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 by Calculation of Ends for
373 $C, \py, C \not\in \py$:
374 $\pendsof{C}{\pn} = \pendsof{A}{\pn}$ so
375 $\baseof{C} = \baseof{A}$. $\qed$
377 \subsection{Tip Contents}
378 We need to consider only $A, C \in \py$. From Tip Contents for $A$:
379 \[ D \isin A \equiv D \isin \baseof{A} \lor ( D \in \py \land D \le A ) \]
380 Substitute into the contents of $C$:
381 \[ D \isin C \equiv D \isin \baseof{A} \lor ( D \in \py \land D \le A )
383 Since $D = C \implies D \in \py$,
384 and substituting in $\baseof{C}$, this gives:
385 \[ D \isin C \equiv D \isin \baseof{C} \lor
386 (D \in \py \land D \le A) \lor
387 (D = C \land D \in \py) \]
388 \[ \equiv D \isin \baseof{C} \lor
389 [ D \in \py \land ( D \le A \lor D = C ) ] \]
390 So by Exact Ancestors:
391 \[ D \isin C \equiv D \isin \baseof{C} \lor ( D \in \py \land D \le C
395 \subsection{Base Acyclic}
397 Need to consider only $A, C \in \pn$.
399 For $D = C$: $D \in \pn$ so $D \not\in \py$. OK.
401 For $D \neq C$: $D \isin C \equiv D \isin A$, so by Base Acyclic for
402 $A$, $D \isin C \implies D \not\in \py$.
406 \subsection{Coherence and patch inclusion}
408 Need to consider $D \in \py$
410 \subsubsection{For $A \haspatch P, D = C$:}
416 $ D \isin C \equiv \ldots \lor \true \text{ so } D \haspatch C $.
418 \subsubsection{For $A \haspatch P, D \neq C$:}
419 Ancestors: $ D \le C \equiv D \le A $.
421 Contents: $ D \isin C \equiv D \isin A \lor f $
422 so $ D \isin C \equiv D \isin A $.
425 \[ A \haspatch P \implies C \haspatch P \]
427 \subsubsection{For $A \nothaspatch P$:}
429 Firstly, $C \not\in \py$ since if it were, $A \in \py$.
432 Now by contents of $A$, $D \notin A$, so $D \notin C$.
435 \[ A \nothaspatch P \implies C \nothaspatch P \]
438 \subsection{Foreign inclusion:}
440 If $D = C$, trivial. For $D \neq C$:
441 $D \isin C \equiv D \isin A \equiv D \le A \equiv D \le C$. $\qed$
445 Given $L, R^+, R^-$ where
446 $R^+ \in \pry, R^- = \baseof{R^+}$.
447 Construct $C$ which has $\pr$ removed.
448 Used for removing a branch dependency.
450 C \hasparents \{ L \}
452 \patchof{C} = \patchof{L}
454 \mergeof{C}{L}{R^+}{R^-}
457 \subsection{Conditions}
459 \[ \eqn{ Into Base }{
462 \[ \eqn{ Unique Tip }{
463 \pendsof{L}{\pry} = \{ R^+ \}
465 \[ \eqn{ Currently Included }{
469 \subsection{Ordering of ${L, R^+, R^-}$:}
471 By Unique Tip, $R^+ \le L$. By definition of $\base$, $R^- \le R^+$
472 so $R^- \le L$. So $R^+ \le C$ and $R^- \le C$.
475 (Note that the merge base $R^+ \not\le R^-$, i.e. the merge base is
476 later than one of the branches to be merged.)
478 \subsection{No Replay}
480 No Replay for Merge Results applies. $\qed$
482 \subsection{Desired Contents}
484 \[ D \isin C \equiv [ D \notin \pry \land D \isin L ] \lor D = C \]
487 \subsubsection{For $D = C$:}
489 Trivially $D \isin C$. OK.
491 \subsubsection{For $D \neq C, D \not\le L$:}
493 By No Replay $D \not\isin L$. Also $D \not\le R^-$ hence
494 $D \not\isin R^-$. Thus $D \not\isin C$. OK.
496 \subsubsection{For $D \neq C, D \le L, D \in \pry$:}
498 By Currently Included, $D \isin L$.
500 By Tip Self Inpatch, $D \isin R^+ \equiv D \le R^+$, but by
501 by Unique Tip, $D \le R^+ \equiv D \le L$.
