chiark / gitweb /
 author Ian Jackson Wed, 21 Mar 2012 21:47:29 +0000 (21:47 +0000) committer Ian Jackson Wed, 21 Mar 2012 21:47:29 +0000 (21:47 +0000)
 merge.tex patch | blob | history

index 1ca4af7..94b38fe 100644 (file)
--- a/merge.tex
+++ b/merge.tex
@@ -111,14 +111,23 @@ $\qed$

\subsection{Coherence and Patch Inclusion}

-Need to determine $C \haspatch \p$ based on $L,M,R \haspatch \p$.
-This involves considering $D \in \py$.
+$$+\begin{cases} + L \nothaspatch \p \land R \nothaspatch \p : & C \nothaspatch \p \\ + L \haspatch \p \land R \haspatch \p : & C \haspatch \p \\ + \text{otherwise} \land M \haspatch \p : & C \nothaspatch \p \\ + \text{otherwise} \land M \nothaspatch \p : & C \haspatch \p +\end{cases} +$$
+\proofstarts
+~ Consider $D \in \py$.

\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 Own Contents for $L$).
So $D \neq C$.
Applying $\merge$ gives $D \not\isin C$ i.e. $C \nothaspatch \p$.
+OK.

\subsubsection{For $L \haspatch \p, R \haspatch \p$:}
$D \isin L \equiv D \le L$ and $D \isin R \equiv D \le R$.
@@ -150,11 +159,6 @@ and this $F \le C$ so indeed $C \haspatch \p$.

\subsubsection{For (wlog) $X \not\haspatch \p, Y \haspatch \p$:}

-$M \haspatch \p \implies C \nothaspatch \p$.
-$M \nothaspatch \p \implies C \haspatch \p$.
-
-\proofstarts
-
One of the Merge Ends conditions applies.
Recall that we are considering $D \in \py$.
$D \isin Y \equiv D \le Y$.  $D \not\isin X$.