The previous analysis already covers a large pan of realistic cases. For instance, the analysis already works for list data structures since products and recursive type are enough is enough to encode them as right associative nested pairs, as it is done in the language CDuce~\cite{BCF03} (e.g., $X =\textsf{Nil}\vee(\Int\times X)$ is the type of the lists of integers). And even more since the presence of union types makes it possible to type heterogeneous lists whose content is described by regular expressions on types as proposed by~\citet{hosoyapierce}.

First thing we add a new type constructor for products.

We add to types the following production

\[

\begin{array}{lrcl}

\textbf{Types}& t & ::=&\pair t t

\end{array}

\]

to those of Definition~\ref{def:types} and we relax contractivity by that every infinite branch contains an infinite number of occurrences of the arrow or product type constructors.

We also modify subtyping ...

The largest product type is $\pair\Any\Any$ and it denotes the set of all pairs of well-typed values

We also add two operators for the new product type constructor, the right and left projection respectively denoted by $\bpi_{\boldsymbol1}$ and $\bpi_{\boldsymbol2}$ and defined as follows.

Let $t$ be a product type (i.e., $t\leq\pair\Any\Any$) then:

\begin{eqnarray}

\bpl t & = &\min\{ u \alt t\leq\pair u\Any\}\\

\bpr t & = &\min\{ u \alt t\leq\pair\Any u\}

\end{eqnarray}

We add to expressions the terms for pairs and projections:

\[

\begin{array}{lrcl}

\textbf{Expressions}& e & ::=&\pi_1 e\alt\pi_2 e \alt(e,e)

\end{array}

\]

We enrich paths to distinguish uniquely the expression they are applied to (this simplifies the following definitions). So besides $0$ and $1$ we will have steps towards $l$eft, $r$ight, $f$irst, and $s$econd.

Let $e$ be an expression and $\varpi\in\{0,1,l,r,f,s\}^*$ a \emph{path}; we denote $\occ e\varpi$ the occurrence of $e$ reached by the path $\varpi$, that is

\[

\begin{array}{r@{\downarrow}l@{\quad=\quad}l}

e&\epsilon& e\\

e_0e_1& i.\varpi&\occ{e_i}\varpi\qquad i=0,1\\

(e_0,e_1)& l.\varpi&\occ{e_0}\varpi\\

(e_0,e_1)& r.\varpi&\occ{e_1}\varpi\\

\pi_1 e& f.\varpi&\occ{e}\varpi\\

\pi_2 e& s.\varpi&\occ{e}\varpi\\

\end{array}

\]

undefined otherwise

Let $e$ be an expression, $t$ a type, $\Gamma$ a type environment, $\varpi\in\{0,1\}^*$ and $p\in\{+,-\}$, we extend $\typep p \varpi{\Gamma,e,t}$ as follows

\[

\begin{array}{lcl}

\typep{p}{\varpi.l}{\Gamma,e,t}&=&\bpl{\Gp p {\Gamma,e,t}(\varpi)}\\

\typep{p}{\varpi.r}{\Gamma,e,t}&=&\bpr{\Gp p {\Gamma,e,t}(\varpi)}\\

\typep{p}{\varpi.f}{\Gamma,e,t}&=&\pair{\Gp p {\Gamma,e,t}(\varpi)}\Any\\

\typep{p}{\varpi.s}{\Gamma,e,t}&=&\pair\Any{\Gp p {\Gamma,e,t}(\varpi)}\\\\

\Gp p {\Gamma,e,t}(\varpi)&=&\typep p \varpi{\Gamma,e,t}\wedge\tyof{\occ e\varpi}\Gamma

\end{array}

\]

and extend the previous definitions $\Gp p {\Gamma,e,t}(\varpi)$ and

$\Gamma^p_{\Gamma,e,t}$ to the new paths. The reason why in the

definition of $\typep{p}{\varpi.l}{\Gamma,e,t}$ and

$\typep{p}{\varpi.l}{\Gamma,e,t}$ we intersected $\typep p

\varpi{\Gamma,e,t}$ with $\pair\Any\Any$ is to ensure that the

operators $\bpl{}$ and $\bpr{}$ are defined. It handles the (admitedly

stupid) cases in which an \texttt{if} tests whether a pair has a type

which contains values other than pairs, such as for instance

thus deducing the type $\Int$ for the occurrences of $x$ and $y$ in the \texttt{then} branch.

All it remains to do is to add the typing rules for the new expressions:

\begin{mathpar}

\Infer[Proj]

{\Gamma\vdash e:t\and t\leq\pair\Any\Any}

{\Gamma\vdash\pi_i e:\bpi_{\mathbf{i}}(t)}

{\pi_i e\not\in\dom\Gamma}

\Infer[Pair]

{\Gamma\vdash e_1:t_1 \and\Gamma\vdash e_2:t_2}

{\Gamma\vdash (e_1,e_2):\pair{t_1}{t_2}}

{(e_1,e_2)\not\in\dom\Gamma}

\end{mathpar}

As an example one can consider TODO

Finally, for what concerns the algorithmic part, the operators $\bpi_{\boldsymbol1}$ and $\bpi_{\boldsymbol2}$ are standard in the semantic subtyping framework and they can be computed as described by~\citet{} (see also~\citep[\S4.4.1]{Cas15} for a detailed description, in particular formulae (4.20) and (4.21))

}

\subsubsection{Lists} The work above on pairs is enough for having also the lists. It suffices to encode them as right associative nested pairs, as it is done in the language CDuce. The interesting part is that thanks to the presence of union and recursive types one can type heterogeneous lists whose content is described by regular expressions on types as proposed by~\citet{hosoyapierce}.

\subsubsection{Records}

Since the main application of occurrence typing is to type dynamic languages, then what is really missing are record types.

Compare with path expressions of ~\citet{THF10}

\beppe{Compare with path expressions of ~\citet{THF10}}