% !TeX root = tcolorbox.tex
% include file of tcolorbox.tex (manual of the LaTeX package tcolorbox)
\clearpage
\section{Library \mylib{theorems}}\label{sec:theorems}%
\tcbset{external/prefix=external/theorems_}%
The library is loaded by a package option or inside the preamble by:
\begin{dispListing}
\tcbuselibrary{theorems}
\end{dispListing}
This also loads the package \refPkg{amsmath}.

Denis Bitouz\'e, Muzimuzhi, and many others provided very valuable input for this library.


\subsection{Macros of the Library}

\begin{docCommands}[
    doc parameter = \oarg{init options}\marg{name}\marg{display name}\marg{options}\marg{prefix},
    doc updated=2022-07-18
    ]
    {
      { doc name = NewTcbTheorem },
      { doc name = newtcbtheorem }
    }
  Creates new environments \meta{name} and \meta{name}|*| based on |tcolorbox| to frame a
  (mathematical) theorem. The \meta{display name} is used in the title line
  with a number, e.\,g. \mbox{\flqq Theorem 5.1\frqq}.
  The \meta{options} are given to the underlying |tcolorbox| to control
  the appearance.
  %The \meta{counter} is used for automatic numbering.
  The \meta{init options} allow setting up automatic numbering,
  see \Vref{sec:initkeys}.\par\medskip

  If \meta{name} is chosen as |footheorem|, the new environments are described by
  \begin{docEnvironments}[
      doc no index,
      doc name = footheorem,
      doc parameter = \oarg{keys}\oarg{short}\marg{title}\marg{marker}
    ]{}
    Here, \meta{title} is the title text for the theorem and
    is also set as \refKey{/tcb/nameref} identifier.
    Optionally, the \meta{short} title used instead as \refKey{/tcb/nameref} identifier
    and in lists of theorems.
    \meta{keys} supplement the \meta{options} and may be
    used only in rare cases. The theorem is
    automatically labeled with \meta{prefix}\meta{separator}\meta{marker}
    where \meta{separator} is predefined as \enquote{|:|}, see \refKey{/tcb/label separator}.
  \end{docEnvironments}
  \begin{docEnvironments}[
      doc no index,
      doc name = footheorem*,
      doc parameter = \oarg{keys}\marg{title}
    ]{}
    This represents an unnumbered variant. Again, \meta{title} is the title text for the theorem
    and \meta{keys} supplement the \meta{options}.
    This variant is not labeled and not listed in lists of theorems.
  \end{docEnvironments}

  \begin{marker}
  \refCom{NewTcbTheorem} and \refCom{newtcbtheorem} are identical and both
  rely on \refCom{NewTColorBox} based on |\NewDocumentCommand|.
  \end{marker}
\clearpage


\inputpreamblelisting{F}
\begin{dispExample}
% usage of `\nameref' needs `nameref' or `hyperref' to be loaded
\begin{mytheo}{This is my title}{theoexample}
  This is the text of the theorem. The counter is automatically assigned and,
  in this example, prefixed with the section number. This theorem is numbered with
  \ref{th:theoexample}, it is given on page~\pageref{th:theoexample},
  and it is titled \flqq\nameref{th:theoexample}\frqq.
\end{mytheo}
\end{dispExample}


\begin{dispExample}
% usage of `\nameref' needs `nameref' or `hyperref' to be loaded
\begin{mytheo}[][Concise title]{This my very long theorem title with
    too much words}{theoexample2}
  Here we have a long and a short title. This theorem is numbered with
  \ref{th:theoexample2}, it is given on page~\pageref{th:theoexample2},
  and it is titled \flqq\nameref{th:theoexample2}\frqq.
\end{mytheo}
\end{dispExample}

\begin{dispExample}
\begin{mytheo}[label=myownlabel]{This is my title}{}
  The label parameter can be left empty without \LaTeX\ error.
  Or you may use an own label to reference Theorem \ref{myownlabel}.
\end{mytheo}
\end{dispExample}

\begin{dispExample}
\begin{mytheo}{}{}
  The title can also be left empty without problem. Note that the \enquote{:}
  vanished magically.
\end{mytheo}
\end{dispExample}

\begin{dispExample}
\begin{mytheo*}{Unnumbered Theorem}
  This theorem is not numbered.
\end{mytheo*}
\end{dispExample}

\begin{dispExample}
\begin{mytheo*}{}
  This theorem has no number and no title.
\end{mytheo*}
\end{dispExample}

\begin{marker}
To switch off the \texttt{nameref} feature permanently, add
\mbox{\texttt{nameref/.style=\{\}}} inside the \meta{options} list.
\end{marker}

\end{docCommands}

\medskip

\begin{docCommands}[
    doc parameter = \oarg{init options}\marg{name}\marg{display name}\marg{options}\marg{prefix},
    doc updated=2022-07-18
    ]
    {
      { doc name = RenewTcbTheorem },
      { doc name = renewtcbtheorem }
    }
  Operates like \refCom{NewTcbTheorem}, but based on |\RenewDocumentEnvironment| instead of |\NewDocumentEnvironment|.
  Existing environments are redefined.
\end{docCommands}


\begin{docCommands}[
    doc parameter = \oarg{init options}\marg{name}\marg{display name}\marg{options}\marg{prefix},
    doc new       = 2022-07-18,
    doc name      = ProvideTcbTheorem,
    ]{}
  Operates like \refCom{NewTcbTheorem}, but based on |\ProvideDocumentEnvironment| instead of |\NewDocumentEnvironment|.
  New environments are is only created if they are not already defined.
\end{docCommands}


\begin{docCommands}[
    doc parameter = \oarg{init options}\marg{name}\marg{display name}\marg{options}\marg{prefix},
    doc new       = 2022-07-18,
    doc name      = DeclareTcbTheorem,
    ]{}
  Operates like \refCom{NewTcbTheorem}, but based on |\DeclareDocumentEnvironment| instead of |\NewDocumentEnvironment|.
  New environments are always created and possibly redefine existing ones.
\end{docCommands}


\clearpage
\begin{docCommand}{tcboxmath}{\oarg{options}\marg{mathematical box content}}
  Creates a \refEnv{tcolorbox} which is fitted to the width of the given
  \meta{mathematical box content}. This box is intended to be applied as
  part of a larger formula and may be used as replacement for the |\boxed|
  macro of |amsmath|.

