
\section{\TamedFJ{}}\label{sec:tifj}
%LetFJ -> Output language!
%TamedFJ -> ANF transformed input langauge
%Input language only described here. It is standard Featherweight Java
% we use the transformation to proof soundness. this could also be moved to the end. 
% the constraint generation step assumes every method argument to be encapsulated in a let statement. This is the way Java is doing capture conversion

The input to our algorithm is a typeless version of Featherweight Java.
The syntax is shown in figure \ref{fig:syntax}
and the respective type rules in figure \ref{fig:expressionTyping} and \ref{fig:typing}.
Method parameters and return types are optional.
We still require type annotations for fields and generic class parameters.
This is a design choice by us,
as we consider them as data declarations which are given by the programmer.
% They are inferred in for example \cite{plue14_3b}
Note the \texttt{new} expression not requiring generic parameters,
which are also inferred by our algorithm.
The method call naturally has type inferred generic parameters aswell.
We add the elvis operator ($\elvis{}$) to the syntax mainly to showcase applications involving wildcard types.
The syntax is described in figure \ref{fig:syntax} with optional type annotations highlighted in yellow.
It is possible to exclude all optional types.
\TamedFJ{} is a subset of the calculus defined by \textit{Bierhoff} \cite{WildcardsNeedWitnessProtection}.
%The language is designed to showcase type inference involving existential types.
The idea is for our type inference algorithm to calculate all missing types and output a fully and correctly typed \TamedFJ{} program,
which by default is also a correct Featherweight Java program (see chapter \ref{sec:soundness}).

\noindent
\textit{Notes:}
\begin{itemize}
\item The calculus does not include method overriding for simplicity reasons.
Type inference for that is described in \cite{TIforFGJ} and can be added to this algorithm accordingly.
%\textit{Note:}
\item The typing rules for expressions shown in figure \ref{fig:expressionTyping} refrain from restricting polymorphic recursion.
Type inference for polymorphic recursion is undecidable \cite{wells1999typability} and when proving completeness like in \cite{TIforFGJ} the calculus
needs to be restricted in that regard.
Our algorithm is not complete (see discussion in chapter \ref{sec:completeness}) and without the respective proof we can keep the \TamedFJ{} calculus as simple as possible
and as close to it's Featherweight Java correspondent \cite{WildcardsNeedWitnessProtection} as possible.
Our soundness proof works either way.
\end{itemize}

%Additional features like overriding methods and method overloading can be added by copying the respective parts from there.
%Additional features can be easily added by generating the respective constraints (Plümicke hier zitieren)

% The type system in \cite{WildcardsNeedWitnessProtection} allows a method to \textit{override} an existing method declaration in one of its super classes,
% but only by a method with the exact same type.
% The type system presented here does not allow the \textit{overriding} of methods.
% Our type inference algorithm consumes the input classes in succession and could only do a type check instead of type inference
% on overriding methods, because their type is already determined.
% Allowing overriding therefore has no implication on our type inference algorithm.

% The output is a valid Featherweight Java program.
% We use the syntax of the version introduced in \cite{WildcardsNeedWitnessProtection}
% calling it \letfj{} for that it is a Featherweight Java variant including \texttt{let} statements.

\newcommand{\highlight}[1]{\begingroup\fboxsep=0pt\colorbox{yellow}{$\displaystyle #1$}\endgroup}
\begin{figure}
\par\noindent\rule{\textwidth}{0.4pt}
\center
$
\begin{array}{lrcl}
%\hline
\text{Parameterized classes} & \mv N & ::= & \exptype{C}{\ol{T}} \\
\text{Types} & \type{S}, \type{T}, \type{U} & ::= & \type{X} \mid \wcNtype{\Delta}{N} \\
\text{Lower bounds} & \type{K}, \type{L} & ::= & \type{T} \mid \bot \\
\text{Type variable contexts} &  \Delta & ::= & \overline{\wildcard{X}{T}{L}} \\
\text{Class declarations} &  D & ::= & \texttt{class}\ \exptype{C}{\ol{X \triangleleft T}} \triangleleft \type{N} \set{\overline{\type{T}\ f}; \ol{M}} \\
\text{Method declarations} &  \texttt{M} & ::= & \highlight{\generics{\overline{\type{X} \triangleleft \type{N}}}}\ \highlight{\type{T}}\ \texttt{m}(\overline{\highlight{\type{T}}\ \expr{x}}) \{ \texttt{return} \ \expr{e}; \} \\
\text{Terms} & \expr{e} & ::= & \expr{x} \\
& & \ \ | & \texttt{new} \ \type{C}\highlight{\generics{\ol{T}}}(\overline{\expr{e}})\\
& & \ \ | & \expr{e}.f\\
& & \ \ | & \expr{e}.\texttt{m}\highlight{\generics{\ol{T}}}(\overline{\expr{e}})\\
& & \ \ | & \texttt{let}\ \expr{x} \highlight{: \wcNtype{\Delta}{N}} = \expr{e} \ \texttt{in} \ \expr{e}\\
& & \ \ | & \expr{e} \elvis{} \expr{e}\\
\text{Variable contexts} & \Gamma & ::= & \overline{\expr{x}:\type{T}}\\
%\hline
\end{array}
$
\par\noindent\rule{\textwidth}{0.4pt}
\caption{Input Syntax with optional type annotations}\label{fig:syntax}
\end{figure}

