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2351
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(*<*)
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theory Slides3
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imports "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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\renewcommand{\slidecaption}{UNIF, Edinburgh, 14.~July 2010}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}<1>[c]
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\frametitle{Quiz}
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Assuming that \smath{a} and \smath{b} are distinct variables,\\
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is it possible to find $\lambda$-terms \smath{M_1} to \smath{M_7}
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that make the following pairs \alert{$\alpha$-equivalent}?
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\begin{tabular}{@ {\hspace{14mm}}p{12cm}}
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\begin{itemize}
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\item \smath{\lambda a.\lambda b. (M_1\,b)\;} and
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\smath{\lambda b.\lambda a. (a\,M_1)\;}
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\item \smath{\lambda a.\lambda b. (M_2\,b)\;} and
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\smath{\lambda b.\lambda a. (a\,M_3)\;}
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\item \smath{\lambda a.\lambda b. (b\,M_4)\;} and
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\smath{\lambda b.\lambda a. (a\,M_5)\;}
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\item \smath{\lambda a.\lambda b. (b\,M_6)\;} and
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\smath{\lambda a.\lambda a. (a\,M_7)\;}
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\end{itemize}
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\end{tabular}
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If there is one solution for a pair, can you describe all its solutions?
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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2351
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text_raw {*
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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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\huge Nominal Unification\\[-2mm]
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\Large Hitting a Sweet Spot\\[5mm]
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2351
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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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\small initial work with Andy Pitts and Jamie Gabbay\\[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}<1-2>
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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 old Nominal 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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\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}<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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$\forall\{x, y\}.\, x \rightarrow y \;\;\approx_\alpha\;\; \forall\{y, x\}.\, y \rightarrow x$
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\bigskip\smallskip\\
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\onslide<2->{%
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$\forall\{x, y\}.\, x \rightarrow y \;\;\not\approx_\alpha\;\; \forall\{z\}.\, z \rightarrow z$
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}\bigskip\smallskip\\
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\onslide<3->{%
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$\forall\{x\}.\, x \rightarrow y \;\;\approx_\alpha\;\; \forall\{x, \alert{z}\}.\, x \rightarrow y$
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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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\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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*}
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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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\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
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\end{itemize}
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\onslide<2->{
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\begin{center}
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\isacommand{bind\_res}\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::}assn \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} assn $=$}\\
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\multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} ANil}\\
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\multicolumn{2}{l}{\hspace{5mm}$|$ ACons name trm assn}\\
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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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*}
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}<1-5>
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\frametitle{\begin{tabular}{c}Inspiration from Ott\end{tabular}}
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\mbox{}\\[-3mm]
