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Cleanup, rephrase introduction. Fix and Cleanup type rules. Remove override

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Andreas Stadelmeier 2024-02-07 17:29:41 +01:00
parent 6c716c5138
commit 26678767c2
7 changed files with 298 additions and 141 deletions
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2
TIforWildFJ.tex
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@ -126,7 +126,7 @@ $\begin{array}{rcl}
\input{tRules} \input{tRules}
\input{tiRules} %\input{tiRules}
\input{constraints} \input{constraints}

13
constraints.tex
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@ -1,14 +1,15 @@
\section{Constraint generation} \section{Constraint generation}
Our type inference algorithm is split into two parts. % Our type inference algorithm is split into two parts.
A constraint generation step \textbf{TYPE} and a \unify{} step. % A constraint generation step \textbf{TYPE} and a \unify{} step.
Method names are not unique. % Method names are not unique.
It is possible to define the same method in multiple classes. % It is possible to define the same method in multiple classes.
The \TYPE{} algorithm accounts for that by generating Or-Constraints. % The \TYPE{} algorithm accounts for that by generating Or-Constraints.
This can lead to multiple possible solutions. % This can lead to multiple possible solutions.
%\subsection{Well-Formedness} %\subsection{Well-Formedness}
The \fjtype{} algorithm assumes capture conversions for every method parameter.
%Why do we need a constraint generation step? %Why do we need a constraint generation step?
%% The problem is NP-Hard %% The problem is NP-Hard

37
introduction.tex
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@ -17,11 +17,21 @@ In Java this is done implicitly by a process called capture conversion \cite{Jav
The type system in \cite{WildcardsNeedWitnessProtection} makes this process explicit by using \texttt{let} statements. The type system in \cite{WildcardsNeedWitnessProtection} makes this process explicit by using \texttt{let} statements.
Our type inference algorithm will accept an input program without let statements and add them where necessary. Our type inference algorithm will accept an input program without let statements and add them where necessary.
The input to the type inference algorithm is a Featherweight Java program without let statements (see figure \ref{fig:syntax}). 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}.
Type inference adds \texttt{let} statements in a fashion similar to the Java capture conversion \cite{WildFJ}. Type inference adds \texttt{let} statements in a fashion similar to the Java capture conversion \cite{WildFJ}.
We wrap every parameter of a method invocation in \texttt{let} statement unknowing if a capture conversion is necessary (see figure \ref{fig:nested-list-example-let}).
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}).
We figured the \texttt{let} statements to be obsolete for our use case. We figured the \texttt{let} statements to be obsolete for our use case.
Once the type inference algorithm found a correct type solution Once the type inference algorithm found a correct type solution they can be inferred by the given type annotations.
% 1. Constraint generation
% 2. Insert typing
% # Showing soundness
% Every program in our calculus can be converted to a WildcardsNeedWitnessProtection program
\begin{figure}[tp] \begin{figure}[tp]
\begin{subfigure}[t]{\linewidth} \begin{subfigure}[t]{\linewidth}
@ -48,6 +58,27 @@ class List<A> {
List<A> add(A v) { ... } List<A> 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}
\begin{lstlisting}[style=tfgj]
class List<A> {
List<A> add(A v) { ... }
}
class Example { class Example {
m((*@$\exptype{List}{\wctype{\wildcard{X}{\type{Object}}{\bot}}{List}{\rwildcard{X}}}$@*) l, List<Integer> la, List<String> lb){ m((*@$\exptype{List}{\wctype{\wildcard{X}{\type{Object}}{\bot}}{List}{\rwildcard{X}}}$@*) l, List<Integer> la, List<String> lb){
return l return l
@ -57,7 +88,7 @@ class Example {
} }
\end{lstlisting} \end{lstlisting}
\caption{Featherweight Java Representation} \caption{Featherweight Java Representation}
\label{fig:nested-list-example-typed} \label{fig:nested-list-example-let}
\end{subfigure} \end{subfigure}
~ ~
\begin{subfigure}[t]{\linewidth} \begin{subfigure}[t]{\linewidth}

4
prolog.tex
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@ -198,8 +198,8 @@
\end{tcolorbox} \end{tcolorbox}
} }
\newcommand{\wcNtype}[2]{#1 .\ntype{#2}} \newcommand{\wcNtype}[2]{\exists #1 .\ntype{#2}}
\newcommand{\wctype}[3]{#1 .\exptype{#2}{#3}} \newcommand{\wctype}[3]{\exists #1 .\exptype{#2}{#3}}
\newcommand{\wtype}[1]{\mathit{#1}} \newcommand{\wtype}[1]{\mathit{#1}}
\newcommand{\ntype}[1]{\mathtt{#1}} \newcommand{\ntype}[1]{\mathtt{#1}}

149
soundness.tex
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@ -420,3 +420,152 @@ $\sigma(\wildcardEnv) = \sigma([\type{T}/\tv{a}]\wildcardEnv)$
%Proof by Lemma 5 \emph{Type substitution preserves subtyping} from \cite{WildcardsNeedWitnessProtection}. %Proof by Lemma 5 \emph{Type substitution preserves subtyping} from \cite{WildcardsNeedWitnessProtection}.
