
diff -r b38f8d5e0b09 -r 840a857354f2 Slides/Slides3.thy
--- a/Slides/Slides3.thy	Sun Jul 11 21:18:02 2010 +0100
+++ b/Slides/Slides3.thy	Mon Jul 12 21:48:39 2010 +0100
@@ -11,6 +11,20 @@
 
 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}?}}
+
+  \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]
@@ -62,40 +76,45 @@
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
 
 *}
-
 text_raw {*
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   \mode<presentation>{
-  \begin{frame}<1-2>
-  \frametitle{\begin{tabular}{c}Binding in Old Nominal\end{tabular}}
-  \mbox{}\\[-6mm]
+  \begin{frame}<1-4>[c]
+  \frametitle{One Motivation}
+
+  \onslide<2->{Typing implemented in Prolog \textcolor{darkgray}{(from a textbook)}}\bigskip\\
 
-  \begin{itemize}
-  \item old Nominal provided a reasoning infrastructure for single binders\medskip
+  \onslide<3->{
+  \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\\
   
-  \begin{center}
-  Lam [a].(Var a)
-  \end{center}\bigskip
-
-  \item<2-> but representing 
-
+  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)}
+  gets the types
   \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\\
+  \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}}
@@ -105,45 +124,16 @@
 text_raw {*
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   \mode<presentation>{
-  \begin{frame}<1-4>
-  \frametitle{\begin{tabular}{c}Binding Sets of Names\end{tabular}}
-  \mbox{}\\[-3mm]
+  \begin{frame}<1>[c]
+  \frametitle{Higher-Order Unification}
 
   \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}
+  \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}
-  
-  \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}}
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
 *}
@@ -151,23 +141,124 @@
 text_raw {*
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   \mode<presentation>{
-  \begin{frame}<1-3>
-  \frametitle{\begin{tabular}{c}Other Binding Modes\end{tabular}}
-  \mbox{}\\[-3mm]
+  \begin{frame}<1-10>[t]
+  \frametitle{Underlying Ideas}
 
   \begin{itemize}
-  \item alpha-equivalence being preserved under vacuous binders is \underline{not} always
-  wanted:\bigskip\bigskip\normalsize
+  \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}} &
   
-  \begin{tabular}{@ {\hspace{-8mm}}l}
-  $\text{let}\;x = 3\;\text{and}\;y = 2\;\text{in}\;x - y\;\text{end}$\medskip\\
-  \onslide<2->{$\;\;\;\only<2>{\approx_\alpha}\only<3>{\alert{\not\approx_\alpha}}
-   \text{let}\;y = 2\;\text{and}\;x = 3\only<3->{\alert{\;\text{and}
-    \;z = \text{loop}}}\;\text{in}\;x - y\;\text{end}$}
+  \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} occurences 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}}
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
@@ -176,23 +267,33 @@
 text_raw {*
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   \mode<presentation>{
-  \begin{frame}<1>
-  \frametitle{\begin{tabular}{c}\LARGE{}Even Another Binding Mode\end{tabular}}
-  \mbox{}\\[-3mm]
+  \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} \\
 
-  \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\\
-  $\;\;\;\not\approx_\alpha
-   \text{let}\;(y, x) = (3, 2)\;\text{in}\;x - y\;\text{end}$
+  \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{itemize}
 
   \end{frame}}
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
@@ -201,25 +302,56 @@
 text_raw {*
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   \mode<presentation>{
-  \begin{frame}<1-2>
-  \frametitle{\begin{tabular}{c}\LARGE{}Three Binding Modes\end{tabular}}
-  \mbox{}\\[-3mm]
+  \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{\mathtt{fn\ } a. X \;\approx\; \mathtt{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}.
 
