1 \documentclass[a4paper,leqno]{strayman}
3 \let\numberwithin=\notdef
15 \let\stdsection\section
16 \renewcommand\section{\newpage\stdsection}
18 \renewcommand{\ge}{\geqslant}
19 \renewcommand{\le}{\leqslant}
20 \newcommand{\nge}{\ngeqslant}
21 \newcommand{\nle}{\nleqslant}
23 \newcommand{\has}{\sqsupseteq}
24 \newcommand{\isin}{\sqsubseteq}
26 \newcommand{\nothaspatch}{\mathrel{\,\not\!\not\relax\haspatch}}
27 \newcommand{\notpatchisin}{\mathrel{\,\not\!\not\relax\patchisin}}
28 \newcommand{\haspatch}{\sqSupset}
29 \newcommand{\patchisin}{\sqSubset}
31 \newif\ifhidehack\hidehackfalse
32 \DeclareRobustCommand\hidefromedef[2]{%
33 \hidehacktrue\ifhidehack#1\else#2\fi\hidehackfalse}
34 \newcommand{\pa}[1]{\hidefromedef{\varmathbb{#1}}{#1}}
36 \newcommand{\set}[1]{\mathbb{#1}}
37 \newcommand{\pay}[1]{\pa{#1}^+}
38 \newcommand{\pan}[1]{\pa{#1}^-}
40 \newcommand{\p}{\pa{P}}
41 \newcommand{\py}{\pay{P}}
42 \newcommand{\pn}{\pan{P}}
44 \newcommand{\pq}{\pa{Q}}
45 \newcommand{\pqy}{\pay{Q}}
46 \newcommand{\pqn}{\pan{Q}}
48 \newcommand{\pr}{\pa{R}}
49 \newcommand{\pry}{\pay{R}}
50 \newcommand{\prn}{\pan{R}}
52 %\newcommand{\hasparents}{\underaccent{1}{>}}
53 %\newcommand{\hasparents}{{%
54 % \declareslashed{}{_{_1}}{0}{-0.8}{>}\slashed{>}}}
55 \newcommand{\hasparents}{>_{\mkern-7.0mu _1}}
56 \newcommand{\areparents}{<_{\mkern-14.0mu _1\mkern+5.0mu}}
58 \renewcommand{\implies}{\Rightarrow}
59 \renewcommand{\equiv}{\Leftrightarrow}
60 \renewcommand{\nequiv}{\nLeftrightarrow}
61 \renewcommand{\land}{\wedge}
62 \renewcommand{\lor}{\vee}
64 \newcommand{\pancs}{{\mathcal A}}
65 \newcommand{\pends}{{\mathcal E}}
67 \newcommand{\pancsof}[2]{\pancs ( #1 , #2 ) }
68 \newcommand{\pendsof}[2]{\pends ( #1 , #2 ) }
70 \newcommand{\merge}{{\mathcal M}}
71 \newcommand{\mergeof}[4]{\merge(#1,#2,#3,#4)}
72 %\newcommand{\merge}[4]{{#2 {{\frac{ #1 }{ #3 } #4}}}}
74 \newcommand{\patch}{{\mathcal P}}
75 \newcommand{\base}{{\mathcal B}}
77 \newcommand{\patchof}[1]{\patch ( #1 ) }
78 \newcommand{\baseof}[1]{\base ( #1 ) }
80 \newcommand{\eqntag}[2]{ #2 \tag*{\mbox{#1}} }
81 \newcommand{\eqn}[2]{ #2 \tag*{\mbox{\bf #1}} }
83 %\newcommand{\bigforall}{\mathop{\hbox{\huge$\forall$}}}
84 \newcommand{\bigforall}{%
86 {\hbox{\huge$\forall$}}%
87 {\hbox{\Large$\forall$}}%
88 {\hbox{\normalsize$\forall$}}%
89 {\hbox{\scriptsize$\forall$}}}%
92 \newcommand{\Largeexists}{\mathop{\hbox{\Large$\exists$}}}
93 \newcommand{\Largenexists}{\mathop{\hbox{\Large$\nexists$}}}
95 \newcommand{\qed}{\square}
96 \newcommand{\proofstarts}{{\it Proof:}}
97 \newcommand{\proof}[1]{\proofstarts #1 $\qed$}
99 \newcommand{\gathbegin}{\begin{gather} \tag*{}}
100 \newcommand{\gathnext}{\\ \tag*{}}
102 \newcommand{\true}{t}
103 \newcommand{\false}{f}
109 \begin{basedescript}{
111 \desclabelstyle{\nextlinelabel}
113 \item[ $ C \hasparents \set X $ ]
114 The parents of commit $C$ are exactly the set
118 $C$ is a descendant of $D$ in the git commit
119 graph. This is a partial order, namely the transitive closure of
120 $ D \in \set X $ where $ C \hasparents \set X $.
