More refining.

refining.tex

-Both TypeScript and Flow deduce  the type \code{(number$\vee$string)$\to$number} for the function \code{foo} we defined in the introduction.\footnote{\label{null}Actually, they deduce as return type \code{number$\vee$null}, since both languages track the possibility that some operation may yield \code{null} results. Considering this would not pose any problem but clutter the presentation, so we omit such details.} Consider, next the following slight modification of that example
-\begin{alltt}\color{darkblue}
-  function foo(x : number | string) \{
-    (typeof(x) === "number")? x++ : x.\textcolor{darkred}{trim()}
-  \}
-\end{alltt}
-where \code{trim()} is the JavaScript library function that removes all blank spaces at the beginiing and end of a string. Now, the type deduced is  \code{(number$\vee$string)$\to$(number$\vee$string)}, but it is clear that a better type for this function would be the following intersection type (Footnote~\ref{null} applies here, too):
+As explained in the introduction, both TypeScript and Flow deduce  the type
+\code{(number$\vee$string)$\to$number} for the function \code{foo} we
+defined.\footnote{\label{null}Actually, they
+  deduce as return type \code{number$\vee$null}, since both languages
+  track the possibility that some operation may yield \code{null}
+  results. Considering this would not pose any problem but clutter the
+  presentation, so we omit such details.}
+As we have seen, a more precise type for this function is
 \begin{equation}\label{tyinter}
 \code{(number$\to$number)$\,\wedge\,$(string$\to$string)}
 \end{equation}
@@ -20,7 +21,11 @@ type \code{number} or not (i.e., \code{(number$\vee$string)$\setminus$number}, t
 should be enough for the system to deduce the type \eqref{tyinter}
 even in the absence of an explicit type annotation.
 
-In this section we show how to do it by using the theory of occurrence typing we developed in the first part of the paper. In particular we collect the different types the parameter of a function is assigned in its body to determine a partition of the domain types againt which check the body of the function. Let us consider an example more general than the one above.
+In this section we show how to do it by using the theory of occurrence
+typing we developed in the first part of the paper. In particular we
+collect the different types the parameter of a function is assigned in
+its body to determine a partition of the domain types againt which
+to check the body of the function. Consider a more involved example
 \begin{alltt}\color{darkblue}
   function (x \textcolor{darkred}{: \(\tau\)}) \{
     (x \(\in\) Real)
@@ -28,25 +33,28 @@ In this section we show how to do it by using the theory of occurrence typing we
       : \{ \(\neg\)x \}
   \}
 \end{alltt}
-When $\tau$ is \code{Real|Bool} (we suppose that \code{Int} is a subtype of \code{Real}) we want to deduce for this function the
+When $\tau$ is \code{Real|Bool} (we assume that \code{Int} is a
+subtype of \code{Real}) we want to deduce for this function the
 type
-\code{$(\Int\to\Int)\wedge(\Real\backslash\Int\to\Complex)\wedge(\Bool\to\Bool)$}.
-If we omit $\tau$ and imagine that this is syntactic sugar for the
-parameter to be typed with \Any\ (we do not consider polymorphism),
+\code{$(\Int\to\Int)\wedge(\Real\backslash\Int\to\Real)\wedge(\Bool\to\Bool)$}.
+When $\tau$ is \Any,
 then the function must be rejected (since it tries to type
 \code{\(\neg\)x} under the assumption that \code x\ has type
-\code{$\neg\Real$}. Notice that this last syntactic sugar (i.e.,
-typing function parameter by $\Any$ when type annotation are omitted)
+\code{$\neg\Real$}. Notice that typing the function under the
+hypothesis that $\tau$ is \Any,
 allows us to capture user-defined discrimination as defined
 by~\citet{THF10} since, for instance
 \begin{alltt}\color{darkblue}
   let is_int x = (x\(\in\)Int)? true : false
    in if is_int z then z+1 else 42
 \end{alltt}
-is well typed since the function \code{is\_int} is given type $(\Int\to\True)\wedge(\neg\Int\to\False)$.
+is well typed since the function \code{is\_int} is given type
+$(\Int\to\True)\wedge(\neg\Int\to\False)$. We choose a more general
+approach allowing the programmer to hint a particular type for the
+argument and deducing, if possible, an intersection type for the
+function.
 
