--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Slides/Slides9.thy Sat Nov 26 09:47:21 2011 +0000
@@ -0,0 +1,1363 @@
+(*<*)
+theory Slides9
+imports "~~/src/HOL/Library/LaTeXsugar" "Nominal"
+begin
+
+notation (latex output)
+ set ("_") and
+ Cons ("_::/_" [66,65] 65)
+
+(*>*)
+
+
+text_raw {*
+ %% shallow, deep, and recursive binders
+ %%
+ %%\renewcommand{\slidecaption}{Cambridge, 8.~June 2010}
+ %%\renewcommand{\slidecaption}{Uppsala, 3.~March 2011}
+ \renewcommand{\slidecaption}{Leicester, 23.~November 2011}
+ \newcommand{\soutt}[1]{\tikz[baseline=(X.base), inner sep=-0.1pt, outer sep=0pt]
+ \node [cross out,red, ultra thick, draw] (X) {\textcolor{black}{#1}};}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1>[t]
+ \frametitle{%
+ \begin{tabular}{@ {\hspace{-3mm}}c@ {}}
+ \\
+ \LARGE General Binding Structures\\[-1mm]
+ \LARGE in Nominal Isabelle 2\\
+ \end{tabular}}
+ \begin{center}
+ Christian Urban
+ \end{center}
+ \begin{center}
+ joint work with {\bf Cezary Kaliszyk}\\[0mm]
+ \end{center}
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1>
+ \frametitle{\begin{tabular}{c}Binding in Old Nominal\end{tabular}}
+ \mbox{}\\[-6mm]
+
+ \begin{itemize}
+ \item the old Nominal Isabelle provided a reasoning infrastructure for single binders\medskip
+
+ \begin{center}
+ Lam [a].(Var a)
+ \end{center}\bigskip
+
+ \item<2-> but representing
+
+ \begin{center}
+ $\forall\{a_1,\ldots,a_n\}.\; T$
+ \end{center}\medskip
+
+ with single binders and reasoning about it is a \alert{\bf major} pain;
+ take my word for it!
+ \end{itemize}
+
+ \only<1>{
+ \begin{textblock}{6}(1.5,11)
+ \small
+ for example\\
+ \begin{tabular}{l@ {\hspace{2mm}}l}
+ & a $\fresh$ Lam [a]. t\\
+ & Lam [a]. (Var a) \alert{$=$} Lam [b]. (Var b)\\
+ & Barendregt-style reasoning about bound variables\\
+ & (variable convention can lead to faulty reasoning)
+ \end{tabular}
+ \end{textblock}}
+
+ \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}[c]
+ \frametitle{\begin{tabular}{c}Binding in Old Nominal\end{tabular}}
+
+ \begin{itemize}
+ \item<1-> but representing
+
+ \begin{center}
+ $\forall\{a_1,\ldots,a_n\}.\; T$
+ \end{center}\medskip
+
+ with single binders and reasoning about it was a \alert{\bf major} pain;
+ take my word for it!
+ \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}<1-4>
+ \frametitle{\begin{tabular}{c}Binding Sets of Names\end{tabular}}
+ \mbox{}\\[-3mm]
+
+ \begin{itemize}
+ \item binding sets of names has some interesting properties:\medskip
+
+ \begin{center}
+ \begin{tabular}{l}
+ \textcolor{blue}{$\forall\{x, y\}.\, x \rightarrow y \;\;\approx_\alpha\;\; \forall\{y, x\}.\, y \rightarrow x$}
+ \bigskip\smallskip\\
+
+ \onslide<2->{%
+ \textcolor{blue}{$\forall\{x, y\}.\, x \rightarrow y \;\;\not\approx_\alpha\;\; \forall\{z\}.\, z \rightarrow z$}
+ }\bigskip\smallskip\\
+
+ \onslide<3->{%
+ \textcolor{blue}{$\forall\{x\}.\, x \rightarrow y \;\;\approx_\alpha\;\; \forall\{x, \alert{z}\}.\, x \rightarrow y$}
+ }\medskip\\
+ \onslide<3->{\hspace{4cm}\small provided $z$ is fresh for the type}
+ \end{tabular}
+ \end{center}
+ \end{itemize}
+
+ \begin{textblock}{8}(2,14.5)
+ \footnotesize $^*$ $x$, $y$, $z$ are assumed to be distinct
+ \end{textblock}
+
+ \only<4>{
+ \begin{textblock}{6}(2.5,4)
+ \begin{tikzpicture}
+ \draw (0,0) node[inner sep=3mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
+ {\normalsize\color{darkgray}
+ \begin{minipage}{8cm}\raggedright
+ For type-schemes the order of bound names does not matter, and
+ $\alpha$-equivalence is preserved under \alert{vacuous} binders.