122 \item[ $ C \has D $ ]
123 Informally, the tree at commit $C$ contains the change
124 made in commit $D$. Does not take account of deliberate reversions by
125 the user or reversion, rebasing or rewinding in
126 non-Topbloke-controlled branches. For merges and Topbloke-generated
127 anticommits or re-commits, the ``change made'' is only to be thought
128 of as any conflict resolution. This is not a partial order because it
131 \item[ $ \p, \py, \pn $ ]
132 A patch $\p$ consists of two sets of commits $\pn$ and $\py$, which
133 are respectively the base and tip git branches. $\p$ may be used
134 where the context requires a set, in which case the statement
135 is to be taken as applying to both $\py$ and $\pn$.
136 All of these sets are disjoint. Hence:
138 \item[ $ \patchof{ C } $ ]
139 Either $\p$ s.t. $ C \in \p $, or $\bot$.
140 A function from commits to patches' sets $\p$.
142 \item[ $ \pancsof{C}{\set P} $ ]
143 $ \{ A \; | \; A \le C \land A \in \set P \} $
144 i.e. all the ancestors of $C$
145 which are in $\set P$.
147 \item[ $ \pendsof{C}{\set P} $ ]
148 $ \{ E \; | \; E \in \pancsof{C}{\set P}
149 \land \mathop{\not\exists}_{A \in \pancsof{C}{\set P}}
150 E \neq A \land E \le A \} $
151 i.e. all $\le$-maximal commits in $\pancsof{C}{\set P}$.
153 \item[ $ \baseof{C} $ ]
154 $ \pendsof{C}{\pn} = \{ \baseof{C} \} $ where $ C \in \py $.
155 A partial function from commits to commits.
156 See Unique Base, below.
158 \item[ $ C \haspatch \p $ ]
159 $\displaystyle \bigforall_{D \in \py} D \isin C \equiv D \le C $.
160 ~ Informally, $C$ has the contents of $\p$.
162 \item[ $ C \nothaspatch \p $ ]
163 $\displaystyle \bigforall_{D \in \py} D \not\isin C $.
164 ~ Informally, $C$ has none of the contents of $\p$.
166 Commits on Non-Topbloke branches are $\nothaspatch \p$ for all $\p$. This
167 includes commits on plain git branches made by applying a Topbloke
169 patch is applied to a non-Topbloke branch and then bubbles back to
170 the relevant Topbloke branches, we hope that
171 if the user still cares about the Topbloke patch,
172 git's merge algorithm will DTRT when trying to re-apply the changes.
174 \item[ $\displaystyle \mergeof{C}{L}{M}{R} $ ]
175 The contents of a git merge result:
177 $\displaystyle D \isin C \equiv
179 (D \isin L \land D \isin R) \lor D = C : & \true \\
180 (D \not\isin L \land D \not\isin R) \land D \neq C : & \false \\
181 \text{otherwise} : & D \not\isin M
189 We maintain these each time we construct a new commit. \\
191 C \has D \implies C \ge D
193 \[\eqn{Unique Base:}{
194 \bigforall_{C \in \py} \pendsof{C}{\pn} = \{ B \}
196 \[\eqn{Tip Contents:}{
197 \bigforall_{C \in \py} D \isin C \equiv
198 { D \isin \baseof{C} \lor \atop
199 (D \in \py \land D \le C) }
201 \[\eqn{Base Acyclic:}{
202 \bigforall_{B \in \pn} D \isin B \implies D \notin \py
205 \bigforall_{C,\p} C \haspatch \p \lor C \nothaspatch \p
207 \[\eqn{Foreign Inclusion:}{
208 \bigforall_{D \text{ s.t. } \patchof{D} = \bot} D \isin C \equiv D \leq C
210 \[\eqn{Foreign Contents:}{
211 \bigforall_{C \text{ s.t. } \patchof{C} = \bot}
212 D \le C \implies \patchof{D} = \bot
215 \section{Some lemmas}
217 \subsection{Alternative (overlapping) formulations of $\mergeof{C}{L}{M}{R}$}
