--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Slides/Slides3.thy Sat Jul 10 11:27:04 2010 +0100
@@ -0,0 +1,997 @@
+(*<*)
+theory Slides3
+imports "LaTeXsugar" "Nominal"
+begin
+
+notation (latex output)
+ set ("_") and
+ Cons ("_::/_" [66,65] 65)
+
+(*>*)
+
+
+text_raw {*
+ \renewcommand{\slidecaption}{Cambridge, 8.~June 2010}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1>[t]
+ \frametitle{%
+ \begin{tabular}{@ {\hspace{-3mm}}c@ {}}
+ \\
+ \huge Nominal 2\\[-2mm]
+ \large Or, How to Reason Conveniently with\\[-5mm]
+ \large General Bindings in Isabelle/HOL\\[5mm]
+ \end{tabular}}
+ \begin{center}
+ Christian Urban
+ \end{center}
+ \begin{center}
+ joint work with {\bf Cezary Kaliszyk}\\[0mm]
+ \end{center}
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1-2>
+ \frametitle{\begin{tabular}{c}Binding in Old Nominal\end{tabular}}
+ \mbox{}\\[-6mm]
+
+ \begin{itemize}
+ \item old Nominal provided a reasoning infrastructure for single binders\medskip
+
+ \begin{center}
+ Lam [a].(Var a)
+ \end{center}\bigskip
+
+ \item<2-> but representing
+
+ \begin{center}
+ $\forall\{a_1,\ldots,a_n\}.\; T$
+ \end{center}\medskip
+
+ with single binders and reasoning about it is a \alert{\bf major} pain;
+ take my word for it!
+ \end{itemize}
+
+ \only<1>{
+ \begin{textblock}{6}(1.5,11)
+ \small
+ for example\\
+ \begin{tabular}{l@ {\hspace{2mm}}l}
+ & a $\fresh$ Lam [a]. t\\
+ & Lam [a]. (Var a) \alert{$=$} Lam [b]. (Var b)\\
+ & Barendregt style reasoning about bound variables\\
+ \end{tabular}
+ \end{textblock}}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1-4>
+ \frametitle{\begin{tabular}{c}Binding Sets of Names\end{tabular}}
+ \mbox{}\\[-3mm]
+
+ \begin{itemize}
+ \item binding sets of names has some interesting properties:\medskip
+
+ \begin{center}
+ \begin{tabular}{l}
+ $\forall\{x, y\}.\, x \rightarrow y \;\;\approx_\alpha\;\; \forall\{y, x\}.\, y \rightarrow x$
+ \bigskip\smallskip\\
+
+ \onslide<2->{%
+ $\forall\{x, y\}.\, x \rightarrow y \;\;\not\approx_\alpha\;\; \forall\{z\}.\, z \rightarrow z$
+ }\bigskip\smallskip\\
+
+ \onslide<3->{%
+ $\forall\{x\}.\, x \rightarrow y \;\;\approx_\alpha\;\; \forall\{x, \alert{z}\}.\, x \rightarrow y$
+ }\medskip\\
+ \onslide<3->{\hspace{4cm}\small provided $z$ is fresh for the type}
+ \end{tabular}
+ \end{center}
+ \end{itemize}
+
+ \begin{textblock}{8}(2,14.5)
+ \footnotesize $^*$ $x$, $y$, $z$ are assumed to be distinct
+ \end{textblock}
+
+ \only<4>{
+ \begin{textblock}{6}(2.5,4)
+ \begin{tikzpicture}
+ \draw (0,0) node[inner sep=3mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
+ {\normalsize\color{darkgray}
+ \begin{minipage}{8cm}\raggedright
+ For type-schemes the order of bound names does not matter, and
+ alpha-equivalence is preserved under \alert{vacuous} binders.
