diff -r 368fc38321fc -r 878de0084b62 Slides/SlidesA.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Slides/SlidesA.thy	Fri Feb 17 02:05:00 2012 +0000
@@ -0,0 +1,1648 @@
+(*<*)
+theory SlidesA
+imports "~~/src/HOL/Library/LaTeXsugar" "Nominal"
+begin
+
+notation (latex output)
+  set ("_") and
+  Cons  ("_::/_" [66,65] 65) 
+
+(*>*)
+
+
+text_raw {*
+  %% was \begin{colormixin}{20!averagebackgroundcolor}
+  %% 
+  %% is \begin{colormixin}{50!averagebackgroundcolor}
+  \renewcommand{\slidecaption}{Warsaw, 9 February 2012}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}
+  \frametitle{%
+  \begin{tabular}{@ {}c@ {}}
+  \Huge Nominal Techniques\\[0mm]
+  \Huge in Isabelle\\
+  \Large or, How Not to be Intimidated by the\\[-3mm]
+  \Large Variable Convention\\[-5mm]
+  \end{tabular}}
+  
+  \begin{center}
+  Christian Urban\\[1mm]
+  King's College London\\[-6mm]\mbox{}
+  \end{center}
+ 
+  \begin{center}
+  \begin{block}{}
+  \color{gray}
+  \small
+  {\bf\mbox{}\hspace{-1.5mm}Variable Convention:}\\[1mm] 
+  If $M_1,\ldots,M_n$ occur in a certain mathematical context
+  (e.g. definition, proof), then in these terms all bound variables 
+  are chosen to be different from the free variables.\\[2mm]
+  
+  \footnotesize\hfill Barendregt in ``The Lambda-Calculus: Its Syntax and Semantics''
+  \end{block}
+  \end{center}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}<1->[c]
+  \frametitle{Nominal Techniques}
+
+  \begin{itemize}
+  \item Andy Pitts found out that permutations\\ preserve $\alpha$-equivalence:
+  \begin{center}
+  \smath{t_1 \approx_{\alpha} t_2 \quad \Rightarrow\quad \pi \act t_1 \approx_{\alpha} \pi \act t_2} 
+  \end{center}
+
+  \item also permutations and substitutions commute, if you suspend permutations
+  in front of variables
+  \begin{center}
+  \smath{\pi\act\sigma(t) = \sigma(\pi\act t)}
+  \end{center}\medskip\medskip
+
+  \item this allowed us to define Nominal Unification\medskip
+  \begin{center}
+  \smath{\nabla \vdash t \approx^?_{\alpha} t'}\hspace{2cm}
+  \smath{\nabla \vdash a \fresh^? t}
+  \end{center}
+  \end{itemize}
+
+  \begin{textblock}{3}(13.1,1.1)
+  \includegraphics[scale=0.26]{andrewpitts.jpg}
+  \end{textblock}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{Nominal Isabelle}
+
+  \begin{itemize}
+  \item a theory about atoms and permutations\medskip
+
+  \item support and freshness
+  \begin{center}
+  \smath{\text{supp}(x) \dn \{ a \mid \text{infinite}\,\{ b \mid \swap{a}{b}\act x \not= x\}\}}
+  \end{center}\bigskip\pause
+
+  \item $\alpha$-equivalence
+  \begin{center}
+  \begin{tabular}{l}
+  \smath{as.x \approx_\alpha bs.y \dn}\\[2mm]
+  \hspace{2cm}\smath{\exists \pi.\;\text{supp}(x) - as = \text{supp}(y) - bs}\\ 
+  \hspace{2cm}\smath{\;\wedge\; \text{supp}(x) - as \fresh \pi}\\
+  \hspace{2cm}\smath{\;\wedge\; \pi \act x = y}
+  \end{tabular}
+  \end{center}
+  \end{itemize}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}<1-6>
+  \frametitle{New Types in HOL}
+
+   \begin{center}
+  \begin{tikzpicture}[scale=1.5]
+  %%%\draw[step=2mm] (-4,-1) grid (4,1);
+  
+  \onslide<2-4,6>{\draw[very thick] (0.7,0.4) circle (4.25mm);}
+  \onslide<1-4,6>{\draw[rounded corners=1mm, very thick] ( 0.0,-0.8) rectangle ( 1.8, 0.9);}
+  \onslide<3-5,6>{\draw[rounded corners=1mm, very thick] (-1.95,0.85) rectangle (-2.85,-0.05);}
+  
+  \onslide<3-4,6>{\draw (-2.0, 0.845) --  (0.7,0.845);}
+  \onslide<3-4,6>{\draw (-2.0,-0.045)  -- (0.7,-0.045);}
+
+  \onslide<4-4,6>{\alert{\draw ( 0.7, 0.4) node {\footnotesize\begin{tabular}{@ {}c@ {}}$\alpha$-\\[-1mm]classes\end{tabular}};}}
+  \onslide<4-5,6>{\alert{\draw (-2.4, 0.4) node {\footnotesize\begin{tabular}{@ {}c@ {}}$\alpha$-eq.\\[-1mm]terms\end{tabular}};}}
+  \onslide<1-4,6>{\draw (1.8, 0.48) node[right=-0.1mm]
+    {\footnotesize\begin{tabular}{@ {}l@ {}}existing\\[-1mm] type\\ \onslide<4-4,6>{\alert{(sets of raw terms)}}\end{tabular}};}
+  \onslide<2-4,6>{\draw (0.9, -0.35) node {\footnotesize\begin{tabular}{@ {}l@ {}}non-empty\\[-1mm]subset\end{tabular}};}
+  \onslide<3-5,6>{\draw (-3.25, 0.55) node {\footnotesize\begin{tabular}{@ {}l@ {}}new\\[-1mm]type\end{tabular}};}
+  
+  \onslide<3-4,6>{\draw[<->, very thick] (-1.8, 0.3) -- (-0.1,0.3);}
+  \onslide<3-4,6>{\draw (-0.95, 0.3) node[above=0mm] {\footnotesize{}isomorphism};}
+
+  \onslide<6>{\draw[->, line width=2mm, red] (-1.0,-0.4) -- (0.35,0.16);}
+  \end{tikzpicture}
+  \end{center}
+  
+  \begin{center}
+  \textcolor{red}{\large\bf\onslide<6>{define $\alpha$-equivalence}}
+  \end{center}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}<1-3>[c]
+  \frametitle{HOL vs.~Nominal}
+
+  \begin{itemize}
+  \item Nominal logic / nominal sets by Pitts are incompatible 
+  with choice principles\medskip
+
+  \item HOL includes Hilbert's epsilon\pause\bigskip
+
+  \item Solution: Do not require that everything has finite support\medskip
+
+  \begin{center}
+  \smath{\onslide<1-2>{\text{finite}(\text{supp}(x))  \quad\Rightarrow\quad} a \fresh a.x}
+  \end{center}
+  \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$31pp}
+  \end{tabular}\\
+
+  \pause
+  \\[0mm]
+  
+  \begin{tabular}{c}
+  \includegraphics[scale=0.36]{appel.jpg}\\[-2mm] 
+  {\footnotesize Andrew Appel}\\[-2.5mm]
+  {\footnotesize (Princeton)}
+  \end{tabular} &
+
+  \begin{tabular}{p{6cm}}
+  \raggedright
+  \color{gray}{relied on their proof in a\\ {\bf security} critical application}
+  \end{tabular}
+  \end{tabular}\medskip\pause
+
+  \small
+  \begin{minipage}{1.0\textwidth}
+  (I also found an {\bf error} in my Ph.D.-thesis about cut-elimination
+  examined by Henk Barendregt and Andy Pitts.)
