\newcommand\comment[1]{} 
\renewcommand\invrule[2]{ 
% for use in non-math mode, args interpreted in math mode
  \begin{tabular}{@{}c@{}}
  $ #1 $ \\ \hline \hline $ #2 $
  \end{tabular}}
\newcommand\slrule[2]{ % similar to \frac, except
% for use in non-math mode, args interpreted in math mode
  \begin{tabular}{@{}c@{}}
  $ #1 $ \\ \hline $ #2 $
  \end{tabular}}

\newcommand\fig[1]{Fig.~\ref{fig:#1}}
\newcommand\bfi[1]{\textbf{#1}\index{#1|textbf}}
%\renewcommand\Kt{\textbf{Kt}}
\newcommand\dKt{{$\delta$}\Kt}
\newcommand\up[1]{\vspace{-#1ex}} 
\newcommand\vpf{\vspace{-3ex}} % to reduce space before proof tree