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