+ \end{minipage}};
+ \end{tikzpicture}
+ \end{textblock}}
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1-3>
+ \frametitle{\begin{tabular}{c}Other Binding Modes\end{tabular}}
+ \mbox{}\\[-3mm]
+
+ \begin{itemize}
+ \item alpha-equivalence being preserved under vacuous binders is \underline{not} always
+ wanted:\bigskip\bigskip\normalsize
+
+ \textcolor{blue}{\begin{tabular}{@ {\hspace{-8mm}}l}
+ $\text{let}\;x = 3\;\text{and}\;y = 2\;\text{in}\;x - y\;\text{end}$\medskip\\
+ \onslide<2->{$\;\;\;\only<2>{\approx_\alpha}\only<3>{\alert{\not\approx_\alpha}}
+ \text{let}\;y = 2\;\text{and}\;x = 3\only<3->{\alert{\;\text{and}
+ \;z = \text{loop}}}\;\text{in}\;x - y\;\text{end}$}
+ \end{tabular}}
+
+
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1>
+ \frametitle{\begin{tabular}{c}\LARGE{}Even Another Binding Mode\end{tabular}}
+ \mbox{}\\[-3mm]
+
+ \begin{itemize}
+ \item sometimes one wants to abstract more than one name, but the order \underline{does} matter\bigskip
+
+ \begin{center}
+ \textcolor{blue}{\begin{tabular}{@ {\hspace{-8mm}}l}
+ $\text{let}\;(x, y) = (3, 2)\;\text{in}\;x - y\;\text{end}$\medskip\\
+ $\;\;\;\not\approx_\alpha
+ \text{let}\;(y, x) = (3, 2)\;\text{in}\;x - y\;\text{end}$
+ \end{tabular}}
+ \end{center}
+
+
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1-2>
+ \frametitle{\begin{tabular}{c}\LARGE{}Three Binding Modes\end{tabular}}
+ \mbox{}\\[-3mm]
+
+ \begin{itemize}
+ \item the order does not matter and alpha-equivelence is preserved under
+ vacuous binders \textcolor{gray}{(restriction)}\medskip
+
+ \item the order does not matter, but the cardinality of the binders
+ must be the same \textcolor{gray}{(abstraction)}\medskip
+
+ \item the order does matter \textcolor{gray}{(iterated single binders)}
+ \end{itemize}
+
+ \onslide<2->{
+ \begin{center}
+ \isacommand{bind (set+)}\hspace{6mm}
+ \isacommand{bind (set)}\hspace{6mm}
+ \isacommand{bind}
+ \end{center}}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1-3>
+ \frametitle{\begin{tabular}{c}Specification of Binding\end{tabular}}
+ \mbox{}\\[-6mm]
+
+ \mbox{}\hspace{10mm}
+ \begin{tabular}{ll}
+ \multicolumn{2}{l}{\isacommand{nominal\_datatype} trm $=$}\\
+ \hspace{5mm}\phantom{$|$} Var name\\
+ \hspace{5mm}$|$ App trm trm\\
+ \hspace{5mm}$|$ Lam \only<2->{x::}name \only<2->{t::}trm
+ & \onslide<2->{\isacommand{bind} x \isacommand{in} t}\\
+ \hspace{5mm}$|$ Let \only<2->{as::}assns \only<2->{t::}trm
+ & \onslide<2->{\isacommand{bind} bn(as) \isacommand{in} t}\\
+ \multicolumn{2}{l}{\isacommand{and} assns $=$}\\
+ \multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} ANil}\\
+ \multicolumn{2}{l}{\hspace{5mm}$|$ ACons name trm assns}\\
+ \multicolumn{2}{l}{\onslide<3->{\isacommand{binder} bn \isacommand{where}}}\\
+ \multicolumn{2}{l}{\onslide<3->{\hspace{5mm}\phantom{$|$} bn(ANil) $=$ []}}\\
+ \multicolumn{2}{l}{\onslide<3->{\hspace{5mm}$|$ bn(ACons a t as) $=$ [a] @ bn(as)}}\\
+ \end{tabular}
+
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1-2,4-8>
+ \frametitle{\begin{tabular}{c}Alpha-Equivalence\end{tabular}}
+ \mbox{}\\[-3mm]
+
+ \begin{itemize}
+ \item lets first look at pairs\bigskip\medskip
+
+ \textcolor{blue}{\begin{tabular}{@ {\hspace{1cm}}l}
+ $(as, x) \onslide<2->{\approx\!}\makebox[5mm][l]{\only<2-6>{${}_{\text{set}}$}%
+ \only<7>{${}_{\text{\alert{list}}}$}%
+ \only<8>{${}_{\text{\alert{set+}}}$}}%
+ \,\onslide<2->{(bs,y)}$
+ \end{tabular}}\bigskip
+ \end{itemize}
+
+ \only<1>{
+ \begin{textblock}{8}(3,8.5)
+ \begin{tabular}{l@ {\hspace{2mm}}p{8cm}}
+ & \textcolor{blue}{$as$} is a set of names\ldots the binders\\
+ & \textcolor{blue}{$x$} is the body (might be a tuple)\\
+ & \textcolor{blue}{$\approx_{\text{set}}$} is where the cardinality
+ of the binders has to be the same\\
+ \end{tabular}
+ \end{textblock}}
+
+ \only<4->{
+ \begin{textblock}{12}(5,8)
+ \textcolor{blue}{
+ \begin{tabular}{ll@ {\hspace{1mm}}l}
+ $\dn$ & \onslide<5->{$\exists \pi.\,$} & $\text{fv}(x) - as = \text{fv}(y) - bs$\\[1mm]
+ & \onslide<5->{$\;\;\;\wedge$} & \onslide<5->{$\text{fv}(x) - as \fresh^* \pi$}\\[1mm]
+ & \onslide<5->{$\;\;\;\wedge$} & \onslide<5->{$(\pi \act x) = y$}\\[1mm]
+ & \only<6-7>{$\;\;\;\wedge$}\only<8>{\textcolor{gray}{\xout{$\;\;\;\wedge$}}} &
+ \only<6-7>{$\pi \act as = bs$}\only<8>{\textcolor{gray}{\xout{$\pi \act as = bs$}}}\\
+ \end{tabular}}
+ \end{textblock}}
+
+ \only<7>{
+ \begin{textblock}{7}(3,13.8)
+ \footnotesize $^*$ $as$ and $bs$ are \alert{lists} of names
+ \end{textblock}}
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1-3>
