merge tip contentss: clarify arbitrariness of D

[topbloke-formulae.git] / article.tex

diff --git a/article.tex b/article.tex
index 71e09608bcb95d8147f06340e81461a70437cdd3..faf41526029fb4b0c489e4b8fedfe072d7f62dad 100644 (file)
--- a/article.tex
+++ b/article.tex
@@ -496,8 +496,7 @@ We will occasionally use $X,Y$ s.t. $\{X,Y\} = \{L,R\}$.
    \begin{cases}
       R \in \py : & \baseof{R} \ge \baseof{L}
               \land [\baseof{L} = M \lor \baseof{L} = \baseof{M}] \\
-      R \in \pn : & R \ge \baseof{L}
-              \land M = \baseof{L} \\
+      R \in \pn : & M = \baseof{L} \\
       \text{otherwise} : & \false
    \end{cases}
 }\]
@@ -552,7 +551,7 @@ That is, $\baseof{C} = \baseof{R}$.
 
 \subsubsection{For $R \in \pn$:}
 
-By Tip Merge condition on $R$,
+By Tip Merge condition on $R$ and since $M \le R$,
 $A \le \baseof{L} \implies A \le R$, so
 $A \le R \lor A \le \baseof{L} \equiv A \le R$.  
 Thus $A \le C \equiv A \le R$.  
@@ -560,7 +559,7 @@ That is, $\baseof{C} = R$.
 
 $\qed$
 
-\subsection{Coherence and patch inclusion}
+\subsection{Coherence and Patch Inclusion}
 
 Need to determine $C \haspatch \p$ based on $L,M,R \haspatch \p$.
 This involves considering $D \in \py$.  
@@ -648,16 +647,17 @@ R$.  And $D \neq C$.  So $D \not\isin C$.  $\qed$
 
 We need worry only about $C \in \py$.  
 And $\patchof{C} = \patchof{L}$
-so $L \in \py$ so $L \haspatch \p$.  We will use the coherence and
-patch inclusion of $C$ as just proved.
+so $L \in \py$ so $L \haspatch \p$.  We will use the Unique Base
+of $C$, and its Coherence and Patch Inclusion, as just proved.
 
 Firstly we show $C \haspatch \p$: If $R \in \py$, then $R \haspatch
-\p$ and by coherence/inclusion $C \haspatch \p$ .  If $R \not\in \py$
+\p$ and by Coherence/Inclusion $C \haspatch \p$ .  If $R \not\in \py$
 then by Tip Merge $M = \baseof{L}$ so by Base Acyclic and definition
-of $\nothaspatch$, $M \nothaspatch \p$.  So by coherence/inclusion $C
+of $\nothaspatch$, $M \nothaspatch \p$.  So by Coherence/Inclusion $C
 \haspatch \p$ (whether $R \haspatch \p$ or $\nothaspatch$).
 
-We will consider some $D$ and prove the Exclusive Tip Contents form.
+We will consider an arbitrary commit $D$
+and prove the Exclusive Tip Contents form.
 
 \subsubsection{For $D \in \py$:}
 $C \haspatch \p$ so by definition of $\haspatch$, $D \isin C \equiv D
@@ -667,11 +667,13 @@ $C \haspatch \p$ so by definition of $\haspatch$, $D \isin C \equiv D
 
 $D \neq C$.  By Tip Contents of $L$,
 $D \isin L \equiv D \isin \baseof{L}$, and by Tip Merge condition,
-$D \isin L \equiv D \isin M$.  xxx up to here
-
+$D \isin L \equiv D \isin M$.  So by definition of $\merge$, $D \isin
+C \equiv D \isin R$.  And $R = \baseof{C}$ by Unique Base of $C$.
+Thus $D \isin C \equiv D \isin \baseof{C}$.  OK.
 
 \subsubsection{For $D \not\in \py, R \in \py$:}
 
+xxx up to here
 
 %D \in \py$:}