--- a/Slides/Slides5.thy Wed Mar 30 22:27:26 2011 +0200
+++ b/Slides/Slides5.thy Thu Mar 31 15:25:35 2011 +0200
@@ -82,11 +82,38 @@
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
- \begin{frame}<1-2>
+ \begin{frame}<1-6>
\frametitle{New Types in HOL}
- picture
+ \begin{center}
+ \begin{tikzpicture}[scale=1.5]
+ %%%\draw[step=2mm] (-4,-1) grid (4,1);
+
+ \onslide<2-4,6>{\draw[very thick] (0.7,0.4) circle (4.25mm);}
+ \onslide<1-4,6>{\draw[rounded corners=1mm, very thick] ( 0.0,-0.8) rectangle ( 1.8, 0.9);}
+ \onslide<3-5,6>{\draw[rounded corners=1mm, very thick] (-1.95,0.85) rectangle (-2.85,-0.05);}
+ \onslide<3-4,6>{\draw (-2.0, 0.845) -- (0.7,0.845);}
+ \onslide<3-4,6>{\draw (-2.0,-0.045) -- (0.7,-0.045);}
+
+ \onslide<4-4,6>{\alert{\draw ( 0.7, 0.4) node {\footnotesize\begin{tabular}{@ {}c@ {}}$\alpha$-\\[-1mm]classes\end{tabular}};}}
+ \onslide<4-5,6>{\alert{\draw (-2.4, 0.4) node {\footnotesize\begin{tabular}{@ {}c@ {}}$\alpha$-eq.\\[-1mm]terms\end{tabular}};}}
+ \onslide<1-4,6>{\draw (1.8, 0.48) node[right=-0.1mm]
+ {\footnotesize\begin{tabular}{@ {}l@ {}}existing\\[-1mm] type\\ \onslide<4-4,6>{\alert{(sets of raw terms)}}\end{tabular}};}
+ \onslide<2-4,6>{\draw (0.9, -0.35) node {\footnotesize\begin{tabular}{@ {}l@ {}}non-empty\\[-1mm]subset\end{tabular}};}
+ \onslide<3-5,6>{\draw (-3.25, 0.55) node {\footnotesize\begin{tabular}{@ {}l@ {}}new\\[-1mm]type\end{tabular}};}
+
+ \onslide<3-4,6>{\draw[<->, very thick] (-1.8, 0.3) -- (-0.1,0.3);}
+ \onslide<3-4,6>{\draw (-0.95, 0.3) node[above=0mm] {\footnotesize{}isomorphism};}
+
+ \onslide<6>{\draw[->, line width=2mm, red] (-1.0,-0.4) -- (0.35,0.16);}
+ \end{tikzpicture}
+ \end{center}
+
+ \begin{center}
+ \textcolor{red}{\large\bf\onslide<6>{define $\alpha$-equivalence}}
+ \end{center}
+
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*}
@@ -105,15 +132,15 @@
\begin{center}
\begin{tabular}{l}
- $\forall\{x, y\}.\, x \rightarrow y \;\;\approx_\alpha\;\; \forall\{y, x\}.\, y \rightarrow x$
+ \textcolor{blue}{$\forall\{x, y\}.\, x \rightarrow y \;\;\approx_\alpha\;\; \forall\{y, x\}.\, y \rightarrow x$}
\bigskip\smallskip\\
\onslide<2->{%
- $\forall\{x, y\}.\, x \rightarrow y \;\;\not\approx_\alpha\;\; \forall\{z\}.\, z \rightarrow z$
+ \textcolor{blue}{$\forall\{x, y\}.\, x \rightarrow y \;\;\not\approx_\alpha\;\; \forall\{z\}.\, z \rightarrow z$}
}\bigskip\smallskip\\
\onslide<3->{%
- $\forall\{x\}.\, x \rightarrow y \;\;\approx_\alpha\;\; \forall\{x, \alert{z}\}.\, x \rightarrow y$
+ \textcolor{blue}{$\forall\{x\}.\, x \rightarrow y \;\;\approx_\alpha\;\; \forall\{x, \alert{z}\}.\, x \rightarrow y$}
}\medskip\\
\onslide<3->{\hspace{4cm}\small provided $z$ is fresh for the type}
\end{tabular}
@@ -131,7 +158,7 @@
{\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.