\begin{figure}[tp]
\begin{center}
$\begin{array}{l}
    \typerule{S-Refl}\\
    \begin{array}{@{}c}
        \Delta \vdash \type{T} <: \type{T}
    \end{array}
\end{array}$
$\begin{array}{l}
    \typerule{S-Trans}\\
    \begin{array}{@{}c}
        \Delta \vdash \type{S} <: \type{T}' \quad \quad
        \Delta \vdash \type{T}' <: \type{T}
        \\
        \hline
    \vspace*{-0.3cm}\\
        \Delta \vdash \type{S} <: \type{T}
    \end{array}
\end{array}$
$\begin{array}{l}
    \typerule{S-Upper}\\
    \begin{array}{@{}c}
        \wildcard{X}{U}{L} \in \Delta \\
    \vspace*{-0.9em}\\
        \hline
    \vspace*{-0.9em}\\
        \Delta \vdash \type{X} <: \type{U}
    \end{array}
\end{array}$
$\begin{array}{l}
    \typerule{S-Lower}\\
    \begin{array}{@{}c}
        \wildcard{X}{U}{L} \in \Delta \\
    \vspace*{-0.9em}\\
        \hline
    \vspace*{-0.9em}\\
        \Delta \vdash \type{L} <: \type{X}
    \end{array}
\end{array}$
\\[1em]
$\begin{array}{l}
    \typerule{S-Extends}\\
    \begin{array}{@{}c}
        \texttt{class}\ \exptype{C}{\overline{\type{X} \triangleleft \type{U}}} \triangleleft \type{N} \ \{ \ldots \} \\
        \ol{X} \cap \text{dom}(\Delta) = \emptyset
        \\
        \hline
    \vspace*{-0.3cm}\\
        \Delta \vdash \wctype{\Delta'}{C}{\ol{T}} <: \wcNtype{\Delta'}{[\ol{T}/\ol{X}]\type{N}}
    \end{array}
\end{array}$
$\begin{array}{l}
    \typerule{S-Exists}\\
    \begin{array}{@{}c}
        \Delta', \Delta \vdash [\ol{T}/\ol{\type{X}}]\ol{L} <: \ol{T} \quad \quad
        \Delta', \Delta \vdash \ol{T} <: [\ol{T}/\ol{\type{X}}]\ol{U} \\
        \text{fv}(\ol{T}) \subseteq \text{dom}(\Delta, \Delta') \quad
        \text{dom}(\Delta') \cap \text{fv}(\wcNtype{\ol{\wildcard{X}{U}{L}}}{N}) = \emptyset
        \\
        \hline
    \vspace*{-0.3cm}\\
        \Delta \vdash \wcNtype{\Delta'}{[\ol{T}/\ol{X}]\type{N}} <:
        \wcNtype{\ol{\wildcard{X}{U}{L}}}{N}
    \end{array}
\end{array}$
\end{center}
\caption{Subtyping}\label{fig:subtyping}
\end{figure}

