improve \adjustlimits

[torik42]

Initially I wanted to have the ability to also use \adjustlimits for three (or even more) consecutive limits. For that reason I looked into the definition, but I am just learning LaTeX programming. While trying to understand the code, I think there are some mistakes.

First thing I noticed is that \adjustlimits seems to be intended only for lower limits. However, I don’t currently need it for upper limits.

For the other things, here is an example with a possible alternative definition. See the comments in the source.

\documentclass[margin=2mm]{standalone}
\usepackage{mathtools}

\ExplSyntaxOn
\makeatletter

\newcommand{\adjustlimitsalt}[6]{
\tl_set:Nn \l_tmpa_tl {\MT_vphantom:Nn \displaystyle{#1#4}}
\tl_set:Nn \l_tmpb_tl {\MT_vphantom:Nn \scriptstyle{\MT_cramped_internal:Nn \scriptstyle{#3#6}}}
\def\finsm@sh{\ht\z@\z@ \box\z@} % I don’t know why this is uesd in the original definition
\mathop{#1\l_tmpa_tl}\limits#2{\mathsm@sh\scriptstyle{\MT_cramped_internal:Nn \scriptstyle{#3}}\l_tmpb_tl}
\mathop{#4\l_tmpa_tl}\limits#5{\mathsm@sh\scriptstyle{\MT_cramped_internal:Nn \scriptstyle{#6}}\l_tmpb_tl}
}

\makeatother
\ExplSyntaxOff

\begin{document}

$\begin{gathered}
% In this first example, they behave the same
    \lim_{n\to\infty} \sup_{p^2\leq nK} \quad
    \adjustlimits \lim_{n\to\infty} \sup_{p^2\leq nK} \quad
    \adjustlimitsalt \lim_{n\to\infty} \sup_{p^2\leq nK}
\\
% In the default implementation the distance to the limits is a little smaller.
% In my alternative, the distance is due to the vertical gap between the sup and p²≤nK which is not really necessary.
% It woul be very nice if this could be handled. Unfortunately I don’t see an easy way at the moment.
    \sup_{n\to\infty} \lim_{p^2\leq nK} \quad
    \adjustlimits \sup_{n\to\infty} \lim_{p^2\leq nK} \quad
    \adjustlimitsalt \sup_{n\to\infty} \lim_{p^2\leq nK}
\\
% Here, they behave the same, but compare to the next one!
    \lim_{n\to\infty} \lim_{p^2\leq nK} \quad
    \adjustlimits \lim_{n\to\infty} \lim_{p^2\leq nK} \quad
    \adjustlimitsalt \lim_{n\to\infty} \lim_{p^2\leq nK}
\\
% I only changed the order of the limits. The default has a weird behaviour, while my alternative looks the same.
    \lim_{p^2\leq nK} \lim_{n\to\infty} \quad
    \adjustlimits \lim_{p^2\leq nK} \lim_{n\to\infty} \quad
    \adjustlimitsalt \lim_{p^2\leq nK} \lim_{n\to\infty}
\end{gathered}$

\end{document}

And the output:

As noted in the source, I don’t exactly know what \finsm@sh and \mathsm@sh do, it is just copied from the source (\finsm@sh is used by \mathsm@sh).

The above definition is also easily extendable, for example the same for three operators would look like

\renewcommand{\adjustlimitsthree}[9]{
\tl_set:Nn \l_tmpa_tl {\MT_vphantom:Nn \displaystyle{#1#4#7}}
\tl_set:Nn \l_tmpb_tl {\MT_vphantom:Nn \scriptstyle{\MT_cramped_internal:Nn \scriptstyle{#3#6#9}}}
\def\finsm@sh{\ht\z@\z@ \box\z@}
\mathop{#1\l_tmpa_tl}\limits#2{\mathsm@sh\scriptstyle{\MT_cramped_internal:Nn \scriptstyle{#3}}\l_tmpb_tl}
\mathop{#4\l_tmpa_tl}\limits#5{\mathsm@sh\scriptstyle{\MT_cramped_internal:Nn \scriptstyle{#6}}\l_tmpb_tl}
\mathop{#7\l_tmpa_tl}\limits#8{\mathsm@sh\scriptstyle{\MT_cramped_internal:Nn \scriptstyle{#9}}\l_tmpb_tl}
}

I am quite sure that someone with more knowledge can expand this to work for any number of operators. Maybe one could add an optional argument to \adjustlimits to specify the number of operators that should be taken care of.