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\begin{itemize}
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\item this way of specifying binding is inspired by
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Ott\onslide<2->{, \alert{\bf but} we made adjustments:}\medskip
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\only<2>{
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\begin{itemize}
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\item Ott allows specifications like\smallskip
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\begin{center}
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$t ::= t\;t\; |\;\lambda x.t$
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\end{center}
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\end{itemize}}
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\only<3-4>{
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\begin{itemize}
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\item whether something is bound can depend in Ott on other bound things\smallskip
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\begin{center}
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\begin{tikzpicture}
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\node (A) at (-0.5,1) {Foo $(\lambda y. \lambda x. t)$};
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\node (B) at ( 1.1,1) {$s$};
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\onslide<4>{\node (C) at (0.5,0) {$\{y, x\}$};}
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\onslide<4>{\draw[->,red,line width=1mm] (A) -- (C);}
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\onslide<4>{\draw[->,red,line width=1mm] (C) -- (B);}
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\end{tikzpicture}
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\end{center}
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\onslide<4>{this might make sense for ``raw'' terms, but not at all
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for $\alpha$-equated terms}
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\end{itemize}}
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\only<5>{
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\begin{itemize}
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\item we allow multiple binders and bodies\smallskip
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\begin{center}
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\isacommand{bind} a b c \isacommand{in} x y z
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\end{center}\bigskip\medskip
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the reason is that in general
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\begin{center}
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\begin{tabular}{rcl}
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\isacommand{bind\_res} as \isacommand{in} x y & $\not\Leftrightarrow$ &
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\begin{tabular}{@ {}l}
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\isacommand{bind\_res} as \isacommand{in} x,\\
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\isacommand{bind\_res} as \isacommand{in} y
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\end{tabular}
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\end{tabular}
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\end{center}\smallskip
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same with \isacommand{bind\_set}
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\end{itemize}}
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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>
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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 in old Nominal, we represented single binders as partial functions:\bigskip
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\begin{center}
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\begin{tabular}{l}
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Lam [$a$].\,$t$ $\;{^\text{``}}\!\dn{}\!^{\text{''}}$\\[2mm]
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\;\;\;\;$\lambda b.$\;$\text{if}\;a = b\;\text{then}\;t\;\text{else}$\\
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\phantom{\;\;\;\;$\lambda b.$\;\;\;\;\;\;}$\text{if}\;b \fresh t\;
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\text{then}\;(a\;b)\act t\;\text{else}\;\text{error}$
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\end{tabular}
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\end{center}
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\end{itemize}
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\begin{textblock}{10}(2,14)
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\footnotesize $^*$ alpha-equality coincides with equality on functions
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\end{textblock}
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\end{frame}}
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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}New Design\end{tabular}}
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\mbox{}\\[4mm]
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\begin{center}
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\alt<2>
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{\draw (0,0) node[inner sep=3mm, ultra thick, draw=red, rounded corners=2mm]