+ \end{minipage}};
+ \end{tikzpicture}
+ \end{textblock}}
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1-3>
+ \frametitle{\begin{tabular}{c}Other Binding Modes\end{tabular}}
+ \mbox{}\\[-3mm]
+
+ \begin{itemize}
+ \item alpha-equivalence being preserved under vacuous binders is \underline{not} always
+ wanted:\bigskip\bigskip\normalsize
+
+ \begin{tabular}{@ {\hspace{-8mm}}l}
+ $\text{let}\;x = 3\;\text{and}\;y = 2\;\text{in}\;x - y\;\text{end}$\medskip\\
+ \onslide<2->{$\;\;\;\only<2>{\approx_\alpha}\only<3>{\alert{\not\approx_\alpha}}
+ \text{let}\;y = 2\;\text{and}\;x = 3\only<3->{\alert{\;\text{and}
+ \;z = \text{loop}}}\;\text{in}\;x - y\;\text{end}$}
+ \end{tabular}
+
+
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1>
+ \frametitle{\begin{tabular}{c}\LARGE{}Even Another Binding Mode\end{tabular}}
+ \mbox{}\\[-3mm]
+
+ \begin{itemize}
+ \item sometimes one wants to abstract more than one name, but the order \underline{does} matter\bigskip
+
+ \begin{center}
+ \begin{tabular}{@ {\hspace{-8mm}}l}
+ $\text{let}\;(x, y) = (3, 2)\;\text{in}\;x - y\;\text{end}$\medskip\\
+ $\;\;\;\not\approx_\alpha
+ \text{let}\;(y, x) = (3, 2)\;\text{in}\;x - y\;\text{end}$
+ \end{tabular}
+ \end{center}
+
+
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1-2>
+ \frametitle{\begin{tabular}{c}\LARGE{}Three Binding Modes\end{tabular}}
+ \mbox{}\\[-3mm]
+
+ \begin{itemize}
+ \item the order does not matter and alpha-equivelence is preserved under
+ vacuous binders \textcolor{gray}{(restriction)}\medskip
+
+ \item the order does not matter, but the cardinality of the binders
+ must be the same \textcolor{gray}{(abstraction)}\medskip
+
+ \item the order does matter
+ \end{itemize}
+
+ \onslide<2->{
+ \begin{center}
+ \isacommand{bind\_res}\hspace{6mm}
+ \isacommand{bind\_set}\hspace{6mm}
+ \isacommand{bind}
+ \end{center}}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1-3>
+ \frametitle{\begin{tabular}{c}Specification of Binding\end{tabular}}
+ \mbox{}\\[-6mm]
+
+ \mbox{}\hspace{10mm}
+ \begin{tabular}{ll}
+ \multicolumn{2}{l}{\isacommand{nominal\_datatype} trm $=$}\\
+ \hspace{5mm}\phantom{$|$} Var name\\
+ \hspace{5mm}$|$ App trm trm\\
+ \hspace{5mm}$|$ Lam \only<2->{x::}name \only<2->{t::}trm
+ & \onslide<2->{\isacommand{bind} x \isacommand{in} t}\\
+ \hspace{5mm}$|$ Let \only<2->{as::}assn \only<2->{t::}trm
+ & \onslide<2->{\isacommand{bind} bn(as) \isacommand{in} t}\\
+ \multicolumn{2}{l}{\isacommand{and} assn $=$}\\
+ \multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} ANil}\\
+ \multicolumn{2}{l}{\hspace{5mm}$|$ ACons name trm assn}\\
+ \multicolumn{2}{l}{\onslide<3->{\isacommand{binder} bn \isacommand{where}}}\\
+ \multicolumn{2}{l}{\onslide<3->{\hspace{5mm}\phantom{$|$} bn(ANil) $=$ []}}\\
+ \multicolumn{2}{l}{\onslide<3->{\hspace{5mm}$|$ bn(ACons a t as) $=$ [a] @ bn(as)}}\\
+ \end{tabular}
+
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1-5>
+ \frametitle{\begin{tabular}{c}Inspiration from Ott\end{tabular}}
+ \mbox{}\\[-3mm]
+
+ \begin{itemize}
+ \item this way of specifying binding is inspired by
+ Ott\onslide<2->{, \alert{\bf but} we made adjustments:}\medskip
+
+
+ \only<2>{
+ \begin{itemize}
+ \item Ott allows specifications like\smallskip
+ \begin{center}
+ $t ::= t\;t\; |\;\lambda x.t$
+ \end{center}
+ \end{itemize}}
+
+ \only<3-4>{
+ \begin{itemize}
+ \item whether something is bound can depend in Ott on other bound things\smallskip
+ \begin{center}
+ \begin{tikzpicture}
