\right]
\implies
\left[
- D \isin C
+ D \le C
\implies
\patchof{D} = \bot
\right]
}\]
-xxx proof tbd
+\proof{
+Consider some $D \le C$. If $D = C$, $\patchof{D} = \bot$ trivially.
+If $D \neq C$ then $D \le A$ where $A \in \set A$. By Foreign
+Contents of $A$, $\patchof{D} = \bot$.
+}
\subsection{No Replay for Merge Results}
\subsection{Foreign Contents:}
-Only relevant if $\patchof{C} = \bot$. Trivial by Foreign Contents of
-$A$. $\qed$
-
-xxx fixme not trivial use Totally Foreign Contents
+Only relevant if $\patchof{C} = \bot$, and in that case Totally
+Foreign Contents applies. $\qed$
\section{Create Base}
\gathbegin
B \hasparents \{ L \}
\gathnext
- \patchof{B} = \pa{B}
+ \patchof{B} = \pan{B}
\gathnext
D \isin B \equiv D \isin L \lor D = B
\end{gather}
If $\patchof{L} = \pa{L}$, trivial by Base Acyclic for $L$.
-If $\patchof{L} = \bot$, xxx
-
-Trivial from Base Acyclic for $L$. $\qed$
+If $\patchof{L} = \bot$, consider some $D \isin B$. $D \neq B$.
+Thus $D \isin L$. So by No Replay of $D$ in $L$, $D \le L$.
+Thus $D \le B$.
\subsection{Unique Base}
\subsection{Base Acyclic}
-xxx
+Consider some $D \isin B$. If $D = B$, $D \in \pn$, OK.
+
+If $D \neq B$, $D \isin L$. By No Replay of $D$ in $L$, $D \le L$.
+Thus by Foreign Contents of $L$, $\patchof{D} = \bot$. OK.
+
+$\qed$
+
+\subsection{Coherence and Patch Inclusion}
+
+Consider some $D \in \p$.
+$B \not\in \py$ so $D \neq B$. So $D \isin B \equiv D \isin L$.
-xxx unfinished
+Thus $L \haspatch \p \implies B \haspatch P$
+and $L \nothaspatch \p \implies B \nothaspatch P$.
+
+$\qed$.
+
+\subsection{Foreign Inclusion}
+
+Consider some $D$ s.t. $\patchof{D} = \bot$. $D \neq B$
+so $D \isin B \equiv D \isin L$.
+
+By Foreign Inclusion of $D$ xxx up to here
\section{Create Tip}
$\qed$
-\subsection{Foreign Contents:}
+\subsection{Foreign Contents}
Not applicable. $\qed$
\bigforall_{E \in \pendsof{X}{\py}} E \le Y
\right]
}\]
+\[ \eqn{ Foreign Merges }{
+ \patchof{L} = \bot \equiv \patchof{R} = \bot
+}\]
\subsection{Non-Topbloke merges}
-We require both $\patchof{L} = \bot$ and $\patchof{R} = \bot$.
+We require both $\patchof{L} = \bot$ and $\patchof{R} = \bot$
+(Foreign Merges, above).
I.e. not only is it forbidden to merge into a Topbloke-controlled
branch without Topbloke's assistance, it is also forbidden to
merge any Topbloke-controlled branch into any plain git branch.
Given those conditions, Tip Merge and Merge Acyclic do not apply.
And $Y \not\in \py$ so $\neg [ Y \haspatch \p ]$ so neither
-Merge Ends condition applies. Good.
+Merge Ends condition applies.
+
+So a plain git merge of non-Topbloke branches meets the conditions and
+is therefore consistent with our scheme.
\subsection{No Replay}
$\qed$
-\subsection{Foreign Contents:}
-
-xxx use Totally Foreign Contents
+\subsection{Foreign Contents}
-If $\patchof{C} = \bot$, by Foreign Merges
-$\patchof{L} = \patchof{R} = \bot$.
+Only relevant if $\patchof{L} = \bot$, in which case
+$\patchof{C} = \bot$ and by Foreign Merges $\patchof{R} = \bot$,
+so Totally Foreign Contents applies. $\qed$
\end{document}