+ $\alpha$-equivalence is preserved under \alert{vacuous} binders.
\end{minipage}};
\end{tikzpicture}
\end{textblock}}
@@ -150,12 +177,12 @@
\item alpha-equivalence being preserved under vacuous binders is \underline{not} always
wanted:\bigskip\bigskip\normalsize
- \begin{tabular}{@ {\hspace{-8mm}}l}
+ \textcolor{blue}{\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{tabular}}
\end{itemize}
@@ -175,11 +202,11 @@
\item sometimes one wants to abstract more than one name, but the order \underline{does} matter\bigskip
\begin{center}
- \begin{tabular}{@ {\hspace{-8mm}}l}
+ \textcolor{blue}{\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{tabular}}
\end{center}
@@ -247,140 +274,6 @@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*}
-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
- {\bf Ott}\onslide<2->{, \alert{\bf but} we made some 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}
- \begin{tabular}{l}
- \isacommand{bind} a b c \ldots \isacommand{in} x y z \ldots\\
- \isacommand{bind (set)} a b c \ldots \isacommand{in} x y z \ldots\\
- \isacommand{bind (set+)} a b c \ldots \isacommand{in} x y z \ldots
- \end{tabular}
- \end{center}\bigskip\medskip
- the reason is that with our definition of $\alpha$-equivalence\medskip
- \begin{center}
- \begin{tabular}{l}
- \isacommand{bind (set+)} as \isacommand{in} x y $\not\Leftrightarrow$\\
- \hspace{8mm}\isacommand{bind (set+)} as \isacommand{in} x, \isacommand{bind (set+)} as \isacommand{in} y
- \end{tabular}
- \end{center}\medskip
-
- 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 the old Nominal Isabelle, 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}
- {\draw (0,0) node[inner sep=3mm, ultra thick, draw=fg, rounded corners=2mm]
- (A) {\begin{minipage}{1.1cm}bind.\\spec.\end{minipage}};}
-
- {\draw (3,0) node[inner sep=3mm, ultra thick, draw=fg, rounded corners=2mm]
- (B) {\begin{minipage}{1.1cm}raw\\terms\end{minipage}};}
-
- \alt<2>
- {\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=fg, rounded corners=2mm]
- (C) {\begin{minipage}{1.1cm}$\alpha$-\\equiv.\end{minipage}};}
-
- {\draw (0,-3) node[inner sep=3mm, ultra thick, draw=fg, rounded corners=2mm]
- (D) {\begin{minipage}{1.1cm}quot.\\type\end{minipage}};}
-
- {\draw (3,-3) node[inner sep=3mm, ultra thick, draw=fg, rounded corners=2mm]
- (E) {\begin{minipage}{1.1cm}lift\\thms\end{minipage}};}
-
- {\draw (6,-3) node[inner sep=3mm, ultra thick, draw=fg, rounded corners=2mm]
- (F) {\begin{minipage}{1.1cm}add.\\thms\end{minipage}};}
-
- \draw[->,fg!50,line width=1mm] (A) -- (B);
- \draw[->,fg!50,line width=1mm] (B) -- (C);
- \draw[->,fg!50,line width=1mm, line join=round, rounded corners=2mm]
- (C) -- (8,0) -- (8,-1.5) -- (-2,-1.5) -- (-2,-3) -- (D);
- \draw[->,fg!50,line width=1mm] (D) -- (E);
- \draw[->,fg!50,line width=1mm] (E) -- (F);
- \end{tikzpicture}
- \end{center}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -392,32 +285,34 @@
