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Fix Well-formed description

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Andreas Stadelmeier 2024-01-05 09:19:10 +01:00
parent 3a499d2e6d
commit 5873fe68d6
1 changed files with 30 additions and 8 deletions
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tRules.tex
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@ -143,7 +143,7 @@ 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.
$\type{T} \ \ok$ 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.
@ -152,14 +152,36 @@ The type rules from \cite{WildcardsNeedWitnessProtection} use a witness type ins
Stating that a type is well formed (\texttt{witnessed}) if there exists atleast one simple type $\type{N}$ as a possible subtype:
$\Delta \vdash \type{N} <: \wctype{\Delta'}{C}{\ol{T}}$.
%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"
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 $\Delta \vdash \wctype{\ol{\wildcard{W}{U}{L}}}{C}{\ol{T}} \ \ok$.
Everything else regarding subtyping stays the same as in \cite{WildcardsNeedWitnessProtection}.
\begin{lemma}
If $\Delta \vdash $
\end{lemma}
%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.