1 \section{Traversal phase --- proofs}
3 For each operation called for by the traversal algorithms, we prove
4 that the commit generation preconditions are met.
6 WIP WHAT ABOUT PROVING ALL THE TRAVERSAL RESULTS
8 \subsection{Traversal Lemmas}
11 \statement{Tip Correct Contents}{
12 \tipcy \haspatch \pa E
14 \pa E = \pc \lor \pa E \isdep \pc
17 For $\pc = \pa E$, Tip Own Contents suffices.
18 For $\pc \neq \pa E$, Exclusive Tip Contents
19 gives $D \isin \tipcy \equiv D \isin \baseof{\tipcy}$
20 which by Correct Base $\equiv D \isin \tipcn$.
23 \subsection{Base Dependency Merge, Base Sibling Merge}
25 We do not prove that the preconditions are met. Instead, we check
26 them at runtime. If they turn out not to be met, we abandon
27 \alg{Merge-Base} and resort to \alg{Recreate-Base}.
29 TODO COMPLETE MERGE-BASE STUFF
31 WIP WHAT ABOUT PROVING ALL THE TRAVERSAL RESULTS
33 \subsection{Recreate Base Beginning}
35 To recap we are executing Create Base with
36 $L = \tipdy$ and $\pq = \pc$.
38 \subsubsection{Create Acyclic}
40 By Tip Correct Contents of $L$,
41 $L \haspatch \pa E \equiv \pa E = \pd \lor \pa E \isdep \pd$.
42 Now $\pd \isdirdep \pc$,
43 so by Coherence, and setting $\pa E = \pc$,
44 $L \nothaspatch \pc$. I.e. $L \nothaspatch \pq$. OK.
46 That's everything for Create Base. $\qed$
48 \subsection{Recreate Base Final Declaration}
50 \subsubsection{Base Only} $\patchof{W} = \patchof{L} = \pn$. OK.
52 \subsubsection{Unique Tips}
54 Want to prove that for any $\p \isin C$, $\tipdy$ is a suitable $T$.
58 \subsection{Tip Base Merge}
60 $L = W$, $R = \tipcn$.
64 Afterwards, $\baseof{W} = \tipcn$.
66 \subsection{Tip Source Merge}
68 In fact, we do this backwards: $L = S$, $R = W$. Since $S \in \pcy$,
69 the resulting $C \in \pcy$ and the remaining properties of the Merge
70 commit construction are symmetrical in $L$ and $R$ so this is fine.
72 By the results of Tip Base Merge, $\baseof{W} = \tipcn$.
74 By Base Ends Supreme, $\tipcn \ge \baseof{S}$ i.e.
75 $\baseof{R} \ge \baseof{L}$.
77 Either $\baseof{L} = \baseof{M}$, or we must choose a different $M$ in
78 which case $M = \baseof{S}$ will suffice.