\section{Type Inference for Java}
%The goal is to find a correct typing for a given Java program.
Type inference for Java has many use cases and could be used to help programmers by inserting correct types for them,
Finding better type solutions for already typed Java programs (for example more generical ones),
or allowing to write typeless Java code which is then type infered and thereby type checked by our algorithm.
The algorithm proposed in this paper can determine a correct typing for the untyped Java source code example shown in figure \ref{fig:intro-example-typeless}.
Our algorithm is also capable of finding solutions involving wildcards as shown in figure \ref{fig:intro-example-typed}.
%This paper extends a type inference algorithm for Featherweight Java \cite{TIforFGJ} by adding wildcards.
%The last step to create a type inference algorithm compatible to the Java type system.
The algorithm presented in this paper is a slightly improved version of the one in \cite{TIforFGJ} including wildcard support.
%a modified version of the \unify{} algorithm presented in \cite{plue09_1}.
The input to the type inference algorithm is a Featherweight Java program (example in figure \ref{fig:nested-list-example-typeless}) conforming to the syntax shown in figure \ref{fig:syntax}.
The \fjtype{} algorithm calculates constraints based on this intermediate representation,
which are then solved by the \unify{} algorithm
resulting in a correctly typed program (see figure \ref{fig:nested-list-example-typed}).
\begin{figure}[tp]
\begin{subfigure}[t]{\linewidth}
\begin{lstlisting}[style=fgj]
class List {
List add(A v) { ... }
}
class Example {
m(l, la, lb){
return l
.add(la.add(1))
.add(lb.add("str"));
}
}
\end{lstlisting}
\caption{Java method with missing return type}
\label{fig:nested-list-example-typeless}
\end{subfigure}
~
% \begin{subfigure}[t]{\linewidth}
% \begin{lstlisting}[style=tfgj]
% class List {
% List add(A v) { ... }
% }
% class Example {
% m(l, la, lb){
% return let r2 : (*@$\wctype{\wildcard{X}{\type{Object}}{\bot}}{List}{\rwildcard{X}}$@*) = {
% let r1 : (*@$\wctype{\wildcard{X}{\type{Object}}{\bot}}{List}{\rwildcard{X}}$@*) = l in {
% let p1 : (*@$\exptype{List}{\type{Integer}}$@*) = {
% let xa = la in xa.add(1)
% } in x1.add(p1)
% } in {
% let p2 = {
% let xb = lb in xb.add("str")
% } in x2.add(p2)
% };
% }
% }
% \end{lstlisting}
% \caption{Featherweight Java Representation}
% \label{fig:nested-list-example-let}
% \end{subfigure}
% ~
\begin{subfigure}[t]{\linewidth}
\begin{lstlisting}[style=tfgj]
class List {
List add(A v) { ... }
}
class Example {
m(List> l, List la, List lb){
return l
.add(la.add(1))
.add(lb.add("str"));
}
}
\end{lstlisting}
\caption{Java Representation}
\label{fig:nested-list-example-typed}
\end{subfigure}
%\caption{Example code}
%\label{fig:intro-example-code}
\end{figure}
%TODO
% The goal is to proof soundness in respect to the type rules introduced by \cite{aModelForJavaWithWildcards}
% and \cite{WildcardsNeedWitnessProtection}.
\begin{figure}[tp]
\begin{subfigure}[t]{0.49\linewidth}
\begin{lstlisting}[style=fgj]
genList() {
if( ... ) {
return new List();
} else {
return new List();
}
}
\end{lstlisting}
\caption{Java method with missing return type}
\label{fig:intro-example-typeless}
\end{subfigure}
~
\begin{subfigure}[t]{0.49\linewidth}
\begin{lstlisting}[style=tfgj]
List genList() {
if( ... ) {
return new List();
} else {
return new List();
}
}
\end{lstlisting}
\caption{Correct type}
\label{fig:intro-example-typed}
\end{subfigure}
%\caption{Example code}
%\label{fig:intro-example-code}
\end{figure}
% \subsection{Wildcards}
% Java subtyping involving generics is invariant.
% For example \texttt{List} is not a subtype of \texttt{List