504 By Base Acyclic, $D \not\isin R^-$.
506 Apply $\merge$: $D \not\isin C$. OK.
508 \subsubsection{For $D \neq C, D \le L, D \notin \pry$:}
510 By Tip Contents for $R^+$, $D \isin R^+ \equiv D \isin R^-$.
512 Apply $\merge$: $D \isin C \equiv D \isin L$. OK.
516 \subsection{Unique Base}
518 Into Base means that $C \in \pn$, so Unique Base is not
521 \subsection{Tip Contents}
523 Again, not applicable. $\qed$
525 \subsection{Base Acyclic}
527 By Base Acyclic for $L$, $D \isin L \implies D \not\in \py$.
528 And by Into Base $C \not\in \py$.
529 Now from Desired Contents, above, $D \isin C
530 \implies D \isin L \lor D = C$, which thus
531 $\implies D \not\in \py$. $\qed$.
533 \subsection{Coherence and Patch Inclusion}
535 Need to consider some $D \in \py$. By Into Base, $D \neq C$.
537 \subsubsection{For $\p = \pr$:}
538 By Desired Contents, above, $D \not\isin C$.
539 So $C \nothaspatch \pr$.
541 \subsubsection{For $\p \neq \pr$:}
542 By Desired Contents, $D \isin C \equiv D \isin L$
543 (since $D \in \py$ so $D \not\in \pry$).
545 If $L \nothaspatch \p$, $D \not\isin L$ so $D \not\isin C$.
546 So $L \nothaspatch \p \implies C \nothaspatch \p$.
548 Whereas if $L \haspatch \p$, $D \isin L \equiv D \le L$.
549 so $L \haspatch \p \implies C \haspatch \p$.
553 \section{Foreign Inclusion}
555 Consider some $D$ s.t. $\patchof{D} = \bot$. $D \neq C$.
556 So by Desired Contents $D \isin C \equiv D \isin L$.
557 By Foreign Inclusion of $D$ in $L$, $D \isin L \equiv D \le L$.
559 And $D \le C \equiv D \le L$.
560 Thus $D \isin C \equiv D \le C$.
566 Merge commits $L$ and $R$ using merge base $M$ ($M < L, M < R$):
568 C \hasparents \{ L, R \}
570 \patchof{C} = \patchof{L}
574 We will occasionally use $X,Y$ s.t. $\{X,Y\} = \{L,R\}$.
576 \subsection{Conditions}
578 \[ \eqn{ Tip Merge }{
581 R \in \py : & \baseof{R} \ge \baseof{L}
582 \land [\baseof{L} = M \lor \baseof{L} = \baseof{M}] \\
583 R \in \pn : & M = \baseof{L} \\
584 \text{otherwise} : & \false
587 \[ \eqn{ Merge Acyclic }{
592 \[ \eqn{ Removal Merge Ends }{
593 X \not\haspatch \p \land
597 \pendsof{Y}{\py} = \pendsof{M}{\py}
599 \[ \eqn{ Addition Merge Ends }{
600 X \not\haspatch \p \land
604 \bigforall_{E \in \pendsof{X}{\py}} E \le Y
608 \subsection{Non-Topbloke merges}
610 We require both $\patchof{L} = \bot$ and $\patchof{R} = \bot$.
611 I.e. not only is it forbidden to merge into a Topbloke-controlled
612 branch without Topbloke's assistance, it is also forbidden to
613 merge any Topbloke-controlled branch into any plain git branch.
615 Given those conditions, Tip Merge and Merge Acyclic do not apply.
616 And $Y \not\in \py$ so $\neg [ Y \haspatch \p ]$ so neither
617 Merge Ends condition applies. Good.
619 \subsection{No Replay}
621 No Replay for Merge Results applies. $\qed$
623 \subsection{Unique Base}
625 Need to consider only $C \in \py$, ie $L \in \py$,
626 and calculate $\pendsof{C}{\pn}$. So we will consider some
627 putative ancestor $A \in \pn$ and see whether $A \le C$.
629 By Exact Ancestors for C, $A \le C \equiv A \le L \lor A \le R \lor A = C$.
630 But $C \in py$ and $A \in \pn$ so $A \neq C$.
631 Thus $A \le C \equiv A \le L \lor A \le R$.
633 By Unique Base of L and Transitive Ancestors,
634 $A \le L \equiv A \le \baseof{L}$.
