Defining a macro with three optional arguments in the form \newmacro{a}{b}[c]{d}[e][f] and \newmacro{a}{b}[c]{d}*[f]

Question

I am defining a macro in LaTeX2e for writing function declarations.

I want that the macro has tree compulsory arguments: the name of the function (v.g f), a depiction of the type of variable it takes (v.g. x), and the space in which the function is defined (v.g. \mathbb R), as well as three optional arguments: the domain (if the domain is not the same as the codomain), the writing of the function applied to its variables, and the asignment/meaning of the function.

For the sake of clarity when using the macro I had decided for the following structure:

\funcdef{functionName}{variables}[domain]{codomain}[notation][definition]

To avoid ambiguity in the last two optional elements, when notation is given but no definition it takes the form:

\funcdef{functionName}{variables}[domain]{codomain}[notation]

And when definition is given but no notation:

\funcdef{functionName}{variables}[domain]{codomain}*[definition]

I came with the following code (both declaration and example), which is working but seems a little too complicated:

\documentclass{article}
\makeatletter
\newcommand\funcdef[2]{\def\func@name{#1}\def\func@var{#2}%
    \begin{array}{r@{\ }c@{\,}c@{\,}l}#1:\func@dom}
\newcommand\func@dom[2][\@empty]{&\ifx#1\@empty#2\else#1\fi%
    &\to&#2\\&\func@var&\mapsto&\func@use}
\newcommand\func@use{\@ifstar\func@use@\func@usei}
\newcommand\func@use@{\func@name(\func@var)\func@def}
\newcommand\func@usei[1][\@empty]{\ifx#1\@empty\func@name(\func@var)%
    \else#1\fi\func@def}
\newcommand\func@def[1][\@empty]{\ifx#1\@empty\relax\else%
    \mathrel{:=}#1\fi\end{array}}
\makeatother
\begin{document}
Simple function: \[
  \funcdef fx{\mathbf R}
\]

Simple function with declaration: \[
  \funcdef fx{\mathbf R}*[x^2]
\]

Function with alternative writing: \[
  \funcdef{\textrm{exp}}x{\mathbf R}[e^x]
\]textrm{exp}

Function with alternative writing and declaration: \[
  \funcdef\exp x{\mathbf R}[e^x][\lim\limits_{n\to\infty}%
    \left(1+\frac xn\right)^n]
\]

Function with different domain and codomain: \[
  \funcdef{\textrm{sqrt}}n[\mathbf N]{\mathbf R}
\]

Function with different domain and codomain, and alternative
writing: \[
  \funcdef{\textrm{sqrt}}n[\mathbf N]{\mathbf R}[\sqrt n]
\]

Function with different domain and codomain, alternative writing
     and declaration: \[
  \funcdef{\textrm{sqrt}}n[\mathbf N]{\mathbf R}[\sqrt n][\exp(\frac12\ln n)]
\]
\end{document}

Is there any other way to define in LaTeX2e these kind of macros that does not require too many supporting macros (5 in this sample) and temporal definitions (2 in this sample)?

How would it work in pain-TeX or LaTeX3? (answers to this will be upvoted but not accepted)

Answer 1

I would use a completely different approach, with a key-value syntax. Six arguments, some of them optional, with a * to denote a missing one are error prone.

\documentclass{article}
\usepackage{amsmath,keyval}

\makeatletter
\newcommand{\funcdef@key}[1]{%
  \define@key{funcdef}{#1}{\@namedef{cet@#1}{##1}}%
  \expandafter\let\csname cet@#1\endcsname\@empty
}
\funcdef@key{name}
\funcdef@key{domain}
\funcdef@key{codomain}
\funcdef@key{variable}
\funcdef@key{notation}
\funcdef@key{definition}
\newcommand{\funcdef@check}[1]{%
  \expandafter\ifx\csname cet@#1\endcsname\@empty
    \@latex@error{Missing `#1'}{Provide `#1'}%
  \fi
}

\newcommand{\funcdef}[1]{%
  \begingroup
  \setkeys{funcdef}{#1}%
  \ifx\cet@codomain\@empty\let\cet@codomain\cet@domain\fi
  \funcdef@check{name}%
  \funcdef@check{domain}%
  \funcdef@check{variable}%
  \begin{array}{l@{}r@{}l@{}l}
  \cet@name\colon{} & 
  \cet@domain &
  {}\to \cet@codomain \\
  &
  \cet@variable &
  {}\mapsto
  \ifx\cet@notation\@empty
    \cet@name(\cet@variable)
  \else
    \cet@notation
  \fi
  \ifx\cet@definition\@empty
    \expandafter\@gobble
  \else
    \expandafter\@firstofone
  \fi
  {& {}\mathrel{:}=\cet@definition}
  \\
  \end{array}
  \endgroup
}
\makeatletter

