exmath.sty

% This file defines the `exmath` package
\NeedsTeXFormat{LaTeX2e}
\ProvidesPackage{exmath}[2020/11/12 Extended math commands]

\DeclareOption{english}{%
    \def\@theorem@mark{Theorem}
    \def\@definition@mark{Definition}
    \def\@proposition@mark{Proposition}
    \def\@lemma@mark{Lemma}
    \def\@corollary@mark{Corollary}
    \def\@proof@mark{Proof}
    \def\@example@mark{Example}
    \def\@note@mark{Note}
    \def\@notation@mark{Notation}
    \def\@remark@mark{Remark}
    \def\@claim@mark{Afirmación}
}

\DeclareOption{spanish}{%
    \def\@theorem@mark{Teorema}
    \def\@definition@mark{Definición}
    \def\@proposition@mark{Proposición}
    \def\@lemma@mark{Lema}
    \def\@corollary@mark{Corolario}
    \def\@proof@mark{Demostración}
    \def\@example@mark{Ejemplo}
    \def\@note@mark{Nota}
    \def\@notation@mark{Notación}
    \def\@remark@mark{Observación}
    \def\@claim@mark{Claim}
}

\DeclareOption{usechapters}{\def\@usechapters{true}}
\DeclareOption{nochapters}{\def\@usechapters{false}}

\DeclareOption{theoremnoframe}{\def\@theoremstyle{noframe}}
\DeclareOption{theoremframed}{\def\@theoremstyle{framed}}
\DeclareOption{theoremhighlight}{\def\@theoremstyle{highlight}}

\DeclareOption{boldvec}{\def\@vectorstyle{bold}}
\DeclareOption{arrowvec}{\def\@vectorstyle{arrow}}
\DeclareOption{boldarrowvec}{\def\@vectorstyle{boldarrow}}

\ExecuteOptions{english, usechapters, theoremframed}
\ProcessOptions\relax

\RequirePackage{amsmath, amssymb, amsthm, amsfonts} % Meth stuff
\RequirePackage{systeme} % Systems of equations
\RequirePackage{cancel} % Cancel an expression with an arrow
\RequirePackage{physics} % Cool things used in physics
\RequirePackage{siunitx} % Units
\RequirePackage{bm} % Bold math
\RequirePackage{bbm} % More bold math
\RequirePackage[osf, sc]{mathpazo} % Fixes stuff
\RequirePackage{mathtools} % Fixes more stuff
\RequirePackage[framemethod=TikZ]{mdframed} % Frames for theorems, proofs, etc.
\RequirePackage{xargs}
\RequirePackage{imakeidx} % Create indexes
\RequirePackage{tocloft} % Tables of content
\RequirePackage{ifthen} % If then else structure
\RequirePackage{xspace} % Fix space after some commands
\RequirePackage{etoolbox} % Fix stuff
\RequirePackage{xstring} % Macros to work with strings
%\RequirePackage{fdsymbol} % Better QED symbol. This breaks the code.

% ----- General corrections and redefinitions
\DeclareSymbolFont{bbold}{U}{bbold}{m}{n}
\DeclareSymbolFontAlphabet{\mathbbold}{bbold}

\let\svlim\lim\def\lim{\svlim\limits} % Put x\to\infty below \lim
\let\epsilon\varepsilon
\let\oldstylenums\oldstyle
\let\oldensuremath\ensuremath
\renewcommand{\ensuremath}[1]{\oldensuremath{#1}\xspace}

\newcommand{\dm}[1]{\(#1\xspace\)}

% Easily define new math operators
\newcommand\argp[1]{\argopen(#1\argclose)}
\newcommand\mop[2]{\operatorname{#1}\argp{#2}}

\let\oldexists\exists\renewcommand\exists{\oldexists\,}
%\let\oldforall\forall\renewcommand\forall{\oldforall\,}


