Alternate version of Nominal_Base: Executable version.
(*<*)
theory Slides3
imports "~~/src/HOL/Library/LaTeXsugar" "Nominal"
begin
declare [[show_question_marks = false]]
notation (latex output)
set ("_") and
Cons ("_::/_" [66,65] 65)
(*>*)
text_raw {*
\renewcommand{\slidecaption}{UNIF, Edinburgh, 14.~July 2010}
\newcommand{\abst}[2]{#1.#2}% atom-abstraction
\newcommand{\pair}[2]{\langle #1,#2\rangle} % pairing
\newcommand{\susp}{{\boldsymbol{\cdot}}}% for suspensions
\newcommand{\unit}{\langle\rangle}% unit
\newcommand{\app}[2]{#1\,#2}% application
\newcommand{\eqprob}{\mathrel{{\approx}?}}
\newcommand{\freshprob}{\mathrel{\#?}}
\newcommand{\redu}[1]{\stackrel{#1}{\Longrightarrow}}% reduction
\newcommand{\id}{\varepsilon}% identity substitution
\pgfdeclareradialshading{smallbluesphere}{\pgfpoint{0.5mm}{0.5mm}}%
{rgb(0mm)=(0,0,0.9);
rgb(0.9mm)=(0,0,0.7);
rgb(1.3mm)=(0,0,0.5);
rgb(1.4mm)=(1,1,1)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}<1>[c]
\frametitle{Quiz}
Assuming that \smath{a} and \smath{b} are distinct variables,\\
is it possible to find $\lambda$-terms \smath{M_1} to \smath{M_7}
that make the following pairs \alert{$\alpha$-equivalent}?
\begin{tabular}{@ {\hspace{14mm}}p{12cm}}
\begin{itemize}
\item \smath{\lambda a.\lambda b. (M_1\,b)\;} and
\smath{\lambda b.\lambda a. (a\,M_1)\;}
\item \smath{\lambda a.\lambda b. (M_2\,b)\;} and
\smath{\lambda b.\lambda a. (a\,M_3)\;}
\item \smath{\lambda a.\lambda b. (b\,M_4)\;} and
\smath{\lambda b.\lambda a. (a\,M_5)\;}
\item \smath{\lambda a.\lambda b. (b\,M_6)\;} and
\smath{\lambda a.\lambda a. (a\,M_7)\;}
\end{itemize}
\end{tabular}
If there is one solution for a pair, can you describe all its solutions?
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*}
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}<1>[t]
\frametitle{%
\begin{tabular}{@ {\hspace{-3mm}}c@ {}}
\\
\huge Nominal Unification\\[-2mm]
\Large Hitting a Sweet Spot\\[5mm]
\end{tabular}}
\begin{center}
Christian Urban
\end{center}
\begin{center}
\small initial spark from Roy Dyckhoff in November 2001\\[0mm]
\small joint work with Andy Pitts and Jamie Gabbay\\[0mm]
\end{center}
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*}
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}<1-4>[c]
\frametitle{One Motivation}
\onslide<2->{Typing implemented in Prolog \textcolor{darkgray}{(from a textbook)}}\bigskip\\
\onslide<3->{\color{darkgray}
\begin{tabular}{l}
type (Gamma, var(X), T) :- member (X,T) Gamma.\smallskip\medskip\\
type (Gamma, app(M, N), T') :-\\
\hspace{3cm}type (Gamma, M, arrow(T, T')),\\
\hspace{3cm}type (Gamma, N, T).\smallskip\medskip\\
type (Gamma, lam(X, M), arrow(T, T')) :-\\
\hspace{3cm}type ((X, T)::Gamma, M, T').\smallskip\medskip\\
member X X::Tail.\\
member X Y::Tail :- member X Tail.\\
\end{tabular}}
\only<4>{
\begin{textblock}{6}(2.5,2)
\begin{tikzpicture}
\draw (0,0) node[inner sep=3mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
{\color{darkgray}
\begin{minipage}{8cm}\raggedright
The problem is that \smath{\lambda x.\lambda x. (x\;x)}
will have the types
\begin{center}
\begin{tabular}{l}
\smath{T\rightarrow (T\rightarrow S) \rightarrow S} and\\
\smath{(T\rightarrow S)\rightarrow T \rightarrow S}\\
\end{tabular}
\end{center}
\end{minipage}};
\end{tikzpicture}
\end{textblock}}
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*}
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}<1>[c]
\frametitle{Higher-Order Unification}
State of the art at the time:
\begin{itemize}
\item Lambda Prolog with full Higher-Order Unification\\
\textcolor{darkgray}{(no mgus, undecidable, modulo $\alpha\beta$)}\bigskip
\item Higher-Order Pattern Unification\\
\textcolor{darkgray}{(has mgus, decidable, some restrictions, modulo $\alpha\beta_0$)}
\end{itemize}
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*}
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}<1-10>[t]
\frametitle{Underlying Ideas}
\begin{itemize}
\item<1-> Unification (\alert{only}) up to $\alpha$
\item<2-> Swappings / Permutations
\only<2-5>{
\begin{center}
\begin{tabular}{r@ {\hspace{1mm}}l@ {\hspace{12mm}}r@ {\hspace{1mm}}l}
\\
\only<2>{\smath{\textcolor{white}{[b\!:=\!a]}}}%
\only<3>{\smath{[b\!:=\!a]}}%
\only<4-5>{\smath{\alert{\swap{a}{b}\,\act}}} &
\onslide<2-5>{\smath{\lambda a.b}} &
\only<2>{\smath{\textcolor{white}{[b\!:=\!a]}}}%
\only<3>{\smath{[b\!:=\!a]}}%
\only<4-5>{\smath{\alert{\swap{a}{b}\,\act}}} &
\onslide<2-5>{\smath{\lambda c.b}}\\
\onslide<3-5>{\smath{=}} & \only<3>{\smath{\lambda a.a}}\only<4-5>{\smath{\lambda b.a}} &
