--- a/Slides/Slides1.thy Thu May 27 18:40:10 2010 +0200
+++ b/Slides/Slides1.thy Mon May 31 19:57:29 2010 +0200
@@ -20,7 +20,7 @@
\\
\huge Nominal 2\\[-2mm]
\large Or, How to Reason Conveniently with\\[-5mm]
- \large General Bindings\\[15mm]
+ \large General Bindings in Isabelle\\[15mm]
\end{tabular}}
\begin{center}
joint work with {\bf Cezary} and Brian Huf\!fman\\[0mm]
@@ -124,16 +124,16 @@
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
- \begin{frame}<1-4>
+ \begin{frame}<1-6>
\frametitle{\begin{tabular}{c}\LARGE{}A Smoother Nominal Theory\end{tabular}}
\mbox{}\\[-3mm]
\begin{itemize}
- \item<1-> $(a\;b) = (b\;a)$\bigskip
+ \item<1-> $(a\;b) = (b\;a) \onslide<3->{= (a\;c) + (b\;c) + (a\;c)}$\bigskip
\item<2-> permutations are an instance of group\_add\\ $0$, $\pi_1 + \pi_2$, $- \pi$\bigskip
- \item<3-> $\_\;\act\;\_ :: \text{perm} \Rightarrow \alpha \Rightarrow \alpha$\medskip
+ \item<5-> $\_\;\act\;\_ :: \text{perm} \Rightarrow \alpha \Rightarrow \alpha$\medskip
\begin{itemize}
\item $0\;\act\;x = x$\\
@@ -141,9 +141,21 @@
\end{itemize}
\small
- \onslide<4->{$\text{finite}(\text{supp}\;x)$, $\forall \pi. P$}
+ \onslide<6->{$\text{finite}(\text{supp}\;x)$, $\forall \pi. P$}
\end{itemize}
+ \only<4>{
+ \begin{textblock}{6}(2.5,11)
+ \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
+ This is slightly odd, since in general:
+ \begin{center}$\pi_1 + \pi_2 \alert{\not=} \pi_2 + \pi_1$\end{center}
+ \end{minipage}};
+ \end{tikzpicture}
+ \end{textblock}}
+
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*}
@@ -163,13 +175,14 @@
consts sort :: "atom \<Rightarrow> string"
(*>*)
-typedef name = "{a :: atom. sort a = ''name''}"
-(*<*)sorry(*>*)
+typedef name = "{a :: atom. sort a = ''name''}" (*<*)sorry(*>*)
+typedef ident = "{a :: atom. sort a = ''ident''}" (*<*)sorry(*>*)
text_raw {*
\mbox{}\bigskip\bigskip
\begin{itemize}
- \item<2-> there is a function \underline{atom}, which injects concrete atoms into generic atoms\medskip
+ \item<2-> there is an overloaded function \underline{atom}, which injects concrete
+ atoms into generic ones\medskip
\begin{center}
\begin{tabular}{l}
$\text{atom}(a) \fresh x$\\
@@ -194,9 +207,8 @@
\mbox{}\\[-3mm]
\begin{itemize}
- \item the formalised version of the nominal theory is much nicer to
- work with (no assumptions, just two type classes; sorts are occasionally
- explicit)\bigskip
+ \item the formalised version of the nominal theory is now much nicer to
+ work with (sorts are occasionally explicit)\bigskip
\item permutations: ``be as abstract as you can'' (group\_add is a slight oddity)\bigskip
@@ -213,26 +225,790 @@
\mode<presentation>{
\begin{frame}<1-2>
\frametitle{\begin{tabular}{c}\LARGE{}Part II: General Bindings\end{tabular}}
- \mbox{}\\[-3mm]
+ \mbox{}\\[-6mm]
\begin{itemize}
- \item old Nominal provided single binders
+ \item old Nominal provided a reasoning infrastructure for single binders\medskip
+
\begin{center}
Lam [a].(Var a)
\end{center}\bigskip
- \item<2-> representing
+ \item<2-> but representing
+
\begin{center}
$\forall\{a_1,\ldots,a_n\}.\; T$
- \end{center}
- is a major pain, take my word for it
+ \end{center}\medskip
+
+ with single binders is a \alert{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}
+ \pgfuseshading{smallspherered} & a $\fresh$ Lam [a]. t\\
+ \pgfuseshading{smallspherered} & Lam [a]. (Var a) \alert{$=$} Lam [b]. (Var b)\\
+ \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 (restriction)\medskip
+
+ \item the order does not matter, but the cardinality of the binders
+ must be the same (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) $=$ $\varnothing$}}\\
+ \multicolumn{2}{l}{\onslide<3->{\hspace{5mm}$|$ bn(ACons a t as) $=$ $\{$a$\}$ $\cup$ bn(as)}}\\
+ \end{tabular}
+
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1-4>
+ \frametitle{\begin{tabular}{c}Ott\end{tabular}}
+ \mbox{}\\[-3mm]
+
+ \begin{itemize}
+ \item this way of specifying binding is pretty much stolen from
+ Ott\onslide<2->{, \alert{\bf but} with adjustments:}\medskip
+
+ \begin{itemize}
+ \item<2-> Ott allows specifications like\smallskip
+ \begin{center}
+ $t ::= t\;t\; |\;\lambda x.t$
+ \end{center}\medskip
+
+ \item<3-> whether something is bound can depend on other bound things\smallskip
+ \begin{center}
+ Foo $(\lambda x. t)\; s$
+ \end{center}\medskip
+ \onslide<4->{this might make sense for ``raw'' terms, but not at all
+ for $\alpha$-equated terms}
+ \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 [$a$].$t$ $\;\dn$\\[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}<1-9>
+ \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}}
+ \pgfuseshading{smallspherered} & $as$ is a set of atoms\ldots the binders\\
+ \pgfuseshading{smallspherered} & $x$ is the body\\
+ \pgfuseshading{smallspherered} & $\approx_{\text{set}}$ 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 $fv = \text{fv}\times\text{fv}$
+
+ \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->
+ \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 (POPL 2011 tutorial)\medskip
+ \item<4-> Thanks goes to Cezary!\\
+ \only<5->{\hspace{3mm}\ldots{}and of course others $\in$ Isabelle-team!}
+ \end{itemize}
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
(*<*)
end