783 lines
30 KiB
TeX
Vendored
783 lines
30 KiB
TeX
Vendored
%%%% MACROS FOR NOTATION %%%%
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% Use these for any notation where there are multiple options.
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% Slightly modified by Floris van Doorn
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%%% Notes and exercise sections
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\makeatletter
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\newcommand{\sectionNotes}{\phantomsection\section*{Notes}\addcontentsline{toc}{section}{Notes}\markright{\textsc{\@chapapp{} \thechapter{} Notes}}}
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\newcommand{\sectionExercises}[1]{\phantomsection\section*{Exercises}\addcontentsline{toc}{section}{Exercises}\markright{\textsc{\@chapapp{} \thechapter{} Exercises}}}
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\makeatother
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%%% Definitional equality (used infix) %%%
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\newcommand{\jdeq}{\equiv} % An equality judgment
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\let\judgeq\jdeq
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%\newcommand{\defeq}{\coloneqq} % An equality currently being defined
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\newcommand{\defeq}{\vcentcolon\equiv} % A judgmental equality currently being defined
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%%% Term being defined
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\newcommand{\define}[1]{\textbf{#1}}
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%%% Vec (for example)
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\newcommand{\Vect}{\ensuremath{\mathsf{Vec}}}
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\newcommand{\Fin}{\ensuremath{\mathsf{Fin}}}
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\newcommand{\fmax}{\ensuremath{\mathsf{fmax}}}
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\newcommand{\seq}[1]{\langle #1\rangle}
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%%% Dependent products %%%
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\def\prdsym{\textstyle\prod}
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%% Call the macro like \prd{x,y:A}{p:x=y} with any number of
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%% arguments. Make sure that whatever comes *after* the call doesn't
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%% begin with an open-brace, or it will be parsed as another argument.
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\makeatletter
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% Currently the macro is configured to produce
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% {\textstyle\prod}(x:A) \; {\textstyle\prod}(y:B),\
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% in display-math mode, and
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% \prod_{(x:A)} \prod_{y:B}
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% in text-math mode.
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\def\prd#1{\@ifnextchar\bgroup{\prd@parens{#1}}{\@ifnextchar\sm{\prd@parens{#1}\@eatsm}{\prd@noparens{#1}}}}
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\def\prd@parens#1{\@ifnextchar\bgroup%
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{\mathchoice{\@dprd{#1}}{\@tprd{#1}}{\@tprd{#1}}{\@tprd{#1}}\prd@parens}%
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{\@ifnextchar\sm%
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{\mathchoice{\@dprd{#1}}{\@tprd{#1}}{\@tprd{#1}}{\@tprd{#1}}\@eatsm}%
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{\mathchoice{\@dprd{#1}}{\@tprd{#1}}{\@tprd{#1}}{\@tprd{#1}}}}}
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\def\@eatsm\sm{\sm@parens}
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\def\prd@noparens#1{\mathchoice{\@dprd@noparens{#1}}{\@tprd{#1}}{\@tprd{#1}}{\@tprd{#1}}}
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% Helper macros for three styles
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\def\lprd#1{\@ifnextchar\bgroup{\@lprd{#1}\lprd}{\@@lprd{#1}}}
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\def\@lprd#1{\mathchoice{{\textstyle\prod}}{\prod}{\prod}{\prod}({\textstyle #1})\;}
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\def\@@lprd#1{\mathchoice{{\textstyle\prod}}{\prod}{\prod}{\prod}({\textstyle #1}),\ }
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\def\tprd#1{\@tprd{#1}\@ifnextchar\bgroup{\tprd}{}}
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\def\@tprd#1{\mathchoice{{\textstyle\prod_{(#1)}}}{\prod_{(#1)}}{\prod_{(#1)}}{\prod_{(#1)}}}
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\def\dprd#1{\@dprd{#1}\@ifnextchar\bgroup{\dprd}{}}
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\def\@dprd#1{\prod_{(#1)}\,}
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\def\@dprd@noparens#1{\prod_{#1}\,}
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%%% Lambda abstractions.
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% Each variable being abstracted over is a separate argument. If
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% there is more than one such argument, they *must* be enclosed in
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% braces. Arguments can be untyped, as in \lam{x}{y}, or typed with a
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% colon, as in \lam{x:A}{y:B}. In the latter case, the colons are
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% automatically noticed and (with current implementation) the space
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% around the colon is reduced. You can even give more than one variable
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% the same type, as in \lam{x,y:A}.
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\def\lam#1{{\lambda}\@lamarg#1:\@endlamarg\@ifnextchar\bgroup{.\,\lam}{.\,}}
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\def\@lamarg#1:#2\@endlamarg{\if\relax\detokenize{#2}\relax #1\else\@lamvar{\@lameatcolon#2},#1\@endlamvar\fi}
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\def\@lamvar#1,#2\@endlamvar{(#2\,{:}\,#1)}
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% \def\@lamvar#1,#2{{#2}^{#1}\@ifnextchar,{.\,{\lambda}\@lamvar{#1}}{\let\@endlamvar\relax}}
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\def\@lameatcolon#1:{#1}
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\let\lamt\lam
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% This version silently eats any typing annotation.
