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3052
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(*<*)
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theory Slides9
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imports "~~/src/HOL/Library/LaTeXsugar" "Nominal"
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begin
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notation (latex output)
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set ("_") and
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Cons ("_::/_" [66,65] 65)
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(*>*)
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text_raw {*
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%% shallow, deep, and recursive binders
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%%
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%%\renewcommand{\slidecaption}{Cambridge, 8.~June 2010}
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%%\renewcommand{\slidecaption}{Uppsala, 3.~March 2011}
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\renewcommand{\slidecaption}{Leicester, 23.~November 2011}
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\newcommand{\soutt}[1]{\tikz[baseline=(X.base), inner sep=-0.1pt, outer sep=0pt]
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\node [cross out,red, ultra thick, draw] (X) {\textcolor{black}{#1}};}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}<1>[t]
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\frametitle{%
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\begin{tabular}{@ {\hspace{-3mm}}c@ {}}
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\\
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\LARGE General Binding Structures\\[-1mm]
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\LARGE in Nominal Isabelle 2\\
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\end{tabular}}
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\begin{center}
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Christian Urban
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\end{center}
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\begin{center}
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joint work with {\bf Cezary Kaliszyk}\\[0mm]
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\end{center}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}<1>
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\frametitle{\begin{tabular}{c}Binding in Old Nominal\end{tabular}}
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\mbox{}\\[-6mm]
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\begin{itemize}
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\item the old Nominal Isabelle provided a reasoning infrastructure for single binders\medskip
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\begin{center}
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Lam [a].(Var a)
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\end{center}\bigskip
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\item<2-> but representing
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\begin{center}
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$\forall\{a_1,\ldots,a_n\}.\; T$
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\end{center}\medskip
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with single binders and reasoning about it is a \alert{\bf major} pain;
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take my word for it!
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\end{itemize}
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\only<1>{
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\begin{textblock}{6}(1.5,11)
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\small
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for example\\
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\begin{tabular}{l@ {\hspace{2mm}}l}
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& a $\fresh$ Lam [a]. t\\
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& Lam [a]. (Var a) \alert{$=$} Lam [b]. (Var b)\\
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& Barendregt-style reasoning about bound variables\\
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& (variable convention can lead to faulty reasoning)
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\end{tabular}
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\end{textblock}}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}[c]
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\frametitle{}
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\begin{tabular}{c@ {\hspace{2mm}}c}
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\\[6mm]
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\begin{tabular}{c}
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\includegraphics[scale=0.11]{harper.jpg}\\[-2mm]
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{\footnotesize Bob Harper}\\[-2.5mm]
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{\footnotesize (CMU)}
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\end{tabular}
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\begin{tabular}{c}
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\includegraphics[scale=0.37]{pfenning.jpg}\\[-2mm]
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{\footnotesize Frank Pfenning}\\[-2.5mm]
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{\footnotesize (CMU)}
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\end{tabular} &
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\begin{tabular}{p{6cm}}
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\raggedright
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\color{gray}{published a proof in\\ {\bf ACM Transactions on Computational Logic}, 2005,
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$\sim$31pp}
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\end{tabular}\\
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\pause
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\\[0mm]
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\begin{tabular}{c}
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\includegraphics[scale=0.36]{appel.jpg}\\[-2mm]
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{\footnotesize Andrew Appel}\\[-2.5mm]
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{\footnotesize (Princeton)}
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\end{tabular} &
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\begin{tabular}{p{6cm}}
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\raggedright
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\color{gray}{relied on their proof in a\\ {\bf security} critical application}
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\end{tabular}
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\end{tabular}\medskip\pause
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\small
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\begin{minipage}{1.0\textwidth}
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(I also found an {\bf error} in my Ph.D.-thesis about cut-elimination
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examined by Henk Barendregt and Andy Pitts.)
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\end{minipage}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}[c]
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\frametitle{\begin{tabular}{c}Binding in Old Nominal\end{tabular}}
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\begin{itemize}
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\item<1-> but representing
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\begin{center}
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$\forall\{a_1,\ldots,a_n\}.\; T$
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\end{center}\medskip
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with single binders and reasoning about it was a \alert{\bf major} pain;
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take my word for it!
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\end{itemize}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}<1-6>
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\frametitle{New Types in HOL}
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\begin{center}
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\begin{tikzpicture}[scale=1.5]
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%%%\draw[step=2mm] (-4,-1) grid (4,1);
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\onslide<2-4,6>{\draw[very thick] (0.7,0.4) circle (4.25mm);}
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\onslide<1-4,6>{\draw[rounded corners=1mm, very thick] ( 0.0,-0.8) rectangle ( 1.8, 0.9);}
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\onslide<3-5,6>{\draw[rounded corners=1mm, very thick] (-1.95,0.85) rectangle (-2.85,-0.05);}
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\onslide<3-4,6>{\draw (-2.0, 0.845) -- (0.7,0.845);}
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\onslide<3-4,6>{\draw (-2.0,-0.045) -- (0.7,-0.045);}
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\onslide<4-4,6>{\alert{\draw ( 0.7, 0.4) node {\footnotesize\begin{tabular}{@ {}c@ {}}$\alpha$-\\[-1mm]classes\end{tabular}};}}
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\onslide<4-5,6>{\alert{\draw (-2.4, 0.4) node {\footnotesize\begin{tabular}{@ {}c@ {}}$\alpha$-eq.\\[-1mm]terms\end{tabular}};}}
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\onslide<1-4,6>{\draw (1.8, 0.48) node[right=-0.1mm]
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{\footnotesize\begin{tabular}{@ {}l@ {}}existing\\[-1mm] type\\ \onslide<4-4,6>{\alert{(sets of raw terms)}}\end{tabular}};}
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\onslide<2-4,6>{\draw (0.9, -0.35) node {\footnotesize\begin{tabular}{@ {}l@ {}}non-empty\\[-1mm]subset\end{tabular}};}
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\onslide<3-5,6>{\draw (-3.25, 0.55) node {\footnotesize\begin{tabular}{@ {}l@ {}}new\\[-1mm]type\end{tabular}};}
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\onslide<3-4,6>{\draw[<->, very thick] (-1.8, 0.3) -- (-0.1,0.3);}
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\onslide<3-4,6>{\draw (-0.95, 0.3) node[above=0mm] {\footnotesize{}isomorphism};}
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\onslide<6>{\draw[->, line width=2mm, red] (-1.0,-0.4) -- (0.35,0.16);}
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\end{tikzpicture}
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\end{center}
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\begin{center}
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\textcolor{red}{\large\bf\onslide<6>{define $\alpha$-equivalence}}
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\end{center}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}<1-4>
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\frametitle{\begin{tabular}{c}Binding Sets of Names\end{tabular}}
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\mbox{}\\[-3mm]
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\begin{itemize}
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\item binding sets of names has some interesting properties:\medskip
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\begin{center}
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\begin{tabular}{l}
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\textcolor{blue}{$\forall\{x, y\}.\, x \rightarrow y \;\;\approx_\alpha\;\; \forall\{y, x\}.\, y \rightarrow x$}
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\bigskip\smallskip\\
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\onslide<2->{%
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\textcolor{blue}{$\forall\{x, y\}.\, x \rightarrow y \;\;\not\approx_\alpha\;\; \forall\{z\}.\, z \rightarrow z$}
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}\bigskip\smallskip\\
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\onslide<3->{%
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\textcolor{blue}{$\forall\{x\}.\, x \rightarrow y \;\;\approx_\alpha\;\; \forall\{x, \alert{z}\}.\, x \rightarrow y$}
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}\medskip\\
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\onslide<3->{\hspace{4cm}\small provided $z$ is fresh for the type}
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\end{tabular}
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\end{center}
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\end{itemize}
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\begin{textblock}{8}(2,14.5)
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\footnotesize $^*$ $x$, $y$, $z$ are assumed to be distinct
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\end{textblock}
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\only<4>{
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\begin{textblock}{6}(2.5,4)
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\begin{tikzpicture}
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\draw (0,0) node[inner sep=3mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
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{\normalsize\color{darkgray}
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\begin{minipage}{8cm}\raggedright
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For type-schemes the order of bound names does not matter, and
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$\alpha$-equivalence is preserved under \alert{vacuous} binders.