-
-We start by considering the system where $\lambda$-abstraction are
+We start by considering the system where $\lambda$-abstractions are
 typed by a single arrow and later generalize it to the  case of
 intersections of arrows. First, we define the auxiliary judgement
 \begin{displaymath}
@@ -153,10 +161,6 @@ for a variable $x$ during typing and then uses them to re-type the body
 of the lambda under this new refined hypothesis for the type of
 $x$. The re-typing ensures that that the type soundness property
 carries over this new rule.
-\kim{We define the rule on the type system not the algorithm. I think
-  we could do the same (by collecting type scheme) in $\psi$ but we
-  then need to choose a candidate in the type scheme \ldots }
-
 
 This system is enough to type our case study for the case $\tau$
 defined as \code{Real|Bool}. Indeed the analysis of the body yields
@@ -167,10 +171,34 @@ $\psi(x)=\{\Bool\}$ for the branch \code{$\neg$x}. So the function
 will be checked for the input types $\Int$, $\Real\setminus\Int$, and
 \Bool, yielding the expected result.
 
-
+It is not too difficult to generalize this rule when the lambda is
+typed by an intersection type. Indeed, even if the programmer has
+specified a particular intersection type for the function we may want
+to refine each of its arrow. This is exactly what the following rule
+does:
+\begin{mathpar}
+  \Infer[AbsInf+] {\forall i\in I~\Gamma,x:s_i\vdash
+    e\triangleright\psi_i
+    \and
+    \Gamma,x:s_i\vdash
+    e : t_i
+    \and
+    T_i = \{ (u, v) ~|~ u\in\psi_i(x) \land \Gamma, x:u\vdash e : v\}
+  } { \Gamma\vdash\lambda^{\bigwedge_{i\in
+        I}s_i\to t_i} x.e:\bigwedge_{i\in I}(u_j\to
+        v_i)\land\bigwedge_{(u, v)\in T_i}(u\to v) } {}
+\end{mathpar}
+Here, for each arrow in the declared interface of the function, we
+first typecheck the body of the function as usual (to check that the
+arrow is valid) and collect the refined types for the parameter $x$.
+Then we deduce all the possible output types for this refined input
+types and add the resulting arrows to the type we deduce for the whole
+function.
 
 {\color{darkred} RANDOM THOUGHTS:
-
+\kim{We define the rule on the type system not the algorithm. I think
+  we could do the same (by collecting type scheme) in $\psi$ but we
+  then need to choose a candidate in the type scheme \ldots }
 A delicate point is the definition of the union
 $\psi_1\cup\psi_2$. The simplest possible definition is the
 component-wise union. This definition is enough to avoid the problem
@@ -210,16 +238,8 @@ of the ``if'' ... but it is probably better to check it by modifying
 the rule of ``if'' (that is, add $\psi_\circ$ only for when it brings new
 information).
 }
-It is not too difficult to generalize this rule when the lambda is typed by an intersection type. Indeed, even if the programmer has specified a particular intersection type for the function we may want to refine each of its arrow. This is exactly what the following rule does:
 
-\begin{mathpar}
-  \Infer[Abs] {\forall i\in I~\Gamma,x:s_i\vdash
-        e\triangleright\psi_i \quad \forall j\in J_i~\Gamma,x:u_j\vdash
-        e:v_j\leq t_i } { \Gamma\vdash\lambda^{\bigwedge_{i\in
-        I}s_i\to t_i} x.e:\bigwedge_{i\in I,j\in J_i}(u_j\to
-        v_i) } {\psi_i(x)=\{u_j\alt j\in
-        J_i\}}
-\end{mathpar}
+
 
 \beppe{Problem 1: is it true that this typing techniques captures all the ``inituitive cases''? That is to say, is it true that this typing and the semantics in which in type case we strip all ``negative'' casts off the resulting values are equivalent? That is they yield the to the same result (probably module the stripping of negatibe casts)?}