chiark / gitweb /
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9 %\usepackage{accents}
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22 \newcommand{\patchisin}{\sqSubset}
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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}}
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42 %\newcommand{\hasparents}{{%
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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}}
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85 \newcommand{\proofstarts}{{\it Proof:}}
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94 \begin{document}
96 \section{Notation}
98 \begin{basedescript}{
99 \desclabelwidth{5em}
100 \desclabelstyle{\nextlinelabel}
101 }
102 \item[ $C \hasparents \set X$ ]
103 The parents of commit $C$ are exactly the set
104 $\set X$.
106 \item[ $C \ge D$ ]
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
118 is not transitive.
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
157 patch.  If 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 167 \begin{cases} 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 171 \end{cases} 172 174 \end{basedescript} 175 \newpage 176 \section{Invariants} 178 We maintain these each time we construct a new commit. \\ 179 $\eqn{No Replay:}{ 180 C \has D \implies C \ge D 181 }$ 182 $\eqn{Unique Base:}{ 183 \bigforall_{C \in \py} \pendsof{C}{\pn} = \{ B \} 184 }$ 185 $\eqn{Tip Contents:}{ 186 \bigforall_{C \in \py} D \isin C \equiv 187 { D \isin \baseof{C} \lor \atop 188 (D \in \py \land D \le C) } 189 }$ 190 $\eqn{Base Acyclic:}{ 191 \bigforall_{B \in \pn} D \isin B \implies D \notin \py 192 }$ 193 $\eqn{Coherence:}{ 194 \bigforall_{C,\p} C \haspatch \p \lor C \nothaspatch \p 195 }$ 196 $\eqn{Foreign Inclusion:}{ 197 \bigforall_{D \text{ s.t. } \patchof{D} = \bot} D \isin C \equiv D \leq C 198 }$ 199 $\eqn{Foreign Contents:}{ 200 \bigforall_{C \text{ s.t. } \patchof{C} = \bot} 201 D \le C \implies \patchof{D} = \bot 202 }$ 204 \section{Some lemmas} 206 $\eqn{Alternative (overlapping) formulations defining 207 \mergeof{C}{L}{M}{R}:}{ 208 D \isin C \equiv 209 \begin{cases} 210 D \isin L \equiv D \isin R : & D = C \lor D \isin L \\ 211 D \isin L \nequiv D \isin R : & D = C \lor D \not\isin M \\ 212 D \isin L \equiv D \isin M : & D = C \lor D \isin R \\ 213 D \isin L \nequiv D \isin M : & D = C \lor D \isin L \\ 214 \text{as above with L and R exchanged} 215 \end{cases} 216 }$ 217 \proof{ 218 Truth table xxx 220 Original definition is symmetrical in$L$and$R$. 221 } 223 $\eqn{Exclusive Tip Contents:}{ 224 \bigforall_{C \in \py} 225 \neg \Bigl[ D \isin \baseof{C} \land ( D \in \py \land D \le C ) 226 \Bigr] 227 }$ 228 Ie, the two limbs of the RHS of Tip Contents are mutually exclusive. 230 \proof{ 231 Let$B = \baseof{C}$in$D \isin \baseof{C}$. Now$B \in \pn$. 232 So by Base Acyclic$D \isin B \implies D \notin \py$. 233 } 234 $\eqntag{{\it Corollary - equivalent to Tip Contents}}{ 235 \bigforall_{C \in \py} D \isin C \equiv 236 \begin{cases} 237 D \in \py : & D \le C \\ 238 D \not\in \py : & D \isin \baseof{C} 239 \end{cases} 240 }$ 242 $\eqn{Tip Self Inpatch:}{ 243 \bigforall_{C \in \py} C \haspatch \p 244 }$ 245 Ie, tip commits contain their own patch. 247 \proof{ 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 $251 } 253 $\eqn{Exact Ancestors:}{ 254 \bigforall_{ C \hasparents \set{R} } 255 D \le C \equiv 256 ( \mathop{\hbox{\huge{\vee}}}_{R \in \set R} D \le R ) 257 \lor D = C 258 }$ 259 xxx proof tbd 261 $\eqn{Transitive Ancestors:}{ 262 \left[ \bigforall_{ E \in \pendsof{C}{\set P} } E \le M \right] \equiv 263 \left[ \bigforall_{ A \in \pancsof{C}{\set P} } A \le M \right] 264 }$ 266 \proof{ 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''$. 