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\end{minipage}};
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\end{tikzpicture}
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\end{textblock}}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}<1-3>
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\frametitle{\begin{tabular}{c}Other Binding Modes\end{tabular}}
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\mbox{}\\[-3mm]
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\begin{itemize}
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\item alpha-equivalence being preserved under vacuous binders is \underline{not} always
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wanted:\bigskip\bigskip\normalsize
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\textcolor{blue}{\begin{tabular}{@ {\hspace{-8mm}}l}
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$\text{let}\;x = 3\;\text{and}\;y = 2\;\text{in}\;x - y\;\text{end}$\medskip\\
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\onslide<2->{$\;\;\;\only<2>{\approx_\alpha}\only<3>{\alert{\not\approx_\alpha}}
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\text{let}\;y = 2\;\text{and}\;x = 3\only<3->{\alert{\;\text{and}
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\;z = \text{loop}}}\;\text{in}\;x - y\;\text{end}$}
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\end{tabular}}
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\end{itemize}
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\end{frame}}
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*}
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}<1>
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\frametitle{\begin{tabular}{c}\LARGE{}Even Another Binding Mode\end{tabular}}
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\mbox{}\\[-3mm]
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\begin{itemize}
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\item sometimes one wants to abstract more than one name, but the order \underline{does} matter\bigskip
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\begin{center}
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\textcolor{blue}{\begin{tabular}{@ {\hspace{-8mm}}l}
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$\text{let}\;(x, y) = (3, 2)\;\text{in}\;x - y\;\text{end}$\medskip\\
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$\;\;\;\not\approx_\alpha
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\text{let}\;(y, x) = (3, 2)\;\text{in}\;x - y\;\text{end}$
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\end{tabular}}
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\end{center}
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\end{itemize}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}<1-2>
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\frametitle{\begin{tabular}{c}\LARGE{}Three Binding Modes\end{tabular}}
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\mbox{}\\[-3mm]
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\begin{itemize}
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\item the order does not matter and alpha-equivelence is preserved under
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vacuous binders \textcolor{gray}{(restriction)}\medskip
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\item the order does not matter, but the cardinality of the binders
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must be the same \textcolor{gray}{(abstraction)}\medskip
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\item the order does matter \textcolor{gray}{(iterated single binders)}
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\end{itemize}
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\onslide<2->{
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\begin{center}
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\isacommand{bind (set+)}\hspace{6mm}
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\isacommand{bind (set)}\hspace{6mm}
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\isacommand{bind}
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\end{center}}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}<1-3>
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\frametitle{\begin{tabular}{c}Specification of Binding\end{tabular}}
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\mbox{}\\[-6mm]
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\mbox{}\hspace{10mm}
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\begin{tabular}{ll}
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\multicolumn{2}{l}{\isacommand{nominal\_datatype} trm $=$}\\
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\hspace{5mm}\phantom{$|$} Var name\\
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\hspace{5mm}$|$ App trm trm\\
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\hspace{5mm}$|$ Lam \only<2->{x::}name \only<2->{t::}trm
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& \onslide<2->{\isacommand{bind} x \isacommand{in} t}\\
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\hspace{5mm}$|$ Let \only<2->{as::}assns \only<2->{t::}trm
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& \onslide<2->{\isacommand{bind} bn(as) \isacommand{in} t}\\
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\multicolumn{2}{l}{\isacommand{and} assns $=$}\\
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\multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} ANil}\\
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\multicolumn{2}{l}{\hspace{5mm}$|$ ACons name trm assns}\\
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\multicolumn{2}{l}{\onslide<3->{\isacommand{binder} bn \isacommand{where}}}\\
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\multicolumn{2}{l}{\onslide<3->{\hspace{5mm}\phantom{$|$} bn(ANil) $=$ []}}\\
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\multicolumn{2}{l}{\onslide<3->{\hspace{5mm}$|$ bn(ACons a t as) $=$ [a] @ bn(as)}}\\
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\end{tabular}
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\end{frame}}
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*}
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text_raw {*
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\mode<presentation>{
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\begin{frame}<1-2,4-8>
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\frametitle{\begin{tabular}{c}Alpha-Equivalence\end{tabular}}
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\mbox{}\\[-3mm]
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\begin{itemize}
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\item lets first look at pairs\bigskip\medskip
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\textcolor{blue}{\begin{tabular}{@ {\hspace{1cm}}l}
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$(as, x) \onslide<2->{\approx\!}\makebox[5mm][l]{\only<2-6>{${}_{\text{set}}$}%
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\only<7>{${}_{\text{\alert{list}}}$}%
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\only<8>{${}_{\text{\alert{set+}}}$}}%
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\,\onslide<2->{(bs,y)}$
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\end{tabular}}\bigskip
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\end{itemize}
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\only<1>{
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\begin{textblock}{8}(3,8.5)
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\begin{tabular}{l@ {\hspace{2mm}}p{8cm}}
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& \textcolor{blue}{$as$} is a set of names\ldots the binders\\
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& \textcolor{blue}{$x$} is the body (might be a tuple)\\
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& \textcolor{blue}{$\approx_{\text{set}}$} is where the cardinality
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of the binders has to be the same\\
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\end{tabular}
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\end{textblock}}
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\only<4->{
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\begin{textblock}{12}(5,8)
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\textcolor{blue}{
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\begin{tabular}{ll@ {\hspace{1mm}}l}
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$\dn$ & \onslide<5->{$\exists \pi.\,$} & $\text{fv}(x) - as = \text{fv}(y) - bs$\\[1mm]
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& \onslide<5->{$\;\;\;\wedge$} & \onslide<5->{$\text{fv}(x) - as \fresh^* \pi$}\\[1mm]
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& \onslide<5->{$\;\;\;\wedge$} & \onslide<5->{$(\pi \act x) = y$}\\[1mm]
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& \only<6-7>{$\;\;\;\wedge$}\only<8>{\textcolor{gray}{\xout{$\;\;\;\wedge$}}} &
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\only<6-7>{$\pi \act as = bs$}\only<8>{\textcolor{gray}{\xout{$\pi \act as = bs$}}}\\
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\end{tabular}}
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\end{textblock}}
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\only<7>{
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\begin{textblock}{7}(3,13.8)
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\footnotesize $^*$ $as$ and $bs$ are \alert{lists} of names
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\end{textblock}}
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\end{frame}}
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*}
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text_raw {*
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\mode<presentation>{
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\begin{frame}<1-3>
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\frametitle{\begin{tabular}{c}Examples\end{tabular}}
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\mbox{}\\[-3mm]
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\begin{itemize}
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\item lets look at type-schemes:\medskip\medskip
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\begin{center}
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\textcolor{blue}{$(as, x) \approx\!\makebox[5mm][l]{${}_{\text{set}}$} (bs, y)$}
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\end{center}\medskip
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\onslide<2->{
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\begin{center}
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\textcolor{blue}{
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\begin{tabular}{l}
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$\text{fv}(x) = \{x\}$\\[1mm]
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$\text{fv}(T_1 \rightarrow T_2) = \text{fv}(T_1) \cup \text{fv}(T_2)$\\
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\end{tabular}}
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\end{center}}
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\end{itemize}
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\only<3->{
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\begin{textblock}{4}(0.3,12)
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\begin{tikzpicture}
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\draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
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{\tiny\color{darkgray}