\onslide<3-5>{\smath{=}} & \only<3>{\smath{\lambda c.a}}\only<4-5>{\smath{\lambda c.a}}\\
\end{tabular}
\end{center}\bigskip
\onslide<4-5>{
\begin{center}
\begin{tikzpicture}
\draw (0,0) node[inner sep=0mm,fill=cream, ultra thick, draw=cream]
{\begin{minipage}{8cm}
\begin{tabular}{r@ {\hspace{3mm}}l}
\smath{\swap{a}{b}\act t} $\;\dn$ & \alert{swap} {\bf all} occurrences of\\
& \smath{b} and \smath{a} in \smath{t}
\end{tabular}
\end{minipage}};
\end{tikzpicture}
\end{center}}\bigskip
\onslide<5>{
Unlike for \smath{[b\!:=\!a]\act(-)}, for \smath{\swap{a}{b}\act (-)} we do
have if \smath{t =_\alpha t'} then \smath{\pi \act t =_\alpha \pi \act t'.}}}
\item<6-> Variables (or holes)\bigskip
\begin{center}
\onslide<7->{\mbox{}\hspace{-25mm}\smath{\lambda x\hspace{-0.5mm}s .}}
\onslide<8-9>{\raisebox{-1.7mm}{\huge\smath{(}}}\raisebox{-4mm}{\begin{tikzpicture}
\fill[blue] (0, 0) circle (5mm);
\end{tikzpicture}}
\onslide<8-9>{\smath{y\hspace{-0.5mm}s}{\raisebox{-1.7mm}{\huge\smath{)}}}}\bigskip
\end{center}
\only<8-9>{\smath{y\hspace{-0.5mm}s} are the parameters the hole can depend on\onslide<9->{, but
then you need $\beta_0$-reduction\medskip
\begin{center}
\smath{(\lambda x. t) y \longrightarrow_{\beta_0} t[x:=y]}
\end{center}}}
\only<10>{we will record the information about which parameters a hole
\alert{\bf cannot} depend on}
\end{itemize}
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*}
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}<1-4>[c]
\frametitle{Terms}
\begin{tabular}{lll @ {\hspace{10mm}}lll}
\onslide<1->{\pgfuseshading{smallbluesphere}} &
\onslide<1->{\colorbox{cream}{\smath{\unit}}} &
\onslide<1->{Units} &
\onslide<2->{\pgfuseshading{smallbluesphere}} &
\onslide<2->{\colorbox{cream}{\smath{a}}} &
\onslide<2->{Atoms} \\[5mm]
\onslide<1->{\pgfuseshading{smallbluesphere}} &
\onslide<1->{\colorbox{cream}{\smath{\pair{t}{t'}}}} &
\onslide<1->{Pairs} &
\onslide<3->{\pgfuseshading{smallbluesphere}} &
\onslide<3->{\colorbox{cream}{\smath{\abst{a}{t}}}} &
\onslide<3->{Abstractions}\\[5mm]
\onslide<1->{\pgfuseshading{smallbluesphere}} &
\onslide<1->{\colorbox{cream}{\smath{\app{F}{t}}}} &
\onslide<1->{Funct.} &
\onslide<4->{\pgfuseshading{smallbluesphere}} &
\onslide<4->{\colorbox{cream}{\smath{\pi\susp X}}} &
\onslide<4->{Suspensions}
\end{tabular}
\only<2>{
\begin{textblock}{13}(1.5,12)
\small Atoms are constants \textcolor{darkgray}{(infinitely many of them)}
\end{textblock}}
\only<3>{
\begin{textblock}{13}(1.5,12)
\small \smath{\ulcorner \lambda\abst{a}{a}\urcorner \mapsto \text{fn\ }\abst{a}{a}}\\
\small constructions like \smath{\text{fn\ }\abst{X}{X}} are not allowed
\end{textblock}}
\only<4>{
\begin{textblock}{13}(1.5,12)
\small \smath{X} is a variable standing for a term\\
\small \smath{\pi} is an explicit permutation \smath{\swap{a_1}{b_1}\ldots\swap{a_n}{b_n}},
waiting to be applied to the term that is substituted for \smath{X}
\end{textblock}}
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*}
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}<1-3>[c]
\frametitle{Permutations}
a permutation applied to a term
\begin{center}
\begin{tabular}{lrcl}
\pgfuseshading{smallbluesphere} &
\smath{[]\act c} & \smath{\dn} & \smath{c} \\
\pgfuseshading{smallbluesphere} &
\smath{\swap{a}{b}\!::\!\pi\act c} & \smath{\dn} &
\smath{\begin{cases}
a & \text{if}\;\pi\act c = b\\
b & \text{if}\;\pi\act c = a\\
\pi\act c & \text{otherwise}
\end{cases}}\\
\onslide<2->{\pgfuseshading{smallbluesphere}} &
\onslide<2->{\smath{\pi\act\abst{a}{t}}} & \onslide<2->{\smath{\dn}} &
\onslide<2->{\smath{\abst{\pi\act a}{\pi\act t}}}\\
\onslide<3->{\pgfuseshading{smallbluesphere}} &
\onslide<3->{\smath{\pi\act\pi'\act X}} & \onslide<3->{\smath{\dn}} &
\onslide<3->{\smath{(\pi @ \pi')\act X}}\\
\end{tabular}
\end{center}
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*}
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}<1-3>[c]
\frametitle{Freshness Constraints}
Recall \smath{\lambda a. \raisebox{-0.7mm}{\tikz \fill[blue] (0, 0) circle (2.5mm);}}
\bigskip\pause
We therefore will identify
\begin{center}
\smath{\text{fn\ } a. X \;\approx\; \text{fn\ } b. \alert<3->{\swap{a}{b}}\act X}
\end{center}
provided that `\smath{b} is fresh for \smath{X} --- (\smath{b\fresh X})',
i.e., does not occur freely in any ground term that might be substituted for
\smath{X}.\bigskip\pause
If we know more about \smath{X}, e.g., if we knew that \smath{a\fresh X} and
\smath{b\fresh X}, then we can replace\\ \smath{\swap{a}{b}\act X} by
\smath{X}.
