wip merge complex - fix bug, improve merge ends condition

[topbloke-formulae.git] / article.tex

diff --git a/article.tex b/article.tex
index be767aa82859183fa1baebc9f5eb060a5ac66d07..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}
@@ -598,4 +601,9 @@ $D \in \pancsof{X}{\py}$, 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}