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\begin{minipage}{3.4cm}\raggedright
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\begin{tabular}{r@ {\hspace{1mm}}l}
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\multicolumn{2}{@ {}l}{set+:}\\
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$\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
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$\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
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$\wedge$ & $\pi \cdot x = y$\\
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\\
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\end{tabular}
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\end{minipage}};
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\end{tikzpicture}
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\end{textblock}}
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\only<3->{
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\begin{textblock}{4}(5.2,12)
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\begin{tikzpicture}
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\draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
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{\tiny\color{darkgray}
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\begin{minipage}{3.4cm}\raggedright
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\begin{tabular}{r@ {\hspace{1mm}}l}
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\multicolumn{2}{@ {}l}{set:}\\
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$\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
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$\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
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$\wedge$ & $\pi \cdot x = y$\\
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$\wedge$ & $\pi \cdot as = bs$\\
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450 |
\end{tabular}
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\end{minipage}};
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\end{tikzpicture}
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\end{textblock}}
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\only<3->{
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\begin{textblock}{4}(10.2,12)
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\begin{tikzpicture}
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\draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
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{\tiny\color{darkgray}
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\begin{minipage}{3.4cm}\raggedright
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\begin{tabular}{r@ {\hspace{1mm}}l}
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461 |
\multicolumn{2}{@ {}l}{list:}\\
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462 |
$\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
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|
463 |
$\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
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464 |
$\wedge$ & $\pi \cdot x = y$\\
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|
465 |
$\wedge$ & $\pi \cdot as = bs$\\
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466 |
\end{tabular}
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467 |
\end{minipage}};
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\end{tikzpicture}
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\end{textblock}}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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473 |
*}
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474 |
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475 |
text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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477 |
\mode<presentation>{
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478 |
\begin{frame}<1-2>
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479 |
\frametitle{\begin{tabular}{c}Examples\end{tabular}}
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|
480 |
\mbox{}\\[-3mm]
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481 |
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|
482 |
\begin{center}
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|
483 |
\textcolor{blue}{
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484 |
\only<1>{$(\{x, y\}, x \rightarrow y) \approx_? (\{x, y\}, y \rightarrow x)$}
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485 |
\only<2>{$([x, y], x \rightarrow y) \approx_? ([x, y], y \rightarrow x)$}}
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|
486 |
\end{center}
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487 |
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|
488 |
\begin{itemize}
|
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|
489 |
\item \textcolor{blue}{$\approx_{\text{set+}}$, $\approx_{\text{set}}$%
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490 |
\only<2>{, \alert{$\not\approx_{\text{list}}$}}}
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|
491 |
\end{itemize}
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492 |
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493 |
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\only<1->{
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495 |
\begin{textblock}{4}(0.3,12)
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496 |
\begin{tikzpicture}
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\draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
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|
498 |
{\tiny\color{darkgray}
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499 |
\begin{minipage}{3.4cm}\raggedright
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500 |
\begin{tabular}{r@ {\hspace{1mm}}l}
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501 |
\multicolumn{2}{@ {}l}{set+:}\\
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|
502 |
$\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
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|
503 |
$\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
|
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|
504 |
$\wedge$ & $\pi \cdot x = y$\\
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|
505 |
\\
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|
506 |
\end{tabular}
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507 |
\end{minipage}};
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|
508 |
\end{tikzpicture}
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|
509 |
\end{textblock}}
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510 |
\only<1->{
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|
511 |
\begin{textblock}{4}(5.2,12)
|
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|
512 |
\begin{tikzpicture}
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513 |
\draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
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514 |
{\tiny\color{darkgray}
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515 |
\begin{minipage}{3.4cm}\raggedright
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516 |
\begin{tabular}{r@ {\hspace{1mm}}l}
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|
517 |
\multicolumn{2}{@ {}l}{set:}\\
|
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|
518 |
$\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
|
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|
519 |
$\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
|
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|
520 |
$\wedge$ & $\pi \cdot x = y$\\
|
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|
521 |
$\wedge$ & $\pi \cdot as = bs$\\
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|
522 |
\end{tabular}
|
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|
523 |
\end{minipage}};
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|
524 |
\end{tikzpicture}
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|
525 |
\end{textblock}}
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|
526 |
\only<1->{
|
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|
527 |
\begin{textblock}{4}(10.2,12)
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|
528 |
\begin{tikzpicture}
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|
529 |
\draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
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|
530 |
{\tiny\color{darkgray}
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|
531 |
\begin{minipage}{3.4cm}\raggedright
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|
532 |
\begin{tabular}{r@ {\hspace{1mm}}l}
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|
533 |
\multicolumn{2}{@ {}l}{list:}\\
|
|
|
534 |
$\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
|
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|
535 |
$\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
|
|
|
536 |
$\wedge$ & $\pi \cdot x = y$\\
|
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|
537 |
$\wedge$ & $\pi \cdot as = bs$\\
|
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|
538 |
\end{tabular}
|
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|
539 |
\end{minipage}};
|
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|
540 |
\end{tikzpicture}
|
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|
541 |
\end{textblock}}
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|
542 |
|
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|
543 |
\end{frame}}
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|
544 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
|
545 |
*}
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|
546 |
|
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|
547 |
text_raw {*
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|
548 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
549 |
\mode<presentation>{
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|
|
550 |
\begin{frame}<1-2>
|
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|
551 |
\frametitle{\begin{tabular}{c}Examples\end{tabular}}
|
|
|
552 |
\mbox{}\\[-3mm]
|
|
|
553 |
|
|
|
554 |
\begin{center}
|
|
|
555 |
\textcolor{blue}{\only<1>{$(\{x\}, x) \approx_? (\{x, y\}, x)$}}
|
|
|
556 |
\end{center}
|
|
|
557 |
|
|
|
558 |
\begin{itemize}
|
|
|
559 |
\item \textcolor{blue}{$\approx_{\text{set+}}$, $\not\approx_{\text{set}}$,
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|
560 |
$\not\approx_{\text{list}}$}
|
|
|
561 |
\end{itemize}
|
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|
562 |
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563 |
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564 |
\only<1->{
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565 |
\begin{textblock}{4}(0.3,12)
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|
566 |
\begin{tikzpicture}
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567 |
\draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
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568 |
{\tiny\color{darkgray}
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|
569 |
\begin{minipage}{3.4cm}\raggedright
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570 |
\begin{tabular}{r@ {\hspace{1mm}}l}
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571 |
\multicolumn{2}{@ {}l}{set+:}\\
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|
572 |
$\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
|
