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{Totally Foreign Contents:}{
297 \bigforall_{C \hasparents \set A}
299 \patchof{C} = \bot \land
300 \bigforall_{A \in \set A} \patchof{A} = \bot
310 Consider some $D \le C$. If $D = C$, $\patchof{D} = \bot$ trivially.
311 If $D \neq C$ then $D \le A$ where $A \in \set A$. By Foreign
312 Contents of $A$, $\patchof{D} = \bot$.
315 \subsection{No Replay for Merge Results}
317 If we are constructing $C$, with,
325 No Replay is preserved. \proofstarts
327 \subsubsection{For $D=C$:} $D \isin C, D \le C$. OK.
329 \subsubsection{For $D \isin L \land D \isin R$:}
330 $D \isin C$. And $D \isin L \implies D \le L \implies D \le C$. OK.
332 \subsubsection{For $D \neq C \land D \not\isin L \land D \not\isin R$:}
335 \subsubsection{For $D \neq C \land (D \isin L \equiv D \not\isin R)
336 \land D \not\isin M$:}
337 $D \isin C$. Also $D \isin L \lor D \isin R$ so $D \le L \lor D \le
340 \subsubsection{For $D \neq C \land (D \isin L \equiv D \not\isin R)
346 \section{Commit annotation}
348 We annotate each Topbloke commit $C$ with:
352 \baseof{C}, \text{ if } C \in \py
355 \text{ either } C \haspatch \pa{Q} \text{ or } C \nothaspatch \pa{Q}
357 \bigforall_{\pay{Q} \not\ni C} \pendsof{C}{\pay{Q}}
360 $\patchof{C}$, for each kind of Topbloke-generated commit, is stated
361 in the summary in the section for that kind of commit.
363 Whether $\baseof{C}$ is required, and if so what the value is, is
364 stated in the proof of Unique Base for each kind of commit.
366 $C \haspatch \pa{Q}$ or $\nothaspatch \pa{Q}$ is represented as the
367 set $\{ \pa{Q} | C \haspatch \pa{Q} \}$. Whether $C \haspatch \pa{Q}$
369 (in terms of $I \haspatch \pa{Q}$ or $I \nothaspatch \pa{Q}$
370 for the ingredients $I$),
371 in the proof of Coherence for each kind of commit.
373 $\pendsof{C}{\pa{Q}^+}$ is computed, for all Topbloke-generated commits,
374 using the lemma Calculation of Ends, above.
375 We do not annotate $\pendsof{C}{\py}$ for $C \in \py$. Doing so would
376 make it wrong to make plain commits with git because the recorded $\pends$
377 would have to be updated. The annotation is not needed in that case
378 because $\forall_{\py \ni C} \; \pendsof{C}{\py} = \{C\}$.
380 \section{Simple commit}
382 A simple single-parent forward commit $C$ as made by git-commit.
384 \tag*{} C \hasparents \{ A \} \\
385 \tag*{} \patchof{C} = \patchof{A} \\
386 \tag*{} D \isin C \equiv D \isin A \lor D = C
388 This also covers Topbloke-generated commits on plain git branches:
389 Topbloke strips the metadata when exporting.
391 \subsection{No Replay}
394 \subsection{Unique Base}
395 If $A, C \in \py$ then by Calculation of Ends for
396 $C, \py, C \not\in \py$:
397 $\pendsof{C}{\pn} = \pendsof{A}{\pn}$ so
398 $\baseof{C} = \baseof{A}$. $\qed$
400 \subsection{Tip Contents}
401 We need to consider only $A, C \in \py$. From Tip Contents for $A$:
402 \[ D \isin A \equiv D \isin \baseof{A} \lor ( D \in \py \land D \le A ) \]
403 Substitute into the contents of $C$:
404 \[ D \isin C \equiv D \isin \baseof{A} \lor ( D \in \py \land D \le A )
406 Since $D = C \implies D \in \py$,
407 and substituting in $\baseof{C}$, this gives:
408 \[ D \isin C \equiv D \isin \baseof{C} \lor
409 (D \in \py \land D \le A) \lor
410 (D = C \land D \in \py) \]
411 \[ \equiv D \isin \baseof{C} \lor
412 [ D \in \py \land ( D \le A \lor D = C ) ] \]
413 So by Exact Ancestors:
414 \[ D \isin C \equiv D \isin \baseof{C} \lor ( D \in \py \land D \le C
418 \subsection{Base Acyclic}
420 Need to consider only $A, C \in \pn$.
422 For $D = C$: $D \in \pn$ so $D \not\in \py$. OK.
424 For $D \neq C$: $D \isin C \equiv D \isin A$, so by Base Acyclic for
425 $A$, $D \isin C \implies D \not\in \py$.
429 \subsection{Coherence and patch inclusion}
431 Need to consider $D \in \py$
433 \subsubsection{For $A \haspatch P, D = C$:}
439 $ D \isin C \equiv \ldots \lor \true \text{ so } D \haspatch C $.
441 \subsubsection{For $A \haspatch P, D \neq C$:}
442 Ancestors: $ D \le C \equiv D \le A $.
444 Contents: $ D \isin C \equiv D \isin A \lor f $
445 so $ D \isin C \equiv D \isin A $.
448 \[ A \haspatch P \implies C \haspatch P \]
450 \subsubsection{For $A \nothaspatch P$:}
452 Firstly, $C \not\in \py$ since if it were, $A \in \py$.
455 Now by contents of $A$, $D \notin A$, so $D \notin C$.
458 \[ A \nothaspatch P \implies C \nothaspatch P \]
461 \subsection{Foreign inclusion:}
463 If $D = C$, trivial. For $D \neq C$:
464 $D \isin C \equiv D \isin A \equiv D \le A \equiv D \le C$. $\qed$
466 \subsection{Foreign Contents:}
468 Only relevant if $\patchof{C} = \bot$, and in that case Totally
469 Foreign Contents applies. $\qed$
471 \section{Create Base}
473 Given $L$, create a Topbloke base branch initial commit $B$.
475 B \hasparents \{ L \}
477 \patchof{B} = \pan{B}
479 D \isin B \equiv D \isin L \lor D = B
482 \subsection{Conditions}
484 \[ \eqn{ Ingredients }{
485 \patchof{L} = \pa{L} \lor \patchof{L} = \bot
487 \[ \eqn{ Non-recursion }{
491 \subsection{No Replay}
493 If $\patchof{L} = \pa{L}$, trivial by Base Acyclic for $L$.
495 If $\patchof{L} = \bot$, consider some $D \isin B$. $D \neq B$.
496 Thus $D \isin L$. So by No Replay of $D$ in $L$, $D \le L$.
499 \subsection{Unique Base}
501 Not applicable. $\qed$
503 \subsection{Tip Contents}
505 Not applicable. $\qed$
507 \subsection{Base Acyclic}
509 Consider some $D \isin B$. If $D = B$, $D \in \pn$, OK.
511 If $D \neq B$, $D \isin L$. By No Replay of $D$ in $L$, $D \le L$.
512 Thus by Foreign Contents of $L$, $\patchof{D} = \bot$. OK.
516 \subsection{Coherence and Patch Inclusion}
518 Consider some $D \in \p$.
519 $B \not\in \py$ so $D \neq B$. So $D \isin B \equiv D \isin L$.
521 Thus $L \haspatch \p \implies B \haspatch P$
522 and $L \nothaspatch \p \implies B \nothaspatch P$.
526 \subsection{Foreign Inclusion}
528 Consider some $D$ s.t. $\patchof{D} = \bot$. $D \neq B$
529 so $D \isin B \equiv D \isin L$.
