Merge branch 'master' of gitlab.math.univ-paris-diderot.fr:beppe/occurrence-typing

proofs.tex

@@ -217,7 +217,7 @@
         \end{definition}
     
           \begin{definition}[(Pre)order on environments]
-            Let $\Gamma$ and $\Gamma'$ two environments. We say that $\Gamma' \leq \Gamma$ iff:
+            Let $\Gamma$ and $\Gamma'$ two environments. We write $\Gamma' \leq \Gamma$ iff:
             \begin{align*}
               &\Gamma'=\bot \text{ or } (\Gamma\neq\bot \text{ and } \forall e \in \dom \Gamma.\ \Gamma' \vdash e : \Gamma(e))
             \end{align*}
@@ -984,7 +984,7 @@
             If $\Gamma = \bot$, we trivially have $\Gamma \vdash e:t$ with the rule \Rule{Efq}.
             Let's assume $\Gamma \neq \bot$.
           
-            If $e=x$ a variable, then the last rule used is \Rule{Var\Aa}.
+            If $e=x$ is a variable, then the last rule used is \Rule{Var\Aa}.
             We can derive $\Gamma \vdash x:t$ by using the rule \Rule{Env} and \Rule{Subs}.
             So let's assume that $e$ is not a variable.
           
@@ -1002,7 +1002,7 @@
               \item[$e=x$] Already treated.
               \item[$e=\lambda^{\bigwedge_{i\in I} \arrow {t_i}{s_i}}x.e'$]
               The last rule is \Rule{Abs\Aa}.
-              We have $\arrow {t_i}{s_i} \leq t$.
+              We have $\bigwedge_{i\in I} \arrow {t_i}{s_i} \leq t$.
               Using the definition of type schemes, let $t'=\bigwedge_{i\in I} \arrow {t_i}{s_i} \land \bigwedge_{j\in J} \neg \arrow {t'_j}{s'_j}$ such that $\Empty \neq t' \leq t$.
               The induction hypothesis gives, for all $i\in I$, $\Gamma,x:s_i\vdash e':t_i$.