diff -r 0a5320c6a7e6 -r 842969a598f2 Slides/Slides3.thy --- /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{ + \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{ + \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{ + \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{ + \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{ + \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{ + \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{ + \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{ + \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{ + \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{ + \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{ + \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{ + \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{ + \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{ + \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{ + \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{ + \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{ + \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{ + \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{ + \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{ + \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{ + \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{ + \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{ + \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{ + \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{ + \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{ + \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