 ### g

parent 9d1d3259
 \documentclass{article} \input{enTeteArticle} \usepackage{geometry} \geometry{left=3cm, right=3cm, bottom=3cm} \DeclareMathOperator{\Kind}{Kind} \DeclareMathOperator{\Type}{Type} \DeclareMathOperator{\support}{supp} ... ...
 ... ... @@ -3,9 +3,13 @@ \begin{defi}[Type constants] $\CTyp=\enscond{C\in\FTyp}{C\text{ is not the head of any rule}}$ We suppose given a well-founded pre-order $\preced$ on $\CTyp$ . \end{defi} \begin{cond}{cond-succ-well-founded} We suppose given a well-founded pre-order $\preced$ on $\CTyp$ . \end{cond} \begin{defi}[Frozen type] For $C\in\CTyp$, we define the following grammars : \begin{align*} ... ... @@ -40,15 +44,3 @@ we define $\Acc(f)=\enscond{i\<\arity(f)}{T_i\in\Froz_{\preced C}}.$ \end{defi} \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 $f\in\CObj$, $i\in\Acc(f)$ and $\bar u$ is an arbitrary sequence of terms. \end{defi} \todo{ This definition is really violent. It is not well-founded. It does not seem to be a problem, but I must stay cautious. }
avo.tex deleted 100644 → 0
 \section{Accessible Only rules} \begin{defi}[Accessible position] We define the function: $\text{If } \left\{\begin{array}{l} t=c\,u_1\dots u_k\\ c\in\CObj\\ \tau(c)=\Pi(y_1:U_1)\dots(y_r:U_r). C\,\bar v\\ \exists W_{k+1}\dots W_{r},\bar v', T\theta\conv \Pi(y_{k+1}:W_{k+1})\dots(y_r:W_r). C\,\bar v' \end{array} \right.$ \begin{align*} \text{then: }\AccPosCstr : (t,T,\theta) &\mapsto \ens{\eps}\cup\bigcup_{i\in\Acc(c)}\enscond{i.p}{p\in\AccPosCstr\paren{u_i,U_i,(\crochet{\overline{\sur{u}{y}}};\theta)}}\\ \text{else: }\AccPosCstr : (t,T,\theta) &\mapsto\ens{\eps} \end{align*} and the function: $\AccPosHd : (f\,\bar t) \mapsto \bigcup_{i}\enscond{i.x}{x\in\AccPosCstr(t_i)}$ \end{defi} \begin{defi}[Inferred type] We now define the functions: \begin{align*} \InferTypeCstr : (p,c\,\bar u,\pi) &\mapsto \left\{\begin{array}{ll} T_i\crochet{\overline{\sur{u}{y}}}\pi&\text{ if }p=i\text{ and }\tau(c)=\Pi\overline{(y:T)}. U\\ \InferTypeCstr(p',u_i,(\crochet{\overline{\sur{u}{y}}};\pi))&\text{ if }p=i.p', p'\neq\eps\text{ and }\tau(c)=\Pi\overline{(y:T)}. U\\ \end{array}\right.\\ \InferTypeHd : (p,f\,\bar t) &\mapsto \left\{\begin{array}{ll} T_i\crochet{\overline{\sur{t}{x}}}&\text{ if }p=i\text{ and }\tau(f)=\Pi\overline{(x:T)}. U\\ \InferTypeCstr(p',t_i,\crochet{\overline{\sur{t}{x}}})&\text{ if }p=i.p', p'\neq\eps\text{ and }\tau(f)=\Pi\overline{(x:T)}. U\\ \end{array}\right.\\ \end{align*} We must note here that if $p\in\AccPosHd(f\,\bar t)$ then $\InferTypeHd(p,f\,\bar t)$ is well-defined. \end{defi} \begin{defi}[AVO rules] A rule $f\,l_1\dots l_n\rul r$ is \emph{Accessible Variables Only (AVO)} if \begin{itemize} \item there is a function $\phi:\FreeVar(r)\to\AccPosHd(f\,\bar l)$ such that $(f\,\bar l)|_{\phi(x)}=x$, for all $x\in\FreeVar(r)$. \item $\D_r\vdash r:T_r$, where \begin{itemize} \item $\D_r=\enscond{x:\InferTypeHd(\phi(x),f\,\bar l)}{x\in\FreeVar(r)}$ ordered by the alphabetical order on $\phi(x)$. \item $T_r=U\crochet{\overline{\sur{l}{x}}}$, where $\tau(f)=\Pi(x_1:T_1)\dots(x_n:T_n). U$. \end{itemize} \end{itemize} \todo{What is the implication of these constraints on the form of rules?