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|
573 |
$\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
|
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|
574 |
$\wedge$ & $\pi \cdot x = y$\\
|
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|
575 |
\\
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|
576 |
\end{tabular}
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|
577 |
\end{minipage}};
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578 |
\end{tikzpicture}
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|
579 |
\end{textblock}}
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|
580 |
\only<1->{
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581 |
\begin{textblock}{4}(5.2,12)
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|
582 |
\begin{tikzpicture}
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583 |
\draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
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|
584 |
{\tiny\color{darkgray}
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\begin{minipage}{3.4cm}\raggedright
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|
586 |
\begin{tabular}{r@ {\hspace{1mm}}l}
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|
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\multicolumn{2}{@ {}l}{set:}\\
|
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|
588 |
$\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
|
|
|
589 |
$\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
|
|
|
590 |
$\wedge$ & $\pi \cdot x = y$\\
|
|
|
591 |
$\wedge$ & $\pi \cdot as = bs$\\
|
|
|
592 |
\end{tabular}
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|
593 |
\end{minipage}};
|
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|
594 |
\end{tikzpicture}
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|
595 |
\end{textblock}}
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|
596 |
\only<1->{
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|
597 |
\begin{textblock}{4}(10.2,12)
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598 |
\begin{tikzpicture}
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|
599 |
\draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
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|
600 |
{\tiny\color{darkgray}
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|
601 |
\begin{minipage}{3.4cm}\raggedright
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|
602 |
\begin{tabular}{r@ {\hspace{1mm}}l}
|
|
|
603 |
\multicolumn{2}{@ {}l}{list:}\\
|
|
|
604 |
$\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
|
|
|
605 |
$\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
|
|
|
606 |
$\wedge$ & $\pi \cdot x = y$\\
|
|
|
607 |
$\wedge$ & $\pi \cdot as = bs$\\
|
|
|
608 |
\end{tabular}
|
|
|
609 |
\end{minipage}};
|
|
|
610 |
\end{tikzpicture}
|
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|
611 |
\end{textblock}}
|
|
|
612 |
|
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|
613 |
\only<2>{
|
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|
614 |
\begin{textblock}{6}(2.5,4)
|
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|
615 |
\begin{tikzpicture}
|
|
|
616 |
\draw (0,0) node[inner sep=5mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
|
|
|
617 |
{\normalsize
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|
|
618 |
\begin{minipage}{8cm}\raggedright
|
|
|
619 |
\begin{itemize}
|
|
|
620 |
\item \color{darkgray}$\alpha$-equivalences coincide when a single name is
|
|
|
621 |
abstracted
|
|
|
622 |
\item \color{darkgray}in that case they are equivalent to ``old-fashioned'' definitions of $\alpha$
|
|
|
623 |
\end{itemize}
|
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|
624 |
\end{minipage}};
|
|
|
625 |
\end{tikzpicture}
|
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|
626 |
\end{textblock}}
|
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|
627 |
|
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|
628 |
\end{frame}}
|
|
|
629 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
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|
630 |
*}
|
|
|
631 |
|
|
|
632 |
text_raw {*
|
|
|
633 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
634 |
\mode<presentation>{
|
|
|
635 |
\begin{frame}<1->
|
|
|
636 |
\frametitle{\begin{tabular}{c}Our Specifications\end{tabular}}
|
|
|
637 |
\mbox{}\\[-6mm]
|
|
|
638 |
|
|
|
639 |
\mbox{}\hspace{10mm}
|
|
|
640 |
\begin{tabular}{ll}
|
|
|
641 |
\multicolumn{2}{l}{\isacommand{nominal\_datatype} trm $=$}\\
|
|
|
642 |
\hspace{5mm}\phantom{$|$} Var name\\
|
|
|
643 |
\hspace{5mm}$|$ App trm trm\\
|
|
|
644 |
\hspace{5mm}$|$ Lam x::name t::trm
|
|
|
645 |
& \isacommand{bind} x \isacommand{in} t\\
|
|
|
646 |
\hspace{5mm}$|$ Let as::assns t::trm
|
|
|
647 |
& \isacommand{bind} bn(as) \isacommand{in} t\\
|
|
|
648 |
\multicolumn{2}{l}{\isacommand{and} assns $=$}\\
|
|
|
649 |
\multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} ANil}\\
|
|
|
650 |
\multicolumn{2}{l}{\hspace{5mm}$|$ ACons name trm assns}\\
|
|
|
651 |
\multicolumn{2}{l}{\isacommand{binder} bn \isacommand{where}}\\
|
|
|
652 |
\multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} bn(ANil) $=$ $[]$}\\
|
|
|
653 |
\multicolumn{2}{l}{\hspace{5mm}$|$ bn(ACons a t as) $=$ $[$a$]$ @ bn(as)}\\
|
|
|
654 |
\end{tabular}
|
|
|
655 |
|
|
|
656 |
\end{frame}}
|
|
|
657 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
658 |
*}
|
|
|
659 |
|
|
|
660 |
|
|
|
661 |
text_raw {*
|
|
|
662 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
663 |
\mode<presentation>{
|
|
|
664 |
\begin{frame}<1->[t]
|
|
|
665 |
\frametitle{\begin{tabular}{c}Binder Clauses\end{tabular}}
|
|
|
666 |
|
|
|
667 |
\begin{itemize}
|
|
|
668 |
\item We need to have a `clear scope' for a bound variable, and bound
|
|
|
669 |
variables should not be free and bound at the same time.\bigskip
|
|
|
670 |
\end{itemize}
|
|
|
671 |
|
|
|
672 |
\begin{center}
|
|
|
673 |
\only<1>{
|
|
|
674 |
\begin{tabular}{@ {\hspace{-5mm}}l}
|
|
|
675 |
\alert{\bf shallow binders}\\
|
|
|
676 |
\hspace{4mm}Lam x::name t::trm\hspace{4mm} \isacommand{bind} x \isacommand{in} t\\
|
|
|
677 |
\hspace{4mm}All xs::name set T::ty\hspace{4mm} \isacommand{bind} xs \isacommand{in} T\\
|
|
|
678 |
\hspace{4mm}Foo x::name t$_1$::trm t$_2$::trm\hspace{4mm}
|
|
|
679 |
\isacommand{bind} x \isacommand{in} t$_1$, \isacommand{bind} x \isacommand{in} t$_2$\\
|
|
|
680 |
\hspace{4mm}Bar x::name t$_1$::trm t$_2$::trm\hspace{4mm}
|
|
|
681 |
\isacommand{bind} x \isacommand{in} t$_1$ t$_2$\\
|
|
|
682 |
\end{tabular}}
|
|
|
683 |
\only<2>{
|
|
|
684 |
\begin{tabular}{@ {\hspace{-5mm}}l}
|
|
|
685 |
\alert{\bf deep binders} \\
|
|
|
686 |
\hspace{4mm}Let as::assns t::trm\hspace{4mm} \isacommand{bind} bn(as) \isacommand{in} t\\
|
|
|
687 |
\hspace{4mm}Foo as::assns t$_1$::trm t$_2$::trm\\
|
|
|
688 |
\hspace{20mm}\isacommand{bind} bn(as) \isacommand{in} t$_1$, \isacommand{bind} bn(as) \isacommand{in} t$_2$\\[4mm]
|
|
|
689 |
\makebox[0mm][l]{\alert{$\times$}}\hspace{4mm}Bar as::assns t$_1$::trm t$_2$::trm\\
|
|
|
690 |
\hspace{20mm}\isacommand{bind} bn$_1$(as) \isacommand{in} t$_1$, \isacommand{bind} bn$_2$(as) \isacommand{in} t$_2$\\
|
|
|
691 |
\end{tabular}}
|
|
|
692 |
\only<3>{
|
|
|
693 |
\begin{tabular}{@ {\hspace{-5mm}}l}
|
|
|
694 |
{\bf deep \alert{recursive} binders} \\
|
|
|
695 |
\hspace{4mm}Let\_rec as::assns t::trm\hspace{4mm} \isacommand{bind} bn(as) \isacommand{in} t as\\[4mm]
|
|
|
696 |
|
|
|
697 |
\makebox[0mm][l]{\alert{$\times$}}\hspace{4mm}Foo\_rec as::assns t$_1$::trm t$_2$::trm\hspace{4mm}\\
|
|
|
698 |
\hspace{20mm}\isacommand{bind} bn(as) \isacommand{in} t$_1$ as, \isacommand{bind} bn(as) \isacommand{in} t$_2$\\
|
|
|
699 |
|
|
|
700 |
\end{tabular}}
|
|
|
701 |
\end{center}
|
|
|
702 |
|
|
|
703 |
\end{frame}}
|
|
|
704 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
705 |
*}
|
|
|
706 |
|
|
|
707 |
text_raw {*
|
|
|
708 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
709 |
\mode<presentation>{
|
|
|
710 |
\begin{frame}<1-5>
|
|
|
711 |
\frametitle{\begin{tabular}{c}Our Work\end{tabular}}
|
|
|
712 |
\mbox{}\\[-6mm]
|
|
|
713 |
|
|
|
714 |
\begin{center}
|
|
|
715 |
\begin{tikzpicture}[scale=1.5]
|
|
|
716 |
%%%\draw[step=2mm] (-4,-1) grid (4,1);
|
|
|
717 |
|
|
|
718 |
\onslide<1>{\draw[very thick] (0.7,0.4) circle (4.25mm);}
|
|
|
719 |
\onslide<1>{\draw[rounded corners=1mm, very thick] ( 0.0,-0.8) rectangle ( 1.8, 0.9);}
|
|
|
720 |
\onslide<1->{\draw[rounded corners=1mm, very thick] (-1.95,0.85) rectangle (-2.85,-0.05);}
|
|
|
721 |
|
|
|
722 |
\onslide<1>{\draw (-2.0, 0.845) -- (0.7,0.845);}
|
|
|
723 |
\onslide<1>{\draw (-2.0,-0.045) -- (0.7,-0.045);}
|
|
|
724 |
|
|
|
725 |
\onslide<1>{\alert{\draw ( 0.7, 0.4) node {\footnotesize\begin{tabular}{@ {}c@ {}}$\alpha$-\\[-1mm]classes\end{tabular}};}}
|
|
|
726 |
\onslide<1->{\alert{\draw (-2.4, 0.4) node {\footnotesize\begin{tabular}{@ {}c@ {}}$\alpha$-eq.\\[-1mm]terms\end{tabular}};}}
|
|
|
727 |
\onslide<1>{\draw (1.8, 0.48) node[right=-0.1mm]
|
|
|
728 |
{\footnotesize\begin{tabular}{@ {}l@ {}}existing\\[-1mm] type\\ \onslide<1>{\alert{(sets of raw terms)}}\end{tabular}};}
|
|
|
729 |
\onslide<1>{\draw (0.9, -0.35) node {\footnotesize\begin{tabular}{@ {}l@ {}}non-empty\\[-1mm]subset\end{tabular}};}
|
|
|
730 |
\onslide<1->{\draw (-3.25, 0.55) node {\footnotesize\begin{tabular}{@ {}l@ {}}new\\[-1mm]type\end{tabular}};}
|
|
|
731 |
|
|
|
732 |
\onslide<1>{\draw[<->, very thick] (-1.8, 0.3) -- (-0.1,0.3);}
|
|
|
733 |
\onslide<1>{\draw (-0.95, 0.3) node[above=0mm] {\footnotesize{}isomorphism};}
|
|
|
734 |
|
|
|
735 |
\onslide<1>{\draw[->, line width=2mm, red] (-1.0,-0.4) -- (0.35,0.16);}
|
|
|
736 |
\end{tikzpicture}
|
|
|
737 |
\end{center}
|
|
|
738 |
|
|
|
739 |
\begin{textblock}{9.5}(6,3.5)
|
|
|
740 |
\begin{itemize}
|
|
|
741 |
\item<1-> defined fv and $\alpha$
|
|
|
742 |
\item<2-> built quotient / new type
|
|
|
743 |
\item<3-> derived a reasoning infrastructure ($\fresh$, distinctness, injectivity, cases,\ldots)
|
|
|
744 |
\item<4-> derive a {\bf stronger} cases lemma
|
|
|
745 |
\item<5-> from this, a {\bf stronger} induction principle (Barendregt variable convention built in)\\
|
|
|
746 |
\begin{center}
|
|
|
747 |
\textcolor{blue}{Foo ($\lambda x. \lambda y. t$) ($\lambda u. \lambda v. s$)}
|
|
|
748 |
\end{center}
|
|
|
749 |
\end{itemize}
|
|
|
750 |
\end{textblock}
|
|
|
751 |
|
|
|
752 |
|
|
|
753 |
\end{frame}}
|
|
|
754 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
755 |
*}
|
|
|
756 |
|
|
|
757 |
|
|
|
758 |
text_raw {*
|
|
|
759 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
760 |
\mode<presentation>{
|
|
|
761 |
\begin{frame}<1->
|
|
|
762 |
\frametitle{\begin{tabular}{c}Part I: Conclusion\end{tabular}}
|
|
|
763 |
\mbox{}\\[-6mm]
|
|
|
764 |
|
|
|
765 |
\begin{itemize}
|
|
|
766 |
\item the user does not see anything of the raw level\medskip
|
|
|
767 |
\only<1>{\begin{center}
|
|
|
768 |
Lam a (Var a) \alert{$=$} Lam b (Var b)
|
|
|
769 |
\end{center}\bigskip}
|
|
|
770 |
|
|
|
771 |
\item<2-> \textcolor{blue}{http://isabelle.in.tum.de/nominal/}
|
|
|
772 |
\end{itemize}
|
|
|
773 |
|
|
|
774 |
|
|
|
775 |
\end{frame}}
|
|
|
776 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
777 |
*}
|
|
|
778 |
|
|
|
779 |
text_raw {*
|
|
|
780 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
781 |
\mode<presentation>{
|
|
|
782 |
\begin{frame}<1->
|
|
|
783 |
\frametitle{\begin{tabular}{c}Part II: $\alpha\beta$-Equal Terms\end{tabular}}
|
|
|
784 |
|
|
|
785 |
\begin{itemize}
|
|
|
786 |
\item we have implemented a quotient package for Isabelle;
|
|
|
787 |
\item can now introduce the type of $\alpha\beta$-equal terms (starting
|
|
|
788 |
from $\alpha$-equal terms).