Same as Subst Same as Subst
\end{description} \end{description}
\subsection{Converting to Wild FJ}
Figure \ref{fig:tletexpr} shows type rules for fields and method calls.
They have been merged with let statements and simplified.
The let statements and the type $\wcNtype{\Delta'}{N}$, which the type inference algorithm can freely choose,
are necessary for the soundness proof.
%TODO: Show that well-formed implies witnessed!
We change the type rules to require the well-formedness instead of the witnessed property.
See figure \ref{fig:well-formedness}.
Our well-formedness criteria is more restrictive than the ones used for \wildFJ{}.
\cite{WildcardsNeedWitnessProtection} works with the two different judgements $\ok$ and \texttt{witnessed}.
With \texttt{witnessed} being the stronger one.
We rephrased the $\ok$ judgement to include \texttt{witnessed} aswell.
$\type{T} \ \ok$ in this paper means $\type{T} \ \ok$ and $\type{T} \ \texttt{witnessed}$ in the sense of the \wildFJ{} type rules.
Java's type system is complex enough as it is. Simplification, when possible, is always appreciated.
Our $\ok$ rule may not be able to express every corner of the Java type system, but is sufficient to show soundness regarding type inference for a Java core calculus.
The \rulename{WF-Class} rule requires upper and lower bounds to be direct subtypes of each other $\type{L} <: \type{U}$.
The type rules from \cite{WildcardsNeedWitnessProtection} use a witness type instead.
Stating that a type is well formed (\texttt{witnessed}) if there exists atleast one simple type $\type{N}$ as a possible subtype:
$\Delta \vdash \type{N} <: \wctype{\Delta'}{C}{\ol{T}}$.
A witness type is easy to find by replacing every wildcard $\ol{W}$ in $\wctype{\ol{\wildcard{W}{U}{L}}}{C}{\ol{T}}$ by its upper bound:
$\Delta \vdash \exptype{C}{[\ol{U}/\ol{W}]\ol{T}} <: \wctype{\ol{\wildcard{W}{U}{L}}}{C}{\ol{T}}$.
$\Delta \vdash \exptype{C}{[\ol{U}/\ol{W}]\ol{T}} \ \texttt{witnessed}$ is given due to $\Delta \vdash \ol{T}, \ol{L}, \ol{U} \ \ok$
and $\Delta \vdash \wctype{\ol{\wildcard{W}{U}{L}}}{C}{\ol{T}} \ \ok$.
\begin{figure}[tp]
$\begin{array}{l}
\typerule{T-Var}\\
\begin{array}{@{}c}
x : \type{T} \in \Gamma
\\
\hline
\vspace*{-0.3cm}\\
\Delta | \Gamma \vdash x : \type{T}
\end{array}
\end{array}$
\\[1em]
$\begin{array}{l}
\typerule{T-New}\\
\begin{array}{@{}c}
\Delta, \overline{\Delta} \vdash \exptype{C}{\ol{T}} \ \ok \quad \quad
\text{fields}(\exptype{C}{\ol{T}}) = \overline{\type{U}\ f} \quad \quad
\Delta | \Gamma \vdash \overline{t : \type{S}} \quad \quad
\Delta \vdash \overline{\type{S}} <: \overline{\wcNtype{\Delta}{N}} \\
\Delta, \overline{\Delta} \vdash \overline{\type{N}} <: \overline{\type{U}} \quad \quad
\Delta, \overline{\Delta} \vdash \exptype{C}{\ol{T}} <: \type{T} \quad \quad
\overline{\text{dom}(\Delta) \subseteq \text{fv}(\type{N})} \quad \quad
\Delta \vdash \type{T}, \overline{\wcNtype{\Delta}{N}} \ \ok
\\
\hline
\vspace*{-0.3cm}\\
\Delta | \Gamma \vdash \letstmt{\ol{x} : \ol{\wcNtype{\Delta}{N}} = \ol{t}}{\texttt{new} \ \exptype{C}{\ol{T}}(\overline{t})} : \type{T}
\end{array}
\end{array}$
\\[1em]
$\begin{array}{l}
\typerule{T-Field}\\
\begin{array}{@{}c}
\Delta | \Gamma \vdash \texttt{t} : \type{T} \quad \quad
\Delta \vdash \type{T} <: \wcNtype{\Delta'}{N} \quad \quad
\textit{fields}(\type{N}) = \ol{U\ f} \\
