\renewcommand{\implies}{\Rightarrow}
\renewcommand{\equiv}{\Leftrightarrow}
+\renewcommand{\nequiv}{\nLeftrightarrow}
\renewcommand{\land}{\wedge}
\renewcommand{\lor}{\vee}
\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{\corrolary}[1]{ #1 \tag*{\mbox{\it Corrolary.}} }
%\newcommand{\bigforall}{\mathop{\hbox{\huge$\forall$}}}
\newcommand{\bigforall}{%
\section{Notation}
+xxx make all conditions be in conditions, not in (or also in) intro
+
\begin{basedescript}{
\desclabelwidth{5em}
\desclabelstyle{\nextlinelabel}
are respectively the base and tip git branches. $\p$ may be used
where the context requires a set, in which case the statement
is to be taken as applying to both $\py$ and $\pn$.
-All these sets are distinct. Hence:
+None of these sets overlap. Hence:
\item[ $ \patchof{ C } $ ]
Either $\p$ s.t. $ C \in \p $, or $\bot$.
$\displaystyle \bigforall_{D \in \py} D \not\isin C $.
~ Informally, $C$ has none of the contents of $\p$.
-Non-Topbloke commits are $\nothaspatch \p$ for all $\p$; if a Topbloke
+Non-Topbloke commits are $\nothaspatch \p$ for all $\p$. This
+includes commits on plain git branches made by applying a Topbloke
+patch. If a Topbloke
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.
+the relevant Topbloke branches, we hope that
+if the user still cares about the Topbloke patch,
+git's merge algorithm will DTRT when trying to re-apply the changes.
\item[ $\displaystyle \mergeof{C}{L}{M}{R} $ ]
The contents of a git merge result:
\section{Some lemmas}
+\[ \eqn{Alternative (overlapping) formulations defining
+ $\mergeof{C}{L}{M}{R}$:}{
+ D \isin C \equiv
+ \begin{cases}
+ D \isin L \equiv D \isin R : & D = C \lor D \isin L \\
+ D \isin L \nequiv D \isin R : & D = C \lor D \not\isin M \\
+ D \isin L \equiv D \isin M : & D = C \lor D \isin R \\
+ D \isin L \nequiv D \isin M : & D = C \lor D \isin L \\
+ \text{as above with L and R exchanged}
+ \end{cases}
+}\]
+\proof{
+ Truth table xxx
+
+ Original definition is symmetrical in $L$ and $R$.
+}
+
\[ \eqn{Exclusive Tip Contents:}{
\bigforall_{C \in \py}
\neg \Bigl[ D \isin \baseof{C} \land ( D \in \py \land D \le C )
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 \\
\tag*{} \patchof{C} = \patchof{A} \\
\tag*{} D \isin C \equiv D \isin A \lor D = C
\end{gather}
+This also covers Topbloke-generated commits on plain git branches:
+Topbloke strips the metadata when exporting.
\subsection{No Replay}
Trivial.
\subsection{Conditions}
+\[ \eqn{ Into Base }{
+ L \in \pn
+}\]
\[ \eqn{ Unique Tip }{
\pendsof{L}{\pry} = \{ R^+ \}
}\]
\[ \eqn{ Currently Included }{
L \haspatch \pry
}\]
-\[ \eqn{ Not Self }{
- L \not\in \{ R^+ \}
-}\]
-\subsection{No Replay}
+\subsection{Ordering of ${L, R^+, R^-}$:}
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. $\qed$
+so $R^- \le L$. So $R^+ \le C$ and $R^- \le C$.
+
+(Note that the merge base $R^+ \not\le R^-$, i.e. the merge base is
+later than one of the branches to be merged.)
+
+\subsection{No Replay}
+
+No Replay for Merge Results applies. $\qed$
\subsection{Desired Contents}
\subsection{Unique Base}
-Need to consider only $C \in \py$, ie $L \in \py$.
