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\begin{document}
\section*{Inference Rules of G4ip (LJT)}

\begin{figure}[h]
  \begin{center}
  \begin{FramedTitled}{\-}
  \centering
  \def\arraystretch{3.0}
  \begin{tabular}{cc}

   \infer[Axiom]{\sequ{A,\Gamm}{A}}{} &

   \infer[${\em false}-L$]{\sequ{\mbox{\em false},\Gamm}{G}}{}\\[-8mm]

   ($A$ being atomic) \\

   \infer[\&$-L$]{\sequ{B \& C,\Gamm}{G}}
                 {\sequ{B,C,\Gamm}{G}} &

   \infer[\&$-R$]{\sequ{\Gamm}{B \& C}}
                 {  \sequ{\Gamm}{B}
                  & \sequ{\Gamm}{C}}\\

   \infer[\OR$-L$]
              {\sequ{B\OR C,\Gamm}{G}}
              { \sequ{B,\Gamm}{G}
               &\sequ{C,\Gamm}{G}} &

   \infer[\OR$-R$_1]
              {\sequ{\Gamm}{B\OR C}}
              {\sequ{\Gamm}{B}}\hspace{3mm}
   \infer[\OR$-R$_2]
              {\sequ{\Gamm}{B\OR C}}
              {\sequ{\Gamm}{C}}\\

   \infer[\mbox{\tt ->}$-L$_1]
                    {\sequ{A\mbox{\tt ->} B,A,\Gamm}{G}}
                    {\sequ{B,A,\Gamm}{G}} &

   \infer[\mbox{\tt ->}$-R$]
                 {\sequ{\Gamm}{B\mbox{\tt ->} C}}
                 {  \sequ{B,\Gamm}{C}}\\[-8mm]
      
   ($A$ being atomic)\\           
       
   \multicolumn{2}{c}{
   \infer[\mbox{\tt ->}$-L$_2]
              {\sequ{(C \& D)\mbox{\tt ->} B,\Gamm}{G}}
              {\sequ{C \mbox{\tt ->} (D \mbox{\tt ->} B)\Gamm}{G}}} \\

   \multicolumn{2}{c}{
   \infer[\mbox{\tt ->}$-L$_3]
              {\sequ{(C\OR D)\mbox{\tt ->} B\Gamm}{G}}
              {\sequ{C\mbox{\tt ->}B,D\mbox{\tt ->}B,\Gamm}{G}}}\\

   \multicolumn{2}{c}{
   \infer[\mbox{\tt ->}$-L$_4]
              {\sequ{(C\mbox{\tt ->} D)\mbox{\tt ->} B\Gamm}{G}}
              {\sequ{D\mbox{\tt ->}B,\Gamm}{C\mbox{\tt ->} D} &
               \sequ{B,\Gamm}{G}} }\\

   %\infer[\mbox{\tt <->}$-L$]
   %                 {\sequ{B\mbox{\tt <->} C,\Gamm}{G}}
   %                 {\sequ{B\mbox{\tt ->} C, C\mbox{\tt ->} B,\Gamm}{G} 
   %                 } &

   %\infer[\mbox{\tt <->}$-R$]
   %              {\sequ{\Gamm}{B\mbox{\tt <->} C}}
   %              {  \sequ{\Gamm}{B\mbox{\tt ->} C} &
   %                 \sequ{\Gamm}{C\mbox{\tt ->} B} }\\

   \multicolumn{2}{c}
     { $B\mbox{\tt <->}C$ is defined as $B\mbox{\tt ->}C \& C\mbox{\tt ->}B$
     }           
  \end{tabular}
  \end{FramedTitled}
  \caption{Inference rules for G4ip; see for example \cite{Dyckhoff92}.}
  \label{fig:CL}
  \end{center}
\end{figure}

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