\begin{dispExample}
\begin{equation}
\tcbset{fonttitle=\scriptsize}
\tcboxmath[colback=LightBlue!25!white,colframe=blue]{ a^2 = 16 }
\quad \Rightarrow \quad
\tcboxmath[colback=Salmon!25!white,colframe=red,title=Implication]%
  { a = 4 ~\vee~ a=-4. }
\end{equation}
\end{dispExample}
\end{docCommand}

\begin{docCommand}{tcbhighmath}{\oarg{options}\marg{mathematical box content}}
  This is a special case of the \refCom{tcboxmath} macro which uses
  the style \refKey{/tcb/highlight math}.
  It is intended to provide context sensitive highlighting of formula parts.
  The color settings via \refKey{/tcb/highlight math style} may be different
  inside theorems or other colored areas and outside.

\begin{dispExample}
\tcbset{myformula/.style={colback=yellow!10!white,colframe=red!50!black,
  every box/.style={highlight math style={colback=LightBlue!50!white,colframe=Navy}}
  }}

\begin{align}
  \tcbhighmath{\sum\limits_{n=1}^{\infty} \frac{1}{n}} &= \infty.\\
  \int x^2 ~\text{d}x                     &= \frac13 x^3 + c.
\end{align}

\begin{tcolorbox}[ams align,myformula]
  \tcbhighmath{\sum\limits_{n=1}^{\infty} \frac{1}{n}} &= \infty.\\
  \int x^2 ~\text{d}x                     &= \frac13 x^3 + c.
\end{tcolorbox}
\end{dispExample}

\clearpage
\refCom{tcbhighmath} can be used in symbiosis with the \refPkg{empheq} package
which allows to specify own boxing commands to mark multiline formulas.

\begin{dispExample}
% \usepackage{empheq}
\begin{empheq}[box=\tcbhighmath]{align}
a&=\sin(z)\\
E&=mc^2 + \int_a^b x\, dx
\end{empheq}

\tcbset{highlight math style={enhanced,
  colframe=red!60!black,colback=yellow!50!white,arc=4pt,boxrule=1pt,
  drop fuzzy shadow}}

\begin{empheq}[box=\tcbhighmath]{align}
a&=\sin(z)\\
E&=mc^2 + \int_a^b x\, dx
\end{empheq}
\end{dispExample}

Besides \refCom{tcbhighmath}, one can easily define an independent new box
based on \refCom{tcbox} which acts like \refCom{tcbhighmath}:

\begin{dispExample}
% \usepackage{empheq}
\newtcbox{\otherbox}[1][]{nobeforeafter,math upper,tcbox raise base,
  enhanced,frame hidden,boxrule=0pt,interior style={top color=green!10!white,
  bottom color=green!10!white,middle color=green!50!yellow},
  fuzzy halo=1pt with green,#1}

\begin{empheq}[box=\otherbox]{align}
a&=\sin(z)\\
E&=mc^2 + \int_a^b x\, dx
\end{empheq}

\begin{equation}
\tcbhighmath{E} = \otherbox{mc^2}
\end{equation}
\end{dispExample}
\end{docCommand}


\clearpage
\subsection{Option Keys of the Library}


\begin{docTcbKey}{separator sign}{=\meta{sign}}{no default, initially |:|}
  The given \meta{sign} is used inside the title text of a theorem
  as separator between display name combined with number and
  the specific title text. It is omitted, if there is no specific title text.

\begin{dispExample}
% \usepackage{amssymb}
\newtcbtheorem[use counter from=mytheo]{sometheorem}{Theorem}%
  {colback=white,colframe=red!50!black,fonttitle=\bfseries,
   separator sign={\ $\blacktriangleright$}}{theo}
\begin{sometheorem}{My example}{}
My theorem text.
\end{sometheorem}
\end{dispExample}
\end{docTcbKey}

\begin{docTcbKey}{separator sign colon}{}{style, no value, initially set}
Sets \refKey{/tcb/separator sign} to the default colon |:| sign.
\end{docTcbKey}

\begin{docTcbKey}{separator sign dash}{}{style, no value}
Sets \refKey{/tcb/separator sign} to an en-dash sign.
\begin{dispExample}
\newtcbtheorem[use counter from=mytheo]{sometheorem}{Theorem}%
  {colback=white,colframe=red!50!black,fonttitle=\bfseries,
   separator sign dash}{theo}
\begin{sometheorem}{My example}{}
My theorem text.
\end{sometheorem}
\end{dispExample}
\end{docTcbKey}

\begin{docTcbKey}{separator sign none}{}{style, no value}
Sets \refKey{/tcb/separator sign} to empty.
\begin{dispExample}
\newtcbtheorem[use counter from=mytheo]{sometheorem}{Theorem}%
  {colback=white,colframe=red!50!black,fonttitle=\bfseries,
   separator sign none}{theo}
\begin{sometheorem}{My example}{}
My theorem text.
\end{sometheorem}
\end{dispExample}
\end{docTcbKey}

\clearpage
\begin{docTcbKey}{description delimiters}{=\marg{left}\marg{right}}{no default, initially empty}
  The given \meta{left} and \meta{right} delimiter signs are used to frame
  the descriptive title text of a theorem.

\begin{dispExample}
\newtcbtheorem[use counter from=mytheo]{sometheorem}{Theorem}%
  {colback=white,colframe=red!50!black,fonttitle=\bfseries,
   description delimiters={\flqq}{\frqq}}{theo}
\begin{sometheorem}{My example}{}
My theorem text.
\end{sometheorem}
\end{dispExample}
\end{docTcbKey}


\begin{docTcbKey}{description delimiters parenthesis}{}{style, no value}
Sets \refKey{/tcb/description delimiters} to |(| and |)|.
\begin{dispExample}
\newtcbtheorem[use counter from=mytheo]{sometheorem}{Theorem}%
  {colback=white,colframe=red!50!black,fonttitle=\bfseries,
   description delimiters parenthesis}{theo}
\begin{sometheorem}{My example}{}
My theorem text.
\end{sometheorem}
\end{dispExample}
\end{docTcbKey}


\begin{docTcbKey}{description delimiters none}{}{style, no value, initially set}
Sets \refKey{/tcb/description delimiters} to the default empty texts.
\end{docTcbKey}


\begin{docTcbKey}{description color}{\colOpt{=\meta{color}}}{default empty, initially empty}
  Sets the \meta{color} of the descriptive title text deviating from \refKey{/tcb/coltitle}.
  The color is reset to \refKey{/tcb/coltitle}, if |description color| is used without value.