|
|
|
789 |
\item on paper this looks easy\pause\bigskip
|
|
|
790 |
\end{itemize}
|
|
|
791 |
|
|
|
792 |
\begin{center}
|
|
|
793 |
\begin{tabular}{lll}
|
|
|
794 |
\smath{x \approx_{\alpha\beta} y} & \smath{\;\not\Rightarrow\;} &
|
|
|
795 |
\smath{\text{supp}(x) = \text{supp}(y)}\\
|
|
|
796 |
& \smath{\;\not\Rightarrow\;} &
|
|
|
797 |
\smath{\text{size}(x) = \text{size}(y)}\\
|
|
|
798 |
\end{tabular}
|
|
|
799 |
\end{center}\pause
|
|
|
800 |
|
|
|
801 |
\small
|
|
|
802 |
\begin{center}
|
|
|
803 |
Andy: \smath{\;\;\text{supp}\mbox{\isasymlbrakk}x\mbox{\isasymrbrakk}_{\approx_{\alpha\beta}} =
|
|
|
804 |
{\text{\large$\bigcap$}} \{ \text{supp}(y) \;|\; y \approx_{\alpha\beta} x\}}
|
|
|
805 |
\end{center}
|
|
|
806 |
|
|
|
807 |
\end{frame}}
|
|
|
808 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
809 |
*}
|
|
|
810 |
|
|
|
811 |
text_raw {*
|
|
|
812 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
813 |
\mode<presentation>{
|
|
|
814 |
\begin{frame}[c]
|
|
|
815 |
\frametitle{}
|
|
|
816 |
|
|
|
817 |
\begin{center}
|
|
|
818 |
\begin{tabular}{rcl}
|
|
|
819 |
\smath{x\;[y := s]} & \smath{\dn} & \smath{\text{if}\;x=y\;\text{then}\;s\;\text{else}\;x}\bigskip\\
|
|
|
820 |
\smath{t_1 t_2\;[y := s]} & \smath{\dn} & \smath{t_1[y := s]\;t_2[y := s]}\bigskip\\
|
|
|
821 |
\smath{\lambda x.t\;[y := s]} & \smath{\dn} & \smath{\lambda x.\; t[y := s]}\\
|
|
|
822 |
\multicolumn{3}{r}{provided \smath{x \fresh (y, s)}}
|
|
|
823 |
\end{tabular}
|
|
|
824 |
\end{center}
|
|
|
825 |
|
|
|
826 |
\end{frame}}
|
|
|
827 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
828 |
*}
|
|
|
829 |
|
|
|
830 |
text_raw {*
|
|
|
831 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
832 |
\mode<presentation>{
|
|
|
833 |
\begin{frame}[t]
|
|
|
834 |
\frametitle{\begin{tabular}{c}Part III: Regular Languages\\[-8mm]\end{tabular}}
|
|
|
835 |
|
|
|
836 |
\begin{center}
|
|
|
837 |
\huge\bf\textcolor{gray}{in Theorem Provers}\\
|
|
|
838 |
\footnotesize\textcolor{gray}{e.g.~Isabelle, Coq, HOL4, \ldots}
|
|
|
839 |
\end{center}
|
|
|
840 |
|
|
|
841 |
\begin{itemize}
|
|
|
842 |
\item automata @{text "\<Rightarrow>"} graphs, matrices, functions
|
|
|
843 |
\item<2-> combining automata/graphs
|
|
|
844 |
|
|
|
845 |
\onslide<2->{
|
|
|
846 |
\begin{center}
|
|
|
847 |
\begin{tabular}{ccc}
|
|
|
848 |
\begin{tikzpicture}[scale=1]
|
|
|
849 |
%\draw[step=2mm] (-1,-1) grid (1,1);
|
|
|
850 |
|
|
|
851 |
\draw[rounded corners=1mm, very thick] (-1.0,-0.3) rectangle (-0.2,0.3);
|
|
|
852 |
\draw[rounded corners=1mm, very thick] ( 0.2,-0.3) rectangle ( 1.0,0.3);
|
|
|
853 |
|
|
|
854 |
\node (A) at (-1.0,0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
|
|
|
855 |
\node (B) at ( 0.2,0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
|
|
|
856 |
|
|
|
857 |
\node (C) at (-0.2, 0.13) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
|
|
|
858 |
\node (D) at (-0.2,-0.13) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
|
|
|
859 |
|
|
|
860 |
\node (E) at (1.0, 0.2) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
|
|
|
861 |
\node (F) at (1.0,-0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
|
|
|
862 |
\node (G) at (1.0,-0.2) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
|
|
|
863 |
|
|
|
864 |
\draw (-0.6,0.0) node {\small$A_1$};
|
|
|
865 |
\draw ( 0.6,0.0) node {\small$A_2$};
|
|
|
866 |
\end{tikzpicture}}
|
|
|
867 |
|
|
|
868 |
&
|
|
|
869 |
|
|
|
870 |
\onslide<3->{\raisebox{1.1mm}{\bf\Large$\;\Rightarrow\,$}}
|
|
|
871 |
|
|
|
872 |
&
|
|
|
873 |
|
|
|
874 |
\onslide<3->{\begin{tikzpicture}[scale=1]
|
|
|
875 |
%\draw[step=2mm] (-1,-1) grid (1,1);
|
|
|
876 |
|
|
|
877 |
\draw[rounded corners=1mm, very thick] (-1.0,-0.3) rectangle (-0.2,0.3);
|
|
|
878 |
\draw[rounded corners=1mm, very thick] ( 0.2,-0.3) rectangle ( 1.0,0.3);
|
|
|
879 |
|
|
|
880 |
\node (A) at (-1.0,0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
|
|
|
881 |
\node (B) at ( 0.2,0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
|
|
|
882 |
|
|
|
883 |
\node (C) at (-0.2, 0.13) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
|
|
|
884 |
\node (D) at (-0.2,-0.13) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
|
|
|
885 |
|
|
|
886 |
\node (E) at (1.0, 0.2) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
|
|
|
887 |
\node (F) at (1.0,-0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
|
|
|
888 |
\node (G) at (1.0,-0.2) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
|
|
|
889 |
|
|
|
890 |
\draw (C) to [red, very thick, bend left=45] (B);
|
|
|
891 |
\draw (D) to [red, very thick, bend right=45] (B);
|
|
|
892 |
|
|
|
893 |
\draw (-0.6,0.0) node {\small$A_1$};
|
|
|
894 |
\draw ( 0.6,0.0) node {\small$A_2$};
|
|
|
895 |
\end{tikzpicture}}
|
|
|
896 |
|
|
|
897 |
\end{tabular}
|
|
|
898 |
\end{center}\medskip
|
|
|
899 |
|
|
|
900 |
\only<4-5>{
|
|
|
901 |
\begin{tabular}{@ {\hspace{-5mm}}l@ {}}
|
|
|
902 |
disjoint union:\\[2mm]
|
|
|
903 |
\smath{A_1\uplus A_2 \dn \{(1, x)\,|\, x \in A_1\} \,\cup\, \{(2, y)\,|\, y \in A_2\}}
|
|
|
904 |
\end{tabular}}
|
|
|
905 |
\end{itemize}
|
|
|
906 |
|
|
|
907 |
\only<5>{
|
|
|
908 |
\begin{textblock}{13.9}(0.7,7.7)
|
|
|
909 |
\begin{block}{}
|
|
|
910 |
\medskip
|
|
|
911 |
\begin{minipage}{14cm}\raggedright
|
|
|
912 |
Problems with definition for regularity:\bigskip\\
|
|
|
913 |
\smath{\;\text{is\_regular}(A) \dn \exists M.\;\text{is\_dfa}(M) \wedge {\cal L} (M) = A}\bigskip
|
|
|
914 |
\end{minipage}
|
|
|
915 |
\end{block}
|
|
|
916 |
\end{textblock}}
|
|
|
917 |
\medskip
|
|
|
918 |
|
|
|
919 |
\only<6->{\underline{A solution}:\;\;use \smath{\text{nat}}s \;@{text "\<Rightarrow>"}\; state nodes\medskip}
|
|
|
920 |
|
|
|
921 |
\only<7->{You have to \alert{rename} states!}
|
|
|
922 |
|
|
|
923 |
\end{frame}}
|
|
|
924 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
925 |
*}
|
|
|
926 |
|
|
|
927 |
text_raw {*
|
|
|
928 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
929 |
\mode<presentation>{
|
|
|
930 |
\begin{frame}[t]
|
|
|
931 |
\frametitle{\normalsize Formal language theory\ldots\hfill\mbox{}}
|
|
|
932 |
\mbox{}\\[-15mm]\mbox{}
|
|
|
933 |
|
|
|
934 |
\begin{center}
|
|
|
935 |
\huge\bf\textcolor{gray}{in Theorem Provers}\\
|
|
|
936 |
\footnotesize\textcolor{gray}{e.g.~Isabelle, Coq, HOL4, \ldots}
|
|
|
937 |
\end{center}
|
|
|
938 |
|
|
|
939 |
\begin{itemize}
|
|
|
940 |
\item Kozen's ``paper'' proof of Myhill-Nerode:\\
|
|
|
941 |
\hspace{2cm}requires absence of \alert{inaccessible states}
|
|
|
942 |
\end{itemize}\bigskip\bigskip
|
|
|
943 |
|
|
|
944 |
\begin{center}
|
|
|
945 |
\smath{\;\text{is\_regular}(A) \dn \exists M.\;\text{is\_dfa}(M) \wedge {\cal L} (M) = A}
|
|
|
946 |
\end{center}
|
|
|
947 |
|
|
|
948 |
|
|
|
949 |
\end{frame}}
|
|
|
950 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
951 |
*}
|
|
|
952 |
|
|
|
953 |
text_raw {*
|
|
|
954 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
