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
200 \section{Some lemmas}
202 \[ \eqn{Alternative (overlapping) formulations defining
203 $\mergeof{C}{L}{M}{R}$:}{
206 D \isin L \equiv D \isin R : & D = C \lor D \isin L \\
207 D \isin L \nequiv D \isin R : & D = C \lor D \not\isin M \\
208 D \isin L \equiv D \isin M : & D = C \lor D \isin R \\
209 D \isin L \nequiv D \isin M : & D = C \lor D \isin L \\
210 \text{as above with L and R exchanged}
216 Original definition is symmetrical in $L$ and $R$.
219 \[ \eqn{Exclusive Tip Contents:}{
220 \bigforall_{C \in \py}
221 \neg \Bigl[ D \isin \baseof{C} \land ( D \in \py \land D \le C )
224 Ie, the two limbs of the RHS of Tip Contents are mutually exclusive.
227 Let $B = \baseof{C}$ in $D \isin \baseof{C}$. Now $B \in \pn$.
228 So by Base Acyclic $D \isin B \implies D \notin \py$.
230 \[ \eqntag{{\it Corollary - equivalent to Tip Contents}}{
231 \bigforall_{C \in \py} D \isin C \equiv
233 D \in \py : & D \le C \\
234 D \not\in \py : & D \isin \baseof{C}
238 \[ \eqn{Tip Self Inpatch:}{
239 \bigforall_{C \in \py} C \haspatch \p
241 Ie, tip commits contain their own patch.
244 Apply Exclusive Tip Contents to some $D \in \py$:
245 $ \bigforall_{C \in \py}\bigforall_{D \in \py}
246 D \isin C \equiv D \le C $
249 \[ \eqn{Exact Ancestors:}{
250 \bigforall_{ C \hasparents \set{R} }
252 ( \mathop{\hbox{\huge{$\vee$}}}_{R \in \set R} D \le R )
257 \[ \eqn{Transitive Ancestors:}{
258 \left[ \bigforall_{ E \in \pendsof{C}{\set P} } E \le M \right] \equiv
259 \left[ \bigforall_{ A \in \pancsof{C}{\set P} } A \le M \right]
263 The implication from right to left is trivial because
264 $ \pends() \subset \pancs() $.
265 For the implication from left to right:
266 by the definition of $\mathcal E$,
267 for every such $A$, either $A \in \pends()$ which implies
268 $A \le M$ by the LHS directly,
269 or $\exists_{A' \in \pancs()} \; A' \neq A \land A \le A' $
270 in which case we repeat for $A'$. Since there are finitely many
271 commits, this terminates with $A'' \in \pends()$, ie $A'' \le M$
272 by the LHS. And $A \le A''$.
275 \[ \eqn{Calculation Of Ends:}{
276 \bigforall_{C \hasparents \set A}
277 \pendsof{C}{\set P} =
281 C \not\in \p : & \displaystyle
283 \Bigl[ \Largeexists_{A \in \set A}
284 E \in \pendsof{A}{\set P} \Bigr] \land
285 \Bigl[ \Largenexists_{B \in \set A}
286 E \neq B \land E \le B \Bigr]
292 \subsection{No Replay for Merge Results}
294 If we are constructing $C$, with,
302 No Replay is preserved. \proofstarts
304 \subsubsection{For $D=C$:} $D \isin C, D \le C$. OK.
306 \subsubsection{For $D \isin L \land D \isin R$:}
307 $D \isin C$. And $D \isin L \implies D \le L \implies D \le C$. OK.