PS: Independent of that, could someone explain to me the use of \z@ in the default definition? To my knowledge it is usually defined to be 0pt. Then, in the default definition it is defined to be a box \sbox\z@{$\m@th \displaystyle #1$}(line 5183) which is used to calculate the difference of the depths of both boxes. Later it is used in \MH_if_dim:w \@tempdima>\z@(line 5196) to compare dimensions again, where I think it should be 0pt again. How is that possible? And what is it expected to be in \def\finsm@sh{\ht\z@\z@ \box\z@}(line 5201)?

[eg9]

I think you could have a look at https://tex.stackexchange.com/a/409671/4427, with a modification I now realize is necessary, namely to add \cramped

\cs_new_protected:Nn \__multiadjustlimits_print:NNn
 {
  \mathop { \vphantom{\l__multiadjustlimits_operator_tl} \mathopen{} #1 }
  \limits
  \sb{ \vphantom{\cramped{\l__multiadjustlimits_limit_tl}} #3 }
 }

The original version hadn't \cramped, now it has.

With

\begin{equation*}
\multiadjustlimits{\lim_{n\to\infty}, \lim_{p^2\leq nK}} \lim_{p^2\leq nK}
\end{equation*}

I get

[torik42]

Thanks a lot. Could this be added to mathtools? (I myself would also prefer not to write the comma.)

Still, the current implementation seems wrong to me. The more realistic

\adjustlimits\lim_{n\to\infty} \lim_{\tilde{N}\to\infty} \quad
\adjustlimits\lim_{\tilde{N}\to\infty} \lim_{n\to\infty}

gives

[FrankMittelbach]

PS: Independent of that, could someone explain to me the use of \z@ in the default definition? To my knowledge it is usually defined to be 0pt. Then, in the default definition it is defined to be a box \sbox\z@{$\m@th \displaystyle #1$}(line 5183) which is used to calculate the difference of the depths of both boxes. Later it is used in \MH_if_dim:w \@tempdima>\z@(line 5196) to compare dimensions again, where I think it should be 0pt again. How is that possible? And what is it expected to be in \def\finsm@sh{\ht\z@\z@ \box\z@}(line 5201)?

you need to read the TeXbook for that. TeX has some optimizations build in when it comes to use of registers. \z@ is defined as

\newdimen\z@ \z@=0pt % can be used both for 0pt and 0

and that means it can be used as a "dimen" or as a number meaning 0 or 0pt depending on the circumstances. So it is not "usually" that it is "always" that. Box registers are really internally numbers as well so

\sbox\z@{...}      % save something in box 0
\ht\z@\z@    % short for set height of box 0 to 0pt
\box\z@        % typeset box 0

[torik42]

@FrankMittelbach: Thank you for the explanation, I did not know, that \z@ is only a register for the box.

@eg9: You may also want to put a \cramped around #3. And in the example, you did not aligned the third limit using the function ;)

The actual problem with the current implementation is that it does not check the height of the limits.

Based on https://tex.stackexchange.com/a/409671 and https://tex.stackexchange.com/a/410052 I made a variant which checks for the smallest needed distance (see last examples in the following). Including something like this would easily allow for a fix of \adjustlimits by just passing the arguments to \multiadjustlimits.