955 |
\mode<presentation>{
|
|
|
956 |
\begin{frame}[t]
|
|
|
957 |
\frametitle{}
|
|
|
958 |
\mbox{}\\[25mm]\mbox{}
|
|
|
959 |
|
|
|
960 |
\begin{textblock}{13.9}(0.7,1.2)
|
|
|
961 |
\begin{block}{}
|
|
|
962 |
\begin{minipage}{13.4cm}\raggedright
|
|
|
963 |
{\bf Definition:}\smallskip\\
|
|
|
964 |
|
|
|
965 |
A language \smath{A} is \alert{regular}, provided there exists a\\
|
|
|
966 |
\alert{regular expression} that matches all strings of \smath{A}.
|
|
|
967 |
\end{minipage}
|
|
|
968 |
\end{block}
|
|
|
969 |
\end{textblock}\pause
|
|
|
970 |
|
|
|
971 |
{\noindent\large\bf\alert{\ldots{}and forget about automata}}\bigskip\bigskip\pause
|
|
|
972 |
|
|
|
973 |
Infrastructure for free. But do we lose anything?\medskip\pause
|
|
|
974 |
|
|
|
975 |
\begin{minipage}{1.1\textwidth}
|
|
|
976 |
\begin{itemize}
|
|
|
977 |
\item pumping lemma\pause
|
|
|
978 |
\item closure under complementation\pause
|
|
|
979 |
\item \only<6>{regular expression matching}%
|
|
|
980 |
\only<7->{\soutt{regular expression matching}
|
|
|
981 |
{\footnotesize(@{text "\<Rightarrow>"}Brozowski'64, Owens et al '09)}}
|
|
|
982 |
\item<8-> most textbooks are about automata
|
|
|
983 |
\end{itemize}
|
|
|
984 |
\end{minipage}
|
|
|
985 |
|
|
|
986 |
|
|
|
987 |
\end{frame}}
|
|
|
988 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
989 |
|
|
|
990 |
*}
|
|
|
991 |
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|
992 |
text_raw {*
|
|
|
993 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
994 |
\mode<presentation>{
|
|
|
995 |
\begin{frame}[c]
|
|
|
996 |
\frametitle{\LARGE The Myhill-Nerode Theorem}
|
|
|
997 |
|
|
|
998 |
\begin{itemize}
|
|
|
999 |
\item provides necessary and suf\!ficient conditions\\ for a language
|
|
|
1000 |
being regular\\ \textcolor{gray}{(pumping lemma only necessary)}\bigskip
|
|
|
1001 |
|
|
|
1002 |
\item key is the equivalence relation:\medskip
|
|
|
1003 |
\begin{center}
|
|
|
1004 |
\smath{x \approx_{A} y \,\dn\, \forall z.\; x @ z \in A \Leftrightarrow y @ z \in A}
|
|
|
1005 |
\end{center}
|
|
|
1006 |
\end{itemize}
|
|
|
1007 |
|
|
|
1008 |
|
|
|
1009 |
\end{frame}}
|
|
|
1010 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1011 |
|
|
|
1012 |
*}
|
|
|
1013 |
|
|
|
1014 |
text_raw {*
|
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|
1015 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1016 |
\mode<presentation>{
|
|
|
1017 |
\begin{frame}[c]
|
|
|
1018 |
\frametitle{\LARGE The Myhill-Nerode Theorem}
|
|
|
1019 |
|
|
|
1020 |
\begin{center}
|
|
|
1021 |
\only<1>{%
|
|
|
1022 |
\begin{tikzpicture}[scale=3]
|
|
|
1023 |
\draw[very thick] (0.5,0.5) circle (.6cm);
|
|
|
1024 |
\end{tikzpicture}}%
|
|
|
1025 |
\only<2->{%
|
|
|
1026 |
\begin{tikzpicture}[scale=3]
|
|
|
1027 |
\draw[very thick] (0.5,0.5) circle (.6cm);
|
|
|
1028 |
\clip[draw] (0.5,0.5) circle (.6cm);
|
|
|
1029 |
\draw[step=2mm, very thick] (-1.4,-1.4) grid (1.4,1.4);
|
|
|
1030 |
\end{tikzpicture}}
|
|
|
1031 |
\end{center}
|
|
|
1032 |
|
|
|
1033 |
\begin{itemize}
|
|
|
1034 |
\item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_A) \;\Leftrightarrow\; A\; \text{is regular}}
|
|
|
1035 |
\end{itemize}
|
|
|
1036 |
|
|
|
1037 |
\begin{textblock}{5}(2.1,5.3)
|
|
|
1038 |
\begin{tikzpicture}
|
|
|
1039 |
\node at (0,0) [single arrow, fill=red,text=white, minimum height=2cm]
|
|
|
1040 |
{$U\!N\!IV$};
|
|
|
1041 |
\draw (-0.3,-1.1) node {\begin{tabular}{l}set of all\\[-1mm] strings\end{tabular}};
|
|
|
1042 |
\end{tikzpicture}
|
|
|
1043 |
\end{textblock}
|
|
|
1044 |
|
|
|
1045 |
\only<2->{%
|
|
|
1046 |
\begin{textblock}{5}(9.1,7.2)
|
|
|
1047 |
\begin{tikzpicture}
|
|
|
1048 |
\node at (0,0) [shape border rotate=180,single arrow, fill=red,text=white, minimum height=2cm]
|
|
|
1049 |
{@{text "\<lbrakk>x\<rbrakk>"}$_{\approx_{A}}$};
|
|
|
1050 |
\draw (0.9,-1.1) node {\begin{tabular}{l}an equivalence class\end{tabular}};
|
|
|
1051 |
\end{tikzpicture}
|
|
|
1052 |
\end{textblock}}
|
|
|
1053 |
|
|
|
1054 |
\only<3->{
|
|
|
1055 |
\begin{textblock}{11.9}(1.7,3)
|
|
|
1056 |
\begin{block}{}
|
|
|
1057 |
\begin{minipage}{11.4cm}\raggedright
|
|
|
1058 |
Two directions:\medskip\\
|
|
|
1059 |
\begin{tabular}{@ {}ll}
|
|
|
1060 |
1.)\;finite $\Rightarrow$ regular\\
|
|
|
1061 |
\;\;\;\smath{\text{finite}\,(U\!N\!IV /\!/ \approx_A) \Rightarrow \exists r.\;A = {\cal L}(r)}\\[3mm]
|
|
|
1062 |
2.)\;regular $\Rightarrow$ finite\\
|
|
|
1063 |
\;\;\;\smath{\text{finite}\, (U\!N\!IV /\!/ \approx_{{\cal L}(r)})}
|
|
|
1064 |
\end{tabular}
|
|
|
1065 |
|
|
|
1066 |
\end{minipage}
|
|
|
1067 |
\end{block}
|
|
|
1068 |
\end{textblock}}
|
|
|
1069 |
|
|
|
1070 |
\end{frame}}
|
|
|
1071 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1072 |
|
|
|
1073 |
*}
|
|
|
1074 |
|
|
|
1075 |
|
|
|
1076 |
|
|
|
1077 |
text_raw {*
|
|
|
1078 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1079 |
\mode<presentation>{
|
|
|
1080 |
\begin{frame}<-1>[c]
|
|
|
1081 |
\frametitle{\begin{tabular}{@ {}l}\LARGE%
|
|
|
1082 |
Transitions between Eq-Classes\end{tabular}}
|
|
|
1083 |
|
|
|
1084 |
\begin{center}
|
|
|
1085 |
\begin{tikzpicture}[scale=3]
|
|
|
1086 |
\draw[very thick] (0.5,0.5) circle (.6cm);
|
|
|
1087 |
\clip[draw] (0.5,0.5) circle (.6cm);
|
|
|
1088 |
\draw[step=2mm, very thick] (-1.4,-1.4) grid (1.4,1.4);
|
|
|
1089 |
\draw[blue, fill] (0.0, 0.6) rectangle (0.2, 0.8);
|
|
|
1090 |
\draw[blue, fill] (0.8, 0.4) rectangle (1.0, 0.6);
|
|
|
1091 |
\draw[white] (0.1,0.7) node (X) {$X$};
|
|
|
1092 |
\draw[white] (0.9,0.5) node (Y) {$Y$};
|
|
|
1093 |
\draw[blue, ->, line width = 2mm, bend left=45] (X) -- (Y);
|
|
|
1094 |
\node [inner sep=1pt,label=above:\textcolor{blue}{$c$}] at ($ (X)!.5!(Y) $) {};
|
|
|
1095 |
\end{tikzpicture}
|
|
|
1096 |
\end{center}
|
|
|
1097 |
|
|
|
1098 |
\begin{center}
|
|
|
1099 |
\smath{X \stackrel{c}{\longrightarrow} Y \;\dn\; X ; c \subseteq Y}
|
|
|
1100 |
\end{center}
|
|
|
1101 |
|
|
|
1102 |
\onslide<8>{
|
|
|
1103 |
\begin{tabular}{c}
|
|
|
1104 |
\begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick]
|
|
|
1105 |
\tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
|
|
|
1106 |
\node[state,initial] (q_0) {$R_1$};
|
|
|
1107 |
\end{tikzpicture}
|
|
|
1108 |
\end{tabular}}
|
|
|
1109 |
|
|
|
1110 |
\end{frame}}
|
|
|
1111 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1112 |
*}
|
|
|
1113 |
|
|
|
1114 |
text_raw {*
|
|
|
1115 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1116 |
\mode<presentation>{
|
|
|
1117 |
\begin{frame}[c]
|
|
|
1118 |
\frametitle{\LARGE The Other Direction}
|
|
|
1119 |
|
|
|
1120 |
One has to prove
|
|
|
1121 |
|
|
|
1122 |
\begin{center}
|
|
|
1123 |
\smath{\text{finite} (U\!N\!IV /\!/ \approx_{{\cal L}(r)})}
|
|
|
1124 |
\end{center}
|
|
|
1125 |
|
|
|
1126 |
by induction on \smath{r}. Not trivial, but after a bit
|
|
|
1127 |
of thinking, one can find a \alert{refined} relation:\bigskip
