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 Prohibit 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 Prohibit 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{B}
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 Prohibit 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.
502 \subsection{Coherence and Patch Inclusion}
504 Consider some $D \in \p$.
505 $B \not\in \py$ so $D \neq B$. So $D \isin B \equiv D \isin L$.
507 Thus $L \haspatch \p \implies B \haspatch P$
508 and $L \nothaspatch \p \implies B \nothaspatch P$.
512 \subsection{Foreign Inclusion}
514 Consider some $D$ s.t. $\patchof{D} = \bot$. $D \neq B$
515 so $D \isin B \equiv D \isin L$.
516 By Foreign Inclusion of $D$ in $L$, $D \isin L \equiv D \le L$.
517 And by Exact Ancestors $D \le L \equiv D \le B$.
518 So $D \isin B \equiv D \le B$. $\qed$
520 \subsection{Foreign Contents}
530 Given $L$ and $\pr$ as represented by $R^+, R^-$.
531 Construct $C$ which has $\pr$ removed.
532 Used for removing a branch dependency.
534 C \hasparents \{ L \}
536 \patchof{C} = \patchof{L}
538 \mergeof{C}{L}{R^+}{R^-}
541 \subsection{Conditions}
543 \[ \eqn{ Ingredients }{
544 R^+ \in \pry \land R^- = \baseof{R^+}
546 \[ \eqn{ Into Base }{
549 \[ \eqn{ Unique Tip }{
550 \pendsof{L}{\pry} = \{ R^+ \}
552 \[ \eqn{ Currently Included }{
556 \subsection{Ordering of Ingredients:}
558 By Unique Tip, $R^+ \le L$. By definition of $\base$, $R^- \le R^+$
559 so $R^- \le L$. So $R^+ \le C$ and $R^- \le C$.
562 (Note that $R^+ \not\le R^-$, i.e. the merge base
563 is a descendant, not an ancestor, of the 2nd parent.)
565 \subsection{No Replay}
567 By definition of $\merge$,
568 $D \isin C \implies D \isin L \lor D \isin R^- \lor D = C$.
569 So, by Ordering of Ingredients,
570 Ingredients Prohibit Replay applies. $\qed$
572 \subsection{Desired Contents}
574 \[ D \isin C \equiv [ D \notin \pry \land D \isin L ] \lor D = C \]
577 \subsubsection{For $D = C$:}
579 Trivially $D \isin C$. OK.
581 \subsubsection{For $D \neq C, D \not\le L$:}
583 By No Replay $D \not\isin L$. Also $D \not\le R^-$ hence
584 $D \not\isin R^-$. Thus $D \not\isin C$. OK.
586 \subsubsection{For $D \neq C, D \le L, D \in \pry$:}
588 By Currently Included, $D \isin L$.
590 By Tip Self Inpatch, $D \isin R^+ \equiv D \le R^+$, but by
591 by Unique Tip, $D \le R^+ \equiv D \le L$.
594 By Base Acyclic, $D \not\isin R^-$.
596 Apply $\merge$: $D \not\isin C$. OK.
598 \subsubsection{For $D \neq C, D \le L, D \notin \pry$:}
600 By Tip Contents for $R^+$, $D \isin R^+ \equiv D \isin R^-$.
602 Apply $\merge$: $D \isin C \equiv D \isin L$. OK.
606 \subsection{Unique Base}
608 Into Base means that $C \in \pn$, so Unique Base is not
611 \subsection{Tip Contents}
613 Again, not applicable. $\qed$
615 \subsection{Base Acyclic}
617 By Base Acyclic for $L$, $D \isin L \implies D \not\in \py$.
618 And by Into Base $C \not\in \py$.
619 Now from Desired Contents, above, $D \isin C
620 \implies D \isin L \lor D = C$, which thus
621 $\implies D \not\in \py$. $\qed$.
623 \subsection{Coherence and Patch Inclusion}
625 Need to consider some $D \in \py$. By Into Base, $D \neq C$.
627 \subsubsection{For $\p = \pr$:}
628 By Desired Contents, above, $D \not\isin C$.
629 So $C \nothaspatch \pr$.
631 \subsubsection{For $\p \neq \pr$:}
632 By Desired Contents, $D \isin C \equiv D \isin L$
633 (since $D \in \py$ so $D \not\in \pry$).