+  \end{minipage}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+
+
+text_raw {*
+
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}<1,2,3,4>[squeeze]
+  \frametitle{Formalisation of LF} 
+  
+  
+  \begin{center}
+  \begin{tabular}{@ {\hspace{-16mm}}lc}
+  \mbox{}\\[-6mm]
+
+  \textcolor{white}{\raisebox{4mm}{1.~Solution}} &
+  \begin{tikzpicture}
+  [node distance=25mm,
+   text height=1.5ex,
+   text depth=.25ex,
+   node1/.style={
+     rectangle, minimum size=10mm, rounded corners=3mm, very thick, 
+     draw=black!50, top color=white, bottom color=black!20},
+  ]
+
+  \node (proof) [node1] {\large Proof};
+  \node (def)   [node1, left of=proof] {\large$\,\;\dn\;\,$};
+  \node (alg)  [node1, right of=proof] {\large\hspace{1mm}Alg.\hspace{1mm}\mbox{}};
+  
+  \draw[->,black!50,line width=2mm] (proof) -- (def);
+  \draw[->,black!50,line width=2mm] (proof) -- (alg);
+
+  \onslide<2->{\draw[white,line width=1mm] (0.1,0.6) -- (-0.1,0.25) -- (0.1,-0.25) -- (-0.1,-0.6);} 
+  \end{tikzpicture}
+  \\[2mm]
+
+  \onslide<3->{%
+  \raisebox{4mm}{\textcolor{white}{1st Solution}} &
+  \begin{tikzpicture}
+  [node distance=25mm,
+   text height=1.5ex,
+   text depth=.25ex,
+   node1/.style={
+     rectangle, minimum size=10mm, rounded corners=3mm, very thick, 
+     draw=black!50, top color=white, bottom color=black!20},
+   node2/.style={
+     rectangle, minimum size=12mm, rounded corners=3mm, very thick, 
+     draw=red!70, top color=white, bottom color=red!50!black!20}
+  ]
+
+  \node (proof) [node1] {\large Proof};
+  \node (def)   [node2, left of=proof] {\large$\dn{}\!\!^\text{+ex}$};
+  \node (alg)   [node1, right of=proof] {\large\hspace{1mm}Alg.\hspace{1mm}\mbox{}};
+  
+  \draw[->,black!50,line width=2mm] (proof) -- (def);
+  \draw[->,black!50,line width=2mm] (proof) -- (alg);
+
+  \end{tikzpicture}
+  \\[2mm]}
+
+  \end{tabular}
+  \end{center}
+
+  \begin{textblock}{3}(13.2,5.1)
+  \onslide<3->{
+  \begin{tikzpicture}
+  \node at (0,0) [single arrow, shape border rotate=270, fill=red,text=white]{2h};
+  \end{tikzpicture}
+  }
+  \end{textblock}
+
+  
+  \begin{textblock}{11}(1.4,14.3)
+  \only<1->{\footnotesize (one needs to check $\sim$31pp~of informal paper proofs from 
+            ACM Transactions on Computational Logic, 2005)}
+  \end{textblock}
+
+  \only<4->{
+  \begin{textblock}{9}(10,11.5)
+  \begin{tikzpicture}[remember picture, overlay]
+  \draw (0,0) node[fill=cream, text width=5.3cm, thick, draw=red, rounded corners=1mm] (n2)
+  {\raggedright I also found \mbox{(fixable)} mistakes in my Ph.D.~thesis. 
+  };
+  \end{tikzpicture}
+  \end{textblock}}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+text_raw {*
+
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{\LARGE\begin{tabular}{c}Nominal Isabelle\end{tabular}}
+  
+
+  \begin{itemize}
+  \item \ldots{}is a tool on top of the theorem prover
+  Isabelle; bound variables have names (no de Bruijn 
+  indices).\medskip
+
+  \item It can be used to, for example, represent lambda terms
+
+  \begin{center}
+  \smath{M ::= x\;\mid\; M\,N \;\mid\; \lambda x.M}
+  \end{center}
+  \end{itemize}
+  
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+text_raw {* 
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}
+  \small
+  \begin{beamercolorbox}[sep=1mm, wd=11cm]{boxcolor}
+  {\bf Substitution Lemma:} 
+  If \smath{x\not\equiv y} and
+  \smath{x\not\in \text{fv}(L)}, then\\
+  \mbox{}\hspace{5mm}\smath{M[x:=N][y:=L]\equiv M[y:=L][x:=N[y:=L]]}
+  \end{beamercolorbox}
+
+  {\bf Proof:} \alert<4>{By induction on the structure of \smath{M}.}
+  \begin{itemize}
+  \item {\bf Case 1:} \smath{M} is a variable.
+
+  \begin{tabular}{@ {}l@ {\hspace{1mm}}p{9cm}@ {}}
+   Case 1.1. & \smath{M\equiv x}. Then both sides \alert<3,4>{equal} 
+               \smath{N[y:=L]} since \smath{x\not\equiv y}.\\[1mm] 
+   Case 1.2. & \smath{M\equiv y}. Then both sides \alert<3,4>{equal}
+               \smath{L}, for \smath{x\not\in \text{fv}(L)}\\
+             & implies \smath{L[x:=\ldots]\equiv L}.\\[1mm]
+   Case 1.3. & \smath{M\equiv z\not\equiv x,y}. Then both sides \alert<3,4>{equal} \smath{z}.\\[1mm]
+  \end{tabular}
+
+  \item {\bf Case 2:} \smath{M\equiv \lambda z.M_1}. 
+  \alert<2>{By the variable convention we may assume that \smath{z\not\equiv x,y} 
+  and \smath{z} is not free in \smath{N,L}.}
+
+  \begin{tabular}{@ {}r@ {\hspace{1mm}}l@ {}}
+  \smath{(\lambda z.M_1)[x\!:=\!N][y\!:=\!L]} 
+  \smath{\equiv} & \smath{\lambda z.(M_1[x\!:=\!N][y\!:=\!L])}\\
+  \smath{\equiv} & \smath{\lambda z.(M_1[y\!:=\!L][x\!:=\!N[y\!:=\!L]])}\\
+  \smath{\equiv} & \smath{(\lambda z.M_1)[y\!:=\!L][x\!:=\!N[y\!:=\!L]]}.\\
+  \end{tabular}
+  
+  \item {\bf Case 3:} \smath{M\equiv M_1 M_2}. 