\begin{dispExample}
\newtcbtheorem[use counter from=mytheo]{sometheorem}{Theorem}%
  {colback=white,colframe=red!50!black,fonttitle=\bfseries,
   description color=red!25!yellow}{theo}
\begin{sometheorem}{My example}{}
My theorem text.
\end{sometheorem}
\end{dispExample}
\end{docTcbKey}

\clearpage
\begin{docTcbKey}{description font}{\colOpt{=\meta{text}}}{default empty, initially empty}
  Sets \meta{text} (e.\,g.\ font settings) before the descriptive title text deviating from \refKey{/tcb/fonttitle}.
  The \meta{text} is removed, if |description font| is used without value.

\begin{dispExample}
\newtcbtheorem[use counter from=mytheo]{sometheorem}{Theorem}%
  {colback=white,colframe=red!50!black,fonttitle=\bfseries,
   description delimiters={\glqq}{\grqq},
   description font=\mdseries\itshape}{theo}
\begin{sometheorem}{My example}{}
My theorem text.
\end{sometheorem}
\end{dispExample}
\end{docTcbKey}


\begin{docTcbKey}{description formatter}{\colOpt{=\meta{macro}}}{default empty, initially empty}
  Sets \meta{macro} as formatter for the descriptive title text. The \meta{macro}
  has to take one mandatory argument (the description text).\\ 
  Note that \refKey{/tcb/description delimiters}, \refKey{/tcb/description color},
  and \refKey{/tcb/description font} are ignored, if this option is used.\\
  If |description formatter| is used without value, the formatter is reset
  to its standard behavior.

\begin{dispExample}
\newtcbox{\formbox}{enhanced,frame empty,size=minimal,boxsep=2pt,arc=1pt,
  on line,interior style image=goldshade.png}

\newtcbtheorem[use counter from=mytheo]{sometheorem}{Theorem}%
  {colback=white,colframe=red!50!black,fonttitle=\bfseries,
   description formatter=\formbox}{theo}
\begin{sometheorem}{My example}{}
My theorem text.
\end{sometheorem}
\end{dispExample}
\end{docTcbKey}


\begin{docTcbKey}{terminator sign}{=\meta{sign}}{no default, initially empty}
  The given \meta{sign} is used as terminator at the end of the title text of a theorem.

\begin{dispExample}
\newtcbtheorem[use counter from=mytheo]{sometheorem}{Theorem}%
  {colback=white,colframe=red!50!black,fonttitle=\bfseries,
   terminator sign={.}}{theo}
\begin{sometheorem}{My example}{}
My theorem text.
\end{sometheorem}
\end{dispExample}
\end{docTcbKey}

\clearpage
\begin{docTcbKey}{terminator sign colon}{}{style, no value, initially set}
Sets \refKey{/tcb/terminator sign} to the colon |:| sign.
\begin{dispExample}
\newtcbtheorem[use counter from=mytheo]{sometheorem}{Theorem}%
  {colback=white,colframe=red!50!black,fonttitle=\bfseries,
   separator sign dash,terminator sign colon}{theo}
\begin{sometheorem}{My example}{}
My theorem text.
\end{sometheorem}
\end{dispExample}
\end{docTcbKey}

\begin{docTcbKey}{terminator sign dash}{}{style, no value}
Sets \refKey{/tcb/terminator sign} to an en-dash sign.
\begin{dispExample}
\newtcbtheorem[use counter from=mytheo]{sometheorem}{Theorem}%
  {colback=white,colframe=red!50!black,fonttitle=\bfseries,
   terminator sign dash}{theo}
\begin{sometheorem}{My example}{}
My theorem text.
\end{sometheorem}
\end{dispExample}
\end{docTcbKey}

\begin{docTcbKey}{terminator sign none}{}{style, no value}
Sets \refKey{/tcb/terminator sign} to the default empty text.
\end{docTcbKey}


\begin{docTcbKey}[][doc new=2016-04-19]{label separator}{=\meta{separator}}{no default, initially |:|}
  The given \meta{separator} is used for labels created with environments which
  are defined themselves by \refCom{newtcbtheorem}. This \meta{separator} is
  put between \meta{prefix} (defined by \refCom{newtcbtheorem})
  and \meta{marker} (defined by an actual theorem environment).
\begin{dispExample}
\newtcbtheorem[use counter from=mytheo]{sometheorem}{Theorem}%
  {colback=white,colframe=red!50!black,fonttitle=\bfseries,
   label separator=*}{theo}
\begin{sometheorem}{My example}{myex}
My theorem text.
\end{sometheorem}
See Example~\ref{theo*myex}.
\end{dispExample}
\end{docTcbKey}


\clearpage

\begin{docTcbKey}[][doc new=2018-01-12]{theorem full label supplement}{=\marg{style}}{no default, initially empty}
  The given \meta{style} is used in connection with labels created with environments which
  are defined themselves by \refCom{newtcbtheorem}.
  This \meta{style} uses one argument which is automatically set to the
  full label marker of the environment, i.e. a text consisting of
  \meta{prefix} (defined by \refCom{newtcbtheorem}),
  \refKey{/tcb/label separator},
  and \meta{marker} (defined by an actual theorem environment).

\begin{dispExample}
% The following adds a hyper target to all environments
% created with \newtcbtheorem
\tcbset{theorem full label supplement={hypertarget={#1}}}

\newtcbtheorem[use counter from=mytheo]{sometheorem}{Theorem}%
   {colback=white,colframe=red!50!black,fonttitle=\bfseries}{theo}
\begin{sometheorem}{My example}{myex2}
My theorem text.
\end{sometheorem}
This automated \hyperlink{theo:myex2}{hyper target can be linked to with a
 hyper link}.
\end{dispExample}

