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Add prepare rule

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Andreas Stadelmeier 2024-01-10 16:03:23 +01:00
parent 93c0b76b9c
commit bf401f1f08
2 changed files with 47 additions and 6 deletions
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24
soundness.tex
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@ -188,12 +188,16 @@ This rule is only applied for the outer wildcard environments for each type.
% Y.Box<Y> <. Box<a?>
% Y =. a?
% Box<Y.Box<Y>> <. Box<? extends Box<a>>
% Y.Box<Y> <. Box<a>
% Problem: für tvs dürfen keine Wildcards eingesetzt werden, außer für die
\begin{lemma}
The \unify{} algorithm only produces correct output for constraints not containing free variables.
\begin{description}
\item[If] $(\sigma, \Delta) = \unify{}( \overline{ \type{S} \lessdot \type{T} } \cup \overline{ \type{S'} \lessdot \type{T'} } )$ %\cup \overline{ \type{S} \doteq \type{S'} })$
\item[and] $fv(\overline{ \type{S} }) = \emptyset$, $fv(\overline{ \type{T} }) = \emptyset$
\item[If] $(\sigma, \Delta) = \unify{}( \overline{ \type{S} \lessdot \type{T} } \cup \overline{ \type{S'} \lessdotCC \type{T'} } )$ %\cup \overline{ \type{S} \doteq \type{S'} })$
%\item[and] $fv(\overline{ \type{S} }) = \emptyset$, $fv(\overline{ \type{T} }) = \emptyset$
\item[Then] there exists a $\Delta'$ with:
$\Delta \vdash \overline{\sigma(\type{S}) <: \sigma(\type{T})}$
and $\Delta, \Delta' \vdash \overline{\CC{}(\sigma(\type{S'})) <: \sigma(\type{T'})}$
@ -265,6 +269,18 @@ $\Delta \vdash \sigma(C') \implies \Delta \vdash \sigma(C)$
\item[Adapt] Assumption, S-Extends, S-Trans
\item[Adopt] Assumption, because $C \subset C'$
%\item[Capture, Reduce] are always applied together. We have to destinct between two cases:
\item[Prepare]
To show
$\Delta \vdash \wctype{\overline{\wildcard{B}{U}{L}}}{C}{\ol{S}} <: \wctype{\overline{\wildcard{A}{\type{U}}{\type{L}}}}{C}{\ol{T}}$ by S-Exists we have to proof:
\begin{gather}
\Delta', \Delta \vdash [\ol{T}/\ol{\type{A}}]\ol{L} <: \ol{T} \\
\Delta', \Delta \vdash \ol{T} <: [\ol{T}/\ol{\type{X}}]\ol{U} \\
\text{fv}(\ol{T}) \subseteq \text{dom}(\Delta) \\
\label{rp:4}
\text{dom}(\overline{\wildcard{B}{U}{L}}) \cap \text{fv}(\wctype{\ol{\wildcard{X}{U}{L}}}{C}{\ol{S}}) = \emptyset
\end{gather}
\ref{rp:4} is always true.
\item[Capture]
If $\text{fv}(\wctype{\Delta}{C}{\ol{T}}) = \emptyset$ the preposition holds by Assumption and S-Exists.
If $\text{fv}(\wctype{\Delta}{C}{\ol{T}}) \neq \emptyset$ the preposition holds because
@ -286,7 +302,9 @@ $\Delta' \Delta \vdash \text{CC}(\sigma(\wctype{\overline{\wildcard{B}{\type{U'}
%, which implies $\text{fv}(\ol{S}) \subseteq \text{dom}(\Delta \cup \set{\overline{\wildcard{B}{\type{U'}}{\type{L'}}}})$.
%We are doing a capture conversion. If $\type{T}$ does not contain free variables, this does not affect the subtype relation.
\item[Reduce] %Assumption and S-Exists.
\item[Reduce]
%Assumption and S-Exists.
% Three different cases of the constraint $\exptype{C}{\ol{S}} \lessdot \wctype{\overline{\wildcard{A}{\type{U}}{\type{L}}}}{C}{\ol{T}}$:
% \begin{description}
% \item[$\text{fv}(\exptype{C}{\ol{S}}) = \emptyset, \text{fv}(\wctype{\overline{\wildcard{A}{\type{U}}{\type{L}}}}{C}{\ol{T}})$:]