Here is my variant

\documentclass[margin=2mm]{standalone}
\usepackage{mathtools}
\usepackage{xparse}

\ExplSyntaxOn

\msg_new:nnnn {multiadjustlimits} {only_lower_limits} {
  You~should~only~use~multiadjustlimits~with~lower~limits.
} {}

\NewDocumentCommand{\multiadjustlimits}{m}{
  \group_begin:
  \multiadjustlimits_measure:n { #1 }
  \multiadjustlimits_print:n { #1 }
  \group_end:
}

% We will later find the operator/limit pair with the largest sum of
% operator depth and limit height. These vales will be stored here.
\dim_new:N \l__multiadjustlimits_operator_depth_dim
\dim_new:N  \l__multiadjustlimits_limit_height_dim

\cs_new_protected:Nn \multiadjustlimits_measure:n
 {
    \__multiadjustlimits_measure_recursion:nnn #1
    \q_recursion_tail \q_recursion_tail \q_recursion_tail \q_recursion_stop
 }

% Function to determine the largest sum of operator depth and limit
% height recursively.
\cs_new_protected:Nn \__multiadjustlimits_measure_recursion:nnn {
  \quark_if_recursion_tail_stop:n {#1}
  \quark_if_recursion_tail_stop:n {#2}
  \quark_if_recursion_tail_stop:n {#3}

  % Store operator and limit in a box
  \hbox_set:Nn \l_tmpa_box { $\displaystyle#1$ }
  \hbox_set:Nn \l_tmpb_box { $\scriptstyle\cramped{#3}$ }

  % Check if the operator depths and limit height sum is larger than
  % the currently stored one and overwrite in that case
  \dim_compare:nNnTF {\box_dp:N \l_tmpa_box + \box_ht:N \l_tmpb_box}>{
    \dim_use:N \l__multiadjustlimits_operator_depth_dim + \dim_use:N \l__multiadjustlimits_limit_height_dim
  }{
    \dim_set:Nn \l__multiadjustlimits_operator_depth_dim {
      \box_dp:N \l_tmpa_box}
    \dim_set:Nn \l__multiadjustlimits_limit_height_dim {
      \box_ht:N \l_tmpb_box}
  } {}
  % Throw a warning if someone uses \multiadjustlimits with upper limits
  \str_if_eq:nnTF {#2} {_} {} {
    \msg_warning:nn {multiadjustlimits} {only_lower_limits}
  }
  % Recursion
  \__multiadjustlimits_measure_recursion:nnn
}

\cs_new_protected:Nn \multiadjustlimits_print:n {
  \__multiadjustlimits_print_recursion:nnn #1
  \q_recursion_tail \q_recursion_tail \q_recursion_tail \q_recursion_stop
}

% Function to print the operators/limits recursively
\cs_new_protected:Nn \__multiadjustlimits_print_recursion:nnn {
  \quark_if_recursion_tail_stop:n {#1}
  \quark_if_recursion_tail_stop_do:nn {#2}{#1}
  \quark_if_recursion_tail_stop_do:nn {#3}{#1#2}
  \__multiadjustlimits_print:NNn {#1}{#2}{#3}
  % Recursion
  \__multiadjustlimits_print_recursion:nnn
}

% Function to print one operator with its limit
\cs_new_protected:Nn \__multiadjustlimits_print:NNn {
  % Create a box for the operator and adjust the depth
  \hbox_set:Nn \l_tmpa_box {$\displaystyle {#1}$}
  \box_set_dp:Nn \l_tmpa_box {
    \dim_use:N \l__multiadjustlimits_operator_depth_dim
  }
  % Create a box for the limit and adjust the height
  \hbox_set:Nn \l_tmpb_box {$\scriptstyle\cramped{#3}$}
  \box_set_ht:Nn \l_tmpb_box {
    \dim_use:N \l__multiadjustlimits_limit_height_dim
  }
  % Print the operator/limit
  \mathop {  \mathopen{} \box_use:N \l_tmpa_box }
  \limits
  #2{ {}\box_use:N \l_tmpb_box }
 }
\ExplSyntaxOff


