k

dp.tex

@@ -23,7 +23,7 @@
 
 \begin{cond}{cond-symb-order-wf}
   $\sqsubset$ is well-founded.
-  
+
   \todo{Note that it is free if $\Func$ is finite.}
 \end{cond}
 
@@ -40,7 +40,7 @@
   \AxiomC{$g\sqsubset f$}
   \BinaryInfC{$\G\vdash_{\sqsubset f}g:\tau(g)$}
  \end{prooftree}
- 
+
   Note that it introduces predicates well-formed${}_{\sqsubset f}$.
 \begin{center}
   \begin{tabular}{rcl}
@@ -168,14 +168,18 @@
  we have that $\rew_{arg}\cup{\call}$ terminates on $\mbb{U}$.
 
  We now prove that, $f\,\bar t\in\interp{T'\pi}$ by induction on ${\rew_{arg}}\cup{\call}$.
- Because of the assumptions we have on rules, $f\,\bar t$ is neutral.
- Hence, it suffices to prove that,
- for all $u$ such that $f\,\bar t\rew u$, we have $u\in\interp{T'\pi}$.
- There are two cases:
-  \begin{itemize}
-   \item $u=f\bar u$ with $\bar t\rew\bar u$.
+ \begin{itemize}
+ \item
+   If $f\in\FObj\setminus\CObj$, $f\,\bar t$ is neutral.
+   Hence, it suffices to prove that,
+   for all $u$ such that $f\,\bar t\rew u$, we have $u\in\interp{T'\pi}$.
+   There are two cases:
+   \begin{itemize}
+   \item
+    $u=f\,\bar u$ with $f\,\bar t\rew_{arg}f\,\bar u$.
     Then, we can conclude by induction hypothesis.
-   \item There are $fl_1\dots l_k\rul r\in\mcal{R}$ and $\sg$ such that
+   \item
+    There are $fl_1\dots l_k\rul r\in\mcal{R}$ and $\sg$ such that
     $u=(r\sg)\,t_{k+1}\dots t_n$ and,
     for all $i\in\{1,\dots,k\}$, $t_i=l_i\sg$.
     Then, $\crochet{\overline{\sur{l}{x}}}\sg\vDash\G$.
@@ -183,11 +187,17 @@
 
     We now prove that, for all $\G$, $u$ and $U$, if $\G\vdash_{f\bar l} u:U$ and
     $\sg\vDash\G$, then $u\sg\in\interp{U\sg}$.
-    The proof is the same as for \indthm{TypImpliInterp} except for (fun) replaced by (fun').
-    In this case, for all $i$, we have $\G\vdash_{f\bar l} u_i:U_i\g$.
-    By induction hypothesis, $u_i\sg\in\interp{U_i\g\sg}$.
-    So, $\g\sg\vDash\Sg$ and $g\,\bar u\sg\in\mbb{U}$.
-    Now, $g\,\bar u\sg\tilde{<}f\,\bar l\sg$ since $g\,\bar u<f\,\bar l$.
-    Therefore, by induction hypothesis, $g\,\bar u\sg\in\interp{V\g\sg}$.
-  \end{itemize}
+    The proof is the same as for \indthm{thm-adequacy} except for (fun) replaced by (dp).
+    \begin{description}
+    \item[(dp)]
+      In this case, for all $i$, we have $\G\vdash_{f\bar l} u_i:U_i\g$.
+      By induction hypothesis, $u_i\sg\in\interp{U_i\g\sg}$.
+      So, $\g\sg\vDash\Sg$ and $g\,\bar u\sg\in\mbb{U}$.
+      Now, $g\,\bar u\sg\tilde{<}f\,\bar l\sg$ since $g\,\bar u<f\,\bar l$.
+      Therefore, by induction hypothesis, $g\,\bar u\sg\in\interp{V\g\sg}$.
+    \end{description}
+   \end{itemize}
+ \item
+   If $f\in\CObj$, then $T'=C\,\bar u$ with $C\in\CTyp$.
+ \end{itemize}
 \end{proof}

enTeteArticle.tex

 \usepackage[T1]{fontenc} %Permet d'imprimer les caractères spéciaux
 \usepackage[utf8]{inputenc} %Permet de taper directement les caractères spéciaux dans le fichier Tex
-\usepackage[francais]{babel} %Écrit Démonstration et non Proof
-\frenchbsetup{StandardLists=true} %Met des puces rodes plutôt que des - pour les listes
+\usepackage[english]{babel} %Écrit Démonstration et non Proof
+%\frenchbsetup{StandardLists=true} %Met des puces rodes plutôt que des - pour les listes
 \usepackage{amsthm} %Permet de définir les différents types de théorèmes, remarques...
 
 \usepackage{amssymb} %Ajoute \mathbb
@@ -23,7 +23,7 @@
 \usepackage[toc,page]{appendix} % Pour gérer les annexes
 \bibliographystyle{apalike} % Type de bibliographie
 \usepackage{multirow}
-\usepackage{imakeidx}
+%\usepackage{imakeidx}
 \usepackage[refpage]{nomencl}
 \usepackage{listings}
 \usepackage{enumitem}
@@ -34,24 +34,24 @@
 \usepackage{diagbox}
 
 \theoremstyle{plain}
-	\newtheorem{thm}{Th\'{e}or\`{e}me}[section]
+	\newtheorem{thm}{Theorem}[section]
 	\newtheorem{propo}[thm]{Proposition}
-	\newtheorem{coro}[thm]{Corollaire}
-	\newtheorem{lem}[thm]{Lemme}
-	\newtheorem{propri}[thm]{Propri\'et\'e}
-	\newtheorem{propris}[thm]{Propri\'et\'es}
-	\newtheorem{defi}[thm]{D\'efinition}
-	\newtheorem{algo}[thm]{Algorithme}
+	\newtheorem{coro}[thm]{Corollary}
+	\newtheorem{lem}[thm]{Lemma}
+	\newtheorem{propri}[thm]{Property}
+	\newtheorem{propris}[thm]{Properties}
+	\newtheorem{defi}[thm]{Definition}
+	\newtheorem{algo}[thm]{Algorithm}
 \theoremstyle{definition}
-	\newtheorem{exo}[thm]{Exercice}
+	\newtheorem{exo}[thm]{Exercise}
 \theoremstyle{remark}
 	\newtheorem*{nots}{Notations}
-	\newtheorem*{rmq}{Remarque}
-	\newtheorem*{expl}{Exemple}
-	\newtheorem*{expls}{Exemples}
-        \newtheorem{sub}{But local}
-        \newtheorem{rest}{But restant}
-        \newtheorem*{locProof}{Démonstration}
+	\newtheorem*{rmq}{Remark}
+	\newtheorem*{expl}{Example}
+	\newtheorem*{expls}{Examples}
+        \newtheorem{sub}{Local goal}
+        \newtheorem{rest}{Remaining goal}
+        \newtheorem*{locProof}{Proof}
 
 
 \newcommand{\sur}[2]{\raisebox{0.4ex}{$#1$} / \raisebox{-0.7ex}{$#2$}}