\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}
$D = C$ &
$\isin L$ &
\end{cases}
}\]
-\subsection{Tip Self Contents}
+\subsection{Tip Own Contents}
Given Base Acyclic for $C$,
$$
- \bigforall_{C \in \py} C \haspatch \p \land \neg[ C \nothaspatch \p ]
+ \bigforall_{C \in \py} C \haspatch \p
$$
Ie, tip commits contain their own patch.
Apply Exclusive Tip Contents to some $D \in \py$:
$ \bigforall_{C \in \py}\bigforall_{D \in \py}
D \isin C \equiv D \le C $.
-Thus $C \haspatch \p$.
-And, since $C \le C$, $C \isin C$. Therefore $\neg[ C \nothaspatch \p ]$
+Thus $C \zhaspatch \p$.
+And we can set $F=C$ giving $F \in \py \land F \le C$, so $C \haspatch \p$.
}
\subsection{Exact Ancestors}
\bigforall_{C \hasparents \set A}
\pendsof{C}{\set P} =
\begin{cases}
- C \in \p : & \{ C \}
+ C \in \set P : & \{ C \}
\\
- C \not\in \p : & \displaystyle
+ C \not\in \set P : & \displaystyle
\left\{ E \Big|
\Bigl[ \Largeexists_{A \in \set A}
E \in \pendsof{A}{\set P} \Bigr] \land
- \Bigl[ \Largenexists_{B \in \set A, F \in \pendsof{B}{\p}}
+ \Bigl[ \Largenexists_{B \in \set A, F \in \pendsof{B}{\set P}}
E \neq F \land E \le F \Bigr]
\right\}
\end{cases}
\proof{
Trivial for $C \in \set P$. For $C \not\in \set P$,
$\pancsof{C}{\set P} = \bigcup_{A \in \set A} \pancsof{A}{\set P}$.
-So $\pendsof{C}{\set P} \subset \bigcup_{E in \set E} \pendsof{E}{\set P}$.
+So $\pendsof{C}{\set P} \subset \bigcup_{E \in \set E} \pendsof{E}{\set P}$.
Consider some $E \in \pendsof{A}{\set P}$. If $\exists_{B,F}$ as
specified, then either $F$ is going to be in our result and
disqualifies $E$, or there is some other $F'$ (or, eventually,
-an $F''$) which disqualifies $F$.
+an $F''$) which disqualifies $F$ and $E$.
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 $A$, $\isforeign{D}$.
}