X-Git-Url: http://www.chiark.greenend.org.uk/ucgi/~ian/git?p=topbloke-formulae.git;a=blobdiff_plain;f=article.tex;h=cbd56840b63d5579073f30786bcac5ed76fa1773;hp=6a5023c3ceaf24813e16dfbe0d309a2a38e22178;hb=d5e22c8cccab5fb2ed73e3944e022edae9429082;hpb=e600e19a18c0f5b4595c68dd305f093b62fed9e4

diff --git a/article.tex b/article.tex
index 6a5023c..cbd5684 100644
--- a/article.tex
+++ b/article.tex
@@ -488,12 +488,20 @@ We will occasionally use $X,Y$ s.t. $\{X,Y\} = \{L,R\}$.
       \text{otherwise} : & \false
    \end{cases}
 }\]
-\[ \eqn{ Merge Ends }{
+\[ \eqn{ Removal Merge Ends }{
     X \not\haspatch \p \land
     Y \haspatch \p \land
-    E \in \pendsof{X}{\py}
+    M \haspatch \p
   \implies
-    E \le Y
+    \pendsof{Y}{\py} = \pendsof{M}{\py}
+}\]
+\[ \eqn{ Addition Merge Ends }{
+    X \not\haspatch \p \land
+    Y \haspatch \p \land
+    M \nothaspatch \p
+   \implies \left[
+    \bigforall_{E \in \pendsof{X}{\py}} E \le Y
+   \right]
 }\]
 
 \subsection{No Replay}
@@ -572,8 +580,40 @@ So indeed $L \haspatch \p \land R \haspatch \p \implies C \haspatch \p$.
 
 \subsubsection{For (wlog) $X \not\haspatch \p, Y \haspatch \p$:}
 
-$C \haspatch \p \equiv C \nothaspatch M$.
+$C \haspatch \p \equiv M \nothaspatch \p$.
 
 \proofstarts
 
+One of the Merge Ends conditions applies.  
+Recall that we are considering $D \in \py$.
+$D \isin Y \equiv D \le Y$.  $D \not\isin X$.
+We will show for each of
+various cases that $D \isin C \equiv M \nothaspatch \p \land D \le C$
+(which suffices by definition of $\haspatch$ and $\nothaspatch$).
+
+Consider $D = C$:  Thus $C \in \py, L \in \py$, and by Tip
+Self Inpatch $L \haspatch \p$, so $L=Y, R=X$.  By Tip Merge,
+$M=\baseof{L}$.  So by Base Acyclic $D \not\isin M$, i.e.
+$M \nothaspatch \p$.  And indeed $D \isin C$ and $D \le C$.  OK.
+
+Consider $D \neq C, M \nothaspatch P, D \isin Y$:
+$D \le Y$ so $D \le C$.  
+$D \not\isin M$ so by $\merge$, $D \isin C$.  OK.
+
+Consider $D \neq C, M \nothaspatch P, D \not\isin Y$:
+$D \not\le Y$.  If $D \le X$ then
+$D \in \pancsof{X}{\py}$, so by Addition Merge Ends and 
+Transitive Ancestors $D \le Y$ --- a contradiction, so $D \not\le X$.
+Thus $D \not\le C$.  By $\merge$, $D \not\isin C$.  OK.
+
+Consider $D \neq C, M \haspatch P, D \isin Y$:
+$D \le Y$ so $D \in \pancsof{Y}{\py}$ so by Removal Merge Ends
+and Transitive Ancestors $D \in \pancsof{M}{\py}$ so $D \le M$.
+Thus $D \isin M$.  By $\merge$, $D \not\isin C$.  OK.
+
+Consider $D \neq C, M \haspatch P, D \not\isin Y$:
+By $\merge$, $D \not\isin C$.  OK.
+
+$\qed$
+
 \end{document}