635 If $L \nothaspatch \p$, $D \not\isin L$ so $D \not\isin C$.
636 So $L \nothaspatch \p \implies C \nothaspatch \p$.
638 Whereas if $L \haspatch \p$, $D \isin L \equiv D \le L$.
639 so $L \haspatch \p \implies C \haspatch \p$.
643 \subsection{Foreign Inclusion}
645 Consider some $D$ s.t. $\patchof{D} = \bot$. $D \neq C$.
646 So by Desired Contents $D \isin C \equiv D \isin L$.
647 By Foreign Inclusion of $D$ in $L$, $D \isin L \equiv D \le L$.
649 And $D \le C \equiv D \le L$.
650 Thus $D \isin C \equiv D \le C$.
654 \subsection{Foreign Contents}
660 Merge commits $L$ and $R$ using merge base $M$:
662 C \hasparents \{ L, R \}
664 \patchof{C} = \patchof{L}
668 We will occasionally use $X,Y$ s.t. $\{X,Y\} = \{L,R\}$.
670 \subsection{Conditions}
671 \[ \eqn{ Ingredients }{
674 \[ \eqn{ Tip Merge }{
677 R \in \py : & \baseof{R} \ge \baseof{L}
678 \land [\baseof{L} = M \lor \baseof{L} = \baseof{M}] \\
679 R \in \pn : & M = \baseof{L} \\
680 \text{otherwise} : & \false
683 \[ \eqn{ Merge Acyclic }{
688 \[ \eqn{ Removal Merge Ends }{
689 X \not\haspatch \p \land
693 \pendsof{Y}{\py} = \pendsof{M}{\py}
695 \[ \eqn{ Addition Merge Ends }{
696 X \not\haspatch \p \land
700 \bigforall_{E \in \pendsof{X}{\py}} E \le Y
703 \[ \eqn{ Foreign Merges }{
704 \patchof{L} = \bot \equiv \patchof{R} = \bot
707 \subsection{Non-Topbloke merges}
709 We require both $\patchof{L} = \bot$ and $\patchof{R} = \bot$
710 (Foreign Merges, above).
711 I.e. not only is it forbidden to merge into a Topbloke-controlled
712 branch without Topbloke's assistance, it is also forbidden to
713 merge any Topbloke-controlled branch into any plain git branch.
715 Given those conditions, Tip Merge and Merge Acyclic do not apply.
716 And $Y \not\in \py$ so $\neg [ Y \haspatch \p ]$ so neither
717 Merge Ends condition applies.
719 So a plain git merge of non-Topbloke branches meets the conditions and
720 is therefore consistent with our scheme.
722 \subsection{No Replay}
724 By definition of $\merge$,
725 $D \isin C \implies D \isin L \lor D \isin R \lor D = C$.
727 Ingredients Prohibit Replay applies. $\qed$
729 \subsection{Unique Base}
731 Need to consider only $C \in \py$, ie $L \in \py$,
732 and calculate $\pendsof{C}{\pn}$. So we will consider some
733 putative ancestor $A \in \pn$ and see whether $A \le C$.
735 By Exact Ancestors for C, $A \le C \equiv A \le L \lor A \le R \lor A = C$.
736 But $C \in py$ and $A \in \pn$ so $A \neq C$.
737 Thus $A \le C \equiv A \le L \lor A \le R$.
739 By Unique Base of L and Transitive Ancestors,
740 $A \le L \equiv A \le \baseof{L}$.
742 \subsubsection{For $R \in \py$:}
744 By Unique Base of $R$ and Transitive Ancestors,
745 $A \le R \equiv A \le \baseof{R}$.
747 But by Tip Merge condition on $\baseof{R}$,
748 $A \le \baseof{L} \implies A \le \baseof{R}$, so
749 $A \le \baseof{R} \lor A \le \baseof{L} \equiv A \le \baseof{R}$.
750 Thus $A \le C \equiv A \le \baseof{R}$.
751 That is, $\baseof{C} = \baseof{R}$.
753 \subsubsection{For $R \in \pn$:}
755 By Tip Merge condition on $R$ and since $M \le R$,
756 $A \le \baseof{L} \implies A \le R$, so
757 $A \le R \lor A \le \baseof{L} \equiv A \le R$.