29
unify.tex
View File

@ -68,7 +68,7 @@ and a name $\mathtt{X}$.
The \rulename{Normalize} rule eliminates wildcards. % TODO
This is done by setting the upper and lower bound to the same value.
We generate wildcards with the \rulename{\generalizeRule} rule.
\unify{} generates wildcards with the \rulename{\generalizeRule} rule.
It is important to generate new wildcards in a standardized fashion.
When having two constraints $\type{T} \lessdot \tv{a}$ and $\type{T} \lessdot \tv{b}$,
then after applying the \rulename{\generalizeRule} rule the freshly generated constraints are
@ -359,10 +359,10 @@ Their upper and lower bounds are fresh type variables.
\begin{array}[c]{@{}ll}
\begin{array}[c]{l}
\wildcardEnv \vdash
C \cup \, \set{ \wctype{\overline{\wildcard{B}{\type{U'}}{\type{L'}}}}{C}{\ol{S}} \lessdotCC \wctype{\Delta}{C}{\ol{T}} } \\
C \cup \, \set{ \wctype{\overline{\wildcard{B}{\type{U}}{\type{L}}}}{C}{\ol{S}} \lessdotCC \wctype{\Delta}{C}{\ol{T}} } \\
\hline
\vspace*{-0.4cm}\\
\wildcardEnv \cup \overline{\wildcard{C}{\type{U'}}{\type{L'}}}
\wildcardEnv \cup \overline{\wildcard{C}{\type{U}}{\type{L}}}
\vdash C \cup \, \set{
[\ol{\rwildcard{C}}/\ol{\rwildcard{B}}] \exptype{C}{\ol{S}} \lessdot \wctype{\Delta}{C}{\ol{T}} }
\end{array}
@ -373,6 +373,29 @@ Their upper and lower bounds are fresh type variables.
\end{array}
\end{array}
$
\\\\
\rulename{Prepare}
&
$
\begin{array}[c]{@{}ll}
\begin{array}[c]{l}
\wildcardEnv \vdash
C \cup \, \set{ \wctype{\overline{\wildcard{B}{\type{U}}{\type{L}}}}{C}{\ol{S}} \lessdot \wctype{\overline{\wildcard{A}{\type{U}}{\type{L}}}}{C}{\ol{T}} } \\
\hline
\vspace*{-0.4cm}\\
\wildcardEnv \cup \overline{\wildcard{B}{\type{U}}{\type{L}}}
\vdash C \cup \, \set{
\ol{\type{S}} \doteq [\ol{\wtv{a}}/\overline{\rwildcard{A}}]\ol{\type{T}},
\ol{\wtv{a}} \lessdot [\ol{\wtv{a}}/\overline{\rwildcard{A}}]\ol{U}, [\ol{\wtv{a}}/\overline{\rwildcard{A}}]\ol{L} \lessdot \ol{\wtv{a}} }
\end{array}
%\quad \ol{Y} = \textit{fresh}(\ol{X})
\quad \begin{array}[c]{l}
\ol{\rwildcard{C}} \ \text{fresh}\\
\text{fv}(\wctype{\overline{\wildcard{A}{\type{U}}{\type{L}}}}{C}{\ol{T}}) = \emptyset\\
\text{fv}(\wctype{\overline{\wildcard{A}{\type{U}}{\type{L}}}}{C}{\ol{T}}) = \emptyset
\end{array}
\end{array}
$
\\\\
\rulename{Adopt}
& $