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 $\patchof{L} = \bot$, xxx
497 Trivial from Base Acyclic for $L$. $\qed$
499 \subsection{Unique Base}
501 Not applicable. $\qed$
503 \subsection{Tip Contents}
505 Not applicable. $\qed$
507 \subsection{Base Acyclic}
519 Given $L$ and $\pr$ as represented by $R^+, R^-$.
520 Construct $C$ which has $\pr$ removed.
521 Used for removing a branch dependency.
523 C \hasparents \{ L \}
525 \patchof{C} = \patchof{L}
527 \mergeof{C}{L}{R^+}{R^-}
530 \subsection{Conditions}
532 \[ \eqn{ Ingredients }{
533 R^+ \in \pry \land R^- = \baseof{R^+}
535 \[ \eqn{ Into Base }{
538 \[ \eqn{ Unique Tip }{
539 \pendsof{L}{\pry} = \{ R^+ \}
541 \[ \eqn{ Currently Included }{
545 \subsection{Ordering of ${L, R^+, R^-}$:}
547 By Unique Tip, $R^+ \le L$. By definition of $\base$, $R^- \le R^+$
548 so $R^- \le L$. So $R^+ \le C$ and $R^- \le C$.
551 (Note that $R^+ \not\le R^-$, i.e. the merge base
552 is a descendant, not an ancestor, of the 2nd parent.)
554 \subsection{No Replay}
556 No Replay for Merge Results applies. $\qed$
558 \subsection{Desired Contents}
560 \[ D \isin C \equiv [ D \notin \pry \land D \isin L ] \lor D = C \]
563 \subsubsection{For $D = C$:}
565 Trivially $D \isin C$. OK.
567 \subsubsection{For $D \neq C, D \not\le L$:}
569 By No Replay $D \not\isin L$. Also $D \not\le R^-$ hence
570 $D \not\isin R^-$. Thus $D \not\isin C$. OK.
572 \subsubsection{For $D \neq C, D \le L, D \in \pry$:}
574 By Currently Included, $D \isin L$.
576 By Tip Self Inpatch, $D \isin R^+ \equiv D \le R^+$, but by
577 by Unique Tip, $D \le R^+ \equiv D \le L$.
580 By Base Acyclic, $D \not\isin R^-$.
582 Apply $\merge$: $D \not\isin C$. OK.
584 \subsubsection{For $D \neq C, D \le L, D \notin \pry$:}
586 By Tip Contents for $R^+$, $D \isin R^+ \equiv D \isin R^-$.
588 Apply $\merge$: $D \isin C \equiv D \isin L$. OK.
592 \subsection{Unique Base}
594 Into Base means that $C \in \pn$, so Unique Base is not
597 \subsection{Tip Contents}
599 Again, not applicable. $\qed$
601 \subsection{Base Acyclic}
603 By Base Acyclic for $L$, $D \isin L \implies D \not\in \py$.
604 And by Into Base $C \not\in \py$.
605 Now from Desired Contents, above, $D \isin C
606 \implies D \isin L \lor D = C$, which thus
607 $\implies D \not\in \py$. $\qed$.
609 \subsection{Coherence and Patch Inclusion}
611 Need to consider some $D \in \py$. By Into Base, $D \neq C$.
613 \subsubsection{For $\p = \pr$:}
614 By Desired Contents, above, $D \not\isin C$.
615 So $C \nothaspatch \pr$.
617 \subsubsection{For $\p \neq \pr$:}
618 By Desired Contents, $D \isin C \equiv D \isin L$
619 (since $D \in \py$ so $D \not\in \pry$).
621 If $L \nothaspatch \p$, $D \not\isin L$ so $D \not\isin C$.
622 So $L \nothaspatch \p \implies C \nothaspatch \p$.
624 Whereas if $L \haspatch \p$, $D \isin L \equiv D \le L$.
625 so $L \haspatch \p \implies C \haspatch \p$.