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\def\lamu#1{{\lambda}\@lamuarg#1:\@endlamuarg\@ifnextchar\bgroup{.\,\lamu}{.\,}}
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\def\@lamuarg#1:#2\@endlamuarg{#1}
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%%% Dependent products written with \forall, in the same style
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\def\fall#1{\forall (#1)\@ifnextchar\bgroup{.\,\fall}{.\,}}
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%%% Existential quantifier %%%
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\def\exis#1{\exists (#1)\@ifnextchar\bgroup{.\,\exis}{.\,}}
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%%% Dependent sums %%%
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\def\smsym{\textstyle\sum}
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% Use in the same way as \prd
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\def\sm#1{\@ifnextchar\bgroup{\sm@parens{#1}}{\@ifnextchar\prd{\sm@parens{#1}\@eatprd}{\sm@noparens{#1}}}}
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\def\sm@parens#1{\@ifnextchar\bgroup%
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{\mathchoice{\@dsm{#1}}{\@tsm{#1}}{\@tsm{#1}}{\@tsm{#1}}\sm@parens}%
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{\@ifnextchar\prd%
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{\mathchoice{\@dsm{#1}}{\@tsm{#1}}{\@tsm{#1}}{\@tsm{#1}}\@eatprd}%
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{\mathchoice{\@dsm{#1}}{\@tsm{#1}}{\@tsm{#1}}{\@tsm{#1}}}}}
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\def\@eatprd\prd{\prd@parens}
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\def\sm@noparens#1{\mathchoice{\@dsm@noparens{#1}}{\@tsm{#1}}{\@tsm{#1}}{\@tsm{#1}}}
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\def\lsm#1{\@ifnextchar\bgroup{\@lsm{#1}\lsm}{\@@lsm{#1}}}
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\def\@lsm#1{\mathchoice{{\textstyle\sum}}{\sum}{\sum}{\sum}({\textstyle #1})\;}
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\def\@@lsm#1{\mathchoice{{\textstyle\sum}}{\sum}{\sum}{\sum}({\textstyle #1}),\ }
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\def\tsm#1{\@tsm{#1}\@ifnextchar\bgroup{\tsm}{}}
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\def\@tsm#1{\mathchoice{{\textstyle\sum_{(#1)}}}{\sum_{(#1)}}{\sum_{(#1)}}{\sum_{(#1)}}}
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\def\dsm#1{\@dsm{#1}\@ifnextchar\bgroup{\dsm}{}}
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\def\@dsm#1{\sum_{(#1)}\,}
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\def\@dsm@noparens#1{\sum_{#1}\,}
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%%% W-types
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\def\wtypesym{{\mathsf{W}}}
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\def\wtype#1{\@ifnextchar\bgroup%
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{\mathchoice{\@twtype{#1}}{\@twtype{#1}}{\@twtype{#1}}{\@twtype{#1}}\wtype}%
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{\mathchoice{\@twtype{#1}}{\@twtype{#1}}{\@twtype{#1}}{\@twtype{#1}}}}
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\def\lwtype#1{\@ifnextchar\bgroup{\@lwtype{#1}\lwtype}{\@@lwtype{#1}}}
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\def\@lwtype#1{\mathchoice{{\textstyle\mathsf{W}}}{\mathsf{W}}{\mathsf{W}}{\mathsf{W}}({\textstyle #1})\;}
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\def\@@lwtype#1{\mathchoice{{\textstyle\mathsf{W}}}{\mathsf{W}}{\mathsf{W}}{\mathsf{W}}({\textstyle #1}),\ }
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\def\twtype#1{\@twtype{#1}\@ifnextchar\bgroup{\twtype}{}}
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\def\@twtype#1{\mathchoice{{\textstyle\mathsf{W}_{(#1)}}}{\mathsf{W}_{(#1)}}{\mathsf{W}_{(#1)}}{\mathsf{W}_{(#1)}}}
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\def\dwtype#1{\@dwtype{#1}\@ifnextchar\bgroup{\dwtype}{}}
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\def\@dwtype#1{\mathsf{W}_{(#1)}\,}
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\newcommand{\suppsym}{{\mathsf{sup}}}
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\newcommand{\supp}{\ensuremath\suppsym\xspace}
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\def\wtypeh#1{\@ifnextchar\bgroup%
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{\mathchoice{\@lwtypeh{#1}}{\@twtypeh{#1}}{\@twtypeh{#1}}{\@twtypeh{#1}}\wtypeh}%
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{\mathchoice{\@@lwtypeh{#1}}{\@twtypeh{#1}}{\@twtypeh{#1}}{\@twtypeh{#1}}}}
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\def\lwtypeh#1{\@ifnextchar\bgroup{\@lwtypeh{#1}\lwtypeh}{\@@lwtypeh{#1}}}
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\def\@lwtypeh#1{\mathchoice{{\textstyle\mathsf{W}^h}}{\mathsf{W}^h}{\mathsf{W}^h}{\mathsf{W}^h}({\textstyle #1})\;}
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\def\@@lwtypeh#1{\mathchoice{{\textstyle\mathsf{W}^h}}{\mathsf{W}^h}{\mathsf{W}^h}{\mathsf{W}^h}({\textstyle #1}),\ }
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\def\twtypeh#1{\@twtypeh{#1}\@ifnextchar\bgroup{\twtypeh}{}}
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\def\@twtypeh#1{\mathchoice{{\textstyle\mathsf{W}^h_{(#1)}}}{\mathsf{W}^h_{(#1)}}{\mathsf{W}^h_{(#1)}}{\mathsf{W}^h_{(#1)}}}
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\def\dwtypeh#1{\@dwtypeh{#1}\@ifnextchar\bgroup{\dwtypeh}{}}
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\def\@dwtypeh#1{\mathsf{W}^h_{(#1)}\,}
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\makeatother
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% Other notations related to dependent sums
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\let\setof\Set % from package 'braket', write \setof{ x:A | P(x) }.