|
|
|
1128 |
|
|
|
1129 |
|
|
|
1130 |
\begin{center}
|
|
|
1131 |
\mbox{\begin{tabular}{c@ {\hspace{7mm}}c@ {\hspace{7mm}}c}
|
|
|
1132 |
\begin{tikzpicture}[scale=1.1]
|
|
|
1133 |
%Circle
|
|
|
1134 |
\draw[thick] (0,0) circle (1.1);
|
|
|
1135 |
\end{tikzpicture}
|
|
|
1136 |
&
|
|
|
1137 |
\begin{tikzpicture}[scale=1.1]
|
|
|
1138 |
%Circle
|
|
|
1139 |
\draw[thick] (0,0) circle (1.1);
|
|
|
1140 |
%Main rays
|
|
|
1141 |
\foreach \a in {0, 90,...,359}
|
|
|
1142 |
\draw[very thick] (0, 0) -- (\a:1.1);
|
|
|
1143 |
\foreach \a / \l in {45/1, 135/2, 225/3, 315/4}
|
|
|
1144 |
\draw (\a: 0.65) node {\small$a_\l$};
|
|
|
1145 |
\end{tikzpicture}
|
|
|
1146 |
&
|
|
|
1147 |
\begin{tikzpicture}[scale=1.1]
|
|
|
1148 |
%Circle
|
|
|
1149 |
\draw[red, thick] (0,0) circle (1.1);
|
|
|
1150 |
%Main rays
|
|
|
1151 |
\foreach \a in {0, 45,...,359}
|
|
|
1152 |
\draw[red, very thick] (0, 0) -- (\a:1.1);
|
|
|
1153 |
\foreach \a / \l in {22.5/1.1, 67.5/1.2, 112.5/2.1, 157.5/2.2, 202.4/3.1, 247.5/3.2, 292.5/4.1, 337.5/4.2}
|
|
|
1154 |
\draw (\a: 0.77) node {\textcolor{red}{\footnotesize$a_{\l}$}};
|
|
|
1155 |
\end{tikzpicture}\\
|
|
|
1156 |
\small\smath{U\!N\!IV} &
|
|
|
1157 |
\small\smath{U\!N\!IV /\!/ \approx_{{\cal L}(r)}} &
|
|
|
1158 |
\small\smath{U\!N\!IV /\!/ \alert{R}}
|
|
|
1159 |
\end{tabular}}
|
|
|
1160 |
\end{center}
|
|
|
1161 |
|
|
|
1162 |
\begin{textblock}{5}(9.8,2.6)
|
|
|
1163 |
\begin{tikzpicture}
|
|
|
1164 |
\node at (0,0) [shape border rotate=270,single arrow, fill=red,text=white, minimum height=0cm]{\textcolor{red}{a}};
|
|
|
1165 |
\end{tikzpicture}
|
|
|
1166 |
\end{textblock}
|
|
|
1167 |
|
|
|
1168 |
|
|
|
1169 |
\end{frame}}
|
|
|
1170 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1171 |
*}
|
|
|
1172 |
|
|
|
1173 |
text_raw {*
|
|
|
1174 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1175 |
\mode<presentation>{
|
|
|
1176 |
\begin{frame}[t]
|
|
|
1177 |
\frametitle{\LARGE\begin{tabular}{c}Derivatives of RExps\end{tabular}}
|
|
|
1178 |
|
|
|
1179 |
\begin{itemize}
|
|
|
1180 |
\item introduced by Brozowski~'64
|
|
|
1181 |
\item a regular expressions after a character has been parsed\\[-18mm]\mbox{}
|
|
|
1182 |
\end{itemize}
|
|
|
1183 |
|
|
|
1184 |
\only<1>{%
|
|
|
1185 |
\textcolor{blue}{%
|
|
|
1186 |
\begin{center}
|
|
|
1187 |
\begin{tabular}{@ {}lc@ {\hspace{3mm}}l@ {}}
|
|
|
1188 |
der c $\varnothing$ & $\dn$ & $\varnothing$\\
|
|
|
1189 |
der c [] & $\dn$ & $\varnothing$\\
|
|
|
1190 |
der c d & $\dn$ & if c $=$ d then [] else $\varnothing$\\
|
|
|
1191 |
der c ($r_1 + r_2$) & $\dn$ & (der c $r_1$) $+$ (der c $r_2$)\\
|
|
|
1192 |
der c ($r^\star$) & $\dn$ & (der c $r$) $\cdot$ $r^\star$\\
|
|
|
1193 |
der c ($r_1 \cdot r_2$) & $\dn$ & if nullable $r_1$\\
|
|
|
1194 |
& & then (der c $r_1$) $\cdot$ $r_2$ $+$ (der c $r_2$)\\
|
|
|
1195 |
& & else (der c $r_1$) $\cdot$ $r_2$\\
|
|
|
1196 |
\end{tabular}
|
|
|
1197 |
\end{center}}}
|
|
|
1198 |
\only<2>{%
|
|
|
1199 |
\textcolor{blue}{%
|
|
|
1200 |
\begin{center}
|
|
|
1201 |
\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}}
|
|
|
1202 |
pder c $\varnothing$ & $\dn$ & \alert{$\{\}$}\\
|
|
|
1203 |
pder c [] & $\dn$ & \alert{$\{\}$}\\
|
|
|
1204 |
pder c d & $\dn$ & if c $=$ d then $\{$[]$\}$ else $\{\}$\\
|
|
|
1205 |
pder c ($r_1 + r_2$) & $\dn$ & (pder c $r_1$) \alert{$\cup$} (der c $r_2$)\\
|
|
|
1206 |
pder c ($r^\star$) & $\dn$ & (pder c $r$) $\cdot$ $r^\star$\\
|
|
|
1207 |
pder c ($r_1 \cdot r_2$) & $\dn$ & if nullable $r_1$\\
|
|
|
1208 |
& & then (pder c $r_1$) $\cdot$ $r_2$ \alert{$\cup$} (pder c $r_2$)\\
|
|
|
1209 |
& & else (pder c $r_1$) $\cdot$ $r_2$\\
|
|
|
1210 |
\end{tabular}
|
|
|
1211 |
\end{center}}}
|
|
|
1212 |
|
|
|
1213 |
\only<2>{
|
|
|
1214 |
\begin{textblock}{6}(8.5,4.7)
|
|
|
1215 |
\begin{block}{}
|
|
|
1216 |
\begin{quote}
|
|
|
1217 |
\begin{minipage}{6cm}\raggedright
|
|
|
1218 |
\begin{itemize}
|
|
|
1219 |
\item partial derivatives
|
|
|
1220 |
\item by Antimirov~'95
|
|
|
1221 |
\end{itemize}
|
|
|
1222 |
\end{minipage}
|
|
|
1223 |
\end{quote}
|
|
|
1224 |
\end{block}
|
|
|
1225 |
\end{textblock}}
|
|
|
1226 |
|
|
|
1227 |
\end{frame}}
|
|
|
1228 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1229 |
*}
|
|
|
1230 |
|
|
|
1231 |
|
|
|
1232 |
text_raw {*
|
|
|
1233 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1234 |
\mode<presentation>{
|
|
|
1235 |
\begin{frame}[t]
|
|
|
1236 |
\frametitle{\LARGE Partial Derivatives}
|
|
|
1237 |
|
|
|
1238 |
\mbox{}\\[0mm]\mbox{}
|
|
|
1239 |
|
|
|
1240 |
\begin{itemize}
|
|
|
1241 |
|
|
|
1242 |
\item \alt<1>{\smath{\text{pders $x$ $r$ \mbox{$=$} pders $y$ $r$}}}
|
|
|
1243 |
{\smath{\underbrace{\text{pders $x$ $r$ \mbox{$=$} pders $y$ $r$}}_{R}}}
|
|
|
1244 |
refines \textcolor{blue}{$x$ $\approx_{{\cal L}(r)}$ $y$}\\[16mm]\pause
|
|
|
1245 |
\item \smath{\text{finite} (U\!N\!IV /\!/ R)} \bigskip\pause
|
|
|
1246 |
\item Therefore \smath{\text{finite} (U\!N\!IV /\!/ \approx_{{\cal L}(r)})}. Qed.
|
|
|
1247 |
\end{itemize}
|
|
|
1248 |
|
|
|
1249 |
\only<2->{%
|
|
|
1250 |
\begin{textblock}{5}(3.9,7.2)
|
|
|
1251 |
\begin{tikzpicture}
|
|
|
1252 |
\node at (0,0) [shape border rotate=270,single arrow, fill=red,text=white, minimum height=0cm]{\textcolor{red}{a}};
|
|
|
1253 |
\draw (2.2,0) node {Antimirov '95};
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\end{tikzpicture}
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1255 |
\end{textblock}}
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1256 |
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1257 |
\end{frame}}
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1258 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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1259 |
*}
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1260 |
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1261 |
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1262 |
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text_raw {*
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1264 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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1265 |
\mode<presentation>{
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1266 |
\begin{frame}[t]
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1267 |
\frametitle{\LARGE What Have We Achieved?}
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1268 |
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1269 |
\begin{itemize}
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1270 |
\item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_A) \;\Leftrightarrow\; A\; \text{is regular}}
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1271 |
\medskip\pause
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1272 |
\item regular languages are closed under complementation; this is now easy
