\subsection{Conditions}
\[ \eqn{ Base Only }{
- \patchof{L} \neq \py
+ L \in \pn
+}\]
+
+\[ \eqn{ Ingredients }{
+ \bigforall_{R \in \set R}
+ R \in \pn
+ \lor
+ R \in \foreign
+ \lor
+ R \in \pqy
}\]
\[ \eqn{ Unique Tips }{
}\]
\[ \eqn{ Foreign Unaffected }{
- \bigforall_{ D \in \foreign }
- \left[ \bigexists_{A \in \set A} D \le A \right]
- \implies
- D \le L
+ \pendsof{C}{\foreign} = \pendsof{L}{\foreign}
}\]
- TODO THAT IS IMPOSSIBLE TO CALCULATE
\subsection{Lemma: Foreign Identical}
$\isforeign{D} \implies \big[ D \le C \equiv D \le L \big]$.
\proof{
-If $D \le L$, trivially $D \le C$; so conversely
-$D \not\le C \implies D \not\le L$.
-Whereas if $D \le C$, either $D \le L$ or
-$\exists{A \in \set A} D \le A$ (since $D \neq C$),
-in which case by Foreign Unaffected $D \le L$.
+Trivial by Foreign Unaffected and the definition of $\pends$
}
+It might seem that bare git commits might also be psuedo-merges ---
+e.g., merges made directly with {\tt git merge -s ours}. However, by
+our definition of $\has$, these are considered simply as normal merges
+(\autoref{commit-merge}).
+
\subsection{No Replay}
Ingredients Prevent Replay applies:
\subsection{Foreign Inclusion}
-True by Foreign Identical, and Foreign Inclusion of $L$.
+We need to consider $D \in \foreign$.
+
+For $D = C$: $D \has C$, $D \le C$; OK.
+
+For $D \neq C$: $D \has C \equiv D \has L$ by construction.
+$D \has L \equiv D \le L$ by Foreign Inclusion of $L$.
+$D \neq C$ so this $D \le L \equiv D \le C$.
+
+$\qed$
-\subsection{Foreign Contents}
+\subsection{Foreign Ancestry}
Not applicable.
+
+\subsection{Bases' Children}
+
+We need to consider this for $D=L$ and also for $D=R$ ($R \in \set
+R$).
+
+For $D=L$: $L \in \pn$ so $\pd = \p$. And $C \in \pn = \pdn$. Bases'
+Children applies and is satisfied.
+
+For $D = R \in \set R, R \in \pn$: $D \in \pn, \pd = \p, C \in \pn$ as
+for $D = L$.
+
+For $D = R \in \set R, R \in \foreign$, or $R \in \pqy$: $D \not\in
+\pdn$ so Bases' Children does not apply.
+
+Other possibilities for $D \in \set R$ are excluded by Ingredients.