some fixes

new_system3.tex

@@ -281,10 +281,11 @@ rules, because one branch is already unreachable and retyping only occurs with s
   \Infer[Abs]
     {\Gamma,(x:s)\avdash{\Gammap}{\ct} e:\{(t_i,\Gamma_i)\}_{i\in I}\\
     \forall i\in I.\ J_i=\{j\,\alt\,j\in I\text{ s.t. }
-    \forall y\in\fv(e).\ \Gamma_j(y)\leq\Gamma_i(y)\}\\
+    \forall y\in\fv(\lambda x:s.e).\ \Gamma_i(y)\leq\Gamma_j(y)\}\\
     \forall i\in I.\ u_i = \bigwedge_{j\in J_i}(\arrow{\Gamma_j(x)}{(
-      t_j\vee\bigvee_{\substack{k\in J_i\text{ s.t. }
-      \Gamma_j(x)\land\Gamma_k(x)\neq\varnothing\\\text{ and }
+      t_j\vee\bigvee_{\substack{k\in I\text{ s.t. }
+      \forall y\in\fv(e).\ \Gamma_j(y)\land\Gamma_k(y)\neq\varnothing\\
+      \text{ and }
       \exists y\in\bv(e).\ \Gamma_k(y)\not\leq\Gamma_j(y)}
       } t_k
     )})
@@ -295,6 +296,11 @@ rules, because one branch is already unreachable and retyping only occurs with s
 
 TODO: Check abs rule
 
+TODO: The current environments returned by the typing rules are
+not adapted at all to type lambda abstractions. It shouldn't contain
+info about bound variables, and the type associated with a given environment
+should cover all the cases for this environment.
+
 \subsection{Backward typing rules}
 
 \begin{mathpar}