X-Git-Url: http://www.chiark.greenend.org.uk/ucgi/~ian/git?p=topbloke-formulae.git;a=blobdiff_plain;f=article.tex;h=a57bbec868dc202f99c959e68d802c566573a906;hp=f5cfd3df9f7b3154d247dc78facf518734b0fa73;hb=3da86b6d60fbd0edfa34366c2c57a5481790883c;hpb=f8637246494c0136651af433d281ed392f8172a5 diff --git a/article.tex b/article.tex index f5cfd3d..a57bbec 100644 --- a/article.tex +++ b/article.tex @@ -34,6 +34,10 @@ \newcommand{\py}{\pay{P}} \newcommand{\pn}{\pan{P}} +\newcommand{\pq}{\pa{Q}} +\newcommand{\pqy}{\pay{Q}} +\newcommand{\pqn}{\pan{Q}} + \newcommand{\pr}{\pa{R}} \newcommand{\pry}{\pay{R}} \newcommand{\prn}{\pan{R}} @@ -105,7 +109,7 @@ $\set X$. \item[ $ C \ge D $ ] $C$ is a descendant of $D$ in the git commit -graph. This is a partial order, namely the transitive closure of +graph. This is a partial order, namely the transitive closure of $ D \in \set X $ where $ C \hasparents \set X $. \item[ $ C \has D $ ] @@ -125,18 +129,18 @@ is to be taken as applying to both $\py$ and $\pn$. None of these sets overlap. Hence: \item[ $ \patchof{ C } $ ] -Either $\p$ s.t. $ C \in \p $, or $\bot$. +Either $\p$ s.t. $ C \in \p $, or $\bot$. A function from commits to patches' sets $\p$. \item[ $ \pancsof{C}{\set P} $ ] -$ \{ A \; | \; A \le C \land A \in \set P \} $ +$ \{ A \; | \; A \le C \land A \in \set P \} $ i.e. all the ancestors of $C$ which are in $\set P$. \item[ $ \pendsof{C}{\set P} $ ] $ \{ E \; | \; E \in \pancsof{C}{\set P} \land \mathop{\not\exists}_{A \in \pancsof{C}{\set P}} - E \neq A \land E \le A \} $ + E \neq A \land E \le A \} $ i.e. all $\le$-maximal commits in $\pancsof{C}{\set P}$. \item[ $ \baseof{C} $ ] @@ -150,13 +154,13 @@ $\displaystyle \bigforall_{D \in \py} D \isin C \equiv D \le C $. \item[ $ C \nothaspatch \p $ ] $\displaystyle \bigforall_{D \in \py} D \not\isin C $. -~ Informally, $C$ has none of the contents of $\p$. +~ Informally, $C$ has none of the contents of $\p$. Non-Topbloke commits are $\nothaspatch \p$ for all $\p$. This includes commits on plain git branches made by applying a Topbloke patch. If a Topbloke patch is applied to a non-Topbloke branch and then bubbles back to -the relevant Topbloke branches, we hope that +the relevant Topbloke branches, we hope that if the user still cares about the Topbloke patch, git's merge algorithm will DTRT when trying to re-apply the changes. @@ -169,7 +173,7 @@ $\displaystyle D \isin C \equiv (D \not\isin L \land D \not\isin R) \land D \neq C : & \false \\ \text{otherwise} : & D \not\isin M \end{cases} -$ +$ \end{basedescript} \newpage @@ -196,6 +200,10 @@ We maintain these each time we construct a new commit. \\ \[\eqn{Foreign Inclusion:}{ \bigforall_{D \text{ s.t. } \patchof{D} = \bot} D \isin C \equiv D \leq C }\] +\[\eqn{Foreign Contents:}{ + \bigforall_{C \text{ s.t. } \patchof{C} = \bot} + D \le C \implies \patchof{D} = \bot +}\] \section{Some lemmas} @@ -210,14 +218,29 @@ We maintain these each time we construct a new commit. \\ \text{as above with L and R exchanged} \end{cases} }\] -\proof{ - Truth table xxx - - Original definition is symmetrical in $L$ and $R$. +\proof{ ~ Truth table (ordered by original definition): \\ + \begin{tabular}{cccc|c|cc} + $D = C$ & + $\isin L$ & + $\isin M$ & + $\isin R$ & $\isin C$ & + $L$ vs. $R$ & $L$ vs. $M$ + \\\hline + y & ? & ? & ? & y & ? & ? \\ + n & y & y & y & y & $\equiv$ & $\equiv$ \\ + n & y & n & y & y & $\equiv$ & $\nequiv$ \\ + n & n & y & n & n & $\equiv$ & $\nequiv$ \\ + n & n & n & n & n & $\equiv$ & $\equiv$ \\ + n & y & y & n & n & $\nequiv$ & $\equiv$ \\ + n & n & y & y & n & $\nequiv$ & $\nequiv$ \\ + n & y & n & n & y & $\nequiv$ & $\nequiv$ \\ + n & n & n & y & y & $\nequiv$ & $\equiv$ \\ + \end{tabular} \\ + And original definition is symmetrical in $L$ and $R$. } \[ \eqn{Exclusive Tip Contents:}{ - \bigforall_{C \in \py} + \bigforall_{C \in \py} \neg \Bigl[ D \isin \baseof{C} \land ( D \in \py \land D \le C ) \Bigr] }\] @@ -252,6 +275,7 @@ $ \bigforall_{C \in \py}\bigforall_{D \in \py} ( \mathop{\hbox{\huge{$\vee$}}}_{R \in \set R} D \le R ) \lor D = C }\] +\proof{ ~ Trivial.} \[ \eqn{Transitive Ancestors:}{ \left[ \bigforall_{ E \in \pendsof{C}{\set P} } E \le M \right] \equiv @@ -261,7 +285,7 @@ $ \bigforall_{C \in \py}\bigforall_{D \in \py} \proof{ The implication from right to left is trivial because $ \pends() \subset \pancs() $. -For the implication from left to right: +For the implication from left to right: by the definition of $\mathcal E$, for every such $A$, either $A \in \pends()$ which implies $A \le M$ by the LHS directly, @@ -270,48 +294,80 @@ in which case we repeat for $A'$. Since there are finitely many commits, this terminates with $A'' \in \pends()$, ie $A'' \le M$ by the LHS. And $A \le A''$. } + \[ \eqn{Calculation Of Ends:}{ \bigforall_{C \hasparents \set A} \pendsof{C}{\set P} = + \begin{cases} + C \in \p : & \{ C \} + \\ + C \not\in \p : & \displaystyle \left\{ E \Big| - \Bigl[ \Largeexists_{A \in \set A} + \Bigl[ \Largeexists_{A \in \set A} E \in \pendsof{A}{\set P} \Bigr] \land - \Bigl[ \Largenexists_{B \in \set A} + \Bigl[ \Largenexists_{B \in \set A} E \neq B \land E \le B \Bigr] \right\} + \end{cases} }\] -XXX proof TBD. - -\subsection{No Replay for Merge Results} - -If we are constructing $C$, with, -\gathbegin - \mergeof{C}{L}{M}{R} -\gathnext - L \le C -\gathnext - R \le C -\end{gather} -No Replay is preserved. \proofstarts - -\subsubsection{For $D=C$:} $D \isin C, D \le C$. OK. - -\subsubsection{For $D \isin L \land D \isin R$:} -$D \isin C$. And $D \isin L \implies D \le L \implies D \le C$. OK. - -\subsubsection{For $D \neq C \land D \not\isin L \land D \not\isin R$:} -$D \not\isin C$. OK. - -\subsubsection{For $D \neq C \land (D \isin L \equiv D \not\isin R) - \land D \not\isin M$:} -$D \isin C$. Also $D \isin L \lor D \isin R$ so $D \le L \lor D \le -R$ so $D \le C$. OK. +xxx proof tbd + +\[ \eqn{Ingredients Prevent Replay:}{ + \left[ + {C \hasparents \set A} \land + \\ + \left( + D \isin C \implies + D = C \lor + \Largeexists_{A \in \set A} D \isin