A second usage of \refKey{/tcb/theorem full label supplement} overwrites
the first setting.
\end{docTcbKey}


\begin{docTcbKey}[][doc new=2018-01-12]{theorem label supplement}{=\marg{style}}{no default, initially empty}
  The given \meta{style} is used in connection with labels created with environments which
  are defined themselves by \refCom{newtcbtheorem}.
  This \meta{style} uses one argument which is automatically set to the
  label \meta{marker} defined by an actual theorem environment.\par
  A second usage of \refKey{/tcb/theorem label supplement} overwrites
  the first setting, but
  \refKey{/tcb/theorem full label supplement}
  and \refKey{/tcb/theorem label supplement} can be used independently.

\begin{dispExample}
% `marginnote' has to be loaded
\newtcbtheorem[use counter from=mytheo]{sometheorem}{Theorem}%
   {colback=white,colframe=red!50!black,fonttitle=\bfseries,
    theorem label supplement={hypertarget={XYZ-##1}},
    theorem full label supplement={code={\marginnote{##1}}}
   }{theo}
\begin{sometheorem}{My example}{myex3}
My theorem text.
\end{sometheorem}
This automated \hyperlink{XYZ-myex3}{hyper target can be linked to with a
 hyper link}.
\end{dispExample}
\end{docTcbKey}


\clearpage

\begin{docTcbKey}[][doc new=2020-10-21]{theorem hanging indent}{\colOpt{=\docValue*{auto}\textbar \meta{length}}}{default \docValue*{auto}, initially \docValue*{auto}}
  Sets the hanging indent of the theorem title to \docValue{auto} or the
  given \meta{length}.
  For \docValue{auto}, the hanging indent matches the display name, number and
  separator sign of the theorem.
  If \meta{length} is negative, the theorem title is indented positively
  without hanging indent.

\begin{dispExample}
\newtcbtheorem[use counter from=mytheo]{sometheorem}{Theorem}%
  {colback=white,colframe=red!50!black,fonttitle=\bfseries}{theo}

\begin{sometheorem}{This is a very long and complicated title for a quite
  short and nearly empty theorem}{myexA1}
My theorem text.
\end{sometheorem}

\begin{sometheorem}[theorem hanging indent=5mm]{This is a very long and
  complicated title for a quite short and nearly empty theorem}{myexA2}
My theorem text.
\end{sometheorem}

\begin{sometheorem}[theorem hanging indent=0pt]{This is a very long and
  complicated title for a quite short and nearly empty theorem}{myexA3}
My theorem text.
\end{sometheorem}

\begin{sometheorem}[theorem hanging indent=-5mm]{This is a very long and
  complicated title for a quite short and nearly empty theorem}{myexA4}
My theorem text.
\end{sometheorem}
\end{dispExample}
\end{docTcbKey}



\clearpage
\begin{docTcbKey}{theorem name and number}{}{style, no value, initially set}
Prints theorem name followed by theorem number inside the title.
\begin{dispExample}
\newtcbtheorem[use counter from=mytheo]{sometheorem}{Theorem}%
  {colback=white,colframe=red!50!black,fonttitle=\bfseries,
   theorem name and number}{theo}
\begin{sometheorem}{My example}{}
My theorem text.
\end{sometheorem}
\end{dispExample}
\end{docTcbKey}


\begin{docTcbKey}{theorem number and name}{}{style, no value}
Prints theorem number followed by theorem name inside the title.
\begin{dispExample}
\newtcbtheorem[use counter from=mytheo]{sometheorem}{Theorem}%
  {colback=white,colframe=red!50!black,fonttitle=\bfseries,
   theorem number and name}{theo}
\begin{sometheorem}{My example}{}
My theorem text.
\end{sometheorem}
\end{dispExample}
\end{docTcbKey}

\begin{docTcbKey}{theorem name}{}{style, no value}
Prints theorem name without number inside the title.
\begin{dispExample}
\newtcbtheorem[use counter from=mytheo]{sometheorem}{Theorem}%
  {enhanced,colback=white,colframe=red!50!black,fonttitle=\bfseries,
   theorem name,watermark text={\thetcbcounter}}{theo}
\begin{sometheorem}{My example}{}
My theorem text.
\end{sometheorem}
\end{dispExample}
\end{docTcbKey}

\enlargethispage*{20mm}

\begin{docTcbKey}[][doc new=2021-12-03]{theorem number}{}{style, no value}
Prints theorem number without name inside the title.
\begin{dispExample}
\newtcbtheorem[use counter from=mytheo]{sometheorem}{Theorem}%
  {enhanced,colback=white,colframe=red!50!black,fonttitle=\bfseries,
   theorem number}{theo}
\begin{sometheorem}{My example}{}
My theorem text.
\end{sometheorem}
\end{dispExample}
\end{docTcbKey}


\clearpage
\begin{docTcbKey}{theorem}{=\marg{display name}\marg{counter}\marg{title}\marg{marker}}{no default}
  This key can be used
  directly in a |tcolorbox| for a more flexible approach to create a
  theorem type box.
  The \meta{display name} is used together with the increased \meta{counter} value
  and the \meta{title} for the title line of the box. Additionally, a
  |\label| with the given \meta{marker} is created.
\begin{dispExample}
% \newcounter{texercise}%  preamble
\begin{tcolorbox}[colback=green!10,colframe=green!50!black,arc=4mm,
                  theorem={Test}{texercise}{Direct usage}{myMarker}]
Here, we see the test \ref{myMarker}.
\end{tcolorbox}
\end{dispExample}
For a common appearance inside the document, the key |theorem| should not be
used directly as in the example above, but as part of a new environment
created by hand or using \refCom{newtcbtheorem}.
\end{docTcbKey}

\clearpage

\begin{docTcbKey}{highlight math}{}{style, no value}
  A style which is used for \refCom{tcbhighmath} and which is
  predefined as |notitle,nophantom,colframe=red,colback=yellow!25!white|.\par
  It can be changed with the usual |pgf| techniques or
  with \refKey{/tcb/highlight math style}.
\begin{dispExample}
\begin{align*}
  \tcbhighmath{1} + 1 &= 2,\\
  \tcbset{highlight math/.append style={left=0mm,right=0mm,top=0mm,bottom=0mm}}
  \tcbhighmath{1} + 1 &= 2.
\end{align*}
\end{dispExample}

