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(*<*)
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theory Slides3
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imports "LaTeXsugar" "Nominal"
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begin
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notation (latex output)
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set ("_") and
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Cons ("_::/_" [66,65] 65)
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(*>*)
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text_raw {*
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\renewcommand{\slidecaption}{UNIF, Edinburgh, 14.~July 2010}
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\newcommand{\abst}[2]{#1.#2}% atom-abstraction
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\newcommand{\pair}[2]{\langle #1,#2\rangle} % pairing
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\newcommand{\susp}{{\boldsymbol{\cdot}}}% for suspensions
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\newcommand{\unit}{\langle\rangle}% unit
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\newcommand{\app}[2]{#1\,#2}% application
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\newcommand{\eqprob}{\mathrel{{\approx}?}}
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\pgfdeclareradialshading{smallbluesphere}{\pgfpoint{0.5mm}{0.5mm}}%
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{rgb(0mm)=(0,0,0.9);
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rgb(0.9mm)=(0,0,0.7);
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rgb(1.3mm)=(0,0,0.5);
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rgb(1.4mm)=(1,1,1)}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}<1>[c]
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\frametitle{Quiz}
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Assuming that \smath{a} and \smath{b} are distinct variables,\\
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is it possible to find $\lambda$-terms \smath{M_1} to \smath{M_7}
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that make the following pairs \alert{$\alpha$-equivalent}?
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\begin{tabular}{@ {\hspace{14mm}}p{12cm}}
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\begin{itemize}
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\item \smath{\lambda a.\lambda b. (M_1\,b)\;} and
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\smath{\lambda b.\lambda a. (a\,M_1)\;}
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\item \smath{\lambda a.\lambda b. (M_2\,b)\;} and
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\smath{\lambda b.\lambda a. (a\,M_3)\;}
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\item \smath{\lambda a.\lambda b. (b\,M_4)\;} and
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\smath{\lambda b.\lambda a. (a\,M_5)\;}
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\item \smath{\lambda a.\lambda b. (b\,M_6)\;} and
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\smath{\lambda a.\lambda a. (a\,M_7)\;}
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\end{itemize}
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\end{tabular}
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If there is one solution for a pair, can you describe all its solutions?
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}<1>[t]
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\frametitle{%
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\begin{tabular}{@ {\hspace{-3mm}}c@ {}}
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\\
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\huge Nominal Unification\\[-2mm]
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\Large Hitting a Sweet Spot\\[5mm]
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\end{tabular}}
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\begin{center}
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Christian Urban
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\end{center}
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\begin{center}
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\small initial work with Andy Pitts and Jamie Gabbay\\[0mm]
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\end{center}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}<1-4>[c]
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\frametitle{One Motivation}
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\onslide<2->{Typing implemented in Prolog \textcolor{darkgray}{(from a textbook)}}\bigskip\\
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\onslide<3->{
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\begin{tabular}{l}
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type (Gamma, var(X), T) :- member (X,T) Gamma.\smallskip\medskip\\
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type (Gamma, app(M, N), T') :-\\
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\hspace{3cm}type (Gamma, M, arrow(T, T')),\\
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\hspace{3cm}type (Gamma, N, T).\smallskip\medskip\\
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type (Gamma, lam(X, M), arrow(T, T')) :-\\
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\hspace{3cm}type ((X, T)::Gamma, M, T').\smallskip\medskip\\
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member X X::Tail.\\
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member X Y::Tail :- member X Tail.\\
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\end{tabular}}
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\only<4>{
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\begin{textblock}{6}(2.5,2)
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\begin{tikzpicture}
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\draw (0,0) node[inner sep=3mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
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{\color{darkgray}
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\begin{minipage}{8cm}\raggedright
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The problem is that \smath{\lambda x.\lambda x. (x\;x)}
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gets the types
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\begin{center}