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(A) {\textcolor{red}{\begin{minipage}{1.1cm}bind.\\spec.\end{minipage}}};}
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{\draw (0,0) node[inner sep=3mm, ultra thick, draw=white, rounded corners=2mm]
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(A) {\begin{minipage}{1.1cm}bind.\\spec.\end{minipage}};}
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\alt<3>
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{\draw (3,0) node[inner sep=3mm, ultra thick, draw=red, rounded corners=2mm]
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(B) {\textcolor{red}{\begin{minipage}{1.1cm}raw\\terms\end{minipage}}};}
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{\draw (3,0) node[inner sep=3mm, ultra thick, draw=white, rounded corners=2mm]
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(B) {\begin{minipage}{1.1cm}raw\\terms\end{minipage}};}
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\alt<4>
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{\draw (6,0) node[inner sep=3mm, ultra thick, draw=red, rounded corners=2mm]
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(C) {\textcolor{red}{\begin{minipage}{1.1cm}$\alpha$-\\equiv.\end{minipage}}};}
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{\draw (6,0) node[inner sep=3mm, ultra thick, draw=white, rounded corners=2mm]
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(C) {\begin{minipage}{1.1cm}$\alpha$-\\equiv.\end{minipage}};}
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\alt<5>
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{\draw (0,-3) node[inner sep=3mm, ultra thick, draw=red, rounded corners=2mm]
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(D) {\textcolor{red}{\begin{minipage}{1.1cm}quot.\\type\end{minipage}}};}
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{\draw (0,-3) node[inner sep=3mm, ultra thick, draw=white, rounded corners=2mm]
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(D) {\begin{minipage}{1.1cm}quot.\\type\end{minipage}};}
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{\draw (3,-3) node[inner sep=3mm, ultra thick, draw=red, rounded corners=2mm]
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(E) {\textcolor{red}{\begin{minipage}{1.1cm}lift\\thms\end{minipage}}};}
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{\draw (3,-3) node[inner sep=3mm, ultra thick, draw=white, rounded corners=2mm]
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(E) {\begin{minipage}{1.1cm}lift\\thms\end{minipage}};}
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\alt<7>
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{\draw (6,-3) node[inner sep=3mm, ultra thick, draw=red, rounded corners=2mm]
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(F) {\textcolor{red}{\begin{minipage}{1.1cm}add.\\thms\end{minipage}}};}
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{\draw (6,-3) node[inner sep=3mm, ultra thick, draw=white, rounded corners=2mm]
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(F) {\begin{minipage}{1.1cm}add.\\thms\end{minipage}};}
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\draw[->,white!50,line width=1mm] (A) -- (B);
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\draw[->,white!50,line width=1mm] (B) -- (C);
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\draw[->,white!50,line width=1mm, line join=round, rounded corners=2mm]
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(C) -- (8,0) -- (8,-1.5) -- (-2,-1.5) -- (-2,-3) -- (D);
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\draw[->,white!50,line width=1mm] (D) -- (E);
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\draw[->,white!50,line width=1mm] (E) -- (F);
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\end{tikzpicture}
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\end{center}
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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}<1-9>
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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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\begin{tabular}{@ {\hspace{1cm}}l}
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$(as, x) \onslide<2->{\approx\!}\makebox[0mm][l]{\only<2-7>{${}_{\text{set}}$}%
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\only<8>{${}_{\text{\alert{list}}}$}%
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\only<9>{${}_{\text{\alert{res}}}$}}%
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\onslide<3->{^{R,\text{fv}}}\,\onslide<2->{(bs,y)}$
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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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& $as$ is a set of atoms\ldots the binders\\
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& $x$ is the body\\
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& $\approx_{\text{set}}$ 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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\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<6->{$\;\;\;\wedge$} & \onslide<6->{$(\pi \act x)\;R\;y$}\\[1mm]
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& \onslide<7-8>{$\;\;\;\wedge$} & \onslide<7-8>{$\pi \act as = bs$}\\
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\end{tabular}
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\end{textblock}}
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\only<8>{
