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lib/Farabi/files/public/assets/codemirror/mode/sql/sql.js  view on Meta::CPAN

    return null;
  };

  // short client keyword token
  function hookClient(stream) {
    // \N means NULL
    // ref: http://dev.mysql.com/doc/refman/5.5/en/null-values.html
    if (stream.eat("N")) {
        return "atom";
    }
    // \g, etc
    // ref: http://dev.mysql.com/doc/refman/5.5/en/mysql-commands.html
    return stream.match(/^[a-zA-Z.#!?]/) ? "variable-2" : null;
  }

  // these keywords are used by all SQL dialects (however, a mode can still overwrite it)
  var sqlKeywords = "alter and as asc between by count create delete desc distinct drop from having in insert into is join like not on or order select set table union update values where ";

  // turn a space-separated list into an array
  function set(str) {
    var obj = {}, words = str.split(" ");

lib/Farabi/files/public/assets/codemirror/mode/stex/index.html  view on Meta::CPAN

\importmodule[balanced-binary-trees]{balanced-binary-trees}
\importmodule[\KWARCslides{dmath/en/cardinality}]{cardinality}

\begin{frame}
  \frametitle{Size Lemma for Balanced Trees}
  \begin{itemize}
  \item
    \begin{assertion}[id=size-lemma,type=lemma] 
    Let $G=\tup{V,E}$ be a \termref[cd=binary-trees]{balanced binary tree} 
    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
    \termref[cd=graphs-intro,name=node]{nodes} at 
    \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

lib/Farabi/files/public/assets/codemirror/mode/stex/index.html  view on Meta::CPAN

          \begin{spfstep}[display=flow]
           then $\livar{V}{i-1}$ contains $\power2{i-1}$ vertexes 
           \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}
  \item 
    \begin{assertion}[id=fbbt,type=corollary]