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\begin{tabular}{l}
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\smath{T\rightarrow (T\rightarrow S) \rightarrow S} and\\
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\smath{(T\rightarrow S)\rightarrow T \rightarrow S}\\
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\end{tabular}
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\end{center}
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\end{minipage}};
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\end{tikzpicture}
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\end{textblock}}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}<1>[c]
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\frametitle{Higher-Order Unification}
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\begin{itemize}
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\item Lambda Prolog with full Higher-Order Unification\\
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\textcolor{darkgray}{(no mgus, undecidable, modulo $\alpha\beta$)}\bigskip
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\item Higher-Order Pattern Unification\\
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\textcolor{darkgray}{(has mgus, decidable, some restrictions, modulo $\alpha\beta_0$)}
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\end{itemize}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}<1-10>[t]
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\frametitle{Underlying Ideas}
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\begin{itemize}
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\item<1-> Unification (\alert{only}) up to $\alpha$
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\item<2-> Swappings / Permutations
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\only<2-5>{
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\begin{center}
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\begin{tabular}{r@ {\hspace{1mm}}l@ {\hspace{12mm}}r@ {\hspace{1mm}}l}
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\only<2>{\smath{\textcolor{white}{[b\!:=\!a]}}}%
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\only<3>{\smath{[b\!:=\!a]}}%
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\only<4-5>{\smath{\alert{\swap{a}{b}\,\act}}} &
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\onslide<2-5>{\smath{\lambda a.b}} &
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\only<2>{\smath{\textcolor{white}{[b\!:=\!a]}}}%
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\only<3>{\smath{[b\!:=\!a]}}%
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\only<4-5>{\smath{\alert{\swap{a}{b}\,\act}}} &
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\onslide<2-5>{\smath{\lambda c.b}}\\
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\onslide<3-5>{\smath{=}} & \only<3>{\smath{\lambda a.a}}\only<4-5>{\smath{\lambda b.a}} &
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\onslide<3-5>{\smath{=}} & \only<3>{\smath{\lambda c.a}}\only<4-5>{\smath{\lambda c.a}}\\
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\end{tabular}
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\end{center}\bigskip
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\onslide<4-5>{
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\begin{center}
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\begin{tikzpicture}
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\draw (0,0) node[inner sep=0mm,fill=cream, ultra thick, draw=cream]
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{\begin{minipage}{8cm}
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\begin{tabular}{r@ {\hspace{3mm}}l}
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\smath{\swap{a}{b}\act t} $\;\dn$ & \alert{swap} {\bf all} occurences of\\
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& \smath{b} and \smath{a} in \smath{t}
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\end{tabular}
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\end{minipage}};
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\end{tikzpicture}
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\end{center}}\bigskip
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\onslide<5>{
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Unlike for \smath{[b\!:=\!a]\act(-)}, for \smath{\swap{a}{b}\act (-)} we do
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have if \smath{t =_\alpha t'} then \smath{\pi \act t =_\alpha \pi \act t'.}}}
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\item<6-> Variables (or holes)\bigskip
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\begin{center}
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\onslide<7->{\mbox{}\hspace{-25mm}\smath{\lambda x\hspace{-0.5mm}s .}}
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\onslide<8-9>{\raisebox{-1.7mm}{\huge\smath{(}}}\raisebox{-4mm}{\begin{tikzpicture}
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\fill[blue] (0, 0) circle (5mm);
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\end{tikzpicture}}
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\onslide<8-9>{\smath{y\hspace{-0.5mm}s}{\raisebox{-1.7mm}{\huge\smath{)}}}}\bigskip
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\end{center}
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\only<8-9>{\smath{y\hspace{-0.5mm}s} are the parameters the hole can depend on\onslide<9->{, but
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then you need $\beta_0$-reduction\medskip
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\begin{center}
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\smath{(\lambda x. t) y \longrightarrow_{\beta_0} t[x:=y]}
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\end{center}}}
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\only<10>{we will record the information about which parameters a hole
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\alert{\bf cannot} depend on}
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\end{itemize}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}<1-4>[c]
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\frametitle{Terms}
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\begin{tabular}{lll @ {\hspace{10mm}}lll}
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\onslide<1->{\pgfuseshading{smallbluesphere}} &
