\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}

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

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

\[ \eqn{ Foreign Unaffected }{
 \bigforall_{ D \in \foreign }
  \left[ \bigexists_{A \in \set A} D \le A \right]
   \implies
  D \le L
}\]
 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 $D \le L$.
}

\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.