-  \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
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}<1-4>[c]
+  \frametitle{Equivalence Judgements}
 
-  \item the order does matter
-  \end{itemize}
+  \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}
-  \isacommand{bind\_res}\hspace{6mm}
-  \isacommand{bind\_set}\hspace{6mm}
-  \isacommand{bind}
+  \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}}
@@ -229,28 +361,39 @@
 text_raw {*
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   \mode<presentation>{
-  \begin{frame}<1-3>
-  \frametitle{\begin{tabular}{c}Specification of Binding\end{tabular}}
-  \mbox{}\\[-6mm]
+  \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}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
 
-  \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)}}\\
+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}}
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
@@ -259,60 +402,61 @@
 text_raw {*
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   \mode<presentation>{
-  \begin{frame}<1-5>
-  \frametitle{\begin{tabular}{c}Inspiration from Ott\end{tabular}}
-  \mbox{}\\[-3mm]
-
-  \begin{itemize}
-  \item this way of specifying binding is inspired by 
-  Ott\onslide<2->{, \alert{\bf but} we made adjustments:}\medskip
-  
+  \begin{frame}<1-3>[c]
+  \frametitle{Rules for Equivalence}
 
-  \only<2>{
-  \begin{itemize}
-  \item Ott allows specifications like\smallskip
   \begin{center}
-  $t ::= t\;t\; |\;\lambda x.t$
+  \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}
-  \end{itemize}}
 
-  \only<3-4>{
-  \begin{itemize}
-  \item whether something is bound can depend in Ott on other bound things\smallskip
+  \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}
-  \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}}
+  \smath{\{a\fresh\!X, c\fresh\!X\} \vdash \swap{a}{c}\swap{a}{b}\act X \approx \swap{b}{c}\act X}
+  \end{center}}
 
-  \only<5>{
-  \begin{itemize}
-  \item we allow multiple binders and bodies\smallskip
+  \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}
-  \isacommand{bind} a b c \isacommand{in} x y z
-  \end{center}\bigskip\medskip
-  the reason is that in general
-  \begin{center}
-  \begin{tabular}{rcl}
-  \isacommand{bind\_res} as \isacommand{in} x y & $\not\Leftrightarrow$ &
-  \begin{tabular}{@ {}l}
-  \isacommand{bind\_res} as \isacommand{in} x,\\
-  \isacommand{bind\_res} as \isacommand{in} y
+  \begin{tabular}{c}
+  Excerpt\\
+  (i.e.~only the interesting rules)
   \end{tabular}
-  \end{tabular}
-  \end{center}\smallskip
-
-  same with \isacommand{bind\_set}
-  \end{itemize}}
-  \end{itemize}
-
+  \end{center}  
 
   \end{frame}}
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
@@ -321,26 +465,26 @@
 text_raw {*
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   \mode<presentation>{
-  \begin{frame}<1>
-  \frametitle{\begin{tabular}{c}Alpha-Equivalence\end{tabular}}
-  \mbox{}\\[-3mm]
+  \begin{frame}<1>[c]
+  \frametitle{Rules for Freshness}
 
-  \begin{itemize}
-  \item in old Nominal, we represented single binders as partial functions:\bigskip
+  \begin{center}
+  \begin{tabular}{c}
+  \colorbox{cream}{%
+  \smath{\infer{\nabla \vdash a\fresh b}{a\not= b}}}\\[5mm]
   
-  \begin{center}
-  \begin{tabular}{l}
-  Lam [$a$].\,$t$ $\;{^\text{``}}\!\dn{}\!^{\text{''}}$\\[2mm]
-  \;\;\;\;$\lambda b.$\;$\text{if}\;a = b\;\text{then}\;t\;\text{else}$\\
-  \phantom{\;\;\;\;$\lambda b.$\;\;\;\;\;\;}$\text{if}\;b \fresh t\;
-  \text{then}\;(a\;b)\act t\;\text{else}\;\text{error}$ 
+  \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{itemize}
 
-  \begin{textblock}{10}(2,14)
-  \footnotesize $^*$ alpha-equality coincides with equality on functions
-  \end{textblock}
   \end{frame}}
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
 *}
@@ -348,105 +492,43 @@
 text_raw {*
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   \mode<presentation>{
-  \begin{frame}<1->
-  \frametitle{\begin{tabular}{c}New Design\end{tabular}}
-  \mbox{}\\[4mm]
+  \begin{frame}<1-4>[t]
+  \frametitle{$\approx$ is an Equivalence}
+  \mbox{}\\[5mm]
 
