Commit ccb3ba8f authored by François Thiré's avatar François Thiré

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}
Markdown is supported
0% or
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment