rename exclusive tip contents corollary

[topbloke-formulae.git] / article.tex

diff --git a/article.tex b/article.tex
index 080e482847aeeffa24fda58ec31b59cfe63205c8..01297a8fb0681c79d7ffc1933fe941f02f77adb5 100644 (file)
--- a/article.tex
+++ b/article.tex
@@ -65,8 +65,8 @@
 \newcommand{\patchof}[1]{\patch ( #1 ) }
 \newcommand{\baseof}[1]{\base ( #1 ) }
 
 \newcommand{\patchof}[1]{\patch ( #1 ) }
 \newcommand{\baseof}[1]{\base ( #1 ) }
 

+\newcommand{\eqntag}[2]{ #2 \tag*{\mbox{#1}} }

 \newcommand{\eqn}[2]{ #2 \tag*{\mbox{\bf #1}} }
 \newcommand{\eqn}[2]{ #2 \tag*{\mbox{\bf #1}} }

-\newcommand{\corrolary}[1]{ #1 \tag*{\mbox{\it Corrolary.}} }

 
 %\newcommand{\bigforall}{\mathop{\hbox{\huge$\forall$}}}
 \newcommand{\bigforall}{%
 
 %\newcommand{\bigforall}{\mathop{\hbox{\huge$\forall$}}}
 \newcommand{\bigforall}{%


@@ -206,7 +206,7 @@ Ie, the two limbs of the RHS of Tip Contents are mutually exclusive.
 Let $B = \baseof{C}$ in $D \isin \baseof{C}$.  Now $B \in \pn$.
 So by Base Acyclic $D \isin B \implies D \notin \py$.
 }
 Let $B = \baseof{C}$ in $D \isin \baseof{C}$.  Now $B \in \pn$.
 So by Base Acyclic $D \isin B \implies D \notin \py$.
 }

-\[ \corrolary{
+\[ \eqntag{{\it Corollary - equivalent to Tip Contents}}{

   \bigforall_{C \in \py} D \isin C \equiv
   \begin{cases}
     D \in \py : & D \le C \\
   \bigforall_{C \in \py} D \isin C \equiv
   \begin{cases}
     D \in \py : & D \le C \\


@@ -488,16 +488,20 @@ We will occasionally use $X,Y$ s.t. $\{X,Y\} = \{L,R\}$.
       \text{otherwise} : & \false
    \end{cases}
 }\]
       \text{otherwise} : & \false
    \end{cases}
 }\]

-\[ \eqn{ Merge Ends }{
+\[ \eqn{ Removal Merge Ends }{

     X \not\haspatch \p \land
     X \not\haspatch \p \land

-    Y \haspatch \p
+    Y \haspatch \p \land
+    M \haspatch \p

   \implies
   \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}
+    \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}
 }\]
 
 \subsection{No Replay}


@@ -580,7 +584,8 @@ $C \haspatch \p \equiv M \nothaspatch \p$.
 
 \proofstarts
 
 
 \proofstarts
 

-Merge Ends applies.  Recall that we are considering $D \in \py$.
+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 $D \isin C \equiv M \nothaspatch \p \land D \le C$
 $D \isin Y \equiv D \le Y$.  $D \not\isin X$.
 We will show for each of
 various cases that $D \isin C \equiv M \nothaspatch \p \land D \le C$


@@ -597,13 +602,18 @@ $D \not\isin M$ 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
 
 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 
+$D \in \pancsof{X}{\py}$, 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 $D \neq C, M \haspatch P, D \isin Y$:
 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 $D \le M$.
+$D \le Y$ 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.
 
 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}
 \end{document}