-\item[ $ C \hasparents \set X $ ]
-The parents of commit $C$ are exactly the set
-$\set X$.
-
-\item[ $ C \ge D $ ]
-$C$ is a descendant of $D$ in the git commit
-graph. This is a partial order, namely the transitive closure of
-$ D \in \set X $ where $ C \hasparents \set X $.
-
-\item[ $ C \has D $ ]
-Informally, the tree at commit $C$ contains the change
-made in commit $D$. Does not take account of deliberate reversions by
-the user or reversion, rebasing or rewinding in
-non-Topbloke-controlled branches. For merges and Topbloke-generated
-anticommits or re-commits, the ``change made'' is only to be thought
-of as any conflict resolution. This is not a partial order because it
-is not transitive.
-
-\item[ $ \p, \py, \pn $ ]
-A patch $\p$ consists of two sets of commits $\pn$ and $\py$, which
-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 $\pn$.
-None of these sets overlap. Hence:
-
-\item[ $ \patchof{ C } $ ]
-Either $\p$ s.t. $ C \in \p $, or $\bot$.
-A function from commits to patches' sets $\p$.
-
-\item[ $ \pancsof{C}{\set P} $ ]
-$ \{ A \; | \; A \le C \land A \in \set P \} $
-i.e. all the ancestors of $C$
-which are in $\set P$.
-
-\item[ $ \pendsof{C}{\set P} $ ]
-$ \{ E \; | \; E \in \pancsof{C}{\set P}
- \land \mathop{\not\exists}_{A \in \pancsof{C}{\set P}}
- E \neq A \land E \le A \} $
-i.e. all $\le$-maximal commits in $\pancsof{C}{\set P}$.
-
-\item[ $ \baseof{C} $ ]
-$ \pendsof{C}{\pn} = \{ \baseof{C} \} $ where $ C \in \py $.
-A partial function from commits to commits.
-See Unique Base, below.
-
-\item[ $ C \haspatch \p $ ]
-$\displaystyle \bigforall_{D \in \py} D \isin C \equiv D \le C $.
-~ Informally, $C$ has the contents of $\p$.
-
-\item[ $ C \nothaspatch \p $ ]
-$\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 $\p$. 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 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:
-
-$\displaystyle D \isin C \equiv
- \begin{cases}
- (D \isin L \land D \isin R) \lor D = C : & \true \\
- (D \not\isin L \land D \not\isin R) \land D \neq C : & \false \\
- \text{otherwise} : & D \not\isin M
- \end{cases}
-$
-
-\end{basedescript}
-\newpage
-\section{Invariants}
-
-We maintain these each time we construct a new commit. \\
-\[ \eqn{No Replay:}{
- C \has D \implies C \ge D
-}\]
-\[\eqn{Unique Base:}{
- \bigforall_{C \in \py} \pendsof{C}{\pn} = \{ B \}
-}\]
-\[\eqn{Tip Contents:}{
- \bigforall_{C \in \py} D \isin C \equiv
- { D \isin \baseof{C} \lor \atop
- (D \in \py \land D \le C) }
-}\]
-\[\eqn{Base Acyclic:}{
- \bigforall_{B \in \pn} D \isin B \implies D \notin \py
-}\]
-\[\eqn{Coherence:}{
- \bigforall_{C,\p} C \haspatch \p \lor C \nothaspatch \p
-}\]
-\[\eqn{Foreign Inclusion:}{
- \bigforall_{D \text{ s.t. } \patchof{D} = \bot} D \isin C \equiv D \leq C
-}\]
-\[\eqn{Foreign Contents:}{
- \bigforall_{C \text{ s.t. } \patchof{C} = \bot}
- D \le C \implies \patchof{D} = \bot
-}\]
-
-\section{Some lemmas}
-
-\subsection{Alternative (overlapping) formulations of $\mergeof{C}{L}{M}{R}$}
-$$
- D \isin C \equiv
- \begin{cases}
- D \isin L \equiv D \isin R : & D = C \lor D \isin L \\
- D \isin L \nequiv D \isin R : & D = C \lor D \not\isin M \\
- D \isin L \equiv D \isin M : & D = C \lor D \isin R \\
- D \isin L \nequiv D \isin M : & D = C \lor D \isin L \\
- \text{as above with L and R exchanged}
- \end{cases}
-$$
-\proof{ ~ Truth table (ordered by original definition): \\
- \begin{tabular}{cccc|c|cc}
- $D = C$ &
- $\isin L$ &
- $\isin M$ &
- $\isin R$ & $\isin C$ &
- $L$ vs. $R$ & $L$ vs. $M$
- \\\hline
- y & ? & ? & ? & y & ? & ? \\
- n & y & y & y & y & $\equiv$ & $\equiv$ \\
- n & y & n & y & y & $\equiv$ & $\nequiv$ \\
- n & n & y & n & n & $\equiv$ & $\nequiv$ \\
- n & n & n & n & n & $\equiv$ & $\equiv$ \\
- n & y & y & n & n & $\nequiv$ & $\equiv$ \\
- n & n & y & y & n & $\nequiv$ & $\nequiv$ \\
- n & y & n & n & y & $\nequiv$ & $\nequiv$ \\
- n & n & n & y & y & $\nequiv$ & $\equiv$ \\
- \end{tabular} \\
- And original definition is symmetrical in $L$ and $R$.