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\onslide<1->{\colorbox{cream}{\smath{\unit}}} &
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\onslide<1->{Units} &
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\onslide<2->{\pgfuseshading{smallbluesphere}} &
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\onslide<2->{\colorbox{cream}{\smath{a}}} &
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\onslide<2->{Atoms} \\[5mm]
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\onslide<1->{\pgfuseshading{smallbluesphere}} &
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\onslide<1->{\colorbox{cream}{\smath{\pair{t}{t'}}}} &
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\onslide<1->{Pairs} &
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\onslide<3->{\pgfuseshading{smallbluesphere}} &
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\onslide<3->{\colorbox{cream}{\smath{\abst{a}{t}}}} &
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\onslide<3->{Abstractions}\\[5mm]
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\onslide<1->{\pgfuseshading{smallbluesphere}} &
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\onslide<1->{\colorbox{cream}{\smath{\app{F}{t}}}} &
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\onslide<1->{Funct.} &
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\onslide<4->{\pgfuseshading{smallbluesphere}} &
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\onslide<4->{\colorbox{cream}{\smath{\pi\susp X}}} &
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\onslide<4->{Suspensions}
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\end{tabular}
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\only<2>{
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\begin{textblock}{13}(1.5,12)
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\small Atoms are constants \textcolor{darkgray}{(infinitely many of them)}
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\end{textblock}}
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\only<3>{
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\begin{textblock}{13}(1.5,12)
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\small \smath{\ulcorner \lambda\abst{a}{a}\urcorner \mapsto \text{fn\ }\abst{a}{a}}\\
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\small constructions like \smath{\text{fn\ }\abst{X}{X}} are not allowed
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\end{textblock}}
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\only<4>{
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\begin{textblock}{13}(1.5,12)
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\small \smath{X} is a variable standing for a term\\
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\small \smath{\pi} is an explicit permutation \smath{\swap{a_1}{b_1}\ldots\swap{a_n}{b_n}},
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waiting to be applied to the term that is substituted for \smath{X}
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\end{textblock}}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}<1-3>[c]
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\frametitle{Permutations}
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a permutation applied to a term
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\begin{center}
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\begin{tabular}{lrcl}
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\pgfuseshading{smallbluesphere} &
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\smath{[]\act c} & \smath{\dn} & \smath{c} \\
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\pgfuseshading{smallbluesphere} &
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\smath{\swap{a}{b}\!::\!\pi\act c} & \smath{\dn} &
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\smath{\begin{cases}
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a & \text{if}\;\pi\act c = b\\
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b & \text{if}\;\pi\act c = a\\
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\pi\act c & \text{otherwise}
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\end{cases}}\\
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\onslide<2->{\pgfuseshading{smallbluesphere}} &
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\onslide<2->{\smath{\pi\act\abst{a}{t}}} & \onslide<2->{\smath{\dn}} &
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\onslide<2->{\smath{\abst{\pi\act a}{\pi\act t}}}\\
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\onslide<3->{\pgfuseshading{smallbluesphere}} &
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\onslide<3->{\smath{\pi\act\pi'\act X}} & \onslide<3->{\smath{\dn}} &
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\onslide<3->{\smath{(\pi @ \pi')\act X}}\\
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\end{tabular}
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\end{center}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}<1-3>[c]
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\frametitle{Freshness Constraints}
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Recall \smath{\lambda a. \raisebox{-0.7mm}{\tikz \fill[blue] (0, 0) circle (2.5mm);}}
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\bigskip\pause
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We therefore will identify
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\begin{center}
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\smath{\mathtt{fn\ } a. X \;\approx\; \mathtt{fn\ } b. \alert<3->{\swap{a}{b}}\act X}
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\end{center}
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provided that `\smath{b} is fresh for \smath{X} --- (\smath{b\fresh X})',
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i.e., does not occur freely in any ground term that might be substituted for
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\smath{X}.\bigskip\pause
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If we know more about \smath{X}, e.g., if we knew that \smath{a\fresh X} and
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\smath{b\fresh X}, then we can replace\\ \smath{\swap{a}{b}\act X} by
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\smath{X}.