629 \subsection{Foreign Inclusion}
631 Consider some $D$ s.t. $\patchof{D} = \bot$. $D \neq C$.
632 So by Desired Contents $D \isin C \equiv D \isin L$.
633 By Foreign Inclusion of $D$ in $L$, $D \isin L \equiv D \le L$.
635 And $D \le C \equiv D \le L$.
636 Thus $D \isin C \equiv D \le C$.
640 \subsection{Foreign Contents}
642 Not applicable. $\qed$
646 Merge commits $L$ and $R$ using merge base $M$:
648 C \hasparents \{ L, R \}
650 \patchof{C} = \patchof{L}
654 We will occasionally use $X,Y$ s.t. $\{X,Y\} = \{L,R\}$.
656 \subsection{Conditions}
657 \[ \eqn{ Ingredients }{
660 \[ \eqn{ Tip Merge }{
663 R \in \py : & \baseof{R} \ge \baseof{L}
664 \land [\baseof{L} = M \lor \baseof{L} = \baseof{M}] \\
665 R \in \pn : & M = \baseof{L} \\
666 \text{otherwise} : & \false
669 \[ \eqn{ Merge Acyclic }{
674 \[ \eqn{ Removal Merge Ends }{
675 X \not\haspatch \p \land
679 \pendsof{Y}{\py} = \pendsof{M}{\py}
681 \[ \eqn{ Addition Merge Ends }{
682 X \not\haspatch \p \land
686 \bigforall_{E \in \pendsof{X}{\py}} E \le Y
689 \[ \eqn{ Foreign Merges }{
690 \patchof{L} = \bot \equiv \patchof{R} = \bot
693 \subsection{Non-Topbloke merges}
695 We require both $\patchof{L} = \bot$ and $\patchof{R} = \bot$
696 (Foreign Merges, above).
697 I.e. not only is it forbidden to merge into a Topbloke-controlled
698 branch without Topbloke's assistance, it is also forbidden to
699 merge any Topbloke-controlled branch into any plain git branch.
701 Given those conditions, Tip Merge and Merge Acyclic do not apply.
702 And $Y \not\in \py$ so $\neg [ Y \haspatch \p ]$ so neither
703 Merge Ends condition applies.
705 So a plain git merge of non-Topbloke branches meets the conditions and
706 is therefore consistent with our scheme.
708 \subsection{No Replay}
710 No Replay for Merge Results applies. $\qed$
712 \subsection{Unique Base}
714 Need to consider only $C \in \py$, ie $L \in \py$,
715 and calculate $\pendsof{C}{\pn}$. So we will consider some
716 putative ancestor $A \in \pn$ and see whether $A \le C$.
718 By Exact Ancestors for C, $A \le C \equiv A \le L \lor A \le R \lor A = C$.
719 But $C \in py$ and $A \in \pn$ so $A \neq C$.
720 Thus $A \le C \equiv A \le L \lor A \le R$.
722 By Unique Base of L and Transitive Ancestors,
723 $A \le L \equiv A \le \baseof{L}$.
725 \subsubsection{For $R \in \py$:}
727 By Unique Base of $R$ and Transitive Ancestors,
728 $A \le R \equiv A \le \baseof{R}$.
730 But by Tip Merge condition on $\baseof{R}$,
731 $A \le \baseof{L} \implies A \le \baseof{R}$, so
732 $A \le \baseof{R} \lor A \le \baseof{L} \equiv A \le \baseof{R}$.
733 Thus $A \le C \equiv A \le \baseof{R}$.
734 That is, $\baseof{C} = \baseof{R}$.
736 \subsubsection{For $R \in \pn$:}
738 By Tip Merge condition on $R$ and since $M \le R$,
739 $A \le \baseof{L} \implies A \le R$, so
740 $A \le R \lor A \le \baseof{L} \equiv A \le R$.
741 Thus $A \le C \equiv A \le R$.
742 That is, $\baseof{C} = R$.
746 \subsection{Coherence and Patch Inclusion}
748 Need to determine $C \haspatch \p$ based on $L,M,R \haspatch \p$.
749 This involves considering $D \in \py$.
751 \subsubsection{For $L \nothaspatch \p, R \nothaspatch \p$:}
752 $D \not\isin L \land D \not\isin R$. $C \not\in \py$ (otherwise $L
753 \in \py$ ie $L \haspatch \p$ by Tip Self Inpatch). So $D \neq C$.
