
% --------------------------------------------------------------
% This is all preamble stuff that you don't have to worry about.
% Head down to where it says "Start here"
% --------------------------------------------------------------

\documentclass[10pt, letterpaper]{article}

% --------------------------------------------------------------
% Package utf8 for normal characters please
% --------------------------------------------------------------
\usepackage[utf8]{inputenc}

% --------------------------------------------------------------
% Package setup for guillemets « » and non-breaking spaces
% see https://tex.stackexchange.com/questions/2681/guillemets-in-latex
% --------------------------------------------------------------
\usepackage[T1]{fontenc}
\usepackage[french]{babel}
\frenchbsetup{og=«, fg=»}
\usepackage[french=guillemets]{csquotes}
\usepackage{xspace}

% --------------------------------------------------------------
% Control paper size and margins
% --------------------------------------------------------------
\usepackage[margin=1in]{geometry} 

% --------------------------------------------------------------
% Make nice lists with letters and shit
% --------------------------------------------------------------
\usepackage{enumitem}

% --------------------------------------------------------------
% Make columns
% --------------------------------------------------------------
\usepackage{multicol}

% --------------------------------------------------------------
% Better tables
% --------------------------------------------------------------
\usepackage{tabularx}

% --------------------------------------------------------------
% Images
% --------------------------------------------------------------
\usepackage{graphicx}

% --------------------------------------------------------------
% Fitch-style trees
% (make sure to include the `fitch.sty` file in the directory)
% --------------------------------------------------------------
\input{fitch.sty}

% Math stuff
% --------------------------------------------------------------
\usepackage{amsmath,amsthm,amssymb}
 
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
 
\newenvironment{theorem}[2][Theorem]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
\newenvironment{lemma}[2][Lemma]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
\newenvironment{exercise}[2][Exercise]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
\newenvironment{problem}[2][Problem]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
\newenvironment{question}[2][Question]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
\newenvironment{corollary}[2][Corollary]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}

\newenvironment{solution}{\begin{proof}[Solution]}{\end{proof}}
 
\begin{document}
 
% --------------------------------------------------------------
%                         Start here
% --------------------------------------------------------------
 
\title{Devoir 6}
\author{Louis-Olivier Brassard\\
PHI1005 -- Logique 1}
\date{Automne 2019}

% maketitle génère automatiquement le titre
\maketitle

\section*{Question 1.}

En utilisant la méthode de la déduction naturelle, démontrez les raisonnements suivants :

\begin{enumerate}[label=\alph*)]
  % a)
  \item $p \supset q, r \supset s, \lnot (q \lor s) \vdash \lnot (p \lor r)$
  
  \begin{fitch}
    \fh  & H \\ %1
    \fa & () \\
    
  \end{fitch}
 
  % b)
  \item $\vdash \lnot (\lnot p \lor \lnot q) \supset (q \land q)$
  
  \begin{fitch}
    \fh  & H \\ %1
    \fa & () \\
    
  \end{fitch}
 
  % c)
  \item $(p \land r), \lnot r \vdash p$
  
  \begin{fitch}
    \fh  & H \\ %1
    \fa & () \\
    
  \end{fitch}
  
  % d)
  \item $\vdash (p \lor (q \supset r)) \equiv ((p \lor q) \supset (p \lor r))$
  
  \begin{fitch}
    \fh   &  \\ %1
    \fa  & () \\ 
    
  \end{fitch}
  
\end{enumerate}

% --------------------------------------------------------------
%     You don't have to mess with anything below this line.
% --------------------------------------------------------------
 
\end{document}
