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| author | Nivesh Rajbhandari | 2012-02-20 11:14:44 -0800 |
|---|---|---|
| committer | Nivesh Rajbhandari | 2012-02-20 11:14:44 -0800 |
| commit | abf78e2d7a97d295ce5a1c425fd359d47379137e (patch) | |
| tree | d08c91bd2aef31e6325e0b499b2ffc390018bec6 /imports/codemirror/mode/stex/index.html | |
| parent | e80a79bff57fecf3aa9b869d8ed2de5fd815287c (diff) | |
| parent | e23708721a71ca4c71365f5f8e8ac7d6113926db (diff) | |
| download | ninja-abf78e2d7a97d295ce5a1c425fd359d47379137e.tar.gz | |
Merge branch 'refs/heads/ninja-internal' into ToolFixes
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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> | ||