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
174 Some (overlapping) alternative formulations:
176 $\displaystyle D \isin C \equiv
178 D \isin L \equiv D \isin R : & D = C \lor D \isin L \\
179 D \isin L \equiv D \isin R : & D = C \lor D \isin R \\
180 D \isin L \nequiv D \isin R : & D = C \lor D \not\isin M \\
181 D \isin M \equiv D \isin L : & D = C \lor D \isin R \\
182 D \isin M \equiv D \isin R : & D = C \lor D \isin L \\
190 We maintain these each time we construct a new commit. \\
192 C \has D \implies C \ge D
194 \[\eqn{Unique Base:}{
195 \bigforall_{C \in \py} \pendsof{C}{\pn} = \{ B \}
197 \[\eqn{Tip Contents:}{
198 \bigforall_{C \in \py} D \isin C \equiv
199 { D \isin \baseof{C} \lor \atop
200 (D \in \py \land D \le C) }
202 \[\eqn{Base Acyclic:}{
203 \bigforall_{B \in \pn} D \isin B \implies D \notin \py
206 \bigforall_{C,\p} C \haspatch \p \lor C \nothaspatch \p
208 \[\eqn{Foreign Inclusion:}{
209 \bigforall_{D \text{ s.t. } \patchof{D} = \bot} D \isin C \equiv D \leq C
212 \section{Some lemmas}
214 \[ \eqn{Exclusive Tip Contents:}{
215 \bigforall_{C \in \py}
216 \neg \Bigl[ D \isin \baseof{C} \land ( D \in \py \land D \le C )
219 Ie, the two limbs of the RHS of Tip Contents are mutually exclusive.
222 Let $B = \baseof{C}$ in $D \isin \baseof{C}$. Now $B \in \pn$.
223 So by Base Acyclic $D \isin B \implies D \notin \py$.
225 \[ \eqntag{{\it Corollary - equivalent to Tip Contents}}{
226 \bigforall_{C \in \py} D \isin C \equiv
228 D \in \py : & D \le C \\
229 D \not\in \py : & D \isin \baseof{C}
233 \[ \eqn{Tip Self Inpatch:}{
234 \bigforall_{C \in \py} C \haspatch \p
236 Ie, tip commits contain their own patch.
239 Apply Exclusive Tip Contents to some $D \in \py$:
240 $ \bigforall_{C \in \py}\bigforall_{D \in \py}
241 D \isin C \equiv D \le C $
244 \[ \eqn{Exact Ancestors:}{
245 \bigforall_{ C \hasparents \set{R} }
247 ( \mathop{\hbox{\huge{$\vee$}}}_{R \in \set R} D \le R )
251 \[ \eqn{Transitive Ancestors:}{
252 \left[ \bigforall_{ E \in \pendsof{C}{\set P} } E \le M \right] \equiv
253 \left[ \bigforall_{ A \in \pancsof{C}{\set P} } A \le M \right]
257 The implication from right to left is trivial because
258 $ \pends() \subset \pancs() $.
259 For the implication from left to right:
260 by the definition of $\mathcal E$,
261 for every such $A$, either $A \in \pends()$ which implies
262 $A \le M$ by the LHS directly,
263 or $\exists_{A' \in \pancs()} \; A' \neq A \land A \le A' $
264 in which case we repeat for $A'$. Since there are finitely many
265 commits, this terminates with $A'' \in \pends()$, ie $A'' \le M$
266 by the LHS. And $A \le A''$.
268 \[ \eqn{Calculation Of Ends:}{
269 \bigforall_{C \hasparents \set A}
270 \pendsof{C}{\set P} =
272 \Bigl[ \Largeexists_{A \in \set A}
273 E \in \pendsof{A}{\set P} \Bigr] \land
274 \Bigl[ \Largenexists_{B \in \set A}
275 E \neq B \land E \le B \Bigr]
280 \subsection{No Replay for Merge Results}
282 If we are constructing $C$, with,
290 No Replay is preserved. \proofstarts
292 \subsubsection{For $D=C$:} $D \isin C, D \le C$. OK.