\Delta, \Delta' \vdash \type{U}_i <: \type{S} \quad \quad
\text{dom}(\Delta') \subseteq \text{fv}(\type{N}) \quad \quad
\Delta \vdash \type{S}, \wcNtype{\Delta'}{N} \ \ok
\\
\hline
\vspace*{-0.3cm}\\
\Delta | \Gamma \vdash \texttt{let}\ x : \wcNtype{\Delta'}{N} = \texttt{t} \ \texttt{in}\ x.\texttt{f}_i : \type{S}
\end{array}
\end{array}$
\\[1em]
$\begin{array}{l}
\typerule{T-Call}\\
\begin{array}{@{}c}
\Delta, \Delta', \overline{\Delta} \vdash \ol{\type{N}} <: [\ol{S}/\ol{X}]\ol{U} \quad \quad
\textit{mtype}(\texttt{m}, \type{N}) = \generics{\ol{X \triangleleft U'}} \ol{U} \to \type{U} \quad \quad
\Delta, \Delta', \overline{\Delta} \vdash \ol{S} <: [\ol{S}/\ol{X}]\ol{U'}
\\
\Delta, \Delta', \overline{\Delta} \vdash \ol{S} \ \ok \quad \quad
\Delta | \Gamma \vdash \texttt{t}_r : \type{T}_r \quad \quad
\Delta | \Gamma \vdash \ol{t} : \ol{T} \quad \quad
\Delta \vdash \type{T}_r <: \wcNtype{\Delta'}{N} \quad \quad
\Delta \vdash \ol{T} <: \ol{\wcNtype{\Delta}{N}}
\\
\Delta \vdash \type{T}, \wcNtype{\Delta'}{N}, \overline{\wcNtype{\Delta}{N}} \ \ok \quad \quad
\Delta, \Delta', \Delta'' \vdash [\ol{S}/\ol{X}]\type{U} <: \type{T} \quad \quad
\text{dom}(\Delta') \subseteq \text{fv}(\type{N}) \quad \quad
\overline{\text{dom}(\Delta) \subseteq \text{fv}(\type{N})}
\\
\hline
\vspace*{-0.3cm}\\
\Delta | \Gamma \vdash \letstmt{x : \wcNtype{\Delta'}{N} = t_r, \ol{x} : \ol{\wcNtype{\Delta}{N}} = \ol{t}}
{\texttt{x}.\generics{\ol{S}}\texttt{m}(\ol{x})} : \type{T}
\end{array}
\end{array}$
\\[1em]
$\begin{array}{l}
\typerule{T-Elvis}\\
\begin{array}{@{}c}
\Delta | \Gamma \vdash t_1 : \type{T}_1 \quad \quad
\Delta | \Gamma \vdash t_2 : \type{T}_2 \quad \quad
\Delta \vdash \type{T}_1 <: \type{T} \quad \quad
\Delta \vdash \type{T}_2 <: \type{T}
\\
\hline
\vspace*{-0.3cm}\\
\Delta | \Gamma \vdash t_1 \elvis{} t_2 : \type{T}
\end{array}
\end{array}$
\\[1em]
$\begin{array}{l}
\typerule{T-Method}\\
\begin{array}{@{}c}
\text{dom}(\Delta)=\ol{X} \quad \quad
\Delta' = \overline{\type{Y} : \bot .. \type{U}} \quad \quad
\Delta, \Delta' \vdash \ol{U}, \type{T}, \ol{T}\ \ok \quad \quad
\texttt{class}\ \exptype{C}{\ol{X \triangleleft \_ }} \triangleleft \type{N} \set{\ldots} \\
\Delta, \Delta' | \overline{x:\type{T}}, \texttt{this} : \exptype{C}{\ol{X}} \vdash t:\type{S} \quad \quad
\Delta, \Delta' \vdash \type{S} <: \type{T} \quad \quad
\text{override}(\texttt{m}, \type{N}, \generics{\ol{Y \triangleleft U}}\ol{T} \to \type{T})
\\
\hline
\vspace*{-0.3cm}\\
\Delta \vdash \generics{\ol{Y \triangleleft U}} \type{T} \ \texttt{m}(\overline{\type{T} \ x}) = t \ \ok \ \texttt{in C}
\end{array}
\end{array}$
\\[1em]
$\begin{array}{l}
\typerule{T-Class}\\
\begin{array}{@{}c}
\Delta = \overline{\type{X} : \bot .. \type{U}} \quad \quad
\Delta \vdash \ol{U}, \ol{T} \ \ok \quad \quad
\Delta \vdash \type{N} \ \ok \quad \quad
\Delta \vdash \ol{M} \ \ok \texttt{ in C}
\\
\hline
\vspace*{-0.3cm}\\
\texttt{class}\ \exptype{C}{\ol{X \triangleleft U}} \triangleleft \type{N} \set{\overline{\type{T}\ f}; \ol{M} } \ \ok
\end{array}
\end{array}$
\caption{T-Call and T-Field} \label{fig:tletexpr}
\end{figure}

215
tRules.tex
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@ -1,4 +1,15 @@
\section{Syntax} \section{Syntax and Typing}
The input syntax for our algorithm is shown in figure \ref{fig:syntax}
and the respective type rules in figure \ref{fig:expressionTyping} and \ref{fig:typing}.
