From: acondolu <andrea.condoluci@unibo.it>
Date: Wed, 13 Jun 2018 10:20:00 +0000 (+0200)
Subject: Yet another attempt at defining distinction with garbage
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=30a5ca005792d224d78e3044847b74356ffee818;p=fireball-separation.git

Yet another attempt at defining distinction with garbage
---

diff --git a/notes.tex b/notes.tex
index 4c1dd4f..489869f 100644
--- a/notes.tex
+++ b/notes.tex
@@ -46,7 +46,7 @@
 % Paths:
 \newcommand{\PathEmpty}{\epsilon}
 \newcommand{\PathAbs}[1]{\mathtt{abs}(#1)}
-\newcommand{\PathArg}[3]{\mathtt{arg}_{#1}^{#2}(#3)}
+\newcommand{\PathArg}[3]{\mathtt{arg}_{#2}^{#1}(#3)}
 \newcommand{\PathHd}{\mathtt{hd}}
 
 \newcommand{\GarbageOf}[1]{\operatorname{Garb}(#1)}
@@ -104,41 +104,51 @@
   \AC{
    Given a path, one can retrieve from a term (if possible) the subterm at that position.
 
-   Since the path may go through abstraction, bound variables that become free
+   Since the path may go through abstractions, bound variables that become free
    are renamed to variables of the form $\NamedBoundVar\pi$
    (where $\pi$ is the path in the original inert leading to the abstraction binding that variable).
   }
+  % \newcommand{\GetSubtm}[2]{\operatorname{GetSubtm}(#1\texttt{;}#2)}
+  \newcommand{\GetSubtm}[2]{{#1}_{#2}}
   \[\begin{array}{ll}
-    \tm_\pi & \defeq \tm_\pi^{\Box} \\
-    t_{\PathEmpty}^{-} & \defeq t \\
-    (x\,t_1\cdots t_n)_{\PathHd}^{-} & \defeq x \\
-    (x\,t_1\cdots t_n)_{\PathArg i \var \pi}^{\rho} & \defeq
-      (t_i)_\pi^{\rho[\PathArg i \var \Box]} \mbox{(if } 1 \leq i\leq n \mbox{)} \\
-    (\lambda x.\, t)_{\PathAbs \pi}^{\rho} & \defeq t_\pi^{\rho[\PathAbs\Box]} \{\var\mapsto\NamedBoundVar{\rho[\PathEmpty]}\} \\
-    (t)_{\PathAbs \pi}^{\rho} & \defeq (\lambda x.\, t\, x)_{\PathAbs \pi}^{\rho} \text{ (with } x \text{ fresh) (eta)}\\
+    \GetSubtm\tm\pi & \defeq \GetSubtm\tm{\underline\pi} \\
+    \GetSubtm\tm{\rho[\underline\PathEmpty]} & \defeq \tm \\
+    \GetSubtm{(x\,t_1\cdots t_n)}{\rho[\underline\PathHd]} & \defeq x \\
+    \GetSubtm{(x\,t_1\cdots t_n)}{\rho[\underline{\PathArg i \var \pi}]} & \defeq
+      \GetSubtm{(t_i)}{\rho[\PathArg i \var {\underline\pi}]} \mbox{(if } 1 \leq i\leq n \mbox{)} \\
+    \GetSubtm{(\lambda x.\, t)}{\rho[\underline {\PathAbs \pi}]} & \defeq
+     \GetSubtm\tm{\rho(\PathAbs{\underline\pi})}
+     \{\var\mapsto\NamedBoundVar{\rho[\PathEmpty]}\} \\
+    \GetSubtm{\tm}{\rho(\underline{\PathAbs \pi})} &
+     \defeq \GetSubtm{(\lambda \var.\, \tm\,\var)}{\rho(\PathAbs {\underline\pi})} \text{ (with } x \text{ fresh) (eta)}\\
     % \Omega_-^- & \defeq \Omega \\
   \end{array}\]
