\gathnext
\mergeof{C}{L}{M}{R}
\end{gather}
+We will occasionally use $X,Y$ s.t. $\{X,Y\} = \{L,R\}$.
\subsection{Conditions}
\text{otherwise} : & \false
\end{cases}
}\]
+\[ \eqn{ Merge Ends }{
+ X \not\haspatch \p \land
+ Y \haspatch \p \land
+ E \in \pendsof{X}{\py}
+ \implies
+ E \le Y
+}\]
\subsection{No Replay}
\subsection{Coherence and patch inclusion}
-Need to determine $C \haspatch P$ based on $L,M,R \haspatch P$.
+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 $L \nothaspatch P, R \nothaspatch P$:}
+\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. $C \nothaspatch P$.
+\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 $L \haspatch P, R \haspatch P$:}
+\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 $C \haspatch P$.
+Consider $D = C$: $D \isin C$, $D \le C$, OK for $C \haspatch \p$.
For $D \neq C$: $D \le C \equiv D \le L \lor D \le R
\equiv D \isin L \lor D \isin R$.
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 $C \haspatch P$.
+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 $C \haspatch P$.
+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 $C \haspatch P$.
+OK for $C \haspatch \p$.
+
+So indeed $L \haspatch \p \land R \haspatch \p \implies C \haspatch \p$.
+
+\subsubsection{For (wlog) $X \not\haspatch \p, Y \haspatch \p$:}
+
+$C \haspatch \p \equiv M \nothaspatch \p$.
+
+\proofstarts
+
+Merge Ends applies.
+
+$D \isin Y \equiv D \le Y$. $D \not\isin X$. Recall that we
+are considering $D \in \py$.
+
+Consider $D = C$. Thus $C \in \py, L \in \py$.
+But $X \not\haspatch \p$ means xxx wip
+But $X \not\haspatch \p$ means $D \not\in X$,
-So indeed $L \haspatch P \land R \haspatch P \implies C \haspatch P$.
+so we have $L = Y, R =
+X$. Thus $R \not\haspatch \p$ and by Tip Self Inpatch $R \not\in
+\py$. Thus by Tip Merge $R \in \pn$ and $M = \baseof{L}$.
+So by Base Acyclic, $M \nothaspatch \py$. Thus we are expecting
+$C \haspatch \py$. And indeed $D \isin C$ and $D \le C$. OK.
\end{document}