531 By Foreign Inclusion of $D$ xxx up to here
539 Given $L$ and $\pr$ as represented by $R^+, R^-$.
540 Construct $C$ which has $\pr$ removed.
541 Used for removing a branch dependency.
543 C \hasparents \{ L \}
545 \patchof{C} = \patchof{L}
547 \mergeof{C}{L}{R^+}{R^-}
550 \subsection{Conditions}
552 \[ \eqn{ Ingredients }{
553 R^+ \in \pry \land R^- = \baseof{R^+}
555 \[ \eqn{ Into Base }{
558 \[ \eqn{ Unique Tip }{
559 \pendsof{L}{\pry} = \{ R^+ \}
561 \[ \eqn{ Currently Included }{
565 \subsection{Ordering of ${L, R^+, R^-}$:}
567 By Unique Tip, $R^+ \le L$. By definition of $\base$, $R^- \le R^+$
568 so $R^- \le L$. So $R^+ \le C$ and $R^- \le C$.
571 (Note that $R^+ \not\le R^-$, i.e. the merge base
572 is a descendant, not an ancestor, of the 2nd parent.)
574 \subsection{No Replay}
576 No Replay for Merge Results applies. $\qed$
578 \subsection{Desired Contents}
580 \[ D \isin C \equiv [ D \notin \pry \land D \isin L ] \lor D = C \]
583 \subsubsection{For $D = C$:}
585 Trivially $D \isin C$. OK.
587 \subsubsection{For $D \neq C, D \not\le L$:}
589 By No Replay $D \not\isin L$. Also $D \not\le R^-$ hence
590 $D \not\isin R^-$. Thus $D \not\isin C$. OK.
592 \subsubsection{For $D \neq C, D \le L, D \in \pry$:}
594 By Currently Included, $D \isin L$.
596 By Tip Self Inpatch, $D \isin R^+ \equiv D \le R^+$, but by
597 by Unique Tip, $D \le R^+ \equiv D \le L$.
600 By Base Acyclic, $D \not\isin R^-$.
602 Apply $\merge$: $D \not\isin C$. OK.
604 \subsubsection{For $D \neq C, D \le L, D \notin \pry$:}
606 By Tip Contents for $R^+$, $D \isin R^+ \equiv D \isin R^-$.
608 Apply $\merge$: $D \isin C \equiv D \isin L$. OK.
612 \subsection{Unique Base}
614 Into Base means that $C \in \pn$, so Unique Base is not
617 \subsection{Tip Contents}
619 Again, not applicable. $\qed$
621 \subsection{Base Acyclic}
623 By Base Acyclic for $L$, $D \isin L \implies D \not\in \py$.
624 And by Into Base $C \not\in \py$.
625 Now from Desired Contents, above, $D \isin C
626 \implies D \isin L \lor D = C$, which thus
627 $\implies D \not\in \py$. $\qed$.
629 \subsection{Coherence and Patch Inclusion}
631 Need to consider some $D \in \py$. By Into Base, $D \neq C$.
633 \subsubsection{For $\p = \pr$:}
634 By Desired Contents, above, $D \not\isin C$.
635 So $C \nothaspatch \pr$.
637 \subsubsection{For $\p \neq \pr$:}
638 By Desired Contents, $D \isin C \equiv D \isin L$
639 (since $D \in \py$ so $D \not\in \pry$).
641 If $L \nothaspatch \p$, $D \not\isin L$ so $D \not\isin C$.
642 So $L \nothaspatch \p \implies C \nothaspatch \p$.
644 Whereas if $L \haspatch \p$, $D \isin L \equiv D \le L$.
645 so $L \haspatch \p \implies C \haspatch \p$.
