github.com/elliott5/community@v0.14.1-0.20160709191136-823126fb026a/app/public/codemirror/mode/stex/index.html (about)

     1  <!doctype html>
     2  
     3  <title>CodeMirror: sTeX mode</title>
     4  <meta charset="utf-8"/>
     5  <link rel=stylesheet href="../../doc/docs.css">
     6  
     7  <link rel="stylesheet" href="../../lib/codemirror.css">
     8  <script src="../../lib/codemirror.js"></script>
     9  <script src="stex.js"></script>
    10  <style>.CodeMirror {background: #f8f8f8;}</style>
    11  <div id=nav>
    12    <a href="http://codemirror.net"><h1>CodeMirror</h1><img id=logo src="../../doc/logo.png"></a>
    13  
    14    <ul>
    15      <li><a href="../../index.html">Home</a>
    16      <li><a href="../../doc/manual.html">Manual</a>
    17      <li><a href="https://github.com/codemirror/codemirror">Code</a>
    18    </ul>
    19    <ul>
    20      <li><a href="../index.html">Language modes</a>
    21      <li><a class=active href="#">sTeX</a>
    22    </ul>
    23  </div>
    24  
    25  <article>
    26  <h2>sTeX mode</h2>
    27  <form><textarea id="code" name="code">
    28  \begin{module}[id=bbt-size]
    29  \importmodule[balanced-binary-trees]{balanced-binary-trees}
    30  \importmodule[\KWARCslides{dmath/en/cardinality}]{cardinality}
    31  
    32  \begin{frame}
    33    \frametitle{Size Lemma for Balanced Trees}
    34    \begin{itemize}
    35    \item
    36      \begin{assertion}[id=size-lemma,type=lemma] 
    37      Let $G=\tup{V,E}$ be a \termref[cd=binary-trees]{balanced binary tree} 
    38      of \termref[cd=graph-depth,name=vertex-depth]{depth}$n>i$, then the set
    39       $\defeq{\livar{V}i}{\setst{\inset{v}{V}}{\gdepth{v} = i}}$ of
    40      \termref[cd=graphs-intro,name=node]{nodes} at 
    41      \termref[cd=graph-depth,name=vertex-depth]{depth} $i$ has
    42      \termref[cd=cardinality,name=cardinality]{cardinality} $\power2i$.
    43     \end{assertion}
    44    \item
    45      \begin{sproof}[id=size-lemma-pf,proofend=,for=size-lemma]{via induction over the depth $i$.}
    46        \begin{spfcases}{We have to consider two cases}
    47          \begin{spfcase}{$i=0$}
    48            \begin{spfstep}[display=flow]
    49              then $\livar{V}i=\set{\livar{v}r}$, where $\livar{v}r$ is the root, so
    50              $\eq{\card{\livar{V}0},\card{\set{\livar{v}r}},1,\power20}$.
    51            \end{spfstep}
    52          \end{spfcase}
    53          \begin{spfcase}{$i>0$}
    54            \begin{spfstep}[display=flow]
    55             then $\livar{V}{i-1}$ contains $\power2{i-1}$ vertexes 
    56             \begin{justification}[method=byIH](IH)\end{justification}
    57            \end{spfstep}
    58            \begin{spfstep}
    59             By the \begin{justification}[method=byDef]definition of a binary
    60                tree\end{justification}, each $\inset{v}{\livar{V}{i-1}}$ is a leaf or has
    61              two children that are at depth $i$.
    62            \end{spfstep}
    63            \begin{spfstep}
    64             As $G$ is \termref[cd=balanced-binary-trees,name=balanced-binary-tree]{balanced} and $\gdepth{G}=n>i$, $\livar{V}{i-1}$ cannot contain
    65              leaves.
    66            \end{spfstep}
    67            \begin{spfstep}[type=conclusion]
    68             Thus $\eq{\card{\livar{V}i},{\atimes[cdot]{2,\card{\livar{V}{i-1}}}},{\atimes[cdot]{2,\power2{i-1}}},\power2i}$.
    69            \end{spfstep}
    70          \end{spfcase}
    71        \end{spfcases}
    72      \end{sproof}
    73    \item 
    74      \begin{assertion}[id=fbbt,type=corollary]	
    75        A fully balanced tree of depth $d$ has $\power2{d+1}-1$ nodes.
    76      \end{assertion}
    77    \item
    78        \begin{sproof}[for=fbbt,id=fbbt-pf]{}
    79          \begin{spfstep}
    80            Let $\defeq{G}{\tup{V,E}}$ be a fully balanced tree
    81          \end{spfstep}
    82          \begin{spfstep}
    83            Then $\card{V}=\Sumfromto{i}1d{\power2i}= \power2{d+1}-1$.
    84          \end{spfstep}
    85        \end{sproof}
    86      \end{itemize}
    87    \end{frame}
    88  \begin{note}
    89    \begin{omtext}[type=conclusion,for=binary-tree]
    90      This shows that balanced binary trees grow in breadth very quickly, a consequence of
    91      this is that they are very shallow (and this compute very fast), which is the essence of
    92      the next result.
    93    \end{omtext}
    94  \end{note}
    95  \end{module}
    96  
    97  %%% Local Variables: 
    98  %%% mode: LaTeX
    99  %%% TeX-master: "all"
   100  %%% End: \end{document}
   101  </textarea></form>
   102      <script>
   103        var editor = CodeMirror.fromTextArea(document.getElementById("code"), {});
   104      </script>
   105  
   106      <p><strong>MIME types defined:</strong> <code>text/x-stex</code>.</p>
   107  
   108      <p><strong>Parsing/Highlighting Tests:</strong> <a href="../../test/index.html#stex_*">normal</a>,  <a href="../../test/index.html#verbose,stex_*">verbose</a>.</p>
   109  
   110    </article>