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*}
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}<1-4>[c]
\frametitle{Equivalence Judgements}
\alt<1>{Our equality is {\bf not} just}{but judgements}
\begin{center}
\begin{tabular}{rl}
\colorbox{cream}{\smath{\onslide<2->{\nabla \vdash} t \approx t'}} & \alert{$\alpha$-equivalence}\\[1mm]
\onslide<4->{\colorbox{cream}{\smath{\onslide<2->{\nabla \vdash} a \fresh t}}} &
\onslide<4->{\alert{freshness}}
\end{tabular}
\end{center}
\onslide<2->{
where
\begin{center}
\smath{\nabla = \{a_1\fresh X_1,\ldots, a_n\fresh X_n\}}
\end{center}
is a finite set of \alert{freshness assumptions}.}
\onslide<3->{
\begin{center}
\smath{\{a\fresh X,b\fresh X\} \vdash \text{fn\ } a. X \approx \text{fn\ } b. X}
\end{center}}
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*}
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}<1>[c]
\frametitle{Rules for Equivalence}
\begin{center}
\begin{tabular}{c}
Excerpt\\
(i.e.~only the interesting rules)
\end{tabular}
\end{center}
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*}
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}<1>[c]
\frametitle{Rules for Equivalence}
\begin{center}
\begin{tabular}{c}
\colorbox{cream}{\smath{\infer{\nabla \vdash a \approx a}{}}}\\[8mm]
\colorbox{cream}{%
\smath{\infer{\nabla \vdash \abst{a}{t} \approx \abst{a}{t'}}
{\nabla \vdash t \approx t'}}}\\[8mm]
\colorbox{cream}{%
\smath{\infer{\nabla \vdash \abst{a}{t} \approx \abst{b}{t'}}
{a\not=b\;\; & \nabla \vdash t \approx \swap{a}{b}\act t'\;\;& \nabla \vdash a\fresh t'}}}
\end{tabular}
\end{center}
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*}
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}<1-3>[c]
\frametitle{Rules for Equivalence}
\begin{center}
\colorbox{cream}{%
\smath{%
\infer{\nabla \vdash \pi\act X \approx \pi'\act X}
{\begin{array}{c}
(a\fresh X)\in\nabla\\
\text{for all}\; a \;\text{with}\;\pi\act a \not= \pi'\act a
\end{array}
}}}
\end{center}
\onslide<2->{
for example\\[4mm]
\alt<2>{%
\begin{center}
\smath{\{a\fresh\!X, b\fresh\!X\} \vdash X \approx \swap{a}{b}\act X}
\end{center}}
{%
\begin{center}
\smath{\{a\fresh\!X, c\fresh\!X\} \vdash \swap{a}{c}\swap{a}{b}\act X \approx \swap{b}{c}\act X}
\end{center}}
\onslide<3->{
\begin{tabular}{@ {}lllll@ {}}
because &
\smath{\swap{a}{c}\swap{a}{b}}: &
\smath{a\mapsto b} &
\smath{\swap{b}{c}}: &
\smath{a\mapsto a}\\
& & \smath{b\mapsto c} & & \smath{b\mapsto c}\\
& & \smath{c\mapsto a} & & \smath{c\mapsto b}\\
\end{tabular}
disagree at \smath{a} and \smath{c}.}
}
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*}
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}<1>[c]
\frametitle{Rules for Freshness}
\begin{center}
\begin{tabular}{c}
Excerpt\\
(i.e.~only the interesting rules)
\end{tabular}
\end{center}
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*}
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}<1>[c]
\frametitle{Rules for Freshness}
\begin{center}
\begin{tabular}{c}
\colorbox{cream}{%
\smath{\infer{\nabla \vdash a\fresh b}{a\not= b}}}\\[5mm]
\colorbox{cream}{%
\smath{\infer{\nabla \vdash a\fresh\abst{a}{t}}{}}}\hspace{7mm}
\colorbox{cream}{%
\smath{\infer{\nabla \vdash a\fresh\abst{b}{t}}
{a\not= b\;\; & \nabla \vdash a\fresh t}}}\\[5mm]