294 \subsubsection{For $D \isin L \land D \isin R$:}
295 $D \isin C$. And $D \isin L \implies D \le L \implies D \le C$. OK.
297 \subsubsection{For $D \neq C \land D \not\isin L \land D \not\isin R$:}
300 \subsubsection{For $D \neq C \land (D \isin L \equiv D \not\isin R)
301 \land D \not\isin M$:}
302 $D \isin C$. Also $D \isin L \lor D \isin R$ so $D \le L \lor D \le
305 \subsubsection{For $D \neq C \land (D \isin L \equiv D \not\isin R)
311 \section{Commit annotation}
313 We annotate each Topbloke commit $C$ with:
317 \baseof{C}, \text{ if } C \in \py
320 \text{ either } C \haspatch \pa{Q} \text{ or } C \nothaspatch \pa{Q}
322 \bigforall_{\pay{Q} \not\ni C} \pendsof{C}{\pay{Q}}
325 We do not annotate $\pendsof{C}{\py}$ for $C \in \py$. Doing so would
326 make it wrong to make plain commits with git because the recorded $\pends$
327 would have to be updated. The annotation is not needed because
328 $\forall_{\py \ni C} \; \pendsof{C}{\py} = \{C\}$.
330 \section{Simple commit}
332 A simple single-parent forward commit $C$ as made by git-commit.
334 \tag*{} C \hasparents \{ A \} \\
335 \tag*{} \patchof{C} = \patchof{A} \\
336 \tag*{} D \isin C \equiv D \isin A \lor D = C
338 This also covers Topbloke-generated commits on plain git branches:
339 Topbloke strips the metadata when exporting.
341 \subsection{No Replay}
344 \subsection{Unique Base}
345 If $A, C \in \py$ then $\baseof{C} = \baseof{A}$. $\qed$
347 \subsection{Tip Contents}
348 We need to consider only $A, C \in \py$. From Tip Contents for $A$:
349 \[ D \isin A \equiv D \isin \baseof{A} \lor ( D \in \py \land D \le A ) \]
350 Substitute into the contents of $C$:
351 \[ D \isin C \equiv D \isin \baseof{A} \lor ( D \in \py \land D \le A )
353 Since $D = C \implies D \in \py$,
354 and substituting in $\baseof{C}$, this gives:
355 \[ D \isin C \equiv D \isin \baseof{C} \lor
356 (D \in \py \land D \le A) \lor
357 (D = C \land D \in \py) \]
358 \[ \equiv D \isin \baseof{C} \lor
359 [ D \in \py \land ( D \le A \lor D = C ) ] \]
360 So by Exact Ancestors:
361 \[ D \isin C \equiv D \isin \baseof{C} \lor ( D \in \py \land D \le C
365 \subsection{Base Acyclic}
367 Need to consider only $A, C \in \pn$.
369 For $D = C$: $D \in \pn$ so $D \not\in \py$. OK.
371 For $D \neq C$: $D \isin C \equiv D \isin A$, so by Base Acyclic for
372 $A$, $D \isin C \implies D \not\in \py$. $\qed$
374 \subsection{Coherence and patch inclusion}
376 Need to consider $D \in \py$
378 \subsubsection{For $A \haspatch P, D = C$:}
384 $ D \isin C \equiv \ldots \lor \true \text{ so } D \haspatch C $.
386 \subsubsection{For $A \haspatch P, D \neq C$:}
387 Ancestors: $ D \le C \equiv D \le A $.
389 Contents: $ D \isin C \equiv D \isin A \lor f $
390 so $ D \isin C \equiv D \isin A $.
