\usepackage{todonotes}
\usepackage{booktabs}



\graphicspath{{Figures/}}

\newtheorem{theo}{Theorem}[section]
\newtheorem{prop}[theo]{Proposition}
%\newtheorem{proper}[theo]{Property}
%\newtheorem{prope}{Property}
\newtheorem{lemma}[theo]{Lemma}
\newtheorem{coro}[theo]{Corollary}

\theoremstyle{definition}
\newtheorem{defn}[theo]{Definition}

\theoremstyle{remark}
\newtheorem{rem}[theo]{Remark}

\newcommand{\EF}[1]{\todo[inline,size=\small,color=green!40!white]{EF: #1}}
\newcommand{\ef}[1]{\todo[size=\scriptsize,color=green!40!white]{EF: #1}}
\newcommand{\WF}[1]{\todo[inline,size=\small,color=blue!30!white]{WF: #1}}
\newcommand{\wf}[1]{\todo[size=\scriptsize,color=blue!30!white]{WF: #1}}
\newcommand{\PN}[1]{\todo[inline,size=\small,color=red!30!white]{PN: #1}}
\newcommand{\pn}[1]{\todo[size=\scriptsize,color=red!30!white]{PN: #1}}

%%%%%%%% utilities

\newcommand{\cacher}[1]{}

\newcommand{\fig}[3]{\begin{figure}[h!]\begin{center}\includegraphics[#1]{#2}\end{center}\caption{#3}\label{fig:#2}\end{figure}}

\newcommand{\tdef}[1]{\textcolor{blue}{\emph{#1}}}

\def\ophi{\overline{\phi}}
\def\oPhi{\overline{\Phi}}


%%%%%%%% Families of objects

% bilabeled graphs
\newcommand{\bilabel}{\mathcal{G}}
% bilabeled graphs with crossing-free meandric representation
\newcommand{\bilabelpl}{\mathcal{G}^{\mathrm{pl}}}
% bicolored blossoming trees
\newcommand{\blossoming}{\mathcal{B}}
% bicolored blossoming trees with one marked edge
\newcommand{\blossomedge}{\mathcal{E}}
% Tamari lattice
\newcommand{\Tam}{\mathrm{Tam}}
% Tamari intervals
\newcommand{\tamint}{\mathcal{I}}
% Minimal bilabeled trees
\newcommand{\bilabelmin}{\mathcal{M}}
\newcommand{\mdiag}{\mathcal{M\;\!\!D}}
\newcommand{\mtree}{\mathcal{M\;\!\!T}}
% see prop:count_inf_modern, local use?
\def\cR{\mathcal{R}}
% interval posets
\newcommand{\intervalposet}{\mathcal{P}}
% binary trees
\newcommand{\bintree}{\mathcal{T}}
% see prop:count_all_intervals, local use?
\def\cS{\mathcal{S}}
\def\cvS{\widehat{\cS}}
\def\vVec{\mathrm{Vec}}


\newcommand{\pair}{\mathcal{X}}

%%%%%%%% operations, statistics and bijections

% canopy of a binary tree
\newcommand{\can}{\operatorname{can}}
% mirror or a binary tree, extended to pairs and intervals
\newcommand{\mir}{\operatorname{mir}}
% $(T, T') \mapsto ((T, \epsilon), (\epsilon, T'))$ for $T, T'$ binary trees
\newcommand{\rise}{\operatorname{rise}}
% Dual of blossoming trees, obtained by exchanging colors
\newcommand{\dual}{\operatorname{dual}}
% bracket vectors and dual bracket vectors
\newcommand{\bvec}{\mathbf{V}}
\newcommand{\dualbvec}{\bvec'} % or \hat{\bvec}
% bijections between bilabeled graphs and blossoming trees
\newcommand{\bijbilblom}{\gamma}
%\newcommand{\bijblombil}{\overline{\gamma}}
\newcommand{\bijblombil}{\delta}
% bijections between Tamari intervals and blossoming trees (the main bijection)
\newcommand{\inttoblom}{\Phi}
\newcommand{\blomtoint}{\Psi} % really needed? Used only once (commented out)
% number of nodes on the (left or right) branch ending at each leaf
\newcommand{\dne}{d_{\nearrow}}
\newcommand{\dsw}{d_{\swarrow}}
% bijection from pairs of binary trees to bilabeled graphs
\newcommand{\treestobil}{\phi}
\newcommand{\biltotrees}{\psi}
% Mysterious involution on Tamari intervals
\newcommand{\invint}{\delta}
\newcommand{\refl}{\operatorname{refl}} % reflection of blossoming trees

\def\nne{n_{\nearrow}}
\def\nsw{n_{\swarrow}}

\def\dne{\mathrm{Deg}_{\nearrow}}

%%%%%%%% shorthands

% clockwise and counterclockwise
\newcommand{\cw}{clockwise}
\newcommand{\ccw}{counterclockwise}

% canopy types
\newcommand{\canzz}{{0 \brack 0}}
\newcommand{\canoz}{{1 \brack 0}}
\newcommand{\canoo}{{1 \brack 1}}
\newcommand{\canzo}{{0 \brack 1}}

\def\hG{\hat{G}}

\def\Pred{\tilde{P}}

\def\oB{\overline{B}}

\def\oC{\overline{C}}

\def\ni{\noindent}
\def\ds{\displaystyle}

\def\bbeta{\overline{\beta}}

\def\hC{\hat{C}}

\newcommand{\preceqb}{\preceq_{\mathrm{b}}}
\newcommand{\preceqr}{\preceq_{\mathrm{r}}}