\begin{document}

\noindent
Simple function:
\[
\funcdef{name=f,variable=x,domain=\mathbf{R}}
\]
Simple function with declaration:
\[
\funcdef{
  name=f,
  variable=x,
  domain=\mathbf{R},
  notation=x^2
}
\]
Function with alternative writing:
\[
\funcdef{
  name=\exp,
  variable=x,
  domain=\mathbf{R},
  notation=e^x
}
\]
Function with alternative writing and declaration:
\[
\funcdef{
  name=\exp,
  variable=x,
  domain=\mathbf{R},
  notation=e^x,
  definition=\lim\limits_{n\to\infty}\left(1+\frac xn\right)^n
}
\]
Function with different domain and codomain:
\[
\funcdef{
  name=\operatorname{sqrt},
  variable=n,
  domain=\mathbf{N},
  codomain=\mathbf{R}
}
\]
Function with different domain and codomain, and alternative
writing:
\[
\funcdef{
  name=\operatorname{sqrt},
  variable=n,
  domain=\mathbf{N},
  codomain=\mathbf{R},
  definition=\sqrt{n}
}
\]
Function with different domain and codomain, alternative writing
and declaration:
\[
\funcdef{
  name=\operatorname{sqrt},
  variable=n,
  domain=\mathbf{N},
  codomain=\mathbf{R},
  notation=\sqrt{n},
  definition=\exp\bigl(\frac{1}{2}\ln n\bigr)
}
\]
\end{document}

---

Here's an implementation in LaTeX3, with also an “inline” version.

\documentclass{article}
\usepackage{amsmath,xparse}

\ExplSyntaxOn
\keys_define:nn { funcdef }
 {
  name       .tl_set:N   = \l_funcdef_name_tl,
  name       .initial:n  = {},
  domain     .tl_set:N   = \l_funcdef_domain_tl,
  domain     .initial:n  = {},
  codomain   .tl_set:N   = \l_funcdef_codomain_tl,
  codomain   .initial:n  = {},
  variable   .tl_set:N   = \l_funcdef_variable_tl,
  variable   .initial:n  = {},
  variables  .tl_set:N   = \l_funcdef_variables_tl,
  variables  .initial:n  = {},
  notation   .tl_set:N   = \l_funcdef_notation_tl,
  notation   .initial:n  = {},
  definition .tl_set:N   = \l_funcdef_definition_tl,
  definition .initial:n  = {},
  inline     .bool_set:N = \l_funcdef_inline_bool,
 }

\msg_new:nnnn { funcdef } { missing }
 {
  Missing~`#1'
 }
 {
  You~have~to~specify~a~value~for~`#1';~%
  I~have~substituted~??~for~it
 }

\NewDocumentCommand{\funcdef}{ m }
 {
  \group_begin:
  \funcdef_print:n { #1 }
  \group_end:
 }

\cs_new_protected:Npn \funcdef_print:n #1
 {
  \keys_set:nn { funcdef } { #1 }
  \funcdef_check:n { name }
  \funcdef_check:n { domain }
  \tl_if_empty:NT \l_funcdef_variables_tl
   {
    \funcdef_check:n { variable }
   }
  \tl_if_empty:NT \l_funcdef_codomain_tl
   {
    \tl_set_eq:NN \l_funcdef_codomain_tl \l_funcdef_domain_tl
   }
  \bool_if:NTF \l_funcdef_inline_bool
   {
    \funcdef_print_inline:
   }
   {
    \funcdef_print_array:
   }
 }

\cs_new_protected:Npn \funcdef_print_array:
 {
  \begin{array}{ l @{} r @{} l }
  % first row
  \l_funcdef_name_tl \colon {}
  & 
  \l_funcdef_domain_tl
  &
  {}\to \l_funcdef_codomain_tl
  \\
  % second row
  &
  \tl_if_empty:NTF \l_funcdef_variable_tl
   {
    (\l_funcdef_variables_tl)
   }
   {
    \l_funcdef_variable_tl
   }
  &
  {}\mapsto
  \tl_if_empty:NTF \l_funcdef_notation_tl
   {
    \l_funcdef_name_tl (
    \tl_if_empty:NTF \l_funcdef_variable_tl
     {
      \l_funcdef_variables_tl
     }
     {
      \l_funcdef_variable_tl
     }
    )
   }
   {
    \l_funcdef_notation_tl
   }
  \tl_if_empty:NF \l_funcdef_definition_tl
    { \mathrel{:}= \l_funcdef_definition_tl }
  \\
  \end{array}
 }