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\end{frame}}
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*}
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text_raw {*
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\mode<presentation>{
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\begin{frame}<1-4>[c]
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\frametitle{Equivalence Judgements}
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\alt<1>{Our equality is {\bf not} just}{but judgements}
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\begin{center}
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\begin{tabular}{rl}
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\colorbox{cream}{\smath{\onslide<2->{\nabla \vdash} t \approx t'}} & \alert{$\alpha$-equivalence}\\[1mm]
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\onslide<4->{\colorbox{cream}{\smath{\onslide<2->{\nabla \vdash} a \fresh t}}} &
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\onslide<4->{\alert{freshness}}
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\end{tabular}
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\end{center}
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\onslide<2->{
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where
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\begin{center}
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\smath{\nabla = \{a_1\fresh X_1,\ldots, a_n\fresh X_n\}}
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\end{center}
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is a finite set of \alert{freshness assumptions}.}
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\onslide<3->{
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\begin{center}
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\smath{\{a\fresh X,b\fresh X\} \vdash \text{fn\ } a. X \approx \text{fn\ } b. X}
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\end{center}}
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\end{frame}}
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*}
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text_raw {*
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\mode<presentation>{
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\begin{frame}<1>[c]
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\frametitle{Rules for Equivalence}
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\begin{center}
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\begin{tabular}{c}
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Excerpt\\
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(i.e.~only the interesting rules)
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\end{tabular}
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\end{center}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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\mode<presentation>{
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\begin{frame}<1>[c]
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\frametitle{Rules for Equivalence}
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\begin{center}
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\begin{tabular}{c}
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\colorbox{cream}{\smath{\infer{\nabla \vdash a \approx a}{}}}\\[8mm]
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\colorbox{cream}{%
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\smath{\infer{\nabla \vdash \abst{a}{t} \approx \abst{a}{t'}}
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{\nabla \vdash t \approx t'}}}\\[8mm]
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\colorbox{cream}{%
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\smath{\infer{\nabla \vdash \abst{a}{t} \approx \abst{b}{t'}}
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{a\not=b\;\; & \nabla \vdash t \approx \swap{a}{b}\act t'\;\;& \nabla \vdash a\fresh t'}}}
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\end{tabular}
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\end{center}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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\mode<presentation>{
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\begin{frame}<1-3>[c]
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\frametitle{Rules for Equivalence}
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|
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|
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\begin{center}
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|
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\colorbox{cream}{%
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\smath{%
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\infer{\nabla \vdash \pi\act X \approx \pi'\act X}
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{\begin{array}{c}
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(a\fresh X)\in\nabla\\
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\text{for all}\; a \;\text{with}\;\pi\act a \not= \pi'\act a
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\end{array}
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}}}
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\end{center}
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\onslide<2->{
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for example\\[4mm]
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421 |
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422 |
\alt<2>{%
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423 |
\begin{center}
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\smath{\{a\fresh\!X, b\fresh\!X\} \vdash X \approx \swap{a}{b}\act X}
|
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425 |
\end{center}}
|
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|
426 |
{%
|
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\begin{center}
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\smath{\{a\fresh\!X, c\fresh\!X\} \vdash \swap{a}{c}\swap{a}{b}\act X \approx \swap{b}{c}\act X}
|
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|
429 |
\end{center}}
|
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|
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|
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\onslide<3->{
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\begin{tabular}{@ {}lllll@ {}}
