From 2fdf30e540acfd5fb9feaee16ae1298bdf84ed8d Mon Sep 17 00:00:00 2001
From: Ian Jackson <ijackson@chiark.greenend.org.uk>
Date: Mon, 12 Mar 2012 14:48:50 +0000
Subject: [PATCH] introduce and use Totally Foreign Contents

---
 article.tex | 24 +++++++++++++++++++-----
 1 file changed, 19 insertions(+), 5 deletions(-)

diff --git a/article.tex b/article.tex
index cc67c04..3f066dc 100644
--- a/article.tex
+++ b/article.tex
@@ -317,6 +317,24 @@ xxx proof tbd
   $A \le C$ so $D \le C$.
 }
 
+\[ \eqn{Simple Foreign Inclusion:}{
+  \left[
+    C \hasparents \{ L \}
+   \land
+    \bigforall_{D} D \isin C \equiv D \isin L \lor D = C
+  \right]
+ \implies
+   \bigforall_{D \text{ s.t. } \patchof{D} = \bot}
+     D \isin C \equiv D \le C
+}\]
+\proof{
+Consider some $D$ s.t. $\patchof{D} = \bot$.
+If $D = C$, trivially true.  For $D \neq C$,
+by Foreign Inclusion of $D$ in $L$, $D \isin L \equiv D \le L$.
+And by Exact Ancestors $D \le L \equiv D \le C$.
+So $D \isin C \equiv D \le C$.
+}
+
 \[ \eqn{Totally Foreign Contents:}{
   \bigforall_{C \hasparents \set A}
    \left[
@@ -513,11 +531,7 @@ $\qed$.
 
 \subsection{Foreign Inclusion}
 
-Consider some $D$ s.t. $\patchof{D} = \bot$.  $D \neq B$
-so $D \isin B \equiv D \isin L$.
-By Foreign Inclusion of $D$ in $L$, $D \isin L \equiv D \le L$.
-And by Exact Ancestors $D \le L \equiv D \le B$.
-So $D \isin B \equiv D \le B$.  $\qed$
+Simple Foreign Inclusion applies. $\qed$
 
 \subsection{Foreign Contents}
 
-- 
2.30.2