+  The statement follows again from the induction hypothesis. \hfill$\,\Box\,$
+  \end{itemize}
+
+  \begin{textblock}{11}(4,3)
+  \begin{block}<5>{}
+  Remember only if \smath{y\not=x} and \smath{x\not\in \text{fv}(N)} then\\[2mm]
+  \smath{\quad(\lambda y.M)[x:=N]=\lambda y.(M[x:=N])}\\[4mm]
+  
+          \begin{tabular}{c@ {\hspace{2mm}}l@ {\hspace{2mm}}l@ {}}
+            & \smath{(\lambda z.M_1)[x:=N][y:=L]}\\[1.3mm]
+            \smath{\equiv} & \smath{(\lambda z.(M_1[x:=N]))[y:=L]} & $\stackrel{1}{\leftarrow}$\\[1.3mm]
+            \smath{\equiv} & \smath{\lambda z.(M_1[x:=N][y:=L])} & $\stackrel{2}{\leftarrow}$\\[1.3mm]
+            \smath{\equiv} & \smath{\lambda z.(M_1[y:=L][x:=N[y:=L]])} & IH\\[1.3mm] 
+            \smath{\equiv} & \smath{(\lambda z.(M_1[y:=L]))[x:=N[y:=L]])} 
+                                 & $\stackrel{2}{\rightarrow}$ \alert{\bf\;!}\\[1.3mm] 
+            \smath{\equiv} & \smath{(\lambda z.M_1)[y:=L][x:=N[y:=L]]}. & 
+                                                             $\stackrel{1}{\rightarrow}$\\[1.3mm]
+          \end{tabular}
+  \end{block}
+  \end{textblock}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}
+  \frametitle{Nominal Isabelle} 
+
+  \begin{itemize}
+  \item Define lambda-terms as:
+  \end{itemize}
+*}
+      
+ atom_decl name   text_raw {*\medskip\isanewline *}
+	nominal_datatype lam =
+	    Var "name"
+	  | App "lam" "lam" 
+	  | Lam "\<guillemotleft>name\<guillemotright>lam" ("Lam _._")
+
+
+text_raw {*
+  \mbox{}\bigskip
+
+  \begin{itemize}
+  \item These are \underline{\bf named} alpha-equivalence classes, for example
+  \end{itemize}
+
+  \begin{center}
+  \gb{@{text "Lam a.(Var a)"}} \alert{$\,=\,$} \gb{@{text "Lam b.(Var b)"}}
+  \end{center}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+(*<*)
+
+nominal_primrec
+  subst :: "lam \<Rightarrow> name \<Rightarrow> lam \<Rightarrow> lam"  ("_[_::=_]" [100,100,100] 100)
+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> (SlidesA.Lam x t)[y::=s] = SlidesA.Lam x (t[y::=s])"
+apply(finite_guess)+
+apply(rule TrueI)+
+apply(simp add: abs_fresh)+
+apply(fresh_guess)+
+done
+
+(*>*)
+
+text_raw {*
+
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}
+  %%\frametitle{\large Formal Proof of the Substitution Lemma} 
+
+  \small
+  \begin{tabular}{@ {\hspace{-4mm}}c @ {}}
+  \begin{minipage}{1.1\textwidth}
+*}
+
+lemma forget: 
+  assumes a: "x \<sharp> L"
+  shows "L[x::=P] = L"
+using a by (nominal_induct L avoiding: x P rule: lam.strong_induct)
+                (auto simp add: abs_fresh fresh_atm)
+
+lemma fresh_fact: 
+  fixes z::"name"
+  assumes a: "z \<sharp> N" "z \<sharp> L"
+  shows "z \<sharp> N[y::=L]"
+using a by (nominal_induct N avoiding: z y L rule: lam.strong_induct)
+                (auto simp add: abs_fresh fresh_atm)
+
+lemma substitution_lemma:  
+  assumes a: "x \<noteq> y" "x \<sharp> L" -- {* \mbox{}\hspace{-2mm}\tikz[remember picture] \node (n1) {}; *}
+  shows "M[x::=N][y::=L] = M[y::=L][x::=N[y::=L]]"
+using a 
+by (nominal_induct M avoiding: x y N L rule: lam.strong_induct)
+     (auto simp add: fresh_fact forget)
+
+text_raw {*
+  \end{minipage}
+  \end{tabular}
+
+  \begin{textblock}{6}(11,9)
+  \only<2>{
+  \begin{tikzpicture}[remember picture, overlay]
+  \draw (0,0) node[fill=cream, text width=5.5cm, thick, draw=red, rounded corners=1mm] (n2)
+  {\setlength\leftmargini{6mm}%
+   \begin{itemize}
+   \item stands for \smath{x\not\in \text{fv}(L)}\\[-2mm]
+   \item reads as ``\smath{x} fresh for \smath{L}'' 
+   \end{itemize}
+  };
+  
+  \path[overlay, ->, very thick, red] (n2) edge[out=-90, in=0] (n1);
+  \end{tikzpicture}}
+  \end{textblock}
+
+  \only<1-3>{}
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+text_raw {*
+
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}
+  \frametitle{\LARGE (Weak) Induction Principles} 
+
+  \begin{itemize}
+  \item The usual induction principle for lambda-terms is as follows:
+
+  \begin{center}
+  \mbox{}\hspace{-1mm}\begin{beamercolorbox}[sep=1mm, wd=9cm]{boxcolor}
+  \centering\smath{%
+  \infer{P\,t}
+  {\begin{array}{l}
+  \forall x.\;P\,x\\[2mm]
+  \forall t_1\,t_2.\;P\,t_1\wedge P\,t_2\Rightarrow P\,(t_1\;t_2)\\[2mm]
+  \forall x\,t.\;P\,t\Rightarrow P\,(\lambda x.t)\\
+  \end{array}
+  }}
+  \end{beamercolorbox}
+  \end{center}
+
+  \item It requires us in the lambda-case to show the property \smath{P} for
+  all binders \smath{x}.\smallskip\\ 
+
+  (This nearly always requires renamings and they can be 
+  tricky to automate.)
+  \end{itemize}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+
+text_raw {*
+
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}
+  \frametitle{\LARGE Strong Induction Principles} 
+
+  \begin{itemize}
+  \item Therefore we will use the following strong induction principle:
+
+  \begin{center}
+  \mbox{}\hspace{-2mm}\begin{beamercolorbox}[sep=1mm, wd=11.5cm]{boxcolor}
+  \centering\smath{%
+  \infer{\tikz[remember picture] \node[inner sep=1mm] (n1a) {\alert<4>{$P$}};%
+         \tikz[remember picture] \node[inner sep=1mm] (n2a) {\alert<3>{$c$}};%
+         \tikz[remember picture] \node[inner sep=1mm] (n3a) {\alert<2>{$t$}};}
+  {\begin{array}{l}
+  \forall x\,c.\;P\,c\;x\\[2mm]
+  \forall t_1\,t_2\,c.\;(\forall d.\,P d\,t_1)\wedge (\forall d. P\,d\,t_2)
+  \Rightarrow P\,c\;(t_1\,t_2)\\[2mm]
+  \forall x\,t\,c.\;\alert<1>{x\fresh \alert<3>{c}} 
+  \wedge (\forall d. P\,d\,t)\Rightarrow P\,c\;(\lambda x.t)
+  \end{array}
+  }}
+  \end{beamercolorbox}
+  \end{center}
+  \end{itemize}
+
+  \begin{textblock}{11}(0.9,10.9)
+  \only<2>{
+  \begin{tikzpicture}[remember picture]
+  \draw (0,0) node[fill=cream, text width=10.5cm, thick, draw=red, rounded corners=1mm] (n3b)
+  { The variable over which the induction proceeds:\\[2mm]
+    \hspace{3mm}``\ldots By induction over the structure of \smath{M}\ldots''};
+
+  \path[overlay, ->, ultra thick, red] (n3b) edge[out=90, in=-110] (n3a);
+  \end{tikzpicture}}
+
+  \only<3>{
+  \begin{tikzpicture}[remember picture]
+  \draw (0,0) node[fill=cream, text width=11cm, thick, draw=red, rounded corners=1mm] (n2b)
+  {The {\bf context} of the induction; i.e.~what the binder should be fresh for
+   $\quad\Rightarrow$ \smath{(x,y,N,L)}:\\[2mm]
+   ``\ldots By the variable convention we can assume \mbox{\smath{z\not\equiv x,y}} 
+     and \smath{z} not free in \smath{N}$\!$,\,\smath{L}\ldots''};
+
+  \path[overlay, ->, ultra thick, red] (n2b) edge[out=90, in=-100] (n2a);
+  \end{tikzpicture}}
+
+  \only<4>{
+  \begin{tikzpicture}[remember picture]
+  \draw (0,0) node[fill=cream, text width=11cm, thick, draw=red, rounded corners=1mm] (n1b)
+  {The property to be proved by induction:\\[-3mm]
+  \begin{center}\small
+  \begin{tabular}{l}
+  \smath{\!\!\lambda
+  (x,\!y,\!N\!,\!L).\,\lambda M.\;\,x\not=y\,\wedge\,x\fresh L\,\Rightarrow}\\[1mm]
+  \hspace{8mm}
+  \smath{M[x\!:=\!N][y\!:=\!L] = M[y\!:=\!L][x\!:=\!N[y\!:=\!L]]}  
+  \end{tabular}
+  \end{center}};
+
+  \path[overlay, ->, ultra thick, red] (n1b) edge[out=90, in=-70] (n1a);
+  \end{tikzpicture}}
+  \end{textblock}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+
+text_raw {*
+
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}
+  \frametitle{\LARGE Strong Induction Principles} 
+
+   \begin{center}
+  \mbox{}\hspace{-2mm}\begin{beamercolorbox}[sep=1mm, wd=11.5cm]{boxcolor}
+  \centering\smath{%
+  \infer{P\,\alert{c}\;t}
+  {\begin{array}{l}
+  \forall x\,c.\;P\,c\;x\\[2mm]
+  \forall t_1\,t_2\,c.\;(\forall d.\,P d\,t_1)\wedge (\forall d. P\,d\,t_2)
+  \Rightarrow P\,c\;(t_1\,t_2)\\[2mm]
+  \forall x\,t\,c.\;x\fresh c \wedge (\forall d. P\,d\,t)\Rightarrow P\,c\;(\lambda x.t)
+  \end{array}
+  }}
+  \end{beamercolorbox}
+  \end{center}
+ 
+
+  \only<1>{
+  \begin{textblock}{14}(1.2,9.2)
+  \begin{itemize}
+  \item There is a condition for when Barendregt's variable convention
+    is applicable---it is almost always satisfied, but not always:\\[2mm]
+    
+    The induction context \smath{c} needs to be finitely supported 
+    (is not allowed to mention all names as free).