636 \subsubsection{For $R \in \py$:}
638 By Unique Base of $R$ and Transitive Ancestors,
639 $A \le R \equiv A \le \baseof{R}$.
641 But by Tip Merge condition on $\baseof{R}$,
642 $A \le \baseof{L} \implies A \le \baseof{R}$, so
643 $A \le \baseof{R} \lor A \le \baseof{L} \equiv A \le \baseof{R}$.
644 Thus $A \le C \equiv A \le \baseof{R}$.
645 That is, $\baseof{C} = \baseof{R}$.
647 \subsubsection{For $R \in \pn$:}
649 By Tip Merge condition on $R$ and since $M \le R$,
650 $A \le \baseof{L} \implies A \le R$, so
651 $A \le R \lor A \le \baseof{L} \equiv A \le R$.
652 Thus $A \le C \equiv A \le R$.
653 That is, $\baseof{C} = R$.
657 \subsection{Coherence and Patch Inclusion}
659 Need to determine $C \haspatch \p$ based on $L,M,R \haspatch \p$.
660 This involves considering $D \in \py$.
662 \subsubsection{For $L \nothaspatch \p, R \nothaspatch \p$:}
663 $D \not\isin L \land D \not\isin R$. $C \not\in \py$ (otherwise $L
664 \in \py$ ie $L \haspatch \p$ by Tip Self Inpatch). So $D \neq C$.
665 Applying $\merge$ gives $D \not\isin C$ i.e. $C \nothaspatch \p$.
667 \subsubsection{For $L \haspatch \p, R \haspatch \p$:}
668 $D \isin L \equiv D \le L$ and $D \isin R \equiv D \le R$.
669 (Likewise $D \isin X \equiv D \le X$ and $D \isin Y \equiv D \le Y$.)
671 Consider $D = C$: $D \isin C$, $D \le C$, OK for $C \haspatch \p$.
673 For $D \neq C$: $D \le C \equiv D \le L \lor D \le R
674 \equiv D \isin L \lor D \isin R$.
675 (Likewise $D \le C \equiv D \le X \lor D \le Y$.)
677 Consider $D \neq C, D \isin X \land D \isin Y$:
678 By $\merge$, $D \isin C$. Also $D \le X$
679 so $D \le C$. OK for $C \haspatch \p$.
681 Consider $D \neq C, D \not\isin X \land D \not\isin Y$:
682 By $\merge$, $D \not\isin C$.
683 And $D \not\le X \land D \not\le Y$ so $D \not\le C$.
684 OK for $C \haspatch \p$.
686 Remaining case, wlog, is $D \not\isin X \land D \isin Y$.
687 $D \not\le X$ so $D \not\le M$ so $D \not\isin M$.
688 Thus by $\merge$, $D \isin C$. And $D \le Y$ so $D \le C$.
689 OK for $C \haspatch \p$.
691 So indeed $L \haspatch \p \land R \haspatch \p \implies C \haspatch \p$.
693 \subsubsection{For (wlog) $X \not\haspatch \p, Y \haspatch \p$:}
695 $M \haspatch \p \implies C \nothaspatch \p$.
696 $M \nothaspatch \p \implies C \haspatch \p$.
700 One of the Merge Ends conditions applies.
701 Recall that we are considering $D \in \py$.
702 $D \isin Y \equiv D \le Y$. $D \not\isin X$.
703 We will show for each of
704 various cases that $D \isin C \equiv M \nothaspatch \p \land D \le C$
705 (which suffices by definition of $\haspatch$ and $\nothaspatch$).
707 Consider $D = C$: Thus $C \in \py, L \in \py$, and by Tip
708 Self Inpatch $L \haspatch \p$, so $L=Y, R=X$. By Tip Merge,
709 $M=\baseof{L}$. So by Base Acyclic $D \not\isin M$, i.e.
710 $M \nothaspatch \p$. And indeed $D \isin C$ and $D \le C$. OK.
712 Consider $D \neq C, M \nothaspatch P, D \isin Y$:
713 $D \le Y$ so $D \le C$.
714 $D \not\isin M$ so by $\merge$, $D \isin C$. OK.
716 Consider $D \neq C, M \nothaspatch P, D \not\isin Y$:
717 $D \not\le Y$. If $D \le X$ then
718 $D \in \pancsof{X}{\py}$, so by Addition Merge Ends and
719 Transitive Ancestors $D \le Y$ --- a contradiction, so $D \not\le X$.
720 Thus $D \not\le C$. By $\merge$, $D \not\isin C$. OK.
722 Consider $D \neq C, M \haspatch P, D \isin Y$:
723 $D \le Y$ so $D \in \pancsof{Y}{\py}$ so by Removal Merge Ends
724 and Transitive Ancestors $D \in \pancsof{M}{\py}$ so $D \le M$.