309 \subsubsection{For $D \neq C \land D \not\isin L \land D \not\isin R$:}
312 \subsubsection{For $D \neq C \land (D \isin L \equiv D \not\isin R)
313 \land D \not\isin M$:}
314 $D \isin C$. Also $D \isin L \lor D \isin R$ so $D \le L \lor D \le
317 \subsubsection{For $D \neq C \land (D \isin L \equiv D \not\isin R)
323 \section{Commit annotation}
325 We annotate each Topbloke commit $C$ with:
329 \baseof{C}, \text{ if } C \in \py
332 \text{ either } C \haspatch \pa{Q} \text{ or } C \nothaspatch \pa{Q}
334 \bigforall_{\pay{Q} \not\ni C} \pendsof{C}{\pay{Q}}
337 $\patchof{C}$, for each kind of Topbloke-generated commit, is stated
338 in the summary in the section for that kind of commit.
340 Whether $\baseof{C}$ is required, and if so what the value is, is
341 stated in the proof of Unique Base for each kind of commit.
343 $C \haspatch \pa{Q}$ or $\nothaspatch \pa{Q}$ is represented as the
344 set $\{ \pa{Q} | C \haspatch \pa{Q} \}$. Whether $C \haspatch \pa{Q}$
346 (in terms of $I \haspatch \pa{Q}$ or $I \nothaspatch \pa{Q}$
347 for the ingredients $I$),
348 in the proof of Coherence for each kind of commit.
350 $\pendsof{C}{\pa{Q}^+}$ is computed, for all Topbloke-generated commits,
351 using the lemma Calculation of Ends, above.
352 We do not annotate $\pendsof{C}{\py}$ for $C \in \py$. Doing so would
353 make it wrong to make plain commits with git because the recorded $\pends$
354 would have to be updated. The annotation is not needed in that case
355 because $\forall_{\py \ni C} \; \pendsof{C}{\py} = \{C\}$.
357 \section{Simple commit}
359 A simple single-parent forward commit $C$ as made by git-commit.
361 \tag*{} C \hasparents \{ A \} \\
362 \tag*{} \patchof{C} = \patchof{A} \\
363 \tag*{} D \isin C \equiv D \isin A \lor D = C
365 This also covers Topbloke-generated commits on plain git branches:
366 Topbloke strips the metadata when exporting.
368 \subsection{No Replay}
371 \subsection{Unique Base}
372 If $A, C \in \py$ then $\baseof{C} = \baseof{A}$. $\qed$
374 \subsection{Tip Contents}
375 We need to consider only $A, C \in \py$. From Tip Contents for $A$:
376 \[ D \isin A \equiv D \isin \baseof{A} \lor ( D \in \py \land D \le A ) \]
377 Substitute into the contents of $C$:
378 \[ D \isin C \equiv D \isin \baseof{A} \lor ( D \in \py \land D \le A )
380 Since $D = C \implies D \in \py$,
381 and substituting in $\baseof{C}$, this gives:
382 \[ D \isin C \equiv D \isin \baseof{C} \lor
383 (D \in \py \land D \le A) \lor
384 (D = C \land D \in \py) \]
385 \[ \equiv D \isin \baseof{C} \lor
386 [ D \in \py \land ( D \le A \lor D = C ) ] \]
387 So by Exact Ancestors:
388 \[ D \isin C \equiv D \isin \baseof{C} \lor ( D \in \py \land D \le C
392 \subsection{Base Acyclic}
394 Need to consider only $A, C \in \pn$.
396 For $D = C$: $D \in \pn$ so $D \not\in \py$. OK.
398 For $D \neq C$: $D \isin C \equiv D \isin A$, so by Base Acyclic for
399 $A$, $D \isin C \implies D \not\in \py$. $\qed$
401 \subsection{Coherence and patch inclusion}
403 Need to consider $D \in \py$
405 \subsubsection{For $A \haspatch P, D = C$:}
411 $ D \isin C \equiv \ldots \lor \true \text{ so } D \haspatch C $.
413 \subsubsection{For $A \haspatch P, D \neq C$:}
414 Ancestors: $ D \le C \equiv D \le A $.
416 Contents: $ D \isin C \equiv D \isin A \lor f $
417 so $ D \isin C \equiv D \isin A $.