\end{docTcbKey}


\begin{docTcbKey}{highlight math style}{=\meta{style definition}}{style, no default}
  Changes the definition for \refKey{/tcb/highlight math} to
  |notitle,nophantom| plus the given \meta{style definition}.
  See \refCom{tcbhighmath} for another example.
\begin{dispExample}
% \tcbuselibrary{skins}
\tcbset{highlight math style={enhanced,%<-- needed for the `remember' options
  colframe=red,colback=red!10!white,boxsep=0pt}}
\begin{align*}
\tcbhighmath[remember as=fx]{f(x)}
     &= \int\limits_{1}^{x} \frac{1}{t^2}~dt
      = \left[ -\frac{1}{t} \right]_{1}^{x}\\
     &= -\frac{1}{x} + \frac{1}{1}\\
     &=
\tcbhighmath[remember,overlay={%
    \draw[blue,very thick,->] (fx.south) to[bend right] ([yshift=2mm]frame.west);}]
  {1-\frac{1}{x}.}
\end{align*}
\end{dispExample}
\end{docTcbKey}

\clearpage
\begin{docTcbKey}{math upper}{}{style, no value}
  Sets the upper part to mathematical mode with font |\displaystyle|.
\end{docTcbKey}

\begin{docTcbKey}{math lower}{}{style, no value}
  Sets the lower part to mathematical mode with font |\displaystyle|.
\end{docTcbKey}

\begin{docTcbKey}{math}{}{style, no value}
  Sets the upper part \emph{and} lower part to mathematical mode with font |\displaystyle|.
\begin{dispExample}
\begin{tcolorbox}[math,colback=yellow!10!white,colframe=red!50!black]
  \sum\limits_{n=1}^{\infty} \frac{1}{n} = \infty.
\end{tcolorbox}
\end{dispExample}
\end{docTcbKey}


\begin{marker}
  The following styles are only tested to work with the original |amsmath| environments.
  If e.g. the |equation| environment is redefined as |gather|, then
  \refKey{/tcb/ams equation} should / could not be used. Obviously, you are encouraged
  to use \refKey{/tcb/ams gather} in this case.
\end{marker}

\begin{docTcbKey}[][doc updated=2014-10-30]{ams equation upper}{}{style, no value}
  Adds an |amsmath| |equation| environment to the start and end
  of the upper part.
\end{docTcbKey}

\begin{docTcbKey}[][doc updated=2014-10-30]{ams equation lower}{}{style, no value}
  Adds an |amsmath| |equation| environment to the start and end
  of the lower part.
\end{docTcbKey}

\begin{docTcbKey}[][doc updated=2014-10-30]{ams equation}{}{style, no value}
  Adds an |amsmath| |equation| environment to the start and end
  of the upper \emph{and} lower part.
\begin{dispExample}
\begin{tcolorbox}[ams equation,colback=yellow!10!white,colframe=red!50!black]
  \sum\limits_{n=1}^{\infty} \frac{1}{n} = \infty.
\end{tcolorbox}
\end{dispExample}
\end{docTcbKey}

\begin{docTcbKey}[][doc updated=2014-10-30]{ams equation* upper}{}{style, no value}
  Adds an |amsmath| |equation*| environment to the start and end
  of the upper part.
\end{docTcbKey}

\begin{docTcbKey}[][doc updated=2014-10-30]{ams equation* lower}{}{style, no value}
  Adds an |amsmath| |equation*| environment to the start and end
  of the lower part.
\end{docTcbKey}

\enlargethispage*{2cm}
\begin{docTcbKey}[][doc updated=2014-10-30]{ams equation*}{}{style, no value}
  Adds an |amsmath| |equation*| environment to the start and end
  of the upper \emph{and} lower part.
\begin{dispExample}
\begin{tcolorbox}[ams equation*,colback=yellow!10!white,colframe=red!50!black]
  \sum\limits_{n=1}^{\infty} \frac{1}{n} = \infty.
\end{tcolorbox}
\end{dispExample}
\end{docTcbKey}

\clearpage
\begin{docTcbKeys}[
    doc parameter   = ,
    doc description = {style, no value},
  ]
  {
    {
      doc name = ams align upper,
    },
    {
      doc name    = ams align,
      doc updated = 2024-06-27,
    }
  }
  Adds an |amsmath| |align| environment to the start and end
  of the upper part.
\begin{dispExample}
\begin{tcolorbox}[ams align,colback=yellow!10!white,colframe=red!50!black]
  \sum\limits_{n=1}^{\infty} \frac{1}{n} &= \infty.\\
  \int x^2 ~\text{d}x                     &= \frac13 x^3 + c.
\end{tcolorbox}
\end{dispExample}
\begin{marker}
  Boxes with \refKey{/tcb/ams align upper} or \refKey{/tcb/ams align}
  cannot have a lower part!
  For a lower part, use \refKey{/tcb/ams nodisplayskip upper} instead.
\end{marker}
\end{docTcbKeys}

\begin{docTcbKey}{ams align lower}{}{style, no value}
  Adds an |amsmath| |align| environment to the start and end
  of the lower part.
\end{docTcbKey}


\begin{docTcbKeys}[
    doc parameter   = ,
    doc description = {style, no value},
  ]
  {
    {
      doc name = ams align* upper,
    },
    {
      doc name    = ams align*,
      doc updated = 2024-06-27,
    }
  }
  Adds an |amsmath| |align*| environment to the start and end
  of the upper part.
\begin{dispExample}
\begin{tcolorbox}[ams align*,colback=yellow!10!white,colframe=red!50!black]
  \sum\limits_{n=1}^{\infty} \frac{1}{n} &= \infty.\\
  \int x^2 ~\text{d}x                     &= \frac13 x^3 + c.
\end{tcolorbox}
\end{dispExample}
\begin{marker}
  Boxes with \refKey{/tcb/ams align* upper} or \refKey{/tcb/ams align*}
  cannot have a lower part!
  For a lower part, use \refKey{/tcb/ams nodisplayskip upper} instead.
\end{marker}
\end{docTcbKeys}

\begin{docTcbKey}{ams align* lower}{}{style, no value}
  Adds an |amsmath| |align*| environment to the start and end
  of the lower part.
\end{docTcbKey}


