--- /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