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\begin{textblock}{8}(3,13.8)
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\footnotesize $^*$ $as$ and $bs$ are \alert{lists} of atoms
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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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456 |
\mode<presentation>{
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457 |
\begin{frame}<1-2>
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\frametitle{\begin{tabular}{c}Examples\end{tabular}}
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\mbox{}\\[-3mm]
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460 |
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461 |
\begin{itemize}
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462 |
\item lets look at ``type-schemes'':\medskip\medskip
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463 |
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\begin{center}
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|
465 |
$(as, x) \approx\!\makebox[0mm][l]{${}_{\text{set}}$}\only<1>{{}^{R,\text{fv}}}\only<2->{{}^{\alert{=},\alert{\text{fv}}}} (bs, y)$
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\end{center}\medskip
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467 |
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\onslide<2->{
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469 |
\begin{center}
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470 |
\begin{tabular}{l}
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471 |
$\text{fv}(x) = \{x\}$\\[1mm]
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472 |
$\text{fv}(T_1 \rightarrow T_2) = \text{fv}(T_1) \cup \text{fv}(T_2)$\\
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473 |
\end{tabular}
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474 |
\end{center}}
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475 |
\end{itemize}
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476 |
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477 |
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\only<2->{
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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}{res:}\\
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486 |
$\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
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487 |
$\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
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|
488 |
$\wedge$ & $\pi \cdot x = y$\\
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|
489 |
\\
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490 |
\end{tabular}
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491 |
\end{minipage}};
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\end{tikzpicture}
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493 |
\end{textblock}}
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\only<2->{
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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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499 |
\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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|
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 |
$\wedge$ & $\pi \cdot as = bs$\\
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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<2->{
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511 |
\begin{textblock}{4}(10.2,12)
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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}{list:}\\
|
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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 |
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527 |
\end{frame}}
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528 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
529 |
*}
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530 |
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531 |
text_raw {*
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532 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
533 |
\mode<presentation>{
|
|
|
534 |
\begin{frame}<1-2>
|
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|
535 |
\frametitle{\begin{tabular}{c}Examples\end{tabular}}
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|
536 |
\mbox{}\\[-3mm]
|
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|
537 |
|
|
|
538 |
\begin{center}
|
|
|
539 |
\only<1>{$(\{x, y\}, x \rightarrow y) \approx_? (\{x, y\}, y \rightarrow x)$}
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|
|
540 |
\only<2>{$([x, y], x \rightarrow y) \approx_? ([x, y], y \rightarrow x)$}
|
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|
541 |
\end{center}
|
|
|
542 |
|
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|
543 |
\begin{itemize}
|
|
|
544 |
\item $\approx_{\text{res}}$, $\approx_{\text{set}}$%
|
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|
545 |
\only<2>{, \alert{$\not\approx_{\text{list}}$}}
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|
546 |
\end{itemize}
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547 |
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548 |
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\only<1->{
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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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555 |