   \begin{center}
-  \begin{tikzpicture}
-  \alt<2>
-  {\draw (0,0) node[inner sep=3mm, ultra thick, draw=red, rounded corners=2mm]
-  (A) {\textcolor{red}{\begin{minipage}{1.1cm}bind.\\spec.\end{minipage}}};}
-  {\draw (0,0) node[inner sep=3mm, ultra thick, draw=white, rounded corners=2mm]
-  (A) {\begin{minipage}{1.1cm}bind.\\spec.\end{minipage}};}
-  
-  \alt<3>
-  {\draw (3,0) node[inner sep=3mm, ultra thick, draw=red, rounded corners=2mm]
-  (B) {\textcolor{red}{\begin{minipage}{1.1cm}raw\\terms\end{minipage}}};}
-  {\draw (3,0) node[inner sep=3mm, ultra thick, draw=white, rounded corners=2mm]
-  (B) {\begin{minipage}{1.1cm}raw\\terms\end{minipage}};}
+  \colorbox{cream}{\alert{Theorem:}
+  $\approx$ is an equivalence relation.}
+  \end{center}\bigskip
 
-  \alt<4>
-  {\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=white, rounded corners=2mm]
-  (C) {\begin{minipage}{1.1cm}$\alpha$-\\equiv.\end{minipage}};}
-  
-  \alt<5>
-  {\draw (0,-3) node[inner sep=3mm, ultra thick, draw=red, rounded corners=2mm]
-  (D) {\textcolor{red}{\begin{minipage}{1.1cm}quot.\\type\end{minipage}}};}
-  {\draw (0,-3) node[inner sep=3mm, ultra thick, draw=white, rounded corners=2mm]
-  (D) {\begin{minipage}{1.1cm}quot.\\type\end{minipage}};}
-
-  \alt<6>
-  {\draw (3,-3) node[inner sep=3mm, ultra thick, draw=red, rounded corners=2mm]
-  (E) {\textcolor{red}{\begin{minipage}{1.1cm}lift\\thms\end{minipage}}};}
-  {\draw (3,-3) node[inner sep=3mm, ultra thick, draw=white, rounded corners=2mm]
-  (E) {\begin{minipage}{1.1cm}lift\\thms\end{minipage}};}
-
-  \alt<7>
-  {\draw (6,-3) node[inner sep=3mm, ultra thick, draw=red, rounded corners=2mm]
-  (F) {\textcolor{red}{\begin{minipage}{1.1cm}add.\\thms\end{minipage}}};}
-  {\draw (6,-3) node[inner sep=3mm, ultra thick, draw=white, rounded corners=2mm]
-  (F) {\begin{minipage}{1.1cm}add.\\thms\end{minipage}};}
-  
-  \draw[->,white!50,line width=1mm] (A) -- (B);
-  \draw[->,white!50,line width=1mm] (B) -- (C);
-  \draw[->,white!50,line width=1mm, line join=round, rounded corners=2mm] 
-  (C) -- (8,0) -- (8,-1.5) -- (-2,-1.5) -- (-2,-3) -- (D);
-  \draw[->,white!50,line width=1mm] (D) -- (E);
-  \draw[->,white!50,line width=1mm] (E) -- (F);
-  \end{tikzpicture}
-  \end{center}
+  \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}}
 
-  \end{frame}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
-*}
-
-
+  \only<2->{%
+  \begin{itemize}
+  \item<2-> \smath{\nabla \vdash t\approx t'} then \smath{\nabla \vdash \pi\act t\approx \pi\act t'}
 
-text_raw {*
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-  \mode<presentation>{
-  \begin{frame}<1-9>
-  \frametitle{\begin{tabular}{c}Alpha-Equivalence\end{tabular}}
-  \mbox{}\\[-3mm]
-
-  \begin{itemize}
-  \item lets first look at pairs\bigskip\medskip
-
-  \begin{tabular}{@ {\hspace{1cm}}l}
-  $(as, x) \onslide<2->{\approx\!}\makebox[0mm][l]{\only<2-7>{${}_{\text{set}}$}%
-           \only<8>{${}_{\text{\alert{list}}}$}%
-           \only<9>{${}_{\text{\alert{res}}}$}}%
-           \onslide<3->{^{R,\text{fv}}}\,\onslide<2->{(bs,y)}$
-  \end{tabular}\bigskip
-  \end{itemize}
+  \item<2-> \smath{\nabla \vdash a\fresh t} then 
+  \smath{\nabla \vdash \pi\act a\fresh \pi\act t}
 
-  \only<1>{
-  \begin{textblock}{8}(3,8.5)
-  \begin{tabular}{l@ {\hspace{2mm}}p{8cm}}
-   & $as$ is a set of atoms\ldots the binders\\
-   & $x$ is the body\\
-   & $\approx_{\text{set}}$ is where the cardinality 
-  of the binders has to be the same\\
-  \end{tabular}
-  \end{textblock}}
+  \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}
 
