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commands.tex
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\newcommand\qedhere{\square}
\renewcommand{\epsilon}{\varepsilon}
\renewcommand{\phi}{\varphi}
\newcommand{\lemmas}{lemmata\xspace}
\newcommand{\Lemmas}{Lemmata\xspace}
\newcommand{\gam}{\gamma}
\newcommand{\fix}[1]{\textcolor{red}{#1}\xspace}
\newcommand{\nt}{number theory\xspace}
\newcommand{\Nt}{Number theory\xspace}
\newcommand{\NT}{Number Theory\xspace}
\newcommand{\ntc}{number theoretic\xspace}
\newcommand{\Ntc}{Number theoretic\xspace}
\newcommand{\om}{o-minimal\xspace}
\newcommand{\Om}{O-minimal\xspace}
\newcommand{\omy}{o-minimality\xspace}
\newcommand{\Omy}{O-minimality\xspace}
\newcommand{\homeomic}{homeomorphic\xspace}
\newcommand{\homeom}{homeomorphism\xspace}
\newcommand{\cd}{cell decomposition\xspace}
\newcommand{\cds}{cell decompositions\xspace}
\newcommand{\CD}{\textrm{Cell Decomposition}\xspace}
\newcommand{\CDs}{Cell Decompositions\xspace}
\newcommand{\scd}{smooth cell decomposition\xspace}
\newcommand{\scds}{smooth cell decompositions\xspace}
\newcommand{\SCD}{Smooth Cell Decomposition\xspace}
\newcommand{\SCDs}{Smooth Cell Decompositions\xspace}
\newcommand{\sa}{semialgebraic\xspace}
\newcommand{\Sa}{Semialgebraic\xspace}
\newcommand{\saty}{semialgebraicity\xspace}
\newcommand{\Saty}{Semialgebraicity\xspace}
\newcommand{\sacd}{semialgebraic cell decomposition\xspace}
\newcommand{\sacds}{semialgebraic cell decompositions\xspace}
\newcommand{\Sacd}{Semialgebraic cell decomposition\xspace}
\newcommand{\Sacds}{Semialgebraic cell decompositions}\xspace
\newcommand{\SaCd}{Semialgebraic Cell decomposition\xspace}
\newcommand{\SaCds}{Semialgebraic Cell decompositions\xspace}
\newcommand{\SaCD}{Semialgebraic Cell Decomposition\xspace}
\newcommand{\SaCDs}{Semialgebraic Cell Decompositions\xspace}
\newcommand{\cmptble}{compatible\xspace}
\newcommand{\cmptblety}{compatibility\xspace}
\newcommand{\cont}{continuous\xspace}
\newcommand{\contty}{continuity\xspace}
\newcommand{\cnctd}{connected\xspace}
\newcommand{\cnctdns}{connectedness\xspace}
\newcommand{\defnb}{definable\xspace}
\newcommand{\defnbly}{definably\xspace}
\newcommand{\defnbty}{definability\xspace}
\newcommand{\defnblycnctd}{definably connected\xspace}
\newcommand{\defnbcnctdns}{definable connectedness\xspace}
\newcommand{\Mlt}{(M, \ <)}
\newcommand{\Mltp}{(M, \ <, \ \hdots)}
\newcommand{\Mltarithplus}{(M, \ <, \ +, \ \cdot, \ 0, \ 1, \ \hdots)}
\newcommand{\M}{\mathcal{M}}
\newcommand{\Lstr}{\mathcal{L}\mathrm{-Structure}\xspace}
\newcommand{\Lstrs}{\mathcal{L}\mathrm{-Structures}\xspace}
\newcommand{\inv}[1]{#1^{-1}}
\newcommand{\bc}{\textrm{basic cell}}
\newcommand{\Bc}{\textrm{Basic cell}}
\newcommand{\Rtilde}{\tilde{\R}}
\newcommand{\Rtildefull}{\tilde{\R} = (\overline{\R}, \ \hdots)}
\newcommand{\dfblty}{\textrm{differentiability}}
\newcommand{\contdfblty}{\cont \dfblty}
\newcommand{\Crcell}[1][r]{C^{#1}\textrm{-cell}}
\newcommand{\Crcells}[1][r]{C^{#1}\textrm{-cells}}
\newcommand{\pbh}{points of bounded height\xspace}
\newcommand{\pbd}{points of bounded degree\xspace}
\newcommand{\bound}[1][H]{c #1^{\eps}}
\newcommand{\lvrv}[1]{\left\vert \ #1 \ \right\vert}
\newcommand{\pw}{\textrm{Pila-Wilkie}\xspace}
\newcommand{\pwt}{\pw\textrm{theorem}\xspace}