758 Thus $A \le C \equiv A \le R$.
759 That is, $\baseof{C} = R$.
763 \subsection{Coherence and Patch Inclusion}
765 Need to determine $C \haspatch \p$ based on $L,M,R \haspatch \p$.
766 This involves considering $D \in \py$.
768 \subsubsection{For $L \nothaspatch \p, R \nothaspatch \p$:}
769 $D \not\isin L \land D \not\isin R$. $C \not\in \py$ (otherwise $L
770 \in \py$ ie $L \haspatch \p$ by Tip Self Inpatch). So $D \neq C$.
771 Applying $\merge$ gives $D \not\isin C$ i.e. $C \nothaspatch \p$.
773 \subsubsection{For $L \haspatch \p, R \haspatch \p$:}
774 $D \isin L \equiv D \le L$ and $D \isin R \equiv D \le R$.
775 (Likewise $D \isin X \equiv D \le X$ and $D \isin Y \equiv D \le Y$.)
777 Consider $D = C$: $D \isin C$, $D \le C$, OK for $C \haspatch \p$.
779 For $D \neq C$: $D \le C \equiv D \le L \lor D \le R
780 \equiv D \isin L \lor D \isin R$.
781 (Likewise $D \le C \equiv D \le X \lor D \le Y$.)
783 Consider $D \neq C, D \isin X \land D \isin Y$:
784 By $\merge$, $D \isin C$. Also $D \le X$
785 so $D \le C$. OK for $C \haspatch \p$.
787 Consider $D \neq C, D \not\isin X \land D \not\isin Y$:
788 By $\merge$, $D \not\isin C$.
789 And $D \not\le X \land D \not\le Y$ so $D \not\le C$.
790 OK for $C \haspatch \p$.
792 Remaining case, wlog, is $D \not\isin X \land D \isin Y$.
793 $D \not\le X$ so $D \not\le M$ so $D \not\isin M$.
794 Thus by $\merge$, $D \isin C$. And $D \le Y$ so $D \le C$.
795 OK for $C \haspatch \p$.
797 So indeed $L \haspatch \p \land R \haspatch \p \implies C \haspatch \p$.
799 \subsubsection{For (wlog) $X \not\haspatch \p, Y \haspatch \p$:}
801 $M \haspatch \p \implies C \nothaspatch \p$.
802 $M \nothaspatch \p \implies C \haspatch \p$.
806 One of the Merge Ends conditions applies.
807 Recall that we are considering $D \in \py$.
808 $D \isin Y \equiv D \le Y$. $D \not\isin X$.
809 We will show for each of
810 various cases that $D \isin C \equiv M \nothaspatch \p \land D \le C$
811 (which suffices by definition of $\haspatch$ and $\nothaspatch$).
813 Consider $D = C$: Thus $C \in \py, L \in \py$, and by Tip
814 Self Inpatch $L \haspatch \p$, so $L=Y, R=X$. By Tip Merge,
815 $M=\baseof{L}$. So by Base Acyclic $D \not\isin M$, i.e.
816 $M \nothaspatch \p$. And indeed $D \isin C$ and $D \le C$. OK.
818 Consider $D \neq C, M \nothaspatch P, D \isin Y$:
819 $D \le Y$ so $D \le C$.
820 $D \not\isin M$ so by $\merge$, $D \isin C$. OK.
822 Consider $D \neq C, M \nothaspatch P, D \not\isin Y$:
823 $D \not\le Y$. If $D \le X$ then
824 $D \in \pancsof{X}{\py}$, so by Addition Merge Ends and
825 Transitive Ancestors $D \le Y$ --- a contradiction, so $D \not\le X$.
826 Thus $D \not\le C$. By $\merge$, $D \not\isin C$. OK.
828 Consider $D \neq C, M \haspatch P, D \isin Y$:
829 $D \le Y$ so $D \in \pancsof{Y}{\py}$ so by Removal Merge Ends
830 and Transitive Ancestors $D \in \pancsof{M}{\py}$ so $D \le M$.