+  \end{itemize}
+  \end{textblock}}
+
+  \only<2>{
+  \begin{itemize}
+  \item In the case of the substitution lemma:\\[2mm]
+
+  \begin{textblock}{16.5}(0.7,11.5)
+  \small
+*}
+
+(*<*)
+lemma 
+  assumes a: "x\<noteq>y" "x \<sharp> L"
+  shows "M[x::=N][y::=L] = M[y::=L][x::=N[y::=L]]"
+using a
+(*>*)
+proof (nominal_induct M avoiding: x y N L rule: lam.strong_induct)
+txt_raw {* \isanewline$\ldots$ *}
+(*<*)oops(*>*)
+
+text_raw {*
+  \end{textblock}
+  \end{itemize}}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+
+text_raw {*
+
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}
+  \frametitle{\Large \mbox{Same Problem with Rule Inductions}} 
+
+  \begin{itemize}
+  \item We can specify typing-rules for lambda-terms as:
+
+  \begin{center}
+  \begin{tabular}{@ {\hspace{-6mm}}c@ {}}
+  \colorbox{cream}{
+  \smath{\infer{\Gamma\vdash x:\tau}{(x\!:\!\tau)\in\Gamma\;\;\text{valid}\;\Gamma}}}
+  \;\;
+  \colorbox{cream}{
+  \smath{\infer{\Gamma\vdash t_1\;t_2:\tau}
+        {\Gamma\vdash t_1:\sigma\!\rightarrow\!\tau & \Gamma\vdash t_2:\sigma}}}\\[4mm]
+
+  \colorbox{cream}{
+  \smath{\infer{\Gamma\vdash \lambda x.t:\sigma\!\rightarrow\!\tau}
+        {x\fresh \Gamma & (x\!:\!\sigma)\!::\!\Gamma\vdash t:\tau}}}\\[6mm]
+
+  \colorbox{cream}{
+  \smath{\infer{\text{valid}\;[]}{}}}
+  \;\;\;\;
+  \colorbox{cream}{
+  \smath{\infer{\text{valid}\;(x\!:\!\tau)\!::\!\Gamma}{x\fresh\Gamma & \text{valid}\;\Gamma}}}\\[8mm]
+  \end{tabular}
+  \end{center}
+
+  \item If \smath{\Gamma_1\vdash t:\tau} and \smath{\text{valid}\;\Gamma_2},
+  \smath{\Gamma_1\subseteq \Gamma_2} then \smath{\Gamma_2\vdash t:\tau}.$\!\!\!\!\!$
+
+  \end{itemize}
+
+  
+  \begin{textblock}{11}(1.3,4)
+  \only<2>{
+  \begin{tikzpicture}
+  \draw (0,0) node[fill=cream, text width=10.5cm, thick, draw=red, rounded corners=1mm] (nn)
+  {The proof of the weakening lemma is said to be trivial / obvious / routine 
+   /\ldots{} in many places.\\[2mm] 
+          
+  (I am actually still looking for a place in the literature where a
+  trivial / obvious / routine /\ldots{} proof is spelled out --- I know of
+  proofs by Gallier, McKinna \& Pollack and Pitts, but I would not
+  call them trivial / obvious / routine /\ldots)};
+  \end{tikzpicture}}
+  \end{textblock}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+text_raw {*
+
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{Recall: Rule Inductions} 
+
+  \begin{center}\large
+  \colorbox{cream}{
+  \smath{\infer[\text{rule}]{\text{concl}}{\text{prem}_1 \ldots \text{prem}_n\;\text{scs}}}}
+  \end{center}\bigskip
+
+  \begin{tabular}[t]{l}
+  Rule Inductions:\\[1mm]
+  \begin{tabular}{l@ {\hspace{2mm}}p{8.4cm}}
+  1.) & Assume the property for the premises. Assume the side-conditions.\\[1mm]
+  2.) & Show the property for the conclusion.\\
+  \end{tabular}
+  \end{tabular}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+text_raw {*
+
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}
+  \frametitle{\LARGE\mbox{Induction Principle for Typing}} 
+
+  \begin{itemize}
+  \item The induction principle that comes with the typing definition is as follows:\\[-13mm]
+  \mbox{}
+  \end{itemize}
+
+  \begin{center}
+  \begin{tabular}{@ {\hspace{-5mm}}c@ {}}
+  \colorbox{cream}{
+  \smath{
+  \infer{\Gamma\vdash t:\tau \Rightarrow P\,\Gamma\,t\,\tau}
+  {\begin{array}{l}
+      \forall \Gamma\,x\,\tau.\,\;(x\!:\!\tau)\in\Gamma\wedge
+         \text{valid}\,\Gamma\Rightarrow P\,\Gamma\,(x)\,\tau\\[4mm]
+      \forall \Gamma\,t_1\,t_2\,\sigma\,\tau.\\
+      P\,\Gamma\,t_1\,(\sigma\!\rightarrow\!\tau)\wedge
+      P\,\Gamma\,t_2\,\sigma
+      \Rightarrow P\,\Gamma\,(t_1\,t_2)\,\tau\\[4mm]
+      \forall \Gamma\,x\,t\,\sigma\,\tau.\\
+      x\fresh\Gamma\wedge
+      P\,((x\!:\!\sigma)\!::\!\Gamma)\,t\,\tau
+      \Rightarrow P\,\Gamma (\lambda x.t)\,(\sigma\!\rightarrow\!\tau)\\[2mm]
+   \end{array}
+  }
+  }}
+  \end{tabular}
+  \end{center}
+
+  \begin{textblock}{4}(9,13.8)
+  \begin{tikzpicture}
+  \draw (0,0) node[fill=cream, text width=3.9cm, thick, draw=red, rounded corners=1mm] (nn)
+  {\small Note the quantifiers!};
+  \end{tikzpicture}
+  \end{textblock}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+text_raw {*
+
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}
+  \frametitle{\LARGE \mbox{Proof of Weakening Lemma}} 
+  \mbox{}\\[-18mm]\mbox{}
+
+  \begin{center}
+  \colorbox{cream}{
+  \smath{\infer{\Gamma\vdash \lambda x.t:\sigma\!\rightarrow\!\tau}
+        {x\fresh \Gamma & (x\!:\!\sigma)\!::\!\Gamma\vdash t:\tau}}}
+  \end{center}
+
+  \begin{minipage}{1.1\textwidth}
+  \begin{itemize}
+  \item If \smath{\Gamma_1\!\vdash\! t\!:\!\tau} then 
+  \smath{\alert<1>{\forall \Gamma_2}.\,\text{valid}\,\Gamma_2 \wedge \Gamma_1\!\subseteq\!
