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\documentclass{article}
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\usepackage{charter}
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\usepackage{proof}
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\begin{document}
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\section*{Inference Rules}
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\begin{enumerate}
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\item Frequently used inference rules:
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\begin{center}
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\mbox{\infer{F,\Gamma \vdash F}{}}\medskip
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\mbox{\infer{\Gamma \vdash F_2}
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{\Gamma \vdash F_1 \Rightarrow F_2 & \Gamma \vdash F_1}}
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\hspace{10mm}
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\mbox{\infer{\Gamma \vdash F_1 \Rightarrow F_2}
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{F_1, \Gamma \vdash F_2}}\medskip
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\mbox{\infer{\Gamma \vdash P \;\textit{says}\; F}
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{\Gamma \vdash F}}
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\hspace{10mm}
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\mbox{\infer[\star]{\Gamma \vdash F}
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{\Gamma \vdash P \;\textit{controls}\; F & \Gamma \vdash P
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\;\textit{says}\; F}}\medskip
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\mbox{\infer{\Gamma \vdash P \;\textit{says}\;F_2}
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{\Gamma \vdash P \;\textit{says}\; (F_1 \Rightarrow F_2) &
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\Gamma \vdash P \;\textit{says}\;F_1}}\medskip
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\mbox{\infer{\Gamma \vdash F[x := t]}
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{\Gamma \vdash \forall x. F}}\medskip
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\mbox{\infer{\Gamma \vdash F_1 \wedge F_2}
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{\Gamma \vdash F_1 & \Gamma \vdash F_2}}
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\hspace{10mm}
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\mbox{\infer{\Gamma \vdash F_1}
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{\Gamma \vdash F_1 \wedge F_2}}
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\hspace{10mm}
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\mbox{\infer{\Gamma \vdash F_2}
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{\Gamma \vdash F_1 \wedge F_2}}\medskip
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\mbox{\infer[\star]{\Gamma \vdash Q\;\textit{says}\; F}{\Gamma \vdash P\mapsto Q &
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\Gamma \vdash P\;\textit{says}\; F}}
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\end{center}
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\item Less frequently used inference rules:
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\begin{center}
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\mbox{\infer{\Gamma \vdash slev(P) < slev(Q)}
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{\Gamma \vdash slev(P) = l_1 & \Gamma \vdash slev(Q) = l_2 &
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\Gamma \vdash l_1 < l_2}}\medskip
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\mbox{\infer{\Gamma \vdash slev(P) = slev(Q)}
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{\Gamma \vdash slev(P) = l & \Gamma \vdash slev(Q) = l}}\medskip
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\mbox{\infer{\Gamma \vdash l_1 < l_3}
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{\Gamma \vdash l_1 < l_2 & \Gamma \vdash l_2 < l_3}}\medskip
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\mbox{\infer[\star]{\Gamma \vdash P \mapsto R}{\Gamma \vdash P\mapsto Q & \Gamma
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\vdash Q \mapsto R}}\medskip
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\mbox{\infer{\Gamma \vdash F_1 \vee F_2}{\Gamma \vdash F_1}}
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\hspace{5mm}
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\mbox{\infer{\Gamma \vdash F_1 \vee F_2}{\Gamma \vdash F_2}}
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\hspace{5mm}
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\mbox{\infer{\Gamma \vdash F_3}{\Gamma \vdash F_1 \vee F_2 & F_1, \Gamma
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\vdash F_3 & F_2, \Gamma \vdash F_3}}\medskip
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\mbox{\infer[c\;\mbox{must be a fresh variable}]{\Gamma \vdash \forall
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x. F}{\Gamma \vdash F[x := c]}}\medskip
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\mbox{\infer{\Gamma \vdash \textit{true}}{}}\medskip
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\mbox{\infer[\star]{\Gamma \vdash P\;\textit{controls}\; F}{\Gamma \vdash P\mapsto Q
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& \Gamma \vdash Q\;\textit{controls} F}}
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\end{center}
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\end{enumerate}
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$\star$ derived rules
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\end{document}
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: t
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%%% End:
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