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.
\begin{figure} \begin{figure}
$ $
\begin{array}{lrcl} \begin{array}{lrcl}
@ -21,13 +32,12 @@ $
\caption{Input Syntax}\label{fig:syntax} \caption{Input Syntax}\label{fig:syntax}
\end{figure} \end{figure}
Each class type has a set of wildcard types $\overline{\Delta}$ attached to it. % Each class type has a set of wildcard types $\overline{\Delta}$ attached to it.
The type $\wctype{\overline{\Delta}}{C}{\ol{T}}$ defines a set of wildcards $\overline{\Delta}$, % The type $\wctype{\overline{\Delta}}{C}{\ol{T}}$ defines a set of wildcards $\overline{\Delta}$,
which can be used inside the type parameters $\ol{T}$. % which can be used inside the type parameters $\ol{T}$.
\section{Type rules}
\begin{figure}[tp] \begin{figure}[tp]
\begin{center}
$\begin{array}{l} $\begin{array}{l}
\typerule{S-Refl}\\ \typerule{S-Refl}\\
\begin{array}{@{}c} \begin{array}{@{}c}
@ -74,7 +84,7 @@ $\begin{array}{l}
\\ \\
\hline \hline
\vspace*{-0.3cm}\\ \vspace*{-0.3cm}\\
\Delta \vdash \wctype{\Delta'}{C}{\ol{T}} <: \wctype{\Delta'}{D}{[\ol{T}/\ol{X}]\ol{S}} \Delta \vdash \wcNtype{\Delta'}{\type{N}} <: \wcNtype{\Delta'}{[\ol{T}/\ol{X}]\type{N}}
\end{array} \end{array}
\end{array}$ \end{array}$
$\begin{array}{l} $\begin{array}{l}
@ -82,15 +92,16 @@ $\begin{array}{l}
\begin{array}{@{}c} \begin{array}{@{}c}
\Delta', \Delta \vdash [\ol{T}/\ol{\type{X}}]\ol{L} <: \ol{T} \quad \quad \Delta', \Delta \vdash [\ol{T}/\ol{\type{X}}]\ol{L} <: \ol{T} \quad \quad
\Delta', \Delta \vdash \ol{T} <: [\ol{T}/\ol{\type{X}}]\ol{U} \\ \Delta', \Delta \vdash \ol{T} <: [\ol{T}/\ol{\type{X}}]\ol{U} \\
\text{fv}(\ol{T}) \subseteq \text{dom}(\Delta, \Delta') \quad \quad \text{fv}(\ol{T}) \subseteq \text{dom}(\Delta, \Delta') \quad
\text{dom}(\Delta') \cap \text{fv}(\wctype{\ol{\wildcard{X}{U}{L}}}{C}{\ol{S}}) = \emptyset \text{dom}(\Delta') \cap \text{fv}(\wctype{\ol{\wildcard{X}{U}{L}}}{C}{\ol{S}}) = \emptyset
\\ \\
\hline \hline
\vspace*{-0.3cm}\\ \vspace*{-0.3cm}\\
\Delta \vdash \wctype{\Delta'}{C}{[\ol{T}/\ol{\type{X}}]\ol{S}} <: \Delta \vdash \wcNtype{\Delta'}{[\ol{T}/\ol{X}]\type{N}} <:
\wctype{\ol{\wildcard{X}{U}{L}}}{C}{\ol{S}} \wcNtype{\ol{\wildcard{X}{U}{L}}}{N}
\end{array} \end{array}
\end{array}$ \end{array}$
\end{center}
\caption{Subtyping}\label{fig:subtyping} \caption{Subtyping}\label{fig:subtyping}
\end{figure} \end{figure}
@ -141,69 +152,32 @@ $\begin{array}{l}
\end{center} \end{center}
\caption{Well-formedness}\label{fig:well-formedness} \caption{Well-formedness}\label{fig:well-formedness}
\end{figure} \end{figure}
%TODO: Proof that well-formed (ok) implies that a type is witnessed
We change the type rules to require the well-formedness instead of the witnessed property.
