From e8ddb2493ac0e3bebb48f9151d492f2fcdc3a3ec Mon Sep 17 00:00:00 2001
From: Ian Jackson <ijackson@chiark.greenend.org.uk>
Date: Sun, 11 Mar 2012 11:51:01 +0000
Subject: [PATCH] merge foreign inclusion

---
 article.tex | 31 ++++++++++++++++++++++++++++++-
 1 file changed, 30 insertions(+), 1 deletion(-)

diff --git a/article.tex b/article.tex
index b778663..d1ce6e8 100644
--- a/article.tex
+++ b/article.tex
@@ -711,6 +711,35 @@ $\baseof{C} = R$ so $D \isin C \equiv D \isin \baseof{C}$.  OK.
 
 $\qed$
 
-xxx up to here, need to prove other things about merges
+\subsection{Foreign Inclusion}
+
+Consider some $D$ s.t. $\patchof{D} = \bot$.
+By Foreign Inclusion of $L, M, R$:
+$D \isin L \equiv D \le L$;
+$D \isin M \equiv D \le M$;
+$D \isin R \equiv D \le R$.
+
+\subsubsection{For $D = C$:}
+
+$D \isin C$ and $D \le C$.  OK.
+
+\subsubsection{For $D \neq C, D \isin M$:}
+
+Thus $D \le M$ so $D \le L$ and $D \le R$ so $D \isin L$ and $D \isin
+R$.  So by $\merge$, $D \isin C$.  And $D \le C$.  OK.
+
+\subsubsection{For $D \neq C, D \not\isin M, D \isin X$:}
+
+By $\merge$, $D \isin C$.
+And $D \isin X$ means $D \le X$ so $D \le C$.
+OK.
+
+\subsubsection{For $D \neq C, D \not\isin M, D \not\isin L, D \not\isin R$:}
+
+By $\merge$, $D \not\isin C$.
+And $D \not\le L, D \not\le R$ so $D \not\le C$.
+OK
+
+$\qed$
 
 \end{document}
-- 
2.30.2