\section{Some lemmas}
-\subsection{Alternative (overlapping) formulations of $\mergeof{C}{L}{M}{R}$}
+\subsection{Alternative (overlapping) formulations of $\commitmergeof{C}{L}{M}{R}$}
$$
D \isin C \equiv
\begin{cases}
\text{as above with L and R exchanged}
\end{cases}
$$
-\proof{ ~ Truth table (ordered by original definition): \\
+\proof{ ~ Truth table (ordered by original definitions): \\
\begin{tabular}{cccc|c|cc}
$D = C$ &
$\isin L$ &
Otherwise, $E$ meets all the conditions for $\pends$.
}
+\subsection{Single Parent Unique Tips}
+
+Unique Tips is satisfied for single-parent commits. Formally,
+given a conformant commit $A$,
+$$
+ \Big[
+ C \hasparents \{ A \}
+ \Big] \implies \left[
+ \bigforall_{P \patchisin C} \pendsof{C}{\py} = \{ T \}
+ \right]
+$$
+\proof{
+ Trivial for $C \in \py$.
+ For $C \not\in \py$, $\pancsof{C}{\py} = \pancsof{A}{\py}$,
+ so Unique Tips of $A$ suffices.
+}
+
\subsection{Ingredients Prevent Replay}
Given conformant commits $A \in \set A$,
$$
\right]
\implies
\left[
- \bigforall_{D \text{ s.t. } \patchof{D} = \bot}
+ \bigforall_{D \in \foreign}
D \isin C \equiv D \le C
\right]
$$
\proof{
-Consider some $D$ s.t. $\patchof{D} = \bot$.
+Consider some $D \in \foreign$.
If $D = C$, trivially true. For $D \neq C$,
by Foreign Inclusion of $D$ in $L$, $D \isin L \equiv D \le L$.
And by Exact Ancestors $D \le L \equiv D \le C$.
$$
\left[
C \hasparents \set A \land
- \patchof{C} = \bot \land
- \bigforall_{A \in \set A} \patchof{A} = \bot
+ \isforeign{C} \land
+ \bigforall_{A \in \set A} \isforeign{A}
\right]
\implies
\left[
\bigforall_{D}
D \le C
\implies
- \patchof{D} = \bot
+ \isforeign{D}
\right]
$$
\proof{
-Consider some $D \le C$. If $D = C$, $\patchof{D} = \bot$ trivially.
+Consider some $D \le C$. If $D = C$, $\isforeign{D}$ trivially.
If $D \neq C$ then $D \le A$ where $A \in \set A$. By Foreign
-Contents of $A$, $\patchof{D} = \bot$.
+Contents of $A$, $\isforeign{D}$.
}