% ----- Operators and quantifiers
\newcommand{\bydef}{\triangleq} % By definition
\newcommand{\qeq}{\overset{?}{=}} % Equal?
\newcommand{\st}{\; | \;} % Such that
%\newcommand{\st*}{\; : \;} % Alternative such that
\newcommand{\uexists}{\exists\, !\;} % Exists and it's unique
\newcommand{\conj}[1]{\overline{#1}} % Conjugate
\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor} % Floor
\newcommand{\rela}{\mathcal{R}} % Relation
%\renewcommand{\abs}[1]{\left\vert #1 \right\vert} % Absolute value
\newcommand{\invers}[1]{#1 ^{-1}} % Inverse
\renewcommand\neg{\lnot} % Logic negation
% Cooler greater (lower) or equal to symbols
\let\leq\leqslant
\let\nleq\nleqslant
\let\geq\geqslant
\let\ngeq\ngeqslant

% ----- Logical operators
\renewcommand\implies{\Longrightarrow} % Implies
\newcommand\impliesdef{\overset{\textrm{def}}{\implies}} % Implies by definition
\newcommand\nimplies{\not\implies} % Not implies
\newcommand\qimplies{\quad\implies\quad} % Implies, but with extra space
\renewcommand\impliedby{\Longleftarrow} % Implied by
\newcommand\impliedbydef{\overset{\textrm{def}}{\impliedby}} % Implied by definition
\newcommand\nimpliedby{\not\impliedby} % Not implied by
\newcommand\qimpliedby{\quad\impliedby\quad} % Implied by, but with extra space
\renewcommand\iff{\Longleftrightarrow} % If and only if / Equivalent
\newcommand\iffdef{\overset{\textrm{def}}{\iff}} % Equivalent by definition
\renewcommand\land{\wedge} % Logical AND
\renewcommand\lor{\vee} % Logical OR

% ----- Discrete math
\newcommand\N{\ensuremath{\mathbb{N}}\xspace} % Naturals
\newcommand\Z{\ensuremath{\mathbb{Z}}\xspace} % Integers
\newcommand\Q{\ensuremath{\mathbb{Q}}\xspace} % Rationals
\newcommand\R{\ensuremath{\mathbb{R}}\xspace} % Reals
\newcommand\C{\ensuremath{\mathbb{C}}\xspace} % Complex
\renewcommand\O{\ensuremath{\varnothing}\xspace} % Empty set

\newcommand\set[1]{\{ #1 \}} % Set
\newcommand\appl[3]{#1 : #2\longrightarrow #3} % Application
\newcommand\stdf{\ensuremath{\appl{f}{X}{Y}}\xspace} % Standard function
\newcommand\vappl[3]{\vec{#1} : #2\longrightarrow #3} % Vector-valued function
\newcommand\dom[1]{\mathcal{D}\argp{#1}} % Cooler domain
\newcommand\image[1]{\mathcal{R}\argp{#1}} % Range / Image 

\newcommand\parts[1]{\mathcal{P}\left( #1 \right)} % Partitions of a set / Power set
\newcommand\uset{\mathcal{U}} % Universal set
\newcommand\SP[1]{\ensuremath{\mathcal{P}}\argp{#1}} % Super set
\newcommand\minuszero{\setminus\set{0}} % Set minus 0
\newcommand\cset[1]{\overline{#1}} % Complementary set
\newcommand\ccset[1]{{#1}^C} % Complementary set
\newcommand\comb[2]{\left( \begin{matrix} #1 \\ #2 \end{matrix} \right)} % Binomial coefficient

\newcommand{\Rtn}{\R^n} % n-dimensional Euclidean space
\newcommand{\field}{\ensuremath{\mathbb{F}}\xspace} % Field

% ----- Vectors and vector analysis macros
\renewcommand\vec[1]{\mathbf{#1}}
\newcommand{\cvec}[1]{\begin{bmatrix} #1 \end{bmatrix}}
\newcommand{\rvec}[1]{\begin{bmatrix} #1 \end{bmatrix}^T}
\newcommand{\tvec}[1]{\left( #1 \right)^T}
\newcommand\uvec[1]{\vec{\hat{#1}}}
\newcommand\stdvf{\appl{f}{\R^n}{\R^m}} % Standard vector-valued function
\newcommand\gmvf{f\left(x_1, \ldots, x_n\right)} % f(x_1, ..., x_n)
\newcommand\fgraph[1]{\mathcal{G}\argopen(#1\argclose)} % Graph of a function
\newcommand\levelset[2]{L_{#1}\argopen(#2\argclose)} % Level set of value #1 for the function #2
\newcommand\ihat{\uvec{i}}
\newcommand\jhat{\uvec{j}}
\newcommand\khat{\uvec{k}}
\newcommand\Jac[1]{\vec{J}\argopen(#1\argclose)} % Jacobian
\newcommand\Hes[1]{\mathbf{H}\argopen(#1\argclose)} % Hessian