} \end{defi} \begin{lem} AVO rules are valid. \end{lem} \begin{proof} Let $f\,\bar l\rul r$ be a AVO rule with $\tau(f)=\Pi\overline{(x:T)}. U$, $\G=\overline{(x:T)}$ and $\pi=\crochet{\overline{\sur{l}{x}}}$. Let $\sg$ be such that $\pi\sg\vDash\G$. \begin{itemize} \item We have to prove that, $\sg\vDash\D_r$, meaning that for all $(y:V)\in\D_r$, $y\sg\in\interp{V\sg}$. \begin{itemize} \item If $\phi(y)=i$, then by hypothesis, $x_i\pi\sg=y\sg\in\interp{T_i\pi\sg}$. Yet $V=\InferTypeHd(i,f\,\bar l)=T_i\pi$. So we have indeed $y\sg\in\interp{V\sg}$. \item If $\phi(y)=i.p'$ with $p'\neq\eps$, then by hypothesis, there is a $c\in\CObj$ and $u_1\dots u_k$ such that $l_i=c\,u_1\dots u_k$ since $p\in\AccPosHd(f\,\bar l)$. Besides, $x_i\pi\sg=(c\,u_1\dots u_k)\sg\in\interp{T_i\pi\sg}$. Since $T_i\pi\conv \Pi(y_{k+1}:V_{k+1})\dots(y_{\arity(c)}:V_{\arity(c)}). C\,\bar v$ for some $\bar v$, where $\tau(c)=\Pi(\overline{z:A}).C\,\bar v'$, we know that $(c\,\bar u)\sg\in\interp{\Pi\paren{(y_{k+1}:V_{k+1})\dots(y_{\arity(c)}:V_{\arity(c)}). C\,\bar v}\sg}$, so for $j\in\Acc(c)$ with $j\  ... ... @@ -10,24 +10,24 @@ \end{itemize} We denote by$\Candidates$the set of reducibility candidates. \end{defi} \begin{lem}$\SN(\rew\cup\surterm_{acc})$is a candidate. \end{lem} \begin{proof} \begin{itemize} \item By definition we have$\SN(\rew\cup\surterm_{acc})\subseteq\SN(\rew)$. \item Let$t\in\SN(\rew\cup\surterm_{acc})$and$t'$such that$t\rew t'$. We have$t'\in\SN(\rew\cup\surterm_{acc})$. \item Let$t$be a neutral term such that$\enscond{u}{t\rew u}\subseteq\SN(\rew\cup\surterm_{acc})$. Since$t$is neutral, it is not headed by an element of$\CObj$, so$\enscond{u}{t(\rew\cup\surterm_{acc}) u}=\enscond{u}{t\rew u}$. Hence,$t\in\SN(\rew\cup\surterm_{acc})$.\qedhere \end{itemize} \end{proof} % % \begin{lem} %$\SN(\rew\cup\surterm_{acc})$is a candidate. % \end{lem} % \begin{proof} % \begin{itemize} % \item % By definition we have$\SN(\rew\cup\surterm_{acc})\subseteq\SN(\rew)$. % \item % Let$t\in\SN(\rew\cup\surterm_{acc})$and$t'$such that$t\rew t'$. % We have$t'\in\SN(\rew\cup\surterm_{acc})$. % \item % Let$t$be a neutral term such that$\enscond{u}{t\rew u}\subseteq\SN(\rew\cup\surterm_{acc})$. % Since$t$is neutral, it is not headed by an element of$\CObj$, % so$\enscond{u}{t(\rew\cup\surterm_{acc}) u}=\enscond{u}{t\rew u}$. % Hence,$t\in\SN(\rew\cup\surterm_{acc})$.\qedhere % \end{itemize} % \end{proof} \begin{lem}[Reducibility of$\mcal{I}_C$] ... ... @@ -41,11 +41,11 @@ By definition of$K_{\bar C}$, we have$K_{\bar C}(\mcal{I}_C)\subseteq\SN(\rew)$. \item Let$t\in\mcal{I}_C$and$u$be such that$t\rew u$. Since$t\in\SN(\rew\cup\surterm_{acc})$,$u\in\SN(\rew\cup\surterm_{acc})$too. Since$t\in\SN(\rew)$,$u\in\SN(\rew)$too. If$u\reww c\,\bar v$, then$t\reww c\,\bar v$so the constraint on the reducibility of the accessible arguments are fulfilled. \item Let$t$be a neutral term such that$\enscond{u}{t\rew u}\subseteq\mcal{I}_C$. Since$t$is neutral,$\enscond{u}{t(\rew\cup\surterm_{acc}) u}=\enscond{u}{t\rew u}$, so$t\in\SN(\rew\cup\surterm_{acc})$.$t\in\SN(\rew)$. If$t\reww c\,\bar v$, then there is a$u\in\enscond{u}{t\rew u}$such that$u\reww c\,\bar v$so the constraint on the reducibility of the accessible arguments are fulfilled.