\cs_new:Npn \funcdef_print_inline:
 {
  \l_funcdef_name_tl \colon
  \l_funcdef_domain_tl
  \to \l_funcdef_codomain_tl
  ,\quad
  \tl_if_empty:NTF \l_funcdef_variable_tl
   {
    (\l_funcdef_variables_tl)
   }
   {
    \l_funcdef_variable_tl
   }
  \mapsto
  \tl_if_empty:NTF \l_funcdef_notation_tl
   {
    \l_funcdef_name_tl (
    \tl_if_empty:NTF \l_funcdef_variable_tl
     {
      \l_funcdef_variables_tl
     }
     {
      \l_funcdef_variable_tl
     }
    )
   }
   {
    \l_funcdef_notation_tl
   }
  \tl_if_empty:NF \l_funcdef_definition_tl
    { \mathrel{:}= \l_funcdef_definition_tl }
 }

\cs_new_protected:Npn \funcdef_check:n #1
 {
  \tl_if_empty:cT { l_funcdef_#1_tl }
   {
    \msg_error:nnn { funcdef } { missing } { #1 }
    \tl_set:cn { l_funcdef_#1_tl } { ?? }
   }
 }

\ExplSyntaxOff

\begin{document}

\noindent
Simple function inline:
\[
\funcdef{inline,name=f,variable=x,domain=\mathbf{R}}
\]
Simple function:
\[
\funcdef{name=f,variable=x,domain=\mathbf{R}}
\]
Simple function with declaration:
\[
\funcdef{
  name=f,
  variable=x,
  domain=\mathbf{R},
  definition=x^2
}
\]
Function with alternative writing:
\[
\funcdef{
  name=\exp,
  variable=x,
  domain=\mathbf{R},
  notation=e^x
}
\]
Function with alternative writing and declaration:
\[
\funcdef{
  name=\exp,
  variable=x,
  domain=\mathbf{R},
  notation=e^x,
  definition=\lim\limits_{n\to\infty}\left(1+\frac xn\right)^n
}
\]
Function with different domain and codomain:
\[
\funcdef{
  name=\operatorname{sqrt},
  variable=n,
  domain=\mathbf{N},
  codomain=\mathbf{R}
}
\]
Function with different domain and codomain, and alternative
writing:
\[
\funcdef{
  name=\operatorname{sqrt},
  variable=n,
  domain=\mathbf{N},
  codomain=\mathbf{R},
  definition=\sqrt{n}
}
\]
Function with different domain and codomain, alternative writing
and declaration:
\[
\funcdef{
  name=\operatorname{sqrt},
  variable=n,
  domain=\mathbf{N},
  codomain=\mathbf{R},
  notation=\sqrt{n},
  definition=\exp\bigl(\frac{1}{2}\ln n\bigr)
}
\]
Function of two variables:
\[
\funcdef{
  name=f,
  variables={a,b},
  domain=A\times B,
  codomain=C,
}
\]
\end{document}

Answer 2

This seems to be very easy with xparse. The only thing that seems hard to do is to enforce the exact syntax that you want for differentiating the two last optional arguments. With the solution below, the following are allowed (as you want them):

\funcdef{f}{x}{R}
\funcdef{f}{x}{R}[2x]
\funcdef{f}{x}{R}[2x][x+x]
\funcdef{f}{x}{R}*[x+x]

but the following is allowed too:

\funcdef{f}{x}{R}[2x]*[x+x]

and is equivalent to the third one (of the first four). I do not consider this to be much of a problem; if you do, it will not be easy to fix.

\documentclass{article}
\usepackage{xparse}

\DeclareDocumentCommand{\funcdef}{m m o m o s o}{%
  \begin{array}{r@{\ }c@{\,}c@{\,}l}
    #1:
  & \IfNoValueTF{#3}{#4}{#3}
  & \to & #4 \\
  & #2
  & \mapsto
  & \IfNoValueTF{#5}{#1(#2)}{#5}
    \IfNoValueTF{#7}{}{\mathrel{:=}#7}
  \end{array}
}

\begin{document}
Simple function: \[
  \funcdef fx{\mathbf R}
\]

Simple function with declaration: \[
  \funcdef fx{\mathbf R}*[x^2]
\]

Function with alternative writing: \[
  \funcdef{\textrm{exp}}x{\mathbf R}[e^x]
\]

Function with alternative writing and declaration: \[
  \funcdef\exp x{\mathbf R}[e^x][\lim\limits_{n\to\infty}%
    \left(1+\frac xn\right)^n]
\]

Function with different domain and codomain: \[
  \funcdef{\textrm{sqrt}}n[\mathbf N]{\mathbf R}
\]

Function with different domain and codomain, and alternative
writing: \[
  \funcdef{\textrm{sqrt}}n[\mathbf N]{\mathbf R}[\sqrt n]
\]

Function with different domain and codomain, alternative writing
     and declaration: \[
  \funcdef{\textrm{sqrt}}n[\mathbf N]{\mathbf R}[\sqrt n][\exp(\frac12\ln n)]
\]
\end{document}