% Calculus / Analysis
% \newcommand{\deriv}[2]{\frac{\dif #1}{\dif #2}} % 1: f; 2: var
\newcommand\partialdv[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\liminft}[1]{\lim_{{#1}\rightarrow\infty}}
\newcommand{\mylim}[3]{\lim_{#1 \rightarrow #2} #3} % 1: vec; 2: to; 3: var;
\DeclareMathOperator{\rot}{rot}
\DeclareMathOperator{\diver}{div}

% ----- Topology
\newcommand\oball[2]{B_{#1}\argp{#2}} % Open ball of radius #1 centered at #2
\newcommand\cball[2]{\overline{B_{#1}}\argp{#2}} % Closed ball of radius #1 centered at #2
\newcommand\interior[1]{\mop{int}{#1}} % Interior of a set
\newcommand\exterior[1]{\mop{ext}{#1}} % Exterior of a set
\newcommand\frontier[1]{\partial #1} % Frontier / boundary of a set
\newcommand\closure[1]{\overline{#1}} % Closure of a set

% ----- Linear algebra macros
\let\kernel\ker\renewcommand\ker[1]{\mop{ker}{#1}} % Kernel
\newcommand\nullity[1]{\mop{nullity}{#1}} % Nullity
\newcommand\nullsp[1]{\mop{Null}{#1}} % Null space
\renewcommand\rank[1]{\mop{rank}{#1}} % Rank
\newcommand\rg[1]{\mop{rg}{#1}} % Range
\newcommand\range[1]{\mop{range}{#1}} % Range
\newcommand\ensp[1]{\mop{End}{#1}} % Space of endomorphisms
\let\dimension\dim\renewcommand\dim[1]{\mop{dim}{#1}} % Dimension
\newcommand\gen[1]{\langle #1 \rangle} % Generated by
\newcommand\stdlt{\ensuremath{\appl{T}{V}{W}}\xspace} % Standard linear transformation
\newcommand\gea[1]{\underset{\longrightarrow}{\scriptsize #1}} % Gaussian elimination arrow
\newcommand\mtrans[1]{{#1}^{T}} % Transposed matrix
\newcommand\ovproj[2]{\operatorname{proj}_{#1}\argp{#2}} % Orthogonal vector projection of #2 over #1


% BACKUP DEFINITIONS FOR TESTING
%\newcommand\gen[1]{\langle #1 \rangle} % Generated by
%\newcommand\stdlt{\ensuremath{\appl{T}{V}{W}}\xspace} % Standard linear transformation
%\let\kernel\ker % Redefine old kernel macro
%\renewcommand\ker[1]{\operatorname{ker}\left( #1 \right)} % Kernel
%\newcommand\nullity[1]{\operatorname{nullity}\left( #1 \right)} % Nullity
%\newcommand\nullsp[1]{\operatorname{null}\left( #1 \right)} % Null space
%\renewcommand\rank[1]{\operatorname{rank}\left( #1 \right)} % Rank
%\newcommand\rg[1]{\operatorname{rg}\left( #1 \right)} % Range
%\newcommand\range[1]{\operatorname{range}\left( #1 \right)} % Range
%\renewcommand\endosp[1]{\operatorname{End}\left( #1 \right)} % Space of endomorphisms
%\newcommand\gea[1]{\underset{\longrightarrow}{\scriptsize #1}} % Gaussian elimination arrow
%\newcommand{\mtrans}[1]{{#1}^{T}}
%\let\dimension\dim % Refine old dimension macro
%\renewcommand{\dim}[1]{\operatorname{dim}\left( #1 \right)} % Dimension

%\DeclareMathOperator{\rg}{Rg}
%\DeclareMathOperator{\kernel}{Ker}
%\DeclareMathOperator{\rankm}{Rank}
%\DeclareMathOperator{\image}{Im}
%\newcommand{\nullity}[1]{\mop{nullity}\left( #1 \right)}
%\renewcommand{\rank}[1]{\mop{rank}\left( #1 \right)}
\newcommand\bigzero{\text{\Large O}}
\newcommand\innerp[1]{\langle #1 \rangle} % Inner product
\newcommand\vnorm[1]{\left\Vert #1 \right\Vert} % Norm

