1 \documentclass[a4paper,leqno]{strayman}
3 \let\numberwithin=\notdef
11 \renewcommand{\ge}{\geqslant}
12 \renewcommand{\le}{\leqslant}
13 \newcommand{\nge}{\ngeqslant}
14 \newcommand{\nle}{\nleqslant}
16 \newcommand{\has}{\sqsupseteq}
17 \newcommand{\isin}{\sqsubseteq}
19 \newcommand{\nothaspatch}{\mathrel{\,\not\!\not\relax\haspatch}}
20 \newcommand{\notpatchisin}{\mathrel{\,\not\!\not\relax\patchisin}}
21 \newcommand{\haspatch}{\sqSupset}
22 \newcommand{\patchisin}{\sqSubset}
24 \newif\ifhidehack\hidehackfalse
25 \DeclareRobustCommand\hidefromedef[2]{%
26 \hidehacktrue\ifhidehack#1\else#2\fi\hidehackfalse}
27 \newcommand{\pa}[1]{\hidefromedef{\varmathbb{#1}}{#1}}
29 \newcommand{\set}[1]{\mathbb{#1}}
30 \newcommand{\pay}[1]{\pa{#1}^+}
31 \newcommand{\pan}[1]{\pa{#1}^-}
33 \newcommand{\p}{\pa{P}}
34 \newcommand{\py}{\pay{P}}
35 \newcommand{\pn}{\pan{P}}
37 \newcommand{\pr}{\pa{R}}
38 \newcommand{\pry}{\pay{R}}
39 \newcommand{\prn}{\pan{R}}
41 %\newcommand{\hasparents}{\underaccent{1}{>}}
42 %\newcommand{\hasparents}{{%
43 % \declareslashed{}{_{_1}}{0}{-0.8}{>}\slashed{>}}}
44 \newcommand{\hasparents}{>_{\mkern-7.0mu _1}}
45 \newcommand{\areparents}{<_{\mkern-14.0mu _1\mkern+5.0mu}}
47 \renewcommand{\implies}{\Rightarrow}
48 \renewcommand{\equiv}{\Leftrightarrow}
49 \renewcommand{\nequiv}{\nLeftrightarrow}
50 \renewcommand{\land}{\wedge}
51 \renewcommand{\lor}{\vee}
53 \newcommand{\pancs}{{\mathcal A}}
54 \newcommand{\pends}{{\mathcal E}}
56 \newcommand{\pancsof}[2]{\pancs ( #1 , #2 ) }
57 \newcommand{\pendsof}[2]{\pends ( #1 , #2 ) }
59 \newcommand{\merge}{{\mathcal M}}
60 \newcommand{\mergeof}[4]{\merge(#1,#2,#3,#4)}
61 %\newcommand{\merge}[4]{{#2 {{\frac{ #1 }{ #3 } #4}}}}
63 \newcommand{\patch}{{\mathcal P}}
64 \newcommand{\base}{{\mathcal B}}
66 \newcommand{\patchof}[1]{\patch ( #1 ) }
67 \newcommand{\baseof}[1]{\base ( #1 ) }
69 \newcommand{\eqntag}[2]{ #2 \tag*{\mbox{#1}} }
70 \newcommand{\eqn}[2]{ #2 \tag*{\mbox{\bf #1}} }
72 %\newcommand{\bigforall}{\mathop{\hbox{\huge$\forall$}}}
73 \newcommand{\bigforall}{%
75 {\hbox{\huge$\forall$}}%
76 {\hbox{\Large$\forall$}}%
77 {\hbox{\normalsize$\forall$}}%
78 {\hbox{\scriptsize$\forall$}}}%
81 \newcommand{\Largeexists}{\mathop{\hbox{\Large$\exists$}}}
82 \newcommand{\Largenexists}{\mathop{\hbox{\Large$\nexists$}}}
84 \newcommand{\qed}{\square}
85 \newcommand{\proofstarts}{{\it Proof:}}
86 \newcommand{\proof}[1]{\proofstarts #1 $\qed$}
88 \newcommand{\gathbegin}{\begin{gather} \tag*{}}
89 \newcommand{\gathnext}{\\ \tag*{}}
92 \newcommand{\false}{f}
100 \desclabelstyle{\nextlinelabel}
102 \item[ $ C \hasparents \set X $ ]
103 The parents of commit $C$ are exactly the set
107 $C$ is a descendant of $D$ in the git commit
108 graph. This is a partial order, namely the transitive closure of
109 $ D \in \set X $ where $ C \hasparents \set X $.
