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1 <!doctype html>
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2
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3 <title>CodeMirror: sTeX mode</title>
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4 <meta charset="utf-8"/>
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5 <link rel=stylesheet href="../../doc/docs.css">
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6
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7 <link rel="stylesheet" href="../../lib/codemirror.css">
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8 <script src="../../lib/codemirror.js"></script>
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9 <script src="stex.js"></script>
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10 <style>.CodeMirror {background: #f8f8f8;}</style>
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11 <div id=nav>
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12 <a href="https://codemirror.net/5"><h1>CodeMirror</h1><img id=logo src="../../doc/logo.png" alt=""></a>
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13
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14 <ul>
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15 <li><a href="../../index.html">Home</a>
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16 <li><a href="../../doc/manual.html">Manual</a>
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17 <li><a href="https://github.com/codemirror/codemirror5">Code</a>
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18 </ul>
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19 <ul>
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20 <li><a href="../index.html">Language modes</a>
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21 <li><a class=active href="#">sTeX</a>
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22 </ul>
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23 </div>
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24
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25 <article>
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26 <h2>sTeX mode</h2>
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27 <form><textarea id="code" name="code">
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28 \begin{module}[id=bbt-size]
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29 \importmodule[balanced-binary-trees]{balanced-binary-trees}
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30 \importmodule[\KWARCslides{dmath/en/cardinality}]{cardinality}
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31
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32 \begin{frame}
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33 \frametitle{Size Lemma for Balanced Trees}
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34 \begin{itemize}
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35 \item
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36 \begin{assertion}[id=size-lemma,type=lemma]
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37 Let $G=\tup{V,E}$ be a \termref[cd=binary-trees]{balanced binary tree}
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38 of \termref[cd=graph-depth,name=vertex-depth]{depth}$n>i$, then the set
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39 $\defeq{\livar{V}i}{\setst{\inset{v}{V}}{\gdepth{v} = i}}$ of
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40 \termref[cd=graphs-intro,name=node]{nodes} at
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41 \termref[cd=graph-depth,name=vertex-depth]{depth} $i$ has
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42 \termref[cd=cardinality,name=cardinality]{cardinality} $\power2i$.
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43 \end{assertion}
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44 \item
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45 \begin{sproof}[id=size-lemma-pf,proofend=,for=size-lemma]{via induction over the depth $i$.}
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46 \begin{spfcases}{We have to consider two cases}
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47 \begin{spfcase}{$i=0$}
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48 \begin{spfstep}[display=flow]
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49 then $\livar{V}i=\set{\livar{v}r}$, where $\livar{v}r$ is the root, so
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50 $\eq{\card{\livar{V}0},\card{\set{\livar{v}r}},1,\power20}$.
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51 \end{spfstep}
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52 \end{spfcase}
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53 \begin{spfcase}{$i>0$}
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54 \begin{spfstep}[display=flow]
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55 then $\livar{V}{i-1}$ contains $\power2{i-1}$ vertexes
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56 \begin{justification}[method=byIH](IH)\end{justification}
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57 \end{spfstep}
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58 \begin{spfstep}
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59 By the \begin{justification}[method=byDef]definition of a binary
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60 tree\end{justification}, each $\inset{v}{\livar{V}{i-1}}$ is a leaf or has
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61 two children that are at depth $i$.
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62 \end{spfstep}
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63 \begin{spfstep}
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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
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65 leaves.
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66 \end{spfstep}
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67 \begin{spfstep}[type=conclusion]
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68 Thus $\eq{\card{\livar{V}i},{\atimes[cdot]{2,\card{\livar{V}{i-1}}}},{\atimes[cdot]{2,\power2{i-1}}},\power2i}$.
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69 \end{spfstep}
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70 \end{spfcase}
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71 \end{spfcases}
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72 \end{sproof}
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73 \item
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74 \begin{assertion}[id=fbbt,type=corollary]
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75 A fully balanced tree of depth $d$ has $\power2{d+1}-1$ nodes.
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76 \end{assertion}
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77 \item
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78 \begin{sproof}[for=fbbt,id=fbbt-pf]{}
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79 \begin{spfstep}
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80 Let $\defeq{G}{\tup{V,E}}$ be a fully balanced tree
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81 \end{spfstep}
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82 \begin{spfstep}
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83 Then $\card{V}=\Sumfromto{i}1d{\power2i}= \power2{d+1}-1$.
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84 \end{spfstep}
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85 \end{sproof}
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86 \end{itemize}
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87 \end{frame}
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88 \begin{note}
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89 \begin{omtext}[type=conclusion,for=binary-tree]
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90 This shows that balanced binary trees grow in breadth very quickly, a consequence of
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91 this is that they are very shallow (and this compute very fast), which is the essence of
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92 the next result.
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93 \end{omtext}
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94 \end{note}
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95 \end{module}
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96
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97 %%% Local Variables:
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98 %%% mode: LaTeX
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99 %%% TeX-master: "all"
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100 %%% End: \end{document}
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101 </textarea></form>
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102 <script>
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103 var editor = CodeMirror.fromTextArea(document.getElementById("code"), {});
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104 </script>
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105
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106 <p>sTeX mode supports this option:</p>
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107 <d1>
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108 <dt><code>inMathMode: boolean</code></dt>
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109 <dd>Whether to start parsing in math mode (default: <code>false</code>).</dd>
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110 </d1>
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111
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112 <p><strong>MIME types defined:</strong> <code>text/x-stex</code>.</p>
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113
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114 <p><strong>Parsing/Highlighting Tests:</strong> <a href="../../test/index.html#stex_*">normal</a>, <a href="../../test/index.html#verbose,stex_*">verbose</a>.</p>
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115
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116 </article>
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