(*<*)
theory Slides1
imports "LaTeXsugar" "Nominal"
begin
notation (latex output)
set ("_") and
Cons ("_::/_" [66,65] 65)
(*>*)
text_raw {*
\renewcommand{\slidecaption}{TU Munich, 28.~May 2010}
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\mode<presentation>{
\begin{frame}<1>[t]
\frametitle{%
\begin{tabular}{@ {\hspace{-3mm}}c@ {}}
\\
\huge Nominal 2\\[-2mm]
\large Or, How to Reason Conveniently with\\[-5mm]
\large General Bindings in Isabelle\\[15mm]
\end{tabular}}
\begin{center}
joint work with {\bf Cezary} and Brian Huf\!fman\\[0mm]
\end{center}
\end{frame}}
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*}
text_raw {*
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\mode<presentation>{
\begin{frame}<1-2>
\frametitle{\begin{tabular}{c}Part I: Nominal Theory\end{tabular}}
\mbox{}\\[-3mm]
\begin{itemize}
\item sorted atoms and sort-respecting permutations\medskip
\onslide<2->{
\item[] in old Nominal: atoms have \underline{dif\!ferent} type\medskip
\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}}
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*}
text_raw {*
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\mode<presentation>{
\begin{frame}<1>
\frametitle{\begin{tabular}{c}Problems\end{tabular}}
\mbox{}\\[-3mm]
\begin{itemize}
\item @{text "_ \<bullet> _ :: \<alpha> perm \<Rightarrow> \<beta> \<Rightarrow> \<beta>"}\bigskip
\item @{text "supp _ :: \<beta> \<Rightarrow> \<alpha> set"}
\begin{center}
$\text{finite} (\text{supp}\;x)_{\,\alpha_1\,\text{set}}$ \ldots
$\text{finite} (\text{supp}\;x)_{\,\alpha_n\,\text{set}}$
\end{center}\bigskip
\item $\forall \pi_{\alpha_1} \ldots \pi_{\alpha_n}\;.\; P$\bigskip
\item type-classes
\end{itemize}
\end{frame}}
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*}
text_raw {*
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\mode<presentation>{
\begin{frame}<1-4>
\frametitle{\begin{tabular}{c}Our New Solution\end{tabular}}
\mbox{}\\[-3mm]
*}
datatype atom = Atom string nat
text_raw {*
\mbox{}\bigskip
\begin{itemize}
\item<2-> 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}($\text{finite} \{a.\;f a \not= a\}$)
\end{itemize}\medskip
\item<3-> swappings:
\small
\[
\begin{array}{l@ {\hspace{1mm}}l}
(a\;b) \dn & \text{if}\;\text{sort}(a) = \text{sort}(b)\\
& \text{then}\;\lambda c. \text{if}\;a = c\;\text{then}\;b\;\text{else}\;
\text{if}\;b = c\;\text{then}\;a\;\text{else}\;c\\
& \text{else}\;\only<3>{\mbox{\textcolor{red}{\bf ?}}}\only<4->{\text{id}}
\end{array}
\]
\end{itemize}
\end{frame}}
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*}
text_raw {*
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\mode<presentation>{
\begin{frame}<1-6>
\frametitle{\begin{tabular}{c}\LARGE{}A Smoother Nominal Theory\end{tabular}}
\mbox{}\\[-3mm]
\begin{itemize}
\item<1-> $(a\;b) = (b\;a) \onslide<3->{= (a\;c) + (b\;c) + (a\;c)}$\bigskip
\item<2-> permutations are an instance of group\_add\\ $0$, $\pi_1 + \pi_2$, $- \pi$\bigskip
\item<5-> $\_\;\act\;\_ :: \text{perm} \Rightarrow \alpha \Rightarrow \alpha$\medskip
\begin{itemize}
\item $0\;\act\;x = x$\\
\item $(\pi_1 + \pi_2)\;\act\;x = \pi_1\;\act\;(\pi_2\;\act\;x)$
\end{itemize}
\small
\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}}
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*}
text_raw {*
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\mode<presentation>{
\begin{frame}<1-3>
\frametitle{\begin{tabular}{c}Very Few Snatches\end{tabular}}
\mbox{}\\[-3mm]
\begin{itemize}
\item \underline{concrete} atoms:
\end{itemize}
*}
(*<*)
consts sort :: "atom \<Rightarrow> string"
(*>*)
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 an overloaded function \underline{atom}, which injects concrete
atoms into generic ones\medskip
\begin{center}
\begin{tabular}{l}
$\text{atom}(a) \fresh x$\\
$(a \leftrightarrow b) \dn (\text{atom}(a)\;\;\text{atom}(b))$
\end{tabular}
\end{center}\bigskip\medskip
\onslide<3->
{I would like to have $a \fresh x$, $(a\; b)$, \ldots}
\end{itemize}
\end{frame}}
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*}
text_raw {*
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\mode<presentation>{
\begin{frame}<1-2>[c]
\frametitle{\begin{tabular}{c}\LARGE{}End of Part I\end{tabular}}
\mbox{}\\[-3mm]
\begin{itemize}
\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
\item allow sort-disrespecting swappings\onslide<2->{: just define them as the identity}
\end{itemize}
\end{frame}}
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*}
text_raw {*
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\mode<presentation>{
\begin{frame}<1-2>
\frametitle{\begin{tabular}{c}\LARGE{}Part II: General Bindings\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 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}}
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*}
text_raw {*
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\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}}
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*}
text_raw {*
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\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}}
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*}
text_raw {*
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\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}}
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*}
text_raw {*
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\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}}
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*}
text_raw {*
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\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}}
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*}
text_raw {*
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\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}}
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*}
text_raw {*
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\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}}
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*}
text_raw {*
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\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}}
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*}
text_raw {*
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\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}}
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*}
text_raw {*
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\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}}
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*}
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}}
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\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}}
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*}
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\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}}
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*}
text_raw {*
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\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}}
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*}
text_raw {*
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\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}}
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*}
text_raw {*
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\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}}
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*}
text_raw {*
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\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}}
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*}
text_raw {*
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\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}}
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*}
text_raw {*
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\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}}
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*}
text_raw {*
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\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}}
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*}
text_raw {*
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\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}}
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*}
(*<*)
end
(*>*)