Alternate version of Nominal_Base: Executable version.
(*<*)
theory Slides1
imports "~~/src/HOL/Library/LaTeXsugar" "Nominal"
begin
notation (latex output)
set ("_") and
Cons ("_::/_" [66,65] 65)
(*>*)
text_raw {*
%%\renewcommand{\slidecaption}{Cambridge, 8.~June 2010}
\renewcommand{\slidecaption}{Uppsala, 3.~March 2011}
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\mode<presentation>{
\begin{frame}<1>[t]
\frametitle{%
\begin{tabular}{@ {\hspace{-3mm}}c@ {}}
\\
\huge Nominal Isabelle 2\\[-2mm]
\large Or, How to Reason Conveniently\\[-5mm]
\large with General Bindings\\[5mm]
\end{tabular}}
\begin{center}
Christian Urban
\end{center}
\begin{center}
joint work with {\bf Cezary Kaliszyk}\\[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}Binding in Old Nominal\end{tabular}}
\mbox{}\\[-6mm]
\begin{itemize}
\item the old Nominal Isabelle provided a reasoning infrastructure for single binders\medskip
\begin{center}
Lam [a].(Var a)
\end{center}\bigskip
\item<2-> but representing
\begin{center}
$\forall\{a_1,\ldots,a_n\}.\; T$
\end{center}\medskip
with single binders and reasoning about it is a \alert{\bf major} pain;
take my word for it!
\end{itemize}
\only<1>{
\begin{textblock}{6}(1.5,11)
\small
for example\\
\begin{tabular}{l@ {\hspace{2mm}}l}
& a $\fresh$ Lam [a]. t\\
& Lam [a]. (Var a) \alert{$=$} Lam [b]. (Var b)\\
& Barendregt-style reasoning about bound variables\\
\end{tabular}
\end{textblock}}
\end{frame}}
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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 \textcolor{gray}{(restriction)}\medskip
\item the order does not matter, but the cardinality of the binders
must be the same \textcolor{gray}{(abstraction)}\medskip
\item the order does matter \textcolor{gray}{(iterated single binders)}
\end{itemize}
\onslide<2->{
\begin{center}
\isacommand{bind (set+)}\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) $=$ []}}\\
\multicolumn{2}{l}{\onslide<3->{\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}<1-5>
\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}}
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*}
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}}
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*}
text_raw {*
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\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}}
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*}
text_raw {*
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\mode<presentation>{
\begin{frame}<1-8>
\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}
$(as, x) \onslide<2->{\approx\!}\makebox[0mm][l]{\only<2-6>{${}_{\text{set}}$}%
\only<7>{${}_{\text{\alert{list}}}$}%
\only<8>{${}_{\text{\alert{set+}}}$}}%
\onslide<3->{^{R,\text{fv}}}\,\onslide<2->{(bs,y)}$
\end{tabular}\bigskip
\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
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<5->{$\;\;\;\wedge$} & \onslide<5->{$(\pi \act x)\;R\;y$}\\[1mm]
& \onslide<6-7>{$\;\;\;\wedge$} & \onslide<6-7>{$\pi \act as = bs$}\\
\end{tabular}
\end{textblock}}
\only<7>{
\begin{textblock}{7}(3,13.8)
\footnotesize $^*$ $as$ and $bs$ are \alert{lists} of names
\end{textblock}}
\end{frame}}
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*}
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}<1-3>
\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<3->{
\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}{set+:}\\
$\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
$\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
$\wedge$ & $\pi \cdot x = y$\\
\\
\end{tabular}
\end{minipage}};
\end{tikzpicture}
\end{textblock}}
\only<3->{
\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<3->{
\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 $\approx_{\text{set+}}$, $\approx_{\text{set}}$%
\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}{set+:}\\
$\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
$\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
$\wedge$ & $\pi \cdot x = y$\\
\\
\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}<1-2>
\frametitle{\begin{tabular}{c}Examples\end{tabular}}
\mbox{}\\[-3mm]
\begin{center}
\only<1>{$(\{x\}, x) \approx_? (\{x, y\}, x)$}
\end{center}
\begin{itemize}
\item $\approx_{\text{set+}}$, $\not\approx_{\text{set}}$,
$\not\approx_{\text{list}}$
\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}{set+:}\\
$\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
$\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
$\wedge$ & $\pi \cdot x = y$\\
\\
\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}}
\only<2>{
\begin{textblock}{6}(2.5,4)
\begin{tikzpicture}
\draw (0,0) node[inner sep=5mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
{\normalsize
\begin{minipage}{8cm}\raggedright
\begin{itemize}
\item \color{darkgray}$\alpha$-equivalences coincide when a single name is
abstracted
\item \color{darkgray}in that case they are equivalent to ``old-fashioned'' definitions of $\alpha$
