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        marcink
    
added codemirror edit mode with autodetection

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      <!doctype html>

      <html>

        <head>

          <meta charset="utf-8">

          <title>CodeMirror: sTeX mode</title>

          <link rel="stylesheet" href="../../lib/codemirror.css">

          <script src="../../lib/codemirror.js"></script>

          <script src="stex.js"></script>

          <style>.CodeMirror {background: #f8f8f8;}</style>

          <link rel="stylesheet" href="../../doc/docs.css">

        </head>

        <body>

          <h1>CodeMirror: sTeX mode</h1>

           <form><textarea id="code" name="code">

      \begin{module}[id=bbt-size]

      \importmodule[balanced-binary-trees]{balanced-binary-trees}

      \importmodule[\KWARCslides{dmath/en/cardinality}]{cardinality}

      \begin{frame}

        \frametitle{Size Lemma for Balanced Trees}

        \begin{itemize}

        \item

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          \begin{assertion}[id=size-lemma,type=lemma]

          Let $G=\tup{V,E}$ be a \termref[cd=binary-trees]{balanced binary tree}

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          of \termref[cd=graph-depth,name=vertex-depth]{depth}$n>i$, then the set

           $\defeq{\livar{V}i}{\setst{\inset{v}{V}}{\gdepth{v} = i}}$ of

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          \termref[cd=graphs-intro,name=node]{nodes} at

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          \termref[cd=graph-depth,name=vertex-depth]{depth} $i$ has

          \termref[cd=cardinality,name=cardinality]{cardinality} $\power2i$.

         \end{assertion}

        \item

          \begin{sproof}[id=size-lemma-pf,proofend=,for=size-lemma]{via induction over the depth $i$.}

            \begin{spfcases}{We have to consider two cases}

              \begin{spfcase}{$i=0$}

                \begin{spfstep}[display=flow]

                  then $\livar{V}i=\set{\livar{v}r}$, where $\livar{v}r$ is the root, so

                  $\eq{\card{\livar{V}0},\card{\set{\livar{v}r}},1,\power20}$.

                \end{spfstep}

              \end{spfcase}

              \begin{spfcase}{$i>0$}

                \begin{spfstep}[display=flow]

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                 then $\livar{V}{i-1}$ contains $\power2{i-1}$ vertexes

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                 \begin{justification}[method=byIH](IH)\end{justification}

                \end{spfstep}

                \begin{spfstep}

                 By the \begin{justification}[method=byDef]definition of a binary

                    tree\end{justification}, each $\inset{v}{\livar{V}{i-1}}$ is a leaf or has

                  two children that are at depth $i$.

                \end{spfstep}

                \begin{spfstep}

                 As $G$ is \termref[cd=balanced-binary-trees,name=balanced-binary-tree]{balanced} and $\gdepth{G}=n>i$, $\livar{V}{i-1}$ cannot contain

                  leaves.

                \end{spfstep}

                \begin{spfstep}[type=conclusion]

                 Thus $\eq{\card{\livar{V}i},{\atimes[cdot]{2,\card{\livar{V}{i-1}}}},{\atimes[cdot]{2,\power2{i-1}}},\power2i}$.

                \end{spfstep}

              \end{spfcase}

            \end{spfcases}

          \end{sproof}

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        \item

          \begin{assertion}[id=fbbt,type=corollary]

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            A fully balanced tree of depth $d$ has $\power2{d+1}-1$ nodes.

          \end{assertion}

        \item

            \begin{sproof}[for=fbbt,id=fbbt-pf]{}

              \begin{spfstep}

                Let $\defeq{G}{\tup{V,E}}$ be a fully balanced tree

              \end{spfstep}

              \begin{spfstep}

                Then $\card{V}=\Sumfromto{i}1d{\power2i}= \power2{d+1}-1$.

              \end{spfstep}

            \end{sproof}

          \end{itemize}

        \end{frame}

      \begin{note}

        \begin{omtext}[type=conclusion,for=binary-tree]

          This shows that balanced binary trees grow in breadth very quickly, a consequence of

          this is that they are very shallow (and this compute very fast), which is the essence of

          the next result.

        \end{omtext}

      \end{note}

      \end{module}

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      %%% Local Variables:

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      %%% mode: LaTeX

      %%% TeX-master: "all"

      %%% End: \end{document}

      </textarea></form>

          <script>

            var editor = CodeMirror.fromTextArea(document.getElementById("code"), {});

          </script>

          <p><strong>MIME types defined:</strong> <code>text/x-stex</code>.</p>

          <p><strong>Parsing/Highlighting Tests:</strong> <a href="../../test/index.html#stex_*">normal</a>,  <a href="../../test/index.html#verbose,stex_*">verbose</a>.</p>

        </body>

      </html>

	Site-wide shortcuts
/	Use quick search box
g h	Goto home page
g g	Goto my private gists page
g G	Goto my public gists page
g 0-9	Goto bookmarked items from 0-9
n r	New repository page
n g	New gist page