111 \item[ $ C \has D $ ]
112 Informally, the tree at commit $C$ contains the change
113 made in commit $D$. Does not take account of deliberate reversions by
114 the user or reversion, rebasing or rewinding in
115 non-Topbloke-controlled branches. For merges and Topbloke-generated
116 anticommits or re-commits, the ``change made'' is only to be thought
117 of as any conflict resolution. This is not a partial order because it
120 \item[ $ \p, \py, \pn $ ]
121 A patch $\p$ consists of two sets of commits $\pn$ and $\py$, which
122 are respectively the base and tip git branches. $\p$ may be used
123 where the context requires a set, in which case the statement
124 is to be taken as applying to both $\py$ and $\pn$.
125 None of these sets overlap. Hence:
127 \item[ $ \patchof{ C } $ ]
128 Either $\p$ s.t. $ C \in \p $, or $\bot$.
129 A function from commits to patches' sets $\p$.
131 \item[ $ \pancsof{C}{\set P} $ ]
132 $ \{ A \; | \; A \le C \land A \in \set P \} $
133 i.e. all the ancestors of $C$
134 which are in $\set P$.
136 \item[ $ \pendsof{C}{\set P} $ ]
137 $ \{ E \; | \; E \in \pancsof{C}{\set P}
138 \land \mathop{\not\exists}_{A \in \pancsof{C}{\set P}}
139 E \neq A \land E \le A \} $
140 i.e. all $\le$-maximal commits in $\pancsof{C}{\set P}$.
142 \item[ $ \baseof{C} $ ]
143 $ \pendsof{C}{\pn} = \{ \baseof{C} \} $ where $ C \in \py $.
144 A partial function from commits to commits.
145 See Unique Base, below.
147 \item[ $ C \haspatch \p $ ]
148 $\displaystyle \bigforall_{D \in \py} D \isin C \equiv D \le C $.
149 ~ Informally, $C$ has the contents of $\p$.
151 \item[ $ C \nothaspatch \p $ ]
152 $\displaystyle \bigforall_{D \in \py} D \not\isin C $.
153 ~ Informally, $C$ has none of the contents of $\p$.
155 Non-Topbloke commits are $\nothaspatch \p$ for all $\p$. This
156 includes commits on plain git branches made by applying a Topbloke
158 patch is applied to a non-Topbloke branch and then bubbles back to
159 the relevant Topbloke branches, we hope that
160 if the user still cares about the Topbloke patch,
161 git's merge algorithm will DTRT when trying to re-apply the changes.
163 \item[ $\displaystyle \mergeof{C}{L}{M}{R} $ ]