\begin{tabular}{r@ {\hspace{1mm}}l}
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\multicolumn{2}{@ {}l}{res:}\\
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|
557 |
$\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
|
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|
558 |
$\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
|
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|
559 |
$\wedge$ & $\pi \cdot x = y$\\
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|
560 |
\\
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561 |
\end{tabular}
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562 |
\end{minipage}};
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\end{tikzpicture}
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\end{textblock}}
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\only<1->{
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566 |
\begin{textblock}{4}(5.2,12)
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\begin{tikzpicture}
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568 |
\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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573 |
$\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
|
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|
574 |
$\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
|
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|
575 |
$\wedge$ & $\pi \cdot x = y$\\
|
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|
576 |
$\wedge$ & $\pi \cdot as = bs$\\
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|
577 |
\end{tabular}
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|
578 |
\end{minipage}};
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579 |
\end{tikzpicture}
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|
580 |
\end{textblock}}
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581 |
\only<1->{
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582 |
\begin{textblock}{4}(10.2,12)
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583 |
\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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|
588 |
\multicolumn{2}{@ {}l}{list:}\\
|
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|
589 |
$\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
|
|
|
590 |
$\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
|
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|
591 |
$\wedge$ & $\pi \cdot x = y$\\
|
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|
592 |
$\wedge$ & $\pi \cdot as = bs$\\
|
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|
593 |
\end{tabular}
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594 |
\end{minipage}};
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\end{tikzpicture}
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\end{textblock}}
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\end{frame}}
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599 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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600 |
*}
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601 |
|
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602 |
text_raw {*
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603 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
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|
604 |
\mode<presentation>{
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|
605 |
\begin{frame}<1>
|
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|
606 |
\frametitle{\begin{tabular}{c}Examples\end{tabular}}
|
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|
607 |
\mbox{}\\[-3mm]
|
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|
608 |
|
|
|
609 |
\begin{center}
|
|
|
610 |
\only<1>{$(\{x\}, x) \approx_? (\{x, y\}, x)$}
|
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|
611 |
\end{center}
|
|
|
612 |
|
|
|
613 |
\begin{itemize}
|
|
|
614 |
\item $\approx_{\text{res}}$, $\not\approx_{\text{set}}$,
|
|
|
615 |
$\not\approx_{\text{list}}$
|
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|
616 |
\end{itemize}
|
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617 |
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618 |
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619 |
\only<1->{
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\begin{textblock}{4}(0.3,12)
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621 |
\begin{tikzpicture}
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622 |
\draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
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623 |
{\tiny\color{darkgray}
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624 |
\begin{minipage}{3.4cm}\raggedright
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625 |
\begin{tabular}{r@ {\hspace{1mm}}l}
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626 |
\multicolumn{2}{@ {}l}{res:}\\
|
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|
627 |
$\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
|
|
|
628 |
$\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
|
|
|
629 |
$\wedge$ & $\pi \cdot x = y$\\
|
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|
630 |
\\
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|
631 |
\end{tabular}
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632 |