\qedhere \end{itemize} \end{proof} ... ... @@ -148,7 +148,7 @@ Since$\mcal{F}_p(\Lambda,\Candidates)$is a strictly inductive poset, By assumption,$J(A)\in\Candidates$and, for$a\in J(A)$,$J(\subst{B}{x}{a})\in\Candidates$. Therefore,$G(J)(T)\in\Candidates$. \item If$T\Downarrow$is neutral, then$G(J)(T)=\SN(\rew\cup\surterm_{acc})\in\Candidates$. If$T\Downarrow$is neutral, then$G(J)(T)=\SN(\rew)\in\Candidates$. \item Otherwise,$T\Downarrow=C\,\bar t$is a fully applied set constructor, and we have already seen that$\mcal{I}_C$is a candidate. ... ...  \section{Dependency pairs} \begin{defi}[Order associated to accessible subterms] We define$\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$,$i\in\Acc(f)$and$\bar u$is an arbitrary sequence of terms. \end{defi} \todo{ This definition is really violent. It is not well-founded. It does not seem to be a problem, but I must stay cautious. } \begin{defi}[Dependency pairs] Let$f\bar l>g\bar m$if there is a rule$f\bar l\rul r\in\mcal{R}$, such that$g\,\bar m$occurs in$r$,$\valabs{\bar m}\<\arity(g)$... ... @@ -119,6 +131,17 @@ \UnaryInfC{$\G\vdash_{f\bar l}g:\tau(g)$} \end{prooftree} \begin{lem}\label{lem-order-typ} For$f,g\in\Func$,$\bar l$,$\G$,$t$and$T$, such that$g$occurs in$t$\begin{itemize} \item if$\G\vdash_{f\bar l}t:T$then$g\sqsubseteq f$, \item if$\G\vdash_{\sqsubset f}t:T$then$g\sqsubset f$. \end{itemize} \end{lem} \begin{proof} By induction on the proof tree. \end{proof} \begin{defi}[Valid rule] ... ... @@ -152,7 +175,55 @@ and for all$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}$, then$u\sg\in\interp{U\sg}$. \end{lem} \begin{proof} The proof is the same as for \indthm{thm-adequacy} except for (fun) replaced by (dp) and (const). \begin{description} \item[(dp)] In this case, for all$i$, we have$\G\vdash_{f\bar l} u_i:U_i\g$. By induction hypothesis,$u_i\sg\in\interp{U_i\g\sg}$. So,$\g\sg\vDash\Sg$and$g\,\bar u\sg\in\mbb{U}$. Now,$g\,\bar u\sg\tilde{<}f\,\bar l\sg$since$g\,\bar u
 ... ... @@ -43,14 +43,15 @@ Here a cast between set and tuple is left implicit. {\Part{\Lambda}^n} {\Part{\Lambda}^n} {(X_i)_{i\\arity(c)\\ \text{for all }j\in\Acc(c),v_j\in R_{\bar C}(T_j,(X_k)_{k\
 ... ... @@ -36,7 +36,7 @@ By Ramsey's theorem, there is an infinite sequence $\phi_i$ and a matrix $M$ such that $M$ corresponds to all the transitions $(g,\bar u_{\phi_i}),(g,\bar u_{\phi_j})$ with $i\neq j$. $M$ is idempotent, since $(g,\bar u_{\phi_i}),(g,\bar u_{\phi_{i+2}})$ is labeled by $M^2$ by definition of the transitive closure and by $M$ due to Ramsey's theorem, so $M=M^2$. Since, by hypothesis $\Rules$ satisfies SCT, there is $j$ such that $M_{j,j}$ is $-1$. So, for all $i$, $u^{(j)}_{\phi_{i}}(\reww\surterm_{acc})^{+} u^{(j)}_{\phi_{i+1}}$, which contradicts the fact that $(\rew\cup\surterm_{acc})$ is well-founded on $\mbb{U}$. Since, by hypothesis $\Rules$ satisfies SCT, there is $j$ such that $M_{j,j}$ is $-1$. So, for all $i$, $u^{(j)}_{\phi_{i}}(\reww\surterm_{acc})^{+} u^{(j)}_{\phi_{i+1}}$, which contradicts the fact that $(\rew_{arg}\cup\surterm_{acc})$ is well-founded on $\mbb{U}$. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ... ...
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