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paper: wip add more lemmas

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Michael Sippel 2024-07-27 13:30:34 +02:00
parent eebb096f8a
commit c7794d8a89
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GPG key ID: F96CF119C34B64A6
1 changed files with 49 additions and 25 deletions
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paper/main.tex
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@ -176,11 +176,11 @@ $$\\$$
\metavariable{x} \quad \valnonterm{\typevars}{\exprvars}
}{Value Conactenation}
%\otherform{
% \exprterminal{\Lambda} \metavariable{\alpha} \quad
% \exprterminal{\mapsto} \quad
% \valnonterm{ \typevars \cup \{ \metavariable{\alpha} \} }
%}{Type-Function Value}
\otherform{
\exprterminal{\Lambda} \metavariable{\alpha} \quad
\exprterminal{\mapsto} \quad
\valnonterm{ \typevars \cup \{ \metavariable{\alpha} \} }
\{Type-Function Value}
\otherform{
\exprterminal{\lambda} \metavariable{x} \quad
@ -371,8 +371,8 @@ As usual, each rule is composed of premises (above the horizontal line) and a co
}
\inferrule[T-TypeApp]{
\Gamma \vdash \metavariable{e} : \metavariable{\tau} \\
\metavariable{\tau} \in \typenonterm{\typevars \cup \metavariable{\alpha}} \\
\metavariable{\tau} \in \typenonterm{\typevars \cup \{\metavariable{\alpha}\}} \\
\Gamma \vdash \metavariable{e} : \typeterminal{\forall} \metavariable{\alpha} \typeterminal{.} \metavariable{\tau} \\
\metavariable{\sigma} \in \typenonterm{\typevars}
}{
\Gamma \vdash ( \metavariable{e} \quad \metavariable{\sigma} ) : \{\metavariable{\alpha} \mapsto \metavariable{\sigma}\} \metavariable{\tau}
@ -520,42 +520,66 @@ which are given in \ref{def:evalrules}.
\begin{lemma}[\(\beta\)-reduction preserves \(\delta\)-normalform]
Assume \metavariable{e} is in \(\delta\)-normalform and \(\metavariable{e} \rightarrow \metavariable{e'}\). Then \(\metavariable{e'}\) is in \(\delta\)-normalform as well.
\label{lemma:preserve-delta-normalform}
Assume \metavariable{e} is in \(\delta\)-normalform and \(\metavariable{e} \rightarrow_\beta \metavariable{e'}\). Then \(\metavariable{e'}\) is in \(\delta\)-normalform as well.
\begin{proof}
\todo{}
\end{proof}
\end{lemma}
\begin{lemma}[\(\delta\)-normalform eliminates compatibility]
\label{lemma:eliminate-compat}
Assume \(\emptyset \vdash \metavariable{e} :\approx \metavariable{\tau}\) and \(\metavariable{e} \rightarrow_{\delta}^* \metavariable{e'}\) such that \(\metavariable{e'}\) is in \(\delta\)-normalform.
Then \(\emptyset \vdash \metavariable{e'} : \metavariable{\tau}\)
\begin{proof}
\end{proof}
\end{lemma}
\subsection{Proof of Syntactic Type Soundness}
\begin{lemma}[\(\beta\)-Preservation]
\label{lemma:beta-preservation}
Assume the expression \(\metavariable{e}\) is \textbf{syntactically well-typed}, i.e. \(\emptyset \vdash \metavariable{e} : \metavariable{\tau}\) for some type \(\metavariable{\tau}\). Then forall \(\metavariable{e'}\) with \(\metavariable{e} \rightarrow_{\beta} \metavariable{e'}\) it holds that \(\emptyset \vdash \metavariable{e'} : \metavariable{\tau}\) as well.
\begin{proof}
\todo{}
\end{proof}
\end{lemma}
\begin{lemma}[\(\delta\)-Preservation]
\label{lemma:delta-preservation}
\begin{proof}
\todo{}
\end{proof}
\end{lemma}
\begin{lemma}[Preservation]
\label{lemma:preservation}
Assume the expression \(\metavariable{e}\) is well typed, i.e. \(\emptyset \vdash \metavariable{e} : \metavariable{\tau}\) for some type \(\metavariable{\tau}\). Then forall \(\metavariable{e'}\) with \(\metavariable{e} \rightarrow_{eval} \metavariable{e'}\) it holds that \(\emptyset \vdash \metavariable{e'} : \metavariable{\tau}\) as well.
\begin{proof}
\todo{}
\end{proof}
\end{lemma}
\begin{lemma}[Progress]
\label{lemma:progress}
If \(\emptyset \vdash \metavariable{e} : \metavariable{\tau}\), then either \(\metavariable{e}\) is a value or there exists some \(\metavariable{e'}\) such that \(\metavariable{e} \rightarrow_{eval} \metavariable{e'}\)
\begin{proof}
\todo{}
\end{proof}
\end{lemma}
\begin{lemma}[Preservation]
\label{lemma:preservation}
\begin{proof}
\todo{}
\end{proof}
\end{lemma}
\begin{theorem}[Type Soundness]
If \(\emptyset \vdash \metavariable{e}:\metavariable{\tau}\), then it never occurs that \(\metavariable{e} \rightarrow_{eval}^{*} \metavariable{e'}\) where \metavariable{e'} is in normal form but not a value.
\begin{theorem}[Soundness]
If \(\emptyset \vdash \metavariable{e}:\approx\metavariable{\tau}\), then it never occurs that \(\metavariable{e} \rightarrow_{eval}^{*} \metavariable{e'}\) where \metavariable{e'} is in normal form but not a value.
\begin{proof}
By \ref{lemma:}
Follows from \ref{lemma:progress} and \ref{lemma:preservation}.
\end{proof}
\end{theorem}