649 \subsection{Foreign Inclusion}
651 Consider some $D$ s.t. $\patchof{D} = \bot$. $D \neq C$.
652 So by Desired Contents $D \isin C \equiv D \isin L$.
653 By Foreign Inclusion of $D$ in $L$, $D \isin L \equiv D \le L$.
655 And $D \le C \equiv D \le L$.
656 Thus $D \isin C \equiv D \le C$.
660 \subsection{Foreign Contents}
662 Not applicable. $\qed$
666 Merge commits $L$ and $R$ using merge base $M$:
668 C \hasparents \{ L, R \}
670 \patchof{C} = \patchof{L}
674 We will occasionally use $X,Y$ s.t. $\{X,Y\} = \{L,R\}$.
676 \subsection{Conditions}
677 \[ \eqn{ Ingredients }{
680 \[ \eqn{ Tip Merge }{
683 R \in \py : & \baseof{R} \ge \baseof{L}
684 \land [\baseof{L} = M \lor \baseof{L} = \baseof{M}] \\
685 R \in \pn : & M = \baseof{L} \\
686 \text{otherwise} : & \false
689 \[ \eqn{ Merge Acyclic }{
694 \[ \eqn{ Removal Merge Ends }{
695 X \not\haspatch \p \land
699 \pendsof{Y}{\py} = \pendsof{M}{\py}
701 \[ \eqn{ Addition Merge Ends }{
702 X \not\haspatch \p \land
706 \bigforall_{E \in \pendsof{X}{\py}} E \le Y
709 \[ \eqn{ Foreign Merges }{
710 \patchof{L} = \bot \equiv \patchof{R} = \bot
713 \subsection{Non-Topbloke merges}
715 We require both $\patchof{L} = \bot$ and $\patchof{R} = \bot$
716 (Foreign Merges, above).
717 I.e. not only is it forbidden to merge into a Topbloke-controlled
718 branch without Topbloke's assistance, it is also forbidden to
719 merge any Topbloke-controlled branch into any plain git branch.
721 Given those conditions, Tip Merge and Merge Acyclic do not apply.
722 And $Y \not\in \py$ so $\neg [ Y \haspatch \p ]$ so neither
723 Merge Ends condition applies.
725 So a plain git merge of non-Topbloke branches meets the conditions and
726 is therefore consistent with our scheme.
728 \subsection{No Replay}
730 No Replay for Merge Results applies. $\qed$
732 \subsection{Unique Base}
734 Need to consider only $C \in \py$, ie $L \in \py$,
735 and calculate $\pendsof{C}{\pn}$. So we will consider some
736 putative ancestor $A \in \pn$ and see whether $A \le C$.
738 By Exact Ancestors for C, $A \le C \equiv A \le L \lor A \le R \lor A = C$.
739 But $C \in py$ and $A \in \pn$ so $A \neq C$.
740 Thus $A \le C \equiv A \le L \lor A \le R$.
742 By Unique Base of L and Transitive Ancestors,
743 $A \le L \equiv A \le \baseof{L}$.
745 \subsubsection{For $R \in \py$:}
747 By Unique Base of $R$ and Transitive Ancestors,
748 $A \le R \equiv A \le \baseof{R}$.
750 But by Tip Merge condition on $\baseof{R}$,
751 $A \le \baseof{L} \implies A \le \baseof{R}$, so
752 $A \le \baseof{R} \lor A \le \baseof{L} \equiv A \le \baseof{R}$.
753 Thus $A \le C \equiv A \le \baseof{R}$.
754 That is, $\baseof{C} = \baseof{R}$.
756 \subsubsection{For $R \in \pn$:}
758 By Tip Merge condition on $R$ and since $M \le R$,
759 $A \le \baseof{L} \implies A \le R$, so
760 $A \le R \lor A \le \baseof{L} \equiv A \le R$.
761 Thus $A \le C \equiv A \le R$.
762 That is, $\baseof{C} = R$.
766 \subsection{Coherence and Patch Inclusion}
768 Need to determine $C \haspatch \p$ based on $L,M,R \haspatch \p$.
769 This involves considering $D \in \py$.
771 \subsubsection{For $L \nothaspatch \p, R \nothaspatch \p$:}
772 $D \not\isin L \land D \not\isin R$. $C \not\in \py$ (otherwise $L
773 \in \py$ ie $L \haspatch \p$ by Tip Self Inpatch). So $D \neq C$.