\clearpage
\begin{docTcbKeys}[
    doc parameter   = ,
    doc description = {style, no value},
  ]
  {
    {
      doc name = ams gather upper,
    },
    {
      doc name    = ams gather,
      doc updated = 2024-06-27,
    }
  }
  Adds an |amsmath| |gather| environment to the start and end
  of the upper part.
\begin{dispExample}
\begin{tcolorbox}[ams gather,colback=yellow!10!white,colframe=red!50!black]
  \sum\limits_{n=1}^{\infty} \frac{1}{n} = \infty.\\
  \int x^2 ~\text{d}x = \frac13 x^3 + c.
\end{tcolorbox}
\end{dispExample}
\begin{marker}
  Boxes with \refKey{/tcb/ams gather upper} or \refKey{/tcb/ams gather}
  cannot have a lower part!
  For a lower part, use \refKey{/tcb/ams nodisplayskip upper} instead.
\end{marker}
\end{docTcbKeys}

\begin{docTcbKey}{ams gather lower}{}{style, no value}
  Adds an |amsmath| |gather| environment to the start and end
  of the lower part.
\end{docTcbKey}

\begin{docTcbKeys}[
    doc parameter   = ,
    doc description = {style, no value},
  ]
  {
    {
      doc name = ams gather* upper,
    },
    {
      doc name    = ams gather*,
      doc updated = 2024-06-27,
    }
  }
  Adds an |amsmath| |gather*| environment to the start and end
  of the upper part.
\begin{dispExample}
\begin{tcolorbox}[ams gather*,colback=yellow!10!white,colframe=red!50!black]
  \sum\limits_{n=1}^{\infty} \frac{1}{n} = \infty.\\
  \int x^2 ~\text{d}x = \frac13 x^3 + c.
\end{tcolorbox}
\end{dispExample}
\begin{marker}
  Boxes with \refKey{/tcb/ams gather* upper} or \refKey{/tcb/ams gather*}
  cannot have a lower part!
  For a lower part, use \refKey{/tcb/ams nodisplayskip upper} instead.
\end{marker}
\end{docTcbKeys}

\begin{docTcbKey}{ams gather* lower}{}{style, no value}
  Adds an |amsmath| |gather*| environment to the start and end
  of the lower part.
\end{docTcbKey}

\clearpage
\begin{docTcbKey}{ams nodisplayskip upper}{}{style, no value}
  Neutralizes the |\abovedisplayskip| of a following |align| or |gather|
  environment for the upper part. Note that the text content has to
  start with such a formula.
\end{docTcbKey}


\begin{docTcbKey}{ams nodisplayskip lower}{}{style, no value}
  Neutralizes the |\abovedisplayskip| of a following |align| or |gather|
  environment for the lower part. Note that the text content has to
  start with such a formula.
\end{docTcbKey}


\begin{docTcbKey}{ams nodisplayskip}{}{style, no value}
  Neutralizes the |\abovedisplayskip| of a following |align| or |gather|
  environment for the upper part \emph{and} lower part.
  Note that the text content has to start with such a formula.
\begin{dispExample}
\begin{tcolorbox}[ams nodisplayskip,colback=yellow!10!white,colframe=red!50!black]
  \begin{gather}
  \sum\limits_{n=1}^{\infty} \frac{1}{n} = \infty.\\
  \int x^2 ~\text{d}x = \frac13 x^3 + c.
  \end{gather}
  And now for something completely different.
\end{tcolorbox}
\end{dispExample}
\end{docTcbKey}

\bigskip
New colored mathematical environments are easily created using
\refCom{newtcolorbox}:

\begin{dispExample}
\newtcolorbox{mymath}{ams gather*,colback=yellow!10!white,colframe=red!50!black}

\begin{mymath}
  \sum\limits_{n=1}^{\infty} \frac{1}{n} = \infty.\\
  \int x^2 ~\text{d}x = \frac13 x^3 + c.
\end{mymath}
\end{dispExample}

\bigskip
\begin{marker}
  All described options like \refKey{/tcb/ams gather upper}, \refKey{/tcb/ams gather lower},
  \refKey{/tcb/ams gather} are (partially) setting (overwriting) the
  keys \refKey{/tcb/before upper}, \refKey{/tcb/after upper},
  \refKey{/tcb/before lower}, \refKey{/tcb/after lower}.\par
  Therefore, e.\,g.\ |\tcbset{ams gather,before upper={\text{Pythagoras:}}}|
  produces an invalid result. For this case, you are invited to use\\
  |\tcbset{ams gather,before upper app={\text{Pythagoras:}}}|,\\
  see \refKey{/tcb/before upper app}.
\end{marker}


\clearpage
\begin{docTcbKey}{theorem style}{=\meta{name}}{no default, initially |standard|}
Applies a predefined style \meta{name} to the theorem environment. Some of
the feasible \meta{name} values resemble style names from the packages \refPkg{theorem}
and \refPkg{ntheorem} to give convenient access to known patterns.
\begin{marker}
The styles alter \refKey{/tcb/separator sign}, \refKey{/tcb/description delimiters},
\refKey{/tcb/terminator sign}, and more. Therefore, one should apply such
keys \emph{after} a theorem style.
\end{marker}

