--- a/Slides/SlidesA.thy Tue Feb 19 05:38:46 2013 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1648 +0,0 @@
-(*<*)
-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