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Soundness Prepare

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JanUlrich 2024-01-16 10:00:47 +01:00
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soundness.tex
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@ -272,24 +272,29 @@ $\Delta \vdash \sigma(C') \implies \Delta \vdash \sigma(C)$
\item[Prepare]
To show
$\Delta \vdash \wctype{\overline{\wildcard{B}{\type{U'}}{\type{L'}}}}{C}{\ol{S}} <: \wctype{\overline{\wildcard{A}{\type{U}}{\type{L}}}}{C}{\ol{T}}$ by S-Exists we have to proof:
We know
\begin{gather}
\ol{S} = \ol{T}\\
\text{fv}(\wctype{\overline{\wildcard{B}{\type{U'}}{\type{L'}}}}{C}{\ol{S}}) = \emptyset\\
\text{fv}(\wctype{\overline{\wildcard{A}{\type{U}}{\type{L}}}}{C}{\ol{T}}) = \emptyset
%TODO
\end{gather}
\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} \\
\label{rp:3}
\text{fv}(\ol{T}) \subseteq \text{dom}(\Delta, \overline{\wildcard{B}{U}{L}}) \\
\label{rp:4}
\text{dom}(\overline{\wildcard{B}{U}{L}}) \cap \text{fv}(\wctype{\ol{\wildcard{A}{U}{L}}}{C}{\ol{T}}) = \emptyset
\text{dom}(\overline{\wildcard{B}{U}{L}}) \cap \text{fv}(\wctype{\ol{\wildcard{A}{U}{L}}}{C}{\ol{T}}) = \emptyset\\
[\ol{A}/\ol{T}] = \ol{T} %TODO: rename T
\end{gather}
\ref{rp:4} is always true.
Due to $\text{fv}(\sigma(\wctype{\overline{\wildcard{B}{\type{U}}{\type{L}}}}{C}{\ol{S}})) = \emptyset$ implies \ref{rp:3}.
We know
\begin{gather}
\ol{S} = [\ol{\wtv{a}}/\ol{A}]\ol{T}\\
\text{fv}(\wctype{\overline{\wildcard{B}{\type{U'}}{\type{L'}}}}{C}{\ol{S}}) = \emptyset\\
\label{rp:fv2}
\text{fv}(\wctype{\overline{\wildcard{A}{\type{U}}{\type{L}}}}{C}{\ol{T}}) = \emptyset
%TODO
\end{gather}
\ref{rp:fv2} implies \ref{rp:4}.
Due to $\text{fv}(\sigma(\wctype{\overline{\wildcard{B}{\type{U}}{\type{L}}}}{C}{\ol{S}})) = \emptyset$ implies
$\text{fv}(\type{T}) \subseteq \text{dom}(\overline{\wildcard{B}{\type{U}}{\type{L}}})$
and therefore \ref{rp:3}.
\item[Capture]
If $\text{fv}(\wctype{\Delta}{C}{\ol{T}}) = \emptyset$ the preposition holds by Assumption and S-Exists.