774 Applying $\merge$ gives $D \not\isin C$ i.e. $C \nothaspatch \p$.
776 \subsubsection{For $L \haspatch \p, R \haspatch \p$:}
777 $D \isin L \equiv D \le L$ and $D \isin R \equiv D \le R$.
778 (Likewise $D \isin X \equiv D \le X$ and $D \isin Y \equiv D \le Y$.)
780 Consider $D = C$: $D \isin C$, $D \le C$, OK for $C \haspatch \p$.
782 For $D \neq C$: $D \le C \equiv D \le L \lor D \le R
783 \equiv D \isin L \lor D \isin R$.
784 (Likewise $D \le C \equiv D \le X \lor D \le Y$.)
786 Consider $D \neq C, D \isin X \land D \isin Y$:
787 By $\merge$, $D \isin C$. Also $D \le X$
788 so $D \le C$. OK for $C \haspatch \p$.
790 Consider $D \neq C, D \not\isin X \land D \not\isin Y$:
791 By $\merge$, $D \not\isin C$.
792 And $D \not\le X \land D \not\le Y$ so $D \not\le C$.
793 OK for $C \haspatch \p$.
795 Remaining case, wlog, is $D \not\isin X \land D \isin Y$.
796 $D \not\le X$ so $D \not\le M$ so $D \not\isin M$.
797 Thus by $\merge$, $D \isin C$. And $D \le Y$ so $D \le C$.
798 OK for $C \haspatch \p$.
800 So indeed $L \haspatch \p \land R \haspatch \p \implies C \haspatch \p$.
802 \subsubsection{For (wlog) $X \not\haspatch \p, Y \haspatch \p$:}
804 $M \haspatch \p \implies C \nothaspatch \p$.
805 $M \nothaspatch \p \implies C \haspatch \p$.
809 One of the Merge Ends conditions applies.
810 Recall that we are considering $D \in \py$.
811 $D \isin Y \equiv D \le Y$. $D \not\isin X$.
812 We will show for each of
813 various cases that $D \isin C \equiv M \nothaspatch \p \land D \le C$
814 (which suffices by definition of $\haspatch$ and $\nothaspatch$).
816 Consider $D = C$: Thus $C \in \py, L \in \py$, and by Tip
817 Self Inpatch $L \haspatch \p$, so $L=Y, R=X$. By Tip Merge,
818 $M=\baseof{L}$. So by Base Acyclic $D \not\isin M$, i.e.
819 $M \nothaspatch \p$. And indeed $D \isin C$ and $D \le C$. OK.
821 Consider $D \neq C, M \nothaspatch P, D \isin Y$:
822 $D \le Y$ so $D \le C$.
823 $D \not\isin M$ so by $\merge$, $D \isin C$. OK.
825 Consider $D \neq C, M \nothaspatch P, D \not\isin Y$:
826 $D \not\le Y$. If $D \le X$ then
827 $D \in \pancsof{X}{\py}$, so by Addition Merge Ends and
828 Transitive Ancestors $D \le Y$ --- a contradiction, so $D \not\le X$.
829 Thus $D \not\le C$. By $\merge$, $D \not\isin C$. OK.
831 Consider $D \neq C, M \haspatch P, D \isin Y$:
832 $D \le Y$ so $D \in \pancsof{Y}{\py}$ so by Removal Merge Ends
833 and Transitive Ancestors $D \in \pancsof{M}{\py}$ so $D \le M$.
834 Thus $D \isin M$. By $\merge$, $D \not\isin C$. OK.
836 Consider $D \neq C, M \haspatch P, D \not\isin Y$:
837 By $\merge$, $D \not\isin C$. OK.
841 \subsection{Base Acyclic}
843 This applies when $C \in \pn$.
844 $C \in \pn$ when $L \in \pn$ so by Merge Acyclic, $R \nothaspatch \p$.