-}
-
-\subsection{Exclusive Tip Contents}
-$$
- \bigforall_{C \in \py}
- \neg \Bigl[ D \isin \baseof{C} \land ( D \in \py \land D \le C )
- \Bigr]
-$$
-Ie, the two limbs of the RHS of Tip Contents are mutually exclusive.
-
-\proof{
-Let $B = \baseof{C}$ in $D \isin \baseof{C}$. Now $B \in \pn$.
-So by Base Acyclic $D \isin B \implies D \notin \py$.
-}
-\[ \eqntag{{\it Corollary - equivalent to Tip Contents}}{
- \bigforall_{C \in \py} D \isin C \equiv
- \begin{cases}
- D \in \py : & D \le C \\
- D \not\in \py : & D \isin \baseof{C}
- \end{cases}
-}\]
-
-\subsection{Tip Self Inpatch}
-$$
- \bigforall_{C \in \py} C \haspatch \p
-$$
-Ie, tip commits contain their own patch.
-
-\proof{
-Apply Exclusive Tip Contents to some $D \in \py$:
-$ \bigforall_{C \in \py}\bigforall_{D \in \py}
- D \isin C \equiv D \le C $
-}
-
-\subsection{Exact Ancestors}
-$$
- \bigforall_{ C \hasparents \set{R} }
- D \le C \equiv
- ( \mathop{\hbox{\huge{$\vee$}}}_{R \in \set R} D \le R )
- \lor D = C
-$$
-\proof{ ~ Trivial.}
-
-\subsection{Transitive Ancestors}
-$$
- \left[ \bigforall_{ E \in \pendsof{C}{\set P} } E \le M \right] \equiv
- \left[ \bigforall_{ A \in \pancsof{C}{\set P} } A \le M \right]
-$$
-
-\proof{
-The implication from right to left is trivial because
-$ \pends() \subset \pancs() $.
-For the implication from left to right:
-by the definition of $\mathcal E$,
-for every such $A$, either $A \in \pends()$ which implies
-$A \le M$ by the LHS directly,
-or $\exists_{A' \in \pancs()} \; A' \neq A \land A \le A' $
-in which case we repeat for $A'$. Since there are finitely many
-commits, this terminates with $A'' \in \pends()$, ie $A'' \le M$
-by the LHS. And $A \le A''$.
-}
-
-\subsection{Calculation Of Ends}
-$$
- \bigforall_{C \hasparents \set A}
- \pendsof{C}{\set P} =
- \begin{cases}
- C \in \p : & \{ C \}
- \\
- C \not\in \p : & \displaystyle
- \left\{ E \Big|
- \Bigl[ \Largeexists_{A \in \set A}
- E \in \pendsof{A}{\set P} \Bigr] \land
- \Bigl[ \Largenexists_{B \in \set A, F \in \pendsof{B}{\p}}
- E \neq F \land E \le F \Bigr]
- \right\}
- \end{cases}
-$$
-\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 $\pends$.
-}
-
-\subsection{Ingredients Prevent Replay}
-$$
- \left[
- {C \hasparents \set A} \land
- \\
- \left(
- D \isin C \implies
- D = C \lor
- \Largeexists_{A \in \set A} D \isin A
- \right)
- \right] \implies \left[
- D \isin C \implies D \le C
- \right]
-$$
-\proof{
- Trivial for $D = C$. Consider some $D \neq C$, $D \isin C$.
- By the preconditions, there is some $A$ s.t. $D \in \set A$
- and $D \isin A$. By No Replay for $A$, $D \le A$. And
- $A \le C$ so $D \le C$.
