strategy: wip reachable etc.

[topbloke-formulae.git] / trav-proofs.tex

diff --git a/trav-proofs.tex b/trav-proofs.tex
index 14f6cd8942edbc6af512430fa6be7042948d2218..06a54cfb0521161bd329823d9d685a877c52e30b 100644 (file)
--- a/trav-proofs.tex
+++ b/trav-proofs.tex
@@ -3,5 +3,77 @@
 For each operation called for by the traversal algorithms, we prove
 that the commit generation preconditions are met.
 
+WIP WHAT ABOUT PROVING ALL THE TRAVERSAL RESULTS
+
+\subsection{Traversal Lemmas}
+
+Firstly, some lemmas.
+\statement{Tip Correct Contents}{
+  \tipcy \haspatch \pa E
+    \equiv
+  \pa E = \pc \lor \pa E \isdep \pc
+}
+\proof{
+  For $\pc = \pa E$, Tip Own Contents suffices.
+  For $\pc \neq \pa E$, Exclusive Tip Contents
+  gives $D \isin \tipcy \equiv D \isin \baseof{\tipcy}$
+  which by Correct Base $\equiv D \isin \tipcn$.
+}
+
+\subsection{Base Dependency Merge, Base Sibling Merge}
+
+We do not prove that the preconditions are met.  Instead, we check
+them at runtime.  If they turn out not to be met, we abandon
+\alg{Merge-Base} and resort to \alg{Recreate-Base}.
+
+TODO COMPLETE MERGE-BASE STUFF
+
+WIP WHAT ABOUT PROVING ALL THE TRAVERSAL RESULTS
+
+\subsection{Recreate Base Beginning}
+
+To recap we are executing Create Base with
+$L = \tipdy$ and $\pq = \pc$.
+
+\subsubsection{Create Acyclic}
+
+By Tip Correct Contents of $L$,
+$L \haspatch \pa E \equiv \pa E = \pd \lor \pa E \isdep \pd$.
+Now $\pd \isdirdep \pc$,
+so by Coherence, and setting $\pa E = \pc$,
+$L \nothaspatch \pc$. I.e. $L \nothaspatch \pq$. OK.
+
+That's everything for Create Base.  $\qed$
+
+\subsection{Recreate Base Final Declaration}
+
+\subsubsection{Base Only} $\patchof{W} = \patchof{L} = \pn$.  OK.
+
+\subsubsection{Unique Tips}
+
+Want to prove that for any $\p \isin C$, $\tipdy$ is a suitable $T$.
+
+WIP
+
 \subsection{Tip Base Merge}
 
+$L = W$, $R = \tipcn$.
+
+TODO TBD
+
+Afterwards, $\baseof{W} = \tipcn$.
+
+\subsection{Tip Source Merge}
+
+In fact, we do this backwards: $L = S$, $R = W$.  Since $S \in \pcy$,
+the resulting $C \in \pcy$ and the remaining properties of the Merge
+commit construction are symmetrical in $L$ and $R$ so this is fine.
+
+By the results of Tip Base Merge, $\baseof{W} = \tipcn$.
+
+By Base Ends Supreme, $\tipcn \ge \baseof{S}$ i.e.
+$\baseof{R} \ge \baseof{L}$.
+
+Either $\baseof{L} = \baseof{M}$, or we must choose a different $M$ in
+which case $M = \baseof{S}$ will suffice.
+