) rectangle (2.5,4);
-  \node[anchor=base] at (1.5,2.3) 
-     {\small\begin{tabular}{@ {}p{1.9cm}@ {}}\centering  developer ---\\ web server\end{tabular}};
-  
-  \onslide<3->{
-  \draw[line width=1mm, red, fill=red] (5.5,0.6) rectangle (7.5,1.8);
-  \node[anchor=base,white] at (6.5,1.1) 
-     {\small\begin{tabular}{@ {}p{1.9cm}@ {}}\bf\centering proof- checker\end{tabular}};}
-
-  \node at (3.8,3.0) [single arrow, fill=red,text=white, minimum height=3cm]{\bf code};
-  \onslide<2->{
-  \node at (3.8,1.3) [single arrow, fill=red,text=white, minimum height=3cm]{\bf certificate};
-  \node at (3.8,1.9) {\small\color{gray}{\mbox{}\hspace{-1mm}a proof in LF}};
-  }
-
-  
-  \end{tikzpicture}
-  \end{center}
-  \end{block}
-  \end{textblock}
-
-  %\begin{textblock}{15}(2,12)
-  %\small
-  %\begin{itemize}
-  %\item<4-> Appel's checker is $\sim$2700 lines of code (1865 loc of\\ LF definitions; 
-  %803 loc in C including 2 library functions)\\[-3mm]
-  %\item<5-> 167 loc in C implement a type-checker
-  %\end{itemize}
-  %\end{textblock}
-
-  \end{frame}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
-
-*}
-
-text {*
-  \tikzstyle{every node}=[node distance=25mm,text height=1.5ex, text depth=.25ex]
-  \tikzstyle{node1}=[rectangle, minimum size=10mm, rounded corners=3mm, very thick, 
-                     draw=black!50, top color=white, bottom color=black!20]
-  \tikzstyle{node2}=[rectangle, minimum size=12mm, rounded corners=3mm, very thick, 
-                     draw=red!70, top color=white, bottom color=red!50!black!20]
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-  \mode<presentation>{
-  \begin{frame}<2->[squeeze]
-  \frametitle{} 
-  
-  \begin{columns}
-  
-  \begin{column}{0.8\textwidth}
-  \begin{textblock}{0}(1,2)
-
-  \begin{tikzpicture}
-  \matrix[ampersand replacement=\&,column sep=7mm, row sep=5mm]
-  { \&[-10mm] 
-    \node (def1)   [node1] {\large\hspace{1mm}Spec\hspace{1mm}\mbox{}}; \&
-    \node (proof1) [node1] {\large Proof}; \&
-    \node (alg1)   [node1] {\large\hspace{1mm}Alg\hspace{1mm}\mbox{}}; \\
-    
-    \onslide<4->{\node {\begin{tabular}{c}\small 1st\\[-2.5mm] \footnotesize solution\end{tabular}};} \&
-    \onslide<4->{\node (def2)   [node2] {\large Spec$^\text{+ex}$};} \&
-    \onslide<4->{\node (proof2) [node1] {\large Proof};} \&
-    \onslide<4->{\node (alg2)   [node1] {\large\hspace{1mm}Alg\hspace{1mm}\mbox{}};} \\
-     
-    \onslide<5->{\node {\begin{tabular}{c}\small 2nd\\[-2.5mm] \footnotesize solution\end{tabular}};} \&
-    \onslide<5->{\node (def3)   [node1] {\large\hspace{1mm}Spec\hspace{1mm}\mbox{}};} \&
-    \onslide<5->{\node (proof3) [node1] {\large Proof};} \&
-    \onslide<5->{\node (alg3)   [node2] {\large Alg$^\text{-ex}$};} \\
-
-    \onslide<6->{\node {\begin{tabular}{c}\small 3rd\\[-2.5mm] \footnotesize solution\end{tabular}};} \&
-    \onslide<6->{\node (def4)   [node1] {\large\hspace{1mm}Spec\hspace{1mm}\mbox{}};} \&
-    \onslide<6->{\node (proof4) [node2] {\large\hspace{1mm}Proof\hspace{1mm}};} \&
-    \onslide<6->{\node (alg4)   [node1] {\large\hspace{1mm}Alg\hspace{1mm}\mbox{}};} \\
-  };
-
-  \draw[->,black!50,line width=2mm] (proof1) -- (def1);
-  \draw[->,black!50,line width=2mm] (proof1) -- (alg1);
-  
-  \onslide<4->{\draw[->,black!50,line width=2mm] (proof2) -- (def2);}
-  \onslide<4->{\draw[->,black!50,line width=2mm] (proof2) -- (alg2);}
-
-  \onslide<5->{\draw[->,black!50,line width=2mm] (proof3) -- (def3);}
-  \onslide<5->{\draw[->,black!50,line width=2mm] (proof3) -- (alg3);}
