X-Git-Url: http://www.chiark.greenend.org.uk/ucgi/~ian/git?a=blobdiff_plain;f=article.tex;h=0688c7ad02524d0d5c0d399682f3ac46946d1e1f;hb=26c18196858d61ec8bd051837aa9ccb08f68ccaf;hp=b778663dd0c814ac84c3edffa6c57a71969f935c;hpb=1aa4dce9dace603365ff1b302c938b6bfe9bbc1e;p=topbloke-formulae.git

diff --git a/article.tex b/article.tex
index b778663..0688c7a 100644
--- a/article.tex
+++ b/article.tex
@@ -429,15 +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}
 
@@ -481,7 +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
 
@@ -711,6 +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}