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}
794 Merge commits $L$ and $R$ using merge base $M$:
796 C \hasparents \{ L, R \}
798 \patchof{C} = \patchof{L}
802 We will occasionally use $X,Y$ s.t. $\{X,Y\} = \{L,R\}$.
804 \subsection{Conditions}
805 \[ \eqn{ Ingredients }{
808 \[ \eqn{ Tip Merge }{
811 R \in \py : & \baseof{R} \ge \baseof{L}
812 \land [\baseof{L} = M \lor \baseof{L} = \baseof{M}] \\
813 R \in \pn : & M = \baseof{L} \\
814 \text{otherwise} : & \false
817 \[ \eqn{ Merge Acyclic }{
822 \[ \eqn{ Removal Merge Ends }{
823 X \not\haspatch \p \land
827 \pendsof{Y}{\py} = \pendsof{M}{\py}
829 \[ \eqn{ Addition Merge Ends }{
830 X \not\haspatch \p \land
834 \bigforall_{E \in \pendsof{X}{\py}} E \le Y
837 \[ \eqn{ Foreign Merges }{
838 \patchof{L} = \bot \equiv \patchof{R} = \bot
841 \subsection{Non-Topbloke merges}
843 We require both $\patchof{L} = \bot$ and $\patchof{R} = \bot$
844 (Foreign Merges, above).
845 I.e. not only is it forbidden to merge into a Topbloke-controlled
846 branch without Topbloke's assistance, it is also forbidden to
847 merge any Topbloke-controlled branch into any plain git branch.
849 Given those conditions, Tip Merge and Merge Acyclic do not apply.
850 And $Y \not\in \py$ so $\neg [ Y \haspatch \p ]$ so neither
851 Merge Ends condition applies.
853 So a plain git merge of non-Topbloke branches meets the conditions and
854 is therefore consistent with our model.
856 \subsection{No Replay}
858 By definition of $\merge$,
859 $D \isin C \implies D \isin L \lor D \isin R \lor D = C$.
861 Ingredients Prevent Replay applies. $\qed$
863 \subsection{Unique Base}
865 Need to consider only $C \in \py$, ie $L \in \py$,
866 and calculate $\pendsof{C}{\pn}$. So we will consider some
867 putative ancestor $A \in \pn$ and see whether $A \le C$.
869 By Exact Ancestors for C, $A \le C \equiv A \le L \lor A \le R \lor A = C$.
870 But $C \in py$ and $A \in \pn$ so $A \neq C$.
871 Thus $A \le C \equiv A \le L \lor A \le R$.
873 By Unique Base of L and Transitive Ancestors,
874 $A \le L \equiv A \le \baseof{L}$.
876 \subsubsection{For $R \in \py$:}
878 By Unique Base of $R$ and Transitive Ancestors,
879 $A \le R \equiv A \le \baseof{R}$.
881 But by Tip Merge condition on $\baseof{R}$,
882 $A \le \baseof{L} \implies A \le \baseof{R}$, so
883 $A \le \baseof{R} \lor A \le \baseof{L} \equiv A \le \baseof{R}$.
884 Thus $A \le C \equiv A \le \baseof{R}$.
885 That is, $\baseof{C} = \baseof{R}$.
887 \subsubsection{For $R \in \pn$:}
889 By Tip Merge condition on $R$ and since $M \le R$,
890 $A \le \baseof{L} \implies A \le R$, so
891 $A \le R \lor A \le \baseof{L} \equiv A \le R$.
892 Thus $A \le C \equiv A \le R$.
893 That is, $\baseof{C} = R$.
897 \subsection{Coherence and Patch Inclusion}
899 Need to determine $C \haspatch \p$ based on $L,M,R \haspatch \p$.
900 This involves considering $D \in \py$.
902 \subsubsection{For $L \nothaspatch \p, R \nothaspatch \p$:}
903 $D \not\isin L \land D \not\isin R$. $C \not\in \py$ (otherwise $L
904 \in \py$ ie $L \haspatch \p$ by Tip Self Inpatch for $L$). So $D \neq C$.