+ \node (A) at (-0.5,1) {Foo $(\lambda y. \lambda x. t)$};
+ \node (B) at ( 1.1,1) {$s$};
+ \onslide<4>{\node (C) at (0.5,0) {$\{y, x\}$};}
+ \onslide<4>{\draw[->,red,line width=1mm] (A) -- (C);}
+ \onslide<4>{\draw[->,red,line width=1mm] (C) -- (B);}
+ \end{tikzpicture}
+ \end{center}
+ \onslide<4>{this might make sense for ``raw'' terms, but not at all
+ for $\alpha$-equated terms}
+ \end{itemize}}
+
+ \only<5>{
+ \begin{itemize}
+ \item we allow multiple binders and bodies\smallskip
+ \begin{center}
+ \isacommand{bind} a b c \isacommand{in} x y z
+ \end{center}\bigskip\medskip
+ the reason is that in general
+ \begin{center}
+ \begin{tabular}{rcl}
+ \isacommand{bind\_res} as \isacommand{in} x y & $\not\Leftrightarrow$ &
+ \begin{tabular}{@ {}l}
+ \isacommand{bind\_res} as \isacommand{in} x,\\
+ \isacommand{bind\_res} as \isacommand{in} y
+ \end{tabular}
+ \end{tabular}
+ \end{center}\smallskip
+
+ same with \isacommand{bind\_set}
+ \end{itemize}}
+ \end{itemize}
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1>
+ \frametitle{\begin{tabular}{c}Alpha-Equivalence\end{tabular}}
+ \mbox{}\\[-3mm]
+
+ \begin{itemize}
+ \item in old Nominal, we represented single binders as partial functions:\bigskip
+
+ \begin{center}
+ \begin{tabular}{l}
+ Lam [$a$].\,$t$ $\;{^\text{``}}\!\dn{}\!^{\text{''}}$\\[2mm]
+ \;\;\;\;$\lambda b.$\;$\text{if}\;a = b\;\text{then}\;t\;\text{else}$\\
+ \phantom{\;\;\;\;$\lambda b.$\;\;\;\;\;\;}$\text{if}\;b \fresh t\;
+ \text{then}\;(a\;b)\act t\;\text{else}\;\text{error}$
+ \end{tabular}
+ \end{center}
+ \end{itemize}
+
+ \begin{textblock}{10}(2,14)
+ \footnotesize $^*$ alpha-equality coincides with equality on functions
+ \end{textblock}
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1->
+ \frametitle{\begin{tabular}{c}New Design\end{tabular}}
+ \mbox{}\\[4mm]
+
+ \begin{center}
+ \begin{tikzpicture}
+ \alt<2>
+ {\draw (0,0) node[inner sep=3mm, ultra thick, draw=red, rounded corners=2mm]
+ (A) {\textcolor{red}{\begin{minipage}{1.1cm}bind.\\spec.\end{minipage}}};}
+ {\draw (0,0) node[inner sep=3mm, ultra thick, draw=white, rounded corners=2mm]
+ (A) {\begin{minipage}{1.1cm}bind.\\spec.\end{minipage}};}
+
+ \alt<3>
+ {\draw (3,0) node[inner sep=3mm, ultra thick, draw=red, rounded corners=2mm]
+ (B) {\textcolor{red}{\begin{minipage}{1.1cm}raw\\terms\end{minipage}}};}
+ {\draw (3,0) node[inner sep=3mm, ultra thick, draw=white, rounded corners=2mm]
+ (B) {\begin{minipage}{1.1cm}raw\\terms\end{minipage}};}
+
+ \alt<4>
+ {\draw (6,0) node[inner sep=3mm, ultra thick, draw=red, rounded corners=2mm]
+ (C) {\textcolor{red}{\begin{minipage}{1.1cm}$\alpha$-\\equiv.\end{minipage}}};}
+ {\draw (6,0) node[inner sep=3mm, ultra thick, draw=white, rounded corners=2mm]
+ (C) {\begin{minipage}{1.1cm}$\alpha$-\\equiv.\end{minipage}};}
+
+ \alt<5>
+ {\draw (0,-3) node[inner sep=3mm, ultra thick, draw=red, rounded corners=2mm]
+ (D) {\textcolor{red}{\begin{minipage}{1.1cm}quot.\\type\end{minipage}}};}
+ {\draw (0,-3) node[inner sep=3mm, ultra thick, draw=white, rounded corners=2mm]
+ (D) {\begin{minipage}{1.1cm}quot.\\type\end{minipage}};}
+
+ \alt<6>
+ {\draw (3,-3) node[inner sep=3mm, ultra thick, draw=red, rounded corners=2mm]
+ (E) {\textcolor{red}{\begin{minipage}{1.1cm}lift\\thms\end{minipage}}};}
+ {\draw (3,-3) node[inner sep=3mm, ultra thick, draw=white, rounded corners=2mm]
+ (E) {\begin{minipage}{1.1cm}lift\\thms\end{minipage}};}
+
+ \alt<7>
+ {\draw (6,-3) node[inner sep=3mm, ultra thick, draw=red, rounded corners=2mm]