+Into Base means that $C \in \pn$, so Unique Base is not
+applicable. $\qed$
-xxx tbd
+\subsection{Tip Contents}
+
+Again, not applicable. $\qed$
+
+\subsection{Base Acyclic}
-xxx need to finish anticommit
+By Base Acyclic for $L$, $D \isin L \implies D \not\in \py$.
+And by Into Base $C \not\in \py$.
+Now from Desired Contents, above, $D \isin C
+\implies D \isin L \lor D = C$, which thus
+$\implies D \not\in \py$. $\qed$.
+
+\subsection{Coherence and Patch Inclusion}
+
+Need to consider some $D \in \py$. By Into Base, $D \neq C$.
+
+\subsubsection{For $\p = \pr$:}
+By Desired Contents, above, $D \not\isin C$.
+So $C \nothaspatch \pr$.
+
+\subsubsection{For $\p \neq \pr$:}
+By Desired Contents, $D \isin C \equiv D \isin L$
+(since $D \in \py$ so $D \not\in \pry$).
+
+If $L \nothaspatch \p$, $D \not\isin L$ so $D \not\isin C$.
+So $L \nothaspatch \p \implies C \nothaspatch \p$.
+
+Whereas if $L \haspatch \p$, $D \isin L \equiv D \le L$.
+so $L \haspatch \p \implies C \haspatch \p$.
+
+\section{Foreign Inclusion}
+
+Consider some $D$ s.t. $\patchof{D} = \bot$. $D \neq C$.
+So by Desired Contents $D \isin C \equiv D \isin L$.
+By Foreign Inclusion of $D$ in $L$, $D \isin L \equiv D \le L$.
+So $D \isin C \equiv D \le L$.
+
+xxx up to here
+
+By Tip Contents of $R^+$, $D \isin R^+ \equiv D \isin \baseof{R^+}$
+i.e. $\equiv D \isin R^-$.
+So by $\merge$, $D \isin C \equiv D \isin L$.
+
+Thus $D \isin C \equiv $
\section{Merge}
\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}
}\]
-\[ \eqn{ Merge Ends }{
+\[ \eqn{ Merge Acyclic }{
+ L \in \pn
+ \implies
+ R \nothaspatch \p
+}\]
+\[ \eqn{ Removal Merge Ends }{
X \not\haspatch \p \land
- Y \haspatch \p
+ Y \haspatch \p \land
+ M \haspatch \p
\implies
- \begin{cases}
- M \haspatch \p : & \displaystyle
- \bigforall_{E \in \pendsof{Y}{\py}} E \le M \\
- M \nothaspatch \p : & \displaystyle
- \bigforall_{E \in \pendsof{X}{\py}} E \le Y
- \end{cases}
+ \pendsof{Y}{\py} = \pendsof{M}{\py}
}\]
+\[ \eqn{ Addition Merge Ends }{
+ X \not\haspatch \p \land
+ Y \haspatch \p \land
+ M \nothaspatch \p
+ \implies \left[
+ \bigforall_{E \in \pendsof{X}{\py}} E \le Y
+ \right]
+}\]
+
+\subsection{Non-Topbloke merges}
+
+We require both $\patchof{L} = \bot$ and $\patchof{R} = \bot$.
+I.e. not only is it forbidden to merge into a Topbloke-controlled
+branch without Topbloke's assistance, it is also forbidden to
+merge any Topbloke-controlled branch into any plain git branch.
+
+Given those conditions, Tip Merge and Merge Acyclic do not apply.
+And $Y \not\in \py$ so $\neg [ Y \haspatch \p ]$ so neither
+Merge Ends condition applies. Good.
\subsection{No Replay}
-See No Replay for Merge Results.
+No Replay for Merge Results applies. $\qed$
\subsection{Unique Base}
\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 $A \le C \equiv A \le R$.
$\qed$
-\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 $D \in \py$.