164 The contents of a git merge result:
166 $\displaystyle D \isin C \equiv
168 (D \isin L \land D \isin R) \lor D = C : & \true \\
169 (D \not\isin L \land D \not\isin R) \land D \neq C : & \false \\
170 \text{otherwise} : & D \not\isin M
178 We maintain these each time we construct a new commit. \\
180 C \has D \implies C \ge D
182 \[\eqn{Unique Base:}{
183 \bigforall_{C \in \py} \pendsof{C}{\pn} = \{ B \}
185 \[\eqn{Tip Contents:}{
186 \bigforall_{C \in \py} D \isin C \equiv
187 { D \isin \baseof{C} \lor \atop
188 (D \in \py \land D \le C) }
190 \[\eqn{Base Acyclic:}{
191 \bigforall_{B \in \pn} D \isin B \implies D \notin \py
194 \bigforall_{C,\p} C \haspatch \p \lor C \nothaspatch \p
196 \[\eqn{Foreign Inclusion:}{
197 \bigforall_{D \text{ s.t. } \patchof{D} = \bot} D \isin C \equiv D \leq C
199 \[\eqn{Foreign Contents:}{
200 \bigforall_{C \text{ s.t. } \patchof{C} = \bot}
201 D \le C \implies \patchof{D} = \bot
204 \section{Some lemmas}
206 \[ \eqn{Alternative (overlapping) formulations defining
207 $\mergeof{C}{L}{M}{R}$:}{
210 D \isin L \equiv D \isin R : & D = C \lor D \isin L \\
211 D \isin L \nequiv D \isin R : & D = C \lor D \not\isin M \\
212 D \isin L \equiv D \isin M : & D = C \lor D \isin R \\
213 D \isin L \nequiv D \isin M : & D = C \lor D \isin L \\
214 \text{as above with L and R exchanged}
220 Original definition is symmetrical in $L$ and $R$.
223 \[ \eqn{Exclusive Tip Contents:}{
224 \bigforall_{C \in \py}
225 \neg \Bigl[ D \isin \baseof{C} \land ( D \in \py \land D \le C )
228 Ie, the two limbs of the RHS of Tip Contents are mutually exclusive.
231 Let $B = \baseof{C}$ in $D \isin \baseof{C}$. Now $B \in \pn$.
232 So by Base Acyclic $D \isin B \implies D \notin \py$.
234 \[ \eqntag{{\it Corollary - equivalent to Tip Contents}}{
235 \bigforall_{C \in \py} D \isin C \equiv
237 D \in \py : & D \le C \\
238 D \not\in \py : & D \isin \baseof{C}
242 \[ \eqn{Tip Self Inpatch:}{
243 \bigforall_{C \in \py} C \haspatch \p
245 Ie, tip commits contain their own patch.
248 Apply Exclusive Tip Contents to some $D \in \py$:
249 $ \bigforall_{C \in \py}\bigforall_{D \in \py}
250 D \isin C \equiv D \le C $
253 \[ \eqn{Exact Ancestors:}{
254 \bigforall_{ C \hasparents \set{R} }
256 ( \mathop{\hbox{\huge{$\vee$}}}_{R \in \set R} D \le R )
261 \[ \eqn{Transitive Ancestors:}{
262 \left[ \bigforall_{ E \in \pendsof{C}{\set P} } E \le M \right] \equiv
263 \left[ \bigforall_{ A \in \pancsof{C}{\set P} } A \le M \right]
267 The implication from right to left is trivial because
268 $ \pends() \subset \pancs() $.
269 For the implication from left to right:
270 by the definition of $\mathcal E$,
271 for every such $A$, either $A \in \pends()$ which implies
272 $A \le M$ by the LHS directly,
273 or $\exists_{A' \in \pancs()} \; A' \neq A \land A \le A' $
274 in which case we repeat for $A'$. Since there are finitely many
275 commits, this terminates with $A'' \in \pends()$, ie $A'' \le M$
276 by the LHS. And $A \le A''$.
279 \[ \eqn{Calculation Of Ends:}{
280 \bigforall_{C \hasparents \set A}
281 \pendsof{C}{\set P} =
285 C \not\in \p : & \displaystyle
287 \Bigl[ \Largeexists_{A \in \set A}
288 E \in \pendsof{A}{\set P} \Bigr] \land
289 \Bigl[ \Largenexists_{B \in \set A}
290 E \neq B \land E \le B \Bigr]
296 \[ \eqn{Ingredients Prevent Replay:}{
298 {C \hasparents \set A} \land
303 \Largeexists_{A \in \set A} D \isin A
305 \right] \implies \left[
306 D \isin C \implies D \le C
310 Trivial for $D = C$. Consider some $D \neq C$, $D \isin C$.
311 By the preconditions, there is some $A$ s.t. $D \in \set A$
312 and $D \isin A$. By No Replay for $A$, $D \le A$. And
313 $A \le C$ so $D \le C$.
316 \[ \eqn{Totally Foreign Contents:}{
317 \bigforall_{C \hasparents \set A}
319 \patchof{C} = \bot \land
320 \bigforall_{A \in \set A} \patchof{A} = \bot
330 Consider some $D \le C$. If $D = C$, $\patchof{D} = \bot$ trivially.
331 If $D \neq C$ then $D \le A$ where $A \in \set A$. By Foreign
332 Contents of $A$, $\patchof{D} = \bot$.
335 \section{Commit annotation}
337 We annotate each Topbloke commit $C$ with:
341 \baseof{C}, \text{ if } C \in \py
344 \text{ either } C \haspatch \pa{Q} \text{ or } C \nothaspatch \pa{Q}
346 \bigforall_{\pay{Q} \not\ni C} \pendsof{C}{\pay{Q}}
349 $\patchof{C}$, for each kind of Topbloke-generated commit, is stated
350 in the summary in the section for that kind of commit.
352 Whether $\baseof{C}$ is required, and if so what the value is, is
353 stated in the proof of Unique Base for each kind of commit.