% ----- Miscellanea
\newcommand\wtf{(\textrm{wtf}?)}
\newcommand\brackets[1]{\left[ #1 \right]}

% ----- Other
\newcommand{\eqreason}[2][=]{\underset{\substack{\vspace{6pt} \uparrow  \\ \mathrlap{\wrapreasontext{#2}}}}{#1}}
\newcommand{\eqreasonup}[2][=]{\overset{\substack{\mathrlap{\wrapreasontext{#2}} \\ \vspace{4pt} \downarrow}}{#1}}

% ----- Theorem-like environments
\newcommand{\IS}{}
\def\toggle@IS@print{\renewcommand{\IS}{ }}
\def\toggle@IS@index{\renewcommand{\IS}{!}}

\ifthenelse{\equal{\@usechapters}{false}}{%
    \newcounter{thm}[section]\setcounter{thm}{0}
    \newcounter{def}[section]\setcounter{def}{0}
    \renewcommand{\thethm}{\thesection.\arabic{thm}}
    \renewcommand{\thedef}{\thesection.\arabic{def}}
    \numberwithin{thm}{section}
    \numberwithin{def}{section}
    \numberwithin{equation}{section}
    \message{Using section numbering for theorems and definitions}
}{%
    \newcounter{thm}[chapter] \setcounter{thm}{0}
    \newcounter{def}[chapter] \setcounter{def}{0}
    \renewcommand{\thethm}{\thechapter.\arabic{thm}}
    \renewcommand{\thedef}{\thechapter.\arabic{def}}
    \numberwithin{thm}{chapter}
    \numberwithin{def}{chapter}
    \numberwithin{equation}{chapter}
    \message{Using chapter numbering for theorems and definitions}
}

% https://tex.stackexchange.com/a/17551
\newtheoremstyle{defi}{\topsep}{\topsep}{}{0pt}{}{}{5pt plus 1pt minus 1pt}{\fontfamily{cmss}\selectfont\textbf{\thmname{#1}\thmnumber{ #2}}\thmnote{ (\textbf{#3})}}
\newtheoremstyle{thm}{\topsep}{\topsep}{\itshape}{0pt}{}{}{5pt plus 1pt minus 1pt}{\fontfamily{cmss}\selectfont\textbf{\thmname{#1}\thmnumber{ #2}}\thmnote{ (\textbf{#3})}}}

\theoremstyle{remark}
%\newtheorem*{note}{\@note@mark}
\newtheorem*{notation}{\@notation@mark}
\newtheorem*{claim}{\@claim@mark}

% Define a List of Theorems. ithm is a special counter which
% is incremented when a new item is added to this list, that
% is saved in the master.ithmc file. To this list is only added
% the theorems, lemmas and propositions that have a special name.
%\newcommand{\listtheoremname}{List of Theorems}
%\newlistof[\ifthenelse{\equal{\@usechapters}{false}}{section}{chapter}]{ithm}{ithmc}{\listtheoremname}
%\newcommand{\listoftheorems}{\listofithm}

% Helper macro to increment the ithm counter and add the entry to the list.
%\newcommand{\ithm}[1]{%
%	\refstepcounter{ithm}
%	\addcontentsline{ithmc}{ithm}{\protect\numberline{\theithm}#1}\par
%}

% This macro let us use references with names in commands such as
% \nref{ref}, which will write in the document something like
% "Theorem name of the theorem (1.15)".
\newcommand{\set@labelname}[1]{%
    \ifthenelse{\equal{#1}{}}{}{%
        \toggle@IS@print\protected@edef\@currentlabelname{#1}%
        \renewcommand{\label@hook}{}
    }
}

\def\@restore@itemlabels{%
    \renewcommand{\labelenumi}{\alph{enumi}. }
    \renewcommand{\theenumi}{\alph{enumi}}
}

\def\@set@theorem@itemlabels{%
    \renewcommand{\labelenumi}{\alph{enumi}. }
    \renewcommand{\theenumi}{\thethm\ - (\alph{enumi})}
}