+ (F) {\textcolor{red}{\begin{minipage}{1.1cm}add.\\thms\end{minipage}}};}
+ {\draw (6,-3) node[inner sep=3mm, ultra thick, draw=white, rounded corners=2mm]
+ (F) {\begin{minipage}{1.1cm}add.\\thms\end{minipage}};}
+
+ \draw[->,white!50,line width=1mm] (A) -- (B);
+ \draw[->,white!50,line width=1mm] (B) -- (C);
+ \draw[->,white!50,line width=1mm, line join=round, rounded corners=2mm]
+ (C) -- (8,0) -- (8,-1.5) -- (-2,-1.5) -- (-2,-3) -- (D);
+ \draw[->,white!50,line width=1mm] (D) -- (E);
+ \draw[->,white!50,line width=1mm] (E) -- (F);
+ \end{tikzpicture}
+ \end{center}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1-9>
+ \frametitle{\begin{tabular}{c}Alpha-Equivalence\end{tabular}}
+ \mbox{}\\[-3mm]
+
+ \begin{itemize}
+ \item lets first look at pairs\bigskip\medskip
+
+ \begin{tabular}{@ {\hspace{1cm}}l}
+ $(as, x) \onslide<2->{\approx\!}\makebox[0mm][l]{\only<2-7>{${}_{\text{set}}$}%
+ \only<8>{${}_{\text{\alert{list}}}$}%
+ \only<9>{${}_{\text{\alert{res}}}$}}%
+ \onslide<3->{^{R,\text{fv}}}\,\onslide<2->{(bs,y)}$
+ \end{tabular}\bigskip
+ \end{itemize}
+
+ \only<1>{
+ \begin{textblock}{8}(3,8.5)
+ \begin{tabular}{l@ {\hspace{2mm}}p{8cm}}
+ & $as$ is a set of atoms\ldots the binders\\
+ & $x$ is the body\\
+ & $\approx_{\text{set}}$ is where the cardinality
+ of the binders has to be the same\\
+ \end{tabular}
+ \end{textblock}}
+
+ \only<4->{
+ \begin{textblock}{12}(5,8)
+ \begin{tabular}{ll@ {\hspace{1mm}}l}
+ $\dn$ & \onslide<5->{$\exists \pi.\,$} & $\text{fv}(x) - as = \text{fv}(y) - bs$\\[1mm]
+ & \onslide<5->{$\;\;\;\wedge$} & \onslide<5->{$\text{fv}(x) - as \fresh^* \pi$}\\[1mm]
+ & \onslide<6->{$\;\;\;\wedge$} & \onslide<6->{$(\pi \act x)\;R\;y$}\\[1mm]
+ & \onslide<7-8>{$\;\;\;\wedge$} & \onslide<7-8>{$\pi \act as = bs$}\\
+ \end{tabular}
+ \end{textblock}}
+
+ \only<8>{
+ \begin{textblock}{8}(3,13.8)
+ \footnotesize $^*$ $as$ and $bs$ are \alert{lists} of atoms
+ \end{textblock}}
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1-2>
+ \frametitle{\begin{tabular}{c}Examples\end{tabular}}
+ \mbox{}\\[-3mm]
+
+ \begin{itemize}
+ \item lets look at ``type-schemes'':\medskip\medskip
+
+ \begin{center}
+ $(as, x) \approx\!\makebox[0mm][l]{${}_{\text{set}}$}\only<1>{{}^{R,\text{fv}}}\only<2->{{}^{\alert{=},\alert{\text{fv}}}} (bs, y)$
+ \end{center}\medskip
+
+ \onslide<2->{
+ \begin{center}
+ \begin{tabular}{l}
+ $\text{fv}(x) = \{x\}$\\[1mm]
+ $\text{fv}(T_1 \rightarrow T_2) = \text{fv}(T_1) \cup \text{fv}(T_2)$\\
+ \end{tabular}
+ \end{center}}
+ \end{itemize}
+
+
+ \only<2->{
+ \begin{textblock}{4}(0.3,12)
+ \begin{tikzpicture}
+ \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
+ {\tiny\color{darkgray}
+ \begin{minipage}{3.4cm}\raggedright
+ \begin{tabular}{r@ {\hspace{1mm}}l}
+ \multicolumn{2}{@ {}l}{res:}\\
+ $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
+ $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
+ $\wedge$ & $\pi \cdot x = y$\\
+ \\
+ \end{tabular}
+ \end{minipage}};
+ \end{tikzpicture}
+ \end{textblock}}
+ \only<2->{
+ \begin{textblock}{4}(5.2,12)
+ \begin{tikzpicture}
+ \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
+ {\tiny\color{darkgray}
+ \begin{minipage}{3.4cm}\raggedright
+ \begin{tabular}{r@ {\hspace{1mm}}l}
+ \multicolumn{2}{@ {}l}{set:}\\
+ $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
+ $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
+ $\wedge$ & $\pi \cdot x = y$\\
+ $\wedge$ & $\pi \cdot as = bs$\\