+ \frametitle{\begin{tabular}{c}Examples\end{tabular}}
+ \mbox{}\\[-3mm]
+
+ \begin{itemize}
+ \item lets look at type-schemes:\medskip\medskip
+
+ \begin{center}
+ \textcolor{blue}{$(as, x) \approx\!\makebox[5mm][l]{${}_{\text{set}}$} (bs, y)$}
+ \end{center}\medskip
+
+ \onslide<2->{
+ \begin{center}
+ \textcolor{blue}{
+ \begin{tabular}{l}
+ $\text{fv}(x) = \{x\}$\\[1mm]
+ $\text{fv}(T_1 \rightarrow T_2) = \text{fv}(T_1) \cup \text{fv}(T_2)$\\
+ \end{tabular}}
+ \end{center}}
+ \end{itemize}
+
+
+ \only<3->{
+ \begin{textblock}{4}(0.3,12)
+ \begin{tikzpicture}
+ \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
+ {\tiny\color{darkgray}
+ \begin{minipage}{3.4cm}\raggedright
+ \begin{tabular}{r@ {\hspace{1mm}}l}
+ \multicolumn{2}{@ {}l}{set+:}\\
+ $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
+ $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
+ $\wedge$ & $\pi \cdot x = y$\\
+ \\
+ \end{tabular}
+ \end{minipage}};
+ \end{tikzpicture}
+ \end{textblock}}
+ \only<3->{
+ \begin{textblock}{4}(5.2,12)
+ \begin{tikzpicture}
+ \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
+ {\tiny\color{darkgray}
+ \begin{minipage}{3.4cm}\raggedright
+ \begin{tabular}{r@ {\hspace{1mm}}l}
+ \multicolumn{2}{@ {}l}{set:}\\
+ $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
+ $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
+ $\wedge$ & $\pi \cdot x = y$\\
+ $\wedge$ & $\pi \cdot as = bs$\\
+ \end{tabular}
+ \end{minipage}};
+ \end{tikzpicture}
+ \end{textblock}}
+ \only<3->{
+ \begin{textblock}{4}(10.2,12)
+ \begin{tikzpicture}
+ \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
+ {\tiny\color{darkgray}
+ \begin{minipage}{3.4cm}\raggedright
+ \begin{tabular}{r@ {\hspace{1mm}}l}
+ \multicolumn{2}{@ {}l}{list:}\\
+ $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
+ $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
+ $\wedge$ & $\pi \cdot x = y$\\
+ $\wedge$ & $\pi \cdot as = bs$\\
+ \end{tabular}
+ \end{minipage}};
+ \end{tikzpicture}
+ \end{textblock}}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1-2>
+ \frametitle{\begin{tabular}{c}Examples\end{tabular}}
+ \mbox{}\\[-3mm]
+
+ \begin{center}
+ \textcolor{blue}{
+ \only<1>{$(\{x, y\}, x \rightarrow y) \approx_? (\{x, y\}, y \rightarrow x)$}
+ \only<2>{$([x, y], x \rightarrow y) \approx_? ([x, y], y \rightarrow x)$}}
+ \end{center}
+
+ \begin{itemize}
+ \item \textcolor{blue}{$\approx_{\text{set+}}$, $\approx_{\text{set}}$%
+ \only<2>{, \alert{$\not\approx_{\text{list}}$}}}
+ \end{itemize}
+
+
+ \only<1->{
+ \begin{textblock}{4}(0.3,12)
+ \begin{tikzpicture}
+ \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
+ {\tiny\color{darkgray}
+ \begin{minipage}{3.4cm}\raggedright
+ \begin{tabular}{r@ {\hspace{1mm}}l}
+ \multicolumn{2}{@ {}l}{set+:}\\
+ $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
+ $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
+ $\wedge$ & $\pi \cdot x = y$\\
+ \\
+ \end{tabular}
+ \end{minipage}};
+ \end{tikzpicture}
+ \end{textblock}}
+ \only<1->{
+ \begin{textblock}{4}(5.2,12)
+ \begin{tikzpicture}
+ \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
+ {\tiny\color{darkgray}
+ \begin{minipage}{3.4cm}\raggedright
+ \begin{tabular}{r@ {\hspace{1mm}}l}
+ \multicolumn{2}{@ {}l}{set:}\\
+ $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
+ $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
+ $\wedge$ & $\pi \cdot x = y$\\
+ $\wedge$ & $\pi \cdot as = bs$\\
+ \end{tabular}
+ \end{minipage}};
+ \end{tikzpicture}
+ \end{textblock}}
+ \only<1->{
+ \begin{textblock}{4}(10.2,12)
+ \begin{tikzpicture}
+ \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
+ {\tiny\color{darkgray}
+ \begin{minipage}{3.4cm}\raggedright
+ \begin{tabular}{r@ {\hspace{1mm}}l}
+ \multicolumn{2}{@ {}l}{list:}\\
+ $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
+ $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
+ $\wedge$ & $\pi \cdot x = y$\\
+ $\wedge$ & $\pi \cdot as = bs$\\
+ \end{tabular}
+ \end{minipage}};
+ \end{tikzpicture}
+ \end{textblock}}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1-2>
+ \frametitle{\begin{tabular}{c}Examples\end{tabular}}
+ \mbox{}\\[-3mm]
+
+ \begin{center}
+ \textcolor{blue}{\only<1>{$(\{x\}, x) \approx_? (\{x, y\}, x)$}}
+ \end{center}
+
+ \begin{itemize}
+ \item \textcolor{blue}{$\approx_{\text{set+}}$, $\not\approx_{\text{set}}$,
+ $\not\approx_{\text{list}}$}
+ \end{itemize}
+
+
+ \only<1->{
+ \begin{textblock}{4}(0.3,12)
+ \begin{tikzpicture}