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because &
|
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\smath{\swap{a}{c}\swap{a}{b}}: &
|
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|
435 |
\smath{a\mapsto b} &
|
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436 |
\smath{\swap{b}{c}}: &
|
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|
437 |
\smath{a\mapsto a}\\
|
|
|
438 |
& & \smath{b\mapsto c} & & \smath{b\mapsto c}\\
|
|
|
439 |
& & \smath{c\mapsto a} & & \smath{c\mapsto b}\\
|
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440 |
\end{tabular}
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|
441 |
disagree at \smath{a} and \smath{c}.}
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442 |
}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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\mode<presentation>{
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\begin{frame}<1>[c]
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452 |
\frametitle{Rules for Freshness}
|
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|
453 |
|
|
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|
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\begin{center}
|
|
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|
455 |
\begin{tabular}{c}
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|
456 |
Excerpt\\
|
|
|
457 |
(i.e.~only the interesting rules)
|
|
2351
|
458 |
\end{tabular}
|
|
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|
459 |
\end{center}
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460 |
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461 |
\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}<1>[c]
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469 |
\frametitle{Rules for Freshness}
|
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2351
|
470 |
|
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|
471 |
\begin{center}
|
|
|
472 |
\begin{tabular}{c}
|
|
|
473 |
\colorbox{cream}{%
|
|
|
474 |
\smath{\infer{\nabla \vdash a\fresh b}{a\not= b}}}\\[5mm]
|
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|
475 |
|
|
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|
476 |
\colorbox{cream}{%
|
|
|
477 |
\smath{\infer{\nabla \vdash a\fresh\abst{a}{t}}{}}}\hspace{7mm}
|
|
|
478 |
\colorbox{cream}{%
|
|
|
479 |
\smath{\infer{\nabla \vdash a\fresh\abst{b}{t}}
|
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|
480 |
{a\not= b\;\; & \nabla \vdash a\fresh t}}}\\[5mm]
|
|
|
481 |
|
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|
482 |
\colorbox{cream}{%
|
|
|
483 |
\smath{\infer{\nabla \vdash a\fresh \pi\act X}
|
|
|
484 |
{(\pi^{-1}\act a\fresh X)\in\nabla}}}
|
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2351
|
485 |
\end{tabular}
|
|
|
486 |
\end{center}
|
|
|
487 |
|
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|
488 |
\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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\mode<presentation>{
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|
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\begin{frame}<1-4>[t]
|
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|
496 |
\frametitle{$\approx$ is an Equivalence}
|
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|
497 |
\mbox{}\\[5mm]
|
|
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|
498 |
|
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|
499 |
\begin{center}
|
|
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|
500 |
\colorbox{cream}{\alert{Theorem:}
|
|
|
501 |
$\approx$ is an equivalence relation.}
|
|
|
502 |
\end{center}\bigskip
|
|
2351
|
503 |
|
|
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|
504 |
\only<1>{%
|
|
|
505 |
\begin{tabular}{ll}
|
|
|
506 |
(Reflexivity) & $\smath{\nabla\vdash t\approx t}$\\[2mm]
|
|
|
507 |
(Symmetry) & if $\smath{\nabla\vdash t_1\approx t_2}\;$
|
|
|
508 |
then $\;\smath{\nabla\vdash t_2\approx t_1}$\\[2mm]
|
|
|
509 |
(Transitivity) & if $\smath{\nabla\vdash t_1\approx t_2}\;$ and
|
|
|
510 |
$\;\smath{\nabla\vdash t_2\approx t_3}$\\
|
|
|
511 |
& then $\smath{\nabla\vdash t_1\approx t_3}$\\
|
|
|
512 |
\end{tabular}}
|
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2351
|
513 |
|
|
2356
|
514 |
\only<2->{%
|
|
|
515 |
\begin{itemize}
|
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|
516 |
\item<2-> \smath{\nabla \vdash t\approx t'} then \smath{\nabla \vdash \pi\act t\approx \pi\act t'}
|
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2351
|
517 |
|
|
2356
|
518 |
\item<2-> \smath{\nabla \vdash a\fresh t} then
|
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|
519 |
\smath{\nabla \vdash \pi\act a\fresh \pi\act t}
|
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2351
|
520 |
|
|
2356
|
521 |
\item<3-> \smath{\nabla \vdash t\approx \pi\act t'} then
|
|
|
522 |
\smath{\nabla \vdash (\pi^{-1})\act t\approx t'}
|
|
|
523 |
|
|
|
524 |
\item<3-> \smath{\nabla \vdash a\fresh \pi\act t} then
|
|
|
525 |
\smath{\nabla \vdash (\pi^{-1})\act a\fresh t}
|
|
2351
|
526 |
|
|
2356
|
527 |
\item<4-> \smath{\nabla \vdash a\fresh t} and \smath{\nabla \vdash t\approx t'} then
|
|
|
528 |
\smath{\nabla \vdash a\fresh t'}
|
|
|
529 |
\end{itemize}
|
|
|
530 |
}
|
|
|
531 |
|
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2351
|
532 |
\end{frame}}
|
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|
533 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
534 |
*}
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535 |
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536 |
text_raw {*
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537 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
538 |
\mode<presentation>{
|
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|
539 |
\begin{frame}<1-4>
|
|
|
540 |
\frametitle{Comparison $=_\alpha$}
|
|
2351
|
541 |
|
|
2356
|
542 |
Traditionally \smath{=_\alpha} is defined as
|
|
2351
|
543 |
|
|
|
544 |
\begin{center}
|
|
2356
|
545 |
\colorbox{cream}{%
|
|
|
546 |
\begin{minipage}{9cm}
|
|
|
547 |
\raggedright least congruence which identifies \smath{\abst{a}{t}}
|
|
|
548 |
with \smath{\abst{b}{[a:=b]t}} provided \smath{b} is not free
|
|
|
549 |
in \smath{t}
|
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|
550 |
\end{minipage}}
|
|
|
551 |
\end{center}
|
|
2351
|
552 |
|
|
2356
|
553 |
where \smath{[a:=b]t} replaces all free occurrences of\\
|
|
|
554 |
\smath{a} by \smath{b} in \smath{t}.