221 D \isin L \equiv D \isin R : & D = C \lor D \isin L \\
222 D \isin L \nequiv D \isin R : & D = C \lor D \not\isin M \\
223 D \isin L \equiv D \isin M : & D = C \lor D \isin R \\
224 D \isin L \nequiv D \isin M : & D = C \lor D \isin L \\
225 \text{as above with L and R exchanged}
228 \proof{ ~ Truth table (ordered by original definition): \\
229 \begin{tabular}{cccc|c|cc}
233 $\isin R$ & $\isin C$ &
234 $L$ vs. $R$ & $L$ vs. $M$
236 y & ? & ? & ? & y & ? & ? \\
237 n & y & y & y & y & $\equiv$ & $\equiv$ \\
238 n & y & n & y & y & $\equiv$ & $\nequiv$ \\
239 n & n & y & n & n & $\equiv$ & $\nequiv$ \\
240 n & n & n & n & n & $\equiv$ & $\equiv$ \\
241 n & y & y & n & n & $\nequiv$ & $\equiv$ \\
242 n & n & y & y & n & $\nequiv$ & $\nequiv$ \\
243 n & y & n & n & y & $\nequiv$ & $\nequiv$ \\
244 n & n & n & y & y & $\nequiv$ & $\equiv$ \\
246 And original definition is symmetrical in $L$ and $R$.
249 \subsection{Exclusive Tip Contents}
250 Given Base Acyclic for $C$,
252 \bigforall_{C \in \py}
253 \neg \Bigl[ D \isin \baseof{C} \land ( D \in \py \land D \le C )
256 Ie, the two limbs of the RHS of Tip Contents are mutually exclusive.
259 Let $B = \baseof{C}$ in $D \isin \baseof{C}$. Now $B \in \pn$.
260 So by Base Acyclic $D \isin B \implies D \notin \py$.
262 \[ \eqntag{{\it Corollary - equivalent to Tip Contents}}{
263 \bigforall_{C \in \py} D \isin C \equiv
265 D \in \py : & D \le C \\
266 D \not\in \py : & D \isin \baseof{C}
270 \subsection{Tip Self Inpatch}
271 Given Exclusive Tip Contents and Base Acyclic for $C$,
273 \bigforall_{C \in \py} C \haspatch \p
275 Ie, tip commits contain their own patch.
278 Apply Exclusive Tip Contents to some $D \in \py$:
279 $ \bigforall_{C \in \py}\bigforall_{D \in \py}
280 D \isin C \equiv D \le C $
283 \subsection{Exact Ancestors}
285 \bigforall_{ C \hasparents \set{R} }
288 ( \mathop{\hbox{\huge{$\vee$}}}_{R \in \set R} D \le R )
294 \subsection{Transitive Ancestors}
296 \left[ \bigforall_{ E \in \pendsof{C}{\set P} } E \le M \right] \equiv
297 \left[ \bigforall_{ A \in \pancsof{C}{\set P} } A \le M \right]
301 The implication from right to left is trivial because
302 $ \pends() \subset \pancs() $.
303 For the implication from left to right:
304 by the definition of $\mathcal E$,
305 for every such $A$, either $A \in \pends()$ which implies
306 $A \le M$ by the LHS directly,
307 or $\exists_{A' \in \pancs()} \; A' \neq A \land A \le A' $
308 in which case we repeat for $A'$. Since there are finitely many
309 commits, this terminates with $A'' \in \pends()$, ie $A'' \le M$
310 by the LHS. And $A \le A''$.
313 \subsection{Calculation of Ends}
315 \bigforall_{C \hasparents \set A}
316 \pendsof{C}{\set P} =
320 C \not\in \p : & \displaystyle
322 \Bigl[ \Largeexists_{A \in \set A}
323 E \in \pendsof{A}{\set P} \Bigr] \land
324 \Bigl[ \Largenexists_{B \in \set A, F \in \pendsof{B}{\p}}
325 E \neq F \land E \le F \Bigr]
330 Trivial for $C \in \set P$. For $C \not\in \set P$,
331 $\pancsof{C}{\set P} = \bigcup_{A \in \set A} \pancsof{A}{\set P}$.
332 So $\pendsof{C}{\set P} \subset \bigcup_{E in \set E} \pendsof{E}{\set P}$.
333 Consider some $E \in \pendsof{A}{\set P}$. If $\exists_{B,F}$ as
334 specified, then either $F$ is going to be in our result and
335 disqualifies $E$, or there is some other $F'$ (or, eventually,
336 an $F''$) which disqualifies $F$.
337 Otherwise, $E$ meets all the conditions for $\pends$.
340 \subsection{Ingredients Prevent Replay}
343 {C \hasparents \set A} \land
349 \Largeexists_{A \in \set A} D \isin A
351 \right] \implies \left[ \bigforall_{D}
352 D \isin C \implies D \le C
356 Trivial for $D = C$. Consider some $D \neq C$, $D \isin C$.
357 By the preconditions, there is some $A$ s.t. $D \in \set A$
358 and $D \isin A$. By No Replay for $A$, $D \le A$. And
359 $A \le C$ so $D \le C$.
362 \subsection{Simple Foreign Inclusion}
365 C \hasparents \{ L \}
367 \bigforall_{D} D \isin C \equiv D \isin L \lor D = C
371 \bigforall_{D \text{ s.t. } \patchof{D} = \bot}
372 D \isin C \equiv D \le C
376 Consider some $D$ s.t. $\patchof{D} = \bot$.
377 If $D = C$, trivially true. For $D \neq C$,
378 by Foreign Inclusion of $D$ in $L$, $D \isin L \equiv D \le L$.
379 And by Exact Ancestors $D \le L \equiv D \le C$.
380 So $D \isin C \equiv D \le C$.
383 \subsection{Totally Foreign Contents}
386 C \hasparents \set A \land
387 \patchof{C} = \bot \land
388 \bigforall_{A \in \set A} \patchof{A} = \bot
399 Consider some $D \le C$. If $D = C$, $\patchof{D} = \bot$ trivially.
400 If $D \neq C$ then $D \le A$ where $A \in \set A$. By Foreign
401 Contents of $A$, $\patchof{D} = \bot$.
404 \section{Commit annotation}
406 We annotate each Topbloke commit $C$ with:
410 \baseof{C}, \text{ if } C \in \py
413 \text{ either } C \haspatch \pq \text{ or } C \nothaspatch \pq
415 \bigforall_{\pqy \not\ni C} \pendsof{C}{\pqy}
418 $\patchof{C}$, for each kind of Topbloke-generated commit, is stated
419 in the summary in the section for that kind of commit.
