Commit 23c38922 authored by Guillaume GENESTIER's avatar Guillaume GENESTIER

Corrections Olivier

parent e10c88ff
......@@ -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.
......
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