\begin{figure}[tp]
\begin{center}
$\begin{array}{l}
    \typerule{WF-Bot}\\
    \begin{array}{@{}c}
        \Delta \vdash \bot \ \ok
    \end{array}
\end{array}$
$\begin{array}{l}
    \typerule{WF-Top}\\
    \begin{array}{@{}c}
        \Delta \vdash \ol{L}, \ol{U} \ \ok
        \\
        \hline
    \vspace*{-0.3cm}\\
        \Delta \vdash \overline{\wildcard{W}{U}{L}}.\texttt{Object}
    \end{array}
\end{array}$
$\begin{array}{l}
    \typerule{WF-Var}\\
    \begin{array}{@{}c}
        \wildcard{W}{U}{L} \in \Delta \quad \quad 
        \Delta \vdash \ol{L}, \ol{U} \ \ok
        \\
        \hline
    \vspace*{-0.3cm}\\
        \Delta \vdash \wildcard{W}{U}{L} \ \ok
    \end{array}
\end{array}$
\\[1em]
$\begin{array}{l}
    \typerule{WF-Class}\\
    \begin{array}{@{}c}
        \Delta' = \ol{\wildcard{W}{U}{L}} \quad \quad
        \Delta, \Delta' \vdash \ol{T}, \ol{L}, \ol{U} \ \ok \quad \quad
        \Delta, \Delta' \vdash \ol{L} <: \ol{U} \\
        \Delta, \Delta' \vdash \ol{T} <: [\ol{T}/\ol{X}] \ol{U'} \quad \quad 
        \texttt{class}\ \exptype{C}{\ol{X \triangleleft U'}} \triangleleft \type{N} \ \{ \ldots \}
        \\
        \hline
    \vspace*{-0.3cm}\\
        \Delta \vdash \wctype{\ol{\wildcard{W}{U}{L}}}{C}{\ol{T}} \ \ok
    \end{array}
\end{array}$
\end{center}
\caption{Well-formedness}\label{fig:well-formedness}
\end{figure}

