Commit 7541c956 by Guillaume GENESTIER

### Redaction de la bonne fondaison de subterm_acc par l'absurde

parent 09bb37e7
 ... ... @@ -32,40 +32,57 @@ \end{defi} \begin{lem}\label{lem-subacc-wf} $\surterm_{acc}$ is well-founded on $\mbb{U}$. There is no infinite sequence $(t_i)_{i\in\N}$ such that $t_0\in\mbb{U}$ and for all $i\in\N$, $t_i\surterm_{acc}t_{i+1}$. \end{lem} \begin{proof} Let $f\,\bar u\in\mbb{U}$. We want to show there is no infinite sequence $(t_i)_{i\in\N}$ such that $t_0=f\,\bar u$ and for all $i\in\N$, $t_i\surterm_{acc}t_{i+1}$. Let us assume that there is such an infinite sequence. First note that every $t_i$ is headed by an element of $\CObj$, otherwise there is no $t$ such that $t_i\surterm_{acc}t$. Hence, among every infinite sequence, let us choose one such that $t_0=f\,\bar u$ where $\tau(f)=\Pi\overline{(x:T)}.C\,\bar v$ and for all $C'\prec C$, there is no infinite sequence starting by a constructor of $C'$. We will prove that for every $i$, $t_i=t'_i\,\bar v$ where $t'_0=t_0$, for all $i$, $t'_i\surterm t'_{i+1}$ and $t'_i$ is headed by a constructor of a type equivalent to $C$. Let $i$ be such that $t_i$ has this property. Hence $t_i=t'_i\,\bar v$. Since $t_i$ is headed by a constructor, $t'_i$ too, $t'_i=c\,\bar u$. Since $t'_i\,\bar v\surterm_{acc}t_{i+1}$, two options: \begin{itemize} \item If $f\notin\CObj$, then there is no $t$ such that $f\,\bar u\surterm_{acc}t$. \item So, let us consider $f\in\CObj$. Then $\tau(f)=\Pi\overline{(x:T)}.C\,\bar v$. By induction on $\succ$, let's assume that for all $C'\prec C$, for all terms headed by a constructor of $C'$, there is no infinite sequence with the expected property. There is a $j\in\Acc(f)$ such that $t_1=u_j\,\bar w$. $u_j\in\interp{T_j}$, hence: \begin{itemize} \item if $T_j$ is not a product and $u_j$ is headed by a constructor, the constructor is fully applied, so $\valabs{\bar w}=0$. \item if $T_j$ is a product, then by definition of accessibility, \$T_j\in\Froz_{\
Markdown is supported
0% or
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!