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author | Armen Kesablyan | 2012-02-22 16:26:41 -0800 |
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committer | Armen Kesablyan | 2012-02-22 16:26:41 -0800 |
commit | 0bd1cefea2ab350fad1a891bdc926053b799aafc (patch) | |
tree | 962f559fcc02a3dfeb297d59907e40fa153453f3 /imports/codemirror/mode/stex/index.html | |
parent | 695bc5082f48dddf66ce31480a4faefc067b38bd (diff) | |
parent | 2d2b1af8b5c0d506fe6a1cf65614101fec145970 (diff) | |
download | ninja-0bd1cefea2ab350fad1a891bdc926053b799aafc.tar.gz |
Merge branch 'refs/heads/master' into new-tool-icons
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-rwxr-xr-x | imports/codemirror/mode/stex/index.html | 95 |
1 files changed, 95 insertions, 0 deletions
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1 | <!doctype html> | ||
2 | <html> | ||
3 | <head> | ||
4 | <title>CodeMirror: 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 | <style>.CodeMirror {background: #f8f8f8;}</style> | ||
9 | <link rel="stylesheet" href="../../doc/docs.css"> | ||
10 | </head> | ||
11 | <body> | ||
12 | <h1>CodeMirror: sTeX mode</h1> | ||
13 | <form><textarea id="code" name="code"> | ||
14 | \begin{module}[id=bbt-size] | ||
15 | \importmodule[balanced-binary-trees]{balanced-binary-trees} | ||
16 | \importmodule[\KWARCslides{dmath/en/cardinality}]{cardinality} | ||
17 | |||
18 | \begin{frame} | ||
19 | \frametitle{Size Lemma for Balanced Trees} | ||
20 | \begin{itemize} | ||
21 | \item | ||
22 | \begin{assertion}[id=size-lemma,type=lemma] | ||
23 | Let $G=\tup{V,E}$ be a \termref[cd=binary-trees]{balanced binary tree} | ||
24 | of \termref[cd=graph-depth,name=vertex-depth]{depth}$n>i$, then the set | ||
25 | $\defeq{\livar{V}i}{\setst{\inset{v}{V}}{\gdepth{v} = i}}$ of | ||
26 | \termref[cd=graphs-intro,name=node]{nodes} at | ||
27 | \termref[cd=graph-depth,name=vertex-depth]{depth} $i$ has | ||
28 | \termref[cd=cardinality,name=cardinality]{cardinality} $\power2i$. | ||
29 | \end{assertion} | ||
30 | \item | ||
31 | \begin{sproof}[id=size-lemma-pf,proofend=,for=size-lemma]{via induction over the depth $i$.} | ||
32 | \begin{spfcases}{We have to consider two cases} | ||
33 | \begin{spfcase}{$i=0$} | ||
34 | \begin{spfstep}[display=flow] | ||
35 | then $\livar{V}i=\set{\livar{v}r}$, where $\livar{v}r$ is the root, so | ||
36 | $\eq{\card{\livar{V}0},\card{\set{\livar{v}r}},1,\power20}$. | ||
37 | \end{spfstep} | ||
38 | \end{spfcase} | ||
39 | \begin{spfcase}{$i>0$} | ||
40 | \begin{spfstep}[display=flow] | ||
41 | then $\livar{V}{i-1}$ contains $\power2{i-1}$ vertexes | ||
42 | \begin{justification}[method=byIH](IH)\end{justification} | ||
43 | \end{spfstep} | ||
44 | \begin{spfstep} | ||
45 | By the \begin{justification}[method=byDef]definition of a binary | ||
46 | tree\end{justification}, each $\inset{v}{\livar{V}{i-1}}$ is a leaf or has | ||
47 | two children that are at depth $i$. | ||
48 | \end{spfstep} | ||
49 | \begin{spfstep} | ||
50 | As $G$ is \termref[cd=balanced-binary-trees,name=balanced-binary-tree]{balanced} and $\gdepth{G}=n>i$, $\livar{V}{i-1}$ cannot contain | ||
51 | leaves. | ||
52 | \end{spfstep} | ||
53 | \begin{spfstep}[type=conclusion] | ||
54 | Thus $\eq{\card{\livar{V}i},{\atimes[cdot]{2,\card{\livar{V}{i-1}}}},{\atimes[cdot]{2,\power2{i-1}}},\power2i}$. | ||
55 | \end{spfstep} | ||
56 | \end{spfcase} | ||
57 | \end{spfcases} | ||
58 | \end{sproof} | ||
59 | \item | ||
60 | \begin{assertion}[id=fbbt,type=corollary] | ||
61 | A fully balanced tree of depth $d$ has $\power2{d+1}-1$ nodes. | ||
62 | \end{assertion} | ||
63 | \item | ||
64 | \begin{sproof}[for=fbbt,id=fbbt-pf]{} | ||
65 | \begin{spfstep} | ||
66 | Let $\defeq{G}{\tup{V,E}}$ be a fully balanced tree | ||
67 | \end{spfstep} | ||
68 | \begin{spfstep} | ||
69 | Then $\card{V}=\Sumfromto{i}1d{\power2i}= \power2{d+1}-1$. | ||
70 | \end{spfstep} | ||
71 | \end{sproof} | ||
72 | \end{itemize} | ||
73 | \end{frame} | ||
74 | \begin{note} | ||
75 | \begin{omtext}[type=conclusion,for=binary-tree] | ||
76 | This shows that balanced binary trees grow in breadth very quickly, a consequence of | ||
77 | this is that they are very shallow (and this compute very fast), which is the essence of | ||
78 | the next result. | ||
79 | \end{omtext} | ||
80 | \end{note} | ||
81 | \end{module} | ||
82 | |||
83 | %%% Local Variables: | ||
84 | %%% mode: LaTeX | ||
85 | %%% TeX-master: "all" | ||
86 | %%% End: \end{document} | ||
87 | </textarea></form> | ||
88 | <script> | ||
89 | var editor = CodeMirror.fromTextArea(document.getElementById("code"), {}); | ||
90 | </script> | ||
91 | |||
92 | <p><strong>MIME types defined:</strong> <code>text/stex</code>.</p> | ||
93 | |||
94 | </body> | ||
95 | </html> | ||