Commit 6d990ebd by François Thiré

clarify last commit

parent b28e6140
 ... ... @@ -110,7 +110,7 @@ Note: $$\mathcal{T}_{\mathcal{\emptyset}}$$ is the set of closed terms. \Gamma, x : A \vdash t : B } {\Gamma \vdash \tabs{x}{A}{t} : \tpi{x}{A}{B}} \and \and \inferrule*[right=app]{ \Gamma \vdash t : \tpi{x}{A}{B} \\ ... ... @@ -137,7 +137,7 @@ Note: $$\mathcal{T}_{\mathcal{\emptyset}}$$ is the set of closed terms. \end{proof} \begin{lemma} For any well-formed context $\Gamma$ and terms $t$ and $A$ such that $\Gamma \vdash t : A$, there exists $t'$ such that $t \convbg t'$ and $\tkind$ is not a subterm of $t'$. For any well-formed context\, $\Gamma$, term $t$, term $A$ such that $\Gamma \vdash t : A$, there exists $t'$ such that $t \convbg t'$ and $\tkind$ is not a subterm of $t'$. \end{lemma} \begin{proof} By induction on the derivation of $\Gamma \vdash t : A$. ... ... @@ -162,81 +162,67 @@ Note: $$\mathcal{T}_{\mathcal{\emptyset}}$$ is the set of closed terms. \begin{definition}[subject conversion] A well-formed context $\Gamma$ satisfies the \emph{subject conversion} property when for all terms $t_1, t_2, T_1, T_2$ such that $t_1 \convbg t_2$, $\Gamma \vdash t_1 : T_1$ and $\Gamma \vdash t_2 : T_2$, then $T_1 \convbg T_2$. when for all terms $t_1, t_2, A, B$ such that $t_1 \convbg t_2$, $\Gamma \vdash t_1 : A$ and $\Gamma \vdash t_2 : B$, then $A \convbg B$. \end{definition} Remark: when $A$ (resp. $B$) is typable with a sort, $\Gamma \vdash A :s$ (resp. $\Gamma \vdash B :s$), then it follows from $A \convbg B$ that $\Gamma \vdash t_2 : A$ (resp. $\Gamma \vdash t_1 : B$). The subjection conversion property is a generalization of a subject reduction in a context where the \emph{computation} is not oriented but it is also weaker as we will see below: \begin{definition}[subject reduction] A well-formed context $\Gamma$ satisfies the \emph{subject reduction} property when for all terms $t_1, t_2, A$ such that $t_1 \hookrightarrow t_2$, $\Gamma \vdash t_1 : A$ then $\Gamma \vdash t_2 : A$. \end{definition} Remark: when $T_1$ (resp. $T_2$) is typable with a sort, $\Gamma \vdash T_1 :s$ (resp. $\Gamma \vdash T_2 :s$), then it follows from $T_1 \convbg T_2$ that $\Gamma \vdash t_2 : T_1$ (resp. $\Gamma \vdash t_2 : T_1$). \begin{definition}[typability preservation] A well-formed context $\Gamma$ satisfies the \emph{typability preservation} property when for all terms $t_1, t_2, A$ such that $t_1 \hookrightarrow t_2$, $\Gamma \vdash t_1 : A$ then $\exists B, \Gamma \vdash t_2 : B$. \end{definition} \begin{lemma} When the relation system is oriented (a rewriting system) then \emph{subject conversion} and \emph{typability preservation} together imply the \emph{subject reduction} property. \end{lemma} \begin{proof} $\Gamma$ verify the \emph{subject reduction} property when \forall t_1, t_2, T, \quad \left\{ \begin{aligned} & t_1 \longrightarrow^{\Gamma} t_2 \\ & \Gamma \vdash t_1 : T \end{aligned} \right. \quad \Longrightarrow \quad \Gamma \vdash t_2 : T $\Gamma$ verify the \emph{typability preservation} property when \forall t_1, t_2, T_1, \quad \left\{ \begin{aligned} & t_1 \longrightarrow^{\Gamma} t_2 \\ & \Gamma \vdash t_1 : T_1 \end{aligned} \right. \quad \Longrightarrow \quad \exists T_2, \ \Gamma \vdash t_2 : T_2 $\Gamma$ verify the \emph{subject conversion} property when \forall t_1, t_2, T_1, T_2 \quad \left\{ \begin{aligned} & t_1 \convbg t_2 \\ & \Gamma \vdash t_1 : T_1 \\ & \Gamma \vdash t_2 : T_2 \end{aligned} \right. \quad \Longrightarrow \quad T_1 \convbg T_2 \end{proof} \begin{lemma}[False] For any well-formed context $$\Gamma$$, and term $t$ such that $$\tkind \convbg t$$ then $$t = \tkind$$. \end{lemma} \begin{proof} By induction on the length of $$\Gamma$$. 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 base case: $$\Gamma$$ is $$\empty$$ and the $$\beta$$ rule cannot be applied \item inductive case: \begin{itemize} \item $$\Gamma = \Gamma, x : A$$, then the congruence relation is the same \item $$\Gamma = \Gamma, (t,u)$$, TODO \end{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} we can prove that there is no $$u$$ such that $$\Gamma \vdash \wf{(\tkind,u)}$$ or the symmetrical case are derivable. \end{proof} \begin{lemma} For any context $$\Gamma$$, if $$\ttype \convbg t$$ then $$t = \ttype$$. \end{lemma} \begin{proof} Same proof as the precedent lemma. \end{proof} % \begin{lemma}[False] % For any well-formed context $$\Gamma$$, and term $t$ such that $$\tkind \convbg t$$ then $$t = \tkind$$. % \end{lemma} % \begin{proof} % By induction on the length of $$\Gamma$$. % \begin{itemize} % \item base case: $$\Gamma$$ is $$\empty$$ and the $$\beta$$ rule cannot be applied % \item inductive case: % \begin{itemize} % \item $$\Gamma = \Gamma, x : A$$, then the congruence relation is the same % \item $$\Gamma = \Gamma, (t,u)$$, TODO % \end{itemize} % \end{itemize} % we can prove that there is no $$u$$ such that $$\Gamma \vdash \wf{(\tkind,u)}$$ or the symmetrical case are derivable. % \end{proof} % \begin{lemma} % For any context $$\Gamma$$, if $$\ttype \convbg t$$ then $$t = \ttype$$. % \end{lemma} % \begin{proof} % Same proof as the precedent lemma. % \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!