754 Applying $\merge$ gives $D \not\isin C$ i.e. $C \nothaspatch \p$.
756 \subsubsection{For $L \haspatch \p, R \haspatch \p$:}
757 $D \isin L \equiv D \le L$ and $D \isin R \equiv D \le R$.
758 (Likewise $D \isin X \equiv D \le X$ and $D \isin Y \equiv D \le Y$.)
760 Consider $D = C$: $D \isin C$, $D \le C$, OK for $C \haspatch \p$.
762 For $D \neq C$: $D \le C \equiv D \le L \lor D \le R
763 \equiv D \isin L \lor D \isin R$.
764 (Likewise $D \le C \equiv D \le X \lor D \le Y$.)
766 Consider $D \neq C, D \isin X \land D \isin Y$:
767 By $\merge$, $D \isin C$. Also $D \le X$
768 so $D \le C$. OK for $C \haspatch \p$.
770 Consider $D \neq C, D \not\isin X \land D \not\isin Y$:
771 By $\merge$, $D \not\isin C$.
772 And $D \not\le X \land D \not\le Y$ so $D \not\le C$.
773 OK for $C \haspatch \p$.
775 Remaining case, wlog, is $D \not\isin X \land D \isin Y$.
776 $D \not\le X$ so $D \not\le M$ so $D \not\isin M$.
777 Thus by $\merge$, $D \isin C$. And $D \le Y$ so $D \le C$.
778 OK for $C \haspatch \p$.
780 So indeed $L \haspatch \p \land R \haspatch \p \implies C \haspatch \p$.
782 \subsubsection{For (wlog) $X \not\haspatch \p, Y \haspatch \p$:}
784 $M \haspatch \p \implies C \nothaspatch \p$.
785 $M \nothaspatch \p \implies C \haspatch \p$.
789 One of the Merge Ends conditions applies.
790 Recall that we are considering $D \in \py$.
791 $D \isin Y \equiv D \le Y$. $D \not\isin X$.
792 We will show for each of
793 various cases that $D \isin C \equiv M \nothaspatch \p \land D \le C$
794 (which suffices by definition of $\haspatch$ and $\nothaspatch$).
796 Consider $D = C$: Thus $C \in \py, L \in \py$, and by Tip
797 Self Inpatch $L \haspatch \p$, so $L=Y, R=X$. By Tip Merge,
798 $M=\baseof{L}$. So by Base Acyclic $D \not\isin M$, i.e.
799 $M \nothaspatch \p$. And indeed $D \isin C$ and $D \le C$. OK.
801 Consider $D \neq C, M \nothaspatch P, D \isin Y$:
802 $D \le Y$ so $D \le C$.
803 $D \not\isin M$ so by $\merge$, $D \isin C$. OK.
805 Consider $D \neq C, M \nothaspatch P, D \not\isin Y$:
806 $D \not\le Y$. If $D \le X$ then
807 $D \in \pancsof{X}{\py}$, so by Addition Merge Ends and
808 Transitive Ancestors $D \le Y$ --- a contradiction, so $D \not\le X$.
809 Thus $D \not\le C$. By $\merge$, $D \not\isin C$. OK.
811 Consider $D \neq C, M \haspatch P, D \isin Y$:
812 $D \le Y$ so $D \in \pancsof{Y}{\py}$ so by Removal Merge Ends
813 and Transitive Ancestors $D \in \pancsof{M}{\py}$ so $D \le M$.
814 Thus $D \isin M$. By $\merge$, $D \not\isin C$. OK.
816 Consider $D \neq C, M \haspatch P, D \not\isin Y$:
817 By $\merge$, $D \not\isin C$. OK.
821 \subsection{Base Acyclic}
823 This applies when $C \in \pn$.
824 $C \in \pn$ when $L \in \pn$ so by Merge Acyclic, $R \nothaspatch \p$.