\newcommand{\pwT}{\pw\textrm{Theorem}\xspace}
\newcommand{\mt}{monotonicity theorem\xspace}
\newcommand{\Mt}{Monotonicity theorem\xspace}
\newcommand{\MT}{Monotonicity Theorem\xspace}
\newcommand{\sm}{smooth monotonicity\xspace}
\newcommand{\Sm}{Smooth monotonicity\xspace}
\newcommand{\SM}{Smooth Monotonicity\xspace}
\newcommand{\smt}{smooth monotonicity theorem\xspace}
\newcommand{\SMt}{Smooth Monotonicity theorem\xspace}
\newcommand{\SMT}{Smooth Monotonicity Theorem\xspace}
\newcommand{\cdt}{\cd\textrm{theorem}\xspace}
\newcommand{\Cdt}{\Cd\textrm{theorem}\xspace}
\newcommand{\CDt}{\CD\textrm{theorem}\xspace}
\newcommand{\CDT}{\CD\textrm{Theorem}\xspace}
\newcommand{\ft}{\textrm{finiteness theorem}\xspace}
\newcommand{\Ft}{\textrm{Finiteness theorem}\xspace}
\newcommand{\FT}{\textrm{Finiteness Theorem}\xspace}
\newcommand{\pft}{\textrm{planar finiteness theorem}\xspace}
\newcommand{\pFt}{\textrm{planar Finiteness theorem}\xspace}
\newcommand{\PFt}{\textrm{Planar Finiteness theorem}\xspace}
\newcommand{\PFT}{\textrm{Planar Finiteness Theorem}\xspace}
\newcommand{\bpft}{\textrm{(planar) finiteness theorem}\xspace}
\newcommand{\bpFt}{\textrm{(planar) Finiteness theorem}\xspace}
\newcommand{\bPFt}{\textrm{(Planar) Finiteness theorem}\xspace}
\newcommand{\bPFT}{\textrm{(Planar) Finiteness Theorem}\xspace}
\newcommand{\eps}{\varepsilon}
\newcommand{\cnr}{c_{n, \ r}}
\newcommand{\deriv}[2][]{\cfrac{d #1}{d #2} \:}
\newcommand{\pderiv}[2][]{\cfrac{\partial #1}{\partial #2} \:}
\newcommand{\XeH}[2][H]{X(#2, \ #1)}
\newcommand{\AeH}[3][X]{#1(#3, \ #2)}
\newcommand{\XtraeH}{\AeH[ {\Xtr[X_a]} ]{H}{e}}
\newcommand{\XtraeHsize}{\card{ \ \AeH[ {\Xtr[X_a]} ]{H}{e} \ }}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\Qbar}{\overline{\Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\zeroset}{\{ 0 \}}
\newcommand{\fancyD}{\mathfrak{D}}
%For tyos
\newcommand{\francyD}{\mathfrak{D}}
\newcommand{\Zgeq}[1]{\mathbb{Z}_{\geq #1}}
\newcommand{\Zleq}[1]{\mathbb{Z}_{\leq #1}}
\newcommand{\Jp}{J^{\prime}}
\newcommand{\Xprime}[1][X]{#1^{\prime}}
\newcommand{\pri}[1]{#1^{\prime}}
\newcommand{\dpri}[1]{#1^{\prime \prime}}
\newcommand{\Xtr}[1][X]{#1^{\textrm{tr}}}
\newcommand{\Xtrans}[1][X]{\Xtr[#1]}
\newcommand{\Xalg}[1][X]{#1^{\textrm{alg}}}
\newcommand{\XQH}[1][H]{X(\Q, \ #1)}
\newcommand{\XtrQH}[1][H]{\Xtr[X](\Q, \ #1)}
\newcommand{\XalgQH}[1][H]{\Xalg[X](\Q, \ #1)}
\newcommand{\MmMn}{M^m \times M^n}
\newcommand{\XMn}[2][X]{#1 \subseteq M^{#2}}
\newcommand{\XRn}[2][X]{#1 \subseteq \R^{#2}}
\newcommand{\CIn}[2][C]{#1 \subseteq I^{#2}}
\newcommand{\XRtn}[2][X]{#1 \subseteq \Rtilde^{#2}}
\newcommand{\aMn}[2][a]{#1 \in M^{#2}}
\newcommand{\AM}[1][A]{#1 \subseteq M}
\newcommand{\In}[1][n]{(\textrm{I})_{#1}}
\newcommand{\IIn}[1][n]{(\textrm{II})_{#1}}
\newcommand{\isupj}{i^{(j)}}
\newcommand{\Minf}{M \cup \{ \pm \infty \}}
\newcommand{\ionein}{i_1, \ \hdots, \ i_n}
\newcommand{\aoneak}{a_1, \ \hdots, \ a_k}
\newcommand{\vonevm}[2]{#1_1, \ \hdots, \ #1_{#2}}
\newcommand{\vmvn}[3]{#1_{#2}, \ \hdots, \ #1_{#3}}
\newcommand{\vmvnsup}[4][,]{#2^{#3}#1 \ \hdots #1 \ #2^{#4}}
\newcommand{\vmvnsupbr}[4][,]{\vmvnsup[#1]{#2}{(#3)}{(#4)}}
\newcommand{\izerocell}{(\ionein, 0)\textrm{-cell}}
\newcommand{\ionecell}{(\ionein, 1)\textrm{-cell}}
\newcommand{\incell}{(\ionein)\textrm{-cell}}
\newcommand{\incells}{(\ionein)\textrm{-cells}}