421 Whether $\baseof{C}$ is required, and if so what the value is, is
422 stated in the proof of Unique Base for each kind of commit.
424 $C \haspatch \pq$ or $\nothaspatch \pq$ is represented as the
425 set $\{ \pq | C \haspatch \pq \}$. Whether $C \haspatch \pq$
427 (in terms of $I \haspatch \pq$ or $I \nothaspatch \pq$
428 for the ingredients $I$)
429 in the proof of Coherence for each kind of commit.
431 $\pendsof{C}{\pq^+}$ is computed, for all Topbloke-generated commits,
432 using the lemma Calculation of Ends, above.
433 We do not annotate $\pendsof{C}{\py}$ for $C \in \py$. Doing so would
434 make it wrong to make plain commits with git because the recorded $\pends$
435 would have to be updated. The annotation is not needed in that case
436 because $\forall_{\py \ni C} \; \pendsof{C}{\py} = \{C\}$.
438 \section{Simple commit}
440 A simple single-parent forward commit $C$ as made by git-commit.
442 \tag*{} C \hasparents \{ L \} \\
443 \tag*{} \patchof{C} = \patchof{L} \\
444 \tag*{} D \isin C \equiv D \isin L \lor D = C
446 This also covers Topbloke-generated commits on plain git branches:
447 Topbloke strips the metadata when exporting.
449 \subsection{No Replay}
451 Ingredients Prevent Replay applies. $\qed$
453 \subsection{Unique Base}
454 If $L, C \in \py$ then by Calculation of Ends,
455 $\pendsof{C}{\pn} = \pendsof{L}{\pn}$ so
456 $\baseof{C} = \baseof{L}$. $\qed$
458 \subsection{Tip Contents}
459 We need to consider only $L, C \in \py$. From Tip Contents for $L$:
460 \[ D \isin L \equiv D \isin \baseof{L} \lor ( D \in \py \land D \le L ) \]
461 Substitute into the contents of $C$:
462 \[ D \isin C \equiv D \isin \baseof{L} \lor ( D \in \py \land D \le L )
464 Since $D = C \implies D \in \py$,
465 and substituting in $\baseof{C}$, from Unique Base above, this gives:
466 \[ D \isin C \equiv D \isin \baseof{C} \lor
467 (D \in \py \land D \le L) \lor
468 (D = C \land D \in \py) \]
469 \[ \equiv D \isin \baseof{C} \lor
470 [ D \in \py \land ( D \le L \lor D = C ) ] \]
471 So by Exact Ancestors:
472 \[ D \isin C \equiv D \isin \baseof{C} \lor ( D \in \py \land D \le C
476 \subsection{Base Acyclic}
478 Need to consider only $L, C \in \pn$.
480 For $D = C$: $D \in \pn$ so $D \not\in \py$. OK.
482 For $D \neq C$: $D \isin C \equiv D \isin L$, so by Base Acyclic for
483 $L$, $D \isin C \implies D \not\in \py$.
487 \subsection{Coherence and patch inclusion}
489 Need to consider $D \in \py$
491 \subsubsection{For $L \haspatch P, D = C$:}
497 $ D \isin C \equiv \ldots \lor \true \text{ so } D \haspatch C $.
499 \subsubsection{For $L \haspatch P, D \neq C$:}
500 Ancestors: $ D \le C \equiv D \le L $.
502 Contents: $ D \isin C \equiv D \isin L \lor f $
503 so $ D \isin C \equiv D \isin L $.
506 \[ L \haspatch P \implies C \haspatch P \]
508 \subsubsection{For $L \nothaspatch P$:}
510 Firstly, $C \not\in \py$ since if it were, $L \in \py$.
513 Now by contents of $L$, $D \notin L$, so $D \notin C$.
516 \[ L \nothaspatch P \implies C \nothaspatch P \]
519 \subsection{Foreign Inclusion:}
521 Simple Foreign Inclusion applies. $\qed$
523 \subsection{Foreign Contents:}
525 Only relevant if $\patchof{C} = \bot$, and in that case Totally
526 Foreign Contents applies. $\qed$
528 \section{Create Base}
530 Given a starting point $L$ and a proposed patch $\pq$,
531 create a Topbloke base branch initial commit $B$.
533 B \hasparents \{ L \}
537 D \isin B \equiv D \isin L \lor D = B
540 \subsection{Conditions}
542 \[ \eqn{ Create Acyclic }{
543 \pendsof{L}{\pqy} = \{ \}
546 \subsection{No Replay}
548 Ingredients Prevent Replay applies. $\qed$
550 \subsection{Unique Base}
554 \subsection{Tip Contents}
558 \subsection{Base Acyclic}
560 Consider some $D \isin B$. If $D = B$, $D \in \pqn$.
561 If $D \neq B$, $D \isin L$, so by No Replay $D \le L$
562 and by Create Acyclic
563 $D \not\in \pqy$. $\qed$
565 \subsection{Coherence and Patch Inclusion}
567 Consider some $D \in \py$.
568 $B \not\in \py$ so $D \neq B$. So $D \isin B \equiv D \isin L$
569 and $D \le B \equiv D \le L$.
