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author Coffee CMS <info@coffee-cms.ru>
date Fri, 11 Oct 2024 22:40:23 +0000
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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="https://codemirror.net/5"><h1>CodeMirror</h1><img id=logo src="../../doc/logo.png" alt=""></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/codemirror5">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>sTeX mode supports this option:</p>
107 <d1>
108 <dt><code>inMathMode: boolean</code></dt>
109 <dd>Whether to start parsing in math mode (default: <code>false</code>).</dd>
110 </d1>
111
112 <p><strong>MIME types defined:</strong> <code>text/x-stex</code>.</p>
113
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>
115
116 </article>