\colorbox{cream}{%
\smath{\infer{\nabla \vdash a\fresh \pi\act X}
{(\pi^{-1}\act a\fresh X)\in\nabla}}}
\end{tabular}
\end{center}
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*}
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}<1-4>[t]
\frametitle{$\approx$ is an Equivalence}
\mbox{}\\[5mm]
\begin{center}
\colorbox{cream}{\alert{Theorem:}
$\approx$ is an equivalence relation.}
\end{center}\bigskip
\only<1>{%
\begin{tabular}{ll}
(Reflexivity) & $\smath{\nabla\vdash t\approx t}$\\[2mm]
(Symmetry) & if $\smath{\nabla\vdash t_1\approx t_2}\;$
then $\;\smath{\nabla\vdash t_2\approx t_1}$\\[2mm]
(Transitivity) & if $\smath{\nabla\vdash t_1\approx t_2}\;$ and
$\;\smath{\nabla\vdash t_2\approx t_3}$\\
& then $\smath{\nabla\vdash t_1\approx t_3}$\\
\end{tabular}}
\only<2->{%
\begin{itemize}
\item<2-> \smath{\nabla \vdash t\approx t'} then \smath{\nabla \vdash \pi\act t\approx \pi\act t'}
\item<2-> \smath{\nabla \vdash a\fresh t} then
\smath{\nabla \vdash \pi\act a\fresh \pi\act t}
\item<3-> \smath{\nabla \vdash t\approx \pi\act t'} then
\smath{\nabla \vdash (\pi^{-1})\act t\approx t'}
\item<3-> \smath{\nabla \vdash a\fresh \pi\act t} then
\smath{\nabla \vdash (\pi^{-1})\act a\fresh t}
\item<4-> \smath{\nabla \vdash a\fresh t} and \smath{\nabla \vdash t\approx t'} then
\smath{\nabla \vdash a\fresh t'}
\end{itemize}
}
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*}
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}<1-4>
\frametitle{Comparison $=_\alpha$}
Traditionally \smath{=_\alpha} is defined as
\begin{center}
\colorbox{cream}{%
\begin{minipage}{9cm}
\raggedright least congruence which identifies \smath{\abst{a}{t}}
with \smath{\abst{b}{[a:=b]t}} provided \smath{b} is not free
in \smath{t}
\end{minipage}}
\end{center}
where \smath{[a:=b]t} replaces all free occurrences of\\
\smath{a} by \smath{b} in \smath{t}.
\bigskip
\only<2>{%
\begin{textblock}{13}(1.2,10)
For \alert{ground} terms:
\begin{center}
\colorbox{cream}{%
\begin{minipage}{9.0cm}
\begin{tabular}{@ {}rl}
\underline{Theorem:}
& \smath{t=_\alpha t'\;\;} if\hspace{-0.5mm}f~\smath{\;\;\emptyset \vdash t\approx t'}\\[2mm]
& \smath{a\not\in F\hspace{-0.9mm}A(t)\;\;} if\hspace{-0.5mm}f~\smath{\;\;\emptyset\vdash a\fresh t}
\end{tabular}
\end{minipage}}
\end{center}
\end{textblock}}
\only<3>{%
\begin{textblock}{13}(1.2,10)
In general \smath{=_\alpha} and \smath{\approx} are distinct!
\begin{center}
\colorbox{cream}{%
\begin{minipage}{6.0cm}
\smath{\abst{a}{X}=_\alpha \abst{b}{X}\;} but not\\[2mm]
\smath{\emptyset \vdash \abst{a}{X} \approx \abst{b}{X}\;} (\smath{a\not=b})
\end{minipage}}
\end{center}
\end{textblock}}
\only<4>{
\begin{textblock}{6}(1,2)
\begin{tikzpicture}
\draw (0,0) node[inner sep=3mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
{\color{darkgray}
\begin{minipage}{10cm}\raggedright
That is a crucial point: if we had\\[-2mm]
\[\smath{\emptyset \vdash \abst{a}{X}\approx \abst{b}{X}}\mbox{,}\]
then applying $\smath{[X:=a]}$, $\smath{[X:=b]}$, $\ldots$\\
give two terms that are {\bf not} $\alpha$-equivalent.\\[3mm]
The freshness constraints $\smath{a\fresh X}$ and $\smath{b\fresh X}$
rule out the problematic substitutions. Therefore
\[\smath{\{a\fresh X,b\fresh X\} \vdash \abst{a}{X}\approx \abst{b}{X}}\]
does hold.