571 Thus $L \haspatch \p \implies B \haspatch P$
572 and $L \nothaspatch \p \implies B \nothaspatch P$.
576 \subsection{Foreign Inclusion}
578 Simple Foreign Inclusion applies. $\qed$
580 \subsection{Foreign Contents}
586 Given a Topbloke base $B$ for a patch $\pq$,
587 create a tip branch initial commit B.
589 C \hasparents \{ B \}
593 D \isin C \equiv D \isin B \lor D = C
596 \subsection{Conditions}
598 \[ \eqn{ Ingredients }{
602 \pendsof{B}{\pqy} = \{ \}
605 \subsection{No Replay}
607 Ingredients Prevent Replay applies. $\qed$
609 \subsection{Unique Base}
611 Trivially, $\pendsof{C}{\pqn} = \{B\}$ so $\baseof{C} = B$. $\qed$
613 \subsection{Tip Contents}
615 Consider some arbitrary commit $D$. If $D = C$, trivially satisfied.
617 If $D \neq C$, $D \isin C \equiv D \isin B$,
618 which by Unique Base, above, $ \equiv D \isin \baseof{B}$.
619 By Base Acyclic of $B$, $D \isin B \implies D \not\in \pqy$.
624 \subsection{Base Acyclic}
628 \subsection{Coherence and Patch Inclusion}
632 \p = \pq \lor B \haspatch \p : & C \haspatch \p \\
633 \p \neq \pq \land B \nothaspatch \p : & C \nothaspatch \p
638 ~ Consider some $D \in \py$.
640 \subsubsection{For $\p = \pq$:}
642 By Base Acyclic, $D \not\isin B$. So $D \isin C \equiv D = C$.
643 By No Sneak, $D \not\le B$ so $D \le C \equiv D = C$. Thus $C \haspatch \pq$.
645 \subsubsection{For $\p \neq \pq$:}
647 $D \neq C$. So $D \isin C \equiv D \isin B$,
648 and $D \le C \equiv D \le B$.
652 \subsection{Foreign Inclusion}
654 Simple Foreign Inclusion applies. $\qed$
656 \subsection{Foreign Contents}
660 \section{Dependency Removal}
662 Given $L$ which contains $\pr$ as represented by $R^+, R^-$.
663 Construct $C$ which has $\pr$ removed by applying a single
664 commit which is the anticommit of $\pr$.
665 Used for removing a branch dependency.
667 C \hasparents \{ L \}
669 \patchof{C} = \patchof{L}
671 \mergeof{C}{L}{R^+}{R^-}
674 \subsection{Conditions}
676 \[ \eqn{ Ingredients }{
677 R^+ \in \pry \land R^- = \baseof{R^+}
679 \[ \eqn{ Into Base }{
682 \[ \eqn{ Unique Tip }{
683 \pendsof{L}{\pry} = \{ R^+ \}
685 \[ \eqn{ Currently Included }{
689 \subsection{Ordering of Ingredients:}
691 By Unique Tip, $R^+ \le L$. By definition of $\base$, $R^- \le R^+$
692 so $R^- \le L$. So $R^+ \le C$ and $R^- \le C$.
695 (Note that $R^+ \not\le R^-$, i.e. the merge base
696 is a descendant, not an ancestor, of the 2nd parent.)
698 \subsection{No Replay}
701 $D \isin C \implies D \isin L \lor D \isin R^- \lor D = C$.
702 So, by Ordering of Ingredients,
703 Ingredients Prevent Replay applies. $\qed$
705 \subsection{Desired Contents}
707 \[ D \isin C \equiv [ D \notin \pry \land D \isin L ] \lor D = C \]
710 \subsubsection{For $D = C$:}
712 Trivially $D \isin C$. OK.
714 \subsubsection{For $D \neq C, D \not\le L$:}
716 By No Replay for $L$, $D \not\isin L$.
717 Also, by Ordering of Ingredients, $D \not\le R^-$ hence
718 $D \not\isin R^-$. Thus $D \not\isin C$. OK.
720 \subsubsection{For $D \neq C, D \le L, D \in \pry$:}
722 By Currently Included, $D \isin L$.
724 By Tip Self Inpatch for $R^+$, $D \isin R^+ \equiv D \le R^+$, but by
725 by Unique Tip, $D \le R^+ \equiv D \le L$.
