\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amssymb}
\usepackage{amsfonts}
\usepackage[T1]{fontenc}
\usepackage{dsfont}
\usepackage[many]{tcolorbox}
\usepackage{soul}
\usepackage{xcolor}
\usepackage{graphicx}
\usepackage{adjustbox}
\usepackage{tikz-cd}
\usepackage{appendix}
\usepackage{enumerate}
\usepackage{multicol}
\usepackage{fancyhdr}
\usepackage{parskip}
\usepackage{geometry}
\usepackage{hyperref}
\usepackage{mathtools}
\usepackage{nameref}
\usepackage{thmtools}
\usepackage{chngcntr}
\usepackage{thm-restate}
\usepackage{setspace}
\usepackage{float}
\usepackage{bm}
\usepackage{cleveref}
\usepackage{adjustbox}
\usepackage{multicol}

\DeclareSymbolFont{bbold}{U}{bbold}{m}{n}
\DeclareSymbolFontAlphabet{\mathbbold}{bbold}


\let\strokeL\L

% Standard symbols.
\newcommand{\limplies}{\ensuremath{\rightarrow}}
\newcommand{\liff}{\leftrightarrow}
\newcommand{\elsub}{\preccurlyeq}
\newcommand{\elext}{\succcurlyeq}
\newcommand{\elequiv}{\equiv}
\newcommand{\iso}{\simeq}
\newcommand{\pow}{\mathcal{P}}
\newcommand{\Aut}{\mathsf{Aut}}
\newcommand{\Av}{\mathsf{Av}}
\newcommand{\Age}{\mathsf{Age}}
\newcommand{\Emb}{\mathsf{Emb}}
\newcommand{\Th}{\mathsf{Th}}
\newcommand{\Diag}{\mathsf{Diag}}
\newcommand{\ElDiag}{\mathsf{ElDiag}}
\newcommand{\AtDiag}{\mathsf{AtDiag}}
\newcommand{\PC}{\ensuremath{\mathsf{PC}}}
\newcommand{\tp}{\ensuremath{\mathsf{tp}}}
\newcommand{\qftp}{\ensuremath{\mathsf{qftp}}}
\newcommand{\stp}{\ensuremath{\mathsf{stp}}}
\newcommand{\lstp}{\ensuremath{\mathsf{Lstp}}}
\newcommand{\Def}{\ensuremath{\mathsf{Def}}}
\newcommand{\VC}{\ensuremath{\mathrm{VC-dim}}}
\newcommand{\id}{\ensuremath{\mathsf{id}}}
\newcommand{\cl}{\ensuremath{\mathsf{cl}}}
\newcommand{\acl}{\ensuremath{\mathsf{acl}}}
\newcommand{\dcl}{\ensuremath{\mathsf{dcl}}}
\newcommand{\scl}{\ensuremath{\mathsf{scl}}}
\newcommand{\EM}{\ensuremath{\mathsf{EM-tp}}}
\newcommand{\restr}{\upharpoonright}

\DeclarePairedDelimiter\ceil{\lceil}{\rceil}
\DeclarePairedDelimiter\floor{\lfloor}{\rfloor}

% Sequences
\newcommand{\xbar}{\bar{x}}
\newcommand{\ybar}{\bar{y}}
\newcommand{\abar}{\bar{a}}
\newcommand{\xseq}[1]{x_0,\dots,x_{#1-1}}
\newcommand{\charseq}[2]{\chi_{#1,#2}}

% Forking
\newcommand{\Ind}[1]
{#1\setbox0=\hbox{$#1x$}\kern\wd0\hbox to 0pt{\hss$#1\mid$\hss} \lower.9\ht0\hbox to 0pt{\hss$#1\smile$\hss}\kern\wd0}

\newcommand{\indep}{\mathop{\mathpalette\Ind{}}}

\newcommand{\notind}[1]
{#1\setbox0=\hbox{$#1x$}\kern\wd0
\hbox to 0pt{\mathchardef\nn=12854\hss$#1\nn$\kern1.4\wd0\hss}
\hbox to 0pt{\hss$#1\mid$\hss}\lower.9\ht0 \hbox to 0pt{\hss$#1\smile$\hss}\kern\wd0}

\newcommand{\nindep}{\mathop{\mathpalette\notind{}}}

% Standard structures.
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}

% Model Theory specifics.
\newcommand{\M}{\mathcal{M}}
\newcommand{\C}{\mathcal{C}}
\renewcommand{\L}{\mathcal{L}}
\newcommand{\U}{\mathcal{U}}
\newcommand{\F}{\mathcal{F}}


