Commit 4db4e4f4 authored by Gaspard Ferey's avatar Gaspard Ferey

Fixed typos and order.

parent 022fbe42
......@@ -189,6 +189,16 @@ The subjection conversion property is a generalization of a subject reduction in
\emph{subject conversion} and \emph{typability preservation} together
imply the \emph{subject reduction} property.
\end{lemma}
\begin{proof}
The idea, is to be able to use the \textsc{conv} rule to deduce this property.
We want to prove that \(\Gamma \vdash t_2 : A\) from the fact that \(t_1 \hookrightarrow t_2\) and \(\Gamma \vdash t_1 : A\).
We use the \emph{typability preservation} property to get a type \(B\) to \(t_2\). Then, from the \emph{subject conversion} property, we know that \(\Gamma \vdash A \convbg B\).
The only thing we need now to apply the \textsc{conv} rule is to show that \(\Gamma \vdash A : s\) for some sort \(s\). However, this is not possible in general, for example when \(A = \tkind\). We have two cases to prove:
\begin{itemize}
\item If \(A = \tkind\), then using Lemma TODO, we can show that \(B = \tkind\), hence subject reduction is satisfied
\item If \(A \neq \tkind\), because \(A\) is inhabited, we can use Lemma TODO to conclude that \(\Gamma \vdash A :s\). Hence we can apply the \textsc{conv}, this closes the proof.
\end{itemize}
\end{proof}
\begin{lemma}[Type convertibility]
For all well-formed context $\Gamma$ satisfying
......@@ -210,16 +220,6 @@ The subjection conversion property is a generalization of a subject reduction in
For any well-formed context \(\Gamma\), and term $t$ such that \(\tkind \convbg t\) then \(t = \tkind\).
\end{lemma}
\begin{proof}
The idea, is to be able to use the \textsc{conv} rule to deduce this property.
We want to prove that \(\Gamma \vdash t_2 : A\) from the fact that \(t_1 \hookrightarrow t_2\) and \(\Gamma \vdash t_1 : A\).
We use the \emph{typability preservation} property to get a type \(B\) to \(t_2\). Then, from the \emph{subject conversion} property, we know that \(\Gamma \vdash A \convbg B\).
The only thing we need now to apply the \textsc{conv} rule is to show that \(\Gamma \vdash A : s\) for some sort \(s\). However, this is not possible in general, for example when \(A = \tkind\). We have two cases to prove:
\begin{itemize}
\item If \(A = kind\), then using Lemma TODO, we can show that \(B = kind\), hence subject reduction is satisfied
\item If \(A \neq kind\), because \(A\) is inhabited, we can use Lemma TODO to conclude that \(\Gamma \vdash A :s\). Hence we can apply the \textsc{conv}, this closes the proof.
\end{itemize}
\end{proof}
\begin{lemma}[False]
......
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