846 Consider some $D \in \py$.
848 By Base Acyclic of $L$, $D \not\isin L$. By the above, $D \not\isin
849 R$. And $D \neq C$. So $D \not\isin C$.
853 \subsection{Tip Contents}
855 We need worry only about $C \in \py$.
856 And $\patchof{C} = \patchof{L}$
857 so $L \in \py$ so $L \haspatch \p$. We will use the Unique Base
858 of $C$, and its Coherence and Patch Inclusion, as just proved.
860 Firstly we show $C \haspatch \p$: If $R \in \py$, then $R \haspatch
861 \p$ and by Coherence/Inclusion $C \haspatch \p$ . If $R \not\in \py$
862 then by Tip Merge $M = \baseof{L}$ so by Base Acyclic and definition
863 of $\nothaspatch$, $M \nothaspatch \p$. So by Coherence/Inclusion $C
864 \haspatch \p$ (whether $R \haspatch \p$ or $\nothaspatch$).
866 We will consider an arbitrary commit $D$
867 and prove the Exclusive Tip Contents form.
869 \subsubsection{For $D \in \py$:}
870 $C \haspatch \p$ so by definition of $\haspatch$, $D \isin C \equiv D
873 \subsubsection{For $D \not\in \py, R \not\in \py$:}
875 $D \neq C$. By Tip Contents of $L$,
876 $D \isin L \equiv D \isin \baseof{L}$, and by Tip Merge condition,
877 $D \isin L \equiv D \isin M$. So by definition of $\merge$, $D \isin
878 C \equiv D \isin R$. And $R = \baseof{C}$ by Unique Base of $C$.
879 Thus $D \isin C \equiv D \isin \baseof{C}$. OK.
881 \subsubsection{For $D \not\in \py, R \in \py$:}
886 $D \isin L \equiv D \isin \baseof{L}$ and
887 $D \isin R \equiv D \isin \baseof{R}$.
889 If $\baseof{L} = M$, trivially $D \isin M \equiv D \isin \baseof{L}.$
890 Whereas if $\baseof{L} = \baseof{M}$, by definition of $\base$,
891 $\patchof{M} = \patchof{L} = \py$, so by Tip Contents of $M$,
892 $D \isin M \equiv D \isin \baseof{M} \equiv D \isin \baseof{L}$.
894 So $D \isin M \equiv D \isin L$ and by $\merge$,
895 $D \isin C \equiv D \isin R$. But from Unique Base,
896 $\baseof{C} = R$ so $D \isin C \equiv D \isin \baseof{C}$. OK.
900 \subsection{Foreign Inclusion}
902 Consider some $D$ s.t. $\patchof{D} = \bot$.
903 By Foreign Inclusion of $L, M, R$:
904 $D \isin L \equiv D \le L$;
905 $D \isin M \equiv D \le M$;
906 $D \isin R \equiv D \le R$.
908 \subsubsection{For $D = C$:}
910 $D \isin C$ and $D \le C$. OK.
912 \subsubsection{For $D \neq C, D \isin M$:}
914 Thus $D \le M$ so $D \le L$ and $D \le R$ so $D \isin L$ and $D \isin
915 R$. So by $\merge$, $D \isin C$. And $D \le C$. OK.
917 \subsubsection{For $D \neq C, D \not\isin M, D \isin X$:}
919 By $\merge$, $D \isin C$.
920 And $D \isin X$ means $D \le X$ so $D \le C$.
923 \subsubsection{For $D \neq C, D \not\isin M, D \not\isin L, D \not\isin R$:}
925 By $\merge$, $D \not\isin C$.
926 And $D \not\le L, D \not\le R$ so $D \not\le C$.
931 \subsection{Foreign Contents}
933 Only relevant if $\patchof{L} = \bot$, in which case
934 $\patchof{C} = \bot$ and by Foreign Merges $\patchof{R} = \bot$,
935 so Totally Foreign Contents applies. $\qed$