|
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|
555 |
\bigskip
|
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|
556 |
|
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|
557 |
\only<2>{%
|
|
|
558 |
\begin{textblock}{13}(1.2,10)
|
|
|
559 |
For \alert{ground} terms:
|
|
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|
560 |
|
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|
561 |
\begin{center}
|
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|
562 |
\colorbox{cream}{%
|
|
|
563 |
\begin{minipage}{9.0cm}
|
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|
564 |
\begin{tabular}{@ {}rl}
|
|
|
565 |
\underline{Theorem:}
|
|
|
566 |
& \smath{t=_\alpha t'\;\;} if\hspace{-0.5mm}f~\smath{\;\;\emptyset \vdash t\approx t'}\\[2mm]
|
|
|
567 |
& \smath{a\not\in FA(t)\;\;} if\hspace{-0.5mm}f~\smath{\;\;\emptyset\vdash a\fresh t}
|
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2351
|
568 |
\end{tabular}
|
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|
569 |
\end{minipage}}
|
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|
570 |
\end{center}
|
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|
571 |
\end{textblock}}
|
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|
572 |
|
|
|
573 |
\only<3>{%
|
|
|
574 |
\begin{textblock}{13}(1.2,10)
|
|
|
575 |
In general \smath{=_\alpha} and \smath{\approx} are distinct!
|
|
|
576 |
\begin{center}
|
|
|
577 |
\colorbox{cream}{%
|
|
|
578 |
\begin{minipage}{6.0cm}
|
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|
579 |
\smath{\abst{a}{X}=_\alpha \abst{b}{X}\;} but not\\[2mm]
|
|
|
580 |
\smath{\emptyset \vdash \abst{a}{X} \approx \abst{b}{X}\;} (\smath{a\not=b})
|
|
|
581 |
\end{minipage}}
|
|
|
582 |
\end{center}
|
|
|
583 |
\end{textblock}}
|
|
|
584 |
|
|
|
585 |
\only<4>{
|
|
|
586 |
\begin{textblock}{6}(1,2)
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|
587 |
\begin{tikzpicture}
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|
588 |
\draw (0,0) node[inner sep=3mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
|
|
|
589 |
{\color{darkgray}
|
|
|
590 |
\begin{minipage}{10cm}\raggedright
|
|
|
591 |
That is a crucial point: if we had\\[-2mm]
|
|
|
592 |
\[\smath{\emptyset \vdash \abst{a}{X}\approx \abst{b}{X}}\mbox{,}\]
|
|
|
593 |
then applying $\smath{[X:=a]}$, $\smath{[X:=b]}$, $\ldots$\\
|
|
|
594 |
give two terms that are {\bf not} $\alpha$-equivalent.\\[3mm]
|
|
|
595 |
The freshness constraints $\smath{a\fresh X}$ and $\smath{b\fresh X}$
|
|
|
596 |
rule out the problematic substitutions. Therefore
|
|
|
597 |
|
|
|
598 |
\[\smath{\{a\fresh X,b\fresh X\} \vdash \abst{a}{X}\approx \abst{b}{X}}\]
|
|
|
599 |
|
|
|
600 |
does hold.
