From b1adf56763bea8bb31f2b2113b0718be0dd0bc3a Mon Sep 17 00:00:00 2001
From: Ian Jackson <ijackson@chiark.greenend.org.uk>
Date: Wed, 21 Mar 2012 21:34:24 +0000
Subject: [PATCH] refactor for coherence cases - simple

---
 simple.tex | 27 ++++++++++++++++++---------
 1 file changed, 18 insertions(+), 9 deletions(-)

diff --git a/simple.tex b/simple.tex
index dfcf907..ae6fc55 100644
--- a/simple.tex
+++ b/simple.tex
@@ -49,7 +49,21 @@ $\qed$
 
 \subsection{Coherence and Patch Inclusion}
 
-Need to consider $D \in \py$
+$$
+\begin{cases}
+  L \haspatch    \p : & C \haspatch    \p \\
+  L \nothaspatch \p : & C \nothaspatch \p
+\end{cases}
+$$
+\proofstarts
+~
+
+Firstly, if $L \haspatch \p$, $\exists_{F \in \py} F \le L$
+and this $F$ is also $\le C$
+so $C \zhaspatch \p \implies C \haspatch \p$.
+We will prove $\zhaspatch$
+
+We need to consider $D \in \py$.
 
 \subsubsection{For $L \haspatch \p, D = C$:}
 
@@ -57,18 +71,14 @@ Ancestors of $C$:
 $ D \le C $.
 
 Contents of $C$:
-$ D \isin C \equiv \ldots \lor \true \text{ so } D \haspatch C $.
+$ D \isin C \equiv \ldots \lor \true$.  So $ D \zhaspatch C $.
+OK.
 
 \subsubsection{For $L \haspatch \p, D \neq C$:}
 Ancestors: $ D \le C \equiv D \le L $.
 
 Contents: $ D \isin C \equiv D \isin L \lor f $
 so $ D \isin C \equiv D \isin L $, i.e. $ C \zhaspatch P $.
-By $\haspatch$ for $L$, $\exists_{F \in \py} F \le L$
-and this $F$ is also $\le C$.  So $\haspatch$.
-
-So:
-\[ L \haspatch \p \implies C \haspatch \p \]
 
 \subsubsection{For $L \nothaspatch \p$:}
 
@@ -76,9 +86,8 @@ Firstly, $C \not\in \py$ since if it were, $L \in \py$.
 Thus $D \neq C$.
 
 Now by contents of $L$, $D \notin L$, so $D \notin C$.
+OK.
 
-So:
-\[ L \nothaspatch \p \implies C \nothaspatch \p \]
 $\qed$
 
 \subsection{Foreign Inclusion:}
-- 
2.30.2