+  \Gamma_2\!\Rightarrow\! \Gamma_2\!\vdash\! t\!:\!\tau}
+  \end{itemize}
+
+  \pause
+
+  \mbox{}\hspace{-5mm}
+  \underline{For all \smath{\Gamma_1}, \smath{x}, \smath{t}, \smath{\sigma} and \smath{\tau}}:
+  
+  \begin{itemize}
+  \item We know:\\
+  \smath{\forall \alert<4->{\Gamma_2}.\,\text{valid}\,\alert<4->{\Gamma_2} \wedge 
+    (x\!:\!\sigma)\!::\!\Gamma_1\!\subseteq\! \alert<4->{\Gamma_2} \Rightarrow \!\!
+    \tikz[remember picture, baseline=(ea.base)] 
+         \node (ea) {\smath{\alert<4->{\Gamma_2}}};\!\vdash\! t\!:\!\tau}\\
+  \smath{x\fresh\Gamma_1}\\
+  \onslide<3->{\smath{\text{valid}\,\Gamma_2 \wedge \Gamma_1\!\subseteq\!\Gamma_2
+                      \only<6->{\Rightarrow (x\!:\!\sigma)\!::\!\Gamma_1\!\subseteq\!
+                                            (x\!:\!\sigma)\!::\!\Gamma_2}}}\\
+  \onslide<3->{\smath{\textcolor{white}{\text{valid}\,\Gamma_2 \wedge \Gamma_1\!\subseteq\!\Gamma_2
+          \Rightarrow} \only<7->{\;\alert{\text{valid}\,(x\!:\!\sigma)\!::\!\Gamma_2\;\;\text{\bf ???}}}}}
+
+  \item We have to show:\\
+  \only<2>{
+  \smath{\forall \Gamma_2.\,\text{valid}\,\Gamma_2 \wedge 
+  \Gamma_1\!\subseteq\!\Gamma_2 \Rightarrow \Gamma_2\!\vdash\! 
+  \lambda x.t\!:\!\sigma\!\rightarrow\!\tau}}
+  \only<3->{
+  \smath{\Gamma_2\!\vdash\!\lambda x.t\!:\!\sigma\!\rightarrow\!\tau}}
+
+  \end{itemize}
+  \end{minipage}
+
+  \begin{textblock}{4}(10,6.5)
+  \only<5->{
+  \begin{tikzpicture}[remember picture]
+  \draw (0,0) node[fill=cream, text width=4cm, thick, draw=red, rounded corners=1mm] (eb)
+  {\smath{\Gamma_2\mapsto (x\!:\!\sigma)\!::\!\Gamma_2}};
+  
+  \path[overlay, ->, ultra thick, red] (eb) edge[out=-90, in=80] (ea);
+  \end{tikzpicture}}
+  \end{textblock}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+text_raw {*
+
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}
+
+  \begin{textblock}{14.8}(0.7,0.5)
+  \begin{itemize}
+  \item The usual proof of strong normalisation for simply- typed lambda-terms 
+  establishes first:\\[1mm]
+
+  \colorbox{cream}{%
+  \begin{tabular}{@ {}p{11cm}}
+  Lemma: If for all reducible \smath{s}, \smath{t[x\!:=\!s]} is reducible, then
+  \smath{\lambda x.t} is reducible.
+  \end{tabular}}\smallskip
+
+  \item Then one shows for a closing (simultaneous) substitution:\\[2mm]
+
+  \colorbox{cream}{%
+  \begin{tabular}{@ {}p{11cm}}
+  Theorem: If \smath{\Gamma\vdash t:\tau}, then for all closing
+  substitutions \smath{\theta} containing reducible terms only, 
+  \smath{\theta(t)} is reducible.
+  \end{tabular}}
+
+  \mbox{}\\[1mm]
+
+  Lambda-Case: By ind.~we know \smath{(x\!\mapsto\! s\cup\theta)(t)}
+  is reducible with \smath{s} being reducible. This is equal\alert{$^*$} to
+  \smath{(\theta(t))[x\!:=\!s]}. Therefore, we can apply the lemma and get \smath{\lambda
+  x.(\theta(t))} is reducible. Because this is equal\alert{$^*$} to
+  \smath{\theta(\lambda x.t)}, we are done.
+  \hfill\footnotesize\alert{$^*$}you have to take a deep breath
+  \end{itemize}
+  \end{textblock}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+
+text_raw {*
+
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}
+  \frametitle{\LARGE \mbox{Proof of Weakening Lemma}} 
+  \mbox{}\\[-18mm]\mbox{}
+
+  \begin{center}
+  \colorbox{cream}{
+  \smath{\infer{\Gamma\vdash \lambda x.t:\sigma\!\rightarrow\!\tau}
+        {x\fresh \Gamma & (x\!:\!\sigma)\!::\!\Gamma\vdash t:\tau}}}
+  \end{center}
+
+  \begin{minipage}{1.1\textwidth}
+  \begin{itemize}
+  \item If \smath{\Gamma_1\!\vdash\! t\!:\!\tau} then 
+  \smath{\text{valid}\,\Gamma_2 \wedge \Gamma_1\!\subseteq\!
+  \Gamma_2\!\Rightarrow\! \Gamma_2\!\vdash\! t\!:\!\tau}
+  \end{itemize}
+
+  \mbox{}\hspace{-5mm}
+  \underline{For all \smath{\Gamma_1}, \smath{x}, \smath{t}, \smath{\sigma} and \smath{\tau}}:
+  
+  \begin{itemize}
+  \item We know:\\
+  \smath{\forall \Gamma_2.\,\text{valid}\,\Gamma_2 \wedge 
+    (x\!:\!\sigma)\!::\!\Gamma_1\!\subseteq\! \Gamma_2 \Rightarrow \!\!
+    \Gamma_2\!\vdash\! t\!:\!\tau}\\
+  \smath{x\fresh\Gamma_1}\\
+  \begin{tabular}{@ {}ll@ {}}
+  \smath{\text{valid}\,\Gamma_2 \wedge \Gamma_1\!\subseteq\!\Gamma_2} &
+  \only<2->{\smath{\alert{\Rightarrow (x\!:\!\sigma)\!::\!\Gamma_1\!\subseteq\! 
+                           (x\!:\!\sigma)\!::\!\Gamma_2}}}\\
+  \smath{\alert{x\fresh\Gamma_2}} & 
+  \only<2->{\smath{\alert{\Rightarrow \text{valid}\,(x\!:\!\sigma)\!::\!\Gamma_2}}}
+  \end{tabular}
+
+  \item We have to show:\\
+  \smath{\Gamma_2\!\vdash\!\lambda x.t\!:\!\sigma\!\rightarrow\!\tau}
+
+  \end{itemize}
+  \end{minipage}
+
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+
+
+text_raw {*
+
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}
+  \frametitle{SN (Again)} 
+  \mbox{}\\[-8mm]
+
+  \colorbox{cream}{%
+  \begin{tabular}{@ {}p{10.5cm}}
+  Theorem: If \smath{\Gamma\vdash t:\tau}, then for all closing
+  substitutions \smath{\theta} containing reducible terms only, 
+  \smath{\theta(t)} is reducible.