\end{minipage}};
\end{tikzpicture}
\end{textblock}}
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*}
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}<1-9>
\frametitle{Substitution}
\begin{tabular}{l@ {\hspace{8mm}}r@ {\hspace{1.5mm}}c@ {\hspace{1.5mm}}l@ {}}
\pgfuseshading{smallbluesphere} &
\smath{\sigma(\abst{a}{t})} & \smath{\dn} & \smath{\abst{a}{\sigma(t)}}\\[2mm]
\pgfuseshading{smallbluesphere} &
\smath{\sigma(\pi\act X)} & \smath{\dn} &
\smath{\begin{cases}%
\pi\;\act\;\sigma(X) & \!\!\text{if\ } \sigma(X)\not=X\\
\pi\act X & \!\!\text{otherwise}%
\end{cases}}\\[6mm]
\end{tabular}\bigskip\bigskip
\pause
\only<2-5>{
\only<2->{for example}
\def\arraystretch{1.3}
\begin{tabular}{@ {\hspace{14mm}}l@ {\hspace{3mm}}l}
\onslide<2->{\textcolor{white}{$\Rightarrow$}} &
\onslide<2->{\alt<3>{\smath{\underline{\abst{a}{\swap{a}{b}\act X}\;\,[X:=\pair{b}{Y}]}}}
{\smath{\abst{a}{\swap{a}{b}\act X}\;\,[X:=\pair{b}{Y}]}}}\\
\onslide<3->{\smath{\Rightarrow}} &
\onslide<3->{\alt<3,4>{\smath{\abst{a}{\underline{\swap{a}{b}\act X[X:=\pair{b}{Y}]}}}}
{\smath{\abst{a}{\swap{a}{b}\act X}[X:=\pair{b}{Y}]}}}\\
\onslide<4->{\smath{\Rightarrow}} &
\onslide<4->{\alt<4>{\smath{\abst{a}{\swap{a}{b}\act \underline{\pair{b}{Y}}}}}
{\smath{\abst{a}{\underline{\swap{a}{b}}\act \pair{b}{Y}}}}}\\
\onslide<5->{\smath{\Rightarrow}} &
\onslide<5->{\smath{\abst{a}{\pair{a}{\swap{a}{b}\act Y}}}}
\end{tabular}}
\only<6->
{\begin{tabular}{l@ {\hspace{8mm}}l@ {}}
\pgfuseshading{smallbluesphere} &
if \smath{\nabla\vdash t\approx t'} and\hspace{-2mm}\mbox{}
\raisebox{-2.7mm}{
\alt<7>{\begin{tikzpicture}
\draw (0,0) node[inner sep=1mm,fill=cream, very thick, draw=red, rounded corners=3mm]
{\smath{\;\nabla'\vdash\sigma(\nabla)\;}};
\end{tikzpicture}}
{\begin{tikzpicture}
\draw (0,0) node[inner sep=1mm,fill=white, very thick, draw=white, rounded corners=3mm]
{\smath{\;\nabla'\vdash\sigma(\nabla)\;}};
\end{tikzpicture}}}\\
& then \smath{\nabla'\vdash\sigma(t)\approx\sigma(t')}
\end{tabular}}
\only<9>
{\begin{tabular}{l@ {\hspace{8mm}}l@ {}}
\\[-4mm]
\pgfuseshading{smallbluesphere} &
\smath{\sigma(\pi\act t)=\pi\act\sigma(t)}
\end{tabular}}
\only<7>{
\begin{textblock}{6}(10,10.5)
\begin{tikzpicture}
\draw (0,0) node[inner sep=1mm,fill=cream, very thick, draw=red, rounded corners=2mm]
{\color{darkgray}
\begin{minipage}{3.8cm}\raggedright
this means\\[1mm]
\smath{\nabla'\vdash a\fresh\sigma(X)}\\[1mm]
holds for all\\[1mm]
\smath{(a\fresh X)\in\nabla}
\end{minipage}};
\end{tikzpicture}
\end{textblock}}
\end{frame}}
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*}
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\mode<presentation>{
\begin{frame}<1->
\frametitle{Equational Problems}
An equational problem
\[
\colorbox{cream}{\smath{t \eqprob t'}}
\]
is \alert{solved} by
\begin{center}
\begin{tabular}{ll}
\pgfuseshading{smallbluesphere} & a substitution \smath{\sigma} (terms for variables)\\[3mm]
\pgfuseshading{smallbluesphere} & {\bf and} a set of freshness assumptions \smath{\nabla}
\end{tabular}
\end{center}
so that \smath{\nabla\vdash \sigma(t)\approx \sigma(t')}.
\end{frame}}
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*}
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\mode<presentation>{
\begin{frame}<1->
Unifying equations may entail solving
\alert{freshness problems}.
\bigskip
E.g.~assuming that \smath{a\not=a'}, then
\[
\smath{\abst{a}{t}\eqprob \abst{a'}{t'}}
\]
can only be solved if
\[
\smath{t\eqprob \swap{a}{a'}\act t'} \quad\text{\emph{and}}\quad
\smath{a\freshprob t'}
\]
can be solved.
\end{frame}}
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*}
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\mode<presentation>{
\begin{frame}<1->
\frametitle{Freshness Problems}
A freshness problem
\[
\colorbox{cream}{\smath{a \freshprob t}}
\]
is \alert{solved} by
\begin{center}
\begin{tabular}{ll}
\pgfuseshading{smallbluesphere} & a substitution \smath{\sigma}\\[3mm]
\pgfuseshading{smallbluesphere} & and a set of freshness assumptions \smath{\nabla}
\end{tabular}
\end{center}
so that \smath{\nabla\vdash a \fresh \sigma(t)}.
