%MACROS:
%Local


%\renewcommand{\labelitemi}{$\circ$}
\renewcommand{\labelitemii}{$\circ$}
\renewcommand{\labelitemiii}{$\bullet$}
%\renewcommand{\labelitemiii}{$\circ$}
%\renewcommand{\labelitemiv}{$\circ$}



\newcommand{\xx}{{\bm{x}}}
\newcommand{\yy}{{\bm{y}}}
\newcommand{\zz}{{\bm{z}}}
\newcommand{\bcF}{\bm{\cF}}



\newcommand{\Lempty}{\mc{L}}
\newcommand{\Lkappalambda}[2]{\Lempty_{#1, #2}}
\newcommand{\Lomega}[1]{\Lkappalambda{#1}{\w}}
\newcommand{\Linfty}[1]{\Lkappalambda{#1}{\infty}}
\newcommand{\Lww}{\Lomega{\w}}
\newcommand{\Lwow}{\Lomega{\omega_1}}
\newcommand{\Lwtw}{\Lomega{\omega_2}}
\newcommand{\Liw}{\Lomega{\infty}}

\DeclareDocumentCommand{\Sym}{m}
{
S_{#1}
}

\DeclareDocumentCommand{\SymDiff}{}
{
\Delta
}




%\newcommand\rest[1][]{\EM{\upharpoonleft_{#1}}}
%\newcommand\rest[1][]{\EM{\upharpoonright_{#1}}}
\newcommand\rest[1][]{\EM{\!\upharpoonright_{#1}}}

\newcommand{\defn}[1]{{\bf{#1}}}
\newcommand{\defas}{\EM{:=}}



%Specific to this document

\def\KP{\text{KP}}
\def\ZF{\text{ZF}}
\def\ZFC{\text{ZFC}}
\def\Set{\text{SET}}
\DeclareDocumentCommand{\Powerset}{}{\raisebox{.15\baselineskip}{\Large\EM{\wp}}}

\DeclareDocumentCommand{\PowersetFinite}{}{\Powerset_{<\w}}






\DeclareMathOperator{\Seq}{Seq}
\DeclareMathOperator{\len}{len}

\DeclareMathOperator{\ORD}{ORD}
\DeclareMathOperator{\CARD}{CARD}
%\DeclareMathOperator{\sup}{sup}
%\DeclareMathOperator{\inf}{inf}
\DeclareMathOperator{\dom}{dom}
\DeclareMathOperator{\range}{range}
\DeclareMathOperator{\obj}{obj}
\DeclareMathOperator{\Obj}{Obj}
\DeclareMathOperator{\morph}{mor}
\DeclareMathOperator{\Morph}{Mor}
\DeclareMathOperator{\cod}{cod}
\DeclareMathOperator{\Graph}{graph}
\DeclareMathOperator{\codom}{codom}
\DeclareMathOperator{\cl}{cl}
%\DeclareMathOperator{\graph}{graph}


\DeclareDocumentCommand{\extent}{d<> d[] d()}
{
\IfNoValueTF{#1}
{
    \IfNoValueTF{#2}
    {
    \llbracket #3 \rrbracket
    }
    {
    \llbracket #3 \rrbracket_{#2}
    }
}
{
    \IfNoValueTF{#2}
    {
    \llbracket #3 \rrbracket_{#1}
    }
    {
    \llbracket #3 \rrbracket_{#1: #2}
    }
}
}


\DeclareDocumentCommand{\OrdLabels}{d[]}{
\IfNoValueTF{#1}
	{
        \{-\infty\} \cup \w_1 \cup \{\infty\}    
	}
	{
        \{-\infty\} \cup #1 \cup \{\infty\}    
	}
}