+ \end{tabular}
+ \end{minipage}};
+ \end{tikzpicture}
+ \end{textblock}}
+ \only<2->{
+ \begin{textblock}{4}(10.2,12)
+ \begin{tikzpicture}
+ \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
+ {\tiny\color{darkgray}
+ \begin{minipage}{3.4cm}\raggedright
+ \begin{tabular}{r@ {\hspace{1mm}}l}
+ \multicolumn{2}{@ {}l}{list:}\\
+ $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
+ $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
+ $\wedge$ & $\pi \cdot x = y$\\
+ $\wedge$ & $\pi \cdot as = bs$\\
+ \end{tabular}
+ \end{minipage}};
+ \end{tikzpicture}
+ \end{textblock}}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1-2>
+ \frametitle{\begin{tabular}{c}Examples\end{tabular}}
+ \mbox{}\\[-3mm]
+
+ \begin{center}
+ \only<1>{$(\{x, y\}, x \rightarrow y) \approx_? (\{x, y\}, y \rightarrow x)$}
+ \only<2>{$([x, y], x \rightarrow y) \approx_? ([x, y], y \rightarrow x)$}
+ \end{center}
+
+ \begin{itemize}
+ \item $\approx_{\text{res}}$, $\approx_{\text{set}}$%
+ \only<2>{, \alert{$\not\approx_{\text{list}}$}}
+ \end{itemize}
+
+
+ \only<1->{
+ \begin{textblock}{4}(0.3,12)
+ \begin{tikzpicture}
+ \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
+ {\tiny\color{darkgray}
+ \begin{minipage}{3.4cm}\raggedright
+ \begin{tabular}{r@ {\hspace{1mm}}l}
+ \multicolumn{2}{@ {}l}{res:}\\
+ $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
+ $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
+ $\wedge$ & $\pi \cdot x = y$\\
+ \\
+ \end{tabular}
+ \end{minipage}};
+ \end{tikzpicture}
+ \end{textblock}}
+ \only<1->{
+ \begin{textblock}{4}(5.2,12)
+ \begin{tikzpicture}
+ \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
+ {\tiny\color{darkgray}
+ \begin{minipage}{3.4cm}\raggedright
+ \begin{tabular}{r@ {\hspace{1mm}}l}
+ \multicolumn{2}{@ {}l}{set:}\\
+ $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
+ $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
+ $\wedge$ & $\pi \cdot x = y$\\
+ $\wedge$ & $\pi \cdot as = bs$\\
+ \end{tabular}
+ \end{minipage}};
+ \end{tikzpicture}
+ \end{textblock}}
+ \only<1->{
+ \begin{textblock}{4}(10.2,12)
+ \begin{tikzpicture}
+ \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
+ {\tiny\color{darkgray}
+ \begin{minipage}{3.4cm}\raggedright
+ \begin{tabular}{r@ {\hspace{1mm}}l}
+ \multicolumn{2}{@ {}l}{list:}\\
+ $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
+ $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
+ $\wedge$ & $\pi \cdot x = y$\\
+ $\wedge$ & $\pi \cdot as = bs$\\
+ \end{tabular}
+ \end{minipage}};
+ \end{tikzpicture}
+ \end{textblock}}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1>
+ \frametitle{\begin{tabular}{c}Examples\end{tabular}}
+ \mbox{}\\[-3mm]
+
+ \begin{center}
+ \only<1>{$(\{x\}, x) \approx_? (\{x, y\}, x)$}
+ \end{center}
+
+ \begin{itemize}
+ \item $\approx_{\text{res}}$, $\not\approx_{\text{set}}$,
+ $\not\approx_{\text{list}}$
+ \end{itemize}
+
+
+ \only<1->{
+ \begin{textblock}{4}(0.3,12)
+ \begin{tikzpicture}
+ \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
+ {\tiny\color{darkgray}
+ \begin{minipage}{3.4cm}\raggedright
+ \begin{tabular}{r@ {\hspace{1mm}}l}
+ \multicolumn{2}{@ {}l}{res:}\\
+ $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
+ $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
+ $\wedge$ & $\pi \cdot x = y$\\
+ \\
+ \end{tabular}
+ \end{minipage}};
+ \end{tikzpicture}
+ \end{textblock}}
+ \only<1->{
+ \begin{textblock}{4}(5.2,12)
+ \begin{tikzpicture}
+ \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