420 \[ A \haspatch P \implies C \haspatch P \]
422 \subsubsection{For $A \nothaspatch P$:}
424 Firstly, $C \not\in \py$ since if it were, $A \in \py$.
427 Now by contents of $A$, $D \notin A$, so $D \notin C$.
430 \[ A \nothaspatch P \implies C \nothaspatch P \]
433 \subsection{Foreign inclusion:}
435 If $D = C$, trivial. For $D \neq C$:
436 $D \isin C \equiv D \isin A \equiv D \le A \equiv D \le C$. $\qed$
440 Given $L, R^+, R^-$ where
441 $R^+ \in \pry, R^- = \baseof{R^+}$.
442 Construct $C$ which has $\pr$ removed.
443 Used for removing a branch dependency.
445 C \hasparents \{ L \}
447 \patchof{C} = \patchof{L}
449 \mergeof{C}{L}{R^+}{R^-}
452 \subsection{Conditions}
454 \[ \eqn{ Into Base }{
457 \[ \eqn{ Unique Tip }{
458 \pendsof{L}{\pry} = \{ R^+ \}
460 \[ \eqn{ Currently Included }{
464 \subsection{Ordering of ${L, R^+, R^-}$:}
466 By Unique Tip, $R^+ \le L$. By definition of $\base$, $R^- \le R^+$
467 so $R^- \le L$. So $R^+ \le C$ and $R^- \le C$.
469 (Note that the merge base $R^+ \not\le R^-$, i.e. the merge base is
470 later than one of the branches to be merged.)
472 \subsection{No Replay}
474 No Replay for Merge Results applies. $\qed$
476 \subsection{Desired Contents}
478 \[ D \isin C \equiv [ D \notin \pry \land D \isin L ] \lor D = C \]
481 \subsubsection{For $D = C$:}
483 Trivially $D \isin C$. OK.
485 \subsubsection{For $D \neq C, D \not\le L$:}
487 By No Replay $D \not\isin L$. Also $D \not\le R^-$ hence
488 $D \not\isin R^-$. Thus $D \not\isin C$. OK.
490 \subsubsection{For $D \neq C, D \le L, D \in \pry$:}
492 By Currently Included, $D \isin L$.
494 By Tip Self Inpatch, $D \isin R^+ \equiv D \le R^+$, but by
495 by Unique Tip, $D \le R^+ \equiv D \le L$.
498 By Base Acyclic, $D \not\isin R^-$.
500 Apply $\merge$: $D \not\isin C$. OK.
502 \subsubsection{For $D \neq C, D \le L, D \notin \pry$:}
504 By Tip Contents for $R^+$, $D \isin R^+ \equiv D \isin R^-$.
506 Apply $\merge$: $D \isin C \equiv D \isin L$. OK.
510 \subsection{Unique Base}
512 Into Base means that $C \in \pn$, so Unique Base is not
515 \subsection{Tip Contents}
517 Again, not applicable. $\qed$
519 \subsection{Base Acyclic}
521 By Base Acyclic for $L$, $D \isin L \implies D \not\in \py$.
522 And by Into Base $C \not\in \py$.
523 Now from Desired Contents, above, $D \isin C
524 \implies D \isin L \lor D = C$, which thus
525 $\implies D \not\in \py$. $\qed$.
527 \subsection{Coherence and Patch Inclusion}
529 Need to consider some $D \in \py$. By Into Base, $D \neq C$.
531 \subsubsection{For $\p = \pr$:}
532 By Desired Contents, above, $D \not\isin C$.
533 So $C \nothaspatch \pr$.
535 \subsubsection{For $\p \neq \pr$:}
536 By Desired Contents, $D \isin C \equiv D \isin L$
537 (since $D \in \py$ so $D \not\in \pry$).
539 If $L \nothaspatch \p$, $D \not\isin L$ so $D \not\isin C$.
540 So $L \nothaspatch \p \implies C \nothaspatch \p$.
542 Whereas if $L \haspatch \p$, $D \isin L \equiv D \le L$.
543 so $L \haspatch \p \implies C \haspatch \p$.
