\documentclass[12pt]{article}

\usepackage{proof}
\usepackage{fullpage}

\newcommand\os[1] {$\mit{#1}$}
%\newcommand\os[1]{#1}
 
\newcommand\imp\Rightarrow

%\input{qobitree}
\pagestyle{empty}

\begin{document}

\begin{center}
\large 
%\vspace*{-1.5cm}
Harvey Mudd College\\
Computer Science \os{80}\\
Logic for Computer Science\\
Spring Semester \os{2001}\\[16pt]

Assignment \#3 -- Propositional Logic: Natural Deduction\\
\large
Due { \os{11}:\os{00}am}, Wednesday February \os{21}, \os{2001}
\end{center}

\vspace{0.25in}


\newcommand{\G}{\Gamma}
\newcommand{\D}{\Delta}
\newcommand{\lra}{\longrightarrow}

\vspace{.5in}
Give Natural Deduction proofs of each of the following formulas. That is,
give a Natural Deduction proof tree with the given formula as its conclusion,
and no undischarged assumptions. (Note, brackets are used in place of parentheses in
some formulas to make the structure clearer):

\vspace{.5in}

\begin{enumerate}

\item $[(p \land q) \imp r] \equiv [p\imp(q\imp r)]$

\item $(p \imp q) \imp [(r \imp p)\imp (r\imp q)]$

\item $([(p\imp q)\imp p]\imp p)$

\item $(p\imp r)\equiv \neg(p\land \neg r)$

\item $(p\imp r)\equiv (\neg p \lor r)$
\end{enumerate}

\end{document}