\end{frame}}
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*}
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\mode<presentation>{
\begin{frame}<1-3>
\frametitle{Existence of MGUs}
\underline{Theorem}: There is an algorithm which, given a nominal
unification problem \smath{P}, decides whether\\
or not it has a solution \smath{(\sigma,\nabla)}, and returns a \\
\alert{most general} one if it does.\bigskip\bigskip
\only<3>{
Proof: one can reduce all the equations to `solved form'
first (creating a substitution), and then solve the freshness
problems (easy).}
\only<2>{
\begin{textblock}{6}(2.5,9.5)
\begin{tikzpicture}
\draw (0,0) node[inner sep=3mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
{\color{darkgray}
\begin{minipage}{8cm}\raggedright
\alert{most general:}\\
straightforward definition\\
``if\hspace{-0.5mm}f there exists a \smath{\tau} such that \ldots''
\end{minipage}};
\end{tikzpicture}
\end{textblock}}
\end{frame}}
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*}
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\mode<presentation>{
\begin{frame}<1>
\frametitle{Remember the Quiz?}
\textcolor{gray}{Assuming that $a$ and $b$ are distinct variables,\\
is it possible to find $\lambda$-terms $M_1$ to $M_7$
that make the following pairs $\alpha$-equivalent?}
\begin{tabular}{@ {\hspace{14mm}}p{12cm}}
\begin{itemize}
\item \smath{\lambda a.\lambda b. (M_1\,b)\;} and
\smath{\lambda b.\lambda a. (a\,M_1)\;}
\item \textcolor{gray}{$\lambda a.\lambda b. (M_2\,b)\;$ and
$\lambda b.\lambda a. (a\,M_3)\;$}
\item \textcolor{gray}{$\lambda a.\lambda b. (b\,M_4)\;$ and
$\lambda b.\lambda a. (a\,M_5)\;$}
\item \smath{\lambda a.\lambda b. (b\,M_6)\;} and
\smath{\lambda a.\lambda a. (a\,M_7)\;}
\end{itemize}
\end{tabular}
\textcolor{gray}{If there is one solution for a pair, can you
describe all its solutions?}
\end{frame}}
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*}
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\mode<presentation>{
\begin{frame}<1->
\frametitle{Answers to the Quiz}
\small
\def\arraystretch{1.6}
\begin{tabular}{c@ {\hspace{2mm}}l}
& \only<1>{\smath{\lambda a.\lambda b. (M_1\,b)\;} and \smath{\;\lambda b.\lambda a. (a\,M_1)}}%
\only<2->{\smath{\abst{a}{\abst{b}{\pair{M_1}{b}}} \;\eqprob\; \abst{b}{\abst{a}{\pair{a}{M_1}}}}}\\
\onslide<3->{\smath{\redu{\id}}} &
\only<3>{\smath{\abst{b}{\pair{M_1}{b}} \eqprob
\alert{\swap{a}{b}} \act \abst{a}{\pair{a}{M_1}}\;,\;a\freshprob \abst{a}{\pair{a}{M_1}}}}%
\only<4->{\smath{\abst{b}{\pair{M_1}{b}} \eqprob \abst{b}{\pair{b}{\swap{a}{b}\act M_1}}\;,\
a\freshprob \abst{a}{\pair{a}{M_1}}}}\\
\onslide<5->{\smath{\redu{\id}}} &
\only<5->{\smath{\pair{M_1}{b} \eqprob \pair{b}{\swap{a}{b}\act M_1}\;,\;%
a\freshprob \abst{a}{\pair{a}{M_1}}}}\\
\onslide<6->{\smath{\redu{\id}}} &
\only<6->{\smath{M_1 \eqprob b \;,\; b \eqprob \swap{a}{b}\act M_1\;,\;%
a\freshprob \abst{a}{\pair{a}{M_1}}}}\\
\onslide<7->{\smath{\redu{[M_1:=b]}}} &
\only<7>{\smath{b \eqprob \swap{a}{b}\act \alert{b}\;,\;%
a\freshprob \abst{a}{\pair{a}{\alert{b}}}}}%
\only<8->{\smath{b \eqprob a\;,\; a\freshprob \abst{a}{\pair{a}{b}}}}\\
\onslide<9->{\smath{\redu{}}} &
\only<9->{\smath{F\hspace{-0.5mm}AIL}}
\end{tabular}
\only<10>{
\begin{textblock}{6}(2,11)
\begin{tikzpicture}
\draw (0,0) node[inner sep=3mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
{\color{darkgray}
\begin{minipage}{9cm}\raggedright
\smath{\lambda a.\lambda b. (M_1\,b)} \smath{=_\alpha}
\smath{\lambda b.\lambda a. (a\,M_1)} has no solution
\end{minipage}};
\end{tikzpicture}
\end{textblock}}
\end{frame}}
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*}
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\mode<presentation>{
\begin{frame}<1->
\frametitle{Answers to the Quiz}
\small
\def\arraystretch{1.6}
\begin{tabular}{c@ {\hspace{2mm}}l}
& \only<1>{\smath{\lambda a.\lambda b. (b\,M_6)\;} and \smath{\;\lambda a.\lambda a. (a\,M_7)}}%
\only<2->{\smath{\abst{a}{\abst{b}{\pair{b}{M_6}}} \;\eqprob\; \abst{a}{\abst{a}{\pair{a}{M_7}}}}}\\
\onslide<3->{\smath{\redu{\id}}} &
\only<3->{\smath{\abst{b}{\pair{b}{M_6}} \eqprob \abst{a}{\pair{a}{M_7}}}}\\
\onslide<4->{\smath{\redu{\id}}} &
\only<4->{\smath{\pair{b}{M_6} \eqprob \pair{b}{\swap{b}{a}\act M_7}\;,\;b\freshprob\pair{a}{M_7}}}\\
\onslide<5->{\smath{\redu{\id}}} &
\only<5->{\smath{b\eqprob b\;,\; M_6 \eqprob \swap{b}{a}\act M_7\;,\;%
b\freshprob \pair{a}{M_7}}}\\
\onslide<6->{\smath{\redu{\id}}} &
\only<6->{\smath{M_6 \eqprob \swap{b}{a}\act M_7\;,\;%
b\freshprob \pair{a}{M_7}}}\\