+  \end{tabular}}\medskip
+
+  \begin{itemize}
+  \item
+  Since we say that the strong induction should avoid \smath{\theta}, we
+  get the assumption \alert{$x\fresh\theta$} then:\\[2mm]
+  
+  \begin{tabular}{@ {}p{10.5cm}}\raggedright
+  Lambda-Case: By ind.~we know \smath{(x\!\mapsto\! s\cup\theta)(t)} is reducible 
+  with
+  \smath{s} being reducible. This is {\bf equal} to
+  \smath{(\theta(t))[x\!:=\!s]}. Therefore, we can apply the lemma and get
+  \smath{\lambda x.(\theta(t))} is reducible. Because this is {\bf equal} to
+  \smath{\theta(\lambda x.t)}, we are done.
+  \end{tabular}\smallskip
+
+  \begin{center}
+  \begin{tabular}{rl}
+  \smath{x\fresh\theta\Rightarrow} & 
+  \smath{(x\!\mapsto\! s\cup\theta)(t) \;\alert{=}\;(\theta(t))[x\!:=\!s]}\\[1mm]
+  &
+  \smath{\theta(\lambda x.t) \;\alert{=}\; \lambda x.(\theta(t))}
+  \end{tabular}
+  \end{center}
+  \end{itemize}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+text_raw {*
+
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}
+  \frametitle{So Far So Good}
+
+  \begin{itemize}
+  \item A Faulty Lemma with the Variable Convention?\\[-8mm]\mbox{}
+  \end{itemize}
+
+  \begin{center}
+  \begin{block}{}
+  \color{gray}
+  \small%
+  {\bf\mbox{}\hspace{-1.5mm}Variable Convention:}\\[1mm] 
+  If $M_1,\ldots,M_n$ occur in a certain mathematical context
+  (e.g. definition, proof), then in these terms all bound variables 
+  are chosen to be different from the free variables.\\[2mm]
+  
+  \footnotesize\hfill Barendregt in ``The Lambda-Calculus: Its Syntax and Semantics''
+  \end{block}
+  \end{center}
+
+  \mbox{}\\[-18mm]\mbox{}
+
+  \begin{columns}
+  \begin{column}[t]{4.7cm}
+  Inductive Definitions:\\
+  \begin{center}
+  \smath{\infer{\text{concl}}{\text{prem}_1 \ldots \text{prem}_n\;\text{scs}}}
+  \end{center}
+  \end{column}
+  \begin{column}[t]{7cm}
+  Rule Inductions:\\[2mm]
+  \begin{tabular}{l@ {\hspace{2mm}}p{5.5cm}}
+  1.) & Assume the property for\\ & the premises. Assume \\ & the side-conditions.\\[1mm]
+  2.) & Show the property for\\ & the conclusion.\\
+  \end{tabular}
+  \end{column}
+  \end{columns}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+text_raw {*
+
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \setbeamerfont{itemize/enumerate subbody}{size=\normalsize}
+  \begin{frame}[sqeeze]
+  \frametitle{Faulty Reasoning} 
+
+  %\mbox{}
+
+  \begin{itemize}
+  \item Consider the two-place relation \smath{\text{foo}}:\medskip
+  \begin{center}
+  \begin{tabular}{ccc}
+  \raisebox{2.5mm}{\colorbox{cream}{%
+  \smath{\;\infer{x\mapsto x}{}}}}\hspace{2mm}
+  &
+  \raisebox{2mm}{\colorbox{cream}{%
+  \smath{\infer{t_1\;t_2\mapsto t_1\;t_2}{}}}}\hspace{2mm}
+  &
+  \colorbox{cream}{%
+  \smath{\infer{\lambda x.t\mapsto t'}{t\mapsto t'}}}\\[5mm]
+  \end{tabular}
+  \end{center}
+
+  \pause
+
+  \item The lemma we going to prove:\smallskip
+  \begin{center}
+  Let \smath{t\mapsto t'}. If \smath{y\fresh t} then \smath{y\fresh t'}.
+  \end{center}\bigskip
+
+  \only<3>{
+  \item Cases 1 and 2 are trivial:\medskip
+  \begin{itemize}
+  \item If \smath{y\fresh x} then \smath{y\fresh x}.
+  \item If \smath{y\fresh t_1\,t_2} then \smath{y\fresh t_1\,t_2}.
+  \end{itemize}
+  }
+
+  \only<4->{
+  \item Case 3:
+  \begin{itemize}
+  \item We know \tikz[remember picture,baseline=(ta.base)] \node (ta) {\smath{y\fresh \lambda x.t}.}; 
+  We have to show \smath{y\fresh t'}.$\!\!\!\!$ 
+  \item The IH says: if \smath{y\fresh t} then \smath{y\fresh t'}.
+  \item<7,8> So we have \smath{y\fresh t}. Hence \smath{y\fresh t'} by IH. Done!
+  \end{itemize}
+  }
+  \end{itemize}
+  
+  \begin{textblock}{11.3}(0.7,0.6)
+  \only<5-7>{
+  \begin{tikzpicture}
+  \draw (0,0) node[fill=cream, text width=11.2cm, thick, draw=red, rounded corners=1mm] (nn)
+  {{\bf Variable Convention:}\\[2mm] 
+  \small
+   If $M_1,\ldots,M_n$ occur in a certain mathematical context
+   (e.g. definition, proof), then in these terms all bound variables 
+   are chosen to be different from the free variables.\smallskip
+
+  \normalsize
+   {\bf In our case:}\\[2mm]
+   The free variables are \smath{y} and \smath{t'}; the bound one is 
+   \smath{x}.\medskip
+          
+   By the variable convention we conclude that \smath{x\not= y}.
+  };
+  \end{tikzpicture}}
+  \end{textblock}
+
+  \begin{textblock}{9.2}(3.6,9)
+  \only<6,7>{
+  \begin{tikzpicture}[remember picture]
+  \draw (0,0) node[fill=cream, text width=9cm, thick, draw=red, rounded corners=1mm] (tb)
+  {\small\smath{y\!\not\in\! \text{fv}(\lambda x.t) \Longleftrightarrow
+          y\!\not\in\! \text{fv}(t)\!-\!\{x\}
+          \stackrel{x\not=y}{\Longleftrightarrow}
+          y\!\not\in\! \text{fv}(t)}};
+
+  \path[overlay, ->, ultra thick, red] (tb) edge[out=-120, in=75] (ta);
+  \end{tikzpicture}}
+  \end{textblock}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+text_raw {*
+
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \setbeamerfont{itemize/enumerate subbody}{size=\normalsize}
+  \begin{frame}
+  \frametitle{VC-Compatibility} 
+
+  \begin{itemize}
+  \item We introduced two conditions that make the VC safe to use in rule inductions:
+  
+  \begin{itemize}
+  \item the relation needs to be \alert{\bf equivariant}, and
+  \item the binder is not allowed to occur in the \alert{\bf support} of 
+  the conclusion (not free in the conclusion)\bigskip
+  \end{itemize}
+  
+  \item Once a relation satisfies these two conditions, then Nominal
+  Isabelle derives the strong induction principle automatically.
+  \end{itemize}
+
+  \begin{textblock}{11.3}(0.7,6)
+  \only<2>{
+  \begin{tikzpicture}
+  \draw (0,0) node[fill=cream, text width=11cm, thick, draw=red, rounded corners=1mm] (nn)
+  {A relation \smath{R} is {\bf equivariant} iff
+  %
+  \begin{center}
+  \smath{%
+  \begin{array}[t]{l}
+  \forall \pi\,t_1\ldots t_n\\[1mm]
+  \;\;\;\;R\,t_1\ldots t_n \Rightarrow R (\pi\act t_1)\ldots(\pi\act t_n)
+  \end{array}}
+  \end{center}
+  %
+  This means the relation has to be invariant under permutative renaming of
+  variables.\smallskip
+      
+  \small
+  (This property can be checked automatically if the inductive definition is composed of
+  equivariant ``things''.)