% Fraktur Letters
\newcommand{\Afrak}{\ensuremath{\mathfrak{A}}}
\newcommand{\Bfrak}{\ensuremath{\mathfrak{B}}}
\newcommand{\Cfrak}{\ensuremath{\mathfrak{C}}}
\newcommand{\Dfrak}{\ensuremath{\mathfrak{D}}}
\newcommand{\Efrak}{\ensuremath{\mathfrak{E}}}
\newcommand{\Ffrak}{\ensuremath{\mathfrak{F}}}
\newcommand{\Gfrak}{\ensuremath{\mathfrak{G}}}
\newcommand{\Hfrak}{\ensuremath{\mathfrak{H}}}
\newcommand{\Ifrak}{\ensuremath{\mathfrak{I}}}
\newcommand{\Jfrak}{\ensuremath{\mathfrak{J}}}
\newcommand{\Kfrak}{\ensuremath{\mathfrak{K}}}
\newcommand{\Lfrak}{\ensuremath{\mathfrak{L}}}
\newcommand{\Mfrak}{\ensuremath{\mathfrak{M}}}
\newcommand{\Nfrak}{\ensuremath{\mathfrak{N}}}
\newcommand{\Ofrak}{\ensuremath{\mathfrak{O}}}
\newcommand{\Pfrak}{\ensuremath{\mathfrak{P}}}
\newcommand{\Qfrak}{\ensuremath{\mathfrak{Q}}}
\newcommand{\Rfrak}{\ensuremath{\mathfrak{R}}}
\newcommand{\Sfrak}{\ensuremath{\mathfrak{S}}}
\newcommand{\Tfrak}{\ensuremath{\mathfrak{T}}}
\newcommand{\Ufrak}{\ensuremath{\mathfrak{U}}}
\newcommand{\Vfrak}{\ensuremath{\mathfrak{V}}}
\newcommand{\Wfrak}{\ensuremath{\mathfrak{W}}}
\newcommand{\Xfrak}{\ensuremath{\mathfrak{X}}}
\newcommand{\Yfrak}{\ensuremath{\mathfrak{Y}}}
\newcommand{\Zfrak}{\ensuremath{\mathfrak{Z}}}

% Calligraphic Letters
\newcommand{\Acal}{\ensuremath{\mathcal{A}}}
\newcommand{\Bcal}{\ensuremath{\mathcal{B}}}
\newcommand{\Ccal}{\ensuremath{\mathcal{C}}}
\newcommand{\Dcal}{\ensuremath{\mathcal{D}}}
\newcommand{\Ecal}{\ensuremath{\mathcal{E}}}
\newcommand{\Fcal}{\ensuremath{\mathcal{F}}}
\newcommand{\Gcal}{\ensuremath{\mathcal{G}}}
\newcommand{\Hcal}{\ensuremath{\mathcal{H}}}
\newcommand{\Ical}{\ensuremath{\mathcal{I}}} 
\newcommand{\Jcal}{\ensuremath{\mathcal{J}}} 
\newcommand{\Kcal}{\ensuremath{\mathcal{K}}} 
\newcommand{\Lcal}{\ensuremath{\mathcal{L}}}
\newcommand{\Mcal}{\ensuremath{\mathcal{M}}}
\newcommand{\Ncal}{\ensuremath{\mathcal{N}}}
\newcommand{\Ocal}{\ensuremath{\mathcal{O}}}
\newcommand{\Pcal}{\ensuremath{\mathcal{P}}}
\newcommand{\Qcal}{\ensuremath{\mathcal{Q}}}
\newcommand{\Rcal}{\ensuremath{\mathcal{R}}}
\newcommand{\Scal}{\ensuremath{\mathcal{S}}}
\newcommand{\Tcal}{\ensuremath{\mathcal{T}}}
\newcommand{\Ucal}{\ensuremath{\mathcal{U}}}
\newcommand{\Vcal}{\ensuremath{\mathcal{V}}}
\newcommand{\Wcal}{\ensuremath{\mathcal{W}}}
\newcommand{\Xcal}{\ensuremath{\mathcal{X}}}
\newcommand{\Ycal}{\ensuremath{\mathcal{Y}}}
\newcommand{\Zcal}{\ensuremath{\mathcal{Z}}}

