stan/Ecoop2024_TIforWildFJ
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Add Wildcard Environment to intro Unify example

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This commit is contained in:
Andreas Stadelmeier 2024-03-27 01:53:23 +01:00
parent 9556f1521e
commit e93a762441
2 changed files with 17 additions and 4 deletions
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18
introduction.tex
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@ -287,10 +287,10 @@ $\begin{array}{c}
\wctype{\wildcard{X}{\type{Object}}{\type{String}}}{List}{\rwildcard{X}} \lessdotCC \exptype{List}{\wtv{a}}, \, \type{String} \lessdotCC \wtv{a} \wctype{\wildcard{X}{\type{Object}}{\type{String}}}{List}{\rwildcard{X}} \lessdotCC \exptype{List}{\wtv{a}}, \, \type{String} \lessdotCC \wtv{a}
\\ \\
\hline \hline
\textit{Capture Conversion:}\ \exptype{List}{\rwildcard{X}} \lessdot \exptype{List}{\wtv{a}}, \, \type{String} \lessdot \wtv{a} \textit{Capture Conversion:}\ \wildcard{Y}{\type{Object}}{\type{String}} \wcSep \exptype{List}{\rwildcard{Y}} \lessdot \exptype{List}{\wtv{a}}, \, \type{String} \lessdot \wtv{a}
\\ \\
\hline \hline
\textit{Solution:}\ \wtv{a} \doteq \rwildcard{X} \implies \exptype{List}{\rwildcard{X}} \lessdot \exptype{List}{\rwildcard{X}}, \, \type{String} \lessdot \rwildcard{X} \textit{Solution:}\ \wtv{a} \doteq \rwildcard{Y} \implies \wildcard{Y}{\type{Object}}{\type{String}} \wcSep \exptype{List}{\rwildcard{Y}} \lessdot \exptype{List}{\rwildcard{Y}}, \, \type{String} \lessdot \rwildcard{Y}
\end{array} \end{array}
$ $
\end{center} \end{center}
@ -304,7 +304,11 @@ which converts a constraint of the form $(\wctype{\rwildcard{X}}{C}{\rwildcard{X
The type of \texttt{l} can be capture converted by a let statement if needed (see listing \ref{lst:addExampleLet}). The type of \texttt{l} can be capture converted by a let statement if needed (see listing \ref{lst:addExampleLet}).
Therefore we assign the constraint $\wctype{\wildcard{X}{\type{Object}}{\type{String}}}{List}{\rwildcard{X}} \lessdotCC \exptype{List}{\wtv{a}}$ Therefore we assign the constraint $\wctype{\wildcard{X}{\type{Object}}{\type{String}}}{List}{\rwildcard{X}} \lessdotCC \exptype{List}{\wtv{a}}$
which allows \unify{} to do a capture conversion to $\exptype{List}{\rwildcard{X}} \lessdot \exptype{List}{\wtv{a}}$. which allows \unify{} to do a capture conversion to $\exptype{List}{\rwildcard{X}} \lessdot \exptype{List}{\wtv{a}}$.
\textit{Note:} The constraint $\type{String} \lessdot \rwildcard{X}$ is satisfied because $\rwildcard{X}$ has $\type{String}$ as lower bound. The captured wildcard $\rwildcard{X}$ gets a fresh name and is stored in the wildcard environment of the \unify{} algorithm.
\textit{Note:} The constraint $\type{String} \lessdot \rwildcard{Y}$ is satisfied
because $\rwildcard{Y}$ has $\type{String}$ as lower bound.
For the example shown in listing \ref{shuffleExample} the method call \texttt{shuffle(l2d)} creates the constraints: For the example shown in listing \ref{shuffleExample} the method call \texttt{shuffle(l2d)} creates the constraints:
@ -696,3 +700,11 @@ But afterwards a capture conversion is applied, which can generate different typ
\item $\text{CC}(\wctype{\rwildcard{X}}{List}{\rwildcard{X}}) \implies \exptype{List}{\rwildcard{Y}}$ \item $\text{CC}(\wctype{\rwildcard{X}}{List}{\rwildcard{X}}) \implies \exptype{List}{\rwildcard{Y}}$
\item $\text{CC}(\wctype{\rwildcard{X}}{List}{\rwildcard{X}}) \implies \exptype{List}{\rwildcard{Z}}$ \item $\text{CC}(\wctype{\rwildcard{X}}{List}{\rwildcard{X}}) \implies \exptype{List}{\rwildcard{Z}}$
\end{itemize} \end{itemize}
Wildcards are not reflexive. A box of type $\wctype{\rwildcard{X}}{Box}{\rwildcard{X}}$
is able to hold a value of any type. It could be a $\exptype{Box}{String}$ or a $\exptype{Box}{Integer}$ etc.
Also two of those boxes do not necessarily contain the same type.
But there are situations where it is possible to assume that.
For example the type $\wctype{\rwildcard{X}}{Pair}{\exptype{Box}{\rwildcard{X}}, \exptype{Box}{\rwildcard{X}}}$
is a pair of two boxes, which always contain the same type.
Inside the scope of the \texttt{Pair} type, the wildcard $\rwildcard{X}$ stays the same.

1
prolog.tex
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@ -20,6 +20,7 @@
\newcommand{\tifj}{\texttt{TamedFJ}} \newcommand{\tifj}{\texttt{TamedFJ}}
\newcommand{\wcSep}{\vdash}
\newcommand\mv[1]{{\tt #1}} \newcommand\mv[1]{{\tt #1}}
\newcommand{\ol}[1]{\overline{\tt #1}} \newcommand{\ol}[1]{\overline{\tt #1}}
\newcommand{\exptype}[2]{\mathtt{#1 \texttt{<} #2 \texttt{>} }} \newcommand{\exptype}[2]{\mathtt{#1 \texttt{<} #2 \texttt{>} }}