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}
662 Given $L$ and $\pr$ as represented by $R^+, R^-$.
663 Construct $C$ which has $\pr$ removed.
664 Used for removing a branch dependency.
666 C \hasparents \{ L \}
668 \patchof{C} = \patchof{L}
670 \mergeof{C}{L}{R^+}{R^-}
673 \subsection{Conditions}
675 \[ \eqn{ Ingredients }{
676 R^+ \in \pry \land R^- = \baseof{R^+}
678 \[ \eqn{ Into Base }{
681 \[ \eqn{ Unique Tip }{
682 \pendsof{L}{\pry} = \{ R^+ \}
684 \[ \eqn{ Currently Included }{
688 \subsection{Ordering of Ingredients:}
690 By Unique Tip, $R^+ \le L$. By definition of $\base$, $R^- \le R^+$
691 so $R^- \le L$. So $R^+ \le C$ and $R^- \le C$.
694 (Note that $R^+ \not\le R^-$, i.e. the merge base
695 is a descendant, not an ancestor, of the 2nd parent.)
697 \subsection{No Replay}
699 By definition of $\merge$,
700 $D \isin C \implies D \isin L \lor D \isin R^- \lor D = C$.
701 So, by Ordering of Ingredients,
702 Ingredients Prevent Replay applies. $\qed$
704 \subsection{Desired Contents}
706 \[ D \isin C \equiv [ D \notin \pry \land D \isin L ] \lor D = C \]
709 \subsubsection{For $D = C$:}
711 Trivially $D \isin C$. OK.
713 \subsubsection{For $D \neq C, D \not\le L$:}
715 By No Replay $D \not\isin L$. Also $D \not\le R^-$ hence
716 $D \not\isin R^-$. Thus $D \not\isin C$. OK.
718 \subsubsection{For $D \neq C, D \le L, D \in \pry$:}
720 By Currently Included, $D \isin L$.
722 By Tip Self Inpatch for $R^+$, $D \isin R^+ \equiv D \le R^+$, but by
723 by Unique Tip, $D \le R^+ \equiv D \le L$.
726 By Base Acyclic, $D \not\isin R^-$.
728 Apply $\merge$: $D \not\isin C$. OK.
730 \subsubsection{For $D \neq C, D \le L, D \notin \pry$:}
732 By Tip Contents for $R^+$, $D \isin R^+ \equiv D \isin R^-$.
734 Apply $\merge$: $D \isin C \equiv D \isin L$. OK.
738 \subsection{Unique Base}
740 Into Base means that $C \in \pn$, so Unique Base is not
743 \subsection{Tip Contents}
745 Again, not applicable. $\qed$
747 \subsection{Base Acyclic}
749 By Base Acyclic for $L$, $D \isin L \implies D \not\in \py$.
750 And by Into Base $C \not\in \py$.
751 Now from Desired Contents, above, $D \isin C
752 \implies D \isin L \lor D = C$, which thus
753 $\implies D \not\in \py$. $\qed$.
755 \subsection{Coherence and Patch Inclusion}
757 Need to consider some $D \in \py$. By Into Base, $D \neq C$.
759 \subsubsection{For $\p = \pr$:}
760 By Desired Contents, above, $D \not\isin C$.
761 So $C \nothaspatch \pr$.
763 \subsubsection{For $\p \neq \pr$:}
764 By Desired Contents, $D \isin C \equiv D \isin L$
765 (since $D \in \py$ so $D \not\in \pry$).
767 If $L \nothaspatch \p$, $D \not\isin L$ so $D \not\isin C$.
768 So $L \nothaspatch \p \implies C \nothaspatch \p$.
770 Whereas if $L \haspatch \p$, $D \isin L \equiv D \le L$.
771 so $L \haspatch \p \implies C \haspatch \p$.