-  
-  \onslide<6->{\draw[->,black!50,line width=2mm] (proof4) -- (def4);}
-  \onslide<6->{\draw[->,black!50,line width=2mm] (proof4) -- (alg4);}
-
-  \onslide<3->{\draw[white,line width=1mm] (1.1,3.2) -- (0.9,2.85) -- (1.1,2.35) -- (0.9,2.0);} 
-  \end{tikzpicture}
-
-  \end{textblock}
-  \end{column}
-  \end{columns}
-
-
-  \begin{textblock}{3}(12,3.6)
-  \onslide<4->{
-  \begin{tikzpicture}
-  \node at (0,0) [single arrow, shape border rotate=270, fill=red,text=white]{2h};
-  \end{tikzpicture}}
-  \end{textblock}
-
-  \end{frame}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
-
-*}
-
-
-(*<*)
-atom_decl name
-
-nominal_datatype lam = 
-    Var "name"
-  | App "lam" "lam"
-  | Lam "\<guillemotleft>name\<guillemotright>lam" ("Lam [_]._" [100,100] 100)
-
-nominal_primrec
-  subst :: "lam \<Rightarrow> name \<Rightarrow> lam \<Rightarrow> lam"  ("_[_::=_]")
-where
-  "(Var x)[y::=s] = (if x=y then s else (Var x))"
-| "(App t\<^isub>1 t\<^isub>2)[y::=s] = App (t\<^isub>1[y::=s]) (t\<^isub>2[y::=s])"
-| "x\<sharp>(y,s) \<Longrightarrow> (Lam [x].t)[y::=s] = Lam [x].(t[y::=s])"
-apply(finite_guess)+
-apply(rule TrueI)+
-apply(simp add: abs_fresh)
-apply(fresh_guess)+
-done
-
-lemma  subst_eqvt[eqvt]:
-  fixes pi::"name prm"
-  shows "pi\<bullet>(t1[x::=t2]) = (pi\<bullet>t1)[(pi\<bullet>x)::=(pi\<bullet>t2)]"
-by (nominal_induct t1 avoiding: x t2 rule: lam.strong_induct)
-   (auto simp add: perm_bij fresh_atm fresh_bij)
-
-lemma fresh_fact:
-  fixes z::"name"
-  shows "\<lbrakk>z\<sharp>s; (z=y \<or> z\<sharp>t)\<rbrakk> \<Longrightarrow> z\<sharp>t[y::=s]"
-by (nominal_induct t avoiding: z y s rule: lam.strong_induct)
-   (auto simp add: abs_fresh fresh_prod fresh_atm)
-
-lemma forget: 
-  assumes asm: "x\<sharp>L"
-  shows "L[x::=P] = L"
-  using asm 
-by (nominal_induct L avoiding: x P rule: lam.strong_induct)
-   (auto simp add: abs_fresh fresh_atm)
-(*>*)
-
-text_raw {*
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-  \mode<presentation>{
-  \begin{frame}
-
-  \begin{textblock}{16}(1,1)
-  \renewcommand{\isasymbullet}{$\cdot$}
-  \tiny\color{black}
-*}
-lemma substitution_lemma_not_to_be_tried_at_home: 
-  assumes asm: "x\<noteq>y" "x\<sharp>L"
-  shows "M[x::=N][y::=L] = M[y::=L][x::=N[y::=L]]"
-using asm
-proof (induct M arbitrary: x y N L rule: lam.induct)
-  case (Lam z M1) 
-  have ih: "\<And>x y N L. \<lbrakk>x\<noteq>y; x\<sharp>L\<rbrakk> \<Longrightarrow> M1[x::=N][y::=L] = M1[y::=L][x::=N[y::=L]]" by fact
-  have "x\<noteq>y" by fact
-  have "x\<sharp>L" by fact
-  obtain z'::"name" where fc: "z'\<sharp>(x,y,z,M1,N,L)" by (rule exists_fresh) (auto simp add: fs_name1)
-  have eq: "Lam [z'].([(z',z)]\<bullet>M1) = Lam [z].M1" using fc 
-    by (auto simp add: lam.inject alpha fresh_prod fresh_atm)
-  have fc': "z'\<sharp>N[y::=L]" using fc by (simp add: fresh_fact fresh_prod)
-  have "([(z',z)]\<bullet>x) \<noteq> ([(z',z)]\<bullet>y)" using `x\<noteq>y` by (auto simp add: calc_atm)
-  moreover
-  have "([(z',z)]\<bullet>x)\<sharp>([(z',z)]\<bullet>L)" using `x\<sharp>L` by (simp add: fresh_bij)
-  ultimately 
-  have "M1[([(z',z)]\<bullet>x)::=([(z',z)]\<bullet>N)][([(z',z)]\<bullet>y)::=([(z',z)]\<bullet>L)] 
-        = M1[([(z',z)]\<bullet>y)::=([(z',z)]\<bullet>L)][([(z',z)]\<bullet>x)::=([(z',z)]\<bullet>N)[([(z',z)]\<bullet>y)::=([(z',z)]\<bullet>L)]]"
-    using ih by simp
-  then have "[(z',z)]\<bullet>(M1[([(z',z)]\<bullet>x)::=([(z',z)]\<bullet>N)][([(z',z)]\<bullet>y)::=([(z',z)]\<bullet>L)] 