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% \newcommand{\pair}{\ensuremath{\mathsf{pair}}\xspace}
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\newcommand{\tup}[2]{(#1,#2)}
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\newcommand{\proj}[1]{\ensuremath{\mathsf{pr}_{#1}}\xspace}
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\newcommand{\fst}{\ensuremath{\proj1}\xspace}
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\newcommand{\snd}{\ensuremath{\proj2}\xspace}
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\newcommand{\ac}{\ensuremath{\mathsf{ac}}\xspace} % not needed in symbol index
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\newcommand{\un}{\ensuremath{\mathsf{upun}}\xspace} % not needed in symbol index, uniqueness principle for unit type
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%%% recursor and induction
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\newcommand{\rec}[1]{\mathsf{rec}_{#1}}
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\newcommand{\ind}[1]{\mathsf{ind}_{#1}}
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\newcommand{\indid}[1]{\ind{=_{#1}}} % (Martin-Lof) path induction principle for identity types
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\newcommand{\indidb}[1]{\ind{=_{#1}}'} % (Paulin-Mohring) based path induction principle for identity types
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%%% the uniqueness principle for product types, formerly called surjective pairing and named \spr:
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\newcommand{\uppt}{\ensuremath{\mathsf{uppt}}\xspace}
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% Paths in pairs
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\newcommand{\pairpath}{\ensuremath{\mathsf{pair}^{\mathord{=}}}\xspace}
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% \newcommand{\projpath}[1]{\proj{#1}^{\mathord{=}}}
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\newcommand{\projpath}[1]{\ensuremath{\apfunc{\proj{#1}}}\xspace}
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%%% For quotients %%%
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%\newcommand{\pairr}[1]{{\langle #1\rangle}}
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\newcommand{\pairr}[1]{{\mathopen{}(#1)\mathclose{}}}
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\newcommand{\Pairr}[1]{{\mathopen{}\left(#1\right)\mathclose{}}}
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% \newcommand{\type}{\ensuremath{\mathsf{Type}}} % this command is overridden below, so it's commented out
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\newcommand{\im}{\ensuremath{\mathsf{im}}} % the image
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%%% 2D path operations
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\newcommand{\leftwhisker}{\mathbin{{\ct}_{\ell}}}
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\newcommand{\rightwhisker}{\mathbin{{\ct}_{r}}}
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\newcommand{\hct}{\star}
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%%% modalities %%%
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\newcommand{\modal}{\ensuremath{\ocircle}}
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\let\reflect\modal
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\newcommand{\modaltype}{\ensuremath{\type_\modal}}
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% \newcommand{\ism}[1]{\ensuremath{\mathsf{is}_{#1}}}
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% \newcommand{\ismodal}{\ism{\modal}}
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% \newcommand{\existsmodal}{\ensuremath{{\exists}_{\modal}}}
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% \newcommand{\existsmodalunique}{\ensuremath{{\exists!}_{\modal}}}
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% \newcommand{\modalfunc}{\textsf{\modal-fun}}
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% \newcommand{\Ecirc}{\ensuremath{\mathsf{E}_\modal}}
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% \newcommand{\Mcirc}{\ensuremath{\mathsf{M}_\modal}}
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\newcommand{\mreturn}{\ensuremath{\eta}}
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\let\project\mreturn
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%\newcommand{\mbind}[1]{\ensuremath{\hat{#1}}}
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\newcommand{\ext}{\mathsf{ext}}
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%\newcommand{\mmap}[1]{\ensuremath{\bar{#1}}}
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%\newcommand{\mjoin}{\ensuremath{\mreturn^{-1}}}
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% Subuniverse
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\renewcommand{\P}{\ensuremath{\type_{P}}\xspace}
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%%% Localizations
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% \newcommand{\islocal}[1]{\ensuremath{\mathsf{islocal}_{#1}}\xspace}
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% \newcommand{\loc}[1]{\ensuremath{\mathcal{L}_{#1}}\xspace}
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%%% Identity types %%%
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\newcommand{\idsym}{{=}}
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% \newcommand{\id}[3][]{\ensuremath{#2 =_{#1} #3}\xspace}
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\newcommand{\idtype}[3][]{\ensuremath{\mathsf{Id}_{#1}(#2,#3)}\xspace}
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\newcommand{\idtypevar}[1]{\ensuremath{\mathsf{Id}_{#1}}\xspace}
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% A propositional equality currently being defined
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\newcommand{\defid}{\coloneqq}
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%%% Dependent paths
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\newcommand{\dpath}[4]{#3 =^{#1}_{#2} #4}
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%%% singleton
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% \newcommand{\sgl}{\ensuremath{\mathsf{sgl}}\xspace}
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% \newcommand{\sctr}{\ensuremath{\mathsf{sctr}}\xspace}