+ {\tiny\color{darkgray}
+ \begin{minipage}{3.4cm}\raggedright
+ \begin{tabular}{r@ {\hspace{1mm}}l}
+ \multicolumn{2}{@ {}l}{set:}\\
+ $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
+ $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
+ $\wedge$ & $\pi \cdot x = y$\\
+ $\wedge$ & $\pi \cdot as = bs$\\
+ \end{tabular}
+ \end{minipage}};
+ \end{tikzpicture}
+ \end{textblock}}
+ \only<1->{
+ \begin{textblock}{4}(10.2,12)
+ \begin{tikzpicture}
+ \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
+ {\tiny\color{darkgray}
+ \begin{minipage}{3.4cm}\raggedright
+ \begin{tabular}{r@ {\hspace{1mm}}l}
+ \multicolumn{2}{@ {}l}{list:}\\
+ $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
+ $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
+ $\wedge$ & $\pi \cdot x = y$\\
+ $\wedge$ & $\pi \cdot as = bs$\\
+ \end{tabular}
+ \end{minipage}};
+ \end{tikzpicture}
+ \end{textblock}}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1-3>
+ \frametitle{\begin{tabular}{c}General Abstractions\end{tabular}}
+ \mbox{}\\[-7mm]
+
+ \begin{itemize}
+ \item we take $(as, x) \approx\!\makebox[0mm][l]{${}_{\star}$}^{=,\text{supp}} (bs, y)$\medskip
+ \item they are equivalence relations\medskip
+ \item we can therefore use the quotient package to introduce the
+ types $\beta\;\text{abs}_\star$\bigskip
+ \begin{center}
+ \only<1>{$[as].\,x$}
+ \only<2>{$\text{supp}([as].x) = \text{supp}(x) - as$}
+ \only<3>{%
+ \begin{tabular}{r@ {\hspace{1mm}}l}
+ \multicolumn{2}{@ {\hspace{-7mm}}l}{$[as]. x \alert{=} [bs].y\;\;\;\text{if\!f}$}\\[2mm]
+ $\exists \pi.$ & $\text{supp}(x) - as = \text{supp}(y) - bs$\\
+ $\wedge$ & $\text{supp}(x) - as \fresh^* \pi$\\
+ $\wedge$ & $\pi \act x = y $\\
+ $(\wedge$ & $\pi \act as = bs)\;^\star$\\
+ \end{tabular}}
+ \end{center}
+ \end{itemize}
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1>
+ \frametitle{\begin{tabular}{c}One Problem\end{tabular}}
+ \mbox{}\\[-3mm]
+
+ \begin{center}
+ $\text{let}\;x_1=t_1 \ldots x_n=t_n\;\text{in}\;s$
+ \end{center}
+
+ \begin{itemize}
+ \item we cannot represent this as\medskip
+ \begin{center}
+ $\text{let}\;[x_1,\ldots,x_n]\alert{.}s\;\;[t_1,\ldots,t_n]$
+ \end{center}\bigskip
+
+ because\medskip
+ \begin{center}
+ $\text{let}\;[x].s\;\;[t_1,t_2]$
+ \end{center}
+ \end{itemize}
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1->
+ \frametitle{\begin{tabular}{c}Our Specifications\end{tabular}}
+ \mbox{}\\[-6mm]
+
+ \mbox{}\hspace{10mm}
+ \begin{tabular}{ll}
+ \multicolumn{2}{l}{\isacommand{nominal\_datatype} trm $=$}\\
+ \hspace{5mm}\phantom{$|$} Var name\\
+ \hspace{5mm}$|$ App trm trm\\
+ \hspace{5mm}$|$ Lam x::name t::trm
+ & \isacommand{bind} x \isacommand{in} t\\
+ \hspace{5mm}$|$ Let as::assn t::trm
+ & \isacommand{bind} bn(as) \isacommand{in} t\\
+ \multicolumn{2}{l}{\isacommand{and} assn $=$}\\
+ \multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} ANil}\\
+ \multicolumn{2}{l}{\hspace{5mm}$|$ ACons name trm assn}\\
+ \multicolumn{2}{l}{\isacommand{binder} bn \isacommand{where}}\\
+ \multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} bn(ANil) $=$ $[]$}\\
+ \multicolumn{2}{l}{\hspace{5mm}$|$ bn(ACons a t as) $=$ $[$a$]$ @ bn(as)}\\
+ \end{tabular}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1-2>
+ \frametitle{\begin{tabular}{c}``Raw'' Definitions\end{tabular}}
+ \mbox{}\\[-6mm]
+
+ \mbox{}\hspace{10mm}
+ \begin{tabular}{ll}
+ \multicolumn{2}{l}{\isacommand{datatype} trm $=$}\\
