Commit cde7debd by Guillaume G

### Merge branch 'str_pos_naif_en' of...

Merge branch 'str_pos_naif_en' of https://git.lsv.fr/genestier/bw_versionpositive into str_pos_naif_en
parents ca8a9517 7541c956
 \section{Dependency pairs} \section{Fully applied signature symbol and structural order} \begin{defi}[Order associated to accessible subterms] We define $\surterm_{acc}$ as the transitive closure of $\paren{f\,t_1\dots t_{\arity(c)}}\surterm_{acc}(t_i\,\bar u)$, where $f\in\CObj$, $\tau(f)=\Pi\overline{(x:T)}.V$, $i\in\Acc(f)$, $T_i=\Pi\overline{(x:U)}.W$ and $\overline{\crochet{\sur{u}{x}}}\vDash\overline{(x:U)}$. where $f\in\CObj$, $\tau(f)=\Pi\overline{(x:T)}.U$, $i\in\Acc(f)$, $T_i=\Pi\overline{(y:V)}.W$ and $\overline{\crochet{\sur{u}{y}}}\vDash\overline{(y:V)}$. \end{defi} \begin{defi}[Function applied to reducible terms] Let $\mbb{U}=\enscond{f\,\bar t}{ \begin{matrix} \vdash\tau(f):s(f),\\ \tau(f)\in\interp{s(f)},\\ \tau(f)=\Pi\overline{(x:T)}. T',\\ T'\text{ is not an arrow}\\ \valabs{\bar t}=\arity(f)\\ \crochet{\overline{\sur{t}{x}}}\vDash\overline{(x:T)} \end{matrix} }$ \end{defi} \begin{defi}[$\rew_{arg}$] Let $f\,\bar t\in\mbb{U}$. $f\,\bar t\rew_{arg}u$ if $u=f\,\bar t'$, there is a $i$ such that $t_i\rew t'_i$ and for all $j\neq i$, $t_j=t'_j$. \end{defi} \begin{lem}\label{lem-subacc-wf} 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 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 $t_{i+1}=u_j\,\bar x$, then $u_j\subterm t'_{i}$ hence there is a $t'_{i+1}$ (namely $u_j$) such that $t_{i+1}=t'_{i+1}\,\bar x$ such that $t'_i\surterm t'_{i+1}$. By definition of accessibility, if $\tau(c)=\Pi\overline{(x:T)}.C'\,\bar w$ with $C'\sim C$, $T_j\in\Froz_{\g\bar m$ if there is a rule $f\bar l\rul r\in\mcal{R}$, ... ... @@ -149,28 +236,6 @@ \end{itemize} \end{defi} \begin{defi}[Function applied to reducible terms] Let $\mbb{U}=\enscond{f\,\bar t}{ \begin{matrix} \vdash\tau(f):s(f),\\ \tau(f)\in\interp{s(f)},\\ \tau(f)=\Pi\overline{(x:T)}. T',\\ T'\text{ is not an arrow}\\ \valabs{\bar t}=\arity(f)\\ \crochet{\overline{\sur{t}{x}}}\vDash\overline{(x:T)} \end{matrix} }$ \end{defi} \begin{defi}[$\rew_{arg}$] Let $f\,\bar t\in\mbb{U}$. $f\,\bar t\rew_{arg}u$ if $u=f\,\bar t'$, there is a $i$ such that $t_i\rew t'_i$ and for all $j\neq i$, $t_j=t'_j$. \end{defi} \begin{lem}[Pseudo-adequacy]\label{lem-pseudo-adequacy} For all $f\,\bar l$, $\G$, $u$ and $U$, if $\G\vdash_{f\bar l} u:U$, $\sg\vDash\G$ and for all $g\sqsubset f$, $g\in\interp{\tau(g)\sg}$, ... ... @@ -190,54 +255,12 @@ Hence, $g\sqsubseteq h$. Furthermore, by \indlem{lem-order-typ}, $h\sqsubset f$ because $\G\vdash_{\sqsubset f}\tau(g):s(g)$. Hence by transitivity, $g\sqsubset f$. We then have $g\in\interp{\tau(g)\sg}$ by hypothesis. \qedhere \end{description} \end{proof} \begin{lem}\label{lem-subacc-wf} Let $f\,\bar u\in\mbb{U}$. There is no infinite sequence $(t_i)_{i\in\N}$ such that $t_0=f\,\bar u$ and for all $i\in\N$, $t_i\surterm_{acc}t_{i+1}$. \end{lem} \begin{proof} \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 $i\in\Acc(f)$ such that $t_1=u_i\,\bar w$. By definition of accessibility, $T_i\in\Froz_{\  ... ... @@ -45,7 +45,7 @@$\rew=(\rew_\beta\cup\rew_{\Rules})$terminates on terms typable in$\lm\Pi/\Rules$if \begin{itemize} \item$\rew$is confluent, \item$\rew$is locally confluent, \item$\rew$preserves typing, \item$\FTyp\cup\FObj$is finite, \item$\Rules\$ is valid (for instance AVO), ... ...
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