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%%% Reflexivity terms %%%
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% \newcommand{\reflsym}{{\mathsf{refl}}}
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% \newcommand{\refl}[1]{\ensuremath{\mathsf{refl}_{#1}}\xspace}
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%%% Path concatenation (used infix, in diagrammatic order) %%%
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\newcommand{\ct}{%
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\mathchoice{\mathbin{\raisebox{0.5ex}{$\displaystyle\centerdot$}}}%
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{\mathbin{\raisebox{0.5ex}{$\centerdot$}}}%
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{\mathbin{\raisebox{0.25ex}{$\scriptstyle\,\centerdot\,$}}}%
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{\mathbin{\raisebox{0.1ex}{$\scriptscriptstyle\,\centerdot\,$}}}
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}
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%%% Path reversal %%%
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\newcommand{\opp}[1]{\mathord{{#1}^{-1}}}
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\let\rev\opp
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%%% Transport (covariant) %%%
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\newcommand{\trans}[2]{\ensuremath{{#1}_{*}\mathopen{}\left({#2}\right)\mathclose{}}\xspace}
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\let\Trans\trans
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%\newcommand{\Trans}[2]{\ensuremath{{#1}_{*}\left({#2}\right)}\xspace}
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\newcommand{\transf}[1]{\ensuremath{{#1}_{*}}\xspace} % Without argument
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%\newcommand{\transport}[2]{\ensuremath{\mathsf{transport}_{*} \: {#2}\xspace}}
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\newcommand{\transfib}[3]{\ensuremath{\mathsf{transport}^{#1}(#2,#3)\xspace}}
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\newcommand{\Transfib}[3]{\ensuremath{\mathsf{transport}^{#1}\Big(#2,\, #3\Big)\xspace}}
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\newcommand{\transfibf}[1]{\ensuremath{\mathsf{transport}^{#1}\xspace}}
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%%% 2D transport
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\newcommand{\transtwo}[2]{\ensuremath{\mathsf{transport}^2\mathopen{}\left({#1},{#2}\right)\mathclose{}}\xspace}
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%%% Constant transport
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\newcommand{\transconst}[3]{\ensuremath{\mathsf{transportconst}}^{#1}_{#2}(#3)\xspace}
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\newcommand{\transconstf}{\ensuremath{\mathsf{transportconst}}\xspace}
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%%% Map on paths %%%
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\newcommand{\mapfunc}[1]{\ensuremath{\mathsf{ap}_{#1}}\xspace} % Without argument
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\newcommand{\map}[2]{\ensuremath{{#1}\mathopen{}\left({#2}\right)\mathclose{}}\xspace}
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\let\Ap\map
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%\newcommand{\Ap}[2]{\ensuremath{{#1}\left({#2}\right)}\xspace}
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\newcommand{\mapdepfunc}[1]{\ensuremath{\mathsf{apd}_{#1}}\xspace} % Without argument
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% \newcommand{\mapdep}[2]{\ensuremath{{#1}\llparenthesis{#2}\rrparenthesis}\xspace}
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\newcommand{\mapdep}[2]{\ensuremath{\mapdepfunc{#1}\mathopen{}\left(#2\right)\mathclose{}}\xspace}
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\let\apfunc\mapfunc
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\let\ap\map
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\let\apdfunc\mapdepfunc
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\let\apd\mapdep
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%%% 2D map on paths
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\newcommand{\aptwofunc}[1]{\ensuremath{\mathsf{ap}^2_{#1}}\xspace}
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\newcommand{\aptwo}[2]{\ensuremath{\aptwofunc{#1}\mathopen{}\left({#2}\right)\mathclose{}}\xspace}
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\newcommand{\apdtwofunc}[1]{\ensuremath{\mathsf{apd}^2_{#1}}\xspace}
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\newcommand{\apdtwo}[2]{\ensuremath{\apdtwofunc{#1}\mathopen{}\left(#2\right)\mathclose{}}\xspace}
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%%% Identity functions %%%
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\newcommand{\idfunc}[1][]{\ensuremath{\mathsf{id}_{#1}}\xspace}
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%%% Homotopies (written infix) %%%
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\newcommand{\htpy}{\sim}
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%%% Other meanings of \sim
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\newcommand{\bisim}{\sim} % bisimulation
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\newcommand{\eqr}{\sim} % an equivalence relation
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%%% Equivalence types %%%
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\newcommand{\eqv}[2]{\ensuremath{#1 \simeq #2}\xspace}
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\newcommand{\eqvspaced}[2]{\ensuremath{#1 \;\simeq\; #2}\xspace}
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\newcommand{\eqvsym}{\simeq} % infix symbol
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\newcommand{\texteqv}[2]{\ensuremath{\mathsf{Equiv}(#1,#2)}\xspace}
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\newcommand{\isequiv}{\ensuremath{\mathsf{isequiv}}}
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\newcommand{\qinv}{\ensuremath{\mathsf{qinv}}}
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\newcommand{\ishae}{\ensuremath{\mathsf{ishae}}}
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\newcommand{\linv}{\ensuremath{\mathsf{linv}}}
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\newcommand{\rinv}{\ensuremath{\mathsf{rinv}}}