393 \[ A \haspatch P \implies C \haspatch P \]
395 \subsubsection{For $A \nothaspatch P$:}
397 Firstly, $C \not\in \py$ since if it were, $A \in \py$.
400 Now by contents of $A$, $D \notin A$, so $D \notin C$.
403 \[ A \nothaspatch P \implies C \nothaspatch P \]
406 \subsection{Foreign inclusion:}
408 If $D = C$, trivial. For $D \neq C$:
409 $D \isin C \equiv D \isin A \equiv D \le A \equiv D \le C$. $\qed$
413 Given $L, R^+, R^-$ where
414 $R^+ \in \pry, R^- = \baseof{R^+}$.
415 Construct $C$ which has $\pr$ removed.
416 Used for removing a branch dependency.
418 C \hasparents \{ L \}
420 \patchof{C} = \patchof{L}
422 \mergeof{C}{L}{R^+}{R^-}
425 \subsection{Conditions}
427 \[ \eqn{ Unique Tip }{
428 \pendsof{L}{\pry} = \{ R^+ \}
430 \[ \eqn{ Currently Included }{
437 \subsection{No Replay}
439 By Unique Tip, $R^+ \le L$. By definition of $\base$, $R^- \le R^+$
440 so $R^- \le L$. So $R^+ \le C$ and $R^- \le C$ and No Replay for
441 Merge Results applies. $\qed$
443 \subsection{Desired Contents}
445 \[ D \isin C \equiv [ D \notin \pry \land D \isin L ] \lor D = C \]
448 \subsubsection{For $D = C$:}
450 Trivially $D \isin C$. OK.
452 \subsubsection{For $D \neq C, D \not\le L$:}
454 By No Replay $D \not\isin L$. Also $D \not\le R^-$ hence
455 $D \not\isin R^-$. Thus $D \not\isin C$. OK.
457 \subsubsection{For $D \neq C, D \le L, D \in \pry$:}
459 By Currently Included, $D \isin L$.
461 By Tip Self Inpatch, $D \isin R^+ \equiv D \le R^+$, but by
462 by Unique Tip, $D \le R^+ \equiv D \le L$.
465 By Base Acyclic, $D \not\isin R^-$.
467 Apply $\merge$: $D \not\isin C$. OK.
469 \subsubsection{For $D \neq C, D \le L, D \notin \pry$:}
471 By Tip Contents for $R^+$, $D \isin R^+ \equiv D \isin R^-$.
473 Apply $\merge$: $D \isin C \equiv D \isin L$. OK.
477 \subsection{Unique Base}
479 Need to consider only $C \in \py$, ie $L \in \py$.
483 xxx need to finish anticommit
487 Merge commits $L$ and $R$ using merge base $M$ ($M < L, M < R$):
489 C \hasparents \{ L, R \}
491 \patchof{C} = \patchof{L}
495 We will occasionally use $X,Y$ s.t. $\{X,Y\} = \{L,R\}$.
497 \subsection{Conditions}
499 \[ \eqn{ Tip Merge }{
502 R \in \py : & \baseof{R} \ge \baseof{L}
503 \land [\baseof{L} = M \lor \baseof{L} = \baseof{M}] \\
504 R \in \pn : & M = \baseof{L} \\
505 \text{otherwise} : & \false
508 \[ \eqn{ Merge Acyclic }{
513 \[ \eqn{ Removal Merge Ends }{
514 X \not\haspatch \p \land
518 \pendsof{Y}{\py} = \pendsof{M}{\py}
520 \[ \eqn{ Addition Merge Ends }{
521 X \not\haspatch \p \land
525 \bigforall_{E \in \pendsof{X}{\py}} E \le Y
529 \subsection{No Replay}
531 See No Replay for Merge Results.
533 \subsection{Unique Base}
535 Need to consider only $C \in \py$, ie $L \in \py$,
536 and calculate $\pendsof{C}{\pn}$. So we will consider some
537 putative ancestor $A \in \pn$ and see whether $A \le C$.
539 By Exact Ancestors for C, $A \le C \equiv A \le L \lor A \le R \lor A = C$.
540 But $C \in py$ and $A \in \pn$ so $A \neq C$.
541 Thus $A \le C \equiv A \le L \lor A \le R$.
543 By Unique Base of L and Transitive Ancestors,
544 $A \le L \equiv A \le \baseof{L}$.
