Indentation level inside proof

Question

I would like to be able to format my proof "like a horizontal tree". And because pictures are better than words, here is what I would like to do :

Proof :
To prove the big theorem, we first
need to prove that XXX:
| To prove XXX, let's begin to prove
| YYY:
| | YYY is trivial when you consider the
| | lemma 42, which is usable because of
| | the hypothesis 1.0. So YYY is true!
| Now, we would like also the property ZZZ
| Let's proove it !
| | To prove ZZZ, we can build a path and "paf !"
| | ZZZ becomes trivial !
| Let's now notice that applying YYY inside the
| | property YYY will let us conclude on the
| | validity of XXX.
□

So the idea is to indent the proof within several levels to always be clear about what we are proving, and show the indentation with a vertical border on the left. The result should looks like what you got when you enter in a While/If block within the algorithm2e package (but of course I don't want the If/While words):

Thank you for your help !

EDIT

See this more modern answer for a semi-working solution Robustly add a border to the left of a text spanning several pages

Answer 1

One solution I found is to use the mdframed package (very complete package and documentation !), and create a special box for that. Here is an example :

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[english]{babel}
\usepackage{amsthm}
\usepackage{thmtools}
\usepackage{enumitem} % To avoid "Too deeply nested error"
\usepackage{mdframed}

% For the subproof
\newmdenv[linecolor=black
          ,topline=false
          ,bottomline=false
          ,rightline=false
          ,leftline=true
          ,leftmargin=0.1cm
          ,linewidth=0.02cm
          ,skipabove=0cm
          ,innerbottommargin=0.05cm
          ,skipbelow=0.05cm
          ]{subproof}
% Avoid alignement problem
\setlength{\parindent}{0cm}

% For the theorem environment
\declaretheorem[name=Theorem]{theorem}

\begin{document}

\begin{theorem}[My great theorem]
  Hello, here is the best theorem ever
\end{theorem}
\begin{proof}
    Let's begin the proof !

    To prove the big theorem, we first need to prove XXX:
    \begin{subproof}
      To prove XXX, let's begin to prove YYY:
      \begin{subproof}
        YYY is trivial when you consider the lemma 42, which is usable because of the hypothesis 1.0. So YYY is true!
      \end{subproof}
      Now, we would like also the property ZZZ
      Let's prove it !
      \begin{subproof}
        To prove ZZZ, we can build a path and "paf !" ZZZ becomes trivial !
      \end{subproof}
      Which ends the proof of XXX.
    \end{subproof}
\end{proof}
\end{document}

Which gives :

Their is an alignment problem if you do not set \setlength{\parindent}{0cm} (which I always do anyway). If you use one, I don't know if it's possible to detect if their is only one line in the above paragraph, so I would use two environments with one containing leftmargin=0.7cm...

Moreover, it also has trouble with proofs that goes on several pages...