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\newcommand{\biinv}{\ensuremath{\mathsf{biinv}}}
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\newcommand{\lcoh}[3]{\mathsf{lcoh}_{#1}(#2,#3)}
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\newcommand{\rcoh}[3]{\mathsf{rcoh}_{#1}(#2,#3)}
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\newcommand{\hfib}[2]{{\mathsf{fib}}_{#1}(#2)}
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%%% Map on total spaces %%%
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\newcommand{\total}[1]{\ensuremath{\mathsf{total}(#1)}}
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%%% Universe types %%%
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%\newcommand{\type}{\ensuremath{\mathsf{Type}}\xspace}
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\newcommand{\UU}{\ensuremath{\mathcal{U}}\xspace}
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\let\bbU\UU
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\let\type\UU
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% Universes of truncated types
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\newcommand{\typele}[1]{\ensuremath{{#1}\text-\mathsf{Type}}\xspace}
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\newcommand{\typeleU}[1]{\ensuremath{{#1}\text-\mathsf{Type}_\UU}\xspace}
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\newcommand{\typelep}[1]{\ensuremath{{(#1)}\text-\mathsf{Type}}\xspace}
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\newcommand{\typelepU}[1]{\ensuremath{{(#1)}\text-\mathsf{Type}_\UU}\xspace}
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\let\ntype\typele
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\let\ntypeU\typeleU
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\let\ntypep\typelep
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\let\ntypepU\typelepU
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\newcommand{\set}{\ensuremath{\mathsf{Set}}\xspace}
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\newcommand{\setU}{\ensuremath{\mathsf{Set}_\UU}\xspace}
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\newcommand{\prop}{\ensuremath{\mathsf{Prop}}\xspace}
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\newcommand{\propU}{\ensuremath{\mathsf{Prop}_\UU}\xspace}
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%Pointed types
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\newcommand{\pointed}[1]{\ensuremath{#1_\bullet}}
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%%% Ordinals and cardinals
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\newcommand{\card}{\ensuremath{\mathsf{Card}}\xspace}
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\newcommand{\ord}{\ensuremath{\mathsf{Ord}}\xspace}
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\newcommand{\ordsl}[2]{{#1}_{/#2}}
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%%% Univalence
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\newcommand{\ua}{\ensuremath{\mathsf{ua}}\xspace} % the inverse of idtoeqv
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\newcommand{\idtoeqv}{\ensuremath{\mathsf{idtoeqv}}\xspace}
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\newcommand{\univalence}{\ensuremath{\mathsf{univalence}}\xspace} % the full axiom
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%%% Truncation levels
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\newcommand{\iscontr}{\ensuremath{\mathsf{isContr}}}
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\newcommand{\contr}{\ensuremath{\mathsf{contr}}} % The path to the center of contraction
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\newcommand{\isset}{\ensuremath{\mathsf{isSet}}}
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\newcommand{\isprop}{\ensuremath{\mathsf{isProp}}}
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% h-propositions
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% \newcommand{\anhprop}{a mere proposition\xspace}
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% \newcommand{\hprops}{mere propositions\xspace}
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%%% Homotopy fibers %%%
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%\newcommand{\hfiber}[2]{\ensuremath{\mathsf{hFiber}(#1,#2)}\xspace}
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\let\hfiber\hfib
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%%% Bracket/squash/truncation types %%%
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% \newcommand{\brck}[1]{\textsf{mere}(#1)}
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% \newcommand{\Brck}[1]{\textsf{mere}\Big(#1\Big)}
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% \newcommand{\trunc}[2]{\tau_{#1}(#2)}
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% \newcommand{\Trunc}[2]{\tau_{#1}\Big(#2\Big)}
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% \newcommand{\truncf}[1]{\tau_{#1}}
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%\newcommand{\trunc}[2]{\Vert #2\Vert_{#1}}
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\newcommand{\trunc}[2]{\mathopen{}\left\Vert #2\right\Vert_{#1}\mathclose{}}
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\newcommand{\ttrunc}[2]{\bigl\Vert #2\bigr\Vert_{#1}}
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\newcommand{\Trunc}[2]{\Bigl\Vert #2\Bigr\Vert_{#1}}
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\newcommand{\truncf}[1]{\Vert \blank \Vert_{#1}}
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\newcommand{\tproj}[3][]{\mathopen{}\left|#3\right|_{#2}^{#1}\mathclose{}}
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\newcommand{\tprojf}[2][]{|\blank|_{#2}^{#1}}
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\def\pizero{\trunc0}
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%\newcommand{\brck}[1]{\trunc{-1}{#1}}
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%\newcommand{\Brck}[1]{\Trunc{-1}{#1}}
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%\newcommand{\bproj}[1]{\tproj{-1}{#1}}
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%\newcommand{\bprojf}{\tprojf{-1}}
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\newcommand{\brck}[1]{\trunc{}{#1}}
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\newcommand{\bbrck}[1]{\ttrunc{}{#1}}