+ \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
+ {\tiny\color{darkgray}
+ \begin{minipage}{3.4cm}\raggedright
+ \begin{tabular}{r@ {\hspace{1mm}}l}
+ \multicolumn{2}{@ {}l}{set+:}\\
+ $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
+ $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
+ $\wedge$ & $\pi \cdot x = y$\\
+ \\
+ \end{tabular}
+ \end{minipage}};
+ \end{tikzpicture}
+ \end{textblock}}
+ \only<1->{
+ \begin{textblock}{4}(5.2,12)
+ \begin{tikzpicture}
+ \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
+ {\tiny\color{darkgray}
+ \begin{minipage}{3.4cm}\raggedright
+ \begin{tabular}{r@ {\hspace{1mm}}l}
+ \multicolumn{2}{@ {}l}{set:}\\
+ $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
+ $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
+ $\wedge$ & $\pi \cdot x = y$\\
+ $\wedge$ & $\pi \cdot as = bs$\\
+ \end{tabular}
+ \end{minipage}};
+ \end{tikzpicture}
+ \end{textblock}}
+ \only<1->{
+ \begin{textblock}{4}(10.2,12)
+ \begin{tikzpicture}
+ \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
+ {\tiny\color{darkgray}
+ \begin{minipage}{3.4cm}\raggedright
+ \begin{tabular}{r@ {\hspace{1mm}}l}
+ \multicolumn{2}{@ {}l}{list:}\\
+ $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
+ $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
+ $\wedge$ & $\pi \cdot x = y$\\
+ $\wedge$ & $\pi \cdot as = bs$\\
+ \end{tabular}
+ \end{minipage}};
+ \end{tikzpicture}
+ \end{textblock}}
+
+ \only<2>{
+ \begin{textblock}{6}(2.5,4)
+ \begin{tikzpicture}
+ \draw (0,0) node[inner sep=5mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
+ {\normalsize
+ \begin{minipage}{8cm}\raggedright
+ \begin{itemize}
+ \item \color{darkgray}$\alpha$-equivalences coincide when a single name is
+ abstracted
+ \item \color{darkgray}in that case they are equivalent to ``old-fashioned'' definitions of $\alpha$
+ \end{itemize}
+ \end{minipage}};
+ \end{tikzpicture}
+ \end{textblock}}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1->
+ \frametitle{\begin{tabular}{c}Our Specifications\end{tabular}}
+ \mbox{}\\[-6mm]
+
+ \mbox{}\hspace{10mm}
+ \begin{tabular}{ll}
+ \multicolumn{2}{l}{\isacommand{nominal\_datatype} trm $=$}\\
+ \hspace{5mm}\phantom{$|$} Var name\\
+ \hspace{5mm}$|$ App trm trm\\
+ \hspace{5mm}$|$ Lam x::name t::trm
+ & \isacommand{bind} x \isacommand{in} t\\
+ \hspace{5mm}$|$ Let as::assns t::trm
+ & \isacommand{bind} bn(as) \isacommand{in} t\\
+ \multicolumn{2}{l}{\isacommand{and} assns $=$}\\
+ \multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} ANil}\\
+ \multicolumn{2}{l}{\hspace{5mm}$|$ ACons name trm assns}\\
+ \multicolumn{2}{l}{\isacommand{binder} bn \isacommand{where}}\\
+ \multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} bn(ANil) $=$ $[]$}\\
+ \multicolumn{2}{l}{\hspace{5mm}$|$ bn(ACons a t as) $=$ $[$a$]$ @ bn(as)}\\
+ \end{tabular}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1->[t]
+ \frametitle{\begin{tabular}{c}Binder Clauses\end{tabular}}
+
+ \begin{itemize}
+ \item We need to have a `clear scope' for a bound variable, and bound
+ variables should not be free and bound at the same time.\bigskip
+ \end{itemize}
+
+ \begin{center}
+ \only<1>{
+ \begin{tabular}{@ {\hspace{-5mm}}l}
+ \alert{\bf shallow binders}\\
+ \hspace{4mm}Lam x::name t::trm\hspace{4mm} \isacommand{bind} x \isacommand{in} t\\
+ \hspace{4mm}All xs::name set T::ty\hspace{4mm} \isacommand{bind} xs \isacommand{in} T\\
+ \hspace{4mm}Foo x::name t$_1$::trm t$_2$::trm\hspace{4mm}
+ \isacommand{bind} x \isacommand{in} t$_1$, \isacommand{bind} x \isacommand{in} t$_2$\\
+ \hspace{4mm}Bar x::name t$_1$::trm t$_2$::trm\hspace{4mm}
+ \isacommand{bind} x \isacommand{in} t$_1$ t$_2$\\
+ \end{tabular}}
+ \only<2>{
+ \begin{tabular}{@ {\hspace{-5mm}}l}
+ \alert{\bf deep binders} \\
+ \hspace{4mm}Let as::assns t::trm\hspace{4mm} \isacommand{bind} bn(as) \isacommand{in} t\\
+ \hspace{4mm}Foo as::assns t$_1$::trm t$_2$::trm\\
+ \hspace{20mm}\isacommand{bind} bn(as) \isacommand{in} t$_1$, \isacommand{bind} bn(as) \isacommand{in} t$_2$\\[4mm]
+ \makebox[0mm][l]{\alert{$\times$}}\hspace{4mm}Bar as::assns t$_1$::trm t$_2$::trm\\
+ \hspace{20mm}\isacommand{bind} bn$_1$(as) \isacommand{in} t$_1$, \isacommand{bind} bn$_2$(as) \isacommand{in} t$_2$\\
+ \end{tabular}}
+ \only<3>{
+ \begin{tabular}{@ {\hspace{-5mm}}l}
+ {\bf deep \alert{recursive} binders} \\
+ \hspace{4mm}Let\_rec as::assns t::trm\hspace{4mm} \isacommand{bind} bn(as) \isacommand{in} t as\\[4mm]
+
+ \makebox[0mm][l]{\alert{$\times$}}\hspace{4mm}Foo\_rec as::assns t$_1$::trm t$_2$::trm\hspace{4mm}\\
