 ### 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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