See figure \ref{fig:well-formedness}.
Our well-formedness criteria is more restrictive than the ones used for \wildFJ{}.
\cite{WildcardsNeedWitnessProtection} works with the two different judgements $\ok$ and \texttt{witnessed}.
With \texttt{witnessed} being the stronger one.
We rephrased the $\ok$ judgement to include \texttt{witnessed} aswell.
$\type{T} \ \ok$ in this paper means $\type{T} \ \ok$ and $\type{T} \ \texttt{witnessed}$ in the sense of the \wildFJ{} type rules.
Java's type system is complex enough as it is. Simplification, when possible, is always appreciated.
Our $\ok$ rule may not be able to express every corner of the Java type system, but is sufficient to show soundness regarding type inference for a Java core calculus.
The \rulename{WF-Class} rule requires upper and lower bounds to be direct subtypes of each other $\type{L} <: \type{U}$.
The type rules from \cite{WildcardsNeedWitnessProtection} use a witness type instead.
Stating that a type is well formed (\texttt{witnessed}) if there exists atleast one simple type $\type{N}$ as a possible subtype:
$\Delta \vdash \type{N} <: \wctype{\Delta'}{C}{\ol{T}}$.
A witness type is easy to find by replacing every wildcard $\ol{W}$ in $\wctype{\ol{\wildcard{W}{U}{L}}}{C}{\ol{T}}$ by its upper bound:
$\Delta \vdash \exptype{C}{[\ol{U}/\ol{W}]\ol{T}} <: \wctype{\ol{\wildcard{W}{U}{L}}}{C}{\ol{T}}$.
$\Delta \vdash \exptype{C}{[\ol{U}/\ol{W}]\ol{T}} \ \texttt{witnessed}$ is given due to $\Delta \vdash \ol{T}, \ol{L}, \ol{U} \ \ok$
and $\Delta \vdash \wctype{\ol{\wildcard{W}{U}{L}}}{C}{\ol{T}} \ \ok$.
%Not necessary!
% Additionally we do not allow nested wildcards. %TODO: Unify cannot create them (or does it?) What is when the input contains them
% The \unify{} algorithm is not capable of generating types like $\wctype{\rwildcard{X}, \wildcard{Y}{\bot}{\rwildcard{X}}}{Pair}{\rwildcard{X}, \rwildcard{Y}}$.
% %But they could be created in intermidiate types (\ol{S}) in method calls by capture conversion:
% class WList<A> extends List<List<? extends A>>
% m(List<X> l) % TODO
% m(X.Wlist<X>) % S = Y^X.List<Y>
% X.WList<X> <. List<x?>
% X.List<Y^X.List<Y>> <. List<x?>
% x =. Y^X.List<Y>
% %TODO: do we need \Delta' \Delta \vdash T, L ,U ok ? or is \Delta \vdashh ... sufficient?
% %<A> m(List<? extends A> l)
% %This is important because it means different things maybe for the proof of our OK being more restrictive than "witnessed"
% % where can nested wildcards occur?
% WBox<X> extends Box<Box<? extends X>>
% WBox<?> => Y.WBox<Y> <: Y.Box<X^Y.Box<X>>
% Everything else regarding subtyping stays the same as in \cite{WildcardsNeedWitnessProtection}.
% \begin{lemma}
% If $\Delta \vdash $
% \end{lemma}
Figure \ref{fig:tletexpr} shows type rules for fields and method calls.
They have been merged with let statements and simplified.
The let statements and the type $\wcNtype{\Delta'}{N}$, which the type inference algorithm can freely choose,
are necessary for the soundness proof.