|
|
2351
|
601 |
\end{minipage}};
|
|
|
602 |
\end{tikzpicture}
|
|
|
603 |
\end{textblock}}
|
|
|
604 |
|
|
|
605 |
\end{frame}}
|
|
|
606 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
607 |
*}
|
|
|
608 |
|
|
|
609 |
text_raw {*
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|
610 |
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|
|
|
611 |
\mode<presentation>{
|
|
2356
|
612 |
\begin{frame}<1-9>
|
|
|
613 |
\frametitle{Substitution}
|
|
|
614 |
|
|
|
615 |
\begin{tabular}{l@ {\hspace{8mm}}r@ {\hspace{1.5mm}}c@ {\hspace{1.5mm}}l@ {}}
|
|
|
616 |
\pgfuseshading{smallbluesphere} &
|
|
|
617 |
\smath{\sigma(\abst{a}{t})} & \smath{\dn} & \smath{\abst{a}{\sigma(t)}}\\[2mm]
|
|
2351
|
618 |
|
|
2356
|
619 |
\pgfuseshading{smallbluesphere} &
|
|
|
620 |
\smath{\sigma(\pi\act X)} & \smath{\dn} &
|
|
|
621 |
\smath{\begin{cases}%
|
|
|
622 |
\pi\;\act\;\sigma(X) & \!\!\text{if\ } \sigma(X)\not=X\\
|
|
|
623 |
\pi\act X & \!\!\text{otherwise}%
|
|
|
624 |
\end{cases}}\\[6mm]
|
|
|
625 |
\end{tabular}\bigskip\bigskip
|
|
2351
|
626 |
|
|
2356
|
627 |
\pause
|
|
|
628 |
\only<2-5>{
|
|
|
629 |
\only<2->{for example}
|
|
|
630 |
\def\arraystretch{1.3}
|
|
|
631 |
\begin{tabular}{@ {\hspace{14mm}}l@ {\hspace{3mm}}l}
|
|
|
632 |
\onslide<2->{\textcolor{white}{$\Rightarrow$}} &
|
|
|
633 |
\onslide<2->{\alt<3>{\smath{\underline{\abst{a}{\swap{a}{b}\act X}\;\,[X:=\pair{b}{Y}]}}}
|
|
|
634 |
{\smath{\abst{a}{\swap{a}{b}\act X}\;\,[X:=\pair{b}{Y}]}}}\\
|
|
|
635 |
\onslide<3->{\smath{\Rightarrow}} &
|
|
|
636 |
\onslide<3->{\alt<3,4>{\smath{\abst{a}{\underline{\swap{a}{b}\act X[X:=\pair{b}{Y}]}}}}
|
|
|
637 |
{\smath{\abst{a}{\swap{a}{b}\act X}[X:=\pair{b}{Y}]}}}\\
|
|
|
638 |
\onslide<4->{\smath{\Rightarrow}} &
|
|
|
639 |
\onslide<4->{\alt<4>{\smath{\abst{a}{\swap{a}{b}\act \underline{\pair{b}{Y}}}}}
|
|
|
640 |
{\smath{\abst{a}{\underline{\swap{a}{b}}\act \pair{b}{Y}}}}}\\
|
|
|
641 |
\onslide<5->{\smath{\Rightarrow}} &
|
|
|
642 |
\onslide<5->{\smath{\abst{a}{\pair{a}{\swap{a}{b}\act Y}}}}
|
|
|
643 |
\end{tabular}}
|
|
2351
|
644 |
|
|
2356
|
645 |
\only<6->
|
|
|
646 |
{\begin{tabular}{l@ {\hspace{8mm}}l@ {}}
|
|
|
647 |
\pgfuseshading{smallbluesphere} &
|
|
|
648 |
if \smath{\nabla\vdash t\approx t'} and\hspace{-2mm}\mbox{}
|
|
|
649 |
\raisebox{-2.7mm}{
|
|
|
650 |
\alt<7>{\begin{tikzpicture}
|
|
|
651 |
\draw (0,0) node[inner sep=1mm,fill=cream, very thick, draw=red, rounded corners=3mm]
|
|
|
652 |
{\smath{\;\nabla'\vdash\sigma(\nabla)\;}};
|
|
|
653 |
\end{tikzpicture}}
|
|
|
654 |
{\begin{tikzpicture}
|
|
|
655 |
\draw (0,0) node[inner sep=1mm,fill=white, very thick, draw=white, rounded corners=3mm]
|
|
|
656 |
{\smath{\;\nabla'\vdash\sigma(\nabla)\;}};
|
|
|
657 |
\end{tikzpicture}}}\\
|
|
|
658 |
& then \smath{\nabla'\vdash\sigma(t)\approx\sigma(t')}
|
|
|
659 |
\end{tabular}}
|
|
2351
|
660 |
|
|
2356
|
661 |
\only<9>