+ \hspace{20mm}\isacommand{bind} bn(as) \isacommand{in} t$_1$ as, \isacommand{bind} bn(as) \isacommand{in} t$_2$\\
+
+ \end{tabular}}
+ \end{center}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1-5>
+ \frametitle{\begin{tabular}{c}Our Work\end{tabular}}
+ \mbox{}\\[-6mm]
+
+ \begin{center}
+ \begin{tikzpicture}[scale=1.5]
+ %%%\draw[step=2mm] (-4,-1) grid (4,1);
+
+ \onslide<1>{\draw[very thick] (0.7,0.4) circle (4.25mm);}
+ \onslide<1>{\draw[rounded corners=1mm, very thick] ( 0.0,-0.8) rectangle ( 1.8, 0.9);}
+ \onslide<1->{\draw[rounded corners=1mm, very thick] (-1.95,0.85) rectangle (-2.85,-0.05);}
+
+ \onslide<1>{\draw (-2.0, 0.845) -- (0.7,0.845);}
+ \onslide<1>{\draw (-2.0,-0.045) -- (0.7,-0.045);}
+
+ \onslide<1>{\alert{\draw ( 0.7, 0.4) node {\footnotesize\begin{tabular}{@ {}c@ {}}$\alpha$-\\[-1mm]classes\end{tabular}};}}
+ \onslide<1->{\alert{\draw (-2.4, 0.4) node {\footnotesize\begin{tabular}{@ {}c@ {}}$\alpha$-eq.\\[-1mm]terms\end{tabular}};}}
+ \onslide<1>{\draw (1.8, 0.48) node[right=-0.1mm]
+ {\footnotesize\begin{tabular}{@ {}l@ {}}existing\\[-1mm] type\\ \onslide<1>{\alert{(sets of raw terms)}}\end{tabular}};}
+ \onslide<1>{\draw (0.9, -0.35) node {\footnotesize\begin{tabular}{@ {}l@ {}}non-empty\\[-1mm]subset\end{tabular}};}
+ \onslide<1->{\draw (-3.25, 0.55) node {\footnotesize\begin{tabular}{@ {}l@ {}}new\\[-1mm]type\end{tabular}};}
+
+ \onslide<1>{\draw[<->, very thick] (-1.8, 0.3) -- (-0.1,0.3);}
+ \onslide<1>{\draw (-0.95, 0.3) node[above=0mm] {\footnotesize{}isomorphism};}
+
+ \onslide<1>{\draw[->, line width=2mm, red] (-1.0,-0.4) -- (0.35,0.16);}
+ \end{tikzpicture}
+ \end{center}
+
+ \begin{textblock}{9.5}(6,3.5)
+ \begin{itemize}
+ \item<1-> defined fv and $\alpha$
+ \item<2-> built quotient / new type
+ \item<3-> derived a reasoning infrastructure ($\fresh$, distinctness, injectivity, cases,\ldots)
+ \item<4-> derive a {\bf stronger} cases lemma
+ \item<5-> from this, a {\bf stronger} induction principle (Barendregt variable convention built in)\\
+ \begin{center}
+ \textcolor{blue}{Foo ($\lambda x. \lambda y. t$) ($\lambda u. \lambda v. s$)}
+ \end{center}
+ \end{itemize}
+ \end{textblock}
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1->
+ \frametitle{\begin{tabular}{c}Part I: Conclusion\end{tabular}}
+ \mbox{}\\[-6mm]
+
+ \begin{itemize}
+ \item the user does not see anything of the raw level\medskip
+ \only<1>{\begin{center}
+ Lam a (Var a) \alert{$=$} Lam b (Var b)
+ \end{center}\bigskip}
+
+ \item<2-> \textcolor{blue}{http://isabelle.in.tum.de/nominal/}
+ \end{itemize}
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1->
+ \frametitle{\begin{tabular}{c}Part II: $\alpha\beta$-Equal Terms\end{tabular}}
+
+ \begin{itemize}
+ \item we have implemented a quotient package for Isabelle;
+ \item can now introduce the type of $\alpha\beta$-equal terms (starting
+ from $\alpha$-equal terms).
+ \item on paper this looks easy\pause\bigskip
+ \end{itemize}
+
+ \begin{center}
+ \begin{tabular}{lll}
+ \smath{x \approx_{\alpha\beta} y} & \smath{\;\not\Rightarrow\;} &
+ \smath{\text{supp}(x) = \text{supp}(y)}\\
+ & \smath{\;\not\Rightarrow\;} &
+ \smath{\text{size}(x) = \text{size}(y)}\\
+ \end{tabular}
+ \end{center}\pause
+
+ \small
+ \begin{center}
+ Andy: \smath{\;\;\text{supp}\mbox{\isasymlbrakk}x\mbox{\isasymrbrakk}_{\approx_{\alpha\beta}} =
+ {\text{\large$\bigcap$}} \{ \text{supp}(y) \;|\; y \approx_{\alpha\beta} x\}}
+ \end{center}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{}
+
+ \begin{center}
+ \begin{tabular}{rcl}
+ \smath{x\;[y := s]} & \smath{\dn} & \smath{\text{if}\;x=y\;\text{then}\;s\;\text{else}\;x}\bigskip\\
+ \smath{t_1 t_2\;[y := s]} & \smath{\dn} & \smath{t_1[y := s]\;t_2[y := s]}\bigskip\\
+ \smath{\lambda x.t\;[y := s]} & \smath{\dn} & \smath{\lambda x.\; t[y := s]}\\
+ \multicolumn{3}{r}{provided \smath{x \fresh (y, s)}}
+ \end{tabular}
+ \end{center}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[t]
+ \frametitle{\begin{tabular}{c}Part III: Regular Languages\\[-8mm]\end{tabular}}
+
+ \begin{center}
+ \huge\bf\textcolor{gray}{in Theorem Provers}\\
+ \footnotesize\textcolor{gray}{e.g.~Isabelle, Coq, HOL4, \ldots}
+ \end{center}
+
+ \begin{itemize}
+ \item automata @{text "\<Rightarrow>"} graphs, matrices, functions
+ \item<2-> combining automata/graphs
+
+ \onslide<2->{
+ \begin{center}
+ \begin{tabular}{ccc}
+ \begin{tikzpicture}[scale=1]
+ %\draw[step=2mm] (-1,-1) grid (1,1);
+
+ \draw[rounded corners=1mm, very thick] (-1.0,-0.3) rectangle (-0.2,0.3);
