Different counter for equations inside environment

Question

Let's say I have a tcolorbox environment called note. Is there a way for equations that appear inside this environment to automatically assume a different counter from those outside? With automatically I mean without defining a new equation environment?

Here's an example (no second counter implemented):

\documentclass{article}

\usepackage[many]{tcolorbox}
\newtcolorbox{note}[1][]{%
    breakable,
    enhanced jigsaw,
    borderline west={1pt}{0pt}{black},
    sharp corners,
    boxrule=0pt,
    frame hidden,
    left=1ex,right=1ex,
    fonttitle={\bfseries},
    coltitle={black},
    title={Note:\ },
    attach title to upper,
    #1}

\begin{document}

Outside text followed by equation.
\begin{equation}
    a = b.
\end{equation}

\begin{note}
    Inside text followed by equation.
    \begin{equation}
        b = c.
    \end{equation}
\end{note}

More outside text followed by conclusion.
\begin{equation}
    a = c.
\end{equation}

\end{document}

Answer 1

Here's a variant based on the preceding code, which should work in any environment and is compatible with cleveref:

\documentclass{article}
\usepackage{ebgaramond, ebgaramond-maths}
\usepackage{chngcntr, mathtools,}
\usepackage{framed,  etoolbox, cleveref}

\makeatletter
\newcounter{framedeqn}
\AtBeginEnvironment{framed}{\let\c@equation\c@framedeqn\def\theequation{N\theframedeqn}}
\makeatother

\begin{document}

Some outside text followed by conclusion in \cref{noteeq:a}:  %
\begin{equation}
    a = c.
\end{equation}

Outside text followed by equation:
\begin{equation}
    a = b. \label{eq-out}
\end{equation}
%
\begin{framed}
\noindent\textsc{Note 1: }\\
 An equation inside some environment,  different from \cref{eq-out,noteeq-b}:
 \belowdisplayskip = 0pt
\begin{align}
  x = y \label{noteeq:a}
\end{align}
\end{framed}
Outside text followed by equation:
\begin{equation}
    c= d.
\end{equation}

\begin{framed}
\noindent\textsc{Note 2: }\\
 Another equation inside the same environment:
 \belowdisplayskip = 0pt
\begin{align}
  z = t \label{noteeq-b}
\end{align}
\end{framed}

\end{document}

Answer 2

There's no need for a whole new equation environment, only a fake counter has to be established which is used inside by saying

\let\c@equation\c@fakeequation

Remember, that \c@foo is the internal representation macro of the counter foo, so this means that unless specified otherwise all operations to the equation counter are actually done with fakeequation, so this trick should be used in a group rather, which is done by the code=.... option in the definition of \newtcolorbox.

cleveref works out of the box too:

\documentclass{article}

\usepackage[many]{tcolorbox}
\newcounter{fakeequation}

\usepackage{cleveref}
\makeatletter

\let\latex@theequation\theequation
\newtcolorbox{note}[1][]{%
  code={\let\c@equation\c@fakeequation\renewcommand{\theequation}{N\latex@theequation}},
  breakable,
  enhanced jigsaw,
  borderline west={1pt}{0pt}{black},
  sharp corners,
  boxrule=0pt,
  frame hidden,
  left=1ex,right=1ex,
  fonttitle={\bfseries},
  coltitle={black},
  title={Note:\ },
  attach title to upper,
  #1}
\makeatother

\begin{document}

\setcounter{fakeequation}{100}

Outside text followed by equation.
\begin{equation}
    a = b.
\end{equation}

\begin{note}
    Inside text followed by equation.
    \begin{equation}
        b = c. \label{innerref}
    \end{equation}
\end{note}

More outside text followed by conclusion.
\begin{equation}
    a = c.
\end{equation}

\begin{note}
    Inside text followed by equation.
    \begin{equation}
        b = c. \label{secondref}
    \end{equation}
\end{note}


The inner equation is \eqref{innerref} and \cref{secondref}

\end{document}

Answer 3

I prefer to avoid monkeying with counter names.

\documentclass{article}

\usepackage[many]{tcolorbox}
\usepackage{cleveref}

\newcounter{savedequation}
\newcounter{tcbequation}
\newtcolorbox{note}[1][]{%
  code={%
    \setcounter{savedequation}{\value{equation}}%
    \setcounter{equation}{\value{tcbequation}}%
    \renewcommand{\theequation}{N\arabic{equation}}%
  },
  after={%
    \setcounter{tcbequation}{\value{equation}}%
    \setcounter{equation}{\value{savedequation}}%
  },
  breakable,
  enhanced jigsaw,
  borderline west={1pt}{0pt}{black},
  sharp corners,
  boxrule=0pt,
  frame hidden,
  left=1ex,right=1ex,
  fonttitle={\bfseries},
  coltitle={black},
  title={Note:\ },
  attach title to upper,
  #1,
}

\begin{document}

Outside text followed by equation.
\begin{equation}\label{A}
    a = b.
\end{equation}

\begin{note}
    Inside text followed by equation.
    \begin{equation}\label{B}
        b = c.
    \end{equation}
\end{note}

More outside text followed by conclusion.
\begin{equation}\label{C}
    a = c.
\end{equation}

\cref{A}, \cref{B}

\end{document}