826 Consider some $D \in \py$.
828 By Base Acyclic of $L$, $D \not\isin L$. By the above, $D \not\isin
829 R$. And $D \neq C$. So $D \not\isin C$.
833 \subsection{Tip Contents}
835 We need worry only about $C \in \py$.
836 And $\patchof{C} = \patchof{L}$
837 so $L \in \py$ so $L \haspatch \p$. We will use the Unique Base
838 of $C$, and its Coherence and Patch Inclusion, as just proved.
840 Firstly we show $C \haspatch \p$: If $R \in \py$, then $R \haspatch
841 \p$ and by Coherence/Inclusion $C \haspatch \p$ . If $R \not\in \py$
842 then by Tip Merge $M = \baseof{L}$ so by Base Acyclic and definition
843 of $\nothaspatch$, $M \nothaspatch \p$. So by Coherence/Inclusion $C
844 \haspatch \p$ (whether $R \haspatch \p$ or $\nothaspatch$).
846 We will consider an arbitrary commit $D$
847 and prove the Exclusive Tip Contents form.
849 \subsubsection{For $D \in \py$:}
850 $C \haspatch \p$ so by definition of $\haspatch$, $D \isin C \equiv D
853 \subsubsection{For $D \not\in \py, R \not\in \py$:}
855 $D \neq C$. By Tip Contents of $L$,
856 $D \isin L \equiv D \isin \baseof{L}$, and by Tip Merge condition,
857 $D \isin L \equiv D \isin M$. So by definition of $\merge$, $D \isin
858 C \equiv D \isin R$. And $R = \baseof{C}$ by Unique Base of $C$.
859 Thus $D \isin C \equiv D \isin \baseof{C}$. OK.
861 \subsubsection{For $D \not\in \py, R \in \py$:}
866 $D \isin L \equiv D \isin \baseof{L}$ and
867 $D \isin R \equiv D \isin \baseof{R}$.
869 If $\baseof{L} = M$, trivially $D \isin M \equiv D \isin \baseof{L}.$
870 Whereas if $\baseof{L} = \baseof{M}$, by definition of $\base$,
871 $\patchof{M} = \patchof{L} = \py$, so by Tip Contents of $M$,
872 $D \isin M \equiv D \isin \baseof{M} \equiv D \isin \baseof{L}$.
874 So $D \isin M \equiv D \isin L$ and by $\merge$,
875 $D \isin C \equiv D \isin R$. But from Unique Base,
876 $\baseof{C} = R$ so $D \isin C \equiv D \isin \baseof{C}$. OK.
880 \subsection{Foreign Inclusion}
882 Consider some $D$ s.t. $\patchof{D} = \bot$.
883 By Foreign Inclusion of $L, M, R$:
884 $D \isin L \equiv D \le L$;
885 $D \isin M \equiv D \le M$;
886 $D \isin R \equiv D \le R$.
888 \subsubsection{For $D = C$:}
890 $D \isin C$ and $D \le C$. OK.
892 \subsubsection{For $D \neq C, D \isin M$:}
894 Thus $D \le M$ so $D \le L$ and $D \le R$ so $D \isin L$ and $D \isin
895 R$. So by $\merge$, $D \isin C$. And $D \le C$. OK.
897 \subsubsection{For $D \neq C, D \not\isin M, D \isin X$:}
899 By $\merge$, $D \isin C$.
900 And $D \isin X$ means $D \le X$ so $D \le C$.
903 \subsubsection{For $D \neq C, D \not\isin M, D \not\isin L, D \not\isin R$:}
905 By $\merge$, $D \not\isin C$.
906 And $D \not\le L, D \not\le R$ so $D \not\le C$.
911 \subsection{Foreign Contents}
913 Only relevant if $\patchof{L} = \bot$, in which case
914 $\patchof{C} = \bot$ and by Foreign Merges $\patchof{R} = \bot$,
915 so Totally Foreign Contents applies. $\qed$