\onslide<7->{\makebox[0mm]{\smath{\redu{[M_6:=\swap{b}{a}\act M_7]}}}} &
\only<7->{\smath{\qquad b\freshprob \pair{a}{M_7}}}\\
\onslide<8->{\smath{\redu{\varnothing}}} &
\only<8->{\smath{b\freshprob a\;,\;b\freshprob M_7}}\\
\onslide<9->{\smath{\redu{\varnothing}}} &
\only<9->{\smath{b\freshprob M_7}}\\
\onslide<10->{\makebox[0mm]{\smath{\redu{\{b\fresh M_7\}}}}} &
\only<10->{\smath{\;\;\varnothing}}\\
\end{tabular}
\only<10>{
\begin{textblock}{6}(6,9)
\begin{tikzpicture}
\draw (0,0) node[inner sep=3mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
{\color{darkgray}
\begin{minipage}{7cm}\raggedright
\smath{\lambda a.\lambda b. (b\,M_6)\;} \smath{=_\alpha}
\smath{\;\lambda a.\lambda a. (a\,M_7)}\\[2mm]
we can take \smath{M_7} to be any $\lambda$-term that does not
contain free occurrences of \smath{b}, so long as we take \smath{M_6} to
be the result of swapping all occurrences of \smath{b} and \smath{a}
throughout \smath{M_7}
\end{minipage}};
\end{tikzpicture}
\end{textblock}}
\end{frame}}
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*}
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\mode<presentation>{
\begin{frame}<1->
\frametitle{Properties}
\begin{itemize}
\item An interesting feature of nominal unification is that it
does not need to create new atoms.\bigskip
\begin{center}\small
\colorbox{cream}{
\smath{\{a.t \eqprob b.t'\}\cup P \redu{\id} \{t \eqprob \swap{a}{b}\act t', a \freshprob t'\} \cup P}}
\end{center}\bigskip\bigskip
\pause
\item The alternative rule
\begin{center}\small
\colorbox{cream}{
\begin{tabular}{@ {}l@ {}}
\smath{\{a.t \eqprob b.t'\}\cup P \redu{\id}}\\
\mbox{}\hspace{2cm}\smath{\{\swap{a}{c}\act t \eqprob
\swap{b}{c}\act t', c \freshprob t, c \freshprob t'\} \cup P}
\end{tabular}}
\end{center}
leads to a more complicated notion of mgu.\medskip\pause
\footnotesize
\smath{\{a.X \eqprob b.Y\} \redu{} (\{a\fresh Y, c\fresh Y\}, [X:=\swap{a}{c}\swap{b}{c}\act Y])}
\end{itemize}
\end{frame}}
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*}
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\mode<presentation>{
\begin{frame}<1-3>
\frametitle{Is it Useful?}
Yes. $\alpha$Prolog by James Cheney (main developer)\bigskip\bigskip
\color{darkgray}
\begin{tabular}{@ {}l}
type (Gamma, var(X), T) :- member (X,T) Gamma.\smallskip\medskip\\
type (Gamma, app(M, N), T') :-\\
\hspace{3cm}type (Gamma, M, arrow(T, T')),\\
\hspace{3cm}type (Gamma, N, T).\smallskip\medskip\\
type (Gamma, lam(\alert{x.M}), arrow(T, T')) / \alert{x \# Gamma} :-\\
\hspace{3cm}type ((x, T)::Gamma, M, T').\smallskip\medskip\\
member X X::Tail.\\
member X Y::Tail :- member X Tail.\\
\end{tabular}
\only<2->{
\begin{textblock}{6}(1.5,0.5)
\begin{tikzpicture}
\draw (0,0) node[inner sep=3mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
{\color{darkgray}
\begin{minipage}{9cm}\raggedright
{\bf One problem:} If we ask whether
\begin{center}
?- type ([(x, T')], lam(x.Var(x)), T)
\end{center}
is typable, we expect an answer for T.\bigskip
\onslide<3>{Solution: Before back-chaining freshen all variables and atoms
in a program (clause).}
\end{minipage}};
\end{tikzpicture}
\end{textblock}}
\end{frame}}
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*}
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\mode<presentation>{
\begin{frame}<1->
\frametitle{Equivariant Unification}
James Cheney proposed
\begin{center}
\colorbox{cream}{
\smath{t \eqprob t' \redu{\nabla, \sigma, \pi}
\nabla \vdash \sigma(t) \approx \pi \act \sigma(t')}}
\end{center}\bigskip\bigskip
\pause
But he also showed this problem is undecidable\\ in general. :(
\end{frame}}
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*}
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\mode<presentation>{
\begin{frame}<1->
\frametitle{Taking Atoms as Variables}
Instead of \smath{a.X}, have \smath{A.X}.\bigskip
\pause
Unfortunately this breaks the mgu-property:
\begin{center}
\smath{a.Z \eqprob X.Y.v(a)}
\end{center}
can be solved by
\begin{center}
\smath{[X:=a, Z:=Y.v(a)]} and
\smath{[Y:=a, Z:=Y.v(Y)]}
\end{center}
\end{frame}}
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*}
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\mode<presentation>{
\begin{frame}<1>[c]
\frametitle{HOPU vs. NOMU}
\begin{itemize}
\item James Cheney showed\bigskip
\begin{center}
\colorbox{cream}{\smath{HOPU \Rightarrow NOMU}}
\end{center}\bigskip
\item Jordi Levy and Mateu Villaret established\bigskip
\begin{center}
\colorbox{cream}{\smath{HOPU \Leftarrow NOMU}}
\end{center}\bigskip
\end{itemize}
The translations `explode' the problems quadratically.
