--- a/Slides/Slides2.thy Sat Jul 10 15:50:33 2010 +0100
+++ b/Slides/Slides2.thy Sat Jul 10 23:36:45 2010 +0100
@@ -18,12 +18,11 @@
\frametitle{%
\begin{tabular}{@ {\hspace{-3mm}}c@ {}}
\\
- \LARGE Proof Pearl:\\[-2mm]
+ \LARGE Proof Pearl:\\[-0mm]
\LARGE A New Foundation for\\[-2mm]
\LARGE Nominal Isabelle\\[12mm]
\end{tabular}}
\begin{center}
- \small
Brian Huf\!fman and {\bf Christian Urban}\\[0mm]
\end{center}
\end{frame}}
@@ -34,28 +33,43 @@
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
- \begin{frame}<1-2>
- \frametitle{\begin{tabular}{c}Nominal Isabelle\end{tabular}}
- \mbox{}\\[-3mm]
+ \begin{frame}<1-2>[c]
+ \frametitle{Nominal Isabelle}
\begin{itemize}
- \item sorted atoms and sort-respecting permutations\medskip
+ \item \ldots is a definitional extension of Isabelle/HOL
+ (let-polymorphism and type classes)\medskip
+
+ \item \ldots provides a convenient reasoning infrastructure for
+ terms involving binders (e.g.~lambda calculus, variable convention)\medskip
+
+ \item<2-> \ldots mainly used to find errors in my own
+ (published) paper proofs and in that of others \alert{\bf ;o)}
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1-3>[c]
+ \frametitle{Nominal Theory}
+
+ by Pitts; at its core are:\bigskip
+
+ \begin{itemize}
+ \item sorted atoms and
+ \item sort-respecting permutations
+ \end{itemize}
\onslide<2->{
- \item[] in old Nominal: atoms have \underline{dif\!ferent} type\medskip
+ \begin{textblock}{8}(4,11)
+ \onslide<3->{$\text{inv\_of\_}\pi \,\act\, ($}\onslide<2->{$\pi \,\act\, x$}%
+ \onslide<3->{$) \,=\, x$}
+ \end{textblock}}
- \begin{center}
- \begin{tabular}{c@ {\hspace{7mm}}c}
- $[]\;\act\;c \dn c$ &
- $(a\;b)\!::\!\pi\;\act\;c \dn$
- $\begin{cases}
- b & \text{if}\; \pi \act c = a\\
- a & \text{if}\; \pi \act c = b\\
- \pi \act c & \text{otherwise}
- \end{cases}$
- \end{tabular}
- \end{center}}
- \end{itemize}
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -64,29 +78,37 @@
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
- \begin{frame}<1-2>
- \frametitle{\begin{tabular}{c}Nominal Theory\end{tabular}}
- \mbox{}\\[-3mm]
-
+ \begin{frame}<1-4>[t]
+ \frametitle{The ``Old Way''}
+
\begin{itemize}
- \item sorted atoms and sort-respecting permutations\medskip
-
+ \item sorted atoms\\ \alert{$\quad\mapsto$ separate types}\; (``copies'' of nat)\bigskip
+ \item sort-respecting permutations\\ \alert{$\quad\mapsto$ lists of pairs of atoms (list swappings)}
\onslide<2->{
- \item[] in old Nominal: atoms have \underline{dif\!ferent} type\medskip
-
- \begin{center}
+ \begin{center}
\begin{tabular}{c@ {\hspace{7mm}}c}
- $[]\;\act\;c \dn c$ &
- $(a\;b)\!::\!\pi\;\act\;c \dn$
- $\begin{cases}
- b & \text{if}\; \pi \act c = a\\
- a & \text{if}\; \pi \act c = b\\
- \pi \act c & \text{otherwise}
+ $\text{[]} \,\act\, c = c$ &
+ $\swap{a}{b}\!::\!\pi \,\act\, c =
+ \begin{cases}
+ b & \text{if}\, \pi\act c = a\\
+ a & \text{if}\, \pi\act c = b\\
+ \pi\act c & \text{otherwise}
\end{cases}$
\end{tabular}
\end{center}}
\end{itemize}
+ \only<3>{
+ \begin{textblock}{14}(1,12.5)
+ \alert{The big benefit:} the type system takes care of the sort-respecting part.
+ \end{textblock}}
+ \only<4>{
+ \begin{textblock}{14}(1,12.5)
+ \alert{A small benefit:} permutation composition is \alert{list append}
+ and permutation inversion is \alert{list reversal}.