355 $C \haspatch \pa{Q}$ or $\nothaspatch \pa{Q}$ is represented as the
356 set $\{ \pa{Q} | C \haspatch \pa{Q} \}$. Whether $C \haspatch \pa{Q}$
358 (in terms of $I \haspatch \pa{Q}$ or $I \nothaspatch \pa{Q}$
359 for the ingredients $I$),
360 in the proof of Coherence for each kind of commit.
362 $\pendsof{C}{\pa{Q}^+}$ is computed, for all Topbloke-generated commits,
363 using the lemma Calculation of Ends, above.
364 We do not annotate $\pendsof{C}{\py}$ for $C \in \py$. Doing so would
365 make it wrong to make plain commits with git because the recorded $\pends$
366 would have to be updated. The annotation is not needed in that case
367 because $\forall_{\py \ni C} \; \pendsof{C}{\py} = \{C\}$.
369 \section{Simple commit}
371 A simple single-parent forward commit $C$ as made by git-commit.
373 \tag*{} C \hasparents \{ A \} \\
374 \tag*{} \patchof{C} = \patchof{A} \\
375 \tag*{} D \isin C \equiv D \isin A \lor D = C
377 This also covers Topbloke-generated commits on plain git branches:
378 Topbloke strips the metadata when exporting.
380 \subsection{No Replay}
382 Ingredients Prevent Replay applies. $\qed$
384 \subsection{Unique Base}
385 If $A, C \in \py$ then by Calculation of Ends for
386 $C, \py, C \not\in \py$:
387 $\pendsof{C}{\pn} = \pendsof{A}{\pn}$ so
388 $\baseof{C} = \baseof{A}$. $\qed$
390 \subsection{Tip Contents}
391 We need to consider only $A, C \in \py$. From Tip Contents for $A$:
392 \[ D \isin A \equiv D \isin \baseof{A} \lor ( D \in \py \land D \le A ) \]
393 Substitute into the contents of $C$:
394 \[ D \isin C \equiv D \isin \baseof{A} \lor ( D \in \py \land D \le A )
396 Since $D = C \implies D \in \py$,
397 and substituting in $\baseof{C}$, this gives:
398 \[ D \isin C \equiv D \isin \baseof{C} \lor
399 (D \in \py \land D \le A) \lor
400 (D = C \land D \in \py) \]
401 \[ \equiv D \isin \baseof{C} \lor
402 [ D \in \py \land ( D \le A \lor D = C ) ] \]
403 So by Exact Ancestors:
404 \[ D \isin C \equiv D \isin \baseof{C} \lor ( D \in \py \land D \le C
408 \subsection{Base Acyclic}
410 Need to consider only $A, C \in \pn$.
412 For $D = C$: $D \in \pn$ so $D \not\in \py$. OK.
414 For $D \neq C$: $D \isin C \equiv D \isin A$, so by Base Acyclic for
415 $A$, $D \isin C \implies D \not\in \py$.
419 \subsection{Coherence and patch inclusion}
421 Need to consider $D \in \py$
423 \subsubsection{For $A \haspatch P, D = C$:}
429 $ D \isin C \equiv \ldots \lor \true \text{ so } D \haspatch C $.
431 \subsubsection{For $A \haspatch P, D \neq C$:}
432 Ancestors: $ D \le C \equiv D \le A $.
434 Contents: $ D \isin C \equiv D \isin A \lor f $
435 so $ D \isin C \equiv D \isin A $.
438 \[ A \haspatch P \implies C \haspatch P \]
440 \subsubsection{For $A \nothaspatch P$:}
442 Firstly, $C \not\in \py$ since if it were, $A \in \py$.
445 Now by contents of $A$, $D \notin A$, so $D \notin C$.
448 \[ A \nothaspatch P \implies C \nothaspatch P \]