|
|
|
662 |
{\begin{tabular}{l@ {\hspace{8mm}}l@ {}}
|
|
|
663 |
\\[-4mm]
|
|
|
664 |
\pgfuseshading{smallbluesphere} &
|
|
|
665 |
\smath{\sigma(\pi\act t)=\pi\act\sigma(t)}
|
|
|
666 |
\end{tabular}}
|
|
2351
|
667 |
|
|
|
668 |
|
|
2356
|
669 |
\only<7>{
|
|
|
670 |
\begin{textblock}{6}(10,10.5)
|
|
2351
|
671 |
\begin{tikzpicture}
|
|
2356
|
672 |
\draw (0,0) node[inner sep=1mm,fill=cream, very thick, draw=red, rounded corners=2mm]
|
|
|
673 |
{\color{darkgray}
|
|
|
674 |
\begin{minipage}{3.8cm}\raggedright
|
|
|
675 |
this means\\[1mm]
|
|
|
676 |
\smath{\nabla'\vdash a\fresh\sigma(X)}\\[1mm]
|
|
|
677 |
holds for all\\[1mm]
|
|
|
678 |
\smath{(a\fresh X)\in\nabla}
|
|
2351
|
679 |
\end{minipage}};
|
|
|
680 |
\end{tikzpicture}
|
|
|
681 |
\end{textblock}}
|
|
|
682 |
|
|
|
683 |
\end{frame}}
|
|
|
684 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
685 |
*}
|
|
|
686 |
|
|
|
687 |
text_raw {*
|
|
|
688 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
689 |
\mode<presentation>{
|
|
2356
|
690 |
\begin{frame}<1->
|
|
|
691 |
\frametitle{Equational Problems}
|
|
2351
|
692 |
|
|
2356
|
693 |
An equational problem
|
|
|
694 |
\[
|
|
|
695 |
\colorbox{cream}{\smath{t \eqprob t'}}
|
|
|
696 |
\]
|
|
|
697 |
is \alert{solved} by
|
|
2351
|
698 |
|
|
|
699 |
\begin{center}
|
|
2356
|
700 |
\begin{tabular}{ll}
|
|
|
701 |
\pgfuseshading{smallbluesphere} & a substitution \smath{\sigma} (terms for variables)\\[3mm]
|
|
|
702 |
\pgfuseshading{smallbluesphere} & {\bf and} a set of freshness assumptions \smath{\nabla}
|
|
|
703 |
\end{tabular}
|
|
2351
|
704 |
\end{center}
|
|
|
705 |
|
|
2356
|
706 |
so that \smath{\nabla\vdash \sigma(t)\approx \sigma(t')}.
|
|
2351
|
707 |
|
|
|
708 |
|
|
|
709 |
\end{frame}}
|
|
|
710 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
711 |
*}
|
|
|
712 |
|
|
|
713 |
text_raw {*
|
|
|
714 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
715 |
\mode<presentation>{
|
|
|
716 |
\begin{frame}<1->
|
|
2356
|
717 |
\frametitle{Conclusion}
|
|
2351
|
718 |
|
|
|
719 |
\begin{itemize}
|
|
|
720 |
\item the user does not see anything of the raw level\medskip
|
|
|
721 |
\only<1>{\begin{center}
|
|
|
722 |
Lam a (Var a) \alert{$=$} Lam b (Var b)
|
|
|
723 |
\end{center}\bigskip}
|
|
|
724 |
|
|
|
725 |
\item<2-> we have not yet done function definitions (will come soon and
|
|
|
726 |
we hope to make improvements over the old way there too)\medskip
|
|
|
727 |
\item<3-> it took quite some time to get here, but it seems worthwhile
|
|
|
728 |
(Barendregt's variable convention is unsound in general,
|
|
|
729 |
found bugs in two paper proofs, quotient package, POPL 2011 tutorial)\medskip
|
|
|
730 |
\end{itemize}
|
|
|
731 |
|
|
|
732 |
|
|
|
733 |
\end{frame}}
|
|
|
734 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
735 |
*}
|
|
|
736 |
|
|
|
737 |
(*<*)
|
|
|
738 |
end
|
|
|
739 |
(*>*) |