+ \end{textblock}
+ }
+
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*}
@@ -94,9 +116,8 @@
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
- \begin{frame}<1>
- \frametitle{\begin{tabular}{c}Problems\end{tabular}}
- \mbox{}\\[-3mm]
+ \begin{frame}<1-4>
+ \frametitle{Problems}
\begin{itemize}
\item @{text "_ \<bullet> _ :: \<alpha> perm \<Rightarrow> \<beta> \<Rightarrow> \<beta>"}\bigskip
@@ -110,9 +131,29 @@
\item $\forall \pi_{\alpha_1} \ldots \pi_{\alpha_n}\;.\; P$\bigskip
- \item type-classes
+ \item type-classes
+ \onslide<3>{\small\hspace{5mm}can only have \alert{\bf one} type parameter}
+ \begin{itemize}
+ \item<2-> $\text{[]}\,\act\, x = x$
+ \item<2-> $(\pi_1 @ \pi_2) \,\act \, x = \pi_1 \,\act\,(\pi_2 \,\act\, x)$
+ \item<2-> if $\pi_1 \sim \pi_2$ then $\pi_1 \,\act\, x = \pi_2 \,\act\, x$
+ \end{itemize}
\end{itemize}
+ \only<4->{
+ \begin{textblock}{9}(3,7)\begin{tikzpicture}
+ \draw (0,0) node[inner sep=3mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
+ {\normalsize\color{darkgray}
+ \begin{quote}
+ \begin{itemize}
+ \item \alert{lots} of ML-code
+ \item \alert{not} pretty
+ \item \alert{not a proof pearl} :o(
+ \end{itemize}
+ \end{quote}};
+ \end{tikzpicture}
+ \end{textblock}}
+
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*}
@@ -120,34 +161,39 @@
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
- \begin{frame}<1-4>
- \frametitle{\begin{tabular}{c}Our New Solution\end{tabular}}
- \mbox{}\\[-3mm]
+ \begin{frame}<1-5>
+ \frametitle{A Better Way}
*}
datatype atom = Atom string nat
text_raw {*
\mbox{}\bigskip
\begin{itemize}
- \item<2-> permutations are (restricted) bijective functions from @{text "atom \<Rightarrow> atom"}
+ \item<3-> permutations are (restricted) bijective functions from @{text "atom \<Rightarrow> atom"}
\begin{itemize}
- \item sort-respecting \hspace{5mm}($\forall a.\;\text{sort}(f a) = \text{sort}(a)$)
- \item finite domain \hspace{5mm}($\text{finite} \{a.\;f a \not= a\}$)
+ \item sort-respecting \hspace{5mm}($\,\forall a.\;\text{sort}(\pi a) = \text{sort}(a)$)
+ \item finite domain \hspace{5mm}($\text{finite} \{a.\;\pi a \not= a\}$)
\end{itemize}\medskip
-
- \item<3-> swappings:
+
+ \item<4-> \alert{What about swappings?}
\small
\[
\begin{array}{l@ {\hspace{1mm}}l}
(a\;b) \dn & \text{if}\;\text{sort}(a) = \text{sort}(b)\\
& \text{then}\;\lambda c. \text{if}\;a = c\;\text{then}\;b\;\text{else}\;
\text{if}\;b = c\;\text{then}\;a\;\text{else}\;c\\
- & \text{else}\;\only<3>{\mbox{\textcolor{red}{\bf ?}}}\only<4->{\text{id}}
+ & \text{else}\;\only<4>{\raisebox{-5mm}{\textcolor{red}{\huge\bf $?$}}}
+ \only<5->{\text{\alert{\bf id}}}
\end{array}
\]
\end{itemize}
+ \only<3>{
+ \begin{textblock}{7}(4,11)
+ @{text "\<pi> \<bullet> _ :: perm \<Rightarrow> \<beta> \<Rightarrow> \<beta>"}
+ \end{textblock}}
+
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*}
@@ -155,27 +201,28 @@
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
- \begin{frame}<1-6>
- \frametitle{\begin{tabular}{c}\LARGE{}A Smoother Nominal Theory\end{tabular}}
- \mbox{}\\[-3mm]
+ \begin{frame}<1-7>
+ \frametitle{A Smoother Nominal Theory}
+
+ From there it is essentially plain sailing:\bigskip
\begin{itemize}
- \item<1-> $(a\;b) = (b\;a) \onslide<3->{= (a\;c) + (b\;c) + (a\;c)}$\bigskip
+ \item<2-> $(a\;b) = (b\;a) \onslide<4->{= (a\;c) + (b\;c) + (a\;c)}$\bigskip
- \item<2-> permutations are an instance of group\_add\\ $0$, $\pi_1 + \pi_2$, $- \pi$\bigskip
+ \item<3-> permutations are an instance of Isabelle's\\
+ group\_add ($0$, $\pi_1 + \pi_2$, $- \pi$)\bigskip
- \item<5-> $\_\;\act\;\_ :: \text{perm} \Rightarrow \alpha \Rightarrow \alpha$\medskip
