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Gradual typing is an approach proposed by~\citet{siek2006gradual} to
combine the safety guarantees of static typing with the programming
flexibility of dynamic typing. The idea is to introduce an \emph{unknown} 
(or \emph{dynamic}) type, denoted $\dyn$, used to inform the compiler that
some static type-checking can be omitted, at the cost of some additional
runtime checks. The use of both static typing and dynamic typing in a same
program creates a boundary between the two, where the compiler automatically
adds ---often costly~\cite{takikawa2016sound}--- dynamic type-checks to ensure 
that a value crossing the barrier is correctly typed.

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Occurrence typing and gradual typing are two complementary disciplines
which have a lot to gain to be integrated, although we are not aware
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of any work in this sense. Moreover, the integration of gradual typing with
set-theoretic types has already been studied by~\citet{castagna2019gradual},
which allows us to keep the same formalism. In a sense, occurrence typing is a
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discipline designed to push forward the frontiers beyond which gradual
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typing is needed, thus reducing the amount of runtime checks needed. For 
instance, the example at the beginning can be
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typed by using gradual typing:
\begin{alltt}\color{darkblue}
  function foo(x\textcolor{darkred}{ : \dyn}) \{
    (typeof(x) === "number")? x++ : x.length
  \}
\end{alltt}
Using ``standard'' gradual typing this is compiled into:
\begin{alltt}\color{darkblue}
  function foo(x) \{
    (typeof(x) === "number")? (\Cast{number}{x})++ : (\Cast{string}{x}).length
  \}
\end{alltt}
where {\Cast{$t$}{$e$}} is a type-cast.\footnote{Intuitively, \code{\Cast{$t$}{$e$}} is
  syntactic sugar for \code{(typeof($e$)==="$t$")? $e$ : (throw "Type
    error")}. Not exactly though, since to implement compilation \emph{à la} sound gradual typing we need cast on function types.}
%
We have already seen in the introduction that by using occurrence
typing combined with a union type instead of the gradual type \dyn
for parameter annotation, we can avoid the insertion of any cast, at the cost
of some additional type annotations. 
But occurrence typing can be used also on the gradually typed code. If we use
occurrence typing to type the gradually-typed version of \code{foo}, this
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 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 of the function.
The reason is that the compiled version of the function
has type \code{\dyn$\to$number}, that is, it expects an argument of type
\dyn, and thus we have to apply a cast (either to the argument or
to the function) whenever this is not the case. In particular, the
application \code{foo(42)} will be compiled as
\code{foo(\Cast{\dyn}{42})}. Now the main problem with such a cast is not
that it produces some unnecessary overhead by performing useless
checks (a cast to \dyn can easily be detected and safely ignored at runtime). 
The main problem is that the combination of such a cast with type-cases 
will lead to unintuitive results under the standard operational
semantics of type-cases and casts.
Indeed, consider the standard semantics
of the type-case \code{(typeof($e$)==="$t$")} which consists in
reducing $e$ to a value and checking whether the type of the value is a
subtype of $t$. In standard gradual semantics, \code{\Cast{\dyn}{42}} is a value. 
And this value is of type \code{\dyn}, which is not a subtype of \code{number}. 
Therefore the check in \code{foo} would fail for \code{\Cast{\dyn}{42}}, and so
would the whole function call. 
Although this behavior is sound, this is the opposite of
what every programmer would expect: one would expect the test
\code{(typeof($e$)==="number")} to return true for \code{\Cast{\dyn}{42}}
and false for, say, \code{\Cast{\dyn}{true}}, whereas
the standard semantics of type-cases would return false in both cases.

A solution is to modify the semantics of type-cases, and in particular of 
\code{typeof}, to strip off all the casts in a value, even nested ones. This
however adds a new overhead at runtime. Another solution is to simply accept
this counter-intuitive result, which has the additional benefit of promoting
the dynamic type to a first class type, instead of just considering it as a 
directive to the front-end. Indeed, this approach allows to dynamically check
whether some argument has the dynamic type \code{\dyn} (i.e., whether it was
applied to a cast to such a type, simply by \code{(typeof($e$)==="\dyn")}. 
Whatever solution we choose it is clear that in both cases it would be much
better if the application \code{foo(42)} were compiled as is, thus getting
rid of a cast that at best is useless and at worse gives a counter-intuitive and
unexpected semantics.

