add some xxx's

[topbloke-formulae.git] / article.tex

diff --git a/article.tex b/article.tex
index 674e5936f60fc6ae27161506b57b458b7517ccab..023a18a9340c7670e3ee89011177eb9010165e69 100644 (file)
--- a/article.tex
+++ b/article.tex
@@ -10,6 +10,8 @@
 
 \renewcommand{\ge}{\geqslant}
 \renewcommand{\le}{\leqslant}
+\newcommand{\nge}{\ngeqslant}
+\newcommand{\nle}{\nleqslant}
 
 \newcommand{\has}{\sqsupseteq}
 \newcommand{\isin}{\sqsubseteq}
@@ -53,11 +55,15 @@
 \newcommand{\pancsof}[2]{\pancs ( #1 , #2 ) }
 \newcommand{\pendsof}[2]{\pends ( #1 , #2 ) }
 
-\newcommand{\merge}[4]{{\mathcal M}(#1,#2,#3,#4)}
+\newcommand{\merge}{{\mathcal M}}
+\newcommand{\mergeof}[4]{\merge(#1,#2,#3,#4)}
 %\newcommand{\merge}[4]{{#2 {{\frac{ #1 }{ #3 } #4}}}}
 
-\newcommand{\patchof}[1]{{\mathcal P} ( #1 ) }
-\newcommand{\baseof}[1]{{\mathcal B} ( #1 ) }
+\newcommand{\patch}{{\mathcal P}}
+\newcommand{\base}{{\mathcal B}}
+
+\newcommand{\patchof}[1]{\patch ( #1 ) }
+\newcommand{\baseof}[1]{\base ( #1 ) }
 
 \newcommand{\eqn}[2]{ #2 \tag*{\mbox{\bf #1}} }
 \newcommand{\corrolary}[1]{ #1 \tag*{\mbox{\it Corrolary.}} }
@@ -149,7 +155,7 @@ patch is applied to a non-Topbloke branch and then bubbles back to
 the Topbloke patch itself, we hope that git's merge algorithm will
 DTRT or that the user will no longer care about the Topbloke patch.
 
-\item[ $\displaystyle \merge{C}{L}{M}{R} $ ]
+\item[ $\displaystyle \mergeof{C}{L}{M}{R} $ ]
 The contents of a git merge result:
 
 $\displaystyle D \isin C \equiv
@@ -245,19 +251,26 @@ by the LHS.  And $A \le A''$.
 \[ \eqn{Calculation Of Ends:}{
   \bigforall_{C \hasparents \set A}
     \pendsof{C}{\set P} =
-       \Bigl\{ E \Big|
+       \left\{ E \Big|
            \Bigl[ \Largeexists_{A \in \set A} 
                        E \in \pendsof{A}{\set P} \Bigr] \land
            \Bigl[ \Largenexists_{B \in \set A} 
                        E \neq B \land E \le B \Bigr]
-       \Bigr\}
+       \right\}
 }\]
 XXX proof TBD.
 
 \subsection{No Replay for Merge Results}
 
-If we are constructing $C$ such that $\merge{C}{L}{M}{R}$, No Replay
-is preserved.  {\it Proof:}
+If we are constructing $C$, given
+\gathbegin
+  \mergeof{C}{L}{M}{R}
+\gathnext
+  L \le C
+\gathnext
+  R \le C
+\end{gather}
+No Replay is preserved.  {\it Proof:}
 
 \subsubsection{For $D=C$:} $D \isin C, D \le C$.  OK.
 
@@ -267,9 +280,6 @@ $D \isin C$.  And $D \isin L \implies D \le L \implies D \le C$.  OK.
 \subsubsection{For $D \neq C \land D \not\isin L \land D \not\isin R$:}
 $D \not\isin C$.  OK.
 
-\subsubsection{For $D \neq C \land D \not\isin L \land D \not\isin R$:}
-$D \not\isin C$.  OK.
-
 \subsubsection{For $D \neq C \land (D \isin L \equiv D \not\isin R)
  \land D \not\isin M$:}
 $D \isin C$.  Also $D \isin L \lor D \isin R$ so $D \le L \lor D \le
@@ -277,8 +287,7 @@ R$ so $D \le C$.  OK.
 
