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{ Inserted's Ends }{
818 E \in \pendsof{L}{\pry} \implies E \le R^+
820 \[ \eqn{ Others' Ends }{
821 \bigforall_{\p \neq \pr, L \haspatch \p}
822 E \in \pendsof{R^+}{\py} \implies E \le L
824 \[ \eqn{ Insertion Acyclic }{
828 \subsection{No Replay}
831 $D \isin C \implies D \isin L \lor D \isin R^+ \lor D = C$.
832 So Ingredients Prevent Replay applies. $\qed$
834 \subsection{Unique Base}
838 \subsection{Tip Contents}
842 \subsection{Base Acyclic}
844 Consider some $D \isin C$. We will show that $D \not\in \pqy$.
845 By $\merge$, $D \isin L \lor D \isin R^+ \lor D = C$.
847 For $D \isin L$, Base Acyclic for L suffices. For $D \isin R^+$,
848 Insertion Acyclic suffices. For $D = C$, trivial. $\qed$.
850 \subsection{Coherence and Patch Inclusion}
854 \p = \pr \lor L \haspatch \p : & C \haspatch \p \\
855 \p \neq \pr \land L \nothaspatch \p : & C \nothaspatch \p
860 ~ Consider some $D \in \py$.
861 $D \neq C$ so $D \le C \equiv D \le L \lor D \le R^+$.
863 \subsubsection{For $\p = \pr$:}
865 $D \not\isin L$ by Currently Excluded.
866 $D \not\isin R^-$ by Base Acyclic.
867 So by $\merge$, $D \isin C \equiv D \isin R^+$,
868 which by Tip Self Inpatch of $R^+$, $\equiv D \le R^+$.
870 And by definition of $\pancs$,
871 $D \le L \equiv D \in \pancsof{L}{R^+}$.
872 Applying Transitive Ancestors to Inserted's Ends gives
873 $A \in \pancsof{L}{R^+} \implies A \le R^+$.
874 So $D \le L \implies D \le R^+$.
875 Thus $D \le C \equiv D \le R^+$.
877 So $D \isin C \equiv D \le C$, i.e. $C \haspatch \pr$.
880 \subsubsection{For $\p \neq \pr$:}
884 By Insertion Acyclic, $D \not\isin R^+$. xxx this is wrong
885 By Tip Contents for $R^+$,
886 $D \isin R^+ \equiv D \isin R^- \lor (D \in \pry \land ...)$
887 but $D \in \py$ so $D \not\in \pry$. So $D \not\isin R^-$.
888 By $\merge$, $D \isin C \equiv D \isin L$.
890 If $L \nothaspatch \p$, $D \not\isin L$ so $C \nothaspatch \p$. OK.
892 If $L \haspatch \p$, Others' Ends applies; by Transitive
893 Ancestors, $A \in \pancsof{R^+}{\py} \implies A \le L$.
894 So $D \le R^+$, which is the same as $D \in \pancsof{R^+}{\py}$,
895 $\implies D \le L$. Thus $D le C \equiv D \le L$.
896 And by $\haspatch$, $D \le L \equiv D \isin L$ so
897 $D \isin C \equiv D \le C$. Thus $C \haspatch \p$.
904 Merge commits $L$ and $R$ using merge base $M$:
906 C \hasparents \{ L, R \}
908 \patchof{C} = \patchof{L}
912 We will occasionally use $X,Y$ s.t. $\{X,Y\} = \{L,R\}$.
914 \subsection{Conditions}
915 \[ \eqn{ Ingredients }{
918 \[ \eqn{ Tip Merge }{
921 R \in \py : & \baseof{R} \ge \baseof{L}
922 \land [\baseof{L} = M \lor \baseof{L} = \baseof{M}] \\
923 R \in \pn : & M = \baseof{L} \\
924 \text{otherwise} : & \false
927 \[ \eqn{ Merge Acyclic }{
932 \[ \eqn{ Removal Merge Ends }{
933 X \not\haspatch \p \land
937 \pendsof{Y}{\py} = \pendsof{M}{\py}
939 \[ \eqn{ Addition Merge Ends }{
940 X \not\haspatch \p \land
944 \bigforall_{E \in \pendsof{X}{\py}} E \le Y
947 \[ \eqn{ Foreign Merges }{
948 \patchof{L} = \bot \equiv \patchof{R} = \bot
951 \subsection{Non-Topbloke merges}
953 We require both $\patchof{L} = \bot$ and $\patchof{R} = \bot$
954 (Foreign Merges, above).
955 I.e. not only is it forbidden to merge into a Topbloke-controlled
956 branch without Topbloke's assistance, it is also forbidden to
957 merge any Topbloke-controlled branch into any plain git branch.
959 Given those conditions, Tip Merge and Merge Acyclic do not apply.
960 And $Y \not\in \py$ so $\neg [ Y \haspatch \p ]$ so neither
961 Merge Ends condition applies.
963 So a plain git merge of non-Topbloke branches meets the conditions and
964 is therefore consistent with our model.
966 \subsection{No Replay}
968 By definition of $\merge$,
969 $D \isin C \implies D \isin L \lor D \isin R \lor D = C$.
