Files
@ ad49bbd8f984
Branch filter:
Location: kallithea/rhodecode/public/js/mode/stex/index.html - annotation
ad49bbd8f984
3.7 KiB
text/html
changelog - should refer to Mercurial repository logs instead
a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 c9bcfe2d2ade c9bcfe2d2ade a60a0e9092c6 a60a0e9092c6 c9bcfe2d2ade a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 c9bcfe2d2ade a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 c9bcfe2d2ade c9bcfe2d2ade a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 c9bcfe2d2ade a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 | <!doctype html>
<html>
<head>
<meta charset="utf-8">
<title>CodeMirror: sTeX mode</title>
<link rel="stylesheet" href="../../lib/codemirror.css">
<script src="../../lib/codemirror.js"></script>
<script src="stex.js"></script>
<style>.CodeMirror {background: #f8f8f8;}</style>
<link rel="stylesheet" href="../../doc/docs.css">
</head>
<body>
<h1>CodeMirror: sTeX mode</h1>
<form><textarea id="code" name="code">
\begin{module}[id=bbt-size]
\importmodule[balanced-binary-trees]{balanced-binary-trees}
\importmodule[\KWARCslides{dmath/en/cardinality}]{cardinality}
\begin{frame}
\frametitle{Size Lemma for Balanced Trees}
\begin{itemize}
\item
\begin{assertion}[id=size-lemma,type=lemma]
Let $G=\tup{V,E}$ be a \termref[cd=binary-trees]{balanced binary tree}
of \termref[cd=graph-depth,name=vertex-depth]{depth}$n>i$, then the set
$\defeq{\livar{V}i}{\setst{\inset{v}{V}}{\gdepth{v} = i}}$ of
\termref[cd=graphs-intro,name=node]{nodes} at
\termref[cd=graph-depth,name=vertex-depth]{depth} $i$ has
\termref[cd=cardinality,name=cardinality]{cardinality} $\power2i$.
\end{assertion}
\item
\begin{sproof}[id=size-lemma-pf,proofend=,for=size-lemma]{via induction over the depth $i$.}
\begin{spfcases}{We have to consider two cases}
\begin{spfcase}{$i=0$}
\begin{spfstep}[display=flow]
then $\livar{V}i=\set{\livar{v}r}$, where $\livar{v}r$ is the root, so
$\eq{\card{\livar{V}0},\card{\set{\livar{v}r}},1,\power20}$.
\end{spfstep}
\end{spfcase}
\begin{spfcase}{$i>0$}
\begin{spfstep}[display=flow]
then $\livar{V}{i-1}$ contains $\power2{i-1}$ vertexes
\begin{justification}[method=byIH](IH)\end{justification}
\end{spfstep}
\begin{spfstep}
By the \begin{justification}[method=byDef]definition of a binary
tree\end{justification}, each $\inset{v}{\livar{V}{i-1}}$ is a leaf or has
two children that are at depth $i$.
\end{spfstep}
\begin{spfstep}
As $G$ is \termref[cd=balanced-binary-trees,name=balanced-binary-tree]{balanced} and $\gdepth{G}=n>i$, $\livar{V}{i-1}$ cannot contain
leaves.
\end{spfstep}
\begin{spfstep}[type=conclusion]
Thus $\eq{\card{\livar{V}i},{\atimes[cdot]{2,\card{\livar{V}{i-1}}}},{\atimes[cdot]{2,\power2{i-1}}},\power2i}$.
\end{spfstep}
\end{spfcase}
\end{spfcases}
\end{sproof}
\item
\begin{assertion}[id=fbbt,type=corollary]
A fully balanced tree of depth $d$ has $\power2{d+1}-1$ nodes.
\end{assertion}
\item
\begin{sproof}[for=fbbt,id=fbbt-pf]{}
\begin{spfstep}
Let $\defeq{G}{\tup{V,E}}$ be a fully balanced tree
\end{spfstep}
\begin{spfstep}
Then $\card{V}=\Sumfromto{i}1d{\power2i}= \power2{d+1}-1$.
\end{spfstep}
\end{sproof}
\end{itemize}
\end{frame}
\begin{note}
\begin{omtext}[type=conclusion,for=binary-tree]
This shows that balanced binary trees grow in breadth very quickly, a consequence of
this is that they are very shallow (and this compute very fast), which is the essence of
the next result.
\end{omtext}
\end{note}
\end{module}
%%% Local Variables:
%%% mode: LaTeX
%%% TeX-master: "all"
%%% End: \end{document}
</textarea></form>
<script>
var editor = CodeMirror.fromTextArea(document.getElementById("code"), {});
</script>
<p><strong>MIME types defined:</strong> <code>text/x-stex</code>.</p>
<p><strong>Parsing/Highlighting Tests:</strong> <a href="../../test/index.html#stex_*">normal</a>, <a href="../../test/index.html#verbose,stex_*">verbose</a>.</p>
</body>
</html>
|