\section{Pseudo-merge}

Given $L$ and some other commits $\set R$, generate a
`fake merge': i.e., a commit which is a descendant of $L$ and $\set R$
but whose contents are exactly those of $L$.

\gathbegin
 C \hasparents \{ L \} \cup \set R
\gathnext
 \patchof{C} = \patchof{L}
\gathnext
 D \isin C \equiv D \isin L \lor D = C
\end{gather}

\subsection{Conditions}

\[ \eqn{ Base Only }{
 \patchof{L} \neq \py
}\]

\[ \eqn{ Unique Tips }{
 C \haspatch \p \implies
  \bigexists_T
    \pendsof{C}{\py} = \{ T \}
}\]

\[ \eqn{ Foreign Unaffected }{
 \pendsof{C}{\foreign} = \pendsof{L}{\foreign}
}\]

\subsection{Lemma: Foreign Identical}

$\isforeign{D} \implies \big[ D \le C \equiv D \le L \big]$.

\proof{
Trivial by Foreign Unaffected and the definition of $\pends$
}

\subsection{No Replay}

Ingredients Prevent Replay applies:
$A = L$ always satisfies the $\exists$.  $\qed$

\subsection{Unique Base}

Not applicable, by Base Only.

\subsection{Tip Contents}

Not applicable, by Base Only.

\subsection{Base Acyclic}

Relevant only if $L \in \pn$.  For $D = C$, $D \in \pn$; OK.
For $D \neq C$, OK by Base Acyclic for $L$. $\qed$

\subsection{Coherence and Patch Inclusion}

$$
\begin{cases}
  L \haspatch    \p : & C \haspatch    \p \\
  L \nothaspatch \p : & C \nothaspatch \p
\end{cases}
$$

\proof{
Consider some $D \in \py$.  $D \neq C$ by Base Only.
So $C \has \p \equiv L \has \p$.
}

\subsection{Unique Tips}

Explicitly dealt with by our Unique Tips condition.

\subsection{Foreign Inclusion}

True by Foreign Identical, and Foreign Inclusion of $L$.

\subsection{Foreign Contents}

Not applicable.

\subsection{Bases' Children}

We need to consider this for $D=L$ and also for $D=R$ ($R \in \set
R$).

For $D=L$, 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
$R$.

Other possibilities for $L \in \pn$ are excluded by Base Merge.

If $L \in \foreign$: not applicable for $L$; nor for $R$, by Foreign Merges.