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\newcommand{\Brck}[1]{\Trunc{}{#1}}
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\newcommand{\bproj}[1]{\tproj{}{#1}}
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\newcommand{\bprojf}{\tprojf{}}
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% Big parentheses
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|
\newcommand{\Parens}[1]{\Bigl(#1\Bigr)}
|
|
|
|
% Projection and extension for truncations
|
|
\let\extendsmb\ext
|
|
\newcommand{\extend}[1]{\extendsmb(#1)}
|
|
|
|
%
|
|
%%% The empty type
|
|
\newcommand{\emptyt}{\ensuremath{\mathbf{0}}\xspace}
|
|
|
|
%%% The unit type
|
|
\newcommand{\unit}{\ensuremath{\mathbf{1}}\xspace}
|
|
\newcommand{\ttt}{\ensuremath{\star}\xspace}
|
|
|
|
%%% The two-element type
|
|
\newcommand{\bool}{\ensuremath{\mathbf{2}}\xspace}
|
|
\newcommand{\btrue}{{1_{\bool}}}
|
|
\newcommand{\bfalse}{{0_{\bool}}}
|
|
|
|
%%% Injections into binary sums and pushouts
|
|
\newcommand{\inlsym}{{\mathsf{inl}}}
|
|
\newcommand{\inrsym}{{\mathsf{inr}}}
|
|
\newcommand{\inl}{\ensuremath\inlsym\xspace}
|
|
\newcommand{\inr}{\ensuremath\inrsym\xspace}
|
|
|
|
%%% The segment of the interval
|
|
\newcommand{\seg}{\ensuremath{\mathsf{seg}}\xspace}
|
|
|
|
%%% Free groups
|
|
\newcommand{\freegroup}[1]{F(#1)}
|
|
\newcommand{\freegroupx}[1]{F'(#1)} % the "other" free group
|
|
|
|
%%% Glue of a pushout
|
|
\newcommand{\glue}{\mathsf{glue}}
|
|
|
|
%%% Circles and spheres
|
|
\newcommand{\Sn}{\mathbb{S}}
|
|
\newcommand{\base}{\ensuremath{\mathsf{base}}\xspace}
|
|
\newcommand{\lloop}{\ensuremath{\mathsf{loop}}\xspace}
|
|
\newcommand{\surf}{\ensuremath{\mathsf{surf}}\xspace}
|
|
|
|
%%% Suspension
|
|
\newcommand{\susp}{\Sigma}
|
|
\newcommand{\north}{\mathsf{N}}
|
|
\newcommand{\south}{\mathsf{S}}
|
|
\newcommand{\merid}{\mathsf{merid}}
|
|
|
|
%%% Blanks (shorthand for lambda abstractions)
|
|
\newcommand{\blank}{\mathord{\hspace{1pt}\text{--}\hspace{1pt}}}
|
|
|
|
%%% Nameless objects
|
|
\newcommand{\nameless}{\mathord{\hspace{1pt}\underline{\hspace{1ex}}\hspace{1pt}}}
|
|
|
|
%%% Some decorations
|
|
%\newcommand{\bbU}{\ensuremath{\mathbb{U}}\xspace}
|
|
% \newcommand{\bbB}{\ensuremath{\mathbb{B}}\xspace}
|
|
\newcommand{\bbP}{\ensuremath{\mathbb{P}}\xspace}
|
|
|
|
%%% Some categories
|
|
\newcommand{\uset}{\ensuremath{\mathcal{S}et}\xspace}
|
|
\newcommand{\ucat}{\ensuremath{{\mathcal{C}at}}\xspace}
|
|
\newcommand{\urel}{\ensuremath{\mathcal{R}el}\xspace}
|
|
\newcommand{\uhilb}{\ensuremath{\mathcal{H}ilb}\xspace}
|
|
\newcommand{\utype}{\ensuremath{\mathcal{T}\!ype}\xspace}
|
|
|
|
% Pullback corner
|
|
%\newbox\pbbox
|
|
%\setbox\pbbox=\hbox{\xy \POS(65,0)\ar@{-} (0,0) \ar@{-} (65,65)\endxy}
|
|
%\def\pb{\save[]+<3.5mm,-3.5mm>*{\copy\pbbox} \restore}
|
|
|
|
% Macros for the categories chapter
|
|
% \newcommand{\inv}[1]{{#1}^{-1}}
|
|
\newcommand{\idtoiso}{\ensuremath{\mathsf{idtoiso}}\xspace}
|
|
\newcommand{\isotoid}{\ensuremath{\mathsf{isotoid}}\xspace}
|
|
\newcommand{\op}{^{\mathrm{op}}}
|
|
\newcommand{\y}{\ensuremath{\mathbf{y}}\xspace}
|
|
\newcommand{\dgr}[1]{{#1}^{\dagger}}
|
|
\newcommand{\unitaryiso}{\mathrel{\cong^\dagger}}
|
|
\newcommand{\cteqv}[2]{\ensuremath{#1 \simeq #2}\xspace}
|
|
\newcommand{\cteqvsym}{\simeq} % Symbol for equivalence of categories
|
|
|
|
%%% Natural numbers
|
|
\newcommand{\N}{\ensuremath{\mathbb{N}}\xspace}
|
|
%\newcommand{\N}{\textbf{N}}
|
|
\let\nat\N
|
|
\newcommand{\natp}{\ensuremath{\nat'}\xspace} % alternative nat in induction chapter
|
|
|
|
\newcommand{\zerop}{\ensuremath{0'}\xspace} % alternative zero in induction chapter
|
|
\newcommand{\suc}{\mathsf{succ}}
|
|
\newcommand{\sucp}{\ensuremath{\suc'}\xspace} % alternative suc in induction chapter
|
|
\newcommand{\add}{\mathsf{add}}
|
|
\newcommand{\ack}{\mathsf{ack}}
|
|
\newcommand{\ite}{\mathsf{iter}}
|
|
\newcommand{\assoc}{\mathsf{assoc}}
|
|
\newcommand{\dbl}{\ensuremath{\mathsf{double}}}
|
|
\newcommand{\dblp}{\ensuremath{\dbl'}\xspace} % alternative double in induction chapter
|
|
|
|
|
|
%%% Lists
|
|
\newcommand{\lst}[1]{\mathsf{List}(#1)}
|
|
\newcommand{\nil}{\mathsf{nil}}
|
|
\newcommand{\cons}{\mathsf{cons}}
|
|
|
|
%%% Vectors of given length, used in induction chapter
|
|
\newcommand{\vect}[2]{\ensuremath{\mathsf{Vec}_{#1}(#2)}\xspace}
|
|
|
|
%%% Integers
|
|
\newcommand{\Z}{\ensuremath{\mathbb{Z}}\xspace}
|
|
\newcommand{\Zsuc}{\mathsf{succ}}
|
|
\newcommand{\Zpred}{\mathsf{pred}}
|
|
|
|
%%% Rationals
|
|
\newcommand{\Q}{\ensuremath{\mathbb{Q}}\xspace}
|
|
|
|
%%% Function extensionality
|
|
\newcommand{\funext}{\mathsf{funext}}
|
|
\newcommand{\happly}{\mathsf{happly}}
|
|
|
|
%%% A naturality lemma
|
|
\newcommand{\com}[3]{\mathsf{swap}_{#1,#2}(#3)}
|
|
|
|
%%% Code/encode/decode
|
|
\newcommand{\code}{\ensuremath{\mathsf{code}}\xspace}
|
|
\newcommand{\encode}{\ensuremath{\mathsf{encode}}\xspace}
|
|
\newcommand{\decode}{\ensuremath{\mathsf{decode}}\xspace}
|
|
|
|
% Function definition with domain and codomain
|
|
\newcommand{\function}[4]{\left\{\begin{array}{rcl}#1 &
|
|
\longrightarrow & #2 \\ #3 & \longmapsto & #4 \end{array}\right.}
|
|
|
|
%%% Cones and cocones
|
|
\newcommand{\cone}[2]{\mathsf{cone}_{#1}(#2)}
|
|
\newcommand{\cocone}[2]{\mathsf{cocone}_{#1}(#2)}
|
|
% Apply a function to a cocone
|
|
\newcommand{\composecocone}[2]{#1\circ#2}
|
|
\newcommand{\composecone}[2]{#2\circ#1}
|
|
%%% Diagrams
|
|
\newcommand{\Ddiag}{\mathscr{D}}
|
|
|
|
%%% (pointed) mapping spaces
|
|
\newcommand{\Map}{\mathsf{Map}}
|
|
|
|
%%% The interval
|
|
\newcommand{\interval}{\ensuremath{I}\xspace}
|
|
\newcommand{\izero}{\ensuremath{0_{\interval}}\xspace}
|
|
\newcommand{\ione}{\ensuremath{1_{\interval}}\xspace}
|
|
|
|
%%% Arrows
|
|
\newcommand{\epi}{\ensuremath{\twoheadrightarrow}}
|
|
\newcommand{\mono}{\ensuremath{\rightarrowtail}}
|
|
|
|
%%% Sets
|
|
\newcommand{\bin}{\ensuremath{\mathrel{\widetilde{\in}}}}
|
|
|
|
%%% Semigroup structure
|
|
\newcommand{\semigroupstrsym}{\ensuremath{\mathsf{SemigroupStr}}}
|
|
\newcommand{\semigroupstr}[1]{\ensuremath{\mathsf{SemigroupStr}}(#1)}
|
|
\newcommand{\semigroup}[0]{\ensuremath{\mathsf{Semigroup}}}
|
|
|
|
%%% Macros for the formal type theory
|
|
\newcommand{\emptyctx}{\ensuremath{\cdot}}
|
|
\newcommand{\production}{\vcentcolon\vcentcolon=}
|
|
\newcommand{\conv}{\downarrow}
|
|
\newcommand{\ctx}{\ensuremath{\mathsf{ctx}}}