\end{frame}}
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*}
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\mode<presentation>{
\begin{frame}<1>
\small\tt
\begin{minipage}{13cm}
\begin{tabular}{@ {\hspace{-2mm}}p{11.5cm}}
\\
From: Zhenyu Qian <zhqian@microsoft.com>\\
To: Christian Urban <urbanc@in.tum.de>\\
Subject: RE: Linear Higher-Order Pattern Unification\\
Date: Mon, 14 Apr 2008 09:56:47 +0800\\
\\
Hi Christian,\\
\\
Thanks for your interests and asking. I know that that paper is complex. As
I told Tobias when we met last time, I have raised the question to myself
many times whether the proof could have some flaws, and so making it through
a theorem prover would definitely bring piece to my mind (no matter what
the result would be). The only problem for me is the time.\\
\ldots\\
Thanks/Zhenyu
\end{tabular}
\end{minipage}
\end{frame}}
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*}
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\mode<presentation>{
\begin{frame}<1>
\frametitle{Complexity}
\begin{itemize}
\item Christiopher Calves and Maribel Fernandez showed first that
it is polynomial and then also quadratic
\item Jordi Levy and Mateu Villaret showed that it is quadratic
by a translation into a subset of NOMU and using ideas from
Martelli/Montenari.
\end{itemize}
\end{frame}}
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*}
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\mode<presentation>{
\begin{frame}<1->[c]
\frametitle{Conclusion}
\begin{itemize}
\item Nominal Unification is a completely first-order
language, but implements unification modulo $\alpha$.
\textcolor{gray}{(verification\ldots Ramana Kumar and Michael Norrish)}
\medskip\pause
\item NOMU has been applied in term-rewriting and
logic programming. \textcolor{gray}{(Maribel Fernandez et
al has a KB-completion procedure.)}
I hope it will also be used in typing
systems.\medskip\pause
\item NOMU and HOPU are `equivalent' (it took a long time
and considerable research to find this out).\medskip\pause
\item The question about complexity is still an ongoing
story.\medskip
\end{itemize}
\end{frame}}
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*}
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\mode<presentation>{
\begin{frame}<1>[c]
\frametitle{
\begin{tabular}{c}
\mbox{}\\[23mm]
\alert{\LARGE Thank you very much!}\\
\alert{\Large Questions?}
\end{tabular}}
\end{frame}}
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*}
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\mode<presentation>{
\begin{frame}<1-3>
\frametitle{Most General Unifiers}
\underline{Definition}: For a unification problem
\smath{P}, a solution \smath{(\sigma_1,\nabla_1)} is
\alert{more general} than another solution
\smath{(\sigma_2,\nabla_2)}, iff~there exists a substitution
\smath{\tau} with
\begin{center}
\begin{tabular}{ll}
\pgfuseshading{smallbluesphere} &
\alt<2>{\smath{\alert{\nabla_2\vdash\tau(\nabla_1)}}}
{\smath{\nabla_2\vdash\tau(\nabla_1)}}\\
\pgfuseshading{smallbluesphere} &
\alt<3>{\smath{\alert{\nabla_2\vdash\sigma_2\approx \tau\circ\sigma_1}}}
{\smath{\nabla_2\vdash\sigma_2\approx \tau\circ\sigma_1}}
\end{tabular}
\end{center}
\only<2>{
\begin{textblock}{13}(1.5,10.5)
\smath{\nabla_2\vdash a\fresh \sigma(X)} holds for all
\smath{(a\fresh X)\in\nabla_1}
\end{textblock}}
\only<3>{
\begin{textblock}{11}(1.5,10.5)
\smath{\nabla_2\vdash \sigma_2(X)\approx
\sigma(\sigma_1(X))}
holds for all
\smath{X\in\text{dom}(\sigma_2)\cup\text{dom}(\sigma\circ\sigma_1)}
\end{textblock}}
\end{frame}}
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*}
(*<*)
end
(*>*)