\gathnext
\patchof{C} = \patchof{L}
\gathnext
- \mergeof{C}{L}{M}{R}
+ \commitmergeof{C}{L}{M}{R}
\end{gather}
We will occasionally use $X,Y$ s.t. $\{X,Y\} = \{L,R\}$.
\subsection{No Replay}
-By definition of $\merge$,
+By definition of \commitmergename,
$D \isin C \implies D \isin L \lor D \isin R \lor D = C$.
So, by Ingredients,
Ingredients Prevent Replay applies. $\qed$
$D \not\isin L \land D \not\isin R$. $C \not\in \py$ (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. $C \nothaspatch \p$.
+Applying \commitmergename gives $D \not\isin C$ i.e. $C \nothaspatch \p$.
OK.
\subsubsection{For $L \haspatch \p, R \haspatch \p$:}
(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 $D \le X$
+By \commitmergename, $D \isin C$. Also $D \le X$
so $D \le C$. OK for $C \zhaspatch \p$.
Consider $D \neq C, D \not\isin X \land D \not\isin Y$:
-By $\merge$, $D \not\isin C$.
+By \commitmergename, $D \not\isin C$.
And $D \not\le X \land D \not\le Y$ so $D \not\le C$.
OK for $C \zhaspatch \p$.
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 $D \le C$.
+Thus by \commitmergename, $D \isin C$. And $D \le Y$ so $D \le C$.
OK for $C \zhaspatch \p$.
So, in all cases, $C \zhaspatch \p$.
therefore we must have $L=Y$, $R=X$.
Conversely $R \not\in \py$
so by Tip Merge $M = \baseof{L}$. Thus $M \in \pn$ so
-by Base Acyclic $M \nothaspatch \p$. By $\merge$, $D \isin C$,
+by Base Acyclic $M \nothaspatch \p$. By \commitmergename, $D \isin C$,
and $D \le C$. OK.
Consider $D \neq C, M \nothaspatch \p, D \isin Y$:
$D \le Y$ so $D \le C$.
-$D \not\isin M$ so by $\merge$, $D \isin C$. OK.
+$D \not\isin M$ so by \commitmergename, $D \isin C$. OK.
Consider $D \neq C, M \nothaspatch \p, D \not\isin Y$:
$D \not\le Y$. If $D \le X$ then
$D \in \pancsof{X}{\py}$, so by Addition Merge Ends and
Transitive Ancestors $D \le Y$ --- a contradiction, so $D \not\le X$.
-Thus $D \not\le C$. By $\merge$, $D \not\isin C$. OK.
+Thus $D \not\le C$. By \commitmergename, $D \not\isin C$. OK.
Consider $D \neq C, M \haspatch \p, D \isin Y$:
$D \le Y$ so $D \in \pancsof{Y}{\py}$ so by Removal Merge Ends
and Transitive Ancestors $D \in \pancsof{M}{\py}$ so $D \le M$.
-Thus $D \isin M$. By $\merge$, $D \not\isin C$. OK.
+Thus $D \isin M$. By \commitmergename, $D \not\isin C$. OK.
Consider $D \neq C, M \haspatch \p, D \not\isin Y$:
-By $\merge$, $D \not\isin C$. OK.
+By \commitmergename, $D \not\isin C$. OK.
$\qed$
$D \neq C$. By Tip Contents of $L$,
$D \isin L \equiv D \isin \baseof{L}$, so by Tip Merge condition,
-$D \isin L \equiv D \isin M$. So by $\merge$, $D \isin
+$D \isin L \equiv D \isin M$. So by \commitmergename, $D \isin
C \equiv D \isin R$. And $R = \baseof{C}$ by Unique Base of $C$.
Thus $D \isin C \equiv D \isin \baseof{C}$. OK.
$\patchof{M} = \patchof{L} = \py$, so by Tip Contents of $M$,
$D \isin M \equiv D \isin \baseof{M} \equiv D \isin \baseof{L}$.
-So $D \isin M \equiv D \isin L$ so by $\merge$,
+So $D \isin M \equiv D \isin L$ so by \commitmergename,
$D \isin C \equiv D \isin R$. But from Unique Base,
$\baseof{C} = \baseof{R}$.
Therefore $D \isin C \equiv D \isin \baseof{C}$. OK.
\subsubsection{For $D \neq C, D \isin M$:}
Thus $D \le M$ so $D \le L$ and $D \le R$ so $D \isin L$ and $D \isin
-R$. So by $\merge$, $D \isin C$. And $D \le C$. OK.
+R$. So by \commitmergename, $D \isin C$. And $D \le C$. OK.
\subsubsection{For $D \neq C, D \not\isin M, D \isin X$:}
-By $\merge$, $D \isin C$.
+By \commitmergename, $D \isin C$.
And $D \isin X$ means $D \le X$ so $D \le C$.
OK.
\subsubsection{For $D \neq C, D \not\isin M, D \not\isin L, D \not\isin R$:}
-By $\merge$, $D \not\isin C$.
+By \commitmergename, $D \not\isin C$.
And $D \not\le L, D \not\le R$ so $D \not\le C$.
OK
~ian / topbloke-formulae.git / blobdiff