strategy: define W in Notation

[topbloke-formulae.git] / traversal.tex

diff --git a/traversal.tex b/traversal.tex
index 407c60d7f84f19e94564928d05c35ef7e82d7659..61d904e13375efbe111a27300a3f7bac430cc16b 100644 (file)
--- a/traversal.tex
+++ b/traversal.tex
@@ -215,10 +215,23 @@ $L = W, \; R = \tipdy, \; M = \baseof{R} = \tipdn$.
 \item TODO CHOOSE/REFINE W AND S as was done during Ranking for bases
 
 \item $\alg{Merge}$ from $\tipcn$.  That is, $L = W, \;
-R = \tipcn$ and choose any suitable $M$.
+R = \tipcn$ and $M = \baseof{W}$.
 
  \commitproof{
-  $L = W$, $R = \tipcn$.
+  \condproof{Ingredients}{
+   $M \le L$ is trivial.  For $M \le R$ we want
+   $\tipcn \ge \baseof{W}$.
+   Well $W \in \set S^{\pcy}$ so $W \in \allreachof{\pcn}$
+   and $W \in \pcy$.  So $W \in \pendsof{\allreachof{\pcn}}{\pcy}$
+   so Base Covers Reachable indeed
+   $\tipcn \ge \baseof{W}$.
+  }
+  \condproof{Tip Merge}{ Trivial. }
+  \condproof{Merge Acyclic}{
+   By Base Acyclic, $\tipcn \nothaspatch \p$.
+  }
+  \condproof{Foreign Merges}{ Not applicable. }
+
   TODO TBD
 
   Afterwards, $\baseof{W} = \tipcn$.
@@ -235,7 +248,7 @@ $\alg{Merge}$ with $L = W, \; R = S$ and any suitable $M$.
 
   By the results of Tip Base Merge, $\baseof{W} = \tipcn$.
 
-  By Base Ends Supreme, $\tipcn \ge \baseof{S}$ i.e.
+  By Base Ends Supreme TODO CHECK THIS, $\tipcn \ge \baseof{S}$ i.e.
   $\baseof{R} \ge \baseof{L}$.
 
   Either $\baseof{L} = \baseof{M}$, or we must choose a different $M$ in