\newcommand{\vmcell}[2]{(\vonevm{#1}{#2})\textrm{-cell}}
\newcommand{\vmncell}[3]{(\vmvn{#1}{#2}{#3})\textrm{-cell}}
\newcommand{\XsubseteqMn}[1][n]{X \subseteq M^{#1}}
\newcommand{\inhb}{non-empty\xspace}
% \newcommand{\inhb}{inhabited\xspace}
\newcommand{\inj}{\textrm{injective}\xspace}
\newcommand{\injtion}{\textrm{injection}\xspace}
\newcommand{\injtvty}{\textrm{injectivity}\xspace}
\newcommand{\surj}{\textrm{surjective}\xspace}
\newcommand{\surjtion}{\textrm{surjection}\xspace}
\newcommand{\surjtvty}{\textrm{surjectivity}\xspace}
\newcommand{\bij}{\textrm{bijective}\xspace}
\newcommand{\bijtion}{\textrm{bijection}\xspace}
\newcommand{\bijtions}{\textrm{bijections}\xspace}
\newcommand{\XsubsetMn}[1][n]{X \subset M^{#1}}
\newcommand{\image}[1]{\operatorname{image}\left(#1\right)}
\newcommand{\graph}[1]{\operatorname{graph}\left(#1\right)}
\newcommand{\dom}[1]{\operatorname{dom}\left(#1\right)}
\newcommand{\domain}[1]{\dom{#1}}
\newcommand{\codom}[1]{\operatorname{codom}\left(#1\right)}
\newcommand{\inter}[1]{\operatorname{int}\left(#1\right)}
\newcommand{\cl}[1]{\operatorname{cl}\left(#1\right)}
\newcommand{\dcl}[1]{\operatorname{dcl}\left(#1\right)}
\newcommand{\dclop}{\operatorname{dcl}}
\newcommand{\acl}[1]{\operatorname{acl}\left(#1\right)}
\newcommand{\aclop}{\operatorname{acl}}
\renewcommand{\dim}[1]{\operatorname{dim}\left(#1\right)}
\newcommand{\defnbdim}[1]{\operatorname{dim_{defn}}\left(#1\right)}
\newcommand{\algdim}[1]{\operatorname{dim_{alg}}\left(#1\right)}
\newcommand{\fr}[1]{\operatorname{fr}\left(#1\right)}
\newcommand{\funcdom}[3]{#1 \colon #2 \to #3}
\newcommand{\funcdef}[3]{#1 \colon #2 \mapsto #3}
\newcommand{\funcrestr}[2]{#1 \vert_{#2}}
\newcommand{\esdefnb}{\mathrel{\emptyset \textrm{-\defnb}}}
\newcommand{\Adefnb}[1][A]{#1\textrm{-\defnb}}
\newcommand{\defnbcl}{\defnb\textrm{closure}\xspace}
\newcommand{\pregeom}{\textrm{pregeometry}\xspace}
\newcommand{\pregeoms}{\textrm{pregeometries}\xspace}
\newcommand{\Pregeom}{\textrm{Pregeometry}\xspace}
\newcommand{\Pregeoms}{\textrm{Pregeometries}\xspace}
\newcommand{\param}{\textrm{parameterization}\xspace}
\newcommand{\params}{\textrm{parameterizations}\xspace}
\newcommand{\Param}{\textrm{parameterization}\xspace}
\newcommand{\Params}{\textrm{parameterizations}\xspace}
\newcommand{\cellr}{\textrm{cellular}\xspace}
\newcommand{\Cellr}{\textrm{Cellular}\xspace}
\newcommand{\cellrparam}[1][r]{\textrm{cellular} #1\textrm{-\param}\xspace}
\newcommand{\cellrparams}[1][r]{\textrm{cellular} #1\textrm{-\params}\xspace}
\newcommand{\Cellrparam}[1][r]{\textrm{Cellular} #1\textrm{-\param}\xspace}
\newcommand{\Cellrparams}[1][r]{\textrm{Cellular} #1\textrm{-\params}\xspace}
\newcommand{\comp}[2]{#1 \circ #2}
%Sets
\DeclarePairedDelimiterX{\Set}[2]\{\}{%
\, #1 \;\delimsize\colon\; #2 \,
}
%cardinality
\DeclarePairedDelimiter\card{\vert}{\vert}
%abs
\DeclarePairedDelimiter\abs{\vert}{\vert}
%norm
\DeclarePairedDelimiter\normdouble{\vert\vert}{\vert\vert}
\DeclarePairedDelimiter\twonormdouble{\vert\vert}{\vert\vert_2}
\DeclarePairedDelimiter\norm{\vert}{\vert}
\DeclarePairedDelimiter\twonorm{\vert}{\vert_2}
\newcommand{\rnorm}[2][r]{\norm{#2}_{#1}}
%subproof
\newenvironment{subproof}[1][\proofname]{%
\renewcommand{\qedsymbol}{$\blacksquare$}%
\begin{proof}[#1]%
}{%
\end{proof}%
}