   \item \textbf{Head restriction:} $\OfHead T \var \defeq \{\tm \in T \mid \HeadOf{\tm} (\defeq \tm_{\PathHd}) = \var \}$
-  \item \textbf{Distinction:} \underline{$S,R$ is $\Div$--distinct} iff:
-   $S$ is empty, or there exists a path $\pi$ \emph{effective} for $\{\Div\}\cup \OfHead{\GarbageOf\Div}{\HeadOf\Div}$, s.t. EITHER:
+  \item \textbf{Telescopic garbage chain:} $\{\langle\tm_1,\pi_1\rangle,\ldots,\langle\tm_n,\pi_n\rangle\}$ is a $-$ if $\forall i$:
+   \[\tm_{i+1} \in \SubtmsOf{\text{garbage of } \GetSubtm{(\tm_i)}{\pi_i}}\]
+  \item \textbf{Distinction:} \underline{$S$ is $\{\langle\Div_1,\pi_1\rangle,\ldots,\langle\Div_n,\pi_n\rangle\}$--distinct} iff (one of the following three):
+  \begin{itemize}
+    \item $\OfHead S {\HeadOf \Div}$ is empty and $n=1$
+  \end{itemize}
+   OR: let $\Div\defeq\Div_1$ in:
  \begin{itemize}
-   \item $\forall \tm\in \OfHead{\SubtmsOf{\Div}}{\HeadOf\Div}$, $\tm_\pi$ is defined;
-   % \item 1. $\exists s\in S$ s.t. $s \not\sim_\pi \Div$;
-   \item $\{s\in S \mid s \sim_\pi \Div\},R$ is $\Div$--distinct.
+   \item there exists a path $\pi$ s.t.
+   \item (Effective) $\pi$ is \emph{effective} for all $\Div_i$ s.t. $\HeadOf{\Div_i} = \HeadOf{\Div}$
+   \item $\forall \tm\in \OfHead{\SubtmsOf{\Div_i}}{\HeadOf\Div}$, $\tm_\pi \neq \Omega$;
+   \item (Useful) $\exists s\in \OfHead S{\HeadOf\Div}$ s.t. $s \not\sim_\pi \Div$;
+   \item $S\setminus\{s\in \OfHead S{\HeadOf\Div} \mid s \not\sim_\pi \Div\}$ is $D$--distinct.
  \end{itemize}
  OR:
 \begin{itemize}
-   \item $\Div_\pi = \lambda\var.\, -, g$
-    (note that $\Div$ has recursively only one garbage);
-   % \item $\Div_\pi \neq \Omega$
-   \item there is $\Div'\in\SubtmsOf g$ s.t.
-    $S',R'$ is $\Div'$--distinct, where:
-    $R'\defeq R\cup \bigcup\{\text{subtms of } s \text{ along } \pi \mid s\in S\}$ and
-    $S' \defeq \OfHead{R'}{\HeadOf{\Div'}}$.
+   \item $S'$ is $\{\langle\Div_2,\pi_2\rangle,\ldots,\langle\Div_n,\pi_n\rangle\}$--distinct, where:
+    \[S' \defeq S \mathrel{\cup} \SubtmsOf{\{\text{garbage of } s \text{ along } \pi_1 \mid s\in \OfHead{S}{\HeadOf\Div}\}} \]
  \end{itemize}
 
  \item \textbf{Semi-$\sigma$-separability: } $(\Div,\Conv)$ are semi-$\sigma$-separable
-  IF $\OfHead{\SubtmsOf{\Conv}}{\HeadOf{\Div'}}, \SubtmsOf{\Conv}$ is $\Div'$--distinct for some $\Div'\in\SubtmsOf\Div$.
+  IFF there is $\Div_1$ (an $\Omega$--approximation of a subterm of $\Div$ with
+  at most one garbage, and without stuck variables)
+  and a telescopic garbage chain $D\defeq\{\langle\Div_1,\pi_1\rangle,\ldots,\langle\Div_n,\pi_n\rangle\}$ s.t.
+  $\SubtmsOf\Conv$ is $D$--distinct.
 
  \clearpage
  \item $\bullet$ $\bullet$ $\bullet$ $\bullet$ $\bullet$