This is where the previous section about refining function types comes in handy.
To get rid of all superfluous casts, we have to fully exploit the information 
provided to us by occurrence typing and deduce for the compiled function the type
\code{(number$\to$number)$\wedge$((\dyn\textbackslash
  number)$\to$number)}, so that no cast is inserted when the
function is applied to a number. 
To achieve this, we simply modify the typing rule for functions that we defined
in the previous section to accommodate for gradual typing. For every gradual type
$\tau$, we define $\tau^*$ as the type obtained from $\tau$ by replacing all
covariant occurrences of \dyn by \Any\ and all contravariant ones by \Empty. The
type $\tau^*$ can be seen as the \emph{maximal} interpretation of $\tau$, that is,
any expression that can safely be cast to $\tau$ is of type $\tau^*$. In
other words, if a function expects an argument of type $\tau$ but can be 
typed under the hypothesis that the argument has type $\tau^*$, then no casts
are needed, since every cast that succeeds will always be to a subtype of
$\tau^*$. Taking advantage of this property, we modify the typing rule for
functions as follows:

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%\begin{mathpar}
%  \Infer[Abs]
%    {\Gamma,x:\tau\vdash e\triangleright\psi\and \forall i\in I\quad \Gamma,x:\sigma_i\vdash e:\tau_i
%     \and \forall j \in J \subseteq I\quad \Gamma,x:\sigma_j^*\vdash e:\tau_j}
%    {
%    \Gamma\vdash\lambda x:\tau.e:\textstyle\bigwedge_{i\in I}\sigma_i\to \tau_i
%      \land \bigwedge_{j\in J}\sigma_j^*\to \tau_j
%    }
%    {\psi(x)=\{\sigma_i\alt i\in I\}}
%\end{mathpar}

\[
  \textsc{[AbsInf+]}
  \frac
  {
    \begin{align*}
    \Gamma,x:&\sigma\vdash e\triangleright\psi \qquad \qquad \Gamma,x:\sigma\vdash e:\tau\\
    T = \{ (\sigma, \tau) \} 
      &\cup \{ (\sigma,\tau) ~|~ \sigma \in \psi(x) \land \Gamma, x: \sigma \vdash e: \tau \}\\
      &\cup \{ (\sigma^*,\tau) ~|~ \sigma \in \psi(x) \land \Gamma, x: \sigma^* \vdash e: \tau \}
    \end{align*}
  }
  {
    \Gamma\vdash\lambda x:\sigma.e:\textstyle\bigwedge_{(\sigma,\tau) \in T}\sigma\to \tau
  }
\]

The main idea behind this rule is the same as before: we first collect all the
information we can into $\psi$ by analyzing the body of the function. We then
retype the function using the new hypothesis $x : \sigma$ for every 
$\sigma \in \psi(x)$. However, we also retype the function using the hypothesis
$x : \sigma^*$, following the explanation we gave in the previous paragraph.
This allows us to eliminate unnecessary gradual types.

Going back to our previous example with function \code{foo}, we first deduce
the refined hypothesis 
$\psi(x) = \{\code{number}\land\dyn, \dyn \textbackslash \code{number}\}$.
Typing the function using this new hypothesis but without considering the
maximal interpretation would yield
$(\dyn \to \code{number}) \land ((\code{number} \land \dyn) \to \code{number})
\land ((\dyn \textbackslash \code{number}) \to \code{number})$. However, as
we stated before, this would introduce an unnecessary cast if the function 
were to be applied to an integer. Hence the need for the second part of
Rule~\textsc{[AbsInf+]}: the maximal interpretation of $\code{number} \land \dyn$
is $\code{number}$, and it is clear that, if $x$ is given type \code{number},
the function type-checks, thanks to occurrence typing. Thus, after some
simplifications, we can actually deduce the desired type
$(\code{number} \to \code{number}) \land ((\dyn \textbackslash \code{number}) \to \code{number})$.