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Flip vertical order of prime-factors example

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Adam Chlipala 2016-03-07 07:51:40 -05:00
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@ -1793,27 +1793,27 @@ As another example, consider a lattice tracking prime factors of numbers, up to
Then the picture version might go like so: Then the picture version might go like so:
\begin{center}\begin{tikzpicture}[node distance=1.5cm] \begin{center}\begin{tikzpicture}[node distance=1.5cm]
\node(top) {$\{2, 3, 5\}$}; \node(top) {$\{\}$};
\node(twothree) [below left of=top] {$\{2, 3\}$}; \node(two) [below left of=top] {$\{2\}$};
\node(twofive) [below of=top] {$\{2, 5\}$}; \node(three) [below of=top] {$\{3\}$};
\node(threefive) [below right of=top] {$\{3, 5\}$}; \node(five) [below right of=top] {$\{5\}$};
\node(two) [below of=twothree] {$\{2\}$}; \node(twothree) [below left of=two] {$\{2, 3\}$};
\node(three) [below of=twofive] {$\{3\}$}; \node(twofive) [below of=three] {$\{2, 5\}$};
\node(five) [below of=threefive] {$\{5\}$}; \node(threefive) [below right of=five] {$\{3, 5\}$};
\node(emp) [below of=three] {$\{\}$}; \node(bot) [below of=twofive] {$\{2, 3, 5\}$};
\draw(top) -- (twothree); \draw(top) -- (two);
\draw(top) -- (twofive); \draw(top) -- (three);
\draw(top) -- (threefive); \draw(top) -- (five);
\draw(twothree) -- (two); \draw(two) -- (twothree);
\draw(twothree) -- (three); \draw(two) -- (twofive);
\draw(twofive) -- (two); \draw(three) -- (twothree);
\draw(twofive) -- (five); \draw(three) -- (threefive);
\draw(threefive) -- (three); \draw(five) -- (twofive);
\draw(threefive) -- (five); \draw(five) -- (threefive);
\draw(two) -- (emp); \draw(twothree) -- (bot);
\draw(three) -- (emp); \draw(twofive) -- (bot);
\draw(five) -- (emp); \draw(threefive) -- (bot);
\end{tikzpicture}\end{center} \end{tikzpicture}\end{center}
Since $\sqsubseteq$ is clearly transitive, upward-moving paths across multiple nodes also imply $\sqsubseteq$ relationships between their endpoints. Since $\sqsubseteq$ is clearly transitive, upward-moving paths across multiple nodes also imply $\sqsubseteq$ relationships between their endpoints.