\begin{itemize}
\item lets first look at pairs\bigskip\medskip
- \begin{tabular}{@ {\hspace{1cm}}l}
- $(as, x) \onslide<2->{\approx\!}\makebox[0mm][l]{\only<2-6>{${}_{\text{set}}$}%
+ \textcolor{blue}{\begin{tabular}{@ {\hspace{1cm}}l}
+ $(as, x) \onslide<2->{\approx\!}\makebox[5mm][l]{\only<2-6>{${}_{\text{set}}$}%
\only<7>{${}_{\text{\alert{list}}}$}%
\only<8>{${}_{\text{\alert{set+}}}$}}%
- \onslide<3->{^{R,\text{fv}}}\,\onslide<2->{(bs,y)}$
- \end{tabular}\bigskip
+ \,\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 names\ldots the binders\\
- & $x$ is the body (might be a tuple)\\
- & $\approx_{\text{set}}$ is where the cardinality
+ & \textcolor{blue}{$as$} is a set of names\ldots the binders\\
+ & \textcolor{blue}{$x$} is the body (might be a tuple)\\
+ & \textcolor{blue}{$\approx_{\text{set}}$} is where the cardinality
of the binders has to be the same\\
\end{tabular}
\end{textblock}}
\only<4->{
\begin{textblock}{12}(5,8)
+ \textcolor{blue}{
\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<5->{$\;\;\;\wedge$} & \onslide<5->{$(\pi \act x)\;R\;y$}\\[1mm]
- & \onslide<6-7>{$\;\;\;\wedge$} & \onslide<6-7>{$\pi \act as = bs$}\\
- \end{tabular}
+ & \onslide<5->{$\;\;\;\wedge$} & \onslide<5->{$(\pi \act x) = y$}\\[1mm]
+ & \only<6-7>{$\;\;\;\wedge$}\only<8>{\textcolor{gray}{\xout{$\;\;\;\wedge$}}} &
+ \only<6-7>{$\pi \act as = bs$}\only<8>{\textcolor{gray}{\xout{$\pi \act as = bs$}}}\\
+ \end{tabular}}
\end{textblock}}
\only<7>{
@@ -436,18 +331,19 @@
\mbox{}\\[-3mm]
\begin{itemize}
- \item lets look at ``type-schemes'':\medskip\medskip
+ \item lets look at type-schemes:\medskip\medskip
\begin{center}
- $(as, x) \approx\!\makebox[0mm][l]{${}_{\text{set}}$}\only<1>{{}^{R,\text{fv}}}\only<2->{{}^{\alert{=},\alert{\text{fv}}}} (bs, y)$
+ \textcolor{blue}{$(as, x) \approx\!\makebox[5mm][l]{${}_{\text{set}}$} (bs, y)$}
\end{center}\medskip
\onslide<2->{
\begin{center}
+ \textcolor{blue}{
\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{tabular}}
\end{center}}
\end{itemize}
@@ -513,13 +409,14 @@
\mbox{}\\[-3mm]
\begin{center}
+ \textcolor{blue}{
\only<1>{$(\{x, y\}, x \rightarrow y) \approx_? (\{x, y\}, y \rightarrow x)$}
- \only<2>{$([x, y], x \rightarrow y) \approx_? ([x, y], y \rightarrow x)$}
+ \only<2>{$([x, y], x \rightarrow y) \approx_? ([x, y], y \rightarrow x)$}}
\end{center}
\begin{itemize}
- \item $\approx_{\text{set+}}$, $\approx_{\text{set}}$%
- \only<2>{, \alert{$\not\approx_{\text{list}}$}}
+ \item \textcolor{blue}{$\approx_{\text{set+}}$, $\approx_{\text{set}}$%
+ \only<2>{, \alert{$\not\approx_{\text{list}}$}}}
\end{itemize}
@@ -584,12 +481,12 @@
\mbox{}\\[-3mm]
\begin{center}
- \only<1>{$(\{x\}, x) \approx_? (\{x, y\}, x)$}
+ \textcolor{blue}{\only<1>{$(\{x\}, x) \approx_? (\{x, y\}, x)$}}
\end{center}
\begin{itemize}
- \item $\approx_{\text{set+}}$, $\not\approx_{\text{set}}$,
- $\not\approx_{\text{list}}$