831 Thus $D \isin M$. By $\merge$, $D \not\isin C$. OK.
833 Consider $D \neq C, M \haspatch P, D \not\isin Y$:
834 By $\merge$, $D \not\isin C$. OK.
838 \subsection{Base Acyclic}
840 This applies when $C \in \pn$.
841 $C \in \pn$ when $L \in \pn$ so by Merge Acyclic, $R \nothaspatch \p$.
843 Consider some $D \in \py$.
845 By Base Acyclic of $L$, $D \not\isin L$. By the above, $D \not\isin
846 R$. And $D \neq C$. So $D \not\isin C$.
850 \subsection{Tip Contents}
852 We need worry only about $C \in \py$.
853 And $\patchof{C} = \patchof{L}$
854 so $L \in \py$ so $L \haspatch \p$. We will use the Unique Base
855 of $C$, and its Coherence and Patch Inclusion, as just proved.
857 Firstly we show $C \haspatch \p$: If $R \in \py$, then $R \haspatch
858 \p$ and by Coherence/Inclusion $C \haspatch \p$ . If $R \not\in \py$
859 then by Tip Merge $M = \baseof{L}$ so by Base Acyclic and definition
860 of $\nothaspatch$, $M \nothaspatch \p$. So by Coherence/Inclusion $C
861 \haspatch \p$ (whether $R \haspatch \p$ or $\nothaspatch$).
863 We will consider an arbitrary commit $D$
864 and prove the Exclusive Tip Contents form.
866 \subsubsection{For $D \in \py$:}
867 $C \haspatch \p$ so by definition of $\haspatch$, $D \isin C \equiv D
870 \subsubsection{For $D \not\in \py, R \not\in \py$:}
872 $D \neq C$. By Tip Contents of $L$,
873 $D \isin L \equiv D \isin \baseof{L}$, and by Tip Merge condition,
874 $D \isin L \equiv D \isin M$. So by definition of $\merge$, $D \isin
875 C \equiv D \isin R$. And $R = \baseof{C}$ by Unique Base of $C$.
876 Thus $D \isin C \equiv D \isin \baseof{C}$. OK.
878 \subsubsection{For $D \not\in \py, R \in \py$:}
883 $D \isin L \equiv D \isin \baseof{L}$ and
884 $D \isin R \equiv D \isin \baseof{R}$.
886 If $\baseof{L} = M$, trivially $D \isin M \equiv D \isin \baseof{L}.$
887 Whereas if $\baseof{L} = \baseof{M}$, by definition of $\base$,
888 $\patchof{M} = \patchof{L} = \py$, so by Tip Contents of $M$,
889 $D \isin M \equiv D \isin \baseof{M} \equiv D \isin \baseof{L}$.
891 So $D \isin M \equiv D \isin L$ and by $\merge$,
892 $D \isin C \equiv D \isin R$. But from Unique Base,
893 $\baseof{C} = R$ so $D \isin C \equiv D \isin \baseof{C}$. OK.
897 \subsection{Foreign Inclusion}
899 Consider some $D$ s.t. $\patchof{D} = \bot$.
900 By Foreign Inclusion of $L, M, R$:
901 $D \isin L \equiv D \le L$;
902 $D \isin M \equiv D \le M$;
903 $D \isin R \equiv D \le R$.
905 \subsubsection{For $D = C$:}
907 $D \isin C$ and $D \le C$. OK.
909 \subsubsection{For $D \neq C, D \isin M$:}
911 Thus $D \le M$ so $D \le L$ and $D \le R$ so $D \isin L$ and $D \isin
912 R$. So by $\merge$, $D \isin C$. And $D \le C$. OK.
914 \subsubsection{For $D \neq C, D \not\isin M, D \isin X$:}
916 By $\merge$, $D \isin C$.
917 And $D \isin X$ means $D \le X$ so $D \le C$.
920 \subsubsection{For $D \neq C, D \not\isin M, D \not\isin L, D \not\isin R$:}
922 By $\merge$, $D \not\isin C$.
923 And $D \not\le L, D \not\le R$ so $D \not\le C$.
928 \subsection{Foreign Contents}
930 Only relevant if $\patchof{L} = \bot$, in which case
931 $\patchof{C} = \bot$ and by Foreign Merges $\patchof{R} = \bot$,
932 so Totally Foreign Contents applies. $\qed$