\DeclareMathOperator{\rankop}{rank}
\DeclareDocumentCommand{\rank}{d<> d[] d()}
{
\IfNoValueTF{#1}
{
    \IfNoValueTF{#2}
    {
        \IfNoValueTF{#3}
        {
            \rankop
        }
        {
            \rankop(#3)
        }
    }
    {
        \IfNoValueTF{#3}
        {
            \rankop_{#2}
        }
        {
            \rankop_{#2}(#3)
        }
    }
}
{
    \IfNoValueTF{#2}
    {
        \IfNoValueTF{#3}
        {
            \rankop^{#1}
        }
        {
            \rankop^{#1}(#3)
        }    
    }
    {
        \IfNoValueTF{#3}
        {
            \rankop_{#2}^{#1}
        }
        {
            \rankop_{#2}^{#1}(#3)
        }    
    }
}
}



\DeclareMathOperator{\truncop}{tr}
\DeclareDocumentCommand{\trunc}{d[] d()}
{
\IfNoValueTF{#1}
{
    \IfNoValueTF{#2}
    {
        \truncop
    }
    {
        \truncop(#2)
    }
}
{
    \IfNoValueTF{#2}
    {
        \truncop_{#1}
    }
    {
        \truncop_{#1}(#2)
    }
}
}


\DeclareMathOperator{\visibleop}{vis}
\DeclareDocumentCommand{\visible}{d<> d[]}
{
\IfNoValueTF{#1}
{
    \IfNoValueTF{#2}
    {
        \visibleop
    }
    {
        \visibleop_{#2}
    }
}
{
    \IfNoValueTF{#2}
    {
        \text{Error}
    }
    {
        \visibleop_{#1:#2}
    }
}
}





\DeclareDocumentCommand{\RenameEq}{}
{
\bumpeq
}


\DeclareMathOperator{\DensityThop}{Den}
\DeclareDocumentCommand{\DensityTh}{d~~ d<> d[] d()}
{
\IfNoValueTF{#1}
{
    \Th_{\DensityThop}(#4, #2, #3)
}
{
    \Th_{\DensityThop}(#4\!:\!#1, #2, #3)
}
}


\DeclareMathOperator{\glabelingop}{glab}
\DeclareDocumentCommand{\glabeling}{d[]}
{
\glabelingop_{#1}
}

\DeclareMathOperator{\labelingop}{lab}
\DeclareDocumentCommand{\labeling}{d[]}
{
\labelingop_{#1}
}


\DeclareMathOperator{\ThPlanop}{pl}
\DeclareDocumentCommand{\ThPlan}{}
{
\Th_{\ThPlanop}
}





%ID
\DeclareDocumentCommand{\ID}{}{\mathcal{I}}

%ID full trees
\DeclareDocumentCommand{\IDtree}{}{\ID_{\mathrm{tr}}}

%ID for templates
\DeclareDocumentCommand{\IDtemp}{d[]}{\ID_{\mathrm{temp}}[#1]}

\DeclareMathOperator{\Str}{Str}



\DeclareMathOperator{\Th}{Th}

\DeclareMathOperator{\ThGeoop}{Geo}
\DeclareDocumentCommand{\ThGeo}{d[]}
{
\Th_{\ThGeoop}(#1)
}

\DeclareMathOperator{\ThStrGeoop}{SGeo}
\DeclareDocumentCommand{\ThStrGeo}{d<> d[] d()}
{
\Th_{\ThStrGeoop}(#1, #2, #3)
}



\DeclareMathOperator{\GradedFaceop}{Geo}
\DeclareDocumentCommand{\GradedFace}{d[]}
{
\mathfrak{GF}(#1)
}




%To Clean Up
\DeclareMathOperator{\Sorts}{Sorts}
\DeclareMathOperator{\Funct}{Funct}
\DeclareMathOperator{\Rel}{Rel}
\DeclareMathOperator{\arity}{ar}