+ \item \textcolor{blue}{$\approx_{\text{set+}}$, $\not\approx_{\text{set}}$,
+ $\not\approx_{\text{list}}$}
\end{itemize}
@@ -664,69 +561,6 @@
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
- \begin{frame}<1-3>
- \frametitle{\begin{tabular}{c}General Abstractions\end{tabular}}
- \mbox{}\\[-7mm]
-
- \begin{itemize}
- \item we take $(as, x) \approx\!\makebox[0mm][l]{${}_{{}*{}}$}^{=,\text{supp}} (bs, y)$\medskip
- \item they are equivalence relations\medskip
- \item we can therefore use the quotient package to introduce the
- types $\beta\;\text{abs}_*$\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 $\\
- $(\wedge$ & $\pi \act as = bs)\;^*$\\
- \end{tabular}}
- \end{center}
- \end{itemize}
-
- \only<1->{
- \begin{textblock}{8}(12,3.8)
- \footnotesize $^*$ set, set+, list
- \end{textblock}}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}<1>
- \frametitle{\begin{tabular}{c}A 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]
@@ -755,80 +589,20 @@
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 assns trm\\
- \multicolumn{2}{l}{\isacommand{and} assns $=$}\\
- \multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} ANil}\\
- \multicolumn{2}{l}{\hspace{5mm}$|$ ACons name trm assns}\\[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{frame}<1>[c]
+ \frametitle{\begin{tabular}{c}Binding Functions\end{tabular}}
\begin{center}
- Lam x::name t::trm \hspace{10mm}\isacommand{bind} x \isacommand{in} t\\
+ \begin{tikzpicture}
+ \node (A) at (-0.5,1) {Foo $(\lambda y. \lambda x. t)$};
+ \node (B) at ( 1.5,1) {$s$};
+ \onslide<1>{\node (C) at (0.5,-0.5) {$\{y, x\}$};}
+ \onslide<1>{\draw[->,red,line width=1mm] (A) -- (C);}
+ \onslide<1>{\draw[->,red,line width=1mm] (C) -- (B);}
+ \end{tikzpicture}
\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}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*}
@@ -836,98 +610,45 @@
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
- \begin{frame}<1-2>
- \frametitle{\begin{tabular}{c}\LARGE``Raw'' Alpha-Equivalence\end{tabular}}
- \mbox{}\\[6mm]
-
- \begin{center}
- Let as::assns t::trm \hspace{10mm}\isacommand{bind} bn(as) \isacommand{in} t\\
- \end{center}
-
-
- \[
- \infer[\text{Let-}\!\approx_\alpha]
- {\text{Let}\;as\;t \approx_\alpha \text{Let}\;as'\;t'}
- {(\text{bn}(as), t) \approx\!\makebox[0mm][l]{${}_{\text{list}}$}
- ^{\approx_\alpha,\text{fv}} (\text{bn}(as'), t') &
- \onslide<2->{as \approx_\alpha^{\text{bn}} as'}}
- \]\bigskip
-
+ \begin{frame}<1->[t]
+ \frametitle{\begin{tabular}{c}Binder Clauses\end{tabular}}
- \onslide<1->{\small{}bn-function $\Rightarrow$ \alert{deep binders}}
- \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{itemize}
+ \item We need for a bound variable to have a `clear scope', and bound
+ variables should not be free and bound at the same time.\bigskip
+ \end{itemize}
\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}\LARGE``Raw'' Alpha-Equivalence\end{tabular}}