728 By Base Acyclic for $R^-$, $D \not\isin R^-$.
730 Apply $\merge$: $D \not\isin C$. OK.
732 \subsubsection{For $D \neq C, D \le L, D \notin \pry$:}
734 By Tip Contents for $R^+$, $D \isin R^+ \equiv D \isin R^-$.
736 Apply $\merge$: $D \isin C \equiv D \isin L$. OK.
740 \subsection{Unique Base}
742 Into Base means that $C \in \pqn$, so Unique Base is not
745 \subsection{Tip Contents}
747 Again, not applicable. $\qed$
749 \subsection{Base Acyclic}
751 By Into Base and Base Acyclic for $L$, $D \isin L \implies D \not\in \pqy$.
752 And by Into Base $C \not\in \pqy$.
753 Now from Desired Contents, above, $D \isin C
754 \implies D \isin L \lor D = C$, which thus
755 $\implies D \not\in \pqy$. $\qed$.
757 \subsection{Coherence and Patch Inclusion}
759 Need to consider some $D \in \py$. By Into Base, $D \neq C$.
761 \subsubsection{For $\p = \pr$:}
762 By Desired Contents, above, $D \not\isin C$.
763 So $C \nothaspatch \pr$.
765 \subsubsection{For $\p \neq \pr$:}
766 By Desired Contents, $D \isin C \equiv D \isin L$
767 (since $D \in \py$ so $D \not\in \pry$).
769 If $L \nothaspatch \p$, $D \not\isin L$ so $D \not\isin C$.
770 So $L \nothaspatch \p \implies C \nothaspatch \p$.
772 Whereas if $L \haspatch \p$, $D \isin L \equiv D \le L$.
773 so $L \haspatch \p \implies C \haspatch \p$.
777 \subsection{Foreign Inclusion}
779 Consider some $D$ s.t. $\patchof{D} = \bot$. $D \neq C$.
780 So by Desired Contents $D \isin C \equiv D \isin L$.
781 By Foreign Inclusion of $D$ in $L$, $D \isin L \equiv D \le L$.
783 And $D \le C \equiv D \le L$.
784 Thus $D \isin C \equiv D \le C$.
788 \subsection{Foreign Contents}
792 \section{Dependency Insertion}
794 Given $L$ construct $C$ which additionally
795 contains $\pr$ as represented by $R^+$ and $R^-$.
796 This may even be used for reintroducing a previous-removed branch
799 C \hasparents \{ L, R^+ \}
801 \patchof{C} = \patchof{L}
803 \mergeof{C}{L}{R^-}{R^+}
806 \subsection{Conditions}
808 \[ \eqn{ Ingredients }{
811 \[ \eqn{ Into Base }{
814 \[ \eqn{ Currently Excluded }{
817 \[ \eqn{ Insertion Acyclic }{
821 \subsection{No Replay}
824 $D \isin C \implies D \isin L \lor D \isin R^+ \lor D = C$.
825 So Ingredients Prevent Replay applies. $\qed$
827 \subsection{Unique Base}
831 \subsection{Tip Contents}
835 \subsection{Base Acyclic}
837 Consider some $D \isin C$. We will show that $D \not\in \pqy$.
838 By $\merge$, $D \isin L \lor D \isin R^+ \lor D = C$.
840 For $D \isin L$, Base Acyclic for L suffices. For $D \isin R^+$,
841 Insertion Acyclic suffices. For $D = C$, trivial. $\qed$.
843 \subsection{Coherence}
845 We consider some $D \in \py$.
847 \subsubsection{For $\p = \pq$:}
851 $D \not\isin L$, $D \not\isin $
855 Merge commits $L$ and $R$ using merge base $M$:
857 C \hasparents \{ L, R \}
859 \patchof{C} = \patchof{L}
863 We will occasionally use $X,Y$ s.t. $\{X,Y\} = \{L,R\}$.
865 \subsection{Conditions}
866 \[ \eqn{ Ingredients }{
869 \[ \eqn{ Tip Merge }{
872 R \in \py : & \baseof{R} \ge \baseof{L}
873 \land [\baseof{L} = M \lor \baseof{L} = \baseof{M}] \\
874 R \in \pn : & M = \baseof{L} \\
875 \text{otherwise} : & \false
878 \[ \eqn{ Merge Acyclic }{
883 \[ \eqn{ Removal Merge Ends }{
884 X \not\haspatch \p \land
888 \pendsof{Y}{\py} = \pendsof{M}{\py}
890 \[ \eqn{ Addition Merge Ends }{
891 X \not\haspatch \p \land
895 \bigforall_{E \in \pendsof{X}{\py}} E \le Y
898 \[ \eqn{ Foreign Merges }{
899 \patchof{L} = \bot \equiv \patchof{R} = \bot
902 \subsection{Non-Topbloke merges}
904 We require both $\patchof{L} = \bot$ and $\patchof{R} = \bot$
905 (Foreign Merges, above).
906 I.e. not only is it forbidden to merge into a Topbloke-controlled
907 branch without Topbloke's assistance, it is also forbidden to
908 merge any Topbloke-controlled branch into any plain git branch.
910 Given those conditions, Tip Merge and Merge Acyclic do not apply.
911 And $Y \not\in \py$ so $\neg [ Y \haspatch \p ]$ so neither
912 Merge Ends condition applies.
914 So a plain git merge of non-Topbloke branches meets the conditions and
915 is therefore consistent with our model.