% Blackboard Letters
\newcommand{\Abb}{\ensuremath{\mathbb{A}}}
\renewcommand{\Bbb}{\ensuremath{\mathbb{B}}}
\newcommand{\Cbb}{\ensuremath{\mathbb{C}}}
\newcommand{\Dbb}{\ensuremath{\mathbb{D}}}
\newcommand{\Ebb}{\ensuremath{\mathbb{E}}}
\newcommand{\Fbb}{\ensuremath{\mathbb{F}}}
\newcommand{\Gbb}{\ensuremath{\mathbb{G}}}
\newcommand{\Hbb}{\ensuremath{\mathbb{H}}}
\newcommand{\Ibb}{\ensuremath{\mathbb{I}}} 
\newcommand{\Jbb}{\ensuremath{\mathbb{J}}} 
\newcommand{\Kbb}{\ensuremath{\mathbb{K}}} 
\newcommand{\Lbb}{\ensuremath{\mathbb{L}}}
\newcommand{\Mbb}{\ensuremath{\mathbb{M}}}
\newcommand{\Nbb}{\ensuremath{\mathbb{N}}}
\newcommand{\Obb}{\ensuremath{\mathbb{O}}}
\newcommand{\Pbb}{\ensuremath{\mathbb{P}}}
\newcommand{\Qbb}{\ensuremath{\mathbb{Q}}}
\newcommand{\Rbb}{\ensuremath{\mathbb{R}}}
\newcommand{\Sbb}{\ensuremath{\mathbb{S}}}
\newcommand{\Tbb}{\ensuremath{\mathbb{T}}}
\newcommand{\Ubb}{\ensuremath{\mathbb{U}}}
\newcommand{\Vbb}{\ensuremath{\mathbb{V}}}
\newcommand{\Wbb}{\ensuremath{\mathbb{W}}}
\newcommand{\Xbb}{\ensuremath{\mathbb{X}}}
\newcommand{\Ybb}{\ensuremath{\mathbb{Y}}}
\newcommand{\Zbb}{\ensuremath{\mathbb{Z}}}

% Sans Letters
\newcommand{\Asf}{\ensuremath{\mathsf{A}}}
\newcommand{\Bsf}{\ensuremath{\mathsf{B}}}
\newcommand{\Csf}{\ensuremath{\mathsf{C}}}
\newcommand{\Dsf}{\ensuremath{\mathsf{D}}}
\newcommand{\Esf}{\ensuremath{\mathsf{E}}}
\newcommand{\Fsf}{\ensuremath{\mathsf{F}}}
\newcommand{\Gsf}{\ensuremath{\mathsf{G}}}
\newcommand{\Hsf}{\ensuremath{\mathsf{H}}}
\newcommand{\Isf}{\ensuremath{\mathsf{I}}} 
\newcommand{\Jsf}{\ensuremath{\mathsf{J}}} 
\newcommand{\Ksf}{\ensuremath{\mathsf{K}}} 
\newcommand{\Lsf}{\ensuremath{\mathsf{L}}}
\newcommand{\Msf}{\ensuremath{\mathsf{M}}}
\newcommand{\Nsf}{\ensuremath{\mathsf{N}}}
\newcommand{\Osf}{\ensuremath{\mathsf{O}}}
\newcommand{\Psf}{\ensuremath{\mathsf{P}}}
\newcommand{\Qsf}{\ensuremath{\mathsf{Q}}}
\newcommand{\Rsf}{\ensuremath{\mathsf{R}}}
\newcommand{\Ssf}{\ensuremath{\mathsf{S}}}
\newcommand{\Tsf}{\ensuremath{\mathsf{T}}}
\newcommand{\Usf}{\ensuremath{\mathsf{U}}}
\newcommand{\Vsf}{\ensuremath{\mathsf{V}}}
\newcommand{\Wsf}{\ensuremath{\mathsf{W}}}
\newcommand{\Xsf}{\ensuremath{\mathsf{X}}}
\newcommand{\Ysf}{\ensuremath{\mathsf{Y}}}
\newcommand{\Zsf}{\ensuremath{\mathsf{Z}}}
\newcommand{\asf}{\ensuremath{\mathsf{a}}}
\newcommand{\bsf}{\ensuremath{\mathsf{b}}}
\newcommand{\csf}{\ensuremath{\mathsf{c}}}
\newcommand{\dsf}{\ensuremath{\mathsf{d}}}
\newcommand{\esf}{\ensuremath{\mathsf{e}}}
\newcommand{\fsf}{\ensuremath{\mathsf{f}}}
\newcommand{\gsf}{\ensuremath{\mathsf{g}}}
\newcommand{\hsf}{\ensuremath{\mathsf{h}}}
\newcommand{\isf}{\ensuremath{\mathsf{i}}} 
\newcommand{\jsf}{\ensuremath{\mathsf{j}}} 
\newcommand{\ksf}{\ensuremath{\mathsf{k}}} 
\newcommand{\lsf}{\ensuremath{\mathsf{l}}}
\newcommand{\msf}{\ensuremath{\mathsf{m}}}
\newcommand{\nsf}{\ensuremath{\mathsf{n}}}
\newcommand{\osf}{\ensuremath{\mathsf{o}}}
\newcommand{\psf}{\ensuremath{\mathsf{p}}}
\newcommand{\qsf}{\ensuremath{\mathsf{q}}}
\newcommand{\rsf}{\ensuremath{\mathsf{r}}}
\newcommand{\ssf}{\ensuremath{\mathsf{s}}}
\newcommand{\tsf}{\ensuremath{\mathsf{t}}}
\newcommand{\usf}{\ensuremath{\mathsf{u}}}
\newcommand{\vsf}{\ensuremath{\mathsf{v}}}
\newcommand{\wsf}{\ensuremath{\mathsf{w}}}
\newcommand{\xsf}{\ensuremath{\mathsf{x}}}
\newcommand{\ysf}{\ensuremath{\mathsf{y}}}
\newcommand{\zsf}{\ensuremath{\mathsf{z}}}