-  \only<4->{
-  \begin{textblock}{12}(5,8)
-  \begin{tabular}{ll@ {\hspace{1mm}}l}
-  $\dn$ & \onslide<5->{$\exists \pi.\,$} & $\text{fv}(x) - as = \text{fv}(y) - bs$\\[1mm]
-        & \onslide<5->{$\;\;\;\wedge$} & \onslide<5->{$\text{fv}(x) - as \fresh^* \pi$}\\[1mm]
-        & \onslide<6->{$\;\;\;\wedge$} & \onslide<6->{$(\pi \act x)\;R\;y$}\\[1mm]
-        & \onslide<7-8>{$\;\;\;\wedge$} & \onslide<7-8>{$\pi \act as = bs$}\\
-  \end{tabular}
-  \end{textblock}}
-  
-  \only<8>{
-  \begin{textblock}{8}(3,13.8)
-  \footnotesize $^*$ $as$ and $bs$ are \alert{lists} of atoms 
-  \end{textblock}}
+  \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}}
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
 *}
@@ -454,72 +536,68 @@
 text_raw {*
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   \mode<presentation>{
-  \begin{frame}<1-2>
-  \frametitle{\begin{tabular}{c}Examples\end{tabular}}
-  \mbox{}\\[-3mm]
+  \begin{frame}<1-4>
+  \frametitle{Comparison $=_\alpha$}
 
-  \begin{itemize}
-  \item lets look at ``type-schemes'':\medskip\medskip
+  Traditionally \smath{=_\alpha} is defined as
 
   \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
+  \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}
 
-  \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}
+  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:
   
-  \only<2->{
-  \begin{textblock}{4}(0.3,12)
-  \begin{tikzpicture}
-  \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] 
-  {\tiny\color{darkgray}
-  \begin{minipage}{3.4cm}\raggedright
-  \begin{tabular}{r@ {\hspace{1mm}}l}
-  \multicolumn{2}{@ {}l}{res:}\\
-  $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
-  $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
-  $\wedge$ & $\pi \cdot x = y$\\
-  \\
+  \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 FA(t)\;\;} if\hspace{-0.5mm}f~\smath{\;\;\emptyset\vdash a\fresh t} 
   \end{tabular}
-  \end{minipage}};
-  \end{tikzpicture}
+  \end{minipage}}
+  \end{center}
   \end{textblock}}
-  \only<2->{
-  \begin{textblock}{4}(5.2,12)
+
+  \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=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] 
-  {\tiny\color{darkgray}
-  \begin{minipage}{3.4cm}\raggedright
-  \begin{tabular}{r@ {\hspace{1mm}}l}
-  \multicolumn{2}{@ {}l}{set:}\\
-  $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
-  $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
-  $\wedge$ & $\pi \cdot x = y$\\
-  $\wedge$ & $\pi \cdot as = bs$\\
-  \end{tabular}
-  \end{minipage}};
-  \end{tikzpicture}
-  \end{textblock}}
-  \only<2->{
-  \begin{textblock}{4}(10.2,12)
-  \begin{tikzpicture}
-  \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] 
-  {\tiny\color{darkgray}
-  \begin{minipage}{3.4cm}\raggedright
-  \begin{tabular}{r@ {\hspace{1mm}}l}
-  \multicolumn{2}{@ {}l}{list:}\\
-  $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
-  $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
-  $\wedge$ & $\pi \cdot x = y$\\
-  $\wedge$ & $\pi \cdot as = bs$\\
-  \end{tabular}
+  \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}}
@@ -531,136 +609,73 @@
 text_raw {*
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   \mode<presentation>{
-  \begin{frame}<1-2>
-  \frametitle{\begin{tabular}{c}Examples\end{tabular}}
-  \mbox{}\\[-3mm]
+  \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]
 
-  \begin{center}
-  \only<1>{$(\{x, y\}, x \rightarrow y) \approx_? (\{x, y\}, y \rightarrow x)$}
-  \only<2>{$([x, y], x \rightarrow y) \approx_? ([x, y], y \rightarrow x)$}
-  \end{center}
-
-  \begin{itemize}
-  \item $\approx_{\text{res}}$, $\approx_{\text{set}}$% 
-  \only<2>{, \alert{$\not\approx_{\text{list}}$}}
-  \end{itemize}
+  \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
 