451 \subsection{Foreign inclusion:}
453 If $D = C$, trivial. For $D \neq C$:
454 $D \isin C \equiv D \isin A \equiv D \le A \equiv D \le C$. $\qed$
456 \subsection{Foreign Contents:}
458 Only relevant if $\patchof{C} = \bot$, and in that case Totally
459 Foreign Contents applies. $\qed$
461 \section{Create Base}
463 Given $L$, create a Topbloke base branch initial commit $B$.
465 B \hasparents \{ L \}
467 \patchof{B} = \pan{Q}
469 D \isin B \equiv D \isin L \lor D = B
472 \subsection{Conditions}
474 \[ \eqn{ Ingredients }{
475 \patchof{L} = \pa{L} \lor \patchof{L} = \bot
477 \[ \eqn{ Non-recursion }{
481 \subsection{No Replay}
483 Ingredients Prevent Replay applies. $\qed$
485 \subsection{Unique Base}
489 \subsection{Tip Contents}
493 \subsection{Base Acyclic}
495 Consider some $D \isin B$. If $D = B$, $D \in \pn$, OK.
497 If $D \neq B$, $D \isin L$. By No Replay of $D$ in $L$, $D \le L$.
498 Thus by Foreign Contents of $L$, $\patchof{D} = \bot$. OK.
504 \subsection{Coherence and Patch Inclusion}
506 Consider some $D \in \p$.
507 $B \not\in \py$ so $D \neq B$. So $D \isin B \equiv D \isin L$.
509 Thus $L \haspatch \p \implies B \haspatch P$
510 and $L \nothaspatch \p \implies B \nothaspatch P$.
514 \subsection{Foreign Inclusion}
516 Consider some $D$ s.t. $\patchof{D} = \bot$. $D \neq B$
517 so $D \isin B \equiv D \isin L$.
518 By Foreign Inclusion of $D$ in $L$, $D \isin L \equiv D \le L$.
519 And by Exact Ancestors $D \le L \equiv D \le B$.
520 So $D \isin B \equiv D \le B$. $\qed$
522 \subsection{Foreign Contents}
532 Given $L$ and $\pr$ as represented by $R^+, R^-$.
533 Construct $C$ which has $\pr$ removed.
534 Used for removing a branch dependency.
536 C \hasparents \{ L \}
538 \patchof{C} = \patchof{L}
540 \mergeof{C}{L}{R^+}{R^-}
543 \subsection{Conditions}
545 \[ \eqn{ Ingredients }{
546 R^+ \in \pry \land R^- = \baseof{R^+}
548 \[ \eqn{ Into Base }{
551 \[ \eqn{ Unique Tip }{
552 \pendsof{L}{\pry} = \{ R^+ \}
554 \[ \eqn{ Currently Included }{
558 \subsection{Ordering of Ingredients:}
560 By Unique Tip, $R^+ \le L$. By definition of $\base$, $R^- \le R^+$
561 so $R^- \le L$. So $R^+ \le C$ and $R^- \le C$.
564 (Note that $R^+ \not\le R^-$, i.e. the merge base
565 is a descendant, not an ancestor, of the 2nd parent.)
567 \subsection{No Replay}
569 By definition of $\merge$,
570 $D \isin C \implies D \isin L \lor D \isin R^- \lor D = C$.
571 So, by Ordering of Ingredients,
572 Ingredients Prevent Replay applies. $\qed$
574 \subsection{Desired Contents}
576 \[ D \isin C \equiv [ D \notin \pry \land D \isin L ] \lor D = C \]
579 \subsubsection{For $D = C$:}
581 Trivially $D \isin C$. OK.
583 \subsubsection{For $D \neq C, D \not\le L$:}
585 By No Replay $D \not\isin L$. Also $D \not\le R^-$ hence
586 $D \not\isin R^-$. Thus $D \not\isin C$. OK.
588 \subsubsection{For $D \neq C, D \le L, D \in \pry$:}
590 By Currently Included, $D \isin L$.
592 By Tip Self Inpatch, $D \isin R^+ \equiv D \le R^+$, but by
593 by Unique Tip, $D \le R^+ \equiv D \le L$.
596 By Base Acyclic, $D \not\isin R^-$.
598 Apply $\merge$: $D \not\isin C$. OK.
600 \subsubsection{For $D \neq C, D \le L, D \notin \pry$:}
602 By Tip Contents for $R^+$, $D \isin R^+ \equiv D \isin R^-$.
604 Apply $\merge$: $D \isin C \equiv D \isin L$. OK.
608 \subsection{Unique Base}
610 Into Base means that $C \in \pn$, so Unique Base is not
613 \subsection{Tip Contents}
615 Again, not applicable. $\qed$
617 \subsection{Base Acyclic}
619 By Base Acyclic for $L$, $D \isin L \implies D \not\in \py$.
620 And by Into Base $C \not\in \py$.
621 Now from Desired Contents, above, $D \isin C
622 \implies D \isin L \lor D = C$, which thus
623 $\implies D \not\in \py$. $\qed$.