\mdfsetup{skipabove=1em, skipbelow=0em}

% mdframed styles for the different frames
\mdfdefinestyle{framed}{nobreak=true}
\mdfdefinestyle{highlight}{nobreak=true, linewidth=2pt, topline=false, rightline=false, bottomline=false}
\mdfdefinestyle{noframe}{%
    nobreak=true, hidealllines=true, innertopmargin=0pt, innerleftmargin=0pt, innerrightmargin=0pt, innerbottommargin=0pt
}

\theoremstyle{thm}

% {theorem} environment definition.
\newmdtheoremenv[stlye=\@theoremstyle]{itheorem}[thm]{\@theorem@mark}
\newenvironment{theorem}[1][]{%
    \vspace{2pt}
    \set@labelname{#1}
    \@set@theorem@itemlabels
    \begin{itheorem}[#1]
}{\end{itheorem}\@restore@itemlabels}

\theoremstyle{defi}

% {defn} environment definition
\newmdtheoremenv[style=\@theoremstyle]{definition}[def]{\@definition@mark}
\newenvironment{defn}[1][]{\set@labelname{#1}\begin{definition}[#1]}{\end{definition}\@restore@itemlabels}

% {prop} environment definition.
\newmdtheoremenv[style=\@theoremstyle]{iprop}[def]{\@proposition@mark}
\newenvironment{prop}[1][]{\set@labelname{#1}\begin{iprop}[#1]}{\end{iprop}\@restore@itemlabels}

% {lemma} environment definition.
\newmdtheoremenv[style=\@theoremstyle]{ilemma}[def]{\@lemma@mark}
\newenvironment{lemma}[1][]{\set@labelname{#1}\begin{ilemma}[#1]}{\end{ilemma}\@restore@itemlabels}

% {coro} environment definition.
\newmdtheoremenv[style=\@theoremstyle]{coro}[def]{\@corollary@mark}

% This macro changes the equation numbering from (\theequation)
% to (\thesection.\theequation). The equation counter is reseted
% when the section counter is incremented.
% Note: If using a LaTeX distribution older than April 2018 this
% macro won't work unless the chngctr package is used.
%
% https://tex.stackexchange.com/questions/207532/reset-equation-numbering-after-each-section
% \counterwithin{equation}{chapter}

\renewcommand{\qedsymbol}{$\blacksquare$} % \medblacksquare with fdsymbol

\newenvironment{example}{{\fontfamily{cmss}\selectfont\par\noindent\textbf{\@example@mark}}~}{\null\hfill$\diamond$}    % Examples
\newenvironment{remark}{{\fontfamily{cmss}\selectfont\par\noindent\textbf{\@remark@mark}}~}{}      % Remarks
\newenvironment{note}{{\fontfamily{cmss}\selectfont\par\noindent\textbf{\@note@mark}}~}{}          % Notes
\renewenvironment{proof}{{\fontfamily{cmss}\selectfont\par\noindent\textbf{\@proof@mark}}~}{\qed}  % Proofs

\newcounter{problem}\setcounter{problem}{0}
\renewcommand{\theproblem}{\arabic{problem}}

\newenvironment{problem}[1]{%
\refstepcounter{problem}\vspace{5pt}\par\noindent\textrm{\textcolor{blue!60!black}{H2-P\theproblem.} \textbf{#1}}}{}

% Fix some weird spacing on boxes created by the mdframed package
% http://tex.stackexchange.com/a/22119
\makeatletter
\def\thm@space@setup{%
    \thm@preskip=\parskip \thm@postskip=0pt
}

% ----- Fancy references
\def\ref@prefix@definition{def:}
\def\ref@name@definition{Definition}
\def\ref@lname@definition{\MakeLowercase{\ref@name@definition}}
\newcommand\exrefdefname[1]{\def\ref@name@definition{#1}}
\newcommand\exrefdefprefix[1]{\def\ref@prefix@definition{#1}}

\def\ref@prefix@theorem{thm:}
\def\ref@name@theorem{Theorem}
\def\ref@lname@theorem{\MakeLowercase{\ref@name@theorem}}
\newcommand\exrefthmname[1]{\def\ref@name@theorem{#1}}
\newcommand\exrefthmprefix[1]{\def\ref@prefix@theorem{#1}}

