From f627166bfcc401287564e9bc7fdc94a72d2d5f1a Mon Sep 17 00:00:00 2001 From: Ian Jackson Date: Tue, 6 Mar 2012 17:11:17 +0000 Subject: [PATCH] wip merge content --- article.tex | 36 ++++++++++++++++++++++++++++++++++-- 1 file changed, 34 insertions(+), 2 deletions(-) diff --git a/article.tex b/article.tex index 023a18a..0018429 100644 --- a/article.tex +++ b/article.tex @@ -527,8 +527,40 @@ $\qed$ \subsection{Coherence and patch inclusion} -xxx tbd +Need to determine $C \haspatch P$ based on $L,M,R \haspatch P$. +This involves considering $D \in \py$. + +We will use $X,Y$ s.t. $\{X,Y\} = \{L,R\}$. + +\subsubsection{For $L \nothaspatch P, R \nothaspatch P$:} +$D \not\isin L \land D \not\isin R$. $C \not\in \py$ (otherwise $L +\in \py$ ie $L \haspatch P$ by Tip Self Inpatch). So $D \neq C$. +Applying $\merge$ gives $D \not\isin C$ i.e. $C \nothaspatch P$. + +\subsubsection{For $L \haspatch P, R \haspatch P$:} +$D \isin L \equiv D \le L$ and $D \isin R \equiv D \le R$. +(Likewise $D \isin X \equiv D \le X$ and $D \isin Y \equiv D \le Y$.) + +Consider $D = C$: $D \isin C$, $D \le C$, OK for $C \haspatch P$. + +For $D \neq C$: $D \le C \equiv D \le L \lor D \le R + \equiv D \isin L \lor D \isin R$. +(Likewise $D \le C \equiv D \le X \lor D \le Y$.) + +Consider $D \neq C, D \isin X \land D \isin Y$: +By $\merge$, $D \isin C$. Also $D \le X$ +so $D \le C$. OK for $C \haspatch P$. + +Consider $D \neq C, D \not\isin X \land D \not\isin Y$: +By $\merge$, $D \not\isin C$. +And $D \not\le X \land D \not\le Y$ so $D \not\le C$. +OK for $C \haspatch P$. + +Remaining case, wlog, is $D \not\isin X \land D \isin Y$. +$D \not\le X$ so $D \not\le M$ so $D \not\isin M$. +Thus by $\merge$, $D \isin C$. And $D \le Y$ so $D \le C$. +OK for $C \haspatch P$. -xxx need to finish merge +So indeed $L \haspatch P \land R \haspatch P \implies C \haspatch P$. \end{document} -- 2.1.4