|
|
\newcommand{\wfctx}[1]{#1\ \ctx}
|
|
\newcommand{\oftp}[3]{#1 \vdash #2 : #3}
|
|
\newcommand{\jdeqtp}[4]{#1 \vdash #2 \jdeq #3 : #4}
|
|
\newcommand{\judg}[2]{#1 \vdash #2}
|
|
\newcommand{\tmtp}[2]{#1 \mathord{:} #2}
|
|
|
|
% rule names
|
|
\newcommand{\form}{\textsc{form}}
|
|
\newcommand{\intro}{\textsc{intro}}
|
|
\newcommand{\elim}{\textsc{elim}}
|
|
\newcommand{\comp}{\textsc{comp}}
|
|
\newcommand{\uniq}{\textsc{uniq}}
|
|
\newcommand{\Weak}{\mathsf{Wkg}}
|
|
\newcommand{\Vble}{\mathsf{Vble}}
|
|
\newcommand{\Exch}{\mathsf{Exch}}
|
|
\newcommand{\Subst}{\mathsf{Subst}}
|
|
|
|
%%% Macros for HITs
|
|
\newcommand{\cc}{\mathsf{c}}
|
|
\newcommand{\pp}{\mathsf{p}}
|
|
\newcommand{\cct}{\widetilde{\mathsf{c}}}
|
|
\newcommand{\ppt}{\widetilde{\mathsf{p}}}
|
|
\newcommand{\Wtil}{\ensuremath{\widetilde{W}}\xspace}
|
|
|
|
%%% Macros for n-types
|
|
\newcommand{\istype}[1]{\mathsf{is}\mbox{-}{#1}\mbox{-}\mathsf{type}}
|
|
\newcommand{\nplusone}{\ensuremath{(n+1)}}
|
|
\newcommand{\nminusone}{\ensuremath{(n-1)}}
|
|
% \newcommand{\fact}{\mathsf{fact}}
|
|
|
|
%%% Macros for homotopy
|
|
\newcommand{\kbar}{\overline{k}} % Used in van Kampen's theorem
|
|
|
|
%%% Macros for induction
|
|
\newcommand{\natw}{\ensuremath{\mathbf{N^w}}\xspace}
|
|
\newcommand{\zerow}{\ensuremath{0^\mathbf{w}}\xspace}
|
|
\newcommand{\sucw}{\ensuremath{\mathbf{s^w}}\xspace}
|
|
\newcommand{\nalg}{\nat\mathsf{Alg}}
|
|
\newcommand{\nhom}{\nat\mathsf{Hom}}
|
|
\newcommand{\ishinitw}{\mathsf{isHinit}_{\mathsf{W}}}
|
|
\newcommand{\ishinitn}{\mathsf{isHinit}_\nat}
|
|
\newcommand{\w}{\mathsf{W}}
|
|
\newcommand{\walg}{\w\mathsf{Alg}}
|
|
\newcommand{\whom}{\w\mathsf{Hom}}
|
|
|
|
%%% Macros for real numbers
|
|
\newcommand{\RC}{\ensuremath{\mathbb{R}_\mathsf{c}}\xspace} % Cauchy
|
|
\newcommand{\RD}{\ensuremath{\mathbb{R}_\mathsf{d}}\xspace} % Dedekind
|
|
\newcommand{\R}{\ensuremath{\mathbb{R}}\xspace} % Either
|
|
\newcommand{\barRD}{\ensuremath{\bar{\mathbb{R}}_\mathsf{d}}\xspace} % Dedekind completion of Dedekind
|
|
|
|
\newcommand{\close}[1]{\sim_{#1}} % Relation of closeness
|
|
\newcommand{\closesym}{\mathord\sim}
|
|
\newcommand{\rclim}{\mathsf{lim}} % HIT constructor for Cauchy reals
|
|
\newcommand{\rcrat}{\mathsf{rat}} % Embedding of rationals into Cauchy reals
|
|
\newcommand{\rceq}{\mathsf{eq}_{\RC}} % HIT path constructor
|
|
\newcommand{\CAP}{\mathcal{C}} % The type of Cauchy approximations
|
|
\newcommand{\Qp}{\Q_{+}}
|
|
\newcommand{\apart}{\mathrel{\#}} % apartness
|
|
\newcommand{\dcut}{\mathsf{isCut}} % Dedekind cut
|
|
\newcommand{\cover}{\triangleleft} % inductive cover
|
|
\newcommand{\intfam}[3]{(#2, \lam{#1} #3)} % family of rational intervals
|
|
|
|
% Macros for the Cauchy reals construction
|
|
\newcommand{\bsim}{\frown}
|
|
\newcommand{\bbsim}{\smile}
|
|
|
|
\newcommand{\hapx}{\diamondsuit\approx}
|
|
\newcommand{\hapname}{\diamondsuit}
|
|
\newcommand{\hapxb}{\heartsuit\approx}
|
|
\newcommand{\hapbname}{\heartsuit}
|
|
\newcommand{\tap}[1]{\bullet\approx_{#1}\triangle}
|
|
\newcommand{\tapname}{\triangle}
|
|
\newcommand{\tapb}[1]{\bullet\approx_{#1}\square}
|
|
\newcommand{\tapbname}{\square}
|
|
|
|
%%% Macros for surreals
|
|
\newcommand{\NO}{\ensuremath{\mathsf{No}}\xspace}
|
|
\newcommand{\surr}[2]{\{\,#1\,\big|\,#2\,\}}
|
|
\newcommand{\LL}{\mathcal{L}}
|
|
\newcommand{\RR}{\mathcal{R}}
|
|
\newcommand{\noeq}{\mathsf{eq}_{\NO}} % HIT path constructor
|
|
|
|
\newcommand{\ble}{\trianglelefteqslant}
|
|
\newcommand{\blt}{\vartriangleleft}
|
|
\newcommand{\bble}{\sqsubseteq}
|
|
\newcommand{\bblt}{\sqsubset}
|
|
|
|
\newcommand{\hle}{\diamondsuit\preceq}
|
|
\newcommand{\hlt}{\diamondsuit\prec}
|
|
\newcommand{\hlname}{\diamondsuit}
|
|
\newcommand{\hleb}{\heartsuit\preceq}
|
|
\newcommand{\hltb}{\heartsuit\prec}
|
|
\newcommand{\hlbname}{\heartsuit}
|
|
% \newcommand{\tle}{(\bullet\preceq\triangle)}
|
|
% \newcommand{\tlt}{(\bullet\prec\triangle)}
|
|
\newcommand{\tle}{\triangle\preceq}
|
|
\newcommand{\tlt}{\triangle\prec}
|
|
\newcommand{\tlname}{\triangle}
|
|
% \newcommand{\tleb}{(\bullet\preceq\square)}
|
|
% \newcommand{\tltb}{(\bullet\prec\square)}
|
|
\newcommand{\tleb}{\square\preceq}
|
|
\newcommand{\tltb}{\square\prec}
|
|
\newcommand{\tlbname}{\square}
|
|
|
|
%%% Macros for set theory
|
|
\newcommand{\vset}{\mathsf{set}} % point constructor for cummulative hierarchy V
|
|
\def\cd{\tproj0}
|
|
\newcommand{\inj}{\ensuremath{\mathsf{inj}}} % type of injections
|
|
\newcommand{\acc}{\ensuremath{\mathsf{acc}}} % accessibility
|
|
|
|
\newcommand{\atMostOne}{\mathsf{atMostOne}}
|
|
|
|
\newcommand{\power}[1]{\mathcal{P}(#1)} % power set
|
|
\newcommand{\powerp}[1]{\mathcal{P}_+(#1)} % inhabited power set
|
|
|
|
%%%% THEOREM ENVIRONMENTS %%%%
|
|
|
|
% Hyperref includes the command \autoref{...} which is like \ref{...}
|
|
% except that it automatically inserts the type of the thing you're
|
|
% referring to, e.g. it produces "Theorem 3.8" instead of just "3.8"
|
|
% (and makes the whole thing a hyperlink). This saves a slight amount
|
|
% of typing, but more importantly it means that if you decide later on
|
|
% that 3.8 should be a Lemma or a Definition instead of a Theorem, you
|
|
% don't have to change the name in all the places you referred to it.
|
|
|
|
% The following hack improves on this by using the same counter for
|
|
% all theorem-type environments, so that after Theorem 1.1 comes
|
|
% Corollary 1.2 rather than Corollary 1.1. This makes it much easier
|
|
% for the reader to find a particular theorem when flipping through
|
|
% the document.
|
|
\makeatletter
|
|
\def\defthm#1#2#3{%
|
|
%% Ensure all theorem types are numbered with the same counter
|
|
\newaliascnt{#1}{thm}
|
|
\newtheorem{#1}[#1]{#2}
|
|
\aliascntresetthe{#1}
|
|
%% This command tells cleveref's \cref what to call things
|
|
\crefname{#1}{#2}{#3}}
|
|
|
|
% Now define a bunch of theorem-type environments.
|
|
\newtheorem{thm}{Theorem}[section]
|
|
\crefname{thm}{Theorem}{Theorems}
|
|
\defthm{propn}{Proposition}{Propositions} % Probably we shouldn't use "Proposition" in this way
|
|
\defthm{cor}{Corollary}{Corollaries}
|
|
\defthm{lem}{Lemma}{Lemmas}
|
|
\defthm{ax}{Axiom}{Axioms}
|
|
\defthm{conj}{Conjecture}{Conjectures}
|
|
% Since definitions and theorems in type theory are synonymous, should
|
|
% we actually use the same theoremstyle for them?
