Corrections Olivier

Slides/SlidesFSCD.tex

@@ -102,7 +102,7 @@
 \definecolor{bleufonce}{RGB}{0,0,130}
 \definecolor{blue-violet}{rgb}{0.54, 0.17, 0.89}
 
-\newcommand{\bto}{{\color{blue} \to}}
+\newcommand{\bto}{~{\color{blue} \to}~}
 
 \newcommand{\interp}[1]{\left\llbracket#1\right\rrbracket}
 
@@ -112,12 +112,6 @@
 
 \lstset{escapeinside={@}{@}}
 
-\AtBeginSection[]{
-   \begin{frame}
-   \frametitle{Contents}
-   \tableofcontents[currentsection,hideothersubsections]
-   \end{frame}
-}
 
 \begin{document}
 
@@ -130,6 +124,12 @@
 
 \section{Context: \Dedukti{}}
 
+\AtBeginSection[]{
+   \begin{frame}
+   \frametitle{Contents}
+   \tableofcontents[currentsection,hideothersubsections]
+   \end{frame}
+}
 
 \begin{frame}[fragile]\frametitle{\emph{Dedukti}}
   \emph{Dedukti} is a type-checker for the $\lambda\Pi$-calculus modulo rewriting.
@@ -187,7 +187,7 @@ def sum : (n: Nat) -> F n
 \end{frame}
 
 
-\begin{frame}{Non-restrictive Rewriting}
+\begin{frame}\frametitle{Non-restrictive Rewriting}
 
   \begin{itemize}\itemsep=6mm
   \item overlapping:\hfill$x+0\bto x$, $0+x\bto x$
@@ -196,13 +196,13 @@ def sum : (n: Nat) -> F n
   \item higher-order:\hfill$lam\,(\lambda x.app\,F\,x)\bto F$
   \item matching modulo $\beta$:\hfill
     $\partial\,(\lambda x. ln\,(F\,x))\bto\lambda x.( \partial\,F\,x ) / (F\,x )$
-  \item rules can be both at the object and type levels
+  \item there can be rules both at the object and type levels
   \end{itemize}
 
 \end{frame}
 
 \begin{frame}
- \frametitle{Expected properties of rewriting}
+ \frametitle{Expected Properties of Rewriting}
  \begin{itemize}
   \item Termination: There is no infinite sequence of reduction starting from a well-typed term;
 
@@ -240,7 +240,7 @@ def sum : (n: Nat) -> F n
  \end{itemize}
 \end{frame}
 
-\begin{frame}{Termination: difficulties}
+\begin{frame}\frametitle{Termination: Difficulties}
 
   \begin{itemize}\itemsep=5mm
   \item The set of terms $\lambda\Pi/\mathcal{R}$ depends on rewriting rules $\mathcal{R}$;
@@ -248,8 +248,8 @@ def sum : (n: Nat) -> F n
     independently of $\beta$-reduction;
   \item Type-level rewriting forbids the use of erasing tricks
     reducing termination to simply-typed $\lambda$-calculus;
-  \item Type-level rewriting allows to encode any functional pure type
-    system (e.g. system F or the calculus of constructions).
+  \item Type-level rewriting allows to encode any functional Pure Type
+    System (e.g. System F or the Calculus of Constructions).
   \end{itemize}
 
 \end{frame}
@@ -265,7 +265,7 @@ def sum : (n: Nat) -> F n
 \newcommand{\paren}[1]{\left(#1\right)}
 \newcommand{\crochet}[1]{\left[#1\right]}
 
-\begin{frame}\frametitle{Typing rules}
+\begin{frame}\frametitle{Typing Rules}
   Abstraction:
   \begin{prooftree}
     \AxiomC{$\Gamma\vdash \rA:\bType$}
@@ -303,7 +303,7 @@ def sum : (n: Nat) -> F n
 
 \subsection{Logical relation}
 
-\begin{frame}\frametitle{Logical relation}
+\begin{frame}\frametitle{Logical Relations}
 \begin{block}{Goal}
 	Define $\interp{T}$ such :
 	\begin{itemize}
@@ -312,7 +312,7 @@ def sum : (n: Nat) -> F n
 	\end{itemize}
 \end{block}
 
-\begin{block}{Reducibility conditions}
+\begin{block}{Reducibility Conditions}
 	\begin{itemize}
 		\item $\interp{T}\subseteq\mathrm{SN}$,
 		\item If $t\in\interp{T}$ and $t\to_{\beta\mathcal{R}}u$, then $u\in\interp{T}$,
@@ -328,7 +328,7 @@ def sum : (n: Nat) -> F n
 \end{block}
 \end{frame}
 
-\begin{frame}\frametitle{Our interpretation}
+\begin{frame}\frametitle{Our Interpretation}
 
 We define $\interp{.}$ as the fixpoint of a monotonic function.
 
