X-Git-Url: http://www.chiark.greenend.org.uk/ucgi/~ian/git?a=blobdiff_plain;f=article.tex;h=0688c7ad02524d0d5c0d399682f3ac46946d1e1f;hb=26c18196858d61ec8bd051837aa9ccb08f68ccaf;hp=afefebd90d7aaf60e5de2f8da98c4fee810a4aa8;hpb=71f18d98f82d4ba454e875c79119879d3dc957ed;p=topbloke-formulae.git

diff --git a/article.tex b/article.tex
index afefebd..0688c7a 100644
--- a/article.tex
+++ b/article.tex
@@ -171,18 +171,6 @@ $\displaystyle D \isin C \equiv
   \end{cases}
 $ 
 
-Some (overlapping) alternative formulations:
-
-$\displaystyle D \isin C \equiv
-  \begin{cases}
-    D \isin L \equiv D \isin R     : & D = C \lor D \isin L \\
-    D \isin L \equiv D \isin R     : & D = C \lor D \isin R \\
-    D \isin L \nequiv D \isin R  : & D = C \lor D \not\isin M \\
-    D \isin M \equiv D \isin L     : & D = C \lor D \isin R \\
-    D \isin M \equiv D \isin R     : & D = C \lor D \isin L \\
-  \end{cases}
-$
-
 \end{basedescript}
 \newpage
 \section{Invariants}
@@ -211,6 +199,23 @@ We maintain these each time we construct a new commit. \\
 
 \section{Some lemmas}
 
+\[ \eqn{Alternative (overlapping) formulations defining
+  $\mergeof{C}{L}{M}{R}$:}{
+ D \isin C \equiv
+  \begin{cases}
+    D \isin L  \equiv D \isin R  : & D = C \lor D \isin L     \\
+    D \isin L \nequiv D \isin R  : & D = C \lor D \not\isin M \\
+    D \isin L  \equiv D \isin M  : & D = C \lor D \isin R     \\
+    D \isin L \nequiv D \isin M  : & D = C \lor D \isin L     \\
+    \text{as above with L and R exchanged}
+  \end{cases}
+}\]
+\proof{
+  Truth table xxx
+
+  Original definition is symmetrical in $L$ and $R$.
+}
+
 \[ \eqn{Exclusive Tip Contents:}{
   \bigforall_{C \in \py} 
     \neg \Bigl[ D \isin \baseof{C} \land ( D \in \py \land D \le C )
@@ -424,15 +429,15 @@ Used for removing a branch dependency.
 
 \subsection{Conditions}
 
+\[ \eqn{ From Base }{
+ L \in \pn
+}\]
 \[ \eqn{ Unique Tip }{
  \pendsof{L}{\pry} = \{ R^+ \}
 }\]
 \[ \eqn{ Currently Included }{
  L \haspatch \pry
 }\]
-\[ \eqn{ Not Self }{
- L \not\in \{ R^+ \}
-}\]
 
 \subsection{No Replay}
 
@@ -476,7 +481,12 @@ $\qed$
 
 \subsection{Unique Base}
 
-Need to consider only $C \in \py$, ie $L \in \py$.
+From Base means that $C \in \pn$, so Unique Base is not
+applicable. $\qed$
+
+\subsection{Tip Contents}
+
+Again, not applicable. $\qed$
 
 xxx tbd
 
@@ -526,6 +536,17 @@ We will occasionally use $X,Y$ s.t. $\{X,Y\} = \{L,R\}$.
    \right]
 }\]
 
+\subsection{Non-Topbloke merges}
+
+We require both $\patchof{L} = \bot$ and $\patchof{R} = \bot$.
+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.
+
 \subsection{No Replay}
 
 See No Replay for Merge Results.
@@ -695,6 +716,35 @@ $\baseof{C} = R$ so $D \isin C \equiv D \isin \baseof{C}$.  OK.
 
 $\qed$
 
-xxx up to here, need to prove other things about merges
+\subsection{Foreign Inclusion}
+
+Consider some $D$ s.t. $\patchof{D} = \bot$.
+By Foreign Inclusion of $L, M, R$:
+$D \isin L \equiv D \le L$;
+$D \isin M \equiv D \le M$;
+$D \isin R \equiv D \le R$.
+
+\subsubsection{For $D = C$:}
+
+$D \isin C$ and $D \le C$.  OK.
+
+\subsubsection{For $D \neq C, D \isin M$:}
+
+Thus $D \le M$ so $D \le L$ and $D \le R$ so $D \isin L$ and $D \isin
+R$.  So by $\merge$, $D \isin C$.  And $D \le C$.  OK.
+
+\subsubsection{For $D \neq C, D \not\isin M, D \isin X$:}
+
+By $\merge$, $D \isin C$.
+And $D \isin X$ means $D \le X$ so $D \le C$.
+OK.
+
+\subsubsection{For $D \neq C, D \not\isin M, D \not\isin L, D \not\isin R$:}
+
+By $\merge$, $D \not\isin C$.
+And $D \not\le L, D \not\le R$ so $D \not\le C$.
+OK
+
+$\qed$
 
 \end{document}