 ### stuff

parent 4db4e4f4
 ... ... @@ -184,6 +184,21 @@ The subjection conversion property is a generalization of a subject reduction in \end{definition} \begin{lemma}[$\tkind$ unicity] \label{lemma:kindunicity} For all well-formed context $\Gamma$, if there exists a term $K$ such that $K \convbg \tkind$ and a term $t$ such that $\Gamma \vdash t : K$ then necessarily $K = \tkind$. \end{lemma} \begin{proof} \todo{PROOF} \end{proof} \begin{lemma}[$\ttype$-terms are $\tkind$] Let $\Gamma$ be a well-formed context and $t$, $K$ a terms such that $\Gamma \vdash t : K$ then $\ttype$ is a subterm of $t$ if and only if $K = \tkind$. \end{lemma} \begin{proof} \todo{PROOF} \end{proof} \begin{lemma} When the relation system is oriented (a rewriting system) then \emph{subject conversion} and \emph{typability preservation} together ... ... @@ -195,7 +210,7 @@ The subjection conversion property is a generalization of a subject reduction in 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 $$A = \tkind$$, then using Lemma~\ref{lemma:kindunicity}, 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} ... ... @@ -206,28 +221,7 @@ The subjection conversion property is a generalization of a subject reduction in for all terms $t, A, B$, if $\Gamma \vdash t : A$ and $\Gamma \vdash t : B$ then $A \convbg B$. \end{lemma} \begin{lemma}[$\tkind$ unicity - Maybe true] For all well-formed context $\Gamma$, if there exists a term $K$ such that $K \convbg \tkind$ and a term $t$ such that $\Gamma \vdash t : K$ then necessarily $K = \tkind$. \end{lemma} \begin{lemma}[$\ttype$-terms are $\tkind$ - Maybe true] Let $\Gamma$ be a well-formed context and $t$, $K$ a terms such that $\Gamma \vdash t : K$ then $\ttype$ is a subterm of $t$ if and only if $K = \tkind$. \end{lemma} \begin{lemma}[False] For any well-formed context $$\Gamma$$, and term $t$ such that $$\tkind \convbg t$$ then $$t = \tkind$$. \end{lemma} \begin{lemma}[False] For any context $$\Gamma$$, if $$\ttype \convbg t$$ then $$t = \ttype$$. \end{lemma} \begin{proof} Same proof as the precedent lemma. \todo{PROOF} \end{proof} \end{document}
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