\documentclass[12pt]{article}

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\usepackage{hyperref}
\usepackage{cmll}
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\usepackage[margin=.25in]{geometry}

%\DeclareMathOperator\parr{\bindnasrepma}

\begin{document}
\title{Linear logic}
\author{<NAME>}
\maketitle


\begin{prooftree}
\def\fCenter{\mbox{\ $\vdash$\ }}
\Axiom   $ \Gamma \fCenter \Delta $
\UnaryInf$ \Gamma \fCenter \Delta $
\end{prooftree}


\begin{prooftree}
\def\fCenter{\mbox{\ $\vdash$\ }}
\Axiom    $ \Gamma \fCenter \Delta $
\Axiom    $ \Gamma \fCenter \Delta $
\BinaryInf$ \Gamma \fCenter \Delta $
\end{prooftree}

\section*{Classical linear logic}
\subsection*{Double-sided form}
\subsubsection*{Structural rules}

Weakening
\begin{prooftree}
\def\fCenter{\mbox{\ $\vdash$\ }}
\Axiom   $\Gamma     \fCenter \Delta $
\UnaryInf$\Gamma, !A \fCenter \Delta $
\end{prooftree}

\begin{prooftree}
\def\fCenter{\mbox{\ $\vdash$\ }}
\Axiom   $\Gamma \fCenter \Delta      $
\UnaryInf$\Gamma \fCenter \Delta, ?A  $
\end{prooftree}

Identity

\begin{prooftree}
\def\fCenter{\mbox{\ $\vdash$\ }}
\Axiom   $ \fCenter $
\UnaryInf$ A \fCenter A $
\end{prooftree}

\subsubsection*{Logical rules}

$\perp$

\begin{prooftree}
\def\fCenter{\mbox{\ $\vdash$\ }}
\Axiom   $\Gamma          \fCenter A, \Delta$
\UnaryInf$\Gamma, A^\perp \fCenter \Delta   $
\end{prooftree}

\begin{prooftree}
\def\fCenter{\mbox{\ $\vdash$\ }}
\Axiom    $\Gamma, A \fCenter \Delta          $
\UnaryInf $\Gamma    \fCenter A^\perp , \Delta$
\end{prooftree}

$\with$

\begin{prooftree}
\def\fCenter{\mbox{\ $\vdash$\ }}
\Axiom   $ \Gamma, A   \fCenter \Delta  $
\UnaryInf$ \Gamma, A \with B \fCenter \Delta $
\end{prooftree}

\begin{prooftree}
\def\fCenter{\mbox{\ $\vdash$\ }}
\Axiom   $ \Gamma, B   \fCenter \Delta  $
\UnaryInf$ \Gamma, A \with B \fCenter \Delta $
\end{prooftree}

\begin{prooftree}
\def\fCenter{\mbox{\ $\vdash$\ }}
\Axiom    $ \Gamma \fCenter \Delta, A    $
\Axiom    $ \Gamma \fCenter \Delta, B    $
\BinaryInf$ \Gamma \fCenter \Delta, A \with B $
\end{prooftree}

$\oplus$

\begin{prooftree}
\def\fCenter{\mbox{\ $\vdash$\ }}
\Axiom   $ \Gamma \fCenter \Delta, A          $
\UnaryInf$ \Gamma \fCenter \Delta, A \oplus B $
\end{prooftree}

\begin{prooftree}
\def\fCenter{\mbox{\ $\vdash$\ }}
\Axiom   $ \Gamma \fCenter \Delta, B          $
\UnaryInf$ \Gamma \fCenter \Delta, A \oplus B $
\end{prooftree}

\begin{prooftree}
\def\fCenter{\mbox{\ $\vdash$\ }}
\Axiom    $ \Gamma, A \fCenter \Delta          $
\Axiom    $ \Gamma, B \fCenter \Delta          $
\BinaryInf$ \Gamma, A \oplus B \fCenter \Delta $
\end{prooftree}

$\otimes$

\begin{prooftree}
\def\fCenter{\mbox{\ $\vdash$\ }}
\Axiom   $ \Gamma, A, B        \fCenter \Delta $
\UnaryInf$ \Gamma, A \otimes B \fCenter \Delta $
\end{prooftree}

\begin{prooftree}
\def\fCenter{\mbox{\ $\vdash$\ }}
\Axiom    $ \Gamma_1           \fCenter \Delta_1, A                     $
\Axiom    $ \Gamma_2           \fCenter \Delta_2, B                     $
\BinaryInf$ \Gamma_1, \Gamma_2 \fCenter \Delta_1, \Delta_2, A \otimes B $
\end{prooftree}

$\parr$

\begin{prooftree}
\def\fCenter{\mbox{\ $\vdash$\ }}
\Axiom   $ \Gamma \fCenter \Delta, A, B      $
\UnaryInf$ \Gamma \fCenter \Delta, A \parr B $
\end{prooftree}

\begin{prooftree}
\def\fCenter{\mbox{\ $\vdash$\ }}
\Axiom    $ \Gamma_1, A                    \fCenter \Delta_1           $
\Axiom    $ \Gamma_2, B                    \fCenter \Delta_2           $
\BinaryInf$ \Gamma_1, \Gamma_2, A \parr B  \fCenter \Delta_1, \Delta_2 $
\end{prooftree}


\end{document}