For the following examples, we use:
\inputpreamblelisting{J}

The predefined styles are:
\begin{itemize}
%
\item\docValue{standard}: This is the initial value.
\begin{dispExample}
\begin{theorem}[theorem style=standard]{standard}{}
This is my theorem. \begin{equation*} a^2 + b^2 = c^2. \end{equation*}
\end{theorem}
\end{dispExample}
%
\item\docValue{change standard}
\begin{dispExample}
\begin{theorem}[theorem style=change standard]{change standard}{}
This is my theorem. \begin{equation*} a^2 + b^2 = c^2. \end{equation*}
\end{theorem}
\end{dispExample}
%
\item\docValue{plain}
\begin{dispExample}
\begin{theorem}[theorem style=plain]{plain}{}
This is my theorem. \begin{equation*} a^2 + b^2 = c^2. \end{equation*}
\end{theorem}
\end{dispExample}
%
\clearpage
\item\docValue{break}
\begin{dispExample}
\begin{theorem}[theorem style=break]{break}{}
This is my theorem. \begin{equation*} a^2 + b^2 = c^2. \end{equation*}
\end{theorem}
\end{dispExample}
%
\item\docValue{plain apart}
\begin{dispExample}
\begin{theorem}[theorem style=plain apart]{plain apart}{}
This is my theorem. \begin{equation*} a^2 + b^2 = c^2. \end{equation*}
\end{theorem}
\end{dispExample}
%
\item\docValue{change}
\begin{dispExample}
\begin{theorem}[theorem style=change]{change}{}
This is my theorem. \begin{equation*} a^2 + b^2 = c^2. \end{equation*}
\end{theorem}
\end{dispExample}
%
\item\docValue{change break}
\begin{dispExample}
\begin{theorem}[theorem style=change break]{change break}{}
This is my theorem. \begin{equation*} a^2 + b^2 = c^2. \end{equation*}
\end{theorem}
\end{dispExample}
%
\item\docValue{change apart}
\begin{dispExample}
\begin{theorem}[theorem style=change apart]{change apart}{}
This is my theorem. \begin{equation*} a^2 + b^2 = c^2. \end{equation*}
\end{theorem}
\end{dispExample}
%
\clearpage
\item\docValue{margin}
\begin{dispExample}
\begin{theorem}[theorem style=margin,left=10mm]{margin}{}
This is my theorem. \begin{equation*} a^2 + b^2 = c^2. \end{equation*}
\end{theorem}
\begin{theorem}[theorem style=margin,left=10mm,oversize]{margin}{}
This is my theorem. \begin{equation*} a^2 + b^2 = c^2. \end{equation*}
\end{theorem}
\end{dispExample}
%
\item\docValue{margin break}
\begin{dispExample}
\begin{theorem}[theorem style=margin break,left=10mm]{margin break}{}
This is my theorem. \begin{equation*} a^2 + b^2 = c^2. \end{equation*}
\end{theorem}
\begin{theorem}[theorem style=margin break,left=10mm,oversize]{margin break}{}
This is my theorem. \begin{equation*} a^2 + b^2 = c^2. \end{equation*}
\end{theorem}
\end{dispExample}
%
\item\docValue{margin apart}
\begin{dispExample}
\begin{theorem}[theorem style=margin apart,left=10mm]{margin apart}{}
This is my theorem. \begin{equation*} a^2 + b^2 = c^2. \end{equation*}
\end{theorem}
\begin{theorem}[theorem style=margin apart,left=10mm,oversize]{margin apart}{}
This is my theorem. \begin{equation*} a^2 + b^2 = c^2. \end{equation*}
\end{theorem}
\end{dispExample}
%
\end{itemize}
\end{docTcbKey}


%
\clearpage
\subsection{Examples for Definitions and Theorems}
In the following, the application of \refCom{newtcbtheorem}
to highlight mathematical definitions, theorems, or the like is demonstrated.

At first, additional |tcb| keys are created for the appearance of
the colored boxes. It is assumed that theorems and corollaries should be
identically colored.
All following environments are numbered with a common counter, but this
can be changed easily. Here, the counter output is supplemented by
the subsection number.
Further, the \refPkg{cleveref} package \cite{cubitt:2018a} is used for clever
references.

\inputpreamblelisting{G}

By \refCom{newtcbtheorem}, commonly numbered theorem environments are
created now. |defstyle| and |theostyle| are used for the appearance.

Now, everything is prepared for the following examples.

\begin{dispExample}
The following theorem is numbered as \Cref{theo:diffbarstetig} and
referenced with the marker \texttt{theo:diffbarstetig}.

\begin{Theorem}{Differenzierbarkeit bedingt Stetigkeit, wobei diese Benennung
  zu Testzwecken ungew\"{o}hnlich lang ist}{diffbarstetig}%
  Eine Funktion $f:I\to\mathbb{R}$ ist in $x_0\in I$ stetig, wenn $f$ in
  $x_0$ differenzierbar ist.
\end{Theorem}
\end{dispExample}


\begin{dispExample}
The following definition is numbered as \Cref{def:diffbarkeit} and
referenced with the marker \texttt{def:diffbarkeit}.\bigskip

\begin{Definition}{Differenzierbarkeit}{diffbarkeit}
  Eine Funktion $f:~I\to\mathbb{R}$ auf einem Intervall $I$ hei\ss{}t in
  $x_0\in I$ differenzierbar oder linear approximierbar,
  wenn der Grenzwert
  \begin{equation*}
  \lim\limits_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}=
  \lim\limits_{h\to 0}\frac{f(x_0+h)-f(x_0)}{h}
  \end{equation*}
  existiert. Bei Existenz hei\ss{}t dieser Grenzwert Ableitung
  oder Differentialquotient von $f$ in $x_0$ und man
  schreibt f\"{u}r ihn
  \begin{equation*}
  f'(x_0)\quad\text{oder}\quad\frac{df}{dx}(x_0).
  \end{equation*}
\end{Definition}
\end{dispExample}


\begin{dispExample}
The following corollary is numbered as \Cref{cor:nullstellen} and
referenced with the marker \texttt{cor:nullstellen}.\bigskip

\begin{Corollary}{Nullstellenexistenz}{nullstellen}
  Ist $f:[a,b]\to\mathbb{R}$ stetig und haben $f(a)$ und $f(b)$ entgegengesetzte
  Vorzeichen, also $f(a)f(b)<0$, so besitzt $f$ eine Nullstelle $x_0\in]a,b[$,
  also $f(x_0)=0$.
\end{Corollary}
\end{dispExample}


\begin{dispExample}
\begin{Theorem}[boxrule=2mm,toptitle=-1.5mm,bottomtitle=-1.5mm]{%
    Hinreichende Bedingung f\"{u}r Wendepunkte}{wendehinreichend}%
  $f$ sei eine auf einem Intervall $]a,b[$ dreimal stetig differenzierbare Funktion.
  Ist $f''(x_0)=0$ in $x_0\in]a,b[$ und $f'''(x_0)\ne 0$, so ist
  $(x_0,f(x_0))$ ein Wendepunkt von $f$.
\end{Theorem}
\end{dispExample}

\begin{dispExample}
% \usepackage{varioref}
% \usepackage{cleveref}
% \tcbuselibrary{skins}
\newtcbtheorem[use counter from=Definition]{YetAnotherTheorem}{Theorem}%
  {theorem style=plain apart,label type=theorem,enhanced,frame hidden,
  boxrule=2mm,titlerule=0mm,toptitle=1mm,bottomtitle=1mm,
  fonttitle=\bfseries\large,fontupper=\normalsize,
  coltitle=green!35!black,colbacktitle=green!15!white,
  colback=green!50!yellow!15!white,borderline={1pt}{0pt}{green!25!blue},
  }{theo}

\begin{YetAnotherTheorem}{Mittelwertsatz f\"{u}r $n$ Variable}{meanvaluetheorem}%
  Es sei $n\in\mathbb{N}$, $D\subseteq\mathbb{R}^n$ eine offene Menge und
  $f\in C^{1}(D,\mathbb{R})$. Dann gibt es auf jeder Strecke
  $[x_0,x]\subset D$ einen Punkt $\xi\in[x_0,x]$, so dass gilt
  \begin{equation*}
  f(x)-f(x_0) = \operatorname{grad} f(\xi)^{\top}(x-x_0)
  \end{equation*}
\end{YetAnotherTheorem}

