From fc75f297123c083b696f8d5639a4242a12e70f33 Mon Sep 17 00:00:00 2001
From: Ian Jackson <ijackson@chiark.greenend.org.uk>
Date: Thu, 1 Mar 2012 14:32:37 +0000
Subject: [PATCH] simple commits

---
 article.tex | 38 ++++++++++++++++++++++++++++++++++----
 1 file changed, 34 insertions(+), 4 deletions(-)

diff --git a/article.tex b/article.tex
index ad68c2c..f40f6a2 100644
--- a/article.tex
+++ b/article.tex
@@ -276,12 +276,42 @@ For $D = C$: $D \in \pn$ so $D \not\in \py$. OK.
 For $D \neq C$: $D \isin C \equiv D \isin A$, so by Base Acyclic for
 $A$, $D \isin C \implies D \not\in \py$. $\qed$
 
-\subsection{Coherence}
+\subsection{Coherence and patch inclusion}
+
+Need to consider $D \in \py$
 
 \subsubsection{For $A \haspatch P, D = C$:}
-\[ D \isin C \equiv \ldots \lor t \text{ so } D \haspatch C \]
-\[ D \le C \]
-OK
+
+Ancestors of $C$:
+$ D \le C $.
+
+Contents of $C$:
+$ D \isin C \equiv \ldots \lor t \text{ so } D \haspatch C $.
+
+\subsubsection{For $A \haspatch P, D \neq C$:}
+Ancestors: $ D \le C \equiv D \le A $.
+
+Contents: $ D \isin C \equiv D \isin A \lor f $
+so $ D \isin C \equiv D \isin A $.
+
+So:
+\[ A \haspatch P \implies C \haspatch P \]
+
+\subsubsection{For $A \nothaspatch P$:}
+
+Firstly, $C \not\in \py$ since if it were, $A \in \py$.  
+Thus $D \neq C$.
+
+Now by contents of $A$, $D \notin A$, so $D \notin C$.
+
+So:
+\[ A \nothaspatch P \implies C \nothaspatch P \]
+$\qed$
+
+\subsection{Foreign inclusion:}
+
+If $D = C$, trivial.  For $D \neq C$:
+$D \isin C \equiv D \isin A \equiv D \le A \equiv D \le C$.  $\qed$
 
 \section{Test more symbols}
 
-- 
2.30.2