- \mbox{}\\[6mm]
-
- \begin{center}
- LetRec as::assns t::trm \hspace{10mm}\isacommand{bind} bn(as) \isacommand{in} t \alert{as}\\
- \end{center}
-
+ \only<1>{
+ \begin{tabular}{@ {\hspace{-5mm}}l}
+ \alert{\bf shallow binders}\\
+ \hspace{4mm}Lam x::name t::trm\hspace{4mm} \isacommand{bind} x \isacommand{in} t\\
+ \hspace{4mm}All xs::name set T::ty\hspace{4mm} \isacommand{bind} xs \isacommand{in} T\\
+ \hspace{4mm}Foo x::name t$_1$::trm t$_2$::trm\hspace{4mm}
+ \isacommand{bind} x \isacommand{in} t$_1$, \isacommand{bind} x \isacommand{in} t$_2$\\
+ \hspace{4mm}Bar x::name t$_1$::trm t$_2$::trm\hspace{4mm}
+ \isacommand{bind} x \isacommand{in} t$_1$ t$_2$\\
+ \end{tabular}}
+ \only<2>{
+ \begin{tabular}{@ {\hspace{-5mm}}l}
+ \alert{\bf deep binders} \\
+ \hspace{4mm}Let as::assns t::trm\hspace{4mm} \isacommand{bind} bn(as) \isacommand{in} t\\
+ \hspace{4mm}Foo as::assns t$_1$::trm t$_2$::trm\\
+ \hspace{20mm}\isacommand{bind} bn(as) \isacommand{in} t$_1$, \isacommand{bind} bn(as) \isacommand{in} t$_2$\\[4mm]
+ \makebox[0mm][l]{\alert{$\times$}}\hspace{4mm}Bar as::assns t$_1$::trm t$_2$::trm\\
+ \hspace{20mm}\isacommand{bind} bn$_1$(as) \isacommand{in} t$_1$, \isacommand{bind} bn$_2$(as) \isacommand{in} t$_2$\\
+ \end{tabular}}
+ \only<3>{
+ \begin{tabular}{@ {\hspace{-5mm}}l}
+ {\bf deep \alert{recursive} binders} \\
+ \hspace{4mm}Let\_rec as::assns t::trm\hspace{4mm} \isacommand{bind} bn(as) \isacommand{in} t as\\[4mm]
- \[\mbox{}\hspace{-4mm}
- \infer[\text{LetRec-}\!\approx_\alpha]
- {\text{LetRec}\;as\;t \approx_\alpha \text{LetRec}\;as'\;t'}
- {(\text{bn}(as), (t, as)) \approx\!\makebox[0mm][l]{${}_{\text{list}}$}
- ^{R,\text{fv}} (\text{bn}(as'), (t', as'))}
- \]\bigskip
-
- \onslide<1->{\alert{deep recursive binders}}
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
+ \makebox[0mm][l]{\alert{$\times$}}\hspace{4mm}Foo\_rec as::assns t$_1$::trm t$_2$::trm\hspace{4mm}\\
+ \hspace{20mm}\isacommand{bind} bn(as) \isacommand{in} t$_1$ as, \isacommand{bind} bn(as) \isacommand{in} t$_2$\\
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}<1->
- \frametitle{\begin{tabular}{c}Restrictions\end{tabular}}
- \mbox{}\\[-6mm]
-
- Our restrictions on binding specifications:
-
- \begin{itemize}
- \item a body can only occur once in a list of binding clauses\medskip
- \item you can only have one binding function for a deep binder\medskip
- \item binding functions can return: the empty set, singletons, unions (similarly for lists)
- \end{itemize}
-
-
+ \end{tabular}}
+ \end{center}
+
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*}
@@ -935,121 +656,52 @@
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
- \begin{frame}<1->
- \frametitle{\begin{tabular}{c}Automatic Proofs\end{tabular}}
+ \begin{frame}<2-5>
+ \frametitle{\begin{tabular}{c}Our Work\end{tabular}}
\mbox{}\\[-6mm]
- \begin{itemize}
- \item we can show that $\alpha$'s are equivalence relations\medskip