971 Ingredients Prevent Replay applies. $\qed$
973 \subsection{Unique Base}
975 Need to consider only $C \in \py$, ie $L \in \py$,
976 and calculate $\pendsof{C}{\pn}$. So we will consider some
977 putative ancestor $A \in \pn$ and see whether $A \le C$.
979 By Exact Ancestors for C, $A \le C \equiv A \le L \lor A \le R \lor A = C$.
980 But $C \in py$ and $A \in \pn$ so $A \neq C$.
981 Thus $A \le C \equiv A \le L \lor A \le R$.
983 By Unique Base of L and Transitive Ancestors,
984 $A \le L \equiv A \le \baseof{L}$.
986 \subsubsection{For $R \in \py$:}
988 By Unique Base of $R$ and Transitive Ancestors,
989 $A \le R \equiv A \le \baseof{R}$.
991 But by Tip Merge condition on $\baseof{R}$,
992 $A \le \baseof{L} \implies A \le \baseof{R}$, so
993 $A \le \baseof{R} \lor A \le \baseof{L} \equiv A \le \baseof{R}$.
994 Thus $A \le C \equiv A \le \baseof{R}$.
995 That is, $\baseof{C} = \baseof{R}$.
997 \subsubsection{For $R \in \pn$:}
999 By Tip Merge condition on $R$ and since $M \le R$,
1000 $A \le \baseof{L} \implies A \le R$, so
1001 $A \le R \lor A \le \baseof{L} \equiv A \le R$.
1002 Thus $A \le C \equiv A \le R$.
1003 That is, $\baseof{C} = R$.
1007 \subsection{Coherence and Patch Inclusion}
1009 Need to determine $C \haspatch \p$ based on $L,M,R \haspatch \p$.
1010 This involves considering $D \in \py$.
1012 \subsubsection{For $L \nothaspatch \p, R \nothaspatch \p$:}
1013 $D \not\isin L \land D \not\isin R$. $C \not\in \py$ (otherwise $L
1014 \in \py$ ie $L \haspatch \p$ by Tip Self Inpatch for $L$). So $D \neq C$.
1015 Applying $\merge$ gives $D \not\isin C$ i.e. $C \nothaspatch \p$.
1017 \subsubsection{For $L \haspatch \p, R \haspatch \p$:}
1018 $D \isin L \equiv D \le L$ and $D \isin R \equiv D \le R$.
1019 (Likewise $D \isin X \equiv D \le X$ and $D \isin Y \equiv D \le Y$.)
1021 Consider $D = C$: $D \isin C$, $D \le C$, OK for $C \haspatch \p$.
1023 For $D \neq C$: $D \le C \equiv D \le L \lor D \le R
1024 \equiv D \isin L \lor D \isin R$.
1025 (Likewise $D \le C \equiv D \le X \lor D \le Y$.)
1027 Consider $D \neq C, D \isin X \land D \isin Y$:
1028 By $\merge$, $D \isin C$. Also $D \le X$
1029 so $D \le C$. OK for $C \haspatch \p$.
1031 Consider $D \neq C, D \not\isin X \land D \not\isin Y$:
1032 By $\merge$, $D \not\isin C$.
1033 And $D \not\le X \land D \not\le Y$ so $D \not\le C$.
1034 OK for $C \haspatch \p$.
1036 Remaining case, wlog, is $D \not\isin X \land D \isin Y$.
1037 $D \not\le X$ so $D \not\le M$ so $D \not\isin M$.
1038 Thus by $\merge$, $D \isin C$. And $D \le Y$ so $D \le C$.
1039 OK for $C \haspatch \p$.
1041 So indeed $L \haspatch \p \land R \haspatch \p \implies C \haspatch \p$.
1043 \subsubsection{For (wlog) $X \not\haspatch \p, Y \haspatch \p$:}
1045 $M \haspatch \p \implies C \nothaspatch \p$.
1046 $M \nothaspatch \p \implies C \haspatch \p$.
1050 One of the Merge Ends conditions applies.
1051 Recall that we are considering $D \in \py$.
1052 $D \isin Y \equiv D \le Y$. $D \not\isin X$.
1053 We will show for each of
1054 various cases that $D \isin C \equiv M \nothaspatch \p \land D \le C$
1055 (which suffices by definition of $\haspatch$ and $\nothaspatch$).
1057 Consider $D = C$: Thus $C \in \py, L \in \py$, and by Tip
1058 Self Inpatch for $L$, $L \haspatch \p$, so $L=Y, R=X$. By Tip Merge,
1059 $M=\baseof{L}$. So by Base Acyclic $D \not\isin M$, i.e.
1060 $M \nothaspatch \p$. And indeed $D \isin C$ and $D \le C$. OK.
1062 Consider $D \neq C, M \nothaspatch P, D \isin Y$:
1063 $D \le Y$ so $D \le C$.
1064 $D \not\isin M$ so by $\merge$, $D \isin C$. OK.