905 Applying $\merge$ gives $D \not\isin C$ i.e. $C \nothaspatch \p$.
907 \subsubsection{For $L \haspatch \p, R \haspatch \p$:}
908 $D \isin L \equiv D \le L$ and $D \isin R \equiv D \le R$.
909 (Likewise $D \isin X \equiv D \le X$ and $D \isin Y \equiv D \le Y$.)
911 Consider $D = C$: $D \isin C$, $D \le C$, OK for $C \haspatch \p$.
913 For $D \neq C$: $D \le C \equiv D \le L \lor D \le R
914 \equiv D \isin L \lor D \isin R$.
915 (Likewise $D \le C \equiv D \le X \lor D \le Y$.)
917 Consider $D \neq C, D \isin X \land D \isin Y$:
918 By $\merge$, $D \isin C$. Also $D \le X$
919 so $D \le C$. OK for $C \haspatch \p$.
921 Consider $D \neq C, D \not\isin X \land D \not\isin Y$:
922 By $\merge$, $D \not\isin C$.
923 And $D \not\le X \land D \not\le Y$ so $D \not\le C$.
924 OK for $C \haspatch \p$.
926 Remaining case, wlog, is $D \not\isin X \land D \isin Y$.
927 $D \not\le X$ so $D \not\le M$ so $D \not\isin M$.
928 Thus by $\merge$, $D \isin C$. And $D \le Y$ so $D \le C$.
929 OK for $C \haspatch \p$.
931 So indeed $L \haspatch \p \land R \haspatch \p \implies C \haspatch \p$.
933 \subsubsection{For (wlog) $X \not\haspatch \p, Y \haspatch \p$:}
935 $M \haspatch \p \implies C \nothaspatch \p$.
936 $M \nothaspatch \p \implies C \haspatch \p$.
940 One of the Merge Ends conditions applies.
941 Recall that we are considering $D \in \py$.
942 $D \isin Y \equiv D \le Y$. $D \not\isin X$.
943 We will show for each of
944 various cases that $D \isin C \equiv M \nothaspatch \p \land D \le C$
945 (which suffices by definition of $\haspatch$ and $\nothaspatch$).
947 Consider $D = C$: Thus $C \in \py, L \in \py$, and by Tip
948 Self Inpatch for $L$, $L \haspatch \p$, so $L=Y, R=X$. By Tip Merge,
949 $M=\baseof{L}$. So by Base Acyclic $D \not\isin M$, i.e.
950 $M \nothaspatch \p$. And indeed $D \isin C$ and $D \le C$. OK.
952 Consider $D \neq C, M \nothaspatch P, D \isin Y$:
953 $D \le Y$ so $D \le C$.
954 $D \not\isin M$ so by $\merge$, $D \isin C$. OK.
956 Consider $D \neq C, M \nothaspatch P, D \not\isin Y$:
957 $D \not\le Y$. If $D \le X$ then
958 $D \in \pancsof{X}{\py}$, so by Addition Merge Ends and
959 Transitive Ancestors $D \le Y$ --- a contradiction, so $D \not\le X$.
960 Thus $D \not\le C$. By $\merge$, $D \not\isin C$. OK.
962 Consider $D \neq C, M \haspatch P, D \isin Y$:
963 $D \le Y$ so $D \in \pancsof{Y}{\py}$ so by Removal Merge Ends
964 and Transitive Ancestors $D \in \pancsof{M}{\py}$ so $D \le M$.
965 Thus $D \isin M$. By $\merge$, $D \not\isin C$. OK.
967 Consider $D \neq C, M \haspatch P, D \not\isin Y$:
968 By $\merge$, $D \not\isin C$. OK.
972 \subsection{Base Acyclic}
974 This applies when $C \in \pn$.
975 $C \in \pn$ when $L \in \pn$ so by Merge Acyclic, $R \nothaspatch \p$.