\begin{figure}[tp] \begin{figure}[tp]
\begin{center}
$\begin{array}{l} $\begin{array}{l}
\typerule{T-Var}\\ \typerule{T-Var}\\
\begin{array}{@{}c} \begin{array}{@{}c}
x : \type{T} \in \Gamma \texttt{x} : \type{T} \in \Gamma
\\ \\
\hline \hline
\vspace*{-0.3cm}\\ \vspace*{-0.3cm}\\
\Delta | \Gamma \vdash x : \type{T} \triangle | \Gamma \vdash \texttt{x} : \type{T}
\end{array}
\end{array}$ \hfill
$\begin{array}{l}
\typerule{T-Field}\\
\begin{array}{@{}c}
\Delta | \Gamma \vdash \texttt{e} : \type{T} \quad \quad
\Delta \vdash \type{T} <: \wcNtype{\Delta'}{N} \quad \quad
\textit{fields}(\type{N}) = \ol{U\ f} \\
\Delta, \Delta' \vdash \type{U}_i <: \type{S} \quad \quad
\text{dom}(\Delta') \subseteq \text{fv}(\type{N}) \quad \quad
\Delta \vdash \type{S}, \wcNtype{\Delta'}{N} \ \ok
\\
\hline
\vspace*{-0.3cm}\\
\Delta | \Gamma \vdash \texttt{e}.\texttt{f}_i : \type{S}
\end{array} \end{array}
\end{array}$ \end{array}$
\\[1em] \\[1em]
@ -221,23 +195,7 @@ $\begin{array}{l}
\\ \\
\hline \hline
\vspace*{-0.3cm}\\ \vspace*{-0.3cm}\\
\Delta | \Gamma \vdash \letstmt{\ol{x} : \ol{\wcNtype{\Delta}{N}} = \ol{t}}{\texttt{new} \ \exptype{C}{\ol{T}}(\overline{t})} : \type{T} \triangle | \Gamma \vdash \texttt{new}\ \type{C}(\ol{t}) : \exptype{C}{\ol{T}}
\end{array}
\end{array}$
\\[1em]
$\begin{array}{l}
\typerule{T-Field}\\
\begin{array}{@{}c}
\Delta | \Gamma \vdash \texttt{t} : \type{T} \quad \quad
\Delta \vdash \type{T} <: \wcNtype{\Delta'}{N} \quad \quad
\textit{fields}(\type{N}) = \ol{U\ f} \\
\Delta, \Delta' \vdash \type{U}_i <: \type{S} \quad \quad
\text{dom}(\Delta') \subseteq \text{fv}(\type{N}) \quad \quad
\Delta \vdash \type{S}, \wcNtype{\Delta'}{N} \ \ok
\\
\hline
\vspace*{-0.3cm}\\
\Delta | \Gamma \vdash \texttt{let}\ x : \wcNtype{\Delta'}{N} = \texttt{t} \ \texttt{in}\ x.\texttt{f}_i : \type{S}
\end{array} \end{array}
\end{array}$ \end{array}$
\\[1em] \\[1em]
@ -245,72 +203,109 @@ $\begin{array}{l}
\typerule{T-Call}\\ \typerule{T-Call}\\
\begin{array}{@{}c} \begin{array}{@{}c}
\Delta, \Delta', \overline{\Delta} \vdash \ol{\type{N}} <: [\ol{S}/\ol{X}]\ol{U} \quad \quad \Delta, \Delta', \overline{\Delta} \vdash \ol{\type{N}} <: [\ol{S}/\ol{X}]\ol{U} \quad \quad
\textit{mtype}(\texttt{m}, \type{N}) = \generics{\ol{X \triangleleft U'}} \ol{U} \to \type{U} \quad \quad \generics{\ol{X \triangleleft U'}} \type{N} \to \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} <: [\ol{S}/\ol{X}]\ol{U'}
\\ \\
\Delta, \Delta', \overline{\Delta} \vdash \ol{S} \ \ok \quad \quad \Delta, \Delta', \overline{\Delta} \vdash \ol{S} \ \ok \quad \quad
\Delta | \Gamma \vdash \texttt{t}_r : \type{T}_r \quad \quad \Delta | \Gamma \vdash \texttt{e} : \type{T}_r \quad \quad
\Delta | \Gamma \vdash \ol{t} : \ol{T} \quad \quad \Delta | \Gamma \vdash \ol{e} : \ol{T} \quad \quad
\Delta \vdash \type{T}_r <: \wcNtype{\Delta'}{N} \quad \quad \Delta \vdash \type{T}_r <: \wcNtype{\Delta'}{N} \quad \quad
\Delta \vdash \ol{T} <: \ol{\wcNtype{\Delta}{N}} \Delta \vdash \ol{T} <: \overline{\wcNtype{\Delta}{N}}
\\ \\
\Delta \vdash \type{T}, \wcNtype{\Delta'}{N}, \overline{\wcNtype{\Delta}{N}} \ \ok \quad \quad \Delta \vdash \type{T}, \wcNtype{\Delta'}{N}, \overline{\wcNtype{\Delta}{N}} \ \ok \quad \quad
