add some explanations in the appendix

proofs.tex

@@ -876,7 +876,12 @@
 
             \subsection{New algorithmic type system with type schemes}
 
-TODO: Why do we introduce type schemes?
+As explained in TODO, we introduce for the proofs the notion of \emph{types schemes}
+and we define a new (more powerful) algorithmic type system that uses them.
+It allows us to have a stronger (but still partial) completeness theorem.
+
+The proofs for the algorithmic type system presented in \ref{sec:algorules} can be derived
+from the proofs of this section (see section \ref{sec:proofs_algorithmic_without_ts}).
 
 \subsubsection{Type schemes}
 
@@ -1735,7 +1740,7 @@ theorem for the algorithmic type system presented in \ref{sec:algorules}.
       We can prove it by induction over the structure of $e_+$.
 
       The main idea of this proof is that, as $e_+$ is a positive expression, the rule \Rule{Abs-} is not needed anymore
-      because the negative part of functional types (i.e. the $N_i$ part of their DNF) become useless:
+      because the negative part of functional types (i.e. the $N_i$ part of their DNF) becomes useless:
   
       \begin{itemize}
         \item When typing an application $e_1 e_2$, the negative part of the type of $e_1$