\end{minipage}};
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\end{tikzpicture}
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634 |
\end{textblock}}
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\only<1->{
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\begin{textblock}{4}(5.2,12)
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\begin{tikzpicture}
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638 |
\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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640 |
\begin{minipage}{3.4cm}\raggedright
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641 |
\begin{tabular}{r@ {\hspace{1mm}}l}
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642 |
\multicolumn{2}{@ {}l}{set:}\\
|
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|
643 |
$\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
|
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|
644 |
$\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
|
|
|
645 |
$\wedge$ & $\pi \cdot x = y$\\
|
|
|
646 |
$\wedge$ & $\pi \cdot as = bs$\\
|
|
|
647 |
\end{tabular}
|
|
|
648 |
\end{minipage}};
|
|
|
649 |
\end{tikzpicture}
|
|
|
650 |
\end{textblock}}
|
|
|
651 |
\only<1->{
|
|
|
652 |
\begin{textblock}{4}(10.2,12)
|
|
|
653 |
\begin{tikzpicture}
|
|
|
654 |
\draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
|
|
|
655 |
{\tiny\color{darkgray}
|
|
|
656 |
\begin{minipage}{3.4cm}\raggedright
|
|
|
657 |
\begin{tabular}{r@ {\hspace{1mm}}l}
|
|
|
658 |
\multicolumn{2}{@ {}l}{list:}\\
|
|
|
659 |
$\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
|
|
|
660 |
$\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
|
|
|
661 |
$\wedge$ & $\pi \cdot x = y$\\
|
|
|
662 |
$\wedge$ & $\pi \cdot as = bs$\\
|
|
|
663 |
\end{tabular}
|
|
|
664 |
\end{minipage}};
|
|
|
665 |
\end{tikzpicture}
|
|
|
666 |
\end{textblock}}
|
|
|
667 |
|
|
|
668 |
\end{frame}}
|
|
|
669 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
670 |
*}
|
|
|
671 |
|
|
|
672 |
text_raw {*
|
|
|
673 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
674 |
\mode<presentation>{
|
|
|
675 |
\begin{frame}<1-3>
|
|
|
676 |
\frametitle{\begin{tabular}{c}General Abstractions\end{tabular}}
|
|
|
677 |
\mbox{}\\[-7mm]
|
|
|
678 |
|
|
|
679 |
\begin{itemize}
|
|
|
680 |
\item we take $(as, x) \approx\!\makebox[0mm][l]{${}_{\star}$}^{=,\text{supp}} (bs, y)$\medskip
|
|
|
681 |
\item they are equivalence relations\medskip
|
|
|
682 |
\item we can therefore use the quotient package to introduce the
|
|
|
683 |
types $\beta\;\text{abs}_\star$\bigskip
|
|
|
684 |
\begin{center}
|
|
|
685 |
\only<1>{$[as].\,x$}
|
|
|
686 |
\only<2>{$\text{supp}([as].x) = \text{supp}(x) - as$}
|
|
|
687 |
\only<3>{%
|
|
|
688 |
\begin{tabular}{r@ {\hspace{1mm}}l}
|
|
|
689 |
\multicolumn{2}{@ {\hspace{-7mm}}l}{$[as]. x \alert{=} [bs].y\;\;\;\text{if\!f}$}\\[2mm]
|
|
|
690 |
$\exists \pi.$ & $\text{supp}(x) - as = \text{supp}(y) - bs$\\
|
|
|
691 |
$\wedge$ & $\text{supp}(x) - as \fresh^* \pi$\\
|
|
|
692 |
$\wedge$ & $\pi \act x = y $\\
|
|
|
693 |
$(\wedge$ & $\pi \act as = bs)\;^\star$\\
|
|
|
694 |
\end{tabular}}
|
|
|
695 |
\end{center}
|
|
|
696 |
\end{itemize}
|
|
|
697 |
|
|
|
698 |
|
|
|
699 |
\end{frame}}
|
|
|
700 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
701 |
*}
|
|
|
702 |
|
|
|
703 |
text_raw {*
|
|
|
704 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
705 |
\mode<presentation>{
|
|
|
706 |
\begin{frame}<1>
|
|
|
707 |
\frametitle{\begin{tabular}{c}One Problem\end{tabular}}
|
|
|
708 |
\mbox{}\\[-3mm]
|
|
|
709 |
|
|
|
710 |
\begin{center}
|
|
|
711 |
$\text{let}\;x_1=t_1 \ldots x_n=t_n\;\text{in}\;s$
|
|
|
712 |
\end{center}
|
|
|
713 |
|
|
|
714 |
\begin{itemize}
|
|
|
715 |
\item we cannot represent this as\medskip
|
|
|
716 |
\begin{center}
|
|
|
717 |
$\text{let}\;[x_1,\ldots,x_n]\alert{.}s\;\;[t_1,\ldots,t_n]$
|
|
|
718 |
\end{center}\bigskip
|
|
|
719 |
|
|
|
720 |
because\medskip
|
|
|
721 |
\begin{center}
|
|
|
722 |
$\text{let}\;[x].s\;\;[t_1,t_2]$
|
|
|
723 |
\end{center}
|
|
|
724 |
\end{itemize}
|
|
|
725 |
|
|
|
726 |
|
|
|
727 |
\end{frame}}
|
|
|
728 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
729 |
*}
|
|
|
730 |
|
|
|
731 |
text_raw {*
|
|
|
732 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
733 |
\mode<presentation>{
|
|
|
734 |
\begin{frame}<1->
|
|
|
735 |
\frametitle{\begin{tabular}{c}Our Specifications\end{tabular}}
|
|
|
736 |
\mbox{}\\[-6mm]
|
|
|
737 |
|
|
|
738 |
\mbox{}\hspace{10mm}
|
|
|
739 |
\begin{tabular}{ll}
|
|
|
740 |
\multicolumn{2}{l}{\isacommand{nominal\_datatype} trm $=$}\\
|
|
|
741 |
\hspace{5mm}\phantom{$|$} Var name\\
|
|
|
742 |
\hspace{5mm}$|$ App trm trm\\
|
|
|
743 |
\hspace{5mm}$|$ Lam x::name t::trm
|
|
|
744 |
& \isacommand{bind} x \isacommand{in} t\\
|
|
|
745 |
\hspace{5mm}$|$ Let as::assn t::trm
|
|
|
746 |
& \isacommand{bind} bn(as) \isacommand{in} t\\
|
|
|
747 |
\multicolumn{2}{l}{\isacommand{and} assn $=$}\\
|
|
|
748 |
\multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} ANil}\\
|
|
|
749 |
\multicolumn{2}{l}{\hspace{5mm}$|$ ACons name trm assn}\\
|
|
|
750 |
\multicolumn{2}{l}{\isacommand{binder} bn \isacommand{where}}\\
|
|
|
751 |
\multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} bn(ANil) $=$ $[]$}\\
|
|
|
752 |
\multicolumn{2}{l}{\hspace{5mm}$|$ bn(ACons a t as) $=$ $[$a$]$ @ bn(as)}\\
|
|
|
753 |
\end{tabular}
|
|
|
754 |
|
|
|
755 |
\end{frame}}
|
|
|