625 \subsection{Coherence and Patch Inclusion}
627 Need to consider some $D \in \py$. By Into Base, $D \neq C$.
629 \subsubsection{For $\p = \pr$:}
630 By Desired Contents, above, $D \not\isin C$.
631 So $C \nothaspatch \pr$.
633 \subsubsection{For $\p \neq \pr$:}
634 By Desired Contents, $D \isin C \equiv D \isin L$
635 (since $D \in \py$ so $D \not\in \pry$).
637 If $L \nothaspatch \p$, $D \not\isin L$ so $D \not\isin C$.
638 So $L \nothaspatch \p \implies C \nothaspatch \p$.
640 Whereas if $L \haspatch \p$, $D \isin L \equiv D \le L$.
641 so $L \haspatch \p \implies C \haspatch \p$.
645 \subsection{Foreign Inclusion}
647 Consider some $D$ s.t. $\patchof{D} = \bot$. $D \neq C$.
648 So by Desired Contents $D \isin C \equiv D \isin L$.
649 By Foreign Inclusion of $D$ in $L$, $D \isin L \equiv D \le L$.
651 And $D \le C \equiv D \le L$.
652 Thus $D \isin C \equiv D \le C$.
656 \subsection{Foreign Contents}
662 Merge commits $L$ and $R$ using merge base $M$:
664 C \hasparents \{ L, R \}
666 \patchof{C} = \patchof{L}
670 We will occasionally use $X,Y$ s.t. $\{X,Y\} = \{L,R\}$.
672 \subsection{Conditions}
673 \[ \eqn{ Ingredients }{
676 \[ \eqn{ Tip Merge }{
679 R \in \py : & \baseof{R} \ge \baseof{L}
680 \land [\baseof{L} = M \lor \baseof{L} = \baseof{M}] \\
681 R \in \pn : & M = \baseof{L} \\
682 \text{otherwise} : & \false
685 \[ \eqn{ Merge Acyclic }{
690 \[ \eqn{ Removal Merge Ends }{
691 X \not\haspatch \p \land
695 \pendsof{Y}{\py} = \pendsof{M}{\py}
697 \[ \eqn{ Addition Merge Ends }{
698 X \not\haspatch \p \land
702 \bigforall_{E \in \pendsof{X}{\py}} E \le Y
705 \[ \eqn{ Foreign Merges }{
706 \patchof{L} = \bot \equiv \patchof{R} = \bot
709 \subsection{Non-Topbloke merges}
711 We require both $\patchof{L} = \bot$ and $\patchof{R} = \bot$
712 (Foreign Merges, above).
713 I.e. not only is it forbidden to merge into a Topbloke-controlled
714 branch without Topbloke's assistance, it is also forbidden to
715 merge any Topbloke-controlled branch into any plain git branch.
717 Given those conditions, Tip Merge and Merge Acyclic do not apply.
718 And $Y \not\in \py$ so $\neg [ Y \haspatch \p ]$ so neither
719 Merge Ends condition applies.
721 So a plain git merge of non-Topbloke branches meets the conditions and
722 is therefore consistent with our scheme.
724 \subsection{No Replay}
726 By definition of $\merge$,
727 $D \isin C \implies D \isin L \lor D \isin R \lor D = C$.
729 Ingredients Prevent Replay applies. $\qed$
731 \subsection{Unique Base}
733 Need to consider only $C \in \py$, ie $L \in \py$,
734 and calculate $\pendsof{C}{\pn}$. So we will consider some
735 putative ancestor $A \in \pn$ and see whether $A \le C$.
737 By Exact Ancestors for C, $A \le C \equiv A \le L \lor A \le R \lor A = C$.
738 But $C \in py$ and $A \in \pn$ so $A \neq C$.
739 Thus $A \le C \equiv A \le L \lor A \le R$.
741 By Unique Base of L and Transitive Ancestors,
742 $A \le L \equiv A \le \baseof{L}$.
744 \subsubsection{For $R \in \py$:}
746 By Unique Base of $R$ and Transitive Ancestors,
747 $A \le R \equiv A \le \baseof{R}$.
749 But by Tip Merge condition on $\baseof{R}$,
750 $A \le \baseof{L} \implies A \le \baseof{R}$, so
751 $A \le \baseof{R} \lor A \le \baseof{L} \equiv A \le \baseof{R}$.