\newcommand{\Fraisse}{Fra\"{i}ss\'{e}}


\newcommand{\Bool}{\EM{\mathfrak{B}}}
\newcommand{\Atomic}{\EM{\mathfrak{At}}}
\DeclareDocumentCommand{\Literal}{d[]}
{
\IfNoValueTF{#1}
	{
        \EM{\mathfrak{Lit}}
	}
	{
        \EM{\mathfrak{Lit}_{#1}}
	}
}





%\DeclareMathOperator{\Lang}{L}
\DeclareDocumentCommand{\Lang}{}{\EM{\mathfrak{L}}}

\DeclareDocumentCommand{\quadiff}{}
{
    \EM{\quad\text{ if and only if }\quad}
}


\def\aa{{\EM{\mathbf{a}}}}
\def\bb{{\EM{\mathbf{b}}}}
\def\cc{{\EM{\mathbf{c}}}}
\def\dd{{\EM{\mathbf{d}}}}


\def\cA{{\EM{\mathcal{A}}}}
\def\cB{{\EM{\mathcal{B}}}}
\def\cC{{\EM{\mathcal{C}}}}
\def\cD{{\EM{\mathcal{D}}}}
\def\cF{{\EM{\mathcal{F}}}}
\def\cG{{\EM{\mathcal{G}}}}
\def\cH{{\EM{\mathcal{H}}}}
\def\cK{{\EM{\mathcal{K}}}}
\def\cM{{\EM{\mathcal{M}}}}
\def\cN{{\EM{\mathcal{N}}}}
\def\cO{{\EM{\mathcal{O}}}}
\def\cP{{\EM{\mathcal{P}}}}
\def\cQ{{\EM{\mathcal{Q}}}}
\def\cR{{\EM{\mathcal{R}}}}
\def\cS{{\EM{\mathcal{S}}}}
\def\cT{{\EM{\mathcal{T}}}}
\def\cW{{\EM{\mathcal{W}}}}
\def\cX{{\EM{\mathcal{X}}}}
\def\cY{{\EM{\mathcal{Y}}}}

%-  -   -   -

\def\w{\EM{\omega}}
\def\Naturals{{\EM{{\mbb{N}}}}}
\def\Nats{\Naturals}
\def\Integers{{\EM{{\mbb{Z}}}}}
\def\Rationals{{\EM{{\mbb{Q}}}}}
\def\PosRationals{\EM{\Rationals^{+}}}
\def\NNRationals{\EM{\Rationals^{\geq 0}}}
\def\Reals{\EM{\mbb{R}}}
\def\RealsQuantale{\EM{\mbf{[0,\infty]}}}

\newcommand{\Cantor}{2^\Nats}


\def\^{\EM{{}^{\And}}}

\def\Or{\EM{\vee}}
\def\And{\EM{\wedge}}

\def\<{\EM{\langle}}
\def\>{\EM{\rangle}}

\def\qu#1{``#1''}

\def\indent{\hspace*{2em}}
\def\newp{\newline\indent}
\def\nl{\newline}
\def\nnll{\newline\newline}

\def\EM#1{\ensuremath{#1}}
\def\mbb#1{\EM{\mathbb{#1}}}
\def\mbf#1{\EM{\mathop{\pmb{#1}}}}
\def\mc#1{\EM{\mathcal{#1}}}
\def\mf#1{\EM{\mathfrak{#1}}}

%\ol:Overline \tol:Text Overline
\def\ol#1{\EM{\overline{#1}}}
\def\tol#1{\EM{\overline{\text{#1}}}}
\def\ul#1{\underline{#1}}

\def\st{\,:\,}
\def\:{\colon}

%For uniform monic/epic arrows
\def\monicrightarrow{\rightarrowtail}
\def\mrightarrow{\rightarrowtail}
\def\monicleftarrow{\leftarrowtail}
\def\mleftarrow{\leftarrowtail}
%
\def\epirightarrow{\twoheadrightarrow}
\def\erightarrow{\twoheadrightarrow}
\def\epileftarrow{\twoheadleftarrow}
\def\eleftarrow{\twoheadleftarrow}





%-  -   -   -

%