775 \subsection{Foreign Inclusion}
777 Consider some $D$ s.t. $\patchof{D} = \bot$. $D \neq C$.
778 So by Desired Contents $D \isin C \equiv D \isin L$.
779 By Foreign Inclusion of $D$ in $L$, $D \isin L \equiv D \le L$.
781 And $D \le C \equiv D \le L$.
782 Thus $D \isin C \equiv D \le C$.
786 \subsection{Foreign Contents}
792 Merge commits $L$ and $R$ using merge base $M$:
794 C \hasparents \{ L, R \}
796 \patchof{C} = \patchof{L}
800 We will occasionally use $X,Y$ s.t. $\{X,Y\} = \{L,R\}$.
802 \subsection{Conditions}
803 \[ \eqn{ Ingredients }{
806 \[ \eqn{ Tip Merge }{
809 R \in \py : & \baseof{R} \ge \baseof{L}
810 \land [\baseof{L} = M \lor \baseof{L} = \baseof{M}] \\
811 R \in \pn : & M = \baseof{L} \\
812 \text{otherwise} : & \false
815 \[ \eqn{ Merge Acyclic }{
820 \[ \eqn{ Removal Merge Ends }{
821 X \not\haspatch \p \land
825 \pendsof{Y}{\py} = \pendsof{M}{\py}
827 \[ \eqn{ Addition Merge Ends }{
828 X \not\haspatch \p \land
832 \bigforall_{E \in \pendsof{X}{\py}} E \le Y
835 \[ \eqn{ Foreign Merges }{
836 \patchof{L} = \bot \equiv \patchof{R} = \bot
839 \subsection{Non-Topbloke merges}
841 We require both $\patchof{L} = \bot$ and $\patchof{R} = \bot$
842 (Foreign Merges, above).
843 I.e. not only is it forbidden to merge into a Topbloke-controlled
844 branch without Topbloke's assistance, it is also forbidden to
845 merge any Topbloke-controlled branch into any plain git branch.
847 Given those conditions, Tip Merge and Merge Acyclic do not apply.
848 And $Y \not\in \py$ so $\neg [ Y \haspatch \p ]$ so neither
849 Merge Ends condition applies.
851 So a plain git merge of non-Topbloke branches meets the conditions and
852 is therefore consistent with our model.
854 \subsection{No Replay}
856 By definition of $\merge$,
857 $D \isin C \implies D \isin L \lor D \isin R \lor D = C$.
859 Ingredients Prevent Replay applies. $\qed$
861 \subsection{Unique Base}
863 Need to consider only $C \in \py$, ie $L \in \py$,
864 and calculate $\pendsof{C}{\pn}$. So we will consider some
865 putative ancestor $A \in \pn$ and see whether $A \le C$.
867 By Exact Ancestors for C, $A \le C \equiv A \le L \lor A \le R \lor A = C$.
868 But $C \in py$ and $A \in \pn$ so $A \neq C$.
869 Thus $A \le C \equiv A \le L \lor A \le R$.
871 By Unique Base of L and Transitive Ancestors,
872 $A \le L \equiv A \le \baseof{L}$.
874 \subsubsection{For $R \in \py$:}
876 By Unique Base of $R$ and Transitive Ancestors,
877 $A \le R \equiv A \le \baseof{R}$.
879 But by Tip Merge condition on $\baseof{R}$,
880 $A \le \baseof{L} \implies A \le \baseof{R}$, so
881 $A \le \baseof{R} \lor A \le \baseof{L} \equiv A \le \baseof{R}$.
882 Thus $A \le C \equiv A \le \baseof{R}$.
883 That is, $\baseof{C} = \baseof{R}$.
885 \subsubsection{For $R \in \pn$:}
887 By Tip Merge condition on $R$ and since $M \le R$,
888 $A \le \baseof{L} \implies A \le R$, so
889 $A \le R \lor A \le \baseof{L} \equiv A \le R$.
890 Thus $A \le C \equiv A \le R$.
891 That is, $\baseof{C} = R$.
895 \subsection{Coherence and Patch Inclusion}
897 Need to determine $C \haspatch \p$ based on $L,M,R \haspatch \p$.
898 This involves considering $D \in \py$.
900 \subsubsection{For $L \nothaspatch \p, R \nothaspatch \p$:}
901 $D \not\isin L \land D \not\isin R$. $C \not\in \py$ (otherwise $L
902 \in \py$ ie $L \haspatch \p$ by Tip Self Inpatch for $L$). So $D \neq C$.