917 \subsection{No Replay}
919 By definition of $\merge$,
920 $D \isin C \implies D \isin L \lor D \isin R \lor D = C$.
922 Ingredients Prevent Replay applies. $\qed$
924 \subsection{Unique Base}
926 Need to consider only $C \in \py$, ie $L \in \py$,
927 and calculate $\pendsof{C}{\pn}$. So we will consider some
928 putative ancestor $A \in \pn$ and see whether $A \le C$.
930 By Exact Ancestors for C, $A \le C \equiv A \le L \lor A \le R \lor A = C$.
931 But $C \in py$ and $A \in \pn$ so $A \neq C$.
932 Thus $A \le C \equiv A \le L \lor A \le R$.
934 By Unique Base of L and Transitive Ancestors,
935 $A \le L \equiv A \le \baseof{L}$.
937 \subsubsection{For $R \in \py$:}
939 By Unique Base of $R$ and Transitive Ancestors,
940 $A \le R \equiv A \le \baseof{R}$.
942 But by Tip Merge condition on $\baseof{R}$,
943 $A \le \baseof{L} \implies A \le \baseof{R}$, so
944 $A \le \baseof{R} \lor A \le \baseof{L} \equiv A \le \baseof{R}$.
945 Thus $A \le C \equiv A \le \baseof{R}$.
946 That is, $\baseof{C} = \baseof{R}$.
948 \subsubsection{For $R \in \pn$:}
950 By Tip Merge condition on $R$ and since $M \le R$,
951 $A \le \baseof{L} \implies A \le R$, so
952 $A \le R \lor A \le \baseof{L} \equiv A \le R$.
953 Thus $A \le C \equiv A \le R$.
954 That is, $\baseof{C} = R$.
958 \subsection{Coherence and Patch Inclusion}
960 Need to determine $C \haspatch \p$ based on $L,M,R \haspatch \p$.
961 This involves considering $D \in \py$.
963 \subsubsection{For $L \nothaspatch \p, R \nothaspatch \p$:}
964 $D \not\isin L \land D \not\isin R$. $C \not\in \py$ (otherwise $L
965 \in \py$ ie $L \haspatch \p$ by Tip Self Inpatch for $L$). So $D \neq C$.
966 Applying $\merge$ gives $D \not\isin C$ i.e. $C \nothaspatch \p$.
968 \subsubsection{For $L \haspatch \p, R \haspatch \p$:}
969 $D \isin L \equiv D \le L$ and $D \isin R \equiv D \le R$.
970 (Likewise $D \isin X \equiv D \le X$ and $D \isin Y \equiv D \le Y$.)
972 Consider $D = C$: $D \isin C$, $D \le C$, OK for $C \haspatch \p$.
974 For $D \neq C$: $D \le C \equiv D \le L \lor D \le R
975 \equiv D \isin L \lor D \isin R$.
976 (Likewise $D \le C \equiv D \le X \lor D \le Y$.)
978 Consider $D \neq C, D \isin X \land D \isin Y$:
979 By $\merge$, $D \isin C$. Also $D \le X$
980 so $D \le C$. OK for $C \haspatch \p$.
982 Consider $D \neq C, D \not\isin X \land D \not\isin Y$:
983 By $\merge$, $D \not\isin C$.
984 And $D \not\le X \land D \not\le Y$ so $D \not\le C$.
985 OK for $C \haspatch \p$.
987 Remaining case, wlog, is $D \not\isin X \land D \isin Y$.
988 $D \not\le X$ so $D \not\le M$ so $D \not\isin M$.
989 Thus by $\merge$, $D \isin C$. And $D \le Y$ so $D \le C$.
990 OK for $C \haspatch \p$.
992 So indeed $L \haspatch \p \land R \haspatch \p \implies C \haspatch \p$.
994 \subsubsection{For (wlog) $X \not\haspatch \p, Y \haspatch \p$:}
996 $M \haspatch \p \implies C \nothaspatch \p$.
997 $M \nothaspatch \p \implies C \haspatch \p$.
1001 One of the Merge Ends conditions applies.
1002 Recall that we are considering $D \in \py$.
1003 $D \isin Y \equiv D \le Y$. $D \not\isin X$.
1004 We will show for each of
1005 various cases that $D \isin C \equiv M \nothaspatch \p \land D \le C$
1006 (which suffices by definition of $\haspatch$ and $\nothaspatch$).
1008 Consider $D = C$: Thus $C \in \py, L \in \py$, and by Tip
1009 Self Inpatch for $L$, $L \haspatch \p$, so $L=Y, R=X$. By Tip Merge,
1010 $M=\baseof{L}$. So by Base Acyclic $D \not\isin M$, i.e.
1011 $M \nothaspatch \p$. And indeed $D \isin C$ and $D \le C$. OK.
1013 Consider $D \neq C, M \nothaspatch P, D \isin Y$:
1014 $D \le Y$ so $D \le C$.
1015 $D \not\isin M$ so by $\merge$, $D \isin C$. OK.