- \item as a result we can use our quotient package to introduce the type(s)
- of $\alpha$-equated terms
+ \begin{center}
+ \begin{tikzpicture}[scale=1.5]
+ %%%\draw[step=2mm] (-4,-1) grid (4,1);
+
+ \onslide<1>{\draw[very thick] (0.7,0.4) circle (4.25mm);}
+ \onslide<1>{\draw[rounded corners=1mm, very thick] ( 0.0,-0.8) rectangle ( 1.8, 0.9);}
+ \onslide<1->{\draw[rounded corners=1mm, very thick] (-1.95,0.85) rectangle (-2.85,-0.05);}
+
+ \onslide<1>{\draw (-2.0, 0.845) -- (0.7,0.845);}
+ \onslide<1>{\draw (-2.0,-0.045) -- (0.7,-0.045);}
- \[
- \infer
- {\text{Lam}\;x\;t \alert{=} \text{Lam}\;x'\;t'}
- {\only<1>{([x], t) \approx\!\makebox[0mm][l]{${}_{\text{list}}$}
- ^{=,\text{supp}} ([x'], t')}%
- \only<2>{[x].t = [x'].t'}}
- \]
+ \onslide<1>{\alert{\draw ( 0.7, 0.4) node {\footnotesize\begin{tabular}{@ {}c@ {}}$\alpha$-\\[-1mm]classes\end{tabular}};}}
+ \onslide<1->{\alert{\draw (-2.4, 0.4) node {\footnotesize\begin{tabular}{@ {}c@ {}}$\alpha$-eq.\\[-1mm]terms\end{tabular}};}}
+ \onslide<1>{\draw (1.8, 0.48) node[right=-0.1mm]
+ {\footnotesize\begin{tabular}{@ {}l@ {}}existing\\[-1mm] type\\ \onslide<1>{\alert{(sets of raw terms)}}\end{tabular}};}
+ \onslide<1>{\draw (0.9, -0.35) node {\footnotesize\begin{tabular}{@ {}l@ {}}non-empty\\[-1mm]subset\end{tabular}};}
+ \onslide<1->{\draw (-3.25, 0.55) node {\footnotesize\begin{tabular}{@ {}l@ {}}new\\[-1mm]type\end{tabular}};}
+
+ \onslide<1>{\draw[<->, very thick] (-1.8, 0.3) -- (-0.1,0.3);}
+ \onslide<1>{\draw (-0.95, 0.3) node[above=0mm] {\footnotesize{}isomorphism};}
-
- \item the properties for support are implied by the properties of $[\_].\_$
- \item we can derive strong induction principles
+ \onslide<1>{\draw[->, line width=2mm, red] (-1.0,-0.4) -- (0.35,0.16);}
+ \end{tikzpicture}
+ \end{center}
+
+ \begin{textblock}{9.5}(6,3.5)
+ \begin{itemize}
+ \item<1-> defined fv and $\alpha$
+ \item<3-> derived a reasoning infrastructure ($\fresh$, distinctness, injectivity, cases,\ldots)
+ \item<4-> a (weak) induction principle
+ \item<5-> derive a {\bf stronger} induction principle (Barendregt variable convention built in)\\
+ \begin{center}
+ \textcolor{blue}{Foo ($\lambda x. \lambda y. t$) ($\lambda u. \lambda v. s$)}
+ \end{center}
\end{itemize}
+ \end{textblock}
\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=fg, rounded corners=2mm]
- (A) {\begin{minipage}{0.8cm}bind.\\spec.\end{minipage}};
-
- \draw (2,0) node[inner sep=2mm, ultra thick, draw=fg, rounded corners=2mm]
- (B) {\begin{minipage}{0.8cm}raw\\terms\end{minipage}};
-
- \draw (4,0) node[inner sep=2mm, ultra thick, draw=fg, rounded corners=2mm]
- (C) {\begin{minipage}{0.8cm}$\alpha$-\\equiv.\end{minipage}};
-
- \draw (0,-2) node[inner sep=2mm, ultra thick, draw=fg, rounded corners=2mm]
- (D) {\begin{minipage}{0.8cm}quot.\\type\end{minipage}};
-
- \draw (2,-2) node[inner sep=2mm, ultra thick, draw=fg, rounded corners=2mm]