756 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
757 |
*}
|
|
|
758 |
|
|
|
759 |
text_raw {*
|
|
|
760 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
761 |
\mode<presentation>{
|
|
|
762 |
\begin{frame}<1-2>
|
|
|
763 |
\frametitle{\begin{tabular}{c}``Raw'' Definitions\end{tabular}}
|
|
|
764 |
\mbox{}\\[-6mm]
|
|
|
765 |
|
|
|
766 |
\mbox{}\hspace{10mm}
|
|
|
767 |
\begin{tabular}{ll}
|
|
|
768 |
\multicolumn{2}{l}{\isacommand{datatype} trm $=$}\\
|
|
|
769 |
\hspace{5mm}\phantom{$|$} Var name\\
|
|
|
770 |
\hspace{5mm}$|$ App trm trm\\
|
|
|
771 |
\hspace{5mm}$|$ Lam name trm\\
|
|
|
772 |
\hspace{5mm}$|$ Let assn trm\\
|
|
|
773 |
\multicolumn{2}{l}{\isacommand{and} assn $=$}\\
|
|
|
774 |
\multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} ANil}\\
|
|
|
775 |
\multicolumn{2}{l}{\hspace{5mm}$|$ ACons name trm assn}\\[5mm]
|
|
|
776 |
\multicolumn{2}{l}{\isacommand{function} bn \isacommand{where}}\\
|
|
|
777 |
\multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} bn(ANil) $=$ $[]$}\\
|
|
|
778 |
\multicolumn{2}{l}{\hspace{5mm}$|$ bn(ACons a t as) $=$ $[$a$]$ @ bn(as)}\\
|
|
|
779 |
\end{tabular}
|
|
|
780 |
|
|
|
781 |
\only<2>{
|
|
|
782 |
\begin{textblock}{5}(10,5)
|
|
|
783 |
$+$ \begin{tabular}{l}automatically\\
|
|
|
784 |
generate fv's\end{tabular}
|
|
|
785 |
\end{textblock}}
|
|
|
786 |
\end{frame}}
|
|
|
787 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
788 |
*}
|
|
|
789 |
|
|
|
790 |
text_raw {*
|
|
|
791 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
792 |
\mode<presentation>{
|
|
|
793 |
\begin{frame}<1>
|
|
|
794 |
\frametitle{\begin{tabular}{c}\LARGE``Raw'' Alpha-Equivalence\end{tabular}}
|
|
|
795 |
\mbox{}\\[6mm]
|
|
|
796 |
|
|
|
797 |
\begin{center}
|
|
|
798 |
Lam x::name t::trm \hspace{10mm}\isacommand{bind} x \isacommand{in} t\\
|
|
|
799 |
\end{center}
|
|
|
800 |
|
|
|
801 |
|
|
|
802 |
\[
|
|
|
803 |
\infer[\text{Lam-}\!\approx_\alpha]
|
|
|
804 |
{\text{Lam}\;x\;t \approx_\alpha \text{Lam}\;x'\;t'}
|
|
|
805 |
{([x], t) \approx\!\makebox[0mm][l]{${}_{\text{list}}$}
|
|
|
806 |
^{\approx_\alpha,\text{fv}} ([x'], t')}
|
|
|
807 |
\]
|
|
|
808 |
|
|
|
809 |
|
|
|
810 |
\end{frame}}
|
|
|
811 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
812 |
*}
|
|
|
813 |
|
|
|
814 |
text_raw {*
|
|
|
815 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
816 |
\mode<presentation>{
|
|
|
817 |
\begin{frame}<1>
|
|
|
818 |
\frametitle{\begin{tabular}{c}\LARGE``Raw'' Alpha-Equivalence\end{tabular}}
|
|
|
819 |
\mbox{}\\[6mm]
|
|
|
820 |
|
|
|
821 |
\begin{center}
|
|
|
822 |
Lam x::name y::name t::trm s::trm \hspace{5mm}\isacommand{bind} x y \isacommand{in} t s\\
|
|
|
823 |
\end{center}
|
|
|
824 |
|
|
|
825 |
|
|
|
826 |
\[
|
|
|
827 |
\infer[\text{Lam-}\!\approx_\alpha]
|
|
|
828 |
{\text{Lam}\;x\;y\;t\;s \approx_\alpha \text{Lam}\;x'\;y'\;t'\;s'}
|
|
|
829 |
{([x, y], (t, s)) \approx\!\makebox[0mm][l]{${}_{\text{list}}$}
|
|
|
830 |
^{R, fv} ([x', y'], (t', s'))}
|
|
|
831 |
\]
|
|
|
832 |
|
|
|
833 |
\footnotesize
|
|
|
834 |
where $R =\;\approx_\alpha\times\approx_\alpha$ and $fv = \text{fv}\cup\text{fv}$
|
|
|
835 |
|
|
|
836 |
\end{frame}}
|
|
|
837 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
838 |
*}
|
|
|
839 |
|
|
|
840 |
text_raw {*
|
|
|
841 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
842 |
\mode<presentation>{
|
|
|
843 |
\begin{frame}<1-2>
|
|
|
844 |
\frametitle{\begin{tabular}{c}\LARGE``Raw'' Alpha-Equivalence\end{tabular}}
|
|
|
845 |
\mbox{}\\[6mm]
|
|
|
846 |
|
|
|
847 |
\begin{center}
|
|
|
848 |
Let as::assn t::trm \hspace{10mm}\isacommand{bind} bn(as) \isacommand{in} t\\
|
|
|
849 |
\end{center}
|
|
|
850 |
|
|
|
851 |
|
|
|
852 |
\[
|
|
|
853 |
\infer[\text{Let-}\!\approx_\alpha]
|
|
|
854 |
{\text{Let}\;as\;t \approx_\alpha \text{Let}\;as'\;t'}
|
|
|
855 |
{(\text{bn}(as), t) \approx\!\makebox[0mm][l]{${}_{\text{list}}$}
|
|
|
856 |
^{\approx_\alpha,\text{fv}} (\text{bn}(as'), t') &
|
|
|
857 |
\onslide<2>{as \approx_\alpha^{\text{bn}} as'}}
|
|
|
858 |
\]
|
|
|
859 |
|
|
|
860 |
|
|
|
861 |
\end{frame}}
|
|
|
862 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
863 |
*}
|
|
|
864 |
|
|
|
865 |
text_raw {*
|
|
|
866 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
867 |
\mode<presentation>{
|
|
|
868 |
\begin{frame}<1->
|
|
|
869 |
\frametitle{\begin{tabular}{c}\LARGE{}$\alpha$ for Binding Functions\end{tabular}}
|
|
|
870 |
\mbox{}\\[-6mm]
|
|
|
871 |
|
|
|
872 |
\mbox{}\hspace{10mm}
|
|
|
873 |
\begin{tabular}{l}
|
|
|
874 |
\ldots\\
|
|
|
875 |
\isacommand{binder} bn \isacommand{where}\\
|
|
|
876 |
\phantom{$|$} bn(ANil) $=$ $[]$\\
|
|
|
877 |
$|$ bn(ACons a t as) $=$ $[$a$]$ @ bn(as)\\
|
|
|
878 |
\end{tabular}\bigskip
|
|
|
879 |
|
|
|
880 |
\begin{center}
|
|
|
881 |
\mbox{\infer{\text{ANil} \approx_\alpha^{\text{bn}} \text{ANil}}{}}\bigskip
|
|
|
882 |
|
|
|
883 |
\mbox{\infer{\text{ACons}\;a\;t\;as \approx_\alpha^{\text{bn}} \text{ACons}\;a'\;t'\;as'}
|
|
|
884 |
{t \approx_\alpha t' & as \approx_\alpha^{bn} as'}}
|
|
|
885 |
\end{center}
|
|
|
886 |
|
|
|
887 |
|
|
|
888 |
\end{frame}}
|
|
|
889 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
890 |
*}
|
|
|
891 |
|
|
|
892 |
text_raw {*
|
|
|
893 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
894 |
\mode<presentation>{
|
|
|
895 |
\begin{frame}<1->
|
|
|
896 |
\frametitle{\begin{tabular}{c}Automatic Proofs\end{tabular}}
|
|
|
897 |
\mbox{}\\[-6mm]
|
|
|
898 |
|
|
|
899 |
\begin{itemize}
|
|
|
900 |