+ \hspace{5mm}\phantom{$|$} Var name\\
+ \hspace{5mm}$|$ App trm trm\\
+ \hspace{5mm}$|$ Lam name trm\\
+ \hspace{5mm}$|$ Let assn trm\\
+ \multicolumn{2}{l}{\isacommand{and} assn $=$}\\
+ \multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} ANil}\\
+ \multicolumn{2}{l}{\hspace{5mm}$|$ ACons name trm assn}\\[5mm]
+ \multicolumn{2}{l}{\isacommand{function} bn \isacommand{where}}\\
+ \multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} bn(ANil) $=$ $[]$}\\
+ \multicolumn{2}{l}{\hspace{5mm}$|$ bn(ACons a t as) $=$ $[$a$]$ @ bn(as)}\\
+ \end{tabular}
+
+ \only<2>{
+ \begin{textblock}{5}(10,5)
+ $+$ \begin{tabular}{l}automatically\\
+ generate fv's\end{tabular}
+ \end{textblock}}
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1>
+ \frametitle{\begin{tabular}{c}\LARGE``Raw'' Alpha-Equivalence\end{tabular}}
+ \mbox{}\\[6mm]
+
+ \begin{center}
+ Lam x::name t::trm \hspace{10mm}\isacommand{bind} x \isacommand{in} t\\
+ \end{center}
+
+
+ \[
+ \infer[\text{Lam-}\!\approx_\alpha]
+ {\text{Lam}\;x\;t \approx_\alpha \text{Lam}\;x'\;t'}
+ {([x], t) \approx\!\makebox[0mm][l]{${}_{\text{list}}$}
+ ^{\approx_\alpha,\text{fv}} ([x'], t')}
+ \]
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1>
+ \frametitle{\begin{tabular}{c}\LARGE``Raw'' Alpha-Equivalence\end{tabular}}
+ \mbox{}\\[6mm]
+
+ \begin{center}
+ Lam x::name y::name t::trm s::trm \hspace{5mm}\isacommand{bind} x y \isacommand{in} t s\\
+ \end{center}
+
+
+ \[
+ \infer[\text{Lam-}\!\approx_\alpha]
+ {\text{Lam}\;x\;y\;t\;s \approx_\alpha \text{Lam}\;x'\;y'\;t'\;s'}
+ {([x, y], (t, s)) \approx\!\makebox[0mm][l]{${}_{\text{list}}$}
+ ^{R, fv} ([x', y'], (t', s'))}
+ \]
+
+ \footnotesize
+ where $R =\;\approx_\alpha\times\approx_\alpha$ and $fv = \text{fv}\cup\text{fv}$
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1-2>
+ \frametitle{\begin{tabular}{c}\LARGE``Raw'' Alpha-Equivalence\end{tabular}}
+ \mbox{}\\[6mm]
+
+ \begin{center}
+ Let as::assn t::trm \hspace{10mm}\isacommand{bind} bn(as) \isacommand{in} t\\
+ \end{center}
+
+
+ \[
+ \infer[\text{Let-}\!\approx_\alpha]
+ {\text{Let}\;as\;t \approx_\alpha \text{Let}\;as'\;t'}
+ {(\text{bn}(as), t) \approx\!\makebox[0mm][l]{${}_{\text{list}}$}
+ ^{\approx_\alpha,\text{fv}} (\text{bn}(as'), t') &
+ \onslide<2>{as \approx_\alpha^{\text{bn}} as'}}
+ \]
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1->
+ \frametitle{\begin{tabular}{c}\LARGE{}$\alpha$ for Binding Functions\end{tabular}}
+ \mbox{}\\[-6mm]
+
+ \mbox{}\hspace{10mm}
+ \begin{tabular}{l}
+ \ldots\\
+ \isacommand{binder} bn \isacommand{where}\\
+ \phantom{$|$} bn(ANil) $=$ $[]$\\
+ $|$ bn(ACons a t as) $=$ $[$a$]$ @ bn(as)\\
+ \end{tabular}\bigskip
+
+ \begin{center}
+ \mbox{\infer{\text{ANil} \approx_\alpha^{\text{bn}} \text{ANil}}{}}\bigskip
+
+ \mbox{\infer{\text{ACons}\;a\;t\;as \approx_\alpha^{\text{bn}} \text{ACons}\;a'\;t'\;as'}
+ {t \approx_\alpha t' & as \approx_\alpha^{bn} as'}}
+ \end{center}
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1->
+ \frametitle{\begin{tabular}{c}Automatic Proofs\end{tabular}}
+ \mbox{}\\[-6mm]
+
+ \begin{itemize}
+ \item we can show that $\alpha$'s are equivalence relations\medskip
+ \item as a result we can use the quotient package to introduce the type(s)
+ of $\alpha$-equated terms
+
+ \[
+ \infer
+ {\text{Lam}\;x\;t \alert{=} \text{Lam}\;x'\;t'}
+ {\only<1>{([x], t) \approx\!\makebox[0mm][l]{${}_{\text{list}}$}
+ ^{=,\text{supp}} ([x'], t')}%
+ \only<2>{[x].t = [x'].t'}}
+ \]
+
+