545 \section{Foreign Inclusion}
547 Consider some $D$ s.t. $\patchof{D} = \bot$. $D \neq C$.
548 So by Desired Contents $D \isin C \equiv D \isin L$.
549 By Foreign Inclusion of $D$ in $L$, $D \isin L \equiv D \le L$.
551 And $D \le C \equiv D \le L$.
552 Thus $D \isin C \equiv D \le C$. $\qed$
556 Merge commits $L$ and $R$ using merge base $M$ ($M < L, M < R$):
558 C \hasparents \{ L, R \}
560 \patchof{C} = \patchof{L}
564 We will occasionally use $X,Y$ s.t. $\{X,Y\} = \{L,R\}$.
566 \subsection{Conditions}
568 \[ \eqn{ Tip Merge }{
571 R \in \py : & \baseof{R} \ge \baseof{L}
572 \land [\baseof{L} = M \lor \baseof{L} = \baseof{M}] \\
573 R \in \pn : & M = \baseof{L} \\
574 \text{otherwise} : & \false
577 \[ \eqn{ Merge Acyclic }{
582 \[ \eqn{ Removal Merge Ends }{
583 X \not\haspatch \p \land
587 \pendsof{Y}{\py} = \pendsof{M}{\py}
589 \[ \eqn{ Addition Merge Ends }{
590 X \not\haspatch \p \land
594 \bigforall_{E \in \pendsof{X}{\py}} E \le Y
598 \subsection{Non-Topbloke merges}
600 We require both $\patchof{L} = \bot$ and $\patchof{R} = \bot$.
601 I.e. not only is it forbidden to merge into a Topbloke-controlled
602 branch without Topbloke's assistance, it is also forbidden to
603 merge any Topbloke-controlled branch into any plain git branch.
605 Given those conditions, Tip Merge and Merge Acyclic do not apply.
606 And $Y \not\in \py$ so $\neg [ Y \haspatch \p ]$ so neither
607 Merge Ends condition applies. Good.
609 \subsection{No Replay}
611 No Replay for Merge Results applies. $\qed$
613 \subsection{Unique Base}
615 Need to consider only $C \in \py$, ie $L \in \py$,
616 and calculate $\pendsof{C}{\pn}$. So we will consider some
617 putative ancestor $A \in \pn$ and see whether $A \le C$.
619 By Exact Ancestors for C, $A \le C \equiv A \le L \lor A \le R \lor A = C$.
620 But $C \in py$ and $A \in \pn$ so $A \neq C$.
621 Thus $A \le C \equiv A \le L \lor A \le R$.
623 By Unique Base of L and Transitive Ancestors,
624 $A \le L \equiv A \le \baseof{L}$.
626 \subsubsection{For $R \in \py$:}
628 By Unique Base of $R$ and Transitive Ancestors,
629 $A \le R \equiv A \le \baseof{R}$.
631 But by Tip Merge condition on $\baseof{R}$,
632 $A \le \baseof{L} \implies A \le \baseof{R}$, so
633 $A \le \baseof{R} \lor A \le \baseof{L} \equiv A \le \baseof{R}$.
634 Thus $A \le C \equiv A \le \baseof{R}$.
635 That is, $\baseof{C} = \baseof{R}$.
637 \subsubsection{For $R \in \pn$:}
639 By Tip Merge condition on $R$ and since $M \le R$,
640 $A \le \baseof{L} \implies A \le R$, so
641 $A \le R \lor A \le \baseof{L} \equiv A \le R$.