A + \right) + \right] \implies \left[ + D \isin C \implies D \le C + \right] +}\] +\proof{ + Trivial for $D = C$. Consider some $D \neq C$, $D \isin C$. + By the preconditions, there is some $A$ s.t. $D \in \set A$ + and $D \isin A$. By No Replay for $A$, $D \le A$. And + $A \le C$ so $D \le C$. +} -\subsubsection{For $D \neq C \land (D \isin L \equiv D \not\isin R) - \land D \isin M$:} -$D \not\isin C$. OK. +\[ \eqn{Simple Foreign Inclusion:}{ + \left[ + C \hasparents \{ L \} + \land + \bigforall_{D} D \isin C \equiv D \isin L \lor D = C + \right] + \implies + \bigforall_{D \text{ s.t. } \patchof{D} = \bot} + D \isin C \equiv D \le C +}\] +\proof{ +Consider some $D$ s.t. $\patchof{D} = \bot$. +If $D = C$, trivially true. For $D \neq C$, +by Foreign Inclusion of $D$ in $L$, $D \isin L \equiv D \le L$. +And by Exact Ancestors $D \le L \equiv D \le C$. +So $D \isin C \equiv D \le C$. +} -$\qed$ +\[ \eqn{Totally Foreign Contents:}{ + \bigforall_{C \hasparents \set A} + \left[ + \patchof{C} = \bot \land + \bigforall_{A \in \set A} \patchof{A} = \bot + \right] + \implies + \left[ + D \le C + \implies + \patchof{D} = \bot + \right] +}\] +\proof{ +Consider some $D \le C$. If $D = C$, $\patchof{D} = \bot$ trivially. +If $D \neq C$ then $D \le A$ where $A \in \set A$. By Foreign +Contents of $A$, $\patchof{D} = \bot$. +} \section{Commit annotation} @@ -321,16 +377,31 @@ We annotate each Topbloke commit $C$ with: \gathnext \baseof{C}, \text{ if } C \in \py \gathnext - \bigforall_{\pa{Q}} - \text{ either } C \haspatch \pa{Q} \text{ or } C \nothaspatch \pa{Q} + \bigforall_{\pq} + \text{ either } C \haspatch \pq \text{ or } C \nothaspatch \pq \gathnext - \bigforall_{\pay{Q} \not\ni C} \pendsof{C}{\pay{Q}} + \bigforall_{\pqy \not\ni C} \pendsof{C}{\pqy} \end{gather} +$\patchof{C}$, for each kind of Topbloke-generated commit, is stated +in the summary in the section for that kind of commit. + +Whether $\baseof{C}$ is required, and if so what the value is, is +stated in the proof of Unique Base for each kind of commit. + +$C \haspatch \pq$ or $\nothaspatch \pq$ is represented as the +set $\{ \pq | C \haspatch \pq \}$. Whether $C \haspatch \pq$ +is in stated +(in terms of $I \haspatch \pq$ or $I \nothaspatch \pq$ +for the ingredients $I$), +in the proof of Coherence for each kind of commit. + +$\pendsof{C}{\pq^+}$ is computed, for all Topbloke-generated commits, +using the lemma Calculation of Ends, above. We do not annotate $\pendsof{C}{\py}$ for $C \in \py$. Doing so would make it wrong to make plain commits with git because the recorded $\pends$ -would have to be updated. The annotation is not needed because -$\forall_{\py \ni C} \; \pendsof{C}{\py} = \{C\}$. +would have to be updated. The annotation is not needed in that case +because $\forall_{\py \ni C} \; \pendsof{C}{\py} = \{C\}$. \section{Simple commit} @@ -344,10 +415,14 @@ This also covers Topbloke-generated commits on plain git branches: Topbloke strips the metadata when exporting. \subsection{No Replay} -Trivial. + +Ingredients