+ \item<6-> $\_\;\act\;\_ :: \text{perm} \Rightarrow \alpha \Rightarrow \alpha$\medskip
\begin{itemize}
\item $0\;\act\;x = x$\\
\item $(\pi_1 + \pi_2)\;\act\;x = \pi_1\;\act\;(\pi_2\;\act\;x)$
- \end{itemize}
+ \end{itemize}\medskip
- \small
- \onslide<6->{$\text{finite}(\text{supp}\;x)$, $\forall \pi. P$}
+ \onslide<7->{\alert{$\;\mapsto\;$}only one type class, $\text{finite}(\text{supp}\;x)$, $\forall \pi. P$}
\end{itemize}
- \only<4>{
+ \only<5>{
\begin{textblock}{6}(2.5,11)
\begin{tikzpicture}
\draw (0,0) node[inner sep=3mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
@@ -195,8 +242,8 @@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}<1-3>
- \frametitle{\begin{tabular}{c}Very Few Snatches\end{tabular}}
- \mbox{}\\[-3mm]
+ \frametitle{One Snatch}
+
\begin{itemize}
\item \underline{concrete} atoms:
@@ -212,7 +259,7 @@
text_raw {*
\mbox{}\bigskip\bigskip
\begin{itemize}
- \item<2-> there is an overloaded function \underline{atom}, which injects concrete
+ \item<2-> there is an overloaded function \alert{\bf atom}, which injects concrete
atoms into generic ones\medskip
\begin{center}
\begin{tabular}{l}
@@ -222,9 +269,46 @@
\end{center}\bigskip\medskip
\onslide<3->
- {I would like to have $a \fresh x$, $(a\; b)$, \ldots}
+ {One would like to have $a \fresh x$, $(a\; b)$, \ldots}
+ \end{itemize}
+
+ \only<1>{
+ \begin{textblock}{11}(2.5,8.5)
+ You like to get the advantages of the old way back: you \alert{cannot mix} atoms
+ of dif$\!$ferent sort:\bigskip
+
+ \small
+ e.g.~LF-objects:\\[-9mm]\mbox{}
+ \begin{center}$\quad M ::= c \mid x \mid \lambda x\!:\!A. M \mid M_1\;M_2$
+ \end{center}
+ \end{textblock}}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
- \end{itemize}
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1-4>
+ \frametitle{Sorts Reloaded}
+*}
+datatype atom\<iota> = Atom\<iota> string nat
+
+text_raw {*
+ \isanewline\isanewline
+ \pause
+ \alert{Problem}: HOL-binders or Church-style lambda-terms
+ \begin{center}
+ $\lambda x_\alpha.\, x_\alpha\;x_\beta$
+ \end{center}
+ \pause
+ \isanewline\isanewline
+
+\isacommand{datatype} ty = TVar string $\mid$ ty $\rightarrow$ ty\\
+\isacommand{datatype} var = Var name ty\\
+\pause
+$(x \leftrightarrow y) \,\act\, (x_\alpha, x_\beta) = (y_\alpha, y_\beta)$
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -233,8 +317,64 @@
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
+ \begin{frame}<1-3>
+ \frametitle{Non-Working Solution}
+
+Instead of\isanewline\isanewline
+*}
+ datatype atom\<iota>\<iota> = Atom\<iota>\<iota> string nat
+
+text_raw {*
+\isanewline\isanewline
+have
+\isanewline\isanewline
+*}
+
+ datatype 'a atom\<iota>\<iota>\<iota> = Atom\<iota>\<iota> 'a nat
+text_raw {*
+ \pause
+ \isanewline\isanewline
+ But then
+ \begin{center}
+ @{text "_ \<bullet> _ :: \<alpha> perm \<Rightarrow> \<beta> \<Rightarrow> \<beta>"}
+ \end{center}
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1-3>
+ \frametitle{A Working Solution}
+
+*}
+ datatype sort = Sort string "sort list"
+ datatype atom\<iota>\<iota> = Atom\<iota>\<iota> string nat
+
+text_raw {*
+\isanewline\isanewline
+have
+\isanewline\isanewline
+*}
+
+ datatype 'a atom\<iota>\<iota>\<iota> = Atom\<iota>\<iota> 'a nat
+text_raw {*
+ \pause
+ \isanewline\isanewline
+ But then
+ \begin{center}
+ @{text "_ \<bullet> _ :: \<alpha> perm \<Rightarrow> \<beta> \<Rightarrow> \<beta>"}
+ \end{center}
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
\begin{frame}<1-2>[c]
- \frametitle{\begin{tabular}{c}\LARGE{}End of Part I\end{tabular}}
+ \frametitle{Conclusion}
\mbox{}\\[-3mm]
\begin{itemize}