small modifications

algorithm.tex

@@ -375,7 +375,7 @@ most important ones can be summarized as follows:
     whether the type deduced for the result of the expression is either
     $\Int{\to}\Int$ or $3{\to}3$ (which are both valid but incomparable).
 
-  \item For gradually-typed languages it would yield an unsound system as
+  \item For gradually-typed languages it would yield a problematic system as
     we explain in Section~\ref{sec:gradual}.
 \end{enumerate}
 Now, if we apply this restriction to the language of

gradual.tex

@@ -14,7 +14,7 @@ of any study in this sense. We study it for the formalism of Section~\ref{sec:la
 In a sense, occurrence typing is a
 discipline designed to push forward the frontiers beyond which gradual
 typing is needed, thus reducing the amount of runtime checks needed. For 
-instance, the the JavaScript code of~\eqref{foo} and~\eqref{foo2} in the introduction can be
+instance, the JavaScript code of~\eqref{foo} and~\eqref{foo2} in the introduction can also be
 typed by using gradual typing:\vspace{-.4mm}
 \begin{alltt}\color{darkblue}\morecompact
   function foo(x\textcolor{darkred}{ : \pmb{\dyn}}) \{
@@ -32,11 +32,11 @@ where {\Cast{$t$}{$e$}} is a type-cast that dynamically checks whether the value
     error")}. Not exactly though, since to implement compilation \emph{à la} sound gradual typing it is necessary to use casts on function types that need special handling.}
 %
 We already saw  that thanks to occurrence typing we can annotate the parameter \code{x} by \code{number|string} instead of \dyn{} and avoid the insertion of any cast. 
-But occurrence typing can be used also on the gradually typed code in order to statically detect the insertion of useless casts. Using
+But occurrence typing can be used also on the gradually typed code above in order to statically detect the insertion of useless casts. Using
 occurrence typing to type the gradually-typed version of \code{foo} in~\eqref{foo3}, allows the system to avoid inserting the first cast
 \code{\Cast{number}{x}} since, thanks to occurrence typing, the
-occurrence of \code{x} at issue is given type \code{number} (the
-second cast is still necessary however). But removing only this cast is far
+occurrence of \code{x} at issue is given type \code{number} (but the
+second cast is still necessary though). But removing only this cast is far
 from being satisfactory, since when this function is applied to an integer
 there are some casts that still need to be inserted outside  the function.
 The reason is that the compiled version of the function

intro.tex

@@ -437,7 +437,7 @@ available online.
    of current approaches which require both the tests and the refinement
    to take place on variables.
 
- \item We define a type-theoretic approach that is alternative to the
+ \item We define a type-theoretic approach alternative to the
    current flow-based approaches. As such it provides different
    results and it can be thus profitably combined with flow-based techniques.
    
@@ -454,8 +454,8 @@ available online.
    gradual typing and apply our results to optimze the compilation of the
    latter.
 
- \item We show how it is possible to extend our framework to
-   exploit the richer information provided by polymorphic types.
+% \item We show how it is possible to extend our framework to
+%   exploit the richer information provided by polymorphic types.
 \end{itemize}