-        = M1[([(z',z)]\<bullet>y)::=([(z',z)]\<bullet>L)][([(z',z)]\<bullet>x)::=([(z',z)]\<bullet>N)[([(z',z)]\<bullet>y)::=([(z',z)]\<bullet>L)]])"
-    by (simp add: perm_bool)
-  then have ih': "([(z',z)]\<bullet>M1)[x::=N][y::=L] = ([(z',z)]\<bullet>M1)[y::=L][x::=N[y::=L]]"
-    by (simp add: eqvts perm_swap)
-  show "(Lam [z].M1)[x::=N][y::=L] = (Lam [z].M1)[y::=L][x::=N[y::=L]]" (is "?LHS=?RHS") 
-  proof - 
-    have "?LHS = (Lam [z'].([(z',z)]\<bullet>M1))[x::=N][y::=L]" using eq by simp
-    also have "\<dots> = Lam [z'].(([(z',z)]\<bullet>M1)[x::=N][y::=L])" using fc by (simp add: fresh_prod)
-    also from ih have "\<dots> = Lam [z'].(([(z',z)]\<bullet>M1)[y::=L][x::=N[y::=L]])" sorry 
-    also have "\<dots> = (Lam [z'].([(z',z)]\<bullet>M1))[y::=L][x::=N[y::=L]]" using fc fc' by (simp add: fresh_prod)
-    also have "\<dots> = ?RHS" using eq by simp
-    finally show "?LHS = ?RHS" .
-  qed
-qed (auto simp add: forget)
-text_raw {*
-  \end{textblock}
-  \mbox{}
-
-  \only<2->{
-  \begin{textblock}{11.5}(4,2.3)
-  \begin{minipage}{9.3cm}
-  \begin{block}{}\footnotesize
-*}
-lemma substitution_lemma\<iota>:
-  assumes asm: "x \<noteq> y" "x \<sharp> L"
-  shows "M[x::=N][y::=L] = M[y::=L][x::=N[y::=L]]"
-  using asm
-by (nominal_induct M avoiding: x y N L rule: lam.strong_induct)
-     (auto simp add: forget fresh_fact)
-text_raw {*  
-  \end{block}
-  \end{minipage}
-  \end{textblock}}
-  \end{frame}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
-*}
-
-
-text_raw {*
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-  \mode<presentation>{
-  \begin{frame}<1->[c]
-  \frametitle{Lesson Learned}
-
-  \begin{textblock}{11.5}(1.2,5)
-  \begin{minipage}{10.5cm}
-  \begin{block}{}
-  Theorem provers can keep large proofs and definitions consistent and
-  make them modifiable.
-   \end{block}
-  \end{minipage}
-  \end{textblock}
-
-  
-  \end{frame}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
-*}
-
-text_raw {*
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-  \mode<presentation>{
-  \begin{frame}
-  \frametitle{}
-
-  \begin{textblock}{11.5}(0.8,2.3)
-  \begin{minipage}{11.2cm}
-  In most papers/books: 
-  \begin{block}{}
-  \color{darkgray}
-  ``\ldots this necessary hygienic discipline is somewhat swept under the carpet via
-  the so-called `{\bf variable convention}' \ldots
-  The {\color{black}{\bf belief}} that this is {\bf sound} came from the calculus 
-  with nameless binders in de Bruijn'' 
-  \end{block}\medskip
-  \end{minipage}
-  \end{textblock}
-
-  \begin{textblock}{11.5}(0.8,10)
-  \includegraphics[scale=0.25]{LambdaBook.jpg}\hspace{-3mm}\includegraphics[scale=0.3]{barendregt.jpg}
-  \end{textblock}
-  \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->[c]
-  \frametitle{Testing}
-
-  \begin{itemize}
-  \item We can only test a {\bf finite} amount of examples:\bigskip
-
-  \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 Equiv.~Classes}
-
-  \mbox{}\\[3cm]
-
-  \begin{itemize}
-  \item \smath{\text{finals}\,L \dn 
-     \{{\lbrack\mkern-2mu\lbrack{s}\rbrack\mkern-2mu\rbrack}_\approx\;|\; s \in L\}}\\
-  \medskip
-
-  \item we can prove: \smath{L = \bigcup (\text{finals}\,L)}
-
-  \end{itemize}
-
-  \end{frame}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
-*}
-
-text_raw {*
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-  \mode<presentation>{
-  \begin{frame}[c]
-  \frametitle{\LARGE Transitions between ECs}
-
-  \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