725 Thus $D \isin M$. By $\merge$, $D \not\isin C$. OK.
727 Consider $D \neq C, M \haspatch P, D \not\isin Y$:
728 By $\merge$, $D \not\isin C$. OK.
732 \subsection{Base Acyclic}
734 This applies when $C \in \pn$.
735 $C \in \pn$ when $L \in \pn$ so by Merge Acyclic, $R \nothaspatch \p$.
737 Consider some $D \in \py$.
739 By Base Acyclic of $L$, $D \not\isin L$. By the above, $D \not\isin
740 R$. And $D \neq C$. So $D \not\isin C$.
744 \subsection{Tip Contents}
746 We need worry only about $C \in \py$.
747 And $\patchof{C} = \patchof{L}$
748 so $L \in \py$ so $L \haspatch \p$. We will use the Unique Base
749 of $C$, and its Coherence and Patch Inclusion, as just proved.
751 Firstly we show $C \haspatch \p$: If $R \in \py$, then $R \haspatch
752 \p$ and by Coherence/Inclusion $C \haspatch \p$ . If $R \not\in \py$
753 then by Tip Merge $M = \baseof{L}$ so by Base Acyclic and definition
754 of $\nothaspatch$, $M \nothaspatch \p$. So by Coherence/Inclusion $C
755 \haspatch \p$ (whether $R \haspatch \p$ or $\nothaspatch$).
757 We will consider an arbitrary commit $D$
758 and prove the Exclusive Tip Contents form.
760 \subsubsection{For $D \in \py$:}
761 $C \haspatch \p$ so by definition of $\haspatch$, $D \isin C \equiv D
764 \subsubsection{For $D \not\in \py, R \not\in \py$:}
766 $D \neq C$. By Tip Contents of $L$,
767 $D \isin L \equiv D \isin \baseof{L}$, and by Tip Merge condition,
768 $D \isin L \equiv D \isin M$. So by definition of $\merge$, $D \isin
769 C \equiv D \isin R$. And $R = \baseof{C}$ by Unique Base of $C$.
770 Thus $D \isin C \equiv D \isin \baseof{C}$. OK.
772 \subsubsection{For $D \not\in \py, R \in \py$:}
777 $D \isin L \equiv D \isin \baseof{L}$ and
778 $D \isin R \equiv D \isin \baseof{R}$.
780 If $\baseof{L} = M$, trivially $D \isin M \equiv D \isin \baseof{L}.$
781 Whereas if $\baseof{L} = \baseof{M}$, by definition of $\base$,
782 $\patchof{M} = \patchof{L} = \py$, so by Tip Contents of $M$,
783 $D \isin M \equiv D \isin \baseof{M} \equiv D \isin \baseof{L}$.
785 So $D \isin M \equiv D \isin L$ and by $\merge$,
786 $D \isin C \equiv D \isin R$. But from Unique Base,
787 $\baseof{C} = R$ so $D \isin C \equiv D \isin \baseof{C}$. OK.
791 \subsection{Foreign Inclusion}
793 Consider some $D$ s.t. $\patchof{D} = \bot$.
794 By Foreign Inclusion of $L, M, R$:
795 $D \isin L \equiv D \le L$;
796 $D \isin M \equiv D \le M$;
797 $D \isin R \equiv D \le R$.
799 \subsubsection{For $D = C$:}
801 $D \isin C$ and $D \le C$. OK.
803 \subsubsection{For $D \neq C, D \isin M$:}
805 Thus $D \le M$ so $D \le L$ and $D \le R$ so $D \isin L$ and $D \isin
806 R$. So by $\merge$, $D \isin C$. And $D \le C$. OK.
808 \subsubsection{For $D \neq C, D \not\isin M, D \isin X$:}
810 By $\merge$, $D \isin C$.
811 And $D \isin X$ means $D \le X$ so $D \le C$.
814 \subsubsection{For $D \neq C, D \not\isin M, D \not\isin L, D \not\isin R$:}
816 By $\merge$, $D \not\isin C$.
817 And $D \not\le L, D \not\le R$ so $D \not\le C$.