\medskip
Here, |cleveref| support is used to reference \Cref{theo:meanvaluetheorem}
on \Cpageref{theo:meanvaluetheorem}. This \namecref{theo:meanvaluetheorem}
can also be referenced by |\Vref| resulting in \Vref{theo:meanvaluetheorem}.
\end{dispExample}

\begin{marker}
Note that \refKey{/tcb/label type} was used in the example above to feed
|cleveref| \cite{cubitt:2018a} with the needed name information.
\end{marker}

\clearpage

\begin{dispExample}
Here, using |\Vref| resulting in \Vref{theo:meanvaluetheorem} is more interesting\ldots
\end{dispExample}


\begin{dispExample}
% \tcbuselibrary{skins}
\newtcbtheorem[use counter from=Definition]{YetAnotherTheorem}{Theorem}%
  {theorem style=change apart,enhanced,arc=0mm,outer arc=0mm,
  boxrule=0mm,toprule=1mm,bottomrule=1mm,left=1mm,right=1mm,
  titlerule=0mm,toptitle=0mm,bottomtitle=1mm,top=0mm,
  colframe=red!50!black,colback=red!5!white,coltitle=red!50!black,
  title style={top color=yellow!50!white,bottom color=red!5!white,
    middle color=yellow!50!white},
  fonttitle=\bfseries\sffamily\normalsize,fontupper=\normalsize\itshape,
  }{theo}

\begin{YetAnotherTheorem}{Mittelwertsatz f\"{u}r $n$ Variable}{mittelwertsatz_n2}%
  Es sei $n\in\mathbb{N}$, $D\subseteq\mathbb{R}^n$ eine offene Menge und
  $f\in C^{1}(D,\mathbb{R})$. Dann gibt es auf jeder Strecke
  $[x_0,x]\subset D$ einen Punkt $\xi\in[x_0,x]$, so dass gilt
  \begin{equation*}
  f(x)-f(x_0) = \operatorname{grad} f(\xi)^{\top}(x-x_0)
  \end{equation*}
\end{YetAnotherTheorem}
\end{dispExample}


\begin{dispExample}
% \usepackage{varwidth}   \tcbuselibrary{skins}
\newtcbtheorem[use counter from=Definition]{YetAnotherTheorem}{Theorem}%
{enhanced,frame empty,interior empty,colframe=ForestGreen!50!white,
  coltitle=ForestGreen!50!black,fonttitle=\bfseries,colbacktitle=ForestGreen!15!white,
  borderline={0.5mm}{0mm}{ForestGreen!15!white},
  borderline={0.5mm}{0mm}{ForestGreen!50!white,dashed},
  attach boxed title to top center={yshift=-2mm},
  boxed title style={boxrule=0.4pt},varwidth boxed title}{theo}

\begin{YetAnotherTheorem}{Mittelwertsatz f\"{u}r $n$ Variable}{mittelwertsatz_n3}%
  Es sei $n\in\mathbb{N}$, $D\subseteq\mathbb{R}^n$ eine offene Menge und
  $f\in C^{1}(D,\mathbb{R})$. Dann gibt es auf jeder Strecke
  $[x_0,x]\subset D$ einen Punkt $\xi\in[x_0,x]$, so dass gilt
  \begin{equation*}
  f(x)-f(x_0) = \operatorname{grad} f(\xi)^{\top}(x-x_0)
  \end{equation*}
\end{YetAnotherTheorem}
\end{dispExample}


\clearpage
You need more attention for your theorems? Here, you are \ldots

\begin{dispExample}
% tcbuselibrary{skins}  % preamble
\begin{Theorem}[enhanced,
    fuzzy halo=3mm with yellow,
    fuzzy halo=2mm with red,
    fuzzy halo=1mm with yellow,
    watermark color=red!35!white,
    watermark text={Overacting\\Fundamental Theorem}]%
  {Fundamental Theorem of Theorems}{fundamental}%
  \lipsum[1-2]
\end{Theorem}
\end{dispExample}

Let's try a more conservative approach:

\begin{dispExample}
% \tcbuselibrary{skins}
\newtcbtheorem[use counter from=Definition]{YetAnotherTheorem}{Theorem}%
  {theorem style=plain,enhanced,colframe=blue!50!black,colback=yellow!20!white,
   coltitle=red!50!black,fonttitle=\upshape\bfseries,fontupper=\itshape,
   drop fuzzy shadow=blue!50!black!50!white,boxrule=0.4pt}{theo}

\begin{YetAnotherTheorem}{Mittelwertsatz f\"{u}r $n$ Variable}{mittelwertsatz_n4}%
  Es sei $n\in\mathbb{N}$, $D\subseteq\mathbb{R}^n$ eine offene Menge und
  $f\in C^{1}(D,\mathbb{R})$. Dann gibt es auf jeder Strecke
  $[x_0,x]\subset D$ einen Punkt $\xi\in[x_0,x]$, so dass gilt
  \begin{equation*}
  f(x)-f(x_0) = \operatorname{grad} f(\xi)^{\top}(x-x_0)
  \end{equation*}
\end{YetAnotherTheorem}
\end{dispExample}


\clearpage
\subsection{Using other theorem environments with \texttt{tcolorbox}}\label{subsec:theorems_other}

Instead of creating theorem environments with the methods described before,
environments from other packages can be boxed with a |tcolorbox|.

Environments may be created e.g. by methods from the \refPkg{theorem} package or
the \refPkg{amsthm} package. \refCom{tcolorboxenvironment} can be used to put a box
around these environments.

\inputpreamblelisting{K}

\begin{dispExample}
\begin{lem}
  \lipsum[2]
\end{lem}

\lipsum[3]

\begin{proof}
  \lipsum*[4]
\end{proof}
\end{dispExample}