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1273 |
\begin{center}
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1274 |
\smath{U\!N\!IV /\!/ \approx_A \;\;=\;\; U\!N\!IV /\!/ \approx_{\overline{A}}}
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1275 |
\end{center}\pause\medskip
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1276 |
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1277 |
\item non-regularity (\smath{a^nb^n})\medskip\pause\pause
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1278 |
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1279 |
\item take \alert{\bf any} language; build the language of substrings\\
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1280 |
\pause
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1281 |
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1282 |
then this language \alert{\bf is} regular\;\; (\smath{a^nb^n} $\Rightarrow$ \smath{a^\star{}b^\star})
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1283 |
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|
1284 |
\end{itemize}
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1285 |
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1286 |
\only<2>{
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1287 |
\begin{textblock}{10}(4,14)
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1288 |
\small
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1289 |
\smath{x \approx_{A} y \,\dn\, \forall z.\; x @ z \in A \Leftrightarrow y @ z \in A}
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1290 |
\end{textblock}}
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1291 |
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1292 |
\only<4>{
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1293 |
\begin{textblock}{5}(2,8.6)
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1294 |
\begin{minipage}{8.8cm}
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1295 |
\begin{block}{}
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1296 |
\begin{minipage}{8.6cm}
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|
1297 |
If there exists a sufficiently large set \smath{B} (for example infinitely large),
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|
1298 |
such that
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|
1299 |
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|
1300 |
\begin{center}
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|
1301 |
\smath{\forall x,y \in B.\; x \not= y \;\Rightarrow\; x \not\approx_{A} y}.
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1302 |
\end{center}
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1303 |
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|
1304 |
then \smath{A} is not regular.\hspace{1.3cm}\small(\smath{B \dn \bigcup_n a^n})
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1305 |
\end{minipage}
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|
1306 |
\end{block}
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|
1307 |
\end{minipage}
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1308 |
\end{textblock}
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1309 |
}
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1310 |
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1311 |
\end{frame}}
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|
1312 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
1313 |
*}
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1314 |
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1315 |
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1316 |
text_raw {*
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1317 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
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|
1318 |
\mode<presentation>{
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|
1319 |
\begin{frame}[b]
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|
1320 |
\frametitle{\mbox{}\\[2cm]\textcolor{red}{Thank you!\\[5mm]Questions?}}
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1321 |
|
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|
1322 |
\end{frame}}
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|
1323 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
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|
1324 |
*}
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|
1325 |
|
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|
1326 |
|
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|
1327 |
|
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|
1328 |
|
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|
1329 |
text_raw {*
|
|
|
1330 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1331 |
\mode<presentation>{
|
|
|
1332 |
\begin{frame}<1-2>[c]
|
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|
1333 |
\frametitle{\begin{tabular}{c}Examples\end{tabular}}
|
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|
1334 |
\mbox{}\\[-6mm]
|
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|
1335 |
|
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|
1336 |
\textcolor{blue}{
|
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|
1337 |
\begin{center}
|
|
|
1338 |
$(\{a,b\}, a \rightarrow b) \approx_\alpha (\{a, b\}, a \rightarrow b)$
|
|
|
1339 |
$(\{a,b\}, a \rightarrow b) \approx_\alpha (\{a, b\}, b \rightarrow a)$
|
|
|
1340 |
\end{center}}
|
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|
1341 |
|
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|
1342 |
\textcolor{blue}{
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|
1343 |
\begin{center}
|
|
|
1344 |
$(\{a,b\}, (a \rightarrow b, a \rightarrow b))$\\
|
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|
1345 |
\hspace{17mm}$\not\approx_\alpha (\{a, b\}, (a \rightarrow b, b \rightarrow a))$
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|
1346 |
\end{center}}
|
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|
1347 |
|
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|
1348 |
\onslide<2->
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|
1349 |
{1.) \hspace{3mm}\isacommand{bind (set)} as \isacommand{in} $\tau_1$,
|
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|
1350 |
\isacommand{bind (set)} as \isacommand{in} $\tau_2$\medskip
|
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|
1351 |
|
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|
1352 |
2.) \hspace{3mm}\isacommand{bind (set)} as \isacommand{in} $\tau_1$ $\tau_2$
|
|
|
1353 |
}
|
|
|
1354 |
|
|
|
1355 |
\end{frame}}
|
|
|
1356 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1357 |
*}
|
|
|
1358 |
|
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|
1359 |
|
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|
1360 |
|
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|
1361 |
(*<*)
|
|
|
1362 |
end
|
|
|
1363 |
(*>*) |