Beispiel für Wildcard Anwendung

vortrag.tex

@@ -19,17 +19,37 @@
 $\begin{array}{l}
     \begin{array}{@{}c}
         \textit{mtype}(\texttt{m}) = \generics{\ol{X}}\ \ol{T} \to \type{T} \quad \quad
-        \Delta \vdash  \expr{v}, \ol{v} : [\ol{T'}/\ol{X}]\ol{T}
+        \Delta \vdash  \ol{v} : [\ol{T'}/\ol{X}]\ol{T}
         \\
         \hline
     \vspace*{-0.3cm}\\
-        \Delta \vdash \expr{v}.\generics{\ol{T'}}\texttt{m}(\ol{v}) : [\ol{T'}/\ol{X}]\type{T}
+        \Delta \vdash \generics{\ol{T'}}\texttt{m}(\ol{v}) : [\ol{T'}/\ol{X}]\type{T}
     \end{array}
 \end{array}$
 \end{center}
 
+\textbf{Example:}
+$\generics{\type{String}}\texttt{emptyList}() : \exptype{List}{\type{String}}$
+\\[2em]
+\rulename{TI-Call} \\[1em]
+\begin{center}
+$\begin{array}{l}
+    \begin{array}{@{}c}
+        \textit{mtype}(\texttt{m}) = \generics{\ol{X}}\ \ol{T} \to \type{T} \quad \quad
+        \Delta \vdash  \expr{v}, \ol{v} : [\ol{T'}/\ol{X}]\ol{T}
+        \\
+        \hline
+    \vspace*{-0.3cm}\\
+        \Delta \vdash \expr{v}.\texttt{m}(\ol{v}) : [\ol{T'}/\ol{X}]\type{T}
+    \end{array}
+\end{array}$
+\end{center}
+
+\textbf{Example:}
+$\texttt{emptyList}() : \exptype{List}{\type{String}}$
 \end{frame}
 
+
 \begin{frame}[fragile]
 \frametitle{Java Method Call}
 \rulename{M-Class} \\[1em]
@@ -89,7 +109,7 @@ $\begin{array}{l}
         \generics{\type{String}}\texttt{head}(\texttt{new List<String>}()) : [\type{String}/\type{X}]\type{X}
     \end{array}
 \end{array}$
-
+\pause
 $\begin{array}{l}
     \begin{array}{@{}c}
         \textit{mtype}(\texttt{m}) = \generics{\ol{X}}\ \ol{T} \to \type{T} \quad \quad
@@ -120,6 +140,43 @@ $
 $
 \end{frame}
 
+\begin{frame}[fragile]
+\frametitle{Java Method Call}
+$
+\begin{array}[b]{c}
+\texttt{emptyList()} : \exptype{List}{\exptype{List}{\type{String}}} \\
+\quad \textit{mtype}(\texttt{get}) = \generics{X}\ \exptype{List}{\type{X}} \to \type{X}
+\\
+\hline
+\generics{\exptype{List}{\exptype{List}{\type{String}}}}\texttt{head(emptyList())} : \exptype{List}{\type{String}}
+\end{array} \rulenameAfter{T-Call}
+$
+
+$
+\begin{array}[b]{c}
+\texttt{emptyList()} : \exptype{List}{\exptype{List}{\type{String}}} \\
+\quad \textit{mtype}(\texttt{get}) = \generics{X}\ \exptype{List}{\type{X}} \to \type{X}
+\\
+\hline
+\generics{\exptype{List}{\type{String}}}\texttt{head(emptyList())} : \exptype{List}{\type{String}}
+\end{array} \rulenameAfter{T-Call}
+$
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{Java Method Call}
+
+$
+\begin{array}[b]{c}
+\texttt{emptyList()} : \exptype{List}{\exptype{List}{\type{String}}} \\
+\quad \textit{mtype}(\texttt{get}) = \generics{X}\ \exptype{List}{\type{X}} \to \type{X}
+\\
+\hline
+\generics{\exptype{List}{\type{String}}}\texttt{head(emptyList())} : \exptype{List}{\type{String}}
+\end{array} \rulenameAfter{T-Call}
+$
+\end{frame}
+
 \begin{frame}[fragile]
 \frametitle{Java Method Call}
 $
@@ -147,4 +204,68 @@ Dieser muss ausgepackt werden.
 Die entstehende freie Typvariable kann nun als Typ benutzt werden.
 
 \end{frame}
+
+
+
+\begin{frame}[fragile]
+%Wildcard Capture
+\begin{verbatim}
+List<List<?>> l;
+
+head(l) : List<?>
+\end{verbatim}
+%why do we need capture constraints?
+% List<List<?>> not a subtype of List<List<X>> for any X
+
+$\wctype{\rwildcard{A}}{List}{\rwildcard{A}}$
+\end{frame}
+
+\begin{frame}[fragile]
+%Explain why ? cannot be used as a regular type
+\begin{verbatim}
+<X> List<X> concat(List<X> l1, List<X> l2){
+    return l1.addAll(l2);
+}
+
+List<?> l1;
+List<?> l2;
+
+concat(l1, l2); //Error
+\end{verbatim}
+
+\begin{verbatim}
+List<String> ls = new List<String>();
+
+List<?> l1 = ls; //korrekt!
+List<?> l2 = new List<Integer>(1);
+
+concat(l1, l2); //Error (würde l2 an ls anfügen!)
+// ls.getLast() -> 1 (ein Int obwohl ls : List<String>)
+\end{verbatim}
+
+
+\end{frame}
+
+\begin{frame}[fragile]
+%Wildcard Capture
+\begin{verbatim}
+<X> void shuffle(List<List<X>> l){ 
+    //randomly exchange every element
+    // in the two dimensional list l
+}
+
+List<List<?>> l;
+
+shuffle(l); // Error
+\end{verbatim}
+
+$\wctype{\rwildcard{A}}{List}{\rwildcard{A}}$
+\end{frame}
+
+% TODO: 
+% Folie zu aktueller Stand Unify: Es gibt <. Constraints. Unify löst diese
+% Folie zu <? constraints: sie werden benötigt, weil unify während der Lösungsfindung Capture Conversion durchführen muss
+%why do we need capture constraints?
+% List<List<?>> not a subtype of List<List<X>> for any X
+
 \end{document}