-}
-
-\subsection{Simple Foreign Inclusion}
-$$
- \left[
- C \hasparents \{ L \}
- \land
- \bigforall_{D} D \isin C \equiv D \isin L \lor D = C
- \right]
- \implies
- \left[
- \bigforall_{D \text{ s.t. } \patchof{D} = \bot}
- D \isin C \equiv D \le C
- \right]
-$$
-\proof{
-Consider some $D$ s.t. $\patchof{D} = \bot$.
-If $D = C$, trivially true. For $D \neq C$,
-by Foreign Inclusion of $D$ in $L$, $D \isin L \equiv D \le L$.
-And by Exact Ancestors $D \le L \equiv D \le C$.
-So $D \isin C \equiv D \le C$.
-}
-
-\subsection{Totally Foreign Contents}
-$$
- \bigforall_{C \hasparents \set A}
- \left[
- \patchof{C} = \bot \land
- \bigforall_{A \in \set A} \patchof{A} = \bot
- \right]
- \implies
- \left[
- D \le C
- \implies
- \patchof{D} = \bot
- \right]
-$$
-\proof{
-Consider some $D \le C$. If $D = C$, $\patchof{D} = \bot$ trivially.
-If $D \neq C$ then $D \le A$ where $A \in \set A$. By Foreign
-Contents of $A$, $\patchof{D} = \bot$.
-}
-
-\section{Commit annotation}
-
-We annotate each Topbloke commit $C$ with:
-\gathbegin
- \patchof{C}
-\gathnext
- \baseof{C}, \text{ if } C \in \py
-\gathnext
- \bigforall_{\pq}
- \text{ either } C \haspatch \pq \text{ or } C \nothaspatch \pq
-\gathnext
- \bigforall_{\pqy \not\ni C} \pendsof{C}{\pqy}
-\end{gather}
-
-$\patchof{C}$, for each kind of Topbloke-generated commit, is stated
-in the summary in the section for that kind of commit.
-
-Whether $\baseof{C}$ is required, and if so what the value is, is
-stated in the proof of Unique Base for each kind of commit.
-
-$C \haspatch \pq$ or $\nothaspatch \pq$ is represented as the
-set $\{ \pq | C \haspatch \pq \}$. Whether $C \haspatch \pq$
-is in stated
-(in terms of $I \haspatch \pq$ or $I \nothaspatch \pq$
-for the ingredients $I$),
-in the proof of Coherence for each kind of commit.
-
-$\pendsof{C}{\pq^+}$ is computed, for all Topbloke-generated commits,
-using the lemma Calculation of Ends, above.
-We do not annotate $\pendsof{C}{\py}$ for $C \in \py$. Doing so would
-make it wrong to make plain commits with git because the recorded $\pends$
-would have to be updated. The annotation is not needed in that case
-because $\forall_{\py \ni C} \; \pendsof{C}{\py} = \{C\}$.
-
-\section{Simple commit}
-
-A simple single-parent forward commit $C$ as made by git-commit.
-\begin{gather}
-\tag*{} C \hasparents \{ L \} \\
-\tag*{} \patchof{C} = \patchof{L} \\
-\tag*{} D \isin C \equiv D \isin L \lor D = C
-\end{gather}
-This also covers Topbloke-generated commits on plain git branches:
-Topbloke strips the metadata when exporting.
-
-\subsection{No Replay}
-
-Ingredients Prevent Replay applies. $\qed$
-
-\subsection{Unique Base}
-If $L, C \in \py$ then by Calculation of Ends for
-$C, \py, C \not\in \py$:
-$\pendsof{C}{\pn} = \pendsof{L}{\pn}$ so
-$\baseof{C} = \baseof{L}$. $\qed$
-
-\subsection{Tip Contents}
-We need to consider only $L, C \in \py$. From Tip Contents for $L$:
-\[ D \isin L \equiv D \isin \baseof{L} \lor ( D \in \py \land D \le L ) \]
-Substitute into the contents of $C$:
-\[ D \isin C \equiv D \isin \baseof{L} \lor ( D \in \py \land D \le L )
- \lor D = C \]
-Since $D = C \implies D \in \py$,
-and substituting in $\baseof{C}$, this gives:
-\[ D \isin C \equiv D \isin \baseof{C} \lor
- (D \in \py \land D \le L) \lor
- (D = C \land D \in \py) \]
-\[ \equiv D \isin \baseof{C} \lor
- [ D \in \py \land ( D \le L \lor D = C ) ] \]
-So by Exact Ancestors:
-\[ D \isin C \equiv D \isin \baseof{C} \lor ( D \in \py \land D \le C
-) \]
-$\qed$
-
-\subsection{Base Acyclic}
-
-Need to consider only $L, C \in \pn$.