\begin{document}

$\begin{aligned}
% It works in all cases where \adjustlimits works (examples from the documentation)
    % &\adjustlimits\lim_{n\to\infty} \max_{p \leq n}
    % & &\multiadjustlimits{\lim_{n\to\infty} \max_{p \leq n}}\\
    % &\adjustlimits\lim_{n\to\infty} \max_{p^2 \leq n}
    % & &\multiadjustlimits{\lim_{n\to\infty} \max_{p^2 \leq n}}\\
    &\adjustlimits\lim_{n\to\infty} \sup_{p^2 \leq nK}
    & &\multiadjustlimits{\lim_{n\to\infty} \sup_{p^2 \leq nK}}\\
    &\adjustlimits\limsup_{n\to\infty} \max_{p \leq n}
    & &\multiadjustlimits{\limsup_{n\to\infty} \max_{p \leq n}}\\
% And also in the cases, where \adjustlimits fails
    &\adjustlimits\lim_{\tilde{N}\to\infty} \lim_{n\to\infty}
    & &\multiadjustlimits{\lim_{\tilde{N}\to\infty} \lim_{n\to\infty}}\\
    &\adjustlimits\lim_{n\to\infty} \lim_{\tilde{N}\to\infty}
    & &\multiadjustlimits{\lim_{n\to\infty} \lim_{\tilde{N}\to\infty}}\\
\\
% Furthermore it also checks the smallest distance, compare
    &&&\multiadjustlimits{\sup_{n<a} \max_{p^2\leq \tilde{N}}}\\
% and
    &&&\multiadjustlimits{\max_{n<a} \sup_{p^2\leq \tilde{N}}}
\\
% It also allows for multiple operators
    &&&\multiadjustlimits{\lim_{\tilde{N}\to\infty} \max_{n\leq\tilde{N}} \sup_{p^2\leq nK}}
\end{aligned}$

\end{document}

And the output (left is the usual \adjustlimits and right my variant of \multiadjustlimits)

[FrankMittelbach]

Am 14.03.21 um 16:46 schrieb torik42:

@FrankMittelbach <https://github.com/FrankMittelbach>: Thank you for the explanation, I did not know, that |\z@| is only a register for the box.

no, \z@ is not a box register it is just away to write the number 0. The box registers (as all TeX registers) are all accessible by number, e.g. \box 0 \box 1 ... If you say \newbox\foo then \foo is simply associated with the next free number so \box\foo is the same as \box 123 (if that happens to be the number allocated) So \box\z@ is the same as saying \box 0 only slightly more efficient.

[torik42]

Thank you for explaining again @FrankMittelbach. This is how I also understood your last comment. I should have written something like “number for the register of the box”.

@daleif: I think in the discussion above, which was very helpful for me, the actual issue got lost. The problem is that \adjustlimits only works in certain situations. Depending on the depth of the operator it chooses which operator dictates the spacing and then forces the other limit to be on the same line. Thus, having two consecutive \lim the second will always dictate the spacing. For example \adjustlimits \lim_{\tilde{N}\to\infty} \lim_{n\to\infty} results in:

If the depth of the first operator is larger than the depth of the second, then the first dictates the spacing, i.e. \adjustlimits \sup_{\tilde{N}\in\mathbb{N}} \lim_{n\to\infty} looks good:

But again, if the height of the second limit is large, we obtain a bad output as with \adjustlimits \sup_{N\in\mathbb{N}} \lim_{\tilde{\tilde{N}}\to\infty}:

An easy fix which only sometimes produces a too large spacing is to always have the other operator/limit as a vertical phantom, i.e. something like

\newcommand{\adjustlimits}[6]{%
\mathop{\vphantom{#4}\mathopen{}#1}\limits#2{\cramped{#3\vphantom{\cramped{#6}}}}
\mathop{\vphantom{#1}\mathopen{}#4}\limits#5{\cramped{#6\vphantom{\cramped{#3}}}}
}

which is similar to the solution you suggested here but only for two limits.