@@ -358,7 +358,7 @@ A rule $f\,\bar l \to r$ gives rise to the \emph{dependency pairs} $f\,\bar l >
  \end{itemize}
 \end{definition}
 
-\begin{theorem}[Arts et Giesl]
+\begin{theorem}[Arts and Giesl]
  First order:
  
  $\to_{\mathcal{R}}$ terminates iff there is no $f\,\bar t>g\,\bar u\to_{arg}^*g\,\bar u'>\dots$ 
@@ -392,7 +392,7 @@ $\color{red}{(3)}$ append _ (cons x p l) q m $\color{red}{>}$ p + q
 \end{frame}
 
 
-\begin{frame}[fragile]\frametitle{Call-graph: example}
+\begin{frame}[fragile]\frametitle{Call-graph: Example}
 \begin{lstlisting}[mathescape=true]
 def plus : Nat -> Nat -> Nat.
 set infix "+" := plus.
@@ -436,20 +436,20 @@ def append: (p: Nat) -> List p ->
 \end{definition}
 
   \begin{definition}[Plain Function Passing]
-        $f\,\bar l\to r$ is \emph{PFP} if every functional type variable occurring in $r$ are equal to one of the $l_i$.
+        $f\,\bar l\to r$ is \emph{PFP} if every functional type variable occurring in $r$ is equal to one of the $l_i$.
   \end{definition}
 \end{frame}
 
-\subsection{Main theorem}
+\subsection{Main Theorem}
 
-\begin{frame}\frametitle{Main result}
+\begin{frame}\frametitle{Main Result}
   \begin{block}{Reminder}
    If for all $f$, $f\in\interp{\Theta_f}$
    and $\Gamma\vdash t:T$, then $t\in\interp{T}$.
   \end{block}
 
   \begin{lemma}
-  	Every $f$ is in the interpretation of $\Theta_f$ if:
+  	Every $f\in\interp{\Theta_f}$, if:
   	\begin{itemize}
   		\item $\mathcal{R}$ is \emph{well-structured},
   		\item $\mathcal{R}$ is \emph{PFP},
@@ -458,7 +458,7 @@ def append: (p: Nat) -> List p ->
   \end{lemma}
 \end{frame}
 
-\begin{frame}\frametitle{Main result}
+\begin{frame}\frametitle{Main Result}
 \begin{theorem}
   $\to_{\beta\mathcal{R}}$ terminates on every typable term in $\lambda\Pi/\mathcal{R}$ if:
   \begin{itemize}
@@ -473,7 +473,7 @@ def append: (p: Nat) -> List p ->
 
 \subsection{Size-Change Termination}
 
-\begin{frame}[fragile]\frametitle{Call-graph: example}
+\begin{frame}[fragile]\frametitle{Call-graph: Example}
 \begin{lstlisting}[mathescape=true]
 def plus : Nat -> Nat -> Nat.
 set infix "+" := plus.
@@ -502,7 +502,7 @@ def append: (p: Nat) -> List p ->
 \end{center}
 \end{frame}
 
-\begin{frame}{Size-Change Termination}
+\begin{frame}\frametitle{Size-Change Termination}
  \begin{block}{Principle}
   Exclude cycles occurring in the call-graph.
   
@@ -525,7 +525,7 @@ def append: (p: Nat) -> List p ->
 
 
 \definecolor{darkgreen}{RGB}{30,160,30}
-\begin{frame}[fragile]\frametitle{Size-Change Termination : example}
+\begin{frame}[fragile]\frametitle{Size-Change Termination : Example}
   
   Evolution of the sizes of the arguments:\vspace{1em}
   \begin{tabular}{|l|c|}
@@ -630,7 +630,7 @@ $\color{red}{(3)}$ append _ (cons x p l) q m
                 \end{tikzpicture}}
 \end{frame}
 
-\begin{frame}\frametitle{Comparisons with other tools}
+\begin{frame}\frametitle{Comparisons with Other Tools}
   \begin{block}{Simply-typed}
   \begin{itemize}
         \item Annual competition, few participants (Wanda, SOL?),
@@ -647,7 +647,7 @@ $\color{red}{(3)}$ append _ (cons x p l) q m
 \end{frame}
 
 
-\begin{frame}\frametitle{Future work}
+\begin{frame}\frametitle{Future Work}
 \begin{block}{Plain Function Passingness}
   Weaken this hypothesis to ``positivity'', requires to use structural ordering rather than subterm.