752 Thus $A \le C \equiv A \le \baseof{R}$.
753 That is, $\baseof{C} = \baseof{R}$.
755 \subsubsection{For $R \in \pn$:}
757 By Tip Merge condition on $R$ and since $M \le R$,
758 $A \le \baseof{L} \implies A \le R$, so
759 $A \le R \lor A \le \baseof{L} \equiv A \le R$.
760 Thus $A \le C \equiv A \le R$.
761 That is, $\baseof{C} = R$.
765 \subsection{Coherence and Patch Inclusion}
767 Need to determine $C \haspatch \p$ based on $L,M,R \haspatch \p$.
768 This involves considering $D \in \py$.
770 \subsubsection{For $L \nothaspatch \p, R \nothaspatch \p$:}
771 $D \not\isin L \land D \not\isin R$. $C \not\in \py$ (otherwise $L
772 \in \py$ ie $L \haspatch \p$ by Tip Self Inpatch). So $D \neq C$.
773 Applying $\merge$ gives $D \not\isin C$ i.e. $C \nothaspatch \p$.
775 \subsubsection{For $L \haspatch \p, R \haspatch \p$:}
776 $D \isin L \equiv D \le L$ and $D \isin R \equiv D \le R$.
777 (Likewise $D \isin X \equiv D \le X$ and $D \isin Y \equiv D \le Y$.)
779 Consider $D = C$: $D \isin C$, $D \le C$, OK for $C \haspatch \p$.
781 For $D \neq C$: $D \le C \equiv D \le L \lor D \le R
782 \equiv D \isin L \lor D \isin R$.
783 (Likewise $D \le C \equiv D \le X \lor D \le Y$.)
785 Consider $D \neq C, D \isin X \land D \isin Y$:
786 By $\merge$, $D \isin C$. Also $D \le X$
787 so $D \le C$. OK for $C \haspatch \p$.
789 Consider $D \neq C, D \not\isin X \land D \not\isin Y$:
790 By $\merge$, $D \not\isin C$.
791 And $D \not\le X \land D \not\le Y$ so $D \not\le C$.
792 OK for $C \haspatch \p$.
794 Remaining case, wlog, is $D \not\isin X \land D \isin Y$.
795 $D \not\le X$ so $D \not\le M$ so $D \not\isin M$.
796 Thus by $\merge$, $D \isin C$. And $D \le Y$ so $D \le C$.
797 OK for $C \haspatch \p$.
799 So indeed $L \haspatch \p \land R \haspatch \p \implies C \haspatch \p$.
801 \subsubsection{For (wlog) $X \not\haspatch \p, Y \haspatch \p$:}
803 $M \haspatch \p \implies C \nothaspatch \p$.
804 $M \nothaspatch \p \implies C \haspatch \p$.
808 One of the Merge Ends conditions applies.
809 Recall that we are considering $D \in \py$.
810 $D \isin Y \equiv D \le Y$. $D \not\isin X$.
811 We will show for each of
812 various cases that $D \isin C \equiv M \nothaspatch \p \land D \le C$
813 (which suffices by definition of $\haspatch$ and $\nothaspatch$).
815 Consider $D = C$: Thus $C \in \py, L \in \py$, and by Tip
816 Self Inpatch $L \haspatch \p$, so $L=Y, R=X$. By Tip Merge,
817 $M=\baseof{L}$. So by Base Acyclic $D \not\isin M$, i.e.
818 $M \nothaspatch \p$. And indeed $D \isin C$ and $D \le C$. OK.
820 Consider $D \neq C, M \nothaspatch P, D \isin Y$:
821 $D \le Y$ so $D \le C$.
822 $D \not\isin M$ so by $\merge$, $D \isin C$. OK.
824 Consider $D \neq C, M \nothaspatch P, D \not\isin Y$:
825 $D \not\le Y$. If $D \le X$ then
826 $D \in \pancsof{X}{\py}$, so by Addition Merge Ends and
827 Transitive Ancestors $D \le Y$ --- a contradiction, so $D \not\le X$.
828 Thus $D \not\le C$. By $\merge$, $D \not\isin C$. OK.
830 Consider $D \neq C, M \haspatch P, D \isin Y$:
831 $D \le Y$ so $D \in \pancsof{Y}{\py}$ so by Removal Merge Ends
832 and Transitive Ancestors $D \in \pancsof{M}{\py}$ so $D \le M$.