 \subsubsection{For $D \neq C \land (D \isin L \equiv D \not\isin R)
  \land D \isin M$:}
-$D \not\isin C$.  Also $D \isin L \lor D \isin R$ so $D \le L \lor D \le
-R$ so $D \le C$.  OK.
+$D \not\isin C$.  OK.
 
 $\qed$
 
@@ -383,7 +392,7 @@ $D \isin C \equiv D \isin A \equiv D \le A \equiv D \le C$.  $\qed$
 \section{Anticommit}
 
 Given $L, R^+, R^-$ where
-$\patchof{R^+} = \pry, \patchof{R^-} = \prn$.  
+$R^+ \in \pry, R^- = \baseof{R^+}$.  
 Construct $C$ which has $\pr$ removed.
 Used for removing a branch dependency.
 \gathbegin
@@ -391,7 +400,7 @@ Used for removing a branch dependency.
 \gathnext
  \patchof{C} = \patchof{L}
 \gathnext
- \merge{C}{L}{R^+}{R^-}
+ \mergeof{C}{L}{R^+}{R^-}
 \end{gather}
 
 \subsection{Conditions}
@@ -399,16 +408,60 @@ Used for removing a branch dependency.
 \[ \eqn{ Unique Tip }{
  \pendsof{L}{\pry} = \{ R^+ \}
 }\]
-\[ \eqn{ Correct Base }{
- \baseof{R^+} = R^-
-}\]
 \[ \eqn{ Currently Included }{
  L \haspatch \pry
 }\]
+\[ \eqn{ Not Self }{
+ L \not\in \{ R^+ \}
+}\]
+
+\subsection{No Replay}
+
+By Unique Tip, $R^+ \le L$.  By definition of $\base$, $R^- \le R^+$
+so $R^- \le L$.  So $R^+ \le C$ and $R^- \le C$ and No Replay for
+Merge Results applies. $\qed$
+
+\subsection{Desired Contents}
+
+\[ D \isin C \equiv [ D \notin \pry \land D \isin L ] \lor D = C \]
+{\it Proof.}
+
+\subsubsection{For $D = C$:}
+
+Trivially $D \isin C$.  OK.
 
+\subsubsection{For $D \neq C, D \not\le L$:}
 
+By No Replay $D \not\isin L$.  Also $D \not\le R^-$ hence
+$D \not\isin R^-$.  Thus $D \not\isin C$.  OK.
 
-xxx want to prove $D \isin C \equiv D \not\in \pry \land D \isin L$.
+\subsubsection{For $D \neq C, D \le L, D \in \pry$:}
+
+By Currently Included, $D \isin L$.
+
+By Tip Self Inpatch, $D \isin R^+ \equiv D \le R^+$, but by
+by Unique Tip, $D \le R^+ \equiv D \le L$.  
+So $D \isin R^+$.
+
+By Base Acyclic, $D \not\isin R^-$.
+
+Apply $\merge$: $D \not\isin C$.  OK.
+
+\subsubsection{For $D \neq C, D \le L, D \notin \pry$:}
+
+By Tip Contents for $R^+$, $D \isin R^+ \equiv D \isin R^-$.
+
+Apply $\merge$: $D \isin C \equiv D \isin L$.  OK.
+
+$\qed$
+
+\subsection{Unique Base}
+
+Need to consider only $C \in \py$, ie $L \in \py$.
+
+xxx tbd
+
+xxx need to finish anticommit
 
 \section{Merge}
 
@@ -418,7 +471,7 @@ Merge commits $L$ and $R$ using merge base $M$ ($M < L, M < R$):
 \gathnext
  \patchof{C} = \patchof{L}
 \gathnext
- \merge{C}{L}{M}{R}
+ \mergeof{C}{L}{M}{R}
 \end{gather}
 
 \subsection{Conditions}
@@ -434,9 +487,9 @@ Merge commits $L$ and $R$ using merge base $M$ ($M < L, M < R$):
    \end{cases}
 }\]
 
-\subsection{Merge Results}
+\subsection{No Replay}
 
-As above.
+See No Replay for Merge Results.
 
 \subsection{Unique Base}
 
@@ -472,4 +525,10 @@ That is, $\baseof{C} = R$.
 
 $\qed$
 
+\subsection{Coherence and patch inclusion}
+
+xxx tbd
+
+xxx need to finish merge
+
 \end{document}