642 Thus $A \le C \equiv A \le R$.
643 That is, $\baseof{C} = R$.
647 \subsection{Coherence and Patch Inclusion}
649 Need to determine $C \haspatch \p$ based on $L,M,R \haspatch \p$.
650 This involves considering $D \in \py$.
652 \subsubsection{For $L \nothaspatch \p, R \nothaspatch \p$:}
653 $D \not\isin L \land D \not\isin R$. $C \not\in \py$ (otherwise $L
654 \in \py$ ie $L \haspatch \p$ by Tip Self Inpatch). So $D \neq C$.
655 Applying $\merge$ gives $D \not\isin C$ i.e. $C \nothaspatch \p$.
657 \subsubsection{For $L \haspatch \p, R \haspatch \p$:}
658 $D \isin L \equiv D \le L$ and $D \isin R \equiv D \le R$.
659 (Likewise $D \isin X \equiv D \le X$ and $D \isin Y \equiv D \le Y$.)
661 Consider $D = C$: $D \isin C$, $D \le C$, OK for $C \haspatch \p$.
663 For $D \neq C$: $D \le C \equiv D \le L \lor D \le R
664 \equiv D \isin L \lor D \isin R$.
665 (Likewise $D \le C \equiv D \le X \lor D \le Y$.)
667 Consider $D \neq C, D \isin X \land D \isin Y$:
668 By $\merge$, $D \isin C$. Also $D \le X$
669 so $D \le C$. OK for $C \haspatch \p$.
671 Consider $D \neq C, D \not\isin X \land D \not\isin Y$:
672 By $\merge$, $D \not\isin C$.
673 And $D \not\le X \land D \not\le Y$ so $D \not\le C$.
674 OK for $C \haspatch \p$.
676 Remaining case, wlog, is $D \not\isin X \land D \isin Y$.
677 $D \not\le X$ so $D \not\le M$ so $D \not\isin M$.
678 Thus by $\merge$, $D \isin C$. And $D \le Y$ so $D \le C$.
679 OK for $C \haspatch \p$.
681 So indeed $L \haspatch \p \land R \haspatch \p \implies C \haspatch \p$.
683 \subsubsection{For (wlog) $X \not\haspatch \p, Y \haspatch \p$:}
685 $M \haspatch \p \implies C \nothaspatch \p$.
686 $M \nothaspatch \p \implies C \haspatch \p$.
690 One of the Merge Ends conditions applies.
691 Recall that we are considering $D \in \py$.
692 $D \isin Y \equiv D \le Y$. $D \not\isin X$.
693 We will show for each of
694 various cases that $D \isin C \equiv M \nothaspatch \p \land D \le C$
695 (which suffices by definition of $\haspatch$ and $\nothaspatch$).
697 Consider $D = C$: Thus $C \in \py, L \in \py$, and by Tip
698 Self Inpatch $L \haspatch \p$, so $L=Y, R=X$. By Tip Merge,
699 $M=\baseof{L}$. So by Base Acyclic $D \not\isin M$, i.e.
700 $M \nothaspatch \p$. And indeed $D \isin C$ and $D \le C$. OK.
702 Consider $D \neq C, M \nothaspatch P, D \isin Y$:
703 $D \le Y$ so $D \le C$.
704 $D \not\isin M$ so by $\merge$, $D \isin C$. OK.
706 Consider $D \neq C, M \nothaspatch P, D \not\isin Y$:
707 $D \not\le Y$. If $D \le X$ then
708 $D \in \pancsof{X}{\py}$, so by Addition Merge Ends and
709 Transitive Ancestors $D \le Y$ --- a contradiction, so $D \not\le X$.
710 Thus $D \not\le C$. By $\merge$, $D \not\isin C$. OK.
712 Consider $D \neq C, M \haspatch P, D \isin Y$:
713 $D \le Y$ so $D \in \pancsof{Y}{\py}$ so by Removal Merge Ends
714 and Transitive Ancestors $D \in \pancsof{M}{\py}$ so $D \le M$.
