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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> |