\pause and \framebreak in the same frame

Question

I'm trying to use \pause and \framebreak in the same frame and it does not work.
The reason why I need that is that I'm writing a proof over several slides but don't want the stuff showing up all at once on each slide.

And since it is a proof I cannot just make a new frame instead of using \framebreak because of the \begin{proof} environment.
Is there any way to do this?

\edit: Here is the code I'm talking about:

\usetheme{Luebeck}  
\usecolortheme{beaver}

\newenvironment{myproof}[1][\proofname]{%  
\par
\def\insertproofname{#1{.}}%
\pushQED{\qed}
\alert{\textbf{
{\insertproofname}}} \hspace*{\fill} \\[5pt]}
{\popQED}
\makeatother  

\frame{
\frametitle{(1) Rate function}
\begin{myproof}
 \begin{itemize}
 \item[(i)]
 \vspace*{7pt}
 By definition of $I(\cdot)$ we have 
 \[
   text
 \] 
 \pause
 \[
  ...
 \].
 \end{itemize}
}

\frame{
\frametitle{(1) Rate function}
\begin{itemize}
\item[(ii)]
  ...
 \pause
 \vspace*{10pt}
\item[(iii)] 
...
 \end{itemize}
\end{myproof}
}  

Where I started a new frame I had \framebreak before, but neither of the two ways gives me the desired. I get errors instead.

Answer 1

As Daniel mentioned in his answer, it's not possible to use frame breaks and overlay specifications; however, using a beamercolorbox (or an appropriate variant, depending on the theme used) you can define a new environment that behaves as the proof environment, except for the title, and then you can use this new environment to write the successive parts of your proof in different frames (or in different slides of the same frame).

Below there's an example showing both approaches for the Berkeley theme (other themes will require a modification of the box used for the continuation of the proof). Locally redefining \qedsymbol you can prevent the symbol from appearing on the first frame/slide and then using \qed you can typeset the symbol one the proof is finished:

\documentclass{beamer}
\usetheme{Berkeley}
\usepackage{lipsum}% just to generate filler text

\setbeamercolor{proofblock}{use=block body,bg=block body.bg}

\newenvironment<>{proofc}[1]
  {\begin{actionenv}#2%
    \begin{beamercolorbox}[sep=1ex]{proofblock}#1%
  }
  {\end{beamercolorbox}\end{actionenv}}

\begin{document}

\begin{frame}
\def\qedsymbol{} % temporarily suppress the \qedsymbol
\frametitle{Proof of the main result}
\begin{proof}
\lipsum*[2]
\end{proof}
\end{frame}

\begin{frame}
\frametitle{Proof of the main result (Cont.)}
\begin{proofc}
\lipsum*[2]
\end{proofc}
\end{frame}

\begin{frame}
\frametitle{Proof of the main result (Cont.)}
\begin{proofc}
\lipsum*[4]\qed
\end{proofc}
\end{frame}

\begin{frame}
\only<1>{
\def\qedsymbol{} % temporarily suppress the \qedsymbol
\frametitle{Proof of the main result}
\begin{proof}
\lipsum*[2]
\end{proof}
}
\only<2>{
\frametitle{Proof of the main result (Cont.)}
\begin{proofc}
\lipsum*[2]
\end{proofc}
}
\only<3>{
\frametitle{Proof of the main result (Cont.)}
\begin{proofc}
\lipsum*[4]\qed
\end{proofc}
}
\end{frame}

\end{document}

Some images showing the result:

Now that the original question has been edited, the solution is simpler since no block is used in the customized myproof environment:

1. Begin and end the myproof environment in every frame.

2. Locally redefine \qedsymbol for those frames in which the proof isn't yet finished.

3. Use the myproofc environment for those frames not needing the title.

A complete example using the code snippet given:

\documentclass{beamer}
\usetheme{Luebeck}  
\usecolortheme{beaver}

\newenvironment{myproof}[1][\proofname]{%  
  \par\def\insertproofname{#1{.}}%
  \pushQED{\qed}\alert{\textbf{{\insertproofname}}} \hspace*{\fill} \\[5pt]}
  {\popQED}

\newenvironment{myproofc}[1][\proofname]{%  
  \par%
  \pushQED{\qed}}
  {\popQED}

\begin{document}

\frame{
\def\qedsymbol{}
\frametitle{(1) Rate function}
\begin{myproof}
 \begin{itemize}
 \item[(i)]
 \vspace*{7pt}
 By definition of $I(\cdot)$ we have 
 \[
   text
 \] 
 \pause
 \[
  ...
 \].
 \end{itemize}
\end{myproof}
}

\frame{
\frametitle{(1) Rate function}
\begin{myproofc}
\begin{itemize}
\item[(ii)]
  ...
 \pause
 \vspace*{10pt}
\item[(iii)] 
...
 \end{itemize}
\end{myproofc}
} 

\end{document}

Answer 2

It is not possible :-(

This is clearly stated on page 60 (§8.1) of the beamer manual in the documentation of the allowframebreaks option:

• allowframebreaks=⟨fraction⟩. When this option is given, the frame will be automatically broken up into several frames if the text does not fit on a single slide. In detail, when this option is given, the following things happen:
  1. Overlays are not supported.
  2. ...

The \pause command internally uses \onslide (see §9.1 of the beamer manual), so it does employ overlay specifications.