+ \draw[rounded corners=1mm, very thick] ( 0.2,-0.3) rectangle ( 1.0,0.3);
+
+ \node (A) at (-1.0,0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
+ \node (B) at ( 0.2,0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
+
+ \node (C) at (-0.2, 0.13) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
+ \node (D) at (-0.2,-0.13) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
+
+ \node (E) at (1.0, 0.2) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
+ \node (F) at (1.0,-0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
+ \node (G) at (1.0,-0.2) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
+
+ \draw (-0.6,0.0) node {\small$A_1$};
+ \draw ( 0.6,0.0) node {\small$A_2$};
+ \end{tikzpicture}}
+
+ &
+
+ \onslide<3->{\raisebox{1.1mm}{\bf\Large$\;\Rightarrow\,$}}
+
+ &
+
+ \onslide<3->{\begin{tikzpicture}[scale=1]
+ %\draw[step=2mm] (-1,-1) grid (1,1);
+
+ \draw[rounded corners=1mm, very thick] (-1.0,-0.3) rectangle (-0.2,0.3);
+ \draw[rounded corners=1mm, very thick] ( 0.2,-0.3) rectangle ( 1.0,0.3);
+
+ \node (A) at (-1.0,0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
+ \node (B) at ( 0.2,0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
+
+ \node (C) at (-0.2, 0.13) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
+ \node (D) at (-0.2,-0.13) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
+
+ \node (E) at (1.0, 0.2) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
+ \node (F) at (1.0,-0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
+ \node (G) at (1.0,-0.2) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
+
+ \draw (C) to [red, very thick, bend left=45] (B);
+ \draw (D) to [red, very thick, bend right=45] (B);
+
+ \draw (-0.6,0.0) node {\small$A_1$};
+ \draw ( 0.6,0.0) node {\small$A_2$};
+ \end{tikzpicture}}
+
+ \end{tabular}
+ \end{center}\medskip
+
+ \only<4-5>{
+ \begin{tabular}{@ {\hspace{-5mm}}l@ {}}
+ disjoint union:\\[2mm]
+ \smath{A_1\uplus A_2 \dn \{(1, x)\,|\, x \in A_1\} \,\cup\, \{(2, y)\,|\, y \in A_2\}}
+ \end{tabular}}
+ \end{itemize}
+
+ \only<5>{
+ \begin{textblock}{13.9}(0.7,7.7)
+ \begin{block}{}
+ \medskip
+ \begin{minipage}{14cm}\raggedright
+ Problems with definition for regularity:\bigskip\\
+ \smath{\;\text{is\_regular}(A) \dn \exists M.\;\text{is\_dfa}(M) \wedge {\cal L} (M) = A}\bigskip
+ \end{minipage}
+ \end{block}
+ \end{textblock}}
+ \medskip
+
+ \only<6->{\underline{A solution}:\;\;use \smath{\text{nat}}s \;@{text "\<Rightarrow>"}\; state nodes\medskip}
+
+ \only<7->{You have to \alert{rename} states!}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[t]
+ \frametitle{\normalsize Formal language theory\ldots\hfill\mbox{}}
+ \mbox{}\\[-15mm]\mbox{}
+
+ \begin{center}
+ \huge\bf\textcolor{gray}{in Theorem Provers}\\
+ \footnotesize\textcolor{gray}{e.g.~Isabelle, Coq, HOL4, \ldots}
+ \end{center}
+
+ \begin{itemize}
+ \item Kozen's ``paper'' proof of Myhill-Nerode:\\
+ \hspace{2cm}requires absence of \alert{inaccessible states}
+ \end{itemize}\bigskip\bigskip
+
+ \begin{center}
+ \smath{\;\text{is\_regular}(A) \dn \exists M.\;\text{is\_dfa}(M) \wedge {\cal L} (M) = A}
+ \end{center}
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[t]
+ \frametitle{}
+ \mbox{}\\[25mm]\mbox{}
+
+ \begin{textblock}{13.9}(0.7,1.2)
+ \begin{block}{}
+ \begin{minipage}{13.4cm}\raggedright
+ {\bf Definition:}\smallskip\\
+
+ A language \smath{A} is \alert{regular}, provided there exists a\\
+ \alert{regular expression} that matches all strings of \smath{A}.
+ \end{minipage}
+ \end{block}
+ \end{textblock}\pause
+
+ {\noindent\large\bf\alert{\ldots{}and forget about automata}}\bigskip\bigskip\pause
+
+ Infrastructure for free. But do we lose anything?\medskip\pause
+
+ \begin{minipage}{1.1\textwidth}
+ \begin{itemize}
+ \item pumping lemma\pause
+ \item closure under complementation\pause
+ \item \only<6>{regular expression matching}%
+ \only<7->{\soutt{regular expression matching}
+ {\footnotesize(@{text "\<Rightarrow>"}Brozowski'64, Owens et al '09)}}
+ \item<8-> most textbooks are about automata
+ \end{itemize}
+ \end{minipage}
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE The Myhill-Nerode Theorem}
+
+ \begin{itemize}
+ \item provides necessary and suf\!ficient conditions\\ for a language
+ being regular\\ \textcolor{gray}{(pumping lemma only necessary)}\bigskip
+
+ \item key is the equivalence relation:\medskip
+ \begin{center}
+ \smath{x \approx_{A} y \,\dn\, \forall z.\; x @ z \in A \Leftrightarrow y @ z \in A}
+ \end{center}
+ \end{itemize}
