wip merge complex - fix bug, improve merge ends condition

[topbloke-formulae.git] / article.tex

diff --git a/article.tex b/article.tex
index fac5c82efd6483ac6cc865cbd2db836f2d29d81e..080e482847aeeffa24fda58ec31b59cfe63205c8 100644 (file)
--- a/article.tex
+++ b/article.tex
@@ -491,10 +491,13 @@ We will occasionally use $X,Y$ s.t. $\{X,Y\} = \{L,R\}$.
 \[ \eqn{ Merge Ends }{
     X \not\haspatch \p \land
     Y \haspatch \p
-  \implies \left[
-  \bigforall_{E \in \pendsof{X}{\py}}
-    E \le Y
-  \right]
+  \implies
+  \begin{cases}
+    M \haspatch \p :    & \displaystyle
+                          \bigforall_{E \in \pendsof{Y}{\py}} E \le M \\
+    M \nothaspatch \p : & \displaystyle
+                          \bigforall_{E \in \pendsof{X}{\py}} E \le Y
+  \end{cases}
 }\]
 
 \subsection{No Replay}
@@ -583,9 +586,24 @@ 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
+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 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 Merge Ends
+and Transitive Ancestors $D \le M$.
+Thus $D \isin M$.  By $\merge$, $D \not\isin C$.  OK.
+
 \end{document}