\item we can show that $\alpha$'s are equivalence relations\medskip
|
|
|
901 |
\item as a result we can use the quotient package to introduce the type(s)
|
|
|
902 |
of $\alpha$-equated terms
|
|
|
903 |
|
|
|
904 |
\[
|
|
|
905 |
\infer
|
|
|
906 |
{\text{Lam}\;x\;t \alert{=} \text{Lam}\;x'\;t'}
|
|
|
907 |
{\only<1>{([x], t) \approx\!\makebox[0mm][l]{${}_{\text{list}}$}
|
|
|
908 |
^{=,\text{supp}} ([x'], t')}%
|
|
|
909 |
\only<2>{[x].t = [x'].t'}}
|
|
|
910 |
\]
|
|
|
911 |
|
|
|
912 |
|
|
|
913 |
\item the properties for support are implied by the properties of $[\_].\_$
|
|
|
914 |
\item we can derive strong induction principles (almost automatic---just a matter of
|
|
|
915 |
another week or two)
|
|
|
916 |
\end{itemize}
|
|
|
917 |
|
|
|
918 |
|
|
|
919 |
\end{frame}}
|
|
|
920 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
921 |
*}
|
|
|
922 |
|
|
|
923 |
text_raw {*
|
|
|
924 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
925 |
\mode<presentation>{
|
|
|
926 |
\begin{frame}<1>[t]
|
|
|
927 |
\frametitle{\begin{tabular}{c}Runtime is Acceptable\end{tabular}}
|
|
|
928 |
\mbox{}\\[-7mm]\mbox{}
|
|
|
929 |
|
|
|
930 |
\footnotesize
|
|
|
931 |
\begin{center}
|
|
|
932 |
\begin{tikzpicture}
|
|
|
933 |
\draw (0,0) node[inner sep=2mm, ultra thick, draw=white, rounded corners=2mm]
|
|
|
934 |
(A) {\begin{minipage}{0.8cm}bind.\\spec.\end{minipage}};
|
|
|
935 |
|
|
|
936 |
\draw (2,0) node[inner sep=2mm, ultra thick, draw=white, rounded corners=2mm]
|
|
|
937 |
(B) {\begin{minipage}{0.8cm}raw\\terms\end{minipage}};
|
|
|
938 |
|
|
|
939 |
\draw (4,0) node[inner sep=2mm, ultra thick, draw=white, rounded corners=2mm]
|
|
|
940 |
(C) {\begin{minipage}{0.8cm}$\alpha$-\\equiv.\end{minipage}};
|
|
|
941 |
|
|
|
942 |
\draw (0,-2) node[inner sep=2mm, ultra thick, draw=white, rounded corners=2mm]
|
|
|
943 |
(D) {\begin{minipage}{0.8cm}quot.\\type\end{minipage}};
|
|
|
944 |
|
|
|
945 |
\draw (2,-2) node[inner sep=2mm, ultra thick, draw=white, rounded corners=2mm]
|
|
|
946 |
(E) {\begin{minipage}{0.8cm}lift\\thms\end{minipage}};
|
|
|
947 |
|
|
|
948 |
\draw (4,-2) node[inner sep=2mm, ultra thick, draw=white, rounded corners=2mm]
|
|
|
949 |
(F) {\begin{minipage}{0.8cm}add.\\thms\end{minipage}};
|
|
|
950 |
|
|
|
951 |
\draw[->,white!50,line width=1mm] (A) -- (B);
|
|
|
952 |
\draw[->,white!50,line width=1mm] (B) -- (C);
|
|
|
953 |
\draw[->,white!50,line width=1mm, line join=round, rounded corners=2mm]
|
|
|
954 |
(C) -- (5,0) -- (5,-1) -- (-1,-1) -- (-1,-2) -- (D);
|
|
|
955 |
\draw[->,white!50,line width=1mm] (D) -- (E);
|
|
|
956 |
\draw[->,white!50,line width=1mm] (E) -- (F);
|
|
|
957 |
\end{tikzpicture}
|
|
|
958 |
\end{center}
|
|
|
959 |
|
|
|
960 |
\begin{itemize}
|
|
|
961 |
\item Core Haskell: 11 types, 49 term-constructors,
|
|
|
962 |
\end{itemize}
|
|
|
963 |
|
|
|
964 |
\end{frame}}
|
|
|
965 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
966 |
*}
|
|
|
967 |
|
|
|
968 |
|
|
|
969 |
text_raw {*
|
|
|
970 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
971 |
\mode<presentation>{
|
|
|
972 |
\begin{frame}<1->
|
|
|
973 |
\frametitle{\begin{tabular}{c}Interesting Phenomenon\end{tabular}}
|
|
|
974 |
\mbox{}\\[-6mm]
|
|
|
975 |
|
|
|
976 |
\small
|
|
|
977 |
\mbox{}\hspace{10mm}
|
|
|
978 |
\begin{tabular}{ll}
|
|
|
979 |
\multicolumn{2}{l}{\isacommand{nominal\_datatype} trm $=$}\\
|
|
|
980 |
\hspace{5mm}\phantom{$|$} Var name\\
|
|
|
981 |
\hspace{5mm}$|$ App trm trm\\
|
|
|
982 |
\hspace{5mm}$|$ Lam x::name t::trm
|
|
|
983 |
& \isacommand{bind} x \isacommand{in} t\\
|
|
|
984 |
\hspace{5mm}$|$ Let as::assn t::trm
|
|
|
985 |
& \isacommand{bind} bn(as) \isacommand{in} t\\
|
|
|
986 |
\multicolumn{2}{l}{\isacommand{and} assn $=$}\\
|
|
|
987 |
\multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} ANil}\\
|
|
|
988 |
\multicolumn{2}{l}{\hspace{5mm}$|$ ACons name trm assn}\\
|
|
|
989 |
\multicolumn{2}{l}{\isacommand{binder} bn \isacommand{where}}\\
|
|
|
990 |
\multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} bn(ANil) $=$ $[]$}\\
|
|
|
991 |
\multicolumn{2}{l}{\hspace{5mm}$|$ bn(ACons a t as) $=$ $[$a$]$ @ bn(as)}\\
|
|
|
992 |
\end{tabular}\bigskip\medskip
|
|
|
993 |
|
|
|
994 |
we cannot quotient assn: ACons a \ldots $\not\approx_\alpha$ ACons b \ldots
|
|
|
995 |
|
|
|
996 |
\end{frame}}
|
|
|
997 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
998 |
*}
|
|
|
999 |
|
|
|
1000 |
text_raw {*
|
|
|
1001 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1002 |
\mode<presentation>{
|
|
|
1003 |
\begin{frame}<1->
|
|
|
1004 |
\frametitle{\begin{tabular}{c}Conclusion\end{tabular}}
|
|
|
1005 |
\mbox{}\\[-6mm]
|
|
|
1006 |
|
|
|
1007 |
\begin{itemize}
|
|
|
1008 |
\item the user does not see anything of the raw level\medskip
|
|
|
1009 |
\only<1>{\begin{center}
|
|
|
1010 |
Lam a (Var a) \alert{$=$} Lam b (Var b)
|
|
|
1011 |
\end{center}\bigskip}
|
|
|
1012 |
|
|
|
1013 |
\item<2-> we have not yet done function definitions (will come soon and
|
|
|
1014 |
we hope to make improvements over the old way there too)\medskip
|
|
|
1015 |
\item<3-> it took quite some time to get here, but it seems worthwhile
|
|
|
1016 |
(Barendregt's variable convention is unsound in general,
|
|
|
1017 |
found bugs in two paper proofs, quotient package, POPL 2011 tutorial)\medskip
|
|
|
1018 |
\end{itemize}
|
|
|
1019 |
|
|
|
1020 |
|
|
|
1021 |
\end{frame}}
|
|
|
1022 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1023 |
*}
|
|
|
1024 |
|
|
|
1025 |
(*<*)
|
|
|
1026 |
end
|
|
|
1027 |
(*>*) |