1017 Consider $D \neq C, M \nothaspatch P, D \not\isin Y$:
1018 $D \not\le Y$. If $D \le X$ then
1019 $D \in \pancsof{X}{\py}$, so by Addition Merge Ends and
1020 Transitive Ancestors $D \le Y$ --- a contradiction, so $D \not\le X$.
1021 Thus $D \not\le C$. By $\merge$, $D \not\isin C$. OK.
1023 Consider $D \neq C, M \haspatch P, D \isin Y$:
1024 $D \le Y$ so $D \in \pancsof{Y}{\py}$ so by Removal Merge Ends
1025 and Transitive Ancestors $D \in \pancsof{M}{\py}$ so $D \le M$.
1026 Thus $D \isin M$. By $\merge$, $D \not\isin C$. OK.
1028 Consider $D \neq C, M \haspatch P, D \not\isin Y$:
1029 By $\merge$, $D \not\isin C$. OK.
1033 \subsection{Base Acyclic}
1035 This applies when $C \in \pn$.
1036 $C \in \pn$ when $L \in \pn$ so by Merge Acyclic, $R \nothaspatch \p$.
1038 Consider some $D \in \py$.
1040 By Base Acyclic of $L$, $D \not\isin L$. By the above, $D \not\isin
1041 R$. And $D \neq C$. So $D \not\isin C$.
1045 \subsection{Tip Contents}
1047 We need worry only about $C \in \py$.
1048 And $\patchof{C} = \patchof{L}$
1049 so $L \in \py$ so $L \haspatch \p$. We will use the Unique Base
1050 of $C$, and its Coherence and Patch Inclusion, as just proved.
1052 Firstly we show $C \haspatch \p$: If $R \in \py$, then $R \haspatch
1053 \p$ and by Coherence/Inclusion $C \haspatch \p$ . If $R \not\in \py$
1054 then by Tip Merge $M = \baseof{L}$ so by Base Acyclic and definition
1055 of $\nothaspatch$, $M \nothaspatch \p$. So by Coherence/Inclusion $C
1056 \haspatch \p$ (whether $R \haspatch \p$ or $\nothaspatch$).
1058 We will consider an arbitrary commit $D$
1059 and prove the Exclusive Tip Contents form.
1061 \subsubsection{For $D \in \py$:}
1062 $C \haspatch \p$ so by definition of $\haspatch$, $D \isin C \equiv D
1065 \subsubsection{For $D \not\in \py, R \not\in \py$:}
1067 $D \neq C$. By Tip Contents of $L$,
1068 $D \isin L \equiv D \isin \baseof{L}$, and by Tip Merge condition,
1069 $D \isin L \equiv D \isin M$. So by definition of $\merge$, $D \isin
1070 C \equiv D \isin R$. And $R = \baseof{C}$ by Unique Base of $C$.
1071 Thus $D \isin C \equiv D \isin \baseof{C}$. OK.
1073 \subsubsection{For $D \not\in \py, R \in \py$:}
1078 $D \isin L \equiv D \isin \baseof{L}$ and
1079 $D \isin R \equiv D \isin \baseof{R}$.
1081 If $\baseof{L} = M$, trivially $D \isin M \equiv D \isin \baseof{L}.$
1082 Whereas if $\baseof{L} = \baseof{M}$, by definition of $\base$,
1083 $\patchof{M} = \patchof{L} = \py$, so by Tip Contents of $M$,
1084 $D \isin M \equiv D \isin \baseof{M} \equiv D \isin \baseof{L}$.
1086 So $D \isin M \equiv D \isin L$ and by $\merge$,
1087 $D \isin C \equiv D \isin R$. But from Unique Base,
1088 $\baseof{C} = R$ so $D \isin C \equiv D \isin \baseof{C}$. OK.
1092 \subsection{Foreign Inclusion}
1094 Consider some $D$ s.t. $\patchof{D} = \bot$.
1095 By Foreign Inclusion of $L, M, R$:
1096 $D \isin L \equiv D \le L$;
1097 $D \isin M \equiv D \le M$;
1098 $D \isin R \equiv D \le R$.
1100 \subsubsection{For $D = C$:}
1102 $D \isin C$ and $D \le C$. OK.
1104 \subsubsection{For $D \neq C, D \isin M$:}
1106 Thus $D \le M$ so $D \le L$ and $D \le R$ so $D \isin L$ and $D \isin
1107 R$. So by $\merge$, $D \isin C$. And $D \le C$. OK.
1109 \subsubsection{For $D \neq C, D \not\isin M, D \isin X$:}
1111 By $\merge$, $D \isin C$.
1112 And $D \isin X$ means $D \le X$ so $D \le C$.
1115 \subsubsection{For $D \neq C, D \not\isin M, D \not\isin L, D \not\isin R$:}
1117 By $\merge$, $D \not\isin C$.
1118 And $D \not\le L, D \not\le R$ so $D \not\le C$.
1123 \subsection{Foreign Contents}
1125 Only relevant if $\patchof{L} = \bot$, in which case
1126 $\patchof{C} = \bot$ and by Foreign Merges $\patchof{R} = \bot$,
1127 so Totally Foreign Contents applies. $\qed$