Help on an expl3 command for building shortcuts for math macros

Question

I am trying to create a command \projlib_math_define_shortcut:nnn for defining shortcuts for math letters. Ideally, it should be able to use like \projlib_math_define_shortcut:nnn { mathcal } { cal, mc } { A, B, C }, which should do:

1. create two short cuts \cal and \mc for \mathcal
2. create shortcuts for the letters given, in this case, \calA and \mcA for \mathcal{A}, etc.

For this, I make use of the clist structure of expl3. I think it is my wrong usage of ## and ### that causes the errors (Illegal parameter number in definition of \__clist_map_1:w.). My current code is listed in the following MWE. Could you please tell me why does it produce errors and how should it be fixed?

\documentclass{article}
\usepackage{amssymb}
\ExplSyntaxOn
% \projlib_math_define_shortcut:nnn { math command } { shortcut name(s) } { shortcut member(s) }
\cs_new:Nn \projlib_math_define_shortcut:nnn
  {
    \clist_set:Nn \l_tmpa_clist { #2 }
    \clist_set:Nn \l_tmpb_clist { #3 }
    \clist_map_inline:Nn \l_tmpa_clist
      {
        % create \shortcut{}
        \tl_if_blank:eF { ##1 }
          {
            \tl_if_exist:cF { ##1 }
              {
                \tl_gset:cn { ##1 }
                  {
                    \ensuremath { \tl_use:c { #1 } { ###1 } }
                  }
              }
          }
        % create \shortcut
        \clist_map_inline:Nn \l_tmpb_clist
          {
            \tl_if_exist:cF { ##1 ###1 }
              {
                \tl_gset:cn { ##1 ###1 }
                  {
                    \ensuremath { \tl_use:c { #1 } { ###1 } }
                  }
              }
          }
      }
  }
\projlib_math_define_shortcut:nnn { mathbb } { bb }
  {
    A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z,
    a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z
  }
\projlib_math_define_shortcut:nnn { mathfrak } { mf, frak }
  {
    A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z,
    a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z
  }
\projlib_math_define_shortcut:nnn { mathcal } { mc, cal }
  {
    A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z,
    a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z
  }
% \projlib_math_define_shortcut:nnn { mathbb } { }
%   {
%     N, Z, Q, R, C, F, A
%   }
\ExplSyntaxOff
\begin{document}
\fraka
\bbA
% \C
\end{document}

Answer 1

I'm not sure why defining several different shortcuts for the same commands, as they will be confusing. Also, you should not provide the shortcuts in the package, as users might have different ideas about what \mfb, say, might mean and they wouldn't be able to use it for obscure reasons (that the package already provides it). So define a user level command for that purpose and document its usage.

In any case I'd leave out \ensuremath: you gain nothing by allowing \calA{} in text mode instead of the clearer and more semantic $\calA$.

You also want to avoid TeX defining so many token list variables and do \tl_use:c { cal } { A } which is also disputable from a conceptual point of view.

Let's see: you want, given \macro that takes one argument, to define a shortcut prefix, say pre, and, for each letter in the list, say X, you want \preX to stand for \macro{X}.

So what you want is something like

\cs_new_protected:Npn \preX { \macro{X} }

which can be accomplished with

\cs_new_protected:cpn { preX } { \macro{X} }

So here's the code: instead of doing everything in the same function, I split the mapping into two parts.

\documentclass{article}
\usepackage{amsmath,amssymb}
\ExplSyntaxOn
\cs_new_protected:Nn \projlib_math_define_shortcut:Nnn
 {% #1 is the math command
  % #2 is a list of prefixes
  % #3 is a list of letters
  \clist_map_inline:nn { #2 }
   {
    __projlib_math_define_shortcut_do:Nnn #1 { ##1 } { #3 }
   }
 }
\cs_new_protected:Nn __projlib_math_define_shortcut_do:Nnn
 {% #1 is the math command
  % #2 is a prefix
  % #3 is a list of letters
  \clist_map_inline:nn { #3 }
   {
    \cs_new_protected:cpn { #2##1 } { #1{##1} }
   }
 }
\projlib_math_define_shortcut:Nnn \mathbb { bb }
  {
    A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z,
%    a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z
  }
\projlib_math_define_shortcut:Nnn \mathfrak { mf, frak }
  {
    A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z,
    a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z
  }
\ExplSyntaxOff
\begin{document}
$\bbA+\bbZ+\mfA+\mfb+\frakY$
\end{document}

If you prefer to use {mathfrak} instead of \mathfrak (but I can't see why), you can use \exp_not:c.