-  
-  \only<1->{
-  \begin{textblock}{4}(0.3,12)
-  \begin{tikzpicture}
-  \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] 
-  {\tiny\color{darkgray}
-  \begin{minipage}{3.4cm}\raggedright
-  \begin{tabular}{r@ {\hspace{1mm}}l}
-  \multicolumn{2}{@ {}l}{res:}\\
-  $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
-  $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
-  $\wedge$ & $\pi \cdot x = y$\\
-  \\
-  \end{tabular}
-  \end{minipage}};
-  \end{tikzpicture}
-  \end{textblock}}
-  \only<1->{
-  \begin{textblock}{4}(5.2,12)
-  \begin{tikzpicture}
-  \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] 
-  {\tiny\color{darkgray}
-  \begin{minipage}{3.4cm}\raggedright
-  \begin{tabular}{r@ {\hspace{1mm}}l}
-  \multicolumn{2}{@ {}l}{set:}\\
-  $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
-  $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
-  $\wedge$ & $\pi \cdot x = y$\\
-  $\wedge$ & $\pi \cdot as = bs$\\
-  \end{tabular}
-  \end{minipage}};
-  \end{tikzpicture}
-  \end{textblock}}
-  \only<1->{
-  \begin{textblock}{4}(10.2,12)
-  \begin{tikzpicture}
-  \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] 
-  {\tiny\color{darkgray}
-  \begin{minipage}{3.4cm}\raggedright
-  \begin{tabular}{r@ {\hspace{1mm}}l}
-  \multicolumn{2}{@ {}l}{list:}\\
-  $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
-  $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
-  $\wedge$ & $\pi \cdot x = y$\\
-  $\wedge$ & $\pi \cdot as = bs$\\
-  \end{tabular}
-  \end{minipage}};
-  \end{tikzpicture}
-  \end{textblock}}
+  \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}}
 
-  \end{frame}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
-*}
-
-text_raw {*
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-  \mode<presentation>{
-  \begin{frame}<1>
-  \frametitle{\begin{tabular}{c}Examples\end{tabular}}
-  \mbox{}\\[-3mm]
+  \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}}
 
-  \begin{center}
-  \only<1>{$(\{x\}, x) \approx_? (\{x, y\}, x)$}
-  \end{center}
+  \only<9>
+  {\begin{tabular}{l@ {\hspace{8mm}}l@ {}}
+  \\[-4mm]
+  \pgfuseshading{smallbluesphere} &
+  \smath{\sigma(\pi\act t)=\pi\act\sigma(t)}
+  \end{tabular}}
 
-  \begin{itemize}
-  \item $\approx_{\text{res}}$, $\not\approx_{\text{set}}$,
-        $\not\approx_{\text{list}}$
-  \end{itemize}
 
-  
-  \only<1->{
-  \begin{textblock}{4}(0.3,12)
+  \only<7>{
+  \begin{textblock}{6}(10,10.5)
   \begin{tikzpicture}
-  \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] 
-  {\tiny\color{darkgray}
-  \begin{minipage}{3.4cm}\raggedright
-  \begin{tabular}{r@ {\hspace{1mm}}l}
-  \multicolumn{2}{@ {}l}{res:}\\
-  $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
-  $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
-  $\wedge$ & $\pi \cdot x = y$\\
-  \\
-  \end{tabular}
-  \end{minipage}};
-  \end{tikzpicture}
-  \end{textblock}}
-  \only<1->{
-  \begin{textblock}{4}(5.2,12)
-  \begin{tikzpicture}
-  \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] 
-  {\tiny\color{darkgray}
-  \begin{minipage}{3.4cm}\raggedright
-  \begin{tabular}{r@ {\hspace{1mm}}l}
-  \multicolumn{2}{@ {}l}{set:}\\
-  $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
-  $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
-  $\wedge$ & $\pi \cdot x = y$\\
-  $\wedge$ & $\pi \cdot as = bs$\\
-  \end{tabular}
-  \end{minipage}};
-  \end{tikzpicture}
-  \end{textblock}}
-  \only<1->{
-  \begin{textblock}{4}(10.2,12)
-  \begin{tikzpicture}
-  \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] 
-  {\tiny\color{darkgray}
-  \begin{minipage}{3.4cm}\raggedright
-  \begin{tabular}{r@ {\hspace{1mm}}l}
-  \multicolumn{2}{@ {}l}{list:}\\
-  $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\
-  $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\
-  $\wedge$ & $\pi \cdot x = y$\\
-  $\wedge$ & $\pi \cdot as = bs$\\
-  \end{tabular}
+  \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}}
@@ -672,190 +687,23 @@
 text_raw {*
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   \mode<presentation>{
-  \begin{frame}<1-3>
-  \frametitle{\begin{tabular}{c}General Abstractions\end{tabular}}
-  \mbox{}\\[-7mm]
+  \begin{frame}<1->
+  \frametitle{Equational Problems}
 
