.

code_table.tex

@@ -133,6 +133,18 @@ let xor_ = fun (x : Any) -> fun (y : Any) ->
 $\True\to((\True\to\False)\land(\lnot\True\to\True))~\land$\newline
 $(\lnot\True\to((\True\to\True) \land (\lnot\True\to\False))$
     \\\hline
+
+ 10 & \begin{lstlisting}
+(* f and g have type:
+   (Int -> Int) & (Any ->Bool) *)
+
+let example10 =
+   fun (x : Any) ->
+     if (f x, g x) is (Int, Bool) then 1 else 2
+\end{lstlisting} &\vfill
+$(\Int\to\Empty)\land(\Any\setminus\Int\to{}2)$
+\texttt{Warning: line 4, charcater 39-40: expression is unreachable}
+    \\\hline
   \end{tabular}
 }
 \caption{Types inferred by implementation}\vspace{-10mm}

intro.tex

@@ -370,13 +370,13 @@ $\ifty{\textsf{false}}{\Bool}{\textsf{false}}{\textsf{true}}$): see Appendix~\re
 from the mutual dependence of different type checks. In particular, there are two cases
 that pose a problem. The first can be shown by two functions $f$ and $g$ both of type $(\Int\to\Int)\wedge(\Any\to\Bool)$, $x$ of type $\Any$ and the test:
 \begin{equation}\label{nest1}
-\ifty{(fx,gx)}{\pair\Int\Bool}{\,...\,}{\,...}
+\ifty{(f\,x,g\,x)}{\pair\Int\Bool}{\,...\,}{\,...}
 \end{equation}
-If we independently check $fx$ against $\Int$ and $gx$ against $\Bool$
+If we independently check $f\,x$ against $\Int$ and $g\,x$ against $\Bool$
 we deduce $\Int$ for the first occurrence of $x$ and $\Any$ for the
 second. Thus we would type the positive branch of \eqref{nest1} under the hypothesis
 that $x$ is of type $\Int$. But if we use the hypothesis generated by
-the test of $fx$, that is, that $x$ is of type \Int, to check $gx$ against \Bool,
+the test of $f\,x$, that is, that $x$ is of type \Int, to check $g\,x$ against \Bool,
 then the type deduced for $x$ is $\Empty$---i.e., the branch is never selected. In other words, we want to produce type
 environmments for occurrence typing by taking into account  all
 the available hypotheses, even when these hypotheses are formulated later

practical.tex

@@ -127,7 +127,7 @@ nodes. This splits, at the type level, the case for the \Keyw{Element}
 type depending on whether the content of the \texttt{childNodes} field
 is the empty list or not.
 
-Our implementation features one last enhancement that allows us
+Our implementation features another enhancement that allows us
 to further improve the precision of the inferred type.
 Consider the definition of the \texttt{xor\_} operator (Code~9).
 Here the rule~[{\sc AbsInf}+] is not sufficient to precisely type the
@@ -172,3 +172,21 @@ We consider \True, \Any, and $\lnot\True$ as candidate types for
   let xor_ = fun (x : Any) -> fun (y : Any) -> and_ (or_ x y) (not_ (and_ x y))
 \end{alltt}
 for which the very same types as in Table~\ref{tab:implem} are deduced.
+
+Last but not least Code~10 (which corresponds to our introductory
+Example~\ref{nest1}) illustrate the need for iterative refinement of
+type environments, as defined in Section~\ref{sec:typenv}. Indeed, has
+we have explained, a single pass analysis would deduce independently
+for {\tt x}
+a type \Int from the {\tt f~x} application and \Any from the {\tt g~x}
+application. Here by iterating a second time, the algorithm deduces
+that {\tt x} has type \Empty, that is that the first branch can never
+be selected (and our implementation warns the user accordingly). In
+hindsight, the only way for a well typed overloaded function to have
+type $(\Int\to\Int)\land(\Any\to\Bool)$ is to diverge when the
+argument is of type \Int (since this intersection types states that
+whenever the input is \Int, {\em both\/} branches can be selected,
+yielding a result that is at the same time an integer and a Boolean.
+This is precisely reflected by the case $\Int\to\Empty$ in the result.
+Indeed our {\tt example10} function can be applied to an integer, but
+at runtime the application of {\tt f ~x} will diverge.