Prevent Replay applies. $\qed$ \subsection{Unique Base} -If $A, C \in \py$ then $\baseof{C} = \baseof{A}$. $\qed$ +If $A, C \in \py$ then by Calculation of Ends for +$C, \py, C \not\in \py$: +$\pendsof{C}{\pn} = \pendsof{A}{\pn}$ so +$\baseof{C} = \baseof{A}$. $\qed$ \subsection{Tip Contents} We need to consider only $A, C \in \py$. From Tip Contents for $A$: @@ -355,7 +430,7 @@ We need to consider only $A, C \in \py$. From Tip Contents for $A$: Substitute into the contents of $C$: \[ D \isin C \equiv D \isin \baseof{A} \lor ( D \in \py \land D \le A ) \lor D = C \] -Since $D = C \implies D \in \py$, +Since $D = C \implies D \in \py$, and substituting in $\baseof{C}$, this gives: \[ D \isin C \equiv D \isin \baseof{C} \lor (D \in \py \land D \le A) \lor @@ -369,12 +444,14 @@ $\qed$ \subsection{Base Acyclic} -Need to consider only $A, C \in \pn$. +Need to consider only $A, C \in \pn$. For $D = C$: $D \in \pn$ so $D \not\in \py$. OK. For $D \neq C$: $D \isin C \equiv D \isin A$, so by Base Acyclic for -$A$, $D \isin C \implies D \not\in \py$. $\qed$ +$A$, $D \isin C \implies D \not\in \py$. + +$\qed$ \subsection{Coherence and patch inclusion} @@ -399,7 +476,7 @@ So: \subsubsection{For $A \nothaspatch P$:} -Firstly, $C \not\in \py$ since if it were, $A \in \py$. +Firstly, $C \not\in \py$ since if it were, $A \in \py$. Thus $D \neq C$. Now by contents of $A$, $D \notin A$, so $D \notin C$. @@ -408,15 +485,149 @@ So: \[ A \nothaspatch P \implies C \nothaspatch P \] $\qed$ -\subsection{Foreign inclusion:} +\subsection{Foreign Inclusion:} + +Simple Foreign Inclusion applies. $\qed$ + +\subsection{Foreign Contents:} + +Only relevant if $\patchof{C} = \bot$, and in that case Totally +Foreign Contents applies. $\qed$ + +\section{Create Base} + +Given $L$, create a Topbloke base branch initial commit $B$. +\gathbegin + B \hasparents \{ L \} +\gathnext + \patchof{B} = \pqn +\gathnext + D \isin B \equiv D \isin L \lor D = B +\end{gather} + +\subsection{Conditions} + +\[ \eqn{ Ingredients }{ + \patchof{L} = \pa{L} \lor \patchof{L} = \bot +}\] +\[ \eqn{ Create Acyclic }{ + L \not\haspatch \pq +}\] + +\subsection{No Replay} + +Ingredients Prevent Replay applies. $\qed$ + +\subsection{Unique Base} + +Not applicable. + +\subsection{Tip Contents} + +Not applicable. + +\subsection{Base Acyclic} + +Consider some $D \isin B$. If $D = B$, $D \in \pqn$. +If $D \neq B$, $D \isin L$, and by Create Acyclic +$D \not\in \pqy$. $\qed$ + +\subsection{Coherence and Patch Inclusion} + +Consider some $D \in \p$. +$B \not\in \py$ so $D \neq B$. So $D \isin B \equiv D \isin L$ +and $D \le B \equiv D \le L$. + +Thus $L \haspatch \p \implies B \haspatch P$ +and $L \nothaspatch \p \implies B \nothaspatch P$. + +$\qed$. + +\subsection{Foreign Inclusion} + +Simple Foreign Inclusion applies. $\qed$ + +\subsection{Foreign Contents} + +Not applicable. + +\section{Create Tip} + +Given a Topbloke base $B$, create a tip branch initial commit B. +\gathbegin + C \hasparents \{ B \} +\gathnext + \patchof{B} = \pqy +\gathnext + D \isin C \equiv D \isin B \lor D = C +\end{gather} + +\subsection{Conditions} + +\[ \eqn{ Ingredients }{ + \patchof{B} = \pqn +}\] +\[ \eqn{ No Sneak }{ + \pendsof{B}{\pqy} = \{ \} +}\] + +\subsection{No Replay} + +Ingredients Prevent Replay applies. $\qed$ + +\subsection{Unique Base} + +Trivially, $\pendsof{C}{\pqn} = \{B\}$ so $\baseof{C} = B$. $\qed$ + +\subsection{Tip Contents} + +Consider some arbitrary commit $D$. If $D = C$, trivially satisfied. + +If $D \neq C$, $D \isin C \equiv D \isin B$. +By Base Acyclic of $B$, $D \isin B \implies D \not\in \pqy$. +So $D \isin C \equiv D \isin \baseof{B}$. + +$\qed$ + +\subsection{Base Acyclic} + +Not applicable. + +\subsection{Coherence and Patch Inclusion} + +$$ +\begin{cases} + \p = \pq \lor B \haspatch \p : & C \haspatch \p \\ + \p \neq \pq \land B \nothaspatch \p : & C \nothaspatch \p +\end{cases} +$$ + +\proofstarts +~ Consider some $D \in \py$. + +\subsubsection{For $\p = \pq$:} + +By Base Acyclic, $D \not\isin B$. So $D \isin C \equiv D = C$. +By No Sneak, $D \le B \equiv D = C$. Thus $C \haspatch \pq$. + +\subsubsection{For $\p \neq \pq$:} + +$D \neq C$. So $D \isin C \equiv D \isin B$, +and $D \le C \equiv D \le B$. + +$\qed$ + +\subsection{Foreign Inclusion} + +Simple Foreign Inclusion applies. $\qed$ + +\subsection{Foreign Contents} -If $D = C$, trivial. For $D \neq C$: -$D \isin C \equiv D \isin A \equiv D \le A \equiv D \le C$. $\qed$ +Not applicable. \section{Anticommit} -Given $L, R^+, R^-$ where -$R^+ \in \pry, R^- = \baseof{R^+}$. +Given $L$ and $\pr$ as represented by $R^+, R^-$. Construct $C$ which has $\pr$ removed. Used for removing a branch dependency. \gathbegin @@ -429,6 +640,9 @@ Used for removing a branch dependency. \subsection{Conditions} +\[ \eqn{ Ingredients }{ +R^+ \in \pry \land R^- = \baseof{R^+} +}\] \[ \eqn{ Into Base }{ L \in \pn }\] @@ -439,11 +653,21 @@ Used for removing a branch dependency. L \haspatch \pry }\] -\subsection{No Replay} +\subsection{Ordering of Ingredients:} By Unique Tip, $R^+ \le L$. By definition of $\base$, $R^- \le R^+$ -so $R^- \le L$. So $R^+ \le C$ and $R^- \le C$ and No Replay for -Merge Results applies. $\qed$ +so $R^- \le L$. So $R^+ \le C$ and $R^- \le C$. +$\qed$ + +(Note that $R^+ \not\le R^-$, i.e. the merge base +is a descendant, not an ancestor, of the 2nd parent.) + +\subsection{No Replay} + +By definition of $\merge$, +$D \isin C \implies D \isin L \lor D \isin R^- \lor D = C$. +So, by Ordering of Ingredients, +Ingredients Prevent Replay applies. $\qed$ \subsection{Desired Contents} @@ -464,7 +688,7 @@ $D \not\isin R^-$. Thus $D \not\isin C$. OK. By Currently Included, $D \isin L$. By Tip Self Inpatch, $D \isin R^+ \equiv D \le R^+$, but by -by Unique Tip, $D \le R^+ \equiv D \le L$. +by Unique Tip, $D \le R^+ \equiv D \le L$. So $D \isin R^+$. By Base Acyclic, $D \not\isin R^-$. @@ -514,9 +738,26 @@ So $L \nothaspatch \p \implies C \nothaspatch \p$. Whereas if $L \haspatch \p$, $D \isin L \equiv D \le L$. so $L \haspatch \p \implies C \haspatch \p$. +$\qed$ + +\subsection{Foreign Inclusion} + +Consider some $D$ s.t. $\patchof{D} = \bot$. $D \neq C$. +So by Desired Contents $D \isin C \equiv D \isin L$. +By Foreign Inclusion of $D$ in $L$, $D \isin L \equiv D \le L$. + +And $D \le C \equiv D \le L$. +Thus $D \isin C \equiv D \le C$. + +$\qed$ + +\subsection{Foreign Contents} + +Not applicable. + \section{Merge} -Merge commits $L$ and $R$ using merge base $M$ ($M < L, M < R$): +Merge commits $L$ and $R$ using merge base $M$: \gathbegin C \hasparents \{ L, R \} \gathnext @@ -527,7 +768,9 @@ Merge commits $L$ and $R$ using merge base $M$ ($M < L, M < R$): We will occasionally use $X,Y$ s.t. $\{X,Y\} = \{L,R\}$. \subsection{Conditions} - +\[ \eqn{ Ingredients }{ + M \le L, M \le R +}\] \[ \eqn{ Tip Merge }{ L \in \py \implies \begin{cases} @@ -557,21 +800,31 @@ We will occasionally use $X,Y$ s.t. $\{X,Y\} = \{L,R\}$. \bigforall_{E \in \pendsof{X}{\py}} E \le Y \right] }\] +\[ \eqn{ Foreign Merges }{ + \patchof{L} = \bot \equiv \patchof{R} = \bot +}\] \subsection{Non-Topbloke merges} -We require both $\patchof{L} = \bot$ and $\patchof{R} = \bot$. +We require both $\patchof{L} = \bot$ and $\patchof{R} = \bot$ +(Foreign Merges, above). I.e. not only is it forbidden to merge into a Topbloke-controlled branch without Topbloke's assistance, it is also forbidden to merge any Topbloke-controlled branch into any plain git branch. Given those conditions, Tip Merge and Merge Acyclic do not apply. And $Y \not\in \py$ so $\neg [ Y \haspatch \p ]$ so neither -Merge Ends condition applies. Good. +Merge Ends condition applies. + +So a plain git merge of non-Topbloke branches meets the conditions and +is therefore consistent with our scheme. \subsection{No Replay} -See No Replay for Merge Results. +By definition of $\merge$, +$D \isin C \implies D \isin L \lor D \isin R \lor D = C$. +So, by Ingredients, +Ingredients Prevent Replay applies. $\qed$ \subsection{Unique Base} @@ -580,7 +833,7 @@ and calculate $\pendsof{C}{\pn}$. So we will consider some putative ancestor $A \in \pn$ and see whether $A \le C$. By Exact Ancestors for C, $A \le C \equiv A \le L \lor A \le R \lor A = C$. -But $C \in py$ and $A \in \pn$ so $A \neq C$. +But $C \in py$ and $A \in \pn$ so $A \neq C$. Thus $A \le C \equiv A \le L \lor A \le R$. By Unique Base of L and Transitive Ancestors, @@ -594,15 +847,15 @@ $A \le R \equiv A \le \baseof{R}$. But by Tip Merge condition on $\baseof{R}$, $A \le \baseof{L} \implies A \le \baseof{R}$, so $A \le \baseof{R} \lor A \le \baseof{L} \equiv A \le \baseof{R}$. -Thus $A \le C \equiv A \le \baseof{R}$. +Thus $A \le C \equiv A \le \baseof{R}$. That is, $\baseof{C} = \baseof{R}$. \subsubsection{For $R \in \pn$:} By Tip Merge condition on $R$ and since $M \le R$, $A \le \baseof{L} \implies A \le R$, so -$A \le R \lor A \le \baseof{L} \equiv A \le R$. -Thus $A \le C \equiv A \le R$. +$A \le R \lor A \le \baseof{L} \equiv A \le R$. +Thus $A \le C \equiv A \le R$. That is, $\baseof{C} = R$. $\qed$ @@ -610,7 +863,7 @@ $\qed$ \subsection{Coherence and Patch Inclusion} Need to determine $C \haspatch \p$ based on $L,M,R \haspatch \p$. -This involves considering $D \in \py$. +This involves considering $D \in \py$. \subsubsection{For $L \nothaspatch \p, R \nothaspatch \p$:} $D \not\isin L \land D \not\isin R$. $C \not\in \py$ (otherwise $L @@ -624,20 +877,20 @@ $D \isin L \equiv D \le L$ and $D \isin R \equiv D \le R$. Consider $D = C$: $D \isin C$, $D \le C$, OK for $C \haspatch \p$. For $D \neq C$: $D \le C \equiv D \le L \lor D \le R - \equiv D \isin L \lor D \isin R$. + \equiv D \isin L \lor D \isin R$. (Likewise $D \le C \equiv D \le X \lor D \le Y$.) Consider $D \neq C, D \isin X \land D \isin Y$: -By $\merge$, $D \isin C$. Also $D \le X$ +By $\merge$, $D \isin C$. Also $D \le X$ so $D \le C$. OK for $C \haspatch \p$. Consider $D \neq C, D \not\isin X \land D \not\isin Y$: -By $\merge$, $D \not\isin C$. -And $D \not\le X \land D \not\le Y$ so $D \not\le C$. +By $\merge$, $D \not\isin C$. +And $D \not\le X \land D \not\le Y$ so $D \not\le C$. OK for $C \haspatch \p$. Remaining case, wlog, is $D \not\isin X \land D \isin Y$. -$D \not\le X$ so $D \not\le M$ so $D \not\isin M$. +$D \not\le X$ so $D \not\le M$ so $D \not\isin M$. Thus by $\merge$, $D \isin C$. And $D \le Y$ so $D \le C$. OK for $C \haspatch \p$. @@ -650,7 +903,7 @@ $M \nothaspatch \p \implies C \haspatch \p$. \proofstarts -One of the Merge Ends conditions applies. +One of the Merge Ends conditions applies. Recall that we are considering $D \in \py$. $D \isin Y \equiv D \le Y$. $D \not\isin X$. We will show for each of @@ -663,12 +916,12 @@ $M=\baseof{L}$. So by Base Acyclic $D \not\isin M$, i.e. $M \nothaspatch \p$. And indeed $D \isin C$ and $D \le C$. OK. Consider $D \neq C, M \nothaspatch P, D \isin Y$: -$D \le Y$ so $D \le C$. +$D \le Y$ so $D \le C$. $D \not\isin M$ so by $\merge$, $D \isin C$. OK. Consider $D \neq C, M \nothaspatch P, D \not\isin Y$: $D \not\le Y$. If $D \le X$ then -$D \in \pancsof{X}{\py}$, so by Addition Merge Ends and +$D \in \pancsof{X}{\py}$, so by Addition Merge Ends and Transitive Ancestors $D \le Y$ --- a contradiction, so $D \not\le X$. Thus $D \not\le C$. By $\merge$, $D \not\isin C$. OK. @@ -690,11 +943,13 @@ $C \in \pn$ when $L \in \pn$ so by Merge Acyclic, $R \nothaspatch \p$. Consider some $D \in \py$. By Base Acyclic of $L$, $D \not\isin L$. By the above, $D \not\isin -R$. And $D \neq C$. So $D \not\isin C$. $\qed$ +R$. And $D \neq C$. So $D \not\isin C$. + +$\qed$ \subsection{Tip Contents} -We need worry only about $C \in \py$. +We need worry only about $C \in \py$. And $\patchof{C} = \patchof{L}$ so $L \in \py$ so $L \haspatch \p$. We will use the Unique Base of $C$, and its Coherence and Patch Inclusion, as just proved. @@ -770,4 +1025,10 @@ OK $\qed$ +\subsection{Foreign Contents} + +Only relevant if $\patchof{L} = \bot$, in which case +$\patchof{C} = \bot$ and by Foreign Merges $\patchof{R} = \bot$, +so Totally Foreign Contents applies. $\qed$ + \end{document}