\documentclass{article}
\usepackage{amsmath,amssymb}
\ExplSyntaxOn
\cs_new_protected:Nn \projlib_math_define_shortcut:nnn
 {% #1 is the math command name
  % #2 is a list of prefixes
  % #3 is a list of letters
  \clist_map_inline:nn { #2 }
   {
    __projlib_math_define_shortcut_do:nnn { #1 } { ##1 } { #3 }
   }
 }
\cs_new_protected:Nn __projlib_math_define_shortcut_do:nnn
 {% #1 is the math command name
  % #2 is a prefix
  % #3 is a list of letters
  \clist_map_inline:nn { #3 }
   {
    \cs_new_protected:cpx { #2##1 } { \exp_not:c { #1 }{##1} }
   }
 }
\projlib_math_define_shortcut:nnn {mathbb} { bb }
  {
    A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z,
%    a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z
  }
\projlib_math_define_shortcut:nnn {mathfrak} { mf, frak }
  {
    A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z,
    a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z
  }
\ExplSyntaxOff
\begin{document}
$\bbA+\bbZ+\mfA+\mfb+\frakY$
\end{document}

Some other points to note.

1. If you nest a mapping inside another, you need to double the number of # characters, not add one.

2. Doing \clist_set:Nn \l_tmpa_clist { #2 } is a waste, because you can use \clist_map_inline:nn { #2 } { ... }.

3. There's no need to check whether the current item during the mapping is empty, because \clist_map_inline:nn will ignore empty items anyway.

Answer 2

If you don't use expl3, macros have more compact form:

\def\mydefmath #1#2 #3{\expandafter\def\csname #2#3\endcsname{#1{#3}}}
\def\Letters{ABCDEFGHIJKLMNOPQRSTUVWXYZ}
\def\letters{abcdefghijklmnopqrstuvwxyz}
\def\mydefall #1#2 #3{\def\tmp{\mydefmath #1#2 }\expandafter\mydefallA#3{}}
\def\mydefallA #1{\ifx^#1^\else \tmp#1\expandafter\mydefallA\fi}
\mydefall \mathbb   bb   \Letters
\mydefall \mathfrak mf   \Letters
\mydefall \mathfrak mf   \letters
\mydefall \mathfrak frak \Letters
\mydefall \mathfrak frak \letters
\mydefall \mathcal  mc   \Letters
\mydefall \mathcal  cal  \Letters
\mydefall \mathscr  ms   \Letters
\mydefall \mathscr  scr  \Letters
%% testing:
\meaning\msC  % -> \mathscr{C}
\meaning\bbD  % -> \mathbb{D}
\meaning\mfa  % -> \mathfrak{a}
\end

Answer 3

Following @egreg's great answer above, here is the completed version:

\documentclass{article}
\usepackage{amsmath,amssymb,mathpazo}
\ExplSyntaxOn
\cs_new_protected:Nn \projlib_math_define_shortcut:Nnn
% #1 is the math command
% #2 is a prefix
% #3 is a list of letters
  {
    \tl_if_blank:eTF { #2 }
      {
        % create \shortcut*
        \clist_map_inline:nn { #3 }
          {
            \cs_if_exist:cF { ##1 }
              {
                \cs_new_protected:cpn { ##1 } { #1 {##1}  }
              }
          }
      }
      {
        \clist_map_inline:nn { #2 }
          {
            % create \shortcut{} and \shortcut*
            __projlib_math_define_shortcut_do:Nnn #1 { ##1 } { #3 }
          }
      }
  }
\cs_new_protected:Nn __projlib_math_define_shortcut_do:Nnn
  {
    % create \shortcut{}
    \cs_if_exist:cF { #2 }
      {
        \cs_new_protected:cpn { #2 } { #1 }
      }
    % create \shortcut*
    \clist_map_inline:nn { #3 }
      {
        \cs_if_exist:cF { #2 ##1 }
          {
            \cs_new_protected:cpn { #2 ##1 } { #1{##1} }
          }
      }
  }
\projlib_math_define_shortcut:Nnn \mathbb { bb }
 {
   A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z, 1
 }
\projlib_math_define_shortcut:Nnn \mathbb { }
 {
   N, Z, Q, R, C, F, A
 }
\projlib_math_define_shortcut:Nnn \mathfrak { mf, frak }
 {
   A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z,
   a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z
 }
\projlib_math_define_shortcut:Nnn \mathcal { mc, cal }
 {
   A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z
 }
\projlib_math_define_shortcut:Nnn \mathscr { ms, scr }
 {
   A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z
 }
\ExplSyntaxOff
\begin{document}
$\bbA+\bbZ+\mfA+\mfb+\frakY$
$\C \N \Q \R \F \bb1$
\end{document}