	Repositories
g s	Goto summary page
g c	Goto changelog page
g f	Goto files page
g F	Goto files page with file search activated
g p	Goto pull requests page
g o	Goto repository settings
g O	Goto repository access permissions settings
t s	Toggle sidebar on some pages

marcink added codemirror edit mode with autodetection	r4026	<!doctype html>
		<html>
		<head>
		<meta charset="utf-8">
		<title>CodeMirror: sTeX mode</title>
		<link rel="stylesheet" href="../../lib/codemirror.css">
		<script src="../../lib/codemirror.js"></script>
		<script src="stex.js"></script>
		<style>.CodeMirror {background: #f8f8f8;}</style>
		<link rel="stylesheet" href="../../doc/docs.css">
		</head>
		<body>
		<h1>CodeMirror: sTeX mode</h1>
		<form><textarea id="code" name="code">
		\begin{module}[id=bbt-size]
		\importmodule[balanced-binary-trees]{balanced-binary-trees}
		\importmodule[\KWARCslides{dmath/en/cardinality}]{cardinality}

		\begin{frame}
		\frametitle{Size Lemma for Balanced Trees}
		\begin{itemize}
		\item
marcink codecleaner	r4029	\begin{assertion}[id=size-lemma,type=lemma]
		Let $G=\tup{V,E}$ be a \termref[cd=binary-trees]{balanced binary tree}
marcink added codemirror edit mode with autodetection	r4026	of \termref[cd=graph-depth,name=vertex-depth]{depth}$n>i$, then the set
		$\defeq{\livar{V}i}{\setst{\inset{v}{V}}{\gdepth{v} = i}}$ of
marcink codecleaner	r4029	\termref[cd=graphs-intro,name=node]{nodes} at
marcink added codemirror edit mode with autodetection	r4026	\termref[cd=graph-depth,name=vertex-depth]{depth} $i$ has
		\termref[cd=cardinality,name=cardinality]{cardinality} $\power2i$.
		\end{assertion}
		\item
		\begin{sproof}[id=size-lemma-pf,proofend=,for=size-lemma]{via induction over the depth $i$.}
		\begin{spfcases}{We have to consider two cases}
		\begin{spfcase}{$i=0$}
		\begin{spfstep}[display=flow]
		then $\livar{V}i=\set{\livar{v}r}$, where $\livar{v}r$ is the root, so
		$\eq{\card{\livar{V}0},\card{\set{\livar{v}r}},1,\power20}$.
		\end{spfstep}
		\end{spfcase}
		\begin{spfcase}{$i>0$}
		\begin{spfstep}[display=flow]
marcink codecleaner	r4029	then $\livar{V}{i-1}$ contains $\power2{i-1}$ vertexes
marcink added codemirror edit mode with autodetection	r4026	\begin{justification}[method=byIH](IH)\end{justification}
		\end{spfstep}
		\begin{spfstep}
		By the \begin{justification}[method=byDef]definition of a binary
		tree\end{justification}, each $\inset{v}{\livar{V}{i-1}}$ is a leaf or has
		two children that are at depth $i$.
		\end{spfstep}
		\begin{spfstep}
		As $G$ is \termref[cd=balanced-binary-trees,name=balanced-binary-tree]{balanced} and $\gdepth{G}=n>i$, $\livar{V}{i-1}$ cannot contain
		leaves.
		\end{spfstep}
		\begin{spfstep}[type=conclusion]
		Thus $\eq{\card{\livar{V}i},{\atimes[cdot]{2,\card{\livar{V}{i-1}}}},{\atimes[cdot]{2,\power2{i-1}}},\power2i}$.
		\end{spfstep}
		\end{spfcase}
		\end{spfcases}
		\end{sproof}
marcink codecleaner	r4029	\item
		\begin{assertion}[id=fbbt,type=corollary]
marcink added codemirror edit mode with autodetection	r4026	A fully balanced tree of depth $d$ has $\power2{d+1}-1$ nodes.
		\end{assertion}
		\item
		\begin{sproof}[for=fbbt,id=fbbt-pf]{}
		\begin{spfstep}
		Let $\defeq{G}{\tup{V,E}}$ be a fully balanced tree
		\end{spfstep}
		\begin{spfstep}
		Then $\card{V}=\Sumfromto{i}1d{\power2i}= \power2{d+1}-1$.
		\end{spfstep}
		\end{sproof}
		\end{itemize}
		\end{frame}
		\begin{note}
		\begin{omtext}[type=conclusion,for=binary-tree]
		This shows that balanced binary trees grow in breadth very quickly, a consequence of
		this is that they are very shallow (and this compute very fast), which is the essence of
		the next result.
		\end{omtext}
		\end{note}
		\end{module}

marcink codecleaner	r4029	%%% Local Variables:
marcink added codemirror edit mode with autodetection	r4026	%%% mode: LaTeX
		%%% TeX-master: "all"
		%%% End: \end{document}
		</textarea></form>
		<script>
		var editor = CodeMirror.fromTextArea(document.getElementById("code"), {});
		</script>

		<p><strong>MIME types defined:</strong> <code>text/x-stex</code>.</p>

		<p><strong>Parsing/Highlighting Tests:</strong> <a href="../../test/index.html#stex_">normal</a>, <a href="../../test/index.html#verbose,stex_">verbose</a>.</p>

		</body>
		</html>