903 Applying $\merge$ gives $D \not\isin C$ i.e. $C \nothaspatch \p$.
905 \subsubsection{For $L \haspatch \p, R \haspatch \p$:}
906 $D \isin L \equiv D \le L$ and $D \isin R \equiv D \le R$.
907 (Likewise $D \isin X \equiv D \le X$ and $D \isin Y \equiv D \le Y$.)
909 Consider $D = C$: $D \isin C$, $D \le C$, OK for $C \haspatch \p$.
911 For $D \neq C$: $D \le C \equiv D \le L \lor D \le R
912 \equiv D \isin L \lor D \isin R$.
913 (Likewise $D \le C \equiv D \le X \lor D \le Y$.)
915 Consider $D \neq C, D \isin X \land D \isin Y$:
916 By $\merge$, $D \isin C$. Also $D \le X$
917 so $D \le C$. OK for $C \haspatch \p$.
919 Consider $D \neq C, D \not\isin X \land D \not\isin Y$:
920 By $\merge$, $D \not\isin C$.
921 And $D \not\le X \land D \not\le Y$ so $D \not\le C$.
922 OK for $C \haspatch \p$.
924 Remaining case, wlog, is $D \not\isin X \land D \isin Y$.
925 $D \not\le X$ so $D \not\le M$ so $D \not\isin M$.
926 Thus by $\merge$, $D \isin C$. And $D \le Y$ so $D \le C$.
927 OK for $C \haspatch \p$.
929 So indeed $L \haspatch \p \land R \haspatch \p \implies C \haspatch \p$.
931 \subsubsection{For (wlog) $X \not\haspatch \p, Y \haspatch \p$:}
933 $M \haspatch \p \implies C \nothaspatch \p$.
934 $M \nothaspatch \p \implies C \haspatch \p$.
938 One of the Merge Ends conditions applies.
939 Recall that we are considering $D \in \py$.
940 $D \isin Y \equiv D \le Y$. $D \not\isin X$.
941 We will show for each of
942 various cases that $D \isin C \equiv M \nothaspatch \p \land D \le C$
943 (which suffices by definition of $\haspatch$ and $\nothaspatch$).
945 Consider $D = C$: Thus $C \in \py, L \in \py$, and by Tip
946 Self Inpatch for $L$, $L \haspatch \p$, so $L=Y, R=X$. By Tip Merge,
947 $M=\baseof{L}$. So by Base Acyclic $D \not\isin M$, i.e.
948 $M \nothaspatch \p$. And indeed $D \isin C$ and $D \le C$. OK.
950 Consider $D \neq C, M \nothaspatch P, D \isin Y$:
951 $D \le Y$ so $D \le C$.
952 $D \not\isin M$ so by $\merge$, $D \isin C$. OK.
954 Consider $D \neq C, M \nothaspatch P, D \not\isin Y$:
955 $D \not\le Y$. If $D \le X$ then
956 $D \in \pancsof{X}{\py}$, so by Addition Merge Ends and
957 Transitive Ancestors $D \le Y$ --- a contradiction, so $D \not\le X$.
958 Thus $D \not\le C$. By $\merge$, $D \not\isin C$. OK.
960 Consider $D \neq C, M \haspatch P, D \isin Y$:
961 $D \le Y$ so $D \in \pancsof{Y}{\py}$ so by Removal Merge Ends
962 and Transitive Ancestors $D \in \pancsof{M}{\py}$ so $D \le M$.
963 Thus $D \isin M$. By $\merge$, $D \not\isin C$. OK.
965 Consider $D \neq C, M \haspatch P, D \not\isin Y$:
966 By $\merge$, $D \not\isin C$. OK.
970 \subsection{Base Acyclic}
972 This applies when $C \in \pn$.
973 $C \in \pn$ when $L \in \pn$ so by Merge Acyclic, $R \nothaspatch \p$.