277 } 279 $\eqn{Calculation Of Ends:}{ 280 \bigforall_{C \hasparents \set A} 281 \pendsof{C}{\set P} = 282 \begin{cases} 283 C \in \p : & \{ C \} 284 \\ 285 C \not\in \p : & \displaystyle 286 \left\{ E \Big| 287 \Bigl[ \Largeexists_{A \in \set A} 288 E \in \pendsof{A}{\set P} \Bigr] \land 289 \Bigl[ \Largenexists_{B \in \set A} 290 E \neq B \land E \le B \Bigr] 291 \right\} 292 \end{cases} 293 }$ 294 xxx proof tbd 296 $\eqn{Totally Foreign Contents:}{ 297 \bigforall_{C \hasparents \set A} 298 \left[ 299 \patchof{C} = \bot \land 300 \bigforall_{A \in \set A} \patchof{A} = \bot 301 \right] 302 \implies 303 \left[ 304 D \le C 305 \implies 306 \patchof{D} = \bot 307 \right] 308 }$ 309 \proof{ 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$A$,$\patchof{D} = \bot$. 313 } 315 \subsection{No Replay for Merge Results} 317 If we are constructing$C$, with, 318 \gathbegin 319 \mergeof{C}{L}{M}{R} 320 \gathnext 321 L \le C 322 \gathnext 323 R \le C 324 \end{gather} 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$D \neq C \land D \not\isin L \land D \not\isin R$:} 333$D \not\isin C$. OK. 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$D \le L \lor D \le
338 R$so$D \le C$. OK. 340 \subsubsection{For$D \neq C \land (D \isin L \equiv D \not\isin R)
341  \land D \isin M$:} 342$D \not\isin C$. OK. 344$\qed$346 \section{Commit annotation} 348 We annotate each Topbloke commit$C$with: 349 \gathbegin 350 \patchof{C} 351 \gathnext 352 \baseof{C}, \text{ if } C \in \py 353 \gathnext 354 \bigforall_{\pa{Q}} 355 \text{ either } C \haspatch \pa{Q} \text{ or } C \nothaspatch \pa{Q} 356 \gathnext 357 \bigforall_{\pay{Q} \not\ni C} \pendsof{C}{\pay{Q}} 358 \end{gather} 360$\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}$368 is in stated 369 (in terms of$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$as made by git-commit. 383 \begin{gather} 384 \tag*{} C \hasparents \{ A \} \\ 385 \tag*{} \patchof{C} = \patchof{A} \\ 386 \tag*{} D \isin C \equiv D \isin A \lor D = C 387 \end{gather} 388 This also covers Topbloke-generated commits on plain git branches: 389 Topbloke strips the metadata when exporting. 391 \subsection{No Replay} 392 Trivial. 394 \subsection{Unique Base} 395 If$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 $D \isin C \equiv D \isin \baseof{A} \lor ( D \in \py \land D \le A ) 405 \lor D = C$ 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 $D \isin C \equiv D \isin \baseof{C} \lor ( D \in \py \land D \le C 415 )$ 416$\qed$418 \subsection{Base Acyclic} 420 Need to consider only$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$. 427$\qed$429 \subsection{Coherence and patch inclusion} 431 Need to consider$D \in \py$433 \subsubsection{For$A \haspatch P, D = C$:} 435 Ancestors of$C$: 436$ D \le C $. 438 Contents of$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 $. 447 So: 448 $A \haspatch P \implies C \haspatch P$ 450 \subsubsection{For$A \nothaspatch P$:} 452 Firstly,$C \not\in \py$since if it were,$A \in \py$. 453 Thus$D \neq C$. 455 Now by contents of$A$,$D \notin A$, so$D \notin C$. 457 So: 458 $A \nothaspatch P \implies C \nothaspatch P$ 459$\qed$461 \subsection{Foreign inclusion:} 463 If$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$. 474 \gathbegin 475 B \hasparents \{ L \} 476 \gathnext 477 \patchof{B} = \pa{B} 478 \gathnext 479 D \isin B \equiv D \isin L \lor D = B 480 \end{gather} 482 \subsection{Conditions} 484 $\eqn{ Ingredients }{ 485 \patchof{L} = \pa{L} \lor \patchof{L} = \bot 486 }$ 487 $\eqn{ Non-recursion }{ 488 L \not\in \pa{B} 489 }$ 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$507 \subsection{Base Acyclic} 509 xxx 511 xxx unfinished 513 \section{Create Tip} 515 xxx tbd 517 \section{Anticommit} 519 Given$L$and$\pr$as represented by$R^+, R^-$. 