add purported proof of calculation of ends

[topbloke-formulae.git] / article.tex

diff --git a/article.tex b/article.tex
index 1dd625fb915d87f34687753e19ec98647e753ebe..b26a120098b953606abe0fdee77d83abe9543735 100644 (file)
--- a/article.tex
+++ b/article.tex
@@ -275,7 +275,7 @@ $ \bigforall_{C \in \py}\bigforall_{D \in \py}
     ( \mathop{\hbox{\huge{$\vee$}}}_{R \in \set R} D \le R )
     \lor D = C
 }\]
-xxx proof tbd
+\proof{ ~ Trivial.}
 
 \[ \eqn{Transitive Ancestors:}{
   \left[ \bigforall_{ E \in \pendsof{C}{\set P} } E \le M \right] \equiv
@@ -305,12 +305,21 @@ by the LHS.  And $A \le A''$.
        \left\{ E \Big|
            \Bigl[ \Largeexists_{A \in \set A}
                        E \in \pendsof{A}{\set P} \Bigr] \land
-           \Bigl[ \Largenexists_{B \in \set A}
-                       E \neq B \land E \le B \Bigr]
+           \Bigl[ \Largenexists_{B \in \set A, F \in \pendsof{B}{\p}}
+                       E \neq F \land E \le F \Bigr]
        \right\}
       \end{cases}
 }\]
-xxx proof tbd
+\proof{
+Trivial for $C \in \set P$.  For $C \not\in \set P$,
+$\pancsof{C}{\set P} = \bigcup_{A \in \set A} \pancsof{A}{\set P}$.
+So $\pendsof{C}{\set P} \subset \bigcup_{E in \set E} \pendsof{E}{\set P}$.
+Consider some $E \in \pendsof{A}{\set P}$.  If $\exists_{B,F}$ as
+specified, then either $F$ is going to be in our result and
+disqualifies $E$, or there is some other $F'$ (or, eventually,
+an $F''$) which disqualifies $F$.
+Otherwise, $E$ meets all the conditions for $\pends$.
+}
 
 \[ \eqn{Ingredients Prevent Replay:}{
   \left[