% Standard Theories
\newcommand{\InfSet}{\ensuremath{\mathsf{InfSet}}}
\newcommand{\DLO}{\ensuremath{\mathsf{DLO}}}
\newcommand{\RG}{\ensuremath{\mathsf{RG}}}

% Classification Theory commands
\newcommand{\NSOP}{\ensuremath{\mathsf{NSOP}}}
\newcommand{\NIP}{\ensuremath{\mathsf{NIP}}}

\newcommand{\Rsbd}{\R_\mathsf{sbd}}
\newcommand{\RCF}{\mathsf{RCF}}
\newcommand{\lin}{\mathsf{lin}}
\newcommand{\ind}{\mathsf{ind}}
\newcommand{\dom}{\ensuremath{\mathsf{dom}}}
\renewcommand{\dim}{\ensuremath{\mathsf{dim}}}
\newcommand{\cldim}{\cl\text{-}\dim}

\newcommand{\eqdef}{\stackrel{\mathclap{\normalfont\mbox{\tiny{\textsf{def}}}}}{=}}

\newcommand{\Sym}{\ensuremath{\mathsf{Sym}}}

\newcommand{\range}{\ensuremath{\mathsf{range}}}

\newcommand{\val}{\ensuremath{\mathsf{val}}}
\newcommand{\Sub}{\ensuremath{\mathsf{Sub}}}
\newcommand{\Var}{\ensuremath{\mathsf{Var}}}
\newcommand{\arity}{\mathsf{arity}}
\newcommand{\Const}{\mathsf{Const}}
\newcommand{\Rel}{\mathsf{Rel}}
\newcommand{\Fun}{\mathsf{Fun}}

\newcommand{\Free}{\mathsf{Free}}
\newcommand{\Bound}{\mathsf{Bound}}


%Theorems
\newtheorem{theorem}{Theorem}[subsection]
\newtheorem*{theorem*}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem*{proposition*}{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem*{corollary*}{Corollary}
\newtheorem*{fact}{Fact}
\newtheorem*{conjecture}{Conjecture}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem*{definition*}{Definition}
\newtheorem{exercise}[theorem]{Exercise}
\newtheorem{example}[theorem]{Example}


\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{claim}{Claim}
\newtheorem{poc}{Proof of Claim}

\newenvironment{claimproof}{%
	\begin{poc}
	}{%
	\hfill$\blacktriangleleft$
	\end{poc}
	}

\counterwithin*{claim}{theorem}
\counterwithin*{poc}{theorem}




\hypersetup{
  colorlinks   = true, %Colours links instead of ugly boxes
  urlcolor     = {blue!50!black}, %Colour for external hyperlinks
  linkcolor    = {blue!50!black}, %Colour of internal links
  citecolor   = {red!50!black} %Colour of citations
}



\title{}
\author{}
\date{}