+  };
+  \end{tikzpicture}}
+  \end{textblock}
+
+  \only<3>{}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+text_raw {*
+
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}
+  \frametitle{\mbox{Honest Toil, No Theft!}} 
+
+  \begin{itemize}
+  \item The \underline{sacred} principle of HOL:
+
+  \begin{block}{}
+  ``The method of `postulating' what we want has many advantages; they are
+  the same as the advantages of theft over honest toil.''\\[2mm] 
+  \hfill{}\footnotesize B.~Russell, Introduction of Mathematical Philosophy
+  \end{block}\bigskip\medskip
+
+  \item I will show next that the \underline{weak} structural induction 
+  principle implies the \underline{strong} structural induction principle.\\[3mm] 
+  
+  \textcolor{gray}{(I am only going to show the lambda-case.)}
+  \end{itemize}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+text_raw {*
+
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}
+  \frametitle{Permutations} 
+
+  A permutation \alert{\bf acts} on variable names as follows:
+
+  \begin{center}
+  \begin{tabular}{rcl}
+    $\smath{{[]}\act a}$ & $\smath{\dn}$ & $\smath{a}$\\
+    $\smath{(\swap{a_1}{a_2}\!::\!\pi)\act a}$ & $\smath{\dn}$ &
+    $\smath{\begin{cases}
+      a_1 &\text{if $\pi\act a = a_2$}\\
+      a_2 &\text{if $\pi\act a = a_1$}\\
+      \pi\act a &\text{otherwise}
+    \end{cases}}$
+   \end{tabular}
+  \end{center}
+
+  \begin{itemize}
+  \item $\smath{[]}$ stands for the empty list (the identity permutation), and\smallskip  
+  \item $\smath{\swap{a_1}{a_2}\!::\!\pi}$ stands for the permutation $\smath{\pi}$ 
+  followed by the swapping $\smath{\swap{a_1}{a_2}}$.
+  \end{itemize}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+text_raw {*
+
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}
+  \frametitle{\Large\mbox{Permutations on Lambda-Terms}} 
+
+  \begin{itemize}
+  \item Permutations act on lambda-terms as follows:
+
+  \begin{center}
+  \begin{tabular}{rcl}
+  $\smath{\pi\act\,x}$ & $\smath{\dn}$ & ``action on variables''\\
+  $\smath{\pi\act\, (t_1~t_2)}$ & $\smath{\dn}$ & $\smath{(\pi\act t_1)~(\pi\act t_2)}$\\
+  $\smath{\pi\act(\lambda x.t)}$ & $\smath{\dn}$ & $\smath{\lambda (\pi\act x).(\pi\act t)}$\\
+  \end{tabular}
+  \end{center}\medskip
+
+  \item Alpha-equivalence can be defined as:
+
+  \begin{center}
+  \begin{tabular}{c}
+  \colorbox{cream}{\smath{\infer{\lambda x.t_1 = \lambda x.t_2}{t_1=t_2}}}\\[3mm]
+  \colorbox{cream}{\smath{\infer{\lambda x.t_1 
+                                 \tikz[baseline=-3pt,remember picture] \node (e1) {\alert<2>{$=$}}; 
+                                 \lambda y.t_2}
+                  {x\not=y & t_1 = \swap{x}{y}\act t_2 & x\fresh t_2}}}
+  \end{tabular}
+  \end{center}
+
+  \end{itemize}
+
+
+  \begin{textblock}{4}(8.3,14.2)
+  \only<2>{
+  \begin{tikzpicture}[remember picture]
+  \draw (0,0) node[fill=cream, text width=5.5cm, thick, draw=red, rounded corners=1mm] (e2)
+  {\small Notice, I wrote equality here!};
+
+  \path[overlay, ->, ultra thick, red] (e2) edge[out=180, in=-90] (e1);
+  \end{tikzpicture}}
+  \end{textblock}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+text_raw {*
+
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}
+  \frametitle{My Claim} 
+
+  \begin{center}
+  \colorbox{cream}{%
+  \smath{%
+  \infer{P\;t}
+  {\begin{array}{l}
+  \forall x.\;P\;x\\[2mm]
+  \forall t_1\,t_2.\;P\;t_1\wedge P\;t_2\Rightarrow P\;(t_1\;t_2)\\[2mm]
+  \forall x\,t.\;P\;t\Rightarrow P\;(\lambda x.t)\\
+  \end{array}
+  }}}\medskip
+
+  \begin{tikzpicture}
+  \node at (0,0) [single arrow, single arrow tip angle=140, 
+                  shape border rotate=270, fill=red,text=white]{implies};
+  \end{tikzpicture}\medskip
+
+  \colorbox{cream}{%
+  \smath{%
+  \infer{P c\,t}%
+  {\begin{array}{@ {}l@ {}}
+  \forall x\,c.\;P c\,x\\[2mm]
+  \forall t_1\,t_2\,c.\;(\forall d.\,P d\,t_1)\wedge (\forall d.\,P d\,t_2)
+  \Rightarrow P c\,(t_1\,t_2)\\[2mm]
+  \forall x\,t\,c.\;
+  x\fresh c \wedge (\forall d.\,P d\,t)\Rightarrow P c\,(\lambda x.t)
+  \end{array}}}}
+  
+  \end{center}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+text_raw {*
+
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}
+  \frametitle{\large\mbox{Proof for the Strong Induction Principle}} 
+
+  \begin{textblock}{14}(1.2,1.7)
+  \begin{itemize}
+  \item<1-> We prove \alt<1>{\smath{P c\,t}}{\smath{\forall \pi\,c.\;P c\,(\pi\act t)}} 
+  by induction on \smath{t}.
+  
+  \item<3-> I.e., we have to show \alt<3>{\smath{P c\,(\pi\act(\lambda x.t))}}
+  {\smath{P c\,\lambda(\pi\act x).(\pi\act t)}}.
+ 
+  \item<5-> We have \smath{\forall \pi\,c.\;P c\,(\pi\act t)} by induction.
+  
+  
+  \item<6-> Our weaker precondition says that:\\
+  \begin{center}
+  \smath{\forall x\,t\,c.\,x\fresh c \wedge (\forall c.\,P c\,t) \Rightarrow P c\,(\lambda x.t)}  
+  \end{center}
+  
+  \item<7-> We choose a fresh \smath{y} such that \smath{y\fresh (\pi\act x,\pi\act t,c)}.
+  
+  \item<8-> Now we can use 
+  \alt<8>{\smath{\forall c.\;P c\,((\swap{y}{\,\pi\act x}\!::\!\pi)\act t)}}
+         {\smath{\forall c.\;P c\,(\swap{y}{\,\pi\act x}\act\pi\act t)}} \only<10->{to infer}
+
+  \only<10->{
+  \begin{center}
+  \smath{P\,c\,\lambda y.(\swap{y}{\,\pi\act x}\act\pi\act t)}  
+  \end{center}}
+ 
+  \item<11-> However 
+  \begin{center}
+  \smath{\lambda y.(\swap{y}{\,\pi\act x}\act\pi\act t)
+         \textcolor{red}{\;=\;}\lambda (\pi\act x).(\pi\act t)}
+  \end{center}
+
+  \item<12> Therefore \smath{P\,c\,\lambda (\pi\act x).(\pi\act t)} and we are done.