977 Consider some $D \in \py$.
979 By Base Acyclic of $L$, $D \not\isin L$. By the above, $D \not\isin
980 R$. And $D \neq C$. So $D \not\isin C$.
984 \subsection{Tip Contents}
986 We need worry only about $C \in \py$.
987 And $\patchof{C} = \patchof{L}$
988 so $L \in \py$ so $L \haspatch \p$. We will use the Unique Base
989 of $C$, and its Coherence and Patch Inclusion, as just proved.
991 Firstly we show $C \haspatch \p$: If $R \in \py$, then $R \haspatch
992 \p$ and by Coherence/Inclusion $C \haspatch \p$ . If $R \not\in \py$
993 then by Tip Merge $M = \baseof{L}$ so by Base Acyclic and definition
994 of $\nothaspatch$, $M \nothaspatch \p$. So by Coherence/Inclusion $C
995 \haspatch \p$ (whether $R \haspatch \p$ or $\nothaspatch$).
997 We will consider an arbitrary commit $D$
998 and prove the Exclusive Tip Contents form.
1000 \subsubsection{For $D \in \py$:}
1001 $C \haspatch \p$ so by definition of $\haspatch$, $D \isin C \equiv D
1004 \subsubsection{For $D \not\in \py, R \not\in \py$:}
1006 $D \neq C$. By Tip Contents of $L$,
1007 $D \isin L \equiv D \isin \baseof{L}$, and by Tip Merge condition,
1008 $D \isin L \equiv D \isin M$. So by definition of $\merge$, $D \isin
1009 C \equiv D \isin R$. And $R = \baseof{C}$ by Unique Base of $C$.
1010 Thus $D \isin C \equiv D \isin \baseof{C}$. OK.
1012 \subsubsection{For $D \not\in \py, R \in \py$:}
1017 $D \isin L \equiv D \isin \baseof{L}$ and
1018 $D \isin R \equiv D \isin \baseof{R}$.
1020 If $\baseof{L} = M$, trivially $D \isin M \equiv D \isin \baseof{L}.$
1021 Whereas if $\baseof{L} = \baseof{M}$, by definition of $\base$,
1022 $\patchof{M} = \patchof{L} = \py$, so by Tip Contents of $M$,
1023 $D \isin M \equiv D \isin \baseof{M} \equiv D \isin \baseof{L}$.
1025 So $D \isin M \equiv D \isin L$ and by $\merge$,
1026 $D \isin C \equiv D \isin R$. But from Unique Base,
1027 $\baseof{C} = R$ so $D \isin C \equiv D \isin \baseof{C}$. OK.
1031 \subsection{Foreign Inclusion}
1033 Consider some $D$ s.t. $\patchof{D} = \bot$.
1034 By Foreign Inclusion of $L, M, R$:
1035 $D \isin L \equiv D \le L$;
1036 $D \isin M \equiv D \le M$;
1037 $D \isin R \equiv D \le R$.
1039 \subsubsection{For $D = C$:}
1041 $D \isin C$ and $D \le C$. OK.
1043 \subsubsection{For $D \neq C, D \isin M$:}
1045 Thus $D \le M$ so $D \le L$ and $D \le R$ so $D \isin L$ and $D \isin
1046 R$. So by $\merge$, $D \isin C$. And $D \le C$. OK.
1048 \subsubsection{For $D \neq C, D \not\isin M, D \isin X$:}
1050 By $\merge$, $D \isin C$.
1051 And $D \isin X$ means $D \le X$ so $D \le C$.
1054 \subsubsection{For $D \neq C, D \not\isin M, D \not\isin L, D \not\isin R$:}
1056 By $\merge$, $D \not\isin C$.
1057 And $D \not\le L, D \not\le R$ so $D \not\le C$.
1062 \subsection{Foreign Contents}
1064 Only relevant if $\patchof{L} = \bot$, in which case
1065 $\patchof{C} = \bot$ and by Foreign Merges $\patchof{R} = \bot$,
1066 so Totally Foreign Contents applies. $\qed$