-  \begin{itemize}
-  \item we take $(as, x) \approx\!\makebox[0mm][l]{${}_{\star}$}^{=,\text{supp}} (bs, y)$\medskip
-  \item they are equivalence relations\medskip
-  \item we can therefore use the quotient package to introduce the 
-  types $\beta\;\text{abs}_\star$\bigskip
-  \begin{center}
-  \only<1>{$[as].\,x$}
-  \only<2>{$\text{supp}([as].x) = \text{supp}(x) - as$}
-  \only<3>{%
-  \begin{tabular}{r@ {\hspace{1mm}}l}
-  \multicolumn{2}{@ {\hspace{-7mm}}l}{$[as]. x \alert{=}  [bs].y\;\;\;\text{if\!f}$}\\[2mm]
-  $\exists \pi.$ & $\text{supp}(x) - as = \text{supp}(y) - bs$\\
-  $\wedge$       & $\text{supp}(x) - as \fresh^* \pi$\\
-  $\wedge$       & $\pi \act x = y $\\
-  $(\wedge$       & $\pi \act as = bs)\;^\star$\\
-  \end{tabular}}
-  \end{center}
-  \end{itemize}
-
-  
-  \end{frame}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
-*}
-
-text_raw {*
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-  \mode<presentation>{
-  \begin{frame}<1>
-  \frametitle{\begin{tabular}{c}One Problem\end{tabular}}
-  \mbox{}\\[-3mm]
+  An equational problem 
+  \[
+    \colorbox{cream}{\smath{t \eqprob t'}}
+  \]
+  is \alert{solved} by
 
   \begin{center}
-  $\text{let}\;x_1=t_1 \ldots x_n=t_n\;\text{in}\;s$
+  \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}
 
-  \begin{itemize}
-  \item we cannot represent this as\medskip
-  \begin{center}
-  $\text{let}\;[x_1,\ldots,x_n]\alert{.}s\;\;[t_1,\ldots,t_n]$
-  \end{center}\bigskip
-
-  because\medskip
-  \begin{center}
-  $\text{let}\;[x].s\;\;[t_1,t_2]$
-  \end{center}
-  \end{itemize}
-
-  
-  \end{frame}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
-*}
-
-text_raw {*
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-  \mode<presentation>{
-  \begin{frame}<1->
-  \frametitle{\begin{tabular}{c}Our Specifications\end{tabular}}
-  \mbox{}\\[-6mm]
-
-  \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}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
-*}
-
-text_raw {*
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-  \mode<presentation>{
-  \begin{frame}<1-2>
-  \frametitle{\begin{tabular}{c}``Raw'' Definitions\end{tabular}}
-  \mbox{}\\[-6mm]
-
-  \mbox{}\hspace{10mm}
-  \begin{tabular}{ll}
-  \multicolumn{2}{l}{\isacommand{datatype} trm $=$}\\
-  \hspace{5mm}\phantom{$|$} Var name\\
-  \hspace{5mm}$|$ App trm trm\\
-  \hspace{5mm}$|$ Lam name trm\\
-  \hspace{5mm}$|$ Let 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}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
-*}
-
-text_raw {*
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-  \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}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
-*}
-
-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}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
-*}
-
-text_raw {*
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-  \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}
-
-
-  \[
-  \infer[\text{Let-}\!\approx_\alpha]
-  {\text{Let}\;as\;t \approx_\alpha \text{Let}\;as'\;t'}
-  {(\text{bn}(as), t) \approx\!\makebox[0mm][l]{${}_{\text{list}}$}
-    ^{\approx_\alpha,\text{fv}} (\text{bn}(as'), t') &
-   \onslide<2>{as \approx_\alpha^{\text{bn}} as'}}
-  \]
+  so that \smath{\nabla\vdash \sigma(t)\approx \sigma(t')}.
 