\def\ref@prefix@proposition{prop:}
\def\ref@name@proposition{Proposition}
\def\ref@lname@proposition{\MakeLowercase{\ref@name@proposition}}
\newcommand\exrefpropname[1]{\def\ref@name@proposition{#1}}
\newcommand\exrefpropprefix[1]{\def\ref@prefix@proposition{#1}}

\def\ref@prefix@lemma{lem:}
\def\ref@name@lemma{Lemma}
\def\ref@lname@lemma{\MakeLowercase{\ref@name@lemma}}
\newcommand\exreflemname[1]{\def\ref@name@lemma{#1}}
\newcommand\exreflemprefix[1]{\def\ref@prefix@proposition{#1}}

\def\ref@prefix@corollary{coro:}
\def\ref@name@corollary{Corollary}
\def\ref@lname@corollary{\MakeLowercase{\ref@name@corollary}}
\newcommand\exrefcoroname[1]{\def\ref@name@corollary{#1}}
\newcommand\exrefcoroprefix[1]{\def\ref@prefix@corollary{#1}}

\newcommand\getrefname[1]{\getrefbykeydefault{#1}{name}{??}}
\newcommand\nref[1]{\refused{#1}\hyperref[#1]{\getrefname{#1}}~(\ref{#1})}
\newcommand\nlref[1]{\MakeLowercase{\nref{#1}}}
\newcommand\reflink[2]{\hyperref[#2]{#1}~\ref{#2}}
\newcommand\fullreflink[2]{\hyperref[#2]{#1}~\ref{#2}~\hyperref[#2]{(\getrefname{#2})}~}

\newcommand\eref[1]{%
    \IfSubStr{#1}{\ref@prefix@definition}{\reflink{\ref@lname@definition}{#1}}{}%
    \IfSubStr{#1}{\ref@prefix@theorem}{\reflink{\ref@lname@theorem}{#1}}{}%
    \IfSubStr{#1}{\ref@prefix@proposition}{\reflink{\ref@lname@proposition}{#1}}{}%
    \IfSubStr{#1}{\ref@prefix@lemma}{\reflink{\ref@lname@lemma}{#1}}{}%
    \IfSubStr{#1}{\ref@prefix@corollary}{\reflink{\ref@lname@corollary}{#1}}{}%
}

\newcommand\Eref[1]{%
    \IfSubStr{#1}{\ref@prefix@definition}{\reflink{\ref@name@definition}{#1}}{}%
    \IfSubStr{#1}{\ref@prefix@theorem}{\reflink{\ref@name@theorem}{#1}}{}%
    \IfSubStr{#1}{\ref@prefix@proposition}{\reflink{\ref@name@proposition}{#1}}{}%
    \IfSubStr{#1}{\ref@prefix@lemma}{\reflink{\ref@name@lemma}{#1}}{}%
    \IfSubStr{#1}{\ref@prefix@corollary}{\reflink{\ref@name@corollary}{#1}}{}%
}

\newcommand\fulleref[1]{%
    \IfSubStr{#1}{\ref@prefix@definition}{\fullreflink{\ref@lname@definition}{#1}}{}%
    \IfSubStr{#1}{\ref@prefix@theorem}{\fullreflink{\ref@lname@theorem}{#1}}{}%
    \IfSubStr{#1}{\ref@prefix@proposition}{\fullreflink{\ref@lname@proposition}{#1}}{}%
    \IfSubStr{#1}{\ref@prefix@lemma}{\fullreflink{\ref@lname@lemma}{#1}}{}%
    \IfSubStr{#1}{\ref@prefix@corollary}{\fullreflink{\ref@lname@corollary}{#1}}{}%
}

\newcommand\Fulleref[1]{%
    \IfSubStr{#1}{\ref@prefix@definition}{\fullreflink{\ref@name@definition}{#1}}{}%
    \IfSubStr{#1}{\ref@prefix@theorem}{\fullreflink{\ref@name@theorem}{#1}}{}%
    \IfSubStr{#1}{\ref@prefix@proposition}{\fullreflink{\ref@name@proposition}{#1}}{}%
    \IfSubStr{#1}{\ref@prefix@lemma}{\fullreflink{\ref@name@lemma}{#1}}{}%
    \IfSubStr{#1}{\ref@prefix@corollary}{\fullreflink{\ref@name@corollary}{#1}}{}%
}