\subsubsection{For (wlog) $X \not\haspatch \p, Y \haspatch \p$:}
-$C \haspatch \p \equiv M \nothaspatch \p$.
+$M \haspatch \p \implies C \nothaspatch \p$.
+$M \nothaspatch \p \implies C \haspatch \p$.
\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 $D \isin C \equiv M \nothaspatch \p \land D \le C$
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 $D \neq C, M \haspatch P, D \isin Y$:
-$D \le Y$ 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.
Consider $D \neq C, M \haspatch P, D \not\isin Y$:
By $\merge$, $D \not\isin C$. OK.
+$\qed$
+
+\subsection{Base Acyclic}
+
+This applies when $C \in \pn$.
+$C \in \pn$ when $L \in \pn$ so by Merge Acyclic, $R \nothaspatch \p$.
+
+Consider some $D \in \py$.
+
+By Base Acyclic of $L$, $D \not\isin L$. By the above, $D \not\isin
+R$. And $D \neq C$. So $D \not\isin C$. $\qed$
+
+\subsection{Tip Contents}
+
+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 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 $R \not\in \py$
+then by Tip Merge $M = \baseof{L}$ so by Base Acyclic and definition
+of $\nothaspatch$, $M \nothaspatch \p$. So by Coherence/Inclusion $C
+\haspatch \p$ (whether $R \haspatch \p$ or $\nothaspatch$).
+
+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 $\haspatch$, $D \isin C \equiv D
+\le C$. OK.
+
+\subsubsection{For $D \not\in \py, R \not\in \py$:}
+
+$D \neq C$. By Tip Contents of $L$,
+$D \isin L \equiv D \isin \baseof{L}$, and by Tip Merge condition,
+$D \isin L \equiv D \isin M$. 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 $D \not\in \py, R \in \py$:}
+
+$D \neq C$.
+
+By Tip Contents
+$D \isin L \equiv D \isin \baseof{L}$ and
+$D \isin R \equiv D \isin \baseof{R}$.
+
+If $\baseof{L} = M$, trivially $D \isin M \equiv D \isin \baseof{L}.$
+Whereas if $\baseof{L} = \baseof{M}$, by definition of $\base$,
+$\patchof{M} = \patchof{L} = \py$, so by Tip Contents of $M$,
+$D \isin M \equiv D \isin \baseof{M} \equiv D \isin \baseof{L}$.
+
+So $D \isin M \equiv D \isin L$ and by $\merge$,
+$D \isin C \equiv D \isin R$. But from Unique Base,
+$\baseof{C} = R$ so $D \isin C \equiv D \isin \baseof{C}$. OK.
+
+$\qed$
+
+\subsection{Foreign Inclusion}
+
+Consider some $D$ s.t. $\patchof{D} = \bot$.
+By Foreign Inclusion of $L, M, R$:
+$D \isin L \equiv D \le L$;
+$D \isin M \equiv D \le M$;
+$D \isin R \equiv D \le R$.
+
+\subsubsection{For $D = C$:}
+
+$D \isin C$ and $D \le C$. OK.
+
+\subsubsection{For $D \neq C, D \isin M$:}
+
+Thus $D \le M$ so $D \le L$ and $D \le R$ so $D \isin L$ and $D \isin
+R$. So by $\merge$, $D \isin C$. And $D \le C$. OK.
+
+\subsubsection{For $D \neq C, D \not\isin M, D \isin X$:}
+
+By $\merge$, $D \isin C$.
+And $D \isin X$ means $D \le X$ so $D \le C$.
+OK.
+
+\subsubsection{For $D \neq C, D \not\isin M, D \not\isin L, D \not\isin R$:}
+
+By $\merge$, $D \not\isin C$.
+And $D \not\le L, D \not\le R$ so $D \not\le C$.
+OK
+
+$\qed$
+
\end{document}
~ian / topbloke-formulae.git / blobdiff