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE The Myhill-Nerode Theorem}
+
+ \begin{center}
+ \only<1>{%
+ \begin{tikzpicture}[scale=3]
+ \draw[very thick] (0.5,0.5) circle (.6cm);
+ \end{tikzpicture}}%
+ \only<2->{%
+ \begin{tikzpicture}[scale=3]
+ \draw[very thick] (0.5,0.5) circle (.6cm);
+ \clip[draw] (0.5,0.5) circle (.6cm);
+ \draw[step=2mm, very thick] (-1.4,-1.4) grid (1.4,1.4);
+ \end{tikzpicture}}
+ \end{center}
+
+ \begin{itemize}
+ \item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_A) \;\Leftrightarrow\; A\; \text{is regular}}
+ \end{itemize}
+
+ \begin{textblock}{5}(2.1,5.3)
+ \begin{tikzpicture}
+ \node at (0,0) [single arrow, fill=red,text=white, minimum height=2cm]
+ {$U\!N\!IV$};
+ \draw (-0.3,-1.1) node {\begin{tabular}{l}set of all\\[-1mm] strings\end{tabular}};
+ \end{tikzpicture}
+ \end{textblock}
+
+ \only<2->{%
+ \begin{textblock}{5}(9.1,7.2)
+ \begin{tikzpicture}
+ \node at (0,0) [shape border rotate=180,single arrow, fill=red,text=white, minimum height=2cm]
+ {@{text "\<lbrakk>x\<rbrakk>"}$_{\approx_{A}}$};
+ \draw (0.9,-1.1) node {\begin{tabular}{l}an equivalence class\end{tabular}};
+ \end{tikzpicture}
+ \end{textblock}}
+
+ \only<3->{
+ \begin{textblock}{11.9}(1.7,3)
+ \begin{block}{}
+ \begin{minipage}{11.4cm}\raggedright
+ Two directions:\medskip\\
+ \begin{tabular}{@ {}ll}
+ 1.)\;finite $\Rightarrow$ regular\\
+ \;\;\;\smath{\text{finite}\,(U\!N\!IV /\!/ \approx_A) \Rightarrow \exists r.\;A = {\cal L}(r)}\\[3mm]
+ 2.)\;regular $\Rightarrow$ finite\\
+ \;\;\;\smath{\text{finite}\, (U\!N\!IV /\!/ \approx_{{\cal L}(r)})}
+ \end{tabular}
+
+ \end{minipage}
+ \end{block}
+ \end{textblock}}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<-1>[c]
+ \frametitle{\begin{tabular}{@ {}l}\LARGE%
+ Transitions between Eq-Classes\end{tabular}}
+
+ \begin{center}
+ \begin{tikzpicture}[scale=3]
+ \draw[very thick] (0.5,0.5) circle (.6cm);
+ \clip[draw] (0.5,0.5) circle (.6cm);
+ \draw[step=2mm, very thick] (-1.4,-1.4) grid (1.4,1.4);
+ \draw[blue, fill] (0.0, 0.6) rectangle (0.2, 0.8);
+ \draw[blue, fill] (0.8, 0.4) rectangle (1.0, 0.6);
+ \draw[white] (0.1,0.7) node (X) {$X$};
+ \draw[white] (0.9,0.5) node (Y) {$Y$};
+ \draw[blue, ->, line width = 2mm, bend left=45] (X) -- (Y);
+ \node [inner sep=1pt,label=above:\textcolor{blue}{$c$}] at ($ (X)!.5!(Y) $) {};
+ \end{tikzpicture}
+ \end{center}
+
+ \begin{center}
+ \smath{X \stackrel{c}{\longrightarrow} Y \;\dn\; X ; c \subseteq Y}
+ \end{center}
+
+ \onslide<8>{
+ \begin{tabular}{c}
+ \begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick]
+ \tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
+ \node[state,initial] (q_0) {$R_1$};
+ \end{tikzpicture}
+ \end{tabular}}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE The Other Direction}
+
+ One has to prove
+
+ \begin{center}
+ \smath{\text{finite} (U\!N\!IV /\!/ \approx_{{\cal L}(r)})}
+ \end{center}
+
+ by induction on \smath{r}. Not trivial, but after a bit
+ of thinking, one can find a \alert{refined} relation:\bigskip
+
+
+ \begin{center}
+ \mbox{\begin{tabular}{c@ {\hspace{7mm}}c@ {\hspace{7mm}}c}
+ \begin{tikzpicture}[scale=1.1]
+ %Circle
+ \draw[thick] (0,0) circle (1.1);
+ \end{tikzpicture}
+ &
+ \begin{tikzpicture}[scale=1.1]
+ %Circle
+ \draw[thick] (0,0) circle (1.1);
+ %Main rays
+ \foreach \a in {0, 90,...,359}
+ \draw[very thick] (0, 0) -- (\a:1.1);
+ \foreach \a / \l in {45/1, 135/2, 225/3, 315/4}
+ \draw (\a: 0.65) node {\small$a_\l$};
+ \end{tikzpicture}
+ &
+ \begin{tikzpicture}[scale=1.1]
+ %Circle
+ \draw[red, thick] (0,0) circle (1.1);
+ %Main rays
+ \foreach \a in {0, 45,...,359}
+ \draw[red, very thick] (0, 0) -- (\a:1.1);
+ \foreach \a / \l in {22.5/1.1, 67.5/1.2, 112.5/2.1, 157.5/2.2, 202.4/3.1, 247.5/3.2, 292.5/4.1, 337.5/4.2}
+ \draw (\a: 0.77) node {\textcolor{red}{\footnotesize$a_{\l}$}};
+ \end{tikzpicture}\\
+ \small\smath{U\!N\!IV} &
+ \small\smath{U\!N\!IV /\!/ \approx_{{\cal L}(r)}} &
+ \small\smath{U\!N\!IV /\!/ \alert{R}}
+ \end{tabular}}
+ \end{center}
+
+ \begin{textblock}{5}(9.8,2.6)
+ \begin{tikzpicture}
+ \node at (0,0) [shape border rotate=270,single arrow, fill=red,text=white, minimum height=0cm]{\textcolor{red}{a}};
+ \end{tikzpicture}
+ \end{textblock}
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[t]
+ \frametitle{\LARGE\begin{tabular}{c}Derivatives of RExps\end{tabular}}
+
+ \begin{itemize}
+ \item introduced by Brozowski~'64
+ \item a regular expressions after a character has been parsed\\[-18mm]\mbox{}
+ \end{itemize}
+
+ \only<1>{%
+ \textcolor{blue}{%