546 \subsubsection{For $R \in \py$:}
548 By Unique Base of $R$ and Transitive Ancestors,
549 $A \le R \equiv A \le \baseof{R}$.
551 But by Tip Merge condition on $\baseof{R}$,
552 $A \le \baseof{L} \implies A \le \baseof{R}$, so
553 $A \le \baseof{R} \lor A \le \baseof{L} \equiv A \le \baseof{R}$.
554 Thus $A \le C \equiv A \le \baseof{R}$.
555 That is, $\baseof{C} = \baseof{R}$.
557 \subsubsection{For $R \in \pn$:}
559 By Tip Merge condition on $R$ and since $M \le R$,
560 $A \le \baseof{L} \implies A \le R$, so
561 $A \le R \lor A \le \baseof{L} \equiv A \le R$.
562 Thus $A \le C \equiv A \le R$.
563 That is, $\baseof{C} = R$.
567 \subsection{Coherence and Patch Inclusion}
569 Need to determine $C \haspatch \p$ based on $L,M,R \haspatch \p$.
570 This involves considering $D \in \py$.
572 \subsubsection{For $L \nothaspatch \p, R \nothaspatch \p$:}
573 $D \not\isin L \land D \not\isin R$. $C \not\in \py$ (otherwise $L
574 \in \py$ ie $L \haspatch \p$ by Tip Self Inpatch). So $D \neq C$.
575 Applying $\merge$ gives $D \not\isin C$ i.e. $C \nothaspatch \p$.
577 \subsubsection{For $L \haspatch \p, R \haspatch \p$:}
578 $D \isin L \equiv D \le L$ and $D \isin R \equiv D \le R$.
579 (Likewise $D \isin X \equiv D \le X$ and $D \isin Y \equiv D \le Y$.)
581 Consider $D = C$: $D \isin C$, $D \le C$, OK for $C \haspatch \p$.
583 For $D \neq C$: $D \le C \equiv D \le L \lor D \le R
584 \equiv D \isin L \lor D \isin R$.
585 (Likewise $D \le C \equiv D \le X \lor D \le Y$.)
587 Consider $D \neq C, D \isin X \land D \isin Y$:
588 By $\merge$, $D \isin C$. Also $D \le X$
589 so $D \le C$. OK for $C \haspatch \p$.
591 Consider $D \neq C, D \not\isin X \land D \not\isin Y$:
592 By $\merge$, $D \not\isin C$.
593 And $D \not\le X \land D \not\le Y$ so $D \not\le C$.
594 OK for $C \haspatch \p$.
596 Remaining case, wlog, is $D \not\isin X \land D \isin Y$.
597 $D \not\le X$ so $D \not\le M$ so $D \not\isin M$.
598 Thus by $\merge$, $D \isin C$. And $D \le Y$ so $D \le C$.
599 OK for $C \haspatch \p$.
601 So indeed $L \haspatch \p \land R \haspatch \p \implies C \haspatch \p$.
603 \subsubsection{For (wlog) $X \not\haspatch \p, Y \haspatch \p$:}
605 $C \haspatch \p \equiv M \nothaspatch \p$.
609 One of the Merge Ends conditions applies.
610 Recall that we are considering $D \in \py$.
611 $D \isin Y \equiv D \le Y$. $D \not\isin X$.
612 We will show for each of
613 various cases that $D \isin C \equiv M \nothaspatch \p \land D \le C$
614 (which suffices by definition of $\haspatch$ and $\nothaspatch$).
616 Consider $D = C$: Thus $C \in \py, L \in \py$, and by Tip
617 Self Inpatch $L \haspatch \p$, so $L=Y, R=X$. By Tip Merge,
618 $M=\baseof{L}$. So by Base Acyclic $D \not\isin M$, i.e.
619 $M \nothaspatch \p$. And indeed $D \isin C$ and $D \le C$. OK.