\Delta, \Delta', \Delta'' \vdash [\ol{S}/\ol{X}]\type{U} <: \type{T} \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 \text{dom}(\Delta') \subseteq \text{fv}(\type{N}) \quad \quad
\overline{\text{dom}(\Delta) \subseteq \text{fv}(\type{N})} \overline{\text{dom}(\Delta) \subseteq \text{fv}(\type{N})}
\\ \\
\hline \hline
\vspace*{-0.3cm}\\ \vspace*{-0.3cm}\\
\Delta | \Gamma \vdash \letstmt{x : \wcNtype{\Delta'}{N} = t_r, \ol{x} : \ol{\wcNtype{\Delta}{N}} = \ol{t}} \Delta | \Gamma \vdash \texttt{e}.\texttt{m}(\ol{e}) : \type{T}
{\texttt{x}.\generics{\ol{S}}\texttt{m}(\ol{x})} : \type{T}
\end{array} \end{array}
\end{array}$ \end{array}$
\\[1em] \\[1em]
$\begin{array}{l} $\begin{array}{l}
\typerule{T-Elvis}\\ \typerule{T-Elvis}\\
\begin{array}{@{}c} \begin{array}{@{}c}
\Delta | \Gamma \vdash t_1 : \type{T}_1 \quad \quad \triangle | \Gamma \vdash \texttt{t} : \type{T}_1 \quad \quad
\Delta | \Gamma \vdash t_2 : \type{T}_2 \quad \quad \triangle | \Gamma \vdash \texttt{t}_2 : \type{T}_2 \quad \quad
\Delta \vdash \type{T}_1 <: \type{T} \quad \quad \triangle \vdash \type{T}_1 <: \type{T} \quad \quad
\Delta \vdash \type{T}_2 <: \type{T} \triangle \vdash \type{T}_2 <: \type{T} \quad \quad
\\ \\
\hline \hline
\vspace*{-0.3cm}\\ \vspace*{-0.3cm}\\
\Delta | \Gamma \vdash t_1 \elvis{} t_2 : \type{T} \triangle | \Gamma \vdash \texttt{t}_1 \ \texttt{?:} \ \texttt{t}_2 : \type{T}
\end{array}
\end{array}$
\end{center}
\caption{Expression Typing}\label{fig:expressionTyping}
\end{figure}
\begin{figure}
\begin{center}
$\begin{array}{l}
\typerule{T-Method}\\
\begin{array}{@{}c}
\exptype{C}{\ol{X}} \to \ol{T} \to \type{T} \in \mathtt{\Pi}(\texttt{m})\quad \quad
%\text{dom}(\triangle) = \ol{X} \quad \quad
\triangle' = \overline{\type{Y} : \bot .. \type{P}} \quad \quad
\triangle, \triangle' \vdash \ol{U}, \type{T}, \ol{T} \ \ok \\
%\texttt{class}\ \exptype{C}{\ol{X \triangleleft \_ }} \triangleleft \type{N} \ \{ \ldots \} \\
\mathtt{\Pi} | \triangle, \triangle' | \ol{x : T}, \texttt{this} : \exptype{C}{\ol{X}} \vdash \texttt{e} : \type{S} \quad \quad
\triangle \vdash \type{S} <: \type{T}
\\
\hline
\vspace*{-0.3cm}\\
\triangle \vdash \texttt{m}(\ol{x}) = \texttt{e} \ \ok \text{ in C with } \generics{\ol{Y \triangleleft P}}
\end{array} \end{array}
\end{array}$ \end{array}$
\\[1em] \\[1em]
$\begin{array}{l} $\begin{array}{l}
\typerule{T-Method}\\ \typerule{T-Class}\\
\begin{array}{@{}c} \begin{array}{@{}c}
\text{dom}(\Delta)=\ol{X} \quad \quad \mathtt{\Pi}' = \mathtt{\Pi} \cup \set{ \texttt{m} \mapsto (\exptype{C}{\ol{X}} \to \ol{T}_\texttt{m} \to \type{T}_\texttt{m}) \mid \texttt{m} \in \ol{M}} \\
\Delta' = \overline{\type{Y} : \bot .. \type{U}} \quad \quad \mathtt{\Pi}'' = \mathtt{\Pi} \cup \set{ \texttt{m} \mapsto
\Delta, \Delta' \vdash \ol{U}, \type{T}, \ol{T}\ \ok \quad \quad \generics{\ol{X \triangleleft \type{N}}, \ol{Y \triangleleft P}}(\exptype{C}{\ol{X}} \to \ol{T}_\texttt{m} \to \type{T}_\texttt{m}) \mid \texttt{m} \in \ol{M} } \\
\texttt{class}\ \exptype{C}{\ol{X \triangleleft \_ }} \triangleleft \type{N} \set{\ldots} \\ \triangle = \overline{\type{X} : \bot .. \type{U}} \quad \quad