 
   \end{frame}}
@@ -866,143 +714,7 @@
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   \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}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
-*}
-
-text_raw {*
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-  \mode<presentation>{
-  \begin{frame}<1->
-  \frametitle{\begin{tabular}{c}Automatic Proofs\end{tabular}}
-  \mbox{}\\[-6mm]
-
-  \begin{itemize}
-  \item we can show that $\alpha$'s are equivalence relations\medskip
-  \item as a result we can use the quotient package to introduce the type(s)
-  of $\alpha$-equated terms
-
-  \[
-  \infer
-  {\text{Lam}\;x\;t \alert{=} \text{Lam}\;x'\;t'}
-  {\only<1>{([x], t) \approx\!\makebox[0mm][l]{${}_{\text{list}}$}
-    ^{=,\text{supp}} ([x'], t')}%
-   \only<2>{[x].t = [x'].t'}}
-  \]
-
-
-  \item the properties for support are implied by the properties of $[\_].\_$
-  \item we can derive strong induction principles (almost automatic---just a matter of
-  another week or two)
-  \end{itemize}
-
-
-  \end{frame}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
-*}
-
-text_raw {*
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-  \mode<presentation>{
-  \begin{frame}<1>[t]
-  \frametitle{\begin{tabular}{c}Runtime is Acceptable\end{tabular}}
-  \mbox{}\\[-7mm]\mbox{}
-
-  \footnotesize
-  \begin{center}
-  \begin{tikzpicture}
-  \draw (0,0) node[inner sep=2mm, ultra thick, draw=white, rounded corners=2mm]
-  (A) {\begin{minipage}{0.8cm}bind.\\spec.\end{minipage}};
-  
-  \draw (2,0) node[inner sep=2mm, ultra thick, draw=white, rounded corners=2mm]
-  (B) {\begin{minipage}{0.8cm}raw\\terms\end{minipage}};
-
-  \draw (4,0) node[inner sep=2mm, ultra thick, draw=white, rounded corners=2mm]
-  (C) {\begin{minipage}{0.8cm}$\alpha$-\\equiv.\end{minipage}};
-  
-  \draw (0,-2) node[inner sep=2mm, ultra thick, draw=white, rounded corners=2mm]
-  (D) {\begin{minipage}{0.8cm}quot.\\type\end{minipage}};
-
-  \draw (2,-2) node[inner sep=2mm, ultra thick, draw=white, rounded corners=2mm]
-  (E) {\begin{minipage}{0.8cm}lift\\thms\end{minipage}};
-
-  \draw (4,-2) node[inner sep=2mm, ultra thick, draw=white, rounded corners=2mm]
-  (F) {\begin{minipage}{0.8cm}add.\\thms\end{minipage}};
-  
-  \draw[->,white!50,line width=1mm] (A) -- (B);
-  \draw[->,white!50,line width=1mm] (B) -- (C);
-  \draw[->,white!50,line width=1mm, line join=round, rounded corners=2mm] 
-  (C) -- (5,0) -- (5,-1) -- (-1,-1) -- (-1,-2) -- (D);
-  \draw[->,white!50,line width=1mm] (D) -- (E);
-  \draw[->,white!50,line width=1mm] (E) -- (F);
-  \end{tikzpicture}
-  \end{center}
-
-  \begin{itemize}
-  \item Core Haskell: 11 types, 49 term-constructors, 
-  \end{itemize}
-
-  \end{frame}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
-*}
-
-
-text_raw {*
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-  \mode<presentation>{
-  \begin{frame}<1->
-  \frametitle{\begin{tabular}{c}Interesting Phenomenon\end{tabular}}
-  \mbox{}\\[-6mm]
-
-  \small
-  \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}\bigskip\medskip
-
-  we cannot quotient assn: ACons a \ldots $\not\approx_\alpha$ ACons b \ldots
-
-  \end{frame}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
-*}
-
-text_raw {*
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-  \mode<presentation>{
-  \begin{frame}<1->
-  \frametitle{\begin{tabular}{c}Conclusion\end{tabular}}
-  \mbox{}\\[-6mm]
+  \frametitle{Conclusion}
 
   \begin{itemize}
   \item the user does not see anything of the raw level\medskip




