update completeness theorem in the proofs

proofs.tex

@@ -954,8 +954,8 @@ can be extended to type schemes (see also~\citep[\S4.4]{Cas15} for a detailed de
 
 We present here a refinement of the algorithmic type system presented in \ref{sec:algorules}
 that associates to an expression a type scheme instead of a regular type.
-This allows to type expressions more precisely and thus to have a more powerful completeness theorem in regards to the
-declarative type system.
+This allows to type expressions more precisely and thus to have a more powerful
+(but still partial) completeness theorem in regards to the declarative type system.
 
 The results about this new type system will be used in \ref{sec:proofs_algorithmic_without_ts} in order to obtain a soundness and completeness
 theorem for the algorithmic type system presented in \ref{sec:algorules}.
@@ -1756,11 +1756,9 @@ theorem for the algorithmic type system presented in \ref{sec:algorules}.
     \end{proof}
   
     \begin{theorem}[Completeness of the algorithmic type system for positive expressions]\label{completenessA}
-      If we restrict the language to positive expressions $e_+$,
-      the algorithmic type system without type schemes is complete.
-
-      More precisely:
-      $\forall \Gamma, e_+, t.\ \Gamma \vdash e_+:t \Rightarrow \exists t'.\ \Gamma \vdashA e_+ : t'$
+      For every type environment $\Gamma$ and positive expression $e_+$, if
+      $\Gamma\vdash e_+: t$, then there exist $n_o$ and  $t'$ such that $\Gamma\vdashA
+      e_+: t'$.
     \end{theorem}
   
     \begin{proof}