520 Construct$C$which has$\pr$removed. 521 Used for removing a branch dependency. 522 \gathbegin 523 C \hasparents \{ L \} 524 \gathnext 525 \patchof{C} = \patchof{L} 526 \gathnext 527 \mergeof{C}{L}{R^+}{R^-} 528 \end{gather} 530 \subsection{Conditions} 532 $\eqn{ Ingredients }{ 533 R^+ \in \pry \land R^- = \baseof{R^+} 534 }$ 535 $\eqn{ Into Base }{ 536 L \in \pn 537 }$ 538 $\eqn{ Unique Tip }{ 539 \pendsof{L}{\pry} = \{ R^+ \} 540 }$ 541 $\eqn{ Currently Included }{ 542 L \haspatch \pry 543 }$ 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$. 549$\qed$551 (Note that$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 $D \isin C \equiv [ D \notin \pry \land D \isin L ] \lor D = C$ 561 \proofstarts 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,$D \le R^+ \equiv D \le L$. 578 So$D \isin R^+$. 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. 590$\qed$592 \subsection{Unique Base} 594 Into Base means that$C \in \pn$, so Unique Base is not 595 applicable.$\qed$597 \subsection{Tip Contents} 599 Again, not applicable.$\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$. 627$\qed$629 \subsection{Foreign Inclusion} 631 Consider some$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$. 638$\qed$640 \subsection{Foreign Contents} 642 Not applicable.$\qed$644 \section{Merge} 646 Merge commits$L$and$R$using merge base$M$: 647 \gathbegin 648 C \hasparents \{ L, R \} 649 \gathnext 650 \patchof{C} = \patchof{L} 651 \gathnext 652 \mergeof{C}{L}{M}{R} 653 \end{gather} 654 We will occasionally use$X,Y$s.t.$\{X,Y\} = \{L,R\}$. 656 \subsection{Conditions} 657 $\eqn{ Ingredients }{ 658 M \le L, M \le R 659 }$ 660 $\eqn{ Tip Merge }{ 661 L \in \py \implies 662 \begin{cases} 663 R \in \py : & \baseof{R} \ge \baseof{L} 664 \land [\baseof{L} = M \lor \baseof{L} = \baseof{M}] \\ 665 R \in \pn : & M = \baseof{L} \\ 666 \text{otherwise} : & \false 667 \end{cases} 668 }$ 669 $\eqn{ Merge Acyclic }{ 670 L \in \pn 671 \implies 672 R \nothaspatch \p 673 }$ 674 $\eqn{ Removal Merge Ends }{ 675 X \not\haspatch \p \land 676 Y \haspatch \p \land 677 M \haspatch \p 678 \implies 679 \pendsof{Y}{\py} = \pendsof{M}{\py} 680 }$ 681 $\eqn{ Addition Merge Ends }{ 682 X \not\haspatch \p \land 683 Y \haspatch \p \land 684 M \nothaspatch \p 685 \implies \left[ 686 \bigforall_{E \in \pendsof{X}{\py}} E \le Y 687 \right] 688 }$ 689 $\eqn{ Foreign Merges }{ 690 \patchof{L} = \bot \equiv \patchof{R} = \bot 691 }$ 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$. 744$\qed$746 \subsection{Coherence and Patch Inclusion} 748 Need to determine$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$. 787 \proofstarts 789 One of the Merge Ends conditions applies. 790 Recall that we are considering$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$. OK. 819$\qed$821 \subsection{Base Acyclic} 823 This applies when$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$. 831$\qed$833 \subsection{Tip Contents} 835 We need worry only about$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
851 \le C$. OK. 853 \subsubsection{For$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$:} 863$D \neq C$. 865 By Tip Contents 866$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}$. OK. 878$\qed$880 \subsection{Foreign Inclusion} 882 Consider some$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$. 901 OK. 903 \subsubsection{For$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$. 907 OK 909$\qed$911 \subsection{Foreign Contents} 913 Only relevant if$\patchof{L} = \bot$, in which case 914$\patchof{C} = \bot$and by Foreign Merges$\patchof{R} = \bot$, 915 so Totally Foreign Contents applies.$\qed\$
917 \end{document}