|
|
\theoremstyle{definition}
|
|
\defthm{defn}{Definition}{Definitions}
|
|
\theoremstyle{remark}
|
|
\defthm{rmk}{Remark}{Remarks}
|
|
\defthm{ex}{Example}{Examples}
|
|
\defthm{exs}{Examples}{Examples}
|
|
\defthm{notes}{Notes}{Notes}
|
|
% Number exercises within chapters, with their own counter.
|
|
%\newtheorem{ex}{Exercise}[chapter]
|
|
%\crefname{ex}{Exercise}{Exercises}
|
|
|
|
% Display format for sections
|
|
\crefformat{section}{\S#2#1#3}
|
|
\Crefformat{section}{Section~#2#1#3}
|
|
\crefrangeformat{section}{\S\S#3#1#4--#5#2#6}
|
|
\Crefrangeformat{section}{Sections~#3#1#4--#5#2#6}
|
|
\crefmultiformat{section}{\S\S#2#1#3}{ and~#2#1#3}{, #2#1#3}{ and~#2#1#3}
|
|
\Crefmultiformat{section}{Sections~#2#1#3}{ and~#2#1#3}{, #2#1#3}{ and~#2#1#3}
|
|
\crefrangemultiformat{section}{\S\S#3#1#4--#5#2#6}{ and~#3#1#4--#5#2#6}{, #3#1#4--#5#2#6}{ and~#3#1#4--#5#2#6}
|
|
\Crefrangemultiformat{section}{Sections~#3#1#4--#5#2#6}{ and~#3#1#4--#5#2#6}{, #3#1#4--#5#2#6}{ and~#3#1#4--#5#2#6}
|
|
|
|
% Display format for appendices
|
|
\crefformat{appendix}{Appendix~#2#1#3}
|
|
\Crefformat{appendix}{Appendix~#2#1#3}
|
|
\crefrangeformat{appendix}{Appendices~#3#1#4--#5#2#6}
|
|
\Crefrangeformat{appendix}{Appendices~#3#1#4--#5#2#6}
|
|
\crefmultiformat{appendix}{Appendices~#2#1#3}{ and~#2#1#3}{, #2#1#3}{ and~#2#1#3}
|
|
\Crefmultiformat{appendix}{Appendices~#2#1#3}{ and~#2#1#3}{, #2#1#3}{ and~#2#1#3}
|
|
\crefrangemultiformat{appendix}{Appendices~#3#1#4--#5#2#6}{ and~#3#1#4--#5#2#6}{, #3#1#4--#5#2#6}{ and~#3#1#4--#5#2#6}
|
|
\Crefrangemultiformat{appendix}{Appendices~#3#1#4--#5#2#6}{ and~#3#1#4--#5#2#6}{, #3#1#4--#5#2#6}{ and~#3#1#4--#5#2#6}
|
|
|
|
\crefname{part}{Part}{Parts}
|
|
|
|
% Number subsubsections
|
|
\setcounter{secnumdepth}{5}
|
|
|
|
% Display format for figures
|
|
\crefname{figure}{Figure}{Figures}
|
|
|
|
% Use cleveref instead of hyperref's \autoref
|
|
\let\autoref\cref
|
|
|
|
|
|
%%%% EQUATION NUMBERING %%%%
|
|
|
|
% The following hack uses the single theorem counter to number
|
|
% equations as well, so that we don't have both Theorem 1.1 and
|
|
% equation (1.1).
|
|
\let\c@equation\c@thm
|
|
\numberwithin{equation}{section}
|
|
|
|
|
|
%%%% ENUMERATE NUMBERING %%%%
|
|
|
|
% Number the first level of enumerates as (i), (ii), ...
|
|
\renewcommand{\theenumi}{(\roman{enumi})}
|
|
\renewcommand{\labelenumi}{\theenumi}
|
|
|
|
|
|
%%%% MARGINS %%%%
|
|
|
|
% This is a matter of personal preference, but I think the left
|
|
% margins on enumerates and itemizes are too wide.
|
|
% \setitemize[1]{leftmargin=2em}
|
|
% \setenumerate[1]{leftmargin=*}
|
|
|
|
% Likewise that they are too spaced out.
|
|
% \setitemize[1]{itemsep=-0.2em}
|
|
% \setenumerate[1]{itemsep=-0.2em}
|
|
|
|
%%% Notes %%%
|
|
\def\noteson{%
|
|
\gdef\note##1{\mbox{}\marginpar{\color{blue}\textasteriskcentered\ ##1}}}
|
|
\gdef\notesoff{\gdef\note##1{\null}}
|
|
\noteson
|
|
|
|
\newcommand{\Coq}{\textsc{Coq}\xspace}
|
|
\newcommand{\Agda}{\textsc{Agda}\xspace}
|
|
\newcommand{\NuPRL}{\textsc{NuPRL}\xspace}
|
|
|
|
%%%% CITATIONS %%%%
|
|
|
|
% \let \cite \citep
|
|
|
|
%%%% INDEX %%%%
|
|
|
|
\newcommand{\footstyle}[1]{{\hyperpage{#1}}n} % If you index something that is in a footnote
|
|
\newcommand{\defstyle}[1]{\textbf{\hyperpage{#1}}} % Style for pageref to a definition
|
|
|
|
\newcommand{\indexdef}[1]{\index{#1|defstyle}} % Index a definition
|
|
\newcommand{\indexfoot}[1]{\index{#1|footstyle}} % Index a term in a footnote
|
|
\newcommand{\indexsee}[2]{\index{#1|see{#2}}} % Index "see also"
|
|
|
|
|
|
%%%% Standard phrasing or spelling of common phrases %%%%
|
|
|
|
\newcommand{\ZF}{Zermelo--Fraenkel}
|
|
\newcommand{\CZF}{Constructive \ZF{} Set Theory}
|
|
|
|
\newcommand{\LEM}[1]{\ensuremath{\mathsf{LEM}_{#1}}\xspace}
|
|
\newcommand{\choice}[1]{\ensuremath{\mathsf{AC}_{#1}}\xspace}
|
|
|
|
%%%% MISC %%%%
|
|
|
|
\newcommand{\mentalpause}{\medskip} % Use for "mental" pause, instead of \smallskip or \medskip
|
|
|
|
%% Use \symlabel instead of \label to mark a pageref that you need in the index of symbols
|
|
\newcounter{symindex}
|
|
\newcommand{\symlabel}[1]{\refstepcounter{symindex}\label{#1}}
|
|
|
|
% Local Variables:
|
|
% mode: latex
|
|
% TeX-master: "hott-online"
|
|
% End:
|