715 Thus $D \isin M$. By $\merge$, $D \not\isin C$. OK.
717 Consider $D \neq C, M \haspatch P, D \not\isin Y$:
718 By $\merge$, $D \not\isin C$. OK.
722 \subsection{Base Acyclic}
724 This applies when $C \in \pn$.
725 $C \in \pn$ when $L \in \pn$ so by Merge Acyclic, $R \nothaspatch \p$.
727 Consider some $D \in \py$.
729 By Base Acyclic of $L$, $D \not\isin L$. By the above, $D \not\isin
730 R$. And $D \neq C$. So $D \not\isin C$. $\qed$
732 \subsection{Tip Contents}
734 We need worry only about $C \in \py$.
735 And $\patchof{C} = \patchof{L}$
736 so $L \in \py$ so $L \haspatch \p$. We will use the Unique Base
737 of $C$, and its Coherence and Patch Inclusion, as just proved.
739 Firstly we show $C \haspatch \p$: If $R \in \py$, then $R \haspatch
740 \p$ and by Coherence/Inclusion $C \haspatch \p$ . If $R \not\in \py$
741 then by Tip Merge $M = \baseof{L}$ so by Base Acyclic and definition
742 of $\nothaspatch$, $M \nothaspatch \p$. So by Coherence/Inclusion $C
743 \haspatch \p$ (whether $R \haspatch \p$ or $\nothaspatch$).
745 We will consider an arbitrary commit $D$
746 and prove the Exclusive Tip Contents form.
748 \subsubsection{For $D \in \py$:}
749 $C \haspatch \p$ so by definition of $\haspatch$, $D \isin C \equiv D
752 \subsubsection{For $D \not\in \py, R \not\in \py$:}
754 $D \neq C$. By Tip Contents of $L$,
755 $D \isin L \equiv D \isin \baseof{L}$, and by Tip Merge condition,
756 $D \isin L \equiv D \isin M$. So by definition of $\merge$, $D \isin
757 C \equiv D \isin R$. And $R = \baseof{C}$ by Unique Base of $C$.
758 Thus $D \isin C \equiv D \isin \baseof{C}$. OK.
760 \subsubsection{For $D \not\in \py, R \in \py$:}
765 $D \isin L \equiv D \isin \baseof{L}$ and
766 $D \isin R \equiv D \isin \baseof{R}$.
768 If $\baseof{L} = M$, trivially $D \isin M \equiv D \isin \baseof{L}.$
769 Whereas if $\baseof{L} = \baseof{M}$, by definition of $\base$,
770 $\patchof{M} = \patchof{L} = \py$, so by Tip Contents of $M$,
771 $D \isin M \equiv D \isin \baseof{M} \equiv D \isin \baseof{L}$.
773 So $D \isin M \equiv D \isin L$ and by $\merge$,
774 $D \isin C \equiv D \isin R$. But from Unique Base,
775 $\baseof{C} = R$ so $D \isin C \equiv D \isin \baseof{C}$. OK.
779 \subsection{Foreign Inclusion}
781 Consider some $D$ s.t. $\patchof{D} = \bot$.
782 By Foreign Inclusion of $L, M, R$:
783 $D \isin L \equiv D \le L$;
784 $D \isin M \equiv D \le M$;
785 $D \isin R \equiv D \le R$.
787 \subsubsection{For $D = C$:}
789 $D \isin C$ and $D \le C$. OK.
791 \subsubsection{For $D \neq C, D \isin M$:}
793 Thus $D \le M$ so $D \le L$ and $D \le R$ so $D \isin L$ and $D \isin
794 R$. So by $\merge$, $D \isin C$. And $D \le C$. OK.
796 \subsubsection{For $D \neq C, D \not\isin M, D \isin X$:}
798 By $\merge$, $D \isin C$.
799 And $D \isin X$ means $D \le X$ so $D \le C$.
802 \subsubsection{For $D \neq C, D \not\isin M, D \not\isin L, D \not\isin R$:}
804 By $\merge$, $D \not\isin C$.
805 And $D \not\le L, D \not\le R$ so $D \not\le C$.