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backport fix
fixed password refill in login form when wrong password was given
It's better to not refill the passwords on wrong credentials given.
Standard behaviour on all pages are making the password blank
fixed password refill in login form when wrong password was given
It's better to not refill the passwords on wrong credentials given.
Standard behaviour on all pages are making the password blank
a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 c9bcfe2d2ade c9bcfe2d2ade a60a0e9092c6 a60a0e9092c6 c9bcfe2d2ade a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 c9bcfe2d2ade a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 c9bcfe2d2ade c9bcfe2d2ade a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 c9bcfe2d2ade a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 a60a0e9092c6 | <!doctype html>
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\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
\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
$\eq{\card{\livar{V}0},\card{\set{\livar{v}r}},1,\power20}$.
\end{spfstep}
\end{spfcase}
\begin{spfcase}{$i>0$}
\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]
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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