+  \end{itemize}
+  \end{textblock}
+
+  \only<11->{
+  \begin{textblock}{9}(7,6)
+  \begin{tikzpicture}[remember picture, overlay]
+  \draw (0,0) node[fill=cream, text width=7cm, thick, draw=red, rounded corners=1mm] (n2)
+  {\centering
+  \smath{\infer{\lambda y.t_1=\lambda x.t_2}{x\not=y & t_1=\swap{x}{y}\act t_2 &
+              y\fresh t_2}}
+  };
+  \end{tikzpicture}
+  \end{textblock}}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+text_raw {*
+
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}<3->[squeeze]
+  \frametitle{Formalisation of LF} 
+  
+  
+  \begin{center}
+  \begin{tabular}{@ {\hspace{-16mm}}lc}
+  \mbox{}\\[-6mm]
+
+  \textcolor{white}{\raisebox{4mm}{1.~Solution}} &
+  \begin{tikzpicture}
+  [node distance=25mm,
+   text height=1.5ex,
+   text depth=.25ex,
+   node1/.style={
+     rectangle, minimum size=10mm, rounded corners=3mm, very thick, 
+     draw=black!50, top color=white, bottom color=black!20},
+  ]
+
+  \node (proof) [node1] {\large Proof};
+  \node (def)   [node1, left of=proof] {\large$\,\;\dn\;\,$};
+  \node (alg)  [node1, right of=proof] {\large\hspace{1mm}Alg.\hspace{1mm}\mbox{}};
+  
+  \draw[->,black!50,line width=2mm] (proof) -- (def);
+  \draw[->,black!50,line width=2mm] (proof) -- (alg);
+
+  \onslide<2->{\draw[white,line width=1mm] (0.1,0.6) -- (-0.1,0.25) -- (0.1,-0.25) -- (-0.1,-0.6);} 
+  \end{tikzpicture}
+  \\[2mm]
+
+  \onslide<3->{%
+  \raisebox{4mm}{1st Solution} &
+  \begin{tikzpicture}
+  [node distance=25mm,
+   text height=1.5ex,
+   text depth=.25ex,
+   node1/.style={
+     rectangle, minimum size=10mm, rounded corners=3mm, very thick, 
+     draw=black!50, top color=white, bottom color=black!20},
+   node2/.style={
+     rectangle, minimum size=12mm, rounded corners=3mm, very thick, 
+     draw=red!70, top color=white, bottom color=red!50!black!20}
+  ]
+
+  \node (proof) [node1] {\large Proof};
+  \node (def)   [node2, left of=proof] {\large$\dn{}\!\!^\text{+ex}$};
+  \node (alg)   [node1, right of=proof] {\large\hspace{1mm}Alg.\hspace{1mm}\mbox{}};
+  
+  \draw[->,black!50,line width=2mm] (proof) -- (def);
+  \draw[->,black!50,line width=2mm] (proof) -- (alg);
+
+  \end{tikzpicture}
+  \\[2mm]}
+
+  \onslide<4->{%
+  \raisebox{4mm}{\hspace{-1mm}2nd Solution} & 
+  \begin{tikzpicture}
+  [node distance=25mm,
+   text height=1.5ex,
+   text depth=.25ex,
+   node1/.style={
+     rectangle, minimum size=10mm, rounded corners=3mm, very thick, 
+     draw=black!50, top color=white, bottom color=black!20},
+   node2/.style={
+     rectangle, minimum size=12mm, rounded corners=3mm, very thick, 
+     draw=red!70, top color=white, bottom color=red!50!black!20}
+  ]
+
+  \node (proof) [node1] {\large Proof};
+  \node (def)   [node1, left of=proof] {\large$\,\;\dn\;\,$};
+  \node (alg)   [node2, right of=proof] {\large Alg.$\!^\text{-ex}$};
+  
+  \draw[->,black!50,line width=2mm] (proof) -- (def);
+  \draw[->,black!50,line width=2mm] (proof) -- (alg);
+
+  \end{tikzpicture}
+  \\[2mm]}
+
+  \onslide<5->{%
+  \raisebox{4mm}{\hspace{-1mm}3rd Solution} & 
+  \begin{tikzpicture}
+  [node distance=25mm,
+   text height=1.5ex,
+   text depth=.25ex,
+   node1/.style={
+     rectangle, minimum size=10mm, rounded corners=3mm, very thick, 
+     draw=black!50, top color=white, bottom color=black!20},
+   node2/.style={
+     rectangle, minimum size=12mm, rounded corners=3mm, very thick, 
+     draw=red!70, top color=white, bottom color=red!50!black!20}
+  ]
+
+  \node (proof) [node2] {\large Proof};
+  \node (def)   [node1, left of=proof] {\large$\,\;\dn\;\,$};
+  \node (alg)   [node1, right of=proof] {\large\hspace{1mm}Alg.\hspace{1mm}\mbox{}};
+  
+  \draw[->,black!50,line width=2mm] (proof) -- (def);
+  \draw[->,black!50,line width=2mm] (proof) -- (alg);
+
+  \end{tikzpicture}
+  \\}
+
+  \end{tabular}
+  \end{center}
+
+  \begin{textblock}{3}(13.2,5.1)
+  \onslide<3->{
+  \begin{tikzpicture}
+  \node at (0,0) [single arrow, shape border rotate=270, fill=red,text=white]{2h};
+  \end{tikzpicture}
+  }
+  \end{textblock}
+
+  
+  \begin{textblock}{13}(1.4,15)
+  \only<3->{\footnotesize (each time one needs to check $\sim$31pp~of informal paper proofs)}
+  \end{textblock}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+
+text_raw {*
+
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}
+  \frametitle{Conclusions} 
+
+  \begin{itemize}
+  \item The Nominal Isabelle automatically derives the strong structural
+  induction principle for \underline{\bf all} nominal datatypes (not just the 
+  lambda-calculus);
+
+  \item also for rule inductions (though they have to satisfy the vc-condition).
+
+  \item They are easy to use: you just have to think carefully what the variable
+  convention should be.
+
+  \item We can explore the \colorbox{black}{\textcolor{white}{dark}} corners 
+  of the variable convention: when and where it can be used safely.
+  
+  \item<2> \alert{\bf Main Point:} Actually these proofs using the
+  variable convention are all trivial / obvious / routine\ldots {\bf provided} 
+  you use Nominal Isabelle. ;o)
+ 
+  \end{itemize}
+
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+text_raw {*
+
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}
+  \frametitle{\begin{tabular}{c}Nominal Meets\\[-2mm] Automata Theory\end{tabular}} 
+
+  \begin{itemize}
+  \item<1-> So what?\bigskip\medskip
+ 
+  \item<2-> I can give you a good argument why regular expressions
+  are much, much better than automata. \textcolor{darkgray}{(over dinner?)}\medskip
+
+  \item<3-> Nominal automata?\bigskip\bigskip\medskip
+  \end{itemize}
+
+
+  \onslide<2->{
+  \footnotesize\textcolor{darkgray}{A Formalisation of the Myhill-Nerode Theorem based on
+  Regular Expressions (by Wu, Zhang and Urban)}
+  }
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}
+  \frametitle{Quiz}
+  %%%\small
+  \mbox{}\\[-9mm]
+
+  Imagine\ldots\\[2mm]
+
+  \begin{tabular}{@ {\hspace{1cm}}l}
+  \textcolor{blue}{Var\;``name''} \\
+  \textcolor{blue}{App\;``lam''\;''lam''}\\
+  \textcolor{blue}{Lam\;``\flqq{}name\frqq{}lam''} \\
+  \textcolor{red}{Foo\;``\flqq{}name\frqq{}\flqq{}name\frqq{}lam''\;``
+                  \flqq{}name\frqq{}\flqq{}name\frqq{}lam''}\\[2mm]
+  \end{tabular}
+
+  That means roughly:\\[2mm]
+ 
+  \begin{tabular}{@ {\hspace{1cm}}l}
+  \alert{Foo\;($\lambda x.y.t_1$)\;($\lambda z.u.t_2$)}
+  \end{tabular}
+
+  \begin{itemize}
+  \item What does the variable convention look like for \alert{Foo}?
+  \item What does the clause for capture-avoiding substitution look like?
+  \end{itemize}
+
+  \footnotesize
+  Answers: Download Nominal Isabelle and try it out\\
+  \textcolor{white}{Answers:} http://isabelle.in.tum.de/nominal\\
+  \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