Suggestions for nested proofs

Question

I would like to write down my lecture notes. The proofs of the theorems in this lecture are usually really big. It usually looks like this:

Theorem 1: Here comes the statement of the Theorem

Proof.

1. Without loss of generality let us assume...
2. Claim 1: Statement.

Proof: Blah blah \endOfInnerProofSymbol

3. Claim 2: Statement.

Proof. Blah blah \endOfInnerProofSymbol

\endOfProofSymbol

So my question is now, how can I make such nested proofs with another QED-symbol than the parent proof (maybe a filled square)?

Answer 1

You can simply define a new environment subproof that mimics the behavior of proof but uses a filled square instead of the default one.

\newenvironment{subproof}[1][\proofname]{%
  \renewcommand{\qedsymbol}{$\blacksquare$}%
  \begin{proof}[#1]%
}{%
  \end{proof}%
}

You can even give it the optional argument as in

\begin{subproof}[Subproof]

so to print "Subproof" instead of "Proof".

MWE (note the \item after the outer proof to let enumerate start on a new line)

\documentclass{article}
\usepackage{amsthm,amssymb}

\newtheorem{theorem}{Theorem}

\newenvironment{subproof}[1][\proofname]{%
  \renewcommand{\qedsymbol}{$\blacksquare$}%
  \begin{proof}[#1]%
}{%
  \end{proof}%
}

\begin{document}
\begin{theorem}
  Here comes the statement of the Theorem
\end{theorem}
\begin{proof}
  \item
  \begin{enumerate}
    \item Without loss of generality let us assume...
    \item Claim 1: Statement.%
    \begin{subproof}[Subproof]
      Blah blah
    \end{subproof}
    \item Claim 2: Statement.%
    \begin{subproof}
      Blah blah
    \end{subproof}
  \end{enumerate} 
\end{proof}
\end{document}