Trivial by Foreign Unaffected and the definition of $\pends$
}
+It might seem that foreign 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{Unique Base}
-Not applicable, by Base Only.
+Not applicable, by Base Only. $\qed$
\subsection{Tip Contents}
-Not applicable, by Base Only.
+Not applicable, by Base Only. $\qed$
\subsection{Base Acyclic}
\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 $D \le L \equiv D \le C$.
-\subsection{Foreign Contents}
+$\qed$
+
+\subsection{Foreign Ancestry}
Not applicable.
We need to consider this for $D=L$ and also for $D=R$ ($R \in \set
R$).
-For $D=L$, if $L \in \pn$ then $C \in \pn$, OK; whereas if
-$L \not \in \pn$ Bases' Children is inapplicable.
-
-For $D=R$,
-xxx up to here?
-
-If $L \in \py, R \in \py$: not applicable for either $D=L$ or $D=R$.
-
-If $L \in \py, R \in \pn$: not applicable for $L$, OK for $R$.
-
-Other possibilities for $L \in \py$ are excluded by Tip Merge.
-
-If $L \in \pn, R \in \pn$: satisfied for both $L$ and $R$.
-
-If $L \in \pn, R \in \foreign$: satisfied for $L$, not applicable for
-$R$.
-
-If $L \in \pn, R \in \pqy$: satisfied for $L$, not applicable for
-$R$.
+For $D=L$: $L \in \pn$ so $\pd = \p$. And $C \in \pn = \pdn$. Bases'
+Children applies and is satisfied.
-Other possibilities for $L \in \pn$ are excluded by Base Merge.
+For $D = R \in \set R, R \in \pn$: $D \in \pn, \pd = \p, C \in \pn$ as
+for $D = L$.
-If $L \in \foreign$: not applicable for $L$; nor for $R$, by Foreign Merges.
+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.
+$\qed$