X-Git-Url: http://www.chiark.greenend.org.uk/ucgi/~ian/git?p=topbloke-formulae.git;a=blobdiff_plain;f=simple.tex;h=297103adf419b066cb5269773de20409c14c0135;hp=82b90762d9c04eaad53142b866e2a3c84587126a;hb=c497f56e8a7f09c4c328bd7560631d8773b7e676;hpb=2daa9f7411a0be9dfd8f1e762e46fc42d4e1c95a diff --git a/simple.tex b/simple.tex index 82b9076..297103a 100644 --- a/simple.tex +++ b/simple.tex @@ -47,9 +47,22 @@ $L$, $D \isin C \implies D \not\in \py$. $\qed$ -\subsection{Coherence and patch inclusion} +\subsection{Coherence and Patch Inclusion} -Need to consider $D \in \py$ +$$+\begin{cases} + L \haspatch \p : & C \haspatch \p \\ + L \nothaspatch \p : & C \nothaspatch \p +\end{cases} +$$ +\proofstarts + +Firstly, if $L \haspatch \p$, $\exists_{F \in \py} F \le L$ +and this $F$ is also $\le C$ +so $C \zhaspatch \p \implies C \haspatch \p$. +We will prove $\zhaspatch$ + +We need to consider $D \in \py$. \subsubsection{For $L \haspatch \p, D = C$:} @@ -57,36 +70,36 @@ Ancestors of $C$: $D \le C$. Contents of $C$: -$D \isin C \equiv \ldots \lor \true \text{ so } D \haspatch C$. +$D \isin C \equiv \ldots \lor \true$. So $D \zhaspatch C$. +OK. \subsubsection{For $L \haspatch \p, D \neq C$:} Ancestors: $D \le C \equiv D \le L$. Contents: $D \isin C \equiv D \isin L \lor f$ so $D \isin C \equiv D \isin L$, i.e. $C \zhaspatch P$. -By $\haspatch$ for $L$, $\exists_{F \in \py} F \le L$ -and this $F$ is also $\le C$. So $\haspatch$. - -So: -$L \haspatch \p \implies C \haspatch \p$ +OK. \subsubsection{For $L \nothaspatch \p$:} Firstly, $C \not\in \py$ since if it were, $L \in \py$. Thus $D \neq C$. -Now by contents of $L$, $D \notin L$, so $D \notin C$. +Now by $\nothaspatch$, $D \not\isin L$ so $D \not\isin C$. +OK. -So: -$L \nothaspatch \p \implies C \nothaspatch \p$ $\qed$ +\subsection{Unique Tips:} + +Single Parent Unique Tips applies. $\qed$ + \subsection{Foreign Inclusion:} Simple Foreign Inclusion applies. $\qed$ \subsection{Foreign Contents:} -Only relevant if $\patchof{C} = \bot$, and in that case Totally +Only relevant if $\isforeign{C}$, and in that case Totally Foreign Contents applies. $\qed$