notation.tex

% Theorem styles

% \theoremstyle{plain}
% \newtheorem{thm}{Theorem}

% \theoremstyle{definition}
% \newtheorem{prop}[thm]{Proposition}

% \theoremstyle{remark}
% \newtheorem{claim}[thm]{Claim}

% \theoremstyle{remark}
% \newtheorem{corollary}[thm]{Corollary}

% \theoremstyle{remark}
% \newtheorem{rem}[thm]{Remark}

% \theoremstyle{definition}
% \newtheorem{defn}[thm]{Definition}

% Notation

% General
\newcommand{\defeq}{\coloneqq}
\newcommand{\cat}[1]{\mathbf{#1}}
\newcommand{\equalizer}[2]{Eq(#1, #2)}
\newcommand{\powerset}[1]{\mathcal{P}(#1)}
\newcommand{\denote}[1]{\llbracket #1 \rrbracket}
\newcommand{\sem}[1]{\denote{#1}} % Small abbreviation
\newcommand{\ev}{\operatorname{ev}}
\newcommand{\id}{\operatorname{id}}
\newcommand{\doubleplus}{\ensuremath{\mathbin{+\mkern-10mu+}}}
\newcommand{\pair}[2]{\left\langle {#1}, {#2} \right\rangle}
\newcommand{\into}[0]{\rightarrow}

% Change structures
% Generic "thing in a circle" operator
\makeatletter
\newcommand\cplussym
{
  \mathpalette\@incircbin
}
\newcommand\@incircbin[2]
{
  \mathbin
  {
    \ooalign{\hidewidth$#1#2$\hidewidth\crcr$#1\bigcirc$}%
  }
}
\makeatother

\newcommand{\cplus}{\oplus}
\newcommand{\cpluss}{\boxplus}
\newcommand{\cplusss}{\odot}
\newcommand{\cplusvee}{\cplussym{\vee}}
\newcommand{\cminus}{\ominus}

\newcommand{\splus}{\cdot}
\newcommand{\mzero}{\mathbf{0}}

\newcommand{\cstruct}[3]{(#1,#2,#3)}
\newcommand{\cstr}[1]{\hat{#1}}
\newcommand{\changes}[1]{\Delta #1}
\newcommand{\change}[1]{\delta #1}

\newcommand{\discrete}{\emptyset}

\newcommand{\derive}[1]{#1'}
\newcommand{\supderive}[1]{#1_\uparrow}
\newcommand{\supderiveM}[1]{#1_{\uparrow\uparrow}}
\newcommand{\subderive}[1]{#1_\downarrow}
\newcommand{\subderiveM}[1]{#1_{\downarrow\downarrow}}

\newcommand{\monotoneDerive}[1]{#1'^{M}}

\newcommand{\NN}{\mathbb{N}}
\newcommand{\ZZ}{\mathbb{Z}}
  
\newcommand{\upsem}[1]{\sem{\updiff{#1}}}
\newcommand{\downsem}[1]{\sem{\downdiff{#1}}}
\newcommand{\bothsem}[1]{\sem{\bothdiff{#1}}}

\newcommand{\semR}[0]{\mathcal{R}}
\newcommand{\diffR}[0]{\change{\semR}}

\newcommand{\kernel}{\sim}
\newcommand{\kernelOrder}{\leq_\kernel}
\newcommand{\pointwiseOrder}{\leq_{pt}}

% Some kinds of function spaces
\newcommand{\ra}{\rightarrow}
\newcommand{\Ra}{\Rightarrow}
\newcommand{\ptfunc}{\Rightarrow_{pt}}

% Algebra
\newcommand{\reach}{\operatorname{Reach}}
\newcommand{\direct}{\operatorname{Direct}}
\newcommand{\superpose}{{\bowtie}}
\newcommand{\curry}[1]{\Lambda{#1}}
\newcommand{\exponential}[2]{{#1} \Rightarrow {#2}}

% Orders
\newcommand{\reachOrder}{\leq_R}
\newcommand{\changeOrder}{\leq_\Delta}
\newcommand{\fineOrder}{\leq_D}
\newcommand{\minusOrder}{\leq_\cminus}

% Lattices
\newcommand{\twist}{\cplus_{\bowtie}}
\newcommand{\disjointTimes}{\bowtie}
\newcommand{\updiff}{\Delta}
\newcommand{\downdiff}{\nabla}

% Fixpoints
\newcommand{\fixpoint}{\mathbf{fix}}
\newcommand{\lfp}{\mathbf{lfp}}
\newcommand{\adjust}{\mathbf{adjust}}
\newcommand{\iter}{\mathbf{iter}}
\newcommand{\nextiter}{\mathbf{recur}}

% Datalog
\newcommand{\Formula}{\mathrm{Formula}}
\newcommand{\Rel}{\cat{Rel}}
\newcommand{\universalRel}{\mathcal{U}}
\newcommand{\consq}{\mathcal{I}}