1066 Consider $D \neq C, M \nothaspatch P, D \not\isin Y$:
1067 $D \not\le Y$. If $D \le X$ then
1068 $D \in \pancsof{X}{\py}$, so by Addition Merge Ends and
1069 Transitive Ancestors $D \le Y$ --- a contradiction, so $D \not\le X$.
1070 Thus $D \not\le C$. By $\merge$, $D \not\isin C$. OK.
1072 Consider $D \neq C, M \haspatch P, D \isin Y$:
1073 $D \le Y$ so $D \in \pancsof{Y}{\py}$ so by Removal Merge Ends
1074 and Transitive Ancestors $D \in \pancsof{M}{\py}$ so $D \le M$.
1075 Thus $D \isin M$. By $\merge$, $D \not\isin C$. OK.
1077 Consider $D \neq C, M \haspatch P, D \not\isin Y$:
1078 By $\merge$, $D \not\isin C$. OK.
1082 \subsection{Base Acyclic}
1084 This applies when $C \in \pn$.
1085 $C \in \pn$ when $L \in \pn$ so by Merge Acyclic, $R \nothaspatch \p$.
1087 Consider some $D \in \py$.
1089 By Base Acyclic of $L$, $D \not\isin L$. By the above, $D \not\isin
1090 R$. And $D \neq C$. So $D \not\isin C$.
1094 \subsection{Tip Contents}
1096 We need worry only about $C \in \py$.
1097 And $\patchof{C} = \patchof{L}$
1098 so $L \in \py$ so $L \haspatch \p$. We will use the Unique Base
1099 of $C$, and its Coherence and Patch Inclusion, as just proved.
1101 Firstly we show $C \haspatch \p$: If $R \in \py$, then $R \haspatch
1102 \p$ and by Coherence/Inclusion $C \haspatch \p$ . If $R \not\in \py$
1103 then by Tip Merge $M = \baseof{L}$ so by Base Acyclic and definition
1104 of $\nothaspatch$, $M \nothaspatch \p$. So by Coherence/Inclusion $C
1105 \haspatch \p$ (whether $R \haspatch \p$ or $\nothaspatch$).
1107 We will consider an arbitrary commit $D$
1108 and prove the Exclusive Tip Contents form.
1110 \subsubsection{For $D \in \py$:}
1111 $C \haspatch \p$ so by definition of $\haspatch$, $D \isin C \equiv D
1114 \subsubsection{For $D \not\in \py, R \not\in \py$:}
1116 $D \neq C$. By Tip Contents of $L$,
1117 $D \isin L \equiv D \isin \baseof{L}$, and by Tip Merge condition,
1118 $D \isin L \equiv D \isin M$. So by definition of $\merge$, $D \isin
1119 C \equiv D \isin R$. And $R = \baseof{C}$ by Unique Base of $C$.
1120 Thus $D \isin C \equiv D \isin \baseof{C}$. OK.
1122 \subsubsection{For $D \not\in \py, R \in \py$:}
1127 $D \isin L \equiv D \isin \baseof{L}$ and
1128 $D \isin R \equiv D \isin \baseof{R}$.
1130 If $\baseof{L} = M$, trivially $D \isin M \equiv D \isin \baseof{L}.$
1131 Whereas if $\baseof{L} = \baseof{M}$, by definition of $\base$,
1132 $\patchof{M} = \patchof{L} = \py$, so by Tip Contents of $M$,
1133 $D \isin M \equiv D \isin \baseof{M} \equiv D \isin \baseof{L}$.
1135 So $D \isin M \equiv D \isin L$ and by $\merge$,
1136 $D \isin C \equiv D \isin R$. But from Unique Base,
1137 $\baseof{C} = R$ so $D \isin C \equiv D \isin \baseof{C}$. OK.
1141 \subsection{Foreign Inclusion}
1143 Consider some $D$ s.t. $\patchof{D} = \bot$.
1144 By Foreign Inclusion of $L, M, R$:
1145 $D \isin L \equiv D \le L$;
1146 $D \isin M \equiv D \le M$;
1147 $D \isin R \equiv D \le R$.
1149 \subsubsection{For $D = C$:}
1151 $D \isin C$ and $D \le C$. OK.
1153 \subsubsection{For $D \neq C, D \isin M$:}
1155 Thus $D \le M$ so $D \le L$ and $D \le R$ so $D \isin L$ and $D \isin
1156 R$. So by $\merge$, $D \isin C$. And $D \le C$. OK.
1158 \subsubsection{For $D \neq C, D \not\isin M, D \isin X$:}
1160 By $\merge$, $D \isin C$.
1161 And $D \isin X$ means $D \le X$ so $D \le C$.
1164 \subsubsection{For $D \neq C, D \not\isin M, D \not\isin L, D \not\isin R$:}
1166 By $\merge$, $D \not\isin C$.
1167 And $D \not\le L, D \not\le R$ so $D \not\le C$.
1172 \subsection{Foreign Contents}
1174 Only relevant if $\patchof{L} = \bot$, in which case
1175 $\patchof{C} = \bot$ and by Foreign Merges $\patchof{R} = \bot$,
1176 so Totally Foreign Contents applies. $\qed$