\begin{figure}[tp]
\begin{center}
$\begin{array}{l}
    \typerule{T-Var}\\
    \begin{array}{@{}c}
        \texttt{x} : \type{T} \in \Gamma
        \\
        \hline
    \vspace*{-0.3cm}\\
        \triangle | \Gamma \vdash \texttt{x} : \type{T}
    \end{array}
\end{array}$ \hfill
$\begin{array}{l}
    \typerule{T-Field}\\
    \begin{array}{@{}c}
        \Delta | \Gamma \vdash \expr{v} : \type{T} \quad \quad 
        \Delta \vdash \type{T} <: \wcNtype{}{N} \quad \quad
        \textit{fields}(\type{N}) = \ol{U\ f} \\
        \hline
    \vspace*{-0.3cm}\\
        \Delta | \Gamma \vdash \expr{v}.\texttt{f}_i : \type{U}_i
    \end{array}
\end{array}$
\\[1em]
% $\begin{array}{l}
%     \typerule{T-Field}\\
%     \begin{array}{@{}c}
%         \Delta | \Gamma \vdash \texttt{e} : \type{T} \quad \quad 
%         \Delta \vdash \type{T} <: \wcNtype{\Delta'}{N} \quad \quad
%         \textit{fields}(\type{N}) = \ol{U\ f} \\
% \Delta, \Delta' \vdash \type{U}_i <: \type{S} \quad \quad
% \text{dom}(\Delta') \subseteq \text{fv}(\type{N}) \quad \quad
% \Delta \vdash \type{S}, \wcNtype{\Delta'}{N} \ \ok
%         \\
%         \hline
%     \vspace*{-0.3cm}\\
%         \Delta | \Gamma \vdash \texttt{e}.\texttt{f}_i : \type{S}
%     \end{array}
% \end{array}$
% \\[1em]
% $\begin{array}{l}
%     \typerule{T-New}\\
%     \begin{array}{@{}c}
%         \Delta, \overline{\Delta} \vdash  \exptype{C}{\ol{T}} \ \ok \quad \quad
%         \text{fields}(\exptype{C}{\ol{T}}) = \overline{\type{U}\ f} \quad \quad
%         \Delta | \Gamma \vdash \overline{t : \type{S}} \quad \quad
%         \Delta \vdash  \overline{\type{S}} <: \overline{\wcNtype{\Delta}{N}} \\
%         \Delta, \overline{\Delta} \vdash  \overline{\type{N}} <: \overline{\type{U}} \quad \quad
%         \Delta, \overline{\Delta} \vdash \exptype{C}{\ol{T}} <: \type{T} \quad \quad
%         \overline{\text{dom}(\Delta) \subseteq \text{fv}(\type{N})} \quad \quad
%         \Delta \vdash \type{T},  \overline{\wcNtype{\Delta}{N}} \ \ok
%         \\
%         \hline
%     \vspace*{-0.3cm}\\
%         \triangle | \Gamma \vdash \texttt{new}\ \type{C}(\ol{t}) : \exptype{C}{\ol{T}}
%     \end{array}
% \end{array}$
% \\[1em] 
$\begin{array}{l}
    \typerule{T-New}\\
    \begin{array}{@{}c}
        \Delta, \overline{\Delta} \vdash  \exptype{C}{\ol{T}} \ \ok \quad \quad
        \Delta \vdash  \overline{\type{S}} <: \overline{\type{U}} \\
        \Delta | \Gamma \vdash \overline{\expr{v} : \type{S}} \quad \quad
        \text{fields}(\exptype{C}{\ol{T}}) = \overline{\type{U}\ f} 
        \\
        \hline
    \vspace*{-0.3cm}\\
        \triangle | \Gamma \vdash \texttt{new}\ \type{C}(\ol{v}) : \exptype{C}{\ol{T}}
    \end{array}
\end{array}$
\hfill
% $\begin{array}{l}
%     \typerule{T-Call}\\
%     \begin{array}{@{}c}
%         \Delta, \Delta', \overline{\Delta} \vdash \ol{\type{N}} <: [\ol{S}/\ol{X}]\ol{U} \quad \quad
%         \generics{\ol{X \triangleleft U'}} \ol{U} \to \type{U} \in \Pi(\texttt{m}) \quad \quad
%         \Delta, \Delta', \overline{\Delta} \vdash \ol{S} <: [\ol{S}/\ol{X}]\ol{U'}
%         \\
%         \Delta, \Delta', \overline{\Delta} \vdash \ol{S} \ \ok \quad \quad
%         \Delta | \Gamma \vdash  \texttt{e}, \ol{e} : \ol{T} \quad \quad
%         \Delta \vdash \ol{T} <: \overline{\wcNtype{\Delta}{N}}
%         \\
%         \Delta \vdash \type{T}, \overline{\wcNtype{\Delta}{N}} \ \ok \quad \quad
%         \Delta, \Delta', \overline{\Delta} \vdash [\ol{S}/\ol{X}]\type{U} <: \type{T} \quad \quad
%         \text{dom}(\Delta') \subseteq \text{fv}(\type{N}) \quad \quad
%         \overline{\text{dom}(\Delta) \subseteq \text{fv}(\type{N})}
%         \\
%         \hline
%     \vspace*{-0.3cm}\\
%         \Delta | \Gamma \vdash \texttt{e}.\texttt{m}(\ol{e}) : \type{T}
%     \end{array}
% \end{array}$
% \\[1em]
$\begin{array}{l}
    \typerule{T-Call}\\
    \begin{array}{@{}c}
        \generics{\ol{X \triangleleft U'}} \ol{U} \to \type{U} \in \Pi(\texttt{m}) \quad \quad
        \Delta \vdash \ol{S} <: [\ol{S}/\ol{X}]\ol{U'}
        \\
        \Delta \vdash \ol{S} \ \ok \quad \quad
        \Delta | \Gamma \vdash  \expr{v}, \ol{v} : \ol{T} \quad \quad
        \Delta \vdash \ol{T} <: [\ol{S}/\ol{X}]\ol{U} 
        \\
        \hline
    \vspace*{-0.3cm}\\
        \Delta | \Gamma \vdash \expr{v}.\texttt{m}(\ol{v}) : \type{T}
    \end{array}
\end{array}$
\\[1em]
$\begin{array}{l}
    \typerule{T-Let}\\
    \begin{array}{@{}c}
        \Delta | \Gamma \vdash \expr{t}_1 : \type{T}_1 \quad \quad
        \Delta \vdash \type{T}_1 <: \wcNtype{\Delta'}{N}
        \\
        \Delta, \Delta' | \Gamma, \expr{x} : \wcNtype{}{N} \vdash \expr{t}_2 : \type{T}_2 \quad \quad
        \Delta, \Delta' \vdash  \type{T}_2 <: \type{T} \quad \quad
        \text{dom}(\Delta') \subseteq \text{fv}(\type{N}) \quad \quad
        \Delta \vdash \type{T}, \wcNtype{\Delta'}{N} \ \ok
        \\
        \hline
    \vspace*{-0.3cm}\\
        \Delta | \Gamma \vdash \texttt{let}\ \expr{x} = \expr{t}_1 \ \texttt{in} \ \expr{t}_2 : \type{T}
    \end{array}
\end{array}$
\\[1em]
$\begin{array}{l}
    \typerule{T-Elvis}\\
    \begin{array}{@{}c}
        \triangle | \Gamma \vdash \texttt{t} : \type{T}_1 \quad \quad
        \triangle | \Gamma \vdash \texttt{t}_2 : \type{T}_2 \quad \quad
        \triangle \vdash \type{T}_1 <: \type{T} \quad \quad
        \triangle \vdash \type{T}_2 <: \type{T} \quad \quad
        \\
        \hline
    \vspace*{-0.3cm}\\
        \triangle | \Gamma \vdash \texttt{t}_1 \ \texttt{?:} \ \texttt{t}_2 : \type{T}
    \end{array}
\end{array}$
\end{center}
\caption{Expression Typing}\label{fig:expressionTyping}
\end{figure}

