\newcommand{\eu}{\mathrm{e}}
\newcommand{\im}{\mathrm{i}}

\newcommand{\degree}{\,^{\circ}}

\newcommand{\transpose}[1]{{#1}^{\mathsf{T}}}

\newcommand{\Int}{\int\limits_{-\infty}^{\infty}}

\newcommand{\rint}[2]{\int{#1}\dd{#2}}

\newcommand{\Rint}[4]{\int\limits_{#1}^{#2}{#3}\dd{#4}}

\newcommand{\HH}{\mathbb{H}}
\DeclareMathOperator{\ot}{ot}
\DeclareMathOperator{\cof}{cof}
\DeclareMathOperator{\TC}{TC}
\newcommand{\ORD}{\text{ORD}}

\makeatletter
\newcommand*\bigcdot{\mathpalette\bigcdot@{.5}}
\newcommand*\bigcdot@[2]{\mathbin{\vcenter{\hbox{\scalebox{#2}{$\m@th#1\bullet$}}}}}
\makeatother

\newcommand{\Ham}{\hat{\mathcal{H}}}

\renewcommand{\Tr}{\mathrm{Tr}}

\newcommand{\christoffelsecond}[4]{\dfrac{1}{2}g^{#3 #4}(\partial_{#1} g_{#2 #4} + \partial_{#2} g_{#1 #4} - \partial_{#4} g_{#1 #2})}

\newcommand{\riemanncurvature}[5]{\partial_{#3} \Gamma_{#4 #2}^{#1} - \partial_{#4} \Gamma_{#3 #2}^{#1} + \Gamma_{#3 #5}^{#1} \Gamma_{#4 #2}^{#5} - \Gamma_{#4 #5}^{#1} \Gamma_{#3 #2}^{#5}}

\newcommand{\covariantriemanncurvature}[5]{g_{#1 #5} R^{#5}{}_{#2 #3 #4}}

\newcommand{\riccitensor}[5]{g_{#1 #5} R^{#5}{}_{#2 #3 #4}}

\renewcommand{\emptyset}{\varnothing}
\renewcommand{\iff}{\longleftrightarrow}
\renewcommand{\implies}{\longrightarrow}
\renewcommand{\impliedby}{\longleftarrow}

\DeclareMathOperator{\dom}{dom}
\DeclareMathOperator{\imm}{imm}
\newcommand{\rel}{\mathcal{R}}
\newcommand{\FF}{\mathcal{F}}
\newcommand{\bigtimes}{\varprod}
\DeclareMathOperator{\Fun}{Fun}
\DeclareMathOperator{\id}{id}
\newcommand{\PP}{\mathcal{P}}

\newcommand{\NN}{\mathbb{N}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\UU}{\mathbb{V}}
\newcommand{\VV}{\mathbb{V}}

\newcommand{\se}{\text{se }}
\newcommand{\altrimenti}{\text{altrimenti}}
\newcommand{\inv}{^{-1}}

\newcommand{\AC}{$\mathsf{AC}$}
\newcommand{\PA}{$\mathsf{PA}$}
\newcommand{\cc}{\mathfrak{c}}
\DeclareMathOperator{\Fin}{Fin}
\newcommand{\PPFin}{\PP^{\mathsf{fin}}}
\DeclareMathOperator{\FSeq}{FSeq}
\DeclareMathOperator{\bige}{Big}
\DeclareMathOperator{\Sym}{\mathfrak{S}}

\newcommand{\OO}{\mathfrak{O}}
\newcommand{\basis}{\mathcal{B}}

\newcommand{\restr}[2]{
	#1\arrowvert_{#2}
}