\end{itemize}
\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}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}}
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*}
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\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}}
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*}
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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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*}
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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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*}
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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 {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}}
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*}
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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}
\[
\infer[\text{Let-}\!\approx_\alpha]
{\text{Let}\;as\;t \approx_\alpha \text{Let}\;as'\;t'}
{(\text{bn}(as), t) \approx\!\makebox[0mm][l]{${}_{\text{list}}$}
^{\approx_\alpha,\text{fv}} (\text{bn}(as'), t') &
\onslide<2->{as \approx_\alpha^{\text{bn}} as'}}
\]\bigskip
\onslide<1->{\small{}bn-function $\Rightarrow$ \alert{deep binders}}
\end{frame}}
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*}
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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}\LARGE``Raw'' Alpha-Equivalence\end{tabular}}
\mbox{}\\[6mm]
\begin{center}
LetRec as::assn t::trm \hspace{10mm}\isacommand{bind} bn(as) \isacommand{in} t \alert{as}\\
\end{center}
\[\mbox{}\hspace{-4mm}
\infer[\text{LetRec-}\!\approx_\alpha]
{\text{LetRec}\;as\;t \approx_\alpha \text{LetRec}\;as'\;t'}
{(\text{bn}(as), (t, as)) \approx\!\makebox[0mm][l]{${}_{\text{list}}$}
^{R,\text{fv}} (\text{bn}(as'), (t', as'))}
\]\bigskip
\onslide<1->{\alert{deep recursive binders}}
\end{frame}}
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*}
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\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{frame}}
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*}
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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 our 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
\end{itemize}
\end{frame}}
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*}
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\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}}
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*}
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\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::assn t::trm
& \isacommand{bind} bn(as) \isacommand{in} t\\
\multicolumn{2}{l}{\isacommand{and} assn $=$}\\
\multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} ANil}\\
\multicolumn{2}{l}{\hspace{5mm}$|$ ACons name trm assn}\\
\multicolumn{2}{l}{\isacommand{binder} bn \isacommand{where}}\\
\multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} bn(ANil) $=$ $[]$}\\
\multicolumn{2}{l}{\hspace{5mm}$|$ bn(ACons a t as) $=$ $[$a$]$ @ bn(as)}\\
\end{tabular}\bigskip\medskip
we cannot quotient assn: ACons a \ldots $\not\approx_\alpha$ ACons b \ldots
\only<1->{
\begin{textblock}{8}(0.2,7.3)
\alert{\begin{tabular}{p{2.6cm}}
\raggedright\footnotesize{}Should a ``naked'' assn be quotient?
\end{tabular}\hspace{-3mm}
$\begin{cases}
\mbox{} \\ \mbox{}
\end{cases}$}
\end{textblock}}
\end{frame}}
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*}
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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
(Barendregt's variable convention is unsound in general,
found bugs in two paper proofs, quotient package, POPL 2011 tutorial)\medskip
\end{itemize}
\end{frame}}
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*}
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\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}}
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*}
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\mode<presentation>{
\begin{frame}<1->[c]
\frametitle{\begin{tabular}{c}Questions?\end{tabular}}
\mbox{}\\[-6mm]
\begin{center}
\alert{\huge{Thanks!}}
\end{center}
\end{frame}}
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*}
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\mode<presentation>{
\begin{frame}<1-2>[c]
\frametitle{\begin{tabular}{c}Examples\end{tabular}}
\mbox{}\\[-6mm]
\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}
\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}
\onslide<2->
{1.) \hspace{3mm}\isacommand{bind (set)} as \isacommand{in} $\tau_1$,
\isacommand{bind (set)} as \isacommand{in} $\tau_2$\medskip
2.) \hspace{3mm}\isacommand{bind (set)} as \isacommand{in} $\tau_1$ $\tau_2$
}
\end{frame}}
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*}
(*<*)
end
(*>*)