+ \begin{center}
+ \begin{tabular}{@ {}lc@ {\hspace{3mm}}l@ {}}
+ der c $\varnothing$ & $\dn$ & $\varnothing$\\
+ der c [] & $\dn$ & $\varnothing$\\
+ der c d & $\dn$ & if c $=$ d then [] else $\varnothing$\\
+ der c ($r_1 + r_2$) & $\dn$ & (der c $r_1$) $+$ (der c $r_2$)\\
+ der c ($r^\star$) & $\dn$ & (der c $r$) $\cdot$ $r^\star$\\
+ der c ($r_1 \cdot r_2$) & $\dn$ & if nullable $r_1$\\
+ & & then (der c $r_1$) $\cdot$ $r_2$ $+$ (der c $r_2$)\\
+ & & else (der c $r_1$) $\cdot$ $r_2$\\
+ \end{tabular}
+ \end{center}}}
+ \only<2>{%
+ \textcolor{blue}{%
+ \begin{center}
+ \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}}
+ pder c $\varnothing$ & $\dn$ & \alert{$\{\}$}\\
+ pder c [] & $\dn$ & \alert{$\{\}$}\\
+ pder c d & $\dn$ & if c $=$ d then $\{$[]$\}$ else $\{\}$\\
+ pder c ($r_1 + r_2$) & $\dn$ & (pder c $r_1$) \alert{$\cup$} (der c $r_2$)\\
+ pder c ($r^\star$) & $\dn$ & (pder c $r$) $\cdot$ $r^\star$\\
+ pder c ($r_1 \cdot r_2$) & $\dn$ & if nullable $r_1$\\
+ & & then (pder c $r_1$) $\cdot$ $r_2$ \alert{$\cup$} (pder c $r_2$)\\
+ & & else (pder c $r_1$) $\cdot$ $r_2$\\
+ \end{tabular}
+ \end{center}}}
+
+ \only<2>{
+ \begin{textblock}{6}(8.5,4.7)
+ \begin{block}{}
+ \begin{quote}
+ \begin{minipage}{6cm}\raggedright
+ \begin{itemize}
+ \item partial derivatives
+ \item by Antimirov~'95
+ \end{itemize}
+ \end{minipage}
+ \end{quote}
+ \end{block}
+ \end{textblock}}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[t]
+ \frametitle{\LARGE Partial Derivatives}
+
+ \mbox{}\\[0mm]\mbox{}
+
+ \begin{itemize}
+
+ \item \alt<1>{\smath{\text{pders $x$ $r$ \mbox{$=$} pders $y$ $r$}}}
+ {\smath{\underbrace{\text{pders $x$ $r$ \mbox{$=$} pders $y$ $r$}}_{R}}}
+ refines \textcolor{blue}{$x$ $\approx_{{\cal L}(r)}$ $y$}\\[16mm]\pause
+ \item \smath{\text{finite} (U\!N\!IV /\!/ R)} \bigskip\pause
+ \item Therefore \smath{\text{finite} (U\!N\!IV /\!/ \approx_{{\cal L}(r)})}. Qed.
+ \end{itemize}
+
+ \only<2->{%
+ \begin{textblock}{5}(3.9,7.2)
+ \begin{tikzpicture}
+ \node at (0,0) [shape border rotate=270,single arrow, fill=red,text=white, minimum height=0cm]{\textcolor{red}{a}};
+ \draw (2.2,0) node {Antimirov '95};
+ \end{tikzpicture}
+ \end{textblock}}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[t]
+ \frametitle{\LARGE What Have We Achieved?}
+
+ \begin{itemize}
+ \item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_A) \;\Leftrightarrow\; A\; \text{is regular}}
+ \medskip\pause
+ \item regular languages are closed under complementation; this is now easy
+ \begin{center}
+ \smath{U\!N\!IV /\!/ \approx_A \;\;=\;\; U\!N\!IV /\!/ \approx_{\overline{A}}}
+ \end{center}\pause\medskip
+
+ \item non-regularity (\smath{a^nb^n})\medskip\pause\pause
+
+ \item take \alert{\bf any} language; build the language of substrings\\
+ \pause
+
+ then this language \alert{\bf is} regular\;\; (\smath{a^nb^n} $\Rightarrow$ \smath{a^\star{}b^\star})
+
+ \end{itemize}
+
+\only<2>{
+\begin{textblock}{10}(4,14)
+\small
+\smath{x \approx_{A} y \,\dn\, \forall z.\; x @ z \in A \Leftrightarrow y @ z \in A}
+\end{textblock}}
+
+\only<4>{
+\begin{textblock}{5}(2,8.6)
+\begin{minipage}{8.8cm}
+\begin{block}{}
+\begin{minipage}{8.6cm}
+If there exists a sufficiently large set \smath{B} (for example infinitely large),
+such that
+
+\begin{center}
+\smath{\forall x,y \in B.\; x \not= y \;\Rightarrow\; x \not\approx_{A} y}.
+\end{center}
+
+then \smath{A} is not regular.\hspace{1.3cm}\small(\smath{B \dn \bigcup_n a^n})
+\end{minipage}
+\end{block}
+\end{minipage}
+\end{textblock}
+}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[b]
+ \frametitle{\mbox{}\\[2cm]\textcolor{red}{Thank you!\\[5mm]Questions?}}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1-2>[c]
+ \frametitle{\begin{tabular}{c}Examples\end{tabular}}
+ \mbox{}\\[-6mm]
+
+ \textcolor{blue}{
+ \begin{center}
+ $(\{a,b\}, a \rightarrow b) \approx_\alpha (\{a, b\}, a \rightarrow b)$
+ $(\{a,b\}, a \rightarrow b) \approx_\alpha (\{a, b\}, b \rightarrow a)$
+ \end{center}}
+
+ \textcolor{blue}{
+ \begin{center}
+ $(\{a,b\}, (a \rightarrow b, a \rightarrow b))$\\
+ \hspace{17mm}$\not\approx_\alpha (\{a, b\}, (a \rightarrow b, b \rightarrow a))$
+ \end{center}}
+
+ \onslide<2->
+ {1.) \hspace{3mm}\isacommand{bind (set)} as \isacommand{in} $\tau_1$,
+ \isacommand{bind (set)} as \isacommand{in} $\tau_2$\medskip
+
+ 2.) \hspace{3mm}\isacommand{bind (set)} as \isacommand{in} $\tau_1$ $\tau_2$
+ }
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+
+(*<*)
+end
+(*>*)
\ No newline at end of file