\Delta, \Delta' | \overline{x:\type{T}}, \texttt{this} : \exptype{C}{\ol{X}} \vdash t:\type{S} \quad \quad \triangle \vdash \ol{U}, \ol{T}, \type{N} \ \ok \quad \quad
\Delta, \Delta' \vdash \type{S} <: \type{T} \quad \quad \triangle \vdash \ol{M} \ \ok \text{ in C with} \ \generics{\ol{Y \triangleleft P}}
\text{override}(\texttt{m}, \type{N}, \generics{\ol{Y \triangleleft U}}\ol{T} \to \type{T})
\\ \\
\hline \hline
\vspace*{-0.3cm}\\ \vspace*{-0.3cm}\\
\Delta \vdash \generics{\ol{Y \triangleleft U}} \type{T} \ \texttt{m}(\overline{\type{T} \ x}) = t \ \ok \ \texttt{in C} \texttt{class}\ \exptype{C}{\ol{X \triangleleft U}} \triangleleft \type{N} \{ \ol{T\ f}; \ol{M} \} : \mathtt{\Pi}''
\end{array} \end{array}
\end{array}$ \end{array}$
\\[1em] \\[1em]
$\begin{array}{l} $\begin{array}{l}
\typerule{T-Class}\\ \typerule{T-Program}\\
\begin{array}{@{}c} \begin{array}{@{}c}
\Delta = \overline{\type{X} : \bot .. \type{U}} \quad \quad \emptyset \vdash \texttt{L}_1 : \mathtt{\Pi}_1 \quad \quad
\Delta \vdash \ol{U}, \ol{T} \ \ok \quad \quad \mathtt{\Pi}_1 \vdash \texttt{L}_2 : \mathtt{\Pi}_1 \quad \quad
\Delta \vdash \type{N} \ \ok \quad \quad \ldots \quad \quad
\Delta \vdash \ol{M} \ \ok \texttt{ in C} \mathtt{\Pi}_{n-1} \vdash \texttt{L}_n : \mathtt{\Pi}_n \quad \quad
\\ \\
\hline \hline
\vspace*{-0.3cm}\\ \vspace*{-0.3cm}\\
\texttt{class}\ \exptype{C}{\ol{X \triangleleft U}} \triangleleft \type{N} \set{\overline{\type{T}\ f}; \ol{M} } \ \ok \vdash \ol{L} : \mathtt{\Pi}_n
\end{array} \end{array}
\end{array}$ \end{array}$
\caption{T-Call and T-Field} \label{fig:tletexpr} \end{center}
\caption{Class and Method Typing rules}\label{fig:typing}
\end{figure} \end{figure}
\begin{figure}
$\text{fields}(\exptype{Object}{}) = \emptyset$
\quad \quad
$\begin{array}{l}
\typerule{F-Class}\\
\begin{array}{@{}c}
\texttt{class}\ \exptype{C}{\ol{X \triangleleft \_ }} \triangleleft \type{N} \set{\ol{S\ f}; \ol{M}} \quad \quad
\text{fields}([\ol{T}/\ol{X}]\type{N}) = \ol{U\ g}
\\
\hline
\vspace*{-0.3cm}\\
\text{fields}(\exptype{C}{\ol{T}}) = \ol{U\ g}, [\ol{T}/\ol{X}]\ol{S\ f}
\end{array}
\end{array}$
\end{figure}

19
tiRules.tex
View File

@ -1,24 +1,5 @@
\section{Typeless FJ} \section{Typeless FJ}
\subsection{Syntax}
We use the TameFJ syntax for the most part (\ref{fig:syntax-tamingfj}).
\begin{figure}[tp]
\begin{align*}
\mv N &::= \exptype{C}{\ol{W}}\\
\mv T &::= \mv X \mid \wctype{\ol{W}}{C}{\ol{W}}\\
\mv W &::= \mv T \mid \wildcard{Z}{T}{T}\\
\mv L &::= \mathtt{class} \ \exptype{C}{\ol{X} \triangleleft \ol{T}} \triangleleft \ \mv N\ \{ \ol{T} \ \ol{f}; \,\mv K \, \ol{M} \} \\
\mv K &::= \mv C(\ol{f})\ \{\mathtt{super}(\ol{f}); \ \mathtt{this}.\ol{f}=\ol{f};\} \\
\mv M &::= \mathtt{m}(\ol{x})\ = \mv e \\
\mv e &::= \mv x \mid \mv e.\mv f \mid
\mv e.\mathtt{m}(\ol{e}) \mid \mathtt{new}\ \mathtt{C}(\ol{e})
\mid \mv e \elvis{} \mv e
\end{align*}
\caption{Syntax of TameFJ + Type Inference}
\label{fig:syntax-tamingfj}
\end{figure}
\subsection{Type System} \subsection{Type System}
The type system is similar to the ones used in \emph{Wildcards need Witness Protection}. The type system is similar to the ones used in \emph{Wildcards need Witness Protection}.