- (E) {\begin{minipage}{0.8cm}lift\\thms\end{minipage}};
-
- \draw (4,-2) node[inner sep=2mm, ultra thick, draw=fg, rounded corners=2mm]
- (F) {\begin{minipage}{0.8cm}add.\\thms\end{minipage}};
-
- \draw[->,fg!50,line width=1mm] (A) -- (B);
- \draw[->,fg!50,line width=1mm] (B) -- (C);
- \draw[->,fg!50,line width=1mm, line join=round, rounded corners=2mm]
- (C) -- (5,0) -- (5,-1) -- (-1,-1) -- (-1,-2) -- (D);
- \draw[->,fg!50,line width=1mm] (D) -- (E);
- \draw[->,fg!50,line width=1mm] (E) -- (F);
- \end{tikzpicture}
- \end{center}
-
- \begin{itemize}
- \item Core Haskell: 11 types, 49 term-constructors, 7 binding functions
- \begin{center}
- $\sim$ 2 mins
- \end{center}
- \end{itemize}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}<1->
- \frametitle{\begin{tabular}{c}Interesting Phenomenon\end{tabular}}
- \mbox{}\\[-6mm]
-
- \small
- \mbox{}\hspace{20mm}
- \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::assns t::trm
- & \isacommand{bind} bn(as) \isacommand{in} t\\
- \multicolumn{2}{l}{\isacommand{and} assns $=$}\\
- \multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} ANil}\\
- \multicolumn{2}{l}{\hspace{5mm}$|$ ACons name trm assns}\\
- \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 assns: ACons a \ldots $\not\approx_\alpha$ ACons b \ldots
-
- \only<1->{
- \begin{textblock}{8}(0.2,7.3)
- \alert{\begin{tabular}{p{2.6cm}}
- \raggedright\footnotesize{}Should a ``naked'' assns be quotient?
- \end{tabular}\hspace{-3mm}
- $\begin{cases}
- \mbox{} \\ \mbox{}
- \end{cases}$}
- \end{textblock}}
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -1064,32 +716,18 @@
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
+ \item<2-> 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
+ found bugs in two paper proofs)\bigskip\medskip
+
+ \item<3-> \textcolor{blue}{http://isabelle.in.tum.de/nominal/}
\end{itemize}
-
+
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*}
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}<1->[c]
- \frametitle{\begin{tabular}{c}Future Work\end{tabular}}
- \mbox{}\\[-6mm]
-
- \begin{itemize}
- \item Function definitions
- \end{itemize}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
text_raw {*
@@ -1116,15 +754,17 @@
\frametitle{\begin{tabular}{c}Examples\end{tabular}}
\mbox{}\\[-6mm]
+ \textcolor{blue}{
\begin{center}
$(\{a,b\}, a \rightarrow b) \approx_\alpha (\{a, b\}, a \rightarrow b)$
$(\{a,b\}, a \rightarrow b) \approx_\alpha (\{a, b\}, b \rightarrow a)$
- \end{center}
+ \end{center}}
+ \textcolor{blue}{
\begin{center}
$(\{a,b\}, (a \rightarrow b, a \rightarrow b))$\\
\hspace{17mm}$\not\approx_\alpha (\{a, b\}, (a \rightarrow b, b \rightarrow a))$
- \end{center}
+ \end{center}}
\onslide<2->
{1.) \hspace{3mm}\isacommand{bind (set)} as \isacommand{in} $\tau_1$,
@@ -1137,6 +777,8 @@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*}
+
+
(*<*)
end
(*>*)
\ No newline at end of file