\begin{figure}
\begin{center}
$\begin{array}{l}
    \typerule{T-Method}\\
    \begin{array}{@{}c}
        \texttt{class}\ \exptype{C}{\ol{X \triangleleft U}} \triangleleft \type{N} \set{\ldots} \quad \quad 
        \generics{\ol{Y \triangleleft \type{N}}}(\exptype{C}{\ol{X}},\ol{T}) \to \type{T} \in \mathtt{\Pi}(\texttt{m}) \quad \quad
        \triangle' =  \overline{\type{Y} : \bot .. \type{P}} \\
        \triangle, \triangle' \vdash \ol{P}, \type{T}, \ol{T} \ \ok \quad \quad
        \text{dom}(\triangle) = \ol{X} \quad \quad
        %\texttt{class}\ \exptype{C}{\ol{X \triangleleft \_ }} \triangleleft \type{N} \ \{ \ldots \} \\
        \mathtt{\Pi} | \triangle, \triangle' | \ol{x : T}, \texttt{this} : \exptype{C}{\ol{X}} \vdash \texttt{e} : \type{S} \quad \quad 
        \triangle \vdash \type{S} <: \type{T}
        \\
        \hline
    \vspace*{-0.3cm}\\
        \mathtt{\Pi} | \triangle \vdash \texttt{m}(\ol{x}) \set{ \texttt{return}\ \texttt{e}; } \ \ok \ \text{in C}
    \end{array}
\end{array}$
\\[1em]
$\begin{array}{l}
    \typerule{T-Class}\\
    \begin{array}{@{}c}
        %\forall \texttt{m} \in \ol{M} : \mathtt{\Pi}(\texttt{m}) = \generics{\ol{X \triangleleft \type{N}}}(\exptype{C}{\ol{X}},\ol{T_\texttt{m}}) \to \type{T}_\texttt{m} \\
        \triangle = \overline{\type{X} : \bot .. \type{U}} \quad \quad
        \triangle \vdash \ol{U}, \ol{T}, \type{N} \ \ok \quad \quad
        \mathtt{\Pi} | \triangle \vdash \ol{M} \ \ok \text{ in C}
        \\
        \hline
    \vspace*{-0.3cm}\\
        \mathtt{\Pi} \vdash \texttt{class}\ \exptype{C}{\ol{X \triangleleft U}} \triangleleft \type{N} \{ \ol{T\ f}; \ol{M} \} 
    \end{array}
\end{array}$
%\\[1em]
\hfill
$\begin{array}{l}
    \typerule{T-Program}\\
    \begin{array}{@{}c}
        \mathtt{\Pi} = \overline{\texttt{m} : \generics{\ol{X \triangleleft \type{N}}}\ol{T} \to \type{T}}
        \\
        \hline
    \vspace*{-0.3cm}\\
    \mathtt{\Pi} \vdash \overline{D}
    \end{array}
\end{array}$
\end{center}
\caption{Class and Method Typing rules}\label{fig:typing}
\end{figure}



\begin{figure}
    $\text{fields}(\exptype{Object}{}) = \emptyset$
    \quad \quad
    $\begin{array}{l}
    \typerule{F-Class}\\
    \begin{array}{@{}c}
        \texttt{class}\ \exptype{C}{\ol{X \triangleleft \_ }} \triangleleft \type{N} \set{\ol{S\ f}; \ol{M}} \quad \quad 
        \text{fields}([\ol{T}/\ol{X}]\type{N}) = \ol{U\ g}
        \\
        \hline
    \vspace*{-0.3cm}\\
        \text{fields}(\exptype{C}{\ol{T}}) = \ol{U\ g}, [\ol{T}/\ol{X}]\ol{S\ f}
    \end{array}
\end{array}$
\caption{Field access}
\end{figure}