833 Thus $D \isin M$. By $\merge$, $D \not\isin C$. OK.
835 Consider $D \neq C, M \haspatch P, D \not\isin Y$:
836 By $\merge$, $D \not\isin C$. OK.
840 \subsection{Base Acyclic}
842 This applies when $C \in \pn$.
843 $C \in \pn$ when $L \in \pn$ so by Merge Acyclic, $R \nothaspatch \p$.
845 Consider some $D \in \py$.
847 By Base Acyclic of $L$, $D \not\isin L$. By the above, $D \not\isin
848 R$. And $D \neq C$. So $D \not\isin C$.
852 \subsection{Tip Contents}
854 We need worry only about $C \in \py$.
855 And $\patchof{C} = \patchof{L}$
856 so $L \in \py$ so $L \haspatch \p$. We will use the Unique Base
857 of $C$, and its Coherence and Patch Inclusion, as just proved.
859 Firstly we show $C \haspatch \p$: If $R \in \py$, then $R \haspatch
860 \p$ and by Coherence/Inclusion $C \haspatch \p$ . If $R \not\in \py$
861 then by Tip Merge $M = \baseof{L}$ so by Base Acyclic and definition
862 of $\nothaspatch$, $M \nothaspatch \p$. So by Coherence/Inclusion $C
863 \haspatch \p$ (whether $R \haspatch \p$ or $\nothaspatch$).
865 We will consider an arbitrary commit $D$
866 and prove the Exclusive Tip Contents form.
868 \subsubsection{For $D \in \py$:}
869 $C \haspatch \p$ so by definition of $\haspatch$, $D \isin C \equiv D
872 \subsubsection{For $D \not\in \py, R \not\in \py$:}
874 $D \neq C$. By Tip Contents of $L$,
875 $D \isin L \equiv D \isin \baseof{L}$, and by Tip Merge condition,
876 $D \isin L \equiv D \isin M$. So by definition of $\merge$, $D \isin
877 C \equiv D \isin R$. And $R = \baseof{C}$ by Unique Base of $C$.
878 Thus $D \isin C \equiv D \isin \baseof{C}$. OK.
880 \subsubsection{For $D \not\in \py, R \in \py$:}
885 $D \isin L \equiv D \isin \baseof{L}$ and
886 $D \isin R \equiv D \isin \baseof{R}$.
888 If $\baseof{L} = M$, trivially $D \isin M \equiv D \isin \baseof{L}.$
889 Whereas if $\baseof{L} = \baseof{M}$, by definition of $\base$,
890 $\patchof{M} = \patchof{L} = \py$, so by Tip Contents of $M$,
891 $D \isin M \equiv D \isin \baseof{M} \equiv D \isin \baseof{L}$.
893 So $D \isin M \equiv D \isin L$ and by $\merge$,
894 $D \isin C \equiv D \isin R$. But from Unique Base,
895 $\baseof{C} = R$ so $D \isin C \equiv D \isin \baseof{C}$. OK.
899 \subsection{Foreign Inclusion}
901 Consider some $D$ s.t. $\patchof{D} = \bot$.
902 By Foreign Inclusion of $L, M, R$:
903 $D \isin L \equiv D \le L$;
904 $D \isin M \equiv D \le M$;
905 $D \isin R \equiv D \le R$.
907 \subsubsection{For $D = C$:}
909 $D \isin C$ and $D \le C$. OK.
911 \subsubsection{For $D \neq C, D \isin M$:}
913 Thus $D \le M$ so $D \le L$ and $D \le R$ so $D \isin L$ and $D \isin
914 R$. So by $\merge$, $D \isin C$. And $D \le C$. OK.
916 \subsubsection{For $D \neq C, D \not\isin M, D \isin X$:}
918 By $\merge$, $D \isin C$.
919 And $D \isin X$ means $D \le X$ so $D \le C$.
922 \subsubsection{For $D \neq C, D \not\isin M, D \not\isin L, D \not\isin R$:}
924 By $\merge$, $D \not\isin C$.
925 And $D \not\le L, D \not\le R$ so $D \not\le C$.
930 \subsection{Foreign Contents}
932 Only relevant if $\patchof{L} = \bot$, in which case
933 $\patchof{C} = \bot$ and by Foreign Merges $\patchof{R} = \bot$,
934 so Totally Foreign Contents applies. $\qed$