975 Consider some $D \in \py$.
977 By Base Acyclic of $L$, $D \not\isin L$. By the above, $D \not\isin
978 R$. And $D \neq C$. So $D \not\isin C$.
982 \subsection{Tip Contents}
984 We need worry only about $C \in \py$.
985 And $\patchof{C} = \patchof{L}$
986 so $L \in \py$ so $L \haspatch \p$. We will use the Unique Base
987 of $C$, and its Coherence and Patch Inclusion, as just proved.
989 Firstly we show $C \haspatch \p$: If $R \in \py$, then $R \haspatch
990 \p$ and by Coherence/Inclusion $C \haspatch \p$ . If $R \not\in \py$
991 then by Tip Merge $M = \baseof{L}$ so by Base Acyclic and definition
992 of $\nothaspatch$, $M \nothaspatch \p$. So by Coherence/Inclusion $C
993 \haspatch \p$ (whether $R \haspatch \p$ or $\nothaspatch$).
995 We will consider an arbitrary commit $D$
996 and prove the Exclusive Tip Contents form.
998 \subsubsection{For $D \in \py$:}
999 $C \haspatch \p$ so by definition of $\haspatch$, $D \isin C \equiv D
1002 \subsubsection{For $D \not\in \py, R \not\in \py$:}
1004 $D \neq C$. By Tip Contents of $L$,
1005 $D \isin L \equiv D \isin \baseof{L}$, and by Tip Merge condition,
1006 $D \isin L \equiv D \isin M$. So by definition of $\merge$, $D \isin
1007 C \equiv D \isin R$. And $R = \baseof{C}$ by Unique Base of $C$.
1008 Thus $D \isin C \equiv D \isin \baseof{C}$. OK.
1010 \subsubsection{For $D \not\in \py, R \in \py$:}
1015 $D \isin L \equiv D \isin \baseof{L}$ and
1016 $D \isin R \equiv D \isin \baseof{R}$.
1018 If $\baseof{L} = M$, trivially $D \isin M \equiv D \isin \baseof{L}.$
1019 Whereas if $\baseof{L} = \baseof{M}$, by definition of $\base$,
1020 $\patchof{M} = \patchof{L} = \py$, so by Tip Contents of $M$,
1021 $D \isin M \equiv D \isin \baseof{M} \equiv D \isin \baseof{L}$.
1023 So $D \isin M \equiv D \isin L$ and by $\merge$,
1024 $D \isin C \equiv D \isin R$. But from Unique Base,
1025 $\baseof{C} = R$ so $D \isin C \equiv D \isin \baseof{C}$. OK.
1029 \subsection{Foreign Inclusion}
1031 Consider some $D$ s.t. $\patchof{D} = \bot$.
1032 By Foreign Inclusion of $L, M, R$:
1033 $D \isin L \equiv D \le L$;
1034 $D \isin M \equiv D \le M$;
1035 $D \isin R \equiv D \le R$.
1037 \subsubsection{For $D = C$:}
1039 $D \isin C$ and $D \le C$. OK.
1041 \subsubsection{For $D \neq C, D \isin M$:}
1043 Thus $D \le M$ so $D \le L$ and $D \le R$ so $D \isin L$ and $D \isin
1044 R$. So by $\merge$, $D \isin C$. And $D \le C$. OK.
1046 \subsubsection{For $D \neq C, D \not\isin M, D \isin X$:}
1048 By $\merge$, $D \isin C$.
1049 And $D \isin X$ means $D \le X$ so $D \le C$.
1052 \subsubsection{For $D \neq C, D \not\isin M, D \not\isin L, D \not\isin R$:}
1054 By $\merge$, $D \not\isin C$.
1055 And $D \not\le L, D \not\le R$ so $D \not\le C$.
1060 \subsection{Foreign Contents}
1062 Only relevant if $\patchof{L} = \bot$, in which case
1063 $\patchof{C} = \bot$ and by Foreign Merges $\patchof{R} = \bot$,
1064 so Totally Foreign Contents applies. $\qed$