621 Consider $D \neq C, M \nothaspatch P, D \isin Y$:
622 $D \le Y$ so $D \le C$.
623 $D \not\isin M$ so by $\merge$, $D \isin C$. OK.
625 Consider $D \neq C, M \nothaspatch P, D \not\isin Y$:
626 $D \not\le Y$. If $D \le X$ then
627 $D \in \pancsof{X}{\py}$, so by Addition Merge Ends and
628 Transitive Ancestors $D \le Y$ --- a contradiction, so $D \not\le X$.
629 Thus $D \not\le C$. By $\merge$, $D \not\isin C$. OK.
631 Consider $D \neq C, M \haspatch P, D \isin Y$:
632 $D \le Y$ so $D \in \pancsof{Y}{\py}$ so by Removal Merge Ends
633 and Transitive Ancestors $D \in \pancsof{M}{\py}$ so $D \le M$.
634 Thus $D \isin M$. By $\merge$, $D \not\isin C$. OK.
636 Consider $D \neq C, M \haspatch P, D \not\isin Y$:
637 By $\merge$, $D \not\isin C$. OK.
641 \subsection{Base Acyclic}
643 This applies when $C \in \pn$.
644 $C \in \pn$ when $L \in \pn$ so by Merge Acyclic, $R \nothaspatch \p$.
646 Consider some $D \in \py$.
648 By Base Acyclic of $L$, $D \not\isin L$. By the above, $D \not\isin
649 R$. And $D \neq C$. So $D \not\isin C$. $\qed$
651 \subsection{Tip Contents}
653 We need worry only about $C \in \py$.
654 And $\patchof{C} = \patchof{L}$
655 so $L \in \py$ so $L \haspatch \p$. We will use the Unique Base
656 of $C$, and its Coherence and Patch Inclusion, as just proved.
658 Firstly we show $C \haspatch \p$: If $R \in \py$, then $R \haspatch
659 \p$ and by Coherence/Inclusion $C \haspatch \p$ . If $R \not\in \py$
660 then by Tip Merge $M = \baseof{L}$ so by Base Acyclic and definition
661 of $\nothaspatch$, $M \nothaspatch \p$. So by Coherence/Inclusion $C
662 \haspatch \p$ (whether $R \haspatch \p$ or $\nothaspatch$).
664 We will consider an arbitrary commit $D$
665 and prove the Exclusive Tip Contents form.
667 \subsubsection{For $D \in \py$:}
668 $C \haspatch \p$ so by definition of $\haspatch$, $D \isin C \equiv D
671 \subsubsection{For $D \not\in \py, R \not\in \py$:}
673 $D \neq C$. By Tip Contents of $L$,
674 $D \isin L \equiv D \isin \baseof{L}$, and by Tip Merge condition,
675 $D \isin L \equiv D \isin M$. So by definition of $\merge$, $D \isin
676 C \equiv D \isin R$. And $R = \baseof{C}$ by Unique Base of $C$.
677 Thus $D \isin C \equiv D \isin \baseof{C}$. OK.
679 \subsubsection{For $D \not\in \py, R \in \py$:}
684 $D \isin L \equiv D \isin \baseof{L}$ and
685 $D \isin R \equiv D \isin \baseof{R}$.
687 If $\baseof{L} = M$, trivially $D \isin M \equiv D \isin \baseof{L}.$
688 Whereas if $\baseof{L} = \baseof{M}$, by definition of $\base$,
689 $\patchof{M} = \patchof{L} = \py$, so by Tip Contents of $M$,
690 $D \isin M \equiv D \isin \baseof{M} \equiv D \isin \baseof{L}$.
692 So $D \isin M \equiv D \isin L$ and by $\merge$,
693 $D \isin C \equiv D \isin R$. But from Unique Base,
694 $\baseof{C} = R$ so $D \isin C \equiv D \isin \baseof{C}$. OK.
698 xxx up to here, need to prove other things about merges