-
-For $D = C$: $D \in \pn$ so $D \not\in \py$. OK.
-
-For $D \neq C$: $D \isin C \equiv D \isin L$, so by Base Acyclic for
-$L$, $D \isin C \implies D \not\in \py$.
-
-$\qed$
-
-\subsection{Coherence and patch inclusion}
-
-Need to consider $D \in \py$
-
-\subsubsection{For $L \haspatch P, D = C$:}
-
-Ancestors of $C$:
-$ D \le C $.
-
-Contents of $C$:
-$ D \isin C \equiv \ldots \lor \true \text{ so } D \haspatch C $.
-
-\subsubsection{For $L \haspatch P, D \neq C$:}
-Ancestors: $ D \le C \equiv D \le L $.
-
-Contents: $ D \isin C \equiv D \isin L \lor f $
-so $ D \isin C \equiv D \isin L $.
-
-So:
-\[ L \haspatch P \implies C \haspatch P \]
-
-\subsubsection{For $L \nothaspatch P$:}
-
-Firstly, $C \not\in \py$ since if it were, $L \in \py$.
-Thus $D \neq C$.
-
-Now by contents of $L$, $D \notin L$, so $D \notin C$.
-
-So:
-\[ L \nothaspatch P \implies C \nothaspatch P \]
-$\qed$
-
-\subsection{Foreign Inclusion:}
-
-Simple Foreign Inclusion applies. $\qed$
-
-\subsection{Foreign Contents:}
-
-Only relevant if $\patchof{C} = \bot$, and in that case Totally
-Foreign Contents applies. $\qed$
-
-\section{Create Base}
-
-Given $L$, create a Topbloke base branch initial commit $B$.
-\gathbegin
- B \hasparents \{ L \}
-\gathnext
- \patchof{B} = \pqn
-\gathnext
- D \isin B \equiv D \isin L \lor D = B
-\end{gather}
-
-\subsection{Conditions}
-
-\[ \eqn{ Create Acyclic }{
- \pendsof{L}{\pqy} = \{ \}
-}\]
-
-\subsection{No Replay}
-
-Ingredients Prevent Replay applies. $\qed$
-
-\subsection{Unique Base}
-
-Not applicable.
-
-\subsection{Tip Contents}
-
-Not applicable.
-
-\subsection{Base Acyclic}
-
-Consider some $D \isin B$. If $D = B$, $D \in \pqn$.
-If $D \neq B$, $D \isin L$, so by No Replay $D \le L$
-and by Create Acyclic
-$D \not\in \pqy$. $\qed$
-
-\subsection{Coherence and Patch Inclusion}
-
-Consider some $D \in \p$.
-$B \not\in \py$ so $D \neq B$. So $D \isin B \equiv D \isin L$
-and $D \le B \equiv D \le L$.
-
-Thus $L \haspatch \p \implies B \haspatch P$
-and $L \nothaspatch \p \implies B \nothaspatch P$.
-
-$\qed$.
-
-\subsection{Foreign Inclusion}
-
-Simple Foreign Inclusion applies. $\qed$
-
-\subsection{Foreign Contents}
-
-Not applicable.
-
-\section{Create Tip}
-
-Given a Topbloke base $B$, create a tip branch initial commit B.
-\gathbegin
- C \hasparents \{ B \}
-\gathnext
- \patchof{B} = \pqy
-\gathnext
- D \isin C \equiv D \isin B \lor D = C
-\end{gather}
-
-\subsection{Conditions}
-
-\[ \eqn{ Ingredients }{
- \patchof{B} = \pqn
-}\]
-\[ \eqn{ No Sneak }{
- \pendsof{B}{\pqy} = \{ \}
-}\]
-
-\subsection{No Replay}
-
-Ingredients Prevent Replay applies. $\qed$
-
-\subsection{Unique Base}
-
-Trivially, $\pendsof{C}{\pqn} = \{B\}$ so $\baseof{C} = B$. $\qed$
-
-\subsection{Tip Contents}
-
-Consider some arbitrary commit $D$. If $D = C$, trivially satisfied.
-
-If $D \neq C$, $D \isin C \equiv D \isin B$.
-By Base Acyclic of $B$, $D \isin B \implies D \not\in \pqy$.
-So $D \isin C \equiv D \isin \baseof{B}$.