+ \item the properties for support are implied by the properties of $[\_].\_$
+ \item we can derive strong induction principles (almost automatic---just a matter of
+ another week or two)
+ \end{itemize}
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1>[t]
+ \frametitle{\begin{tabular}{c}Runtime is Acceptable\end{tabular}}
+ \mbox{}\\[-7mm]\mbox{}
+
+ \footnotesize
+ \begin{center}
+ \begin{tikzpicture}
+ \draw (0,0) node[inner sep=2mm, ultra thick, draw=white, rounded corners=2mm]
+ (A) {\begin{minipage}{0.8cm}bind.\\spec.\end{minipage}};
+
+ \draw (2,0) node[inner sep=2mm, ultra thick, draw=white, rounded corners=2mm]
+ (B) {\begin{minipage}{0.8cm}raw\\terms\end{minipage}};
+
+ \draw (4,0) node[inner sep=2mm, ultra thick, draw=white, rounded corners=2mm]
+ (C) {\begin{minipage}{0.8cm}$\alpha$-\\equiv.\end{minipage}};
+
+ \draw (0,-2) node[inner sep=2mm, ultra thick, draw=white, rounded corners=2mm]
+ (D) {\begin{minipage}{0.8cm}quot.\\type\end{minipage}};
+
+ \draw (2,-2) node[inner sep=2mm, ultra thick, draw=white, rounded corners=2mm]
+ (E) {\begin{minipage}{0.8cm}lift\\thms\end{minipage}};
+
+ \draw (4,-2) node[inner sep=2mm, ultra thick, draw=white, rounded corners=2mm]
+ (F) {\begin{minipage}{0.8cm}add.\\thms\end{minipage}};
+
+ \draw[->,white!50,line width=1mm] (A) -- (B);
+ \draw[->,white!50,line width=1mm] (B) -- (C);
+ \draw[->,white!50,line width=1mm, line join=round, rounded corners=2mm]
+ (C) -- (5,0) -- (5,-1) -- (-1,-1) -- (-1,-2) -- (D);
+ \draw[->,white!50,line width=1mm] (D) -- (E);
+ \draw[->,white!50,line width=1mm] (E) -- (F);
+ \end{tikzpicture}
+ \end{center}
+
+ \begin{itemize}
+ \item Core Haskell: 11 types, 49 term-constructors,
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1->
+ \frametitle{\begin{tabular}{c}Interesting Phenomenon\end{tabular}}
+ \mbox{}\\[-6mm]
+
+ \small
+ \mbox{}\hspace{10mm}
+ \begin{tabular}{ll}
+ \multicolumn{2}{l}{\isacommand{nominal\_datatype} trm $=$}\\
+ \hspace{5mm}\phantom{$|$} Var name\\
+ \hspace{5mm}$|$ App trm trm\\
+ \hspace{5mm}$|$ Lam x::name t::trm
+ & \isacommand{bind} x \isacommand{in} t\\
+ \hspace{5mm}$|$ Let as::assn t::trm
+ & \isacommand{bind} bn(as) \isacommand{in} t\\
+ \multicolumn{2}{l}{\isacommand{and} assn $=$}\\
+ \multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} ANil}\\
+ \multicolumn{2}{l}{\hspace{5mm}$|$ ACons name trm assn}\\
+ \multicolumn{2}{l}{\isacommand{binder} bn \isacommand{where}}\\
+ \multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} bn(ANil) $=$ $[]$}\\
+ \multicolumn{2}{l}{\hspace{5mm}$|$ bn(ACons a t as) $=$ $[$a$]$ @ bn(as)}\\
+ \end{tabular}\bigskip\medskip
+
+ we cannot quotient assn: ACons a \ldots $\not\approx_\alpha$ ACons b \ldots
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1->
+ \frametitle{\begin{tabular}{c}Conclusion\end{tabular}}
+ \mbox{}\\[-6mm]
+
+ \begin{itemize}
+ \item the user does not see anything of the raw level\medskip
+ \only<1>{\begin{center}
+ Lam a (Var a) \alert{$=$} Lam b (Var b)
+ \end{center}\bigskip}
+
+ \item<2-> we have not yet done function definitions (will come soon and
+ we hope to make improvements over the old way there too)\medskip
+ \item<3-> it took quite some time to get here, but it seems worthwhile
+ (Barendregt's variable convention is unsound in general,
+ found bugs in two paper proofs, quotient package, POPL 2011 tutorial)\medskip
+ \end{itemize}
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+(*<*)
+end
+(*>*)
\ No newline at end of file