% Style helpers for use in Math M413

\usepackage[leqno]{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{latexsym}

\numberwithin{equation}{section}

\DeclareMathOperator{\suchthat}{s.t.}
\DeclareMathOperator{\st}{\;|\;}
\DeclareMathOperator{\lub}{lub}
\DeclareMathOperator{\glb}{glb}
\DeclareMathOperator{\C}{C}
\DeclareMathOperator{\Rl}{R_L}
\DeclareMathOperator{\Rm}{R_M}
\DeclareMathOperator{\Poly}{\mathcal{P}}
\DeclareMathOperator{\NPoly}{\mathcal{NP}}
\DeclareMathOperator{\outdegree}{out-degree}
\DeclareMathOperator{\indegree}{in-degree}

% Abbreviations for Commonly used sets
\newcommand{\field}[1]{\mathbb{#1}}
\newcommand{\Q}{\field{Q}} % Rational numbers
\newcommand{\N}{\field{N}} % Natural numbers
\newcommand{\R}{\field{R}} % Real numbers
\newcommand{\Z}{\field{Z}} % Integers
\newcommand{\Zp}{\field{Z}^+} % Positive Integers

% Theorem declarations
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{correllary}{Correllary}[section]
\newtheorem{proposition}{Proposition}[section]

\theoremstyle{definition}
\newtheorem{definition}{Definition}[section]
\newtheorem{example}{Example}[section]

\theoremstyle{remark}
\newtheorem*{remark}{Remark}
\newtheorem*{note}{Note}
\newtheorem{case}{Case}


% Meaningful abbreviations, etc
\newcommand{\onlyif}{\Rightarrow}
\newcommand{\thus}{\Longrightarrow}
\newcommand{\contradicts}{\rightarrow\leftarrow}
\newcommand{\Union}{\bigcup} 
\newcommand{\union}{\cup}
\newcommand{\Intersect}{\bigcap}
\newcommand{\intersect}{\cap}
\newcommand{\setdiff}{\setminus}
\newcommand{\divides}{|}
\newcommand{\Land}{\;\bigwedge\;}
\newcommand{\Lor}{\;\bigvee\;}
\newcommand{\floor}[1]{\lfloor#1\rfloor}
\newcommand{\inverse}[1]{\ensuremath\frac{1}{#1}}
\newcommand{\abs}[1]{\ensuremath \lvert#1\rvert}
\newcommand{\norm}[1]{\ensuremath \lVert#1\rVert}
\newcommand{\inner}[2]{\ensuremath #1 \cdot #2}

\newcommand{\cross}{\times}
\newcommand{\setcomplement}[1]{\overline{#1}}
\newcommand{\emptystring}{\epsilon}
\newcommand{\concat}[2]{#1 \cdot #2}
\newcommand{\pair}[2]{\langle #1, #2 \rangle}
\newcommand{\closure}[1]{#1^*}
\newcommand{\strlen}[1]{\abs{#1}}

\newcommand{\produce}{\rightarrow}
\newcommand{\derive}{\Rightarrow}
\newcommand{\derives}{\stackrel{*}{\Rightarrow}}